# Properties

 Label 99.6.a.b.1.1 Level $99$ Weight $6$ Character 99.1 Self dual yes Analytic conductor $15.878$ Analytic rank $0$ 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] = [99,6,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, 6, names="a")

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

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$15.8779981615$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -28.0000 q^{4} -46.0000 q^{5} +148.000 q^{7} -120.000 q^{8} +O(q^{10})$$ $$q+2.00000 q^{2} -28.0000 q^{4} -46.0000 q^{5} +148.000 q^{7} -120.000 q^{8} -92.0000 q^{10} -121.000 q^{11} +574.000 q^{13} +296.000 q^{14} +656.000 q^{16} +722.000 q^{17} +2160.00 q^{19} +1288.00 q^{20} -242.000 q^{22} +2536.00 q^{23} -1009.00 q^{25} +1148.00 q^{26} -4144.00 q^{28} -4650.00 q^{29} +5032.00 q^{31} +5152.00 q^{32} +1444.00 q^{34} -6808.00 q^{35} +8118.00 q^{37} +4320.00 q^{38} +5520.00 q^{40} +5138.00 q^{41} +8304.00 q^{43} +3388.00 q^{44} +5072.00 q^{46} -24728.0 q^{47} +5097.00 q^{49} -2018.00 q^{50} -16072.0 q^{52} +28746.0 q^{53} +5566.00 q^{55} -17760.0 q^{56} -9300.00 q^{58} +5860.00 q^{59} -53658.0 q^{61} +10064.0 q^{62} -10688.0 q^{64} -26404.0 q^{65} +30908.0 q^{67} -20216.0 q^{68} -13616.0 q^{70} +69648.0 q^{71} -18446.0 q^{73} +16236.0 q^{74} -60480.0 q^{76} -17908.0 q^{77} -25300.0 q^{79} -30176.0 q^{80} +10276.0 q^{82} +17556.0 q^{83} -33212.0 q^{85} +16608.0 q^{86} +14520.0 q^{88} -132570. q^{89} +84952.0 q^{91} -71008.0 q^{92} -49456.0 q^{94} -99360.0 q^{95} +70658.0 q^{97} +10194.0 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.353553 0.176777 0.984251i $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ 0 0
$$4$$ −28.0000 −0.875000
$$5$$ −46.0000 −0.822873 −0.411437 0.911438i $$-0.634973\pi$$
−0.411437 + 0.911438i $$0.634973\pi$$
$$6$$ 0 0
$$7$$ 148.000 1.14161 0.570803 0.821087i $$-0.306632\pi$$
0.570803 + 0.821087i $$0.306632\pi$$
$$8$$ −120.000 −0.662913
$$9$$ 0 0
$$10$$ −92.0000 −0.290930
$$11$$ −121.000 −0.301511
$$12$$ 0 0
$$13$$ 574.000 0.942006 0.471003 0.882132i $$-0.343892\pi$$
0.471003 + 0.882132i $$0.343892\pi$$
$$14$$ 296.000 0.403619
$$15$$ 0 0
$$16$$ 656.000 0.640625
$$17$$ 722.000 0.605919 0.302960 0.953003i $$-0.402025\pi$$
0.302960 + 0.953003i $$0.402025\pi$$
$$18$$ 0 0
$$19$$ 2160.00 1.37268 0.686341 0.727280i $$-0.259216\pi$$
0.686341 + 0.727280i $$0.259216\pi$$
$$20$$ 1288.00 0.720014
$$21$$ 0 0
$$22$$ −242.000 −0.106600
$$23$$ 2536.00 0.999608 0.499804 0.866139i $$-0.333405\pi$$
0.499804 + 0.866139i $$0.333405\pi$$
$$24$$ 0 0
$$25$$ −1009.00 −0.322880
$$26$$ 1148.00 0.333049
$$27$$ 0 0
$$28$$ −4144.00 −0.998906
$$29$$ −4650.00 −1.02673 −0.513367 0.858169i $$-0.671602\pi$$
−0.513367 + 0.858169i $$0.671602\pi$$
$$30$$ 0 0
$$31$$ 5032.00 0.940451 0.470226 0.882546i $$-0.344172\pi$$
0.470226 + 0.882546i $$0.344172\pi$$
$$32$$ 5152.00 0.889408
$$33$$ 0 0
$$34$$ 1444.00 0.214225
$$35$$ −6808.00 −0.939398
$$36$$ 0 0
$$37$$ 8118.00 0.974866 0.487433 0.873161i $$-0.337933\pi$$
0.487433 + 0.873161i $$0.337933\pi$$
$$38$$ 4320.00 0.485316
$$39$$ 0 0
$$40$$ 5520.00 0.545493
$$41$$ 5138.00 0.477347 0.238674 0.971100i $$-0.423287\pi$$
0.238674 + 0.971100i $$0.423287\pi$$
$$42$$ 0 0
$$43$$ 8304.00 0.684883 0.342441 0.939539i $$-0.388746\pi$$
0.342441 + 0.939539i $$0.388746\pi$$
$$44$$ 3388.00 0.263822
$$45$$ 0 0
$$46$$ 5072.00 0.353415
$$47$$ −24728.0 −1.63284 −0.816421 0.577457i $$-0.804045\pi$$
−0.816421 + 0.577457i $$0.804045\pi$$
$$48$$ 0 0
$$49$$ 5097.00 0.303266
$$50$$ −2018.00 −0.114155
$$51$$ 0 0
$$52$$ −16072.0 −0.824255
$$53$$ 28746.0 1.40568 0.702842 0.711346i $$-0.251914\pi$$
0.702842 + 0.711346i $$0.251914\pi$$
$$54$$ 0 0
$$55$$ 5566.00 0.248106
$$56$$ −17760.0 −0.756786
$$57$$ 0 0
$$58$$ −9300.00 −0.363005
$$59$$ 5860.00 0.219163 0.109582 0.993978i $$-0.465049\pi$$
0.109582 + 0.993978i $$0.465049\pi$$
$$60$$ 0 0
$$61$$ −53658.0 −1.84633 −0.923166 0.384401i $$-0.874408\pi$$
−0.923166 + 0.384401i $$0.874408\pi$$
$$62$$ 10064.0 0.332500
$$63$$ 0 0
$$64$$ −10688.0 −0.326172
$$65$$ −26404.0 −0.775151
$$66$$ 0 0
$$67$$ 30908.0 0.841170 0.420585 0.907253i $$-0.361825\pi$$
0.420585 + 0.907253i $$0.361825\pi$$
$$68$$ −20216.0 −0.530180
$$69$$ 0 0
$$70$$ −13616.0 −0.332127
$$71$$ 69648.0 1.63969 0.819847 0.572583i $$-0.194058\pi$$
0.819847 + 0.572583i $$0.194058\pi$$
$$72$$ 0 0
$$73$$ −18446.0 −0.405131 −0.202565 0.979269i $$-0.564928\pi$$
−0.202565 + 0.979269i $$0.564928\pi$$
$$74$$ 16236.0 0.344667
$$75$$ 0 0
$$76$$ −60480.0 −1.20110
$$77$$ −17908.0 −0.344207
$$78$$ 0 0
