Atlas › Modular forms › Weight 4 › Level 223 › 223.4.a › 223.4.a.b

Newform orbit 223.4.a.b

Trace form

The 31 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$31q + 15q^{2} + 10q^{3} + 135q^{4} + 50q^{5} + 46q^{6} + 51q^{7} + 180q^{8} + 381q^{9} + 39q^{10} + 182q^{11} + 53q^{12} + 168q^{13} + 75q^{14} + 76q^{15} + 631q^{16} + 619q^{17} + 422q^{18} + 19q^{19} + 595q^{20} + 256q^{21} + 362q^{22} + 596q^{23} + 767q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 31. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{31} - 25x^{30} - 534x^{29} + 17139x^{28} + 95137x^{27} - 4935216x^{26} - 3288668x^{25} + 799976620x^{24} - 1180760182x^{23} - 82282799516x^{22} + 204437421503x^{21} + 5714064354746x^{20} - 16440889274317x^{19} - 277021835160278x^{18} + 796530753294439x^{17} + 9518573462720808x^{16} - 24804071174949744x^{15} - 231886084788760231x^{14} + 502370182660311413x^{13} + 3949959875116006735x^{12} - 6485432426242617738x^{11} - 45658063344273582254x^{10} + 50345939822631790831x^{9} + 340138341453365245992x^{8} - 204479294589134954925x^{7} - 1496252500729924077584x^{6} + 260091821726843081248x^{5} + 3302326020111355102600x^{4} + 490865589689854667792x^{3} - 2524151216195599868640x^{2} - 869928183437712630912x + 189423682021911177216$

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
$\operatorname{tr} a_p$	15	10	50	51	182	168	619	19	596	411	123	337	1659	693	1025	2931	−122	−234	−966	800	311	−1580	2843	3493	710

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.151 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, 223]: [1, 57] (certified at ell = 2147482591)
T = [(2, 1), (3, 1)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 8 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, 4)[1]   # 223.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 223:4 1000