# Properties

 Label 363.6.a.c.1.1 Level $363$ Weight $6$ Character 363.1 Self dual yes Analytic conductor $58.219$ 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] = [363,6,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, 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("363.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 363.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: $$58.2193265921$$ 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 $$\chi$$ $$=$$ 363.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -9.00000 q^{3} -28.0000 q^{4} +46.0000 q^{5} -18.0000 q^{6} -148.000 q^{7} -120.000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -9.00000 q^{3} -28.0000 q^{4} +46.0000 q^{5} -18.0000 q^{6} -148.000 q^{7} -120.000 q^{8} +81.0000 q^{9} +92.0000 q^{10} +252.000 q^{12} -574.000 q^{13} -296.000 q^{14} -414.000 q^{15} +656.000 q^{16} +722.000 q^{17} +162.000 q^{18} -2160.00 q^{19} -1288.00 q^{20} +1332.00 q^{21} -2536.00 q^{23} +1080.00 q^{24} -1009.00 q^{25} -1148.00 q^{26} -729.000 q^{27} +4144.00 q^{28} -4650.00 q^{29} -828.000 q^{30} +5032.00 q^{31} +5152.00 q^{32} +1444.00 q^{34} -6808.00 q^{35} -2268.00 q^{36} +8118.00 q^{37} -4320.00 q^{38} +5166.00 q^{39} -5520.00 q^{40} +5138.00 q^{41} +2664.00 q^{42} -8304.00 q^{43} +3726.00 q^{45} -5072.00 q^{46} +24728.0 q^{47} -5904.00 q^{48} +5097.00 q^{49} -2018.00 q^{50} -6498.00 q^{51} +16072.0 q^{52} -28746.0 q^{53} -1458.00 q^{54} +17760.0 q^{56} +19440.0 q^{57} -9300.00 q^{58} -5860.00 q^{59} +11592.0 q^{60} +53658.0 q^{61} +10064.0 q^{62} -11988.0 q^{63} -10688.0 q^{64} -26404.0 q^{65} +30908.0 q^{67} -20216.0 q^{68} +22824.0 q^{69} -13616.0 q^{70} -69648.0 q^{71} -9720.00 q^{72} +18446.0 q^{73} +16236.0 q^{74} +9081.00 q^{75} +60480.0 q^{76} +10332.0 q^{78} +25300.0 q^{79} +30176.0 q^{80} +6561.00 q^{81} +10276.0 q^{82} +17556.0 q^{83} -37296.0 q^{84} +33212.0 q^{85} -16608.0 q^{86} +41850.0 q^{87} +132570. q^{89} +7452.00 q^{90} +84952.0 q^{91} +71008.0 q^{92} -45288.0 q^{93} +49456.0 q^{94} -99360.0 q^{95} -46368.0 q^{96} +70658.0 q^{97} +10194.0 q^{98} +O(q^{100})$$

## 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 $$0.443433\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ −28.0000 −0.875000
$$5$$ 46.0000 0.822873 0.411437 0.911438i $$-0.365027\pi$$
0.411437 + 0.911438i $$0.365027\pi$$
$$6$$ −18.0000 −0.204124
$$7$$ −148.000 −1.14161 −0.570803 0.821087i $$-0.693368\pi$$
−0.570803 + 0.821087i $$0.693368\pi$$
$$8$$ −120.000 −0.662913
$$9$$ 81.0000 0.333333
$$10$$ 92.0000 0.290930
$$11$$ 0 0
$$12$$ 252.000 0.505181
$$13$$ −574.000 −0.942006 −0.471003 0.882132i $$-0.656108\pi$$
−0.471003 + 0.882132i $$0.656108\pi$$
$$14$$ −296.000 −0.403619
$$15$$ −414.000 −0.475086
$$16$$ 656.000 0.640625
$$17$$ 722.000 0.605919 0.302960 0.953003i $$-0.402025\pi$$
0.302960 + 0.953003i $$0.402025\pi$$
$$18$$ 162.000 0.117851
$$19$$ −2160.00 −1.37268 −0.686341 0.727280i $$-0.740784\pi$$
−0.686341 + 0.727280i $$0.740784\pi$$
$$20$$ −1288.00 −0.720014
$$21$$ 1332.00 0.659107
$$22$$ 0 0
$$23$$ −2536.00 −0.999608 −0.499804 0.866139i $$-0.666595\pi$$
−0.499804 + 0.866139i $$0.666595\pi$$
$$24$$ 1080.00 0.382733
$$25$$ −1009.00 −0.322880
$$26$$ −1148.00 −0.333049
$$27$$ −729.000 −0.192450
$$28$$ 4144.00 0.998906
$$29$$ −4650.00 −1.02673 −0.513367 0.858169i $$-0.671602\pi$$
−0.513367 + 0.858169i $$0.671602\pi$$
$$30$$ −828.000 −0.167968
$$31$$ 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
$$36$$ −2268.00 −0.291667
$$37$$ 8118.00 0.974866 0.487433 0.873161i $$-0.337933\pi$$
0.487433 + 0.873161i $$0.337933\pi$$
$$38$$ −4320.00 −0.485316
$$39$$ 5166.00 0.543867
$$40$$ −5520.00 −0.545493
$$41$$ 5138.00 0.477347 0.238674 0.971100i $$-0.423287\pi$$
0.238674 + 0.971100i $$0.423287\pi$$
$$42$$ 2664.00 0.233030
$$43$$ −8304.00 −0.684883 −0.342441 0.939539i $$-0.611254\pi$$
−0.342441 + 0.939539i $$0.611254\pi$$
$$44$$ 0 0
$$45$$ 3726.00 0.274291
$$46$$ −5072.00 −0.353415
$$47$$ 24728.0 1.63284 0.816421 0.577457i $$-0.195955\pi$$
0.816421 + 0.577457i $$0.195955\pi$$
$$48$$ −5904.00 −0.369865
$$49$$ 5097.00 0.303266
$$50$$ −2018.00 −0.114155
$$51$$ −6498.00 −0.349828
$$52$$ 16072.0 0.824255
$$53$$ −28746.0 −1.40568 −0.702842 0.711346i $$-0.748086\pi$$
−0.702842 + 0.711346i $$0.748086\pi$$
$$54$$ −1458.00 −0.0680414
$$55$$ 0 0
$$56$$ 17760.0 0.756786
$$57$$ 19440.0 0.792518
$$58$$ −9300.00 −0.363005
$$59$$ −5860.00 −0.219163 −0.109582 0.993978i $$-0.534951\pi$$
−0.109582 + 0.993978i $$0.534951\pi$$
$$60$$ 11592.0 0.415700
$$61$$ 53658.0 1.84633 0.923166 0.384401i $$-0.125592\pi$$
0.923166 + 0.384401i $$0.125592\pi$$
$$62$$ 10064.0 0.332500
$$63$$ −11988.0 −0.380536
