Properties

 Label 8670.2.a.r.1.1 Level $8670$ Weight $2$ Character 8670.1 Self dual yes Analytic conductor $69.230$ 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] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8670.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: $$69.2302985525$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 510) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} -8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -8.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} -8.00000 q^{67} +4.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000 q^{83} -8.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -14.0000 q^{89} +1.00000 q^{90} -4.00000 q^{92} -4.00000 q^{93} -4.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 0 0
$$18$$ 1.00000 0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 4.00000 0.696311
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ −2.00000 −0.320256
$$40$$ 1.00000 0.158114
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 1.00000 0.149071
$$46$$ −4.00000 −0.589768
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ −2.00000 −0.262613
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 2.00000 0.248069
$$66$$ 4.00000 0.492366
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 6.00000 0.697486
$$75$$ −1.00000 −0.115470
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 2.00000 0.214423
$$88$$ −4.00000 −0.426401
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ −1.00000 −0.102062
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ −4.00000 −0.402015
$$100$$ 1.00000 0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −4.00000 −0.373002
$$116$$ −2.00000 −0.185695
$$117$$ 2.00000 0.184900
$$118$$ −8.00000 −0.736460
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ 5.00000 0.454545
$$122$$ −10.0000 −0.905357
$$123$$ −10.0000 −0.901670
$$124$$ 4.00000 0.359211
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 2.00000 0.175412
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ −8.00000 −0.668994
$$144$$ 1.00000 0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ 2.00000 0.165521
$$147$$ 7.00000 0.577350
$$148$$ 6.00000 0.493197
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ −2.00000 −0.160128
$$157$$ −6.00000 −0.478852 −0.239426 0.970915i $$-0.576959\pi$$
−0.239426 + 0.970915i $$0.576959\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 4.00000 0.311400
$$166$$ 4.00000 0.310460
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ −8.00000 −0.609994
$$173$$ −2.00000 −0.152057 −0.0760286 0.997106i $$-0.524224\pi$$
−0.0760286 + 0.997106i $$0.524224\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 8.00000 0.601317
$$178$$ −14.0000 −1.04934
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ −4.00000 −0.294884
$$185$$ 6.00000 0.441129
$$186$$ −4.00000 −0.293294
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 10.0000 0.717958
$$195$$ −2.00000 −0.143223
$$196$$ −7.00000 −0.500000
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 8.00000 0.564276
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 10.0000 0.698430
$$206$$ 16.0000 1.11477
$$207$$ −4.00000 −0.278019
$$208$$ 2.00000 0.138675
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 8.00000 0.548151
$$214$$ −12.0000 −0.820303
$$215$$ −8.00000 −0.545595
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 14.0000 0.948200
$$219$$ −2.00000 −0.135147
$$220$$ −4.00000 −0.269680
$$221$$ 0 0
$$222$$ −6.00000 −0.402694
$$223$$ 24.0000 1.60716 0.803579 0.595198i $$-0.202926\pi$$
0.803579 + 0.595198i $$0.202926\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ −18.0000 −1.19734
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ 30.0000 1.96537 0.982683 0.185296i $$-0.0593245\pi$$
0.982683 + 0.185296i $$0.0593245\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ −8.00000 −0.520756
$$237$$ 4.00000 0.259828
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ −7.00000 −0.447214
$$246$$ −10.0000 −0.637577
$$247$$ −8.00000 −0.509028
$$248$$ 4.00000 0.254000
$$249$$ −4.00000 −0.253490
$$250$$ 1.00000 0.0632456
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 8.00000 0.498058
$$259$$ 0 0
$$260$$ 2.00000 0.124035
$$261$$ −2.00000 −0.123797
$$262$$ −4.00000 −0.247121
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ 14.0000 0.856786
$$268$$ −8.00000 −0.488678
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −4.00000 −0.241209
