Atlas › Modular forms › Weight 2 › Level 103 › 103.2.a › 103.2.a.b

Newform orbit 103.2.a.b

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 4q^{2} + 6q^{4} + 3q^{5} - 3q^{6} - 2q^{7} + 9q^{8} + 8q^{9} - 10q^{10} - q^{11} - 13q^{12} - q^{13} - 9q^{14} - 9q^{15} + 2q^{16} + 21q^{17} - 3q^{18} - 7q^{19} - 9q^{20} - 14q^{21} - 11q^{22} + 12q^{23} - 36q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} - 4x^{5} - 11x^{4} + 61x^{3} - 67x^{2} + 22x - 1$

Atkin–Lehner signs

$p$	$w_p$
103	$-1$

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
$\operatorname{tr} a_p$	4	0	3	−2	−1	−1	21	−7	12	12	−16	0	14	−6	1	19	3	1	−12	−27	−7	−21	−9	−14	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.157 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, 103]: [0, 9] (certified at ell = 2147482501)
T = [(2, 1), (3, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 1 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(103, 2)[1]   # 103.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 103:2 1000