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Properties

Label 162.2.g.b.13.2
Level $162$
Weight $2$
Character 162.13
Analytic conductor $1.294$
Analytic rank $0$
Dimension $90$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,2,Mod(7,162)] 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("162.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [90] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(5\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 13.2
Character \(\chi\) \(=\) 162.13
Dual form 162.2.g.b.25.2

$q$-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+(0.893633 + 0.448799i) q^{2} +(-0.977769 - 1.42967i) q^{3} +(0.597159 + 0.802123i) q^{4} +(-1.15296 - 3.85116i) q^{5} +(-0.232129 - 1.71643i) q^{6} +(-0.663579 - 1.53835i) q^{7} +(0.173648 + 0.984808i) q^{8} +(-1.08794 + 2.79578i) q^{9} +(0.698073 - 3.95897i) q^{10} +(0.0131573 - 0.00311835i) q^{11} +(0.562892 - 1.63803i) q^{12} +(3.84080 + 2.52614i) q^{13} +(0.0974140 - 1.67253i) q^{14} +(-4.37857 + 5.41390i) q^{15} +(-0.286803 + 0.957990i) q^{16} +(4.83704 - 1.76054i) q^{17} +(-2.22696 + 2.01014i) q^{18} +(2.66092 + 0.968497i) q^{19} +(2.40060 - 3.22457i) q^{20} +(-1.55051 + 2.45285i) q^{21} +(0.0131573 + 0.00311835i) q^{22} +(-0.253001 + 0.586523i) q^{23} +(1.23817 - 1.21117i) q^{24} +(-9.32465 + 6.13292i) q^{25} +(2.29854 + 3.98119i) q^{26} +(5.06081 - 1.17823i) q^{27} +(0.837684 - 1.45091i) q^{28} +(-0.262835 - 4.51270i) q^{29} +(-6.34259 + 2.87294i) q^{30} +(-9.48738 + 1.10892i) q^{31} +(-0.686242 + 0.727374i) q^{32} +(-0.0173231 - 0.0157617i) q^{33} +(5.11267 + 0.597585i) q^{34} +(-5.15935 + 4.32920i) q^{35} +(-2.89223 + 0.796865i) q^{36} +(6.03321 + 5.06246i) q^{37} +(1.94323 + 2.05970i) q^{38} +(-0.143865 - 7.96107i) q^{39} +(3.59244 - 1.80419i) q^{40} +(6.30010 - 3.16403i) q^{41} +(-2.48643 + 1.49608i) q^{42} +(4.34864 + 4.60929i) q^{43} +(0.0103583 + 0.00869166i) q^{44} +(12.0213 + 0.966392i) q^{45} +(-0.489321 + 0.410589i) q^{46} +(-11.3730 - 1.32931i) q^{47} +(1.65004 - 0.526657i) q^{48} +(2.87751 - 3.04998i) q^{49} +(-11.0853 + 1.29568i) q^{50} +(-7.24650 - 5.19399i) q^{51} +(0.267296 + 4.58930i) q^{52} +(-2.96943 + 5.14321i) q^{53} +(5.05129 + 1.21838i) q^{54} +(-0.0271791 - 0.0470757i) q^{55} +(1.39975 - 0.920630i) q^{56} +(-1.21713 - 4.75122i) q^{57} +(1.79042 - 4.15065i) q^{58} +(-2.42002 - 0.573555i) q^{59} +(-6.95732 - 0.279198i) q^{60} +(1.39556 - 1.87456i) q^{61} +(-8.97591 - 3.26697i) q^{62} +(5.02282 - 0.181594i) q^{63} +(-0.939693 + 0.342020i) q^{64} +(5.30025 - 17.7041i) q^{65} +(-0.00840662 - 0.0218597i) q^{66} +(0.232730 - 3.99581i) q^{67} +(4.30065 + 2.82858i) q^{68} +(1.08591 - 0.211774i) q^{69} +(-6.55350 + 1.55321i) q^{70} +(-1.01165 + 5.73737i) q^{71} +(-2.94223 - 0.585927i) q^{72} +(1.45371 + 8.24441i) q^{73} +(3.11944 + 7.23168i) q^{74} +(17.8854 + 7.33464i) q^{75} +(0.812139 + 2.71273i) q^{76} +(-0.0135280 - 0.0181713i) q^{77} +(3.44436 - 7.17884i) q^{78} +(-4.39144 - 2.20546i) q^{79} +4.02004 q^{80} +(-6.63278 - 6.08327i) q^{81} +7.04998 q^{82} +(13.5542 + 6.80719i) q^{83} +(-2.89339 + 0.221040i) q^{84} +(-12.3570 - 16.5984i) q^{85} +(1.81744 + 6.07068i) q^{86} +(-6.19470 + 4.78814i) q^{87} +(0.00535572 + 0.0124160i) q^{88} +(1.34899 + 7.65049i) q^{89} +(10.3089 + 6.25877i) q^{90} +(1.33740 - 7.58479i) q^{91} +(-0.621545 + 0.147309i) q^{92} +(10.8619 + 12.4796i) q^{93} +(-9.56667 - 6.29210i) q^{94} +(0.661893 - 11.3643i) q^{95} +(1.71089 + 0.269899i) q^{96} +(0.923670 - 3.08527i) q^{97} +(3.94027 - 1.43414i) q^{98} +(-0.00559615 + 0.0401776i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 90 q - 9 q^{6} - 18 q^{13} - 9 q^{18} - 9 q^{20} - 54 q^{21} + 27 q^{23} - 18 q^{25} - 27 q^{26} - 27 q^{27} - 18 q^{28} - 27 q^{29} + 9 q^{30} + 54 q^{31} - 63 q^{33} - 27 q^{35} - 9 q^{36} - 18 q^{38}+ \cdots - 81 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{4}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.893633 + 0.448799i 0.631894 + 0.317349i
\(3\) −0.977769 − 1.42967i −0.564515 − 0.825423i
\(4\) 0.597159 + 0.802123i 0.298579 + 0.401062i
\(5\) −1.15296 − 3.85116i −0.515620 − 1.72229i −0.677540 − 0.735486i \(-0.736954\pi\)
0.161920 − 0.986804i \(-0.448231\pi\)
\(6\) −0.232129 − 1.71643i −0.0947664 − 0.700728i
\(7\) −0.663579 − 1.53835i −0.250809 − 0.581441i 0.745573 − 0.666424i \(-0.232176\pi\)
−0.996382 + 0.0849824i \(0.972917\pi\)
\(8\) 0.173648 + 0.984808i 0.0613939 + 0.348182i
\(9\) −1.08794 + 2.79578i −0.362646 + 0.931927i
\(10\) 0.698073 − 3.95897i 0.220750 − 1.25194i
\(11\) 0.0131573 − 0.00311835i 0.00396709 − 0.000940217i −0.228632 − 0.973513i \(-0.573425\pi\)
0.232599 + 0.972573i \(0.425277\pi\)
\(12\) 0.562892 − 1.63803i 0.162493 − 0.472859i
\(13\) 3.84080 + 2.52614i 1.06525 + 0.700624i 0.955993 − 0.293390i \(-0.0947834\pi\)
0.109254 + 0.994014i \(0.465154\pi\)
\(14\) 0.0974140 − 1.67253i 0.0260350 − 0.447003i
\(15\) −4.37857 + 5.41390i −1.13054 + 1.39786i
\(16\) −0.286803 + 0.957990i −0.0717008 + 0.239497i
\(17\) 4.83704 − 1.76054i 1.17315 − 0.426993i 0.319375 − 0.947628i \(-0.396527\pi\)
0.853779 + 0.520635i \(0.174305\pi\)
\(18\) −2.22696 + 2.01014i −0.524900 + 0.473794i
\(19\) 2.66092 + 0.968497i 0.610457 + 0.222188i 0.628703 − 0.777645i \(-0.283586\pi\)
−0.0182459 + 0.999834i \(0.505808\pi\)
\(20\) 2.40060 − 3.22457i 0.536791 − 0.721035i
\(21\) −1.55051 + 2.45285i −0.338349 + 0.535256i
\(22\) 0.0131573 + 0.00311835i 0.00280516 + 0.000664834i
