Atlas › Modular forms › Weight 2 › Level 832 › 832.2.a

Newspace 832.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(832))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
832.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + q^{5} - q^{7} + O(q^{8})$	−	832j1 rank 1
832.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + 3q^{7} + O(q^{8})$	+	832b1 rank 1
832.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + q^{5} + 5q^{7} + O(q^{8})$	−	832e1 rank 0
832.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + 3q^{5} - q^{7} + O(q^{8})$	+	832c1 rank 1
832.2.a.e	1	$\mathbb{Q}$	$q - 2q^{5} - 2q^{7} + O(q^{8})$	−	832d1 rank 0
832.2.a.f	1	$\mathbb{Q}$	$q - 2q^{5} + 2q^{7} + O(q^{8})$	−	832h1 rank 1
832.2.a.g	1	$\mathbb{Q}$	$q + q^{3} - q^{5} - 3q^{7} + O(q^{8})$	+	832a1 rank 1
832.2.a.h	1	$\mathbb{Q}$	$q + q^{3} + q^{5} - 5q^{7} + O(q^{8})$	−	832i1 rank 1
832.2.a.i	1	$\mathbb{Q}$	$q + q^{3} + 3q^{5} + q^{7} + O(q^{8})$	+	832g1 rank 0
832.2.a.j	1	$\mathbb{Q}$	$q + 3q^{3} + q^{5} + q^{7} + O(q^{8})$	−	832f1 rank 0
832.2.a.k	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} - 3q^{5} - q^{7} + O(q^{8})$	+
832.2.a.l	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} + 3q^{5} + 3q^{7} + O(q^{8})$	−
832.2.a.m	2	$\mathbb{Q}(\sqrt{5})$	$2q - 6q^{5} + O(q^{8})$	−
832.2.a.n	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} - 3q^{5} + q^{7} + O(q^{8})$	+
832.2.a.o	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{3} + 3q^{5} - 3q^{7} + O(q^{8})$	−
832.2.a.p	4	degree 4	$4q + 2q^{5} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17} + 9T_{19}$ 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 $T$
832.2.a.a	$x - 23$
832.2.a.b	$x + 53$
832.2.a.c	$x - 9$
832.2.a.d	$x + 61$
832.2.a.e	$x - 100$
832.2.a.f	$x + 28$
832.2.a.g	$x - 5$
832.2.a.h	$x + 45$
832.2.a.i	$x - 37$
832.2.a.j	$x + 59$
832.2.a.k	$x^{2} + 25x - 800$
832.2.a.l	$x^{2} + 99x + 1936$
832.2.a.m	$x^{2} + 60x - 2745$
832.2.a.n	$x^{2} + 7x - 4616$
832.2.a.o	$x^{2} - 159x + 5976$
832.2.a.p	$x^{4} - 104x^{3} + 3315x^{2} - 39480x + 135632$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(832))$	24
Old forms	77
Cusp forms $S_{2}(\Gamma_0(832))$	101
Eisenstein series $E_{2}(\Gamma_0(832))$	23
Modular forms $M_{2}(\Gamma_0(832))$	124

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(832)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(104))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(208))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(416))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.449 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, 8, 13, 16, 26, 32, 52, 64, 104, 208, 416, 832]: [0, 1, 2, 3, 1, 4, 5, 6, 10, 10, 18, 32, 60, 116] (certified at ell = 2147483137)
T = [(3, 1), (5, 3), (11, 5), (17, 7), (19, 9)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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(832, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 832:2 1000

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