$$64$$ −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
$$69$$ 22824.0 0.577124
$$70$$ −13616.0 −0.332127
$$71$$ −69648.0 −1.63969 −0.819847 0.572583i $$-0.805942\pi$$
−0.819847 + 0.572583i $$0.805942\pi$$
$$72$$ −9720.00 −0.220971
$$73$$ 18446.0 0.405131 0.202565 0.979269i $$-0.435072\pi$$
0.202565 + 0.979269i $$0.435072\pi$$
$$74$$ 16236.0 0.344667
$$75$$ 9081.00 0.186415
$$76$$ 60480.0 1.20110
$$77$$ 0 0
$$78$$ 10332.0 0.192286
$$79$$ 25300.0 0.456092 0.228046 0.973650i $$-0.426766\pi$$
0.228046 + 0.973650i $$0.426766\pi$$
$$80$$ 30176.0 0.527153
$$81$$ 6561.00 0.111111
$$82$$ 10276.0 0.168768
$$83$$ 17556.0 0.279724 0.139862 0.990171i $$-0.455334\pi$$
0.139862 + 0.990171i $$0.455334\pi$$
$$84$$ −37296.0 −0.576719
$$85$$ 33212.0 0.498595
$$86$$ −16608.0 −0.242143
$$87$$ 41850.0 0.592785
$$88$$ 0 0
$$89$$ 132570. 1.77407 0.887034 0.461704i $$-0.152762\pi$$
0.887034 + 0.461704i $$0.152762\pi$$
$$90$$ 7452.00 0.0969765
$$91$$ 84952.0 1.07540
$$92$$ 71008.0 0.874657
$$93$$ −45288.0 −0.542970
$$94$$ 49456.0 0.577297
$$95$$ −99360.0 −1.12954
$$96$$ −46368.0 −0.513500
$$97$$ 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 $$0.334266\pi$$
$$102$$ −12996.0 −0.123683
$$103$$ 130904. 1.21579 0.607897 0.794016i $$-0.292013\pi$$
0.607897 + 0.794016i $$0.292013\pi$$
$$104$$ 68880.0 0.624467
$$105$$ 61272.0 0.542361
$$106$$ −57492.0 −0.496984
$$107$$ 141612. 1.19575 0.597875 0.801589i $$-0.296012\pi$$
0.597875 + 0.801589i $$0.296012\pi$$
$$108$$ 20412.0 0.168394
$$109$$ 239810. 1.93331 0.966654 0.256086i $$-0.0824330\pi$$
0.966654 + 0.256086i $$0.0824330\pi$$
$$110$$ 0 0
$$111$$ −73062.0 −0.562839
$$112$$ −97088.0 −0.731342
$$113$$ −42726.0 −0.314772 −0.157386 0.987537i $$-0.550307\pi$$
−0.157386 + 0.987537i $$0.550307\pi$$
$$114$$ 38880.0 0.280197
$$115$$ −116656. −0.822550
$$116$$ 130200. 0.898392
$$117$$ −46494.0 −0.314002
$$118$$ −11720.0 −0.0774859
$$119$$ −106856. −0.691722
$$120$$ 49680.0 0.314940
$$121$$ 0 0
$$122$$ 107316. 0.652777
$$123$$ −46242.0 −0.275597
$$124$$ −140896. −0.822895
$$125$$ −190164. −1.08856
$$126$$ −23976.0 −0.134540
$$127$$ −51788.0 −0.284918 −0.142459 0.989801i $$-0.545501\pi$$
−0.142459 + 0.989801i $$0.545501\pi$$
$$128$$ −186240. −1.00473
$$129$$ 74736.0 0.395417
$$130$$ −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
$$135$$ −33534.0 −0.158362
$$136$$ −86640.0 −0.401672
$$137$$ −228862. −1.04177 −0.520886 0.853627i $$-0.674398\pi$$
−0.520886 + 0.853627i $$0.674398\pi$$
$$138$$ 45648.0 0.204044
$$139$$ −374920. −1.64589 −0.822947 0.568119i $$-0.807671\pi$$
−0.822947 + 0.568119i $$0.807671\pi$$
$$140$$ 190624. 0.821973
$$141$$ −222552. −0.942722
$$142$$ −139296. −0.579719
$$143$$ 0 0
$$144$$ 53136.0 0.213542
$$145$$ −213900. −0.844872
$$146$$ 36892.0 0.143235
$$147$$ −45873.0 −0.175091
$$148$$ −227304. −0.853007
$$149$$ 65830.0 0.242917 0.121459 0.992597i $$-0.461243\pi$$
0.121459 + 0.992597i $$0.461243\pi$$
$$150$$ 18162.0 0.0659076
$$151$$ −154052. −0.549826 −0.274913 0.961469i $$-0.588649\pi$$
−0.274913 + 0.961469i $$0.588649\pi$$
$$152$$ 259200. 0.909968
$$153$$ 58482.0 0.201973
$$154$$ 0 0
$$155$$ 231472. 0.773872
$$156$$ −144648. −0.475884
$$157$$ 287678. 0.931446 0.465723 0.884931i $$-0.345794\pi$$
0.465723 + 0.884931i $$0.345794\pi$$
$$158$$ 50600.0 0.161253
$$159$$ 258714. 0.811572
$$160$$ 236992. 0.731870
$$161$$ 375328. 1.14116
$$162$$ 13122.0 0.0392837
$$163$$ 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 $$0.566966\pi$$
$$168$$ −159840. −0.436930
$$169$$ −41817.0 −0.112625
$$170$$ 66424.0 0.176280
$$171$$ −174960. −0.457560
$$172$$ 232512. 0.599272
$$173$$ 2166.00 0.00550229 0.00275114 0.999996i $$-0.499124\pi$$
0.00275114 + 0.999996i $$0.499124\pi$$
$$174$$ 83700.0 0.209581
$$175$$ 149332. 0.368602
$$176$$ 0 0
$$177$$ 52740.0 0.126534
$$178$$ 265140. 0.627228
$$179$$ 672780. 1.56942 0.784712 0.619860i $$-0.212811\pi$$
0.784712 + 0.619860i $$0.212811\pi$$
$$180$$ −104328. −0.240005
$$181$$ −526778. −1.19517 −0.597587 0.801804i $$-0.703874\pi$$
−0.597587 + 0.801804i $$0.703874\pi$$
$$182$$ 169904. 0.380211
$$183$$ −482922. −1.06598
$$184$$ 304320. 0.662653
$$185$$ 373428. 0.802191
$$186$$ −90576.0 −0.191969
$$187$$ 0 0
$$188$$ −692384. −1.42874
$$189$$ 107892. 0.219702
$$190$$ −198720. −0.399354
$$191$$ −305608. −0.606152 −0.303076 0.952966i $$-0.598014\pi$$
−0.303076 + 0.952966i $$0.598014\pi$$
$$192$$ 96192.0 0.188315
$$193$$ −116434. −0.225002 −0.112501 0.993652i $$-0.535886\pi$$
−0.112501 + 0.993652i $$0.535886\pi$$
$$194$$ 141316. 0.269580
$$195$$ 237636. 0.447534
$$196$$ −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
$$201$$ −278172. −0.485650
$$202$$ 203996. 0.351757
$$203$$ 688200. 1.17213