$$276$$ 4.00000 0.240772
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 4.00000 0.236940
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 0 0
$$290$$ −2.00000 −0.117444
$$291$$ −10.0000 −0.586210
$$292$$ 2.00000 0.117041
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 7.00000 0.408248
$$295$$ −8.00000 −0.465778
$$296$$ 6.00000 0.348743
$$297$$ 4.00000 0.232104
$$298$$ 2.00000 0.115857
$$299$$ −8.00000 −0.462652
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ −4.00000 −0.229416
$$305$$ −10.0000 −0.572598
$$306$$ 0 0
$$307$$ 16.0000 0.913168 0.456584 0.889680i $$-0.349073\pi$$
0.456584 + 0.889680i $$0.349073\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 4.00000 0.227185
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ −26.0000 −1.46031 −0.730153 0.683284i $$-0.760551\pi$$
−0.730153 + 0.683284i $$0.760551\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 8.00000 0.447914
$$320$$ 1.00000 0.0559017
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 2.00000 0.110940
$$326$$ −12.0000 −0.664619
$$327$$ −14.0000 −0.774202
$$328$$ 10.0000 0.552158
$$329$$ 0 0
$$330$$ 4.00000 0.220193
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 6.00000 0.328798
$$334$$ −12.0000 −0.656611
$$335$$ −8.00000 −0.437087
$$336$$ 0 0
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 18.0000 0.977626
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ −4.00000 −0.216295
$$343$$ 0 0
$$344$$ −8.00000 −0.431331
$$345$$ 4.00000 0.215353
$$346$$ −2.00000 −0.107521
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 2.00000 0.107211
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ −4.00000 −0.213201
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 8.00000 0.425195
$$355$$ −8.00000 −0.424596
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 32.0000 1.68890 0.844448 0.535638i $$-0.179929\pi$$
0.844448 + 0.535638i $$0.179929\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ 14.0000 0.735824
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 10.0000 0.522708
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 10.0000 0.520579
$$370$$ 6.00000 0.311925
$$371$$ 0 0
$$372$$ −4.00000 −0.207390
$$373$$ 18.0000 0.932005 0.466002 0.884783i $$-0.345694\pi$$
0.466002 + 0.884783i $$0.345694\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 16.0000 0.819705
$$382$$ 0 0
$$383$$ −24.0000 −1.22634 −0.613171 0.789950i $$-0.710106\pi$$
−0.613171 + 0.789950i $$0.710106\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ −8.00000 −0.406663
$$388$$ 10.0000 0.507673
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ −2.00000 −0.101274
$$391$$ 0 0
$$392$$ −7.00000 −0.353553
$$393$$ 4.00000 0.201773
$$394$$ −18.0000 −0.906827
$$395$$ −4.00000 −0.201262
$$396$$ −4.00000 −0.201008
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 8.00000 0.398508
$$404$$ −6.00000 −0.298511
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 10.0000 0.493865
$$411$$ 6.00000 0.295958
$$412$$ 16.0000 0.788263
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 4.00000 0.196352
$$416$$ 2.00000 0.0980581
$$417$$ −4.00000 −0.195881
$$418$$ 16.0000 0.782586
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 8.00000 0.386244
$$430$$ −8.00000 −0.385794
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −6.00000 −0.288342 −0.144171 0.989553i $$-0.546051\pi$$
−0.144171 + 0.989553i $$0.546051\pi$$
$$434$$ 0 0
$$435$$ 2.00000 0.0958927
$$436$$ 14.0000 0.670478
$$437$$ 16.0000 0.765384
$$438$$ −2.00000 −0.0955637
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ −7.00000 −0.333333
$$442$$ 0 0
$$443$$ −28.0000 −1.33032 −0.665160 0.746701i $$-0.731637\pi$$
−0.665160 + 0.746701i $$0.731637\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ −14.0000 −0.663664
$$446$$ 24.0000 1.13643
$$447$$ −2.00000 −0.0945968
$$448$$ 0 0
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −40.0000 −1.88353
$$452$$ −18.0000 −0.846649
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 0 0
$$460$$ −4.00000 −0.186501
$$461$$ −38.0000 −1.76984 −0.884918 0.465746i $$-0.845786\pi$$
−0.884918 + 0.465746i $$0.845786\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ −4.00000 −0.185496
$$466$$ 30.0000 1.38972
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ −8.00000 −0.368230
$$473$$ 32.0000 1.47136
$$474$$ 4.00000 0.183726
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 8.00000 0.365911
