# Properties

 Label 8670.2.a.y.1.1 Level $8670$ Weight $2$ Character 8670.1 Self dual yes Analytic conductor $69.230$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} -2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} +4.00000 q^{41} -2.00000 q^{42} +10.0000 q^{43} +1.00000 q^{45} -4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +4.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} -2.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} -2.00000 q^{59} +1.00000 q^{60} +14.0000 q^{61} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +2.00000 q^{67} -4.00000 q^{69} -2.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{78} +12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} +8.00000 q^{83} -2.00000 q^{84} +10.0000 q^{86} -2.00000 q^{87} -10.0000 q^{89} +1.00000 q^{90} -8.00000 q^{91} -4.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} +1.00000 q^{96} -8.00000 q^{97} -3.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.00000 0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 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 $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 1.00000 0.223607
$$21$$ −2.00000 −0.436436
$$22$$ 0 0
$$23$$ −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
$$26$$ 4.00000 0.784465
$$27$$ 1.00000 0.192450
$$28$$ −2.00000 −0.377964
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 4.00000 0.640513
$$40$$ 1.00000 0.158114
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 10.0000 1.52499 0.762493 0.646997i $$-0.223975\pi$$
0.762493 + 0.646997i $$0.223975\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ −4.00000 −0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 4.00000 0.554700
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 4.00000 0.529813
$$58$$ −2.00000 −0.262613
$$59$$ −2.00000 −0.260378 −0.130189 0.991489i $$-0.541558\pi$$
−0.130189 + 0.991489i $$0.541558\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 0 0
$$69$$ −4.00000 −0.481543
$$70$$ −2.00000 −0.239046
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 1.00000 0.115470
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 4.00000 0.452911
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 4.00000 0.441726
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 10.0000 1.07833
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 1.00000 0.105409
$$91$$ −8.00000 −0.838628
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 4.00000 0.410391
$$96$$ 1.00000 0.102062
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 4.00000 0.392232
$$105$$ −2.00000 −0.195180
$$106$$ 2.00000 0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ −2.00000 −0.188982
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −4.00000 −0.373002
$$116$$ −2.00000 −0.185695
$$117$$ 4.00000 0.369800
$$118$$ −2.00000 −0.184115
$$119$$ 0 0
$$120$$ 1.00000 0.0912871
$$121$$ −11.0000 −1.00000
$$122$$ 14.0000 1.26750
$$123$$ 4.00000 0.360668
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 10.0000 0.880451
$$130$$ 4.00000 0.350823
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −8.00000 −0.693688
$$134$$ 2.00000 0.172774
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −8.00000 −0.673722
$$142$$ 6.00000 0.503509
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ 4.00000 0.331042
$$147$$ −3.00000 −0.247436
$$148$$ 2.00000 0.164399
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 4.00000 0.320256
