# Properties

 Label 99.14.a.a.1.1 Level $99$ Weight $14$ Character 99.1 Self dual yes Analytic conductor $106.159$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,14,Mod(1,99)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 14, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 99.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$106.158619662$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 99.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-140.000 q^{2} +11408.0 q^{4} -48740.0 q^{5} +487486. q^{7} -450240. q^{8} +O(q^{10})$$ $$q-140.000 q^{2} +11408.0 q^{4} -48740.0 q^{5} +487486. q^{7} -450240. q^{8} +6.82360e6 q^{10} +1.77156e6 q^{11} -1.83883e7 q^{13} -6.82480e7 q^{14} -3.04207e7 q^{16} +9.62333e7 q^{17} -1.49547e7 q^{19} -5.56026e8 q^{20} -2.48019e8 q^{22} -1.53804e8 q^{23} +1.15488e9 q^{25} +2.57436e9 q^{26} +5.56124e9 q^{28} -5.21901e9 q^{29} +1.18381e9 q^{31} +7.94727e9 q^{32} -1.34727e10 q^{34} -2.37601e10 q^{35} -1.76722e10 q^{37} +2.09365e9 q^{38} +2.19447e10 q^{40} +1.94617e10 q^{41} -3.41230e9 q^{43} +2.02100e10 q^{44} +2.15326e10 q^{46} +1.00328e11 q^{47} +1.40754e11 q^{49} -1.61684e11 q^{50} -2.09774e11 q^{52} -2.75469e11 q^{53} -8.63459e10 q^{55} -2.19486e11 q^{56} +7.30661e11 q^{58} +2.67677e11 q^{59} +5.63487e11 q^{61} -1.65734e11 q^{62} -8.63411e11 q^{64} +8.96246e11 q^{65} +1.08084e12 q^{67} +1.09783e12 q^{68} +3.32641e12 q^{70} +1.15056e12 q^{71} -3.45915e11 q^{73} +2.47411e12 q^{74} -1.70603e11 q^{76} +8.63611e11 q^{77} -2.00408e12 q^{79} +1.48271e12 q^{80} -2.72464e12 q^{82} +3.33673e12 q^{83} -4.69041e12 q^{85} +4.77723e11 q^{86} -7.97628e11 q^{88} +5.69624e12 q^{89} -8.96404e12 q^{91} -1.75460e12 q^{92} -1.40459e13 q^{94} +7.28890e11 q^{95} -6.55011e12 q^{97} -1.97055e13 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −140.000 −1.54680 −0.773398 0.633921i $$-0.781445\pi$$
−0.773398 + 0.633921i $$0.781445\pi$$
$$3$$ 0 0
$$4$$ 11408.0 1.39258
$$5$$ −48740.0 −1.39502 −0.697510 0.716575i $$-0.745709\pi$$
−0.697510 + 0.716575i $$0.745709\pi$$
$$6$$ 0 0
$$7$$ 487486. 1.56612 0.783060 0.621947i $$-0.213658\pi$$
0.783060 + 0.621947i $$0.213658\pi$$
$$8$$ −450240. −0.607238
$$9$$ 0 0
$$10$$ 6.82360e6 2.15781
$$11$$ 1.77156e6 0.301511
$$12$$ 0 0
$$13$$ −1.83883e7 −1.05660 −0.528299 0.849058i $$-0.677170\pi$$
−0.528299 + 0.849058i $$0.677170\pi$$
$$14$$ −6.82480e7 −2.42247
$$15$$ 0 0
$$16$$ −3.04207e7 −0.453304
$$17$$ 9.62333e7 0.966957 0.483479 0.875356i $$-0.339373\pi$$
0.483479 + 0.875356i $$0.339373\pi$$
$$18$$ 0 0
$$19$$ −1.49547e7 −0.0729252 −0.0364626 0.999335i $$-0.511609\pi$$
−0.0364626 + 0.999335i $$0.511609\pi$$
$$20$$ −5.56026e8 −1.94267
$$21$$ 0 0
$$22$$ −2.48019e8 −0.466377
$$23$$ −1.53804e8 −0.216640 −0.108320 0.994116i $$-0.534547\pi$$
−0.108320 + 0.994116i $$0.534547\pi$$
$$24$$ 0 0
$$25$$ 1.15488e9 0.946081
$$26$$ 2.57436e9 1.63434
$$27$$ 0 0
$$28$$ 5.56124e9 2.18094
$$29$$ −5.21901e9 −1.62930 −0.814650 0.579953i $$-0.803071\pi$$
−0.814650 + 0.579953i $$0.803071\pi$$
$$30$$ 0 0
$$31$$ 1.18381e9 0.239570 0.119785 0.992800i $$-0.461780\pi$$
0.119785 + 0.992800i $$0.461780\pi$$
$$32$$ 7.94727e9 1.30841
$$33$$ 0 0
$$34$$ −1.34727e10 −1.49569
$$35$$ −2.37601e10 −2.18477
$$36$$ 0 0
$$37$$ −1.76722e10 −1.13235 −0.566173 0.824286i $$-0.691577\pi$$
−0.566173 + 0.824286i $$0.691577\pi$$
$$38$$ 2.09365e9 0.112800
$$39$$ 0 0
$$40$$ 2.19447e10 0.847110
$$41$$ 1.94617e10 0.639862 0.319931 0.947441i $$-0.396340\pi$$
0.319931 + 0.947441i $$0.396340\pi$$
$$42$$ 0 0
$$43$$ −3.41230e9 −0.0823195 −0.0411598 0.999153i $$-0.513105\pi$$
−0.0411598 + 0.999153i $$0.513105\pi$$
$$44$$ 2.02100e10 0.419878
$$45$$ 0 0
$$46$$ 2.15326e10 0.335097
$$47$$ 1.00328e11 1.35764 0.678820 0.734304i $$-0.262491\pi$$
0.678820 + 0.734304i $$0.262491\pi$$
$$48$$ 0 0
$$49$$ 1.40754e11 1.45273
$$50$$ −1.61684e11 −1.46339
$$51$$ 0 0
$$52$$ −2.09774e11 −1.47140
$$53$$ −2.75469e11 −1.70718 −0.853591 0.520944i $$-0.825580\pi$$
−0.853591 + 0.520944i $$0.825580\pi$$
$$54$$ 0 0
$$55$$ −8.63459e10 −0.420614
$$56$$ −2.19486e11 −0.951008
$$57$$ 0 0
$$58$$ 7.30661e11 2.52020
$$59$$ 2.67677e11 0.826176 0.413088 0.910691i $$-0.364450\pi$$
0.413088 + 0.910691i $$0.364450\pi$$
$$60$$ 0 0
$$61$$ 5.63487e11 1.40036 0.700180 0.713966i $$-0.253103\pi$$
0.700180 + 0.713966i $$0.253103\pi$$
$$62$$ −1.65734e11 −0.370565
$$63$$ 0 0
$$64$$ −8.63411e11 −1.57054
$$65$$ 8.96246e11 1.47398
$$66$$ 0 0
$$67$$ 1.08084e12 1.45974 0.729871 0.683585i $$-0.239580\pi$$
0.729871 + 0.683585i $$0.239580\pi$$
$$68$$ 1.09783e12 1.34656
$$69$$ 0 0
$$70$$ 3.32641e12 3.37939
$$71$$ 1.15056e12 1.06594 0.532968 0.846136i $$-0.321077\pi$$
0.532968 + 0.846136i $$0.321077\pi$$
$$72$$ 0 0
