Atlas › Modular forms › Weight 8 › Level 44 › 44.8.a › 44.8.a.a

Newform orbit 44.8.a.a

Trace form

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

$3q - 62q^{3} + 288q^{5} - 196q^{7} + 1873q^{9} - 3993q^{11} - 7602q^{13} - 40330q^{15} - 47338q^{17} - 81780q^{19} - 37028q^{21} - 71794q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} - 226x^{2} - 53616x - 2265120$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 44$ )
11	$+1$

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	−62	288	−196	−3993	−7602	−47338	−81780	−71794	−171938	−193866	369348	−29614	357280	599080	346802	−168858	−441050	−2697402	−6589318	−2051782	−7495540	−981432	10112280	−402284

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 11, 22, 44]: [1, 3, 5, 8, 23, 45] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(44, 8)[0]   # 44.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 44:8 1000