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

Newform orbit 203.4.a.f

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 3q^{2} + 6q^{3} + 85q^{4} - 31q^{6} + 84q^{7} + 111q^{8} + 106q^{9} - 87q^{10} + 104q^{11} + 83q^{12} + 136q^{13} + 21q^{14} + 126q^{15} + 429q^{16} + 156q^{17} - 56q^{18} - 70q^{19} + 23q^{20} + 42q^{21} + 293q^{22} - 24q^{23} + 125q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{12} - 21x^{11} - 1712x^{10} + 35992x^{9} + 1031576x^{8} - 21651554x^{7} - 265003614x^{6} + 5582036764x^{5} + 27750793687x^{4} - 607333894825x^{3} - 882520114042x^{2} + 23037537147660x - 11773978663944$

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
$\operatorname{tr} a_p$	3	6	0	84	104	136	156	−70	−24	348	−524	960	688	1128	678	342	426	752	1490	−3042	1564	1378	1954	−2624	1432

$\operatorname{tr} a_n$ 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)[5]   # 203.4.a.f

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

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