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Properties

Label 322.2.k.b.111.2
Level $322$
Weight $2$
Character 322.111
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 111.2
Character \(\chi\) \(=\) 322.111
Dual form 322.2.k.b.293.2

$q$-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+(-0.415415 - 0.909632i) q^{2} +(-0.725956 - 2.47238i) q^{3} +(-0.654861 + 0.755750i) q^{4} +(-2.94826 - 1.89473i) q^{5} +(-1.94738 + 1.68742i) q^{6} +(2.01573 - 1.71372i) q^{7} +(0.959493 + 0.281733i) q^{8} +(-3.06188 + 1.96775i) q^{9} +(-0.498756 + 3.46893i) q^{10} +(-3.67583 - 1.67869i) q^{11} +(2.34390 + 1.07042i) q^{12} +(4.91009 + 0.705965i) q^{13} +(-2.39622 - 1.12167i) q^{14} +(-2.54418 + 8.66469i) q^{15} +(-0.142315 - 0.989821i) q^{16} +(0.871577 + 1.00585i) q^{17} +(3.06188 + 1.96775i) q^{18} +(-4.50188 + 5.19545i) q^{19} +(3.36264 - 0.987359i) q^{20} +(-5.70030 - 3.73956i) q^{21} +4.04100i q^{22} +(2.42305 - 4.13870i) q^{23} -2.57676i q^{24} +(3.02514 + 6.62413i) q^{25} +(-1.39756 - 4.75965i) q^{26} +(1.24567 + 1.07938i) q^{27} +(-0.0248781 + 2.64563i) q^{28} +(2.66889 + 3.08007i) q^{29} +(8.93857 - 1.28517i) q^{30} +(0.659936 - 2.24754i) q^{31} +(-0.841254 + 0.540641i) q^{32} +(-1.48188 + 10.3067i) q^{33} +(0.552890 - 1.21066i) q^{34} +(-9.18992 + 1.23323i) q^{35} +(0.517979 - 3.60262i) q^{36} +(-1.28305 - 1.99646i) q^{37} +(6.59609 + 1.93679i) q^{38} +(-1.81910 - 12.6521i) q^{39} +(-2.29502 - 2.64860i) q^{40} +(0.695827 - 1.08273i) q^{41} +(-1.03363 + 6.73864i) q^{42} +(-1.74309 - 5.93643i) q^{43} +(3.67583 - 1.67869i) q^{44} +12.7556 q^{45} +(-4.77126 - 0.484801i) q^{46} -1.04984i q^{47} +(-2.34390 + 1.07042i) q^{48} +(1.12632 - 6.90879i) q^{49} +(4.76883 - 5.50352i) q^{50} +(1.85412 - 2.88507i) q^{51} +(-3.74896 + 3.24849i) q^{52} +(-13.7020 + 1.97004i) q^{53} +(0.464370 - 1.58150i) q^{54} +(7.65660 + 11.9139i) q^{55} +(2.41689 - 1.07641i) q^{56} +(16.1133 + 7.35869i) q^{57} +(1.69303 - 3.70722i) q^{58} +(-5.70640 - 0.820456i) q^{59} +(-4.88225 - 7.59693i) q^{60} +(-10.6569 - 3.12915i) q^{61} +(-2.31858 + 0.333361i) q^{62} +(-2.79975 + 9.21367i) q^{63} +(0.841254 + 0.540641i) q^{64} +(-13.1386 - 11.3847i) q^{65} +(9.99089 - 2.93359i) q^{66} +(9.86338 - 4.50446i) q^{67} -1.33093 q^{68} +(-11.9915 - 2.98617i) q^{69} +(4.93941 + 7.84714i) q^{70} +(-2.19067 - 4.79691i) q^{71} +(-3.49223 + 1.02541i) q^{72} +(0.539878 + 0.467807i) q^{73} +(-1.28305 + 1.99646i) q^{74} +(14.1812 - 12.2881i) q^{75} +(-0.978352 - 6.80459i) q^{76} +(-10.2863 + 