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Properties

Label 162.2.g.a.31.1
Level $162$
Weight $2$
Character 162.31
Analytic conductor $1.294$
Analytic rank $0$
Dimension $72$
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 = [72] 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: \(72\)
Relative dimension: \(4\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 31.1
Character \(\chi\) \(=\) 162.31
Dual form 162.2.g.a.115.1

$q$-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+(-0.396080 - 0.918216i) q^{2} +(-1.62371 + 0.602974i) q^{3} +(-0.686242 + 0.727374i) q^{4} +(-0.0263678 + 0.452718i) q^{5} +(1.19678 + 1.25209i) q^{6} +(4.03057 - 0.955263i) q^{7} +(0.939693 + 0.342020i) q^{8} +(2.27284 - 1.95811i) q^{9} +(0.426137 - 0.155101i) q^{10} +(0.410794 - 0.270184i) q^{11} +(0.675668 - 1.59483i) q^{12} +(3.42955 - 0.400857i) q^{13} +(-2.47357 - 3.32258i) q^{14} +(-0.230164 - 0.750981i) q^{15} +(-0.0581448 - 0.998308i) q^{16} +(2.51871 + 2.11345i) q^{17} +(-2.69819 - 1.31140i) q^{18} +(-1.21368 + 1.01840i) q^{19} +(-0.311201 - 0.329854i) q^{20} +(-5.96847 + 3.98140i) q^{21} +(-0.410794 - 0.270184i) q^{22} +(-6.89877 - 1.63504i) q^{23} +(-1.73201 + 0.0112699i) q^{24} +(4.76193 + 0.556590i) q^{25} +(-1.72645 - 2.99029i) q^{26} +(-2.50975 + 4.54985i) q^{27} +(-2.07111 + 3.58727i) q^{28} +(2.14438 - 2.88040i) q^{29} +(-0.598400 + 0.508788i) q^{30} +(1.00695 + 3.36344i) q^{31} +(-0.893633 + 0.448799i) q^{32} +(-0.504095 + 0.686397i) q^{33} +(0.942994 - 3.14982i) q^{34} +(0.326188 + 1.84990i) q^{35} +(-0.135446 + 2.99694i) q^{36} +(-0.757807 + 4.29774i) q^{37} +(1.41582 + 0.711053i) q^{38} +(-5.32687 + 2.71880i) q^{39} +(-0.179617 + 0.416398i) q^{40} +(3.78165 - 8.76684i) q^{41} +(6.01977 + 3.90339i) q^{42} +(-11.3624 - 5.70639i) q^{43} +(-0.0853797 + 0.484212i) q^{44} +(0.826540 + 1.08059i) q^{45} +(1.23114 + 6.98217i) q^{46} +(-0.460706 + 1.53886i) q^{47} +(0.696364 + 1.58590i) q^{48} +(9.07756 - 4.55892i) q^{49} +(-1.37504 - 4.59294i) q^{50} +(-5.36401 - 1.91291i) q^{51} +(-2.06193 + 2.76965i) q^{52} +(-6.96808 + 12.0691i) q^{53} +(5.17181 + 0.502384i) q^{54} +(0.111485 + 0.193098i) q^{55} +(4.11422 + 0.480883i) q^{56} +(1.35659 - 2.38540i) q^{57} +(-3.49418 - 0.828136i) q^{58} +(1.26512 + 0.832083i) q^{59} +(0.704192 + 0.347939i) q^{60} +(-3.67663 - 3.89700i) q^{61} +(2.68954 - 2.25679i) q^{62} +(7.29036 - 10.0634i) q^{63} +(0.766044 + 0.642788i) q^{64} +(0.0910454 + 1.56319i) q^{65} +(0.829923 + 0.191001i) q^{66} +(-0.930994 - 1.25054i) q^{67} +(-3.26572 + 0.381707i) q^{68} +(12.1875 - 1.50496i) q^{69} +(1.56941 - 1.03222i) q^{70} +(4.24906 - 1.54653i) q^{71} +(2.80549 - 1.06266i) q^{72} +(-9.97468 - 3.63049i) q^{73} +(4.24640 - 1.00642i) q^{74} +(-8.06759 + 1.96758i) q^{75} +(0.0921215 - 1.58167i) q^{76} +(1.39764 - 1.48141i) q^{77} +(4.60631 + 3.81436i) q^{78} +(-5.14102 - 11.9182i) q^{79} +0.453486 q^{80} +(1.33165 - 8.90094i) q^{81} -9.54769 q^{82} +(2.70032 + 6.26005i) q^{83} +(1.19985 - 7.07351i) q^{84} +(-1.02321 + 1.08454i) q^{85} +(-0.739300 + 12.6933i) q^{86} +(-1.74504 + 5.96994i) q^{87} +(0.478429 - 0.113390i) q^{88} +(7.03445 + 2.56033i) q^{89} +(0.664839 - 1.18694i) q^{90} +(13.4401 - 4.89180i) q^{91} +(5.92351 - 3.89595i) q^{92} +(-3.66306 - 4.85408i) q^{93} +(1.59549 - 0.186486i) q^{94} +(-0.429045 - 0.576308i) q^{95} +(1.18038 - 1.26756i) q^{96} +(0.517706 + 8.88867i) q^{97} +(-7.78151 - 6.52946i) q^{98} +(0.404624 - 1.41846i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 9 q^{6} + 18 q^{13} - 9 q^{20} - 81 q^{23} + 18 q^{25} - 27 q^{26} - 27 q^{27} + 18 q^{28} - 27 q^{29} - 63 q^{30} - 54 q^{31} + 9 q^{33} - 27 q^{35} - 9 q^{36} + 9 q^{38} - 9 q^{41} + 9 q^{42} + 36 q^{43}+ \cdots + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{10}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.396080 − 0.918216i −0.280071 − 0.649277i
\(3\) −1.62371 + 0.602974i −0.937447 + 0.348127i
\(4\) −0.686242 + 0.727374i −0.343121 + 0.363687i
\(5\) −0.0263678 + 0.452718i −0.0117921 + 0.202462i 0.987266 + 0.159076i \(0.0508515\pi\)
−0.999058 + 0.0433858i \(0.986186\pi\)
\(6\) 1.19678 + 1.25209i 0.488582 + 0.511163i
\(7\) 4.03057 − 0.955263i 1.52341 − 0.361056i 0.618203 − 0.786019i \(-0.287861\pi\)
0.905211 + 0.424963i \(0.139713\pi\)
\(8\) 0.939693 + 0.342020i 0.332232 + 0.120922i
\(9\) 2.27284 − 1.95811i 0.757615 − 0.652702i
\(10\) 0.426137 − 0.155101i 0.134756 − 0.0490473i
\(11\) 0.410794 − 0.270184i 0.123859 − 0.0814634i −0.486068 − 0.873921i \(-0.661569\pi\)
0.609927 + 0.792458i \(0.291199\pi\)
\(12\) 0.675668 − 1.59483i 0.195048 − 0.460387i
\(13\) 3.42955 − 0.400857i 0.951185 − 0.111178i 0.373674 − 0.927560i \(-0.378098\pi\)
0.577512 + 0.816383i \(0.304024\pi\)
\(14\) −2.47357 − 3.32258i −0.661088 − 0.887996i
\(15\) −0.230164 − 0.750981i −0.0594280 − 0.193902i
\(16\) −0.0581448 − 0.998308i −0.0145362 − 0.249577i
\(17\) 2.51871 + 2.11345i 0.610878 + 0.512587i 0.894921 − 0.446224i \(-0.147232\pi\)
−0.284043 + 0.958811i \(0.591676\pi\)
\(18\) −2.69819 − 1.31140i −0.635970 − 0.309099i
\(19\) −1.21368 + 1.01840i −0.278437 + 0.233636i −0.771302 − 0.636469i \(-0.780394\pi\)
0.492865 + 0.870106i \(0.335950\pi\)
\(20\) −0.311201 − 0.329854i −0.0695866 − 0.0737575i
\(21\) −5.96847 + 3.98140i −1.30243 + 0.868812i
\(22\) −0.410794 − 0.270184i −0.0875816 − 0.0576033i
