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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 = [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 103.1
Character \(\chi\) \(=\) 162.103
Dual form 162.2.g.b.151.1

$q$-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+(-0.686242 - 0.727374i) q^{2} +(-1.70169 + 0.322861i) q^{3} +(-0.0581448 + 0.998308i) q^{4} +(0.269488 - 0.0314987i) q^{5} +(1.40261 + 1.01621i) q^{6} +(1.70173 + 0.854643i) q^{7} +(0.766044 - 0.642788i) q^{8} +(2.79152 - 1.09882i) q^{9} +(-0.207845 - 0.174403i) q^{10} +(0.493535 + 1.14414i) q^{11} +(-0.223370 - 1.71759i) q^{12} +(6.47151 + 1.53378i) q^{13} +(-0.546156 - 1.82429i) q^{14} +(-0.448417 + 0.140608i) q^{15} +(-0.993238 - 0.116093i) q^{16} +(0.874267 - 4.95822i) q^{17} +(-2.71491 - 1.27642i) q^{18} +(0.683318 + 3.87529i) q^{19} +(0.0157760 + 0.270864i) q^{20} +(-3.17176 - 0.904918i) q^{21} +(0.493535 - 1.14414i) q^{22} +(1.82524 - 0.916669i) q^{23} +(-1.09604 + 1.34115i) q^{24} +(-4.79359 + 1.13610i) q^{25} +(-3.32539 - 5.75975i) q^{26} +(-4.39555 + 2.77113i) q^{27} +(-0.952145 + 1.64916i) q^{28} +(-1.75411 + 5.85915i) q^{29} +(0.409997 + 0.229675i) q^{30} +(-1.57574 - 1.03638i) q^{31} +(0.597159 + 0.802123i) q^{32} +(-1.20924 - 1.78764i) q^{33} +(-4.20643 + 2.76662i) q^{34} +(0.485518 + 0.176714i) q^{35} +(0.934648 + 2.85069i) q^{36} +(7.10876 - 2.58738i) q^{37} +(2.34986 - 3.15641i) q^{38} +(-11.5077 - 0.520621i) q^{39} +(0.186193 - 0.197353i) q^{40} +(-0.0646786 + 0.0685554i) q^{41} +(1.51838 + 2.92805i) q^{42} +(0.298821 - 0.401387i) q^{43} +(-1.17090 + 0.426174i) q^{44} +(0.717671 - 0.384048i) q^{45} +(-1.91932 - 0.698574i) q^{46} +(-8.22043 + 5.40667i) q^{47} +(1.72767 - 0.123123i) q^{48} +(-2.01462 - 2.70611i) q^{49} +(4.11593 + 2.70709i) q^{50} +(0.113078 + 8.71963i) q^{51} +(-1.90747 + 6.37138i) q^{52} +(5.41495 - 9.37896i) q^{53} +(5.03205 + 1.29554i) q^{54} +(0.169041 + 0.292787i) q^{55} +(1.85296 - 0.439159i) q^{56} +(-2.41398 - 6.37394i) q^{57} +(5.46554 - 2.74490i) q^{58} +(-0.219677 + 0.509269i) q^{59} +(-0.114297 - 0.455834i) q^{60} +(-0.127382 - 2.18707i) q^{61} +(0.327503 + 1.85736i) q^{62} +(5.68953 + 0.515856i) q^{63} +(0.173648 - 0.984808i) q^{64} +(1.79231 + 0.209491i) q^{65} +(-0.470447 + 2.10632i) q^{66} +(-3.40189 - 11.3631i) q^{67} +(4.89899 + 1.16108i) q^{68} +(-2.81004 + 2.14919i) q^{69} +(-0.204645 - 0.474421i) q^{70} +(3.56320 + 2.98988i) q^{71} +(1.43212 - 2.63610i) q^{72} +(0.592380 - 0.497066i) q^{73} +(-6.76031 - 3.39516i) q^{74} +(7.79042 - 3.48096i) q^{75} +(-3.90846 + 0.456834i) q^{76} +(-0.137968 + 2.36882i) q^{77} +(7.51839 + 8.72768i) q^{78} +(-9.82654 - 10.4155i) q^{79} -0.271323 q^{80} +(6.58519 - 6.13476i) q^{81} +0.0942505 q^{82} +(5.24496 + 5.55933i) q^{83} +(1.08781 - 3.11378i) q^{84} +(0.0794276 - 1.36372i) q^{85} +(-0.497022 + 0.0580935i) q^{86} +(1.09328 - 10.5368i) q^{87} +(1.11351 + 0.559226i) q^{88} +(-8.83168 + 7.41066i) q^{89} +(-0.771842 - 0.258465i) q^{90} +(9.70196 + 8.14091i) q^{91} +(0.808990 + 1.87545i) q^{92} +(3.01604 + 1.25486i) q^{93} +(9.57387 + 2.26905i) q^{94} +(0.306212 + 1.02282i) q^{95} +(-1.27515 - 1.17217i) q^{96} +(3.13232 + 0.366116i) q^{97} +(-0.585834 + 3.32243i) q^{98} +(2.63492 + 2.65159i) 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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{7}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.686242 − 0.727374i −0.485246 − 0.514331i
\(3\) −1.70169 + 0.322861i −0.982473 + 0.186404i
\(4\) −0.0581448 + 0.998308i −0.0290724 + 0.499154i
\(5\) 0.269488 − 0.0314987i 0.120519 − 0.0140866i −0.0556197 − 0.998452i \(-0.517713\pi\)
0.176138 + 0.984365i \(0.443639\pi\)
\(6\) 1.40261 + 1.01621i 0.572614 + 0.414865i
\(7\) 1.70173 + 0.854643i 0.643195 + 0.323025i 0.740327 − 0.672247i \(-0.234671\pi\)
−0.0971319 + 0.995272i \(0.530967\pi\)
\(8\) 0.766044 − 0.642788i 0.270838 − 0.227260i
\(9\) 2.79152 − 1.09882i 0.930507 − 0.366273i
\(10\) −0.207845 − 0.174403i −0.0657265 − 0.0551510i
\(11\) 0.493535 + 1.14414i 0.148806 + 0.344972i 0.976273 − 0.216543i \(-0.0694781\pi\)
−0.827467 + 0.561515i \(0.810219\pi\)
\(12\) −0.223370 − 1.71759i −0.0644813 − 0.495825i
\(13\) 6.47151 + 1.53378i 1.79487 + 0.425393i 0.986565 − 0.163367i \(-0.0522354\pi\)
0.808308 + 0.588760i \(0.200384\pi\)
\(14\) −0.546156 − 1.82429i −0.145966 − 0.487562i
\(15\) −0.448417 + 0.140608i −0.115781 + 0.0363049i
\(16\) −0.993238 − 0.116093i −0.248310 − 0.0290232i
\(17\) 0.874267 − 4.95822i 0.212041 − 1.20254i −0.673927 − 0.738798i \(-0.735394\pi\)
0.885968 − 0.463746i \(-0.153495\pi\)
\(18\) −2.71491 − 1.27642i −0.639911 − 0.300856i
\(19\) 0.683318 + 3.87529i 0.156764 + 0.889052i 0.957155 + 0.289575i \(0.0935138\pi\)
−0.800392 + 0.599477i \(0.795375\pi\)
\(20\) 0.0157760 + 0.270864i 0.00352762 + 0.0605670i
\(21\) −3.17176 − 0.904918i −0.692135 − 0.197469i
\(22\) 0.493535 − 1.14414i 0.105222 − 0.243932i
\(23\) 1.82524 − 0.916669i 0.380588 − 0.191139i −0.248211 − 0.968706i \(-0.579843\pi\)
0.628800 + 0.777567i \(0.283546\pi\)
\(24\) −1.09604 + 1.34115i −0.223729 + 0.273762i
\(25\) −4.79359 + 1.13610i −0.958719 + 0.227220i
\(26\) −3.32539 − 5.75975i −0.652163 − 1.12958i
