# Properties

 Label 192.9.g.b.127.2 Level $192$ Weight $9$ Character 192.127 Analytic conductor $78.217$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,9,Mod(127,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 192.g (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$78.2166931317$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 127.2 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 192.127 Dual form 192.9.g.b.127.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+46.7654i q^{3} +90.0000 q^{5} -921.451i q^{7} -2187.00 q^{9} +O(q^{10})$$ $$q+46.7654i q^{3} +90.0000 q^{5} -921.451i q^{7} -2187.00 q^{9} +5175.37i q^{11} +16358.0 q^{13} +4208.88i q^{15} -42678.0 q^{17} -114447. i q^{19} +43092.0 q^{21} +250662. i q^{23} -382525. q^{25} -102276. i q^{27} +1.27053e6 q^{29} +512943. i q^{31} -242028. q^{33} -82930.6i q^{35} +2.26214e6 q^{37} +764988. i q^{39} -872694. q^{41} -1.66208e6i q^{43} -196830. q^{45} -797007. i q^{47} +4.91573e6 q^{49} -1.99585e6i q^{51} -1.06169e6 q^{53} +465783. i q^{55} +5.35216e6 q^{57} +2.33723e7i q^{59} -1.53010e7 q^{61} +2.01521e6i q^{63} +1.47222e6 q^{65} +9.18022e6i q^{67} -1.17223e7 q^{69} -2.28310e7i q^{71} +1.89164e7 q^{73} -1.78889e7i q^{75} +4.76885e6 q^{77} +5.41593e7i q^{79} +4.78297e6 q^{81} +6.52089e7i q^{83} -3.84102e6 q^{85} +5.94168e7i q^{87} -8.98132e7 q^{89} -1.50731e7i q^{91} -2.39880e7 q^{93} -1.03002e7i q^{95} -7.57782e7 q^{97} -1.13185e7i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 180 q^{5} - 4374 q^{9}+O(q^{10})$$ 2 * q + 180 * q^5 - 4374 * q^9 $$2 q + 180 q^{5} - 4374 q^{9} + 32716 q^{13} - 85356 q^{17} + 86184 q^{21} - 765050 q^{25} + 2541060 q^{29} - 484056 q^{33} + 4524284 q^{37} - 1745388 q^{41} - 393660 q^{45} + 9831458 q^{49} - 2123388 q^{53} + 10704312 q^{57} - 30602020 q^{61} + 2944440 q^{65} - 23444640 q^{69} + 37832708 q^{73} + 9537696 q^{77} + 9565938 q^{81} - 7682040 q^{85} - 179626428 q^{89} - 47975976 q^{93} - 151556476 q^{97}+O(q^{100})$$ 2 * q + 180 * q^5 - 4374 * q^9 + 32716 * q^13 - 85356 * q^17 + 86184 * q^21 - 765050 * q^25 + 2541060 * q^29 - 484056 * q^33 + 4524284 * q^37 - 1745388 * q^41 - 393660 * q^45 + 9831458 * q^49 - 2123388 * q^53 + 10704312 * q^57 - 30602020 * q^61 + 2944440 * q^65 - 23444640 * q^69 + 37832708 * q^73 + 9537696 * q^77 + 9565938 * q^81 - 7682040 * q^85 - 179626428 * q^89 - 47975976 * q^93 - 151556476 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 46.7654i 0.577350i
$$4$$ 0 0
$$5$$ 90.0000 0.144000 0.0720000 0.997405i $$-0.477062\pi$$
0.0720000 + 0.997405i $$0.477062\pi$$
$$6$$ 0 0
$$7$$ − 921.451i − 0.383778i −0.981417 0.191889i $$-0.938539\pi$$
0.981417 0.191889i $$-0.0614614\pi$$
$$8$$ 0 0
$$9$$ −2187.00 −0.333333
$$10$$ 0 0
$$11$$ 5175.37i 0.353485i 0.984257 + 0.176742i $$0.0565559\pi$$
−0.984257 + 0.176742i $$0.943444\pi$$
$$12$$ 0 0
$$13$$ 16358.0 0.572739 0.286370 0.958119i $$-0.407552\pi$$
0.286370 + 0.958119i $$0.407552\pi$$
$$14$$ 0 0
$$15$$ 4208.88i 0.0831384i
$$16$$ 0 0
$$17$$ −42678.0 −0.510985 −0.255493 0.966811i $$-0.582238\pi$$
−0.255493 + 0.966811i $$0.582238\pi$$
$$18$$ 0 0
$$19$$ − 114447.i − 0.878193i −0.898440 0.439096i $$-0.855299\pi$$
0.898440 0.439096i $$-0.144701\pi$$
$$20$$ 0 0
$$21$$ 43092.0 0.221574
$$22$$ 0 0
$$23$$ 250662.i 0.895731i 0.894101 + 0.447866i $$0.147816\pi$$
−0.894101 + 0.447866i $$0.852184\pi$$
$$24$$ 0 0
$$25$$ −382525. −0.979264
$$26$$ 0 0
$$27$$ − 102276.i − 0.192450i
$$28$$ 0 0
$$29$$ 1.27053e6 1.79636 0.898179 0.439630i $$-0.144890\pi$$
0.898179 + 0.439630i $$0.144890\pi$$
$$30$$ 0 0
$$31$$ 512943.i 0.555421i 0.960665 + 0.277711i $$0.0895757\pi$$
−0.960665 + 0.277711i $$0.910424\pi$$
$$32$$ 0 0
$$33$$ −242028. −0.204084
$$34$$ 0 0
$$35$$ − 82930.6i − 0.0552640i
$$36$$ 0 0
$$37$$ 2.26214e6 1.20702 0.603508 0.797357i $$-0.293769\pi$$
0.603508 + 0.797357i $$0.293769\pi$$
$$38$$ 0 0
$$39$$ 764988.i 0.330671i
$$40$$ 0 0
$$41$$ −872694. −0.308835 −0.154418 0.988006i $$-0.549350\pi$$
−0.154418 + 0.988006i $$0.549350\pi$$
$$42$$ 0 0
$$43$$ − 1.66208e6i − 0.486160i −0.970006 0.243080i $$-0.921842\pi$$
0.970006 0.243080i $$-0.0781577\pi$$
$$44$$ 0 0
$$45$$ −196830. −0.0480000
$$46$$ 0 0
$$47$$ − 797007.i − 0.163332i −0.996660 0.0816659i $$-0.973976\pi$$
0.996660 0.0816659i $$-0.0260240\pi$$
$$48$$ 0 0
$$49$$ 4.91573e6 0.852714
$$50$$ 0 0
$$51$$ − 1.99585e6i − 0.295017i
$$52$$ 0 0
$$53$$ −1.06169e6 −0.134554 −0.0672769 0.997734i $$-0.521431\pi$$
−0.0672769 + 0.997734i $$0.521431\pi$$
$$54$$ 0 0
$$55$$ 465783.i 0.0509018i
$$56$$ 0 0
$$57$$ 5.35216e6 0.507025
