Atlas › Modular forms › Weight 6 › Level 25 › 25.6.a › 25.6.a.b

Newform orbit 25.6.a.b

Trace form

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

$2q - 5q^{2} - 20q^{3} + 69q^{4} - 191q^{6} - 200q^{7} - 615q^{8} + 196q^{9} - 196q^{11} + 515q^{12} + 360q^{13} + 18q^{14} + 1137q^{16} - 1490q^{17} + 4330q^{18} - 3180q^{19} + 2964q^{21} + 6515q^{22} - 1560q^{23} + 2535q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{241})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} + 25x + 96$

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
$\operatorname{tr} a_p$	−5	−20	0	−200	−196	360	−1490	−3180	−1560	−3920	−1096	−2020	27754	3000	−25760	26980	11960	−24396	40060	−87296	70290	65480	−92580	−72810	126140

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.217 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]: [1, 3, 13] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (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(25, 6)[1]   # 25.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 25:6 1000