Atlas › Modular forms › Weight 2 › Level 863 › 863.2.a

Newspace 863.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(863))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
863.2.a.a	4	degree 4	$4q + q^{2} - 4q^{3} + q^{4} - 9q^{5} + 3q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
863.2.a.b	22	degree 22	$22q - 6q^{2} - 2q^{3} + 16q^{4} - 2q^{5} - 13q^{6} - 20q^{7} + O(q^{8})$	+	$-1$
863.2.a.c	46	degree 46	$46q + 4q^{2} + 6q^{3} + 52q^{4} + 11q^{5} + 2q^{6} + 28q^{7} + O(q^{8})$	−	$+1$

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 $T$
863.2.a.a	$x^{4} + 3x^{3} - 9x^{2} - 17x + 31$
863.2.a.b	$x^{22} + 8x^{21} - 24x^{20} - 294x^{19} + 95x^{18} + 4233x^{17} + 1956x^{16} - 30485x^{15} - 23619x^{14} + 115486x^{13} + 103037x^{12} - 221288x^{11} - 197488x^{10} + 195264x^{9} + 149773x^{8} - 84157x^{7} - 42439x^{6} + 18059x^{5} + 2876x^{4} - 1538x^{3} + 86x^{2} + 14x - 1$
863.2.a.c	$x^{46} - 10x^{45} - 124x^{44} + 1518x^{43} + 5974x^{42} - 102374x^{41} - 116011x^{40} + 4069871x^{39} - 1045480x^{38} - 106795767x^{37} + 115727513x^{36} + 1964724210x^{35} - 3197611595x^{34} - 26307233865x^{33} + 52729459109x^{32} + 263094973657x^{31} - 595996027435x^{30} - 2003851775771x^{29} + 4868481282296x^{28} + 11802759340169x^{27} - 29490933472232x^{26} - 54389704222550x^{25} + 134173571270776x^{24} + 197482820670140x^{23} - 460596021108031x^{22} - 565390694459455x^{21} + 1191114920767312x^{20} + 1267220721736176x^{19} - 2304707919704650x^{18} - 2188859618624828x^{17} + 3299038798178660x^{16} + 2846073259055039x^{15} - 3440488844053013x^{14} - 2703559144016765x^{13} + 2564462362757947x^{12} + 1810273594276592x^{11} - 1333163444531387x^{10} - 818132279475743x^{9} + 466521564724692x^{8} + 235716757607846x^{7} - 103277330159144x^{6} - 39723058511038x^{5} + 12744786693544x^{4} + 3367373989531x^{3} - 630482044248x^{2} - 102022043123x - 3274265657$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(863))$	72
Old forms	0
Cusp forms $S_{2}(\Gamma_0(863))$	72
Eisenstein series $E_{2}(\Gamma_0(863))$	1
Modular forms $M_{2}(\Gamma_0(863))$	73

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 863]: [0, 73] (certified at ell = 2147466121)
T = [(2, 1), (3, 1)]; h of degree 72 = 1 x 72 (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(863, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 863:2 1000

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