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 97.5
Character \(\chi\) \(=\) 162.97
Dual form 162.2.g.b.157.5

$q$-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+(0.973045 + 0.230616i) q^{2} +(1.70747 + 0.290767i) q^{3} +(0.893633 + 0.448799i) q^{4} +(0.537372 - 0.721815i) q^{5} +(1.59439 + 0.676699i) q^{6} +(-3.95679 - 2.60242i) q^{7} +(0.766044 + 0.642788i) q^{8} +(2.83091 + 0.992953i) q^{9} +(0.689349 - 0.578432i) q^{10} +(-4.21098 + 0.492194i) q^{11} +(1.39536 + 1.02615i) q^{12} +(-1.75605 + 5.86562i) q^{13} +(-3.24997 - 3.44477i) q^{14} +(1.12743 - 1.07623i) q^{15} +(0.597159 + 0.802123i) q^{16} +(-0.432180 - 2.45101i) q^{17} +(2.52561 + 1.61904i) q^{18} +(-0.284021 + 1.61076i) q^{19} +(0.804163 - 0.403866i) q^{20} +(-5.99940 - 5.59406i) q^{21} +(-4.21098 - 0.492194i) q^{22} +(6.35325 - 4.17860i) q^{23} +(1.12110 + 1.32028i) q^{24} +(1.20177 + 4.01418i) q^{25} +(-3.06142 + 5.30254i) q^{26} +(4.54497 + 2.51857i) q^{27} +(-2.36795 - 4.10141i) q^{28} +(2.09368 - 2.21918i) q^{29} +(1.34523 - 0.787216i) q^{30} +(-0.107618 - 1.84773i) q^{31} +(0.396080 + 0.918216i) q^{32} +(-7.33324 - 0.384010i) q^{33} +(0.144712 - 2.48461i) q^{34} +(-4.00473 + 1.45760i) q^{35} +(2.08416 + 2.15784i) q^{36} +(-8.42259 - 3.06557i) q^{37} +(-0.647833 + 1.50184i) q^{38} +(-4.70394 + 9.50478i) q^{39} +(0.875624 - 0.207527i) q^{40} +(5.04755 - 1.19629i) q^{41} +(-4.54760 - 6.82682i) q^{42} +(1.06337 - 2.46516i) q^{43} +(-3.98397 - 1.45005i) q^{44} +(2.23798 - 1.50981i) q^{45} +(7.14564 - 2.60080i) q^{46} +(0.486795 - 8.35794i) q^{47} +(0.786399 + 1.54324i) q^{48} +(6.11101 + 14.1669i) q^{49} +(0.243639 + 4.18313i) q^{50} +(-0.0252598 - 4.31070i) q^{51} +(-4.20175 + 4.45360i) q^{52} +(-1.11544 - 1.93201i) q^{53} +(3.84164 + 3.49883i) q^{54} +(-1.90759 + 3.30404i) q^{55} +(-1.35827 - 4.53694i) q^{56} +(-0.953314 + 2.66775i) q^{57} +(2.54903 - 1.67652i) q^{58} +(-3.71218 - 0.433892i) q^{59} +(1.49051 - 0.455765i) q^{60} +(2.81018 - 1.41133i) q^{61} +(0.321399 - 1.82275i) q^{62} +(-8.61722 - 11.2961i) q^{63} +(0.173648 + 0.984808i) q^{64} +(3.29024 + 4.41957i) q^{65} +(-7.04702 - 2.06482i) q^{66} +(3.28793 + 3.48500i) q^{67} +(0.713803 - 2.38427i) q^{68} +(12.0630 - 5.28751i) q^{69} +(-4.23293 + 0.494758i) q^{70} +(-3.18551 + 2.67296i) q^{71} +(1.53034 + 2.58032i) q^{72} +(1.09995 + 0.922965i) q^{73} +(-7.48859 - 4.92532i) q^{74} +(0.884789 + 7.20353i) q^{75} +(-0.976720 + 1.31196i) q^{76} +(17.9429 + 9.01124i) q^{77} +(-6.76910 + 8.16377i) q^{78} +(-5.03759 - 