Atlas › Modular forms › Weight 4 › Level 194 › 194.4.a

Newspace 194.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(194))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
194.4.a.a	3	degree 3	$3q + 6q^{2} - 8q^{3} + 12q^{4} - 11q^{5} - 16q^{6} - 23q^{7} + O(q^{8})$	− +	$-1$
194.4.a.b	4	degree 4	$4q - 8q^{2} - 4q^{3} + 16q^{4} + 9q^{5} + 8q^{6} - 23q^{7} + O(q^{8})$	+ −	$-1$
194.4.a.c	8	degree 8	$8q - 16q^{2} + 2q^{3} + 32q^{4} - q^{5} - 4q^{6} + 19q^{7} + O(q^{8})$	+ +	$+1$
194.4.a.d	9	degree 9	$9q + 18q^{2} + 10q^{3} + 36q^{4} + 29q^{5} + 20q^{6} + 19q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
194.4.a.a	$x^{3} + 19x^{2} + 84x - 45$
194.4.a.b	$x^{4} - 5x^{3} - 156x^{2} + 153x + 3186$
194.4.a.c	$x^{8} - x^{7} - 1036x^{6} + 2999x^{5} + 330684x^{4} - 1452932x^{3} - 31069784x^{2} + 161480068x + 289222064$
194.4.a.d	$x^{9} - 39x^{8} - 288x^{7} + 26819x^{6} - 226468x^{5} - 3175186x^{4} + 57843464x^{3} - 267690036x^{2} + 134631904x + 1021518704$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(194))$	24
Old forms	48
Cusp forms $S_{4}(\Gamma_0(194))$	72
Eisenstein series $E_{4}(\Gamma_0(194))$	4
Modular forms $M_{4}(\Gamma_0(194))$	76

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(194)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(97))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.133 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]: [1, 2, 26, 76] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 24 = 1 x 24 (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(194, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 194:4 1000

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