Atlas › Modular forms › Weight 2 › Level 499 › 499.2.a › 499.2.a.c

Newform orbit 499.2.a.c

Trace form

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

$23q + 4q^{2} + q^{3} + 22q^{4} + 25q^{5} + 7q^{6} + 2q^{7} + 9q^{8} + 26q^{9} + 5q^{10} + 10q^{11} - 4q^{12} + 14q^{13} + 16q^{14} - 4q^{15} + 12q^{16} + 18q^{17} - 8q^{18} + 4q^{19} + 25q^{20} + 26q^{21} - 15q^{22} + 4q^{23} + 2q^{24} + O(q^{25})$

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

$x^{23} - 5x^{22} - 76x^{21} + 413x^{20} + 2183x^{19} - 13368x^{18} - 30099x^{17} + 221930x^{16} + 213247x^{15} - 2120635x^{14} - 742125x^{13} + 12369792x^{12} + 927282x^{11} - 45228179x^{10} + 317812x^{9} + 103288800x^{8} + 3155052x^{7} - 141212714x^{6} - 19347436x^{5} + 103479426x^{4} + 27732357x^{3} - 30205912x^{2} - 12093780x - 437688$

Atkin–Lehner signs

$p$	$w_p$
499	$-1$

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	1	25	2	10	14	18	4	4	57	5	11	56	−13	19	16	30	31	−26	8	−4	−9	25	62	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.196 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, 499]: [0, 42] (certified at ell = 2147475601)
T = [(2, 1), (3, 1)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 4 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(499, 2)[2]   # 499.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 499:2 1000