Atlas › Modular forms › Weight 2 › Level 914 › 914.2.a

Newspace 914.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(914))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
914.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + O(q^{8})$	+ +	$-1$	914a1 rank 1
914.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$	914b1 rank 0
914.2.a.c	5	degree 5	$5q - 5q^{2} - q^{3} + 5q^{4} + 3q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
914.2.a.d	5	degree 5	$5q + 5q^{2} - 5q^{3} + 5q^{4} - 7q^{5} - 5q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
914.2.a.e	12	degree 12	$12q - 12q^{2} + 12q^{4} - 8q^{5} + 7q^{7} + O(q^{8})$	+ −	$+1$
914.2.a.f	15	degree 15	$15q + 15q^{2} + 5q^{3} + 15q^{4} + 9q^{5} + 5q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$

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 $T$
914.2.a.a	$x$
914.2.a.b	$x - 4$
914.2.a.c	$x^{5} - 2x^{4} - 9x^{3} + 9x^{2} + 21x + 5$
914.2.a.d	$x^{5} + 12x^{4} + 47x^{3} + 59x^{2} - 31x - 83$
914.2.a.e	$x^{12} + 8x^{11} - 37x^{10} - 417x^{9} - 50x^{8} + 5799x^{7} + 7840x^{6} - 21653x^{5} - 36898x^{4} + 13942x^{3} + 21651x^{2} - 3141x - 2978$
914.2.a.f	$x^{15} - 14x^{14} + 17x^{13} + 609x^{12} - 2982x^{11} - 2877x^{10} + 49728x^{9} - 97465x^{8} - 47730x^{7} + 273278x^{6} - 98247x^{5} - 160775x^{4} + 98452x^{3} + 4954x^{2} - 9776x + 1232$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(914))$	39
Old forms	74
Cusp forms $S_{2}(\Gamma_0(914))$	113
Eisenstein series $E_{2}(\Gamma_0(914))$	3
Modular forms $M_{2}(\Gamma_0(914))$	116

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(914)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(457))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 457, 914]: [0, 1, 38, 116] (certified at ell = 2147478649)
T = [(3, 1), (5, 1)]; h of degree 39 = 1 x 39 (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(914, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 914:2 1000

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