Atlas › Modular forms › Weight 4 › Level 170 › 170.4.a › 170.4.a.g

Newform orbit 170.4.a.g

Trace form

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

$3q - 6q^{2} + 7q^{3} + 12q^{4} - 15q^{5} - 14q^{6} - 10q^{7} - 24q^{8} + 52q^{9} + 30q^{10} + 44q^{11} + 28q^{12} + 99q^{13} + 20q^{14} - 35q^{15} + 48q^{16} - 51q^{17} - 104q^{18} + 225q^{19} - 60q^{20} - 42q^{21} - 88q^{22} + 46q^{23} - 56q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 23x^{2} - 8856x - 391000$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$+1$
17	$+1$

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	7	−15	−10	44	99	−51	225	46	317	275	372	178	−394	123	−1075	−827	−253	522	−585	−593	428	−1568	−1297	−899

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.156 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, 2, 5, 10, 17, 34, 85, 170]: [1, 2, 3, 7, 6, 16, 30, 86] (certified at ell = 2147483489)
T = [(3, 1), (7, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 4 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(170, 4)[6]   # 170.4.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 170:4 1000