# Properties

 Label 234.4.a.f.1.1 Level $234$ Weight $4$ Character 234.1 Self dual yes Analytic conductor $13.806$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,4,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 234.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: $$13.8064469413$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 234.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +4.00000 q^{4} +20.0000 q^{5} -32.0000 q^{7} -8.00000 q^{8} +O(q^{10})$$ $$q-2.00000 q^{2} +4.00000 q^{4} +20.0000 q^{5} -32.0000 q^{7} -8.00000 q^{8} -40.0000 q^{10} -50.0000 q^{11} -13.0000 q^{13} +64.0000 q^{14} +16.0000 q^{16} +30.0000 q^{17} -120.000 q^{19} +80.0000 q^{20} +100.000 q^{22} +20.0000 q^{23} +275.000 q^{25} +26.0000 q^{26} -128.000 q^{28} -82.0000 q^{29} -44.0000 q^{31} -32.0000 q^{32} -60.0000 q^{34} -640.000 q^{35} -306.000 q^{37} +240.000 q^{38} -160.000 q^{40} -108.000 q^{41} -356.000 q^{43} -200.000 q^{44} -40.0000 q^{46} +178.000 q^{47} +681.000 q^{49} -550.000 q^{50} -52.0000 q^{52} -198.000 q^{53} -1000.00 q^{55} +256.000 q^{56} +164.000 q^{58} -94.0000 q^{59} -62.0000 q^{61} +88.0000 q^{62} +64.0000 q^{64} -260.000 q^{65} -140.000 q^{67} +120.000 q^{68} +1280.00 q^{70} +778.000 q^{71} +62.0000 q^{73} +612.000 q^{74} -480.000 q^{76} +1600.00 q^{77} -1096.00 q^{79} +320.000 q^{80} +216.000 q^{82} +462.000 q^{83} +600.000 q^{85} +712.000 q^{86} +400.000 q^{88} -1224.00 q^{89} +416.000 q^{91} +80.0000 q^{92} -356.000 q^{94} -2400.00 q^{95} +614.000 q^{97} -1362.00 q^{98} +O(q^{100})$$

## 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 −0.707107
$$3$$ 0 0
$$4$$ 4.00000 0.500000
$$5$$ 20.0000 1.78885 0.894427 0.447214i $$-0.147584\pi$$
0.894427 + 0.447214i $$0.147584\pi$$
$$6$$ 0 0
$$7$$ −32.0000 −1.72784 −0.863919 0.503631i $$-0.831997\pi$$
−0.863919 + 0.503631i $$0.831997\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ 0 0
$$10$$ −40.0000 −1.26491
$$11$$ −50.0000 −1.37051 −0.685253 0.728305i $$-0.740308\pi$$
−0.685253 + 0.728305i $$0.740308\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 64.0000 1.22177
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ 30.0000 0.428004 0.214002 0.976833i $$-0.431350\pi$$
0.214002 + 0.976833i $$0.431350\pi$$
$$18$$ 0 0
$$19$$ −120.000 −1.44894 −0.724471 0.689306i $$-0.757916\pi$$
−0.724471 + 0.689306i $$0.757916\pi$$
$$20$$ 80.0000 0.894427
$$21$$ 0 0
$$22$$ 100.000 0.969094
$$23$$ 20.0000 0.181317 0.0906584 0.995882i $$-0.471103\pi$$
0.0906584 + 0.995882i $$0.471103\pi$$
$$24$$ 0 0
$$25$$ 275.000 2.20000
$$26$$ 26.0000 0.196116
$$27$$ 0 0
$$28$$ −128.000 −0.863919
$$29$$ −82.0000 −0.525070 −0.262535 0.964923i $$-0.584558\pi$$
−0.262535 + 0.964923i $$0.584558\pi$$
$$30$$ 0 0
$$31$$ −44.0000 −0.254924 −0.127462 0.991843i $$-0.540683\pi$$
−0.127462 + 0.991843i $$0.540683\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 0 0
$$34$$ −60.0000 −0.302645
$$35$$ −640.000 −3.09085
$$36$$ 0 0
$$37$$ −306.000 −1.35962 −0.679812 0.733386i $$-0.737939\pi$$
−0.679812 + 0.733386i $$0.737939\pi$$
$$38$$ 240.000 1.02456
$$39$$ 0 0
$$40$$ −160.000 −0.632456
$$41$$ −108.000 −0.411385 −0.205692 0.978617i $$-0.565945\pi$$
−0.205692 + 0.978617i $$0.565945\pi$$
$$42$$ 0 0
$$43$$ −356.000 −1.26255 −0.631273 0.775561i $$-0.717467\pi$$
−0.631273 + 0.775561i $$0.717467\pi$$
$$44$$ −200.000 −0.685253
$$45$$ 0 0
$$46$$ −40.0000 −0.128210
$$47$$ 178.000 0.552425 0.276212 0.961097i $$-0.410921\pi$$
0.276212 + 0.961097i $$0.410921\pi$$
$$48$$ 0 0
$$49$$ 681.000 1.98542
$$50$$ −550.000 −1.55563
$$51$$ 0 0
$$52$$ −52.0000 −0.138675
$$53$$ −198.000 −0.513158 −0.256579 0.966523i $$-0.582595\pi$$
−0.256579 + 0.966523i $$0.582595\pi$$
$$54$$ 0 0
$$55$$ −1000.00 −2.45164
$$56$$ 256.000 0.610883
$$57$$ 0 0
$$58$$ 164.000 0.371280
$$59$$ −94.0000 −0.207420 −0.103710 0.994608i $$-0.533071\pi$$
−0.103710 + 0.994608i $$0.533071\pi$$
$$60$$ 0 0
$$61$$ −62.0000 −0.130136 −0.0650679 0.997881i $$-0.520726\pi$$
−0.0650679 + 0.997881i $$0.520726\pi$$
$$62$$ 88.0000 0.180258
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ −260.000 −0.496139
$$66$$ 0 0
$$67$$ −140.000 −0.255279 −0.127640 0.991821i $$-0.540740\pi$$
−0.127640 + 0.991821i $$0.540740\pi$$
$$68$$ 120.000 0.214002
$$69$$ 0 0
$$70$$ 1280.00 2.18556
$$71$$ 778.000 1.30045 0.650223 0.759744i $$-0.274676\pi$$
0.650223 + 0.759744i $$0.274676\pi$$
$$72$$ 0 0
$$73$$ 62.0000 0.0994048 0.0497024 0.998764i $$-0.484173\pi$$
0.0497024 + 0.998764i $$0.484173\pi$$
$$74$$ 612.000 0.961399