$$157$$ 4.00000 0.319235 0.159617 0.987179i $$-0.448974\pi$$
0.159617 + 0.987179i $$0.448974\pi$$
$$158$$ 12.0000 0.954669
$$159$$ 2.00000 0.158610
$$160$$ 1.00000 0.0790569
$$161$$ 8.00000 0.630488
$$162$$ 1.00000 0.0785674
$$163$$ −24.0000 −1.87983 −0.939913 0.341415i $$-0.889094\pi$$
−0.939913 + 0.341415i $$0.889094\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ 8.00000 0.620920
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 10.0000 0.762493
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ −2.00000 −0.151186
$$176$$ 0 0
$$177$$ −2.00000 −0.150329
$$178$$ −10.0000 −0.749532
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ −8.00000 −0.592999
$$183$$ 14.0000 1.03491
$$184$$ −4.00000 −0.294884
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −8.00000 −0.583460
$$189$$ −2.00000 −0.145479
$$190$$ 4.00000 0.290191
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 8.00000 0.575853 0.287926 0.957653i $$-0.407034\pi$$
0.287926 + 0.957653i $$0.407034\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 4.00000 0.286446
$$196$$ −3.00000 −0.214286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 2.00000 0.141069
$$202$$ −4.00000 −0.281439
$$203$$ 4.00000 0.280745
$$204$$ 0 0
$$205$$ 4.00000 0.279372
$$206$$ 8.00000 0.557386
$$207$$ −4.00000 −0.278019
$$208$$ 4.00000 0.277350
$$209$$ 0 0
$$210$$ −2.00000 −0.138013
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 6.00000 0.411113
$$214$$ 12.0000 0.820303
$$215$$ 10.0000 0.681994
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −6.00000 −0.406371
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 2.00000 0.134231
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 1.00000 0.0666667
$$226$$ 6.00000 0.399114
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 4.00000 0.261488
$$235$$ −8.00000 −0.521862
$$236$$ −2.00000 −0.130189
$$237$$ 12.0000 0.779484
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 30.0000 1.93247 0.966235 0.257663i $$-0.0829523\pi$$
0.966235 + 0.257663i $$0.0829523\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 1.00000 0.0641500
$$244$$ 14.0000 0.896258
$$245$$ −3.00000 −0.191663
$$246$$ 4.00000 0.255031
$$247$$ 16.0000 1.01806
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 1.00000 0.0632456
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 0 0
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 10.0000 0.622573
$$259$$ −4.00000 −0.248548
$$260$$ 4.00000 0.248069
$$261$$ −2.00000 −0.123797
$$262$$ −12.0000 −0.741362
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ −8.00000 −0.490511
$$267$$ −10.0000 −0.611990
$$268$$ 2.00000 0.122169
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 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
$$273$$ −8.00000 −0.484182
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 0 0
$$280$$ −2.00000 −0.119523
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 4.00000 0.236940
$$286$$ 0 0
$$287$$ −8.00000 −0.472225
$$288$$ 1.00000 0.0589256
$$289$$ 0 0
$$290$$ −2.00000 −0.117444
$$291$$ −8.00000 −0.468968
$$292$$ 4.00000 0.234082
$$293$$ 22.0000 1.28525 0.642627 0.766179i $$-0.277845\pi$$
0.642627 + 0.766179i $$0.277845\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ −2.00000 −0.116445
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ −12.0000 −0.695141
$$299$$ −16.0000 −0.925304
$$300$$ 1.00000 0.0577350
$$301$$ −20.0000 −1.15278
$$302$$ 8.00000 0.460348
$$303$$ −4.00000 −0.229794
$$304$$ 4.00000 0.229416
$$305$$ 14.0000 0.801638
