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Properties

Label 322.2.i.d.141.4
Level $322$
Weight $2$
Character 322.141
Analytic conductor $2.571$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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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(29,322)] 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("322.29"); 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([0, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(4\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 141.4
Character \(\chi\) \(=\) 322.141
Dual form 322.2.i.d.169.4

$q$-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+(-0.142315 - 0.989821i) q^{2} +(1.27568 - 2.79334i) q^{3} +(-0.959493 + 0.281733i) q^{4} +(0.685739 - 0.791385i) q^{5} +(-2.94645 - 0.865157i) q^{6} +(-0.841254 + 0.540641i) q^{7} +(0.415415 + 0.909632i) q^{8} +(-4.21081 - 4.85954i) q^{9} +(-0.880921 - 0.566133i) q^{10} +(0.0338493 - 0.235427i) q^{11} +(-0.437027 + 3.03959i) q^{12} +(0.199918 + 0.128480i) q^{13} +(0.654861 + 0.755750i) q^{14} +(-1.33583 - 2.92505i) q^{15} +(0.841254 - 0.540641i) q^{16} +(0.132620 + 0.0389409i) q^{17} +(-4.21081 + 4.85954i) q^{18} +(1.12718 - 0.330970i) q^{19} +(-0.435003 + 0.952524i) q^{20} +(0.437027 + 3.03959i) q^{21} -0.237848 q^{22} +(3.62925 - 3.13506i) q^{23} +3.07084 q^{24} +(0.555522 + 3.86374i) q^{25} +(0.0987207 - 0.216168i) q^{26} +(-10.1066 + 2.96756i) q^{27} +(0.654861 - 0.755750i) q^{28} +(2.08205 + 0.611344i) q^{29} +(-2.70517 + 1.73851i) q^{30} +(3.55049 + 7.77448i) q^{31} +(-0.654861 - 0.755750i) q^{32} +(-0.614447 - 0.394881i) q^{33} +(0.0196707 - 0.136812i) q^{34} +(-0.149025 + 1.03649i) q^{35} +(5.40933 + 3.47637i) q^{36} +(-7.11194 - 8.20762i) q^{37} +(-0.488015 - 1.06860i) q^{38} +(0.613919 - 0.394542i) q^{39} +(1.00474 + 0.295017i) q^{40} +(-3.94279 + 4.55022i) q^{41} +(2.94645 - 0.865157i) q^{42} +(1.06930 - 2.34144i) q^{43} +(0.0338493 + 0.235427i) q^{44} -6.73328 q^{45} +(-3.61964 - 3.14614i) q^{46} +8.37101 q^{47} +(-0.437027 - 3.03959i) q^{48} +(0.415415 - 0.909632i) q^{49} +(3.74535 - 1.09973i) q^{50} +(0.277956 - 0.320778i) q^{51} +(-0.228017 - 0.0669519i) q^{52} +(8.46563 - 5.44053i) q^{53} +(4.37568 + 9.58140i) q^{54} +(-0.163102 - 0.188230i) q^{55} +(-0.841254 - 0.540641i) q^{56} +(0.513404 - 3.57080i) q^{57} +(0.308815 - 2.14786i) q^{58} +(2.31594 + 1.48836i) q^{59} +(2.10580 + 2.43022i) q^{60} +(-3.25445 - 7.12625i) q^{61} +(7.19006 - 4.62077i) q^{62} +(6.16962 + 1.81157i) q^{63} +(-0.654861 + 0.755750i) q^{64} +(0.238769 - 0.0701089i) q^{65} +(-0.303417 + 0.664391i) q^{66} +(-0.867612 - 6.03437i) q^{67} -0.138219 q^{68} +(-4.12754 - 