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

Newform orbit 203.4.a.c

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 2q^{2} - 2q^{3} + 10q^{4} - 24q^{5} - 39q^{6} + 49q^{7} - 24q^{8} - 17q^{9} + 25q^{10} - 4q^{11} + 31q^{12} - 172q^{13} - 14q^{14} - 154q^{15} - 178q^{16} - 46q^{17} - 83q^{18} - 106q^{19} - 127q^{20} - 14q^{21} - 299q^{22} - 284q^{23} - 87q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{7} + 8x^{6} - 658x^{5} - 1240x^{4} + 107857x^{3} - 309332x^{2} - 379384x + 639724$

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
$\operatorname{tr} a_p$	−2	−2	−24	49	−4	−172	−46	−106	−284	−203	−436	−854	154	−980	118	−358	654	−158	−1862	46	−82	−422	442	1946	−554

$\operatorname{tr} a_n$ 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)[2]   # 203.4.a.c

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

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