Atlas › Modular forms › Weight 2 › Level 830 › 830.2.a › 830.2.a.f

Newform orbit 830.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{3} + 187x^{2} + 10419x + 154361$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$-1$
83	$-1$

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 odd 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	−1	3	−5	3	−6	−15	−4	0	−5	3	−7	6	−12	−10	4	9	−19	−4	10	−28	12	3	−18	−30

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.792 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, 5, 10, 83, 166, 415, 830]: [0, 1, 1, 3, 8, 23, 44, 130] (certified at ell = 2147483077)
T = [(3, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 8 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(830, 2)[5]   # 830.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 830:2 1000