$$75$$ 0 0
$$76$$ −480.000 −0.724471
$$77$$ 1600.00 2.36801
$$78$$ 0 0
$$79$$ −1096.00 −1.56088 −0.780441 0.625230i $$-0.785005\pi$$
−0.780441 + 0.625230i $$0.785005\pi$$
$$80$$ 320.000 0.447214
$$81$$ 0 0
$$82$$ 216.000 0.290893
$$83$$ 462.000 0.610977 0.305488 0.952196i $$-0.401180\pi$$
0.305488 + 0.952196i $$0.401180\pi$$
$$84$$ 0 0
$$85$$ 600.000 0.765637
$$86$$ 712.000 0.892755
$$87$$ 0 0
$$88$$ 400.000 0.484547
$$89$$ −1224.00 −1.45779 −0.728897 0.684623i $$-0.759967\pi$$
−0.728897 + 0.684623i $$0.759967\pi$$
$$90$$ 0 0
$$91$$ 416.000 0.479216
$$92$$ 80.0000 0.0906584
$$93$$ 0 0
$$94$$ −356.000 −0.390623
$$95$$ −2400.00 −2.59195
$$96$$ 0 0
$$97$$ 614.000 0.642704 0.321352 0.946960i $$-0.395863\pi$$
0.321352 + 0.946960i $$0.395863\pi$$
$$98$$ −1362.00 −1.40391
$$99$$ 0 0
$$100$$ 1100.00 1.10000
$$101$$ −1058.00 −1.04233 −0.521163 0.853457i $$-0.674502\pi$$
−0.521163 + 0.853457i $$0.674502\pi$$
$$102$$ 0 0
$$103$$ 1768.00 1.69132 0.845661 0.533720i $$-0.179206\pi$$
0.845661 + 0.533720i $$0.179206\pi$$
$$104$$ 104.000 0.0980581
$$105$$ 0 0
$$106$$ 396.000 0.362858
$$107$$ 1808.00 1.63351 0.816757 0.576982i $$-0.195770\pi$$
0.816757 + 0.576982i $$0.195770\pi$$
$$108$$ 0 0
$$109$$ −1886.00 −1.65730 −0.828652 0.559765i $$-0.810891\pi$$
−0.828652 + 0.559765i $$0.810891\pi$$
$$110$$ 2000.00 1.73357
$$111$$ 0 0
$$112$$ −512.000 −0.431959
$$113$$ −1246.00 −1.03729 −0.518645 0.854990i $$-0.673563\pi$$
−0.518645 + 0.854990i $$0.673563\pi$$
$$114$$ 0 0
$$115$$ 400.000 0.324349
$$116$$ −328.000 −0.262535
$$117$$ 0 0
$$118$$ 188.000 0.146668
$$119$$ −960.000 −0.739521
$$120$$ 0 0
$$121$$ 1169.00 0.878287
$$122$$ 124.000 0.0920199
$$123$$ 0 0
$$124$$ −176.000 −0.127462
$$125$$ 3000.00 2.14663
$$126$$ 0 0
$$127$$ 1624.00 1.13470 0.567349 0.823477i $$-0.307969\pi$$
0.567349 + 0.823477i $$0.307969\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 0 0
$$130$$ 520.000 0.350823
$$131$$ 2072.00 1.38192 0.690960 0.722893i $$-0.257188\pi$$
0.690960 + 0.722893i $$0.257188\pi$$
$$132$$ 0 0
$$133$$ 3840.00 2.50354
$$134$$ 280.000 0.180510
$$135$$ 0 0
$$136$$ −240.000 −0.151322
$$137$$ 756.000 0.471456 0.235728 0.971819i $$-0.424253\pi$$
0.235728 + 0.971819i $$0.424253\pi$$
$$138$$ 0 0
$$139$$ 172.000 0.104956 0.0524779 0.998622i $$-0.483288\pi$$
0.0524779 + 0.998622i $$0.483288\pi$$
$$140$$ −2560.00 −1.54542
$$141$$ 0 0
$$142$$ −1556.00 −0.919554
$$143$$ 650.000 0.380110
$$144$$ 0 0
$$145$$ −1640.00 −0.939273
$$146$$ −124.000 −0.0702898
$$147$$ 0 0
$$148$$ −1224.00 −0.679812
$$149$$ −1272.00 −0.699371 −0.349686 0.936867i $$-0.613712\pi$$
−0.349686 + 0.936867i $$0.613712\pi$$
$$150$$ 0 0
$$151$$ 1404.00 0.756662 0.378331 0.925670i $$-0.376498\pi$$
0.378331 + 0.925670i $$0.376498\pi$$
$$152$$ 960.000 0.512278
$$153$$ 0 0
$$154$$ −3200.00 −1.67444
$$155$$ −880.000 −0.456021
$$156$$ 0 0
$$157$$ −2170.00 −1.10309 −0.551544 0.834146i $$-0.685961\pi$$
−0.551544 + 0.834146i $$0.685961\pi$$
$$158$$ 2192.00 1.10371
$$159$$ 0 0
$$160$$ −640.000 −0.316228
$$161$$ −640.000 −0.313286
$$162$$ 0 0
$$163$$ 248.000 0.119171 0.0595855 0.998223i $$-0.481022\pi$$
0.0595855 + 0.998223i $$0.481022\pi$$
$$164$$ −432.000 −0.205692
$$165$$ 0 0
$$166$$ −924.000 −0.432026
$$167$$ −102.000 −0.0472635 −0.0236317 0.999721i $$-0.507523\pi$$
−0.0236317 + 0.999721i $$0.507523\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ −1200.00 −0.541387
$$171$$ 0 0
$$172$$ −1424.00 −0.631273
$$173$$ −682.000 −0.299720 −0.149860 0.988707i $$-0.547882\pi$$
−0.149860 + 0.988707i $$0.547882\pi$$
$$174$$ 0 0
$$175$$ −8800.00 −3.80124
$$176$$ −800.000 −0.342627
$$177$$ 0 0
$$178$$ 2448.00 1.03082
$$179$$ 612.000 0.255548 0.127774 0.991803i $$-0.459217\pi$$
0.127774 + 0.991803i $$0.459217\pi$$
$$180$$ 0 0
$$181$$ −66.0000 −0.0271035 −0.0135518 0.999908i $$-0.504314\pi$$
−0.0135518 + 0.999908i $$0.504314\pi$$
$$182$$ −832.000 −0.338857
$$183$$ 0 0
$$184$$ −160.000 −0.0641052
$$185$$ −6120.00 −2.43217
$$186$$ 0 0
$$187$$ −1500.00 −0.586582
$$188$$ 712.000 0.276212
$$189$$ 0 0
$$190$$ 4800.00 1.83278
$$191$$ −608.000 −0.230332 −0.115166 0.993346i $$-0.536740\pi$$
−0.115166 + 0.993346i $$0.536740\pi$$
$$192$$ 0 0
$$193$$ 1370.00 0.510957 0.255479 0.966815i $$-0.417767\pi$$
0.255479 + 0.966815i $$0.417767\pi$$
$$194$$ −1228.00 −0.454460
$$195$$ 0 0
$$196$$ 2724.00 0.992711
$$197$$ 4908.00 1.77503 0.887514 0.460781i $$-0.152431\pi$$
0.887514 + 0.460781i $$0.152431\pi$$
$$198$$ 0 0
$$199$$ −328.000 −0.116841 −0.0584204 0.998292i $$-0.518606\pi$$
