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Properties

Label 164.3.n.c.39.16
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 39.16
Character \(\chi\) \(=\) 164.39
Dual form 164.3.n.c.143.16

$q$-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+(-0.730370 + 1.86187i) q^{2} +(-0.989033 + 0.989033i) q^{3} +(-2.93312 - 2.71971i) q^{4} +(-3.65044 - 5.02440i) q^{5} +(-1.11909 - 2.56381i) q^{6} +(0.740377 + 1.45307i) q^{7} +(7.20600 - 3.47470i) q^{8} +7.04363i q^{9} +(12.0209 - 3.12698i) q^{10} +(17.0306 - 2.69739i) q^{11} +(5.59084 - 0.211073i) q^{12} +(10.4108 - 20.4324i) q^{13} +(-3.24618 + 0.317206i) q^{14} +(8.57970 + 1.35889i) q^{15} +(1.20639 + 15.9545i) q^{16} +(-14.1867 + 2.24695i) q^{17} +(-13.1143 - 5.14445i) q^{18} +(6.69586 - 3.41171i) q^{19} +(-2.95771 + 24.6653i) q^{20} +(-2.16939 - 0.704879i) q^{21} +(-7.41647 + 33.6789i) q^{22} +(31.5277 - 10.2440i) q^{23} +(-3.69039 + 10.5636i) q^{24} +(-4.19345 + 12.9061i) q^{25} +(30.4387 + 34.3067i) q^{26} +(-15.8677 - 15.8677i) q^{27} +(1.78032 - 6.27564i) q^{28} +(-20.6817 - 3.27565i) q^{29} +(-8.79644 + 14.9818i) q^{30} +(12.9217 - 17.7852i) q^{31} +(-30.5862 - 9.40651i) q^{32} +(-14.1761 + 19.5117i) q^{33} +(6.17799 - 28.0548i) q^{34} +(4.59811 - 9.02430i) q^{35} +(19.1566 - 20.6598i) q^{36} +(31.5392 - 22.9146i) q^{37} +(1.46171 + 14.9586i) q^{38} +(9.91165 + 30.5049i) q^{39} +(-43.7633 - 23.5217i) q^{40} +(-24.3057 - 33.0187i) q^{41} +(2.89685 - 3.52431i) q^{42} +(10.0273 + 30.8609i) q^{43} +(-57.2890 - 38.4066i) q^{44} +(35.3900 - 25.7123i) q^{45} +(-3.95394 + 66.1824i) q^{46} +(-13.3388 + 26.1788i) q^{47} +(-16.9726 - 14.5863i) q^{48} +(27.2382 - 37.4902i) q^{49} +(-20.9667 - 17.2339i) q^{50} +(11.8088 - 16.2534i) q^{51} +(-86.1062 + 31.6162i) q^{52} +(63.7146 + 10.0914i) q^{53} +(41.1328 - 17.9543i) q^{54} +(-75.7221 - 75.7221i) q^{55} +(10.3841 + 7.89826i) q^{56} +(-3.24813 + 9.99672i) q^{57} +(21.2041 - 36.1141i) q^{58} +(-97.1828 + 31.5766i) q^{59} +(-21.4695 - 27.3201i) q^{60} +(-8.81354 - 2.86369i) q^{61} +(23.6762 + 37.0484i) q^{62} +(-10.2349 + 5.21494i) q^{63} +(39.8530 - 50.0774i) q^{64} +(-140.664 + 22.2790i) q^{65} +(-25.9744 - 40.6447i) q^{66} +(110.085 + 17.4357i) q^{67} +(47.7222 + 31.9930i) q^{68} +(-21.0503 + 41.3136i) q^{69} +(13.4438 + 15.1522i) q^{70} +(-119.387 + 18.9090i) q^{71} +(24.4745 + 50.7564i) q^{72} -53.2985i q^{73} +(19.6287 + 75.4580i) q^{74} +(-8.61711 - 16.9120i) q^{75} +(-28.9186 - 8.20381i) q^{76} +(16.5286 + 