Atlas › Modular forms › Weight 2 › Level 653 › 653.2.a › 653.2.a.a

Newform orbit 653.2.a.a

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 9q^{3} + 2q^{4} + 5q^{5} + 4q^{6} - 16q^{7} + 4q^{9} - 7q^{10} - 9q^{11} + q^{12} + 7q^{13} - 8q^{14} - 14q^{15} - 16q^{16} + 3q^{17} - 12q^{18} - 9q^{19} - 2q^{20} + 27q^{21} + 3q^{22} + 8q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} + 9x^{6} + 16x^{5} - 48x^{4} - 127x^{3} + 62x^{2} + 193x - 29$

Atkin–Lehner signs

$p$	$w_p$
653	$+1$

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
$\operatorname{tr} a_p$	0	−9	5	−16	−9	7	3	−9	8	−8	5	16	−19	−15	−35	−18	10	−28	−33	−8	−37	−4	5	10	−28

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.219 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, 653]: [0, 55] (certified at ell = 2147482621)
T = [(2, 1), (3, 1)]; h of degree 54 = 1 x 54 (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(653, 2)[0]   # 653.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 653:2 1000