−0.0584204 + 0.998292i $$0.518606\pi$$
$$200$$ −2200.00 −0.777817
$$201$$ 0 0
$$202$$ 2116.00 0.737036
$$203$$ 2624.00 0.907235
$$204$$ 0 0
$$205$$ −2160.00 −0.735907
$$206$$ −3536.00 −1.19595
$$207$$ 0 0
$$208$$ −208.000 −0.0693375
$$209$$ 6000.00 1.98578
$$210$$ 0 0
$$211$$ 1316.00 0.429371 0.214685 0.976683i $$-0.431127\pi$$
0.214685 + 0.976683i $$0.431127\pi$$
$$212$$ −792.000 −0.256579
$$213$$ 0 0
$$214$$ −3616.00 −1.15507
$$215$$ −7120.00 −2.25851
$$216$$ 0 0
$$217$$ 1408.00 0.440467
$$218$$ 3772.00 1.17189
$$219$$ 0 0
$$220$$ −4000.00 −1.22582
$$221$$ −390.000 −0.118707
$$222$$ 0 0
$$223$$ −1932.00 −0.580163 −0.290081 0.957002i $$-0.593682\pi$$
−0.290081 + 0.957002i $$0.593682\pi$$
$$224$$ 1024.00 0.305441
$$225$$ 0 0
$$226$$ 2492.00 0.733475
$$227$$ −4998.00 −1.46136 −0.730680 0.682720i $$-0.760797\pi$$
−0.730680 + 0.682720i $$0.760797\pi$$
$$228$$ 0 0
$$229$$ −78.0000 −0.0225082 −0.0112541 0.999937i $$-0.503582\pi$$
−0.0112541 + 0.999937i $$0.503582\pi$$
$$230$$ −800.000 −0.229350
$$231$$ 0 0
$$232$$ 656.000 0.185640
$$233$$ 1282.00 0.360458 0.180229 0.983625i $$-0.442316\pi$$
0.180229 + 0.983625i $$0.442316\pi$$
$$234$$ 0 0
$$235$$ 3560.00 0.988208
$$236$$ −376.000 −0.103710
$$237$$ 0 0
$$238$$ 1920.00 0.522921
$$239$$ −294.000 −0.0795702 −0.0397851 0.999208i $$-0.512667\pi$$
−0.0397851 + 0.999208i $$0.512667\pi$$
$$240$$ 0 0
$$241$$ −4962.00 −1.32627 −0.663134 0.748501i $$-0.730774\pi$$
−0.663134 + 0.748501i $$0.730774\pi$$
$$242$$ −2338.00 −0.621043
$$243$$ 0 0
$$244$$ −248.000 −0.0650679
$$245$$ 13620.0 3.55163
$$246$$ 0 0
$$247$$ 1560.00 0.401864
$$248$$ 352.000 0.0901291
$$249$$ 0 0
$$250$$ −6000.00 −1.51789
$$251$$ −744.000 −0.187095 −0.0935475 0.995615i $$-0.529821\pi$$
−0.0935475 + 0.995615i $$0.529821\pi$$
$$252$$ 0 0
$$253$$ −1000.00 −0.248496
$$254$$ −3248.00 −0.802353
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ 1026.00 0.249028 0.124514 0.992218i $$-0.460263\pi$$
0.124514 + 0.992218i $$0.460263\pi$$
$$258$$ 0 0
$$259$$ 9792.00 2.34921
$$260$$ −1040.00 −0.248069
$$261$$ 0 0
$$262$$ −4144.00 −0.977165
$$263$$ 5532.00 1.29703 0.648513 0.761204i $$-0.275391\pi$$
0.648513 + 0.761204i $$0.275391\pi$$
$$264$$ 0 0
$$265$$ −3960.00 −0.917966
$$266$$ −7680.00 −1.77027
$$267$$ 0 0
$$268$$ −560.000 −0.127640
$$269$$ 3534.00 0.801010 0.400505 0.916294i $$-0.368835\pi$$
0.400505 + 0.916294i $$0.368835\pi$$
$$270$$ 0 0
$$271$$ 2392.00 0.536176 0.268088 0.963394i $$-0.413608\pi$$
0.268088 + 0.963394i $$0.413608\pi$$
$$272$$ 480.000 0.107001
$$273$$ 0 0
$$274$$ −1512.00 −0.333370
$$275$$ −13750.0 −3.01511
$$276$$ 0 0
$$277$$ 6102.00 1.32359 0.661794 0.749686i $$-0.269796\pi$$
0.661794 + 0.749686i $$0.269796\pi$$
$$278$$ −344.000 −0.0742149
$$279$$ 0 0
$$280$$ 5120.00 1.09278
$$281$$ 7540.00 1.60071 0.800354 0.599528i $$-0.204645\pi$$
0.800354 + 0.599528i $$0.204645\pi$$
$$282$$ 0 0
$$283$$ −2756.00 −0.578895 −0.289447 0.957194i $$-0.593472\pi$$
−0.289447 + 0.957194i $$0.593472\pi$$
$$284$$ 3112.00 0.650223
$$285$$ 0 0
$$286$$ −1300.00 −0.268778
$$287$$ 3456.00 0.710806
$$288$$ 0 0
$$289$$ −4013.00 −0.816813
$$290$$ 3280.00 0.664166
$$291$$ 0 0
$$292$$ 248.000 0.0497024
$$293$$ −968.000 −0.193007 −0.0965037 0.995333i $$-0.530766\pi$$
−0.0965037 + 0.995333i $$0.530766\pi$$
$$294$$ 0 0
$$295$$ −1880.00 −0.371043
$$296$$ 2448.00 0.480700
$$297$$ 0 0
$$298$$ 2544.00 0.494530
$$299$$ −260.000 −0.0502883
$$300$$ 0 0
$$301$$ 11392.0 2.18147
$$302$$ −2808.00 −0.535041
$$303$$ 0 0
$$304$$ −1920.00 −0.362235
$$305$$ −1240.00 −0.232794
$$306$$ 0 0
$$307$$ −6436.00 −1.19649 −0.598244 0.801314i $$-0.704135\pi$$
−0.598244 + 0.801314i $$0.704135\pi$$
$$308$$ 6400.00 1.18401
$$309$$ 0 0
$$310$$ 1760.00 0.322456
$$311$$ −7932.00 −1.44625 −0.723123 0.690719i $$-0.757294\pi$$
−0.723123 + 0.690719i $$0.757294\pi$$
$$312$$ 0 0
$$313$$ 10358.0 1.87051 0.935254 0.353978i $$-0.115171\pi$$
0.935254 + 0.353978i $$0.115171\pi$$
$$314$$ 4340.00 0.780001
$$315$$ 0 0
$$316$$ −4384.00 −0.780441
$$317$$ 2820.00 0.499643 0.249822 0.968292i $$-0.419628\pi$$
0.249822 + 0.968292i $$0.419628\pi$$
$$318$$ 0 0
$$319$$ 4100.00 0.719611
$$320$$ 1280.00 0.223607
$$321$$ 0 0
$$322$$ 1280.00 0.221527
$$323$$ −3600.00 −0.620153
$$324$$ 0 0
$$325$$ −3575.00 −0.610170
$$326$$ −496.000 −0.0842666
$$327$$ 0 0
$$328$$ 864.000 0.145446
$$329$$ −5696.00 −0.954500
$$330$$ 0 0
$$331$$ −4180.00 −0.694120 −0.347060 0.937843i $$-0.612820\pi$$
−0.347060 + 0.937843i $$0.612820\pi$$
$$332$$ 1848.00 0.305488
$$333$$ 0 0
