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Properties

Label 322.2.k.b.83.1
Level $322$
Weight $2$
Character 322.83
Analytic conductor $2.571$
Analytic rank $0$
Dimension $80$
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] = [322,2,Mod(83,322)] 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("322.83"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 21])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.k (of order \(22\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [80,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 83.1
Character \(\chi\) \(=\) 322.83
Dual form 322.2.k.b.97.1

$q$-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+(0.959493 - 0.281733i) q^{2} +(-2.27797 + 1.97387i) q^{3} +(0.841254 - 0.540641i) q^{4} +(-0.347463 - 2.41666i) q^{5} +(-1.62959 + 2.53570i) q^{6} +(-2.04477 - 1.67896i) q^{7} +(0.654861 - 0.755750i) q^{8} +(0.866033 - 6.02339i) q^{9} +(-1.01424 - 2.22088i) q^{10} +(0.301971 - 1.02842i) q^{11} +(-0.849195 + 2.89209i) q^{12} +(3.13129 - 1.43001i) q^{13} +(-2.43496 - 1.03487i) q^{14} +(5.56169 + 4.81924i) q^{15} +(0.415415 - 0.909632i) q^{16} +(-4.15199 - 2.66832i) q^{17} +(-0.866033 - 6.02339i) q^{18} +(2.76601 - 1.77760i) q^{19} +(-1.59885 - 1.84517i) q^{20} +(7.97199 - 0.211495i) q^{21} -1.07184i q^{22} +(1.90833 + 4.39980i) q^{23} +3.01419i q^{24} +(-0.922053 + 0.270739i) q^{25} +(2.60157 - 2.25428i) q^{26} +(5.02784 + 7.82347i) q^{27} +(-2.62789 - 0.306945i) q^{28} +(-6.30313 - 4.05078i) q^{29} +(6.69414 + 3.05711i) q^{30} +(-4.67510 - 4.05100i) q^{31} +(0.142315 - 0.989821i) q^{32} +(1.34209 + 2.93876i) q^{33} +(-4.73556 - 1.39049i) q^{34} +(-3.34700 + 5.52490i) q^{35} +(-2.52794 - 5.53541i) q^{36} +(-9.12956 - 1.31263i) q^{37} +(2.15316 - 2.48487i) q^{38} +(-4.31033 + 9.43831i) q^{39} +(-2.05393 - 1.31998i) q^{40} +(8.07379 - 1.16084i) q^{41} +(7.58948 - 2.44890i) q^{42} +(2.46022 - 2.13179i) q^{43} +(-0.301971 - 1.02842i) q^{44} -14.8574 q^{45} +(3.07060 + 3.68394i) q^{46} +6.40308i q^{47} +(0.849195 + 2.89209i) q^{48} +(1.36218 + 6.86618i) q^{49} +(-0.808427 + 0.519545i) q^{50} +(14.7250 - 2.11714i) q^{51} +(1.86109 - 2.89591i) q^{52} +(4.57257 + 2.08822i) q^{53} +(7.02830 + 6.09006i) q^{54} +(-2.59027 - 0.372424i) q^{55} +(-2.60791 + 0.445849i) q^{56} +(-2.79212 + 9.50908i) q^{57} +(-7.18905 - 2.11089i) q^{58} +(-9.82076 + 4.48499i) q^{59} +(7.28427 + 1.04732i) q^{60} +(6.33716 - 7.31348i) q^{61} +(-5.62703 - 2.56978i) q^{62} +(-11.8839 + 10.8624i) q^{63} +(-0.142315 - 0.989821i) q^{64} +(-4.54387 - 7.07040i) q^{65} +(2.11567 + 2.44161i) q^{66} +(1.52848 + 5.20553i) q^{67} -4.93548 q^{68} +(-13.0318 - 