$$73$$ −3.45915e11 −0.267529 −0.133764 0.991013i $$-0.542707\pi$$
−0.133764 + 0.991013i $$0.542707\pi$$
$$74$$ 2.47411e12 1.75151
$$75$$ 0 0
$$76$$ −1.70603e11 −0.101554
$$77$$ 8.63611e11 0.472203
$$78$$ 0 0
$$79$$ −2.00408e12 −0.927554 −0.463777 0.885952i $$-0.653506\pi$$
−0.463777 + 0.885952i $$0.653506\pi$$
$$80$$ 1.48271e12 0.632369
$$81$$ 0 0
$$82$$ −2.72464e12 −0.989737
$$83$$ 3.33673e12 1.12025 0.560123 0.828409i $$-0.310754\pi$$
0.560123 + 0.828409i $$0.310754\pi$$
$$84$$ 0 0
$$85$$ −4.69041e12 −1.34892
$$86$$ 4.77723e11 0.127332
$$87$$ 0 0
$$88$$ −7.97628e11 −0.183089
$$89$$ 5.69624e12 1.21494 0.607468 0.794344i $$-0.292185\pi$$
0.607468 + 0.794344i $$0.292185\pi$$
$$90$$ 0 0
$$91$$ −8.96404e12 −1.65476
$$92$$ −1.75460e12 −0.301688
$$93$$ 0 0
$$94$$ −1.40459e13 −2.09999
$$95$$ 7.28890e11 0.101732
$$96$$ 0 0
$$97$$ −6.55011e12 −0.798422 −0.399211 0.916859i $$-0.630716\pi$$
−0.399211 + 0.916859i $$0.630716\pi$$
$$98$$ −1.97055e13 −2.24708
$$99$$ 0 0
$$100$$ 1.31749e13 1.31749
$$101$$ −2.00225e13 −1.87685 −0.938424 0.345487i $$-0.887714\pi$$
−0.938424 + 0.345487i $$0.887714\pi$$
$$102$$ 0 0
$$103$$ 8.50230e12 0.701608 0.350804 0.936449i $$-0.385908\pi$$
0.350804 + 0.936449i $$0.385908\pi$$
$$104$$ 8.27915e12 0.641607
$$105$$ 0 0
$$106$$ 3.85657e13 2.64066
$$107$$ −2.60740e13 −1.67963 −0.839814 0.542875i $$-0.817336\pi$$
−0.839814 + 0.542875i $$0.817336\pi$$
$$108$$ 0 0
$$109$$ −2.65871e12 −0.151844 −0.0759222 0.997114i $$-0.524190\pi$$
−0.0759222 + 0.997114i $$0.524190\pi$$
$$110$$ 1.20884e13 0.650605
$$111$$ 0 0
$$112$$ −1.48297e13 −0.709929
$$113$$ 3.76049e13 1.69916 0.849580 0.527460i $$-0.176856\pi$$
0.849580 + 0.527460i $$0.176856\pi$$
$$114$$ 0 0
$$115$$ 7.49643e12 0.302217
$$116$$ −5.95385e13 −2.26893
$$117$$ 0 0
$$118$$ −3.74748e13 −1.27793
$$119$$ 4.69124e13 1.51437
$$120$$ 0 0
$$121$$ 3.13843e12 0.0909091
$$122$$ −7.88881e13 −2.16607
$$123$$ 0 0
$$124$$ 1.35049e13 0.333619
$$125$$ 3.20800e12 0.0752176
$$126$$ 0 0
$$127$$ −6.93279e13 −1.46617 −0.733084 0.680138i $$-0.761920\pi$$
−0.733084 + 0.680138i $$0.761920\pi$$
$$128$$ 5.57735e13 1.12089
$$129$$ 0 0
$$130$$ −1.25474e14 −2.27994
$$131$$ 5.31970e13 0.919653 0.459827 0.888009i $$-0.347912\pi$$
0.459827 + 0.888009i $$0.347912\pi$$
$$132$$ 0 0
$$133$$ −7.29018e12 −0.114210
$$134$$ −1.51318e14 −2.25792
$$135$$ 0 0
$$136$$ −4.33281e13 −0.587173
$$137$$ −1.21037e14 −1.56399 −0.781994 0.623286i $$-0.785797\pi$$
−0.781994 + 0.623286i $$0.785797\pi$$
$$138$$ 0 0
$$139$$ 1.03707e14 1.21958 0.609791 0.792562i $$-0.291253\pi$$
0.609791 + 0.792562i $$0.291253\pi$$
$$140$$ −2.71055e14 −3.04246
$$141$$ 0 0
$$142$$ −1.61079e14 −1.64878
$$143$$ −3.25760e13 −0.318576
$$144$$ 0 0
$$145$$ 2.54375e14 2.27291
$$146$$ 4.84280e13 0.413812
$$147$$ 0 0
$$148$$ −2.01604e14 −1.57688
$$149$$ 2.94068e13 0.220159 0.110080 0.993923i $$-0.464889\pi$$
0.110080 + 0.993923i $$0.464889\pi$$
$$150$$ 0 0
$$151$$ −4.86579e13 −0.334043 −0.167022 0.985953i $$-0.553415\pi$$
−0.167022 + 0.985953i $$0.553415\pi$$
$$152$$ 6.73318e12 0.0442830
$$153$$ 0 0
$$154$$ −1.20906e14 −0.730401
$$155$$ −5.76990e13 −0.334204
$$156$$ 0 0
$$157$$ −2.31548e14 −1.23394 −0.616969 0.786988i $$-0.711640\pi$$
−0.616969 + 0.786988i $$0.711640\pi$$
$$158$$ 2.80571e14 1.43474
$$159$$ 0 0
$$160$$ −3.87350e14 −1.82526
$$161$$ −7.49775e13 −0.339283
$$162$$ 0 0
$$163$$ 1.44315e14 0.602688 0.301344 0.953516i $$-0.402565\pi$$
0.301344 + 0.953516i $$0.402565\pi$$
$$164$$ 2.22020e14 0.891058
$$165$$ 0 0
$$166$$ −4.67143e14 −1.73279
$$167$$ −4.17920e14 −1.49086 −0.745429 0.666585i $$-0.767756\pi$$
−0.745429 + 0.666585i $$0.767756\pi$$
$$168$$ 0 0
$$169$$ 3.52546e13 0.116400
$$170$$ 6.56657e14 2.08651
$$171$$ 0 0
$$172$$ −3.89276e13 −0.114636
$$173$$ −4.17435e14 −1.18383 −0.591914 0.806001i $$-0.701628\pi$$
−0.591914 + 0.806001i $$0.701628\pi$$
$$174$$ 0 0
$$175$$ 5.62990e14 1.48168
$$176$$ −5.38922e13 −0.136676
$$177$$ 0 0
$$178$$ −7.97473e14 −1.87926
$$179$$ −4.89549e14 −1.11238 −0.556188 0.831056i $$-0.687737\pi$$
−0.556188 + 0.831056i $$0.687737\pi$$
$$180$$ 0 0
$$181$$ 3.82069e14 0.807664 0.403832 0.914833i $$-0.367678\pi$$
0.403832 + 0.914833i $$0.367678\pi$$
$$182$$ 1.25497e15 2.55957
$$183$$ 0 0
$$184$$ 6.92489e13 0.131552
$$185$$ 8.61343e14 1.57965
$$186$$ 0 0
$$187$$ 1.70483e14 0.291549
$$188$$ 1.14454e15 1.89062
$$189$$ 0 0
$$190$$ −1.02045e14 −0.157359
$$191$$ −2.81803e14 −0.419980 −0.209990 0.977704i $$-0.567343\pi$$
−0.209990 + 0.977704i $$0.567343\pi$$
$$192$$ 0 0
$$193$$ 6.26551e14 0.872638 0.436319 0.899792i $$-0.356282\pi$$
0.436319 + 0.899792i $$0.356282\pi$$
$$194$$ 9.17016e14 1.23500
$$195$$ 0 0
$$196$$ 1.60572e15 2.02304
$$197$$ 6.60082e14 0.804576 0.402288 0.915513i $$-0.368215\pi$$
