Atlas › Modular forms › Weight 10 › Level 26 › 26.10.a › 26.10.a.e

Newform orbit 26.10.a.e

Trace form

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

$3q + 48q^{2} + 156q^{3} + 768q^{4} - 1272q^{5} + 2496q^{6} + 17058q^{7} + 12288q^{8} + 42273q^{9} - 20352q^{10} + 73974q^{11} + 39936q^{12} - 85683q^{13} + 272928q^{14} + 393756q^{15} + 196608q^{16} + 374976q^{17} + 676368q^{18} + 418338q^{19} - 325632q^{20} - 284694q^{21} + 1183584q^{22} + 1026168q^{23} + 638976q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 1116x^{2} - 4420020x - 2206425872$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
13	$+1$

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
$\operatorname{tr} a_p$	48	156	−1272	17058	73974	−85683	374976	418338	1026168	−3075834	9286482	−17647776	−47257110	−60023760	−40824726	8072046	138035310	264203886	203167074	123067110	−433013250	406418748	105365610	1365375798	669691662

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 13, 26]: [1, 3, 11, 33] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (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(26, 10)[4]   # 26.10.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 26:10 1000