Atlas › Modular forms › Weight 2 › Level 814 › 814.2.a

Newspace 814.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(814))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
814.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - 3q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ − −	$-1$	814a1 rank 1
814.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - 4q^{7} + O(q^{8})$	− + −	$-1$	814b1 rank 1
814.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	+ − −	$-1$
814.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 4q^{3} + 2q^{4} - 2q^{5} - 4q^{6} + O(q^{8})$	− + −	$-1$
814.2.a.e	3	degree 3	$3q - 3q^{2} + 3q^{4} - 5q^{5} + 2q^{7} + O(q^{8})$	+ + +	$-1$
814.2.a.f	3	degree 3	$3q + 3q^{2} - 2q^{3} + 3q^{4} - 7q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$
814.2.a.g	4	degree 4	$4q - 4q^{2} - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ − +	$+1$
814.2.a.h	4	degree 4	$4q + 4q^{2} + 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$
814.2.a.i	4	degree 4	$4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$
814.2.a.j	5	degree 5	$5q - 5q^{2} + 5q^{4} + 6q^{5} - 6q^{7} + O(q^{8})$	+ + −	$+1$

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 $T$
814.2.a.a	$x + 11$
814.2.a.b	$x + 3$
814.2.a.c	$x^{2} - 6x + 1$
814.2.a.d	$x^{2} + 10x + 17$
814.2.a.e	$x^{3} + 15x^{2} + 47x - 67$
814.2.a.f	$x^{3} + 23x^{2} + 119x - 147$
814.2.a.g	$x^{4} - 4x^{3} - 90x^{2} - 92x + 13$
814.2.a.h	$x^{4} - 8x^{3} - 58x^{2} + 468x - 31$
814.2.a.i	$x^{4} - 16x^{3} + 34x^{2} + 224x + 49$
814.2.a.j	$x^{5} - 18x^{4} - 54x^{3} + 2164x^{2} - 8331x + 4442$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(814))$	29
Old forms	82
Cusp forms $S_{2}(\Gamma_0(814))$	111
Eisenstein series $E_{2}(\Gamma_0(814))$	7
Modular forms $M_{2}(\Gamma_0(814))$	118

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(814)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(74))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(407))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.403 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, 22, 37, 74, 407, 814]: [0, 1, 2, 5, 3, 11, 40, 118] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 29 = 1 x 29 (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(814, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 814:2 1000

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