Atlas › Modular forms › Weight 2 › Level 834 › 834.2.a › 834.2.a.l

Newform orbit 834.2.a.l

Trace form

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

$3q + 3q^{2} + 3q^{3} + 3q^{4} + 7q^{5} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + 7q^{10} - 3q^{11} + 3q^{12} + 7q^{13} - 3q^{14} + 7q^{15} + 3q^{16} + 4q^{17} + 3q^{18} - 4q^{19} + 7q^{20} - 3q^{21} - 3q^{22} - 8q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{17} + 7T_{19} + 9T_{23}$ , whose minimal polynomial is

$x^{3} + 82x^{2} - 7110x + 71096$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
139	$-1$

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
$\operatorname{tr} a_p$	3	3	7	−3	−3	7	4	−4	−8	5	−5	2	−2	−2	−6	−6	−14	−4	−15	9	−2	−11	−1	31	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.524 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, 6, 139, 278, 417, 834]: [0, 1, 1, 3, 12, 37, 48, 144] (certified at ell = 2147483137)
T = [(5, 1), (7, 3), (17, 5), (19, 7), (23, 9)]; h of degree 23 = 1 x 23 (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(834, 2)[11]   # 834.2.a.l

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

cargo run --release -p sagebrush-web --example atlas -- 834:2 1000