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Properties

Label 164.3.n.c.39.19
Level $164$
Weight $3$
Character 164.39
Analytic conductor $4.469$
Analytic rank $0$
Dimension $304$
Inner twists $4$

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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] = [164,3,Mod(39,164)] 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("164.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 164.n (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [304,-10,0,-22,-20,-6,0,-10,0,-46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.46867633551\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(38\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 39.19
Character \(\chi\) \(=\) 164.39
Dual form 164.3.n.c.143.19

$q$-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+(-0.0663434 - 1.99890i) q^{2} +(3.12766 - 3.12766i) q^{3} +(-3.99120 + 0.265228i) q^{4} +(-3.20481 - 4.41104i) q^{5} +(-6.45938 - 6.04438i) q^{6} +(0.253371 + 0.497269i) q^{7} +(0.794953 + 7.96041i) q^{8} -10.5646i q^{9} +(-8.60460 + 6.69873i) q^{10} +(-1.62523 + 0.257411i) q^{11} +(-11.6536 + 13.3127i) q^{12} +(0.00630771 - 0.0123796i) q^{13} +(0.977181 - 0.539454i) q^{14} +(-23.8198 - 3.77269i) q^{15} +(15.8593 - 2.11715i) q^{16} +(-2.55322 + 0.404391i) q^{17} +(-21.1175 + 0.700889i) q^{18} +(31.6268 - 16.1147i) q^{19} +(13.9609 + 16.7553i) q^{20} +(2.34775 + 0.762830i) q^{21} +(0.622361 + 3.23159i) q^{22} +(-11.9240 + 3.87433i) q^{23} +(27.3838 + 22.4111i) q^{24} +(-1.46105 + 4.49664i) q^{25} +(-0.0251640 - 0.0117872i) q^{26} +(-4.89341 - 4.89341i) q^{27} +(-1.14314 - 1.91750i) q^{28} +(-25.9394 - 4.10840i) q^{29} +(-5.96093 + 47.8637i) q^{30} +(20.3144 - 27.9603i) q^{31} +(-5.28413 - 31.5607i) q^{32} +(-4.27807 + 5.88825i) q^{33} +(0.977726 + 5.07681i) q^{34} +(1.38147 - 2.71128i) q^{35} +(2.80201 + 42.1652i) q^{36} +(-30.0974 + 21.8670i) q^{37} +(-34.3099 - 62.1498i) q^{38} +(-0.0189908 - 0.0584476i) q^{39} +(32.5660 - 29.0181i) q^{40} +(-11.5160 + 39.3495i) q^{41} +(1.36906 - 4.74352i) q^{42} +(0.252092 + 0.775859i) q^{43} +(6.41833 - 1.45843i) q^{44} +(-46.6007 + 33.8574i) q^{45} +(8.53546 + 23.5777i) q^{46} +(30.7467 - 60.3438i) q^{47} +(42.9808 - 56.2243i) q^{48} +(28.6184 - 39.3898i) q^{49} +(9.08526 + 2.62216i) q^{50} +(-6.72082 + 9.25042i) q^{51} +(-0.0218919 + 0.0510824i) q^{52} +(69.6475 + 11.0311i) q^{53} +(-9.45678 + 10.1061i) q^{54} +(6.34398 + 6.34398i) q^{55} +(-3.75704 + 2.41224i) q^{56} +(48.5168 - 149.319i) q^{57} +(-6.49137 + 52.1228i) q^{58} +(12.3943 - 4.02716i) q^{59} +(96.0701 + 8.73986i) q^{60} +(59.1326 + 19.2133i) q^{61} +(-57.2376 - 38.7514i) q^{62} +(5.25342 - 2.67675i) q^{63} +(-62.7361 + 12.6563i) q^{64} +(-0.0748218 + 0.0118506i) q^{65} +(12.0538 + 8.16078i) q^{66} +(103.669 + 16.4195i) q^{67} +(10.0832 - 2.29119i) q^{68} +(-25.1765 + 49.4117i) q^{69} +(-5.51123 - 2.58154i) q^{70} +(-120.240 + 19.0442i) q^{71} +(84.0982 - 8.39832i) q^{72} +76.2235i q^{73} +(45.7068 + 58.7110i) q^{74} +(9.49431 + 18.6336i) q^{75} +(-121.955 + 72.7052i) q^{76} +(-0.539787 - 0.742954i) q^{77} +(-0.115571 + 0.0418383i) q^{78} +(25.6123 - 25.6123i) q^{79} +(-60.1649 - 63.1710i) q^{80} +64.4711 q^{81} +(79.4197 + 20.4087i) q^{82} +9.31374i q^{83} +(-9.57265 - 2.42192i) q^{84} +(9.96637 + 9.96637i) q^{85} +(1.53414 - 0.555379i) q^{86} +(-93.9794 + 68.2800i) q^{87} +(-3.34107 - 12.7328i) q^{88} +(-18.0468 + 9.19528i) q^{89} +(70.7691 + 90.9038i) q^{90} +0.00775417 q^{91} +(46.5633 - 18.6258i) q^{92} +(-23.9140 - 150.987i) q^{93} +(-122.661 - 57.4561i) q^{94} +(-172.440 - 87.8628i) q^{95} +(-115.238 - 82.1843i) q^{96} +(19.0991 - 120.587i) q^{97} +(-80.6350 - 54.5920i) q^{98} +(2.71943 + 17.1698i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 304 q - 10 q^{2} - 22 q^{4} - 20 q^{5} - 6 q^{6} - 10 q^{8} - 46 q^{10} - 20 q^{12} - 48 q^{13} - 30 q^{14} + 58 q^{16} - 80 q^{17} + 18 q^{18} - 220 q^{20} - 20 q^{21} - 68 q^{22} - 26 q^{24} + 384 q^{25}+ \cdots + 124 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0663434 − 1.99890i −0.0331717 − 0.999450i
\(3\) 3.12766 − 3.12766i 1.04255 − 1.04255i 0.0435011 − 0.999053i \(-0.486149\pi\)
0.999053 − 0.0435011i \(-0.0138512\pi\)
\(4\) −3.99120 + 0.265228i −0.997799 + 0.0663069i
\(5\) −3.20481 − 4.41104i −0.640961 − 0.882208i 0.357705 − 0.933835i \(-0.383559\pi\)
−0.998666 + 0.0516270i \(0.983559\pi\)
\(6\) −6.45938 − 6.04438i −1.07656 − 1.00740i
\(7\) 0.253371 + 0.497269i 0.0361959 + 0.0710384i 0.908396 − 0.418110i \(-0.137307\pi\)
−0.872200 + 0.489149i \(0.837307\pi\)
\(8\) 0.794953 + 7.96041i 0.0993691 + 0.995051i
\(9\) − 10.5646i − 1.17384i
\(10\) −8.60460 + 6.69873i −0.860460 + 0.669873i
\(11\) −1.62523 + 0.257411i −0.147748 + 0.0234010i −0.229870 − 0.973221i \(-0.573830\pi\)
0.0821224 + 0.996622i \(0.473830\pi\)
\(12\) −11.6536 + 13.3127i −0.971132 + 1.10939i
\(13\) 0.00630771 − 0.0123796i 0.000485209 − 0.000952276i −0.890764 − 0.454467i \(-0.849830\pi\)
0.891249 + 0.453514i \(0.149830\pi\)
\(14\) 0.977181 − 0.539454i 0.0697986 − 0.0385324i
\(15\) −23.8198 − 3.77269i −1.58799 − 0.251512i
\(16\) 15.8593 − 2.11715i 0.991207 − 0.132322i
\(17\) −2.55322 + 0.404391i −0.150190 + 0.0237877i −0.231076 − 0.972936i \(-0.574225\pi\)
0.0808869 + 0.996723i \(0.474225\pi\)
\(18\) −21.1175 + 0.700889i −1.17319 + 0.0389383i
\(19\) 31.6268 − 16.1147i 1.66457 − 0.848141i 0.670201 − 0.742180i \(-0.266208\pi\)
0.994370 − 0.105961i \(-0.0337920\pi\)
\(20\) 13.9609 + 16.7553i 0.698047 + 0.837766i
