# Properties

 Label 147.6.a.f.1.1 Level $147$ Weight $6$ Character 147.1 Self dual yes Analytic conductor $23.576$ 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] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, 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("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.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: $$23.5764215125$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 147.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{2} -9.00000 q^{3} -7.00000 q^{4} -94.0000 q^{5} -45.0000 q^{6} -195.000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+5.00000 q^{2} -9.00000 q^{3} -7.00000 q^{4} -94.0000 q^{5} -45.0000 q^{6} -195.000 q^{8} +81.0000 q^{9} -470.000 q^{10} +52.0000 q^{11} +63.0000 q^{12} +770.000 q^{13} +846.000 q^{15} -751.000 q^{16} +2022.00 q^{17} +405.000 q^{18} -1732.00 q^{19} +658.000 q^{20} +260.000 q^{22} -576.000 q^{23} +1755.00 q^{24} +5711.00 q^{25} +3850.00 q^{26} -729.000 q^{27} +5518.00 q^{29} +4230.00 q^{30} -6336.00 q^{31} +2485.00 q^{32} -468.000 q^{33} +10110.0 q^{34} -567.000 q^{36} -7338.00 q^{37} -8660.00 q^{38} -6930.00 q^{39} +18330.0 q^{40} +3262.00 q^{41} +5420.00 q^{43} -364.000 q^{44} -7614.00 q^{45} -2880.00 q^{46} -864.000 q^{47} +6759.00 q^{48} +28555.0 q^{50} -18198.0 q^{51} -5390.00 q^{52} +4182.00 q^{53} -3645.00 q^{54} -4888.00 q^{55} +15588.0 q^{57} +27590.0 q^{58} +11220.0 q^{59} -5922.00 q^{60} +45602.0 q^{61} -31680.0 q^{62} +36457.0 q^{64} -72380.0 q^{65} -2340.00 q^{66} +1396.00 q^{67} -14154.0 q^{68} +5184.00 q^{69} +18720.0 q^{71} -15795.0 q^{72} -46362.0 q^{73} -36690.0 q^{74} -51399.0 q^{75} +12124.0 q^{76} -34650.0 q^{78} +97424.0 q^{79} +70594.0 q^{80} +6561.00 q^{81} +16310.0 q^{82} +81228.0 q^{83} -190068. q^{85} +27100.0 q^{86} -49662.0 q^{87} -10140.0 q^{88} +3182.00 q^{89} -38070.0 q^{90} +4032.00 q^{92} +57024.0 q^{93} -4320.00 q^{94} +162808. q^{95} -22365.0 q^{96} -4914.00 q^{97} +4212.00 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.00000 0.883883 0.441942 0.897044i $$-0.354290\pi$$
0.441942 + 0.897044i $$0.354290\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ −7.00000 −0.218750
$$5$$ −94.0000 −1.68152 −0.840762 0.541406i $$-0.817892\pi$$
−0.840762 + 0.541406i $$0.817892\pi$$
$$6$$ −45.0000 −0.510310
$$7$$ 0 0
$$8$$ −195.000 −1.07723
$$9$$ 81.0000 0.333333
$$10$$ −470.000 −1.48627
$$11$$ 52.0000 0.129575 0.0647876 0.997899i $$-0.479363\pi$$
0.0647876 + 0.997899i $$0.479363\pi$$
$$12$$ 63.0000 0.126295
$$13$$ 770.000 1.26367 0.631833 0.775104i $$-0.282303\pi$$
0.631833 + 0.775104i $$0.282303\pi$$
$$14$$ 0 0
$$15$$ 846.000 0.970828
$$16$$ −751.000 −0.733398
$$17$$ 2022.00 1.69691 0.848455 0.529267i $$-0.177533\pi$$
0.848455 + 0.529267i $$0.177533\pi$$
$$18$$ 405.000 0.294628
$$19$$ −1732.00 −1.10069 −0.550344 0.834938i $$-0.685503\pi$$
−0.550344 + 0.834938i $$0.685503\pi$$
$$20$$ 658.000 0.367833
$$21$$ 0 0
$$22$$ 260.000 0.114529
$$23$$ −576.000 −0.227040 −0.113520 0.993536i $$-0.536213\pi$$
−0.113520 + 0.993536i $$0.536213\pi$$
$$24$$ 1755.00 0.621941
$$25$$ 5711.00 1.82752
$$26$$ 3850.00 1.11693
$$27$$ −729.000 −0.192450
$$28$$ 0 0
$$29$$ 5518.00 1.21839 0.609196 0.793020i $$-0.291492\pi$$
0.609196 + 0.793020i $$0.291492\pi$$
$$30$$ 4230.00 0.858099
$$31$$ −6336.00 −1.18416 −0.592081 0.805879i $$-0.701693\pi$$
−0.592081 + 0.805879i $$0.701693\pi$$
$$32$$ 2485.00 0.428994
$$33$$ −468.000 −0.0748102
$$34$$ 10110.0 1.49987
$$35$$ 0 0
$$36$$ −567.000 −0.0729167
$$37$$ −7338.00 −0.881198 −0.440599 0.897704i $$-0.645234\pi$$
−0.440599 + 0.897704i $$0.645234\pi$$
$$38$$ −8660.00 −0.972879
$$39$$ −6930.00 −0.729578
$$40$$ 18330.0 1.81139
$$41$$ 3262.00 0.303057 0.151528 0.988453i $$-0.451580\pi$$
0.151528 + 0.988453i $$0.451580\pi$$
$$42$$ 0 0
$$43$$ 5420.00 0.447021 0.223511 0.974701i $$-0.428248\pi$$
0.223511 + 0.974701i $$0.428248\pi$$
$$44$$ −364.000 −0.0283446
$$45$$ −7614.00 −0.560508
$$46$$ −2880.00 −0.200677
$$47$$ −864.000 −0.0570518 −0.0285259 0.999593i $$-0.509081\pi$$
−0.0285259 + 0.999593i $$0.509081\pi$$
$$48$$ 6759.00 0.423428
$$49$$ 0 0
$$50$$ 28555.0 1.61531
$$51$$ −18198.0 −0.979712
$$52$$ −5390.00 −0.276427
$$53$$ 4182.00 0.204500 0.102250 0.994759i $$-0.467396\pi$$
0.102250 + 0.994759i $$0.467396\pi$$
$$54$$ −3645.00 −0.170103
$$55$$ −4888.00 −0.217884
$$56$$ 0 0
$$57$$ 15588.0 0.635482
$$58$$ 27590.0 1.07692
$$59$$ 11220.0 0.419626 0.209813 0.977741i $$-0.432714\pi$$
0.209813 + 0.977741i $$0.432714\pi$$
$$60$$ −5922.00 −0.212369
$$61$$ 45602.0 1.56913 0.784566 0.620046i $$-0.212886\pi$$
0.784566 + 0.620046i $$0.212886\pi$$
$$62$$ −31680.0 −1.04666
$$63$$ 0 0
$$64$$ 36457.0 1.11258
$$65$$ −72380.0 −2.12488
$$66$$ −2340.00 −0.0661235
$$67$$ 1396.00 0.0379925 0.0189963 0.999820i $$-0.493953\pi$$
0.0189963 + 0.999820i $$0.493953\pi$$
$$68$$ −14154.0 −0.371199
$$69$$ 5184.00 0.131082
$$70$$ 0 0
$$71$$ 18720.0 0.440717 0.220359 0.975419i $$-0.429277\pi$$
0.220359 + 0.975419i $$0.429277\pi$$
$$72$$ −15795.0 −0.359078