$$79$$ −25300.0 −0.456092 −0.228046 0.973650i $$-0.573234\pi$$
−0.228046 + 0.973650i $$0.573234\pi$$
$$80$$ −30176.0 −0.527153
$$81$$ 0 0
$$82$$ 10276.0 0.168768
$$83$$ 17556.0 0.279724 0.139862 0.990171i $$-0.455334\pi$$
0.139862 + 0.990171i $$0.455334\pi$$
$$84$$ 0 0
$$85$$ −33212.0 −0.498595
$$86$$ 16608.0 0.242143
$$87$$ 0 0
$$88$$ 14520.0 0.199876
$$89$$ −132570. −1.77407 −0.887034 0.461704i $$-0.847238\pi$$
−0.887034 + 0.461704i $$0.847238\pi$$
$$90$$ 0 0
$$91$$ 84952.0 1.07540
$$92$$ −71008.0 −0.874657
$$93$$ 0 0
$$94$$ −49456.0 −0.577297
$$95$$ −99360.0 −1.12954
$$96$$ 0 0
$$97$$ 70658.0 0.762486 0.381243 0.924475i $$-0.375496\pi$$
0.381243 + 0.924475i $$0.375496\pi$$
$$98$$ 10194.0 0.107221
$$99$$ 0 0
$$100$$ 28252.0 0.282520
$$101$$ 101998. 0.994920 0.497460 0.867487i $$-0.334266\pi$$
0.497460 + 0.867487i $$0.334266\pi$$
$$102$$ 0 0
$$103$$ 130904. 1.21579 0.607897 0.794016i $$-0.292013\pi$$
0.607897 + 0.794016i $$0.292013\pi$$
$$104$$ −68880.0 −0.624467
$$105$$ 0 0
$$106$$ 57492.0 0.496984
$$107$$ 141612. 1.19575 0.597875 0.801589i $$-0.296012\pi$$
0.597875 + 0.801589i $$0.296012\pi$$
$$108$$ 0 0
$$109$$ −239810. −1.93331 −0.966654 0.256086i $$-0.917567\pi$$
−0.966654 + 0.256086i $$0.917567\pi$$
$$110$$ 11132.0 0.0877186
$$111$$ 0 0
$$112$$ 97088.0 0.731342
$$113$$ 42726.0 0.314772 0.157386 0.987537i $$-0.449693\pi$$
0.157386 + 0.987537i $$0.449693\pi$$
$$114$$ 0 0
$$115$$ −116656. −0.822550
$$116$$ 130200. 0.898392
$$117$$ 0 0
$$118$$ 11720.0 0.0774859
$$119$$ 106856. 0.691722
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ −107316. −0.652777
$$123$$ 0 0
$$124$$ −140896. −0.822895
$$125$$ 190164. 1.08856
$$126$$ 0 0
$$127$$ 51788.0 0.284918 0.142459 0.989801i $$-0.454499\pi$$
0.142459 + 0.989801i $$0.454499\pi$$
$$128$$ −186240. −1.00473
$$129$$ 0 0
$$130$$ −52808.0 −0.274057
$$131$$ −53652.0 −0.273154 −0.136577 0.990629i $$-0.543610\pi$$
−0.136577 + 0.990629i $$0.543610\pi$$
$$132$$ 0 0
$$133$$ 319680. 1.56706
$$134$$ 61816.0 0.297399
$$135$$ 0 0
$$136$$ −86640.0 −0.401672
$$137$$ 228862. 1.04177 0.520886 0.853627i $$-0.325602\pi$$
0.520886 + 0.853627i $$0.325602\pi$$
$$138$$ 0 0
$$139$$ 374920. 1.64589 0.822947 0.568119i $$-0.192329\pi$$
0.822947 + 0.568119i $$0.192329\pi$$
$$140$$ 190624. 0.821973
$$141$$ 0 0
$$142$$ 139296. 0.579719
$$143$$ −69454.0 −0.284025
$$144$$ 0 0
$$145$$ 213900. 0.844872
$$146$$ −36892.0 −0.143235
$$147$$ 0 0
$$148$$ −227304. −0.853007
$$149$$ 65830.0 0.242917 0.121459 0.992597i $$-0.461243\pi$$
0.121459 + 0.992597i $$0.461243\pi$$
$$150$$ 0 0
$$151$$ 154052. 0.549826 0.274913 0.961469i $$-0.411351\pi$$
0.274913 + 0.961469i $$0.411351\pi$$
$$152$$ −259200. −0.909968
$$153$$ 0 0
$$154$$ −35816.0 −0.121696
$$155$$ −231472. −0.773872
$$156$$ 0 0
$$157$$ 287678. 0.931446 0.465723 0.884931i $$-0.345794\pi$$
0.465723 + 0.884931i $$0.345794\pi$$
$$158$$ −50600.0 −0.161253
$$159$$ 0 0
$$160$$ −236992. −0.731870
$$161$$ 375328. 1.14116
$$162$$ 0 0
$$163$$ 105124. 0.309908 0.154954 0.987922i $$-0.450477\pi$$
0.154954 + 0.987922i $$0.450477\pi$$
$$164$$ −143864. −0.417679
$$165$$ 0 0
$$166$$ 35112.0 0.0988975
$$167$$ −150528. −0.417663 −0.208832 0.977952i $$-0.566966\pi$$
−0.208832 + 0.977952i $$0.566966\pi$$
$$168$$ 0 0
$$169$$ −41817.0 −0.112625
$$170$$ −66424.0 −0.176280
$$171$$ 0 0
$$172$$ −232512. −0.599272
$$173$$ 2166.00 0.00550229 0.00275114 0.999996i $$-0.499124\pi$$
0.00275114 + 0.999996i $$0.499124\pi$$
$$174$$ 0 0
$$175$$ −149332. −0.368602
$$176$$ −79376.0 −0.193156
$$177$$ 0 0
$$178$$ −265140. −0.627228
$$179$$ −672780. −1.56942 −0.784712 0.619860i $$-0.787189\pi$$
−0.784712 + 0.619860i $$0.787189\pi$$
$$180$$ 0 0
$$181$$ −526778. −1.19517 −0.597587 0.801804i $$-0.703874\pi$$
−0.597587 + 0.801804i $$0.703874\pi$$
$$182$$ 169904. 0.380211
$$183$$ 0 0
$$184$$ −304320. −0.662653
$$185$$ −373428. −0.802191
$$186$$ 0 0
$$187$$ −87362.0 −0.182692
$$188$$ 692384. 1.42874
$$189$$ 0 0
$$190$$ −198720. −0.399354
$$191$$ 305608. 0.606152 0.303076 0.952966i $$-0.401986\pi$$
0.303076 + 0.952966i $$0.401986\pi$$
$$192$$ 0 0
$$193$$ 116434. 0.225002 0.112501 0.993652i $$-0.464114\pi$$
0.112501 + 0.993652i $$0.464114\pi$$
$$194$$ 141316. 0.269580
$$195$$ 0 0
$$196$$ −142716. −0.265358
$$197$$ 247742. 0.454814 0.227407 0.973800i $$-0.426975\pi$$
0.227407 + 0.973800i $$0.426975\pi$$
$$198$$ 0 0
$$199$$ −513360. −0.918945 −0.459472 0.888192i $$-0.651961\pi$$
−0.459472 + 0.888192i $$0.651961\pi$$
$$200$$ 121080. 0.214041
$$201$$ 0 0
$$202$$ 203996. 0.351757
$$203$$ −688200. −1.17213
$$204$$ 0 0
$$205$$ −236348. −0.392796
$$206$$ 261808. 0.429848
