Atlas › Modular forms › Weight 4 › Level 191 › 191.4.a › 191.4.a.b

Newform orbit 191.4.a.b

Trace form

The 30 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$30q + 4q^{2} + 13q^{3} + 138q^{4} + 19q^{5} + 39q^{6} + 46q^{7} + 51q^{8} + 377q^{9} + 134q^{10} + 56q^{11} + 195q^{12} + 259q^{13} + 140q^{14} + 78q^{15} + 822q^{16} + 239q^{17} + 281q^{18} + 474q^{19} + 97q^{20} + 438q^{21} + 423q^{22} + 97q^{23} + 618q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 30. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{30} - 17x^{29} - 677x^{28} + 12081x^{27} + 196387x^{26} - 3672939x^{25} - 33128579x^{24} + 638607781x^{23} + 3722592469x^{22} - 71183893175x^{21} - 301760899082x^{20} + 5375862098123x^{19} + 18446323821259x^{18} - 282241284086102x^{17} - 859605186899258x^{16} + 10374358478803487x^{15} + 29991639103380332x^{14} - 265120659806949770x^{13} - 755588327075005241x^{12} + 4623767774844119475x^{11} + 13132808221806275813x^{10} - 53499525234274807255x^{9} - 149290325876158790535x^{8} + 396732181834367611214x^{7} + 1038620318497592468294x^{6} - 1817575956678153469068x^{5} - 4027169894902049891999x^{4} + 4825904052887599232793x^{3} + 7174141031095876094093x^{2} - 5958073443800316127100x - 2854520840695076990389$

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
$\operatorname{tr} a_p$	4	13	19	46	56	259	239	474	97	612	966	606	1622	935	328	−740	77	2710	565	−98	1240	2737	180	2986	3757

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.134 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, 191]: [1, 49] (certified at ell = 2147483171)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (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(191, 4)[1]   # 191.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 191:4 1000