\(23\) −6.89877 − 1.63504i −1.43849 − 0.340929i −0.563923 − 0.825828i \(-0.690708\pi\)
−0.874570 + 0.484898i \(0.838857\pi\)
\(24\) −1.73201 + 0.0112699i −0.353546 + 0.00230046i
\(25\) 4.76193 + 0.556590i 0.952387 + 0.111318i
\(26\) −1.72645 − 2.99029i −0.338584 − 0.586445i
\(27\) −2.50975 + 4.54985i −0.483001 + 0.875620i
\(28\) −2.07111 + 3.58727i −0.391404 + 0.677931i
\(29\) 2.14438 − 2.88040i 0.398202 − 0.534878i −0.557125 − 0.830428i \(-0.688096\pi\)
0.955327 + 0.295551i \(0.0955031\pi\)
\(30\) −0.598400 + 0.508788i −0.109252 + 0.0928916i
\(31\) 1.00695 + 3.36344i 0.180853 + 0.604092i 0.999554 + 0.0298715i \(0.00950979\pi\)
−0.818700 + 0.574221i \(0.805305\pi\)
\(32\) −0.893633 + 0.448799i −0.157973 + 0.0793372i
\(33\) −0.504095 + 0.686397i −0.0877518 + 0.119486i
\(34\) 0.942994 − 3.14982i 0.161722 − 0.540190i
\(35\) 0.326188 + 1.84990i 0.0551358 + 0.312691i
\(36\) −0.135446 + 2.99694i −0.0225744 + 0.499490i
\(37\) −0.757807 + 4.29774i −0.124583 + 0.706544i 0.856972 + 0.515363i \(0.172343\pi\)
−0.981555 + 0.191181i \(0.938768\pi\)
\(38\) 1.41582 + 0.711053i 0.229677 + 0.115348i
\(39\) −5.32687 + 2.71880i −0.852982 + 0.435357i
\(40\) −0.179617 + 0.416398i −0.0283999 + 0.0658383i
\(41\) 3.78165 − 8.76684i 0.590594 − 1.36915i −0.316649 − 0.948543i \(-0.602558\pi\)
0.907243 − 0.420608i \(-0.138183\pi\)
\(42\) 6.01977 + 3.90339i 0.928871 + 0.602306i
\(43\) −11.3624 − 5.70639i −1.73274 − 0.870217i −0.976378 − 0.216068i \(-0.930677\pi\)
−0.756366 − 0.654149i \(-0.773027\pi\)
\(44\) −0.0853797 + 0.484212i −0.0128715 + 0.0729977i
\(45\) 0.826540 + 1.08059i 0.123213 + 0.161085i
\(46\) 1.23114 + 6.98217i 0.181522 + 1.02946i
\(47\) −0.460706 + 1.53886i −0.0672009 + 0.224467i −0.984993 − 0.172595i \(-0.944785\pi\)
0.917792 + 0.397061i \(0.129970\pi\)
\(48\) 0.696364 + 1.58590i 0.100511 + 0.228905i
\(49\) 9.07756 − 4.55892i 1.29679 − 0.651274i
\(50\) −1.37504 − 4.59294i −0.194459 − 0.649539i
\(51\) −5.36401 − 1.91291i −0.751112 − 0.267861i
\(52\) −2.06193 + 2.76965i −0.285938 + 0.384081i
\(53\) −6.96808 + 12.0691i −0.957140 + 1.65782i −0.227747 + 0.973720i \(0.573136\pi\)
−0.729393 + 0.684095i \(0.760197\pi\)
\(54\) 5.17181 + 0.502384i 0.703794 + 0.0683658i
\(55\) 0.111485 + 0.193098i 0.0150327 + 0.0260374i
\(56\) 4.11422 + 0.480883i 0.549786 + 0.0642607i
\(57\) 1.35659 − 2.38540i 0.179685 − 0.315953i
\(58\) −3.49418 − 0.828136i −0.458808 − 0.108740i
\(59\) 1.26512 + 0.832083i 0.164705 + 0.108328i 0.629185 − 0.777255i \(-0.283389\pi\)
−0.464481 + 0.885583i \(0.653759\pi\)
\(60\) 0.704192 + 0.347939i 0.0909108 + 0.0449188i
\(61\) −3.67663 − 3.89700i −0.470744 − 0.498959i 0.447879 − 0.894094i \(-0.352179\pi\)
−0.918623 + 0.395135i \(0.870698\pi\)
