Atlas › Modular forms › Weight 2 › Level 852 › 852.2.a › 852.2.a.c

Newform orbit 852.2.a.c

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + T_{7}$ , whose minimal polynomial is

$x^{4} - 3x^{3} - 37x^{2} + 70x - 30$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 852$ )
3	$+1$
71	$+1$

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	−4	−1	4	−6	5	4	6	3	5	8	19	15	7	3	13	2	12	−1	−4	26	19	−12	8	−3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.395 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, 3, 4, 6, 12, 71, 142, 213, 284, 426, 852]: [0, 1, 1, 2, 3, 5, 7, 20, 26, 39, 76, 150] (certified at ell = 2147483171)
T = [(5, 1), (7, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(852, 2)[2]   # 852.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 852:2 1000