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Properties

Label 162.2.g.a.43.1
Level $162$
Weight $2$
Character 162.43
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 43.1
Character \(\chi\) \(=\) 162.43
Dual form 162.2.g.a.49.1

$q$-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+(0.286803 + 0.957990i) q^{2} +(-1.47699 - 0.904708i) q^{3} +(-0.835488 + 0.549509i) q^{4} +(0.423867 + 0.982634i) q^{5} +(0.443094 - 1.67442i) q^{6} +(-0.264903 + 4.54820i) q^{7} +(-0.766044 - 0.642788i) q^{8} +(1.36301 + 2.67249i) q^{9} +(-0.819786 + 0.687882i) q^{10} +(1.50212 + 2.01769i) q^{11} +(1.73115 - 0.0557480i) q^{12} +(0.0221543 + 0.0234822i) q^{13} +(-4.43310 + 1.05066i) q^{14} +(0.262948 - 1.83482i) q^{15} +(0.396080 - 0.918216i) q^{16} +(-1.28317 - 7.27724i) q^{17} +(-2.16930 + 2.07223i) q^{18} +(-1.22783 + 6.96338i) q^{19} +(-0.894102 - 0.588060i) q^{20} +(4.50605 - 6.47800i) q^{21} +(-1.50212 + 2.01769i) q^{22} +(-0.167912 - 2.88293i) q^{23} +(0.549906 + 1.64244i) q^{24} +(2.64530 - 2.80386i) q^{25} +(-0.0161418 + 0.0279584i) q^{26} +(0.404671 - 5.18037i) q^{27} +(-2.27795 - 3.94553i) q^{28} +(0.245417 + 0.0581649i) q^{29} +(1.83315 - 0.274330i) q^{30} +(-2.86918 - 1.44096i) q^{31} +(0.993238 + 0.116093i) q^{32} +(-0.393192 - 4.33909i) q^{33} +(6.60350 - 3.31640i) q^{34} +(-4.58150 + 1.66753i) q^{35} +(-2.60733 - 1.48385i) q^{36} +(10.7770 + 3.92249i) q^{37} +(-7.02299 + 0.820869i) q^{38} +(-0.0114772 - 0.0547262i) q^{39} +(0.306924 - 1.02520i) q^{40} +(-1.21748 + 4.06667i) q^{41} +(7.49820 + 2.45884i) q^{42} +(3.10495 - 0.362916i) q^{43} +(-2.36374 - 0.860332i) q^{44} +(-2.04835 + 2.47212i) q^{45} +(2.71366 - 0.987691i) q^{46} +(6.73190 - 3.38089i) q^{47} +(-1.41572 + 0.997861i) q^{48} +(-13.6633 - 1.59701i) q^{49} +(3.44475 + 1.73002i) q^{50} +(-4.68853 + 11.9093i) q^{51} +(-0.0314133 - 0.00744510i) q^{52} +(-0.516900 - 0.895297i) q^{53} +(5.07880 - 1.09808i) q^{54} +(-1.34596 + 2.33126i) q^{55} +(3.12645 - 3.31385i) q^{56} +(8.11332 - 9.17402i) q^{57} +(0.0146650 + 0.251789i) q^{58} +(0.314656 - 0.422657i) q^{59} +(0.788558 + 1.67746i) q^{60} +(4.82343 + 3.17242i) q^{61} +(0.557532 - 3.16192i) q^{62} +(-12.5161 + 5.49129i) q^{63} +(0.173648 + 0.984808i) q^{64} +(-0.0136839 + 0.0317229i) q^{65} +(4.04404 - 1.62114i) q^{66} +(-1.10768 + 0.262525i) q^{67} +(5.07098 + 5.37493i) q^{68} +(-2.36020 + 4.40997i) q^{69} +(-2.91146 - 3.91078i) q^{70} +(4.41828 - 3.70738i) q^{71} +(0.673719 - 2.92337i) q^{72} +(-8.95122 - 7.51097i) q^{73} +(-0.666840 + 11.4492i) q^{74} +(-6.44376 + 1.74805i) q^{75} +(-2.80060 - 