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

Newform orbit 653.2.a.c

Trace form

The 30 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$30q + 2q^{2} + 19q^{3} + 32q^{4} + q^{5} + 2q^{6} + 23q^{7} + 33q^{9} + 17q^{10} + 8q^{11} + 24q^{12} + 28q^{13} - 4q^{14} + 2q^{15} + 32q^{16} + 14q^{17} + q^{18} + 43q^{19} - 10q^{20} + 9q^{21} + 15q^{22} + 11q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 30. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{30} - 21x^{29} + 111x^{28} + 626x^{27} - 8157x^{26} + 10642x^{25} + 175958x^{24} - 673049x^{23} - 1353543x^{22} + 11320243x^{21} - 4002569x^{20} - 93149673x^{19} + 144171090x^{18} + 394043027x^{17} - 1081061924x^{16} - 652872229x^{15} + 4014231609x^{14} - 951851945x^{13} - 7958282394x^{12} + 5718949288x^{11} + 7972620418x^{10} - 9023129280x^{9} - 3154243854x^{8} + 6209477072x^{7} - 268301183x^{6} - 1755092961x^{5} + 366755152x^{4} + 152159901x^{3} - 38861416x^{2} - 1649586x + 190689$

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 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
$\operatorname{tr} a_p$	2	19	1	23	8	28	14	43	11	2	29	28	7	82	2	−19	11	27	51	6	41	29	22	−24	57

$\operatorname{tr} a_n$ 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)[2]   # 653.2.a.c

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

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