Atlas › Modular forms › Weight 2 › Level 432 › 432.2.a

Newspace 432.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(432))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
432.2.a.a	1	$\mathbb{Q}$	$q - 4q^{5} + 3q^{7} + O(q^{8})$		432h1 rank 0
432.2.a.b	1	$\mathbb{Q}$	$q - 3q^{5} + q^{7} + O(q^{8})$		432f1 rank 1
432.2.a.c	1	$\mathbb{Q}$	$q - q^{5} - 3q^{7} + O(q^{8})$		432d1 rank 1
432.2.a.d	1	$\mathbb{Q}$	$q - 5q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	432b1 rank 1
432.2.a.e	1	$\mathbb{Q}$	$q + q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	432a1 rank 0
432.2.a.f	1	$\mathbb{Q}$	$q + q^{5} - 3q^{7} + O(q^{8})$		432c1 rank 0
432.2.a.g	1	$\mathbb{Q}$	$q + 3q^{5} + q^{7} + O(q^{8})$		432e1 rank 0
432.2.a.h	1	$\mathbb{Q}$	$q + 4q^{5} + 3q^{7} + O(q^{8})$		432g1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{11} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43} + 13T_{47} + 14T_{53} + 15T_{59} + 16T_{61}$ 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 $T$
432.2.a.a	$x - 56$
432.2.a.b	$x - 290$
432.2.a.c	$x + 148$
432.2.a.d	$x + 304$
432.2.a.e	$x - 86$
432.2.a.f	$x + 62$
432.2.a.g	$x - 116$
432.2.a.h	$x + 104$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(432))$	8
Old forms	47
Cusp forms $S_{2}(\Gamma_0(432))$	55
Eisenstein series $E_{2}(\Gamma_0(432))$	35
Modular forms $M_{2}(\Gamma_0(432))$	90

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.126 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, 36, 48, 54, 72, 108, 144, 216, 432]: [0, 1, 1, 2, 3, 3, 2, 5, 4, 5, 8, 4, 9, 12, 11, 16, 21, 28, 40, 76] (certified at ell = 2147483629)
T = [(5, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12), (47, 13), (53, 14), (59, 15), (61, 16)]; h of degree 8 = 1 x 8 (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(432, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 432:2 1000

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