$$334$$ 204.000 0.0334203
$$335$$ −2800.00 −0.456658
$$336$$ 0 0
$$337$$ −5026.00 −0.812414 −0.406207 0.913781i $$-0.633149\pi$$
−0.406207 + 0.913781i $$0.633149\pi$$
$$338$$ −338.000 −0.0543928
$$339$$ 0 0
$$340$$ 2400.00 0.382818
$$341$$ 2200.00 0.349374
$$342$$ 0 0
$$343$$ −10816.0 −1.70265
$$344$$ 2848.00 0.446378
$$345$$ 0 0
$$346$$ 1364.00 0.211934
$$347$$ 7332.00 1.13430 0.567150 0.823614i $$-0.308046\pi$$
0.567150 + 0.823614i $$0.308046\pi$$
$$348$$ 0 0
$$349$$ −8162.00 −1.25187 −0.625934 0.779876i $$-0.715282\pi$$
−0.625934 + 0.779876i $$0.715282\pi$$
$$350$$ 17600.0 2.68788
$$351$$ 0 0
$$352$$ 1600.00 0.242274
$$353$$ −1244.00 −0.187568 −0.0937839 0.995593i $$-0.529896\pi$$
−0.0937839 + 0.995593i $$0.529896\pi$$
$$354$$ 0 0
$$355$$ 15560.0 2.32631
$$356$$ −4896.00 −0.728897
$$357$$ 0 0
$$358$$ −1224.00 −0.180699
$$359$$ −9558.00 −1.40516 −0.702579 0.711605i $$-0.747968\pi$$
−0.702579 + 0.711605i $$0.747968\pi$$
$$360$$ 0 0
$$361$$ 7541.00 1.09943
$$362$$ 132.000 0.0191651
$$363$$ 0 0
$$364$$ 1664.00 0.239608
$$365$$ 1240.00 0.177821
$$366$$ 0 0
$$367$$ −11032.0 −1.56912 −0.784558 0.620055i $$-0.787110\pi$$
−0.784558 + 0.620055i $$0.787110\pi$$
$$368$$ 320.000 0.0453292
$$369$$ 0 0
$$370$$ 12240.0 1.71980
$$371$$ 6336.00 0.886654
$$372$$ 0 0
$$373$$ 5474.00 0.759874 0.379937 0.925012i $$-0.375946\pi$$
0.379937 + 0.925012i $$0.375946\pi$$
$$374$$ 3000.00 0.414776
$$375$$ 0 0
$$376$$ −1424.00 −0.195312
$$377$$ 1066.00 0.145628
$$378$$ 0 0
$$379$$ −7040.00 −0.954144 −0.477072 0.878864i $$-0.658302\pi$$
−0.477072 + 0.878864i $$0.658302\pi$$
$$380$$ −9600.00 −1.29597
$$381$$ 0 0
$$382$$ 1216.00 0.162869
$$383$$ 1830.00 0.244148 0.122074 0.992521i $$-0.461045\pi$$
0.122074 + 0.992521i $$0.461045\pi$$
$$384$$ 0 0
$$385$$ 32000.0 4.23603
$$386$$ −2740.00 −0.361301
$$387$$ 0 0
$$388$$ 2456.00 0.321352
$$389$$ −10158.0 −1.32399 −0.661994 0.749509i $$-0.730289\pi$$
−0.661994 + 0.749509i $$0.730289\pi$$
$$390$$ 0 0
$$391$$ 600.000 0.0776044
$$392$$ −5448.00 −0.701953
$$393$$ 0 0
$$394$$ −9816.00 −1.25513
$$395$$ −21920.0 −2.79219
$$396$$ 0 0
$$397$$ −12658.0 −1.60022 −0.800109 0.599854i $$-0.795225\pi$$
−0.800109 + 0.599854i $$0.795225\pi$$
$$398$$ 656.000 0.0826189
$$399$$ 0 0
$$400$$ 4400.00 0.550000
$$401$$ −15720.0 −1.95765 −0.978827 0.204689i $$-0.934382\pi$$
−0.978827 + 0.204689i $$0.934382\pi$$
$$402$$ 0 0
$$403$$ 572.000 0.0707031
$$404$$ −4232.00 −0.521163
$$405$$ 0 0
$$406$$ −5248.00 −0.641512
$$407$$ 15300.0 1.86337
$$408$$ 0 0
$$409$$ 7654.00 0.925345 0.462672 0.886529i $$-0.346891\pi$$
0.462672 + 0.886529i $$0.346891\pi$$
$$410$$ 4320.00 0.520365
$$411$$ 0 0
$$412$$ 7072.00 0.845661
$$413$$ 3008.00 0.358387
$$414$$ 0 0
$$415$$ 9240.00 1.09295
$$416$$ 416.000 0.0490290
$$417$$ 0 0
$$418$$ −12000.0 −1.40416
$$419$$ 1848.00 0.215467 0.107734 0.994180i $$-0.465641\pi$$
0.107734 + 0.994180i $$0.465641\pi$$
$$420$$ 0 0
$$421$$ −12542.0 −1.45192 −0.725962 0.687735i $$-0.758605\pi$$
−0.725962 + 0.687735i $$0.758605\pi$$
$$422$$ −2632.00 −0.303611
$$423$$ 0 0
$$424$$ 1584.00 0.181429
$$425$$ 8250.00 0.941609
$$426$$ 0 0
$$427$$ 1984.00 0.224854
$$428$$ 7232.00 0.816757
$$429$$ 0 0
$$430$$ 14240.0 1.59701
$$431$$ 5238.00 0.585396 0.292698 0.956205i $$-0.405447\pi$$
0.292698 + 0.956205i $$0.405447\pi$$
$$432$$ 0 0
$$433$$ −8258.00 −0.916522 −0.458261 0.888818i $$-0.651528\pi$$
−0.458261 + 0.888818i $$0.651528\pi$$
$$434$$ −2816.00 −0.311457
$$435$$ 0 0
$$436$$ −7544.00 −0.828652
$$437$$ −2400.00 −0.262718
$$438$$ 0 0
$$439$$ −6304.00 −0.685361 −0.342681 0.939452i $$-0.611335\pi$$
−0.342681 + 0.939452i $$0.611335\pi$$
$$440$$ 8000.00 0.866784
$$441$$ 0 0
$$442$$ 780.000 0.0839385
$$443$$ −12744.0 −1.36678 −0.683392 0.730051i $$-0.739496\pi$$
−0.683392 + 0.730051i $$0.739496\pi$$
$$444$$ 0 0
$$445$$ −24480.0 −2.60778
$$446$$ 3864.00 0.410237
$$447$$ 0 0
$$448$$ −2048.00 −0.215980
$$449$$ 11776.0 1.23774 0.618868 0.785495i $$-0.287591\pi$$
0.618868 + 0.785495i $$0.287591\pi$$
$$450$$ 0 0
$$451$$ 5400.00 0.563805
$$452$$ −4984.00 −0.518645
$$453$$ 0 0
$$454$$ 9996.00 1.03334
$$455$$ 8320.00 0.857248
$$456$$ 0 0
$$457$$ 2134.00 0.218434 0.109217 0.994018i $$-0.465166\pi$$
0.109217 + 0.994018i $$0.465166\pi$$
$$458$$ 156.000 0.0159157
$$459$$ 0 0
$$460$$ 1600.00 0.162175
$$461$$ −2724.00 −0.275205 −0.137602 0.990488i $$-0.543940\pi$$
−0.137602 + 0.990488i $$0.543940\pi$$
$$462$$ 0 0
$$463$$ −5648.00 −0.566922 −0.283461 0.958984i $$-0.591483\pi$$
−0.283461 + 0.958984i $$0.591483\pi$$