14.1370i) q^{69} +1.04715 q^{70} +(1.19790 + 8.33156i) q^{71} +(2.67115 - 5.84901i) q^{72} +(-13.9823 + 4.10557i) q^{73} +(-7.11194 + 8.20762i) q^{74} +(11.5014 + 3.37712i) q^{75} +(-0.988275 + 0.635126i) q^{76} +(0.0988057 + 0.216354i) q^{77} +(-0.477895 - 0.551521i) q^{78} +(11.0310 + 7.08918i) q^{79} +(0.149025 - 1.03649i) q^{80} +(-1.85803 + 12.9229i) q^{81} +(5.06502 + 3.25509i) q^{82} +(2.61067 + 3.01287i) q^{83} +(-1.27568 - 2.79334i) q^{84} +(0.121760 - 0.0782506i) q^{85} +(-2.46979 - 0.725196i) q^{86} +(4.36370 - 5.03598i) q^{87} +(0.228214 - 0.0670096i) q^{88} +(0.706662 - 1.54737i) q^{89} +(0.958246 + 6.66475i) q^{90} -0.237644 q^{91} +(-2.59899 + 4.03054i) q^{92} +26.2460 q^{93} +(-1.19132 - 8.28581i) q^{94} +(0.511026 - 1.11899i) q^{95} +(-2.94645 + 0.865157i) q^{96} +(-3.29593 + 3.80371i) q^{97} +(-0.959493 - 0.281733i) q^{98} +(-1.28660 + 0.826848i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 4 q^{2} - 4 q^{4} + 9 q^{5} + 4 q^{7} - 4 q^{8} - 8 q^{9} - 2 q^{10} + 6 q^{11} + 2 q^{13} + 4 q^{14} - 3 q^{15} - 4 q^{16} - 13 q^{17} - 8 q^{18} - 22 q^{19} - 2 q^{20} - 16 q^{22} - 9 q^{23} + 22 q^{24}+ \cdots + 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(185\) \(281\)
\(\chi(n)\) \(1\) \(e\left(\frac{8}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.142315 − 0.989821i −0.100632 − 0.699909i
\(3\) 1.27568 − 2.79334i 0.736511 − 1.61273i −0.0526980 − 0.998610i \(-0.516782\pi\)
0.789209 − 0.614124i \(-0.210491\pi\)
\(4\) −0.959493 + 0.281733i −0.479746 + 0.140866i
\(5\) 0.685739 − 0.791385i 0.306672 − 0.353918i −0.581404 − 0.813615i \(-0.697496\pi\)
0.888076 + 0.459697i \(0.152042\pi\)
\(6\) −2.94645 − 0.865157i −1.20288 − 0.353199i
\(7\) −0.841254 + 0.540641i −0.317964 + 0.204343i
\(8\) 0.415415 + 0.909632i 0.146871 + 0.321603i
\(9\) −4.21081 − 4.85954i −1.40360 − 1.61985i
\(10\) −0.880921 − 0.566133i −0.278572 − 0.179027i
\(11\) 0.0338493 − 0.235427i 0.0102060 − 0.0709840i −0.984082 − 0.177715i \(-0.943129\pi\)
0.994288 + 0.106731i \(0.0340385\pi\)
\(12\) −0.437027 + 3.03959i −0.126159 + 0.877453i
\(13\) 0.199918 + 0.128480i 0.0554474 + 0.0356339i 0.568071 − 0.822980i \(-0.307690\pi\)
−0.512623 + 0.858614i \(0.671326\pi\)
\(14\) 0.654861 + 0.755750i 0.175019 + 0.201983i
\(15\) −1.33583 − 2.92505i −0.344909 − 0.755245i
\(16\) 0.841254 − 0.540641i 0.210313 − 0.135160i
\(17\) 0.132620 + 0.0389409i 0.0321652 + 0.00944455i 0.297776 − 0.954636i \(-0.403755\pi\)
−0.265610 + 0.964080i \(0.585573\pi\)
\(18\) −4.21081 + 4.85954i −0.992498 + 1.14540i
\(19\) 1.12718 − 0.330970i 0.258593 − 0.0759296i −0.149867 − 0.988706i \(-0.547884\pi\)
0.408459 + 0.912777i \(0.366066\pi\)
\(20\) −0.435003 + 0.952524i −0.0972696 + 0.212991i
