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

Newform orbit 203.4.a.d

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 4q^{2} - 8q^{3} + 46q^{4} - 24q^{5} - 29q^{6} - 70q^{7} - 48q^{8} + 96q^{9} - 113q^{10} - 26q^{11} - 111q^{12} - 236q^{13} + 28q^{14} + 62q^{15} + 54q^{16} - 288q^{17} - 307q^{18} - 84q^{19} - 131q^{20} + 56q^{21} - 31q^{22} - 120q^{23} - 541q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{10} + 28x^{9} - 1231x^{8} - 28236x^{7} + 555833x^{6} + 8295138x^{5} - 106911303x^{4} - 844985954x^{3} + 8116982132x^{2} + 28366531368x - 199698087232$

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 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
$\operatorname{tr} a_p$	−4	−8	−24	−70	−26	−236	−288	−84	−120	290	−326	−1028	−300	−66	−1104	458	686	−716	−602	2286	−2088	−652	−490	−796	−3192

$\operatorname{tr} a_n$ 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)[3]   # 203.4.a.d

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

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