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

Newform orbit 213.4.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 2q^{2} - 27q^{3} + 38q^{4} - 21q^{5} + 6q^{6} + 49q^{7} - 9q^{8} + 81q^{9} - 84q^{10} - 5q^{11} - 114q^{12} + 116q^{13} + 90q^{14} + 63q^{15} + 250q^{16} - 42q^{17} - 18q^{18} + 259q^{19} - 197q^{20} - 147q^{21} + 458q^{22} + 44q^{23} + 27q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{9} + 23x^{8} - 544x^{7} - 14929x^{6} + 69593x^{5} + 3140288x^{4} + 3619924x^{3} - 223224072x^{2} - 665535744x + 2939693472$

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 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	−27	−21	49	−5	116	−42	259	44	−67	441	502	−537	1727	756	−218	−635	−367	2550	−639	1149	150	−250	−3952	892

$\operatorname{tr} a_n$ 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)[3]   # 213.4.a.d

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

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