0.402288 + 0.915513i $$0.368215\pi$$
$$198$$ 0 0
$$199$$ −1.45047e15 −1.65564 −0.827818 0.560996i $$-0.810418\pi$$
−0.827818 + 0.560996i $$0.810418\pi$$
$$200$$ −5.19975e14 −0.574497
$$201$$ 0 0
$$202$$ 2.80315e15 2.90310
$$203$$ −2.54419e15 −2.55168
$$204$$ 0 0
$$205$$ −9.48565e14 −0.892621
$$206$$ −1.19032e15 −1.08524
$$207$$ 0 0
$$208$$ 5.59386e14 0.478961
$$209$$ −2.64931e13 −0.0219878
$$210$$ 0 0
$$211$$ −1.86139e15 −1.45212 −0.726059 0.687633i $$-0.758650\pi$$
−0.726059 + 0.687633i $$0.758650\pi$$
$$212$$ −3.14255e15 −2.37738
$$213$$ 0 0
$$214$$ 3.65036e15 2.59804
$$215$$ 1.66316e14 0.114837
$$216$$ 0 0
$$217$$ 5.77092e14 0.375195
$$218$$ 3.72219e14 0.234872
$$219$$ 0 0
$$220$$ −9.85034e14 −0.585738
$$221$$ −1.76957e15 −1.02169
$$222$$ 0 0
$$223$$ 2.64472e15 1.44012 0.720059 0.693913i $$-0.244115\pi$$
0.720059 + 0.693913i $$0.244115\pi$$
$$224$$ 3.87418e15 2.04912
$$225$$ 0 0
$$226$$ −5.26468e15 −2.62825
$$227$$ −3.07427e15 −1.49133 −0.745666 0.666319i $$-0.767869\pi$$
−0.745666 + 0.666319i $$0.767869\pi$$
$$228$$ 0 0
$$229$$ −2.98623e15 −1.36834 −0.684168 0.729324i $$-0.739835\pi$$
−0.684168 + 0.729324i $$0.739835\pi$$
$$230$$ −1.04950e15 −0.467467
$$231$$ 0 0
$$232$$ 2.34981e15 0.989374
$$233$$ 1.86637e15 0.764159 0.382079 0.924130i $$-0.375208\pi$$
0.382079 + 0.924130i $$0.375208\pi$$
$$234$$ 0 0
$$235$$ −4.88997e15 −1.89394
$$236$$ 3.05366e15 1.15052
$$237$$ 0 0
$$238$$ −6.56773e15 −2.34242
$$239$$ 3.11970e14 0.108275 0.0541373 0.998534i $$-0.482759\pi$$
0.0541373 + 0.998534i $$0.482759\pi$$
$$240$$ 0 0
$$241$$ 5.25176e15 1.72661 0.863305 0.504683i $$-0.168391\pi$$
0.863305 + 0.504683i $$0.168391\pi$$
$$242$$ −4.39380e14 −0.140618
$$243$$ 0 0
$$244$$ 6.42826e15 1.95011
$$245$$ −6.86033e15 −2.02659
$$246$$ 0 0
$$247$$ 2.74991e14 0.0770527
$$248$$ −5.32999e14 −0.145476
$$249$$ 0 0
$$250$$ −4.49120e14 −0.116346
$$251$$ 2.46350e15 0.621833 0.310916 0.950437i $$-0.399364\pi$$
0.310916 + 0.950437i $$0.399364\pi$$
$$252$$ 0 0
$$253$$ −2.72474e14 −0.0653193
$$254$$ 9.70590e15 2.26786
$$255$$ 0 0
$$256$$ −7.35229e14 −0.163254
$$257$$ −2.41514e15 −0.522851 −0.261425 0.965224i $$-0.584192\pi$$
−0.261425 + 0.965224i $$0.584192\pi$$
$$258$$ 0 0
$$259$$ −8.61495e15 −1.77339
$$260$$ 1.02244e16 2.05263
$$261$$ 0 0
$$262$$ −7.44759e15 −1.42252
$$263$$ 1.68025e15 0.313084 0.156542 0.987671i $$-0.449965\pi$$
0.156542 + 0.987671i $$0.449965\pi$$
$$264$$ 0 0
$$265$$ 1.34264e16 2.38155
$$266$$ 1.02063e15 0.176659
$$267$$ 0 0
$$268$$ 1.23303e16 2.03281
$$269$$ 5.87926e14 0.0946091 0.0473046 0.998881i $$-0.484937\pi$$
0.0473046 + 0.998881i $$0.484937\pi$$
$$270$$ 0 0
$$271$$ 3.69154e14 0.0566119 0.0283059 0.999599i $$-0.490989\pi$$
0.0283059 + 0.999599i $$0.490989\pi$$
$$272$$ −2.92749e15 −0.438326
$$273$$ 0 0
$$274$$ 1.69451e16 2.41917
$$275$$ 2.04595e15 0.285254
$$276$$ 0 0
$$277$$ 3.44319e15 0.457976 0.228988 0.973429i $$-0.426458\pi$$
0.228988 + 0.973429i $$0.426458\pi$$
$$278$$ −1.45190e16 −1.88645
$$279$$ 0 0
$$280$$ 1.06977e16 1.32667
$$281$$ −1.73559e15 −0.210309 −0.105154 0.994456i $$-0.533534\pi$$
−0.105154 + 0.994456i $$0.533534\pi$$
$$282$$ 0 0
$$283$$ 1.72244e15 0.199311 0.0996556 0.995022i $$-0.468226\pi$$
0.0996556 + 0.995022i $$0.468226\pi$$
$$284$$ 1.31256e16 1.48440
$$285$$ 0 0
$$286$$ 4.56064e15 0.492773
$$287$$ 9.48733e15 1.00210
$$288$$ 0 0
$$289$$ −6.43739e14 −0.0649941
$$290$$ −3.56124e16 −3.51572
$$291$$ 0 0
$$292$$ −3.94619e15 −0.372554
$$293$$ 9.32911e15 0.861392 0.430696 0.902497i $$-0.358268\pi$$
0.430696 + 0.902497i $$0.358268\pi$$
$$294$$ 0 0
$$295$$ −1.30466e16 −1.15253
$$296$$ 7.95673e15 0.687604
$$297$$ 0 0
$$298$$ −4.11695e15 −0.340541
$$299$$ 2.82820e15 0.228901
$$300$$ 0 0
$$301$$ −1.66345e15 −0.128922
$$302$$ 6.81210e15 0.516697
$$303$$ 0 0
$$304$$ 4.54932e14 0.0330573
$$305$$ −2.74643e16 −1.95353
$$306$$ 0 0
$$307$$ −1.85795e16 −1.26658 −0.633292 0.773913i $$-0.718297\pi$$
−0.633292 + 0.773913i $$0.718297\pi$$
$$308$$ 9.85208e15 0.657579
$$309$$ 0 0
$$310$$ 8.07786e15 0.516946
$$311$$ −1.10219e16 −0.690739 −0.345369 0.938467i $$-0.612246\pi$$
−0.345369 + 0.938467i $$0.612246\pi$$
$$312$$ 0 0
$$313$$ 2.44634e16 1.47055 0.735273 0.677771i $$-0.237054\pi$$
0.735273 + 0.677771i $$0.237054\pi$$
$$314$$ 3.24167e16 1.90865
$$315$$ 0 0
$$316$$ −2.28626e16 −1.29169
$$317$$ −7.28014e15 −0.402953 −0.201477 0.979493i $$-0.564574\pi$$
−0.201477 + 0.979493i $$0.564574\pi$$
$$318$$ 0 0
$$319$$ −9.24580e15 −0.491253
$$320$$ 4.20827e16 2.19093
$$321$$ 0 0
$$322$$ 1.04968e16 0.524802
$$323$$ −1.43913e15 −0.0705156
$$324$$ 0 0
$$325$$ −2.12364e16 −0.999628
$$326$$ −2.02041e16 −0.932235
$$327$$ 0 0
$$328$$ −8.76245e15 −0.388549
