Atlas › Modular forms › Weight 2 › Level 942 › 942.2.a

Newspace 942.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(942))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
942.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$	942a1 rank 0
942.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	942b1 rank 0
942.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 4q^{5} + q^{6} - q^{7} + O(q^{8})$	− − +	$-1$	942d1 rank 1
942.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} - 5q^{7} + O(q^{8})$	− − +	$-1$	942c1 rank 1
942.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	+ + +	$-1$
942.2.a.f	2	$\mathbb{Q}(\sqrt{7})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ − +	$+1$
942.2.a.g	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} - 2q^{6} + 5q^{7} + O(q^{8})$	− + +	$+1$
942.2.a.h	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 6q^{5} - 3q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$
942.2.a.i	3	degree 3	$3q + 3q^{2} - 3q^{3} + 3q^{4} - 4q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− + −	$-1$
942.2.a.j	4	degree 4	$4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + q^{7} + O(q^{8})$	− − −	$+1$
942.2.a.k	5	degree 5	$5q - 5q^{2} - 5q^{3} + 5q^{4} + 5q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
942.2.a.a	$x - 8$
942.2.a.b	$x + 5$
942.2.a.c	$x + 7$
942.2.a.d	$x + 17$
942.2.a.e	$x^{2} + 6x + 7$
942.2.a.f	$x^{2} + 4x - 3$
942.2.a.g	$x^{2} - 19x + 52$
942.2.a.h	$x^{3} + 3x^{2} - 31x - 95$
942.2.a.i	$x^{3} + 19x^{2} + 59x - 17$
942.2.a.j	$x^{4} - 7x^{3} - 65x^{2} + 433x - 604$
942.2.a.k	$x^{5} - 9x^{4} - 185x^{3} + 1477x^{2} + 8478x - 60688$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(942))$	25
Old forms	130
Cusp forms $S_{2}(\Gamma_0(942))$	155
Eisenstein series $E_{2}(\Gamma_0(942))$	7
Modular forms $M_{2}(\Gamma_0(942))$	162

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(942)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(157))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(314))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(471))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.402 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, 157, 314, 471, 942]: [0, 1, 1, 3, 13, 41, 54, 162] (certified at ell = 2147483053)
T = [(5, 1), (7, 3)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 4 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(942, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 942:2 1000

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