\(23\) −0.253001 + 0.586523i −0.0527544 + 0.122298i −0.942555 − 0.334051i \(-0.891584\pi\)
0.889801 + 0.456350i \(0.150843\pi\)
\(24\) 1.23817 − 1.21117i 0.252740 − 0.247230i
\(25\) −9.32465 + 6.13292i −1.86493 + 1.22658i
\(26\) 2.29854 + 3.98119i 0.450781 + 0.780775i
\(27\) 5.06081 − 1.17823i 0.973953 − 0.226751i
\(28\) 0.837684 − 1.45091i 0.158307 − 0.274196i
\(29\) −0.262835 − 4.51270i −0.0488072 − 0.837987i −0.930529 − 0.366218i \(-0.880653\pi\)
0.881722 − 0.471769i \(-0.156384\pi\)
\(30\) −6.34259 + 2.87294i −1.15799 + 0.524524i
\(31\) −9.48738 + 1.10892i −1.70398 + 0.199167i −0.911663 − 0.410938i \(-0.865201\pi\)
−0.792321 + 0.610105i \(0.791127\pi\)
\(32\) −0.686242 + 0.727374i −0.121312 + 0.128583i
\(33\) −0.0173231 − 0.0157617i −0.00301556 − 0.00274376i
\(34\) 5.11267 + 0.597585i 0.876815 + 0.102485i
\(35\) −5.15935 + 4.32920i −0.872088 + 0.731769i
\(36\) −2.89223 + 0.796865i −0.482039 + 0.132811i
\(37\) 6.03321 + 5.06246i 0.991853 + 0.832264i 0.985835 − 0.167718i \(-0.0536398\pi\)
0.00601844 + 0.999982i \(0.498084\pi\)
\(38\) 1.94323 + 2.05970i 0.315233 + 0.334127i
\(39\) −0.143865 − 7.96107i −0.0230368 − 1.27479i
\(40\) 3.59244 − 1.80419i 0.568015 − 0.285268i
\(41\) 6.30010 − 3.16403i 0.983910 − 0.494138i 0.117285 − 0.993098i \(-0.462581\pi\)
0.866625 + 0.498960i \(0.166285\pi\)
\(42\) −2.48643 + 1.49608i −0.383664 + 0.230850i
\(43\) 4.34864 + 4.60929i 0.663162 + 0.702911i 0.968410 − 0.249362i \(-0.0802209\pi\)
−0.305248 + 0.952273i \(0.598739\pi\)
\(44\) 0.0103583 + 0.00869166i 0.00156158 + 0.00131032i
\(45\) 12.0213 + 0.966392i 1.79204 + 0.144061i
\(46\) −0.489321 + 0.410589i −0.0721465 + 0.0605381i
\(47\) −11.3730 − 1.32931i −1.65892 − 0.193900i −0.765598 − 0.643319i \(-0.777557\pi\)
−0.893321 + 0.449419i \(0.851631\pi\)
\(48\) 1.65004 − 0.526657i 0.238163 − 0.0760164i
\(49\) 2.87751 − 3.04998i 0.411073 − 0.435712i
\(50\) −11.0853 + 1.29568i −1.56769 + 0.183237i
\(51\) −7.24650 − 5.19399i −1.01471 − 0.727305i
\(52\) 0.267296 + 4.58930i 0.0370673 + 0.636421i
\(53\) −2.96943 + 5.14321i −0.407883 + 0.706474i −0.994652 − 0.103280i \(-0.967066\pi\)
0.586769 + 0.809754i \(0.300399\pi\)
\(54\) 5.05129 + 1.21838i 0.687394 + 0.165801i
\(55\) −0.0271791 − 0.0470757i −0.00366484 − 0.00634768i
\(56\) 1.39975 − 0.920630i 0.187049 − 0.123024i
\(57\) −1.21713 − 4.75122i −0.161213 − 0.629314i
\(58\) 1.79042 − 4.15065i 0.235093 − 0.545007i
\(59\) −2.42002 − 0.573555i −0.315060 − 0.0746705i 0.0700438 − 0.997544i \(-0.477686\pi\)
−0.385103 + 0.922873i \(0.625834\pi\)
\(60\) −6.95732 − 0.279198i −0.898186 − 0.0360443i
\(61\) 1.39556 − 1.87456i 0.178683 − 0.240013i −0.703754 − 0.710444i \(-0.748494\pi\)
0.882437 + 0.470431i \(0.155902\pi\)