2.91555i) q^{77} +(-10.7531 + 6.91059i) q^{78} +(4.55223 + 0.654512i) q^{79} +(-1.45586 + 3.18789i) q^{80} +(-2.77158 + 6.06891i) q^{81} +(-1.27394 - 0.183165i) q^{82} +(2.51221 - 1.61450i) q^{83} +(6.55907 - 1.85911i) q^{84} +(-0.663812 - 4.61691i) q^{85} +(-4.67586 + 4.05165i) q^{86} +(5.67759 - 8.83450i) q^{87} +(-3.05399 - 2.64629i) q^{88} +(15.8500 - 4.65398i) q^{89} +(-5.29885 - 11.6029i) q^{90} +(11.1072 - 6.99149i) q^{91} +(1.54106 + 4.54149i) q^{92} -6.03585 q^{93} +(-0.954966 + 0.436119i) q^{94} +(23.1167 - 6.78766i) q^{95} +(1.94738 + 1.68742i) q^{96} +(-4.68834 - 3.01301i) q^{97} +(-6.75235 + 1.84547i) q^{98} +(14.5582 - 2.09315i) 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 \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-1\) \(e\left(\frac{15}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.415415 − 0.909632i −0.293743 − 0.643207i
\(3\) −0.725956 − 2.47238i −0.419131 − 1.42743i −0.850844 − 0.525418i \(-0.823909\pi\)
0.431714 − 0.902011i \(-0.357909\pi\)
\(4\) −0.654861 + 0.755750i −0.327430 + 0.377875i
\(5\) −2.94826 − 1.89473i −1.31850 − 0.847348i −0.323404 − 0.946261i \(-0.604827\pi\)
−0.995096 + 0.0989128i \(0.968464\pi\)
\(6\) −1.94738 + 1.68742i −0.795015 + 0.688885i
\(7\) 2.01573 − 1.71372i 0.761874 − 0.647725i
\(8\) 0.959493 + 0.281733i 0.339232 + 0.0996075i
\(9\) −3.06188 + 1.96775i −1.02063 + 0.655917i
\(10\) −0.498756 + 3.46893i −0.157721 + 1.09697i
\(11\) −3.67583 − 1.67869i −1.10830 − 0.506145i −0.224722 − 0.974423i \(-0.572147\pi\)
−0.883581 + 0.468278i \(0.844875\pi\)
\(12\) 2.34390 + 1.07042i 0.676625 + 0.309004i
\(13\) 4.91009 + 0.705965i 1.36181 + 0.195799i 0.784199 − 0.620509i \(-0.213074\pi\)
0.577616 + 0.816309i \(0.303983\pi\)
\(14\) −2.39622 − 1.12167i −0.640416 − 0.299778i
\(15\) −2.54418 + 8.66469i −0.656905 + 2.23721i
\(16\) −0.142315 − 0.989821i −0.0355787 − 0.247455i
\(17\) 0.871577 + 1.00585i 0.211388 + 0.243955i 0.851535 − 0.524297i \(-0.175672\pi\)
−0.640147 + 0.768253i \(0.721126\pi\)
\(18\) 3.06188 + 1.96775i 0.721693 + 0.463804i
\(19\) −4.50188 + 5.19545i −1.03280 + 1.19192i −0.0516530 + 0.998665i \(0.516449\pi\)
−0.981149 + 0.193252i \(0.938096\pi\)
\(20\) 3.36264 − 0.987359i 0.751908 − 0.220780i
\(21\) −5.70030 − 3.73956i −1.24391 − 0.816039i
\(22\) 4.04100i 0.861545i
\(23\) 2.42305 − 4.13870i 0.505240 − 0.862979i
\(24\) − 2.57676i − 0.525978i
\(25\) 3.02514 + 6.62413i 0.605027 + 1.32483i
\(26\) −1.39756 − 4.75965i −0.274084 − 0.933443i
\(27\) 1.24567 + 1.07938i 0.239730 + 0.207727i
