Atlas › Modular forms › Weight 2 › Level 821 › 821.2.a

Newspace 821.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(821))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
821.2.a.a	27	degree 27	$27q - 3q^{2} - 14q^{3} + 15q^{4} - 11q^{5} - 12q^{6} - 21q^{7} + O(q^{8})$	+	$-1$
821.2.a.b	41	degree 41	$41q + q^{2} + 16q^{3} + 53q^{4} + 7q^{5} + 6q^{6} + 19q^{7} + O(q^{8})$	−	$+1$

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 $T$
821.2.a.a	$x^{27} + 17x^{26} + 73x^{25} - 290x^{24} - 2945x^{23} - 2097x^{22} + 38583x^{21} + 84584x^{20} - 233566x^{19} - 828275x^{18} + 629024x^{17} + 4122941x^{16} - 61801x^{15} - 11850347x^{14} - 3951484x^{13} + 20248079x^{12} + 10044249x^{11} - 20250387x^{10} - 11047719x^{9} + 11369113x^{8} + 5812453x^{7} - 3443032x^{6} - 1470014x^{5} + 522077x^{4} + 162182x^{3} - 32956x^{2} - 5652x + 316$
821.2.a.b	$x^{41} - 17x^{40} - 15x^{39} + 1792x^{38} - 6510x^{37} - 73696x^{36} + 473507x^{35} + 1374667x^{34} - 15676623x^{33} - 4757655x^{32} + 307007377x^{31} - 316966444x^{30} - 3896849142x^{29} + 7632171026x^{28} + 33287182350x^{27} - 93421951977x^{26} - 191842890173x^{25} + 739624729725x^{24} + 712088602843x^{23} - 4060245695066x^{22} - 1375503117137x^{21} + 15865009357500x^{20} - 863975228698x^{19} - 44428010182783x^{18} + 14010730165654x^{17} + 88684193718613x^{16} - 43734098339404x^{15} - 124180565845559x^{14} + 76620274163186x^{13} + 118555826394918x^{12} - 84086264728619x^{11} - 73514634499016x^{10} + 58484791837377x^{9} + 26985008121460x^{8} - 25012125813896x^{7} - 4577140461159x^{6} + 6063887625624x^{5} - 79515545788x^{4} - 678206673220x^{3} + 100630356200x^{2} + 14863371272x - 2928198464$

Orbit

Characteristic polynomial of

T

821.2.a.a

x^{27} + 17x^{26} + 73x^{25} - 290x^{24} - 2945x^{23} - 2097x^{22} + 38583x^{21} + 84584x^{20} - 233566x^{19} - 828275x^{18} + 629024x^{17} + 4122941x^{16} - 61801x^{15} - 11850347x^{14} - 3951484x^{13} + 20248079x^{12} + 10044249x^{11} - 20250387x^{10} - 11047719x^{9} + 11369113x^{8} + 5812453x^{7} - 3443032x^{6} - 1470014x^{5} + 522077x^{4} + 162182x^{3} - 32956x^{2} - 5652x + 316

821.2.a.b

x^{41} - 17x^{40} - 15x^{39} + 1792x^{38} - 6510x^{37} - 73696x^{36} + 473507x^{35} + 1374667x^{34} - 15676623x^{33} - 4757655x^{32} + 307007377x^{31} - 316966444x^{30} - 3896849142x^{29} + 7632171026x^{28} + 33287182350x^{27} - 93421951977x^{26} - 191842890173x^{25} + 739624729725x^{24} + 712088602843x^{23} - 4060245695066x^{22} - 1375503117137x^{21} + 15865009357500x^{20} - 863975228698x^{19} - 44428010182783x^{18} + 14010730165654x^{17} + 88684193718613x^{16} - 43734098339404x^{15} - 124180565845559x^{14} + 76620274163186x^{13} + 118555826394918x^{12} - 84086264728619x^{11} - 73514634499016x^{10} + 58484791837377x^{9} + 26985008121460x^{8} - 25012125813896x^{7} - 4577140461159x^{6} + 6063887625624x^{5} - 79515545788x^{4} - 678206673220x^{3} + 100630356200x^{2} + 14863371272x - 2928198464

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(821))$	68
Old forms	0
Cusp forms $S_{2}(\Gamma_0(821))$	68
Eisenstein series $E_{2}(\Gamma_0(821))$	1
Modular forms $M_{2}(\Gamma_0(821))$	69

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.257 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, 821]: [0, 69] (certified at ell = 2147455361)
T = [(2, 1), (3, 1)]; h of degree 68 = 1 x 68 (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(821, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 821:2 1000

← 820.2.a · 822.2.a →