Atlas › Modular forms › Weight 2 › Level 601 › 601.2.a › 601.2.a.b

Newform orbit 601.2.a.b

Trace form

The 29 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$29q + 4q^{2} + 5q^{3} + 34q^{4} + 10q^{5} - 4q^{6} + 7q^{7} + 9q^{8} + 40q^{9} - 5q^{10} + 38q^{11} - 4q^{12} - 5q^{13} + 22q^{14} + 28q^{15} + 40q^{16} + 9q^{17} - q^{18} + 2q^{19} + 27q^{20} + q^{22} + 61q^{23} - 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 29. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{29} - 9x^{28} - 65x^{27} + 790x^{26} + 1052x^{25} - 27646x^{24} + 15400x^{23} + 508441x^{22} - 744695x^{21} - 5484258x^{20} + 10911525x^{19} + 37059788x^{18} - 85965289x^{17} - 164374558x^{16} + 410757074x^{15} + 496184580x^{14} - 1237203075x^{13} - 1048612813x^{12} + 2344667809x^{11} + 1568175610x^{10} - 2686071131x^{9} - 1606180735x^{8} + 1682262386x^{7} + 1007931879x^{6} - 448796225x^{5} - 300480054x^{4} + 18188047x^{3} + 21033136x^{2} + 115442x - 47831$

Atkin–Lehner signs

$p$	$w_p$
601	$-1$

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 even 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	5	10	7	38	−5	9	2	61	27	16	−5	13	2	43	18	19	−24	14	126	−38	38	7	−8	−45

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 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, 601]: [0, 50] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (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(601, 2)[1]   # 601.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 601:2 1000