Atlas › Modular forms › Weight 2 › Level 607 › 607.2.a › 607.2.a.d

Newform orbit 607.2.a.d

Trace form

The 31 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$31q + 9q^{2} + 7q^{3} + 35q^{4} + 16q^{5} - 2q^{6} + 2q^{7} + 21q^{8} + 50q^{9} - 4q^{10} + 17q^{11} + 11q^{12} + 6q^{13} + 10q^{14} + 7q^{15} + 35q^{16} + 52q^{17} + 17q^{18} - q^{19} + 28q^{20} - 16q^{21} + 9q^{22} + 46q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 31. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{31} - 16x^{30} + 10x^{29} + 1098x^{28} - 4650x^{27} - 26695x^{26} + 193656x^{25} + 217924x^{24} - 3776843x^{23} + 1882247x^{22} + 42407349x^{21} - 57905357x^{20} - 296394143x^{19} + 581595415x^{18} + 1334506490x^{17} - 3285297437x^{16} - 3931184788x^{15} + 11525531694x^{14} + 7673061049x^{13} - 25648581939x^{12} - 10184042893x^{11} + 35494149042x^{10} + 9557862606x^{9} - 28675844912x^{8} - 6112499392x^{7} + 11986257558x^{6} + 1967440155x^{5} - 2277034921x^{4} - 154343408x^{3} + 158954469x^{2} - 15427601x + 393127$

Atkin–Lehner signs

$p$	$w_p$
607	$-1$

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
$\operatorname{tr} a_p$	9	7	16	2	17	6	52	−1	46	25	−4	19	24	6	21	75	−10	−2	−14	−16	14	−14	5	55	39

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.21 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, 607]: [0, 51] (certified at ell = 2147480989)
T = [(2, 1), (3, 1)]; h of degree 50 = 1 x 50 (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(607, 2)[3]   # 607.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 607:2 1000