Atlas › Modular forms › Weight 2 › Level 921 › 921.2.a › 921.2.a.h

Newform orbit 921.2.a.h

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 3q^{2} + 14q^{3} + 21q^{4} + 3q^{5} - 3q^{6} + 6q^{7} - 12q^{8} + 14q^{9} + 7q^{10} + 18q^{11} + 21q^{12} + 6q^{13} - 8q^{14} + 3q^{15} + 31q^{16} - 2q^{17} - 3q^{18} + 23q^{19} + 20q^{20} + 6q^{21} - 8q^{22} - 2q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{14} - 74x^{12} + 55x^{11} + 1960x^{10} - 2624x^{9} - 21838x^{8} + 39381x^{7} + 93119x^{6} - 214438x^{5} - 87331x^{4} + 398971x^{3} - 183780x^{2} - 59912x + 37952$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
307	$+1$

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
$\operatorname{tr} a_p$	−3	14	3	6	18	6	−2	23	−2	24	48	31	0	13	7	−14	14	11	4	35	13	29	−21	10	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.275 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, 3, 307, 921]: [0, 1, 26, 104] (certified at ell = 2147478013)
T = [(2, 1), (5, 1)]; h of degree 51 = 1 x 51 (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(921, 2)[7]   # 921.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 921:2 1000