\(27\) −4.39555 + 2.77113i −0.845924 + 0.533304i
\(28\) −0.952145 + 1.64916i −0.179938 + 0.311662i
\(29\) −1.75411 + 5.85915i −0.325731 + 1.08802i 0.625677 + 0.780082i \(0.284823\pi\)
−0.951408 + 0.307934i \(0.900362\pi\)
\(30\) 0.409997 + 0.229675i 0.0748548 + 0.0419328i
\(31\) −1.57574 − 1.03638i −0.283012 − 0.186140i 0.400064 − 0.916487i \(-0.368988\pi\)
−0.683076 + 0.730347i \(0.739358\pi\)
\(32\) 0.597159 + 0.802123i 0.105564 + 0.141797i
\(33\) −1.20924 − 1.78764i −0.210502 − 0.311188i
\(34\) −4.20643 + 2.76662i −0.721398 + 0.474471i
\(35\) 0.485518 + 0.176714i 0.0820674 + 0.0298701i
\(36\) 0.934648 + 2.85069i 0.155775 + 0.475115i
\(37\) 7.10876 − 2.58738i 1.16867 − 0.425362i 0.316485 − 0.948598i \(-0.397497\pi\)
0.852188 + 0.523236i \(0.175275\pi\)
\(38\) 2.34986 − 3.15641i 0.381198 − 0.512038i
\(39\) −11.5077 − 0.520621i −1.84271 − 0.0833661i
\(40\) 0.186193 − 0.197353i 0.0294397 − 0.0312043i
\(41\) −0.0646786 + 0.0685554i −0.0101011 + 0.0107065i −0.732404 − 0.680870i \(-0.761602\pi\)
0.722303 + 0.691576i \(0.243083\pi\)
\(42\) 1.51838 + 2.92805i 0.234291 + 0.451808i
\(43\) 0.298821 − 0.401387i 0.0455698 − 0.0612109i −0.778754 − 0.627329i \(-0.784148\pi\)
0.824324 + 0.566118i \(0.191555\pi\)
\(44\) −1.17090 + 0.426174i −0.176520 + 0.0642482i
\(45\) 0.717671 − 0.384048i 0.106984 − 0.0572505i
\(46\) −1.91932 − 0.698574i −0.282988 − 0.102999i
\(47\) −8.22043 + 5.40667i −1.19907 + 0.788643i −0.982252 − 0.187568i \(-0.939940\pi\)
−0.216822 + 0.976211i \(0.569569\pi\)
\(48\) 1.72767 − 0.123123i 0.249368 − 0.0177713i
\(49\) −2.01462 − 2.70611i −0.287803 − 0.386587i
\(50\) 4.11593 + 2.70709i 0.582081 + 0.382841i
\(51\) 0.113078 + 8.71963i 0.0158340 + 1.22099i
\(52\) −1.90747 + 6.37138i −0.264518 + 0.883551i
\(53\) 5.41495 − 9.37896i 0.743800 − 1.28830i −0.206953 − 0.978351i \(-0.566355\pi\)
0.950753 − 0.309948i \(-0.100312\pi\)
\(54\) 5.03205 + 1.29554i 0.684776 + 0.176301i
\(55\) 0.169041 + 0.292787i 0.0227935 + 0.0394794i
\(56\) 1.85296 − 0.439159i 0.247612 − 0.0586851i
\(57\) −2.41398 − 6.37394i −0.319739 − 0.844248i
\(58\) 5.46554 − 2.74490i 0.717660 − 0.360422i
\(59\) −0.219677 + 0.509269i −0.0285995 + 0.0663011i −0.931917 − 0.362671i \(-0.881865\pi\)
0.903318 + 0.428972i \(0.141124\pi\)
\(60\) −0.114297 − 0.455834i −0.0147557 − 0.0588479i
\(61\) −0.127382 − 2.18707i −0.0163096 − 0.280025i −0.996596 − 0.0824426i \(-0.973728\pi\)
0.980286 − 0.197583i \(-0.0633092\pi\)
\(62\) 0.327503 + 1.85736i 0.0415930 + 0.235885i
\(63\) 5.68953 + 0.515856i 0.716813 + 0.0649918i
\(64\) 0.173648 − 0.984808i 0.0217060 − 0.123101i
