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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,2,Mod(14,171)] 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("171.14"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([15, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.x (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 110.17
Character \(\chi\) \(=\) 171.110
Dual form 171.2.x.a.14.17

$q$-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+(1.98542 - 1.66596i) q^{2} +(0.915994 + 1.47002i) q^{3} +(0.819153 - 4.64565i) q^{4} +(-1.06492 + 2.92585i) q^{5} +(4.26763 + 1.39259i) q^{6} +(-1.52472 - 2.64090i) q^{7} +(-3.52134 - 6.09914i) q^{8} +(-1.32191 + 2.69306i) q^{9} +(2.76004 + 7.58315i) q^{10} -3.42745i q^{11} +(7.57953 - 3.05122i) q^{12} +(0.864317 + 2.37469i) q^{13} +(-7.42686 - 2.70316i) q^{14} +(-5.27651 + 1.11461i) q^{15} +(-8.28661 - 3.01608i) q^{16} +(-1.79294 + 4.92607i) q^{17} +(1.86199 + 7.54909i) q^{18} +(-3.57649 - 2.49173i) q^{19} +(12.7201 + 7.34397i) q^{20} +(2.48553 - 4.66042i) q^{21} +(-5.71001 - 6.80493i) q^{22} +(2.96343 + 0.522533i) q^{23} +(5.74032 - 10.7632i) q^{24} +(-3.59631 - 3.01766i) q^{25} +(5.67218 + 3.27483i) q^{26} +(-5.16970 + 0.523595i) q^{27} +(-13.5177 + 4.92003i) q^{28} +(-0.536153 + 3.04067i) q^{29} +(-8.61919 + 11.0034i) q^{30} -0.794029i q^{31} +(-8.24116 + 2.99954i) q^{32} +(5.03842 - 3.13953i) q^{33} +(4.64691 + 12.7673i) q^{34} +(9.35059 - 1.64876i) q^{35} +(11.4282 + 8.34715i) q^{36} -7.02028i q^{37} +(-11.2520 + 1.01117i) q^{38} +(-2.69913 + 3.44577i) q^{39} +(21.5951 - 3.80780i) q^{40} +(4.12573 - 3.46190i) q^{41} +(-2.82927 - 13.3937i) q^{42} +(-0.331522 - 1.88015i) q^{43} +(-15.9227 - 2.80761i) q^{44} +(-6.47175 - 6.73560i) q^{45} +(6.75417 - 3.89952i) q^{46} +(12.8465 + 2.26518i) q^{47} +(-3.15680 - 14.9442i) q^{48} +(-1.14957 + 1.99111i) q^{49} -12.1675 q^{50} +(-8.88374 + 1.87659i) q^{51} +(11.7400 - 2.07008i) q^{52} +(3.01965 + 2.53379i) q^{53} +(-9.39173 + 9.65209i) q^{54} +(10.0282 + 3.64997i) q^{55} +(-10.7381 + 18.5990i) q^{56} +(0.386850 - 7.53992i) q^{57} +(4.00116 + 6.93022i) q^{58} +(-0.194589 - 1.10357i) q^{59} +(0.855795 + 25.4259i) q^{60} +(-1.80227 + 0.655973i) q^{61} +(-1.32282 - 1.57648i) q^{62} +(9.12764 - 0.615141i) q^{63} +(-2.54660 + 4.41084i) q^{64} -7.86842 q^{65} +(4.77303 - 14.6271i) q^{66} +(-0.412278 + 0.491334i) q^{67} +(21.4161 + 12.3646i) q^{68} +(1.94635 + 4.83494i) q^{69} +(15.8180 - 18.8512i) q^{70} +(-5.62967 + 4.72386i) q^{71} +(21.0802 - 1.42066i) q^{72} +(0.808108 + 4.58301i) q^{73} +(-11.6955 - 13.9382i) q^{74} +(1.14182 - 8.05081i) q^{75} +(-14.5054 + 14.5740i) q^{76} +(-9.05156 + 