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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,8,Mod(361,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("588.361"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-27,0,100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(183.682394985\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.373
Dual form 588.8.i.c.361.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-13.5000 - 23.3827i) q^{3} +(50.0000 - 86.6025i) q^{5} +(-364.500 + 631.333i) q^{9} +(-1387.00 - 2402.35i) q^{11} +3294.00 q^{13} -2700.00 q^{15} +(2950.00 + 5109.55i) q^{17} +(3322.00 - 5753.87i) q^{19} +(-991.000 + 1716.46i) q^{23} +(34062.5 + 58998.0i) q^{25} +19683.0 q^{27} -208106. q^{29} +(-58896.0 - 102011. i) q^{31} +(-37449.0 + 64863.6i) q^{33} +(167843. - 290713. i) q^{37} +(-44469.0 - 77022.6i) q^{39} +265488. q^{41} -93292.0 q^{43} +(36450.0 + 63133.3i) q^{45} +(-328758. + 569426. i) q^{47} +(79650.0 - 137958. i) q^{51} +(304359. + 527165. i) q^{53} -277400. q^{55} -179388. q^{57} +(-268060. - 464294. i) q^{59} +(-898545. + 1.55633e6i) q^{61} +(164700. - 285269. i) q^{65} +(-1.06159e6 - 1.83872e6i) q^{67} +53514.0 q^{69} -1.19121e6 q^{71} +(528215. + 914895. i) q^{73} +(919688. - 1.59295e6i) q^{75} +(-499242. + 864713. i) q^{79} +(-265720. - 460241. i) q^{81} -3.89800e6 q^{83} +590000. q^{85} +(2.80943e6 + 4.86608e6i) q^{87} +(-2.31118e6 + 4.00307e6i) q^{89} +(-1.59019e6 + 2.75429e6i) q^{93} +(-332200. - 575387. i) q^{95} -1.52877e7 q^{97} +2.02225e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 27 q^{3} + 100 q^{5} - 729 q^{9} - 2774 q^{11} + 6588 q^{13} - 5400 q^{15} + 5900 q^{17} + 6644 q^{19} - 1982 q^{23} + 68125 q^{25} + 39366 q^{27} - 416212 q^{29} - 117792 q^{31} - 74898 q^{33} + 335686 q^{37}+ \cdots + 4044492 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −13.5000 − 23.3827i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) 50.0000 − 86.6025i 0.178885 − 0.309839i −0.762614 − 0.646854i \(-0.776084\pi\)
0.941499 + 0.337016i \(0.109418\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −364.500 + 631.333i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −1387.00 − 2402.35i −0.314197 − 0.544205i 0.665069 − 0.746782i \(-0.268402\pi\)
−0.979266 + 0.202576i \(0.935069\pi\)
\(12\) 0 0
\(13\) 3294.00 0.415836 0.207918 − 0.978146i \(-0.433331\pi\)
0.207918 + 0.978146i \(0.433331\pi\)
\(14\) 0 0
\(15\) −2700.00 −0.206559
\(16\) 0 0
\(17\) 2950.00 + 5109.55i 0.145630 + 0.252239i 0.929608 − 0.368550i \(-0.120146\pi\)
−0.783978 + 0.620789i \(0.786812\pi\)
\(18\) 0 0
\(19\) 3322.00 − 5753.87i 0.111112 − 0.192452i −0.805107 − 0.593130i \(-0.797892\pi\)
0.916219 + 0.400678i \(0.131225\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −991.000 + 1716.46i −0.0169835 + 0.0294162i −0.874392 − 0.485220i \(-0.838740\pi\)
0.857409 + 0.514636i \(0.172073\pi\)
\(24\) 0 0
\(25\) 34062.5 + 58998.0i 0.436000 + 0.755174i
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) −208106. −1.58450 −0.792249 − 0.610198i \(-0.791090\pi\)
−0.792249 + 0.610198i \(0.791090\pi\)
\(30\) 0 0
\(31\) −58896.0 − 102011.i −0.355075 − 0.615008i 0.632056 − 0.774923i \(-0.282211\pi\)
−0.987131 + 0.159915i \(0.948878\pi\)
\(32\) 0 0
