Atlas › Modular forms › Weight 4 › Level 43 › 43.4.a › 43.4.a.a

Newform orbit 43.4.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 4q^{2} - 11q^{3} + 2q^{4} - 27q^{5} - 27q^{6} - 20q^{7} - 66q^{8} - 9q^{9} - 3q^{10} - 62q^{11} + 61q^{12} - 2q^{13} + 112q^{14} + 92q^{15} + 202q^{16} - 207q^{17} + 299q^{18} + 99q^{19} + 81q^{20} - 90q^{21} + 202q^{22} - 103q^{23} + 209q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 15x^{3} + 73x^{2} + 120x + 18$

Atkin–Lehner signs

$p$	$w_p$
43	$-1$

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	−11	−27	−20	−62	−2	−207	99	−103	−99	131	−449	−491	172	19	−1220	816	372	110	468	628	1095	−980	−738	−1765

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.092 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, 43]: [1, 12] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (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(43, 4)[0]   # 43.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 43:4 1000