5.22592i) q^{77} +(0.381617 + 11.3379i) q^{78} +(-2.33565 + 6.41714i) q^{79} +(17.6492 - 21.0335i) q^{80} +(-5.50511 - 7.11995i) q^{81} +(2.42390 - 13.7466i) q^{82} +(-1.59043 + 0.918238i) q^{83} +(-19.6147 - 15.3645i) q^{84} +(-12.5036 - 10.4918i) q^{85} +(-3.79048 - 3.18059i) q^{86} +(-4.96096 + 1.99709i) q^{87} +(-20.9045 + 12.0692i) q^{88} +(2.03020 - 11.5138i) q^{89} +(-24.0704 - 2.59129i) q^{90} +(4.95348 - 5.90333i) q^{91} +(4.85501 - 13.3390i) q^{92} +(1.16724 - 0.727326i) q^{93} +(29.2793 - 16.9044i) q^{94} +(11.0991 - 7.81076i) q^{95} +(-11.9582 - 9.36710i) q^{96} +(-9.52510 - 11.3516i) q^{97} +(1.03475 + 5.86834i) q^{98} +(9.23033 + 4.53078i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q - 9 q^{2} - 3 q^{4} - 9 q^{5} + 3 q^{7} - 24 q^{9} - 12 q^{10} - 9 q^{12} - 6 q^{13} - 9 q^{14} - 36 q^{15} - 9 q^{16} + 27 q^{17} + 36 q^{18} - 15 q^{19} - 18 q^{20} + 3 q^{21} + 30 q^{22} - 45 q^{23}+ \cdots + 39 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.98542 − 1.66596i 1.40390 − 1.17801i 0.444566 − 0.895746i \(-0.353358\pi\)
0.959336 − 0.282268i \(-0.0910867\pi\)
\(3\) 0.915994 + 1.47002i 0.528850 + 0.848716i
\(4\) 0.819153 − 4.64565i 0.409577 − 2.32282i
\(5\) −1.06492 + 2.92585i −0.476248 + 1.30848i 0.436408 + 0.899749i \(0.356251\pi\)
−0.912655 + 0.408730i \(0.865972\pi\)
\(6\) 4.26763 + 1.39259i 1.74225 + 0.568522i
\(7\) −1.52472 − 2.64090i −0.576292 − 0.998167i −0.995900 − 0.0904613i \(-0.971166\pi\)
0.419608 − 0.907705i \(-0.362167\pi\)
\(8\) −3.52134 − 6.09914i −1.24498 − 2.15637i
\(9\) −1.32191 + 2.69306i −0.440636 + 0.897686i
\(10\) 2.76004 + 7.58315i 0.872802 + 2.39800i
\(11\) − 3.42745i − 1.03342i −0.856162 − 0.516708i \(-0.827157\pi\)
0.856162 − 0.516708i \(-0.172843\pi\)
\(12\) 7.57953 − 3.05122i 2.18802 − 0.880811i
\(13\) 0.864317 + 2.37469i 0.239718 + 0.658621i 0.999960 + 0.00898021i \(0.00285853\pi\)
−0.760241 + 0.649641i \(0.774919\pi\)
\(14\) −7.42686 − 2.70316i −1.98491 − 0.722449i
\(15\) −5.27651 + 1.11461i −1.36239 + 0.287790i
\(16\) −8.28661 − 3.01608i −2.07165 − 0.754020i
\(17\) −1.79294 + 4.92607i −0.434853 + 1.19475i 0.507947 + 0.861388i \(0.330404\pi\)
−0.942800 + 0.333359i \(0.891818\pi\)
\(18\) 1.86199 + 7.54909i 0.438876 + 1.77934i
\(19\) −3.57649 − 2.49173i −0.820503 − 0.571643i
\(20\) 12.7201 + 7.34397i 2.84431 + 1.64216i
\(21\) 2.48553 − 4.66042i 0.542388 − 1.01699i
\(22\) −5.71001 − 6.80493i −1.21738 − 1.45082i
\(23\) 2.96343 + 0.522533i 0.617918 + 0.108956i 0.473840 − 0.880611i \(-0.342867\pi\)
0.144078 + 0.989566i \(0.453978\pi\)
\(24\) 5.74032 − 10.7632i 1.17174 − 2.19703i
\(25\) −3.59631 − 3.01766i −0.719262 − 0.603533i
\(26\) 5.67218 + 3.27483i 1.11241 + 0.642248i