\(33\) −37449.0 + 64863.6i −0.181402 + 0.314197i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 167843. − 290713.i 0.544750 − 0.943535i −0.453873 − 0.891067i \(-0.649958\pi\)
0.998623 − 0.0524680i \(-0.0167088\pi\)
\(38\) 0 0
\(39\) −44469.0 − 77022.6i −0.120041 − 0.207918i
\(40\) 0 0
\(41\) 265488. 0.601591 0.300796 − 0.953689i \(-0.402748\pi\)
0.300796 + 0.953689i \(0.402748\pi\)
\(42\) 0 0
\(43\) −93292.0 −0.178939 −0.0894695 − 0.995990i \(-0.528517\pi\)
−0.0894695 + 0.995990i \(0.528517\pi\)
\(44\) 0 0
\(45\) 36450.0 + 63133.3i 0.0596285 + 0.103280i
\(46\) 0 0
\(47\) −328758. + 569426.i −0.461885 + 0.800008i −0.999055 − 0.0434658i \(-0.986160\pi\)
0.537170 + 0.843474i \(0.319493\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 79650.0 − 137958.i 0.0840795 − 0.145630i
\(52\) 0 0
\(53\) 304359. + 527165.i 0.280815 + 0.486386i 0.971586 − 0.236688i \(-0.0760619\pi\)
−0.690771 + 0.723074i \(0.742729\pi\)
\(54\) 0 0
\(55\) −277400. −0.224821
\(56\) 0 0
\(57\) −179388. −0.128301
\(58\) 0 0
\(59\) −268060. − 464294.i −0.169922 − 0.294314i 0.768470 − 0.639886i \(-0.221018\pi\)
−0.938392 + 0.345572i \(0.887685\pi\)
\(60\) 0 0
\(61\) −898545. + 1.55633e6i −0.506857 + 0.877902i 0.493112 + 0.869966i \(0.335859\pi\)
−0.999969 + 0.00793591i \(0.997474\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 164700. − 285269.i 0.0743870 − 0.128842i
\(66\) 0 0
\(67\) −1.06159e6 − 1.83872e6i −0.431215 − 0.746887i 0.565763 − 0.824568i \(-0.308582\pi\)
−0.996978 + 0.0776811i \(0.975248\pi\)
\(68\) 0 0
\(69\) 53514.0 0.0196108
\(70\) 0 0
\(71\) −1.19121e6 −0.394990 −0.197495 − 0.980304i \(-0.563281\pi\)
−0.197495 + 0.980304i \(0.563281\pi\)
\(72\) 0 0
\(73\) 528215. + 914895.i 0.158921 + 0.275259i 0.934480 − 0.356016i \(-0.115865\pi\)
−0.775559 + 0.631275i \(0.782532\pi\)
\(74\) 0 0
\(75\) 919688. − 1.59295e6i 0.251725 − 0.436000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −499242. + 864713.i −0.113924 + 0.197323i −0.917349 − 0.398083i \(-0.869675\pi\)
0.803425 + 0.595406i \(0.203009\pi\)
\(80\) 0 0
\(81\) −265720. − 460241.i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −3.89800e6 −0.748288 −0.374144 − 0.927371i \(-0.622063\pi\)
−0.374144 + 0.927371i \(0.622063\pi\)
\(84\) 0 0
\(85\) 590000. 0.104204
\(86\) 0 0
\(87\) 2.80943e6 + 4.86608e6i 0.457405 + 0.792249i
\(88\) 0 0
\(89\) −2.31118e6 + 4.00307e6i −0.347511 + 0.601906i −0.985807 − 0.167885i \(-0.946306\pi\)
0.638296 + 0.769791i \(0.279640\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.59019e6 + 2.75429e6i −0.205003 + 0.355075i
\(94\) 0 0
\(95\) −332200. − 575387.i −0.0397527 − 0.0688538i
\(96\) 0 0
\(97\) −1.52877e7 −1.70075 −0.850377 − 0.526174i \(-0.823626\pi\)
−0.850377 + 0.526174i \(0.823626\pi\)
\(98\) 0 0
\(99\) 2.02225e6 0.209465
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 588.8.i.c.373.1 2
7.2 even 3 588.8.a.c.1.1 1
7.3 odd 6 588.8.i.f.361.1 2
7.4 even 3 inner 588.8.i.c.361.1 2
7.5 odd 6 84.8.a.a.1.1 ✓ 1
7.6 odd 2 588.8.i.f.373.1 2
21.5 even 6 252.8.a.a.1.1 1
28.19 even 6 336.8.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.a.1.1 ✓ 1 7.5 odd 6
252.8.a.a.1.1 1 21.5 even 6
336.8.a.j.1.1 1 28.19 even 6
588.8.a.c.1.1 1 7.2 even 3
588.8.i.c.361.1 2 7.4 even 3 inner
588.8.i.c.373.1 2 1.1 even 1 trivial
588.8.i.f.361.1 2 7.3 odd 6
588.8.i.f.373.1 2 7.6 odd 2