Atlas › Modular forms › Weight 2 › Level 392 › 392.2.a

Newspace 392.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(392))$

Newforms

Label	Dim	Coefficient field	Trace form	Elliptic curve
392.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} + q^{5} + O(q^{8})$	392f1 rank 1
392.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + 4q^{5} + O(q^{8})$	392d1 rank 0
392.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + O(q^{8})$	392c1 rank 1
392.2.a.d	1	$\mathbb{Q}$	$q - 2q^{5} + O(q^{8})$	392a1 rank 1
392.2.a.e	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + O(q^{8})$	392b1 rank 0
392.2.a.f	1	$\mathbb{Q}$	$q + 3q^{3} - q^{5} + O(q^{8})$	392e1 rank 0
392.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$
392.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
392.2.a.a	$x + 126$
392.2.a.b	$x - 100$
392.2.a.c	$x + 70$
392.2.a.d	$x - 6$
392.2.a.e	$x - 22$
392.2.a.f	$x + 18$
392.2.a.g	$x^{2} + 4x - 1148$
392.2.a.h	$x^{2} - 8x - 2296$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(392))$	10
Old forms	31
Cusp forms $S_{2}(\Gamma_0(392))$	41
Eisenstein series $E_{2}(\Gamma_0(392))$	31
Modular forms $M_{2}(\Gamma_0(392))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(392)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(98))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(196))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.413 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, 7, 8, 14, 28, 49, 56, 98, 196, 392]: [0, 1, 2, 1, 3, 4, 7, 5, 12, 16, 31, 60] (certified at ell = 2147483647)
T = [(3, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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(392, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 392:2 1000

← 391.2.a · 393.2.a →