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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,2,Mod(11,168)] 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("168.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.2
Character \(\chi\) \(=\) 168.11
Dual form 168.2.v.a.107.2

$q$-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+(-1.40124 + 0.191140i) q^{2} +(1.66017 - 0.493786i) q^{3} +(1.92693 - 0.535666i) q^{4} +(1.09212 + 1.89161i) q^{5} +(-2.23191 + 1.00924i) q^{6} +(-0.451847 + 2.60688i) q^{7} +(-2.59770 + 1.11891i) q^{8} +(2.51235 - 1.63954i) q^{9} +(-1.89188 - 2.44184i) q^{10} +(-1.45052 - 0.837460i) q^{11} +(2.93453 - 1.84079i) q^{12} -1.56085i q^{13} +(0.134864 - 3.73923i) q^{14} +(2.74716 + 2.60112i) q^{15} +(3.42612 - 2.06438i) q^{16} +(-0.278795 - 0.160963i) q^{17} +(-3.20702 + 2.77760i) q^{18} +(2.87437 + 4.97856i) q^{19} +(3.11771 + 3.05999i) q^{20} +(0.537096 + 4.55099i) q^{21} +(2.19260 + 0.896227i) q^{22} +(-3.26717 - 5.65891i) q^{23} +(-3.76013 + 3.14029i) q^{24} +(0.114544 - 0.198397i) q^{25} +(0.298342 + 2.18713i) q^{26} +(3.36136 - 3.96248i) q^{27} +(0.525740 + 5.26532i) q^{28} +7.04205 q^{29} +(-4.34660 - 3.11970i) q^{30} +(-7.76983 - 4.48591i) q^{31} +(-4.40622 + 3.54756i) q^{32} +(-2.82165 - 0.674082i) q^{33} +(0.421425 + 0.172258i) q^{34} +(-5.42467 + 1.99231i) q^{35} +(3.96288 - 4.50506i) q^{36} +(0.946173 - 0.546273i) q^{37} +(-4.97929 - 6.42674i) q^{38} +(-0.770727 - 2.59129i) q^{39} +(-4.95354 - 3.69185i) q^{40} -3.44933i q^{41} +(-1.62248 - 6.27436i) q^{42} -11.6570 q^{43} +(-3.24366 - 0.836731i) q^{44} +(5.84516 + 2.96181i) q^{45} +(5.65973 + 7.30498i) q^{46} +(-2.25371 - 3.90354i) q^{47} +(4.66860 - 5.11900i) q^{48} +(-6.59167 - 2.35583i) q^{49} +(-0.122582 + 0.299895i) q^{50} +(-0.542330 - 0.129561i) q^{51} +(-0.836097 - 3.00766i) q^{52} +(-4.42876 + 7.67084i) q^{53} +(-3.95267 + 6.19487i) q^{54} -3.65843i q^{55} +(-1.74310 - 7.27747i) q^{56} +(7.23030 + 6.84595i) q^{57} +(-9.86758 + 1.34602i) q^{58} +(5.37016 + 3.10046i) q^{59} +(6.68692 + 3.54063i) q^{60} +(3.80578 - 2.19727i) q^{61} +(11.7448 + 4.80070i) q^{62} +(3.13889 + 7.29023i) q^{63} +(5.49608 - 5.81318i) q^{64} +(2.95253 - 1.70464i) q^{65} +(4.08264 + 0.405217i) q^{66} +(0.716072 - 1.24027i) q^{67} +(-0.623442 - 0.160822i) q^{68} +(-8.21836 - 7.78149i) q^{69} +(7.22044 - 3.82858i) q^{70} -6.37080 q^{71} +(-4.69184 + 7.07013i) q^{72} +(-4.49012 + 7.77711i) q^{73} +(-1.22140 + 0.946310i) q^{74} +(0.0921980 - 0.385933i) q^{75} +(8.20557 + 8.05364i) q^{76} +(2.83858 - 3.40294i) q^{77} +(1.57527 + 3.48369i) q^{78} +(-3.58799 + 2.07153i) q^{79} +(7.64675 + 4.22633i) q^{80} +(3.62382 - 8.23820i) q^{81} +(0.659306 + 4.83333i) q^{82} +9.06459i q^{83} +(3.47276 + 