\(21\) 0.437027 + 3.03959i 0.0953671 + 0.663292i
\(22\) −0.237848 −0.0507094
\(23\) 3.62925 − 3.13506i 0.756750 − 0.653705i
\(24\) 3.07084 0.626834
\(25\) 0.555522 + 3.86374i 0.111104 + 0.772748i
\(26\) 0.0987207 − 0.216168i 0.0193607 − 0.0423941i
\(27\) −10.1066 + 2.96756i −1.94502 + 0.571108i
\(28\) 0.654861 − 0.755750i 0.123757 − 0.142823i
\(29\) 2.08205 + 0.611344i 0.386626 + 0.113524i 0.469268 − 0.883056i \(-0.344518\pi\)
−0.0826423 + 0.996579i \(0.526336\pi\)
\(30\) −2.70517 + 1.73851i −0.493894 + 0.317407i
\(31\) 3.55049 + 7.77448i 0.637686 + 1.39634i 0.901930 + 0.431882i \(0.142150\pi\)
−0.264243 + 0.964456i \(0.585122\pi\)
\(32\) −0.654861 − 0.755750i −0.115764 − 0.133599i
\(33\) −0.614447 − 0.394881i −0.106962 − 0.0687400i
\(34\) 0.0196707 − 0.136812i 0.00337349 − 0.0234631i
\(35\) −0.149025 + 1.03649i −0.0251899 + 0.175199i
\(36\) 5.40933 + 3.47637i 0.901556 + 0.579395i
\(37\) −7.11194 − 8.20762i −1.16920 − 1.34932i −0.925169 − 0.379555i \(-0.876077\pi\)
−0.244027 − 0.969768i \(-0.578468\pi\)
\(38\) −0.488015 − 1.06860i −0.0791665 − 0.173350i
\(39\) 0.613919 − 0.394542i 0.0983057 − 0.0631772i
\(40\) 1.00474 + 0.295017i 0.158863 + 0.0466463i
\(41\) −3.94279 + 4.55022i −0.615760 + 0.710625i −0.974896 − 0.222661i \(-0.928526\pi\)
0.359136 + 0.933285i \(0.383071\pi\)
\(42\) 2.94645 − 0.865157i 0.454648 − 0.133497i
\(43\) 1.06930 − 2.34144i 0.163067 − 0.357067i −0.810406 − 0.585869i \(-0.800753\pi\)
0.973473 + 0.228802i \(0.0734807\pi\)
\(44\) 0.0338493 + 0.235427i 0.00510298 + 0.0354920i
\(45\) −6.73328 −1.00374
\(46\) −3.61964 − 3.14614i −0.533687 − 0.463873i
\(47\) 8.37101 1.22104 0.610519 − 0.792002i \(-0.290961\pi\)
0.610519 + 0.792002i \(0.290961\pi\)
\(48\) −0.437027 − 3.03959i −0.0630794 − 0.438727i
\(49\) 0.415415 − 0.909632i 0.0593450 − 0.129947i
\(50\) 3.74535 − 1.09973i 0.529673 − 0.155526i
\(51\) 0.277956 − 0.320778i 0.0389216 − 0.0449179i
\(52\) −0.228017 − 0.0669519i −0.0316203 − 0.00928456i
\(53\) 8.46563 − 5.44053i 1.16284 − 0.747314i 0.190688 − 0.981651i \(-0.438928\pi\)
0.972156 + 0.234336i \(0.0752917\pi\)
\(54\) 4.37568 + 9.58140i 0.595454 + 1.30386i
\(55\) −0.163102 − 0.188230i −0.0219926 − 0.0253809i
\(56\) −0.841254 − 0.540641i −0.112417 − 0.0722462i
\(57\) 0.513404 − 3.57080i 0.0680020 − 0.472964i
\(58\) 0.308815 − 2.14786i 0.0405494 − 0.282027i
\(59\) 2.31594 + 1.48836i 0.301509 + 0.193768i 0.682644 − 0.730751i \(-0.260830\pi\)
−0.381135 + 0.924520i \(0.624467\pi\)
\(60\) 2.10580 + 2.43022i 0.271857 + 0.313740i
\(61\) −3.25445 − 7.12625i −0.416690 − 0.912423i −0.995302 − 0.0968202i \(-0.969133\pi\)
0.578612 − 0.815603i \(-0.303594\pi\)
\(62\) 7.19006 − 4.62077i 0.913139 − 0.586839i