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 12.0000 0.547153
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 10.0000 0.454077
$$486$$ −1.00000 −0.0453609
$$487$$ 40.0000 1.81257 0.906287 0.422664i $$-0.138905\pi$$
0.906287 + 0.422664i $$0.138905\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 12.0000 0.542659
$$490$$ −7.00000 −0.316228
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ 0 0
$$494$$ −8.00000 −0.359937
$$495$$ −4.00000 −0.179787
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ −4.00000 −0.179244
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ 28.0000 1.24846 0.624229 0.781241i $$-0.285413\pi$$
0.624229 + 0.781241i $$0.285413\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 16.0000 0.711287
$$507$$ 9.00000 0.399704
$$508$$ −16.0000 −0.709885
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 4.00000 0.176604
$$514$$ 18.0000 0.793946
$$515$$ 16.0000 0.705044
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 2.00000 0.0877903
$$520$$ 2.00000 0.0877058
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ −40.0000 −1.74908 −0.874539 0.484955i $$-0.838836\pi$$
−0.874539 + 0.484955i $$0.838836\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 16.0000 0.697633
$$527$$ 0 0
$$528$$ 4.00000 0.174078
$$529$$ −7.00000 −0.304348
$$530$$ 6.00000 0.260623
$$531$$ −8.00000 −0.347170
$$532$$ 0 0
$$533$$ 20.0000 0.866296
$$534$$ 14.0000 0.605839
$$535$$ −12.0000 −0.518805
$$536$$ −8.00000 −0.345547
$$537$$ 0 0
$$538$$ 14.0000 0.603583
$$539$$ 28.0000 1.20605
$$540$$ −1.00000 −0.0430331
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ −14.0000 −0.600798
$$544$$ 0 0
$$545$$ 14.0000 0.599694
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ −10.0000 −0.426790
$$550$$ −4.00000 −0.170561
$$551$$ 8.00000 0.340811
$$552$$ 4.00000 0.170251
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ −6.00000 −0.254686
$$556$$ 4.00000 0.169638
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −30.0000 −1.26547
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 0 0
$$565$$ −18.0000 −0.757266
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 4.00000 0.167542
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 1.00000 0.0416667
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ 0 0
$$579$$ 6.00000 0.249351
$$580$$ −2.00000 −0.0830455
$$581$$ 0 0
$$582$$ −10.0000 −0.414513
$$583$$ −24.0000 −0.993978
$$584$$ 2.00000 0.0827606
$$585$$ 2.00000 0.0826898
$$586$$ 6.00000 0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 7.00000 0.288675
$$589$$ −16.0000 −0.659269
$$590$$ −8.00000 −0.329355
$$591$$ 18.0000 0.740421
$$592$$ 6.00000 0.246598
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ −4.00000 −0.163709
$$598$$ −8.00000 −0.327144
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −34.0000 −1.38689 −0.693444 0.720510i $$-0.743908\pi$$
−0.693444 + 0.720510i $$0.743908\pi$$
$$602$$ 0 0
$$603$$ −8.00000 −0.325785
$$604$$ 0 0
$$605$$ 5.00000 0.203279
$$606$$ 6.00000 0.243733
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ −10.0000 −0.404888
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 10.0000 0.403896 0.201948 0.979396i $$-0.435273\pi$$
0.201948 + 0.979396i $$0.435273\pi$$
$$614$$ 16.0000 0.645707
$$615$$ −10.0000 −0.403239
$$616$$ 0 0
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ −16.0000 −0.643614
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 4.00000 0.160514
$$622$$ 8.00000 0.320771
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ 2.00000 0.0799361
$$627$$ −16.0000 −0.638978
$$628$$ −6.00000 −0.239426
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ 12.0000 0.476957
$$634$$ −26.0000 −1.03259
$$635$$ −16.0000 −0.634941
$$636$$ −6.00000 −0.237915
$$637$$ −14.0000 −0.554700
$$638$$ 8.00000 0.316723
$$639$$ −8.00000 −0.316475
$$640$$ 1.00000 0.0395285
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 20.0000 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ 0 0
$$647$$ −32.0000 −1.25805 −0.629025 0.777385i $$-0.716546\pi$$
−0.629025 + 0.777385i $$0.716546\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 32.0000 1.25611
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ −4.00000 −0.156293
$$656$$ 10.0000 0.390434
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 4.00000 0.155700
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 0 0
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 8.00000 0.309761
$$668$$ −12.0000 −0.464294
