Atlas › Modular forms › Weight 2 › Level 832 › 832.2.a › 832.2.a.o

Newform orbit 832.2.a.o

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + q^{3} + 3q^{5} - 3q^{7} + 3q^{9} + 4q^{11} + 2q^{13} - 7q^{15} + 3q^{17} + 12q^{19} + 7q^{21} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{2} - 159x + 5976$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 832$ )
13	$-1$

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	3	−3	4	2	3	12	0	0	−6	7	6	3	9	18	−12	−2	12	−9	20	−24	−10	0	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.449 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, 8, 13, 16, 26, 32, 52, 64, 104, 208, 416, 832]: [0, 1, 2, 3, 1, 4, 5, 6, 10, 10, 18, 32, 60, 116] (certified at ell = 2147483137)
T = [(3, 1), (5, 3), (11, 5), (17, 7), (19, 9)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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(832, 2)[14]   # 832.2.a.o

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

cargo run --release -p sagebrush-web --example atlas -- 832:2 1000