$$207$$ 0 0
$$208$$ 376544. 0.603472
$$209$$ −261360. −0.413879
$$210$$ 0 0
$$211$$ −620688. −0.959770 −0.479885 0.877331i $$-0.659322\pi$$
−0.479885 + 0.877331i $$0.659322\pi$$
$$212$$ −804888. −1.22997
$$213$$ 0 0
$$214$$ 283224. 0.422762
$$215$$ −381984. −0.563571
$$216$$ 0 0
$$217$$ 744736. 1.07363
$$218$$ −479620. −0.683528
$$219$$ 0 0
$$220$$ −155848. −0.217092
$$221$$ 414428. 0.570780
$$222$$ 0 0
$$223$$ −1.31802e6 −1.77484 −0.887419 0.460964i $$-0.847504\pi$$
−0.887419 + 0.460964i $$0.847504\pi$$
$$224$$ 762496. 1.01535
$$225$$ 0 0
$$226$$ 85452.0 0.111289
$$227$$ 887412. 1.14304 0.571519 0.820589i $$-0.306354\pi$$
0.571519 + 0.820589i $$0.306354\pi$$
$$228$$ 0 0
$$229$$ −237450. −0.299215 −0.149608 0.988745i $$-0.547801\pi$$
−0.149608 + 0.988745i $$0.547801\pi$$
$$230$$ −233312. −0.290815
$$231$$ 0 0
$$232$$ 558000. 0.680635
$$233$$ 914706. 1.10380 0.551902 0.833909i $$-0.313902\pi$$
0.551902 + 0.833909i $$0.313902\pi$$
$$234$$ 0 0
$$235$$ 1.13749e6 1.34362
$$236$$ −164080. −0.191768
$$237$$ 0 0
$$238$$ 213712. 0.244561
$$239$$ −1.40892e6 −1.59548 −0.797740 0.603001i $$-0.793971\pi$$
−0.797740 + 0.603001i $$0.793971\pi$$
$$240$$ 0 0
$$241$$ −826358. −0.916486 −0.458243 0.888827i $$-0.651521\pi$$
−0.458243 + 0.888827i $$0.651521\pi$$
$$242$$ 29282.0 0.0321412
$$243$$ 0 0
$$244$$ 1.50242e6 1.61554
$$245$$ −234462. −0.249550
$$246$$ 0 0
$$247$$ 1.23984e6 1.29307
$$248$$ −603840. −0.623437
$$249$$ 0 0
$$250$$ 380328. 0.384865
$$251$$ 1.60387e6 1.60688 0.803442 0.595384i $$-0.203000\pi$$
0.803442 + 0.595384i $$0.203000\pi$$
$$252$$ 0 0
$$253$$ −306856. −0.301393
$$254$$ 103576. 0.100734
$$255$$ 0 0
$$256$$ −30464.0 −0.0290527
$$257$$ −397618. −0.375520 −0.187760 0.982215i $$-0.560123\pi$$
−0.187760 + 0.982215i $$0.560123\pi$$
$$258$$ 0 0
$$259$$ 1.20146e6 1.11291
$$260$$ 739312. 0.678257
$$261$$ 0 0
$$262$$ −107304. −0.0965745
$$263$$ −2.13166e6 −1.90033 −0.950166 0.311745i $$-0.899087\pi$$
−0.950166 + 0.311745i $$0.899087\pi$$
$$264$$ 0 0
$$265$$ −1.32232e6 −1.15670
$$266$$ 639360. 0.554040
$$267$$ 0 0
$$268$$ −865424. −0.736024
$$269$$ 725810. 0.611564 0.305782 0.952101i $$-0.401082\pi$$
0.305782 + 0.952101i $$0.401082\pi$$
$$270$$ 0 0
$$271$$ −1.46787e6 −1.21413 −0.607063 0.794654i $$-0.707652\pi$$
−0.607063 + 0.794654i $$0.707652\pi$$
$$272$$ 473632. 0.388167
$$273$$ 0 0
$$274$$ 457724. 0.368322
$$275$$ 122089. 0.0973520
$$276$$ 0 0
$$277$$ 1.52100e6 1.19105 0.595524 0.803338i $$-0.296944\pi$$
0.595524 + 0.803338i $$0.296944\pi$$
$$278$$ 749840. 0.581911
$$279$$ 0 0
$$280$$ 816960. 0.622738
$$281$$ −464382. −0.350840 −0.175420 0.984494i $$-0.556128\pi$$
−0.175420 + 0.984494i $$0.556128\pi$$
$$282$$ 0 0
$$283$$ −415136. −0.308123 −0.154062 0.988061i $$-0.549235\pi$$
−0.154062 + 0.988061i $$0.549235\pi$$
$$284$$ −1.95014e6 −1.43473
$$285$$ 0 0
$$286$$ −138908. −0.100418
$$287$$ 760424. 0.544943
$$288$$ 0 0
$$289$$ −898573. −0.632862
$$290$$ 427800. 0.298707
$$291$$ 0 0
$$292$$ 516488. 0.354489
$$293$$ 2.59321e6 1.76469 0.882344 0.470605i $$-0.155964\pi$$
0.882344 + 0.470605i $$0.155964\pi$$
$$294$$ 0 0
$$295$$ −269560. −0.180343
$$296$$ −974160. −0.646251
$$297$$ 0 0
$$298$$ 131660. 0.0858842
$$299$$ 1.45566e6 0.941636
$$300$$ 0 0
$$301$$ 1.22899e6 0.781867
$$302$$ 308104. 0.194393
$$303$$ 0 0
$$304$$ 1.41696e6 0.879374
$$305$$ 2.46827e6 1.51930
$$306$$ 0 0
$$307$$ −930832. −0.563671 −0.281835 0.959463i $$-0.590943\pi$$
−0.281835 + 0.959463i $$0.590943\pi$$
$$308$$ 501424. 0.301182
$$309$$ 0 0
$$310$$ −462944. −0.273605
$$311$$ −2.48527e6 −1.45704 −0.728522 0.685022i $$-0.759793\pi$$
−0.728522 + 0.685022i $$0.759793\pi$$
$$312$$ 0 0
$$313$$ 1.31719e6 0.759957 0.379978 0.924995i $$-0.375931\pi$$
0.379978 + 0.924995i $$0.375931\pi$$
$$314$$ 575356. 0.329316
$$315$$ 0 0
$$316$$ 708400. 0.399081
$$317$$ −2.25540e6 −1.26059 −0.630297 0.776354i $$-0.717067\pi$$
−0.630297 + 0.776354i $$0.717067\pi$$
$$318$$ 0 0
$$319$$ 562650. 0.309572
$$320$$ 491648. 0.268398
$$321$$ 0 0
$$322$$ 750656. 0.403461
$$323$$ 1.55952e6 0.831734
$$324$$ 0 0
$$325$$ −579166. −0.304155
$$326$$ 210248. 0.109569
$$327$$ 0 0
$$328$$ −616560. −0.316440
$$329$$ −3.65974e6 −1.86406
$$330$$ 0 0
$$331$$ −3.17071e6 −1.59069 −0.795346 0.606155i $$-0.792711\pi$$
−0.795346 + 0.606155i $$0.792711\pi$$
$$332$$ −491568. −0.244759
$$333$$ 0 0
$$334$$ −301056. −0.147666
$$335$$ −1.42177e6 −0.692176
$$336$$ 0 0
$$337$$ 1.27630e6 0.612177 0.306089 0.952003i $$-0.400980\pi$$
0.306089 + 0.952003i $$0.400980\pi$$
$$338$$ −83634.0 −0.0398191
$$339$$ 0 0
$$340$$ 929936. 0.436270
