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

Newform orbit 170.4.a.h

Trace form

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

$3q + 6q^{2} - q^{3} + 12q^{4} - 15q^{5} - 2q^{6} + 20q^{7} + 24q^{8} + 84q^{9} - 30q^{10} - 6q^{11} - 4q^{12} + 177q^{13} + 40q^{14} + 5q^{15} + 48q^{16} + 51q^{17} + 168q^{18} + 121q^{19} - 60q^{20} + 194q^{21} - 12q^{22} - 44q^{23} - 8q^{24} + O(q^{25})$

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

$x^{3} - 59x^{2} - 2884x + 128720$

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	−1	−15	20	−6	177	51	121	−44	−351	73	−76	214	310	9	−449	−1227	471	−414	301	−367	310	−376	−473	2979

$\operatorname{tr} a_n$ 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)[7]   # 170.4.a.h

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

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