Atlas › Modular forms › Weight 12 › Level 25 › 25.12.a › 25.12.a.f

Newform orbit 25.12.a.f

Trace form

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

$4q + 55q^{2} - 20q^{3} + 5557q^{4} - 31187q^{6} + 96400q^{7} + 142785q^{8} + 199168q^{9} + 801548q^{11} - 2113765q^{12} + 1515160q^{13} - 720366q^{14} + 8479729q^{16} + 10717540q^{17} + 37843930q^{18} - 6821740q^{19} + 26235408q^{21} + 53656935q^{22} + 93086040q^{23} - 228344745q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} - 35x^{3} - 428953x^{2} - 55701685x + 10321346496$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 25$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	55	−20	0	96400	801548	1515160	10717540	−6821740	93086040	−318527960	139971128	−404954480	−63105932	−2158093400	1304992360	−1158989720	−3547066120	−4228428352	22046267740	7443462088	45321383740	−21578383360	48664610220	139446307620	209139237160

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.524 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, 5, 25]: [2, 7, 29] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 6 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(25, 12)[5]   # 25.12.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 25:12 1000