Atlas › Modular forms › Weight 2 › Level 190 › 190.2.a

Newspace 190.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(190))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
190.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	190b1 rank 1
190.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} - q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− + −	$-1$	190a1 rank 1
190.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + O(q^{8})$	− − −	$+1$	190c1 rank 0
190.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - q^{3} + 2q^{4} + 2q^{5} + q^{6} - q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
190.2.a.a	$x + 25$
190.2.a.b	$x + 45$
190.2.a.c	$x + 9$
190.2.a.d	$x^{2} - 29x - 1018$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(190))$	5
Old forms	22
Cusp forms $S_{2}(\Gamma_0(190))$	27
Eisenstein series $E_{2}(\Gamma_0(190))$	7
Modular forms $M_{2}(\Gamma_0(190))$	34

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(190)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(95))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.316 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, 5, 10, 19, 38, 95, 190]: [0, 1, 1, 3, 2, 7, 12, 34] (certified at ell = 2147483629)
T = [(3, 1), (7, 3), (11, 5), (13, 7)]; h of degree 5 = 1 x 5 (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(190, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 190:2 1000

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