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

Newform orbit 194.4.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 16q^{2} + 2q^{3} + 32q^{4} - q^{5} - 4q^{6} + 19q^{7} - 64q^{8} + 76q^{9} + 2q^{10} - 7q^{11} + 8q^{12} + 66q^{13} - 38q^{14} + 151q^{15} + 128q^{16} - 77q^{17} - 152q^{18} + 27q^{19} - 4q^{20} + 50q^{21} + 14q^{22} + 136q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{8} - x^{7} - 1036x^{6} + 2999x^{5} + 330684x^{4} - 1452932x^{3} - 31069784x^{2} + 161480068x + 289222064$

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 even 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
$\operatorname{tr} a_p$	−16	2	−1	19	−7	66	−77	27	136	183	1202	672	421	893	827	−556	−749	493	−359	−927	−669	1282	−1268	−2044	−776

$\operatorname{tr} a_n$ 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)[2]   # 194.4.a.c

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

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