6.25583i) q^{69} +(-1.65488 + 6.24406i) q^{70} +(-3.54057 + 1.03961i) q^{71} +(-3.98505 - 4.59899i) q^{72} +(6.55646 + 10.2021i) q^{73} +(-9.12956 + 1.31263i) q^{74} +(1.56601 - 2.43675i) q^{75} +(1.36587 - 2.99083i) q^{76} +(-2.34414 + 1.59589i) q^{77} +(-1.47665 + 10.2704i) q^{78} +(13.4345 - 6.13532i) q^{79} +(-2.34261 - 0.687853i) q^{80} +(-9.37931 - 2.75401i) q^{81} +(7.41970 - 3.38846i) q^{82} +(0.204037 - 1.41911i) q^{83} +(6.59212 - 4.48790i) q^{84} +(-5.00576 + 10.9611i) q^{85} +(1.75997 - 2.73856i) q^{86} +(22.3541 - 3.21403i) q^{87} +(-0.579479 - 0.901687i) q^{88} +(4.23728 + 4.89008i) q^{89} +(-14.2556 + 4.18582i) q^{90} +(-8.80372 - 2.33327i) q^{91} +(3.98410 + 2.66963i) q^{92} +18.6459 q^{93} +(1.80396 + 6.14371i) q^{94} +(-5.25695 - 6.06685i) q^{95} +(1.62959 + 2.53570i) q^{96} +(-0.0499051 - 0.347098i) q^{97} +(3.24143 + 6.20428i) q^{98} +(-5.93306 - 2.70954i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 8 q^{2} - 8 q^{4} + 11 q^{7} + 8 q^{8} + 14 q^{9} - 11 q^{14} - 8 q^{16} - 14 q^{18} - 33 q^{21} + 18 q^{23} + 6 q^{25} - 11 q^{28} + 8 q^{29} + 22 q^{30} + 8 q^{32} - 33 q^{35} - 8 q^{36} - 22 q^{37}+ \cdots + 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1\) \(e\left(\frac{21}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.959493 − 0.281733i 0.678464 − 0.199215i
\(3\) −2.27797 + 1.97387i −1.31519 + 1.13962i −0.334854 + 0.942270i \(0.608687\pi\)
−0.980334 + 0.197346i \(0.936768\pi\)
\(4\) 0.841254 − 0.540641i 0.420627 − 0.270320i
\(5\) −0.347463 − 2.41666i −0.155390 − 1.08076i −0.906992 − 0.421147i \(-0.861628\pi\)
0.751602 − 0.659617i \(-0.229281\pi\)
\(6\) −1.62959 + 2.53570i −0.665279 + 1.03519i
\(7\) −2.04477 − 1.67896i −0.772851 − 0.634588i
\(8\) 0.654861 − 0.755750i 0.231528 − 0.267198i
\(9\) 0.866033 − 6.02339i 0.288678 − 2.00780i
\(10\) −1.01424 − 2.22088i −0.320731 − 0.702303i
\(11\) 0.301971 − 1.02842i 0.0910478 − 0.310080i −0.901360 − 0.433070i \(-0.857430\pi\)
0.992408 + 0.122990i \(0.0392483\pi\)
\(12\) −0.849195 + 2.89209i −0.245141 + 0.834875i
\(13\) 3.13129 − 1.43001i 0.868465 − 0.396615i 0.0692067 − 0.997602i \(-0.477953\pi\)
0.799258 + 0.600988i \(0.205226\pi\)
\(14\) −2.43496 − 1.03487i −0.650771 − 0.276581i
\(15\) 5.56169 + 4.81924i 1.43602 + 1.24432i
\(16\) 0.415415 − 0.909632i 0.103854 − 0.227408i
\(17\) −4.15199 − 2.66832i −1.00701 − 0.647163i −0.0703891 − 0.997520i \(-0.522424\pi\)
−0.936616 + 0.350356i \(0.886060\pi\)
\(18\) −0.866033 − 6.02339i −0.204126 − 1.41973i
\(19\) 2.76601 − 1.77760i 0.634566 − 0.407811i −0.183432 − 0.983032i \(-0.558721\pi\)
0.817997 + 0.575222i \(0.195084\pi\)
\(20\) −1.59885 − 1.84517i −0.357514 − 0.412593i