$$58$$ 0 0
$$59$$ 2.33723e7i 1.92883i 0.264402 + 0.964413i $$0.414826\pi$$
−0.264402 + 0.964413i $$0.585174\pi$$
$$60$$ 0 0
$$61$$ −1.53010e7 −1.10510 −0.552549 0.833480i $$-0.686345\pi$$
−0.552549 + 0.833480i $$0.686345\pi$$
$$62$$ 0 0
$$63$$ 2.01521e6i 0.127926i
$$64$$ 0 0
$$65$$ 1.47222e6 0.0824744
$$66$$ 0 0
$$67$$ 9.18022e6i 0.455569i 0.973712 + 0.227784i $$0.0731481\pi$$
−0.973712 + 0.227784i $$0.926852\pi$$
$$68$$ 0 0
$$69$$ −1.17223e7 −0.517151
$$70$$ 0 0
$$71$$ − 2.28310e7i − 0.898444i −0.893420 0.449222i $$-0.851701\pi$$
0.893420 0.449222i $$-0.148299\pi$$
$$72$$ 0 0
$$73$$ 1.89164e7 0.666110 0.333055 0.942907i $$-0.391921\pi$$
0.333055 + 0.942907i $$0.391921\pi$$
$$74$$ 0 0
$$75$$ − 1.78889e7i − 0.565378i
$$76$$ 0 0
$$77$$ 4.76885e6 0.135660
$$78$$ 0 0
$$79$$ 5.41593e7i 1.39048i 0.718778 + 0.695240i $$0.244702\pi$$
−0.718778 + 0.695240i $$0.755298\pi$$
$$80$$ 0 0
$$81$$ 4.78297e6 0.111111
$$82$$ 0 0
$$83$$ 6.52089e7i 1.37403i 0.726645 + 0.687013i $$0.241078\pi$$
−0.726645 + 0.687013i $$0.758922\pi$$
$$84$$ 0 0
$$85$$ −3.84102e6 −0.0735819
$$86$$ 0 0
$$87$$ 5.94168e7i 1.03713i
$$88$$ 0 0
$$89$$ −8.98132e7 −1.43146 −0.715732 0.698375i $$-0.753907\pi$$
−0.715732 + 0.698375i $$0.753907\pi$$
$$90$$ 0 0
$$91$$ − 1.50731e7i − 0.219805i
$$92$$ 0 0
$$93$$ −2.39880e7 −0.320673
$$94$$ 0 0
$$95$$ − 1.03002e7i − 0.126460i
$$96$$ 0 0
$$97$$ −7.57782e7 −0.855968 −0.427984 0.903786i $$-0.640776\pi$$
−0.427984 + 0.903786i $$0.640776\pi$$
$$98$$ 0 0
$$99$$ − 1.13185e7i − 0.117828i
$$100$$ 0 0
$$101$$ 1.68976e8 1.62383 0.811914 0.583778i $$-0.198426\pi$$
0.811914 + 0.583778i $$0.198426\pi$$
$$102$$ 0 0
$$103$$ 1.59060e8i 1.41323i 0.707601 + 0.706613i $$0.249778\pi$$
−0.707601 + 0.706613i $$0.750222\pi$$
$$104$$ 0 0
$$105$$ 3.87828e6 0.0319067
$$106$$ 0 0
$$107$$ 1.75980e8i 1.34254i 0.741213 + 0.671270i $$0.234251\pi$$
−0.741213 + 0.671270i $$0.765749\pi$$
$$108$$ 0 0
$$109$$ 1.37980e7 0.0977483 0.0488742 0.998805i $$-0.484437\pi$$
0.0488742 + 0.998805i $$0.484437\pi$$
$$110$$ 0 0
$$111$$ 1.05790e8i 0.696871i
$$112$$ 0 0
$$113$$ −9.25151e7 −0.567412 −0.283706 0.958911i $$-0.591564\pi$$
−0.283706 + 0.958911i $$0.591564\pi$$
$$114$$ 0 0
$$115$$ 2.25596e7i 0.128985i
$$116$$ 0 0
$$117$$ −3.57749e7 −0.190913
$$118$$ 0 0
$$119$$ 3.93257e7i 0.196105i
$$120$$ 0 0
$$121$$ 1.87574e8 0.875049
$$122$$ 0 0
$$123$$ − 4.08119e7i − 0.178306i
$$124$$ 0 0
$$125$$ −6.95835e7 −0.285014
$$126$$ 0 0
$$127$$ 2.21744e8i 0.852389i 0.904632 + 0.426195i $$0.140146\pi$$
−0.904632 + 0.426195i $$0.859854\pi$$
$$128$$ 0 0
$$129$$ 7.77279e7 0.280684
$$130$$ 0 0
$$131$$ 3.60662e8i 1.22466i 0.790603 + 0.612329i $$0.209767\pi$$
−0.790603 + 0.612329i $$0.790233\pi$$
$$132$$ 0 0
$$133$$ −1.05457e8 −0.337031
$$134$$ 0 0
$$135$$ − 9.20483e6i − 0.0277128i
$$136$$ 0 0
$$137$$ 4.40725e8 1.25108 0.625541 0.780191i $$-0.284878\pi$$
0.625541 + 0.780191i $$0.284878\pi$$
$$138$$ 0 0
$$139$$ 5.44242e8i 1.45792i 0.684558 + 0.728958i $$0.259995\pi$$
−0.684558 + 0.728958i $$0.740005\pi$$
$$140$$ 0 0
$$141$$ 3.72723e7 0.0942996
$$142$$ 0 0
$$143$$ 8.46587e7i 0.202454i
$$144$$ 0 0
$$145$$ 1.14348e8 0.258676
$$146$$ 0 0
$$147$$ 2.29886e8i 0.492315i
$$148$$ 0 0
$$149$$ −4.40746e7 −0.0894217 −0.0447109 0.999000i $$-0.514237\pi$$
−0.0447109 + 0.999000i $$0.514237\pi$$
$$150$$ 0 0
$$151$$ − 1.11931e8i − 0.215300i −0.994189 0.107650i $$-0.965667\pi$$
0.994189 0.107650i $$-0.0343326\pi$$
$$152$$ 0 0
$$153$$ 9.33368e7 0.170328
$$154$$ 0 0
$$155$$ 4.61649e7i 0.0799807i
$$156$$ 0 0
$$157$$ −2.89510e8 −0.476503 −0.238251 0.971204i $$-0.576574\pi$$
−0.238251 + 0.971204i $$0.576574\pi$$
$$158$$ 0 0
$$159$$ − 4.96505e7i − 0.0776847i
$$160$$ 0 0
$$161$$ 2.30973e8 0.343762
$$162$$ 0 0
$$163$$ 3.06936e8i 0.434808i 0.976082 + 0.217404i $$0.0697589\pi$$
−0.976082 + 0.217404i $$0.930241\pi$$
$$164$$ 0 0
$$165$$ −2.17825e7 −0.0293882
$$166$$ 0 0
$$167$$ 1.19626e9i 1.53801i 0.639243 + 0.769004i $$0.279248\pi$$
−0.639243 + 0.769004i $$0.720752\pi$$
$$168$$ 0 0
$$169$$ −5.48147e8 −0.671970
$$170$$ 0 0
$$171$$ 2.50296e8i 0.292731i
$$172$$ 0 0
$$173$$ −8.26090e7 −0.0922237 −0.0461119 0.998936i $$-0.514683\pi$$
−0.0461119 + 0.998936i $$0.514683\pi$$
$$174$$ 0 0
$$175$$ 3.52478e8i 0.375820i
$$176$$ 0 0
$$177$$ −1.09301e9 −1.11361
$$178$$ 0 0
$$179$$ − 1.13619e9i − 1.10673i −0.832940 0.553364i $$-0.813344\pi$$
0.832940 0.553364i $$-0.186656\pi$$
$$180$$ 0 0
$$181$$ 8.20024e8 0.764033 0.382017 0.924155i $$-0.375230\pi$$
0.382017 + 0.924155i $$0.375230\pi$$
$$182$$ 0 0
$$183$$ − 7.15557e8i − 0.638029i
$$184$$ 0 0
