Atlas › Modular forms › Weight 2 › Level 597 › 597.2.a › 597.2.a.d

Newform orbit 597.2.a.d

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{11} + 20x^{10} - 148x^{9} - 4621x^{8} - 2300x^{7} + 303189x^{6} + 467312x^{5} - 8113765x^{4} - 6927012x^{3} + 90936676x^{2} - 12798101x - 233649995$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
199	$+1$

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
$\operatorname{tr} a_p$	−5	−11	−5	−5	−10	−9	−19	−11	−16	−14	−5	−5	−19	−34	−33	−22	−27	−14	−16	11	−13	11	−53	−5	17

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.221 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, 3, 199, 597]: [0, 1, 17, 68] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 5 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(597, 2)[3]   # 597.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 597:2 1000