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Properties

Label 164.3.n.c.39.10
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 39.10
Character \(\chi\) \(=\) 164.39
Dual form 164.3.n.c.143.10

$q$-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+(-1.44283 - 1.38501i) q^{2} +(-2.92028 + 2.92028i) q^{3} +(0.163493 + 3.99666i) q^{4} +(3.53020 + 4.85890i) q^{5} +(8.25806 - 0.168838i) q^{6} +(-0.522596 - 1.02565i) q^{7} +(5.29952 - 5.99292i) q^{8} -8.05603i q^{9} +(1.63616 - 11.8999i) q^{10} +(-11.3597 + 1.79919i) q^{11} +(-12.1488 - 11.1939i) q^{12} +(-2.93860 + 5.76733i) q^{13} +(-0.666523 + 2.20364i) q^{14} +(-24.4985 - 3.88018i) q^{15} +(-15.9465 + 1.30685i) q^{16} +(7.39159 - 1.17071i) q^{17} +(-11.1577 + 11.6235i) q^{18} +(-8.33754 + 4.24819i) q^{19} +(-18.8422 + 14.9034i) q^{20} +(4.52131 + 1.46906i) q^{21} +(18.8819 + 13.1373i) q^{22} +(-10.5722 + 3.43513i) q^{23} +(2.02493 + 32.9771i) q^{24} +(-3.42118 + 10.5293i) q^{25} +(12.2277 - 4.25126i) q^{26} +(-2.75664 - 2.75664i) q^{27} +(4.01374 - 2.25632i) q^{28} +(-48.6388 - 7.70362i) q^{29} +(29.9729 + 39.5291i) q^{30} +(20.1391 - 27.7191i) q^{31} +(24.8181 + 20.2006i) q^{32} +(27.9192 - 38.4275i) q^{33} +(-12.2862 - 8.54829i) q^{34} +(3.13867 - 6.15999i) q^{35} +(32.1972 - 1.31711i) q^{36} +(-31.8448 + 23.1366i) q^{37} +(17.9134 + 5.41818i) q^{38} +(-8.26067 - 25.4237i) q^{39} +(47.8273 + 4.59363i) q^{40} +(-31.8910 - 25.7675i) q^{41} +(-4.48880 - 8.38166i) q^{42} +(23.5998 + 72.6328i) q^{43} +(-9.04799 - 45.1065i) q^{44} +(39.1434 - 28.4394i) q^{45} +(20.0116 + 9.68636i) q^{46} +(-9.30998 + 18.2719i) q^{47} +(42.7519 - 50.3847i) q^{48} +(28.0226 - 38.5698i) q^{49} +(19.5194 - 10.4536i) q^{50} +(-18.1667 + 25.0043i) q^{51} +(-23.5305 - 10.8017i) q^{52} +(-28.0843 - 4.44812i) q^{53} +(0.159377 + 7.79533i) q^{54} +(-48.8439 - 48.8439i) q^{55} +(-8.91615 - 2.30359i) q^{56} +(11.9420 - 36.7538i) q^{57} +(59.5077 + 78.4802i) q^{58} +(-12.4926 + 4.05908i) q^{59} +(11.5024 - 98.5464i) q^{60} +(69.6517 + 22.6312i) q^{61} +(-67.4484 + 12.1010i) q^{62} +(-8.26268 + 4.21005i) q^{63} +(-7.83021 - 63.5192i) q^{64} +(-38.3967 + 6.08144i) q^{65} +(-93.5050 + 16.7758i) q^{66} +(123.107 + 19.4982i) q^{67} +(5.88742 + 29.3503i) q^{68} +(20.8423 - 40.9054i) q^{69} +(-13.0602 + 4.54070i) q^{70} +(72.6929 - 11.5134i) q^{71} +(-48.2792 - 42.6931i) q^{72} +88.1394i q^{73} +(77.9909 + 10.7233i) q^{74} +(-20.7577 - 40.7393i) q^{75} +(-18.3417 - 32.6277i) q^{76} +(7.78185 + 