Atlas › Modular forms › Weight 2 › Level 864 › 864.2.a

Newspace 864.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(864))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
864.2.a.a	1	$\mathbb{Q}$	$q - 2q^{5} - 3q^{7} + O(q^{8})$	864i1 rank 0
864.2.a.b	1	$\mathbb{Q}$	$q - 2q^{5} - q^{7} + O(q^{8})$	864k1 rank 1
864.2.a.c	1	$\mathbb{Q}$	$q - 2q^{5} + q^{7} + O(q^{8})$	864c1 rank 1
864.2.a.d	1	$\mathbb{Q}$	$q - 2q^{5} + 3q^{7} + O(q^{8})$	864l1 rank 1
864.2.a.e	1	$\mathbb{Q}$	$q - q^{5} - 3q^{7} + O(q^{8})$	864a1 rank 1
864.2.a.f	1	$\mathbb{Q}$	$q - q^{5} + 3q^{7} + O(q^{8})$	864d1 rank 0
864.2.a.g	1	$\mathbb{Q}$	$q + q^{5} - 3q^{7} + O(q^{8})$	864j1 rank 1
864.2.a.h	1	$\mathbb{Q}$	$q + q^{5} + 3q^{7} + O(q^{8})$	864g1 rank 0
864.2.a.i	1	$\mathbb{Q}$	$q + 2q^{5} - 3q^{7} + O(q^{8})$	864b1 rank 1
864.2.a.j	1	$\mathbb{Q}$	$q + 2q^{5} - q^{7} + O(q^{8})$	864e1 rank 0
864.2.a.k	1	$\mathbb{Q}$	$q + 2q^{5} + q^{7} + O(q^{8})$	864h1 rank 0
864.2.a.l	1	$\mathbb{Q}$	$q + 2q^{5} + 3q^{7} + O(q^{8})$	864f1 rank 0
864.2.a.m	2	$\mathbb{Q}(\sqrt{13})$	$2q + O(q^{8})$
864.2.a.n	2	$\mathbb{Q}(\sqrt{13})$	$2q + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = 7159T_{5} + 2426T_{7} + 9370T_{13} + 5254T_{17} + 3871T_{19} + 2211T_{11} + 9392T_{23} + 2182T_{29}$ 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 $T$
864.2.a.a	$x - 59489$
864.2.a.b	$x + 88929$
864.2.a.c	$x + 18927$
864.2.a.d	$x + 88417$
864.2.a.e	$x + 70730$
864.2.a.f	$x - 89716$
864.2.a.g	$x - 9722$
864.2.a.h	$x + 28708$
864.2.a.i	$x + 107039$
864.2.a.j	$x - 141527$
864.2.a.k	$x - 3809$
864.2.a.l	$x - 23527$
864.2.a.m	$x^{2} - 183242x + 7539159468$
864.2.a.n	$x^{2} + 33322x + 194968188$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(864))$	16
Old forms	105
Cusp forms $S_{2}(\Gamma_0(864))$	121
Eisenstein series $E_{2}(\Gamma_0(864))$	47
Modular forms $M_{2}(\Gamma_0(864))$	168

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.161 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, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 96, 108, 144, 216, 288, 432, 864]: [0, 1, 1, 2, 3, 3, 2, 5, 4, 5, 8, 4, 6, 9, 12, 11, 16, 20, 21, 28, 40, 52, 76, 148] (certified at ell = 2147482873)
T = [(5, 7159), (7, 2426), (13, 9370), (17, 5254), (19, 3871), (11, 2211), (23, 9392), (29, 2182)]; h of degree 16 = 1 x 16 (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(864, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 864:2 1000

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