Atlas › Modular forms › Weight 2 › Level 801 › 801.2.a

Newspace 801.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(801))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
801.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	+	801b1 rank 0
801.2.a.b	1	$\mathbb{Q}$	$q - 2q^{4} - 4q^{5} - 2q^{7} + O(q^{8})$	+	801a1 rank 0
801.2.a.c	1	$\mathbb{Q}$	$q - 2q^{4} + 2q^{7} + O(q^{8})$	−	801c1 rank 1
801.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - 4q^{7} + O(q^{8})$	−	801d1 rank 1
801.2.a.e	3	degree 3	$3q - 2q^{2} + 4q^{4} - 5q^{5} - 4q^{7} + O(q^{8})$	−
801.2.a.f	3	degree 3	$3q + 3q^{5} - 6q^{7} + O(q^{8})$	−
801.2.a.g	3	degree 3	$3q + 4q^{2} + 4q^{4} + 7q^{5} - 4q^{7} + O(q^{8})$	+
801.2.a.h	4	degree 4	$4q - q^{2} + 7q^{4} - 3q^{5} + 6q^{7} + O(q^{8})$	+
801.2.a.i	5	degree 5	$5q + q^{2} + 11q^{4} + q^{5} + 8q^{7} + O(q^{8})$	+
801.2.a.j	7	degree 7	$7q - 4q^{2} + 6q^{4} - 4q^{5} + O(q^{8})$	+
801.2.a.k	7	degree 7	$7q + 4q^{2} + 6q^{4} + 4q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ 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 $T$
801.2.a.a	$x + 31$
801.2.a.b	$x + 20$
801.2.a.c	$x + 60$
801.2.a.d	$x + 100$
801.2.a.e	$x^{3} + 49x^{2} + 120x - 10375$
801.2.a.f	$x^{3} + 141x^{2} + 1818x - 171449$
801.2.a.g	$x^{3} - 63x^{2} - 5446x - 64673$
801.2.a.h	$x^{4} - 130x^{3} + 3353x^{2} + 64029x - 458959$
801.2.a.i	$x^{5} - 164x^{4} - 5050x^{3} + 1503422x^{2} - 42435402x - 347623574$
801.2.a.j	$x^{7} + 312x^{6} + 27809x^{5} - 145304x^{4} - 132605208x^{3} - 5371070292x^{2} - 9883960471x + 1863538633576$
801.2.a.k	$x^{7} - 208x^{6} + 2893x^{5} + 1825872x^{4} - 152590792x^{3} + 5013725552x^{2} - 73997231891x + 404158348504$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(801))$	36
Old forms	51
Cusp forms $S_{2}(\Gamma_0(801))$	87
Eisenstein series $E_{2}(\Gamma_0(801))$	7
Modular forms $M_{2}(\Gamma_0(801))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(801)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(89))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(267))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.473 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, 9, 89, 267, 801]: [0, 1, 2, 8, 32, 92] (certified at ell = 2147482921)
T = [(2, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 10 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(801, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 801:2 1000

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