$$464$$ −1312.00 −0.131267
$$465$$ 0 0
$$466$$ −2564.00 −0.254882
$$467$$ 18224.0 1.80579 0.902897 0.429856i $$-0.141436\pi$$
0.902897 + 0.429856i $$0.141436\pi$$
$$468$$ 0 0
$$469$$ 4480.00 0.441081
$$470$$ −7120.00 −0.698768
$$471$$ 0 0
$$472$$ 752.000 0.0733339
$$473$$ 17800.0 1.73033
$$474$$ 0 0
$$475$$ −33000.0 −3.18767
$$476$$ −3840.00 −0.369761
$$477$$ 0 0
$$478$$ 588.000 0.0562646
$$479$$ −9066.00 −0.864794 −0.432397 0.901683i $$-0.642332\pi$$
−0.432397 + 0.901683i $$0.642332\pi$$
$$480$$ 0 0
$$481$$ 3978.00 0.377092
$$482$$ 9924.00 0.937813
$$483$$ 0 0
$$484$$ 4676.00 0.439144
$$485$$ 12280.0 1.14970
$$486$$ 0 0
$$487$$ 8948.00 0.832593 0.416296 0.909229i $$-0.363328\pi$$
0.416296 + 0.909229i $$0.363328\pi$$
$$488$$ 496.000 0.0460100
$$489$$ 0 0
$$490$$ −27240.0 −2.51138
$$491$$ −8720.00 −0.801483 −0.400741 0.916191i $$-0.631247\pi$$
−0.400741 + 0.916191i $$0.631247\pi$$
$$492$$ 0 0
$$493$$ −2460.00 −0.224732
$$494$$ −3120.00 −0.284161
$$495$$ 0 0
$$496$$ −704.000 −0.0637309
$$497$$ −24896.0 −2.24696
$$498$$ 0 0
$$499$$ 6604.00 0.592456 0.296228 0.955117i $$-0.404271\pi$$
0.296228 + 0.955117i $$0.404271\pi$$
$$500$$ 12000.0 1.07331
$$501$$ 0 0
$$502$$ 1488.00 0.132296
$$503$$ −3404.00 −0.301743 −0.150872 0.988553i $$-0.548208\pi$$
−0.150872 + 0.988553i $$0.548208\pi$$
$$504$$ 0 0
$$505$$ −21160.0 −1.86457
$$506$$ 2000.00 0.175713
$$507$$ 0 0
$$508$$ 6496.00 0.567349
$$509$$ 76.0000 0.00661815 0.00330908 0.999995i $$-0.498947\pi$$
0.00330908 + 0.999995i $$0.498947\pi$$
$$510$$ 0 0
$$511$$ −1984.00 −0.171755
$$512$$ −512.000 −0.0441942
$$513$$ 0 0
$$514$$ −2052.00 −0.176089
$$515$$ 35360.0 3.02553
$$516$$ 0 0
$$517$$ −8900.00 −0.757102
$$518$$ −19584.0 −1.66114
$$519$$ 0 0
$$520$$ 2080.00 0.175412
$$521$$ −12054.0 −1.01362 −0.506809 0.862058i $$-0.669175\pi$$
−0.506809 + 0.862058i $$0.669175\pi$$
$$522$$ 0 0
$$523$$ 276.000 0.0230758 0.0115379 0.999933i $$-0.496327\pi$$
0.0115379 + 0.999933i $$0.496327\pi$$
$$524$$ 8288.00 0.690960
$$525$$ 0 0
$$526$$ −11064.0 −0.917136
$$527$$ −1320.00 −0.109108
$$528$$ 0 0
$$529$$ −11767.0 −0.967124
$$530$$ 7920.00 0.649100
$$531$$ 0 0
$$532$$ 15360.0 1.25177
$$533$$ 1404.00 0.114098
$$534$$ 0 0
$$535$$ 36160.0 2.92212
$$536$$ 1120.00 0.0902549
$$537$$ 0 0
$$538$$ −7068.00 −0.566400
$$539$$ −34050.0 −2.72103
$$540$$ 0 0
$$541$$ 13778.0 1.09494 0.547470 0.836825i $$-0.315591\pi$$
0.547470 + 0.836825i $$0.315591\pi$$
$$542$$ −4784.00 −0.379134
$$543$$ 0 0
$$544$$ −960.000 −0.0756611
$$545$$ −37720.0 −2.96467
$$546$$ 0 0
$$547$$ −10844.0 −0.847634 −0.423817 0.905748i $$-0.639310\pi$$
−0.423817 + 0.905748i $$0.639310\pi$$
$$548$$ 3024.00 0.235728
$$549$$ 0 0
$$550$$ 27500.0 2.13201
$$551$$ 9840.00 0.760795
$$552$$ 0 0
$$553$$ 35072.0 2.69695
$$554$$ −12204.0 −0.935917
$$555$$ 0 0
$$556$$ 688.000 0.0524779
$$557$$ −20544.0 −1.56280 −0.781398 0.624033i $$-0.785493\pi$$
−0.781398 + 0.624033i $$0.785493\pi$$
$$558$$ 0 0
$$559$$ 4628.00 0.350167
$$560$$ −10240.0 −0.772712
$$561$$ 0 0
$$562$$ −15080.0 −1.13187
$$563$$ −6988.00 −0.523107 −0.261553 0.965189i $$-0.584235\pi$$
−0.261553 + 0.965189i $$0.584235\pi$$
$$564$$ 0 0
$$565$$ −24920.0 −1.85556
$$566$$ 5512.00 0.409340
$$567$$ 0 0
$$568$$ −6224.00 −0.459777
$$569$$ 706.000 0.0520159 0.0260080 0.999662i $$-0.491720\pi$$
0.0260080 + 0.999662i $$0.491720\pi$$
$$570$$ 0 0
$$571$$ −17532.0 −1.28492 −0.642462 0.766318i $$-0.722087\pi$$
−0.642462 + 0.766318i $$0.722087\pi$$
$$572$$ 2600.00 0.190055
$$573$$ 0 0
$$574$$ −6912.00 −0.502616
$$575$$ 5500.00 0.398897
$$576$$ 0 0
$$577$$ −14814.0 −1.06883 −0.534415 0.845222i $$-0.679468\pi$$
−0.534415 + 0.845222i $$0.679468\pi$$
$$578$$ 8026.00 0.577574
$$579$$ 0 0
$$580$$ −6560.00 −0.469637
$$581$$ −14784.0 −1.05567
$$582$$ 0 0
$$583$$ 9900.00 0.703287
$$584$$ −496.000 −0.0351449
$$585$$ 0 0
$$586$$ 1936.00 0.136477
$$587$$ −14170.0 −0.996352 −0.498176 0.867076i $$-0.665997\pi$$
−0.498176 + 0.867076i $$0.665997\pi$$
$$588$$ 0 0
$$589$$ 5280.00 0.369369
$$590$$ 3760.00 0.262367
$$591$$ 0 0
$$592$$ −4896.00 −0.339906
$$593$$ 11744.0 0.813269 0.406634 0.913591i $$-0.366702\pi$$
0.406634 + 0.913591i $$0.366702\pi$$
$$594$$ 0 0
$$595$$ −19200.0 −1.32290
$$596$$ −5088.00 −0.349686
$$597$$ 0 0
$$598$$ 520.000 0.0355592
$$599$$ 15076.0 1.02836 0.514181 0.857682i $$-0.328096\pi$$
0.514181 + 0.857682i $$0.328096\pi$$
$$600$$ 0 0
$$601$$ 20230.0 1.37304 0.686522 0.727109i $$-0.259137\pi$$
0.686522 + 0.727109i $$0.259137\pi$$
$$602$$ −22784.0 −1.54254
$$603$$ 0 0