\(27\) −5.16970 + 0.523595i −0.994910 + 0.100766i
\(28\) −13.5177 + 4.92003i −2.55460 + 0.929799i
\(29\) −0.536153 + 3.04067i −0.0995611 + 0.564639i 0.893693 + 0.448679i \(0.148105\pi\)
−0.993254 + 0.115960i \(0.963006\pi\)
\(30\) −8.61919 + 11.0034i −1.57364 + 2.00894i
\(31\) − 0.794029i − 0.142612i −0.997454 − 0.0713059i \(-0.977283\pi\)
0.997454 − 0.0713059i \(-0.0227167\pi\)
\(32\) −8.24116 + 2.99954i −1.45685 + 0.530248i
\(33\) 5.03842 − 3.13953i 0.877076 − 0.546522i
\(34\) 4.64691 + 12.7673i 0.796939 + 2.18957i
\(35\) 9.35059 − 1.64876i 1.58054 − 0.278691i
\(36\) 11.4282 + 8.34715i 1.90469 + 1.39119i
\(37\) − 7.02028i − 1.15413i −0.816699 − 0.577064i \(-0.804198\pi\)
0.816699 − 0.577064i \(-0.195802\pi\)
\(38\) −11.2520 + 1.01117i −1.82531 + 0.164033i
\(39\) −2.69913 + 3.44577i −0.432207 + 0.551764i
\(40\) 21.5951 − 3.80780i 3.41449 − 0.602066i
\(41\) 4.12573 − 3.46190i 0.644330 − 0.540657i −0.261014 − 0.965335i \(-0.584057\pi\)
0.905345 + 0.424678i \(0.139613\pi\)
\(42\) −2.82927 − 13.3937i −0.436566 − 2.06669i
\(43\) −0.331522 − 1.88015i −0.0505566 − 0.286721i 0.949039 − 0.315159i \(-0.102058\pi\)
−0.999596 + 0.0284381i \(0.990947\pi\)
\(44\) −15.9227 − 2.80761i −2.40044 − 0.423263i
\(45\) −6.47175 − 6.73560i −0.964751 − 1.00408i
\(46\) 6.75417 − 3.89952i 0.995848 − 0.574953i
\(47\) 12.8465 + 2.26518i 1.87385 + 0.330410i 0.990412 − 0.138143i \(-0.0441135\pi\)
0.883435 + 0.468553i \(0.155225\pi\)
\(48\) −3.15680 − 14.9442i −0.455644 − 2.15701i
\(49\) −1.14957 + 1.99111i −0.164224 + 0.284445i
\(50\) −12.1675 −1.72074
\(51\) −8.88374 + 1.87659i −1.24397 + 0.262776i
\(52\) 11.7400 − 2.07008i 1.62804 − 0.287068i
\(53\) 3.01965 + 2.53379i 0.414781 + 0.348043i 0.826174 − 0.563416i \(-0.190513\pi\)
−0.411392 + 0.911458i \(0.634957\pi\)
\(54\) −9.39173 + 9.65209i −1.27805 + 1.31348i
\(55\) 10.0282 + 3.64997i 1.35220 + 0.492162i
\(56\) −10.7381 + 18.5990i −1.43494 + 2.48540i
\(57\) 0.386850 − 7.53992i 0.0512395 − 0.998686i
\(58\) 4.00116 + 6.93022i 0.525379 + 0.909982i
\(59\) −0.194589 − 1.10357i −0.0253333 − 0.143672i 0.969518 − 0.245022i \(-0.0787951\pi\)
−0.994851 + 0.101349i \(0.967684\pi\)
\(60\) 0.855795 + 25.4259i 0.110483 + 3.28246i
\(61\) −1.80227 + 0.655973i −0.230757 + 0.0839888i −0.454811 − 0.890588i \(-0.650293\pi\)
0.224054 + 0.974577i \(0.428071\pi\)
\(62\) −1.32282 − 1.57648i −0.167999 − 0.200213i
\(63\) 9.12764 − 0.615141i 1.14997 − 0.0775005i
\(64\) −2.54660 + 4.41084i −0.318325 + 0.551354i
\(65\) −7.86842 −0.975958
\(66\) 4.77303 − 14.6271i 0.587519 − 1.80047i
\(67\) −0.412278 + 0.491334i −0.0503678 + 0.0600260i −0.790640 − 0.612281i \(-0.790252\pi\)
0.740272 + 0.672307i \(0.234697\pi\)