$$306$$ 0 0
$$307$$ 10.0000 0.570730 0.285365 0.958419i $$-0.407885\pi$$
0.285365 + 0.958419i $$0.407885\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 10.0000 0.567048 0.283524 0.958965i $$-0.408496\pi$$
0.283524 + 0.958965i $$0.408496\pi$$
$$312$$ 4.00000 0.226455
$$313$$ 16.0000 0.904373 0.452187 0.891923i $$-0.350644\pi$$
0.452187 + 0.891923i $$0.350644\pi$$
$$314$$ 4.00000 0.225733
$$315$$ −2.00000 −0.112687
$$316$$ 12.0000 0.675053
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ 2.00000 0.112154
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 12.0000 0.669775
$$322$$ 8.00000 0.445823
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 4.00000 0.221880
$$326$$ −24.0000 −1.32924
$$327$$ −6.00000 −0.331801
$$328$$ 4.00000 0.220863
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 2.00000 0.109599
$$334$$ 8.00000 0.437741
$$335$$ 2.00000 0.109272
$$336$$ −2.00000 −0.109109
$$337$$ −12.0000 −0.653682 −0.326841 0.945079i $$-0.605984\pi$$
−0.326841 + 0.945079i $$0.605984\pi$$
$$338$$ 3.00000 0.163178
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000 0.216295
$$343$$ 20.0000 1.07990
$$344$$ 10.0000 0.539164
$$345$$ −4.00000 −0.215353
$$346$$ 2.00000 0.107521
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 4.00000 0.213504
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ −2.00000 −0.106299
$$355$$ 6.00000 0.318447
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 18.0000 0.951330
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ 10.0000 0.525588
$$363$$ −11.0000 −0.577350
$$364$$ −8.00000 −0.419314
$$365$$ 4.00000 0.209370
$$366$$ 14.0000 0.731792
$$367$$ 2.00000 0.104399 0.0521996 0.998637i $$-0.483377\pi$$
0.0521996 + 0.998637i $$0.483377\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 4.00000 0.208232
$$370$$ 2.00000 0.103975
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ 8.00000 0.414224 0.207112 0.978317i $$-0.433593\pi$$
0.207112 + 0.978317i $$0.433593\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ −8.00000 −0.412568
$$377$$ −8.00000 −0.412021
$$378$$ −2.00000 −0.102869
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 16.0000 0.819705
$$382$$ 12.0000 0.613973
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 8.00000 0.407189
$$387$$ 10.0000 0.508329
$$388$$ −8.00000 −0.406138
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 4.00000 0.202548
$$391$$ 0 0
$$392$$ −3.00000 −0.151523
$$393$$ −12.0000 −0.605320
$$394$$ −6.00000 −0.302276
$$395$$ 12.0000 0.603786
$$396$$ 0 0
$$397$$ 10.0000 0.501886 0.250943 0.968002i $$-0.419259\pi$$
0.250943 + 0.968002i $$0.419259\pi$$
$$398$$ −16.0000 −0.802008
$$399$$ −8.00000 −0.400501
$$400$$ 1.00000 0.0500000
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 2.00000 0.0997509
$$403$$ 0 0
$$404$$ −4.00000 −0.199007
$$405$$ 1.00000 0.0496904
$$406$$ 4.00000 0.198517
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 4.00000 0.197546
$$411$$ −18.0000 −0.887875
$$412$$ 8.00000 0.394132
$$413$$ 4.00000 0.196827
$$414$$ −4.00000 −0.196589
$$415$$ 8.00000 0.392705
$$416$$ 4.00000 0.196116
$$417$$ 12.0000 0.587643
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ −8.00000 −0.388973
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ 6.00000 0.290701
$$427$$ −28.0000 −1.35501
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 10.0000 0.482243
$$431$$ 34.0000 1.63772 0.818861 0.573992i $$-0.194606\pi$$
0.818861 + 0.573992i $$0.194606\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ −2.00000 −0.0958927
