Atlas › Modular forms › Weight 2 › Level 988 › 988.2.a › 988.2.a.f

Newform orbit 988.2.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - q^{3} - q^{5} + 7q^{7} + 4q^{9} + 8q^{11} - 3q^{13} - 6q^{15} + 4q^{17} - 3q^{19} + 4q^{21} - 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{17} + 6T_{23} + 7T_{29} + 8T_{31}$ , whose minimal polynomial is

$x^{3} - 105x^{2} + 3504x - 36377$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 988$ )
13	$+1$
19	$+1$

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	−1	−1	7	8	−3	4	−3	−4	5	3	12	−3	−5	29	13	6	12	7	−3	−11	0	26	−3	−22

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.659 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, 13, 19, 26, 38, 52, 76, 247, 494, 988]: [0, 1, 2, 1, 2, 5, 7, 10, 13, 24, 74, 146] (certified at ell = 2147483629)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (17, 5), (23, 6), (29, 7), (31, 8)]; h of degree 18 = 1 x 18 (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(988, 2)[5]   # 988.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 988:2 1000