Atlas › Modular forms › Weight 4 › Level 69 › 69.4.a › 69.4.a.c

Newform orbit 69.4.a.c

Trace form

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

$4q - 2q^{2} - 12q^{3} + 26q^{4} - 2q^{5} + 6q^{6} + 32q^{7} - 18q^{8} + 36q^{9} + 150q^{10} + 62q^{11} - 78q^{12} + 32q^{13} + 180q^{14} + 6q^{15} + 258q^{16} - 28q^{17} - 18q^{18} + 42q^{19} - 490q^{20} - 96q^{21} - 322q^{22} - 92q^{23} + 54q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{4} + 8x^{3} - 4335x^{2} - 30400x + 1191450$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
23	$+1$

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
$\operatorname{tr} a_p$	−2	−12	−2	32	62	32	−28	42	−92	−20	444	162	−528	1230	36	358	892	294	−126	1032	−1220	1832	1178	−872	−1324

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 3, 23, 69]: [1, 2, 7, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 3)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(69, 4)[2]   # 69.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 69:4 1000