↑

Properties

Label 135.2.p.a.124.8
Level $135$
Weight $2$
Character 135.124
Analytic conductor $1.078$
Analytic rank $0$
Dimension $96$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [135,2,Mod(4,135)] 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("135.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([2, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 135.p (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: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 124.8
Character \(\chi\) \(=\) 135.124
Dual form 135.2.p.a.49.8

$q$-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+(-0.133405 + 0.158986i) q^{2} +(-1.17329 - 1.27412i) q^{3} +(0.339817 + 1.92720i) q^{4} +(-0.452543 + 2.18980i) q^{5} +(0.359090 - 0.0165625i) q^{6} +(2.83580 + 0.500029i) q^{7} +(-0.711201 - 0.410612i) q^{8} +(-0.246774 + 2.98983i) q^{9} +(-0.287775 - 0.364077i) q^{10} +(1.38169 - 0.502894i) q^{11} +(2.05678 - 2.69413i) q^{12} +(1.55650 + 1.85496i) q^{13} +(-0.457807 + 0.384146i) q^{14} +(3.32103 - 1.99267i) q^{15} +(-3.51766 + 1.28032i) q^{16} +(-1.21975 + 0.704220i) q^{17} +(-0.442420 - 0.438091i) q^{18} +(2.34516 - 4.06194i) q^{19} +(-4.37395 - 0.128009i) q^{20} +(-2.69013 - 4.19984i) q^{21} +(-0.104371 + 0.286757i) q^{22} +(-2.36796 + 0.417535i) q^{23} +(0.311276 + 1.38792i) q^{24} +(-4.59041 - 1.98195i) q^{25} -0.502557 q^{26} +(4.09895 - 3.19353i) q^{27} +5.63507i q^{28} +(-6.73596 - 5.65214i) q^{29} +(-0.126235 + 0.793828i) q^{30} +(1.00865 + 5.72033i) q^{31} +(0.827470 - 2.27346i) q^{32} +(-2.26187 - 1.17040i) q^{33} +(0.0507589 - 0.287868i) q^{34} +(-2.37828 + 5.98355i) q^{35} +(-5.84585 + 0.540413i) q^{36} +(7.57034 - 4.37074i) q^{37} +(0.332934 + 0.914728i) q^{38} +(0.537223 - 4.15958i) q^{39} +(1.22101 - 1.37157i) q^{40} +(8.32538 - 6.98582i) q^{41} +(1.02659 + 0.132587i) q^{42} +(-2.63439 - 7.23793i) q^{43} +(1.43870 + 2.49190i) q^{44} +(-6.43545 - 1.89341i) q^{45} +(0.249515 - 0.432172i) q^{46} +(6.68918 + 1.17948i) q^{47} +(5.75853 + 2.97974i) q^{48} +(1.21391 + 0.441827i) q^{49} +(0.927484 - 0.465407i) q^{50} +(2.32838 + 0.727849i) q^{51} +(-3.04596 + 3.63003i) q^{52} +5.43934i q^{53} +(-0.0390949 + 1.07771i) q^{54} +(0.475961 + 3.25320i) q^{55} +(-1.81151 - 1.52004i) q^{56} +(-7.92696 + 1.77782i) q^{57} +(1.79722 - 0.316898i) q^{58} +(6.83272 + 2.48691i) q^{59} +(4.96882 + 5.72314i) q^{60} +(-1.03952 + 5.89541i) q^{61} +(-1.04401 - 0.602759i) q^{62} +(-2.19481 + 8.35519i) q^{63} +(-3.49236 - 6.04894i) q^{64} +(-4.76637 + 2.56897i) q^{65} +(0.487821 - 0.203468i) q^{66} +(-4.95280 - 5.90251i) q^{67} +(-1.77166 - 2.11138i) q^{68} +(3.31030 + 2.52718i) q^{69} +(-0.634024 - 1.17635i) q^{70} +(4.51802 + 7.82544i) q^{71} +(1.40317 - 2.02504i) q^{72} +(-10.8910 - 6.28791i) q^{73} +(-0.315035 + 1.78665i) q^{74} +(2.86064 + 