Atlas › Modular forms › Weight 2 › Level 949 › 949.2.a › 949.2.a.f

Newform orbit 949.2.a.f

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 4q^{2} + 5q^{3} + 20q^{4} + 8q^{5} + 8q^{6} + 5q^{7} + 12q^{8} + 24q^{9} + 6q^{10} + 27q^{11} + 9q^{12} - 19q^{13} + 5q^{15} + 14q^{16} + 12q^{17} + 5q^{18} + 26q^{19} + 31q^{20} + 12q^{21} + q^{22} - 2q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{19} - 19x^{18} - 237x^{17} + 6282x^{16} + 9903x^{15} - 774724x^{14} + 1382051x^{13} + 45499740x^{12} - 142285086x^{11} - 1411591966x^{10} + 5019648875x^{9} + 24120596351x^{8} - 80189773201x^{7} - 220014454622x^{6} + 566955717513x^{5} + 886550975000x^{4} - 1684702395733x^{3} - 1084862432689x^{2} + 1496400510370x + 151159350959$

Atkin–Lehner signs

$p$	$w_p$
13	$+1$
73	$-1$

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	5	8	5	27	−19	12	26	−2	6	13	−7	31	4	19	−29	65	33	13	29	19	−11	50	23	−16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 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, 13, 73, 949]: [0, 1, 6, 86] (certified at ell = 2147482873)
T = [(2, 1), (3, 3)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 10 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(949, 2)[5]   # 949.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 949:2 1000