22.7497i) q^{77} +(-64.0354 - 3.82567i) q^{78} +(56.9544 - 56.9544i) q^{79} +(75.7577 - 64.3021i) q^{80} -32.0053 q^{81} +(79.2286 - 21.1382i) q^{82} +46.6273i q^{83} +(4.44603 + 7.96761i) q^{84} +(63.0771 + 63.0771i) q^{85} +(-64.7827 - 3.87031i) q^{86} +(23.6946 - 17.2151i) q^{87} +(113.350 - 78.6137i) q^{88} +(38.2192 - 19.4737i) q^{89} +(22.0252 + 84.6711i) q^{90} +37.3976 q^{91} +(-120.335 - 55.6993i) q^{92} +(4.81017 + 30.3702i) q^{93} +(-38.9993 - 43.9553i) q^{94} +(-41.5846 - 21.1884i) q^{95} +(39.5542 - 20.9475i) q^{96} +(-11.0996 + 70.0801i) q^{97} +(49.9079 + 78.0957i) q^{98} +(18.9994 + 119.957i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.730370 + 1.86187i −0.365185 + 0.930935i
\(3\) −0.989033 + 0.989033i −0.329678 + 0.329678i −0.852464 − 0.522786i \(-0.824893\pi\)
0.522786 + 0.852464i \(0.324893\pi\)
\(4\) −2.93312 − 2.71971i −0.733280 − 0.679927i
\(5\) −3.65044 − 5.02440i −0.730088 − 1.00488i −0.999128 − 0.0417508i \(-0.986706\pi\)
0.269040 − 0.963129i \(-0.413294\pi\)
\(6\) −1.11909 − 2.56381i −0.186515 − 0.427302i
\(7\) 0.740377 + 1.45307i 0.105768 + 0.207582i 0.937825 − 0.347109i \(-0.112837\pi\)
−0.832057 + 0.554691i \(0.812837\pi\)
\(8\) 7.20600 − 3.47470i 0.900750 − 0.434337i
\(9\) 7.04363i 0.782625i
\(10\) 12.0209 − 3.12698i 1.20209 − 0.312698i
\(11\) 17.0306 − 2.69739i 1.54824 − 0.245217i 0.676966 − 0.736014i \(-0.263294\pi\)
0.871274 + 0.490797i \(0.163294\pi\)
\(12\) 5.59084 − 0.211073i 0.465903 − 0.0175894i
\(13\) 10.4108 − 20.4324i 0.800831 − 1.57172i −0.0194851 − 0.999810i \(-0.506203\pi\)
0.820316 − 0.571910i \(-0.193797\pi\)
\(14\) −3.24618 + 0.317206i −0.231870 + 0.0226576i
\(15\) 8.57970 + 1.35889i 0.571980 + 0.0905928i
\(16\) 1.20639 + 15.9545i 0.0753993 + 0.997153i
\(17\) −14.1867 + 2.24695i −0.834510 + 0.132173i −0.559044 − 0.829138i \(-0.688832\pi\)
−0.275465 + 0.961311i \(0.588832\pi\)
\(18\) −13.1143 − 5.14445i −0.728573 − 0.285803i
\(19\) 6.69586 − 3.41171i 0.352414 − 0.179564i −0.268817 − 0.963191i \(-0.586633\pi\)
0.621231 + 0.783627i \(0.286633\pi\)
\(20\) −2.95771 + 24.6653i −0.147886 + 1.23326i
\(21\) −2.16939 − 0.704879i −0.103305 − 0.0335657i
\(22\) −7.41647 + 33.6789i −0.337112 + 1.53086i
\(23\) 31.5277 − 10.2440i 1.37077 − 0.445390i 0.471146 − 0.882055i \(-0.343841\pi\)
0.899624 + 0.436665i \(0.143841\pi\)
\(24\) −3.69039 + 10.5636i −0.153766 + 0.440149i
\(25\) −4.19345 + 12.9061i −0.167738 + 0.516244i
\(26\) 30.4387 + 34.3067i 1.17072 + 1.31949i
\(27\) −15.8677 − 15.8677i −0.587692 − 0.587692i