$$185$$ 2.03593e8 0.173810
$$186$$ 0 0
$$187$$ − 2.20874e8i − 0.180625i
$$188$$ 0 0
$$189$$ −9.42422e7 −0.0738581
$$190$$ 0 0
$$191$$ 1.24962e9i 0.938956i 0.882944 + 0.469478i $$0.155558\pi$$
−0.882944 + 0.469478i $$0.844442\pi$$
$$192$$ 0 0
$$193$$ −2.01195e9 −1.45007 −0.725033 0.688714i $$-0.758176\pi$$
−0.725033 + 0.688714i $$0.758176\pi$$
$$194$$ 0 0
$$195$$ 6.88489e7i 0.0476166i
$$196$$ 0 0
$$197$$ 6.32559e8 0.419988 0.209994 0.977703i $$-0.432656\pi$$
0.209994 + 0.977703i $$0.432656\pi$$
$$198$$ 0 0
$$199$$ − 1.45714e9i − 0.929156i −0.885532 0.464578i $$-0.846206\pi$$
0.885532 0.464578i $$-0.153794\pi$$
$$200$$ 0 0
$$201$$ −4.29317e8 −0.263023
$$202$$ 0 0
$$203$$ − 1.17073e9i − 0.689403i
$$204$$ 0 0
$$205$$ −7.85425e7 −0.0444722
$$206$$ 0 0
$$207$$ − 5.48199e8i − 0.298577i
$$208$$ 0 0
$$209$$ 5.92305e8 0.310428
$$210$$ 0 0
$$211$$ − 3.57401e8i − 0.180312i −0.995928 0.0901562i $$-0.971263\pi$$
0.995928 0.0901562i $$-0.0287366\pi$$
$$212$$ 0 0
$$213$$ 1.06770e9 0.518717
$$214$$ 0 0
$$215$$ − 1.49587e8i − 0.0700070i
$$216$$ 0 0
$$217$$ 4.72652e8 0.213159
$$218$$ 0 0
$$219$$ 8.84630e8i 0.384579i
$$220$$ 0 0
$$221$$ −6.98127e8 −0.292661
$$222$$ 0 0
$$223$$ − 1.66879e9i − 0.674813i −0.941359 0.337406i $$-0.890450\pi$$
0.941359 0.337406i $$-0.109550\pi$$
$$224$$ 0 0
$$225$$ 8.36582e8 0.326421
$$226$$ 0 0
$$227$$ − 3.34498e9i − 1.25977i −0.776690 0.629883i $$-0.783103\pi$$
0.776690 0.629883i $$-0.216897\pi$$
$$228$$ 0 0
$$229$$ −1.42427e9 −0.517906 −0.258953 0.965890i $$-0.583378\pi$$
−0.258953 + 0.965890i $$0.583378\pi$$
$$230$$ 0 0
$$231$$ 2.23017e8i 0.0783231i
$$232$$ 0 0
$$233$$ 3.82745e9 1.29863 0.649316 0.760519i $$-0.275055\pi$$
0.649316 + 0.760519i $$0.275055\pi$$
$$234$$ 0 0
$$235$$ − 7.17306e7i − 0.0235198i
$$236$$ 0 0
$$237$$ −2.53278e9 −0.802794
$$238$$ 0 0
$$239$$ − 3.55074e9i − 1.08825i −0.839006 0.544123i $$-0.816863\pi$$
0.839006 0.544123i $$-0.183137\pi$$
$$240$$ 0 0
$$241$$ 2.46612e9 0.731048 0.365524 0.930802i $$-0.380890\pi$$
0.365524 + 0.930802i $$0.380890\pi$$
$$242$$ 0 0
$$243$$ 2.23677e8i 0.0641500i
$$244$$ 0 0
$$245$$ 4.42416e8 0.122791
$$246$$ 0 0
$$247$$ − 1.87212e9i − 0.502975i
$$248$$ 0 0
$$249$$ −3.04952e9 −0.793294
$$250$$ 0 0
$$251$$ 2.51003e9i 0.632389i 0.948694 + 0.316195i $$0.102405\pi$$
−0.948694 + 0.316195i $$0.897595\pi$$
$$252$$ 0 0
$$253$$ −1.29727e9 −0.316627
$$254$$ 0 0
$$255$$ − 1.79627e8i − 0.0424825i
$$256$$ 0 0
$$257$$ 3.24232e9 0.743231 0.371615 0.928387i $$-0.378804\pi$$
0.371615 + 0.928387i $$0.378804\pi$$
$$258$$ 0 0
$$259$$ − 2.08445e9i − 0.463226i
$$260$$ 0 0
$$261$$ −2.77865e9 −0.598786
$$262$$ 0 0
$$263$$ − 8.13109e9i − 1.69952i −0.527172 0.849759i $$-0.676748\pi$$
0.527172 0.849759i $$-0.323252\pi$$
$$264$$ 0 0
$$265$$ −9.55525e7 −0.0193757
$$266$$ 0 0
$$267$$ − 4.20015e9i − 0.826456i
$$268$$ 0 0
$$269$$ 7.44829e9 1.42248 0.711242 0.702947i $$-0.248133\pi$$
0.711242 + 0.702947i $$0.248133\pi$$
$$270$$ 0 0
$$271$$ 7.41657e7i 0.0137507i 0.999976 + 0.00687537i $$0.00218851\pi$$
−0.999976 + 0.00687537i $$0.997811\pi$$
$$272$$ 0 0
$$273$$ 7.04899e8 0.126904
$$274$$ 0 0
$$275$$ − 1.97971e9i − 0.346155i
$$276$$ 0 0
$$277$$ −2.00834e9 −0.341128 −0.170564 0.985347i $$-0.554559\pi$$
−0.170564 + 0.985347i $$0.554559\pi$$
$$278$$ 0 0
$$279$$ − 1.12181e9i − 0.185140i
$$280$$ 0 0
$$281$$ −1.37037e9 −0.219793 −0.109896 0.993943i $$-0.535052\pi$$
−0.109896 + 0.993943i $$0.535052\pi$$
$$282$$ 0 0
$$283$$ − 6.48210e9i − 1.01058i −0.862950 0.505289i $$-0.831386\pi$$
0.862950 0.505289i $$-0.168614\pi$$
$$284$$ 0 0
$$285$$ 4.81694e8 0.0730116
$$286$$ 0 0
$$287$$ 8.04145e8i 0.118524i
$$288$$ 0 0
$$289$$ −5.15435e9 −0.738894
$$290$$ 0 0
$$291$$ − 3.54380e9i − 0.494193i
$$292$$ 0 0
$$293$$ 1.10320e10 1.49687 0.748436 0.663207i $$-0.230805\pi$$
0.748436 + 0.663207i $$0.230805\pi$$
$$294$$ 0 0
$$295$$ 2.10350e9i 0.277751i
$$296$$ 0 0
$$297$$ 5.29315e8 0.0680281
$$298$$ 0 0
$$299$$ 4.10034e9i 0.513020i
$$300$$ 0 0
$$301$$ −1.53153e9 −0.186577
$$302$$ 0 0
$$303$$ 7.90223e9i 0.937517i
$$304$$ 0 0
$$305$$ −1.37709e9 −0.159134
$$306$$ 0 0
$$307$$ − 9.80026e9i − 1.10328i −0.834084 0.551638i $$-0.814003\pi$$
0.834084 0.551638i $$-0.185997\pi$$
$$308$$ 0 0
$$309$$ −7.43849e9 −0.815926
$$310$$ 0 0
$$311$$ − 6.60957e9i − 0.706531i −0.935523 0.353266i $$-0.885071\pi$$
0.935523 0.353266i $$-0.114929\pi$$
$$312$$ 0 0
$$313$$ −8.98046e9 −0.935667 −0.467834 0.883817i $$-0.654965\pi$$
−0.467834 + 0.883817i $$0.654965\pi$$
$$314$$ 0 0
$$315$$ 1.81369e8i 0.0184213i
$$316$$ 0 0
$$317$$ 1.29468e10 1.28211 0.641057 0.767493i $$-0.278496\pi$$