\(68\) 21.4161 + 12.3646i 2.59708 + 1.49943i
\(69\) 1.94635 + 4.83494i 0.234313 + 0.582058i
\(70\) 15.8180 − 18.8512i 1.89062 − 2.25315i
\(71\) −5.62967 + 4.72386i −0.668119 + 0.560619i −0.912508 − 0.409059i \(-0.865857\pi\)
0.244389 + 0.969677i \(0.421413\pi\)
\(72\) 21.0802 − 1.42066i 2.48433 − 0.167427i
\(73\) 0.808108 + 4.58301i 0.0945819 + 0.536401i 0.994875 + 0.101116i \(0.0322413\pi\)
−0.900293 + 0.435285i \(0.856648\pi\)
\(74\) −11.6955 − 13.9382i −1.35958 − 1.62028i
\(75\) 1.14182 − 8.05081i 0.131846 − 0.929627i
\(76\) −14.5054 + 14.5740i −1.66388 + 1.67175i
\(77\) −9.05156 + 5.22592i −1.03152 + 0.595549i
\(78\) 0.381617 + 11.3379i 0.0432096 + 1.28377i
\(79\) −2.33565 + 6.41714i −0.262781 + 0.721985i 0.736196 + 0.676768i \(0.236620\pi\)
−0.998977 + 0.0452165i \(0.985602\pi\)
\(80\) 17.6492 − 21.0335i 1.97324 − 2.35162i
\(81\) −5.50511 − 7.11995i −0.611679 − 0.791106i
\(82\) 2.42390 − 13.7466i 0.267675 − 1.51806i
\(83\) −1.59043 + 0.918238i −0.174573 + 0.100790i −0.584740 − 0.811221i \(-0.698804\pi\)
0.410167 + 0.912010i \(0.365470\pi\)
\(84\) −19.6147 − 15.3645i −2.14013 − 1.67641i
\(85\) −12.5036 − 10.4918i −1.35621 − 1.13799i
\(86\) −3.79048 − 3.18059i −0.408737 − 0.342971i
\(87\) −4.96096 + 1.99709i −0.531871 + 0.214110i
\(88\) −20.9045 + 12.0692i −2.22843 + 1.28658i
\(89\) 2.03020 − 11.5138i 0.215201 − 1.22046i −0.665358 − 0.746525i \(-0.731721\pi\)
0.880558 − 0.473938i \(-0.157168\pi\)
\(90\) −24.0704 − 2.59129i −2.53724 − 0.273145i
\(91\) 4.95348 − 5.90333i 0.519266 − 0.618837i
\(92\) 4.85501 − 13.3390i 0.506170 − 1.39069i
\(93\) 1.16724 − 0.727326i 0.121037 − 0.0754202i
\(94\) 29.2793 − 16.9044i 3.01993 − 1.74355i
\(95\) 11.0991 − 7.81076i 1.13874 − 0.801368i
\(96\) −11.9582 − 9.36710i −1.22048 − 0.956026i
\(97\) −9.52510 − 11.3516i −0.967127 − 1.15258i −0.988257 − 0.152801i \(-0.951171\pi\)
0.0211298 − 0.999777i \(-0.493274\pi\)
\(98\) 1.03475 + 5.86834i 0.104525 + 0.592791i
\(99\) 9.23033 + 4.53078i 0.927683 + 0.455361i
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 171.2.x.a.110.17 yes 108
3.2 odd 2 513.2.bo.a.224.2 108
9.4 even 3 513.2.cd.a.395.17 108
9.5 odd 6 171.2.bd.a.167.2 yes 108
19.14 odd 18 171.2.bd.a.128.2 yes 108
57.14 even 18 513.2.cd.a.413.17 108
171.14 even 18 inner 171.2.x.a.14.17 ✓ 108
171.166 odd 18 513.2.bo.a.71.2 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.x.a.14.17 ✓ 108 171.14 even 18 inner
171.2.x.a.110.17 yes 108 1.1 even 1 trivial
171.2.bd.a.128.2 yes 108 19.14 odd 18
171.2.bd.a.167.2 yes 108 9.5 odd 6
513.2.bo.a.71.2 108 171.166 odd 18
513.2.bo.a.224.2 108 3.2 odd 2
513.2.cd.a.395.17 108 9.4 even 3
513.2.cd.a.413.17 108 57.14 even 18