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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 13.2
Character \(\chi\) \(=\) 162.13
Dual form 162.2.g.a.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} +(-1.41362 - 1.00084i) q^{3} +(0.597159 + 0.802123i) q^{4} +(0.308002 + 1.02880i) q^{5} +(0.814085 + 1.52881i) q^{6} +(-2.06499 - 4.78718i) q^{7} +(-0.173648 - 0.984808i) q^{8} +(0.996657 + 2.82961i) q^{9} +(0.186483 - 1.05760i) q^{10} +(-5.51869 + 1.30795i) q^{11} +(-0.0413635 - 1.73156i) q^{12} +(-2.75916 - 1.81473i) q^{13} +(-0.303142 + 5.20475i) q^{14} +(0.594259 - 1.76259i) q^{15} +(-0.286803 + 0.957990i) q^{16} +(-1.32917 + 0.483778i) q^{17} +(0.379280 - 2.97593i) q^{18} +(-0.526489 - 0.191626i) q^{19} +(-0.641296 + 0.861410i) q^{20} +(-1.87207 + 8.83399i) q^{21} +(5.51869 + 1.30795i) q^{22} +(0.185413 - 0.429835i) q^{23} +(-0.740157 + 1.56594i) q^{24} +(3.21388 - 2.11380i) q^{25} +(1.65123 + 2.86001i) q^{26} +(1.42307 - 4.99749i) q^{27} +(2.60678 - 4.51508i) q^{28} +(-0.413128 - 7.09314i) q^{29} +(-1.32210 + 1.30840i) q^{30} +(2.69492 - 0.314991i) q^{31} +(0.686242 - 0.727374i) q^{32} +(9.11038 + 3.67435i) q^{33} +(1.40491 + 0.164210i) q^{34} +(4.28902 - 3.59892i) q^{35} +(-1.67453 + 2.48917i) q^{36} +(-3.28779 - 2.75878i) q^{37} +(0.384486 + 0.407532i) q^{38} +(2.08417 + 5.32681i) q^{39} +(0.959683 - 0.481971i) q^{40} +(5.78531 - 2.90549i) q^{41} +(5.63763 - 7.05416i) q^{42} +(6.69559 + 7.09691i) q^{43} +(-4.34467 - 3.64561i) q^{44} +(-2.60412 + 1.89688i) q^{45} +(-0.358600 + 0.300901i) q^{46} +(1.37768 + 0.161028i) q^{47} +(1.36422 - 1.06719i) q^{48} +(-13.8493 + 14.6794i) q^{49} +(-3.82070 + 0.446576i) q^{50} +(2.36312 + 0.646400i) q^{51} +(-0.192021 - 3.29687i) q^{52} +(-3.96592 + 6.86918i) q^{53} +(-3.51457 + 3.82724i) q^{54} +(-3.04538 - 5.27476i) q^{55} +(-4.35587 + 2.86490i) q^{56} +(0.552471 + 0.797817i) q^{57} +(-2.81421 + 6.52407i) q^{58} +(-1.06624 - 0.252703i) q^{59} +(1.76868 - 0.575877i) q^{60} +(5.03022 - 6.75675i) q^{61} +(-2.54964 - 0.927991i) q^{62} +(11.4878 - 10.6143i) q^{63} +(-0.939693 + 0.342020i) q^{64} +(1.01716 - 3.39756i) q^{65} +(-6.49229 - 7.37225i) q^{66} +(-0.100933 + 1.73295i) q^{67} +(-1.18177 - 0.777265i) q^{68} +(-0.692297 + 0.422057i) q^{69} +(-5.44800 + 1.29120i) q^{70} +(1.32440 - 7.51103i) q^{71} +(2.61355 - 1.47287i) q^{72} +(-1.86793 - 10.5936i) q^{73} +(1.69994 + 3.94090i) q^{74} +(-6.65878 - 0.228447i) q^{75} +(-0.160690 - 0.536741i) q^{76} +(17.6574 + 23.7181i) q^{77} +(0.528187 - 5.69558i) q^{78} +(-1.99908 - 1.00398i) q^{79} -1.07391 