$$204$$ 181944. 0.306099
$$205$$ 236348. 0.392796
$$206$$ 261808. 0.429848
$$207$$ −205416. −0.333203
$$208$$ −376544. −0.603472
$$209$$ 0 0
$$210$$ 122544. 0.191754
$$211$$ 620688. 0.959770 0.479885 0.877331i $$-0.340678\pi$$
0.479885 + 0.877331i $$0.340678\pi$$
$$212$$ 804888. 1.22997
$$213$$ 626832. 0.946678
$$214$$ 283224. 0.422762
$$215$$ −381984. −0.563571
$$216$$ 87480.0 0.127578
$$217$$ −744736. −1.07363
$$218$$ 479620. 0.683528
$$219$$ −166014. −0.233902
$$220$$ 0 0
$$221$$ −414428. −0.570780
$$222$$ −146124. −0.198994
$$223$$ −1.31802e6 −1.77484 −0.887419 0.460964i $$-0.847504\pi$$
−0.887419 + 0.460964i $$0.847504\pi$$
$$224$$ −762496. −1.01535
$$225$$ −81729.0 −0.107627
$$226$$ −85452.0 −0.111289
$$227$$ 887412. 1.14304 0.571519 0.820589i $$-0.306354\pi$$
0.571519 + 0.820589i $$0.306354\pi$$
$$228$$ −544320. −0.693453
$$229$$ −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 $$0.313902\pi$$
$$234$$ −92988.0 −0.111016
$$235$$ 1.13749e6 1.34362
$$236$$ 164080. 0.191768
$$237$$ −227700. −0.263325
$$238$$ −213712. −0.244561
$$239$$ −1.40892e6 −1.59548 −0.797740 0.603001i $$-0.793971\pi$$
−0.797740 + 0.603001i $$0.793971\pi$$
$$240$$ −271584. −0.304352
$$241$$ 826358. 0.916486 0.458243 0.888827i $$-0.348479\pi$$
0.458243 + 0.888827i $$0.348479\pi$$
$$242$$ 0 0
$$243$$ −59049.0 −0.0641500
$$244$$ −1.50242e6 −1.61554
$$245$$ 234462. 0.249550
$$246$$ −92484.0 −0.0974381
$$247$$ 1.23984e6 1.29307
$$248$$ −603840. −0.623437
$$249$$ −158004. −0.161499
$$250$$ −380328. −0.384865
$$251$$ −1.60387e6 −1.60688 −0.803442 0.595384i $$-0.797000\pi$$
−0.803442 + 0.595384i $$0.797000\pi$$
$$252$$ 335664. 0.332969
$$253$$ 0 0
$$254$$ −103576. −0.100734
$$255$$ −298908. −0.287864
$$256$$ −30464.0 −0.0290527
$$257$$ 397618. 0.375520 0.187760 0.982215i $$-0.439877\pi$$
0.187760 + 0.982215i $$0.439877\pi$$
$$258$$ 149472. 0.139801
$$259$$ −1.20146e6 −1.11291
$$260$$ 739312. 0.678257
$$261$$ −376650. −0.342245
$$262$$ −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
$$267$$ −1.19313e6 −1.02426
$$268$$ −865424. −0.736024
$$269$$ −725810. −0.611564 −0.305782 0.952101i $$-0.598918\pi$$
−0.305782 + 0.952101i $$0.598918\pi$$
$$270$$ −67068.0 −0.0559894
$$271$$ 1.46787e6 1.21413 0.607063 0.794654i $$-0.292348\pi$$
0.607063 + 0.794654i $$0.292348\pi$$
$$272$$ 473632. 0.388167
$$273$$ −764568. −0.620883
$$274$$ −457724. −0.368322
$$275$$ 0 0
$$276$$ −639072. −0.504983
$$277$$ −1.52100e6 −1.19105 −0.595524 0.803338i $$-0.703056\pi$$
−0.595524 + 0.803338i $$0.703056\pi$$
$$278$$ −749840. −0.581911
$$279$$ 407592. 0.313484
$$280$$ 816960. 0.622738
$$281$$ −464382. −0.350840 −0.175420 0.984494i $$-0.556128\pi$$
−0.175420 + 0.984494i $$0.556128\pi$$
$$282$$ −445104. −0.333303
$$283$$ 415136. 0.308123 0.154062 0.988061i $$-0.450765\pi$$
0.154062 + 0.988061i $$0.450765\pi$$
$$284$$ 1.95014e6 1.43473
$$285$$ 894240. 0.652142
$$286$$ 0 0
$$287$$ −760424. −0.544943
$$288$$ 417312. 0.296469
$$289$$ −898573. −0.632862
$$290$$ −427800. −0.298707
$$291$$ −635922. −0.440222
$$292$$ −516488. −0.354489
$$293$$ 2.59321e6 1.76469 0.882344 0.470605i $$-0.155964\pi$$
0.882344 + 0.470605i $$0.155964\pi$$
$$294$$ −91746.0 −0.0619040
$$295$$ −269560. −0.180343
$$296$$ −974160. −0.646251
$$297$$ 0 0
$$298$$ 131660. 0.0858842
$$299$$ 1.45566e6 0.941636
$$300$$ −254268. −0.163113
$$301$$ 1.22899e6 0.781867
$$302$$ −308104. −0.194393
$$303$$ −917982. −0.574417
$$304$$ −1.41696e6 −0.879374
$$305$$ 2.46827e6 1.51930
$$306$$ 116964. 0.0714083
$$307$$ 930832. 0.563671 0.281835 0.959463i $$-0.409057\pi$$
0.281835 + 0.959463i $$0.409057\pi$$
$$308$$ 0 0
$$309$$ −1.17814e6 −0.701939
$$310$$ 462944. 0.273605
$$311$$ 2.48527e6 1.45704 0.728522 0.685022i $$-0.240207\pi$$
0.728522 + 0.685022i $$0.240207\pi$$
$$312$$ −619920. −0.360536
$$313$$ 1.31719e6 0.759957 0.379978 0.924995i $$-0.375931\pi$$
0.379978 + 0.924995i $$0.375931\pi$$
$$314$$ 575356. 0.329316
$$315$$ −551448. −0.313133
$$316$$ −708400. −0.399081
$$317$$ 2.25540e6 1.26059 0.630297 0.776354i $$-0.282933\pi$$
0.630297 + 0.776354i $$0.282933\pi$$
$$318$$ 517428. 0.286934
$$319$$ 0 0
$$320$$ −491648. −0.268398
$$321$$ −1.27451e6 −0.690367
$$322$$ 750656. 0.403461
$$323$$ −1.55952e6 −0.831734
$$324$$ −183708. −0.0972222
$$325$$ 579166. 0.304155
$$326$$ 210248. 0.109569
$$327$$ −2.15829e6 −1.11620
$$328$$ −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
$$333$$ 657558. 0.324955
$$334$$ −301056. −0.147666
$$335$$ 1.42177e6 0.692176
$$336$$ 873792. 0.422240
$$337$$ −1.27630e6 −0.612177 −0.306089 0.952003i $$-0.599020\pi$$
−0.306089 + 0.952003i $$0.599020\pi$$
$$338$$ −83634.0 −0.0398191
$$339$$ 384534. 0.181734
$$340$$ −929936. −0.436270
