Atlas › Modular forms › Weight 2 › Level 553 › 553.2.a › 553.2.a.d

Newform orbit 553.2.a.d

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 8x^{12} - 13x^{11} + 227x^{10} - 223x^{9} - 1651x^{8} + 2263x^{7} + 5053x^{6} - 5638x^{5} - 7243x^{4} + 3116x^{3} + 2369x^{2} - 482x - 123$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
79	$+1$

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
$\operatorname{tr} a_p$	7	1	4	13	15	−1	2	−4	29	16	−3	−10	−8	11	36	8	−11	−17	−14	22	−15	−13	62	−22	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 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, 7, 79, 553]: [0, 1, 7, 54] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 39 = 1 x 39 (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(553, 2)[3]   # 553.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 553:2 1000