Atlas › Modular forms › Weight 2 › Level 953 › 953.2.a › 953.2.a.a

Newform orbit 953.2.a.a

Trace form

The 32 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$32q - 5q^{2} - 16q^{3} + 17q^{4} - 6q^{5} - 5q^{6} - 38q^{7} - 9q^{8} + 22q^{9} - 15q^{10} - 2q^{11} - 26q^{12} - 30q^{13} - 19q^{15} - 9q^{16} - 18q^{17} - 17q^{18} - 19q^{19} - 13q^{20} + 4q^{21} - 12q^{22} - 36q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 32. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{32} + 21x^{31} + 126x^{30} - 292x^{29} - 5821x^{28} - 11546x^{27} + 90473x^{26} + 378191x^{25} - 559356x^{24} - 4896184x^{23} - 658815x^{22} + 36031053x^{21} + 33252387x^{20} - 167349217x^{19} - 241055128x^{18} + 514504223x^{17} + 954619838x^{16} - 1077199086x^{15} - 2400280738x^{14} + 1573736320x^{13} + 4011005237x^{12} - 1656382724x^{11} - 4489790960x^{10} + 1312298637x^{9} + 3299193981x^{8} - 805688820x^{7} - 1503207823x^{6} + 364590009x^{5} + 372378859x^{4} - 102186390x^{3} - 34079546x^{2} + 12176560x - 902775$

Atkin–Lehner signs

$p$	$w_p$
953	$+1$

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
$\operatorname{tr} a_p$	−5	−16	−6	−38	−2	−30	−18	−19	−36	−1	−56	−23	−7	−42	−43	−22	−24	−24	−64	−13	−44	−28	−32	16	−35

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.303 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, 953]: [0, 80] (certified at ell = 2147477809)
T = [(2, 1), (3, 1)]; h of degree 79 = 1 x 79 (phi(ord eps) x dim S_k^new); 8 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(953, 2)[0]   # 953.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 953:2 1000