Atlas › Modular forms › Weight 2 › Level 895 › 895.2.a › 895.2.a.g

Newform orbit 895.2.a.g

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 7q^{2} + 4q^{3} + 19q^{4} + 18q^{5} + 12q^{7} + 18q^{8} + 26q^{9} + 7q^{10} + 13q^{11} + 22q^{13} - 8q^{14} + 4q^{15} + 13q^{16} + 27q^{17} + q^{18} - 6q^{19} + 19q^{20} - 13q^{21} - 4q^{22} + 11q^{23} - 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{18} - 170x^{17} + 7193x^{16} + 303693x^{15} - 30676452x^{14} + 320911288x^{13} + 34466135600x^{12} - 987184404628x^{11} - 10009598712030x^{10} + 693573289572363x^{9} - 4063881921358306x^{8} - 170284407225287487x^{7} + 2335533698932688994x^{6} + 10858690264348221938x^{5} - 332001187446964218610x^{4} + 830270162063493070264x^{3} + 13408545807770393546469x^{2} - 75024107685370235138789x + 56535536270500133211613$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
179	$+1$

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
$\operatorname{tr} a_p$	7	4	18	12	13	22	27	−6	11	18	−7	25	34	8	3	36	−5	−33	−2	11	38	−9	31	17	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 0.429 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, 5, 179, 895]: [0, 1, 16, 92] (certified at ell = 2147483137)
T = [(2, 1), (3, 3), (7, 5), (11, 7)]; h of degree 59 = 1 x 59 (phi(ord eps) x dim S_k^new); 13 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(895, 2)[6]   # 895.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 895:2 1000