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

Newform orbit 949.2.a.e

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q - 3q^{2} - 3q^{3} + 17q^{4} - 12q^{5} - 14q^{6} - 7q^{7} - 9q^{8} + 19q^{9} - 4q^{10} - 35q^{11} - 3q^{12} - 18q^{13} - 22q^{14} - 19q^{15} + 7q^{16} - 10q^{17} - 16q^{18} - 24q^{19} - 29q^{20} - 16q^{21} + 3q^{22} + 10q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{18} + 12x^{17} - 241x^{16} - 2885x^{15} + 23440x^{14} + 265096x^{13} - 1280687x^{12} - 12261903x^{11} + 43260233x^{10} + 305704839x^{9} - 870775120x^{8} - 4044975911x^{7} + 9024986388x^{6} + 26953918266x^{5} - 36280350729x^{4} - 77362220937x^{3} + 6777053538x^{2} + 24969017119x - 5566743311$

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 odd 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
$\operatorname{tr} a_p$	−3	−3	−12	−7	−35	−18	−10	−24	10	−18	−11	−3	−41	−22	−41	−5	−65	−11	−13	−23	−18	13	−76	−23	−14

$\operatorname{tr} a_n$ 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)[4]   # 949.2.a.e

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

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