Atlas › Modular forms › Weight 2 › Level 970 › 970.2.a › 970.2.a.h

Newform orbit 970.2.a.h

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{5} + 12x^{4} + 25x^{3} - 156x^{2} - 715x - 794$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$-1$
97	$-1$

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	−6	5	−6	−4	−4	−4	−8	−10	6	4	−8	6	−22	−14	−30	−28	16	−14	−30	−6	0	−42	−44	5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.347 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, 2, 5, 10, 97, 194, 485, 970]: [0, 1, 1, 3, 8, 26, 50, 150] (certified at ell = 2147483137)
T = [(3, 1), (7, 1)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 4 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(970, 2)[7]   # 970.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 970:2 1000