$$436$$ −6.00000 −0.287348
$$437$$ −16.0000 −0.765384
$$438$$ 4.00000 0.191127
$$439$$ 12.0000 0.572729 0.286364 0.958121i $$-0.407553\pi$$
0.286364 + 0.958121i $$0.407553\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −8.00000 −0.380091 −0.190046 0.981775i $$-0.560864\pi$$
−0.190046 + 0.981775i $$0.560864\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ −10.0000 −0.474045
$$446$$ −16.0000 −0.757622
$$447$$ −12.0000 −0.567581
$$448$$ −2.00000 −0.0944911
$$449$$ −40.0000 −1.88772 −0.943858 0.330350i $$-0.892833\pi$$
−0.943858 + 0.330350i $$0.892833\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ 8.00000 0.375873
$$454$$ 12.0000 0.563188
$$455$$ −8.00000 −0.375046
$$456$$ 4.00000 0.187317
$$457$$ −34.0000 −1.59045 −0.795226 0.606313i $$-0.792648\pi$$
−0.795226 + 0.606313i $$0.792648\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ 0 0
$$460$$ −4.00000 −0.186501
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −14.0000 −0.648537
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 4.00000 0.184900
$$469$$ −4.00000 −0.184703
$$470$$ −8.00000 −0.369012
$$471$$ 4.00000 0.184310
$$472$$ −2.00000 −0.0920575
$$473$$ 0 0
$$474$$ 12.0000 0.551178
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 2.00000 0.0915737
$$478$$ −24.0000 −1.09773
$$479$$ −2.00000 −0.0913823 −0.0456912 0.998956i $$-0.514549\pi$$
−0.0456912 + 0.998956i $$0.514549\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 8.00000 0.364769
$$482$$ 30.0000 1.36646
$$483$$ 8.00000 0.364013
$$484$$ −11.0000 −0.500000
$$485$$ −8.00000 −0.363261
$$486$$ 1.00000 0.0453609
$$487$$ 22.0000 0.996915 0.498458 0.866914i $$-0.333900\pi$$
0.498458 + 0.866914i $$0.333900\pi$$
$$488$$ 14.0000 0.633750
$$489$$ −24.0000 −1.08532
$$490$$ −3.00000 −0.135526
$$491$$ −14.0000 −0.631811 −0.315906 0.948791i $$-0.602308\pi$$
−0.315906 + 0.948791i $$0.602308\pi$$
$$492$$ 4.00000 0.180334
$$493$$ 0 0
$$494$$ 16.0000 0.719874
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −12.0000 −0.538274
$$498$$ 8.00000 0.358489
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 8.00000 0.357414
$$502$$ −2.00000 −0.0892644
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ −4.00000 −0.177998
$$506$$ 0 0
$$507$$ 3.00000 0.133235
$$508$$ 16.0000 0.709885
$$509$$ 4.00000 0.177297 0.0886484 0.996063i $$-0.471745\pi$$
0.0886484 + 0.996063i $$0.471745\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 1.00000 0.0441942
$$513$$ 4.00000 0.176604
$$514$$ −14.0000 −0.617514
$$515$$ 8.00000 0.352522
$$516$$ 10.0000 0.440225
$$517$$ 0 0
$$518$$ −4.00000 −0.175750
$$519$$ 2.00000 0.0877903
$$520$$ 4.00000 0.175412
$$521$$ −16.0000 −0.700973 −0.350486 0.936568i $$-0.613984\pi$$
−0.350486 + 0.936568i $$0.613984\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ −18.0000 −0.787085 −0.393543 0.919306i $$-0.628751\pi$$
−0.393543 + 0.919306i $$0.628751\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ −2.00000 −0.0872872
$$526$$ 24.0000 1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 2.00000 0.0868744
$$531$$ −2.00000 −0.0867926
$$532$$ −8.00000 −0.346844
$$533$$ 16.0000 0.693037
$$534$$ −10.0000 −0.432742
$$535$$ 12.0000 0.518805
$$536$$ 2.00000 0.0863868
$$537$$ 18.0000 0.776757
$$538$$ −10.0000 −0.431131
$$539$$ 0 0
$$540$$ 1.00000 0.0430331
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ 10.0000 0.429141
$$544$$ 0 0
$$545$$ −6.00000 −0.257012
$$546$$ −8.00000 −0.342368
