Atlas › Modular forms › Weight 2 › Level 194 › 194.2.a

Newspace 194.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(194))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
194.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 4q^{5} - 4q^{7} + O(q^{8})$	− +	$+1$	194a1 rank 0
194.2.a.b	4	degree 4	$4q - 4q^{2} + 2q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
194.2.a.c	4	degree 4	$4q + 4q^{2} + 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$

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 $T$
194.2.a.a	$x - 12$
194.2.a.b	$x^{4} + 4x^{3} - 186x^{2} - 576x + 2116$
194.2.a.c	$x^{4} + 4x^{3} - 24x^{2} - 24x + 92$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(194))$	9
Old forms	14
Cusp forms $S_{2}(\Gamma_0(194))$	23
Eisenstein series $E_{2}(\Gamma_0(194))$	3
Modular forms $M_{2}(\Gamma_0(194))$	26

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(194)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(97))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.088 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, 97, 194]: [0, 1, 8, 26] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 9 = 1 x 9 (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(194, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 194:2 1000

← 193.2.a · 195.2.a →