$$604$$ 5616.00 0.378331
$$605$$ 23380.0 1.57113
$$606$$ 0 0
$$607$$ −28056.0 −1.87604 −0.938021 0.346577i $$-0.887344\pi$$
−0.938021 + 0.346577i $$0.887344\pi$$
$$608$$ 3840.00 0.256139
$$609$$ 0 0
$$610$$ 2480.00 0.164610
$$611$$ −2314.00 −0.153215
$$612$$ 0 0
$$613$$ 27446.0 1.80837 0.904187 0.427136i $$-0.140478\pi$$
0.904187 + 0.427136i $$0.140478\pi$$
$$614$$ 12872.0 0.846045
$$615$$ 0 0
$$616$$ −12800.0 −0.837219
$$617$$ −8804.00 −0.574450 −0.287225 0.957863i $$-0.592733\pi$$
−0.287225 + 0.957863i $$0.592733\pi$$
$$618$$ 0 0
$$619$$ 3508.00 0.227784 0.113892 0.993493i $$-0.463668\pi$$
0.113892 + 0.993493i $$0.463668\pi$$
$$620$$ −3520.00 −0.228011
$$621$$ 0 0
$$622$$ 15864.0 1.02265
$$623$$ 39168.0 2.51883
$$624$$ 0 0
$$625$$ 25625.0 1.64000
$$626$$ −20716.0 −1.32265
$$627$$ 0 0
$$628$$ −8680.00 −0.551544
$$629$$ −9180.00 −0.581925
$$630$$ 0 0
$$631$$ 22084.0 1.39326 0.696632 0.717428i $$-0.254681\pi$$
0.696632 + 0.717428i $$0.254681\pi$$
$$632$$ 8768.00 0.551855
$$633$$ 0 0
$$634$$ −5640.00 −0.353301
$$635$$ 32480.0 2.02981
$$636$$ 0 0
$$637$$ −8853.00 −0.550657
$$638$$ −8200.00 −0.508842
$$639$$ 0 0
$$640$$ −2560.00 −0.158114
$$641$$ 7342.00 0.452405 0.226202 0.974080i $$-0.427369\pi$$
0.226202 + 0.974080i $$0.427369\pi$$
$$642$$ 0 0
$$643$$ 2996.00 0.183749 0.0918746 0.995771i $$-0.470714\pi$$
0.0918746 + 0.995771i $$0.470714\pi$$
$$644$$ −2560.00 −0.156643
$$645$$ 0 0
$$646$$ 7200.00 0.438514
$$647$$ −9344.00 −0.567775 −0.283888 0.958858i $$-0.591624\pi$$
−0.283888 + 0.958858i $$0.591624\pi$$
$$648$$ 0 0
$$649$$ 4700.00 0.284270
$$650$$ 7150.00 0.431455
$$651$$ 0 0
$$652$$ 992.000 0.0595855
$$653$$ 16686.0 0.999960 0.499980 0.866037i $$-0.333341\pi$$
0.499980 + 0.866037i $$0.333341\pi$$
$$654$$ 0 0
$$655$$ 41440.0 2.47205
$$656$$ −1728.00 −0.102846
$$657$$ 0 0
$$658$$ 11392.0 0.674934
$$659$$ −31356.0 −1.85350 −0.926750 0.375679i $$-0.877410\pi$$
−0.926750 + 0.375679i $$0.877410\pi$$
$$660$$ 0 0
$$661$$ 590.000 0.0347176 0.0173588 0.999849i $$-0.494474\pi$$
0.0173588 + 0.999849i $$0.494474\pi$$
$$662$$ 8360.00 0.490817
$$663$$ 0 0
$$664$$ −3696.00 −0.216013
$$665$$ 76800.0 4.47846
$$666$$ 0 0
$$667$$ −1640.00 −0.0952040
$$668$$ −408.000 −0.0236317
$$669$$ 0 0
$$670$$ 5600.00 0.322906
$$671$$ 3100.00 0.178352
$$672$$ 0 0
$$673$$ 5938.00 0.340109 0.170054 0.985435i $$-0.445606\pi$$
0.170054 + 0.985435i $$0.445606\pi$$
$$674$$ 10052.0 0.574464
$$675$$ 0 0
$$676$$ 676.000 0.0384615
$$677$$ −9486.00 −0.538518 −0.269259 0.963068i $$-0.586779\pi$$
−0.269259 + 0.963068i $$0.586779\pi$$
$$678$$ 0 0
$$679$$ −19648.0 −1.11049
$$680$$ −4800.00 −0.270694
$$681$$ 0 0
$$682$$ −4400.00 −0.247045
$$683$$ −26162.0 −1.46568 −0.732841 0.680400i $$-0.761806\pi$$
−0.732841 + 0.680400i $$0.761806\pi$$
$$684$$ 0 0
$$685$$ 15120.0 0.843366
$$686$$ 21632.0 1.20396
$$687$$ 0 0
$$688$$ −5696.00 −0.315637
$$689$$ 2574.00 0.142325
$$690$$ 0 0
$$691$$ −17348.0 −0.955064 −0.477532 0.878614i $$-0.658468\pi$$
−0.477532 + 0.878614i $$0.658468\pi$$
$$692$$ −2728.00 −0.149860
$$693$$ 0 0
$$694$$ −14664.0 −0.802072
$$695$$ 3440.00 0.187751
$$696$$ 0 0
$$697$$ −3240.00 −0.176074
$$698$$ 16324.0 0.885204
$$699$$ 0 0
$$700$$ −35200.0 −1.90062
$$701$$ −30.0000 −0.00161638 −0.000808191 1.00000i $$-0.500257\pi$$
−0.000808191 1.00000i $$0.500257\pi$$
$$702$$ 0 0
$$703$$ 36720.0 1.97002
$$704$$ −3200.00 −0.171313
$$705$$ 0 0
$$706$$ 2488.00 0.132630
$$707$$ 33856.0 1.80097
$$708$$ 0 0
$$709$$ 31466.0 1.66676 0.833378 0.552703i $$-0.186404\pi$$
0.833378 + 0.552703i $$0.186404\pi$$
$$710$$ −31120.0 −1.64495
$$711$$ 0 0
$$712$$ 9792.00 0.515408
$$713$$ −880.000 −0.0462220
$$714$$ 0 0
$$715$$ 13000.0 0.679961
$$716$$ 2448.00 0.127774
$$717$$ 0 0
$$718$$ 19116.0 0.993597
$$719$$ 28892.0 1.49859 0.749297 0.662234i $$-0.230391\pi$$
0.749297 + 0.662234i $$0.230391\pi$$
$$720$$ 0 0
$$721$$ −56576.0 −2.92233
$$722$$ −15082.0 −0.777415
$$723$$ 0 0
$$724$$ −264.000 −0.0135518
$$725$$ −22550.0 −1.15515
$$726$$ 0 0
$$727$$ 13384.0 0.682786 0.341393 0.939921i $$-0.389101\pi$$
0.341393 + 0.939921i $$0.389101\pi$$
$$728$$ −3328.00 −0.169428
$$729$$ 0 0
$$730$$ −2480.00 −0.125738
$$731$$ −10680.0 −0.540375
$$732$$ 0 0
$$733$$ 7130.00 0.359280 0.179640 0.983732i $$-0.442507\pi$$
0.179640 + 0.983732i $$0.442507\pi$$
$$734$$ 22064.0 1.10953
$$735$$ 0 0
$$736$$ −640.000 −0.0320526
$$737$$ 7000.00 0.349862
$$738$$ 0 0
$$739$$ −29268.0 −1.45689 −0.728444 0.685105i $$-0.759756\pi$$
−0.728444 + 0.685105i $$0.759756\pi$$
$$740$$ −24480.0 −1.21608
