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

Newform orbit 194.4.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 6q^{2} - 8q^{3} + 12q^{4} - 11q^{5} - 16q^{6} - 23q^{7} + 24q^{8} - 27q^{9} - 22q^{10} - 73q^{11} - 32q^{12} - 70q^{13} - 46q^{14} + q^{15} + 48q^{16} - 75q^{17} - 54q^{18} - 103q^{19} - 44q^{20} - 70q^{21} - 146q^{22} - 94q^{23} - 64q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{3} + 19x^{2} + 84x - 45$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
97	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	6	−8	−11	−23	−73	−70	−75	−103	−94	−127	82	−302	−17	111	−33	−14	797	167	743	813	−501	−666	−70	−352	−291

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

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)[0]   # 194.4.a.a

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

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