$$73$$ −46362.0 −1.01825 −0.509126 0.860692i $$-0.670031\pi$$
−0.509126 + 0.860692i $$0.670031\pi$$
$$74$$ −36690.0 −0.778876
$$75$$ −51399.0 −1.05512
$$76$$ 12124.0 0.240775
$$77$$ 0 0
$$78$$ −34650.0 −0.644862
$$79$$ 97424.0 1.75630 0.878149 0.478387i $$-0.158778\pi$$
0.878149 + 0.478387i $$0.158778\pi$$
$$80$$ 70594.0 1.23323
$$81$$ 6561.00 0.111111
$$82$$ 16310.0 0.267867
$$83$$ 81228.0 1.29423 0.647114 0.762394i $$-0.275976\pi$$
0.647114 + 0.762394i $$0.275976\pi$$
$$84$$ 0 0
$$85$$ −190068. −2.85339
$$86$$ 27100.0 0.395115
$$87$$ −49662.0 −0.703438
$$88$$ −10140.0 −0.139583
$$89$$ 3182.00 0.0425819 0.0212910 0.999773i $$-0.493222\pi$$
0.0212910 + 0.999773i $$0.493222\pi$$
$$90$$ −38070.0 −0.495424
$$91$$ 0 0
$$92$$ 4032.00 0.0496651
$$93$$ 57024.0 0.683676
$$94$$ −4320.00 −0.0504271
$$95$$ 162808. 1.85083
$$96$$ −22365.0 −0.247680
$$97$$ −4914.00 −0.0530281 −0.0265140 0.999648i $$-0.508441\pi$$
−0.0265140 + 0.999648i $$0.508441\pi$$
$$98$$ 0 0
$$99$$ 4212.00 0.0431917
$$100$$ −39977.0 −0.399770
$$101$$ 166354. 1.62267 0.811334 0.584583i $$-0.198742\pi$$
0.811334 + 0.584583i $$0.198742\pi$$
$$102$$ −90990.0 −0.865951
$$103$$ −157160. −1.45965 −0.729825 0.683634i $$-0.760399\pi$$
−0.729825 + 0.683634i $$0.760399\pi$$
$$104$$ −150150. −1.36126
$$105$$ 0 0
$$106$$ 20910.0 0.180755
$$107$$ −6764.00 −0.0571142 −0.0285571 0.999592i $$-0.509091\pi$$
−0.0285571 + 0.999592i $$0.509091\pi$$
$$108$$ 5103.00 0.0420985
$$109$$ 178398. 1.43821 0.719107 0.694899i $$-0.244551\pi$$
0.719107 + 0.694899i $$0.244551\pi$$
$$110$$ −24440.0 −0.192584
$$111$$ 66042.0 0.508760
$$112$$ 0 0
$$113$$ −45134.0 −0.332512 −0.166256 0.986083i $$-0.553168\pi$$
−0.166256 + 0.986083i $$0.553168\pi$$
$$114$$ 77940.0 0.561692
$$115$$ 54144.0 0.381773
$$116$$ −38626.0 −0.266523
$$117$$ 62370.0 0.421222
$$118$$ 56100.0 0.370901
$$119$$ 0 0
$$120$$ −164970. −1.04581
$$121$$ −158347. −0.983210
$$122$$ 228010. 1.38693
$$123$$ −29358.0 −0.174970
$$124$$ 44352.0 0.259035
$$125$$ −243084. −1.39149
$$126$$ 0 0
$$127$$ −205056. −1.12814 −0.564070 0.825727i $$-0.690765\pi$$
−0.564070 + 0.825727i $$0.690765\pi$$
$$128$$ 102765. 0.554396
$$129$$ −48780.0 −0.258088
$$130$$ −361900. −1.87815
$$131$$ −72964.0 −0.371476 −0.185738 0.982599i $$-0.559468\pi$$
−0.185738 + 0.982599i $$0.559468\pi$$
$$132$$ 3276.00 0.0163647
$$133$$ 0 0
$$134$$ 6980.00 0.0335810
$$135$$ 68526.0 0.323609
$$136$$ −394290. −1.82797
$$137$$ −94182.0 −0.428713 −0.214356 0.976756i $$-0.568765\pi$$
−0.214356 + 0.976756i $$0.568765\pi$$
$$138$$ 25920.0 0.115861
$$139$$ 47796.0 0.209824 0.104912 0.994482i $$-0.466544\pi$$
0.104912 + 0.994482i $$0.466544\pi$$
$$140$$ 0 0
$$141$$ 7776.00 0.0329389
$$142$$ 93600.0 0.389543
$$143$$ 40040.0 0.163740
$$144$$ −60831.0 −0.244466
$$145$$ −518692. −2.04875
$$146$$ −231810. −0.900016
$$147$$ 0 0
$$148$$ 51366.0 0.192762
$$149$$ −124266. −0.458550 −0.229275 0.973362i $$-0.573636\pi$$
−0.229275 + 0.973362i $$0.573636\pi$$
$$150$$ −256995. −0.932602
$$151$$ −446296. −1.59287 −0.796436 0.604723i $$-0.793284\pi$$
−0.796436 + 0.604723i $$0.793284\pi$$
$$152$$ 337740. 1.18570
$$153$$ 163782. 0.565637
$$154$$ 0 0
$$155$$ 595584. 1.99119
$$156$$ 48510.0 0.159595
$$157$$ 159746. 0.517227 0.258613 0.965981i $$-0.416734\pi$$
0.258613 + 0.965981i $$0.416734\pi$$
$$158$$ 487120. 1.55236
$$159$$ −37638.0 −0.118068
$$160$$ −233590. −0.721364
$$161$$ 0 0
$$162$$ 32805.0 0.0982093
$$163$$ 247252. 0.728905 0.364452 0.931222i $$-0.381256\pi$$
0.364452 + 0.931222i $$0.381256\pi$$
$$164$$ −22834.0 −0.0662937
$$165$$ 43992.0 0.125795
$$166$$ 406140. 1.14395
$$167$$ 684488. 1.89922 0.949609 0.313438i $$-0.101481\pi$$
0.949609 + 0.313438i $$0.101481\pi$$
$$168$$ 0 0
$$169$$ 221607. 0.596852
$$170$$ −950340. −2.52207
$$171$$ −140292. −0.366896
$$172$$ −37940.0 −0.0977859
$$173$$ 610474. 1.55079 0.775393 0.631479i $$-0.217552\pi$$
0.775393 + 0.631479i $$0.217552\pi$$
$$174$$ −248310. −0.621758
$$175$$ 0 0
$$176$$ −39052.0 −0.0950302
$$177$$ −100980. −0.242271
$$178$$ 15910.0 0.0376374
$$179$$ 662252. 1.54487 0.772433 0.635097i $$-0.219040\pi$$
0.772433 + 0.635097i $$0.219040\pi$$
$$180$$ 53298.0 0.122611
$$181$$ −154630. −0.350830 −0.175415 0.984495i $$-0.556127\pi$$
−0.175415 + 0.984495i $$0.556127\pi$$
$$182$$ 0 0
$$183$$ −410418. −0.905938
$$184$$ 112320. 0.244575
$$185$$ 689772. 1.48175
$$186$$ 285120. 0.604290
$$187$$ 105144. 0.219877
$$188$$ 6048.00 0.0124801
$$189$$ 0 0
$$190$$ 814040. 1.63592
$$191$$ 486904. 0.965739 0.482870 0.875692i $$-0.339594\pi$$
0.482870 + 0.875692i $$0.339594\pi$$
$$192$$ −328113. −0.642348
$$193$$ 620546. 1.19917 0.599585 0.800311i $$-0.295332\pi$$
0.599585 + 0.800311i $$0.295332\pi$$
$$194$$ −24570.0 −0.0468706
$$195$$ 651420. 1.22680
$$196$$ 0 0
$$197$$ −236570. −0.434304 −0.217152 0.976138i $$-0.569677\pi$$
−0.217152 + 0.976138i $$0.569677\pi$$
$$198$$ 21060.0 0.0381764
$$199$$ −82104.0 −0.146971 −0.0734855 0.997296i $$-0.523412\pi$$
−0.0734855 + 0.997296i $$0.523412\pi$$
$$200$$ −1.11364e6 −1.96866
$$201$$ −12564.0 −0.0219350
$$202$$ 831770. 1.43425