1.19393i) q^{79} +0.899881 q^{80} +(7.02809 + 5.62192i) q^{81} +5.18737 q^{82} +(-2.63513 - 0.624537i) q^{83} +(-2.85065 - 7.69155i) q^{84} +(-2.00142 - 1.00515i) q^{85} +(1.60321 - 2.15349i) q^{86} +(4.22017 - 3.18040i) q^{87} +(-3.54218 - 2.32973i) q^{88} +(12.5970 + 10.5701i) q^{89} +(2.52584 - 0.952998i) q^{90} +(22.2131 - 18.6390i) q^{91} +(7.55282 - 0.882798i) q^{92} +(0.353505 - 3.18624i) q^{93} +(2.40115 - 8.02039i) q^{94} +(1.01005 + 1.07059i) q^{95} +(0.409307 + 1.68299i) q^{96} +(2.79423 + 3.75330i) q^{97} +(2.67917 + 15.1943i) q^{98} +(-12.4096 - 2.78795i) 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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{2}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.973045 + 0.230616i 0.688047 + 0.163070i
\(3\) 1.70747 + 0.290767i 0.985808 + 0.167875i
\(4\) 0.893633 + 0.448799i 0.446816 + 0.224400i
\(5\) 0.537372 − 0.721815i 0.240320 − 0.322806i −0.665557 − 0.746347i \(-0.731806\pi\)
0.905877 + 0.423542i \(0.139213\pi\)
\(6\) 1.59439 + 0.676699i 0.650907 + 0.276261i
\(7\) −3.95679 − 2.60242i −1.49552 − 0.983622i −0.992879 − 0.119129i \(-0.961990\pi\)
−0.502646 − 0.864493i \(-0.667640\pi\)
\(8\) 0.766044 + 0.642788i 0.270838 + 0.227260i
\(9\) 2.83091 + 0.992953i 0.943636 + 0.330984i
\(10\) 0.689349 − 0.578432i 0.217991 − 0.182916i
\(11\) −4.21098 + 0.492194i −1.26966 + 0.148402i −0.724115 − 0.689680i \(-0.757751\pi\)
−0.545545 + 0.838082i \(0.683677\pi\)
\(12\) 1.39536 + 1.02615i 0.402804 + 0.296224i
\(13\) −1.75605 + 5.86562i −0.487041 + 1.62683i 0.262541 + 0.964921i \(0.415439\pi\)
−0.749583 + 0.661911i \(0.769746\pi\)
\(14\) −3.24997 − 3.44477i −0.868591 − 0.920653i
\(15\) 1.12743 − 1.07623i 0.291100 − 0.277881i
\(16\) 0.597159 + 0.802123i 0.149290 + 0.200531i
\(17\) −0.432180 − 2.45101i −0.104819 − 0.594458i −0.991292 − 0.131679i \(-0.957963\pi\)
0.886473 − 0.462780i \(-0.153148\pi\)
\(18\) 2.52561 + 1.61904i 0.595292 + 0.381611i
\(19\) −0.284021 + 1.61076i −0.0651589 + 0.369534i 0.934740 + 0.355332i \(0.115632\pi\)
−0.999899 + 0.0142026i \(0.995479\pi\)
\(20\) 0.804163 − 0.403866i 0.179816 − 0.0903071i
\(21\) −5.99940 − 5.59406i −1.30918 − 1.22072i
\(22\) −4.21098 − 0.492194i −0.897785 − 0.104936i
\(23\) 6.35325 − 4.17860i 1.32474 − 0.871297i 0.327453 − 0.944867i \(-0.393810\pi\)
0.997290 + 0.0735700i \(0.0234392\pi\)
\(24\) 1.12110 + 1.32028i 0.228843 + 0.269501i
\(25\) 1.20177 + 4.01418i 0.240353 + 0.802836i
\(26\) −3.06142 + 5.30254i −0.600395 + 1.03991i
\(27\) 4.54497 + 2.51857i 0.874681 + 0.484699i