$$669$$ −24.0000 −0.927894
$$670$$ −8.00000 −0.309067
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ 10.0000 0.385472 0.192736 0.981251i $$-0.438264\pi$$
0.192736 + 0.981251i $$0.438264\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ −1.00000 −0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 18.0000 0.691286
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ −16.0000 −0.612672
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ −6.00000 −0.228914
$$688$$ −8.00000 −0.304997
$$689$$ 12.0000 0.457164
$$690$$ 4.00000 0.152277
$$691$$ 36.0000 1.36950 0.684752 0.728776i $$-0.259910\pi$$
0.684752 + 0.728776i $$0.259910\pi$$
$$692$$ −2.00000 −0.0760286
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 4.00000 0.151729
$$696$$ 2.00000 0.0758098
$$697$$ 0 0
$$698$$ −34.0000 −1.28692
$$699$$ −30.0000 −1.13470
$$700$$ 0 0
$$701$$ 10.0000 0.377695 0.188847 0.982006i $$-0.439525\pi$$
0.188847 + 0.982006i $$0.439525\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −24.0000 −0.905177
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 0 0
$$708$$ 8.00000 0.300658
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ −4.00000 −0.150012
$$712$$ −14.0000 −0.524672
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ −8.00000 −0.299183
$$716$$ 0 0
$$717$$ −8.00000 −0.298765
$$718$$ 32.0000 1.19423
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ 10.0000 0.371904
$$724$$ 14.0000 0.520306
$$725$$ −2.00000 −0.0742781
$$726$$ −5.00000 −0.185567
$$727$$ 40.0000 1.48352 0.741759 0.670667i $$-0.233992\pi$$
0.741759 + 0.670667i $$0.233992\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000 0.0740233
$$731$$ 0 0
$$732$$ 10.0000 0.369611
$$733$$ 26.0000 0.960332 0.480166 0.877178i $$-0.340576\pi$$
0.480166 + 0.877178i $$0.340576\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 7.00000 0.258199
$$736$$ −4.00000 −0.147442
$$737$$ 32.0000 1.17874
$$738$$ 10.0000 0.368105
$$739$$ −44.0000 −1.61857 −0.809283 0.587419i $$-0.800144\pi$$
−0.809283 + 0.587419i $$0.800144\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ −28.0000 −1.02722 −0.513610 0.858024i $$-0.671692\pi$$
−0.513610 + 0.858024i $$0.671692\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 2.00000 0.0732743
$$746$$ 18.0000 0.659027
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −14.0000 −0.508839 −0.254419 0.967094i $$-0.581884\pi$$
−0.254419 + 0.967094i $$0.581884\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ −16.0000 −0.580763
$$760$$ −4.00000 −0.145095
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ −16.0000 −0.577727
$$768$$ −1.00000 −0.0360844
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ −6.00000 −0.215945
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 4.00000 0.143684
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ −30.0000 −1.07555
$$779$$ −40.0000 −1.43315
$$780$$ −2.00000 −0.0716115
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ −7.00000 −0.250000
$$785$$ −6.00000 −0.214149
$$786$$ 4.00000 0.142675
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ −16.0000 −0.569615
$$790$$ −4.00000 −0.142314
$$791$$ 0 0
$$792$$ −4.00000 −0.142134
$$793$$ −20.0000 −0.710221
$$794$$ −18.0000 −0.638796
$$795$$ −6.00000 −0.212798
$$796$$ 4.00000 0.141776
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ −14.0000 −0.494666
$$802$$ −6.00000 −0.211867
$$803$$ −8.00000 −0.282314
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ −14.0000 −0.492823
$$808$$ −6.00000 −0.211079
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ 8.00000 0.280572
$$814$$ −24.0000 −0.841200
$$815$$ −12.0000 −0.420342
$$816$$ 0 0
$$817$$ 32.0000 1.11954
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ 10.0000 0.349215
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 54.0000 1.87550 0.937749 0.347314i $$-0.112906\pi$$
0.937749 + 0.347314i $$0.112906\pi$$
$$830$$ 4.00000 0.138842
$$831$$ 10.0000 0.346896
$$832$$ 2.00000 0.0693375
$$833$$ 0 0
$$834$$ −4.00000 −0.138509
$$835$$ −12.0000 −0.415277
$$836$$ 16.0000 0.553372
$$837$$ −4.00000 −0.138260
$$838$$ −28.0000 −0.967244
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 30.0000 1.03387
$$843$$ 30.0000 1.03325
$$844$$ −12.0000 −0.413057
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ 8.00000 0.274075
$$853$$ 54.0000 1.84892 0.924462 0.381273i $$-0.124514\pi$$
0.924462 + 0.381273i $$0.124514\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ −12.0000 −0.410152