\(21\) 7.97199 − 0.211495i 1.73963 − 0.0461520i
\(22\) − 1.07184i − 0.228516i
\(23\) 1.90833 + 4.39980i 0.397914 + 0.917423i
\(24\) 3.01419i 0.615269i
\(25\) −0.922053 + 0.270739i −0.184411 + 0.0541478i
\(26\) 2.60157 − 2.25428i 0.510211 − 0.442100i
\(27\) 5.02784 + 7.82347i 0.967608 + 1.50563i
\(28\) −2.62789 − 0.306945i −0.496624 − 0.0580071i
\(29\) −6.30313 − 4.05078i −1.17046 − 0.752211i −0.196853 − 0.980433i \(-0.563072\pi\)
−0.973609 + 0.228222i \(0.926709\pi\)
\(30\) 6.69414 + 3.05711i 1.22218 + 0.558150i
\(31\) −4.67510 − 4.05100i −0.839673 − 0.727581i 0.124681 − 0.992197i \(-0.460209\pi\)
−0.964354 + 0.264616i \(0.914755\pi\)
\(32\) 0.142315 − 0.989821i 0.0251579 − 0.174977i
\(33\) 1.34209 + 2.93876i 0.233628 + 0.511573i
\(34\) −4.73556 − 1.39049i −0.812142 − 0.238466i
\(35\) −3.34700 + 5.52490i −0.565746 + 0.933878i
\(36\) −2.52794 − 5.53541i −0.421323 − 0.922569i
\(37\) −9.12956 − 1.31263i −1.50089 − 0.215796i −0.657647 − 0.753326i \(-0.728448\pi\)
−0.843244 + 0.537531i \(0.819357\pi\)
\(38\) 2.15316 − 2.48487i 0.349288 − 0.403100i
\(39\) −4.31033 + 9.43831i −0.690206 + 1.51134i
\(40\) −2.05393 − 1.31998i −0.324755 − 0.208707i
\(41\) 8.07379 − 1.16084i 1.26091 − 0.181292i 0.520745 − 0.853712i \(-0.325654\pi\)
0.740170 + 0.672420i \(0.234745\pi\)
\(42\) 7.58948 − 2.44890i 1.17108 − 0.377873i
\(43\) 2.46022 − 2.13179i 0.375179 − 0.325095i −0.446777 − 0.894645i \(-0.647428\pi\)
0.821956 + 0.569551i \(0.192883\pi\)
\(44\) −0.301971 − 1.02842i −0.0455239 − 0.155040i
\(45\) −14.8574 −2.21481
\(46\) 3.07060 + 3.68394i 0.452735 + 0.543168i
\(47\) 6.40308i 0.933985i 0.884261 + 0.466992i \(0.154662\pi\)
−0.884261 + 0.466992i \(0.845338\pi\)
\(48\) 0.849195 + 2.89209i 0.122571 + 0.417438i
\(49\) 1.36218 + 6.86618i 0.194597 + 0.980883i
\(50\) −0.808427 + 0.519545i −0.114329 + 0.0734747i
\(51\) 14.7250 − 2.11714i 2.06192 − 0.296459i
\(52\) 1.86109 − 2.89591i 0.258087 − 0.401591i
\(53\) 4.57257 + 2.08822i 0.628091 + 0.286840i 0.703918 − 0.710281i \(-0.251432\pi\)
−0.0758267 + 0.997121i \(0.524160\pi\)
\(54\) 7.02830 + 6.09006i 0.956431 + 0.828752i
\(55\) −2.59027 − 0.372424i −0.349271 − 0.0502176i
\(56\) −2.60791 + 0.445849i −0.348497 + 0.0595791i
\(57\) −2.79212 + 9.50908i −0.369825 + 1.25951i
\(58\) −7.18905 − 2.11089i −0.943968 − 0.277174i
\(59\) −9.82076 + 4.48499i −1.27855 + 0.583896i −0.934804 − 0.355165i \(-0.884425\pi\)
−0.343750 + 0.939061i \(0.611697\pi\)
\(60\) 7.28427 + 1.04732i 0.940395 + 0.135208i
\(61\) 6.33716 − 7.31348i 0.811391 − 0.936395i −0.187557 − 0.982254i \(-0.560057\pi\)
0.998948 + 0.0458589i \(0.0146025\pi\)
\(62\) −5.62703 − 2.56978i −0.714633 − 0.326362i
\(63\) −11.8839 + 10.8624i −1.49723 + 1.36854i
