Atlas › Modular forms › Weight 2 › Level 946 › 946.2.a

Newspace 946.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(946))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
946.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} + O(q^{8})$	+ + −	$+1$	946a1 rank 0
946.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - 4q^{7} + O(q^{8})$	+ − −	$-1$	946b1 rank 1
946.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 4q^{5} + q^{6} + O(q^{8})$	− + +	$+1$	946c1 rank 0
946.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 3q^{3} + 2q^{4} - 5q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	+ − −	$-1$
946.2.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} + q^{3} + 2q^{4} - q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$
946.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 3q^{3} + 2q^{4} - 3q^{5} - 3q^{6} + O(q^{8})$	− − +	$-1$
946.2.a.g	4	degree 4	$4q + 4q^{2} - 4q^{3} + 4q^{4} - 6q^{5} - 4q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
946.2.a.h	4	degree 4	$4q + 4q^{2} + q^{3} + 4q^{4} + q^{5} + q^{6} + O(q^{8})$	− + +	$+1$
946.2.a.i	5	degree 5	$5q - 5q^{2} + q^{3} + 5q^{4} - 2q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
946.2.a.j	6	degree 6	$6q - 6q^{2} + 2q^{3} + 6q^{4} + 4q^{5} - 2q^{6} + O(q^{8})$	+ − +	$+1$
946.2.a.k	7	degree 7	$7q + 7q^{2} + 3q^{3} + 7q^{4} + 8q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
946.2.a.a	$x - 6$
946.2.a.b	$x - 1$
946.2.a.c	$x - 13$
946.2.a.d	$x^{2} + 18x + 61$
946.2.a.e	$x^{2} + 2x - 51$
946.2.a.f	$x^{2} + 12x + 31$
946.2.a.g	$x^{4} + 22x^{3} + 82x^{2} - 366x + 249$
946.2.a.h	$x^{4} - 4x^{3} - 53x^{2} + 260x - 104$
946.2.a.i	$x^{5} + 5x^{4} - 100x^{3} - 308x^{2} + 1815x + 1331$
946.2.a.j	$x^{6} - 14x^{5} - 118x^{4} + 1596x^{3} + 3483x^{2} - 26894x - 37706$
946.2.a.k	$x^{7} - 27x^{6} + 136x^{5} + 2038x^{4} - 21357x^{3} + 9563x^{2} + 463920x - 1218506$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(946))$	35
Old forms	94
Cusp forms $S_{2}(\Gamma_0(946))$	129
Eisenstein series $E_{2}(\Gamma_0(946))$	7
Modular forms $M_{2}(\Gamma_0(946))$	136

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(946)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(473))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.444 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, 11, 22, 43, 86, 473, 946]: [0, 1, 2, 5, 4, 13, 46, 136] (certified at ell = 2147482681)
T = [(3, 1), (5, 3)]; h of degree 35 = 1 x 35 (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(946, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 946:2 1000

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