$$203$$ 0 0
$$204$$ 127386. 0.214312
$$205$$ −306628. −0.509597
$$206$$ −785800. −1.29016
$$207$$ −46656.0 −0.0756801
$$208$$ −578270. −0.926771
$$209$$ −90064.0 −0.142622
$$210$$ 0 0
$$211$$ 99892.0 0.154463 0.0772315 0.997013i $$-0.475392\pi$$
0.0772315 + 0.997013i $$0.475392\pi$$
$$212$$ −29274.0 −0.0447345
$$213$$ −168480. −0.254448
$$214$$ −33820.0 −0.0504823
$$215$$ −509480. −0.751677
$$216$$ 142155. 0.207314
$$217$$ 0 0
$$218$$ 891990. 1.27121
$$219$$ 417258. 0.587888
$$220$$ 34216.0 0.0476620
$$221$$ 1.55694e6 2.14433
$$222$$ 330210. 0.449684
$$223$$ 186704. 0.251415 0.125708 0.992067i $$-0.459880\pi$$
0.125708 + 0.992067i $$0.459880\pi$$
$$224$$ 0 0
$$225$$ 462591. 0.609173
$$226$$ −225670. −0.293902
$$227$$ −336372. −0.433267 −0.216633 0.976253i $$-0.569508\pi$$
−0.216633 + 0.976253i $$0.569508\pi$$
$$228$$ −109116. −0.139012
$$229$$ 926314. 1.16727 0.583633 0.812018i $$-0.301631\pi$$
0.583633 + 0.812018i $$0.301631\pi$$
$$230$$ 270720. 0.337443
$$231$$ 0 0
$$232$$ −1.07601e6 −1.31249
$$233$$ 1.25711e6 1.51700 0.758499 0.651675i $$-0.225933\pi$$
0.758499 + 0.651675i $$0.225933\pi$$
$$234$$ 311850. 0.372311
$$235$$ 81216.0 0.0959339
$$236$$ −78540.0 −0.0917933
$$237$$ −876816. −1.01400
$$238$$ 0 0
$$239$$ −347016. −0.392966 −0.196483 0.980507i $$-0.562952\pi$$
−0.196483 + 0.980507i $$0.562952\pi$$
$$240$$ −635346. −0.712004
$$241$$ −99170.0 −0.109986 −0.0549930 0.998487i $$-0.517514\pi$$
−0.0549930 + 0.998487i $$0.517514\pi$$
$$242$$ −791735. −0.869043
$$243$$ −59049.0 −0.0641500
$$244$$ −319214. −0.343247
$$245$$ 0 0
$$246$$ −146790. −0.154653
$$247$$ −1.33364e6 −1.39090
$$248$$ 1.23552e6 1.27562
$$249$$ −731052. −0.747222
$$250$$ −1.21542e6 −1.22992
$$251$$ −344428. −0.345076 −0.172538 0.985003i $$-0.555197\pi$$
−0.172538 + 0.985003i $$0.555197\pi$$
$$252$$ 0 0
$$253$$ −29952.0 −0.0294188
$$254$$ −1.02528e6 −0.997145
$$255$$ 1.71061e6 1.64741
$$256$$ −652799. −0.622558
$$257$$ −295130. −0.278728 −0.139364 0.990241i $$-0.544506\pi$$
−0.139364 + 0.990241i $$0.544506\pi$$
$$258$$ −243900. −0.228120
$$259$$ 0 0
$$260$$ 506660. 0.464818
$$261$$ 446958. 0.406130
$$262$$ −364820. −0.328341
$$263$$ −1.27246e6 −1.13437 −0.567187 0.823589i $$-0.691968\pi$$
−0.567187 + 0.823589i $$0.691968\pi$$
$$264$$ 91260.0 0.0805881
$$265$$ −393108. −0.343872
$$266$$ 0 0
$$267$$ −28638.0 −0.0245847
$$268$$ −9772.00 −0.00831087
$$269$$ −276774. −0.233209 −0.116604 0.993178i $$-0.537201\pi$$
−0.116604 + 0.993178i $$0.537201\pi$$
$$270$$ 342630. 0.286033
$$271$$ 1.28994e6 1.06695 0.533476 0.845815i $$-0.320885\pi$$
0.533476 + 0.845815i $$0.320885\pi$$
$$272$$ −1.51852e6 −1.24451
$$273$$ 0 0
$$274$$ −470910. −0.378932
$$275$$ 296972. 0.236801
$$276$$ −36288.0 −0.0286741
$$277$$ 1.71655e6 1.34418 0.672089 0.740470i $$-0.265397\pi$$
0.672089 + 0.740470i $$0.265397\pi$$
$$278$$ 238980. 0.185460
$$279$$ −513216. −0.394720
$$280$$ 0 0
$$281$$ −1.47218e6 −1.11223 −0.556116 0.831104i $$-0.687709\pi$$
−0.556116 + 0.831104i $$0.687709\pi$$
$$282$$ 38880.0 0.0291141
$$283$$ −1.02881e6 −0.763607 −0.381804 0.924244i $$-0.624697\pi$$
−0.381804 + 0.924244i $$0.624697\pi$$
$$284$$ −131040. −0.0964069
$$285$$ −1.46527e6 −1.06858
$$286$$ 200200. 0.144727
$$287$$ 0 0
$$288$$ 201285. 0.142998
$$289$$ 2.66863e6 1.87950
$$290$$ −2.59346e6 −1.81086
$$291$$ 44226.0 0.0306158
$$292$$ 324534. 0.222742
$$293$$ 1.18607e6 0.807123 0.403562 0.914952i $$-0.367772\pi$$
0.403562 + 0.914952i $$0.367772\pi$$
$$294$$ 0 0
$$295$$ −1.05468e6 −0.705612
$$296$$ 1.43091e6 0.949255
$$297$$ −37908.0 −0.0249367
$$298$$ −621330. −0.405305
$$299$$ −443520. −0.286903
$$300$$ 359793. 0.230807
$$301$$ 0 0
$$302$$ −2.23148e6 −1.40791
$$303$$ −1.49719e6 −0.936848
$$304$$ 1.30073e6 0.807242
$$305$$ −4.28659e6 −2.63853
$$306$$ 818910. 0.499957
$$307$$ −1.51892e6 −0.919788 −0.459894 0.887974i $$-0.652113\pi$$
−0.459894 + 0.887974i $$0.652113\pi$$
$$308$$ 0 0
$$309$$ 1.41444e6 0.842730
$$310$$ 2.97792e6 1.75998
$$311$$ −212808. −0.124763 −0.0623817 0.998052i $$-0.519870\pi$$
−0.0623817 + 0.998052i $$0.519870\pi$$
$$312$$ 1.35135e6 0.785925
$$313$$ 1894.00 0.00109275 0.000546373 1.00000i $$-0.499826\pi$$
0.000546373 1.00000i $$0.499826\pi$$
$$314$$ 798730. 0.457168
$$315$$ 0 0
$$316$$ −681968. −0.384190
$$317$$ −1.57898e6 −0.882527 −0.441263 0.897378i $$-0.645470\pi$$
−0.441263 + 0.897378i $$0.645470\pi$$
$$318$$ −188190. −0.104359
$$319$$ 286936. 0.157873
$$320$$ −3.42696e6 −1.87083
$$321$$ 60876.0 0.0329749
$$322$$ 0 0
$$323$$ −3.50210e6 −1.86777
$$324$$ −45927.0 −0.0243056
$$325$$ 4.39747e6 2.30938
$$326$$ 1.23626e6 0.644267
$$327$$ −1.60558e6 −0.830354
$$328$$ −636090. −0.326463
$$329$$ 0 0
$$330$$ 219960. 0.111188
$$331$$ −3.39471e6 −1.70307 −0.851535 0.524298i $$-0.824328\pi$$
−0.851535 + 0.524298i $$0.824328\pi$$
$$332$$ −568596. −0.283112
$$333$$ −594378. −0.293733
$$334$$ 3.42244e6 1.67869
$$335$$ −131224. −0.0638853
$$336$$ 0 0
$$337$$ 2.02731e6 0.972403 0.486201 0.873847i $$-0.338382\pi$$
0.486201 + 0.873847i $$0.338382\pi$$
$$338$$ 1.10804e6 0.527548