\(21\) 2.34775 + 0.762830i 0.111798 + 0.0363252i
\(22\) 0.622361 + 3.23159i 0.0282891 + 0.146890i
\(23\) −11.9240 + 3.87433i −0.518433 + 0.168449i −0.556534 − 0.830825i \(-0.687869\pi\)
0.0381013 + 0.999274i \(0.487869\pi\)
\(24\) 27.3838 + 22.4111i 1.14099 + 0.933797i
\(25\) −1.46105 + 4.49664i −0.0584419 + 0.179866i
\(26\) −0.0251640 − 0.0117872i −0.000967847 − 0.000453353i
\(27\) −4.89341 − 4.89341i −0.181237 − 0.181237i
\(28\) −1.14314 − 1.91750i −0.0408265 − 0.0684820i
\(29\) −25.9394 − 4.10840i −0.894462 − 0.141669i −0.307753 − 0.951466i \(-0.599577\pi\)
−0.586709 + 0.809798i \(0.699577\pi\)
\(30\) −5.96093 + 47.8637i −0.198698 + 1.59546i
\(31\) 20.3144 − 27.9603i 0.655302 − 0.901945i −0.344013 − 0.938965i \(-0.611786\pi\)
0.999315 + 0.0370196i \(0.0117864\pi\)
\(32\) −5.28413 − 31.5607i −0.165129 − 0.986272i
\(33\) −4.27807 + 5.88825i −0.129638 + 0.178432i
\(34\) 0.977726 + 5.07681i 0.0287566 + 0.149318i
\(35\) 1.38147 − 2.71128i 0.0394705 − 0.0774651i
\(36\) 2.80201 + 42.1652i 0.0778337 + 1.17126i
\(37\) −30.0974 + 21.8670i −0.813443 + 0.591001i −0.914827 − 0.403846i \(-0.867673\pi\)
0.101383 + 0.994847i \(0.467673\pi\)
\(38\) −34.3099 − 62.1498i −0.902891 − 1.63552i
\(39\) −0.0189908 − 0.0584476i −0.000486943 − 0.00149866i
\(40\) 32.5660 − 29.0181i 0.814150 − 0.725453i
\(41\) −11.5160 + 39.3495i −0.280878 + 0.959744i
\(42\) 1.36906 − 4.74352i 0.0325967 − 0.112941i
\(43\) 0.252092 + 0.775859i 0.00586260 + 0.0180432i 0.953945 − 0.299981i \(-0.0969805\pi\)
−0.948082 + 0.318025i \(0.896981\pi\)
\(44\) 6.41833 − 1.45843i 0.145871 − 0.0331462i
\(45\) −46.6007 + 33.8574i −1.03557 + 0.752386i
\(46\) 8.53546 + 23.5777i 0.185554 + 0.512560i
\(47\) 30.7467 − 60.3438i 0.654185 − 1.28391i −0.290796 − 0.956785i \(-0.593920\pi\)
0.944981 − 0.327125i \(-0.106080\pi\)
\(48\) 42.9808 − 56.2243i 0.895434 − 1.17134i
\(49\) 28.6184 − 39.3898i 0.584049 − 0.803874i
\(50\) 9.08526 + 2.62216i 0.181705 + 0.0524433i
\(51\) −6.72082 + 9.25042i −0.131781 + 0.181381i
\(52\) −0.0218919 + 0.0510824i −0.000420999 + 0.000982353i
\(53\) 69.6475 + 11.0311i 1.31410 + 0.208133i 0.773850 − 0.633369i \(-0.218328\pi\)
0.540253 + 0.841503i \(0.318328\pi\)
\(54\) −9.45678 + 10.1061i −0.175126 + 0.187150i
\(55\) 6.34398 + 6.34398i 0.115345 + 0.115345i
\(56\) −3.75704 + 2.41224i −0.0670900 + 0.0430757i
\(57\) 48.5168 − 149.319i 0.851172 − 2.61964i
\(58\) −6.49137 + 52.1228i −0.111920 + 0.898669i
\(59\) 12.3943 − 4.02716i 0.210073 − 0.0682569i −0.202090 − 0.979367i \(-0.564773\pi\)
0.412163 + 0.911110i \(0.364773\pi\)
\(60\) 96.0701 + 8.73986i 1.60117 + 0.145664i
\(61\) 59.1326 + 19.2133i 0.969387 + 0.314973i 0.750568 − 0.660793i \(-0.229780\pi\)
0.218818 + 0.975766i \(0.429780\pi\)
