Atlas › Modular forms › Weight 4 › Level 222 › 222.4.a › 222.4.a.h

Newform orbit 222.4.a.h

Trace form

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

$3q - 6q^{2} + 9q^{3} + 12q^{4} + 6q^{5} - 18q^{6} - 3q^{7} - 24q^{8} + 27q^{9} - 12q^{10} + 93q^{11} + 36q^{12} + 3q^{13} + 6q^{14} + 18q^{15} + 48q^{16} + 159q^{17} - 54q^{18} - 33q^{19} + 24q^{20} - 9q^{21} - 186q^{22} + 171q^{23} - 72q^{24} + O(q^{25})$

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

$x^{3} - 3x^{2} - 1374x + 8180$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$-1$
37	$-1$

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	6	−3	93	3	159	−33	171	150	18	111	966	432	858	753	174	−798	54	−444	−819	−1194	813	−813	−678

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.193 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, 37, 74, 111, 222]: [1, 2, 2, 5, 11, 31, 40, 118] (certified at ell = 2147483629)
T = [(5, 1), (7, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 4 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(222, 4)[7]   # 222.4.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 222:4 1000