\(64\) −0.142315 − 0.989821i −0.0177894 − 0.123728i
\(65\) −4.54387 − 7.07040i −0.563598 − 0.876975i
\(66\) 2.11567 + 2.44161i 0.260421 + 0.300542i
\(67\) 1.52848 + 5.20553i 0.186734 + 0.635957i 0.998638 + 0.0521666i \(0.0166127\pi\)
−0.811905 + 0.583790i \(0.801569\pi\)
\(68\) −4.93548 −0.598515
\(69\) −13.0318 − 6.25583i −1.56884 − 0.753113i
\(70\) −1.65488 + 6.24406i −0.197795 + 0.746307i
\(71\) −3.54057 + 1.03961i −0.420189 + 0.123378i −0.484990 − 0.874520i \(-0.661177\pi\)
0.0648015 + 0.997898i \(0.479359\pi\)
\(72\) −3.98505 − 4.59899i −0.469642 − 0.541996i
\(73\) 6.55646 + 10.2021i 0.767376 + 1.19406i 0.976362 + 0.216142i \(0.0693474\pi\)
−0.208986 + 0.977919i \(0.567016\pi\)
\(74\) −9.12956 + 1.31263i −1.06129 + 0.152590i
\(75\) 1.56601 − 2.43675i 0.180827 − 0.281372i
\(76\) 1.36587 − 2.99083i 0.156676 − 0.343072i
\(77\) −2.34414 + 1.59589i −0.267139 + 0.181868i
\(78\) −1.47665 + 10.2704i −0.167198 + 1.16289i
\(79\) 13.4345 − 6.13532i 1.51150 − 0.690278i 0.524559 − 0.851374i \(-0.324230\pi\)
0.986939 + 0.161097i \(0.0515031\pi\)
\(80\) −2.34261 − 0.687853i −0.261912 − 0.0769043i
\(81\) −9.37931 − 2.75401i −1.04215 − 0.306001i
\(82\) 7.41970 − 3.38846i 0.819369 − 0.374193i
\(83\) 0.204037 − 1.41911i 0.0223960 − 0.155767i −0.975555 − 0.219757i \(-0.929473\pi\)
0.997951 + 0.0639899i \(0.0203825\pi\)
\(84\) 6.59212 − 4.48790i 0.719259 − 0.489670i
\(85\) −5.00576 + 10.9611i −0.542951 + 1.18890i
\(86\) 1.75997 − 2.73856i 0.189782 − 0.295307i
\(87\) 22.3541 − 3.21403i 2.39661 − 0.344580i
\(88\) −0.579479 − 0.901687i −0.0617726 − 0.0961201i
\(89\) 4.23728 + 4.89008i 0.449151 + 0.518348i 0.934495 − 0.355975i \(-0.115851\pi\)
−0.485345 + 0.874323i \(0.661306\pi\)
\(90\) −14.2556 + 4.18582i −1.50267 + 0.441224i
\(91\) −8.80372 − 2.33327i −0.922881 − 0.244593i
\(92\) 3.98410 + 2.66963i 0.415371 + 0.278328i
\(93\) 18.6459 1.93349
\(94\) 1.80396 + 6.14371i 0.186064 + 0.633675i
\(95\) −5.25695 − 6.06685i −0.539352 − 0.622445i
\(96\) 1.62959 + 2.53570i 0.166320 + 0.258798i
\(97\) −0.0499051 − 0.347098i −0.00506710 − 0.0352425i 0.987131 − 0.159914i \(-0.0511215\pi\)
−0.992198 + 0.124671i \(0.960212\pi\)
\(98\) 3.24143 + 6.20428i 0.327434 + 0.626727i
\(99\) −5.93306 − 2.70954i −0.596295 − 0.272319i
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 322.2.k.b.83.1 ✓ 80
7.6 odd 2 inner 322.2.k.b.83.8 yes 80
23.5 odd 22 inner 322.2.k.b.97.8 yes 80
161.97 even 22 inner 322.2.k.b.97.1 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.83.1 ✓ 80 1.1 even 1 trivial
322.2.k.b.83.8 yes 80 7.6 odd 2 inner
322.2.k.b.97.1 yes 80 161.97 even 22 inner
322.2.k.b.97.8 yes 80 23.5 odd 22 inner