8.17415i) q^{75} +(8.62507 + 3.13927i) q^{76} +(4.16966 - 0.735224i) q^{77} +(0.589646 + 0.640319i) q^{78} +(-1.14861 - 0.963795i) q^{79} +(-1.21176 - 8.28236i) q^{80} +(-8.87821 - 1.47563i) q^{81} +2.25556i q^{82} +(7.46590 - 8.89751i) q^{83} +(7.17977 - 6.61158i) q^{84} +(-0.990112 - 2.98968i) q^{85} +(1.50217 + 0.546744i) q^{86} +(0.701727 + 15.2140i) q^{87} +(-1.18915 - 0.209680i) q^{88} +(-5.96766 + 10.3363i) q^{89} +(1.15954 - 0.770553i) q^{90} +(3.48639 + 6.03861i) q^{91} +(-1.60934 - 4.42164i) q^{92} +(6.10496 - 7.99676i) q^{93} +(-1.07989 + 0.906134i) q^{94} +(7.83353 + 6.97362i) q^{95} +(-3.86752 + 1.61313i) q^{96} +(1.92260 + 5.28231i) q^{97} +(-0.232185 + 0.134052i) q^{98} +(1.16260 + 4.25512i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 12 q^{4} - 9 q^{5} - 6 q^{6} - 18 q^{9} - 3 q^{10} - 6 q^{11} - 18 q^{14} - 21 q^{15} - 24 q^{16} - 6 q^{19} - 57 q^{20} + 24 q^{21} - 30 q^{24} + 3 q^{25} + 48 q^{26} - 30 q^{29} - 51 q^{30} - 30 q^{31}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.133405 + 0.158986i −0.0943314 + 0.112420i −0.811145 − 0.584845i \(-0.801155\pi\)
0.716814 + 0.697265i \(0.245600\pi\)
\(3\) −1.17329 − 1.27412i −0.677400 − 0.735615i
\(4\) 0.339817 + 1.92720i 0.169908 + 0.963598i
\(5\) −0.452543 + 2.18980i −0.202383 + 0.979306i
\(6\) 0.359090 − 0.0165625i 0.146598 − 0.00676162i
\(7\) 2.83580 + 0.500029i 1.07183 + 0.188993i 0.681602 − 0.731724i \(-0.261284\pi\)
0.390232 + 0.920717i \(0.372395\pi\)
\(8\) −0.711201 − 0.410612i −0.251448 − 0.145173i
\(9\) −0.246774 + 2.98983i −0.0822580 + 0.996611i
\(10\) −0.287775 − 0.364077i −0.0910023 − 0.115131i
\(11\) 1.38169 − 0.502894i 0.416595 − 0.151628i −0.125214 − 0.992130i \(-0.539962\pi\)
0.541809 + 0.840502i \(0.317740\pi\)
\(12\) 2.05678 − 2.69413i 0.593741 − 0.777729i
\(13\) 1.55650 + 1.85496i 0.431695 + 0.514474i 0.937411 − 0.348226i \(-0.113216\pi\)
−0.505715 + 0.862700i \(0.668771\pi\)
\(14\) −0.457807 + 0.384146i −0.122354 + 0.102667i
\(15\) 3.32103 − 1.99267i 0.857487 − 0.514506i
\(16\) −3.51766 + 1.28032i −0.879415 + 0.320081i
\(17\) −1.21975 + 0.704220i −0.295832 + 0.170799i −0.640569 − 0.767901i \(-0.721301\pi\)
0.344737 + 0.938699i \(0.387968\pi\)
\(18\) −0.442420 − 0.438091i −0.104279 − 0.103259i
\(19\) 2.34516 − 4.06194i 0.538017 − 0.931872i −0.460994 − 0.887403i \(-0.652507\pi\)
0.999011 − 0.0444689i \(-0.0141595\pi\)
\(20\) −4.37395 − 0.128009i −0.978045 − 0.0286237i
\(21\) −2.69013 − 4.19984i −0.587034 − 0.916480i
\(22\) −0.104371 + 0.286757i −0.0222520 + 0.0611368i
\(23\) −2.36796 + 0.417535i −0.493753 + 0.0870620i −0.414980 − 0.909831i \(-0.636211\pi\)
−0.0787733 + 0.996893i \(0.525100\pi\)
\(24\) 0.311276 + 1.38792i 0.0635390 + 0.283309i
\(25\) −4.59041 − 1.98195i −0.918082 − 0.396390i
\(26\) −0.502557 −0.0985595
\(27\) 4.09895 − 3.19353i 0.788843 − 0.614594i
