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

Newform orbit 709.2.a.b

Trace form

The 27 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$27q - 8q^{2} - 13q^{3} + 22q^{4} - 18q^{5} - 14q^{6} - 10q^{7} - 18q^{8} + 20q^{9} - 3q^{10} - 45q^{11} - 18q^{12} - 11q^{13} - 19q^{14} - 8q^{15} + 16q^{16} - 14q^{17} - 5q^{18} - 39q^{19} - 39q^{20} - 17q^{21} + 6q^{22} - 12q^{23} - 38q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 27. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{27} + 21x^{26} + 146x^{25} + 108x^{24} - 3333x^{23} - 12768x^{22} + 16544x^{21} + 175211x^{20} + 123715x^{19} - 1085217x^{18} - 1771973x^{17} + 3594852x^{16} + 8839012x^{15} - 6529854x^{14} - 24021457x^{13} + 6050457x^{12} + 39024539x^{11} - 2251415x^{10} - 38677045x^{9} + 497428x^{8} + 23029157x^{7} - 1145411x^{6} - 7793692x^{5} + 927808x^{4} + 1310538x^{3} - 245745x^{2} - 75090x + 17011$

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 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
$\operatorname{tr} a_p$	−8	−13	−18	−10	−45	−11	−14	−39	−12	−39	−27	−8	−48	−3	−26	−2	−121	−15	1	−75	4	−6	−83	−28	24

$\operatorname{tr} a_n$ 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)[1]   # 709.2.a.b

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

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