q^{80} +(-7.01335 + 5.64029i) q^{81} -6.47392 q^{82} +(-10.3349 - 5.19037i) q^{83} +(-8.20387 + 3.77366i) q^{84} +(-0.907095 - 1.21844i) q^{85} +(-2.79831 - 9.34701i) q^{86} +(-6.51505 + 10.4405i) q^{87} +(2.24639 + 5.20772i) q^{88} +(-0.973720 - 5.52224i) q^{89} +(3.17844 - 0.526388i) q^{90} +(-2.98980 + 16.9560i) q^{91} +(0.455501 - 0.107956i) q^{92} +(-4.12485 - 2.25189i) q^{93} +(-1.15887 - 0.762203i) q^{94} +(0.0349852 - 0.600672i) q^{95} +(-1.69807 + 0.341417i) q^{96} +(-0.471322 + 1.57433i) q^{97} +(18.9642 - 6.90241i) q^{98} +(-9.20123 - 14.3121i) 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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

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

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
\(3\) −1.41362 − 1.00084i −0.816155 − 0.577833i
\(4\) 0.597159 + 0.802123i 0.298579 + 0.401062i
\(5\) 0.308002 + 1.02880i 0.137743 + 0.460092i 0.998830 − 0.0483653i \(-0.0154012\pi\)
−0.861087 + 0.508457i \(0.830216\pi\)
\(6\) 0.814085 + 1.52881i 0.332349 + 0.624135i
\(7\) −2.06499 − 4.78718i −0.780493 − 1.80939i −0.547256 − 0.836966i \(-0.684327\pi\)
−0.233237 − 0.972420i \(-0.574932\pi\)
\(8\) −0.173648 − 0.984808i −0.0613939 − 0.348182i
\(9\) 0.996657 + 2.82961i 0.332219 + 0.943202i
\(10\) 0.186483 − 1.05760i 0.0589711 − 0.334442i
\(11\) −5.51869 + 1.30795i −1.66395 + 0.394363i −0.951569 − 0.307434i \(-0.900530\pi\)
−0.712377 + 0.701797i \(0.752381\pi\)
\(12\) −0.0413635 − 1.73156i −0.0119406 − 0.499857i
\(13\) −2.75916 − 1.81473i −0.765253 − 0.503315i 0.105870 − 0.994380i \(-0.466237\pi\)
−0.871123 + 0.491065i \(0.836608\pi\)
\(14\) −0.303142 + 5.20475i −0.0810181 + 1.39103i
\(15\) 0.594259 − 1.76259i 0.153437 − 0.455099i
\(16\) −0.286803 + 0.957990i −0.0717008 + 0.239497i
\(17\) −1.32917 + 0.483778i −0.322371 + 0.117333i −0.498136 − 0.867099i \(-0.665982\pi\)
0.175766 + 0.984432i \(0.443760\pi\)
\(18\) 0.379280 − 2.97593i 0.0893971 − 0.701433i
\(19\) −0.526489 − 0.191626i −0.120785 − 0.0439621i 0.280921 − 0.959731i \(-0.409360\pi\)
−0.401706 + 0.915769i \(0.631582\pi\)
\(20\) −0.641296 + 0.861410i −0.143398 + 0.192617i
\(21\) −1.87207 + 8.83399i −0.408519 + 1.92773i
\(22\) 5.51869 + 1.30795i 1.17659 + 0.278856i
\(23\) 0.185413 − 0.429835i 0.0386612 − 0.0896268i −0.897791 − 0.440421i \(-0.854829\pi\)
0.936453 + 0.350794i \(0.114088\pi\)
\(24\) −0.740157 + 1.56594i −0.151084 + 0.319646i
\(25\) 3.21388 − 2.11380i 0.642776 − 0.422761i
\(26\) 1.65123 + 2.86001i 0.323832 + 0.560894i
\(27\) 1.42307 − 4.99749i 0.273871 − 0.961767i
