Atlas › Modular forms › Weight 2 › Level 812 › 812.2.a

Newspace 812.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(812))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
812.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + q^{7} + O(q^{8})$	− +	812b1 rank 1
812.2.a.b	1	$\mathbb{Q}$	$q + 3q^{3} - q^{7} + O(q^{8})$	+ +	812a1 rank 0
812.2.a.c	2	$\mathbb{Q}(\sqrt{33})$	$2q - 2q^{3} - q^{5} - 2q^{7} + O(q^{8})$	+ +
812.2.a.d	4	degree 4	$4q - 3q^{3} - q^{5} - 4q^{7} + O(q^{8})$	+ −
812.2.a.e	6	degree 6	$6q + 3q^{3} - q^{5} + 6q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{11} + 7T_{13} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
812.2.a.a	$x + 231$
812.2.a.b	$x - 40$
812.2.a.c	$x^{2} - 333x + 26724$
812.2.a.d	$x^{4} + 531x^{3} + 77561x^{2} - 108583x - 465889206$
812.2.a.e	$x^{6} - 645x^{5} + 104383x^{4} - 988449x^{3} - 342648472x^{2} - 4933703822x + 30023822908$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(812))$	14
Old forms	101
Cusp forms $S_{2}(\Gamma_0(812))$	115
Eisenstein series $E_{2}(\Gamma_0(812))$	11
Modular forms $M_{2}(\Gamma_0(812))$	126

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(812)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(116))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(203))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(406))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.016 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, 7, 14, 28, 29, 58, 116, 203, 406, 812]: [0, 1, 2, 1, 4, 7, 3, 9, 18, 22, 64, 126] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (17, 3), (19, 4), (23, 5), (11, 6), (13, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 14 = 1 x 14 (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(812, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 812:2 1000

← 811.2.a · 813.2.a →