Atlas › Modular forms › Weight 2 › Level 922 › 922.2.a

Newspace 922.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(922))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
922.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + 2q^{5} - 2q^{6} - q^{7} + O(q^{8})$	− +	$+1$	922a1 rank 0
922.2.a.b	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} - 6q^{5} + 3q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
922.2.a.c	4	degree 4	$4q + 4q^{2} - 2q^{3} + 4q^{4} - 5q^{5} - 2q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
922.2.a.d	8	degree 8	$8q - 8q^{2} - 2q^{3} + 8q^{4} + 2q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
922.2.a.e	8	degree 8	$8q - 8q^{2} + 2q^{3} + 8q^{4} + 3q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$
922.2.a.f	14	degree 14	$14q + 14q^{2} + 3q^{3} + 14q^{4} + 6q^{5} + 3q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
922.2.a.a	$x$
922.2.a.b	$x^{3} + 9x^{2} + 11x - 29$
922.2.a.c	$x^{4} + 7x^{3} + 15x^{2} + 9x - 1$
922.2.a.d	$x^{8} - 19x^{6} - 10x^{5} + 87x^{4} + 90x^{3} + 5x^{2} - 10x + 1$
922.2.a.e	$x^{8} - 5x^{7} - 23x^{6} + 113x^{5} + 103x^{4} - 680x^{3} + 172x^{2} + 872x - 416$
922.2.a.f	$x^{14} - 9x^{13} - 54x^{12} + 748x^{11} - 454x^{10} - 18368x^{9} + 61570x^{8} + 66532x^{7} - 743926x^{6} + 1642036x^{5} - 1529445x^{4} + 335197x^{3} + 434368x^{2} - 300292x + 52072$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(922))$	38
Old forms	76
Cusp forms $S_{2}(\Gamma_0(922))$	114
Eisenstein series $E_{2}(\Gamma_0(922))$	3
Modular forms $M_{2}(\Gamma_0(922))$	117

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(922)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(461))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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, 2, 461, 922]: [0, 1, 39, 117] (certified at ell = 2147478721)
T = [(3, 1), (5, 1)]; h of degree 38 = 1 x 38 (phi(ord eps) x dim S_k^new); 5 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(922, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 922:2 1000

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