\(28\) 5.63507i 1.06493i
\(29\) −6.73596 − 5.65214i −1.25084 − 1.04958i −0.996596 − 0.0824353i \(-0.973730\pi\)
−0.254240 − 0.967141i \(-0.581825\pi\)
\(30\) −0.126235 + 0.793828i −0.0230472 + 0.144933i
\(31\) 1.00865 + 5.72033i 0.181159 + 1.02740i 0.930792 + 0.365549i \(0.119119\pi\)
−0.749634 + 0.661853i \(0.769770\pi\)
\(32\) 0.827470 − 2.27346i 0.146277 − 0.401894i
\(33\) −2.26187 − 1.17040i −0.393741 − 0.203740i
\(34\) 0.0507589 − 0.287868i 0.00870509 − 0.0493690i
\(35\) −2.37828 + 5.98355i −0.402003 + 1.01140i
\(36\) −5.84585 + 0.540413i −0.974309 + 0.0900689i
\(37\) 7.57034 − 4.37074i 1.24456 − 0.718545i 0.274538 − 0.961576i \(-0.411475\pi\)
0.970019 + 0.243031i \(0.0781417\pi\)
\(38\) 0.332934 + 0.914728i 0.0540090 + 0.148388i
\(39\) 0.537223 − 4.15958i 0.0860245 − 0.666066i
\(40\) 1.22101 − 1.37157i 0.193058 − 0.216864i
\(41\) 8.32538 − 6.98582i 1.30021 − 1.09100i 0.310096 − 0.950705i \(-0.399639\pi\)
0.990110 − 0.140297i \(-0.0448056\pi\)
\(42\) 1.02659 + 0.132587i 0.158406 + 0.0204586i
\(43\) −2.63439 − 7.23793i −0.401741 − 1.10377i −0.961425 − 0.275068i \(-0.911300\pi\)
0.559684 − 0.828706i \(-0.310923\pi\)
\(44\) 1.43870 + 2.49190i 0.216892 + 0.375667i
\(45\) −6.43545 − 1.89341i −0.959340 − 0.282253i
\(46\) 0.249515 − 0.432172i 0.0367889 − 0.0637203i
\(47\) 6.68918 + 1.17948i 0.975717 + 0.172045i 0.638702 − 0.769454i \(-0.279472\pi\)
0.337015 + 0.941499i \(0.390583\pi\)
\(48\) 5.75853 + 2.97974i 0.831172 + 0.430088i
\(49\) 1.21391 + 0.441827i 0.173416 + 0.0631181i
\(50\) 0.927484 − 0.465407i 0.131166 − 0.0658185i
\(51\) 2.32838 + 0.727849i 0.326038 + 0.101919i
\(52\) −3.04596 + 3.63003i −0.422398 + 0.503394i
\(53\) 5.43934i 0.747151i 0.927600 + 0.373575i \(0.121868\pi\)
−0.927600 + 0.373575i \(0.878132\pi\)
\(54\) −0.0390949 + 1.07771i −0.00532014 + 0.146657i
\(55\) 0.475961 + 3.25320i 0.0641786 + 0.438661i
\(56\) −1.81151 − 1.52004i −0.242073 − 0.203123i
\(57\) −7.92696 + 1.77782i −1.04995 + 0.235477i
\(58\) 1.79722 − 0.316898i 0.235986 − 0.0416108i
\(59\) 6.83272 + 2.48691i 0.889545 + 0.323768i 0.746055 − 0.665884i \(-0.231945\pi\)
0.143489 + 0.989652i \(0.454168\pi\)
\(60\) 4.96882 + 5.72314i 0.641472 + 0.738854i
\(61\) −1.03952 + 5.89541i −0.133097 + 0.754830i 0.843069 + 0.537805i \(0.180746\pi\)
−0.976166 + 0.217025i \(0.930365\pi\)
\(62\) −1.04401 − 0.602759i −0.132589 − 0.0765504i
\(63\) −2.19481 + 8.35519i −0.276520 + 1.05265i
\(64\) −3.49236 − 6.04894i −0.436545 − 0.756118i
\(65\) −4.76637 + 2.56897i −0.591196 + 0.318641i
\(66\) 0.487821 − 0.203468i 0.0600466 − 0.0250452i
\(67\) −4.95280 − 5.90251i −0.605081 − 0.721107i 0.373348 − 0.927691i \(-0.378210\pi\)
−0.978429 + 0.206584i \(0.933765\pi\)
\(68\) −1.77166 − 2.11138i −0.214845 − 0.256043i
\(69\) 3.31030 + 2.52718i 0.398513 + 0.304236i
