Atlas › Modular forms › Weight 4 › Level 148 › 148.4.a › 148.4.a.a

Newform orbit 148.4.a.a

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - q^{3} - 19q^{5} - 6q^{7} - 7q^{9} - 77q^{11} - 115q^{13} - 172q^{15} - 180q^{17} + 4q^{19} - 208q^{21} - 189q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} + 20x^{3} + 62x^{2} - 148x - 479$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 148$ )
37	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	−1	−19	−6	−77	−115	−180	4	−189	−99	−251	−148	−497	694	−50	−290	586	991	823	−96	121	475	820	−880	−904

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.114 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, 4, 37, 74, 148]: [1, 2, 3, 11, 31, 60] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 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(148, 4)[0]   # 148.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 148:4 1000