\(62\) −8.97591 − 3.26697i −1.13994 − 0.414905i
\(63\) 5.02282 − 0.181594i 0.632816 − 0.0228787i
\(64\) −0.939693 + 0.342020i −0.117462 + 0.0427525i
\(65\) 5.30025 − 17.7041i 0.657415 − 2.19592i
\(66\) −0.00840662 − 0.0218597i −0.00103478 − 0.00269075i
\(67\) 0.232730 − 3.99581i 0.0284324 − 0.488166i −0.953793 − 0.300463i \(-0.902859\pi\)
0.982226 − 0.187703i \(-0.0601042\pi\)
\(68\) 4.30065 + 2.82858i 0.521530 + 0.343016i
\(69\) 1.08591 − 0.211774i 0.130729 − 0.0254946i
\(70\) −6.55350 + 1.55321i −0.783293 + 0.185644i
\(71\) −1.01165 + 5.73737i −0.120061 + 0.680901i 0.864058 + 0.503392i \(0.167915\pi\)
−0.984119 + 0.177509i \(0.943196\pi\)
\(72\) −2.94223 − 0.585927i −0.346745 − 0.0690521i
\(73\) 1.45371 + 8.24441i 0.170144 + 0.964935i 0.943601 + 0.331086i \(0.107415\pi\)
−0.773457 + 0.633849i \(0.781474\pi\)
\(74\) 3.11944 + 7.23168i 0.362628 + 0.840666i
\(75\) 17.8854 + 7.33464i 2.06523 + 0.846931i
\(76\) 0.812139 + 2.71273i 0.0931587 + 0.311172i
\(77\) −0.0135280 − 0.0181713i −0.00154166 − 0.00207081i
\(78\) 3.44436 − 7.17884i 0.389997 − 0.812844i
\(79\) −4.39144 − 2.20546i −0.494076 − 0.248134i 0.184272 − 0.982875i \(-0.441007\pi\)
−0.678348 + 0.734741i \(0.737304\pi\)
\(80\) 4.02004 0.449454
\(81\) −6.63278 − 6.08327i −0.736976 − 0.675919i
\(82\) 7.04998 0.778541
\(83\) 13.5542 + 6.80719i 1.48777 + 0.747186i 0.992579 − 0.121598i \(-0.0388018\pi\)
0.495191 + 0.868784i \(0.335098\pi\)
\(84\) −2.89339 + 0.221040i −0.315695 + 0.0241174i
\(85\) −12.3570 − 16.5984i −1.34031 − 1.80035i
\(86\) 1.81744 + 6.07068i 0.195980 + 0.654619i
\(87\) −6.19470 + 4.78814i −0.664141 + 0.513343i
\(88\) 0.00535572 + 0.0124160i 0.000570922 + 0.00132355i
\(89\) 1.34899 + 7.65049i 0.142992 + 0.810950i 0.968957 + 0.247229i \(0.0795199\pi\)
−0.825965 + 0.563722i \(0.809369\pi\)
\(90\) 10.3089 + 6.25877i 1.08666 + 0.659732i
\(91\) 1.33740 − 7.58479i 0.140198 − 0.795102i
\(92\) −0.621545 + 0.147309i −0.0648006 + 0.0153580i
\(93\) 10.8619 + 12.4796i 1.12632 + 1.29407i
\(94\) −9.56667 − 6.29210i −0.986726 − 0.648980i
\(95\) 0.661893 − 11.3643i 0.0679088 − 1.16595i
\(96\) 1.71089 + 0.269899i 0.174617 + 0.0275464i
\(97\) 0.923670 − 3.08527i 0.0937845 − 0.313262i −0.898206 − 0.439575i \(-0.855129\pi\)
0.991990 + 0.126313i \(0.0403143\pi\)
\(98\) 3.94027 − 1.43414i 0.398027 − 0.144870i
\(99\) −0.00559615 + 0.0401776i −0.000562434 + 0.00403800i
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 162.2.g.b.13.2 ✓ 90
3.2 odd 2 486.2.g.b.253.5 90
81.25 even 27 inner 162.2.g.b.25.2 yes 90
81.56 odd 54 486.2.g.b.73.5 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.13.2 ✓ 90 1.1 even 1 trivial
162.2.g.b.25.2 yes 90 81.25 even 27 inner
486.2.g.b.73.5 90 81.56 odd 54
486.2.g.b.253.5 90 3.2 odd 2