8.48174i) q^{84} -0.703162i q^{85} +(16.3343 - 2.22813i) q^{86} +(11.6910 - 3.47726i) q^{87} +(4.70507 + 0.552465i) q^{88} +(4.57586 - 2.64187i) q^{89} +(-8.75657 - 3.03295i) q^{90} +(4.06896 + 0.705268i) q^{91} +(-9.32690 - 9.15421i) q^{92} +(-15.1143 - 3.61076i) q^{93} +(3.90411 + 5.03901i) q^{94} +(-6.27833 + 10.8744i) q^{95} +(-5.56336 + 8.06530i) q^{96} +4.23153 q^{97} +(9.68678 + 2.04113i) q^{98} +(-5.01728 + 0.274197i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{3} - 2 q^{4} - 8 q^{6} - 2 q^{9} + 6 q^{10} + 10 q^{12} - 10 q^{16} - 10 q^{18} - 4 q^{19} - 20 q^{22} - 8 q^{24} - 16 q^{25} - 8 q^{27} - 22 q^{28} - 12 q^{30} - 14 q^{33} - 56 q^{34} + 4 q^{36}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40124 + 0.191140i −0.990824 + 0.135157i
\(3\) 1.66017 − 0.493786i 0.958502 − 0.285087i
\(4\) 1.92693 − 0.535666i 0.963465 − 0.267833i
\(5\) 1.09212 + 1.89161i 0.488411 + 0.845953i 0.999911 − 0.0133302i \(-0.00424327\pi\)
−0.511500 + 0.859283i \(0.670910\pi\)
\(6\) −2.23191 + 1.00924i −0.911175 + 0.412019i
\(7\) −0.451847 + 2.60688i −0.170782 + 0.985309i
\(8\) −2.59770 + 1.11891i −0.918425 + 0.395594i
\(9\) 2.51235 − 1.63954i 0.837450 − 0.546513i
\(10\) −1.89188 − 2.44184i −0.598266 − 0.772179i
\(11\) −1.45052 − 0.837460i −0.437349 − 0.252504i 0.265123 − 0.964215i \(-0.414587\pi\)
−0.702473 + 0.711711i \(0.747921\pi\)
\(12\) 2.93453 − 1.84079i 0.847127 − 0.531390i
\(13\) − 1.56085i − 0.432903i −0.976293 − 0.216452i \(-0.930552\pi\)
0.976293 − 0.216452i \(-0.0694483\pi\)
\(14\) 0.134864 − 3.73923i 0.0360440 − 0.999350i
\(15\) 2.74716 + 2.60112i 0.709313 + 0.671608i
\(16\) 3.42612 − 2.06438i 0.856531 − 0.516096i
\(17\) −0.278795 − 0.160963i −0.0676178 − 0.0390392i 0.465810 − 0.884885i \(-0.345763\pi\)
−0.533428 + 0.845846i \(0.679096\pi\)
\(18\) −3.20702 + 2.77760i −0.755901 + 0.654686i
\(19\) 2.87437 + 4.97856i 0.659427 + 1.14216i 0.980764 + 0.195196i \(0.0625343\pi\)
−0.321337 + 0.946965i \(0.604132\pi\)
\(20\) 3.11771 + 3.05999i 0.697142 + 0.684234i
\(21\) 0.537096 + 4.55099i 0.117204 + 0.993108i
\(22\) 2.19260 + 0.896227i 0.467464 + 0.191076i
\(23\) −3.26717 − 5.65891i −0.681252 − 1.17996i −0.974599 − 0.223958i \(-0.928102\pi\)
0.293346 − 0.956006i \(-0.405231\pi\)
\(24\) −3.76013 + 3.14029i −0.767533 + 0.641009i
\(25\) 0.114544 − 0.198397i 0.0229089 − 0.0396793i
\(26\) 0.298342 + 2.18713i 0.0585098 + 0.428931i
\(27\) 3.36136 − 3.96248i 0.646894 − 0.762580i
\(28\) 0.525740 + 5.26532i 0.0993556 + 0.995052i
\(29\) 7.04205 1.30768 0.653838 − 0.756635i \(-0.273158\pi\)
0.653838 + 0.756635i \(0.273158\pi\)
\(30\) −4.34660 − 3.11970i −0.793577 − 0.569577i
\(31\) −7.76983 − 4.48591i −1.39550 − 0.805694i −0.401585 − 0.915822i \(-0.631541\pi\)
−0.993917 + 0.110128i \(0.964874\pi\)