\(28\) 1.78032 − 6.27564i 0.0635827 − 0.224130i
\(29\) −20.6817 − 3.27565i −0.713160 − 0.112954i −0.210696 − 0.977552i \(-0.567573\pi\)
−0.502464 + 0.864598i \(0.667573\pi\)
\(30\) −8.79644 + 14.9818i −0.293215 + 0.499393i
\(31\) 12.9217 − 17.7852i 0.416830 − 0.573717i −0.548038 − 0.836454i \(-0.684625\pi\)
0.964868 + 0.262736i \(0.0846250\pi\)
\(32\) −30.5862 − 9.40651i −0.955820 − 0.293953i
\(33\) −14.1761 + 19.5117i −0.429578 + 0.591263i
\(34\) 6.17799 − 28.0548i 0.181706 − 0.825142i
\(35\) 4.59811 − 9.02430i 0.131375 − 0.257837i
\(36\) 19.1566 − 20.6598i 0.532128 − 0.573883i
\(37\) 31.5392 − 22.9146i 0.852411 − 0.619313i −0.0733990 − 0.997303i \(-0.523385\pi\)
0.925810 + 0.377990i \(0.123385\pi\)
\(38\) 1.46171 + 14.9586i 0.0384660 + 0.393648i
\(39\) 9.91165 + 30.5049i 0.254145 + 0.782178i
\(40\) −43.7633 − 23.5217i −1.09408 − 0.588042i
\(41\) −24.3057 − 33.0187i −0.592821 − 0.805334i
\(42\) 2.89685 − 3.52431i 0.0689727 − 0.0839121i
\(43\) 10.0273 + 30.8609i 0.233194 + 0.717696i 0.997356 + 0.0726722i \(0.0231527\pi\)
−0.764162 + 0.645024i \(0.776847\pi\)
\(44\) −57.2890 − 38.4066i −1.30202 − 0.872877i
\(45\) 35.3900 − 25.7123i 0.786444 − 0.571385i
\(46\) −3.95394 + 66.1824i −0.0859551 + 1.43875i
\(47\) −13.3388 + 26.1788i −0.283804 + 0.556996i −0.988265 − 0.152746i \(-0.951188\pi\)
0.704462 + 0.709742i \(0.251188\pi\)
\(48\) −16.9726 − 14.5863i −0.353597 − 0.303882i
\(49\) 27.2382 − 37.4902i 0.555882 − 0.765106i
\(50\) −20.9667 − 17.2339i −0.419335 − 0.344678i
\(51\) 11.8088 − 16.2534i 0.231545 − 0.318694i
\(52\) −86.1062 + 31.6162i −1.65589 + 0.608004i
\(53\) 63.7146 + 10.0914i 1.20216 + 0.190404i 0.725211 − 0.688526i \(-0.241742\pi\)
0.476951 + 0.878930i \(0.341742\pi\)
\(54\) 41.1328 − 17.9543i 0.761719 − 0.332487i
\(55\) −75.7221 − 75.7221i −1.37676 − 1.37676i
\(56\) 10.3841 + 7.89826i 0.185431 + 0.141040i
\(57\) −3.24813 + 9.99672i −0.0569848 + 0.175381i
\(58\) 21.2041 − 36.1141i 0.365588 − 0.622657i
\(59\) −97.1828 + 31.5766i −1.64717 + 0.535197i −0.978123 − 0.208027i \(-0.933296\pi\)
−0.669043 + 0.743224i \(0.733296\pi\)
\(60\) −21.4695 − 27.3201i −0.357825 − 0.455335i
\(61\) −8.81354 − 2.86369i −0.144484 − 0.0469458i 0.235882 − 0.971782i \(-0.424202\pi\)
−0.380366 + 0.924836i \(0.624202\pi\)
\(62\) 23.6762 + 37.0484i 0.381873 + 0.597554i
\(63\) −10.2349 + 5.21494i −0.162459 + 0.0827768i
\(64\) 39.8530 − 50.0774i 0.622702 − 0.782459i
\(65\) −140.664 + 22.2790i −2.16407 + 0.342755i
