Atlas › Modular forms › Weight 4 › Level 233 › 233.4.a › 233.4.a.b

Newform orbit 233.4.a.b

Trace form

The 32 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$32q + 4q^{2} + 28q^{3} + 152q^{4} + 24q^{5} + 20q^{6} + 158q^{7} + 48q^{8} + 356q^{9} + 116q^{10} + 16q^{11} + 213q^{12} + 228q^{13} + 135q^{14} + 328q^{15} + 836q^{16} + 138q^{17} + 159q^{18} + 386q^{19} + 209q^{20} + 182q^{21} + 457q^{22} + 558q^{23} + 91q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 32. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{32} - 32x^{31} - 322x^{30} + 19570x^{29} - 30177x^{28} - 4923593x^{27} + 27463024x^{26} + 680445195x^{25} - 5351457971x^{24} - 58262265578x^{23} + 554754454138x^{22} + 3328514110378x^{21} - 35854716967065x^{20} - 135073768135489x^{19} + 1536595297672331x^{18} + 4140372516879701x^{17} - 44728161405594284x^{16} - 100211489761803059x^{15} + 883363314504585705x^{14} + 1901529771039067887x^{13} - 11525963772708978702x^{12} - 26452458507385264337x^{11} + 93379997536546515131x^{10} + 243509392852657150433x^{9} - 414378824803687540298x^{8} - 1303518296268064658436x^{7} + 777652039682976213373x^{6} + 3417300894262480068400x^{5} - 560251960786837864872x^{4} - 4188850222289705061496x^{3} + 459912636308901498060x^{2} + 2041329934726067403328x - 533453661373543916192$

Atkin–Lehner signs

$p$	$w_p$
233	$+1$

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
$\operatorname{tr} a_p$	4	28	24	158	16	228	138	386	558	342	1430	1378	230	882	1554	1104	432	1248	2880	1112	1534	1128	−148	−1564	2226

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 233]: [1, 60] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 58 = 1 x 58 (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(233, 4)[1]   # 233.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 233:4 1000