Atlas › Modular forms › Weight 2 › Level 912 › 912.2.a

Newspace 912.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(912))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
912.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + 3q^{7} + O(q^{8})$	+ +	912a1 rank 1
912.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + O(q^{8})$	+ −	912g1 rank 1
912.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 4q^{7} + O(q^{8})$	+ +	912e1 rank 0
912.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - 3q^{7} + O(q^{8})$	+ −	912f1 rank 1
912.2.a.e	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} + O(q^{8})$	+ −	912b1 rank 0
912.2.a.f	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} - q^{7} + O(q^{8})$	− +	912i1 rank 1
912.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} + 5q^{7} + O(q^{8})$	− −	912l1 rank 0
912.2.a.h	1	$\mathbb{Q}$	$q + q^{3} - 4q^{7} + O(q^{8})$	− +	912h1 rank 1
912.2.a.i	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + 3q^{7} + O(q^{8})$	− +	912c1 rank 0
912.2.a.j	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + O(q^{8})$	− −	912j1 rank 0
912.2.a.k	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + O(q^{8})$	− −	912k1 rank 0
912.2.a.l	1	$\mathbb{Q}$	$q + q^{3} + 4q^{5} - 4q^{7} + O(q^{8})$	− +	912d1 rank 0
912.2.a.m	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{3} + q^{5} - q^{7} + O(q^{8})$	+ +
912.2.a.n	2	$\mathbb{Q}(\sqrt{33})$	$2q - 2q^{3} + 3q^{5} - q^{7} + O(q^{8})$	+ +
912.2.a.o	2	$\mathbb{Q}(\sqrt{41})$	$2q + 2q^{3} - q^{5} + 3q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = 2555T_{5} + 3163T_{7} + 2311T_{13} + 3911T_{17} + 4960T_{23} + 3573T_{29} + 9540T_{11} + 2170T_{31}$ 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 $T$
912.2.a.a	$x + 10071$
912.2.a.b	$x + 44764$
912.2.a.c	$x - 73732$
912.2.a.d	$x + 40357$
912.2.a.e	$x - 33380$
912.2.a.f	$x + 7971$
912.2.a.g	$x - 25035$
912.2.a.h	$x + 83172$
912.2.a.i	$x - 31633$
912.2.a.j	$x - 35850$
912.2.a.k	$x - 28440$
912.2.a.l	$x - 82516$
912.2.a.m	$x^{2} + 109019x + 1773335876$
912.2.a.n	$x^{2} - 43295x - 366149462$
912.2.a.o	$x^{2} + 28451x - 1291647772$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(912))$	18
Old forms	131
Cusp forms $S_{2}(\Gamma_0(912))$	149
Eisenstein series $E_{2}(\Gamma_0(912))$	23
Modular forms $M_{2}(\Gamma_0(912))$	172

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.263 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, 3, 4, 6, 8, 12, 16, 19, 24, 38, 48, 57, 76, 114, 152, 228, 304, 456, 912]: [0, 1, 1, 2, 3, 3, 5, 4, 2, 8, 7, 12, 8, 13, 24, 24, 46, 44, 88, 168] (certified at ell = 2147483629)
T = [(5, 2555), (7, 3163), (13, 2311), (17, 3911), (23, 4960), (29, 3573), (11, 9540), (31, 2170)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 11 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(912, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 912:2 1000

← 911.2.a · 913.2.a →