\(32\) −4.40622 + 3.54756i −0.778918 + 0.627126i
\(33\) −2.82165 − 0.674082i −0.491186 − 0.117343i
\(34\) 0.421425 + 0.172258i 0.0722738 + 0.0295419i
\(35\) −5.42467 + 1.99231i −0.916937 + 0.336762i
\(36\) 3.96288 − 4.50506i 0.660480 − 0.750844i
\(37\) 0.946173 − 0.546273i 0.155550 − 0.0898068i −0.420205 − 0.907429i \(-0.638042\pi\)
0.575755 + 0.817623i \(0.304708\pi\)
\(38\) −4.97929 − 6.42674i −0.807747 − 1.04255i
\(39\) −0.770727 − 2.59129i −0.123415 − 0.414938i
\(40\) −4.95354 − 3.69185i −0.783224 − 0.583732i
\(41\) − 3.44933i − 0.538694i −0.963043 − 0.269347i \(-0.913192\pi\)
0.963043 − 0.269347i \(-0.0868079\pi\)
\(42\) −1.62248 − 6.27436i −0.250354 − 0.968154i
\(43\) −11.6570 −1.77768 −0.888841 − 0.458215i \(-0.848489\pi\)
−0.888841 + 0.458215i \(0.848489\pi\)
\(44\) −3.24366 − 0.836731i −0.489000 − 0.126142i
\(45\) 5.84516 + 2.96181i 0.871345 + 0.441521i
\(46\) 5.65973 + 7.30498i 0.834481 + 1.07706i
\(47\) −2.25371 − 3.90354i −0.328737 − 0.569390i 0.653524 − 0.756906i \(-0.273290\pi\)
−0.982262 + 0.187516i \(0.939956\pi\)
\(48\) 4.66860 − 5.11900i 0.673854 − 0.738865i
\(49\) −6.59167 − 2.35583i −0.941667 − 0.336546i
\(50\) −0.122582 + 0.299895i −0.0173357 + 0.0424115i
\(51\) −0.542330 − 0.129561i −0.0759413 − 0.0181421i
\(52\) −0.836097 − 3.00766i −0.115946 − 0.417087i
\(53\) −4.42876 + 7.67084i −0.608337 + 1.05367i 0.383177 + 0.923675i \(0.374830\pi\)
−0.991515 + 0.129996i \(0.958504\pi\)
\(54\) −3.95267 + 6.19487i −0.537890 + 0.843015i
\(55\) − 3.65843i − 0.493303i
\(56\) −1.74310 − 7.27747i −0.232932 − 0.972493i
\(57\) 7.23030 + 6.84595i 0.957677 + 0.906769i
\(58\) −9.86758 + 1.34602i −1.29568 + 0.176741i
\(59\) 5.37016 + 3.10046i 0.699135 + 0.403646i 0.807025 − 0.590517i \(-0.201076\pi\)
−0.107890 + 0.994163i \(0.534409\pi\)
\(60\) 6.68692 + 3.54063i 0.863278 + 0.457093i
\(61\) 3.80578 − 2.19727i 0.487280 − 0.281331i −0.236165 − 0.971713i \(-0.575891\pi\)
0.723445 + 0.690382i \(0.242557\pi\)
\(62\) 11.7448 + 4.80070i 1.49159 + 0.609690i
\(63\) 3.13889 + 7.29023i 0.395463 + 0.918482i
\(64\) 5.49608 − 5.81318i 0.687010 − 0.726648i
\(65\) 2.95253 − 1.70464i 0.366216 − 0.211435i
\(66\) 4.08264 + 0.405217i 0.502538 + 0.0498788i
\(67\) 0.716072 − 1.24027i 0.0874821 − 0.151523i −0.818964 − 0.573845i \(-0.805451\pi\)
0.906446 + 0.422321i \(0.138785\pi\)
\(68\) −0.623442 − 0.160822i −0.0756034 − 0.0195026i
\(69\) −8.21836 − 7.78149i −0.989374 − 0.936781i
\(70\) 7.22044 − 3.82858i 0.863008 − 0.457602i
\(71\) −6.37080 −0.756074 −0.378037 − 0.925790i \(-0.623401\pi\)
−0.378037 + 0.925790i \(0.623401\pi\)
\(72\) −4.69184 + 7.07013i −0.552938 + 0.833222i
\(73\) −4.49012 + 7.77711i −0.525528 + 0.910242i 0.474030 + 0.880509i \(0.342799\pi\)
−0.999558 + 0.0297328i \(0.990534\pi\)
