Atlas › Modular forms › Weight 6 › Level 26 › 26.6.a › 26.6.a.b

Newform orbit 26.6.a.b

Trace form

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

$2q - 8q^{2} + 9q^{3} + 32q^{4} - 37q^{5} - 36q^{6} + 327q^{7} - 128q^{8} + 947q^{9} + 148q^{10} + 86q^{11} + 144q^{12} + 338q^{13} - 1308q^{14} - 1559q^{15} + 512q^{16} - 481q^{17} - 3788q^{18} + 1242q^{19} - 592q^{20} + 79q^{21} - 344q^{22} - 2688q^{23} - 576q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2785})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 102x - 184$

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
$\operatorname{tr} a_p$	−8	9	−37	327	86	338	−481	1242	−2688	1644	−512	13311	−11950	15495	−33109	−10786	5050	−306	38806	73045	50020	80032	−77700	39572	−150192

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.223 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, 2, 7, 19] (certified at ell = 2147483629)
T = [(3, 1), (5, 3)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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, 6)[1]   # 26.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 26:6 1000