Atlas › Modular forms › Weight 2 › Level 755 › 755.2.a › 755.2.a.k

Newform orbit 755.2.a.k

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q - 2q^{2} + 2q^{3} + 32q^{4} - 18q^{5} + 10q^{6} + 4q^{7} + 32q^{9} + 2q^{10} + 11q^{11} + 12q^{13} + q^{14} - 2q^{15} + 60q^{16} - 25q^{17} - 9q^{18} + 8q^{19} - 32q^{20} + 20q^{21} + 29q^{22} + 12q^{23} + 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{18} - 4x^{17} - 443x^{16} + 1766x^{15} + 77928x^{14} - 292435x^{13} - 7104675x^{12} + 23260438x^{11} + 368493989x^{10} - 923171351x^{9} - 11298093764x^{8} + 16313850778x^{7} + 205503871384x^{6} - 50432127932x^{5} - 2033497525136x^{4} - 1831294924424x^{3} + 7781252419472x^{2} + 16023366691424x + 8217727555008$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
151	$-1$

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
$\operatorname{tr} a_p$	−2	2	−18	4	11	12	−25	8	12	24	11	45	12	19	−16	−4	−8	27	9	9	2	20	−20	13	−27

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 151, 755]: [0, 1, 13, 78] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 7 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(755, 2)[10]   # 755.2.a.k

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

cargo run --release -p sagebrush-web --example atlas -- 755:2 1000