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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 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.361
Dual form 588.8.i.f.373.1

$q$-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 \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 \(-0.223916\pi\)
−0.941499 + 0.337016i \(0.890582\pi\)
\(6\) 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 −0.979266 − 0.202576i \(-0.935069\pi\)
0.665069 + 0.746782i \(0.268402\pi\)
\(12\) 0 0
\(13\) −3294.00 −0.415836 −0.207918 − 0.978146i \(-0.566669\pi\)
−0.207918 + 0.978146i \(0.566669\pi\)
\(14\) 0 0
\(15\) −2700.00 −0.206559
\(16\) 0 0
\(17\) −2950.00 + 5109.55i −0.145630 + 0.252239i −0.929608 − 0.368550i \(-0.879854\pi\)
0.783978 + 0.620789i \(0.213188\pi\)
\(18\) 0 0
\(19\) −3322.00 − 5753.87i −0.111112 − 0.192452i 0.805107 − 0.593130i \(-0.202108\pi\)
−0.916219 + 0.400678i \(0.868775\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −991.000 − 1716.46i −0.0169835 − 0.0294162i 0.857409 − 0.514636i \(-0.172073\pi\)
−0.874392 + 0.485220i \(0.838740\pi\)
\(24\) 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 \(-0.717789\pi\)
0.987131 + 0.159915i \(0.0511221\pi\)
\(32\) 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 0.998623 + 0.0524680i \(0.0167088\pi\)
−0.453873 + 0.891067i \(0.649958\pi\)
\(38\) 0 0
\(39\) −44469.0 + 77022.6i −0.120041 + 0.207918i
\(40\) 0 0
\(41\) −265488. −0.601591 −0.300796 − 0.953689i \(-0.597252\pi\)
−0.300796 + 0.953689i \(0.597252\pi\)
\(42\) 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 \(-0.0138400\pi\)
−0.537170 + 0.843474i \(0.680507\pi\)
\(48\) 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 −0.690771 − 0.723074i \(-0.742729\pi\)
0.971586 + 0.236688i \(0.0760619\pi\)
\(54\) 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 \(-0.778982\pi\)
0.938392 + 0.345572i \(0.112315\pi\)
\(60\) 0 0
\(61\) 898545. + 1.55633e6i 0.506857 + 0.877902i 0.999969 + 0.00793591i \(0.00252610\pi\)
−0.493112 + 0.869966i \(0.664141\pi\)
\(62\) 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 −0.996978 − 0.0776811i \(-0.975248\pi\)
0.565763 + 0.824568i \(0.308582\pi\)
\(68\) 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 \(-0.884135\pi\)
0.775559 + 0.631275i \(0.217468\pi\)
\(74\) 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 0.803425 − 0.595406i \(-0.203009\pi\)
−0.917349 + 0.398083i \(0.869675\pi\)
\(80\) 0 0
\(81\) −265720. + 460241.i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 3.89800e6 0.748288 0.374144 − 0.927371i \(-0.377937\pi\)
0.374144 + 0.927371i \(0.377937\pi\)
\(84\) 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 \(-0.0536938\pi\)
−0.638296 + 0.769791i \(0.720360\pi\)
\(90\) 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 \(-0.176374\pi\)
0.850377 + 0.526174i \(0.176374\pi\)
\(98\) 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.f.361.1 2
7.2 even 3 inner 588.8.i.f.373.1 2
7.3 odd 6 588.8.a.c.1.1 1
7.4 even 3 84.8.a.a.1.1 ✓ 1
7.5 odd 6 588.8.i.c.373.1 2
7.6 odd 2 588.8.i.c.361.1 2
21.11 odd 6 252.8.a.a.1.1 1
28.11 odd 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 7.4 even 3
252.8.a.a.1.1 1 21.11 odd 6
336.8.a.j.1.1 1 28.11 odd 6
588.8.a.c.1.1 1 7.3 odd 6
588.8.i.c.361.1 2 7.6 odd 2
588.8.i.c.373.1 2 7.5 odd 6
588.8.i.f.361.1 2 1.1 even 1 trivial
588.8.i.f.373.1 2 7.2 even 3 inner