Atlas › Modular forms › Weight 2 › Level 677 › 677.2.a › 677.2.a.d

Newform orbit 677.2.a.d

Trace form

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

$35q + 4q^{2} + 18q^{3} + 46q^{4} + 3q^{5} + q^{6} + 23q^{7} + 12q^{8} + 45q^{9} + 8q^{10} + 16q^{11} + 32q^{12} + 10q^{13} + q^{15} + 64q^{16} + 5q^{17} + 8q^{18} + 20q^{19} - 10q^{20} + 3q^{21} + 18q^{22} + 31q^{23} - 19q^{24} + O(q^{25})$

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

$x^{35} - 22x^{34} + 108x^{33} + 1018x^{32} - 11364x^{31} + 2432x^{30} + 365666x^{29} - 1046858x^{28} - 5376124x^{27} + 28574013x^{26} + 30091547x^{25} - 385855751x^{24} + 170982348x^{23} + 3086763145x^{22} - 4030758589x^{21} - 15507323737x^{20} + 30961191168x^{19} + 49530469840x^{18} - 137162780357x^{17} - 97347061777x^{16} + 390015660321x^{15} + 101609091200x^{14} - 735125062914x^{13} - 10614957139x^{12} + 918864605755x^{11} - 108172690001x^{10} - 742235507276x^{9} + 129700042314x^{8} + 364159061001x^{7} - 62013577996x^{6} - 94777466701x^{5} + 8855084815x^{4} + 9078575714x^{3} + 1235985340x^{2} + 54282088x + 407584$

Atkin–Lehner signs

$p$	$w_p$
677	$-1$

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
$\operatorname{tr} a_p$	4	18	3	23	16	10	5	20	31	0	31	37	−15	105	16	9	2	0	118	−17	20	25	10	−22	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 677]: [0, 57] (certified at ell = 2147480297)
T = [(2, 1), (3, 1)]; h of degree 56 = 1 x 56 (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(677, 2)[3]   # 677.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 677:2 1000