\(28\) 2.60678 − 4.51508i 0.492636 − 0.853271i
\(29\) −0.413128 − 7.09314i −0.0767160 − 1.31716i −0.788618 − 0.614883i \(-0.789203\pi\)
0.711902 − 0.702279i \(-0.247834\pi\)
\(30\) −1.32210 + 1.30840i −0.241381 + 0.238881i
\(31\) 2.69492 − 0.314991i 0.484022 − 0.0565740i 0.129417 − 0.991590i \(-0.458689\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(32\) 0.686242 − 0.727374i 0.121312 − 0.128583i
\(33\) 9.11038 + 3.67435i 1.58591 + 0.639621i
\(34\) 1.40491 + 0.164210i 0.240940 + 0.0281618i
\(35\) 4.28902 − 3.59892i 0.724977 − 0.608328i
\(36\) −1.67453 + 2.48917i −0.279088 + 0.414861i
\(37\) −3.28779 − 2.75878i −0.540510 − 0.453541i 0.331203 − 0.943560i \(-0.392546\pi\)
−0.871712 + 0.490018i \(0.836990\pi\)
\(38\) 0.384486 + 0.407532i 0.0623719 + 0.0661104i
\(39\) 2.08417 + 5.32681i 0.333734 + 0.852972i
\(40\) 0.959683 − 0.481971i 0.151739 − 0.0762063i
\(41\) 5.78531 − 2.90549i 0.903513 − 0.453761i 0.0645280 − 0.997916i \(-0.479446\pi\)
0.838985 + 0.544155i \(0.183150\pi\)
\(42\) 5.63763 − 7.05416i 0.869905 − 1.08848i
\(43\) 6.69559 + 7.09691i 1.02107 + 1.08227i 0.996585 + 0.0825691i \(0.0263125\pi\)
0.0244831 + 0.999700i \(0.492206\pi\)
\(44\) −4.34467 − 3.64561i −0.654984 − 0.549597i
\(45\) −2.60412 + 1.89688i −0.388199 + 0.282770i
\(46\) −0.358600 + 0.300901i −0.0528727 + 0.0443655i
\(47\) 1.37768 + 0.161028i 0.200956 + 0.0234883i 0.215975 − 0.976399i \(-0.430707\pi\)
−0.0150195 + 0.999887i \(0.504781\pi\)
\(48\) 1.36422 − 1.06719i 0.196908 − 0.154036i
\(49\) −13.8493 + 14.6794i −1.97847 + 2.09705i
\(50\) −3.82070 + 0.446576i −0.540329 + 0.0631554i
\(51\) 2.36312 + 0.646400i 0.330904 + 0.0905141i
\(52\) −0.192021 − 3.29687i −0.0266285 − 0.457193i
\(53\) −3.96592 + 6.86918i −0.544761 + 0.943554i 0.453861 + 0.891073i \(0.350046\pi\)
−0.998622 + 0.0524814i \(0.983287\pi\)
\(54\) −3.51457 + 3.82724i −0.478273 + 0.520822i
\(55\) −3.04538 − 5.27476i −0.410639 − 0.711248i
\(56\) −4.35587 + 2.86490i −0.582078 + 0.382839i
\(57\) 0.552471 + 0.797817i 0.0731765 + 0.105673i
\(58\) −2.81421 + 6.52407i −0.369524 + 0.856652i
\(59\) −1.06624 − 0.252703i −0.138812 − 0.0328991i 0.160622 − 0.987016i \(-0.448650\pi\)
−0.299435 + 0.954117i \(0.596798\pi\)
\(60\) 1.76868 − 0.575877i 0.228336 − 0.0743454i
\(61\) 5.03022 − 6.75675i 0.644053 − 0.865114i −0.353470 − 0.935446i \(-0.614999\pi\)
0.997524 + 0.0703322i \(0.0224059\pi\)
\(62\) −2.54964 − 0.927991i −0.323804 − 0.117855i
\(63\) 11.4878 − 10.6143i 1.44732 − 1.33727i
\(64\) −0.939693 + 0.342020i −0.117462 + 0.0427525i
\(65\) 1.01716 − 3.39756i 0.126163 − 0.421415i
