Atlas › Modular forms › Weight 4 › Level 185 › 185.4.a

Newspace 185.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(185))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
185.4.a.a	7	degree 7	$7q - 5q^{2} - 6q^{3} + 13q^{4} + 35q^{5} - 40q^{6} - 49q^{7} + O(q^{8})$	− +	$-1$
185.4.a.b	7	degree 7	$7q + q^{2} - 8q^{3} + 17q^{4} - 35q^{5} + 24q^{6} - 77q^{7} + O(q^{8})$	+ −	$-1$
185.4.a.c	10	degree 10	$10q + 7q^{2} + 4q^{3} + 53q^{4} - 50q^{5} + 12q^{6} + 49q^{7} + O(q^{8})$	+ +	$+1$
185.4.a.d	12	degree 12	$12q + q^{2} + 6q^{3} + 73q^{4} + 60q^{5} + 20q^{6} + 77q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
185.4.a.a	$x^{7} + 11x^{6} - 29x^{5} - 353x^{4} + 217x^{3} + 2803x^{2} - 1329x - 4681$
185.4.a.b	$x^{7} + 7x^{6} - 173x^{5} - 453x^{4} + 8881x^{3} - 19533x^{2} - 5505x + 25847$
185.4.a.c	$x^{10} - 11x^{9} - 227x^{8} + 1967x^{7} + 20337x^{6} - 119423x^{5} - 837107x^{4} + 3029001x^{3} + 15453638x^{2} - 27486494x - 99709778$
185.4.a.d	$x^{12} - 7x^{11} - 319x^{10} + 2475x^{9} + 28857x^{8} - 230363x^{7} - 870655x^{6} + 6677881x^{5} + 9870410x^{4} - 60087166x^{3} - 49870498x^{2} + 149122556x + 87557596$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(185))$	36
Old forms	20
Cusp forms $S_{4}(\Gamma_0(185))$	56
Eisenstein series $E_{4}(\Gamma_0(185))$	4
Modular forms $M_{4}(\Gamma_0(185))$	60

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(185)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.127 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, 5, 37, 185]: [1, 3, 11, 60] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 5 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(185, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 185:4 1000

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