Atlas › Modular forms › Weight 2 › Level 374 › 374.2.a

Newspace 374.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(374))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
374.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{7} + O(q^{8})$	+ + +	$-1$	374a1 rank 1
374.2.a.b	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	+ + −	$+1$
374.2.a.c	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} - q^{5} + 3q^{6} + 7q^{7} + O(q^{8})$	− + +	$+1$
374.2.a.d	4	degree 4	$4q - 4q^{2} + q^{3} + 4q^{4} + 5q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$
374.2.a.e	4	degree 4	$4q + 4q^{2} + q^{3} + 4q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	− − −	$+1$

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 $T$
374.2.a.a	$x$
374.2.a.b	$x^{3} + 4x^{2} - 93x - 263$
374.2.a.c	$x^{3} - 65x + 193$
374.2.a.d	$x^{4} - 16x^{3} - 49x^{2} + 1671x - 5804$
374.2.a.e	$x^{4} + 2x^{3} - 157x^{2} + 73x + 710$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(374))$	15
Old forms	36
Cusp forms $S_{2}(\Gamma_0(374))$	51
Eisenstein series $E_{2}(\Gamma_0(374))$	7
Modular forms $M_{2}(\Gamma_0(374))$	58

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(374)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(187))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 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, 17, 22, 34, 187, 374]: [0, 1, 2, 2, 5, 6, 20, 58] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 3 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(374, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 374:2 1000

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