Atlas › Modular forms › Weight 2 › Level 203 › 203.2.a › 203.2.a.g

Newform orbit 203.2.a.g

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 2q^{2} - 2q^{3} + 10q^{4} + 5q^{5} - 9q^{6} + 5q^{7} + 6q^{8} + 9q^{9} - 9q^{10} + 3q^{11} - 11q^{12} + 15q^{13} + 2q^{14} - 17q^{15} + 24q^{16} - 4q^{17} + 9q^{18} - 15q^{19} + 11q^{20} - 2q^{21} + 3q^{22} - 5q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} - 13x^{3} + 40x - 4$

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	−2	5	5	3	15	−4	−15	−5	−5	9	14	−11	19	4	−16	−12	20	−3	−1	35	−13	−10	−25	−25

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 7, 29, 203]: [0, 1, 3, 22] (certified at ell = 2147483353)
T = [(2, 1), (3, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 2 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, 2)[6]   # 203.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 203:2 1000