Atlas › Modular forms › Weight 2 › Level 598 › 598.2.a › 598.2.a.i

Newform orbit 598.2.a.i

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{17} + 4T_{19} + 5T_{29} + 6T_{7} + 7T_{11} + 8T_{31}$ , whose minimal polynomial is

$x^{4} + 108x^{3} - 2184x^{2} - 560520x - 15408972$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
13	$+1$
23	$+1$

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	0	−4	−2	−6	−4	−8	4	−4	−14	4	−20	−18	6	−6	−6	−18	−8	−6	−10	10	0	−24	−18	40

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.4 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, 2, 13, 23, 26, 46, 299, 598]: [0, 1, 1, 3, 5, 8, 30, 88] (certified at ell = 2147483053)
T = [(3, 1), (5, 2), (17, 3), (19, 4), (29, 5), (7, 6), (11, 7), (31, 8)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 6 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(598, 2)[8]   # 598.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 598:2 1000