$$341$$ −608872. −0.283557
$$342$$ 0 0
$$343$$ −1.73308e6 −0.795396
$$344$$ −996480. −0.454017
$$345$$ 0 0
$$346$$ 4332.00 0.00194535
$$347$$ −3.69303e6 −1.64649 −0.823245 0.567687i $$-0.807838\pi$$
−0.823245 + 0.567687i $$0.807838\pi$$
$$348$$ 0 0
$$349$$ 1.70919e6 0.751150 0.375575 0.926792i $$-0.377445\pi$$
0.375575 + 0.926792i $$0.377445\pi$$
$$350$$ −298664. −0.130321
$$351$$ 0 0
$$352$$ −623392. −0.268167
$$353$$ −4.36859e6 −1.86597 −0.932986 0.359914i $$-0.882806\pi$$
−0.932986 + 0.359914i $$0.882806\pi$$
$$354$$ 0 0
$$355$$ −3.20381e6 −1.34926
$$356$$ 3.71196e6 1.55231
$$357$$ 0 0
$$358$$ −1.34556e6 −0.554875
$$359$$ 3.51284e6 1.43854 0.719271 0.694730i $$-0.244476\pi$$
0.719271 + 0.694730i $$0.244476\pi$$
$$360$$ 0 0
$$361$$ 2.18950e6 0.884254
$$362$$ −1.05356e6 −0.422558
$$363$$ 0 0
$$364$$ −2.37866e6 −0.940975
$$365$$ 848516. 0.333371
$$366$$ 0 0
$$367$$ −2.15259e6 −0.834251 −0.417125 0.908849i $$-0.636962\pi$$
−0.417125 + 0.908849i $$0.636962\pi$$
$$368$$ 1.66362e6 0.640374
$$369$$ 0 0
$$370$$ −746856. −0.283617
$$371$$ 4.25441e6 1.60474
$$372$$ 0 0
$$373$$ −2.24247e6 −0.834553 −0.417276 0.908780i $$-0.637015\pi$$
−0.417276 + 0.908780i $$0.637015\pi$$
$$374$$ −174724. −0.0645912
$$375$$ 0 0
$$376$$ 2.96736e6 1.08243
$$377$$ −2.66910e6 −0.967189
$$378$$ 0 0
$$379$$ −2.40986e6 −0.861775 −0.430887 0.902406i $$-0.641799\pi$$
−0.430887 + 0.902406i $$0.641799\pi$$
$$380$$ 2.78208e6 0.988350
$$381$$ 0 0
$$382$$ 611216. 0.214307
$$383$$ 1.01066e6 0.352052 0.176026 0.984386i $$-0.443676\pi$$
0.176026 + 0.984386i $$0.443676\pi$$
$$384$$ 0 0
$$385$$ 823768. 0.283239
$$386$$ 232868. 0.0795503
$$387$$ 0 0
$$388$$ −1.97842e6 −0.667175
$$389$$ −1.27779e6 −0.428140 −0.214070 0.976818i $$-0.568672\pi$$
−0.214070 + 0.976818i $$0.568672\pi$$
$$390$$ 0 0
$$391$$ 1.83099e6 0.605682
$$392$$ −611640. −0.201039
$$393$$ 0 0
$$394$$ 495484. 0.160801
$$395$$ 1.16380e6 0.375306
$$396$$ 0 0
$$397$$ 5.45400e6 1.73676 0.868378 0.495903i $$-0.165163\pi$$
0.868378 + 0.495903i $$0.165163\pi$$
$$398$$ −1.02672e6 −0.324896
$$399$$ 0 0
$$400$$ −661904. −0.206845
$$401$$ 1.48980e6 0.462665 0.231332 0.972875i $$-0.425692\pi$$
0.231332 + 0.972875i $$0.425692\pi$$
$$402$$ 0 0
$$403$$ 2.88837e6 0.885911
$$404$$ −2.85594e6 −0.870555
$$405$$ 0 0
$$406$$ −1.37640e6 −0.414409
$$407$$ −982278. −0.293933
$$408$$ 0 0
$$409$$ −4.39899e6 −1.30030 −0.650152 0.759804i $$-0.725295\pi$$
−0.650152 + 0.759804i $$0.725295\pi$$
$$410$$ −472696. −0.138874
$$411$$ 0 0
$$412$$ −3.66531e6 −1.06382
$$413$$ 867280. 0.250198
$$414$$ 0 0
$$415$$ −807576. −0.230178
$$416$$ 2.95725e6 0.837827
$$417$$ 0 0
$$418$$ −522720. −0.146328
$$419$$ 280420. 0.0780322 0.0390161 0.999239i $$-0.487578\pi$$
0.0390161 + 0.999239i $$0.487578\pi$$
$$420$$ 0 0
$$421$$ 817462. 0.224782 0.112391 0.993664i $$-0.464149\pi$$
0.112391 + 0.993664i $$0.464149\pi$$
$$422$$ −1.24138e6 −0.339330
$$423$$ 0 0
$$424$$ −3.44952e6 −0.931846
$$425$$ −728498. −0.195639
$$426$$ 0 0
$$427$$ −7.94138e6 −2.10779
$$428$$ −3.96514e6 −1.04628
$$429$$ 0 0
$$430$$ −763968. −0.199253
$$431$$ −1.88599e6 −0.489043 −0.244521 0.969644i $$-0.578631\pi$$
−0.244521 + 0.969644i $$0.578631\pi$$
$$432$$ 0 0
$$433$$ 5.84067e6 1.49707 0.748537 0.663093i $$-0.230757\pi$$
0.748537 + 0.663093i $$0.230757\pi$$
$$434$$ 1.48947e6 0.379584
$$435$$ 0 0
$$436$$ 6.71468e6 1.69164
$$437$$ 5.47776e6 1.37214
$$438$$ 0 0
$$439$$ −509540. −0.126188 −0.0630938 0.998008i $$-0.520097\pi$$
−0.0630938 + 0.998008i $$0.520097\pi$$
$$440$$ −667920. −0.164472
$$441$$ 0 0
$$442$$ 828856. 0.201801
$$443$$ −4.10268e6 −0.993250 −0.496625 0.867965i $$-0.665428\pi$$
−0.496625 + 0.867965i $$0.665428\pi$$
$$444$$ 0 0
$$445$$ 6.09822e6 1.45983
$$446$$ −2.63603e6 −0.627500
$$447$$ 0 0
$$448$$ −1.58182e6 −0.372360
$$449$$ −513410. −0.120185 −0.0600923 0.998193i $$-0.519139\pi$$
−0.0600923 + 0.998193i $$0.519139\pi$$
$$450$$ 0 0
$$451$$ −621698. −0.143926
$$452$$ −1.19633e6 −0.275426
$$453$$ 0 0
$$454$$ 1.77482e6 0.404125
$$455$$ −3.90779e6 −0.884918
$$456$$ 0 0
$$457$$ 1.22738e6 0.274908 0.137454 0.990508i $$-0.456108\pi$$
0.137454 + 0.990508i $$0.456108\pi$$
$$458$$ −474900. −0.105789
$$459$$ 0 0
$$460$$ 3.26637e6 0.719732
$$461$$ 6.41000e6 1.40477 0.702386 0.711797i $$-0.252118\pi$$
0.702386 + 0.711797i $$0.252118\pi$$
$$462$$ 0 0
$$463$$ 6.63030e6 1.43741 0.718705 0.695315i $$-0.244735\pi$$
0.718705 + 0.695315i $$0.244735\pi$$
$$464$$ −3.05040e6 −0.657751
$$465$$ 0 0
$$466$$ 1.82941e6 0.390253
$$467$$ 4.14769e6 0.880064 0.440032 0.897982i $$-0.354967\pi$$
0.440032 + 0.897982i $$0.354967\pi$$
$$468$$ 0 0