$$339$$ 406206. 0.191976
$$340$$ 1.33048e6 0.624180
$$341$$ −329472. −0.153438
$$342$$ −701460. −0.324293
$$343$$ 0 0
$$344$$ −1.05690e6 −0.481546
$$345$$ −487296. −0.220417
$$346$$ 3.05237e6 1.37071
$$347$$ 3.48885e6 1.55546 0.777730 0.628598i $$-0.216371\pi$$
0.777730 + 0.628598i $$0.216371\pi$$
$$348$$ 347634. 0.153877
$$349$$ −965566. −0.424344 −0.212172 0.977232i $$-0.568054\pi$$
−0.212172 + 0.977232i $$0.568054\pi$$
$$350$$ 0 0
$$351$$ −561330. −0.243193
$$352$$ 129220. 0.0555870
$$353$$ −1.15393e6 −0.492882 −0.246441 0.969158i $$-0.579261\pi$$
−0.246441 + 0.969158i $$0.579261\pi$$
$$354$$ −504900. −0.214140
$$355$$ −1.75968e6 −0.741076
$$356$$ −22274.0 −0.00931479
$$357$$ 0 0
$$358$$ 3.31126e6 1.36548
$$359$$ 1.61110e6 0.659762 0.329881 0.944022i $$-0.392991\pi$$
0.329881 + 0.944022i $$0.392991\pi$$
$$360$$ 1.48473e6 0.603797
$$361$$ 523725. 0.211512
$$362$$ −773150. −0.310093
$$363$$ 1.42512e6 0.567657
$$364$$ 0 0
$$365$$ 4.35803e6 1.71221
$$366$$ −2.05209e6 −0.800744
$$367$$ −3.67747e6 −1.42523 −0.712614 0.701557i $$-0.752489\pi$$
−0.712614 + 0.701557i $$0.752489\pi$$
$$368$$ 432576. 0.166511
$$369$$ 264222. 0.101019
$$370$$ 3.44886e6 1.30970
$$371$$ 0 0
$$372$$ −399168. −0.149554
$$373$$ 649766. 0.241816 0.120908 0.992664i $$-0.461419\pi$$
0.120908 + 0.992664i $$0.461419\pi$$
$$374$$ 525720. 0.194346
$$375$$ 2.18776e6 0.803379
$$376$$ 168480. 0.0614580
$$377$$ 4.24886e6 1.53964
$$378$$ 0 0
$$379$$ 320700. 0.114683 0.0573417 0.998355i $$-0.481738\pi$$
0.0573417 + 0.998355i $$0.481738\pi$$
$$380$$ −1.13966e6 −0.404869
$$381$$ 1.84550e6 0.651332
$$382$$ 2.43452e6 0.853601
$$383$$ 2.36189e6 0.822740 0.411370 0.911469i $$-0.365050\pi$$
0.411370 + 0.911469i $$0.365050\pi$$
$$384$$ −924885. −0.320081
$$385$$ 0 0
$$386$$ 3.10273e6 1.05993
$$387$$ 439020. 0.149007
$$388$$ 34398.0 0.0115999
$$389$$ −3.53390e6 −1.18408 −0.592039 0.805910i $$-0.701677\pi$$
−0.592039 + 0.805910i $$0.701677\pi$$
$$390$$ 3.25710e6 1.08435
$$391$$ −1.16467e6 −0.385267
$$392$$ 0 0
$$393$$ 656676. 0.214472
$$394$$ −1.18285e6 −0.383874
$$395$$ −9.15786e6 −2.95326
$$396$$ −29484.0 −0.00944819
$$397$$ −4.04811e6 −1.28907 −0.644534 0.764575i $$-0.722949\pi$$
−0.644534 + 0.764575i $$0.722949\pi$$
$$398$$ −410520. −0.129905
$$399$$ 0 0
$$400$$ −4.28896e6 −1.34030
$$401$$ 2.07645e6 0.644853 0.322426 0.946595i $$-0.395502\pi$$
0.322426 + 0.946595i $$0.395502\pi$$
$$402$$ −62820.0 −0.0193880
$$403$$ −4.87872e6 −1.49638
$$404$$ −1.16448e6 −0.354959
$$405$$ −616734. −0.186836
$$406$$ 0 0
$$407$$ −381576. −0.114181
$$408$$ 3.54861e6 1.05538
$$409$$ −2.57431e6 −0.760945 −0.380472 0.924792i $$-0.624239\pi$$
−0.380472 + 0.924792i $$0.624239\pi$$
$$410$$ −1.53314e6 −0.450425
$$411$$ 847638. 0.247517
$$412$$ 1.10012e6 0.319299
$$413$$ 0 0
$$414$$ −233280. −0.0668924
$$415$$ −7.63543e6 −2.17627
$$416$$ 1.91345e6 0.542105
$$417$$ −430164. −0.121142
$$418$$ −450320. −0.126061
$$419$$ −848148. −0.236013 −0.118007 0.993013i $$-0.537650\pi$$
−0.118007 + 0.993013i $$0.537650\pi$$
$$420$$ 0 0
$$421$$ 1.43682e6 0.395092 0.197546 0.980294i $$-0.436703\pi$$
0.197546 + 0.980294i $$0.436703\pi$$
$$422$$ 499460. 0.136527
$$423$$ −69984.0 −0.0190173
$$424$$ −815490. −0.220295
$$425$$ 1.15476e7 3.10114
$$426$$ −842400. −0.224903
$$427$$ 0 0
$$428$$ 47348.0 0.0124937
$$429$$ −360360. −0.0945352
$$430$$ −2.54740e6 −0.664394
$$431$$ 2.35438e6 0.610496 0.305248 0.952273i $$-0.401261\pi$$
0.305248 + 0.952273i $$0.401261\pi$$
$$432$$ 547479. 0.141143
$$433$$ 3.78808e6 0.970955 0.485478 0.874249i $$-0.338646\pi$$
0.485478 + 0.874249i $$0.338646\pi$$
$$434$$ 0 0
$$435$$ 4.66823e6 1.18285
$$436$$ −1.24879e6 −0.314609
$$437$$ 997632. 0.249900
$$438$$ 2.08629e6 0.519624
$$439$$ 3.64322e6 0.902245 0.451123 0.892462i $$-0.351024\pi$$
0.451123 + 0.892462i $$0.351024\pi$$
$$440$$ 953160. 0.234711
$$441$$ 0 0
$$442$$ 7.78470e6 1.89534
$$443$$ 2.48389e6 0.601345 0.300672 0.953728i $$-0.402789\pi$$
0.300672 + 0.953728i $$0.402789\pi$$
$$444$$ −462294. −0.111291
$$445$$ −299108. −0.0716025
$$446$$ 933520. 0.222222
$$447$$ 1.11839e6 0.264744
$$448$$ 0 0
$$449$$ −2.63177e6 −0.616074 −0.308037 0.951374i $$-0.599672\pi$$
−0.308037 + 0.951374i $$0.599672\pi$$
$$450$$ 2.31295e6 0.538438
$$451$$ 169624. 0.0392686
$$452$$ 315938. 0.0727371
$$453$$ 4.01666e6 0.919645
$$454$$ −1.68186e6 −0.382957
$$455$$ 0 0
$$456$$ −3.03966e6 −0.684562
$$457$$ −1.16130e6 −0.260109 −0.130054 0.991507i $$-0.541515\pi$$
−0.130054 + 0.991507i $$0.541515\pi$$
$$458$$ 4.63157e6 1.03173
$$459$$ −1.47404e6 −0.326571
$$460$$ −379008. −0.0835129
$$461$$ −2.81385e6 −0.616663 −0.308332 0.951279i $$-0.599771\pi$$
−0.308332 + 0.951279i $$0.599771\pi$$
$$462$$ 0 0
$$463$$ 6.84299e6 1.48352 0.741760 0.670665i $$-0.233991\pi$$
0.741760 + 0.670665i $$0.233991\pi$$
$$464$$ −4.14402e6 −0.893566
$$465$$ −5.36026e6 −1.14962
$$466$$ 6.28557e6 1.34085
$$467$$ −3.34314e6 −0.709353 −0.354676 0.934989i $$-0.615409\pi$$
−0.354676 + 0.934989i $$0.615409\pi$$
$$468$$ −436590. −0.0921423
$$469$$ 0 0
$$470$$ 406080. 0.0847944