\(62\) −57.2376 − 38.7514i −0.923186 − 0.625022i
\(63\) 5.25342 − 2.67675i 0.0833877 − 0.0424881i
\(64\) −62.7361 + 12.6563i −0.980252 + 0.197755i
\(65\) −0.0748218 + 0.0118506i −0.00115111 + 0.000182317i
\(66\) 12.0538 + 8.16078i 0.182634 + 0.123648i
\(67\) 103.669 + 16.4195i 1.54730 + 0.245068i 0.870897 − 0.491466i \(-0.163539\pi\)
0.676401 + 0.736534i \(0.263539\pi\)
\(68\) 10.0832 − 2.29119i 0.148282 − 0.0336939i
\(69\) −25.1765 + 49.4117i −0.364877 + 0.716111i
\(70\) −5.51123 − 2.58154i −0.0787318 − 0.0368791i
\(71\) −120.240 + 19.0442i −1.69353 + 0.268228i −0.927293 − 0.374335i \(-0.877871\pi\)
−0.766233 + 0.642563i \(0.777871\pi\)
\(72\) 84.0982 − 8.39832i 1.16803 − 0.116643i
\(73\) 76.2235i 1.04416i 0.852897 + 0.522079i \(0.174843\pi\)
−0.852897 + 0.522079i \(0.825157\pi\)
\(74\) 45.7068 + 58.7110i 0.617659 + 0.793391i
\(75\) 9.49431 + 18.6336i 0.126591 + 0.248449i
\(76\) −121.955 + 72.7052i −1.60467 + 0.956647i
\(77\) −0.539787 − 0.742954i −0.00701023 − 0.00964875i
\(78\) −0.115571 + 0.0418383i −0.00148168 + 0.000536388i
\(79\) 25.6123 − 25.6123i 0.324207 − 0.324207i −0.526172 − 0.850378i \(-0.676373\pi\)
0.850378 + 0.526172i \(0.176373\pi\)
\(80\) −60.1649 − 63.1710i −0.752061 − 0.789637i
\(81\) 64.4711 0.795940
\(82\) 79.4197 + 20.4087i 0.968533 + 0.248887i
\(83\) 9.31374i 0.112214i 0.998425 + 0.0561069i \(0.0178688\pi\)
−0.998425 + 0.0561069i \(0.982131\pi\)
\(84\) −9.57265 − 2.42192i −0.113960 − 0.0288323i
\(85\) 9.96637 + 9.96637i 0.117251 + 0.117251i
\(86\) 1.53414 − 0.555379i 0.0178388 − 0.00645790i
\(87\) −93.9794 + 68.2800i −1.08022 + 0.784828i
\(88\) −3.34107 − 12.7328i −0.0379667 − 0.144691i
\(89\) −18.0468 + 9.19528i −0.202773 + 0.103318i −0.552428 − 0.833561i \(-0.686299\pi\)
0.349656 + 0.936878i \(0.386299\pi\)
\(90\) 70.7691 + 90.9038i 0.786323 + 1.01004i
\(91\) 0.00775417 8.52107e−5
\(92\) 46.5633 − 18.6258i 0.506122 − 0.202454i
\(93\) −23.9140 − 150.987i −0.257139 − 1.62351i
\(94\) −122.661 − 57.4561i −1.30490 − 0.611236i
\(95\) −172.440 − 87.8628i −1.81516 − 0.924871i
\(96\) −115.238 − 82.1843i −1.20040 − 0.856086i
\(97\) 19.0991 − 120.587i 0.196898 − 1.24317i −0.669120 − 0.743155i \(-0.733329\pi\)
0.866018 − 0.500013i \(-0.166671\pi\)
\(98\) −80.6350 − 54.5920i −0.822806 − 0.557062i
\(99\) 2.71943 + 17.1698i 0.0274690 + 0.173432i
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 164.3.n.c.39.19 ✓ 304
4.3 odd 2 inner 164.3.n.c.39.27 yes 304
41.20 even 20 inner 164.3.n.c.143.27 yes 304
164.143 odd 20 inner 164.3.n.c.143.19 yes 304
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.3.n.c.39.19 ✓ 304 1.1 even 1 trivial
164.3.n.c.39.27 yes 304 4.3 odd 2 inner
164.3.n.c.143.19 yes 304 164.143 odd 20 inner
164.3.n.c.143.27 yes 304 41.20 even 20 inner