6.49252i) q^{76} +(-9.57479 + 6.29744i) q^{77} +(0.0491354 - 0.0266907i) q^{78} +(-0.614615 - 2.05296i) q^{79} +1.07016 q^{80} +(-5.28442 + 7.28526i) q^{81} -4.24500 q^{82} +(-2.70592 - 9.03840i) q^{83} +(-0.205034 + 7.88840i) q^{84} +(6.60696 - 4.34547i) q^{85} +(1.23818 + 2.87042i) q^{86} +(-0.309857 - 0.307940i) q^{87} +(0.146260 - 2.51119i) q^{88} +(0.755271 + 0.633748i) q^{89} +(-2.95574 - 1.25328i) q^{90} +(-0.112670 + 0.0945417i) q^{91} +(1.72448 + 2.31638i) q^{92} +(2.93412 + 4.72406i) q^{93} +(5.16958 + 5.47944i) q^{94} +(-7.36288 + 1.74504i) q^{95} +(-1.36197 - 1.07006i) q^{96} +(-0.903213 + 2.09388i) q^{97} +(-2.38876 - 13.5473i) q^{98} +(-3.34487 + 6.76453i) 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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{11}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.286803 + 0.957990i 0.202801 + 0.677401i
\(3\) −1.47699 − 0.904708i −0.852741 − 0.522333i
\(4\) −0.835488 + 0.549509i −0.417744 + 0.274754i
\(5\) 0.423867 + 0.982634i 0.189559 + 0.439447i 0.986319 − 0.164849i \(-0.0527137\pi\)
−0.796760 + 0.604296i \(0.793454\pi\)
\(6\) 0.443094 − 1.67442i 0.180893 − 0.683577i
\(7\) −0.264903 + 4.54820i −0.100124 + 1.71906i 0.457706 + 0.889103i \(0.348671\pi\)
−0.557830 + 0.829955i \(0.688366\pi\)
\(8\) −0.766044 − 0.642788i −0.270838 − 0.227260i
\(9\) 1.36301 + 2.67249i 0.454336 + 0.890830i
\(10\) −0.819786 + 0.687882i −0.259239 + 0.217528i
\(11\) 1.50212 + 2.01769i 0.452905 + 0.608358i 0.968664 − 0.248377i \(-0.0798971\pi\)
−0.515758 + 0.856734i \(0.672490\pi\)
\(12\) 1.73115 − 0.0557480i 0.499741 − 0.0160931i
\(13\) 0.0221543 + 0.0234822i 0.00614450 + 0.00651279i 0.730439 − 0.682978i \(-0.239316\pi\)
−0.724294 + 0.689491i \(0.757834\pi\)
\(14\) −4.43310 + 1.05066i −1.18480 + 0.280802i
\(15\) 0.262948 − 1.83482i 0.0678930 − 0.473748i
\(16\) 0.396080 − 0.918216i 0.0990199 − 0.229554i
\(17\) −1.28317 − 7.27724i −0.311215 − 1.76499i −0.592700 − 0.805423i \(-0.701938\pi\)
0.281485 − 0.959566i \(-0.409173\pi\)
\(18\) −2.16930 + 2.07223i −0.511310 + 0.488429i
\(19\) −1.22783 + 6.96338i −0.281684 + 1.59751i 0.435212 + 0.900328i \(0.356673\pi\)
−0.716896 + 0.697180i \(0.754438\pi\)
\(20\) −0.894102 − 0.588060i −0.199927 − 0.131494i
\(21\) 4.50605 − 6.47800i 0.983301 − 1.41361i
\(22\) −1.50212 + 2.01769i −0.320253 + 0.430174i
\(23\) −0.167912 − 2.88293i −0.0350120 − 0.601132i −0.969493 − 0.245119i \(-0.921173\pi\)
0.934481 − 0.356013i \(-0.115864\pi\)
\(24\) 0.549906 + 1.64244i 0.112249 + 0.335261i
\(25\) 2.64530 − 2.80386i 0.529060 − 0.560771i
\(26\) −0.0161418 + 0.0279584i −0.00316566 + 0.00548309i
