Atlas › Modular forms › Weight 12 › Level 26 › 26.12.a › 26.12.a.d

Newform orbit 26.12.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 128q^{2} + 4096q^{4} + 694q^{5} - 5844q^{7} + 131072q^{8} + 650198q^{9} + 22208q^{10} - 261644q^{11} + 1485172q^{13} - 187008q^{14} + 11881816q^{15} + 4194304q^{16} + 4832302q^{17} + 20806336q^{18} + 2762436q^{19} + 710656q^{20} + 63022462q^{21} - 8372608q^{22} + 26525496q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} - 694x^{3} - 177859540x^{2} - 289416160712x + 691689639135360$

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
$\operatorname{tr} a_p$	128	0	694	−5844	−261644	1485172	4832302	2762436	26525496	114398784	170870588	−129457314	−67263908	565894584	−4547521124	−10533452684	−16921519420	−3601744572	21420671204	59097402980	16938424496	38736725720	−67937054892	34556088472	75994027560

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.546 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]: [2, 4, 15, 41] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 11 = 1 x 11 (phi(ord eps) x dim S_k^new); 5 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, 12)[3]   # 26.12.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 26:12 1000