$$471$$ −1.43771e6 −0.298621
$$472$$ −2.18790e6 −0.452035
$$473$$ 281840. 0.0579228
$$474$$ −4.38408e6 −0.896257
$$475$$ −9.89145e6 −2.01153
$$476$$ 0 0
$$477$$ 338742. 0.0681668
$$478$$ −1.73508e6 −0.347336
$$479$$ 4.28248e6 0.852818 0.426409 0.904530i $$-0.359778\pi$$
0.426409 + 0.904530i $$0.359778\pi$$
$$480$$ 2.10231e6 0.416480
$$481$$ −5.65026e6 −1.11354
$$482$$ −495850. −0.0972149
$$483$$ 0 0
$$484$$ 1.10843e6 0.215077
$$485$$ 461916. 0.0891679
$$486$$ −295245. −0.0567012
$$487$$ −8.93175e6 −1.70653 −0.853266 0.521477i $$-0.825381\pi$$
−0.853266 + 0.521477i $$0.825381\pi$$
$$488$$ −8.89239e6 −1.69032
$$489$$ −2.22527e6 −0.420833
$$490$$ 0 0
$$491$$ 2.75306e6 0.515361 0.257681 0.966230i $$-0.417042\pi$$
0.257681 + 0.966230i $$0.417042\pi$$
$$492$$ 205506. 0.0382747
$$493$$ 1.11574e7 2.06750
$$494$$ −6.66820e6 −1.22939
$$495$$ −395928. −0.0726279
$$496$$ 4.75834e6 0.868462
$$497$$ 0 0
$$498$$ −3.65526e6 −0.660458
$$499$$ 4.80408e6 0.863693 0.431846 0.901947i $$-0.357862\pi$$
0.431846 + 0.901947i $$0.357862\pi$$
$$500$$ 1.70159e6 0.304389
$$501$$ −6.16039e6 −1.09651
$$502$$ −1.72214e6 −0.305007
$$503$$ 6.02465e6 1.06172 0.530862 0.847458i $$-0.321868\pi$$
0.530862 + 0.847458i $$0.321868\pi$$
$$504$$ 0 0
$$505$$ −1.56373e7 −2.72855
$$506$$ −149760. −0.0260028
$$507$$ −1.99446e6 −0.344593
$$508$$ 1.43539e6 0.246781
$$509$$ 8.42987e6 1.44220 0.721101 0.692830i $$-0.243636\pi$$
0.721101 + 0.692830i $$0.243636\pi$$
$$510$$ 8.55306e6 1.45612
$$511$$ 0 0
$$512$$ −6.55248e6 −1.10466
$$513$$ 1.26263e6 0.211827
$$514$$ −1.47565e6 −0.246363
$$515$$ 1.47730e7 2.45444
$$516$$ 341460. 0.0564567
$$517$$ −44928.0 −0.00739249
$$518$$ 0 0
$$519$$ −5.49427e6 −0.895347
$$520$$ 1.41141e7 2.28899
$$521$$ −9.25058e6 −1.49305 −0.746525 0.665357i $$-0.768279\pi$$
−0.746525 + 0.665357i $$0.768279\pi$$
$$522$$ 2.23479e6 0.358972
$$523$$ −5.84494e6 −0.934385 −0.467192 0.884156i $$-0.654734\pi$$
−0.467192 + 0.884156i $$0.654734\pi$$
$$524$$ 510748. 0.0812603
$$525$$ 0 0
$$526$$ −6.36232e6 −1.00265
$$527$$ −1.28114e7 −2.00942
$$528$$ 351468. 0.0548657
$$529$$ −6.10457e6 −0.948453
$$530$$ −1.96554e6 −0.303943
$$531$$ 908820. 0.139875
$$532$$ 0 0
$$533$$ 2.51174e6 0.382963
$$534$$ −143190. −0.0217300
$$535$$ 635816. 0.0960389
$$536$$ −272220. −0.0409268
$$537$$ −5.96027e6 −0.891929
$$538$$ −1.38387e6 −0.206129
$$539$$ 0 0
$$540$$ −479682. −0.0707895
$$541$$ 9.22533e6 1.35515 0.677577 0.735452i $$-0.263030\pi$$
0.677577 + 0.735452i $$0.263030\pi$$
$$542$$ 6.44968e6 0.943061
$$543$$ 1.39167e6 0.202552
$$544$$ 5.02467e6 0.727965
$$545$$ −1.67694e7 −2.41839
$$546$$ 0 0
$$547$$ −6.44337e6 −0.920757 −0.460378 0.887723i $$-0.652286\pi$$
−0.460378 + 0.887723i $$0.652286\pi$$
$$548$$ 659274. 0.0937809
$$549$$ 3.69376e6 0.523044
$$550$$ 1.48486e6 0.209305
$$551$$ −9.55718e6 −1.34107
$$552$$ −1.01088e6 −0.141206
$$553$$ 0 0
$$554$$ 8.58275e6 1.18810
$$555$$ −6.20795e6 −0.855491
$$556$$ −334572. −0.0458989
$$557$$ 3.74213e6 0.511070 0.255535 0.966800i $$-0.417748\pi$$
0.255535 + 0.966800i $$0.417748\pi$$
$$558$$ −2.56608e6 −0.348887
$$559$$ 4.17340e6 0.564886
$$560$$ 0 0
$$561$$ −946296. −0.126946
$$562$$ −7.36091e6 −0.983084
$$563$$ 1.46384e7 1.94635 0.973176 0.230060i $$-0.0738923\pi$$
0.973176 + 0.230060i $$0.0738923\pi$$
$$564$$ −54432.0 −0.00720537
$$565$$ 4.24260e6 0.559127
$$566$$ −5.14406e6 −0.674940
$$567$$ 0 0
$$568$$ −3.65040e6 −0.474755
$$569$$ 1.41805e7 1.83616 0.918078 0.396400i $$-0.129741\pi$$
0.918078 + 0.396400i $$0.129741\pi$$
$$570$$ −7.32636e6 −0.944498
$$571$$ −1.25160e6 −0.160648 −0.0803242 0.996769i $$-0.525596\pi$$
−0.0803242 + 0.996769i $$0.525596\pi$$
$$572$$ −280280. −0.0358181
$$573$$ −4.38214e6 −0.557570
$$574$$ 0 0
$$575$$ −3.28954e6 −0.414921
$$576$$ 2.95302e6 0.370860
$$577$$ −5.94378e6 −0.743230 −0.371615 0.928387i $$-0.621196\pi$$
−0.371615 + 0.928387i $$0.621196\pi$$
$$578$$ 1.33431e7 1.66126
$$579$$ −5.58491e6 −0.692341
$$580$$ 3.63084e6 0.448165
$$581$$ 0 0
$$582$$ 221130. 0.0270608
$$583$$ 217464. 0.0264982
$$584$$ 9.04059e6 1.09689
$$585$$ −5.86278e6 −0.708295
$$586$$ 5.93033e6 0.713403
$$587$$ 6.46192e6 0.774046 0.387023 0.922070i $$-0.373503\pi$$
0.387023 + 0.922070i $$0.373503\pi$$
$$588$$ 0 0
$$589$$ 1.09740e7 1.30339
$$590$$ −5.27340e6 −0.623678
$$591$$ 2.12913e6 0.250746
$$592$$ 5.51084e6 0.646269
$$593$$ 2.34605e6 0.273969 0.136984 0.990573i $$-0.456259\pi$$
0.136984 + 0.990573i $$0.456259\pi$$
$$594$$ −189540. −0.0220412
$$595$$ 0 0
$$596$$ 869862. 0.100308
$$597$$ 738936. 0.0848537
$$598$$ −2.21760e6 −0.253589
$$599$$ −1.34959e7 −1.53686 −0.768432 0.639931i $$-0.778963\pi$$
−0.768432 + 0.639931i $$0.778963\pi$$
$$600$$ 1.00228e7 1.13661
$$601$$ −3.87849e6 −0.438002 −0.219001 0.975725i $$-0.570280\pi$$
−0.219001 + 0.975725i $$0.570280\pi$$
$$602$$ 0 0
$$603$$ 113076. 0.0126642
$$604$$ 3.12407e6 0.348441
$$605$$ 1.48846e7 1.65329
$$606$$ −7.48593e6 −0.828065
$$607$$ 533488. 0.0587696 0.0293848 0.999568i $$-0.490645\pi$$
0.0293848 + 0.999568i $$0.490645\pi$$
$$608$$ −4.30402e6 −0.472188
$$609$$ 0 0