\(28\) −2.36795 − 4.10141i −0.447500 − 0.775093i
\(29\) 2.09368 − 2.21918i 0.388787 − 0.412091i −0.503186 − 0.864178i \(-0.667839\pi\)
0.891974 + 0.452087i \(0.149321\pi\)
\(30\) 1.34523 − 0.787216i 0.245605 − 0.143725i
\(31\) −0.107618 − 1.84773i −0.0193288 − 0.331863i −0.994086 − 0.108595i \(-0.965365\pi\)
0.974757 − 0.223267i \(-0.0716723\pi\)
\(32\) 0.396080 + 0.918216i 0.0700177 + 0.162319i
\(33\) −7.33324 − 0.384010i −1.27655 − 0.0668476i
\(34\) 0.144712 − 2.48461i 0.0248180 − 0.426108i
\(35\) −4.00473 + 1.45760i −0.676923 + 0.246380i
\(36\) 2.08416 + 2.15784i 0.347359 + 0.359641i
\(37\) −8.42259 − 3.06557i −1.38467 − 0.503977i −0.461076 − 0.887361i \(-0.652537\pi\)
−0.923589 + 0.383384i \(0.874759\pi\)
\(38\) −0.647833 + 1.50184i −0.105092 + 0.243631i
\(39\) −4.70394 + 9.50478i −0.753233 + 1.52198i
\(40\) 0.875624 − 0.207527i 0.138448 − 0.0328129i
\(41\) 5.04755 − 1.19629i 0.788295 − 0.186829i 0.183298 − 0.983057i \(-0.441323\pi\)
0.604996 + 0.796228i \(0.293175\pi\)
\(42\) −4.54760 − 6.82682i −0.701710 − 1.05340i
\(43\) 1.06337 − 2.46516i 0.162162 − 0.375934i −0.817690 − 0.575659i \(-0.804746\pi\)
0.979852 + 0.199725i \(0.0640049\pi\)
\(44\) −3.98397 − 1.45005i −0.600606 − 0.218603i
\(45\) 2.23798 − 1.50981i 0.333618 − 0.225069i
\(46\) 7.14564 − 2.60080i 1.05357 − 0.383467i
\(47\) 0.486795 − 8.35794i 0.0710063 − 1.21913i −0.755097 − 0.655613i \(-0.772410\pi\)
0.826103 − 0.563519i \(-0.190553\pi\)
\(48\) 0.786399 + 1.54324i 0.113507 + 0.222747i
\(49\) 6.11101 + 14.1669i 0.873002 + 2.02384i
\(50\) 0.243639 + 4.18313i 0.0344558 + 0.591583i
\(51\) −0.0252598 − 4.31070i −0.00353708 − 0.603618i
\(52\) −4.20175 + 4.45360i −0.582678 + 0.617603i
\(53\) −1.11544 − 1.93201i −0.153218 − 0.265381i 0.779191 − 0.626787i \(-0.215630\pi\)
−0.932409 + 0.361406i \(0.882297\pi\)
\(54\) 3.84164 + 3.49883i 0.522781 + 0.476130i
\(55\) −1.90759 + 3.30404i −0.257219 + 0.445517i
\(56\) −1.35827 − 4.53694i −0.181507 − 0.606274i
\(57\) −0.953314 + 2.66775i −0.126270 + 0.353352i
\(58\) 2.54903 − 1.67652i 0.334703 − 0.220138i
\(59\) −3.71218 − 0.433892i −0.483285 − 0.0564879i −0.129038 − 0.991640i \(-0.541189\pi\)
−0.354247 + 0.935152i \(0.615263\pi\)
\(60\) 1.49051 − 0.455765i 0.192425 − 0.0588390i
\(61\) 2.81018 − 1.41133i 0.359807 − 0.180702i −0.259707 − 0.965687i \(-0.583626\pi\)
0.619514 + 0.784986i \(0.287330\pi\)
\(62\) 0.321399 − 1.82275i 0.0408178 − 0.231489i
\(63\) −8.61722 − 11.2961i −1.08567 − 1.42318i
\(64\) 0.173648 + 0.984808i 0.0217060 + 0.123101i
