Atlas › Modular forms › Weight 4 › Level 203 › 203.4.a › 203.4.a.e

Newform orbit 203.4.a.e

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 2q^{2} + 4q^{3} + 50q^{4} + 16q^{5} + 7q^{6} - 77q^{7} + 24q^{8} + 163q^{9} + 177q^{10} - 20q^{11} + 33q^{12} + 244q^{13} - 14q^{14} + 62q^{15} + 174q^{16} + 12q^{17} + 249q^{18} + 220q^{19} + 441q^{20} - 28q^{21} + 101q^{22} - 84q^{23} + 51q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{11} - 14x^{10} - 2062x^{9} + 26050x^{8} + 1532584x^{7} - 16854222x^{6} - 505537818x^{5} + 4665386354x^{4} + 73172841815x^{3} - 547863096324x^{2} - 3555745884952x + 21747374959852$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
29	$+1$

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	4	16	−77	−20	244	12	220	−84	−319	262	730	0	776	432	118	−22	2008	−1758	926	2272	−3186	838	20	1268

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.149 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, 7, 29, 203]: [1, 3, 9, 62] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 8 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(203, 4)[4]   # 203.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 203:4 1000