$$610$$ −2.14329e7 −2.33215
$$611$$ −665280. −0.0720944
$$612$$ −1.14647e6 −0.123733
$$613$$ 5.14610e6 0.553130 0.276565 0.960995i $$-0.410804\pi$$
0.276565 + 0.960995i $$0.410804\pi$$
$$614$$ −7.59458e6 −0.812986
$$615$$ 2.75965e6 0.294216
$$616$$ 0 0
$$617$$ −2.37860e6 −0.251541 −0.125770 0.992059i $$-0.540140\pi$$
−0.125770 + 0.992059i $$0.540140\pi$$
$$618$$ 7.07220e6 0.744875
$$619$$ −1.60023e7 −1.67863 −0.839317 0.543642i $$-0.817045\pi$$
−0.839317 + 0.543642i $$0.817045\pi$$
$$620$$ −4.16909e6 −0.435574
$$621$$ 419904. 0.0436939
$$622$$ −1.06404e6 −0.110276
$$623$$ 0 0
$$624$$ 5.20443e6 0.535071
$$625$$ 5.00302e6 0.512309
$$626$$ 9470.00 0.000965860 0
$$627$$ 810576. 0.0823427
$$628$$ −1.11822e6 −0.113143
$$629$$ −1.48374e7 −1.49531
$$630$$ 0 0
$$631$$ 1.23459e7 1.23439 0.617193 0.786812i $$-0.288270\pi$$
0.617193 + 0.786812i $$0.288270\pi$$
$$632$$ −1.89977e7 −1.89194
$$633$$ −899028. −0.0891793
$$634$$ −7.89489e6 −0.780051
$$635$$ 1.92753e7 1.89699
$$636$$ 263466. 0.0258275
$$637$$ 0 0
$$638$$ 1.43468e6 0.139541
$$639$$ 1.51632e6 0.146906
$$640$$ −9.65991e6 −0.932230
$$641$$ −3.43755e6 −0.330449 −0.165224 0.986256i $$-0.552835\pi$$
−0.165224 + 0.986256i $$0.552835\pi$$
$$642$$ 304380. 0.0291460
$$643$$ 1.62191e7 1.54703 0.773515 0.633778i $$-0.218497\pi$$
0.773515 + 0.633778i $$0.218497\pi$$
$$644$$ 0 0
$$645$$ 4.58532e6 0.433981
$$646$$ −1.75105e7 −1.65089
$$647$$ −1.19929e7 −1.12632 −0.563160 0.826348i $$-0.690415\pi$$
−0.563160 + 0.826348i $$0.690415\pi$$
$$648$$ −1.27940e6 −0.119693
$$649$$ 583440. 0.0543731
$$650$$ 2.19873e7 2.04122
$$651$$ 0 0
$$652$$ −1.73076e6 −0.159448
$$653$$ 1.58009e6 0.145011 0.0725053 0.997368i $$-0.476901\pi$$
0.0725053 + 0.997368i $$0.476901\pi$$
$$654$$ −8.02791e6 −0.733936
$$655$$ 6.85862e6 0.624645
$$656$$ −2.44976e6 −0.222262
$$657$$ −3.75532e6 −0.339417
$$658$$ 0 0
$$659$$ 6.98358e6 0.626419 0.313209 0.949684i $$-0.398596\pi$$
0.313209 + 0.949684i $$0.398596\pi$$
$$660$$ −307944. −0.0275177
$$661$$ −3.69602e6 −0.329027 −0.164513 0.986375i $$-0.552605\pi$$
−0.164513 + 0.986375i $$0.552605\pi$$
$$662$$ −1.69735e7 −1.50532
$$663$$ −1.40125e7 −1.23803
$$664$$ −1.58395e7 −1.39418
$$665$$ 0 0
$$666$$ −2.97189e6 −0.259625
$$667$$ −3.17837e6 −0.276624
$$668$$ −4.79142e6 −0.415454
$$669$$ −1.68034e6 −0.145155
$$670$$ −656120. −0.0564672
$$671$$ 2.37130e6 0.203320
$$672$$ 0 0
$$673$$ 1.84688e6 0.157182 0.0785908 0.996907i $$-0.474958\pi$$
0.0785908 + 0.996907i $$0.474958\pi$$
$$674$$ 1.01366e7 0.859491
$$675$$ −4.16332e6 −0.351706
$$676$$ −1.55125e6 −0.130561
$$677$$ 7.68501e6 0.644426 0.322213 0.946667i $$-0.395573\pi$$
0.322213 + 0.946667i $$0.395573\pi$$
$$678$$ 2.03103e6 0.169684
$$679$$ 0 0
$$680$$ 3.70633e7 3.07377
$$681$$ 3.02735e6 0.250147
$$682$$ −1.64736e6 −0.135621
$$683$$ 7.12180e6 0.584168 0.292084 0.956393i $$-0.405651\pi$$
0.292084 + 0.956393i $$0.405651\pi$$
$$684$$ 982044. 0.0802584
$$685$$ 8.85311e6 0.720891
$$686$$ 0 0
$$687$$ −8.33683e6 −0.673921
$$688$$ −4.07042e6 −0.327845
$$689$$ 3.22014e6 0.258420
$$690$$ −2.43648e6 −0.194823
$$691$$ 3.23787e6 0.257967 0.128983 0.991647i $$-0.458829\pi$$
0.128983 + 0.991647i $$0.458829\pi$$
$$692$$ −4.27332e6 −0.339234
$$693$$ 0 0
$$694$$ 1.74443e7 1.37485
$$695$$ −4.49282e6 −0.352823
$$696$$ 9.68409e6 0.757767
$$697$$ 6.59576e6 0.514260
$$698$$ −4.82783e6 −0.375071
$$699$$ −1.13140e7 −0.875839
$$700$$ 0 0
$$701$$ −7.39163e6 −0.568127 −0.284063 0.958805i $$-0.591683\pi$$
−0.284063 + 0.958805i $$0.591683\pi$$
$$702$$ −2.80665e6 −0.214954
$$703$$ 1.27094e7 0.969923
$$704$$ 1.89576e6 0.144163
$$705$$ −730944. −0.0553874
$$706$$ −5.76965e6 −0.435650
$$707$$ 0 0
$$708$$ 706860. 0.0529969
$$709$$ −5.33361e6 −0.398479 −0.199240 0.979951i $$-0.563847\pi$$
−0.199240 + 0.979951i $$0.563847\pi$$
$$710$$ −8.79840e6 −0.655025
$$711$$ 7.89134e6 0.585433
$$712$$ −620490. −0.0458706
$$713$$ 3.64954e6 0.268852
$$714$$ 0 0
$$715$$ −3.76376e6 −0.275332
$$716$$ −4.63576e6 −0.337939
$$717$$ 3.12314e6 0.226879
$$718$$ 8.05552e6 0.583153
$$719$$ 1.14564e7 0.826468 0.413234 0.910625i $$-0.364399\pi$$
0.413234 + 0.910625i $$0.364399\pi$$
$$720$$ 5.71811e6 0.411075
$$721$$ 0 0
$$722$$ 2.61862e6 0.186952
$$723$$ 892530. 0.0635005
$$724$$ 1.08241e6 0.0767442
$$725$$ 3.15133e7 2.22663
$$726$$ 7.12562e6 0.501742
$$727$$ 2.49540e7 1.75107 0.875536 0.483153i $$-0.160508\pi$$
0.875536 + 0.483153i $$0.160508\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ 2.17901e7 1.51340
$$731$$ 1.09592e7 0.758555
$$732$$ 2.87293e6 0.198174
$$733$$ 1.43398e7 0.985789 0.492894 0.870089i $$-0.335939\pi$$
0.492894 + 0.870089i $$0.335939\pi$$
$$734$$ −1.83874e7 −1.25974
$$735$$ 0 0
$$736$$ −1.43136e6 −0.0973990
$$737$$ 72592.0 0.00492289
$$738$$ 1.32111e6 0.0892890
$$739$$ 922932. 0.0621668 0.0310834 0.999517i $$-0.490104\pi$$
0.0310834 + 0.999517i $$0.490104\pi$$
$$740$$ −4.82840e6 −0.324134
$$741$$ 1.20028e7 0.803037
$$742$$ 0 0
$$743$$ −9.38995e6 −0.624010 −0.312005 0.950081i $$-0.601001\pi$$
−0.312005 + 0.950081i $$0.601001\pi$$
$$744$$ −1.11197e7 −0.736478
