Atlas › Modular forms › Weight 2 › Level 857 › 857.2.a

Newspace 857.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(857))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
857.2.a.a	28	degree 28	$28q - 4q^{2} - 12q^{3} + 14q^{4} - 6q^{5} - 6q^{6} - 38q^{7} + O(q^{8})$	+	$-1$
857.2.a.b	43	degree 43	$43q + 3q^{2} + 12q^{3} + 55q^{4} + 6q^{5} + 44q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
857.2.a.a	$x^{28} + 16x^{27} + 49x^{26} - 472x^{25} - 3057x^{24} + 2812x^{23} + 56742x^{22} + 49899x^{21} - 529735x^{20} - 957506x^{19} + 2812424x^{18} + 7390483x^{17} - 8408608x^{16} - 32730605x^{15} + 10560123x^{14} + 89744142x^{13} + 14583964x^{12} - 151961020x^{11} - 79128792x^{10} + 146999099x^{9} + 129287832x^{8} - 59889842x^{7} - 97438842x^{6} - 11761675x^{5} + 26335136x^{4} + 13087401x^{3} + 1791728x^{2} - 62695x - 11525$
857.2.a.b	$x^{43} - 15x^{42} - 48x^{41} + 1746x^{40} - 2901x^{39} - 85853x^{38} + 321881x^{37} + 2309306x^{36} - 12845663x^{35} - 35867387x^{34} + 297602379x^{33} + 280840352x^{32} - 4523710146x^{31} + 297443467x^{30} + 47612063584x^{29} - 33089816508x^{28} - 357161605423x^{27} + 399283266341x^{26} + 1944204405722x^{25} - 2744341388615x^{24} - 7781943143734x^{23} + 12525595185076x^{22} + 23156423965350x^{21} - 39869676848162x^{20} - 51653370058821x^{19} + 90114832012744x^{18} + 86619263898478x^{17} - 144930312408124x^{16} - 108392719077929x^{15} + 164213192312556x^{14} + 98796922834491x^{13} - 128368186058761x^{12} - 62461349008389x^{11} + 67153187779015x^{10} + 25148285584475x^{9} - 22689172581668x^{8} - 5543347066468x^{7} + 4763875165428x^{6} + 464124953038x^{5} - 560295669005x^{4} + 24229209199x^{3} + 26828128784x^{2} - 4466560336x + 178816301$

Orbit

Characteristic polynomial of

T

857.2.a.a

x^{28} + 16x^{27} + 49x^{26} - 472x^{25} - 3057x^{24} + 2812x^{23} + 56742x^{22} + 49899x^{21} - 529735x^{20} - 957506x^{19} + 2812424x^{18} + 7390483x^{17} - 8408608x^{16} - 32730605x^{15} + 10560123x^{14} + 89744142x^{13} + 14583964x^{12} - 151961020x^{11} - 79128792x^{10} + 146999099x^{9} + 129287832x^{8} - 59889842x^{7} - 97438842x^{6} - 11761675x^{5} + 26335136x^{4} + 13087401x^{3} + 1791728x^{2} - 62695x - 11525

857.2.a.b

x^{43} - 15x^{42} - 48x^{41} + 1746x^{40} - 2901x^{39} - 85853x^{38} + 321881x^{37} + 2309306x^{36} - 12845663x^{35} - 35867387x^{34} + 297602379x^{33} + 280840352x^{32} - 4523710146x^{31} + 297443467x^{30} + 47612063584x^{29} - 33089816508x^{28} - 357161605423x^{27} + 399283266341x^{26} + 1944204405722x^{25} - 2744341388615x^{24} - 7781943143734x^{23} + 12525595185076x^{22} + 23156423965350x^{21} - 39869676848162x^{20} - 51653370058821x^{19} + 90114832012744x^{18} + 86619263898478x^{17} - 144930312408124x^{16} - 108392719077929x^{15} + 164213192312556x^{14} + 98796922834491x^{13} - 128368186058761x^{12} - 62461349008389x^{11} + 67153187779015x^{10} + 25148285584475x^{9} - 22689172581668x^{8} - 5543347066468x^{7} + 4763875165428x^{6} + 464124953038x^{5} - 560295669005x^{4} + 24229209199x^{3} + 26828128784x^{2} - 4466560336x + 178816301

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(857))$	71
Old forms	0
Cusp forms $S_{2}(\Gamma_0(857))$	71
Eisenstein series $E_{2}(\Gamma_0(857))$	1
Modular forms $M_{2}(\Gamma_0(857))$	72

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.266 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 857]: [0, 72] (certified at ell = 2147478017)
T = [(2, 1), (3, 1)]; h of degree 71 = 1 x 71 (phi(ord eps) x dim S_k^new); 7 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(857, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 857:2 1000

← 856.2.a · 858.2.a →