Atlas › Modular forms › Weight 4 › Level 174 › 174.4.a › 174.4.a.g

Newform orbit 174.4.a.g

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{2} - 6q^{3} + 8q^{4} - 5q^{5} - 12q^{6} + 14q^{7} + 16q^{8} + 18q^{9} - 10q^{10} + 61q^{11} - 24q^{12} + 57q^{13} + 28q^{14} + 15q^{15} + 32q^{16} + 140q^{17} + 36q^{18} + 39q^{19} - 20q^{20} - 42q^{21} + 122q^{22} + 214q^{23} - 48q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{457})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + T_{7}$ , whose minimal polynomial is

$x^{2} - 9x - 94$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
29	$-1$

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
$\operatorname{tr} a_p$	4	−6	−5	14	61	57	140	39	214	58	−98	115	−151	241	−114	−378	377	−320	−595	750	−1288	234	1122	417	−2556

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.158 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, 3, 6, 29, 58, 87, 174]: [1, 2, 2, 5, 9, 25, 32, 94] (certified at ell = 2147483549)
T = [(5, 1), (7, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 3 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(174, 4)[6]   # 174.4.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 174:4 1000