$$329$$ 4.89084e16 2.12623
$$330$$ 0 0
$$331$$ 4.91941e15 0.205604 0.102802 0.994702i $$-0.467219\pi$$
0.102802 + 0.994702i $$0.467219\pi$$
$$332$$ 3.80654e16 1.56003
$$333$$ 0 0
$$334$$ 5.85089e16 2.30605
$$335$$ −5.26803e16 −2.03637
$$336$$ 0 0
$$337$$ −3.61707e15 −0.134512 −0.0672562 0.997736i $$-0.521424\pi$$
−0.0672562 + 0.997736i $$0.521424\pi$$
$$338$$ −4.93565e15 −0.180047
$$339$$ 0 0
$$340$$ −5.35082e16 −1.87848
$$341$$ 2.09719e15 0.0722329
$$342$$ 0 0
$$343$$ 2.13834e16 0.709030
$$344$$ 1.53636e15 0.0499876
$$345$$ 0 0
$$346$$ 5.84408e16 1.83114
$$347$$ −2.03135e16 −0.624660 −0.312330 0.949974i $$-0.601110\pi$$
−0.312330 + 0.949974i $$0.601110\pi$$
$$348$$ 0 0
$$349$$ −1.16091e16 −0.343902 −0.171951 0.985106i $$-0.555007\pi$$
−0.171951 + 0.985106i $$0.555007\pi$$
$$350$$ −7.88186e16 −2.29185
$$351$$ 0 0
$$352$$ 1.40791e16 0.394500
$$353$$ −1.93894e16 −0.533370 −0.266685 0.963784i $$-0.585928\pi$$
−0.266685 + 0.963784i $$0.585928\pi$$
$$354$$ 0 0
$$355$$ −5.60784e16 −1.48700
$$356$$ 6.49827e16 1.69189
$$357$$ 0 0
$$358$$ 6.85369e16 1.72062
$$359$$ 1.45895e16 0.359689 0.179844 0.983695i $$-0.442441\pi$$
0.179844 + 0.983695i $$0.442441\pi$$
$$360$$ 0 0
$$361$$ −4.18293e16 −0.994682
$$362$$ −5.34896e16 −1.24929
$$363$$ 0 0
$$364$$ −1.02262e17 −2.30438
$$365$$ 1.68599e16 0.373208
$$366$$ 0 0
$$367$$ −5.04342e16 −1.07745 −0.538724 0.842482i $$-0.681093\pi$$
−0.538724 + 0.842482i $$0.681093\pi$$
$$368$$ 4.67884e15 0.0982037
$$369$$ 0 0
$$370$$ −1.20588e17 −2.44339
$$371$$ −1.34287e17 −2.67365
$$372$$ 0 0
$$373$$ 6.41403e15 0.123317 0.0616586 0.998097i $$-0.480361\pi$$
0.0616586 + 0.998097i $$0.480361\pi$$
$$374$$ −2.38676e16 −0.450966
$$375$$ 0 0
$$376$$ −4.51716e16 −0.824411
$$377$$ 9.59688e16 1.72152
$$378$$ 0 0
$$379$$ 6.85830e15 0.118867 0.0594335 0.998232i $$-0.481071\pi$$
0.0594335 + 0.998232i $$0.481071\pi$$
$$380$$ 8.31517e15 0.141670
$$381$$ 0 0
$$382$$ 3.94524e16 0.649623
$$383$$ −5.27865e16 −0.854537 −0.427268 0.904125i $$-0.640524\pi$$
−0.427268 + 0.904125i $$0.640524\pi$$
$$384$$ 0 0
$$385$$ −4.20924e16 −0.658732
$$386$$ −8.77171e16 −1.34979
$$387$$ 0 0
$$388$$ −7.47237e16 −1.11187
$$389$$ −6.57216e16 −0.961692 −0.480846 0.876805i $$-0.659670\pi$$
−0.480846 + 0.876805i $$0.659670\pi$$
$$390$$ 0 0
$$391$$ −1.48011e16 −0.209481
$$392$$ −6.33729e16 −0.882153
$$393$$ 0 0
$$394$$ −9.24114e16 −1.24452
$$395$$ 9.76789e16 1.29396
$$396$$ 0 0
$$397$$ −1.04942e17 −1.34528 −0.672639 0.739971i $$-0.734839\pi$$
−0.672639 + 0.739971i $$0.734839\pi$$
$$398$$ 2.03066e17 2.56093
$$399$$ 0 0
$$400$$ −3.51324e16 −0.428863
$$401$$ −2.14759e16 −0.257936 −0.128968 0.991649i $$-0.541166\pi$$
−0.128968 + 0.991649i $$0.541166\pi$$
$$402$$ 0 0
$$403$$ −2.17683e16 −0.253129
$$404$$ −2.28416e17 −2.61366
$$405$$ 0 0
$$406$$ 3.56187e17 3.94693
$$407$$ −3.13074e16 −0.341415
$$408$$ 0 0
$$409$$ 1.57661e17 1.66542 0.832708 0.553713i $$-0.186790\pi$$
0.832708 + 0.553713i $$0.186790\pi$$
$$410$$ 1.32799e17 1.38070
$$411$$ 0 0
$$412$$ 9.69943e16 0.977044
$$413$$ 1.30489e17 1.29389
$$414$$ 0 0
$$415$$ −1.62632e17 −1.56277
$$416$$ −1.46137e17 −1.38246
$$417$$ 0 0
$$418$$ 3.70903e15 0.0340106
$$419$$ −8.20257e16 −0.740558 −0.370279 0.928921i $$-0.620738\pi$$
−0.370279 + 0.928921i $$0.620738\pi$$
$$420$$ 0 0
$$421$$ −5.59695e16 −0.489912 −0.244956 0.969534i $$-0.578774\pi$$
−0.244956 + 0.969534i $$0.578774\pi$$
$$422$$ 2.60595e17 2.24613
$$423$$ 0 0
$$424$$ 1.24027e17 1.03667
$$425$$ 1.11138e17 0.914820
$$426$$ 0 0
$$427$$ 2.74692e17 2.19313
$$428$$ −2.97452e17 −2.33901
$$429$$ 0 0
$$430$$ −2.32842e16 −0.177630
$$431$$ −1.77267e17 −1.33207 −0.666034 0.745921i $$-0.732009\pi$$
−0.666034 + 0.745921i $$0.732009\pi$$
$$432$$ 0 0
$$433$$ −5.06598e16 −0.369396 −0.184698 0.982795i $$-0.559131\pi$$
−0.184698 + 0.982795i $$0.559131\pi$$
$$434$$ −8.07928e16 −0.580349
$$435$$ 0 0
$$436$$ −3.03306e16 −0.211455
$$437$$ 2.30009e15 0.0157985
$$438$$ 0 0
$$439$$ −6.76669e16 −0.451187 −0.225594 0.974221i $$-0.572432\pi$$
−0.225594 + 0.974221i $$0.572432\pi$$
$$440$$ 3.88764e16 0.255413
$$441$$ 0 0
$$442$$ 2.47739e17 1.58034
$$443$$ −1.70873e17 −1.07411 −0.537057 0.843546i $$-0.680464\pi$$
−0.537057 + 0.843546i $$0.680464\pi$$
$$444$$ 0 0
$$445$$ −2.77635e17 −1.69486
$$446$$ −3.70261e17 −2.22757
$$447$$ 0 0
$$448$$ −4.20901e17 −2.45965
$$449$$ −2.13667e14 −0.00123066 −0.000615329 1.00000i $$-0.500196\pi$$
−0.000615329 1.00000i $$0.500196\pi$$
$$450$$ 0 0
$$451$$ 3.44777e16 0.192926
$$452$$ 4.28996e17 2.36621
$$453$$ 0 0
$$454$$ 4.30398e17 2.30679
$$455$$ 4.36907e17 2.30842
$$456$$ 0 0
$$457$$ 3.31395e17 1.70173 0.850864 0.525386i $$-0.176079\pi$$
0.850864 + 0.525386i $$0.176079\pi$$
$$458$$ 4.18073e17 2.11654
$$459$$ 0 0
