Atlas › Modular forms › Weight 2 › Level 191 › 191.2.a › 191.2.a.b

Newform orbit 191.2.a.b

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 2q^{3} + 18q^{4} + q^{5} + 5q^{6} + 3q^{7} - 3q^{8} + 22q^{9} + 8q^{10} - 3q^{11} - 3q^{12} + 19q^{13} - 10q^{14} - 7q^{15} + 18q^{16} + 14q^{17} - 17q^{18} + 20q^{19} - 19q^{20} + 9q^{21} - 7q^{22} - 13q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 2x^{13} - 58x^{12} + 131x^{11} + 1174x^{10} - 3033x^{9} - 10116x^{8} + 31155x^{7} + 31485x^{6} - 140174x^{5} + 15063x^{4} + 215706x^{3} - 141607x^{2} - 12225x + 12211$

Atkin–Lehner signs

$p$	$w_p$
191	$-1$

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
$\operatorname{tr} a_p$	0	2	1	3	−3	19	14	20	−13	6	15	−8	16	4	−25	−5	8	44	−16	−25	58	24	−16	14	22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.166 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 191]: [0, 17] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (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(191, 2)[1]   # 191.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 191:2 1000