Atlas › Modular forms › Weight 2 › Level 916 › 916.2.a

Newspace 916.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(916))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
916.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 3q^{5} - 4q^{7} + O(q^{8})$	+	916c1 rank 2
916.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + 2q^{7} + O(q^{8})$	−	916e1 rank 1
916.2.a.c	1	$\mathbb{Q}$	$q - 2q^{5} - 2q^{7} + O(q^{8})$	+	916a1 rank 0
916.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} + 2q^{7} + O(q^{8})$	−	916d1 rank 1
916.2.a.e	1	$\mathbb{Q}$	$q + 3q^{3} + q^{5} + 4q^{7} + O(q^{8})$	+	916b1 rank 0
916.2.a.f	5	degree 5	$5q - q^{3} - 4q^{5} - 9q^{7} + O(q^{8})$	−
916.2.a.g	9	degree 9	$9q - q^{3} + 10q^{5} + 3q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ 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 $T$
916.2.a.a	$x + 89$
916.2.a.b	$x + 41$
916.2.a.c	$x - 30$
916.2.a.d	$x + 1$
916.2.a.e	$x + 45$
916.2.a.f	$x^{5} + 72x^{4} - 3824x^{3} - 384149x^{2} - 6091838x + 34325471$
916.2.a.g	$x^{9} - 164x^{8} + 5388x^{7} + 316579x^{6} - 21519791x^{5} + 262400139x^{4} + 5703458214x^{3} - 148799161253x^{2} + 1084873795924x - 2573290549681$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(916))$	19
Old forms	94
Cusp forms $S_{2}(\Gamma_0(916))$	113
Eisenstein series $E_{2}(\Gamma_0(916))$	5
Modular forms $M_{2}(\Gamma_0(916))$	118

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(916)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(229))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(458))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.451 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, 4, 229, 458, 916]: [0, 1, 2, 19, 59, 118] (certified at ell = 2147478649)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 19 = 1 x 19 (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(916, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 916:2 1000

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