10.7108i) q^{77} +(-23.2934 + 48.1231i) q^{78} +(-62.8336 + 62.8336i) q^{79} +(-62.6443 - 72.8691i) q^{80} +88.6046 q^{81} +(10.3250 + 81.3474i) q^{82} +31.9399i q^{83} +(-5.13214 + 18.3103i) q^{84} +(31.7821 + 31.7821i) q^{85} +(66.5467 - 137.482i) q^{86} +(164.535 - 119.542i) q^{87} +(-49.4183 + 77.6124i) q^{88} +(-110.819 + 56.4652i) q^{89} +(-95.8660 - 13.1810i) q^{90} +7.45097 q^{91} +(-15.4575 - 41.6920i) q^{92} +(22.1357 + 139.759i) q^{93} +(38.7394 - 13.4687i) q^{94} +(-50.0747 - 25.5143i) q^{95} +(-131.467 + 13.4845i) q^{96} +(-15.6987 + 99.1174i) q^{97} +(-93.8514 + 16.8379i) q^{98} +(14.4944 + 91.5138i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.44283 − 1.38501i −0.721413 − 0.692505i
\(3\) −2.92028 + 2.92028i −0.973426 + 0.973426i −0.999656 − 0.0262303i \(-0.991650\pi\)
0.0262303 + 0.999656i \(0.491650\pi\)
\(4\) 0.163493 + 3.99666i 0.0408734 + 0.999164i
\(5\) 3.53020 + 4.85890i 0.706039 + 0.971779i 0.999873 + 0.0159230i \(0.00506868\pi\)
−0.293834 + 0.955856i \(0.594931\pi\)
\(6\) 8.25806 − 0.168838i 1.37634 − 0.0281397i
\(7\) −0.522596 − 1.02565i −0.0746565 − 0.146522i 0.850661 − 0.525714i \(-0.176202\pi\)
−0.925318 + 0.379192i \(0.876202\pi\)
\(8\) 5.29952 − 5.99292i 0.662440 − 0.749115i
\(9\) − 8.05603i − 0.895115i
\(10\) 1.63616 − 11.8999i 0.163616 − 1.18999i
\(11\) −11.3597 + 1.79919i −1.03270 + 0.163563i −0.649703 − 0.760188i \(-0.725107\pi\)
−0.382993 + 0.923751i \(0.625107\pi\)
\(12\) −12.1488 − 11.1939i −1.01240 − 0.932825i
\(13\) −2.93860 + 5.76733i −0.226046 + 0.443641i −0.975976 − 0.217876i \(-0.930087\pi\)
0.749930 + 0.661517i \(0.230087\pi\)
\(14\) −0.666523 + 2.20364i −0.0476088 + 0.157403i
\(15\) −24.4985 − 3.88018i −1.63323 − 0.258678i
\(16\) −15.9465 + 1.30685i −0.996659 + 0.0816784i
\(17\) 7.39159 − 1.17071i 0.434799 − 0.0688655i 0.0648018 − 0.997898i \(-0.479358\pi\)
0.369998 + 0.929033i \(0.379358\pi\)
\(18\) −11.1577 + 11.6235i −0.619872 + 0.645747i
\(19\) −8.33754 + 4.24819i −0.438818 + 0.223589i −0.659420 − 0.751775i \(-0.729198\pi\)
0.220602 + 0.975364i \(0.429198\pi\)
\(20\) −18.8422 + 14.9034i −0.942109 + 0.745169i
\(21\) 4.52131 + 1.46906i 0.215300 + 0.0699554i
\(22\) 18.8819 + 13.1373i 0.858269 + 0.597151i
\(23\) −10.5722 + 3.43513i −0.459662 + 0.149353i −0.529689 − 0.848192i \(-0.677691\pi\)
0.0700271 + 0.997545i \(0.477691\pi\)
\(24\) 2.02493 + 32.9771i 0.0843720 + 1.37404i
\(25\) −3.42118 + 10.5293i −0.136847 + 0.421172i
\(26\) 12.2277 − 4.25126i 0.470296 − 0.163510i