$$460$$ 8.55192e16 0.420860
$$461$$ 2.65757e17 1.28952 0.644761 0.764384i $$-0.276957\pi$$
0.644761 + 0.764384i $$0.276957\pi$$
$$462$$ 0 0
$$463$$ −4.27497e15 −0.0201677 −0.0100839 0.999949i $$-0.503210\pi$$
−0.0100839 + 0.999949i $$0.503210\pi$$
$$464$$ 1.58766e17 0.738569
$$465$$ 0 0
$$466$$ −2.61291e17 −1.18200
$$467$$ 2.56398e16 0.114381 0.0571907 0.998363i $$-0.481786\pi$$
0.0571907 + 0.998363i $$0.481786\pi$$
$$468$$ 0 0
$$469$$ 5.26896e17 2.28613
$$470$$ 6.84596e17 2.92953
$$471$$ 0 0
$$472$$ −1.20519e17 −0.501686
$$473$$ −6.04511e15 −0.0248203
$$474$$ 0 0
$$475$$ −1.72709e16 −0.0689932
$$476$$ 5.35176e17 2.10888
$$477$$ 0 0
$$478$$ −4.36758e16 −0.167479
$$479$$ −3.69707e17 −1.39855 −0.699274 0.714854i $$-0.746493\pi$$
−0.699274 + 0.714854i $$0.746493\pi$$
$$480$$ 0 0
$$481$$ 3.24962e17 1.19644
$$482$$ −7.35246e17 −2.67071
$$483$$ 0 0
$$484$$ 3.58032e16 0.126598
$$485$$ 3.19253e17 1.11382
$$486$$ 0 0
$$487$$ −5.17795e17 −1.75881 −0.879406 0.476073i $$-0.842060\pi$$
−0.879406 + 0.476073i $$0.842060\pi$$
$$488$$ −2.53704e17 −0.850352
$$489$$ 0 0
$$490$$ 9.60446e17 3.13472
$$491$$ −1.72056e17 −0.554165 −0.277082 0.960846i $$-0.589368\pi$$
−0.277082 + 0.960846i $$0.589368\pi$$
$$492$$ 0 0
$$493$$ −5.02242e17 −1.57546
$$494$$ −3.84987e16 −0.119185
$$495$$ 0 0
$$496$$ −3.60124e16 −0.108598
$$497$$ 5.60883e17 1.66938
$$498$$ 0 0
$$499$$ 1.31941e17 0.382584 0.191292 0.981533i $$-0.438732\pi$$
0.191292 + 0.981533i $$0.438732\pi$$
$$500$$ 3.65969e16 0.104746
$$501$$ 0 0
$$502$$ −3.44890e17 −0.961848
$$503$$ −2.26740e17 −0.624218 −0.312109 0.950046i $$-0.601035\pi$$
−0.312109 + 0.950046i $$0.601035\pi$$
$$504$$ 0 0
$$505$$ 9.75896e17 2.61824
$$506$$ 3.81463e16 0.101036
$$507$$ 0 0
$$508$$ −7.90892e17 −2.04175
$$509$$ 8.67070e16 0.220998 0.110499 0.993876i $$-0.464755\pi$$
0.110499 + 0.993876i $$0.464755\pi$$
$$510$$ 0 0
$$511$$ −1.68628e17 −0.418982
$$512$$ −3.53965e17 −0.868371
$$513$$ 0 0
$$514$$ 3.38120e17 0.808743
$$515$$ −4.14402e17 −0.978758
$$516$$ 0 0
$$517$$ 1.77737e17 0.409344
$$518$$ 1.20609e18 2.74307
$$519$$ 0 0
$$520$$ −4.03526e17 −0.895055
$$521$$ −5.57568e17 −1.22138 −0.610692 0.791868i $$-0.709109\pi$$
−0.610692 + 0.791868i $$0.709109\pi$$
$$522$$ 0 0
$$523$$ −4.80021e17 −1.02565 −0.512826 0.858493i $$-0.671401\pi$$
−0.512826 + 0.858493i $$0.671401\pi$$
$$524$$ 6.06872e17 1.28069
$$525$$ 0 0
$$526$$ −2.35235e17 −0.484277
$$527$$ 1.13922e17 0.231653
$$528$$ 0 0
$$529$$ −4.80381e17 −0.953067
$$530$$ −1.87969e18 −3.68378
$$531$$ 0 0
$$532$$ −8.31664e16 −0.159046
$$533$$ −3.57868e17 −0.676077
$$534$$ 0 0
$$535$$ 1.27085e18 2.34311
$$536$$ −4.86639e17 −0.886412
$$537$$ 0 0
$$538$$ −8.23097e16 −0.146341
$$539$$ 2.49354e17 0.438015
$$540$$ 0 0
$$541$$ −3.07367e17 −0.527078 −0.263539 0.964649i $$-0.584890\pi$$
−0.263539 + 0.964649i $$0.584890\pi$$
$$542$$ −5.16816e16 −0.0875670
$$543$$ 0 0
$$544$$ 7.64792e17 1.26517
$$545$$ 1.29585e17 0.211826
$$546$$ 0 0
$$547$$ 6.25426e17 0.998294 0.499147 0.866517i $$-0.333647\pi$$
0.499147 + 0.866517i $$0.333647\pi$$
$$548$$ −1.38079e18 −2.17798
$$549$$ 0 0
$$550$$ −2.86433e17 −0.441230
$$551$$ 7.80485e16 0.118817
$$552$$ 0 0
$$553$$ −9.76961e17 −1.45266
$$554$$ −4.82047e17 −0.708396
$$555$$ 0 0
$$556$$ 1.18309e18 1.69836
$$557$$ −5.97923e17 −0.848372 −0.424186 0.905575i $$-0.639440\pi$$
−0.424186 + 0.905575i $$0.639440\pi$$
$$558$$ 0 0
$$559$$ 6.27465e16 0.0869787
$$560$$ 7.22799e17 0.990365
$$561$$ 0 0
$$562$$ 2.42983e17 0.325305
$$563$$ 9.48274e16 0.125496 0.0627479 0.998029i $$-0.480014\pi$$
0.0627479 + 0.998029i $$0.480014\pi$$
$$564$$ 0 0
$$565$$ −1.83286e18 −2.37036
$$566$$ −2.41141e17 −0.308294
$$567$$ 0 0
$$568$$ −5.18029e17 −0.647277
$$569$$ −7.42364e17 −0.917038 −0.458519 0.888685i $$-0.651620\pi$$
−0.458519 + 0.888685i $$0.651620\pi$$
$$570$$ 0 0
$$571$$ 9.72970e17 1.17480 0.587401 0.809296i $$-0.300151\pi$$
0.587401 + 0.809296i $$0.300151\pi$$
$$572$$ −3.71627e17 −0.443642
$$573$$ 0 0
$$574$$ −1.32823e18 −1.55005
$$575$$ −1.77626e17 −0.204959
$$576$$ 0 0
$$577$$ −4.88742e17 −0.551362 −0.275681 0.961249i $$-0.588903\pi$$
−0.275681 + 0.961249i $$0.588903\pi$$
$$578$$ 9.01234e16 0.100533
$$579$$ 0 0
$$580$$ 2.90191e18 3.16520
$$581$$ 1.62661e18 1.75444
$$582$$ 0 0
$$583$$ −4.88010e17 −0.514735
$$584$$ 1.55745e17 0.162454
$$585$$ 0 0
$$586$$ −1.30608e18 −1.33240
$$587$$ −1.09582e18 −1.10558 −0.552792 0.833320i $$-0.686437\pi$$
−0.552792 + 0.833320i $$0.686437\pi$$
$$588$$ 0 0
$$589$$ −1.77035e16 −0.0174707
$$590$$ 1.82652e18 1.78273
$$591$$ 0 0
$$592$$ 5.37601e17 0.513298
$$593$$ 1.82087e18 1.71959 0.859794 0.510641i $$-0.170592\pi$$
0.859794 + 0.510641i $$0.170592\pi$$
$$594$$ 0 0
$$595$$ −2.28651e18 −2.11258
