Atlas › Modular forms › Weight 2 › Level 906 › 906.2.a

Newspace 906.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(906))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
906.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$	906a1 rank 1
906.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 3q^{5} + q^{6} + O(q^{8})$	+ + +	$-1$	906b1 rank 1
906.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − −	$-1$	906d1 rank 1
906.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} + O(q^{8})$	+ − −	$-1$	906c1 rank 1
906.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$	906g1 rank 1
906.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + O(q^{8})$	− + +	$+1$	906e1 rank 0
906.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 4q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + +	$+1$	906f1 rank 0
906.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$	906h1 rank 1
906.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$	906i1 rank 0
906.2.a.j	4	degree 4	$4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$
906.2.a.k	4	degree 4	$4q - 4q^{2} + 4q^{3} + 4q^{4} + 2q^{5} - 4q^{6} - q^{7} + O(q^{8})$	+ − +	$+1$
906.2.a.l	4	degree 4	$4q + 4q^{2} - 4q^{3} + 4q^{4} - q^{5} - 4q^{6} + 7q^{7} + O(q^{8})$	− + +	$+1$
906.2.a.m	4	degree 4	$4q + 4q^{2} + 4q^{3} + 4q^{4} + 5q^{5} + 4q^{6} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
906.2.a.a	$x - 24$
906.2.a.b	$x + 57$
906.2.a.c	$x + 19$
906.2.a.d	$x + 46$
906.2.a.e	$x + 33$
906.2.a.f	$x - 7$
906.2.a.g	$x - 10$
906.2.a.h	$x + 23$
906.2.a.i	$x + 15$
906.2.a.j	$x^{4} - 62x^{3} - 1040x^{2} + 108630x - 1450041$
906.2.a.k	$x^{4} - 12x^{3} - 1382x^{2} + 13492x + 356413$
906.2.a.l	$x^{4} - 81x^{3} + 1395x^{2} + 10233x - 1620$
906.2.a.m	$x^{4} - 49x^{3} - 77x^{2} + 23433x - 143924$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(906))$	25
Old forms	124
Cusp forms $S_{2}(\Gamma_0(906))$	149
Eisenstein series $E_{2}(\Gamma_0(906))$	7
Modular forms $M_{2}(\Gamma_0(906))$	156

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(906)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(151))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(302))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(453))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.599 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, 6, 151, 302, 453, 906]: [0, 1, 1, 3, 13, 40, 52, 156] (certified at ell = 2147482951)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 6 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(906, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 906:2 1000

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