$$469$$ 4.57438e6 0.960286
$$470$$ 2.27498e6 0.475042
$$471$$ 0 0
$$472$$ −703200. −0.145286
$$473$$ −1.00478e6 −0.206500
$$474$$ 0 0
$$475$$ −2.17944e6 −0.443211
$$476$$ −2.99197e6 −0.605257
$$477$$ 0 0
$$478$$ −2.81784e6 −0.564088
$$479$$ 5.05132e6 1.00593 0.502963 0.864308i $$-0.332243\pi$$
0.502963 + 0.864308i $$0.332243\pi$$
$$480$$ 0 0
$$481$$ 4.65973e6 0.918329
$$482$$ −1.65272e6 −0.324027
$$483$$ 0 0
$$484$$ −409948. −0.0795455
$$485$$ −3.25027e6 −0.627429
$$486$$ 0 0
$$487$$ 2.66221e6 0.508651 0.254325 0.967119i $$-0.418147\pi$$
0.254325 + 0.967119i $$0.418147\pi$$
$$488$$ 6.43896e6 1.22396
$$489$$ 0 0
$$490$$ −468924. −0.0882292
$$491$$ 5.54659e6 1.03830 0.519149 0.854684i $$-0.326249\pi$$
0.519149 + 0.854684i $$0.326249\pi$$
$$492$$ 0 0
$$493$$ −3.35730e6 −0.622118
$$494$$ 2.47968e6 0.457171
$$495$$ 0 0
$$496$$ 3.30099e6 0.602477
$$497$$ 1.03079e7 1.87189
$$498$$ 0 0
$$499$$ −6820.00 −0.00122612 −0.000613060 1.00000i $$-0.500195\pi$$
−0.000613060 1.00000i $$0.500195\pi$$
$$500$$ −5.32459e6 −0.952492
$$501$$ 0 0
$$502$$ 3.20774e6 0.568119
$$503$$ 451136. 0.0795037 0.0397519 0.999210i $$-0.487343\pi$$
0.0397519 + 0.999210i $$0.487343\pi$$
$$504$$ 0 0
$$505$$ −4.69191e6 −0.818693
$$506$$ −613712. −0.106559
$$507$$ 0 0
$$508$$ −1.45006e6 −0.249303
$$509$$ −393390. −0.0673021 −0.0336511 0.999434i $$-0.510713\pi$$
−0.0336511 + 0.999434i $$0.510713\pi$$
$$510$$ 0 0
$$511$$ −2.73001e6 −0.462500
$$512$$ 5.89875e6 0.994455
$$513$$ 0 0
$$514$$ −795236. −0.132766
$$515$$ −6.02158e6 −1.00044
$$516$$ 0 0
$$517$$ 2.99209e6 0.492321
$$518$$ 2.40293e6 0.393474
$$519$$ 0 0
$$520$$ 3.16848e6 0.513857
$$521$$ −3.28432e6 −0.530092 −0.265046 0.964236i $$-0.585387\pi$$
−0.265046 + 0.964236i $$0.585387\pi$$
$$522$$ 0 0
$$523$$ −1.68266e6 −0.268993 −0.134497 0.990914i $$-0.542942\pi$$
−0.134497 + 0.990914i $$0.542942\pi$$
$$524$$ 1.50226e6 0.239010
$$525$$ 0 0
$$526$$ −4.26333e6 −0.671869
$$527$$ 3.63310e6 0.569838
$$528$$ 0 0
$$529$$ −5047.00 −0.000784141 0
$$530$$ −2.64463e6 −0.408955
$$531$$ 0 0
$$532$$ −8.95104e6 −1.37118
$$533$$ 2.94921e6 0.449664
$$534$$ 0 0
$$535$$ −6.51415e6 −0.983951
$$536$$ −3.70896e6 −0.557622
$$537$$ 0 0
$$538$$ 1.45162e6 0.216221
$$539$$ −616737. −0.0914383
$$540$$ 0 0
$$541$$ 9.48158e6 1.39280 0.696398 0.717656i $$-0.254785\pi$$
0.696398 + 0.717656i $$0.254785\pi$$
$$542$$ −2.93574e6 −0.429258
$$543$$ 0 0
$$544$$ 3.71974e6 0.538909
$$545$$ 1.10313e7 1.59087
$$546$$ 0 0
$$547$$ −6.09239e6 −0.870602 −0.435301 0.900285i $$-0.643358\pi$$
−0.435301 + 0.900285i $$0.643358\pi$$
$$548$$ −6.40814e6 −0.911550
$$549$$ 0 0
$$550$$ 244178. 0.0344191
$$551$$ −1.00440e7 −1.40938
$$552$$ 0 0
$$553$$ −3.74440e6 −0.520678
$$554$$ 3.04200e6 0.421099
$$555$$ 0 0
$$556$$ −1.04978e7 −1.44016
$$557$$ −8.49594e6 −1.16031 −0.580154 0.814507i $$-0.697008\pi$$
−0.580154 + 0.814507i $$0.697008\pi$$
$$558$$ 0 0
$$559$$ 4.76650e6 0.645163
$$560$$ −4.46605e6 −0.601802
$$561$$ 0 0
$$562$$ −928764. −0.124041
$$563$$ 7.02216e6 0.933683 0.466842 0.884341i $$-0.345392\pi$$
0.466842 + 0.884341i $$0.345392\pi$$
$$564$$ 0 0
$$565$$ −1.96540e6 −0.259017
$$566$$ −830272. −0.108938
$$567$$ 0 0
$$568$$ −8.35776e6 −1.08697
$$569$$ −9.41847e6 −1.21955 −0.609775 0.792574i $$-0.708740\pi$$
−0.609775 + 0.792574i $$0.708740\pi$$
$$570$$ 0 0
$$571$$ 7.29699e6 0.936599 0.468299 0.883570i $$-0.344867\pi$$
0.468299 + 0.883570i $$0.344867\pi$$
$$572$$ 1.94471e6 0.248522
$$573$$ 0 0
$$574$$ 1.52085e6 0.192666
$$575$$ −2.55882e6 −0.322753
$$576$$ 0 0
$$577$$ −3.29590e6 −0.412131 −0.206065 0.978538i $$-0.566066\pi$$
−0.206065 + 0.978538i $$0.566066\pi$$
$$578$$ −1.79715e6 −0.223750
$$579$$ 0 0
$$580$$ −5.98920e6 −0.739263
$$581$$ 2.59829e6 0.319335
$$582$$ 0 0
$$583$$ −3.47827e6 −0.423830
$$584$$ 2.21352e6 0.268566
$$585$$ 0 0
$$586$$ 5.18641e6 0.623911
$$587$$ −4.39827e6 −0.526849 −0.263425 0.964680i $$-0.584852\pi$$
−0.263425 + 0.964680i $$0.584852\pi$$
$$588$$ 0 0
$$589$$ 1.08691e7 1.29094
$$590$$ −539120. −0.0637610
$$591$$ 0 0
$$592$$ 5.32541e6 0.624523
$$593$$ −9.21781e6 −1.07644 −0.538222 0.842803i $$-0.680904\pi$$
−0.538222 + 0.842803i $$0.680904\pi$$
$$594$$ 0 0
$$595$$ −4.91538e6 −0.569199
$$596$$ −1.84324e6 −0.212553
$$597$$ 0 0
$$598$$ 2.91133e6 0.332919
$$599$$ −3.77140e6 −0.429473 −0.214736 0.976672i $$-0.568889\pi$$
−0.214736 + 0.976672i $$0.568889\pi$$
$$600$$ 0 0
$$601$$ 4.19724e6 0.473999 0.237000 0.971510i $$-0.423836\pi$$
0.237000 + 0.971510i $$0.423836\pi$$
$$602$$ 2.45798e6 0.276432
$$603$$ 0 0
$$604$$ −4.31346e6 −0.481097
$$605$$ −673486. −0.0748066
$$606$$ 0 0
