Atlas › Modular forms › Weight 2 › Level 602 › 602.2.a › 602.2.a.e

Newform orbit 602.2.a.e

Trace form

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

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

Coefficient field

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

$x^{3} - 61x - 32$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$+1$
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
$\operatorname{tr} a_p$	−3	−3	1	−3	−2	−6	−1	−7	−3	15	−7	−15	3	−3	−1	−14	−12	−6	−26	10	−24	−11	0	24	5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.247 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, 7, 14, 43, 86, 301, 602]: [0, 1, 1, 4, 4, 13, 30, 92] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 3 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(602, 2)[4]   # 602.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 602:2 1000