Atlas › Modular forms › Weight 2 › Level 740 › 740.2.a

Newspace 740.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(740))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
740.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + q^{7} + O(q^{8})$	− +	740c1 rank 1
740.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - q^{7} + O(q^{8})$	+ −	740b1 rank 1
740.2.a.c	1	$\mathbb{Q}$	$q + 3q^{3} - q^{5} - 3q^{7} + O(q^{8})$	+ +	740a1 rank 0
740.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{5} + 6q^{7} + O(q^{8})$	− −
740.2.a.e	3	degree 3	$3q - 3q^{5} + 2q^{7} + O(q^{8})$	+ +
740.2.a.f	4	degree 4	$4q + 3q^{3} + 4q^{5} - 5q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17} + 11T_{19}$ 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 $T$
740.2.a.a	$x + 55$
740.2.a.b	$x + 89$
740.2.a.c	$x - 25$
740.2.a.d	$x^{2} - 2x - 1082$
740.2.a.e	$x^{3} - 316x^{2} + 33172x - 1156622$
740.2.a.f	$x^{4} - 61x^{3} - 6960x^{2} + 199976x + 4570078$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(740))$	12
Old forms	97
Cusp forms $S_{2}(\Gamma_0(740))$	109
Eisenstein series $E_{2}(\Gamma_0(740))$	11
Modular forms $M_{2}(\Gamma_0(740))$	120

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(740)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(74))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(148))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(185))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(370))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.395 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, 4, 5, 10, 20, 37, 74, 148, 185, 370, 740]: [0, 1, 2, 1, 3, 6, 3, 11, 22, 20, 60, 120] (certified at ell = 2147483629)
T = [(3, 1), (7, 3), (11, 5), (13, 7), (17, 9), (19, 11)]; h of degree 12 = 1 x 12 (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(740, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 740:2 1000

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