$$341$$ 0 0
$$342$$ −349920. −0.161772
$$343$$ 1.73308e6 0.795396
$$344$$ 996480. 0.454017
$$345$$ 1.04990e6 0.474900
$$346$$ 4332.00 0.00194535
$$347$$ −3.69303e6 −1.64649 −0.823245 0.567687i $$-0.807838\pi$$
−0.823245 + 0.567687i $$0.807838\pi$$
$$348$$ −1.17180e6 −0.518687
$$349$$ −1.70919e6 −0.751150 −0.375575 0.926792i $$-0.622555\pi$$
−0.375575 + 0.926792i $$0.622555\pi$$
$$350$$ 298664. 0.130321
$$351$$ 418446. 0.181289
$$352$$ 0 0
$$353$$ 4.36859e6 1.86597 0.932986 0.359914i $$-0.117194\pi$$
0.932986 + 0.359914i $$0.117194\pi$$
$$354$$ 105480. 0.0447365
$$355$$ −3.20381e6 −1.34926
$$356$$ −3.71196e6 −1.55231
$$357$$ 961704. 0.399366
$$358$$ 1.34556e6 0.554875
$$359$$ 3.51284e6 1.43854 0.719271 0.694730i $$-0.244476\pi$$
0.719271 + 0.694730i $$0.244476\pi$$
$$360$$ −447120. −0.181831
$$361$$ 2.18950e6 0.884254
$$362$$ −1.05356e6 −0.422558
$$363$$ 0 0
$$364$$ −2.37866e6 −0.940975
$$365$$ 848516. 0.333371
$$366$$ −965844. −0.376881
$$367$$ −2.15259e6 −0.834251 −0.417125 0.908849i $$-0.636962\pi$$
−0.417125 + 0.908849i $$0.636962\pi$$
$$368$$ −1.66362e6 −0.640374
$$369$$ 416178. 0.159116
$$370$$ 746856. 0.283617
$$371$$ 4.25441e6 1.60474
$$372$$ 1.26806e6 0.475099
$$373$$ 2.24247e6 0.834553 0.417276 0.908780i $$-0.362985\pi$$
0.417276 + 0.908780i $$0.362985\pi$$
$$374$$ 0 0
$$375$$ 1.71148e6 0.628482
$$376$$ −2.96736e6 −1.08243
$$377$$ 2.66910e6 0.967189
$$378$$ 215784. 0.0776765
$$379$$ −2.40986e6 −0.861775 −0.430887 0.902406i $$-0.641799\pi$$
−0.430887 + 0.902406i $$0.641799\pi$$
$$380$$ 2.78208e6 0.988350
$$381$$ 466092. 0.164497
$$382$$ −611216. −0.214307
$$383$$ −1.01066e6 −0.352052 −0.176026 0.984386i $$-0.556324\pi$$
−0.176026 + 0.984386i $$0.556324\pi$$
$$384$$ 1.67616e6 0.580079
$$385$$ 0 0
$$386$$ −232868. −0.0795503
$$387$$ −672624. −0.228294
$$388$$ −1.97842e6 −0.667175
$$389$$ 1.27779e6 0.428140 0.214070 0.976818i $$-0.431328\pi$$
0.214070 + 0.976818i $$0.431328\pi$$
$$390$$ 475272. 0.158227
$$391$$ −1.83099e6 −0.605682
$$392$$ −611640. −0.201039
$$393$$ 482868. 0.157706
$$394$$ 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
$$399$$ −2.87712e6 −0.904744
$$400$$ −661904. −0.206845
$$401$$ −1.48980e6 −0.462665 −0.231332 0.972875i $$-0.574308\pi$$
−0.231332 + 0.972875i $$0.574308\pi$$
$$402$$ −556344. −0.171703
$$403$$ −2.88837e6 −0.885911
$$404$$ −2.85594e6 −0.870555
$$405$$ 301806. 0.0914303
$$406$$ 1.37640e6 0.414409
$$407$$ 0 0
$$408$$ 779760. 0.231905
$$409$$ 4.39899e6 1.30030 0.650152 0.759804i $$-0.274705\pi$$
0.650152 + 0.759804i $$0.274705\pi$$
$$410$$ 472696. 0.138874
$$411$$ 2.05976e6 0.601467
$$412$$ −3.66531e6 −1.06382
$$413$$ 867280. 0.250198
$$414$$ −410832. −0.117805
$$415$$ 807576. 0.230178
$$416$$ −2.95725e6 −0.837827
$$417$$ 3.37428e6 0.950257
$$418$$ 0 0
$$419$$ −280420. −0.0780322 −0.0390161 0.999239i $$-0.512422\pi$$
−0.0390161 + 0.999239i $$0.512422\pi$$
$$420$$ −1.71562e6 −0.474566
$$421$$ 817462. 0.224782 0.112391 0.993664i $$-0.464149\pi$$
0.112391 + 0.993664i $$0.464149\pi$$
$$422$$ 1.24138e6 0.339330
$$423$$ 2.00297e6 0.544281
$$424$$ 3.44952e6 0.931846
$$425$$ −728498. −0.195639
$$426$$ 1.25366e6 0.334701
$$427$$ −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 $$0.578631\pi$$
$$432$$ −478224. −0.123288
$$433$$ 5.84067e6 1.49707 0.748537 0.663093i $$-0.230757\pi$$
0.748537 + 0.663093i $$0.230757\pi$$
$$434$$ −1.48947e6 −0.379584
$$435$$ 1.92510e6 0.487787
$$436$$ −6.71468e6 −1.69164
$$437$$ 5.47776e6 1.37214
$$438$$ −332028. −0.0826969
$$439$$ 509540. 0.126188 0.0630938 0.998008i $$-0.479903\pi$$
0.0630938 + 0.998008i $$0.479903\pi$$
$$440$$ 0 0
$$441$$ 412857. 0.101089
$$442$$ −828856. −0.201801
$$443$$ 4.10268e6 0.993250 0.496625 0.867965i $$-0.334572\pi$$
0.496625 + 0.867965i $$0.334572\pi$$
$$444$$ 2.04574e6 0.492484
$$445$$ 6.09822e6 1.45983
$$446$$ −2.63603e6 −0.627500
$$447$$ −592470. −0.140248
$$448$$ 1.58182e6 0.372360
$$449$$ 513410. 0.120185 0.0600923 0.998193i $$-0.480861\pi$$
0.0600923 + 0.998193i $$0.480861\pi$$
$$450$$ −163458. −0.0380518
$$451$$ 0 0
$$452$$ 1.19633e6 0.275426
$$453$$ 1.38647e6 0.317442
$$454$$ 1.77482e6 0.404125
$$455$$ 3.90779e6 0.884918
$$456$$ −2.33280e6 −0.525370
$$457$$ −1.22738e6 −0.274908 −0.137454 0.990508i $$-0.543892\pi$$
−0.137454 + 0.990508i $$0.543892\pi$$
$$458$$ −474900. −0.105789
$$459$$ −526338. −0.116609
$$460$$ 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
$$465$$ −2.08325e6 −0.446795
$$466$$ 1.82941e6 0.390253
$$467$$ −4.14769e6 −0.880064 −0.440032 0.897982i $$-0.645033\pi$$
−0.440032 + 0.897982i $$0.645033\pi$$
$$468$$ 1.30183e6 0.274752
$$469$$ −4.57438e6 −0.960286
$$470$$ 2.27498e6 0.475042
$$471$$ −2.58910e6 −0.537770
$$472$$ 703200. 0.145286
