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

Newform orbit 174.4.a.h

Trace form

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

$3q - 6q^{2} + 9q^{3} + 12q^{4} - 5q^{5} - 18q^{6} + 18q^{7} - 24q^{8} + 27q^{9} + 10q^{10} + 57q^{11} + 36q^{12} + 79q^{13} - 36q^{14} - 15q^{15} + 48q^{16} - 80q^{17} - 54q^{18} + 223q^{19} - 20q^{20} + 54q^{21} - 114q^{22} - 132q^{23} - 72q^{24} + O(q^{25})$

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

$x^{3} - 13x^{2} - 1760x + 31356$

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	−5	18	57	79	−80	223	−132	87	540	385	1	519	−116	354	−403	24	−55	−1780	−400	−720	−206	−293	300

$\operatorname{tr} a_n$ 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)[7]   # 174.4.a.h

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

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