\(66\) −6.49229 − 7.37225i −0.799146 − 0.907461i
\(67\) −0.100933 + 1.73295i −0.0123309 + 0.211714i 0.986514 + 0.163674i \(0.0523345\pi\)
−0.998845 + 0.0480399i \(0.984703\pi\)
\(68\) −1.18177 − 0.777265i −0.143311 − 0.0942572i
\(69\) −0.692297 + 0.422057i −0.0833428 + 0.0508096i
\(70\) −5.44800 + 1.29120i −0.651160 + 0.154328i
\(71\) 1.32440 − 7.51103i 0.157177 − 0.891395i −0.799591 − 0.600544i \(-0.794951\pi\)
0.956768 − 0.290851i \(-0.0939384\pi\)
\(72\) 2.61355 − 1.47287i 0.308010 − 0.173580i
\(73\) −1.86793 − 10.5936i −0.218625 − 1.23988i −0.874504 − 0.485018i \(-0.838813\pi\)
0.655879 − 0.754866i \(-0.272298\pi\)
\(74\) 1.69994 + 3.94090i 0.197614 + 0.458120i
\(75\) −6.65878 − 0.228447i −0.768890 − 0.0263788i
\(76\) −0.160690 − 0.536741i −0.0184324 − 0.0615684i
\(77\) 17.6574 + 23.7181i 2.01225 + 2.70292i
\(78\) 0.528187 − 5.69558i 0.0598054 − 0.644898i
\(79\) −1.99908 − 1.00398i −0.224914 − 0.112956i 0.332774 − 0.943007i \(-0.392015\pi\)
−0.557689 + 0.830050i \(0.688312\pi\)
\(80\) −1.07391 −0.120067
\(81\) −7.01335 + 5.64029i −0.779261 + 0.626699i
\(82\) −6.47392 −0.714925
\(83\) −10.3349 − 5.19037i −1.13440 − 0.569717i −0.220391 − 0.975412i \(-0.570733\pi\)
−0.914009 + 0.405694i \(0.867030\pi\)
\(84\) −8.20387 + 3.77366i −0.895115 + 0.411740i
\(85\) −0.907095 − 1.21844i −0.0983883 − 0.132158i
\(86\) −2.79831 − 9.34701i −0.301750 − 1.00791i
\(87\) −6.51505 + 10.4405i −0.698487 + 1.11934i
\(88\) 2.24639 + 5.20772i 0.239466 + 0.555145i
\(89\) −0.973720 − 5.52224i −0.103214 − 0.585356i −0.991919 − 0.126876i \(-0.959505\pi\)
0.888704 − 0.458480i \(-0.151606\pi\)
\(90\) 3.17844 − 0.526388i 0.335037 − 0.0554862i
\(91\) −2.98980 + 16.9560i −0.313416 + 1.77747i
\(92\) 0.455501 − 0.107956i 0.0474893 − 0.0112552i
\(93\) −4.12485 − 2.25189i −0.427727 − 0.233510i
\(94\) −1.15887 − 0.762203i −0.119529 − 0.0786152i
\(95\) 0.0349852 − 0.600672i 0.00358940 − 0.0616276i
\(96\) −1.69807 + 0.341417i −0.173308 + 0.0348457i
\(97\) −0.471322 + 1.57433i −0.0478555 + 0.159848i −0.978577 − 0.205879i \(-0.933995\pi\)
0.930722 + 0.365728i \(0.119180\pi\)
\(98\) 18.9642 − 6.90241i 1.91568 − 0.697249i
\(99\) −9.20123 − 14.3121i −0.924758 − 1.43842i
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.13.2 ✓ 72
3.2 odd 2 486.2.g.a.253.1 72
81.25 even 27 inner 162.2.g.a.25.2 yes 72
81.56 odd 54 486.2.g.a.73.1 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.a.13.2 ✓ 72 1.1 even 1 trivial
162.2.g.a.25.2 yes 72 81.25 even 27 inner
486.2.g.a.73.1 72 81.56 odd 54
486.2.g.a.253.1 72 3.2 odd 2