Atlas › Modular forms › Weight 2 › Level 959 › 959.2.a › 959.2.a.d

Newform orbit 959.2.a.d

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q - q^{2} + 7q^{3} + 29q^{4} + 9q^{5} + 3q^{6} - 24q^{7} - 6q^{8} + 39q^{9} + 24q^{10} + 2q^{11} + 26q^{12} + 14q^{13} + q^{14} - 5q^{15} + 47q^{16} + 13q^{17} - 15q^{18} + 20q^{19} + 12q^{20} - 7q^{21} - 3q^{22} - q^{23} + 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} - 6x^{23} - 79x^{22} + 503x^{21} + 2397x^{20} - 16452x^{19} - 37986x^{18} + 282573x^{17} + 354387x^{16} - 2829031x^{15} - 2048831x^{14} + 17035625x^{13} + 7306617x^{12} - 60759939x^{11} - 14349947x^{10} + 120887315x^{9} + 8862223x^{8} - 118856063x^{7} + 9660010x^{6} + 42915013x^{5} - 7847278x^{4} - 3323492x^{3} + 603222x^{2} + 40115x - 4231$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
137	$-1$

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
$\operatorname{tr} a_p$	−1	7	9	−24	2	14	13	20	−1	13	44	11	41	−15	8	19	12	57	−21	−1	57	−1	26	42	90

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 7, 137, 959]: [0, 1, 12, 94] (certified at ell = 2147477809)
T = [(2, 1), (3, 1)]; h of degree 69 = 1 x 69 (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(959, 2)[3]   # 959.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 959:2 1000