$$473$$ 0 0
$$474$$ −455400. −0.0930995
$$475$$ 2.17944e6 0.443211
$$476$$ 2.99197e6 0.605257
$$477$$ −2.32843e6 −0.468561
$$478$$ −2.81784e6 −0.564088
$$479$$ 5.05132e6 1.00593 0.502963 0.864308i $$-0.332243\pi$$
0.502963 + 0.864308i $$0.332243\pi$$
$$480$$ −2.13293e6 −0.422545
$$481$$ −4.65973e6 −0.918329
$$482$$ 1.65272e6 0.324027
$$483$$ −3.37795e6 −0.658849
$$484$$ 0 0
$$485$$ 3.25027e6 0.627429
$$486$$ −118098. −0.0226805
$$487$$ 2.66221e6 0.508651 0.254325 0.967119i $$-0.418147\pi$$
0.254325 + 0.967119i $$0.418147\pi$$
$$488$$ −6.43896e6 −1.22396
$$489$$ −946116. −0.178925
$$490$$ 468924. 0.0882292
$$491$$ 5.54659e6 1.03830 0.519149 0.854684i $$-0.326249\pi$$
0.519149 + 0.854684i $$0.326249\pi$$
$$492$$ 1.29478e6 0.241147
$$493$$ −3.35730e6 −0.622118
$$494$$ 2.47968e6 0.457171
$$495$$ 0 0
$$496$$ 3.30099e6 0.602477
$$497$$ 1.03079e7 1.87189
$$498$$ −316008. −0.0570985
$$499$$ −6820.00 −0.00122612 −0.000613060 1.00000i $$-0.500195\pi$$
−0.000613060 1.00000i $$0.500195\pi$$
$$500$$ 5.32459e6 0.952492
$$501$$ 1.35475e6 0.241138
$$502$$ −3.20774e6 −0.568119
$$503$$ 451136. 0.0795037 0.0397519 0.999210i $$-0.487343\pi$$
0.0397519 + 0.999210i $$0.487343\pi$$
$$504$$ 1.43856e6 0.252262
$$505$$ 4.69191e6 0.818693
$$506$$ 0 0
$$507$$ 376353. 0.0650243
$$508$$ 1.45006e6 0.249303
$$509$$ 393390. 0.0673021 0.0336511 0.999434i $$-0.489287\pi$$
0.0336511 + 0.999434i $$0.489287\pi$$
$$510$$ −597816. −0.101775
$$511$$ −2.73001e6 −0.462500
$$512$$ 5.89875e6 0.994455
$$513$$ 1.57464e6 0.264173
$$514$$ 795236. 0.132766
$$515$$ 6.02158e6 1.00044
$$516$$ −2.09261e6 −0.345990
$$517$$ 0 0
$$518$$ −2.40293e6 −0.393474
$$519$$ −19494.0 −0.00317675
$$520$$ 3.16848e6 0.513857
$$521$$ 3.28432e6 0.530092 0.265046 0.964236i $$-0.414613\pi$$
0.265046 + 0.964236i $$0.414613\pi$$
$$522$$ −753300. −0.121002
$$523$$ 1.68266e6 0.268993 0.134497 0.990914i $$-0.457058\pi$$
0.134497 + 0.990914i $$0.457058\pi$$
$$524$$ 1.50226e6 0.239010
$$525$$ −1.34399e6 −0.212812
$$526$$ −4.26333e6 −0.671869
$$527$$ 3.63310e6 0.569838
$$528$$ 0 0
$$529$$ −5047.00 −0.000784141 0
$$530$$ −2.64463e6 −0.408955
$$531$$ −474660. −0.0730544
$$532$$ −8.95104e6 −1.37118
$$533$$ −2.94921e6 −0.449664
$$534$$ −2.38626e6 −0.362130
$$535$$ 6.51415e6 0.983951
$$536$$ −3.70896e6 −0.557622
$$537$$ −6.05502e6 −0.906108
$$538$$ −1.45162e6 −0.216221
$$539$$ 0 0
$$540$$ 938952. 0.138567
$$541$$ −9.48158e6 −1.39280 −0.696398 0.717656i $$-0.745215\pi$$
−0.696398 + 0.717656i $$0.745215\pi$$
$$542$$ 2.93574e6 0.429258
$$543$$ 4.74100e6 0.690034
$$544$$ 3.71974e6 0.538909
$$545$$ 1.10313e7 1.59087
$$546$$ −1.52914e6 −0.219515
$$547$$ 6.09239e6 0.870602 0.435301 0.900285i $$-0.356642\pi$$
0.435301 + 0.900285i $$0.356642\pi$$
$$548$$ 6.40814e6 0.911550
$$549$$ 4.34630e6 0.615444
$$550$$ 0 0
$$551$$ 1.00440e7 1.40938
$$552$$ −2.73888e6 −0.382583
$$553$$ −3.74440e6 −0.520678
$$554$$ −3.04200e6 −0.421099
$$555$$ −3.36085e6 −0.463145
$$556$$ 1.04978e7 1.44016
$$557$$ −8.49594e6 −1.16031 −0.580154 0.814507i $$-0.697008\pi$$
−0.580154 + 0.814507i $$0.697008\pi$$
$$558$$ 815184. 0.110833
$$559$$ 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 $$0.345392\pi$$
$$564$$ 6.23146e6 0.824882
$$565$$ −1.96540e6 −0.259017
$$566$$ 830272. 0.108938
$$567$$ −971028. −0.126845
$$568$$ 8.35776e6 1.08697
$$569$$ −9.41847e6 −1.21955 −0.609775 0.792574i $$-0.708740\pi$$
−0.609775 + 0.792574i $$0.708740\pi$$
$$570$$ 1.78848e6 0.230567
$$571$$ −7.29699e6 −0.936599 −0.468299 0.883570i $$-0.655133\pi$$
−0.468299 + 0.883570i $$0.655133\pi$$
$$572$$ 0 0
$$573$$ 2.75047e6 0.349962
$$574$$ −1.52085e6 −0.192666
$$575$$ 2.55882e6 0.322753
$$576$$ −865728. −0.108724
$$577$$ −3.29590e6 −0.412131 −0.206065 0.978538i $$-0.566066\pi$$
−0.206065 + 0.978538i $$0.566066\pi$$
$$578$$ −1.79715e6 −0.223750
$$579$$ 1.04791e6 0.129905
$$580$$ 5.98920e6 0.739263
$$581$$ −2.59829e6 −0.319335
$$582$$ −1.27184e6 −0.155642
$$583$$ 0 0
$$584$$ −2.21352e6 −0.268566
$$585$$ −2.13872e6 −0.258384
$$586$$ 5.18641e6 0.623911
$$587$$ 4.39827e6 0.526849 0.263425 0.964680i $$-0.415148\pi$$
0.263425 + 0.964680i $$0.415148\pi$$
$$588$$ 1.28444e6 0.153205
$$589$$ −1.08691e7 −1.29094
$$590$$ −539120. −0.0637610
$$591$$ −2.22968e6 −0.262587
$$592$$ 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
$$597$$ 4.62024e6 0.530553
$$598$$ 2.91133e6 0.332919
$$599$$ 3.77140e6 0.429473 0.214736 0.976672i $$-0.431111\pi$$
0.214736 + 0.976672i $$0.431111\pi$$
$$600$$ −1.08972e6 −0.123577
$$601$$ −4.19724e6 −0.473999 −0.237000 0.971510i $$-0.576164\pi$$
−0.237000 + 0.971510i $$0.576164\pi$$
$$602$$ 2.45798e6 0.276432
$$603$$ 2.50355e6 0.280390
$$604$$ 4.31346e6 0.481097
$$605$$ 0 0
