Atlas › Modular forms › Weight 2 › Level 469 › 469.2.a › 469.2.a.i

Newform orbit 469.2.a.i

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{9} - 23x^{8} + 71x^{7} + 1709x^{6} - 11218x^{5} - 24902x^{4} + 275754x^{3} - 94044x^{2} - 1624608x + 1884208$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
67	$-1$

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
$\operatorname{tr} a_p$	−1	8	8	−9	18	4	7	13	1	−7	6	−18	22	−1	12	−6	44	−16	9	14	−7	28	32	−2	11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 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, 7, 67, 469]: [0, 1, 6, 46] (certified at ell = 2147483647)
T = [(2, 1), (3, 3)]; h of degree 33 = 1 x 33 (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(469, 2)[8]   # 469.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 469:2 1000