0.641057 + 0.767493i $$0.278496\pi$$
$$318$$ 0 0
$$319$$ 6.57546e9i 0.634985i
$$320$$ 0 0
$$321$$ −8.22975e9 −0.775115
$$322$$ 0 0
$$323$$ 4.88437e9i 0.448744i
$$324$$ 0 0
$$325$$ −6.25734e9 −0.560863
$$326$$ 0 0
$$327$$ 6.45267e8i 0.0564350i
$$328$$ 0 0
$$329$$ −7.34403e8 −0.0626831
$$330$$ 0 0
$$331$$ 8.27574e9i 0.689438i 0.938706 + 0.344719i $$0.112026\pi$$
−0.938706 + 0.344719i $$0.887974\pi$$
$$332$$ 0 0
$$333$$ −4.94730e9 −0.402339
$$334$$ 0 0
$$335$$ 8.26220e8i 0.0656019i
$$336$$ 0 0
$$337$$ −1.22715e9 −0.0951436 −0.0475718 0.998868i $$-0.515148\pi$$
−0.0475718 + 0.998868i $$0.515148\pi$$
$$338$$ 0 0
$$339$$ − 4.32650e9i − 0.327596i
$$340$$ 0 0
$$341$$ −2.65467e9 −0.196333
$$342$$ 0 0
$$343$$ − 9.84159e9i − 0.711031i
$$344$$ 0 0
$$345$$ −1.05501e9 −0.0744697
$$346$$ 0 0
$$347$$ 1.34897e10i 0.930429i 0.885198 + 0.465214i $$0.154023\pi$$
−0.885198 + 0.465214i $$0.845977\pi$$
$$348$$ 0 0
$$349$$ −9.63945e9 −0.649756 −0.324878 0.945756i $$-0.605323\pi$$
−0.324878 + 0.945756i $$0.605323\pi$$
$$350$$ 0 0
$$351$$ − 1.67303e9i − 0.110224i
$$352$$ 0 0
$$353$$ 2.39489e10 1.54237 0.771183 0.636614i $$-0.219666\pi$$
0.771183 + 0.636614i $$0.219666\pi$$
$$354$$ 0 0
$$355$$ − 2.05479e9i − 0.129376i
$$356$$ 0 0
$$357$$ −1.83908e9 −0.113221
$$358$$ 0 0
$$359$$ 1.22556e9i 0.0737831i 0.999319 + 0.0368915i $$0.0117456\pi$$
−0.999319 + 0.0368915i $$0.988254\pi$$
$$360$$ 0 0
$$361$$ 3.88545e9 0.228777
$$362$$ 0 0
$$363$$ 8.77199e9i 0.505210i
$$364$$ 0 0
$$365$$ 1.70247e9 0.0959198
$$366$$ 0 0
$$367$$ 2.19414e10i 1.20948i 0.796422 + 0.604742i $$0.206724\pi$$
−0.796422 + 0.604742i $$0.793276\pi$$
$$368$$ 0 0
$$369$$ 1.90858e9 0.102945
$$370$$ 0 0
$$371$$ 9.78299e8i 0.0516388i
$$372$$ 0 0
$$373$$ −2.32459e10 −1.20091 −0.600457 0.799657i $$-0.705014\pi$$
−0.600457 + 0.799657i $$0.705014\pi$$
$$374$$ 0 0
$$375$$ − 3.25410e9i − 0.164553i
$$376$$ 0 0
$$377$$ 2.07833e10 1.02884
$$378$$ 0 0
$$379$$ 3.77864e10i 1.83138i 0.401887 + 0.915689i $$0.368354\pi$$
−0.401887 + 0.915689i $$0.631646\pi$$
$$380$$ 0 0
$$381$$ −1.03700e10 −0.492127
$$382$$ 0 0
$$383$$ 3.04157e10i 1.41352i 0.707452 + 0.706761i $$0.249845\pi$$
−0.707452 + 0.706761i $$0.750155\pi$$
$$384$$ 0 0
$$385$$ 4.29196e8 0.0195350
$$386$$ 0 0
$$387$$ 3.63498e9i 0.162053i
$$388$$ 0 0
$$389$$ 1.53539e10 0.670532 0.335266 0.942123i $$-0.391174\pi$$
0.335266 + 0.942123i $$0.391174\pi$$
$$390$$ 0 0
$$391$$ − 1.06978e10i − 0.457706i
$$392$$ 0 0
$$393$$ −1.68665e10 −0.707057
$$394$$ 0 0
$$395$$ 4.87434e9i 0.200229i
$$396$$ 0 0
$$397$$ 2.10954e10 0.849231 0.424615 0.905374i $$-0.360409\pi$$
0.424615 + 0.905374i $$0.360409\pi$$
$$398$$ 0 0
$$399$$ − 4.93175e9i − 0.194585i
$$400$$ 0 0
$$401$$ 3.89388e10 1.50593 0.752966 0.658059i $$-0.228622\pi$$
0.752966 + 0.658059i $$0.228622\pi$$
$$402$$ 0 0
$$403$$ 8.39073e9i 0.318112i
$$404$$ 0 0
$$405$$ 4.30467e8 0.0160000
$$406$$ 0 0
$$407$$ 1.17074e10i 0.426661i
$$408$$ 0 0
$$409$$ −1.41946e10 −0.507258 −0.253629 0.967302i $$-0.581624\pi$$
−0.253629 + 0.967302i $$0.581624\pi$$
$$410$$ 0 0
$$411$$ 2.06107e10i 0.722312i
$$412$$ 0 0
$$413$$ 2.15364e10 0.740241
$$414$$ 0 0
$$415$$ 5.86881e9i 0.197860i
$$416$$ 0 0
$$417$$ −2.54517e10 −0.841729
$$418$$ 0 0
$$419$$ − 3.39978e10i − 1.10305i −0.834159 0.551524i $$-0.814047\pi$$
0.834159 0.551524i $$-0.185953\pi$$
$$420$$ 0 0
$$421$$ 1.11314e10 0.354342 0.177171 0.984180i $$-0.443305\pi$$
0.177171 + 0.984180i $$0.443305\pi$$
$$422$$ 0 0
$$423$$ 1.74305e9i 0.0544439i
$$424$$ 0 0
$$425$$ 1.63254e10 0.500389
$$426$$ 0 0
$$427$$ 1.40991e10i 0.424112i
$$428$$ 0 0
$$429$$ −3.95909e9 −0.116887
$$430$$ 0 0
$$431$$ 4.91968e10i 1.42570i 0.701318 + 0.712849i $$0.252595\pi$$
−0.701318 + 0.712849i $$0.747405\pi$$
$$432$$ 0 0
$$433$$ −4.76011e10 −1.35414 −0.677072 0.735916i $$-0.736752\pi$$
−0.677072 + 0.735916i $$0.736752\pi$$
$$434$$ 0 0
$$435$$ 5.34751e9i 0.149346i
$$436$$ 0 0
$$437$$ 2.86876e10 0.786625
$$438$$ 0 0
$$439$$ 1.30610e10i 0.351657i 0.984421 + 0.175829i $$0.0562605\pi$$
−0.984421 + 0.175829i $$0.943740\pi$$
$$440$$ 0 0
$$441$$ −1.07507e10 −0.284238
$$442$$ 0 0
$$443$$ − 1.48073e10i − 0.384468i −0.981349 0.192234i $$-0.938427\pi$$
0.981349 0.192234i $$-0.0615732\pi$$
$$444$$ 0 0
$$445$$ −8.08319e9 −0.206131
$$446$$ 0 0
$$447$$ − 2.06116e9i − 0.0516276i
$$448$$ 0 0
$$449$$ 2.97486e9 0.0731949 0.0365975 0.999330i $$-0.488348\pi$$
0.0365975 + 0.999330i $$0.488348\pi$$
$$450$$ 0 0
$$451$$ − 4.51651e9i − 0.109168i
$$452$$ 0 0
$$453$$ 5.23451e9 0.124304
$$454$$ 0 0
$$455$$ − 1.35658e9i − 0.0316519i