$$745$$ 1.16810e7 0.771062
$$746$$ 3.24883e6 0.213737
$$747$$ 6.57947e6 0.431409
$$748$$ −736008. −0.0480982
$$749$$ 0 0
$$750$$ 1.09388e7 0.710094
$$751$$ −408032. −0.0263994 −0.0131997 0.999913i $$-0.504202\pi$$
−0.0131997 + 0.999913i $$0.504202\pi$$
$$752$$ 648864. 0.0418417
$$753$$ 3.09985e6 0.199229
$$754$$ 2.12443e7 1.36086
$$755$$ 4.19518e7 2.67845
$$756$$ 0 0
$$757$$ 2.59605e7 1.64654 0.823271 0.567649i $$-0.192147\pi$$
0.823271 + 0.567649i $$0.192147\pi$$
$$758$$ 1.60350e6 0.101367
$$759$$ 269568. 0.0169849
$$760$$ −3.17476e7 −1.99378
$$761$$ −1.83554e7 −1.14895 −0.574477 0.818521i $$-0.694794\pi$$
−0.574477 + 0.818521i $$0.694794\pi$$
$$762$$ 9.22752e6 0.575702
$$763$$ 0 0
$$764$$ −3.40833e6 −0.211255
$$765$$ −1.53955e7 −0.951131
$$766$$ 1.18094e7 0.727206
$$767$$ 8.63940e6 0.530268
$$768$$ 5.87519e6 0.359434
$$769$$ 747166. 0.0455618 0.0227809 0.999740i $$-0.492748\pi$$
0.0227809 + 0.999740i $$0.492748\pi$$
$$770$$ 0 0
$$771$$ 2.65617e6 0.160924
$$772$$ −4.34382e6 −0.262318
$$773$$ −2.02692e7 −1.22008 −0.610038 0.792372i $$-0.708846\pi$$
−0.610038 + 0.792372i $$0.708846\pi$$
$$774$$ 2.19510e6 0.131705
$$775$$ −3.61849e7 −2.16408
$$776$$ 958230. 0.0571236
$$777$$ 0 0
$$778$$ −1.76695e7 −1.04659
$$779$$ −5.64978e6 −0.333571
$$780$$ −4.55994e6 −0.268363
$$781$$ 973440. 0.0571060
$$782$$ −5.82336e6 −0.340531
$$783$$ −4.02262e6 −0.234479
$$784$$ 0 0
$$785$$ −1.50161e7 −0.869729
$$786$$ 3.28338e6 0.189568
$$787$$ 4.69982e6 0.270486 0.135243 0.990812i $$-0.456819\pi$$
0.135243 + 0.990812i $$0.456819\pi$$
$$788$$ 1.65599e6 0.0950041
$$789$$ 1.14522e7 0.654931
$$790$$ −4.57893e7 −2.61033
$$791$$ 0 0
$$792$$ −821340. −0.0465275
$$793$$ 3.51135e7 1.98286
$$794$$ −2.02406e7 −1.13939
$$795$$ 3.53797e6 0.198535
$$796$$ 574728. 0.0321499
$$797$$ −584710. −0.0326058 −0.0163029 0.999867i $$-0.505190\pi$$
−0.0163029 + 0.999867i $$0.505190\pi$$
$$798$$ 0 0
$$799$$ −1.74701e6 −0.0968117
$$800$$ 1.41918e7 0.783996
$$801$$ 257742. 0.0141940
$$802$$ 1.03822e7 0.569975
$$803$$ −2.41082e6 −0.131940
$$804$$ 87948.0 0.00479828
$$805$$ 0 0
$$806$$ −2.43936e7 −1.32263
$$807$$ 2.49097e6 0.134643
$$808$$ −3.24390e7 −1.74799
$$809$$ −1.64013e7 −0.881061 −0.440531 0.897738i $$-0.645210\pi$$
−0.440531 + 0.897738i $$0.645210\pi$$
$$810$$ −3.08367e6 −0.165141
$$811$$ 304948. 0.0162807 0.00814036 0.999967i $$-0.497409\pi$$
0.00814036 + 0.999967i $$0.497409\pi$$
$$812$$ 0 0
$$813$$ −1.16094e7 −0.616005
$$814$$ −1.90788e6 −0.100923
$$815$$ −2.32417e7 −1.22567
$$816$$ 1.36667e7 0.718519
$$817$$ −9.38744e6 −0.492031
$$818$$ −1.28716e7 −0.672587
$$819$$ 0 0
$$820$$ 2.14640e6 0.111474
$$821$$ 3.43428e7 1.77819 0.889095 0.457722i $$-0.151335\pi$$
0.889095 + 0.457722i $$0.151335\pi$$
$$822$$ 4.23819e6 0.218777
$$823$$ 1.56684e7 0.806351 0.403176 0.915123i $$-0.367906\pi$$
0.403176 + 0.915123i $$0.367906\pi$$
$$824$$ 3.06462e7 1.57238
$$825$$ −2.67275e6 −0.136717
$$826$$ 0 0
$$827$$ −2.96886e7 −1.50948 −0.754738 0.656026i $$-0.772236\pi$$
−0.754738 + 0.656026i $$0.772236\pi$$
$$828$$ 326592. 0.0165550
$$829$$ 2.30708e7 1.16594 0.582970 0.812494i $$-0.301890\pi$$
0.582970 + 0.812494i $$0.301890\pi$$
$$830$$ −3.81772e7 −1.92357
$$831$$ −1.54490e7 −0.776062
$$832$$ 2.80719e7 1.40593
$$833$$ 0 0
$$834$$ −2.15082e6 −0.107075
$$835$$ −6.43419e7 −3.19358
$$836$$ 630448. 0.0311985
$$837$$ 4.61894e6 0.227892
$$838$$ −4.24074e6 −0.208608
$$839$$ −2.32642e7 −1.14100 −0.570498 0.821299i $$-0.693250\pi$$
−0.570498 + 0.821299i $$0.693250\pi$$
$$840$$ 0 0
$$841$$ 9.93718e6 0.484477
$$842$$ 7.18411e6 0.349215
$$843$$ 1.32496e7 0.642148
$$844$$ −699244. −0.0337888
$$845$$ −2.08311e7 −1.00362
$$846$$ −349920. −0.0168090
$$847$$ 0 0
$$848$$ −3.14068e6 −0.149980
$$849$$ 9.25931e6 0.440869
$$850$$ 5.77382e7 2.74104
$$851$$ 4.22669e6 0.200067
$$852$$ 1.17936e6 0.0556605
$$853$$ −1.91515e7 −0.901219 −0.450610 0.892721i $$-0.648793\pi$$
−0.450610 + 0.892721i $$0.648793\pi$$
$$854$$ 0 0
$$855$$ 1.31874e7 0.616944
$$856$$ 1.31898e6 0.0615253
$$857$$ 5.34683e6 0.248682 0.124341 0.992240i $$-0.460318\pi$$
0.124341 + 0.992240i $$0.460318\pi$$
$$858$$ −1.80180e6 −0.0835581
$$859$$ −3.95858e7 −1.83045 −0.915223 0.402948i $$-0.867986\pi$$
−0.915223 + 0.402948i $$0.867986\pi$$
$$860$$ 3.56636e6 0.164429
$$861$$ 0 0
$$862$$ 1.17719e7 0.539607
$$863$$ −2.50284e7 −1.14395 −0.571973 0.820272i $$-0.693822\pi$$
−0.571973 + 0.820272i $$0.693822\pi$$
$$864$$ −1.81156e6 −0.0825600
$$865$$ −5.73846e7 −2.60768
$$866$$ 1.89404e7 0.858211
$$867$$ −2.40176e7 −1.08513
$$868$$ 0 0
$$869$$ 5.06605e6 0.227573
$$870$$ 2.33411e7 1.04550
$$871$$ 1.07492e6 0.0480099
$$872$$ −3.47876e7 −1.54929
$$873$$ −398034. −0.0176760
$$874$$ 4.98816e6 0.220883
$$875$$ 0 0
$$876$$ −2.92081e6 −0.128600
$$877$$ −5.02589e6 −0.220655 −0.110328 0.993895i $$-0.535190\pi$$
−0.110328 + 0.993895i $$0.535190\pi$$
$$878$$ 1.82161e7 0.797480
$$879$$ −1.06746e7 −0.465993
$$880$$ 3.67089e6 0.159795
$$881$$ 2.60490e7 1.13071 0.565356 0.824847i $$-0.308739\pi$$
0.565356 + 0.824847i $$0.308739\pi$$
$$882$$ 0 0