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ −4.00000 −0.170251
$$553$$ −24.0000 −1.02058
$$554$$ −10.0000 −0.424859
$$555$$ 2.00000 0.0848953
$$556$$ 12.0000 0.508913
$$557$$ −46.0000 −1.94908 −0.974541 0.224208i $$-0.928020\pi$$
−0.974541 + 0.224208i $$0.928020\pi$$
$$558$$ 0 0
$$559$$ 40.0000 1.69182
$$560$$ −2.00000 −0.0845154
$$561$$ 0 0
$$562$$ 22.0000 0.928014
$$563$$ −32.0000 −1.34864 −0.674320 0.738440i $$-0.735563\pi$$
−0.674320 + 0.738440i $$0.735563\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 6.00000 0.252422
$$566$$ 28.0000 1.17693
$$567$$ −2.00000 −0.0839921
$$568$$ 6.00000 0.251754
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ 4.00000 0.167542
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ −8.00000 −0.333914
$$575$$ −4.00000 −0.166812
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000 0.915872 0.457936 0.888985i $$-0.348589\pi$$
0.457936 + 0.888985i $$0.348589\pi$$
$$578$$ 0 0
$$579$$ 8.00000 0.332469
$$580$$ −2.00000 −0.0830455
$$581$$ −16.0000 −0.663792
$$582$$ −8.00000 −0.331611
$$583$$ 0 0
$$584$$ 4.00000 0.165521
$$585$$ 4.00000 0.165380
$$586$$ 22.0000 0.908812
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ 0 0
$$590$$ −2.00000 −0.0823387
$$591$$ −6.00000 −0.246807
$$592$$ 2.00000 0.0821995
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −12.0000 −0.491539
$$597$$ −16.0000 −0.654836
$$598$$ −16.0000 −0.654289
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ −20.0000 −0.815139
$$603$$ 2.00000 0.0814463
$$604$$ 8.00000 0.325515
$$605$$ −11.0000 −0.447214
$$606$$ −4.00000 −0.162489
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 4.00000 0.162088
$$610$$ 14.0000 0.566843
$$611$$ −32.0000 −1.29458
$$612$$ 0 0
$$613$$ 24.0000 0.969351 0.484675 0.874694i $$-0.338938\pi$$
0.484675 + 0.874694i $$0.338938\pi$$
$$614$$ 10.0000 0.403567
$$615$$ 4.00000 0.161296
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 10.0000 0.400963
$$623$$ 20.0000 0.801283
$$624$$ 4.00000 0.160128
$$625$$ 1.00000 0.0400000
$$626$$ 16.0000 0.639489
$$627$$ 0 0
$$628$$ 4.00000 0.159617
$$629$$ 0 0
$$630$$ −2.00000 −0.0796819
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 12.0000 0.477334
$$633$$ −12.0000 −0.476957
$$634$$ −30.0000 −1.19145
$$635$$ 16.0000 0.634941
$$636$$ 2.00000 0.0793052
$$637$$ −12.0000 −0.475457
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 1.00000 0.0395285
$$641$$ −40.0000 −1.57991 −0.789953 0.613168i $$-0.789895\pi$$
−0.789953 + 0.613168i $$0.789895\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 10.0000 0.393750
$$646$$ 0 0
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 4.00000 0.156893
$$651$$ 0 0
$$652$$ −24.0000 −0.939913
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ −12.0000 −0.468879
$$656$$ 4.00000 0.156174
$$657$$ 4.00000 0.156055
$$658$$ 16.0000 0.623745
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ −16.0000 −0.621858
$$663$$ 0 0
$$664$$ 8.00000 0.310460
$$665$$ −8.00000 −0.310227
$$666$$ 2.00000 0.0774984
$$667$$ 8.00000 0.309761
$$668$$ 8.00000 0.309529
$$669$$ −16.0000 −0.618596
$$670$$ 2.00000 0.0772667
$$671$$ 0 0
$$672$$ −2.00000 −0.0771517
$$673$$ −8.00000 −0.308377 −0.154189 0.988041i $$-0.549276\pi$$
−0.154189 + 0.988041i $$0.549276\pi$$
$$674$$ −12.0000 −0.462223
$$675$$ 1.00000 0.0384900
$$676$$ 3.00000 0.115385
$$677$$ −38.0000 −1.46046 −0.730229 0.683202i $$-0.760587\pi$$
−0.730229 + 0.683202i $$0.760587\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 16.0000 0.614024
