Atlas › Modular forms › Weight 2 › Level 947 › 947.2.a › 947.2.a.a

Newform orbit 947.2.a.a

Trace form

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

$35q - 7q^{2} - 10q^{3} + 29q^{4} - 19q^{5} - 12q^{6} - 27q^{7} - 18q^{8} + 29q^{9} - 28q^{10} - 7q^{11} - 30q^{12} - 68q^{13} + q^{14} - 10q^{15} + 21q^{16} - 26q^{17} - 31q^{18} - 32q^{19} - 25q^{20} - 35q^{21} - 32q^{22} - 7q^{23} - 28q^{24} + O(q^{25})$

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

$x^{35} + 17x^{34} + 40x^{33} - 811x^{32} - 4476x^{31} + 13458x^{30} + 133271x^{29} - 47199x^{28} - 2099196x^{27} - 1509922x^{26} + 20645358x^{25} + 26859030x^{24} - 136437160x^{23} - 226747796x^{22} + 628836461x^{21} + 1194216022x^{20} - 2055317651x^{19} - 4228980814x^{18} + 4771106625x^{17} + 10344630714x^{16} - 7773949026x^{15} - 17581147702x^{14} + 8672190478x^{13} + 20576845559x^{12} - 6359400028x^{11} - 16195376277x^{10} + 2880121529x^{9} + 8208108487x^{8} - 743498169x^{7} - 2488107808x^{6} + 113475987x^{5} + 397190632x^{4} - 17346268x^{3} - 26074304x^{2} + 1610826x + 196541$

Atkin–Lehner signs

$p$	$w_p$
947	$+1$

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
$\operatorname{tr} a_p$	−7	−10	−19	−27	−7	−68	−26	−32	−7	−20	−26	−98	−19	−35	−5	−20	−2	−97	−29	15	−113	−13	0	15	−133

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 947]: [0, 80] (certified at ell = 2147481491)
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(947, 2)[0]   # 947.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 947:2 1000