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

Newform orbit 223.2.a.c

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 7q^{2} + 13q^{4} + 7q^{5} - 4q^{6} + 2q^{7} + 18q^{8} + 18q^{9} - q^{10} + 6q^{11} - 7q^{12} + q^{13} - q^{14} - 8q^{15} + 15q^{16} + 27q^{17} + 8q^{18} - 5q^{19} - q^{20} - 9q^{21} - 14q^{22} + 12q^{23} - 31q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{12} - 7x^{11} - 225x^{10} + 1623x^{9} + 17746x^{8} - 141078x^{7} - 539762x^{6} + 5440217x^{5} + 2349059x^{4} - 81368851x^{3} + 108730841x^{2} + 159919856x - 167332732$

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 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
$\operatorname{tr} a_p$	7	0	7	2	6	1	27	−5	12	3	−12	2	22	0	12	26	3	−21	6	−6	16	−12	39	13	2

$\operatorname{tr} a_n$ 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)[2]   # 223.2.a.c

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

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