$$596$$ 3.35473e17 0.306589
$$597$$ 0 0
$$598$$ −3.95948e17 −0.354063
$$599$$ −5.79052e17 −0.512204 −0.256102 0.966650i $$-0.582438\pi$$
−0.256102 + 0.966650i $$0.582438\pi$$
$$600$$ 0 0
$$601$$ −1.68040e18 −1.45455 −0.727273 0.686348i $$-0.759213\pi$$
−0.727273 + 0.686348i $$0.759213\pi$$
$$602$$ 2.32883e17 0.199416
$$603$$ 0 0
$$604$$ −5.55089e17 −0.465182
$$605$$ −1.52967e17 −0.126820
$$606$$ 0 0
$$607$$ −2.18412e17 −0.177235 −0.0886176 0.996066i $$-0.528245\pi$$
−0.0886176 + 0.996066i $$0.528245\pi$$
$$608$$ −1.18849e17 −0.0954159
$$609$$ 0 0
$$610$$ 3.84501e18 3.02171
$$611$$ −1.84486e18 −1.43448
$$612$$ 0 0
$$613$$ −8.91819e16 −0.0678865 −0.0339433 0.999424i $$-0.510807\pi$$
−0.0339433 + 0.999424i $$0.510807\pi$$
$$614$$ 2.60113e18 1.95915
$$615$$ 0 0
$$616$$ −3.88832e17 −0.286740
$$617$$ −1.19832e18 −0.874419 −0.437210 0.899360i $$-0.644033\pi$$
−0.437210 + 0.899360i $$0.644033\pi$$
$$618$$ 0 0
$$619$$ 2.57248e18 1.83807 0.919037 0.394170i $$-0.128968\pi$$
0.919037 + 0.394170i $$0.128968\pi$$
$$620$$ −6.58230e17 −0.465406
$$621$$ 0 0
$$622$$ 1.54307e18 1.06843
$$623$$ 2.77684e18 1.90273
$$624$$ 0 0
$$625$$ −1.56613e18 −1.05101
$$626$$ −3.42488e18 −2.27463
$$627$$ 0 0
$$628$$ −2.64150e18 −1.71835
$$629$$ −1.70065e18 −1.09493
$$630$$ 0 0
$$631$$ 1.23798e18 0.780767 0.390383 0.920652i $$-0.372342\pi$$
0.390383 + 0.920652i $$0.372342\pi$$
$$632$$ 9.02317e17 0.563246
$$633$$ 0 0
$$634$$ 1.01922e18 0.623286
$$635$$ 3.37904e18 2.04533
$$636$$ 0 0
$$637$$ −2.58822e18 −1.53495
$$638$$ 1.29441e18 0.759868
$$639$$ 0 0
$$640$$ −2.71840e18 −1.56366
$$641$$ 3.15860e17 0.179853 0.0899264 0.995948i $$-0.471337\pi$$
0.0899264 + 0.995948i $$0.471337\pi$$
$$642$$ 0 0
$$643$$ 1.62866e18 0.908782 0.454391 0.890802i $$-0.349857\pi$$
0.454391 + 0.890802i $$0.349857\pi$$
$$644$$ −8.55343e17 −0.472479
$$645$$ 0 0
$$646$$ 2.01479e17 0.109073
$$647$$ 1.20409e18 0.645327 0.322664 0.946514i $$-0.395422\pi$$
0.322664 + 0.946514i $$0.395422\pi$$
$$648$$ 0 0
$$649$$ 4.74206e17 0.249102
$$650$$ 2.97309e18 1.54622
$$651$$ 0 0
$$652$$ 1.64635e18 0.839290
$$653$$ 2.58301e18 1.30374 0.651869 0.758332i $$-0.273985\pi$$
0.651869 + 0.758332i $$0.273985\pi$$
$$654$$ 0 0
$$655$$ −2.59282e18 −1.28293
$$656$$ −5.92040e17 −0.290052
$$657$$ 0 0
$$658$$ −6.84717e18 −3.28884
$$659$$ −2.90720e18 −1.38267 −0.691337 0.722532i $$-0.742978\pi$$
−0.691337 + 0.722532i $$0.742978\pi$$
$$660$$ 0 0
$$661$$ −3.29270e18 −1.53547 −0.767737 0.640765i $$-0.778617\pi$$
−0.767737 + 0.640765i $$0.778617\pi$$
$$662$$ −6.88717e17 −0.318027
$$663$$ 0 0
$$664$$ −1.50233e18 −0.680257
$$665$$ 3.55324e17 0.159325
$$666$$ 0 0
$$667$$ 8.02707e17 0.352971
$$668$$ −4.76764e18 −2.07614
$$669$$ 0 0
$$670$$ 7.37524e18 3.14985
$$671$$ 9.98251e17 0.422224
$$672$$ 0 0
$$673$$ −2.98149e18 −1.23690 −0.618451 0.785823i $$-0.712240\pi$$
−0.618451 + 0.785823i $$0.712240\pi$$
$$674$$ 5.06390e17 0.208063
$$675$$ 0 0
$$676$$ 4.02185e17 0.162096
$$677$$ 6.19017e17 0.247102 0.123551 0.992338i $$-0.460572\pi$$
0.123551 + 0.992338i $$0.460572\pi$$
$$678$$ 0 0
$$679$$ −3.19309e18 −1.25042
$$680$$ 2.11181e18 0.819119
$$681$$ 0 0
$$682$$ −2.93607e17 −0.111730
$$683$$ −1.07700e18 −0.405960 −0.202980 0.979183i $$-0.565063\pi$$
−0.202980 + 0.979183i $$0.565063\pi$$
$$684$$ 0 0
$$685$$ 5.89933e18 2.18180
$$686$$ −2.99367e18 −1.09672
$$687$$ 0 0
$$688$$ 1.03805e17 0.0373158
$$689$$ 5.06541e18 1.80381
$$690$$ 0 0
$$691$$ −4.77336e17 −0.166808 −0.0834039 0.996516i $$-0.526579\pi$$
−0.0834039 + 0.996516i $$0.526579\pi$$
$$692$$ −4.76209e18 −1.64857
$$693$$ 0 0
$$694$$ 2.84390e18 0.966222
$$695$$ −5.05467e18 −1.70134
$$696$$ 0 0
$$697$$ 1.87287e18 0.618719
$$698$$ 1.62528e18 0.531946
$$699$$ 0 0
$$700$$ 6.42259e18 2.06335
$$701$$ −4.37219e18 −1.39166 −0.695828 0.718208i $$-0.744962\pi$$
−0.695828 + 0.718208i $$0.744962\pi$$
$$702$$ 0 0
$$703$$ 2.64282e17 0.0825767
$$704$$ −1.52959e18 −0.473534
$$705$$ 0 0
$$706$$ 2.71451e18 0.825014
$$707$$ −9.76068e18 −2.93937
$$708$$ 0 0
$$709$$ −2.41723e18 −0.714689 −0.357344 0.933973i $$-0.616318\pi$$
−0.357344 + 0.933973i $$0.616318\pi$$
$$710$$ 7.85098e18 2.30009
$$711$$ 0 0
$$712$$ −2.56467e18 −0.737755
$$713$$ −1.82075e17 −0.0519002
$$714$$ 0 0
$$715$$ 1.58775e18 0.444420
$$716$$ −5.58478e18 −1.54907
$$717$$ 0 0
$$718$$ −2.04253e18 −0.556365
$$719$$ 1.88829e18 0.509719 0.254860 0.966978i $$-0.417971\pi$$
0.254860 + 0.966978i $$0.417971\pi$$
$$720$$ 0 0
$$721$$ 4.14475e18 1.09880
$$722$$ 5.85611e18 1.53857
$$723$$ 0 0
$$724$$ 4.35864e18 1.12474
$$725$$ −6.02735e18 −1.54145
$$726$$ 0 0
$$727$$ −5.58551e17 −0.140310 −0.0701551 0.997536i $$-0.522349\pi$$
−0.0701551 + 0.997536i $$0.522349\pi$$
$$728$$ 4.03597e18 1.00483
$$729$$ 0 0
