Atlas › Modular forms › Weight 2 › Level 836 › 836.2.a

Newspace 836.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(836))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
836.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + O(q^{8})$	+ −	836a1 rank 1
836.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{5} + O(q^{8})$	− −	836b1 rank 0
836.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} + O(q^{8})$	− +
836.2.a.d	6	degree 6	$6q - 2q^{3} + 2q^{7} + O(q^{8})$	− −
836.2.a.e	6	degree 6	$6q + q^{3} + 5q^{5} - 2q^{7} + O(q^{8})$	+ +

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{17} + 9T_{23} + 11T_{29}$ 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 $T$
836.2.a.a	$x + 39$
836.2.a.b	$x - 46$
836.2.a.c	$x^{2} + 114x + 2857$
836.2.a.d	$x^{6} - 118x^{5} - 10501x^{4} + 1124924x^{3} + 44895719x^{2} - 2578090150x - 92100848579$
836.2.a.e	$x^{6} - 147x^{5} - 5860x^{4} + 2192634x^{3} - 146326887x^{2} + 3846597193x - 34151150182$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(836))$	16
Old forms	99
Cusp forms $S_{2}(\Gamma_0(836))$	115
Eisenstein series $E_{2}(\Gamma_0(836))$	11
Modular forms $M_{2}(\Gamma_0(836))$	126

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(836)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(76))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(209))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(418))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.462 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, 11, 19, 22, 38, 44, 76, 209, 418, 836]: [0, 1, 2, 2, 2, 5, 7, 9, 13, 22, 64, 126] (certified at ell = 2147482801)
T = [(3, 1), (5, 3), (7, 5), (17, 7), (23, 9), (29, 11)]; h of degree 16 = 1 x 16 (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(836, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 836:2 1000

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