Atlas › Modular forms › Weight 4 › Level 241 › 241.4.a › 241.4.a.b

Newform orbit 241.4.a.b

Trace form

The 33 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$33q + 9q^{2} + 10q^{3} + 161q^{4} + 50q^{5} + 33q^{6} + 58q^{7} + 111q^{8} + 405q^{9} + q^{10} + 274q^{11} + 115q^{12} + 24q^{13} + 220q^{14} + 362q^{15} + 789q^{16} + 140q^{17} + 135q^{18} + 164q^{19} + 630q^{20} + 204q^{21} + 270q^{22} + 980q^{23} + 279q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 33. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{33} - 19x^{32} - 713x^{31} + 14888x^{30} + 206737x^{29} - 4906767x^{28} - 32410626x^{27} + 908516434x^{26} + 3117824283x^{25} - 106443625141x^{24} - 204061960084x^{23} + 8393484961463x^{22} + 10519918980897x^{21} - 460073357068844x^{20} - 502792361372960x^{19} + 17738644709032611x^{18} + 22244123433782914x^{17} - 477609633472578551x^{16} - 779122366967965080x^{15} + 8705572088486500877x^{14} + 18955452337289080172x^{13} - 99973323077140701720x^{12} - 294687612451532750995x^{11} + 602766191094844667165x^{10} + 2674786481938500317758x^{9} - 594843531330169046940x^{8} - 11985913575926843595600x^{7} - 10628640627235027452976x^{6} + 15892056079018589613856x^{5} + 30149055600762330642880x^{4} + 9731199845878694075392x^{3} - 8080600033709457220096x^{2} - 5088708972789220769792x - 412625090630475948032$

Atkin–Lehner signs

$p$	$w_p$
241	$+1$

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
$\operatorname{tr} a_p$	9	10	50	58	274	24	140	164	980	772	804	342	978	230	2312	1352	3546	156	−236	5966	−2020	−198	502	−192	−846

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.164 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, 241]: [1, 62] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (phi(ord eps) x dim S_k^new); 9 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(241, 4)[1]   # 241.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 241:4 1000