\(27\) 0.404671 − 5.18037i 0.0778790 − 0.996963i
\(28\) −2.27795 − 3.94553i −0.430493 − 0.745636i
\(29\) 0.245417 + 0.0581649i 0.0455728 + 0.0108010i 0.253339 − 0.967378i \(-0.418471\pi\)
−0.207766 + 0.978179i \(0.566619\pi\)
\(30\) 1.83315 − 0.274330i 0.334686 − 0.0500855i
\(31\) −2.86918 − 1.44096i −0.515321 − 0.258804i 0.172079 − 0.985083i \(-0.444952\pi\)
−0.687400 + 0.726279i \(0.741248\pi\)
\(32\) 0.993238 + 0.116093i 0.175581 + 0.0205225i
\(33\) −0.393192 − 4.33909i −0.0684459 − 0.755339i
\(34\) 6.60350 − 3.31640i 1.13249 − 0.568758i
\(35\) −4.58150 + 1.66753i −0.774415 + 0.281864i
\(36\) −2.60733 − 1.48385i −0.434556 − 0.247308i
\(37\) 10.7770 + 3.92249i 1.77172 + 0.644854i 0.999960 + 0.00894017i \(0.00284578\pi\)
0.771760 + 0.635913i \(0.219376\pi\)
\(38\) −7.02299 + 0.820869i −1.13928 + 0.133163i
\(39\) −0.0114772 − 0.0547262i −0.00183782 − 0.00876320i
\(40\) 0.306924 − 1.02520i 0.0485289 − 0.162098i
\(41\) −1.21748 + 4.06667i −0.190139 + 0.635107i 0.808758 + 0.588142i \(0.200140\pi\)
−0.998897 + 0.0469653i \(0.985045\pi\)
\(42\) 7.49820 + 2.45884i 1.15700 + 0.379407i
\(43\) 3.10495 − 0.362916i 0.473500 − 0.0553442i 0.124003 − 0.992282i \(-0.460427\pi\)
0.349497 + 0.936938i \(0.386353\pi\)
\(44\) −2.36374 − 0.860332i −0.356348 − 0.129700i
\(45\) −2.04835 + 2.47212i −0.305349 + 0.368522i
\(46\) 2.71366 − 0.987691i 0.400107 − 0.145627i
\(47\) 6.73190 − 3.38089i 0.981948 − 0.493153i 0.115980 − 0.993252i \(-0.462999\pi\)
0.865968 + 0.500099i \(0.166703\pi\)
\(48\) −1.41572 + 0.997861i −0.204342 + 0.144029i
\(49\) −13.6633 − 1.59701i −1.95190 − 0.228144i
\(50\) 3.44475 + 1.73002i 0.487161 + 0.244661i
\(51\) −4.68853 + 11.9093i −0.656526 + 1.66764i
\(52\) −0.0314133 − 0.00744510i −0.00435624 − 0.00103245i
\(53\) −0.516900 − 0.895297i −0.0710017 − 0.122979i 0.828339 − 0.560227i \(-0.189286\pi\)
−0.899341 + 0.437249i \(0.855953\pi\)
\(54\) 5.07880 − 1.09808i 0.691137 − 0.149429i
\(55\) −1.34596 + 2.33126i −0.181489 + 0.314348i
\(56\) 3.12645 − 3.31385i 0.417790 − 0.442832i
\(57\) 8.11332 − 9.17402i 1.07463 − 1.21513i
\(58\) 0.0146650 + 0.251789i 0.00192561 + 0.0330615i
\(59\) 0.314656 − 0.422657i 0.0409647 − 0.0550252i −0.781169 − 0.624320i \(-0.785376\pi\)
0.822134 + 0.569295i \(0.192784\pi\)
\(60\) 0.788558 + 1.67746i 0.101802 + 0.216559i
\(61\) 4.82343 + 3.17242i 0.617577 + 0.406187i 0.819389 − 0.573237i \(-0.194313\pi\)
−0.201812 + 0.979424i \(0.564683\pi\)
\(62\) 0.557532 − 3.16192i 0.0708066 − 0.401564i
\(63\) −12.5161 + 5.49129i −1.57688 + 0.691837i
\(64\) 0.173648 + 0.984808i 0.0217060 + 0.123101i