$$456$$ 0 0
$$457$$ −3.07215e10 −0.704333 −0.352166 0.935937i $$-0.614555\pi$$
−0.352166 + 0.935937i $$0.614555\pi$$
$$458$$ 0 0
$$459$$ 4.36493e9i 0.0983392i
$$460$$ 0 0
$$461$$ −8.09097e9 −0.179142 −0.0895709 0.995980i $$-0.528550\pi$$
−0.0895709 + 0.995980i $$0.528550\pi$$
$$462$$ 0 0
$$463$$ 4.56191e10i 0.992711i 0.868119 + 0.496356i $$0.165329\pi$$
−0.868119 + 0.496356i $$0.834671\pi$$
$$464$$ 0 0
$$465$$ −2.15892e9 −0.0461769
$$466$$ 0 0
$$467$$ − 4.09800e10i − 0.861598i −0.902448 0.430799i $$-0.858232\pi$$
0.902448 0.430799i $$-0.141768\pi$$
$$468$$ 0 0
$$469$$ 8.45913e9 0.174837
$$470$$ 0 0
$$471$$ − 1.35391e10i − 0.275109i
$$472$$ 0 0
$$473$$ 8.60189e9 0.171850
$$474$$ 0 0
$$475$$ 4.37788e10i 0.859983i
$$476$$ 0 0
$$477$$ 2.32192e9 0.0448513
$$478$$ 0 0
$$479$$ 4.58901e10i 0.871720i 0.900014 + 0.435860i $$0.143556\pi$$
−0.900014 + 0.435860i $$0.856444\pi$$
$$480$$ 0 0
$$481$$ 3.70041e10 0.691305
$$482$$ 0 0
$$483$$ 1.08015e10i 0.198471i
$$484$$ 0 0
$$485$$ −6.82004e9 −0.123259
$$486$$ 0 0
$$487$$ − 1.01277e11i − 1.80051i −0.435360 0.900256i $$-0.643379\pi$$
0.435360 0.900256i $$-0.356621\pi$$
$$488$$ 0 0
$$489$$ −1.43540e10 −0.251036
$$490$$ 0 0
$$491$$ 8.24892e10i 1.41929i 0.704559 + 0.709645i $$0.251145\pi$$
−0.704559 + 0.709645i $$0.748855\pi$$
$$492$$ 0 0
$$493$$ −5.42237e10 −0.917913
$$494$$ 0 0
$$495$$ − 1.01867e9i − 0.0169673i
$$496$$ 0 0
$$497$$ −2.10376e10 −0.344803
$$498$$ 0 0
$$499$$ − 9.46533e10i − 1.52663i −0.646026 0.763315i $$-0.723570\pi$$
0.646026 0.763315i $$-0.276430\pi$$
$$500$$ 0 0
$$501$$ −5.59434e10 −0.887970
$$502$$ 0 0
$$503$$ − 9.80310e10i − 1.53141i −0.643192 0.765705i $$-0.722390\pi$$
0.643192 0.765705i $$-0.277610\pi$$
$$504$$ 0 0
$$505$$ 1.52078e10 0.233831
$$506$$ 0 0
$$507$$ − 2.56343e10i − 0.387962i
$$508$$ 0 0
$$509$$ −1.42476e10 −0.212261 −0.106131 0.994352i $$-0.533846\pi$$
−0.106131 + 0.994352i $$0.533846\pi$$
$$510$$ 0 0
$$511$$ − 1.74305e10i − 0.255638i
$$512$$ 0 0
$$513$$ −1.17052e10 −0.169008
$$514$$ 0 0
$$515$$ 1.43154e10i 0.203504i
$$516$$ 0 0
$$517$$ 4.12480e9 0.0577352
$$518$$ 0 0
$$519$$ − 3.86324e9i − 0.0532454i
$$520$$ 0 0
$$521$$ 8.85973e10 1.20246 0.601229 0.799077i $$-0.294678\pi$$
0.601229 + 0.799077i $$0.294678\pi$$
$$522$$ 0 0
$$523$$ 3.72959e10i 0.498488i 0.968441 + 0.249244i $$0.0801821\pi$$
−0.968441 + 0.249244i $$0.919818\pi$$
$$524$$ 0 0
$$525$$ −1.64838e10 −0.216980
$$526$$ 0 0
$$527$$ − 2.18914e10i − 0.283812i
$$528$$ 0 0
$$529$$ 1.54794e10 0.197665
$$530$$ 0 0
$$531$$ − 5.11152e10i − 0.642942i
$$532$$ 0 0
$$533$$ −1.42755e10 −0.176882
$$534$$ 0 0
$$535$$ 1.58382e10i 0.193326i
$$536$$ 0 0
$$537$$ 5.31346e10 0.638969
$$538$$ 0 0
$$539$$ 2.54407e10i 0.301421i
$$540$$ 0 0
$$541$$ −8.90331e10 −1.03935 −0.519676 0.854364i $$-0.673947\pi$$
−0.519676 + 0.854364i $$0.673947\pi$$
$$542$$ 0 0
$$543$$ 3.83487e10i 0.441115i
$$544$$ 0 0
$$545$$ 1.24182e9 0.0140758
$$546$$ 0 0
$$547$$ − 1.11679e11i − 1.24745i −0.781645 0.623723i $$-0.785619\pi$$
0.781645 0.623723i $$-0.214381\pi$$
$$548$$ 0 0
$$549$$ 3.34633e10 0.368366
$$550$$ 0 0
$$551$$ − 1.45408e11i − 1.57755i
$$552$$ 0 0
$$553$$ 4.99051e10 0.533636
$$554$$ 0 0
$$555$$ 9.52109e9i 0.100349i
$$556$$ 0 0
$$557$$ 1.51601e11 1.57501 0.787503 0.616310i $$-0.211373\pi$$
0.787503 + 0.616310i $$0.211373\pi$$
$$558$$ 0 0
$$559$$ − 2.71884e10i − 0.278443i
$$560$$ 0 0
$$561$$ 1.03293e10 0.104284
$$562$$ 0 0
$$563$$ − 9.17147e10i − 0.912863i −0.889759 0.456431i $$-0.849127\pi$$
0.889759 0.456431i $$-0.150873\pi$$
$$564$$ 0 0
$$565$$ −8.32636e9 −0.0817074
$$566$$ 0 0
$$567$$ − 4.40727e9i − 0.0426420i
$$568$$ 0 0
$$569$$ −7.26500e10 −0.693085 −0.346543 0.938034i $$-0.612644\pi$$
−0.346543 + 0.938034i $$0.612644\pi$$
$$570$$ 0 0
$$571$$ − 6.84484e10i − 0.643901i −0.946757 0.321950i $$-0.895662\pi$$
0.946757 0.321950i $$-0.104338\pi$$
$$572$$ 0 0
$$573$$ −5.84391e10 −0.542107
$$574$$ 0 0
$$575$$ − 9.58846e10i − 0.877158i
$$576$$ 0 0
$$577$$ −1.62814e11 −1.46889 −0.734445 0.678668i $$-0.762557\pi$$
−0.734445 + 0.678668i $$0.762557\pi$$
$$578$$ 0 0
$$579$$ − 9.40896e10i − 0.837196i
$$580$$ 0 0
$$581$$ 6.00869e10 0.527321
$$582$$ 0 0
$$583$$ − 5.49466e9i − 0.0475627i
$$584$$ 0 0
$$585$$ −3.21975e9 −0.0274915
$$586$$ 0 0
$$587$$ 3.09698e10i 0.260847i 0.991458 + 0.130423i $$0.0416337\pi$$
−0.991458 + 0.130423i $$0.958366\pi$$
$$588$$ 0 0
$$589$$ 5.87048e10 0.487767
$$590$$ 0 0
$$591$$ 2.95819e10i 0.242480i
$$592$$ 0 0
$$593$$ −1.25721e11 −1.01669 −0.508347 0.861152i $$-0.669743\pi$$
−0.508347 + 0.861152i $$0.669743\pi$$
$$594$$ 0 0