\(27\) −2.75664 − 2.75664i −0.102098 − 0.102098i
\(28\) 4.01374 − 2.25632i 0.143348 − 0.0805830i
\(29\) −48.6388 − 7.70362i −1.67720 − 0.265642i −0.755954 − 0.654625i \(-0.772827\pi\)
−0.921245 + 0.388983i \(0.872827\pi\)
\(30\) 29.9729 + 39.5291i 0.999098 + 1.31764i
\(31\) 20.1391 − 27.7191i 0.649648 − 0.894164i −0.349436 − 0.936960i \(-0.613627\pi\)
0.999084 + 0.0427964i \(0.0136267\pi\)
\(32\) 24.8181 + 20.2006i 0.775565 + 0.631267i
\(33\) 27.9192 − 38.4275i 0.846036 − 1.16447i
\(34\) −12.2862 − 8.54829i −0.361360 − 0.251420i
\(35\) 3.13867 − 6.15999i 0.0896763 − 0.176000i
\(36\) 32.1972 − 1.31711i 0.894367 − 0.0365864i
\(37\) −31.8448 + 23.1366i −0.860670 + 0.625313i −0.928067 − 0.372413i \(-0.878531\pi\)
0.0673973 + 0.997726i \(0.478531\pi\)
\(38\) 17.9134 + 5.41818i 0.471405 + 0.142584i
\(39\) −8.26067 − 25.4237i −0.211812 − 0.651890i
\(40\) 47.8273 + 4.59363i 1.19568 + 0.114841i
\(41\) −31.8910 − 25.7675i −0.777830 − 0.628475i
\(42\) −4.48880 − 8.38166i −0.106876 − 0.199563i
\(43\) 23.5998 + 72.6328i 0.548833 + 1.68913i 0.711697 + 0.702486i \(0.247927\pi\)
−0.162864 + 0.986649i \(0.552073\pi\)
\(44\) −9.04799 − 45.1065i −0.205636 − 1.02515i
\(45\) 39.1434 − 28.4394i 0.869854 − 0.631986i
\(46\) 20.0116 + 9.68636i 0.435034 + 0.210573i
\(47\) −9.30998 + 18.2719i −0.198085 + 0.388763i −0.968587 − 0.248674i \(-0.920005\pi\)
0.770503 + 0.637437i \(0.220005\pi\)
\(48\) 42.7519 − 50.3847i 0.890665 − 1.04968i
\(49\) 28.0226 − 38.5698i 0.571890 − 0.787139i
\(50\) 19.5194 − 10.4536i 0.390387 − 0.209072i
\(51\) −18.1667 + 25.0043i −0.356210 + 0.490280i
\(52\) −23.5305 − 10.8017i −0.452509 − 0.207724i
\(53\) −28.0843 − 4.44812i −0.529893 − 0.0839268i −0.114245 − 0.993453i \(-0.536445\pi\)
−0.415647 + 0.909526i \(0.636445\pi\)
\(54\) 0.159377 + 7.79533i 0.00295143 + 0.144358i
\(55\) −48.8439 − 48.8439i −0.888071 − 0.888071i
\(56\) −8.91615 − 2.30359i −0.159217 − 0.0411355i
\(57\) 11.9420 − 36.7538i 0.209509 − 0.644804i
\(58\) 59.5077 + 78.4802i 1.02599 + 1.35311i
\(59\) −12.4926 + 4.05908i −0.211738 + 0.0687979i −0.412966 − 0.910747i \(-0.635507\pi\)
0.201227 + 0.979545i \(0.435507\pi\)
\(60\) 11.5024 − 98.5464i 0.191707 − 1.64244i
\(61\) 69.6517 + 22.6312i 1.14183 + 0.371003i 0.818060 − 0.575132i \(-0.195049\pi\)
0.323771 + 0.946136i \(0.395049\pi\)
\(62\) −67.4484 + 12.1010i −1.08788 + 0.195177i
\(63\) −8.26268 + 4.21005i −0.131154 + 0.0668261i
\(64\) −7.83021 − 63.5192i −0.122347 − 0.992487i
\(65\) −38.3967 + 6.08144i −0.590718 + 0.0935606i