$$857$$ −26.0000 −0.888143 −0.444072 0.895991i $$-0.646466\pi$$
−0.444072 + 0.895991i $$0.646466\pi$$
$$858$$ 8.00000 0.273115
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ 16.0000 0.544962
$$863$$ −8.00000 −0.272323 −0.136162 0.990687i $$-0.543477\pi$$
−0.136162 + 0.990687i $$0.543477\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −2.00000 −0.0680020
$$866$$ −6.00000 −0.203888
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 2.00000 0.0678064
$$871$$ −16.0000 −0.542139
$$872$$ 14.0000 0.474100
$$873$$ 10.0000 0.338449
$$874$$ 16.0000 0.541208
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ −50.0000 −1.68838 −0.844190 0.536044i $$-0.819918\pi$$
−0.844190 + 0.536044i $$0.819918\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ −6.00000 −0.202375
$$880$$ −4.00000 −0.134840
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 8.00000 0.268917
$$886$$ −28.0000 −0.940678
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ 0 0
$$890$$ −14.0000 −0.469281
$$891$$ −4.00000 −0.134005
$$892$$ 24.0000 0.803579
$$893$$ 0 0
$$894$$ −2.00000 −0.0668900
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 8.00000 0.267112
$$898$$ 26.0000 0.867631
$$899$$ −8.00000 −0.266815
$$900$$ 1.00000 0.0333333
$$901$$ 0 0
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 14.0000 0.465376
$$906$$ 0 0
$$907$$ 36.0000 1.19536 0.597680 0.801735i $$-0.296089\pi$$
0.597680 + 0.801735i $$0.296089\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −16.0000 −0.529523
$$914$$ 2.00000 0.0661541
$$915$$ 10.0000 0.330590
$$916$$ 6.00000 0.198246
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ −16.0000 −0.527218
$$922$$ −38.0000 −1.25146
$$923$$ −16.0000 −0.526646
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ −24.0000 −0.788689
$$927$$ 16.0000 0.525509
$$928$$ −2.00000 −0.0656532
$$929$$ 50.0000 1.64045 0.820223 0.572043i $$-0.193849\pi$$
0.820223 + 0.572043i $$0.193849\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ 28.0000 0.917663
$$932$$ 30.0000 0.982683
$$933$$ −8.00000 −0.261908
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ −2.00000 −0.0652675
$$940$$ 0 0
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 6.00000 0.195491
$$943$$ −40.0000 −1.30258
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ 32.0000 1.04041
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 4.00000 0.129914
$$949$$ 4.00000 0.129845
$$950$$ −4.00000 −0.129777
$$951$$ 26.0000 0.843108
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 8.00000 0.258738
$$957$$ −8.00000 −0.258603
$$958$$ 0 0
$$959$$ 0 0
$$960$$ −1.00000 −0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ 12.0000 0.386896
$$963$$ −12.0000 −0.386695
$$964$$ −10.0000 −0.322078
$$965$$ −6.00000 −0.193147
$$966$$ 0 0
$$967$$ 24.0000 0.771788 0.385894 0.922543i $$-0.373893\pi$$
0.385894 + 0.922543i $$0.373893\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ 10.0000 0.321081
$$971$$ 32.0000 1.02693 0.513464 0.858111i $$-0.328362\pi$$
0.513464 + 0.858111i $$0.328362\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 40.0000 1.28168
$$975$$ −2.00000 −0.0640513
$$976$$ −10.0000 −0.320092
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 12.0000 0.383718
$$979$$ 56.0000 1.78977
$$980$$ −7.00000 −0.223607
$$981$$ 14.0000 0.446986
$$982$$ 24.0000 0.765871
$$983$$ 44.0000 1.40338 0.701691 0.712481i $$-0.252429\pi$$
0.701691 + 0.712481i $$0.252429\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ −18.0000 −0.573528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 32.0000 1.01754
$$990$$ −4.00000 −0.127128
$$991$$ 44.0000 1.39771 0.698853 0.715265i $$-0.253694\pi$$
0.698853 + 0.715265i $$0.253694\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 4.00000 0.126809
$$996$$ −4.00000 −0.126745
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ −6.00000 −0.189832
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 8670.2.a.r.1.1 1
17.16 even 2 510.2.a.f.1.1 1
51.50 odd 2 1530.2.a.f.1.1 1
68.67 odd 2 4080.2.a.c.1.1 1
85.33 odd 4 2550.2.d.s.2449.1 2
85.67 odd 4 2550.2.d.s.2449.2 2
85.84 even 2 2550.2.a.d.1.1 1
255.254 odd 2 7650.2.a.bw.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.f.1.1 1 17.16 even 2
1530.2.a.f.1.1 1 51.50 odd 2
2550.2.a.d.1.1 1 85.84 even 2
2550.2.d.s.2449.1 2 85.33 odd 4
2550.2.d.s.2449.2 2 85.67 odd 4
4080.2.a.c.1.1 1 68.67 odd 2
7650.2.a.bw.1.1 1 255.254 odd 2
8670.2.a.r.1.1 1 1.1 even 1 trivial