Atlas › Modular forms › Weight 6 › Level 44 › 44.6.a › 44.6.a.b

Newform orbit 44.6.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 6q^{3} - 22q^{5} + 268q^{7} + 524q^{9} - 242q^{11} + 1232q^{13} + 2050q^{15} - 124q^{17} + 1944q^{19} - 3780q^{21} + 3346q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{31})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 28x - 4268$

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	−6	−22	268	−242	1232	−124	1944	3346	−6576	2498	−14674	−6504	11628	36816	−3292	12126	73128	29334	−46122	8240	−14780	−74564	−33698	126162

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.23 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, 2, 4, 6, 17, 33] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 3 = 1 x 3 (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(44, 6)[1]   # 44.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 44:6 1000