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

Newform orbit 174.4.a.i

Trace form

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

$3q + 6q^{2} + 9q^{3} + 12q^{4} + 7q^{5} + 18q^{6} + 34q^{7} + 24q^{8} + 27q^{9} + 14q^{10} + 3q^{11} + 36q^{12} + 51q^{13} + 68q^{14} + 21q^{15} + 48q^{16} + 8q^{17} + 54q^{18} + 77q^{19} + 28q^{20} + 102q^{21} + 6q^{22} + 22q^{23} + 72q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 41x^{2} + 270x + 2400$

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
$\operatorname{tr} a_p$	6	9	7	34	3	51	8	77	22	−87	−202	−105	−121	−149	−308	−344	−801	−374	−103	−1042	394	−346	−614	−2347	322

$\operatorname{tr} a_n$ 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)[8]   # 174.4.a.i

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

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