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

Newform orbit 21.10.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 13q^{2} + 243q^{3} + 1677q^{4} + 2398q^{5} - 1053q^{6} - 7203q^{7} - 15291q^{8} + 19683q^{9} + 24522q^{10} + 9352q^{11} + 135837q^{12} + 203034q^{13} + 31213q^{14} + 194238q^{15} + 970497q^{16} + 303162q^{17} - 85293q^{18} + 1016772q^{19} + 534182q^{20} - 583443q^{21} + 1053948q^{22} + 2851260q^{23} - 1238571q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} - 2385x^{2} + 1537722x - 136361232$

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
$\operatorname{tr} a_p$	−13	243	2398	−7203	9352	203034	303162	1016772	2851260	−277406	109560	4456098	−6379654	30222468	39062520	−38081742	32289756	131122194	203976852	289658820	341334582	−248369688	−105413076	849077098	489867054

$\operatorname{tr} a_n$ 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)[3]   # 21.10.a.d

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

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