Atlas › Modular forms › Weight 2 › Level 804 › 804.2.a

Newspace 804.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(804))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
804.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 3q^{5} + 3q^{7} + O(q^{8})$	+ −	804c1 rank 1
804.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + O(q^{8})$	+ −	804b1 rank 1
804.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 4q^{5} + O(q^{8})$	+ +	804a1 rank 0
804.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 3q^{7} + O(q^{8})$	− +	804d1 rank 1
804.2.a.e	3	degree 3	$3q - 3q^{3} - 3q^{5} - 5q^{7} + O(q^{8})$	+ +
804.2.a.f	5	degree 5	$5q + 5q^{3} + 3q^{5} + 5q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ 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 $T$
804.2.a.a	$x + 266$
804.2.a.b	$x + 134$
804.2.a.c	$x + 20$
804.2.a.d	$x + 204$
804.2.a.e	$x^{3} - 578x^{2} + 104962x - 6115782$
804.2.a.f	$x^{5} - 458x^{4} + 41850x^{3} + 3578090x^{2} - 314444604x - 16300444352$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(804))$	12
Old forms	119
Cusp forms $S_{2}(\Gamma_0(804))$	131
Eisenstein series $E_{2}(\Gamma_0(804))$	11
Modular forms $M_{2}(\Gamma_0(804))$	142

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(804)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(134))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(201))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(268))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(402))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.046 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, 3, 4, 6, 12, 67, 134, 201, 268, 402, 804]: [0, 1, 1, 2, 3, 5, 6, 19, 24, 37, 72, 142] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 12 = 1 x 12 (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(804, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 804:2 1000

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