$$606$$ −1.83596e6 −0.203087
$$607$$ 1.00133e6 0.110308 0.0551539 0.998478i $$-0.482435\pi$$
0.0551539 + 0.998478i $$0.482435\pi$$
$$608$$ −1.11283e7 −1.22087
$$609$$ −6.19380e6 −0.676728
$$610$$ 4.93654e6 0.537153
$$611$$ −1.41939e7 −1.53815
$$612$$ −1.63750e6 −0.176727
$$613$$ 7.38239e6 0.793498 0.396749 0.917927i $$-0.370138\pi$$
0.396749 + 0.917927i $$0.370138\pi$$
$$614$$ 1.86166e6 0.199288
$$615$$ −2.12713e6 −0.226781
$$616$$ 0 0
$$617$$ −1.54025e7 −1.62884 −0.814418 0.580279i $$-0.802944\pi$$
−0.814418 + 0.580279i $$0.802944\pi$$
$$618$$ −2.35627e6 −0.248173
$$619$$ −7.12402e6 −0.747306 −0.373653 0.927569i $$-0.621895\pi$$
−0.373653 + 0.927569i $$0.621895\pi$$
$$620$$ −6.48122e6 −0.677138
$$621$$ 1.84874e6 0.192375
$$622$$ 4.97054e6 0.515143
$$623$$ −1.96204e7 −2.02529
$$624$$ 3.38890e6 0.348415
$$625$$ −5.59442e6 −0.572869
$$626$$ 2.63439e6 0.268685
$$627$$ 0 0
$$628$$ −8.05498e6 −0.815015
$$629$$ 5.86120e6 0.590690
$$630$$ −1.10290e6 −0.110709
$$631$$ 1.16696e7 1.16677 0.583383 0.812197i $$-0.301729\pi$$
0.583383 + 0.812197i $$0.301729\pi$$
$$632$$ −3.03600e6 −0.302349
$$633$$ −5.58619e6 −0.554124
$$634$$ 4.51080e6 0.445687
$$635$$ −2.38225e6 −0.234451
$$636$$ −7.24399e6 −0.710126
$$637$$ −2.92568e6 −0.285679
$$638$$ 0 0
$$639$$ −5.64149e6 −0.546565
$$640$$ −8.56704e6 −0.826763
$$641$$ −1.10271e7 −1.06003 −0.530014 0.847989i $$-0.677813\pi$$
−0.530014 + 0.847989i $$0.677813\pi$$
$$642$$ −2.54902e6 −0.244082
$$643$$ −9.56024e6 −0.911887 −0.455944 0.890009i $$-0.650698\pi$$
−0.455944 + 0.890009i $$0.650698\pi$$
$$644$$ −1.05092e7 −0.998514
$$645$$ 3.43786e6 0.325378
$$646$$ −3.11904e6 −0.294063
$$647$$ −1.09942e7 −1.03253 −0.516263 0.856430i $$-0.672677\pi$$
−0.516263 + 0.856430i $$0.672677\pi$$
$$648$$ −787320. −0.0736570
$$649$$ 0 0
$$650$$ 1.15833e6 0.107535
$$651$$ 6.70262e6 0.619858
$$652$$ −2.94347e6 −0.271170
$$653$$ −295346. −0.0271049 −0.0135525 0.999908i $$-0.504314\pi$$
−0.0135525 + 0.999908i $$0.504314\pi$$
$$654$$ −4.31658e6 −0.394635
$$655$$ −2.46799e6 −0.224771
$$656$$ 3.37053e6 0.305801
$$657$$ 1.49413e6 0.135044
$$658$$ −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
$$663$$ 3.72985e6 0.329540
$$664$$ −2.10672e6 −0.185433
$$665$$ 1.47053e7 1.28949
$$666$$ 1.31512e6 0.114889
$$667$$ 1.17924e7 1.02633
$$668$$ 4.21478e6 0.365455
$$669$$ 1.18621e7 1.02470
$$670$$ 2.84354e6 0.244721
$$671$$ 0 0
$$672$$ 6.86246e6 0.586215
$$673$$ 1.63733e6 0.139347 0.0696735 0.997570i $$-0.477804\pi$$
0.0696735 + 0.997570i $$0.477804\pi$$
$$674$$ −2.55260e6 −0.216437
$$675$$ 735561. 0.0621383
$$676$$ 1.17088e6 0.0985472
$$677$$ 6.35878e6 0.533215 0.266607 0.963805i $$-0.414097\pi$$
0.266607 + 0.963805i $$0.414097\pi$$
$$678$$ 769068. 0.0642526
$$679$$ −1.04574e7 −0.870460
$$680$$ −3.98544e6 −0.330525
$$681$$ −7.98671e6 −0.659933
$$682$$ 0 0
$$683$$ 1.11033e7 0.910751 0.455376 0.890299i $$-0.349505\pi$$
0.455376 + 0.890299i $$0.349505\pi$$
$$684$$ 4.89888e6 0.400365
$$685$$ −1.05277e7 −0.857245
$$686$$ 3.46616e6 0.281215
$$687$$ 2.13705e6 0.172752
$$688$$ −5.44742e6 −0.438753
$$689$$ 1.65002e7 1.32416
$$690$$ 2.09981e6 0.167902
$$691$$ 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
$$696$$ −5.02200e6 −0.392965
$$697$$ 3.70964e6 0.289234
$$698$$ −3.41838e6 −0.265572
$$699$$ −8.23235e6 −0.637281
$$700$$ −4.18130e6 −0.322527
$$701$$ −1.58021e7 −1.21456 −0.607280 0.794488i $$-0.707740\pi$$
−0.607280 + 0.794488i $$0.707740\pi$$
$$702$$ 836892. 0.0640954
$$703$$ −1.75349e7 −1.33818
$$704$$ 0 0
$$705$$ −1.02374e7 −0.775741
$$706$$ 8.73719e6 0.659720
$$707$$ −1.50957e7 −1.13581
$$708$$ −1.47672e6 −0.110717
$$709$$ 1.24834e7 0.932643 0.466322 0.884615i $$-0.345579\pi$$
0.466322 + 0.884615i $$0.345579\pi$$
$$710$$ −6.40762e6 −0.477035
$$711$$ 2.04930e6 0.152031
$$712$$ −1.59084e7 −1.17605
$$713$$ −1.27612e7 −0.940083
$$714$$ 1.92341e6 0.141197
$$715$$ 0 0
$$716$$ −1.88378e7 −1.37325
$$717$$ 1.26803e7 0.921151
$$718$$ 7.02568e6 0.508601
$$719$$ 2.00724e6 0.144803 0.0724014 0.997376i $$-0.476934\pi$$
0.0724014 + 0.997376i $$0.476934\pi$$
$$720$$ 2.44426e6 0.175718
$$721$$ −1.93738e7 −1.38796
$$722$$ 4.37900e6 0.312631
$$723$$ −7.43722e6 −0.529133
$$724$$ 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
$$729$$ 531441. 0.0370370
$$730$$ 1.69703e6 0.117864
$$731$$ −5.99549e6 −0.414984
$$732$$ 1.35218e7 0.932733
$$733$$ 2.34965e7 1.61527 0.807633 0.589685i $$-0.200748\pi$$
0.807633 + 0.589685i $$0.200748\pi$$
$$734$$ −4.30518e6 −0.294952
$$735$$ −2.11016e6 −0.144078
$$736$$ −1.30655e7 −0.889059
$$737$$ 0 0
$$738$$ 832356. 0.0562559
$$739$$ 1.39901e7 0.942346 0.471173 0.882041i $$-0.343831\pi$$
0.471173 + 0.882041i $$0.343831\pi$$
$$740$$ −1.04560e7 −0.701917