$$741$$ 0 0
$$742$$ −12672.0 −0.626959
$$743$$ 9898.00 0.488725 0.244362 0.969684i $$-0.421421\pi$$
0.244362 + 0.969684i $$0.421421\pi$$
$$744$$ 0 0
$$745$$ −25440.0 −1.25107
$$746$$ −10948.0 −0.537312
$$747$$ 0 0
$$748$$ −6000.00 −0.293291
$$749$$ −57856.0 −2.82245
$$750$$ 0 0
$$751$$ −15120.0 −0.734669 −0.367335 0.930089i $$-0.619730\pi$$
−0.367335 + 0.930089i $$0.619730\pi$$
$$752$$ 2848.00 0.138106
$$753$$ 0 0
$$754$$ −2132.00 −0.102975
$$755$$ 28080.0 1.35356
$$756$$ 0 0
$$757$$ −5454.00 −0.261861 −0.130931 0.991392i $$-0.541797\pi$$
−0.130931 + 0.991392i $$0.541797\pi$$
$$758$$ 14080.0 0.674682
$$759$$ 0 0
$$760$$ 19200.0 0.916391
$$761$$ 11988.0 0.571044 0.285522 0.958372i $$-0.407833\pi$$
0.285522 + 0.958372i $$0.407833\pi$$
$$762$$ 0 0
$$763$$ 60352.0 2.86355
$$764$$ −2432.00 −0.115166
$$765$$ 0 0
$$766$$ −3660.00 −0.172639
$$767$$ 1222.00 0.0575279
$$768$$ 0 0
$$769$$ 1338.00 0.0627432 0.0313716 0.999508i $$-0.490012\pi$$
0.0313716 + 0.999508i $$0.490012\pi$$
$$770$$ −64000.0 −2.99532
$$771$$ 0 0
$$772$$ 5480.00 0.255479
$$773$$ 14408.0 0.670401 0.335200 0.942147i $$-0.391196\pi$$
0.335200 + 0.942147i $$0.391196\pi$$
$$774$$ 0 0
$$775$$ −12100.0 −0.560832
$$776$$ −4912.00 −0.227230
$$777$$ 0 0
$$778$$ 20316.0 0.936200
$$779$$ 12960.0 0.596072
$$780$$ 0 0
$$781$$ −38900.0 −1.78227
$$782$$ −1200.00 −0.0548746
$$783$$ 0 0
$$784$$ 10896.0 0.496356
$$785$$ −43400.0 −1.97326
$$786$$ 0 0
$$787$$ −10660.0 −0.482831 −0.241415 0.970422i $$-0.577612\pi$$
−0.241415 + 0.970422i $$0.577612\pi$$
$$788$$ 19632.0 0.887514
$$789$$ 0 0
$$790$$ 43840.0 1.97438
$$791$$ 39872.0 1.79227
$$792$$ 0 0
$$793$$ 806.000 0.0360932
$$794$$ 25316.0 1.13153
$$795$$ 0 0
$$796$$ −1312.00 −0.0584204
$$797$$ −1974.00 −0.0877323 −0.0438662 0.999037i $$-0.513968\pi$$
−0.0438662 + 0.999037i $$0.513968\pi$$
$$798$$ 0 0
$$799$$ 5340.00 0.236440
$$800$$ −8800.00 −0.388909
$$801$$ 0 0
$$802$$ 31440.0 1.38427
$$803$$ −3100.00 −0.136235
$$804$$ 0 0
$$805$$ −12800.0 −0.560423
$$806$$ −1144.00 −0.0499946
$$807$$ 0 0
$$808$$ 8464.00 0.368518
$$809$$ −31734.0 −1.37912 −0.689560 0.724229i $$-0.742196\pi$$
−0.689560 + 0.724229i $$0.742196\pi$$
$$810$$ 0 0
$$811$$ −38824.0 −1.68100 −0.840502 0.541808i $$-0.817740\pi$$
−0.840502 + 0.541808i $$0.817740\pi$$
$$812$$ 10496.0 0.453617
$$813$$ 0 0
$$814$$ −30600.0 −1.31760
$$815$$ 4960.00 0.213179
$$816$$ 0 0
$$817$$ 42720.0 1.82936
$$818$$ −15308.0 −0.654317
$$819$$ 0 0
$$820$$ −8640.00 −0.367954
$$821$$ 16736.0 0.711438 0.355719 0.934593i $$-0.384236\pi$$
0.355719 + 0.934593i $$0.384236\pi$$
$$822$$ 0 0
$$823$$ 42096.0 1.78296 0.891479 0.453062i $$-0.149668\pi$$
0.891479 + 0.453062i $$0.149668\pi$$
$$824$$ −14144.0 −0.597973
$$825$$ 0 0
$$826$$ −6016.00 −0.253418
$$827$$ −24858.0 −1.04522 −0.522610 0.852572i $$-0.675042\pi$$
−0.522610 + 0.852572i $$0.675042\pi$$
$$828$$ 0 0
$$829$$ 922.000 0.0386277 0.0193139 0.999813i $$-0.493852\pi$$
0.0193139 + 0.999813i $$0.493852\pi$$
$$830$$ −18480.0 −0.772832
$$831$$ 0 0
$$832$$ −832.000 −0.0346688
$$833$$ 20430.0 0.849769
$$834$$ 0 0
$$835$$ −2040.00 −0.0845474
$$836$$ 24000.0 0.992892
$$837$$ 0 0
$$838$$ −3696.00 −0.152358
$$839$$ 14294.0 0.588181 0.294090 0.955778i $$-0.404983\pi$$
0.294090 + 0.955778i $$0.404983\pi$$
$$840$$ 0 0
$$841$$ −17665.0 −0.724302
$$842$$ 25084.0 1.02666
$$843$$ 0 0
$$844$$ 5264.00 0.214685
$$845$$ 3380.00 0.137604
$$846$$ 0 0
$$847$$ −37408.0 −1.51754
$$848$$ −3168.00 −0.128290
$$849$$ 0 0
$$850$$ −16500.0 −0.665818
$$851$$ −6120.00 −0.246523
$$852$$ 0 0
$$853$$ 37966.0 1.52395 0.761976 0.647605i $$-0.224229\pi$$
0.761976 + 0.647605i $$0.224229\pi$$
$$854$$ −3968.00 −0.158996
$$855$$ 0 0
$$856$$ −14464.0 −0.577534
$$857$$ −39038.0 −1.55602 −0.778012 0.628249i $$-0.783772\pi$$
−0.778012 + 0.628249i $$0.783772\pi$$
$$858$$ 0 0
$$859$$ 20564.0 0.816804 0.408402 0.912802i $$-0.366086\pi$$
0.408402 + 0.912802i $$0.366086\pi$$
$$860$$ −28480.0 −1.12926
$$861$$ 0 0
$$862$$ −10476.0 −0.413937
$$863$$ −39866.0 −1.57248 −0.786242 0.617918i $$-0.787976\pi$$
−0.786242 + 0.617918i $$0.787976\pi$$
$$864$$ 0 0
$$865$$ −13640.0 −0.536155
$$866$$ 16516.0 0.648079
$$867$$ 0 0
$$868$$ 5632.00 0.220233
$$869$$ 54800.0 2.13920
$$870$$ 0 0
$$871$$ 1820.00 0.0708018
$$872$$ 15088.0 0.585945
$$873$$ 0 0
$$874$$ 4800.00 0.185769
$$875$$ −96000.0 −3.70902
$$876$$ 0 0
$$877$$ 30990.0 1.19322 0.596612 0.802530i $$-0.296513\pi$$
0.596612 + 0.802530i $$0.296513\pi$$
$$878$$ 12608.0 0.484623
$$879$$ 0 0
$$880$$ −16000.0 −0.612909