\(66\) −93.5050 + 16.7758i −1.41674 + 0.254179i
\(67\) 123.107 + 19.4982i 1.83742 + 0.291019i 0.976145 − 0.217121i \(-0.0696668\pi\)
0.861274 + 0.508140i \(0.169667\pi\)
\(68\) 5.88742 + 29.3503i 0.0865796 + 0.431621i
\(69\) 20.8423 − 40.9054i 0.302063 − 0.592831i
\(70\) −13.0602 + 4.54070i −0.186574 + 0.0648671i
\(71\) 72.6929 − 11.5134i 1.02384 − 0.162161i 0.378139 − 0.925749i \(-0.376564\pi\)
0.645704 + 0.763588i \(0.276564\pi\)
\(72\) −48.2792 − 42.6931i −0.670544 − 0.592960i
\(73\) 88.1394i 1.20739i 0.797216 + 0.603695i \(0.206305\pi\)
−0.797216 + 0.603695i \(0.793695\pi\)
\(74\) 77.9909 + 10.7233i 1.05393 + 0.144909i
\(75\) −20.7577 − 40.7393i −0.276769 − 0.543190i
\(76\) −18.3417 − 32.6277i −0.241338 − 0.429312i
\(77\) 7.78185 + 10.7108i 0.101063 + 0.139101i
\(78\) −23.2934 + 48.1231i −0.298633 + 0.616963i
\(79\) −62.8336 + 62.8336i −0.795362 + 0.795362i −0.982360 − 0.186998i \(-0.940124\pi\)
0.186998 + 0.982360i \(0.440124\pi\)
\(80\) −62.6443 − 72.8691i −0.783053 − 0.910864i
\(81\) 88.6046 1.09388
\(82\) 10.3250 + 81.3474i 0.125914 + 0.992041i
\(83\) 31.9399i 0.384818i 0.981315 + 0.192409i \(0.0616300\pi\)
−0.981315 + 0.192409i \(0.938370\pi\)
\(84\) −5.13214 + 18.3103i −0.0610968 + 0.217980i
\(85\) 31.7821 + 31.7821i 0.373907 + 0.373907i
\(86\) 66.5467 − 137.482i 0.773799 − 1.59863i
\(87\) 164.535 − 119.542i 1.89121 − 1.37405i
\(88\) −49.4183 + 77.6124i −0.561572 + 0.881959i
\(89\) −110.819 + 56.4652i −1.24516 + 0.634440i −0.947355 − 0.320186i \(-0.896254\pi\)
−0.297805 + 0.954627i \(0.596254\pi\)
\(90\) −95.8660 − 13.1810i −1.06518 − 0.146456i
\(91\) 7.45097 0.0818787
\(92\) −15.4575 − 41.6920i −0.168016 − 0.453173i
\(93\) 22.1357 + 139.759i 0.238018 + 1.50279i
\(94\) 38.7394 − 13.4687i 0.412121 − 0.143284i
\(95\) −50.0747 − 25.5143i −0.527102 − 0.268572i
\(96\) −131.467 + 13.4845i −1.36945 + 0.140463i
\(97\) −15.6987 + 99.1174i −0.161842 + 1.02183i 0.764357 + 0.644793i \(0.223057\pi\)
−0.926199 + 0.377036i \(0.876943\pi\)
\(98\) −93.8514 + 16.8379i −0.957667 + 0.171816i
\(99\) 14.4944 + 91.5138i 0.146408 + 0.924382i
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.39.10 ✓ 304
4.3 odd 2 inner 164.3.n.c.39.18 yes 304
41.20 even 20 inner 164.3.n.c.143.18 yes 304
164.143 odd 20 inner 164.3.n.c.143.10 yes 304
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.3.n.c.39.10 ✓ 304 1.1 even 1 trivial
164.3.n.c.39.18 yes 304 4.3 odd 2 inner
164.3.n.c.143.10 yes 304 164.143 odd 20 inner
164.3.n.c.143.18 yes 304 41.20 even 20 inner