Atlas › Modular forms › Weight 2 › Level 185 › 185.2.a

Newspace 185.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(185))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
185.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - q^{5} - 2q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$	185a1 rank 1
185.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + q^{5} - 3q^{7} + O(q^{8})$	− −	$-1$	185b1 rank 1
185.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} - q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$	185c1 rank 1
185.2.a.d	5	degree 5	$5q - q^{3} + 6q^{4} + 5q^{5} - 2q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$
185.2.a.e	5	degree 5	$5q + 2q^{2} + 3q^{3} + 10q^{4} - 5q^{5} - 6q^{6} + 11q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
185.2.a.a	$x - 1$
185.2.a.b	$x + 3$
185.2.a.c	$x + 5$
185.2.a.d	$x^{5} + 3x^{4} - 74x^{3} - 52x^{2} + 1055x - 701$
185.2.a.e	$x^{5} - 11x^{4} - 26x^{3} + 452x^{2} + 107x - 4707$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(185))$	13
Old forms	4
Cusp forms $S_{2}(\Gamma_0(185))$	17
Eisenstein series $E_{2}(\Gamma_0(185))$	3
Modular forms $M_{2}(\Gamma_0(185))$	20

Decomposition of the old subspace

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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]: [0, 1, 3, 20] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 2 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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 185:2 1000

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