$$730$$ −2.36038e18 −0.577276
$$731$$ −3.28377e17 −0.0795994
$$732$$ 0 0
$$733$$ −1.54585e18 −0.368122 −0.184061 0.982915i $$-0.558924\pi$$
−0.184061 + 0.982915i $$0.558924\pi$$
$$734$$ 7.06079e18 1.66659
$$735$$ 0 0
$$736$$ −1.22232e18 −0.283453
$$737$$ 1.91478e18 0.440129
$$738$$ 0 0
$$739$$ −6.42538e18 −1.45114 −0.725571 0.688147i $$-0.758424\pi$$
−0.725571 + 0.688147i $$0.758424\pi$$
$$740$$ 9.82620e18 2.19978
$$741$$ 0 0
$$742$$ 1.88002e19 4.13559
$$743$$ −4.38566e18 −0.956330 −0.478165 0.878270i $$-0.658698\pi$$
−0.478165 + 0.878270i $$0.658698\pi$$
$$744$$ 0 0
$$745$$ −1.43329e18 −0.307127
$$746$$ −8.97965e17 −0.190747
$$747$$ 0 0
$$748$$ 1.94487e18 0.406004
$$749$$ −1.27107e19 −2.63050
$$750$$ 0 0
$$751$$ 1.75685e18 0.357334 0.178667 0.983910i $$-0.442821\pi$$
0.178667 + 0.983910i $$0.442821\pi$$
$$752$$ −3.05204e18 −0.615424
$$753$$ 0 0
$$754$$ −1.34356e19 −2.66283
$$755$$ 2.37158e18 0.465997
$$756$$ 0 0
$$757$$ −1.97367e18 −0.381199 −0.190600 0.981668i $$-0.561043\pi$$
−0.190600 + 0.981668i $$0.561043\pi$$
$$758$$ −9.60161e17 −0.183863
$$759$$ 0 0
$$760$$ −3.28175e17 −0.0617757
$$761$$ −1.91433e18 −0.357286 −0.178643 0.983914i $$-0.557171\pi$$
−0.178643 + 0.983914i $$0.557171\pi$$
$$762$$ 0 0
$$763$$ −1.29608e18 −0.237806
$$764$$ −3.21481e18 −0.584855
$$765$$ 0 0
$$766$$ 7.39011e18 1.32179
$$767$$ −4.92212e18 −0.872936
$$768$$ 0 0
$$769$$ −9.69949e18 −1.69133 −0.845664 0.533716i $$-0.820795\pi$$
−0.845664 + 0.533716i $$0.820795\pi$$
$$770$$ 5.89294e18 1.01892
$$771$$ 0 0
$$772$$ 7.14769e18 1.21522
$$773$$ 1.83115e18 0.308714 0.154357 0.988015i $$-0.450669\pi$$
0.154357 + 0.988015i $$0.450669\pi$$
$$774$$ 0 0
$$775$$ 1.36717e18 0.226652
$$776$$ 2.94912e18 0.484833
$$777$$ 0 0
$$778$$ 9.20103e18 1.48754
$$779$$ −2.91044e17 −0.0466621
$$780$$ 0 0
$$781$$ 2.03829e18 0.321392
$$782$$ 2.07215e18 0.324025
$$783$$ 0 0
$$784$$ −4.28183e18 −0.658529
$$785$$ 1.12856e19 1.72137
$$786$$ 0 0
$$787$$ 4.83945e17 0.0726039 0.0363019 0.999341i $$-0.488442\pi$$
0.0363019 + 0.999341i $$0.488442\pi$$
$$788$$ 7.53021e18 1.12044
$$789$$ 0 0
$$790$$ −1.36750e19 −2.00149
$$791$$ 1.83318e19 2.66109
$$792$$ 0 0
$$793$$ −1.03616e19 −1.47962
$$794$$ 1.46919e19 2.08087
$$795$$ 0 0
$$796$$ −1.65470e19 −2.30560
$$797$$ 1.28612e19 1.77747 0.888734 0.458423i $$-0.151585\pi$$
0.888734 + 0.458423i $$0.151585\pi$$
$$798$$ 0 0
$$799$$ 9.65486e18 1.31278
$$800$$ 9.17818e18 1.23786
$$801$$ 0 0
$$802$$ 3.00662e18 0.398975
$$803$$ −6.12809e17 −0.0806629
$$804$$ 0 0
$$805$$ 3.65440e18 0.473307
$$806$$ 3.04756e18 0.391539
$$807$$ 0 0
$$808$$ 9.01492e18 1.13969
$$809$$ −6.97044e18 −0.874168 −0.437084 0.899421i $$-0.643989\pi$$
−0.437084 + 0.899421i $$0.643989\pi$$
$$810$$ 0 0
$$811$$ 1.21111e19 1.49468 0.747338 0.664444i $$-0.231332\pi$$
0.747338 + 0.664444i $$0.231332\pi$$
$$812$$ −2.90242e19 −3.55341
$$813$$ 0 0
$$814$$ 4.38303e18 0.528100
$$815$$ −7.03392e18 −0.840762
$$816$$ 0 0
$$817$$ 5.10298e16 0.00600317
$$818$$ −2.20725e19 −2.57606
$$819$$ 0 0
$$820$$ −1.08212e19 −1.24304
$$821$$ −1.13387e19 −1.29220 −0.646102 0.763251i $$-0.723602\pi$$
−0.646102 + 0.763251i $$0.723602\pi$$
$$822$$ 0 0
$$823$$ −4.21754e17 −0.0473108 −0.0236554 0.999720i $$-0.507530\pi$$
−0.0236554 + 0.999720i $$0.507530\pi$$
$$824$$ −3.82808e18 −0.426043
$$825$$ 0 0
$$826$$ −1.82684e19 −2.00139
$$827$$ −9.62297e18 −1.04598 −0.522990 0.852339i $$-0.675183\pi$$
−0.522990 + 0.852339i $$0.675183\pi$$
$$828$$ 0 0
$$829$$ −1.01074e19 −1.08152 −0.540759 0.841178i $$-0.681863\pi$$
−0.540759 + 0.841178i $$0.681863\pi$$
$$830$$ 2.27685e19 2.41728
$$831$$ 0 0
$$832$$ 1.58767e19 1.65943
$$833$$ 1.35452e19 1.40473
$$834$$ 0 0
$$835$$ 2.03694e19 2.07978
$$836$$ −3.02233e17 −0.0306197
$$837$$ 0 0
$$838$$ 1.14836e19 1.14549
$$839$$ 9.83113e17 0.0973085 0.0486543 0.998816i $$-0.484507\pi$$
0.0486543 + 0.998816i $$0.484507\pi$$
$$840$$ 0 0
$$841$$ 1.69774e19 1.65462
$$842$$ 7.83573e18 0.757794
$$843$$ 0 0
$$844$$ −2.12348e19 −2.02219
$$845$$ −1.71831e18 −0.162380
$$846$$ 0 0
$$847$$ 1.52994e18 0.142374
$$848$$ 8.37997e18 0.773873
$$849$$ 0 0
$$850$$ −1.55594e19 −1.41504
$$851$$ 2.71806e18 0.245311
$$852$$ 0 0
$$853$$ 1.20347e19 1.06971 0.534854 0.844944i $$-0.320367\pi$$
0.534854 + 0.844944i $$0.320367\pi$$
$$854$$ −3.84569e19 −3.39233
$$855$$ 0 0
$$856$$ 1.17396e19 1.01993
$$857$$ 5.60320e18 0.483127 0.241563 0.970385i $$-0.422340\pi$$
0.241563 + 0.970385i $$0.422340\pi$$
$$858$$ 0 0
$$859$$ 1.16060e17 0.00985656 0.00492828 0.999988i $$-0.498431\pi$$
0.00492828 + 0.999988i $$0.498431\pi$$
$$860$$ 1.89733e18 0.159920
$$861$$ 0 0
$$862$$ 2.48174e19 2.06044
$$863$$ 1.14438e18 0.0942978 0.0471489 0.998888i $$-0.484986\pi$$
0.0471489 + 0.998888i $$0.484986\pi$$
