Atlas › Modular forms › Weight 2 › Level 709 › 709.2.a › 709.2.a.c

Newform orbit 709.2.a.c

Trace form

The 30 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$30q + 8q^{2} + 12q^{3} + 28q^{4} + 15q^{5} + 10q^{6} + 10q^{7} + 18q^{8} + 34q^{9} - q^{10} + 40q^{11} + 12q^{12} + 7q^{13} + 5q^{14} - 3q^{15} + 24q^{16} + 14q^{17} + 3q^{18} + 37q^{19} + 33q^{20} - 5q^{21} - 10q^{22} + 18q^{23} + 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 30. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{30} - 20x^{29} + 84x^{28} + 874x^{27} - 8512x^{26} + 1883x^{25} + 228233x^{24} - 655190x^{23} - 2375382x^{22} + 13468071x^{21} + 4058712x^{20} - 126881813x^{19} + 122418142x^{18} + 647574067x^{17} - 1187800057x^{16} - 1794616340x^{15} + 5217068967x^{14} + 2208355738x^{13} - 12957906012x^{12} + 722606799x^{11} + 19156888135x^{10} - 5296357619x^{9} - 17122782188x^{8} + 5927310772x^{7} + 9174733969x^{6} - 2833237027x^{5} - 2790789719x^{4} + 561393106x^{3} + 414016036x^{2} - 29365645x - 19751209$

Atkin–Lehner signs

$p$	$w_p$
709	$-1$

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
$\operatorname{tr} a_p$	8	12	15	10	40	7	14	37	18	26	29	−12	64	13	25	0	99	−9	10	49	−14	−6	57	44	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.231 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, 709]: [0, 59] (certified at ell = 2147482237)
T = [(2, 1), (3, 1)]; h of degree 58 = 1 x 58 (phi(ord eps) x dim S_k^new); 6 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(709, 2)[2]   # 709.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 709:2 1000