$$607$$ −1.00133e6 −0.110308 −0.0551539 0.998478i $$-0.517565\pi$$
−0.0551539 + 0.998478i $$0.517565\pi$$
$$608$$ 1.11283e7 1.22087
$$609$$ 0 0
$$610$$ 4.93654e6 0.537153
$$611$$ −1.41939e7 −1.53815
$$612$$ 0 0
$$613$$ −7.38239e6 −0.793498 −0.396749 0.917927i $$-0.629862\pi$$
−0.396749 + 0.917927i $$0.629862\pi$$
$$614$$ −1.86166e6 −0.199288
$$615$$ 0 0
$$616$$ 2.14896e6 0.228179
$$617$$ 1.54025e7 1.62884 0.814418 0.580279i $$-0.197056\pi$$
0.814418 + 0.580279i $$0.197056\pi$$
$$618$$ 0 0
$$619$$ −7.12402e6 −0.747306 −0.373653 0.927569i $$-0.621895\pi$$
−0.373653 + 0.927569i $$0.621895\pi$$
$$620$$ 6.48122e6 0.677138
$$621$$ 0 0
$$622$$ −4.97054e6 −0.515143
$$623$$ −1.96204e7 −2.02529
$$624$$ 0 0
$$625$$ −5.59442e6 −0.572869
$$626$$ 2.63439e6 0.268685
$$627$$ 0 0
$$628$$ −8.05498e6 −0.815015
$$629$$ 5.86120e6 0.590690
$$630$$ 0 0
$$631$$ 1.16696e7 1.16677 0.583383 0.812197i $$-0.301729\pi$$
0.583383 + 0.812197i $$0.301729\pi$$
$$632$$ 3.03600e6 0.302349
$$633$$ 0 0
$$634$$ −4.51080e6 −0.445687
$$635$$ −2.38225e6 −0.234451
$$636$$ 0 0
$$637$$ 2.92568e6 0.285679
$$638$$ 1.12530e6 0.109450
$$639$$ 0 0
$$640$$ 8.56704e6 0.826763
$$641$$ 1.10271e7 1.06003 0.530014 0.847989i $$-0.322187\pi$$
0.530014 + 0.847989i $$0.322187\pi$$
$$642$$ 0 0
$$643$$ −9.56024e6 −0.911887 −0.455944 0.890009i $$-0.650698\pi$$
−0.455944 + 0.890009i $$0.650698\pi$$
$$644$$ −1.05092e7 −0.998514
$$645$$ 0 0
$$646$$ 3.11904e6 0.294063
$$647$$ 1.09942e7 1.03253 0.516263 0.856430i $$-0.327323\pi$$
0.516263 + 0.856430i $$0.327323\pi$$
$$648$$ 0 0
$$649$$ −709060. −0.0660802
$$650$$ −1.15833e6 −0.107535
$$651$$ 0 0
$$652$$ −2.94347e6 −0.271170
$$653$$ 295346. 0.0271049 0.0135525 0.999908i $$-0.495686\pi$$
0.0135525 + 0.999908i $$0.495686\pi$$
$$654$$ 0 0
$$655$$ 2.46799e6 0.224771
$$656$$ 3.37053e6 0.305801
$$657$$ 0 0
$$658$$ −7.31949e6 −0.659046
$$659$$ 1.65613e7 1.48553 0.742766 0.669551i $$-0.233514\pi$$
0.742766 + 0.669551i $$0.233514\pi$$
$$660$$ 0 0
$$661$$ 1.97042e6 0.175411 0.0877053 0.996146i $$-0.472047\pi$$
0.0877053 + 0.996146i $$0.472047\pi$$
$$662$$ −6.34142e6 −0.562395
$$663$$ 0 0
$$664$$ −2.10672e6 −0.185433
$$665$$ −1.47053e7 −1.28949
$$666$$ 0 0
$$667$$ −1.17924e7 −1.02633
$$668$$ 4.21478e6 0.365455
$$669$$ 0 0
$$670$$ −2.84354e6 −0.244721
$$671$$ 6.49262e6 0.556690
$$672$$ 0 0
$$673$$ −1.63733e6 −0.139347 −0.0696735 0.997570i $$-0.522196\pi$$
−0.0696735 + 0.997570i $$0.522196\pi$$
$$674$$ 2.55260e6 0.216437
$$675$$ 0 0
$$676$$ 1.17088e6 0.0985472
$$677$$ 6.35878e6 0.533215 0.266607 0.963805i $$-0.414097\pi$$
0.266607 + 0.963805i $$0.414097\pi$$
$$678$$ 0 0
$$679$$ 1.04574e7 0.870460
$$680$$ 3.98544e6 0.330525
$$681$$ 0 0
$$682$$ −1.21774e6 −0.100252
$$683$$ −1.11033e7 −0.910751 −0.455376 0.890299i $$-0.650495\pi$$
−0.455376 + 0.890299i $$0.650495\pi$$
$$684$$ 0 0
$$685$$ −1.05277e7 −0.857245
$$686$$ −3.46616e6 −0.281215
$$687$$ 0 0
$$688$$ 5.44742e6 0.438753
$$689$$ 1.65002e7 1.32416
$$690$$ 0 0
$$691$$ 1.70189e7 1.35592 0.677962 0.735097i $$-0.262864\pi$$
0.677962 + 0.735097i $$0.262864\pi$$
$$692$$ −60648.0 −0.00481450
$$693$$ 0 0
$$694$$ −7.38606e6 −0.582122
$$695$$ −1.72463e7 −1.35436
$$696$$ 0 0
$$697$$ 3.70964e6 0.289234
$$698$$ 3.41838e6 0.265572
$$699$$ 0 0
$$700$$ 4.18130e6 0.322527
$$701$$ −1.58021e7 −1.21456 −0.607280 0.794488i $$-0.707740\pi$$
−0.607280 + 0.794488i $$0.707740\pi$$
$$702$$ 0 0
$$703$$ 1.75349e7 1.33818
$$704$$ 1.29325e6 0.0983445
$$705$$ 0 0
$$706$$ −8.73719e6 −0.659720
$$707$$ 1.50957e7 1.13581
$$708$$ 0 0
$$709$$ 1.24834e7 0.932643 0.466322 0.884615i $$-0.345579\pi$$
0.466322 + 0.884615i $$0.345579\pi$$
$$710$$ −6.40762e6 −0.477035
$$711$$ 0 0
$$712$$ 1.59084e7 1.17605
$$713$$ 1.27612e7 0.940083
$$714$$ 0 0
$$715$$ 3.19488e6 0.233717
$$716$$ 1.88378e7 1.37325
$$717$$ 0 0
$$718$$ 7.02568e6 0.508601
$$719$$ −2.00724e6 −0.144803 −0.0724014 0.997376i $$-0.523066\pi$$
−0.0724014 + 0.997376i $$0.523066\pi$$
$$720$$ 0 0
$$721$$ 1.93738e7 1.38796
$$722$$ 4.37900e6 0.312631
$$723$$ 0 0
$$724$$ 1.47498e7 1.04578
$$725$$ 4.69185e6 0.331512
$$726$$ 0 0
$$727$$ 6.97301e6 0.489310 0.244655 0.969610i $$-0.421325\pi$$
0.244655 + 0.969610i $$0.421325\pi$$
$$728$$ −1.01942e7 −0.712896
$$729$$ 0 0
$$730$$ 1.69703e6 0.117864
$$731$$ 5.99549e6 0.414984
$$732$$ 0 0
$$733$$ −2.34965e7 −1.61527 −0.807633 0.589685i $$-0.799252\pi$$
−0.807633 + 0.589685i $$0.799252\pi$$
$$734$$ −4.30518e6 −0.294952
$$735$$ 0 0
$$736$$ 1.30655e7 0.889059
$$737$$ −3.73987e6 −0.253622
$$738$$ 0 0
$$739$$ −1.39901e7 −0.942346 −0.471173 0.882041i $$-0.656169\pi$$
−0.471173 + 0.882041i $$0.656169\pi$$