$$741$$ −1.11586e7 −0.746556
$$742$$ 8.50882e6 0.567361
$$743$$ −2.42745e7 −1.61316 −0.806582 0.591123i $$-0.798685\pi$$
−0.806582 + 0.591123i $$0.798685\pi$$
$$744$$ 5.43456e6 0.359942
$$745$$ 3.02818e6 0.199890
$$746$$ 4.48493e6 0.295059
$$747$$ 1.42204e6 0.0932415
$$748$$ 0 0
$$749$$ −2.09586e7 −1.36508
$$750$$ 3.42295e6 0.222202
$$751$$ 1.53660e7 0.994170 0.497085 0.867702i $$-0.334404\pi$$
0.497085 + 0.867702i $$0.334404\pi$$
$$752$$ 1.62216e7 1.04604
$$753$$ 1.44348e7 0.927734
$$754$$ 5.33820e6 0.341953
$$755$$ −7.08639e6 −0.452437
$$756$$ −3.02098e6 −0.192240
$$757$$ 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 $$0.558489\pi$$
$$762$$ 932184. 0.0581586
$$763$$ −3.54919e7 −2.20708
$$764$$ 8.55702e6 0.530383
$$765$$ 2.69017e6 0.166198
$$766$$ −2.02131e6 −0.124469
$$767$$ 3.36364e6 0.206453
$$768$$ 274176. 0.0167736
$$769$$ 1.39197e7 0.848818 0.424409 0.905471i $$-0.360482\pi$$
0.424409 + 0.905471i $$0.360482\pi$$
$$770$$ 0 0
$$771$$ −3.57856e6 −0.216807
$$772$$ 3.26015e6 0.196877
$$773$$ −4.17883e6 −0.251539 −0.125770 0.992059i $$-0.540140\pi$$
−0.125770 + 0.992059i $$0.540140\pi$$
$$774$$ −1.34525e6 −0.0807142
$$775$$ −5.07729e6 −0.303653
$$776$$ −8.47896e6 −0.505462
$$777$$ 1.08132e7 0.642541
$$778$$ 2.55558e6 0.151370
$$779$$ −1.10981e7 −0.655246
$$780$$ −6.65381e6 −0.391592
$$781$$ 0 0
$$782$$ −3.66198e6 −0.214141
$$783$$ 3.38985e6 0.197595
$$784$$ 3.34363e6 0.194280
$$785$$ 1.32332e7 0.766461
$$786$$ 965736. 0.0557573
$$787$$ −9.66705e6 −0.556361 −0.278181 0.960529i $$-0.589731\pi$$
−0.278181 + 0.960529i $$0.589731\pi$$
$$788$$ −6.93678e6 −0.397962
$$789$$ 1.91850e7 1.09716
$$790$$ 2.32760e6 0.132691
$$791$$ 6.32345e6 0.359346
$$792$$ 0 0
$$793$$ −3.07997e7 −1.73926
$$794$$ 1.09080e7 0.614036
$$795$$ 1.19008e7 0.667821
$$796$$ 1.43741e7 0.804077
$$797$$ −5.79884e6 −0.323367 −0.161683 0.986843i $$-0.551692\pi$$
−0.161683 + 0.986843i $$0.551692\pi$$
$$798$$ −5.75424e6 −0.319875
$$799$$ 1.78536e7 0.989371
$$800$$ −5.19837e6 −0.287172
$$801$$ 1.07382e7 0.591356
$$802$$ −2.97960e6 −0.163577
$$803$$ 0 0
$$804$$ 7.78882e6 0.424944
$$805$$ 1.72651e7 0.939029
$$806$$ −5.77674e6 −0.313217
$$807$$ 6.53229e6 0.353087
$$808$$ −1.22398e7 −0.659545
$$809$$ 1.92543e7 1.03433 0.517163 0.855887i $$-0.326988\pi$$
0.517163 + 0.855887i $$0.326988\pi$$
$$810$$ 603612. 0.0323255
$$811$$ 1.31938e7 0.704396 0.352198 0.935926i $$-0.385434\pi$$
0.352198 + 0.935926i $$0.385434\pi$$
$$812$$ −1.92696e7 −1.02561
$$813$$ −1.32108e7 −0.700976
$$814$$ 0 0
$$815$$ 4.83570e6 0.255015
$$816$$ −4.26269e6 −0.224108
$$817$$ 1.79366e7 0.940126
$$818$$ 8.79798e6 0.459727
$$819$$ 6.88111e6 0.358467
$$820$$ −6.61774e6 −0.343697
$$821$$ −1.33779e7 −0.692677 −0.346338 0.938110i $$-0.612575\pi$$
−0.346338 + 0.938110i $$0.612575\pi$$
$$822$$ 4.11952e6 0.212651
$$823$$ −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 $$0.635715\pi$$
$$828$$ 5.75165e6 0.291552
$$829$$ −2.18098e7 −1.10221 −0.551107 0.834435i $$-0.685794\pi$$
−0.551107 + 0.834435i $$0.685794\pi$$
$$830$$ 1.61515e6 0.0813801
$$831$$ 1.36890e7 0.687652
$$832$$ 6.13491e6 0.307256
$$833$$ 3.68003e6 0.183755
$$834$$ 6.74856e6 0.335967
$$835$$ −6.92429e6 −0.343684
$$836$$ 0 0
$$837$$ −3.66833e6 −0.180990
$$838$$ −560840. −0.0275886
$$839$$ 1.17771e7 0.577607 0.288804 0.957388i $$-0.406743\pi$$
0.288804 + 0.957388i $$0.406743\pi$$
$$840$$ −7.35264e6 −0.359538
$$841$$ 1.11135e6 0.0541828
$$842$$ 1.63492e6 0.0794726
$$843$$ 4.17944e6 0.202558
$$844$$ −1.73793e7 −0.839799
$$845$$ −1.92358e6 −0.0926764
$$846$$ 4.00594e6 0.192432
$$847$$ 0 0
$$848$$ −1.88574e7 −0.900516
$$849$$ −3.73622e6 −0.177895
$$850$$ −1.45700e6 −0.0691689
$$851$$ −2.05872e7 −0.974483
$$852$$ −1.75513e7 −0.828343
$$853$$ 1.43993e7 0.677591 0.338796 0.940860i $$-0.389980\pi$$
0.338796 + 0.940860i $$0.389980\pi$$
$$854$$ −1.58828e7 −0.745215
$$855$$ −8.04816e6 −0.376514
$$856$$ −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
$$861$$ 6.84382e6 0.314623
$$862$$ −3.77198e6 −0.172903
$$863$$ 7.53926e6 0.344589 0.172295 0.985045i $$-0.444882\pi$$
0.172295 + 0.985045i $$0.444882\pi$$
$$864$$ −3.75581e6 −0.171167
$$865$$ 99636.0 0.00452768
$$866$$ 1.16813e7 0.529296
$$867$$ 8.08716e6 0.365383
$$868$$ 2.08526e7 0.939423
$$869$$ 0 0
$$870$$ 3.85020e6 0.172459
$$871$$ −1.77412e7 −0.792387
$$872$$ −2.87772e7 −1.28161
$$873$$ 5.72330e6 0.254162
$$874$$ 1.09555e7 0.485126
$$875$$ 2.81443e7 1.24271
$$876$$ 4.64839e6 0.204664
$$877$$ 1.04331e7 0.458051 0.229025 0.973420i $$-0.426446\pi$$
0.229025 + 0.973420i $$0.426446\pi$$
$$878$$ 1.01908e6 0.0446141
$$879$$ −2.33389e7 −1.01884
$$880$$ 0 0
$$881$$ 3.91076e7 1.69755 0.848774 0.528756i $$-0.177342\pi$$