\(70\) −0.634024 − 1.17635i −0.0757803 − 0.140600i
\(71\) 4.51802 + 7.82544i 0.536190 + 0.928708i 0.999105 + 0.0423056i \(0.0134703\pi\)
−0.462915 + 0.886403i \(0.653196\pi\)
\(72\) 1.40317 − 2.02504i 0.165365 − 0.238654i
\(73\) −10.8910 − 6.28791i −1.27469 − 0.735944i −0.298825 − 0.954308i \(-0.596595\pi\)
−0.975867 + 0.218364i \(0.929928\pi\)
\(74\) −0.315035 + 1.78665i −0.0366221 + 0.207694i
\(75\) 2.86064 + 8.17415i 0.330318 + 0.943870i
\(76\) 8.62507 + 3.13927i 0.989364 + 0.360099i
\(77\) 4.16966 − 0.735224i 0.475177 − 0.0837865i
\(78\) 0.589646 + 0.640319i 0.0667642 + 0.0725018i
\(79\) −1.14861 − 0.963795i −0.129228 − 0.108435i 0.575883 − 0.817532i \(-0.304658\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(80\) −1.21176 − 8.28236i −0.135478 − 0.925996i
\(81\) −8.87821 − 1.47563i −0.986467 − 0.163958i
\(82\) 2.25556i 0.249085i
\(83\) 7.46590 − 8.89751i 0.819489 − 0.976629i −0.180487 − 0.983577i \(-0.557767\pi\)
0.999976 + 0.00694857i \(0.00221182\pi\)
\(84\) 7.17977 − 6.61158i 0.783377 − 0.721383i
\(85\) −0.990112 − 2.98968i −0.107393 − 0.324277i
\(86\) 1.50217 + 0.546744i 0.161983 + 0.0589569i
\(87\) 0.701727 + 15.2140i 0.0752330 + 1.63112i
\(88\) −1.18915 − 0.209680i −0.126764 − 0.0223519i
\(89\) −5.96766 + 10.3363i −0.632571 + 1.09564i 0.354454 + 0.935074i \(0.384667\pi\)
−0.987024 + 0.160571i \(0.948666\pi\)
\(90\) 1.15954 − 0.770553i 0.122227 − 0.0812235i
\(91\) 3.48639 + 6.03861i 0.365473 + 0.633018i
\(92\) −1.60934 − 4.42164i −0.167786 − 0.460987i
\(93\) 6.10496 − 7.99676i 0.633055 − 0.829225i
\(94\) −1.07989 + 0.906134i −0.111382 + 0.0934606i
\(95\) 7.83353 + 6.97362i 0.803703 + 0.715478i
\(96\) −3.86752 + 1.61313i −0.394727 + 0.164639i
\(97\) 1.92260 + 5.28231i 0.195211 + 0.536337i 0.998221 − 0.0596285i \(-0.0189916\pi\)
−0.803010 + 0.595966i \(0.796769\pi\)
\(98\) −0.232185 + 0.134052i −0.0234543 + 0.0135413i
\(99\) 1.16260 + 4.25512i 0.116846 + 0.427656i
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 135.2.p.a.124.8 yes 96
3.2 odd 2 405.2.p.a.289.9 96
5.2 odd 4 675.2.l.h.151.8 96
5.3 odd 4 675.2.l.h.151.9 96
5.4 even 2 inner 135.2.p.a.124.9 yes 96
15.14 odd 2 405.2.p.a.289.8 96
27.5 odd 18 405.2.p.a.199.8 96
27.22 even 9 inner 135.2.p.a.49.9 yes 96
135.22 odd 36 675.2.l.h.76.8 96
135.49 even 18 inner 135.2.p.a.49.8 ✓ 96
135.59 odd 18 405.2.p.a.199.9 96
135.103 odd 36 675.2.l.h.76.9 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.2.p.a.49.8 ✓ 96 135.49 even 18 inner
135.2.p.a.49.9 yes 96 27.22 even 9 inner
135.2.p.a.124.8 yes 96 1.1 even 1 trivial
135.2.p.a.124.9 yes 96 5.4 even 2 inner
405.2.p.a.199.8 96 27.5 odd 18
405.2.p.a.199.9 96 135.59 odd 18
405.2.p.a.289.8 96 15.14 odd 2
405.2.p.a.289.9 96 3.2 odd 2
675.2.l.h.76.8 96 135.22 odd 36
675.2.l.h.76.9 96 135.103 odd 36
675.2.l.h.151.8 96 5.2 odd 4
675.2.l.h.151.9 96 5.3 odd 4