Atlas › Modular forms › Weight 4 › Level 181 › 181.4.a › 181.4.a.c

Newform orbit 181.4.a.c

Trace form

The 25 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$25q + 7q^{2} + 19q^{3} + 125q^{4} + 26q^{5} + 31q^{6} + 31q^{7} + 87q^{8} + 284q^{9} + 49q^{10} + 339q^{11} + 140q^{12} + 28q^{13} + 257q^{14} + 170q^{15} + 605q^{16} + 242q^{17} + 170q^{18} + 248q^{19} + 306q^{20} + 144q^{21} + 200q^{22} + 367q^{23} + 380q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 25. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{25} - 26x^{24} - 335x^{23} + 13356x^{22} + 13793x^{21} - 2761194x^{20} + 7175070x^{19} + 312123736x^{18} - 1310490834x^{17} - 21949902219x^{16} + 107909056847x^{15} + 1028594412811x^{14} - 5216991294707x^{13} - 33271575342744x^{12} + 159277395665534x^{11} + 749407470361645x^{10} - 3121876357611145x^{9} - 11554392233833243x^{8} + 38635214612499724x^{7} + 116704408862224470x^{6} - 286867673425176732x^{5} - 716308030002589104x^{4} + 1138532105646240224x^{3} + 2345117519308738016x^{2} - 1773278413614323648x - 3070578049436269312$

Atkin–Lehner signs

$p$	$w_p$
181	$+1$

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
$\operatorname{tr} a_p$	7	19	26	31	339	28	242	248	367	750	153	376	572	1319	337	697	2225	−865	−615	3829	−1301	634	2500	638	−4152

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.132 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, 181]: [1, 47] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 7 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(181, 4)[2]   # 181.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 181:4 1000