Atlas › Modular forms › Weight 12 › Level 10 › 10.12.a › 10.12.a.d

Newform orbit 10.12.a.d

Trace form

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

$2q + 64q^{2} + 604q^{3} + 2048q^{4} + 6250q^{5} + 19328q^{6} + 14092q^{7} + 65536q^{8} + 221914q^{9} + 200000q^{10} + 421584q^{11} + 618496q^{12} + 1730524q^{13} + 450944q^{14} + 1887500q^{15} + 2097152q^{16} - 6323628q^{17} + 7101248q^{18} - 28897400q^{19} + 6400000q^{20} - 65446816q^{21} + 13490688q^{22} - 45236076q^{23} + 19791872q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1969})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} - 14696x - 6045181296$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$

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
$\operatorname{tr} a_p$	64	604	6250	14092	421584	1730524	−6323628	−28897400	−45236076	58226220	41413384	377255452	−785271036	−1452987236	−1288127748	−30490836	8677102440	1115498764	−12673769708	13799832984	−17842079516	−12636930320	41986488924	13208740020	−61787462828

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.332 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, 2, 5, 10]: [2, 4, 7, 19] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 3 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(10, 12)[3]   # 10.12.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 10:12 1000