\(63\) 6.16962 + 1.81157i 0.777300 + 0.228236i
\(64\) −0.654861 + 0.755750i −0.0818576 + 0.0944687i
\(65\) 0.238769 − 0.0701089i 0.0296156 − 0.00869594i
\(66\) −0.303417 + 0.664391i −0.0373481 + 0.0817808i
\(67\) −0.867612 − 6.03437i −0.105996 − 0.737216i −0.971624 − 0.236529i \(-0.923990\pi\)
0.865629 − 0.500687i \(-0.166919\pi\)
\(68\) −0.138219 −0.0167615
\(69\) −4.12754 − 14.1370i −0.496897 − 1.70190i
\(70\) 1.04715 0.125159
\(71\) 1.19790 + 8.33156i 0.142164 + 0.988774i 0.928595 + 0.371095i \(0.121018\pi\)
−0.786431 + 0.617679i \(0.788073\pi\)
\(72\) 2.67115 − 5.84901i 0.314799 − 0.689313i
\(73\) −13.9823 + 4.10557i −1.63650 + 0.480520i −0.965384 − 0.260831i \(-0.916003\pi\)
−0.671117 + 0.741351i \(0.734185\pi\)
\(74\) −7.11194 + 8.20762i −0.826746 + 0.954116i
\(75\) 11.5014 + 3.37712i 1.32807 + 0.389956i
\(76\) −0.988275 + 0.635126i −0.113363 + 0.0728539i
\(77\) 0.0988057 + 0.216354i 0.0112600 + 0.0246559i
\(78\) −0.477895 − 0.551521i −0.0541110 − 0.0624474i
\(79\) 11.0310 + 7.08918i 1.24108 + 0.797595i 0.985579 − 0.169213i \(-0.0541227\pi\)
0.255503 + 0.966808i \(0.417759\pi\)
\(80\) 0.149025 − 1.03649i 0.0166615 − 0.115884i
\(81\) −1.85803 + 12.9229i −0.206448 + 1.43588i
\(82\) 5.06502 + 3.25509i 0.559338 + 0.359465i
\(83\) 2.61067 + 3.01287i 0.286558 + 0.330706i 0.880718 − 0.473641i \(-0.157061\pi\)
−0.594160 + 0.804347i \(0.702515\pi\)
\(84\) −1.27568 − 2.79334i −0.139188 − 0.304778i
\(85\) 0.121760 − 0.0782506i 0.0132068 − 0.00848746i
\(86\) −2.46979 − 0.725196i −0.266324 − 0.0781999i
\(87\) 4.36370 − 5.03598i 0.467838 − 0.539914i
\(88\) 0.228214 − 0.0670096i 0.0243277 − 0.00714325i
\(89\) 0.706662 − 1.54737i 0.0749061 − 0.164021i −0.868474 − 0.495734i \(-0.834899\pi\)
0.943381 + 0.331713i \(0.107626\pi\)
\(90\) 0.958246 + 6.66475i 0.101008 + 0.702526i
\(91\) −0.237644 −0.0249118
\(92\) −2.59899 + 4.03054i −0.270963 + 0.420213i
\(93\) 26.2460 2.72159
\(94\) −1.19132 − 8.28581i −0.122875 − 0.854616i
\(95\) 0.511026 − 1.11899i 0.0524302 − 0.114806i
\(96\) −2.94645 + 0.865157i −0.300721 + 0.0882997i
\(97\) −3.29593 + 3.80371i −0.334651 + 0.386208i −0.897989 − 0.440019i \(-0.854972\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(98\) −0.959493 − 0.281733i −0.0969234 − 0.0284593i
\(99\) −1.28660 + 0.826848i −0.129308 + 0.0831013i
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.i.d.141.4 ✓ 40
23.8 even 11 inner 322.2.i.d.169.4 yes 40
23.10 odd 22 7406.2.a.bv.1.18 20
23.13 even 11 7406.2.a.bu.1.18 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.i.d.141.4 ✓ 40 1.1 even 1 trivial
322.2.i.d.169.4 yes 40 23.8 even 11 inner
7406.2.a.bu.1.18 20 23.13 even 11
7406.2.a.bv.1.18 20 23.10 odd 22