Atlas › Modular forms › Weight 2 › Level 223 › 223.2.a › 223.2.a.b

Newform orbit 223.2.a.b

Trace form

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

$4q - 4q^{2} + 4q^{4} - 3q^{5} - 8q^{6} - 6q^{7} - 9q^{8} - 4q^{9} + 7q^{10} - 10q^{11} + 13q^{12} - 9q^{13} + 11q^{14} - 4q^{15} + 12q^{16} - 17q^{17} + 7q^{18} + 7q^{19} - 13q^{20} - 5q^{21} + 10q^{22} - 2q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} + 4x^{3} - 10x^{2} - 20x + 9$

Atkin–Lehner signs

$p$	$w_p$
223	$+1$

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 odd 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
$\operatorname{tr} a_p$	−4	0	−3	−6	−10	−9	−17	7	−2	7	0	−2	2	16	16	−26	−15	1	4	4	0	−12	−13	9	−34

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.173 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, 223]: [0, 19] (certified at ell = 2147482591)
T = [(2, 1), (3, 3)]; h of degree 18 = 1 x 18 (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(223, 2)[1]   # 223.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 223:2 1000