Atlas › Modular forms › Weight 2 › Level 617 › 617.2.a › 617.2.a.a

Newform orbit 617.2.a.a

Trace form

The 23 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$23q - 6q^{2} - 13q^{3} + 18q^{4} - 6q^{5} - 5q^{6} - 37q^{7} - 15q^{8} + 22q^{9} - 22q^{10} - 8q^{11} - 29q^{12} - 29q^{13} + 2q^{14} - 22q^{15} + 12q^{16} - 12q^{17} - 23q^{18} - 29q^{19} - 14q^{20} + 4q^{21} - 22q^{22} - 22q^{23} - 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 23. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{23} + 19x^{22} + 108x^{21} - 87x^{20} - 2861x^{19} - 6203x^{18} + 22432x^{17} + 89110x^{16} - 60756x^{15} - 534442x^{14} - 49836x^{13} + 1784172x^{12} + 609684x^{11} - 3654933x^{10} - 1214769x^{9} + 4715758x^{8} + 766941x^{7} - 3646092x^{6} + 297069x^{5} + 1387418x^{4} - 448255x^{3} - 117350x^{2} + 70326x - 7983$

Atkin–Lehner signs

$p$	$w_p$
617	$+1$

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
$\operatorname{tr} a_p$	−6	−13	−6	−37	−8	−29	−12	−29	−22	5	−64	−26	−12	−35	−43	9	10	−15	−40	−9	−67	−40	−33	31	−30

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.209 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, 617]: [0, 52] (certified at ell = 2147481337)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 5 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(617, 2)[0]   # 617.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 617:2 1000