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Properties

Label 322.2.k.b.97.8
Level $322$
Weight $2$
Character 322.97
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 97.8
Character \(\chi\) \(=\) 322.97
Dual form 322.2.k.b.83.8

$q$-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} +(-2.60791 + 0.445849i) 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} +(-2.62789 - 0.306945i) 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} +(-6.82081 - 4.13206i) 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} +(-2.43496 - 1.03487i) 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} +(0.171312 + 6.45736i) 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} +(-5.38038 - 5.88633i) 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} +(6.60244 - 2.32547i) 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} +(-2.04477 - 1.67896i) 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} +(-4.94407 - 15.3224i) 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} +(-1.24604 - 2.54740i) 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} +(-3.50407 - 7.16372i) 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} +(6.99015 - 0.371155i) 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 \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

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

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 0.980334 + 0.197346i \(0.0632324\pi\)
0.334854 + 0.942270i \(0.391313\pi\)
\(4\) 0.841254 + 0.540641i 0.420627 + 0.270320i
\(5\) 0.347463 − 2.41666i 0.155390 − 1.08076i −0.751602 − 0.659617i \(-0.770719\pi\)
0.906992 − 0.421147i \(-0.138372\pi\)
\(6\) 1.62959 + 2.53570i 0.665279 + 1.03519i
\(7\) −2.60791 + 0.445849i −0.985699 + 0.168515i
\(8\) 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 0.992408 − 0.122990i \(-0.0392483\pi\)
−0.901360 + 0.433070i \(0.857430\pi\)
\(12\) 0.849195 + 2.89209i 0.245141 + 0.834875i
\(13\) −3.13129 − 1.43001i −0.868465 − 0.396615i −0.0692067 − 0.997602i \(-0.522047\pi\)
−0.799258 + 0.600988i \(0.794774\pi\)
\(14\) −2.62789 − 0.306945i −0.702332 − 0.0820345i
\(15\) 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 \(-0.477576\pi\)
0.936616 + 0.350356i \(0.113940\pi\)
\(18\) −0.866033 + 6.02339i −0.204126 + 1.41973i
\(19\) −2.76601 − 1.77760i −0.634566 − 0.407811i 0.183432 − 0.983032i \(-0.441279\pi\)
−0.817997 + 0.575222i \(0.804916\pi\)
\(20\) 1.59885 − 1.84517i 0.357514 − 0.412593i
\(21\) −6.82081 − 4.13206i −1.48842 − 0.901690i
\(22\) 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
\(28\) −2.43496 − 1.03487i −0.460165 − 0.195573i
\(29\) −6.30313 + 4.05078i −1.17046 + 0.752211i −0.973609 − 0.228222i \(-0.926709\pi\)
−0.196853 + 0.980433i \(0.563072\pi\)
\(30\) 6.69414 − 3.05711i 1.22218 − 0.558150i
\(31\) 4.67510 − 4.05100i 0.839673 − 0.727581i −0.124681 − 0.992197i \(-0.539791\pi\)
0.964354 + 0.264616i \(0.0852452\pi\)
\(32\) 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
\(35\) 0.171312 + 6.45736i 0.0289570 + 1.09149i
\(36\) −2.52794 + 5.53541i −0.421323 + 0.922569i
\(37\) −9.12956 + 1.31263i −1.50089 + 0.215796i −0.843244 − 0.537531i \(-0.819357\pi\)
−0.657647 + 0.753326i \(0.728448\pi\)
\(38\) −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 \(-0.674346\pi\)
−0.740170 + 0.672420i \(0.765255\pi\)
\(42\) −5.38038 − 5.88633i −0.830211 − 0.908280i
\(43\) 2.46022 + 2.13179i 0.375179 + 0.325095i 0.821956 − 0.569551i \(-0.192883\pi\)
−0.446777 + 0.894645i \(0.647428\pi\)
\(44\) −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
\(49\) 6.60244 − 2.32547i 0.943205 − 0.332211i
\(50\) −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 −0.0758267 − 0.997121i \(-0.524160\pi\)
0.703918 + 0.710281i \(0.251432\pi\)
\(54\) −7.02830 + 6.09006i −0.956431 + 0.828752i
\(55\) 2.59027 − 0.372424i 0.349271 − 0.0502176i
\(56\) −2.04477 − 1.67896i −0.273244 − 0.224361i
\(57\) −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 \(-0.115575\pi\)
0.343750 + 0.939061i \(0.388303\pi\)
\(60\) 7.28427 − 1.04732i 0.940395 − 0.135208i
\(61\) −6.33716 − 7.31348i −0.811391 − 0.936395i 0.187557 − 0.982254i \(-0.439943\pi\)
−0.998948 + 0.0458589i \(0.985398\pi\)
\(62\) 5.62703 − 2.56978i 0.714633 − 0.326362i
\(63\) −4.94407 − 15.3224i −0.622894 − 1.93044i
\(64\) −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 −0.811905 − 0.583790i \(-0.801569\pi\)
0.998638 − 0.0521666i \(-0.0166127\pi\)
\(68\) 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 0.0648015 − 0.997898i \(-0.479359\pi\)
−0.484990 + 0.874520i \(0.661177\pi\)
\(72\) −3.98505 + 4.59899i −0.469642 + 0.541996i
\(73\) −6.55646 + 10.2021i −0.767376 + 1.19406i 0.208986 + 0.977919i \(0.432984\pi\)
−0.976362 + 0.216142i \(0.930653\pi\)
\(74\) −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
\(77\) −1.24604 − 2.54740i −0.141999 − 0.290303i
\(78\) −1.47665 − 10.2704i −0.167198 − 1.16289i
\(79\) 13.4345 + 6.13532i 1.51150 + 0.690278i 0.986939 − 0.161097i \(-0.0515031\pi\)
0.524559 + 0.851374i \(0.324230\pi\)
\(80\) 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 \(-0.0705266\pi\)
−0.997951 + 0.0639899i \(0.979617\pi\)
\(84\) −3.50407 − 7.16372i −0.382325 − 0.781626i
\(85\) −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 \(-0.884149\pi\)
0.485345 + 0.874323i \(0.338694\pi\)
\(90\) 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 \(-0.948878\pi\)
0.992198 + 0.124671i \(0.0397876\pi\)
\(98\) 6.99015 − 0.371155i 0.706112 − 0.0374924i
\(99\) −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.97.8 yes 80
7.6 odd 2 inner 322.2.k.b.97.1 yes 80
23.14 odd 22 inner 322.2.k.b.83.1 ✓ 80
161.83 even 22 inner 322.2.k.b.83.8 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.83.1 ✓ 80 23.14 odd 22 inner
322.2.k.b.83.8 yes 80 161.83 even 22 inner
322.2.k.b.97.1 yes 80 7.6 odd 2 inner
322.2.k.b.97.8 yes 80 1.1 even 1 trivial