Atlas › Modular forms › Weight 2 › Level 967 › 967.2.a › 967.2.a.a

Newform orbit 967.2.a.a

Trace form

The 35 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$35q - 12q^{2} - 11q^{3} + 32q^{4} - 27q^{5} - 6q^{6} - 12q^{7} - 33q^{8} + 20q^{9} + q^{10} - 13q^{11} - 28q^{12} - 22q^{13} - 23q^{14} - 14q^{15} + 30q^{16} - 66q^{17} - 23q^{18} - 14q^{19} - 56q^{20} - 10q^{21} - 22q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 35. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{35} + 23x^{34} + 157x^{33} - 275x^{32} - 7931x^{31} - 21604x^{30} + 125751x^{29} + 715221x^{28} - 540993x^{27} - 10055055x^{26} - 8084194x^{25} + 81121124x^{24} + 142259923x^{23} - 406849256x^{22} - 1085585938x^{21} + 1265676935x^{20} + 5110662291x^{19} - 2144821487x^{18} - 16250296278x^{17} + 334126191x^{16} + 36172644813x^{15} + 7345112212x^{14} - 57131568928x^{13} - 18104326038x^{12} + 63947323549x^{11} + 22429004747x^{10} - 49896548154x^{9} - 16072876325x^{8} + 26103177304x^{7} + 6465397725x^{6} - 8448205162x^{5} - 1266298976x^{4} + 1418270490x^{3} + 90855099x^{2} - 73124168x - 7701533$

Atkin–Lehner signs

$p$	$w_p$
967	$+1$

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
$\operatorname{tr} a_p$	−12	−11	−27	−12	−13	−22	−66	−14	−22	−24	−4	−22	−74	−2	−66	−48	−46	−3	−19	4	−36	6	−67	−134	−95

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.305 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, 967]: [0, 81] (certified at ell = 2147480791)
T = [(2, 1), (3, 1)]; h of degree 80 = 1 x 80 (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(967, 2)[0]   # 967.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 967:2 1000