\(28\) −0.0248781 + 2.64563i −0.00470151 + 0.499978i
\(29\) 2.66889 + 3.08007i 0.495601 + 0.571954i 0.947354 − 0.320189i \(-0.103746\pi\)
−0.451753 + 0.892143i \(0.649201\pi\)
\(30\) 8.93857 − 1.28517i 1.63195 − 0.234639i
\(31\) 0.659936 − 2.24754i 0.118528 − 0.403670i −0.878760 − 0.477264i \(-0.841629\pi\)
0.997288 + 0.0735940i \(0.0234469\pi\)
\(32\) −0.841254 + 0.540641i −0.148714 + 0.0955727i
\(33\) −1.48188 + 10.3067i −0.257962 + 1.79416i
\(34\) 0.552890 − 1.21066i 0.0948199 − 0.207627i
\(35\) −9.18992 + 1.23323i −1.55338 + 0.208453i
\(36\) 0.517979 − 3.60262i 0.0863298 − 0.600437i
\(37\) −1.28305 − 1.99646i −0.210932 − 0.328216i 0.719626 − 0.694362i \(-0.244313\pi\)
−0.930557 + 0.366146i \(0.880677\pi\)
\(38\) 6.59609 + 1.93679i 1.07003 + 0.314188i
\(39\) −1.81910 − 12.6521i −0.291289 − 2.02596i
\(40\) −2.29502 − 2.64860i −0.362875 − 0.418780i
\(41\) 0.695827 − 1.08273i 0.108670 − 0.169094i −0.782659 − 0.622450i \(-0.786137\pi\)
0.891329 + 0.453357i \(0.149774\pi\)
\(42\) −1.03363 + 6.73864i −0.159493 + 1.03979i
\(43\) −1.74309 − 5.93643i −0.265819 − 0.905296i −0.978922 − 0.204237i \(-0.934529\pi\)
0.713103 − 0.701060i \(-0.247289\pi\)
\(44\) 3.67583 − 1.67869i 0.554152 − 0.253073i
\(45\) 12.7556 1.90149
\(46\) −4.77126 − 0.484801i −0.703485 − 0.0714800i
\(47\) − 1.04984i − 0.153135i −0.997064 − 0.0765673i \(-0.975604\pi\)
0.997064 − 0.0765673i \(-0.0243960\pi\)
\(48\) −2.34390 + 1.07042i −0.338313 + 0.154502i
\(49\) 1.12632 − 6.90879i 0.160903 − 0.986970i
\(50\) 4.76883 − 5.50352i 0.674414 − 0.778316i
\(51\) 1.85412 − 2.88507i 0.259629 − 0.403991i
\(52\) −3.74896 + 3.24849i −0.519887 + 0.450485i
\(53\) −13.7020 + 1.97004i −1.88211 + 0.270606i −0.985174 − 0.171556i \(-0.945121\pi\)
−0.896935 + 0.442163i \(0.854211\pi\)
\(54\) 0.464370 − 1.58150i 0.0631927 − 0.215215i
\(55\) 7.65660 + 11.9139i 1.03242 + 1.60647i
\(56\) 2.41689 − 1.07641i 0.322970 − 0.143841i
\(57\) 16.1133 + 7.35869i 2.13426 + 0.974682i
\(58\) 1.69303 − 3.70722i 0.222306 − 0.486781i
\(59\) −5.70640 − 0.820456i −0.742909 − 0.106814i −0.239541 − 0.970886i \(-0.576997\pi\)
−0.503368 + 0.864072i \(0.667906\pi\)
\(60\) −4.88225 − 7.59693i −0.630296 − 0.980760i
\(61\) −10.6569 − 3.12915i −1.36448 − 0.400647i −0.484138 − 0.874992i \(-0.660867\pi\)
−0.880339 + 0.474345i \(0.842685\pi\)
\(62\) −2.31858 + 0.333361i −0.294460 + 0.0423369i
\(63\) −2.79975 + 9.21367i −0.352735 + 1.16081i
\(64\) 0.841254 + 0.540641i 0.105157 + 0.0675801i
\(65\) −13.1386 − 11.3847i −1.62964 − 1.41209i
