Atlas › Modular forms › Weight 4 › Level 146 › 146.4.a

Newspace 146.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(146))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
146.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{3} + 4q^{4} - 6q^{5} - 8q^{6} - 8q^{7} + O(q^{8})$	+ −	$-1$
146.4.a.b	2	$\mathbb{Q}(\sqrt{61})$	$2q - 4q^{2} - 3q^{3} + 8q^{4} + 7q^{5} + 6q^{6} - 20q^{7} + O(q^{8})$	+ −	$-1$
146.4.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q + 4q^{2} - 3q^{3} + 8q^{4} - 19q^{5} - 6q^{6} - 28q^{7} + O(q^{8})$	− +	$-1$
146.4.a.d	6	degree 6	$6q - 12q^{2} + q^{3} + 24q^{4} - 9q^{5} - 2q^{6} + 28q^{7} + O(q^{8})$	+ +	$+1$
146.4.a.e	7	degree 7	$7q + 14q^{2} + 9q^{3} + 28q^{4} + 21q^{5} + 18q^{6} + 28q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
146.4.a.a	$x + 14$
146.4.a.b	$x^{2} - 18x - 163$
146.4.a.c	$x^{2} + 60x + 263$
146.4.a.d	$x^{6} + 26x^{5} - 7033x^{4} - 130020x^{3} + 12472798x^{2} + 60849762x - 4211157314$
146.4.a.e	$x^{7} - 72x^{6} - 1753x^{5} + 175336x^{4} + 218150x^{3} - 84020762x^{2} - 496498304x + 1285640968$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(146))$	18
Old forms	36
Cusp forms $S_{4}(\Gamma_0(146))$	54
Eisenstein series $E_{4}(\Gamma_0(146))$	4
Modular forms $M_{4}(\Gamma_0(146))$	58

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(146)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.123 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, 73, 146]: [1, 2, 20, 58] (certified at ell = 2147482873)
T = [(3, 1), (5, 3)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 4 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(146, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 146:4 1000

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