$$595$$ 3.53931e9i 0.0282391i
$$596$$ 0 0
$$597$$ 6.81437e10 0.536449
$$598$$ 0 0
$$599$$ − 8.69133e10i − 0.675117i −0.941305 0.337559i $$-0.890399\pi$$
0.941305 0.337559i $$-0.109601\pi$$
$$600$$ 0 0
$$601$$ 1.19412e11 0.915269 0.457634 0.889140i $$-0.348697\pi$$
0.457634 + 0.889140i $$0.348697\pi$$
$$602$$ 0 0
$$603$$ − 2.00771e10i − 0.151856i
$$604$$ 0 0
$$605$$ 1.68817e10 0.126007
$$606$$ 0 0
$$607$$ 2.21728e11i 1.63330i 0.577136 + 0.816648i $$0.304170\pi$$
−0.577136 + 0.816648i $$0.695830\pi$$
$$608$$ 0 0
$$609$$ 5.47497e10 0.398027
$$610$$ 0 0
$$611$$ − 1.30374e10i − 0.0935464i
$$612$$ 0 0
$$613$$ 2.09579e11 1.48425 0.742123 0.670263i $$-0.233819\pi$$
0.742123 + 0.670263i $$0.233819\pi$$
$$614$$ 0 0
$$615$$ − 3.67307e9i − 0.0256761i
$$616$$ 0 0
$$617$$ −1.95007e11 −1.34558 −0.672788 0.739835i $$-0.734904\pi$$
−0.672788 + 0.739835i $$0.734904\pi$$
$$618$$ 0 0
$$619$$ − 1.27224e11i − 0.866574i −0.901256 0.433287i $$-0.857354\pi$$
0.901256 0.433287i $$-0.142646\pi$$
$$620$$ 0 0
$$621$$ 2.56367e10 0.172384
$$622$$ 0 0
$$623$$ 8.27585e10i 0.549364i
$$624$$ 0 0
$$625$$ 1.43161e11 0.938222
$$626$$ 0 0
$$627$$ 2.76994e10i 0.179226i
$$628$$ 0 0
$$629$$ −9.65437e10 −0.616767
$$630$$ 0 0
$$631$$ − 2.89620e11i − 1.82689i −0.406966 0.913443i $$-0.633413\pi$$
0.406966 0.913443i $$-0.366587\pi$$
$$632$$ 0 0
$$633$$ 1.67140e10 0.104103
$$634$$ 0 0
$$635$$ 1.99570e10i 0.122744i
$$636$$ 0 0
$$637$$ 8.04115e10 0.488383
$$638$$ 0 0
$$639$$ 4.99314e10i 0.299481i
$$640$$ 0 0
$$641$$ 1.34888e11 0.798991 0.399496 0.916735i $$-0.369185\pi$$
0.399496 + 0.916735i $$0.369185\pi$$
$$642$$ 0 0
$$643$$ − 1.40627e11i − 0.822670i −0.911484 0.411335i $$-0.865063\pi$$
0.911484 0.411335i $$-0.134937\pi$$
$$644$$ 0 0
$$645$$ 6.99551e9 0.0404185
$$646$$ 0 0
$$647$$ 9.22875e10i 0.526655i 0.964707 + 0.263327i $$0.0848199\pi$$
−0.964707 + 0.263327i $$0.915180\pi$$
$$648$$ 0 0
$$649$$ −1.20960e11 −0.681810
$$650$$ 0 0
$$651$$ 2.21038e10i 0.123067i
$$652$$ 0 0
$$653$$ −1.79607e11 −0.987805 −0.493902 0.869517i $$-0.664430\pi$$
−0.493902 + 0.869517i $$0.664430\pi$$
$$654$$ 0 0
$$655$$ 3.24596e10i 0.176351i
$$656$$ 0 0
$$657$$ −4.13701e10 −0.222037
$$658$$ 0 0
$$659$$ − 5.95800e10i − 0.315907i −0.987447 0.157953i $$-0.949510\pi$$
0.987447 0.157953i $$-0.0504896\pi$$
$$660$$ 0 0
$$661$$ −3.13906e11 −1.64435 −0.822174 0.569236i $$-0.807239\pi$$
−0.822174 + 0.569236i $$0.807239\pi$$
$$662$$ 0 0
$$663$$ − 3.26482e10i − 0.168968i
$$664$$ 0 0
$$665$$ −9.49116e9 −0.0485325
$$666$$ 0 0
$$667$$ 3.18474e11i 1.60905i
$$668$$ 0 0
$$669$$ 7.80418e10 0.389603
$$670$$ 0 0
$$671$$ − 7.91884e10i − 0.390635i
$$672$$ 0 0
$$673$$ 5.89957e10 0.287581 0.143791 0.989608i $$-0.454071\pi$$
0.143791 + 0.989608i $$0.454071\pi$$
$$674$$ 0 0
$$675$$ 3.91231e10i 0.188459i
$$676$$ 0 0
$$677$$ −8.71335e10 −0.414792 −0.207396 0.978257i $$-0.566499\pi$$
−0.207396 + 0.978257i $$0.566499\pi$$
$$678$$ 0 0
$$679$$ 6.98259e10i 0.328502i
$$680$$ 0 0
$$681$$ 1.56429e11 0.727326
$$682$$ 0 0
$$683$$ 4.38375e10i 0.201448i 0.994914 + 0.100724i $$0.0321159\pi$$
−0.994914 + 0.100724i $$0.967884\pi$$
$$684$$ 0 0
$$685$$ 3.96653e10 0.180156
$$686$$ 0 0
$$687$$ − 6.66066e10i − 0.299013i
$$688$$ 0 0
$$689$$ −1.73672e10 −0.0770642
$$690$$ 0 0
$$691$$ 2.55445e11i 1.12043i 0.828347 + 0.560216i $$0.189282\pi$$
−0.828347 + 0.560216i $$0.810718\pi$$
$$692$$ 0 0
$$693$$ −1.04295e10 −0.0452199
$$694$$ 0 0
$$695$$ 4.89818e10i 0.209940i
$$696$$ 0 0
$$697$$ 3.72448e10 0.157810
$$698$$ 0 0
$$699$$ 1.78992e11i 0.749766i
$$700$$ 0 0
$$701$$ −4.08255e11 −1.69067 −0.845337 0.534233i $$-0.820600\pi$$
−0.845337 + 0.534233i $$0.820600\pi$$
$$702$$ 0 0
$$703$$ − 2.58895e11i − 1.05999i
$$704$$ 0 0
$$705$$ 3.35451e9 0.0135791
$$706$$ 0 0
$$707$$ − 1.55703e11i − 0.623189i
$$708$$ 0 0
$$709$$ 9.56878e10 0.378679 0.189340 0.981912i $$-0.439365\pi$$
0.189340 + 0.981912i $$0.439365\pi$$
$$710$$ 0 0
$$711$$ − 1.18446e11i − 0.463493i
$$712$$ 0 0
$$713$$ −1.28576e11 −0.497508
$$714$$ 0 0
$$715$$ 7.61928e9i 0.0291534i
$$716$$ 0 0
$$717$$ 1.66051e11 0.628299
$$718$$ 0 0
$$719$$ 3.37132e11i 1.26149i 0.775990 + 0.630746i $$0.217251\pi$$
−0.775990 + 0.630746i $$0.782749\pi$$
$$720$$ 0 0
$$721$$ 1.46566e11 0.542365
$$722$$ 0 0
$$723$$ 1.15329e11i 0.422071i
$$724$$ 0 0
$$725$$ −4.86009e11 −1.75911
$$726$$ 0 0
$$727$$ 2.71347e11i 0.971377i 0.874132 + 0.485688i $$0.161431\pi$$
−0.874132 + 0.485688i $$0.838569\pi$$
$$728$$ 0 0
$$729$$ −1.04604e10 −0.0370370
$$730$$ 0 0
$$731$$ 7.09344e10i 0.248420i
$$732$$ 0 0
$$733$$ −3.31640e9 −0.0114882 −0.00574409 0.999984i $$-0.501828\pi$$
