Atlas › Modular forms › Weight 10 › Level 21 › 21.10.a › 21.10.a.c

Newform orbit 21.10.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 30q^{2} - 162q^{3} + 116q^{4} + 1128q^{5} - 2430q^{6} + 4802q^{7} + 7080q^{8} + 13122q^{9} + 65220q^{10} + 73284q^{11} - 9396q^{12} + 141100q^{13} + 72030q^{14} - 91368q^{15} + 44048q^{16} - 101784q^{17} + 196830q^{18} + 481744q^{19} + 1514424q^{20} - 388962q^{21} + 2284680q^{22} - 982212q^{23} - 573480q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{345})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 1158x - 1403904$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
7	$-1$

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
$\operatorname{tr} a_p$	30	−162	1128	4802	73284	141100	−101784	481744	−982212	−2550924	−4935848	−16256516	48707856	7989640	85572408	26565324	115200960	−142820204	27521392	38070180	−2095316	435097048	264288744	642673776	345361228

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 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 engine's checks

dim M^+ at levels [1, 3, 7, 21]: [1, 4, 7, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(21, 10)[2]   # 21.10.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 21:10 1000