$$740$$ 1.04560e7 0.701917
$$741$$ 0 0
$$742$$ 8.50882e6 0.567361
$$743$$ −2.42745e7 −1.61316 −0.806582 0.591123i $$-0.798685\pi$$
−0.806582 + 0.591123i $$0.798685\pi$$
$$744$$ 0 0
$$745$$ −3.02818e6 −0.199890
$$746$$ −4.48493e6 −0.295059
$$747$$ 0 0
$$748$$ 2.44614e6 0.159855
$$749$$ 2.09586e7 1.36508
$$750$$ 0 0
$$751$$ 1.53660e7 0.994170 0.497085 0.867702i $$-0.334404\pi$$
0.497085 + 0.867702i $$0.334404\pi$$
$$752$$ −1.62216e7 −1.04604
$$753$$ 0 0
$$754$$ −5.33820e6 −0.341953
$$755$$ −7.08639e6 −0.452437
$$756$$ 0 0
$$757$$ 2.07605e7 1.31674 0.658368 0.752697i $$-0.271247\pi$$
0.658368 + 0.752697i $$0.271247\pi$$
$$758$$ −4.81972e6 −0.304683
$$759$$ 0 0
$$760$$ 1.19232e7 0.748788
$$761$$ −5.83810e6 −0.365435 −0.182717 0.983165i $$-0.558489\pi$$
−0.182717 + 0.983165i $$0.558489\pi$$
$$762$$ 0 0
$$763$$ −3.54919e7 −2.20708
$$764$$ −8.55702e6 −0.530383
$$765$$ 0 0
$$766$$ 2.02131e6 0.124469
$$767$$ 3.36364e6 0.206453
$$768$$ 0 0
$$769$$ −1.39197e7 −0.848818 −0.424409 0.905471i $$-0.639518\pi$$
−0.424409 + 0.905471i $$0.639518\pi$$
$$770$$ 1.64754e6 0.100140
$$771$$ 0 0
$$772$$ −3.26015e6 −0.196877
$$773$$ 4.17883e6 0.251539 0.125770 0.992059i $$-0.459860\pi$$
0.125770 + 0.992059i $$0.459860\pi$$
$$774$$ 0 0
$$775$$ −5.07729e6 −0.303653
$$776$$ −8.47896e6 −0.505462
$$777$$ 0 0
$$778$$ −2.55558e6 −0.151370
$$779$$ 1.10981e7 0.655246
$$780$$ 0 0
$$781$$ −8.42741e6 −0.494386
$$782$$ 3.66198e6 0.214141
$$783$$ 0 0
$$784$$ 3.34363e6 0.194280
$$785$$ −1.32332e7 −0.766461
$$786$$ 0 0
$$787$$ 9.66705e6 0.556361 0.278181 0.960529i $$-0.410269\pi$$
0.278181 + 0.960529i $$0.410269\pi$$
$$788$$ −6.93678e6 −0.397962
$$789$$ 0 0
$$790$$ 2.32760e6 0.132691
$$791$$ 6.32345e6 0.359346
$$792$$ 0 0
$$793$$ −3.07997e7 −1.73926
$$794$$ 1.09080e7 0.614036
$$795$$ 0 0
$$796$$ 1.43741e7 0.804077
$$797$$ 5.79884e6 0.323367 0.161683 0.986843i $$-0.448308\pi$$
0.161683 + 0.986843i $$0.448308\pi$$
$$798$$ 0 0
$$799$$ −1.78536e7 −0.989371
$$800$$ −5.19837e6 −0.287172
$$801$$ 0 0
$$802$$ 2.97960e6 0.163577
$$803$$ 2.23197e6 0.122151
$$804$$ 0 0
$$805$$ −1.72651e7 −0.939029
$$806$$ 5.77674e6 0.313217
$$807$$ 0 0
$$808$$ −1.22398e7 −0.659545
$$809$$ 1.92543e7 1.03433 0.517163 0.855887i $$-0.326988\pi$$
0.517163 + 0.855887i $$0.326988\pi$$
$$810$$ 0 0
$$811$$ −1.31938e7 −0.704396 −0.352198 0.935926i $$-0.614566\pi$$
−0.352198 + 0.935926i $$0.614566\pi$$
$$812$$ 1.92696e7 1.02561
$$813$$ 0 0
$$814$$ −1.96456e6 −0.103921
$$815$$ −4.83570e6 −0.255015
$$816$$ 0 0
$$817$$ 1.79366e7 0.940126
$$818$$ −8.79798e6 −0.459727
$$819$$ 0 0
$$820$$ 6.61774e6 0.343697
$$821$$ −1.33779e7 −0.692677 −0.346338 0.938110i $$-0.612575\pi$$
−0.346338 + 0.938110i $$0.612575\pi$$
$$822$$ 0 0
$$823$$ −1.88613e7 −0.970673 −0.485336 0.874327i $$-0.661303\pi$$
−0.485336 + 0.874327i $$0.661303\pi$$
$$824$$ −1.57085e7 −0.805965
$$825$$ 0 0
$$826$$ 1.73456e6 0.0884584
$$827$$ −1.62680e7 −0.827123 −0.413561 0.910476i $$-0.635715\pi$$
−0.413561 + 0.910476i $$0.635715\pi$$
$$828$$ 0 0
$$829$$ −2.18098e7 −1.10221 −0.551107 0.834435i $$-0.685794\pi$$
−0.551107 + 0.834435i $$0.685794\pi$$
$$830$$ −1.61515e6 −0.0813801
$$831$$ 0 0
$$832$$ −6.13491e6 −0.307256
$$833$$ 3.68003e6 0.183755
$$834$$ 0 0
$$835$$ 6.92429e6 0.343684
$$836$$ 7.31808e6 0.362144
$$837$$ 0 0
$$838$$ 560840. 0.0275886
$$839$$ −1.17771e7 −0.577607 −0.288804 0.957388i $$-0.593257\pi$$
−0.288804 + 0.957388i $$0.593257\pi$$
$$840$$ 0 0
$$841$$ 1.11135e6 0.0541828
$$842$$ 1.63492e6 0.0794726
$$843$$ 0 0
$$844$$ 1.73793e7 0.839799
$$845$$ 1.92358e6 0.0926764
$$846$$ 0 0
$$847$$ 2.16687e6 0.103782
$$848$$ 1.88574e7 0.900516
$$849$$ 0 0
$$850$$ −1.45700e6 −0.0691689
$$851$$ 2.05872e7 0.974483
$$852$$ 0 0
$$853$$ −1.43993e7 −0.677591 −0.338796 0.940860i $$-0.610020\pi$$
−0.338796 + 0.940860i $$0.610020\pi$$
$$854$$ −1.58828e7 −0.745215
$$855$$ 0 0
$$856$$ −1.69934e7 −0.792678
$$857$$ −6.27604e6 −0.291900 −0.145950 0.989292i $$-0.546624\pi$$
−0.145950 + 0.989292i $$0.546624\pi$$
$$858$$ 0 0
$$859$$ −4.71738e6 −0.218131 −0.109066 0.994035i $$-0.534786\pi$$
−0.109066 + 0.994035i $$0.534786\pi$$
$$860$$ 1.06956e7 0.493125
$$861$$ 0 0
$$862$$ −3.77198e6 −0.172903
$$863$$ −7.53926e6 −0.344589 −0.172295 0.985045i $$-0.555118\pi$$
−0.172295 + 0.985045i $$0.555118\pi$$
$$864$$ 0 0
$$865$$ −99636.0 −0.00452768
$$866$$ 1.16813e7 0.529296
$$867$$ 0 0
$$868$$ −2.08526e7 −0.939423
$$869$$ 3.06130e6 0.137517
$$870$$ 0 0
$$871$$ 1.77412e7 0.792387
$$872$$ 2.87772e7 1.28161
$$873$$ 0 0
$$874$$ 1.09555e7 0.485126
$$875$$ 2.81443e7 1.24271
$$876$$ 0 0