$$881$$ 4458.00 0.170481 0.0852405 0.996360i $$-0.472834\pi$$
0.0852405 + 0.996360i $$0.472834\pi$$
$$882$$ 0 0
$$883$$ −3164.00 −0.120586 −0.0602928 0.998181i $$-0.519203\pi$$
−0.0602928 + 0.998181i $$0.519203\pi$$
$$884$$ −1560.00 −0.0593535
$$885$$ 0 0
$$886$$ 25488.0 0.966463
$$887$$ 32512.0 1.23072 0.615359 0.788247i $$-0.289011\pi$$
0.615359 + 0.788247i $$0.289011\pi$$
$$888$$ 0 0
$$889$$ −51968.0 −1.96057
$$890$$ 48960.0 1.84398
$$891$$ 0 0
$$892$$ −7728.00 −0.290081
$$893$$ −21360.0 −0.800431
$$894$$ 0 0
$$895$$ 12240.0 0.457138
$$896$$ 4096.00 0.152721
$$897$$ 0 0
$$898$$ −23552.0 −0.875212
$$899$$ 3608.00 0.133853
$$900$$ 0 0
$$901$$ −5940.00 −0.219634
$$902$$ −10800.0 −0.398670
$$903$$ 0 0
$$904$$ 9968.00 0.366738
$$905$$ −1320.00 −0.0484843
$$906$$ 0 0
$$907$$ 10500.0 0.384396 0.192198 0.981356i $$-0.438438\pi$$
0.192198 + 0.981356i $$0.438438\pi$$
$$908$$ −19992.0 −0.730680
$$909$$ 0 0
$$910$$ −16640.0 −0.606166
$$911$$ −9840.00 −0.357864 −0.178932 0.983861i $$-0.557264\pi$$
−0.178932 + 0.983861i $$0.557264\pi$$
$$912$$ 0 0
$$913$$ −23100.0 −0.837348
$$914$$ −4268.00 −0.154456
$$915$$ 0 0
$$916$$ −312.000 −0.0112541
$$917$$ −66304.0 −2.38773
$$918$$ 0 0
$$919$$ −35040.0 −1.25774 −0.628870 0.777511i $$-0.716482\pi$$
−0.628870 + 0.777511i $$0.716482\pi$$
$$920$$ −3200.00 −0.114675
$$921$$ 0 0
$$922$$ 5448.00 0.194599
$$923$$ −10114.0 −0.360679
$$924$$ 0 0
$$925$$ −84150.0 −2.99117
$$926$$ 11296.0 0.400874
$$927$$ 0 0
$$928$$ 2624.00 0.0928201
$$929$$ −44172.0 −1.56000 −0.779998 0.625782i $$-0.784780\pi$$
−0.779998 + 0.625782i $$0.784780\pi$$
$$930$$ 0 0
$$931$$ −81720.0 −2.87676
$$932$$ 5128.00 0.180229
$$933$$ 0 0
$$934$$ −36448.0 −1.27689
$$935$$ −30000.0 −1.04931
$$936$$ 0 0
$$937$$ −54018.0 −1.88334 −0.941671 0.336535i $$-0.890745\pi$$
−0.941671 + 0.336535i $$0.890745\pi$$
$$938$$ −8960.00 −0.311892
$$939$$ 0 0
$$940$$ 14240.0 0.494104
$$941$$ 1672.00 0.0579231 0.0289616 0.999581i $$-0.490780\pi$$
0.0289616 + 0.999581i $$0.490780\pi$$
$$942$$ 0 0
$$943$$ −2160.00 −0.0745910
$$944$$ −1504.00 −0.0518549
$$945$$ 0 0
$$946$$ −35600.0 −1.22353
$$947$$ −5238.00 −0.179738 −0.0898691 0.995954i $$-0.528645\pi$$
−0.0898691 + 0.995954i $$0.528645\pi$$
$$948$$ 0 0
$$949$$ −806.000 −0.0275699
$$950$$ 66000.0 2.25402
$$951$$ 0 0
$$952$$ 7680.00 0.261460
$$953$$ 50042.0 1.70096 0.850482 0.526004i $$-0.176310\pi$$
0.850482 + 0.526004i $$0.176310\pi$$
$$954$$ 0 0
$$955$$ −12160.0 −0.412030
$$956$$ −1176.00 −0.0397851
$$957$$ 0 0
$$958$$ 18132.0 0.611501
$$959$$ −24192.0 −0.814599
$$960$$ 0 0
$$961$$ −27855.0 −0.935014
$$962$$ −7956.00 −0.266644
$$963$$ 0 0
$$964$$ −19848.0 −0.663134
$$965$$ 27400.0 0.914028
$$966$$ 0 0
$$967$$ 37676.0 1.25293 0.626463 0.779452i $$-0.284502\pi$$
0.626463 + 0.779452i $$0.284502\pi$$
$$968$$ −9352.00 −0.310521
$$969$$ 0 0
$$970$$ −24560.0 −0.812963
$$971$$ 17364.0 0.573880 0.286940 0.957949i $$-0.407362\pi$$
0.286940 + 0.957949i $$0.407362\pi$$
$$972$$ 0 0
$$973$$ −5504.00 −0.181346
$$974$$ −17896.0 −0.588732
$$975$$ 0 0
$$976$$ −992.000 −0.0325340
$$977$$ 14904.0 0.488046 0.244023 0.969769i $$-0.421533\pi$$
0.244023 + 0.969769i $$0.421533\pi$$
$$978$$ 0 0
$$979$$ 61200.0 1.99792
$$980$$ 54480.0 1.77582
$$981$$ 0 0
$$982$$ 17440.0 0.566734
$$983$$ 18038.0 0.585272 0.292636 0.956224i $$-0.405467\pi$$
0.292636 + 0.956224i $$0.405467\pi$$
$$984$$ 0 0
$$985$$ 98160.0 3.17527
$$986$$ 4920.00 0.158909
$$987$$ 0 0
$$988$$ 6240.00 0.200932
$$989$$ −7120.00 −0.228921
$$990$$ 0 0
$$991$$ 46176.0 1.48015 0.740075 0.672524i $$-0.234790\pi$$
0.740075 + 0.672524i $$0.234790\pi$$
$$992$$ 1408.00 0.0450646
$$993$$ 0 0
$$994$$ 49792.0 1.58884
$$995$$ −6560.00 −0.209011
$$996$$ 0 0
$$997$$ 55838.0 1.77373 0.886864 0.462030i $$-0.152879\pi$$
0.886864 + 0.462030i $$0.152879\pi$$
$$998$$ −13208.0 −0.418930
$$999$$ 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 234.4.a.f.1.1 1
3.2 odd 2 78.4.a.d.1.1 1
4.3 odd 2 1872.4.a.r.1.1 1
12.11 even 2 624.4.a.e.1.1 1
15.14 odd 2 1950.4.a.h.1.1 1
24.5 odd 2 2496.4.a.r.1.1 1
24.11 even 2 2496.4.a.i.1.1 1
39.5 even 4 1014.4.b.e.337.1 2
39.8 even 4 1014.4.b.e.337.2 2
39.38 odd 2 1014.4.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.d.1.1 1 3.2 odd 2
234.4.a.f.1.1 1 1.1 even 1 trivial
624.4.a.e.1.1 1 12.11 even 2
1014.4.a.d.1.1 1 39.38 odd 2
1014.4.b.e.337.1 2 39.5 even 4
1014.4.b.e.337.2 2 39.8 even 4
1872.4.a.r.1.1 1 4.3 odd 2
1950.4.a.h.1.1 1 15.14 odd 2
2496.4.a.i.1.1 1 24.11 even 2
2496.4.a.r.1.1 1 24.5 odd 2