\(65\) −0.0136839 + 0.0317229i −0.00169728 + 0.00393474i
\(66\) 4.04404 − 1.62114i 0.497787 − 0.199549i
\(67\) −1.10768 + 0.262525i −0.135324 + 0.0320725i −0.297720 − 0.954653i \(-0.596226\pi\)
0.162395 + 0.986726i \(0.448078\pi\)
\(68\) 5.07098 + 5.37493i 0.614947 + 0.651806i
\(69\) −2.36020 + 4.40997i −0.284135 + 0.530898i
\(70\) −2.91146 − 3.91078i −0.347987 − 0.467427i
\(71\) 4.41828 − 3.70738i 0.524353 − 0.439985i −0.341793 − 0.939775i \(-0.611034\pi\)
0.866146 + 0.499791i \(0.166590\pi\)
\(72\) 0.673719 − 2.92337i 0.0793986 − 0.344523i
\(73\) −8.95122 − 7.51097i −1.04766 − 0.879092i −0.0548158 − 0.998496i \(-0.517457\pi\)
−0.992846 + 0.119404i \(0.961902\pi\)
\(74\) −0.666840 + 11.4492i −0.0775185 + 1.33094i
\(75\) −6.44376 + 1.74805i −0.744061 + 0.201847i
\(76\) −2.80060 − 6.49252i −0.321251 − 0.744743i
\(77\) −9.57479 + 6.29744i −1.09115 + 0.717660i
\(78\) 0.0491354 − 0.0266907i 0.00556349 − 0.00302212i
\(79\) −0.614615 − 2.05296i −0.0691496 − 0.230976i 0.916428 − 0.400199i \(-0.131059\pi\)
−0.985578 + 0.169224i \(0.945874\pi\)
\(80\) 1.07016 0.119647
\(81\) −5.28442 + 7.28526i −0.587157 + 0.809473i
\(82\) −4.24500 −0.468782
\(83\) −2.70592 − 9.03840i −0.297013 − 0.992093i −0.968068 − 0.250686i \(-0.919344\pi\)
0.671055 − 0.741407i \(-0.265841\pi\)
\(84\) −0.205034 + 7.88840i −0.0223710 + 0.860695i
\(85\) 6.60696 − 4.34547i 0.716626 − 0.471332i
\(86\) 1.23818 + 2.87042i 0.133516 + 0.309525i
\(87\) −0.309857 − 0.307940i −0.0332201 − 0.0330146i
\(88\) 0.146260 − 2.51119i 0.0155914 − 0.267693i
\(89\) 0.755271 + 0.633748i 0.0800586 + 0.0671771i 0.681939 − 0.731409i \(-0.261137\pi\)
−0.601880 + 0.798586i \(0.705582\pi\)
\(90\) −2.95574 − 1.25328i −0.311562 − 0.132108i
\(91\) −0.112670 + 0.0945417i −0.0118111 + 0.00991067i
\(92\) 1.72448 + 2.31638i 0.179790 + 0.241500i
\(93\) 2.93412 + 4.72406i 0.304254 + 0.489862i
\(94\) 5.16958 + 5.47944i 0.533202 + 0.565161i
\(95\) −7.36288 + 1.74504i −0.755416 + 0.179037i
\(96\) −1.36197 − 1.07006i −0.139006 − 0.109212i
\(97\) −0.903213 + 2.09388i −0.0917074 + 0.212602i −0.957896 − 0.287116i \(-0.907304\pi\)
0.866188 + 0.499717i \(0.166563\pi\)
\(98\) −2.38876 − 13.5473i −0.241301 − 1.36849i
\(99\) −3.34487 + 6.76453i −0.336172 + 0.679861i
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.43.1 ✓ 72
3.2 odd 2 486.2.g.a.289.1 72
81.32 odd 54 486.2.g.a.37.1 72
81.49 even 27 inner 162.2.g.a.49.1 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.a.43.1 ✓ 72 1.1 even 1 trivial
162.2.g.a.49.1 yes 72 81.49 even 27 inner
486.2.g.a.37.1 72 81.32 odd 54
486.2.g.a.289.1 72 3.2 odd 2