\(74\) −1.22140 + 0.946310i −0.141985 + 0.110006i
\(75\) 0.0921980 − 0.385933i 0.0106461 − 0.0445637i
\(76\) 8.20557 + 8.05364i 0.941243 + 0.923816i
\(77\) 2.83858 − 3.40294i 0.323486 − 0.387801i
\(78\) 1.57527 + 3.48369i 0.178364 + 0.394451i
\(79\) −3.58799 + 2.07153i −0.403681 + 0.233065i −0.688071 − 0.725643i \(-0.741542\pi\)
0.284390 + 0.958709i \(0.408209\pi\)
\(80\) 7.64675 + 4.22633i 0.854932 + 0.472518i
\(81\) 3.62382 − 8.23820i 0.402647 − 0.915356i
\(82\) 0.659306 + 4.83333i 0.0728082 + 0.533751i
\(83\) 9.06459i 0.994968i 0.867473 + 0.497484i \(0.165743\pi\)
−0.867473 + 0.497484i \(0.834257\pi\)
\(84\) 3.47276 + 8.48174i 0.378909 + 0.925434i
\(85\) − 0.703162i − 0.0762687i
\(86\) 16.3343 − 2.22813i 1.76137 − 0.240266i
\(87\) 11.6910 − 3.47726i 1.25341 − 0.372802i
\(88\) 4.70507 + 0.552465i 0.501562 + 0.0588929i
\(89\) 4.57586 − 2.64187i 0.485040 − 0.280038i −0.237474 − 0.971394i \(-0.576320\pi\)
0.722515 + 0.691356i \(0.242986\pi\)
\(90\) −8.75657 − 3.03295i −0.923024 − 0.319701i
\(91\) 4.06896 + 0.705268i 0.426543 + 0.0739322i
\(92\) −9.32690 − 9.15421i −0.972397 − 0.954392i
\(93\) −15.1143 − 3.61076i −1.56728 − 0.374419i
\(94\) 3.90411 + 5.03901i 0.402678 + 0.519734i
\(95\) −6.27833 + 10.8744i −0.644143 + 1.11569i
\(96\) −5.56336 + 8.06530i −0.567808 + 0.823161i
\(97\) 4.23153 0.429647 0.214823 − 0.976653i \(-0.431082\pi\)
0.214823 + 0.976653i \(0.431082\pi\)
\(98\) 9.68678 + 2.04113i 0.978513 + 0.206186i
\(99\) −5.01728 + 0.274197i −0.504255 + 0.0275578i
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 168.2.v.a.11.2 ✓ 56
3.2 odd 2 inner 168.2.v.a.11.27 yes 56
4.3 odd 2 672.2.bd.a.431.4 56
7.2 even 3 inner 168.2.v.a.107.20 yes 56
8.3 odd 2 inner 168.2.v.a.11.9 yes 56
8.5 even 2 672.2.bd.a.431.3 56
12.11 even 2 672.2.bd.a.431.17 56
21.2 odd 6 inner 168.2.v.a.107.9 yes 56
24.5 odd 2 672.2.bd.a.431.18 56
24.11 even 2 inner 168.2.v.a.11.20 yes 56
28.23 odd 6 672.2.bd.a.527.18 56
56.37 even 6 672.2.bd.a.527.17 56
56.51 odd 6 inner 168.2.v.a.107.27 yes 56
84.23 even 6 672.2.bd.a.527.3 56
168.107 even 6 inner 168.2.v.a.107.2 yes 56
168.149 odd 6 672.2.bd.a.527.4 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.2 ✓ 56 1.1 even 1 trivial
168.2.v.a.11.9 yes 56 8.3 odd 2 inner
168.2.v.a.11.20 yes 56 24.11 even 2 inner
168.2.v.a.11.27 yes 56 3.2 odd 2 inner
168.2.v.a.107.2 yes 56 168.107 even 6 inner
168.2.v.a.107.9 yes 56 21.2 odd 6 inner
168.2.v.a.107.20 yes 56 7.2 even 3 inner
168.2.v.a.107.27 yes 56 56.51 odd 6 inner
672.2.bd.a.431.3 56 8.5 even 2
672.2.bd.a.431.4 56 4.3 odd 2
672.2.bd.a.431.17 56 12.11 even 2
672.2.bd.a.431.18 56 24.5 odd 2
672.2.bd.a.527.3 56 84.23 even 6
672.2.bd.a.527.4 56 168.149 odd 6
672.2.bd.a.527.17 56 56.37 even 6
672.2.bd.a.527.18 56 28.23 odd 6