Atlas › Modular forms › Weight 2 › Level 178 › 178.2.a › 178.2.a.d

Newform orbit 178.2.a.d

Trace form

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

$3q + 3q^{2} + q^{3} + 3q^{4} - q^{5} + q^{6} + 3q^{8} + 8q^{9} - q^{10} + 6q^{11} + q^{12} - 2q^{13} - 17q^{15} + 3q^{16} - 5q^{17} + 8q^{18} - 11q^{19} - q^{20} - 12q^{21} + 6q^{22} + 5q^{23} + q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 2x^{2} - 32x - 32$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
89	$+1$

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
$\operatorname{tr} a_p$	3	1	−1	0	6	−2	−5	−11	5	4	−19	10	−4	−9	12	21	−8	2	16	10	−1	−10	0	−3	−19

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.175 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 89, 178]: [0, 1, 8, 24] (certified at ell = 2147482921)
T = [(3, 1), (5, 3)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 1 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(178, 2)[3]   # 178.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 178:2 1000