Atlas › Modular forms › Weight 2 › Level 595 › 595.2.a › 595.2.a.f

Newform orbit 595.2.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{11} + 7T_{13} + 9T_{19}$ , whose minimal polynomial is

$x^{3} + 65x^{2} - 1638x - 44603$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
7	$-1$
17	$-1$

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	−3	3	−9	−3	3	1	−5	−23	1	−4	−11	17	−3	−19	−1	−13	1	−7	27	−20	−10	13	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 5, 7, 17, 35, 85, 119, 595]: [0, 1, 1, 2, 6, 10, 14, 76] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (11, 5), (13, 7), (19, 9)]; h of degree 31 = 1 x 31 (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(595, 2)[5]   # 595.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 595:2 1000