Atlas › Modular forms › Weight 2 › Level 746 › 746.2.a

Newspace 746.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(746))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
746.2.a.a	4	degree 4	$4q + 4q^{2} - 3q^{3} + 4q^{4} - 7q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
746.2.a.b	4	degree 4	$4q + 4q^{2} - 3q^{3} + 4q^{4} + 3q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$
746.2.a.c	6	degree 6	$6q - 6q^{2} - 3q^{3} + 6q^{4} + q^{5} + 3q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
746.2.a.d	8	degree 8	$8q + 8q^{2} + 3q^{3} + 8q^{4} + 3q^{5} + 3q^{6} + 8q^{7} + O(q^{8})$	− +	$+1$
746.2.a.e	10	degree 10	$10q - 10q^{2} + 4q^{3} + 10q^{4} - 4q^{5} - 4q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
746.2.a.a	$x^{4} + 10x^{3} + 34x^{2} + 45x + 19$
746.2.a.b	$x^{4} - 22x^{2} - 45x - 23$
746.2.a.c	$x^{6} + 2x^{5} - 18x^{4} - 5x^{3} + 83x^{2} - 84x + 20$
746.2.a.d	$x^{8} - 6x^{7} - 25x^{6} + 288x^{5} - 821x^{4} + 749x^{3} + 353x^{2} - 736x + 116$
746.2.a.e	$x^{10} - 57x^{8} - 17x^{7} + 1044x^{6} + 362x^{5} - 7216x^{4} - 1699x^{3} + 16292x^{2} + 2308x - 11009$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(746))$	32
Old forms	60
Cusp forms $S_{2}(\Gamma_0(746))$	92
Eisenstein series $E_{2}(\Gamma_0(746))$	3
Modular forms $M_{2}(\Gamma_0(746))$	95

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(746)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(373))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.236 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, 373, 746]: [0, 1, 31, 95] (certified at ell = 2147478997)
T = [(3, 1), (5, 1)]; h of degree 32 = 1 x 32 (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(746, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 746:2 1000

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