$$877$$ −1.04331e7 −0.458051 −0.229025 0.973420i $$-0.573554\pi$$
−0.229025 + 0.973420i $$0.573554\pi$$
$$878$$ −1.01908e6 −0.0446141
$$879$$ 0 0
$$880$$ 3.65130e6 0.158943
$$881$$ −3.91076e7 −1.69755 −0.848774 0.528756i $$-0.822658\pi$$
−0.848774 + 0.528756i $$0.822658\pi$$
$$882$$ 0 0
$$883$$ 1.29282e7 0.558003 0.279001 0.960291i $$-0.409997\pi$$
0.279001 + 0.960291i $$0.409997\pi$$
$$884$$ −1.16040e7 −0.499432
$$885$$ 0 0
$$886$$ −8.20537e6 −0.351167
$$887$$ −3.36466e7 −1.43592 −0.717962 0.696082i $$-0.754925\pi$$
−0.717962 + 0.696082i $$0.754925\pi$$
$$888$$ 0 0
$$889$$ 7.66462e6 0.325264
$$890$$ 1.21964e7 0.516129
$$891$$ 0 0
$$892$$ 3.69044e7 1.55298
$$893$$ −5.34125e7 −2.24137
$$894$$ 0 0
$$895$$ 3.09479e7 1.29144
$$896$$ −2.75635e7 −1.14700
$$897$$ 0 0
$$898$$ −1.02682e6 −0.0424916
$$899$$ −2.33988e7 −0.965594
$$900$$ 0 0
$$901$$ 2.07546e7 0.851731
$$902$$ −1.24340e6 −0.0508854
$$903$$ 0 0
$$904$$ −5.12712e6 −0.208666
$$905$$ 2.42318e7 0.983477
$$906$$ 0 0
$$907$$ −4.19629e7 −1.69374 −0.846872 0.531797i $$-0.821517\pi$$
−0.846872 + 0.531797i $$0.821517\pi$$
$$908$$ −2.48475e7 −1.00016
$$909$$ 0 0
$$910$$ −7.81558e6 −0.312866
$$911$$ 1.92521e6 0.0768567 0.0384283 0.999261i $$-0.487765\pi$$
0.0384283 + 0.999261i $$0.487765\pi$$
$$912$$ 0 0
$$913$$ −2.12428e6 −0.0843401
$$914$$ 2.45476e6 0.0971948
$$915$$ 0 0
$$916$$ 6.64860e6 0.261813
$$917$$ −7.94050e6 −0.311835
$$918$$ 0 0
$$919$$ −1.72481e7 −0.673678 −0.336839 0.941562i $$-0.609358\pi$$
−0.336839 + 0.941562i $$0.609358\pi$$
$$920$$ 1.39987e7 0.545279
$$921$$ 0 0
$$922$$ 1.28200e7 0.496662
$$923$$ 3.99780e7 1.54460
$$924$$ 0 0
$$925$$ −8.19106e6 −0.314765
$$926$$ 1.32606e7 0.508202
$$927$$ 0 0
$$928$$ −2.39568e7 −0.913185
$$929$$ −2.51145e6 −0.0954740 −0.0477370 0.998860i $$-0.515201\pi$$
−0.0477370 + 0.998860i $$0.515201\pi$$
$$930$$ 0 0
$$931$$ 1.10095e7 0.416288
$$932$$ −2.56118e7 −0.965828
$$933$$ 0 0
$$934$$ 8.29538e6 0.311150
$$935$$ 4.01865e6 0.150332
$$936$$ 0 0
$$937$$ 1.79853e7 0.669221 0.334611 0.942357i $$-0.391395\pi$$
0.334611 + 0.942357i $$0.391395\pi$$
$$938$$ 9.14877e6 0.339512
$$939$$ 0 0
$$940$$ −3.18497e7 −1.17567
$$941$$ 3.22586e7 1.18760 0.593802 0.804611i $$-0.297626\pi$$
0.593802 + 0.804611i $$0.297626\pi$$
$$942$$ 0 0
$$943$$ 1.30300e7 0.477160
$$944$$ 3.84416e6 0.140401
$$945$$ 0 0
$$946$$ −2.00957e6 −0.0730087
$$947$$ −4.41659e7 −1.60034 −0.800169 0.599774i $$-0.795257\pi$$
−0.800169 + 0.599774i $$0.795257\pi$$
$$948$$ 0 0
$$949$$ −1.05880e7 −0.381635
$$950$$ −4.35888e6 −0.156699
$$951$$ 0 0
$$952$$ −1.28227e7 −0.458551
$$953$$ −1.87488e7 −0.668714 −0.334357 0.942446i $$-0.608519\pi$$
−0.334357 + 0.942446i $$0.608519\pi$$
$$954$$ 0 0
$$955$$ −1.40580e7 −0.498786
$$956$$ 3.94498e7 1.39605
$$957$$ 0 0
$$958$$ 1.01026e7 0.355649
$$959$$ 3.38716e7 1.18929
$$960$$ 0 0
$$961$$ −3.30813e6 −0.115551
$$962$$ 9.31946e6 0.324678
$$963$$ 0 0
$$964$$ 2.31380e7 0.801925
$$965$$ −5.35596e6 −0.185148
$$966$$ 0 0
$$967$$ 1.08673e7 0.373730 0.186865 0.982386i $$-0.440167\pi$$
0.186865 + 0.982386i $$0.440167\pi$$
$$968$$ −1.75692e6 −0.0602648
$$969$$ 0 0
$$970$$ −6.50054e6 −0.221830
$$971$$ 4.79123e7 1.63079 0.815397 0.578902i $$-0.196519\pi$$
0.815397 + 0.578902i $$0.196519\pi$$
$$972$$ 0 0
$$973$$ 5.54882e7 1.87896
$$974$$ 5.32442e6 0.179835
$$975$$ 0 0
$$976$$ −3.51996e7 −1.18281
$$977$$ −4.01385e7 −1.34532 −0.672658 0.739954i $$-0.734847\pi$$
−0.672658 + 0.739954i $$0.734847\pi$$
$$978$$ 0 0
$$979$$ 1.60410e7 0.534902
$$980$$ 6.56494e6 0.218356
$$981$$ 0 0
$$982$$ 1.10932e7 0.367094
$$983$$ −3.22682e6 −0.106510 −0.0532551 0.998581i $$-0.516960\pi$$
−0.0532551 + 0.998581i $$0.516960\pi$$
$$984$$ 0 0
$$985$$ −1.13961e7 −0.374254
$$986$$ −6.71460e6 −0.219952
$$987$$ 0 0
$$988$$ −3.47155e7 −1.13144
$$989$$ 2.10589e7 0.684614
$$990$$ 0 0
$$991$$ −5.95345e6 −0.192568 −0.0962841 0.995354i $$-0.530696\pi$$
−0.0962841 + 0.995354i $$0.530696\pi$$
$$992$$ 2.59249e7 0.836445
$$993$$ 0 0
$$994$$ 2.06158e7 0.661812
$$995$$ 2.36146e7 0.756175
$$996$$ 0 0
$$997$$ 3.20783e7 1.02205 0.511027 0.859565i $$-0.329265\pi$$
0.511027 + 0.859565i $$0.329265\pi$$
$$998$$ −13640.0 −0.000433499 0
$$999$$ 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 99.6.a.b.1.1 1
3.2 odd 2 33.6.a.a.1.1 1
11.10 odd 2 1089.6.a.d.1.1 1
12.11 even 2 528.6.a.i.1.1 1
15.14 odd 2 825.6.a.b.1.1 1
33.32 even 2 363.6.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.a.1.1 1 3.2 odd 2
99.6.a.b.1.1 1 1.1 even 1 trivial
363.6.a.c.1.1 1 33.32 even 2
528.6.a.i.1.1 1 12.11 even 2
825.6.a.b.1.1 1 15.14 odd 2
1089.6.a.d.1.1 1 11.10 odd 2