$$883$$ −6.82462e6 −0.294562 −0.147281 0.989095i $$-0.547052\pi$$
−0.147281 + 0.989095i $$0.547052\pi$$
$$884$$ −1.08986e7 −0.469072
$$885$$ 9.49212e6 0.407385
$$886$$ 1.24195e7 0.531519
$$887$$ −2.33835e7 −0.997931 −0.498965 0.866622i $$-0.666287\pi$$
−0.498965 + 0.866622i $$0.666287\pi$$
$$888$$ −1.28782e7 −0.548053
$$889$$ 0 0
$$890$$ −1.49554e6 −0.0632882
$$891$$ 341172. 0.0143972
$$892$$ −1.30693e6 −0.0549971
$$893$$ 1.49645e6 0.0627961
$$894$$ 5.59197e6 0.234003
$$895$$ −6.22517e7 −2.59773
$$896$$ 0 0
$$897$$ 3.99168e6 0.165644
$$898$$ −1.31589e7 −0.544537
$$899$$ −3.49620e7 −1.44277
$$900$$ −3.23814e6 −0.133257
$$901$$ 8.45600e6 0.347019
$$902$$ 848120. 0.0347089
$$903$$ 0 0
$$904$$ 8.80113e6 0.358193
$$905$$ 1.45352e7 0.589930
$$906$$ 2.00833e7 0.812859
$$907$$ 3.95959e7 1.59820 0.799102 0.601196i $$-0.205309\pi$$
0.799102 + 0.601196i $$0.205309\pi$$
$$908$$ 2.35460e6 0.0947771
$$909$$ 1.34747e7 0.540890
$$910$$ 0 0
$$911$$ −4.67570e6 −0.186660 −0.0933300 0.995635i $$-0.529751\pi$$
−0.0933300 + 0.995635i $$0.529751\pi$$
$$912$$ −1.17066e7 −0.466062
$$913$$ 4.22386e6 0.167700
$$914$$ −5.80651e6 −0.229906
$$915$$ 3.85793e7 1.52336
$$916$$ −6.48420e6 −0.255339
$$917$$ 0 0
$$918$$ −7.37019e6 −0.288650
$$919$$ −4.92594e6 −0.192398 −0.0961990 0.995362i $$-0.530669\pi$$
−0.0961990 + 0.995362i $$0.530669\pi$$
$$920$$ −1.05581e7 −0.411259
$$921$$ 1.36702e7 0.531040
$$922$$ −1.40692e7 −0.545058
$$923$$ 1.44144e7 0.556919
$$924$$ 0 0
$$925$$ −4.19073e7 −1.61041
$$926$$ 3.42150e7 1.31126
$$927$$ −1.27300e7 −0.486550
$$928$$ 1.37122e7 0.522683
$$929$$ 3.23688e7 1.23052 0.615258 0.788326i $$-0.289052\pi$$
0.615258 + 0.788326i $$0.289052\pi$$
$$930$$ −2.68013e7 −1.01613
$$931$$ 0 0
$$932$$ −8.79980e6 −0.331843
$$933$$ 1.91527e6 0.0720321
$$934$$ −1.67157e7 −0.626985
$$935$$ −9.88354e6 −0.369729
$$936$$ −1.21622e7 −0.453754
$$937$$ 3.32337e7 1.23660 0.618301 0.785941i $$-0.287821\pi$$
0.618301 + 0.785941i $$0.287821\pi$$
$$938$$ 0 0
$$939$$ −17046.0 −0.000630897 0
$$940$$ −568512. −0.0209855
$$941$$ 2.66426e7 0.980852 0.490426 0.871483i $$-0.336841\pi$$
0.490426 + 0.871483i $$0.336841\pi$$
$$942$$ −7.18857e6 −0.263946
$$943$$ −1.87891e6 −0.0688061
$$944$$ −8.42622e6 −0.307753
$$945$$ 0 0
$$946$$ 1.40920e6 0.0511970
$$947$$ 3.14663e7 1.14017 0.570086 0.821585i $$-0.306910\pi$$
0.570086 + 0.821585i $$0.306910\pi$$
$$948$$ 6.13771e6 0.221812
$$949$$ −3.56987e7 −1.28673
$$950$$ −4.94573e7 −1.77796
$$951$$ 1.42108e7 0.509527
$$952$$ 0 0
$$953$$ −1.34516e7 −0.479779 −0.239890 0.970800i $$-0.577111\pi$$
−0.239890 + 0.970800i $$0.577111\pi$$
$$954$$ 1.69371e6 0.0602515
$$955$$ −4.57690e7 −1.62391
$$956$$ 2.42911e6 0.0859613
$$957$$ −2.58242e6 −0.0911481
$$958$$ 2.14124e7 0.753792
$$959$$ 0 0
$$960$$ 3.08426e7 1.08012
$$961$$ 1.15157e7 0.402238
$$962$$ −2.82513e7 −0.984239
$$963$$ −547884. −0.0190381
$$964$$ 694190. 0.0240595
$$965$$ −5.83313e7 −2.01643
$$966$$ 0 0
$$967$$ −2.84963e7 −0.979992 −0.489996 0.871725i $$-0.663002\pi$$
−0.489996 + 0.871725i $$0.663002\pi$$
$$968$$ 3.08777e7 1.05915
$$969$$ 3.15189e7 1.07836
$$970$$ 2.30958e6 0.0788141
$$971$$ −1.81858e7 −0.618990 −0.309495 0.950901i $$-0.600160\pi$$
−0.309495 + 0.950901i $$0.600160\pi$$
$$972$$ 413343. 0.0140328
$$973$$ 0 0
$$974$$ −4.46588e7 −1.50837
$$975$$ −3.95772e7 −1.33332
$$976$$ −3.42471e7 −1.15080
$$977$$ 3.20941e7 1.07569 0.537847 0.843042i $$-0.319238\pi$$
0.537847 + 0.843042i $$0.319238\pi$$
$$978$$ −1.11263e7 −0.371968
$$979$$ 165464. 0.00551756
$$980$$ 0 0
$$981$$ 1.44502e7 0.479405
$$982$$ 1.37653e7 0.455519
$$983$$ −1.56154e7 −0.515429 −0.257715 0.966221i $$-0.582969\pi$$
−0.257715 + 0.966221i $$0.582969\pi$$
$$984$$ 5.72481e6 0.188483
$$985$$ 2.22376e7 0.730293
$$986$$ 5.57870e7 1.82743
$$987$$ 0 0
$$988$$ 9.33548e6 0.304260
$$989$$ −3.12192e6 −0.101492
$$990$$ −1.97964e6 −0.0641946
$$991$$ 4.84499e7 1.56714 0.783572 0.621301i $$-0.213395\pi$$
0.783572 + 0.621301i $$0.213395\pi$$
$$992$$ −1.57450e7 −0.507998
$$993$$ 3.05524e7 0.983268
$$994$$ 0 0
$$995$$ 7.71778e6 0.247135
$$996$$ 5.11736e6 0.163455
$$997$$ 4.54336e7 1.44757 0.723784 0.690027i $$-0.242401\pi$$
0.723784 + 0.690027i $$0.242401\pi$$
$$998$$ 2.40204e7 0.763404
$$999$$ 5.34940e6 0.169587
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 147.6.a.f.1.1 1
3.2 odd 2 441.6.a.c.1.1 1
7.2 even 3 147.6.e.d.67.1 2
7.3 odd 6 147.6.e.c.79.1 2
7.4 even 3 147.6.e.d.79.1 2
7.5 odd 6 147.6.e.c.67.1 2
7.6 odd 2 21.6.a.c.1.1 1
21.20 even 2 63.6.a.b.1.1 1
28.27 even 2 336.6.a.i.1.1 1
35.13 even 4 525.6.d.c.274.1 2
35.27 even 4 525.6.d.c.274.2 2
35.34 odd 2 525.6.a.b.1.1 1
84.83 odd 2 1008.6.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.c.1.1 1 7.6 odd 2
63.6.a.b.1.1 1 21.20 even 2
147.6.a.f.1.1 1 1.1 even 1 trivial
147.6.e.c.67.1 2 7.5 odd 6
147.6.e.c.79.1 2 7.3 odd 6
147.6.e.d.67.1 2 7.2 even 3
147.6.e.d.79.1 2 7.4 even 3
336.6.a.i.1.1 1 28.27 even 2
441.6.a.c.1.1 1 3.2 odd 2
525.6.a.b.1.1 1 35.34 odd 2
525.6.d.c.274.1 2 35.13 even 4
525.6.d.c.274.2 2 35.27 even 4
1008.6.a.a.1.1 1 84.83 odd 2