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −18.0000 −0.687745
$$686$$ 20.0000 0.763604
$$687$$ −2.00000 −0.0763048
$$688$$ 10.0000 0.381246
$$689$$ 8.00000 0.304776
$$690$$ −4.00000 −0.152277
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ 0 0
$$694$$ 20.0000 0.759190
$$695$$ 12.0000 0.455186
$$696$$ −2.00000 −0.0758098
$$697$$ 0 0
$$698$$ 2.00000 0.0757011
$$699$$ −14.0000 −0.529529
$$700$$ −2.00000 −0.0755929
$$701$$ 24.0000 0.906467 0.453234 0.891392i $$-0.350270\pi$$
0.453234 + 0.891392i $$0.350270\pi$$
$$702$$ 4.00000 0.150970
$$703$$ 8.00000 0.301726
$$704$$ 0 0
$$705$$ −8.00000 −0.301297
$$706$$ 18.0000 0.677439
$$707$$ 8.00000 0.300871
$$708$$ −2.00000 −0.0751646
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 6.00000 0.225176
$$711$$ 12.0000 0.450035
$$712$$ −10.0000 −0.374766
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 18.0000 0.672692
$$717$$ −24.0000 −0.896296
$$718$$ 20.0000 0.746393
$$719$$ −26.0000 −0.969636 −0.484818 0.874615i $$-0.661114\pi$$
−0.484818 + 0.874615i $$0.661114\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ −16.0000 −0.595871
$$722$$ −3.00000 −0.111648
$$723$$ 30.0000 1.11571
$$724$$ 10.0000 0.371647
$$725$$ −2.00000 −0.0742781
$$726$$ −11.0000 −0.408248
$$727$$ −24.0000 −0.890111 −0.445055 0.895503i $$-0.646816\pi$$
−0.445055 + 0.895503i $$0.646816\pi$$
$$728$$ −8.00000 −0.296500
$$729$$ 1.00000 0.0370370
$$730$$ 4.00000 0.148047
$$731$$ 0 0
$$732$$ 14.0000 0.517455
$$733$$ −16.0000 −0.590973 −0.295487 0.955347i $$-0.595482\pi$$
−0.295487 + 0.955347i $$0.595482\pi$$
$$734$$ 2.00000 0.0738213
$$735$$ −3.00000 −0.110657
$$736$$ −4.00000 −0.147442
$$737$$ 0 0
$$738$$ 4.00000 0.147242
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 16.0000 0.587775
$$742$$ −4.00000 −0.146845
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −12.0000 −0.439646
$$746$$ 8.00000 0.292901
$$747$$ 8.00000 0.292705
$$748$$ 0 0
$$749$$ −24.0000 −0.876941
$$750$$ 1.00000 0.0365148
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −2.00000 −0.0728841
$$754$$ −8.00000 −0.291343
$$755$$ 8.00000 0.291150
$$756$$ −2.00000 −0.0727393
$$757$$ 52.0000 1.88997 0.944986 0.327111i $$-0.106075\pi$$
0.944986 + 0.327111i $$0.106075\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ 0 0
$$760$$ 4.00000 0.145095
$$761$$ −54.0000 −1.95750 −0.978749 0.205061i $$-0.934261\pi$$
−0.978749 + 0.205061i $$0.934261\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 12.0000 0.434429
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 16.0000 0.578103
$$767$$ −8.00000 −0.288863
$$768$$ 1.00000 0.0360844
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −14.0000 −0.504198
$$772$$ 8.00000 0.287926
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ 10.0000 0.359443
$$775$$ 0 0
$$776$$ −8.00000 −0.287183
$$777$$ −4.00000 −0.143499
$$778$$ 12.0000 0.430221
$$779$$ 16.0000 0.573259
$$780$$ 4.00000 0.143223
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ −3.00000 −0.107143
$$785$$ 4.00000 0.142766
$$786$$ −12.0000 −0.428026
$$787$$ −8.00000 −0.285169 −0.142585 0.989783i $$-0.545541\pi$$
−0.142585 + 0.989783i $$0.545541\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 24.0000 0.854423
$$790$$ 12.0000 0.426941
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 56.0000 1.98862
$$794$$ 10.0000 0.354887
$$795$$ 2.00000 0.0709327
$$796$$ −16.0000 −0.567105
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ −8.00000 −0.283197
