Atlas › Modular forms › Weight 2 › Level 894 › 894.2.a › 894.2.a.l

Newform orbit 894.2.a.l

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 5q^{2} + 5q^{3} + 5q^{4} - 5q^{5} - 5q^{6} + q^{7} - 5q^{8} + 5q^{9} + 5q^{10} + 4q^{11} + 5q^{12} + 8q^{13} - q^{14} - 5q^{15} + 5q^{16} - q^{17} - 5q^{18} + 11q^{19} - 5q^{20} + q^{21} - 4q^{22} - 8q^{23} - 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7}$ , whose minimal polynomial is

$x^{5} + 2x^{4} - 283x^{3} + 907x^{2} + 6462x + 6368$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$-1$
149	$+1$

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
$\operatorname{tr} a_p$	−5	5	−5	1	4	8	−1	11	−8	15	25	4	11	19	−2	−9	16	0	−3	−18	18	−9	20	1	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.384 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 3, 6, 149, 298, 447, 894]: [0, 1, 1, 3, 13, 39, 52, 154] (certified at ell = 2147481629)
T = [(5, 1), (7, 3)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 4 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(894, 2)[11]   # 894.2.a.l

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

cargo run --release -p sagebrush-web --example atlas -- 894:2 1000