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

Newform orbit 213.4.a.c

Trace form

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

$8q - 5q^{2} - 24q^{3} + 31q^{4} + 12q^{5} + 15q^{6} - 42q^{7} - 60q^{8} + 72q^{9} - 99q^{10} - 20q^{11} - 93q^{12} - 64q^{13} + 83q^{14} - 36q^{15} + 115q^{16} + 112q^{17} - 45q^{18} - 356q^{19} + 184q^{20} + 126q^{21} - 477q^{22} - 254q^{23} + 180q^{24} + O(q^{25})$

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

$x^{8} - 7x^{7} - 362x^{6} + 1387x^{5} + 31607x^{4} + 5614x^{3} - 593984x^{2} - 750232x + 1087600$

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
$\operatorname{tr} a_p$	−5	−24	12	−42	−20	−64	112	−356	−254	−102	−310	−574	296	−1348	−72	−618	292	−872	−1280	568	−334	−1840	−2236	414	−1634

$\operatorname{tr} a_n$ 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)[2]   # 213.4.a.c

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

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