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

Newform orbit 26.12.a.c

Trace form

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

$3q - 96q^{2} + 3072q^{4} - 1464q^{5} + 68514q^{7} - 98304q^{8} - 93855q^{9} + 46848q^{10} - 438294q^{11} - 1113879q^{13} - 2192448q^{14} - 5661432q^{15} + 3145728q^{16} - 2807328q^{17} + 3003360q^{18} + 24538998q^{19} - 1499136q^{20} + 50376762q^{21} + 14025408q^{22} + 53983392q^{23} + O(q^{25})$

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

$x^{3} + 1464x^{2} - 119280756x - 333302387264$

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
$\operatorname{tr} a_p$	−96	0	−1464	68514	−438294	−1113879	−2807328	24538998	53983392	318710502	428514462	626364456	116802402	480902316	120653970	223827342	−1266091902	−10838445090	13311040134	9055933542	25154831574	33146281572	7360301238	25049360502	−115625275818

$\operatorname{tr} a_n$ 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)[2]   # 26.12.a.c

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

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