−0.00574409 + 0.999984i $$0.501828\pi$$
$$734$$ 0 0
$$735$$ 2.06897e10i 0.0708933i
$$736$$ 0 0
$$737$$ −4.75110e10 −0.161037
$$738$$ 0 0
$$739$$ 3.59220e11i 1.20443i 0.798333 + 0.602216i $$0.205716\pi$$
−0.798333 + 0.602216i $$0.794284\pi$$
$$740$$ 0 0
$$741$$ 8.75506e10 0.290393
$$742$$ 0 0
$$743$$ 4.22337e11i 1.38581i 0.721028 + 0.692906i $$0.243670\pi$$
−0.721028 + 0.692906i $$0.756330\pi$$
$$744$$ 0 0
$$745$$ −3.96671e9 −0.0128767
$$746$$ 0 0
$$747$$ − 1.42612e11i − 0.458009i
$$748$$ 0 0
$$749$$ 1.62157e11 0.515237
$$750$$ 0 0
$$751$$ 2.60036e11i 0.817473i 0.912653 + 0.408736i $$0.134030\pi$$
−0.912653 + 0.408736i $$0.865970\pi$$
$$752$$ 0 0
$$753$$ −1.17383e11 −0.365110
$$754$$ 0 0
$$755$$ − 1.00738e10i − 0.0310032i
$$756$$ 0 0
$$757$$ 6.09599e11 1.85635 0.928177 0.372140i $$-0.121376\pi$$
0.928177 + 0.372140i $$0.121376\pi$$
$$758$$ 0 0
$$759$$ − 6.06673e10i − 0.182805i
$$760$$ 0 0
$$761$$ −8.45871e10 −0.252212 −0.126106 0.992017i $$-0.540248\pi$$
−0.126106 + 0.992017i $$0.540248\pi$$
$$762$$ 0 0
$$763$$ − 1.27142e10i − 0.0375137i
$$764$$ 0 0
$$765$$ 8.40031e9 0.0245273
$$766$$ 0 0
$$767$$ 3.82324e11i 1.10471i
$$768$$ 0 0
$$769$$ 6.82479e10 0.195157 0.0975784 0.995228i $$-0.468890\pi$$
0.0975784 + 0.995228i $$0.468890\pi$$
$$770$$ 0 0
$$771$$ 1.51628e11i 0.429104i
$$772$$ 0 0
$$773$$ 3.06208e10 0.0857627 0.0428813 0.999080i $$-0.486346\pi$$
0.0428813 + 0.999080i $$0.486346\pi$$
$$774$$ 0 0
$$775$$ − 1.96214e11i − 0.543904i
$$776$$ 0 0
$$777$$ 9.74802e10 0.267444
$$778$$ 0 0
$$779$$ 9.98772e10i 0.271217i
$$780$$ 0 0
$$781$$ 1.18159e11 0.317586
$$782$$ 0 0
$$783$$ − 1.29945e11i − 0.345709i
$$784$$ 0 0
$$785$$ −2.60559e10 −0.0686164
$$786$$ 0 0
$$787$$ − 4.62193e11i − 1.20483i −0.798185 0.602413i $$-0.794206\pi$$
0.798185 0.602413i $$-0.205794\pi$$
$$788$$ 0 0
$$789$$ 3.80253e11 0.981217
$$790$$ 0 0
$$791$$ 8.52481e10i 0.217760i
$$792$$ 0 0
$$793$$ −2.50294e11 −0.632933
$$794$$ 0 0
$$795$$ − 4.46855e9i − 0.0111866i
$$796$$ 0 0
$$797$$ 3.34741e11 0.829614 0.414807 0.909909i $$-0.363849\pi$$
0.414807 + 0.909909i $$0.363849\pi$$
$$798$$ 0 0
$$799$$ 3.40146e10i 0.0834601i
$$800$$ 0 0
$$801$$ 1.96421e11 0.477154
$$802$$ 0 0
$$803$$ 9.78991e10i 0.235460i
$$804$$ 0 0
$$805$$ 2.07876e10 0.0495017
$$806$$ 0 0
$$807$$ 3.48322e11i 0.821271i
$$808$$ 0 0
$$809$$ −5.80925e11 −1.35621 −0.678104 0.734966i $$-0.737198\pi$$
−0.678104 + 0.734966i $$0.737198\pi$$
$$810$$ 0 0
$$811$$ 6.93631e11i 1.60341i 0.597719 + 0.801706i $$0.296074\pi$$
−0.597719 + 0.801706i $$0.703926\pi$$
$$812$$ 0 0
$$813$$ −3.46839e9 −0.00793899
$$814$$ 0 0
$$815$$ 2.76242e10i 0.0626123i
$$816$$ 0 0
$$817$$ −1.90220e11 −0.426942
$$818$$ 0 0
$$819$$ 3.29649e10i 0.0732682i
$$820$$ 0 0
$$821$$ −7.20568e11 −1.58600 −0.792999 0.609223i $$-0.791481\pi$$
−0.792999 + 0.609223i $$0.791481\pi$$
$$822$$ 0 0
$$823$$ 5.20292e11i 1.13409i 0.823686 + 0.567046i $$0.191914\pi$$
−0.823686 + 0.567046i $$0.808086\pi$$
$$824$$ 0 0
$$825$$ 9.25818e10 0.199853
$$826$$ 0 0
$$827$$ − 2.76065e11i − 0.590187i −0.955468 0.295093i $$-0.904649\pi$$
0.955468 0.295093i $$-0.0953508\pi$$
$$828$$ 0 0
$$829$$ 5.58542e11 1.18260 0.591300 0.806452i $$-0.298615\pi$$
0.591300 + 0.806452i $$0.298615\pi$$
$$830$$ 0 0
$$831$$ − 9.39206e10i − 0.196950i
$$832$$ 0 0
$$833$$ −2.09793e11 −0.435725
$$834$$ 0 0
$$835$$ 1.07663e11i 0.221473i
$$836$$ 0 0
$$837$$ 5.24617e10 0.106891
$$838$$ 0 0
$$839$$ − 1.37279e11i − 0.277049i −0.990359 0.138525i $$-0.955764\pi$$
0.990359 0.138525i $$-0.0442360\pi$$
$$840$$ 0 0
$$841$$ 1.11400e12 2.22690
$$842$$ 0 0
$$843$$ − 6.40859e10i − 0.126897i
$$844$$ 0 0
$$845$$ −4.93332e10 −0.0967637
$$846$$ 0 0
$$847$$ − 1.72841e11i − 0.335824i
$$848$$ 0 0
$$849$$ 3.03138e11 0.583457
$$850$$ 0 0
$$851$$ 5.67034e11i 1.08116i
$$852$$ 0 0
$$853$$ −5.08270e11 −0.960060 −0.480030 0.877252i $$-0.659374\pi$$
−0.480030 + 0.877252i $$0.659374\pi$$
$$854$$ 0 0
$$855$$ 2.25266e10i 0.0421533i
$$856$$ 0 0
$$857$$ −5.97042e11 −1.10683 −0.553416 0.832905i $$-0.686676\pi$$
−0.553416 + 0.832905i $$0.686676\pi$$
$$858$$ 0 0
$$859$$ − 8.74635e11i − 1.60640i −0.595708 0.803201i $$-0.703128\pi$$
0.595708 0.803201i $$-0.296872\pi$$
$$860$$ 0 0
$$861$$ −3.76061e10 −0.0684299
$$862$$ 0 0
$$863$$ − 8.44217e11i − 1.52199i −0.648760 0.760993i $$-0.724712\pi$$
0.648760 0.760993i $$-0.275288\pi$$
$$864$$ 0 0
$$865$$ −7.43481e9 −0.0132802
$$866$$ 0 0
$$867$$ − 2.41045e11i − 0.426601i
$$868$$ 0 0
$$869$$ −2.80294e11 −0.491513
$$870$$ 0 0
$$871$$ 1.50170e11i 0.260922i
$$872$$ 0 0
