Atlas › Modular forms › Weight 4 › Level 213 › 213.4.a › 213.4.a.b

Newform orbit 213.4.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 9q^{2} + 24q^{3} + 27q^{4} - 32q^{5} - 27q^{6} - 56q^{7} - 108q^{8} + 72q^{9} - 23q^{10} - 136q^{11} + 81q^{12} - 116q^{13} - 175q^{14} - 96q^{15} - 21q^{16} - 318q^{17} - 81q^{18} - 32q^{19} - 210q^{20} - 168q^{21} + 341q^{22} - 22q^{23} - 324q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} + 41x^{7} + 200x^{6} - 8983x^{5} - 99961x^{4} - 71770x^{3} + 841252x^{2} - 454032x - 173664$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
71	$+1$

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
$\operatorname{tr} a_p$	−9	24	−32	−56	−136	−116	−318	−32	−22	−378	−376	−402	−1014	−640	−868	−1080	−1040	168	−2114	−568	−322	924	−1376	−1270	2414

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.141 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 engine's checks

dim M^+ at levels [1, 3, 71, 213]: [1, 2, 19, 74] (certified at ell = 2147483171)
T = [(2, 1), (5, 1)]; h of degree 36 = 1 x 36 (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(213, 4)[1]   # 213.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 213:4 1000