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ −10.0000 −0.353333
$$802$$ 24.0000 0.847469
$$803$$ 0 0
$$804$$ 2.00000 0.0705346
$$805$$ 8.00000 0.281963
$$806$$ 0 0
$$807$$ −10.0000 −0.352017
$$808$$ −4.00000 −0.140720
$$809$$ 28.0000 0.984428 0.492214 0.870474i $$-0.336188\pi$$
0.492214 + 0.870474i $$0.336188\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −52.0000 −1.82597 −0.912983 0.407997i $$-0.866228\pi$$
−0.912983 + 0.407997i $$0.866228\pi$$
$$812$$ 4.00000 0.140372
$$813$$ −8.00000 −0.280572
$$814$$ 0 0
$$815$$ −24.0000 −0.840683
$$816$$ 0 0
$$817$$ 40.0000 1.39942
$$818$$ −26.0000 −0.909069
$$819$$ −8.00000 −0.279543
$$820$$ 4.00000 0.139686
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ −18.0000 −0.627822
$$823$$ 50.0000 1.74289 0.871445 0.490493i $$-0.163183\pi$$
0.871445 + 0.490493i $$0.163183\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ −52.0000 −1.80822 −0.904109 0.427303i $$-0.859464\pi$$
−0.904109 + 0.427303i $$0.859464\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 8.00000 0.277684
$$831$$ −10.0000 −0.346896
$$832$$ 4.00000 0.138675
$$833$$ 0 0
$$834$$ 12.0000 0.415526
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 4.00000 0.138178
$$839$$ −14.0000 −0.483334 −0.241667 0.970359i $$-0.577694\pi$$
−0.241667 + 0.970359i $$0.577694\pi$$
$$840$$ −2.00000 −0.0690066
$$841$$ −25.0000 −0.862069
$$842$$ −34.0000 −1.17172
$$843$$ 22.0000 0.757720
$$844$$ −12.0000 −0.413057
$$845$$ 3.00000 0.103203
$$846$$ −8.00000 −0.275046
$$847$$ 22.0000 0.755929
$$848$$ 2.00000 0.0686803
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 6.00000 0.205557
$$853$$ −42.0000 −1.43805 −0.719026 0.694983i $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ −28.0000 −0.958140
$$855$$ 4.00000 0.136797
$$856$$ 12.0000 0.410152
$$857$$ 2.00000 0.0683187 0.0341593 0.999416i $$-0.489125\pi$$
0.0341593 + 0.999416i $$0.489125\pi$$
$$858$$ 0 0
$$859$$ 24.0000 0.818869 0.409435 0.912339i $$-0.365726\pi$$
0.409435 + 0.912339i $$0.365726\pi$$
$$860$$ 10.0000 0.340997
$$861$$ −8.00000 −0.272639
$$862$$ 34.0000 1.15804
$$863$$ 32.0000 1.08929 0.544646 0.838666i $$-0.316664\pi$$
0.544646 + 0.838666i $$0.316664\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 2.00000 0.0680020
$$866$$ −34.0000 −1.15537
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −2.00000 −0.0678064
$$871$$ 8.00000 0.271070
$$872$$ −6.00000 −0.203186
$$873$$ −8.00000 −0.270759
$$874$$ −16.0000 −0.541208
$$875$$ −2.00000 −0.0676123
$$876$$ 4.00000 0.135147
$$877$$ −10.0000 −0.337676 −0.168838 0.985644i $$-0.554001\pi$$
−0.168838 + 0.985644i $$0.554001\pi$$
$$878$$ 12.0000 0.404980
$$879$$ 22.0000 0.742042
$$880$$ 0 0
$$881$$ −48.0000 −1.61716 −0.808581 0.588386i $$-0.799764\pi$$
−0.808581 + 0.588386i $$0.799764\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −2.00000 −0.0673054 −0.0336527 0.999434i $$-0.510714\pi$$
−0.0336527 + 0.999434i $$0.510714\pi$$
$$884$$ 0 0
$$885$$ −2.00000 −0.0672293
$$886$$ −8.00000 −0.268765
$$887$$ −20.0000 −0.671534 −0.335767 0.941945i $$-0.608996\pi$$
−0.335767 + 0.941945i $$0.608996\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ −32.0000 −1.07325
$$890$$ −10.0000 −0.335201
$$891$$ 0 0
$$892$$ −16.0000 −0.535720
$$893$$ −32.0000 −1.07084
$$894$$ −12.0000 −0.401340
$$895$$ 18.0000 0.601674
$$896$$ −2.00000 −0.0668153
$$897$$ −16.0000 −0.534224
$$898$$ −40.0000 −1.33482
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ 0 0
$$902$$ 0 0