$$864$$ 0 0
$$865$$ 2.03458e19 1.65146
$$866$$ 7.09237e18 0.571380
$$867$$ 0 0
$$868$$ 6.58346e18 0.522488
$$869$$ −3.55035e18 −0.279668
$$870$$ 0 0
$$871$$ −1.98749e19 −1.54236
$$872$$ 1.19706e18 0.0922057
$$873$$ 0 0
$$874$$ −3.22013e17 −0.0244371
$$875$$ 1.56386e18 0.117800
$$876$$ 0 0
$$877$$ 5.25918e17 0.0390320 0.0195160 0.999810i $$-0.493787\pi$$
0.0195160 + 0.999810i $$0.493787\pi$$
$$878$$ 9.47337e18 0.697895
$$879$$ 0 0
$$880$$ 2.62671e18 0.190666
$$881$$ 1.02154e19 0.736056 0.368028 0.929815i $$-0.380033\pi$$
0.368028 + 0.929815i $$0.380033\pi$$
$$882$$ 0 0
$$883$$ −1.67904e19 −1.19211 −0.596055 0.802943i $$-0.703266\pi$$
−0.596055 + 0.802943i $$0.703266\pi$$
$$884$$ −2.01872e19 −1.42278
$$885$$ 0 0
$$886$$ 2.39223e19 1.66143
$$887$$ 9.98165e18 0.688175 0.344087 0.938938i $$-0.388188\pi$$
0.344087 + 0.938938i $$0.388188\pi$$
$$888$$ 0 0
$$889$$ −3.37964e19 −2.29619
$$890$$ 3.88688e19 2.62160
$$891$$ 0 0
$$892$$ 3.01710e19 2.00548
$$893$$ −1.50037e18 −0.0990063
$$894$$ 0 0
$$895$$ 2.38606e19 1.55179
$$896$$ 2.71888e19 1.75545
$$897$$ 0 0
$$898$$ 2.99134e16 0.00190358
$$899$$ −6.17833e18 −0.390331
$$900$$ 0 0
$$901$$ −2.65093e19 −1.65077
$$902$$ −4.82687e18 −0.298417
$$903$$ 0 0
$$904$$ −1.69312e19 −1.03179
$$905$$ −1.86220e19 −1.12671
$$906$$ 0 0
$$907$$ −1.90931e19 −1.13875 −0.569377 0.822076i $$-0.692816\pi$$
−0.569377 + 0.822076i $$0.692816\pi$$
$$908$$ −3.50713e19 −2.07680
$$909$$ 0 0
$$910$$ −6.11670e19 −3.57066
$$911$$ −2.12403e19 −1.23109 −0.615547 0.788100i $$-0.711065\pi$$
−0.615547 + 0.788100i $$0.711065\pi$$
$$912$$ 0 0
$$913$$ 5.91122e18 0.337767
$$914$$ −4.63953e19 −2.63223
$$915$$ 0 0
$$916$$ −3.40669e19 −1.90552
$$917$$ 2.59328e19 1.44029
$$918$$ 0 0
$$919$$ 2.14539e19 1.17478 0.587388 0.809305i $$-0.300156\pi$$
0.587388 + 0.809305i $$0.300156\pi$$
$$920$$ −3.37519e18 −0.183517
$$921$$ 0 0
$$922$$ −3.72060e19 −1.99463
$$923$$ −2.11569e19 −1.12627
$$924$$ 0 0
$$925$$ −2.04093e19 −1.07129
$$926$$ 5.98496e17 0.0311953
$$927$$ 0 0
$$928$$ −4.14769e19 −2.13179
$$929$$ −6.27434e17 −0.0320233 −0.0160116 0.999872i $$-0.505097\pi$$
−0.0160116 + 0.999872i $$0.505097\pi$$
$$930$$ 0 0
$$931$$ −2.10492e18 −0.105941
$$932$$ 2.12915e19 1.06415
$$933$$ 0 0
$$934$$ −3.58957e18 −0.176925
$$935$$ −8.30935e18 −0.406716
$$936$$ 0 0
$$937$$ 1.14580e19 0.553098 0.276549 0.961000i $$-0.410809\pi$$
0.276549 + 0.961000i $$0.410809\pi$$
$$938$$ −7.37654e19 −3.53618
$$939$$ 0 0
$$940$$ −5.57848e19 −2.63745
$$941$$ −8.37033e18 −0.393016 −0.196508 0.980502i $$-0.562960\pi$$
−0.196508 + 0.980502i $$0.562960\pi$$
$$942$$ 0 0
$$943$$ −2.99330e18 −0.138620
$$944$$ −8.14293e18 −0.374509
$$945$$ 0 0
$$946$$ 8.46315e17 0.0383919
$$947$$ 1.66309e19 0.749275 0.374637 0.927171i $$-0.377767\pi$$
0.374637 + 0.927171i $$0.377767\pi$$
$$948$$ 0 0
$$949$$ 6.36078e18 0.282670
$$950$$ 2.41793e18 0.106718
$$951$$ 0 0
$$952$$ −2.11218e19 −0.919584
$$953$$ 2.54297e19 1.09961 0.549803 0.835294i $$-0.314703\pi$$
0.549803 + 0.835294i $$0.314703\pi$$
$$954$$ 0 0
$$955$$ 1.37351e19 0.585880
$$956$$ 3.55895e18 0.150781
$$957$$ 0 0
$$958$$ 5.17590e19 2.16327
$$959$$ −5.90037e19 −2.44939
$$960$$ 0 0
$$961$$ −2.30161e19 −0.942606
$$962$$ −4.54947e19 −1.85064
$$963$$ 0 0
$$964$$ 5.99121e19 2.40444
$$965$$ −3.05381e19 −1.21735
$$966$$ 0 0
$$967$$ 5.78322e18 0.227456 0.113728 0.993512i $$-0.463721\pi$$
0.113728 + 0.993512i $$0.463721\pi$$
$$968$$ −1.41305e18 −0.0552035
$$969$$ 0 0
$$970$$ −4.46954e19 −1.72284
$$971$$ 4.12182e19 1.57821 0.789104 0.614259i $$-0.210545\pi$$
0.789104 + 0.614259i $$0.210545\pi$$
$$972$$ 0 0
$$973$$ 5.05556e19 1.91001
$$974$$ 7.24912e19 2.72052
$$975$$ 0 0
$$976$$ −1.71417e19 −0.634789
$$977$$ 4.33217e19 1.59364 0.796822 0.604215i $$-0.206513\pi$$
0.796822 + 0.604215i $$0.206513\pi$$
$$978$$ 0 0
$$979$$ 1.00912e19 0.366317
$$980$$ −7.82626e19 −2.82218
$$981$$ 0 0
$$982$$ 2.40878e19 0.857180
$$983$$ 2.52492e19 0.892584 0.446292 0.894887i $$-0.352744\pi$$
0.446292 + 0.894887i $$0.352744\pi$$
$$984$$ 0 0
$$985$$ −3.21724e19 −1.12240
$$986$$ 7.03139e19 2.43692
$$987$$ 0 0
$$988$$ 3.13709e18 0.107302
$$989$$ 5.24827e17 0.0178337
$$990$$ 0 0
$$991$$ 1.90400e19 0.638539 0.319269 0.947664i $$-0.396562\pi$$
0.319269 + 0.947664i $$0.396562\pi$$
$$992$$ 9.40807e18 0.313455
$$993$$ 0 0
$$994$$ −7.85236e19 −2.58219
$$995$$ 7.06961e19 2.30965
$$996$$ 0 0
$$997$$ 3.63643e19 1.17262 0.586309 0.810087i $$-0.300580\pi$$
0.586309 + 0.810087i $$0.300580\pi$$
$$998$$ −1.84717e19 −0.591779
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 99.14.a.a.1.1 1
3.2 odd 2 33.14.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.14.a.a.1.1 1 3.2 odd 2
99.14.a.a.1.1 1 1.1 even 1 trivial