\(66\) −25.9744 − 40.6447i −0.393552 − 0.615829i
\(67\) 110.085 + 17.4357i 1.64306 + 0.260234i 0.908368 − 0.418171i \(-0.137329\pi\)
0.734687 + 0.678406i \(0.237329\pi\)
\(68\) 47.7222 + 31.9930i 0.701798 + 0.470485i
\(69\) −21.0503 + 41.3136i −0.305077 + 0.598748i
\(70\) 13.4438 + 15.1522i 0.192054 + 0.216459i
\(71\) −119.387 + 18.9090i −1.68151 + 0.266325i −0.922848 − 0.385164i \(-0.874145\pi\)
−0.758659 + 0.651488i \(0.774145\pi\)
\(72\) 24.4745 + 50.7564i 0.339923 + 0.704950i
\(73\) − 53.2985i − 0.730116i −0.930985 − 0.365058i \(-0.881049\pi\)
0.930985 − 0.365058i \(-0.118951\pi\)
\(74\) 19.6287 + 75.4580i 0.265252 + 1.01970i
\(75\) −8.61711 − 16.9120i −0.114895 − 0.225494i
\(76\) −28.9186 − 8.20381i −0.380508 − 0.107945i
\(77\) 16.5286 + 22.7497i 0.214657 + 0.295450i
\(78\) −64.0354 − 3.82567i −0.820966 − 0.0490470i
\(79\) 56.9544 − 56.9544i 0.720942 − 0.720942i −0.247855 − 0.968797i \(-0.579726\pi\)
0.968797 + 0.247855i \(0.0797258\pi\)
\(80\) 75.7577 − 64.3021i 0.946971 − 0.803777i
\(81\) −32.0053 −0.395127
\(82\) 79.2286 − 21.1382i 0.966203 − 0.257782i
\(83\) 46.6273i 0.561775i 0.959741 + 0.280888i \(0.0906288\pi\)
−0.959741 + 0.280888i \(0.909371\pi\)
\(84\) 4.44603 + 7.96761i 0.0529289 + 0.0948525i
\(85\) 63.0771 + 63.0771i 0.742084 + 0.742084i
\(86\) −64.7827 − 3.87031i −0.753287 − 0.0450037i
\(87\) 23.6946 − 17.2151i 0.272351 − 0.197875i
\(88\) 113.350 − 78.6137i 1.28807 − 0.893337i
\(89\) 38.2192 − 19.4737i 0.429429 − 0.218805i −0.225897 − 0.974151i \(-0.572531\pi\)
0.655326 + 0.755346i \(0.272531\pi\)
\(90\) 22.0252 + 84.6711i 0.244725 + 0.940790i
\(91\) 37.3976 0.410963
\(92\) −120.335 − 55.6993i −1.30799 − 0.605427i
\(93\) 4.81017 + 30.3702i 0.0517222 + 0.326561i
\(94\) −38.9993 − 43.9553i −0.414886 − 0.467609i
\(95\) −41.5846 − 21.1884i −0.437733 − 0.223036i
\(96\) 39.5542 − 20.9475i 0.412022 − 0.218203i
\(97\) −11.0996 + 70.0801i −0.114429 + 0.722475i 0.862044 + 0.506834i \(0.169184\pi\)
−0.976473 + 0.215641i \(0.930816\pi\)
\(98\) 49.9079 + 78.0957i 0.509264 + 0.796895i
\(99\) 18.9994 + 119.957i 0.191913 + 1.21169i
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.39.16 yes 304
4.3 odd 2 inner 164.3.n.c.39.7 ✓ 304
41.20 even 20 inner 164.3.n.c.143.7 yes 304
164.143 odd 20 inner 164.3.n.c.143.16 yes 304
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.3.n.c.39.7 ✓ 304 4.3 odd 2 inner
164.3.n.c.39.16 yes 304 1.1 even 1 trivial
164.3.n.c.143.7 yes 304 41.20 even 20 inner
164.3.n.c.143.16 yes 304 164.143 odd 20 inner