Atlas › Modular forms › Weight 6 › Level 35 › 35.6.a › 35.6.a.c

Newform orbit 35.6.a.c

Trace form

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

$3q - 6q^{2} + 26q^{3} + 112q^{4} - 75q^{5} - 96q^{6} + 147q^{7} - 120q^{8} + 489q^{9} + 150q^{10} - 194q^{11} + 2956q^{12} + 1892q^{13} - 294q^{14} - 650q^{15} + 1496q^{16} - 184q^{17} - 6078q^{18} + 1212q^{19} - 2800q^{20} + 1274q^{21} - 872q^{22} + 3188q^{23} - 13920q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{3} - 20x^{2} - 417x + 8074$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
7	$-1$

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
$\operatorname{tr} a_p$	−6	26	−75	147	−194	1892	−184	1212	3188	−11332	9200	6042	10442	28112	2330	7190	23760	8722	−97572	−52816	34870	−71546	−20920	−192622	116320

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.228 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 engine's checks

dim M^+ at levels [1, 5, 7, 35]: [1, 3, 5, 22] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 10 = 1 x 10 (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(35, 6)[2]   # 35.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 35:6 1000