\(65\) 1.79231 + 0.209491i 0.222308 + 0.0259841i
\(66\) −0.470447 + 2.10632i −0.0579080 + 0.259270i
\(67\) −3.40189 − 11.3631i −0.415607 − 1.38823i −0.868070 − 0.496442i \(-0.834639\pi\)
0.452462 − 0.891784i \(-0.350546\pi\)
\(68\) 4.89899 + 1.16108i 0.594090 + 0.140802i
\(69\) −2.81004 + 2.14919i −0.338289 + 0.258732i
\(70\) −0.204645 − 0.474421i −0.0244598 − 0.0567042i
\(71\) 3.56320 + 2.98988i 0.422874 + 0.354833i 0.829255 − 0.558870i \(-0.188765\pi\)
−0.406381 + 0.913704i \(0.633209\pi\)
\(72\) 1.43212 − 2.63610i 0.168777 − 0.310667i
\(73\) 0.592380 − 0.497066i 0.0693329 − 0.0581772i −0.607463 − 0.794348i \(-0.707813\pi\)
0.676796 + 0.736171i \(0.263368\pi\)
\(74\) −6.76031 − 3.39516i −0.785871 − 0.394679i
\(75\) 7.79042 − 3.48096i 0.899561 − 0.401947i
\(76\) −3.90846 + 0.456834i −0.448331 + 0.0524024i
\(77\) −0.137968 + 2.36882i −0.0157229 + 0.269953i
\(78\) 7.51839 + 8.72768i 0.851290 + 0.988216i
\(79\) −9.82654 − 10.4155i −1.10557 − 1.17184i −0.983798 − 0.179280i \(-0.942623\pi\)
−0.121774 − 0.992558i \(-0.538858\pi\)
\(80\) −0.271323 −0.0303348
\(81\) 6.58519 − 6.13476i 0.731688 − 0.681640i
\(82\) 0.0942505 0.0104082
\(83\) 5.24496 + 5.55933i 0.575709 + 0.610216i 0.948144 − 0.317843i \(-0.102958\pi\)
−0.372434 + 0.928059i \(0.621477\pi\)
\(84\) 1.08781 − 3.11378i 0.118690 − 0.339741i
\(85\) 0.0794276 − 1.36372i 0.00861513 − 0.147916i
\(86\) −0.497022 + 0.0580935i −0.0535952 + 0.00626439i
\(87\) 1.09328 − 10.5368i 0.117212 − 1.12966i
\(88\) 1.11351 + 0.559226i 0.118701 + 0.0596137i
\(89\) −8.83168 + 7.41066i −0.936156 + 0.785528i −0.976912 − 0.213641i \(-0.931468\pi\)
0.0407561 + 0.999169i \(0.487023\pi\)
\(90\) −0.771842 − 0.258465i −0.0813593 − 0.0272446i
\(91\) 9.70196 + 8.14091i 1.01704 + 0.853399i
\(92\) 0.808990 + 1.87545i 0.0843430 + 0.195529i
\(93\) 3.01604 + 1.25486i 0.312749 + 0.130123i
\(94\) 9.57387 + 2.26905i 0.987469 + 0.234035i
\(95\) 0.306212 + 1.02282i 0.0314167 + 0.104939i
\(96\) −1.27515 − 1.17217i −0.130145 − 0.119634i
\(97\) 3.13232 + 0.366116i 0.318039 + 0.0371734i 0.273615 − 0.961839i \(-0.411781\pi\)
0.0444240 + 0.999013i \(0.485855\pi\)
\(98\) −0.585834 + 3.32243i −0.0591782 + 0.335616i
\(99\) 2.63492 + 2.65159i 0.264819 + 0.266495i
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.103.1 ✓ 90
3.2 odd 2 486.2.g.b.199.3 90
81.11 odd 54 486.2.g.b.127.3 90
81.70 even 27 inner 162.2.g.b.151.1 yes 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.103.1 ✓ 90 1.1 even 1 trivial
162.2.g.b.151.1 yes 90 81.70 even 27 inner
486.2.g.b.127.3 90 81.11 odd 54
486.2.g.b.199.3 90 3.2 odd 2