Atlas › Modular forms › Weight 2 › Level 753 › 753.2.a

Newspace 753.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(753))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
753.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{5} - 5q^{7} + O(q^{8})$	− −	$-1$	753c1 rank 1
753.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 3q^{5} - q^{7} + O(q^{8})$	− +	$+1$	753b1 rank 0
753.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} + 3q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	753a1 rank 0
753.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 6q^{5} - 2q^{7} + O(q^{8})$	− −	$-1$
753.2.a.e	4	degree 4	$4q + 2q^{2} + 4q^{3} + 6q^{4} + 4q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$
753.2.a.f	6	degree 6	$6q - 6q^{2} + 6q^{3} + 8q^{4} - 4q^{5} - 6q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
753.2.a.g	6	degree 6	$6q + 2q^{2} + 6q^{3} + 4q^{4} + 2q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$
753.2.a.h	9	degree 9	$9q - 9q^{3} + 6q^{4} + q^{5} - 14q^{7} + O(q^{8})$	+ +	$-1$
753.2.a.i	11	degree 11	$11q - 3q^{2} - 11q^{3} + 13q^{4} - 4q^{5} + 3q^{6} + 19q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
753.2.a.a	$x - 1$
753.2.a.b	$x - 3$
753.2.a.c	$x - 5$
753.2.a.d	$x^{2} + 6x + 1$
753.2.a.e	$x^{4} - 6x^{3} + 2x^{2} + 18x - 7$
753.2.a.f	$x^{6} + 10x^{5} + 32x^{4} + 21x^{3} - 74x^{2} - 138x - 67$
753.2.a.g	$x^{6} - 2x^{5} - 16x^{4} + 27x^{3} + 56x^{2} - 94x + 11$
753.2.a.h	$x^{9} - x^{8} - 23x^{7} - 2x^{6} + 75x^{5} - 7x^{4} - 64x^{3} + 31x^{2} - x - 1$
753.2.a.i	$x^{11} + 7x^{10} - 30x^{9} - 261x^{8} + 196x^{7} + 3209x^{6} + 897x^{5} - 14852x^{4} - 9709x^{3} + 18704x^{2} + 10949x + 1385$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(753))$	41
Old forms	42
Cusp forms $S_{2}(\Gamma_0(753))$	83
Eisenstein series $E_{2}(\Gamma_0(753))$	3
Modular forms $M_{2}(\Gamma_0(753))$	86

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(753)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(251))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.244 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, 3, 251, 753]: [0, 1, 22, 86] (certified at ell = 2147482501)
T = [(2, 1), (5, 1)]; h of degree 41 = 1 x 41 (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(753, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 753:2 1000

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