\(62\) 2.68954 − 2.25679i 0.341571 − 0.286613i
\(63\) 7.29036 − 10.0634i 0.918499 − 1.26788i
\(64\) 0.766044 + 0.642788i 0.0957556 + 0.0803485i
\(65\) 0.0910454 + 1.56319i 0.0112928 + 0.193890i
\(66\) 0.829923 + 0.191001i 0.102156 + 0.0235106i
\(67\) −0.930994 − 1.25054i −0.113739 − 0.152778i 0.741608 − 0.670833i \(-0.234063\pi\)
−0.855347 + 0.518055i \(0.826656\pi\)
\(68\) −3.26572 + 0.381707i −0.396026 + 0.0462888i
\(69\) 12.1875 − 1.50496i 1.46720 − 0.181175i
\(70\) 1.56941 − 1.03222i 0.187581 − 0.123374i
\(71\) 4.24906 − 1.54653i 0.504271 − 0.183539i −0.0773433 − 0.997005i \(-0.524644\pi\)
0.581614 + 0.813465i \(0.302422\pi\)
\(72\) 2.80549 − 1.06266i 0.330630 − 0.125236i
\(73\) −9.97468 − 3.63049i −1.16745 − 0.424916i −0.315695 − 0.948861i \(-0.602238\pi\)
−0.851753 + 0.523944i \(0.824460\pi\)
\(74\) 4.24640 − 1.00642i 0.493634 − 0.116993i
\(75\) −8.06759 + 1.96758i −0.931565 + 0.227197i
\(76\) 0.0921215 − 1.58167i 0.0105671 − 0.181429i
\(77\) 1.39764 − 1.48141i 0.159276 − 0.168822i
\(78\) 4.60631 + 3.81436i 0.521562 + 0.431891i
\(79\) −5.14102 − 11.9182i −0.578410 − 1.34090i −0.916592 − 0.399824i \(-0.869071\pi\)
0.338182 − 0.941081i \(-0.390188\pi\)
\(80\) 0.453486 0.0507012
\(81\) 1.33165 − 8.90094i 0.147961 − 0.988993i
\(82\) −9.54769 −1.05437
\(83\) 2.70032 + 6.26005i 0.296399 + 0.687129i 0.999718 − 0.0237446i \(-0.00755885\pi\)
−0.703319 + 0.710874i \(0.748300\pi\)
\(84\) 1.19985 − 7.07351i 0.130914 − 0.771783i
\(85\) −1.02321 + 1.08454i −0.110983 + 0.117635i
\(86\) −0.739300 + 12.6933i −0.0797208 + 1.36875i
\(87\) −1.74504 + 5.96994i −0.187088 + 0.640044i
\(88\) 0.478429 − 0.113390i 0.0510007 − 0.0120874i
\(89\) 7.03445 + 2.56033i 0.745650 + 0.271394i 0.686774 − 0.726871i \(-0.259026\pi\)
0.0588760 + 0.998265i \(0.481248\pi\)
\(90\) 0.664839 − 1.18694i 0.0700802 − 0.125115i
\(91\) 13.4401 − 4.89180i 1.40891 − 0.512800i
\(92\) 5.92351 − 3.89595i 0.617568 − 0.406181i
\(93\) −3.66306 − 4.85408i −0.379842 − 0.503345i
\(94\) 1.59549 − 0.186486i 0.164562 − 0.0192345i
\(95\) −0.429045 − 0.576308i −0.0440191 − 0.0591279i
\(96\) 1.18038 − 1.26756i 0.120472 − 0.129369i
\(97\) 0.517706 + 8.88867i 0.0525651 + 0.902508i 0.916782 + 0.399388i \(0.130777\pi\)
−0.864217 + 0.503119i \(0.832186\pi\)
\(98\) −7.78151 − 6.52946i −0.786051 − 0.659575i
\(99\) 0.404624 − 1.41846i 0.0406662 − 0.142561i
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.a.31.1 ✓ 72
3.2 odd 2 486.2.g.a.145.2 72
81.34 even 27 inner 162.2.g.a.115.1 yes 72
81.47 odd 54 486.2.g.a.181.2 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.a.31.1 ✓ 72 1.1 even 1 trivial
162.2.g.a.115.1 yes 72 81.34 even 27 inner
486.2.g.a.145.2 72 3.2 odd 2
486.2.g.a.181.2 72 81.47 odd 54