\(66\) 9.99089 − 2.93359i 1.22979 − 0.361100i
\(67\) 9.86338 − 4.50446i 1.20500 − 0.550307i 0.291278 − 0.956638i \(-0.405919\pi\)
0.913726 + 0.406331i \(0.133192\pi\)
\(68\) −1.33093 −0.161400
\(69\) −11.9915 − 2.98617i −1.44360 − 0.359493i
\(70\) 4.93941 + 7.84714i 0.590373 + 0.937913i
\(71\) −2.19067 − 4.79691i −0.259985 − 0.569288i 0.733957 − 0.679196i \(-0.237672\pi\)
−0.993942 + 0.109908i \(0.964944\pi\)
\(72\) −3.49223 + 1.02541i −0.411564 + 0.120846i
\(73\) 0.539878 + 0.467807i 0.0631880 + 0.0547527i 0.685882 − 0.727713i \(-0.259416\pi\)
−0.622694 + 0.782465i \(0.713962\pi\)
\(74\) −1.28305 + 1.99646i −0.149151 + 0.232084i
\(75\) 14.1812 − 12.2881i 1.63751 − 1.41891i
\(76\) −0.978352 − 6.80459i −0.112225 − 0.780540i
\(77\) −10.2863 + 2.91555i −1.17223 + 0.332257i
\(78\) −10.7531 + 6.91059i −1.21755 + 0.782470i
\(79\) 4.55223 + 0.654512i 0.512166 + 0.0736384i 0.393552 − 0.919302i \(-0.371246\pi\)
0.118614 + 0.992940i \(0.462155\pi\)
\(80\) −1.45586 + 3.18789i −0.162770 + 0.356417i
\(81\) −2.77158 + 6.06891i −0.307953 + 0.674323i
\(82\) −1.27394 − 0.183165i −0.140683 − 0.0202272i
\(83\) 2.51221 − 1.61450i 0.275751 − 0.177214i −0.395455 − 0.918485i \(-0.629413\pi\)
0.671206 + 0.741271i \(0.265777\pi\)
\(84\) 6.55907 − 1.85911i 0.715653 − 0.202845i
\(85\) −0.663812 − 4.61691i −0.0720005 − 0.500775i
\(86\) −4.67586 + 4.05165i −0.504211 + 0.436901i
\(87\) 5.67759 − 8.83450i 0.608702 − 0.947158i
\(88\) −3.05399 − 2.64629i −0.325556 − 0.282096i
\(89\) 15.8500 − 4.65398i 1.68010 − 0.493321i 0.703916 − 0.710283i \(-0.251433\pi\)
0.976180 + 0.216962i \(0.0696148\pi\)
\(90\) −5.29885 − 11.6029i −0.558548 − 1.22305i
\(91\) 11.1072 − 6.99149i 1.16436 − 0.732908i
\(92\) 1.54106 + 4.54149i 0.160667 + 0.473483i
\(93\) −6.03585 −0.625888
\(94\) −0.954966 + 0.436119i −0.0984973 + 0.0449822i
\(95\) 23.1167 − 6.78766i 2.37172 − 0.696399i
\(96\) 1.94738 + 1.68742i 0.198754 + 0.172221i
\(97\) −4.68834 − 3.01301i −0.476029 − 0.305925i 0.280536 − 0.959843i \(-0.409488\pi\)
−0.756565 + 0.653918i \(0.773124\pi\)
\(98\) −6.75235 + 1.84547i −0.682090 + 0.186421i
\(99\) 14.5582 − 2.09315i 1.46315 − 0.210370i
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.111.2 ✓ 80
7.6 odd 2 inner 322.2.k.b.111.7 yes 80
23.17 odd 22 inner 322.2.k.b.293.7 yes 80
161.132 even 22 inner 322.2.k.b.293.2 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.k.b.111.2 ✓ 80 1.1 even 1 trivial
322.2.k.b.111.7 yes 80 7.6 odd 2 inner
322.2.k.b.293.2 yes 80 161.132 even 22 inner
322.2.k.b.293.7 yes 80 23.17 odd 22 inner