0.848774 + 0.528756i $$0.177342\pi$$
$$882$$ 825714. 0.0357403
$$883$$ 1.29282e7 0.558003 0.279001 0.960291i $$-0.409997\pi$$
0.279001 + 0.960291i $$0.409997\pi$$
$$884$$ 1.16040e7 0.499432
$$885$$ 2.42604e6 0.104121
$$886$$ 8.20537e6 0.351167
$$887$$ −3.36466e7 −1.43592 −0.717962 0.696082i $$-0.754925\pi$$
−0.717962 + 0.696082i $$0.754925\pi$$
$$888$$ 8.76744e6 0.373113
$$889$$ 7.66462e6 0.325264
$$890$$ 1.21964e7 0.516129
$$891$$ 0 0
$$892$$ 3.69044e7 1.55298
$$893$$ −5.34125e7 −2.24137
$$894$$ −1.18494e6 −0.0495853
$$895$$ 3.09479e7 1.29144
$$896$$ 2.75635e7 1.14700
$$897$$ −1.31010e7 −0.543654
$$898$$ 1.02682e6 0.0424916
$$899$$ −2.33988e7 −0.965594
$$900$$ 2.28841e6 0.0941733
$$901$$ −2.07546e7 −0.851731
$$902$$ 0 0
$$903$$ −1.10609e7 −0.451411
$$904$$ 5.12712e6 0.208666
$$905$$ −2.42318e7 −0.983477
$$906$$ 2.77294e6 0.112233
$$907$$ −4.19629e7 −1.69374 −0.846872 0.531797i $$-0.821517\pi$$
−0.846872 + 0.531797i $$0.821517\pi$$
$$908$$ −2.48475e7 −1.00016
$$909$$ 8.26184e6 0.331640
$$910$$ 7.81558e6 0.312866
$$911$$ −1.92521e6 −0.0768567 −0.0384283 0.999261i $$-0.512235\pi$$
−0.0384283 + 0.999261i $$0.512235\pi$$
$$912$$ 1.27526e7 0.507707
$$913$$ 0 0
$$914$$ −2.45476e6 −0.0971948
$$915$$ −2.22144e7 −0.877167
$$916$$ 6.64860e6 0.261813
$$917$$ 7.94050e6 0.311835
$$918$$ −1.05268e6 −0.0412276
$$919$$ 1.72481e7 0.673678 0.336839 0.941562i $$-0.390642\pi$$
0.336839 + 0.941562i $$0.390642\pi$$
$$920$$ 1.39987e7 0.545279
$$921$$ −8.37749e6 −0.325435
$$922$$ 1.28200e7 0.496662
$$923$$ 3.99780e7 1.54460
$$924$$ 0 0
$$925$$ −8.19106e6 −0.314765
$$926$$ 1.32606e7 0.508202
$$927$$ 1.06032e7 0.405265
$$928$$ −2.39568e7 −0.913185
$$929$$ 2.51145e6 0.0954740 0.0477370 0.998860i $$-0.484799\pi$$
0.0477370 + 0.998860i $$0.484799\pi$$
$$930$$ −4.16650e6 −0.157966
$$931$$ −1.10095e7 −0.416288
$$932$$ −2.56118e7 −0.965828
$$933$$ −2.23674e7 −0.841225
$$934$$ −8.29538e6 −0.311150
$$935$$ 0 0
$$936$$ 5.57928e6 0.208156
$$937$$ −1.79853e7 −0.669221 −0.334611 0.942357i $$-0.608605\pi$$
−0.334611 + 0.942357i $$0.608605\pi$$
$$938$$ −9.14877e6 −0.339512
$$939$$ −1.18547e7 −0.438761
$$940$$ −3.18497e7 −1.17567
$$941$$ 3.22586e7 1.18760 0.593802 0.804611i $$-0.297626\pi$$
0.593802 + 0.804611i $$0.297626\pi$$
$$942$$ −5.17820e6 −0.190131
$$943$$ −1.30300e7 −0.477160
$$944$$ −3.84416e6 −0.140401
$$945$$ 4.96303e6 0.180787
$$946$$ 0 0
$$947$$ 4.41659e7 1.60034 0.800169 0.599774i $$-0.204743\pi$$
0.800169 + 0.599774i $$0.204743\pi$$
$$948$$ 6.37560e6 0.230409
$$949$$ −1.05880e7 −0.381635
$$950$$ 4.35888e6 0.156699
$$951$$ −2.02986e7 −0.727804
$$952$$ 1.28227e7 0.458551
$$953$$ −1.87488e7 −0.668714 −0.334357 0.942446i $$-0.608519\pi$$
−0.334357 + 0.942446i $$0.608519\pi$$
$$954$$ −4.65685e6 −0.165661
$$955$$ −1.40580e7 −0.498786
$$956$$ 3.94498e7 1.39605
$$957$$ 0 0
$$958$$ 1.01026e7 0.355649
$$959$$ 3.38716e7 1.18929
$$960$$ 4.42483e6 0.154960
$$961$$ −3.30813e6 −0.115551
$$962$$ −9.31946e6 −0.324678
$$963$$ 1.14706e7 0.398584
$$964$$ −2.31380e7 −0.801925
$$965$$ −5.35596e6 −0.185148
$$966$$ −6.75590e6 −0.232938
$$967$$ −1.08673e7 −0.373730 −0.186865 0.982386i $$-0.559833\pi$$
−0.186865 + 0.982386i $$0.559833\pi$$
$$968$$ 0 0
$$969$$ 1.40357e7 0.480202
$$970$$ 6.50054e6 0.221830
$$971$$ −4.79123e7 −1.63079 −0.815397 0.578902i $$-0.803481\pi$$
−0.815397 + 0.578902i $$0.803481\pi$$
$$972$$ 1.65337e6 0.0561313
$$973$$ 5.54882e7 1.87896
$$974$$ 5.32442e6 0.179835
$$975$$ −5.21249e6 −0.175604
$$976$$ 3.51996e7 1.18281
$$977$$ 4.01385e7 1.34532 0.672658 0.739954i $$-0.265153\pi$$
0.672658 + 0.739954i $$0.265153\pi$$
$$978$$ −1.89223e6 −0.0632597
$$979$$ 0 0
$$980$$ −6.56494e6 −0.218356
$$981$$ 1.94246e7 0.644436
$$982$$ 1.10932e7 0.367094
$$983$$ 3.22682e6 0.106510 0.0532551 0.998581i $$-0.483040\pi$$
0.0532551 + 0.998581i $$0.483040\pi$$
$$984$$ 5.54904e6 0.182696
$$985$$ 1.13961e7 0.374254
$$986$$ −6.71460e6 −0.219952
$$987$$ 3.29377e7 1.07622
$$988$$ −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
$$993$$ 2.85364e7 0.918387
$$994$$ 2.06158e7 0.661812
$$995$$ −2.36146e7 −0.756175
$$996$$ 4.42411e6 0.141312
$$997$$ −3.20783e7 −1.02205 −0.511027 0.859565i $$-0.670735\pi$$
−0.511027 + 0.859565i $$0.670735\pi$$
$$998$$ −13640.0 −0.000433499 0
$$999$$ −5.91802e6 −0.187613
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 363.6.a.c.1.1 1
3.2 odd 2 1089.6.a.d.1.1 1
11.10 odd 2 33.6.a.a.1.1 1
33.32 even 2 99.6.a.b.1.1 1
44.43 even 2 528.6.a.i.1.1 1
55.54 odd 2 825.6.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.a.1.1 1 11.10 odd 2
99.6.a.b.1.1 1 33.32 even 2
363.6.a.c.1.1 1 1.1 even 1 trivial
528.6.a.i.1.1 1 44.43 even 2
825.6.a.b.1.1 1 55.54 odd 2
1089.6.a.d.1.1 1 3.2 odd 2