$$873$$ 1.65727e11 0.285323
$$874$$ 0 0
$$875$$ 6.41178e10i 0.109382i
$$876$$ 0 0
$$877$$ −6.03933e11 −1.02092 −0.510458 0.859903i $$-0.670524\pi$$
−0.510458 + 0.859903i $$0.670524\pi$$
$$878$$ 0 0
$$879$$ 5.15917e11i 0.864220i
$$880$$ 0 0
$$881$$ 7.65645e11 1.27094 0.635468 0.772127i $$-0.280807\pi$$
0.635468 + 0.772127i $$0.280807\pi$$
$$882$$ 0 0
$$883$$ 7.59490e11i 1.24934i 0.780891 + 0.624668i $$0.214766\pi$$
−0.780891 + 0.624668i $$0.785234\pi$$
$$884$$ 0 0
$$885$$ −9.83712e10 −0.160360
$$886$$ 0 0
$$887$$ − 2.76654e11i − 0.446932i −0.974712 0.223466i $$-0.928263\pi$$
0.974712 0.223466i $$-0.0717372\pi$$
$$888$$ 0 0
$$889$$ 2.04327e11 0.327128
$$890$$ 0 0
$$891$$ 2.47536e10i 0.0392761i
$$892$$ 0 0
$$893$$ −9.12150e10 −0.143437
$$894$$ 0 0
$$895$$ − 1.02258e11i − 0.159369i
$$896$$ 0 0
$$897$$ −1.91754e11 −0.296192
$$898$$ 0 0
$$899$$ 6.51710e11i 0.997736i
$$900$$ 0 0
$$901$$ 4.53110e10 0.0687550
$$902$$ 0 0
$$903$$ − 7.16225e10i − 0.107720i
$$904$$ 0 0
$$905$$ 7.38022e10 0.110021
$$906$$ 0 0
$$907$$ − 1.00823e12i − 1.48980i −0.667174 0.744902i $$-0.732496\pi$$
0.667174 0.744902i $$-0.267504\pi$$
$$908$$ 0 0
$$909$$ −3.69551e11 −0.541276
$$910$$ 0 0
$$911$$ − 1.30302e12i − 1.89181i −0.324437 0.945907i $$-0.605175\pi$$
0.324437 0.945907i $$-0.394825\pi$$
$$912$$ 0 0
$$913$$ −3.37480e11 −0.485697
$$914$$ 0 0
$$915$$ − 6.44002e10i − 0.0918761i
$$916$$ 0 0
$$917$$ 3.32332e11 0.469997
$$918$$ 0 0
$$919$$ − 8.83053e11i − 1.23801i −0.785387 0.619006i $$-0.787536\pi$$
0.785387 0.619006i $$-0.212464\pi$$
$$920$$ 0 0
$$921$$ 4.58313e11 0.636977
$$922$$ 0 0
$$923$$ − 3.73469e11i − 0.514574i
$$924$$ 0 0
$$925$$ −8.65326e11 −1.18199
$$926$$ 0 0
$$927$$ − 3.47864e11i − 0.471075i
$$928$$ 0 0
$$929$$ −6.58431e11 −0.883990 −0.441995 0.897017i $$-0.645729\pi$$
−0.441995 + 0.897017i $$0.645729\pi$$
$$930$$ 0 0
$$931$$ − 5.62590e11i − 0.748848i
$$932$$ 0 0
$$933$$ 3.09099e11 0.407916
$$934$$ 0 0
$$935$$ − 1.98787e10i − 0.0260101i
$$936$$ 0 0
$$937$$ −3.25948e11 −0.422854 −0.211427 0.977394i $$-0.567811\pi$$
−0.211427 + 0.977394i $$0.567811\pi$$
$$938$$ 0 0
$$939$$ − 4.19975e11i − 0.540208i
$$940$$ 0 0
$$941$$ 5.07812e11 0.647656 0.323828 0.946116i $$-0.395030\pi$$
0.323828 + 0.946116i $$0.395030\pi$$
$$942$$ 0 0
$$943$$ − 2.18752e11i − 0.276633i
$$944$$ 0 0
$$945$$ −8.48180e9 −0.0106356
$$946$$ 0 0
$$947$$ 1.00994e11i 0.125573i 0.998027 + 0.0627867i $$0.0199988\pi$$
−0.998027 + 0.0627867i $$0.980001\pi$$
$$948$$ 0 0
$$949$$ 3.09434e11 0.381507
$$950$$ 0 0
$$951$$ 6.05464e11i 0.740229i
$$952$$ 0 0
$$953$$ −4.54997e11 −0.551616 −0.275808 0.961213i $$-0.588945\pi$$
−0.275808 + 0.961213i $$0.588945\pi$$
$$954$$ 0 0
$$955$$ 1.12466e11i 0.135210i
$$956$$ 0 0
$$957$$ −3.07504e11 −0.366609
$$958$$ 0 0
$$959$$ − 4.06107e11i − 0.480138i
$$960$$ 0 0
$$961$$ 5.89780e11 0.691507
$$962$$ 0 0
$$963$$ − 3.84867e11i − 0.447513i
$$964$$ 0 0
$$965$$ −1.81076e11 −0.208810
$$966$$ 0 0
$$967$$ 1.62872e12i 1.86269i 0.364134 + 0.931346i $$0.381365\pi$$
−0.364134 + 0.931346i $$0.618635\pi$$
$$968$$ 0 0
$$969$$ −2.28419e11 −0.259082
$$970$$ 0 0
$$971$$ − 5.60458e11i − 0.630472i −0.949013 0.315236i $$-0.897916\pi$$
0.949013 0.315236i $$-0.102084\pi$$
$$972$$ 0 0
$$973$$ 5.01492e11 0.559516
$$974$$ 0 0
$$975$$ − 2.92627e11i − 0.323814i
$$976$$ 0 0
$$977$$ −5.16303e11 −0.566665 −0.283332 0.959022i $$-0.591440\pi$$
−0.283332 + 0.959022i $$0.591440\pi$$
$$978$$ 0 0
$$979$$ − 4.64816e11i − 0.506000i
$$980$$ 0 0
$$981$$ −3.01762e10 −0.0325828
$$982$$ 0 0
$$983$$ 6.31070e11i 0.675870i 0.941169 + 0.337935i $$0.109728\pi$$
−0.941169 + 0.337935i $$0.890272\pi$$
$$984$$ 0 0
$$985$$ 5.69304e10 0.0604782
$$986$$ 0 0
$$987$$ − 3.43446e10i − 0.0361901i
$$988$$ 0 0
$$989$$ 4.16622e11 0.435468
$$990$$ 0 0
$$991$$ − 1.28282e12i − 1.33006i −0.746815 0.665032i $$-0.768418\pi$$
0.746815 0.665032i $$-0.231582\pi$$
$$992$$ 0 0
$$993$$ −3.87018e11 −0.398047
$$994$$ 0 0
$$995$$ − 1.31143e11i − 0.133799i
$$996$$ 0 0
$$997$$ 1.44928e12 1.46680 0.733402 0.679795i $$-0.237931\pi$$
0.733402 + 0.679795i $$0.237931\pi$$
$$998$$ 0 0
$$999$$ − 2.31363e11i − 0.232290i
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 192.9.g.b.127.2 2
4.3 odd 2 inner 192.9.g.b.127.1 2
8.3 odd 2 48.9.g.a.31.2 yes 2
8.5 even 2 48.9.g.a.31.1 2
24.5 odd 2 144.9.g.f.127.1 2
24.11 even 2 144.9.g.f.127.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.a.31.1 2 8.5 even 2
48.9.g.a.31.2 yes 2 8.3 odd 2
144.9.g.f.127.1 2 24.5 odd 2
144.9.g.f.127.2 2 24.11 even 2
192.9.g.b.127.1 2 4.3 odd 2 inner
192.9.g.b.127.2 2 1.1 even 1 trivial