$$903$$ −20.0000 −0.665558
$$904$$ 6.00000 0.199557
$$905$$ 10.0000 0.332411
$$906$$ 8.00000 0.265782
$$907$$ 8.00000 0.265636 0.132818 0.991140i $$-0.457597\pi$$
0.132818 + 0.991140i $$0.457597\pi$$
$$908$$ 12.0000 0.398234
$$909$$ −4.00000 −0.132672
$$910$$ −8.00000 −0.265197
$$911$$ −18.0000 −0.596367 −0.298183 0.954509i $$-0.596381\pi$$
−0.298183 + 0.954509i $$0.596381\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 0 0
$$914$$ −34.0000 −1.12462
$$915$$ 14.0000 0.462826
$$916$$ −2.00000 −0.0660819
$$917$$ 24.0000 0.792550
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ −4.00000 −0.131876
$$921$$ 10.0000 0.329511
$$922$$ 0 0
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 16.0000 0.525793
$$927$$ 8.00000 0.262754
$$928$$ −2.00000 −0.0656532
$$929$$ 20.0000 0.656179 0.328089 0.944647i $$-0.393595\pi$$
0.328089 + 0.944647i $$0.393595\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ −14.0000 −0.458585
$$933$$ 10.0000 0.327385
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ 4.00000 0.130744
$$937$$ 54.0000 1.76410 0.882052 0.471153i $$-0.156162\pi$$
0.882052 + 0.471153i $$0.156162\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 16.0000 0.522140
$$940$$ −8.00000 −0.260931
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 4.00000 0.130327
$$943$$ −16.0000 −0.521032
$$944$$ −2.00000 −0.0650945
$$945$$ −2.00000 −0.0650600
$$946$$ 0 0
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 12.0000 0.389742
$$949$$ 16.0000 0.519382
$$950$$ 4.00000 0.129777
$$951$$ −30.0000 −0.972817
$$952$$ 0 0
$$953$$ −34.0000 −1.10137 −0.550684 0.834714i $$-0.685633\pi$$
−0.550684 + 0.834714i $$0.685633\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 12.0000 0.388311
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ −2.00000 −0.0646171
$$959$$ 36.0000 1.16250
$$960$$ 1.00000 0.0322749
$$961$$ −31.0000 −1.00000
$$962$$ 8.00000 0.257930
$$963$$ 12.0000 0.386695
$$964$$ 30.0000 0.966235
$$965$$ 8.00000 0.257529
$$966$$ 8.00000 0.257396
$$967$$ 20.0000 0.643157 0.321578 0.946883i $$-0.395787\pi$$
0.321578 + 0.946883i $$0.395787\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ 0 0
$$970$$ −8.00000 −0.256865
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −24.0000 −0.769405
$$974$$ 22.0000 0.704925
$$975$$ 4.00000 0.128103
$$976$$ 14.0000 0.448129
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ −24.0000 −0.767435
$$979$$ 0 0
$$980$$ −3.00000 −0.0958315
$$981$$ −6.00000 −0.191565
$$982$$ −14.0000 −0.446758
$$983$$ −56.0000 −1.78612 −0.893061 0.449935i $$-0.851447\pi$$
−0.893061 + 0.449935i $$0.851447\pi$$
$$984$$ 4.00000 0.127515
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 16.0000 0.509286
$$988$$ 16.0000 0.509028
$$989$$ −40.0000 −1.27193
$$990$$ 0 0
$$991$$ −24.0000 −0.762385 −0.381193 0.924496i $$-0.624487\pi$$
−0.381193 + 0.924496i $$0.624487\pi$$
$$992$$ 0 0
$$993$$ −16.0000 −0.507745
$$994$$ −12.0000 −0.380617
$$995$$ −16.0000 −0.507234
$$996$$ 8.00000 0.253490
$$997$$ −14.0000 −0.443384 −0.221692 0.975117i $$-0.571158\pi$$
−0.221692 + 0.975117i $$0.571158\pi$$
$$998$$ 12.0000 0.379853
$$999$$ 2.00000 0.0632772
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.y.1.1 1
17.16 even 2 510.2.a.d.1.1 1
51.50 odd 2 1530.2.a.h.1.1 1
68.67 odd 2 4080.2.a.r.1.1 1
85.33 odd 4 2550.2.d.f.2449.1 2
85.67 odd 4 2550.2.d.f.2449.2 2
85.84 even 2 2550.2.a.i.1.1 1
255.254 odd 2 7650.2.a.bn.1.1 1

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