Atlas › Modular forms › Weight 2 › Level 692 › 692.2.a › 692.2.a.b

Newform orbit 692.2.a.b

Trace form

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

$4q - 3q^{5} - 5q^{7} - 4q^{9} - q^{11} - 9q^{13} - 4q^{15} - 6q^{17} - q^{19} - 9q^{21} - 9q^{23} + O(q^{25})$

Coefficient field

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

$x^{4} + 9x^{3} - x^{2} - 155x - 293$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 692$ )
173	$-1$

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
$\operatorname{tr} a_p$	0	0	−3	−5	−1	−9	−6	−1	−9	−3	−5	−22	−3	2	−2	−10	18	−7	12	10	−15	−5	8	18	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.228 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, 4, 173, 346, 692]: [0, 1, 2, 15, 45, 90] (certified at ell = 2147482093)
T = [(3, 1), (5, 3)]; h of degree 15 = 1 x 15 (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(692, 2)[1]   # 692.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 692:2 1000