Atlas › Modular forms › Weight 2 › Level 401 › 401.2.a › 401.2.a.b

Newform orbit 401.2.a.b

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q + 3q^{3} + 28q^{4} + 3q^{5} + 9q^{6} + 24q^{7} + 20q^{9} + 7q^{10} + q^{11} - 7q^{12} + 9q^{13} - 11q^{14} + q^{15} + 30q^{16} - q^{17} - 10q^{18} + 38q^{19} + q^{20} + q^{21} - q^{22} + 3q^{23} + 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{21} - 3x^{20} - 81x^{19} + 255x^{18} + 2530x^{17} - 8171x^{16} - 40338x^{15} + 131197x^{14} + 364234x^{13} - 1158726x^{12} - 1977901x^{11} + 5779265x^{10} + 6846660x^{9} - 16084028x^{8} - 15566480x^{7} + 22964538x^{6} + 21959243x^{5} - 12194854x^{4} - 15013101x^{3} - 1862556x^{2} + 1329605x + 299663$

Atkin–Lehner signs

$p$	$w_p$
401	$-1$

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
$\operatorname{tr} a_p$	0	3	3	24	1	9	−1	38	3	−4	70	3	−2	10	9	−17	7	32	6	15	18	49	−3	−35	−6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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, 401]: [0, 34] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 4 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(401, 2)[1]   # 401.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 401:2 1000