Atlas › Modular forms › Weight 2 › Level 910 › 910.2.a

Newspace 910.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(910))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
910.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ + − −	$-1$	910c1 rank 1
910.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + q^{7} + O(q^{8})$	+ + − +	$+1$	910a1 rank 0
910.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + q^{7} + O(q^{8})$	+ − − +	$-1$	910d1 rank 1
910.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + O(q^{8})$	+ + − −	$-1$	910b1 rank 1
910.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − − −	$+1$	910e1 rank 0
910.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} - q^{5} - 3q^{6} - q^{7} + O(q^{8})$	− + + −	$-1$	910g1 rank 1
910.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− + − −	$+1$	910j1 rank 0
910.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + q^{5} - 2q^{6} - q^{7} + O(q^{8})$	− − + +	$-1$	910k1 rank 1
910.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + O(q^{8})$	− + − +	$-1$	910h1 rank 1
910.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - q^{7} + O(q^{8})$	− + + −	$-1$	910f1 rank 1
910.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} - q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− + − −	$+1$	910i1 rank 0
910.2.a.l	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - q^{3} + 2q^{4} - 2q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ + + +	$-1$
910.2.a.m	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	+ − − −	$+1$
910.2.a.n	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ + + −	$+1$
910.2.a.o	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− − + −	$+1$
910.2.a.p	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ − + +	$+1$
910.2.a.q	3	degree 3	$3q + 3q^{2} + q^{3} + 3q^{4} + 3q^{5} + q^{6} + 3q^{7} + O(q^{8})$	− − − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{11} + 5T_{17} + 7T_{19} + 9T_{23} + 11T_{29}$ 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 $T$
910.2.a.a	$x - 24$
910.2.a.b	$x - 44$
910.2.a.c	$x + 112$
910.2.a.d	$x + 117$
910.2.a.e	$x - 183$
910.2.a.f	$x + 139$
910.2.a.g	$x - 132$
910.2.a.h	$x + 8$
910.2.a.i	$x + 185$
910.2.a.j	$x - 8$
910.2.a.k	$x + 4$
910.2.a.l	$x^{2} + 39x - 2276$
910.2.a.m	$x^{2} - 8x - 784$
910.2.a.n	$x^{2} - 92x + 2096$
910.2.a.o	$x^{2} - 97x + 2144$
910.2.a.p	$x^{3} - 17x^{2} - 21944x - 264496$
910.2.a.q	$x^{3} - 63x^{2} - 2524x + 43184$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(910))$	25
Old forms	136
Cusp forms $S_{2}(\Gamma_0(910))$	161
Eisenstein series $E_{2}(\Gamma_0(910))$	15
Modular forms $M_{2}(\Gamma_0(910))$	176

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.251 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, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910]: [0, 1, 1, 1, 3, 1, 4, 5, 6, 8, 16, 10, 24, 32, 60, 176] (certified at ell = 2147483629)
T = [(3, 1), (11, 3), (17, 5), (19, 7), (23, 9), (29, 11)]; h of degree 25 = 1 x 25 (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(910, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 910:2 1000

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