\(65\) 3.29024 + 4.41957i 0.408105 + 0.548180i
\(66\) −7.04702 − 2.06482i −0.867428 − 0.254162i
\(67\) 3.28793 + 3.48500i 0.401685 + 0.425761i 0.896367 − 0.443313i \(-0.146197\pi\)
−0.494682 + 0.869074i \(0.664716\pi\)
\(68\) 0.713803 − 2.38427i 0.0865614 − 0.289135i
\(69\) 12.0630 − 5.28751i 1.45221 − 0.636542i
\(70\) −4.23293 + 0.494758i −0.505932 + 0.0591349i
\(71\) −3.18551 + 2.67296i −0.378051 + 0.317222i −0.811937 − 0.583746i \(-0.801587\pi\)
0.433886 + 0.900968i \(0.357142\pi\)
\(72\) 1.53034 + 2.58032i 0.180353 + 0.304094i
\(73\) 1.09995 + 0.922965i 0.128739 + 0.108025i 0.704884 − 0.709323i \(-0.250999\pi\)
−0.576145 + 0.817348i \(0.695444\pi\)
\(74\) −7.48859 − 4.92532i −0.870531 − 0.572557i
\(75\) 0.884789 + 7.20353i 0.102167 + 0.831792i
\(76\) −0.976720 + 1.31196i −0.112037 + 0.150492i
\(77\) 17.9429 + 9.01124i 2.04478 + 1.02693i
\(78\) −6.76910 + 8.16377i −0.766449 + 0.924365i
\(79\) −5.03759 − 1.19393i −0.566773 − 0.134328i −0.0627675 − 0.998028i \(-0.519993\pi\)
−0.504005 + 0.863701i \(0.668141\pi\)
\(80\) 0.899881 0.100610
\(81\) 7.02809 + 5.62192i 0.780899 + 0.624657i
\(82\) 5.18737 0.572850
\(83\) −2.63513 − 0.624537i −0.289243 − 0.0685519i 0.0834329 − 0.996513i \(-0.473412\pi\)
−0.372676 + 0.927962i \(0.621560\pi\)
\(84\) −2.85065 − 7.69155i −0.311031 − 0.839217i
\(85\) −2.00142 − 1.00515i −0.217085 − 0.109024i
\(86\) 1.60321 − 2.15349i 0.172879 − 0.232216i
\(87\) 4.22017 − 3.18040i 0.452449 − 0.340975i
\(88\) −3.54218 − 2.32973i −0.377597 − 0.248350i
\(89\) 12.5970 + 10.5701i 1.33528 + 1.12043i 0.982813 + 0.184604i \(0.0591003\pi\)
0.352463 + 0.935826i \(0.385344\pi\)
\(90\) 2.52584 − 0.952998i 0.266247 − 0.100455i
\(91\) 22.2131 − 18.6390i 2.32857 − 1.95390i
\(92\) 7.55282 − 0.882798i 0.787436 − 0.0920380i
\(93\) 0.353505 − 3.18624i 0.0366568 − 0.330398i
\(94\) 2.40115 − 8.02039i 0.247659 − 0.827240i
\(95\) 1.01005 + 1.07059i 0.103629 + 0.109840i
\(96\) 0.409307 + 1.68299i 0.0417747 + 0.171770i
\(97\) 2.79423 + 3.75330i 0.283711 + 0.381090i 0.921018 − 0.389520i \(-0.127359\pi\)
−0.637307 + 0.770610i \(0.719952\pi\)
\(98\) 2.67917 + 15.1943i 0.270637 + 1.53486i
\(99\) −12.4096 − 2.78795i −1.24722 − 0.280200i
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.97.5 ✓ 90
3.2 odd 2 486.2.g.b.451.2 90
81.5 odd 54 486.2.g.b.361.2 90
81.76 even 27 inner 162.2.g.b.157.5 yes 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.97.5 ✓ 90 1.1 even 1 trivial
162.2.g.b.157.5 yes 90 81.76 even 27 inner
486.2.g.b.361.2 90 81.5 odd 54
486.2.g.b.451.2 90 3.2 odd 2