# Properties

 Label 462.2.a.g.1.1 Level $462$ Weight $2$ Character 462.1 Self dual yes Analytic conductor $3.689$ 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] = [462,2,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, 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("462.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.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: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 462.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{39} +2.00000 q^{40} -10.0000 q^{41} -1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} +4.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -1.00000 q^{77} -2.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -8.00000 q^{83} -1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +1.00000 q^{88} -14.0000 q^{89} +2.00000 q^{90} +2.00000 q^{91} +4.00000 q^{93} +4.00000 q^{94} +1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 1.00000 0.408248
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ 1.00000 0.301511
$$12$$ 1.00000 0.288675
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 2.00000 0.447214
$$21$$ −1.00000 −0.218218
$$22$$ 1.00000 0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −1.00000 −0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 2.00000 0.365148
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.00000 0.174078
$$34$$ −2.00000 −0.342997
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 2.00000 0.316228
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ −2.00000 −0.280056
$$52$$ −2.00000 −0.277350
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 2.00000 0.269680
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ −2.00000 −0.262613
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ 1.00000 0.123091
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ −2.00000 −0.239046
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −1.00000 −0.115470
$$76$$ 0 0
$$77$$ −1.00000 −0.113961
$$78$$ −2.00000 −0.226455
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ −10.0000 −1.10432
$$83$$ −8.00000 −0.878114 −0.439057 0.898459i $$-0.644687\pi$$
−0.439057 + 0.898459i $$0.644687\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ −4.00000 −0.433861
$$86$$ 4.00000 0.431331
$$87$$ −2.00000 −0.214423
$$88$$ 1.00000 0.106600
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 2.00000 0.210819
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 1.00000 0.100504
$$100$$ −1.00000 −0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ −2.00000 −0.195180
$$106$$ −2.00000 −0.194257
$$107$$ −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 $$0.266089\pi$$
$$110$$ 2.00000 0.190693
$$111$$ −2.00000 −0.189832
$$112$$ −1.00000 −0.0944911
$$113$$ 18.0000 1.69330 0.846649 0.532152i $$-0.178617\pi$$
0.846649 + 0.532152i $$0.178617\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ −2.00000 −0.184900
$$118$$ −12.0000 −1.10469
$$119$$ 2.00000 0.183340
$$120$$ 2.00000 0.182574
$$121$$ 1.00000 0.0909091
$$122$$ −2.00000 −0.181071
$$123$$ −10.0000 −0.901670
$$124$$ 4.00000 0.359211
$$125$$ −12.0000 −1.07331
$$126$$ −1.00000 −0.0890871
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ −4.00000 −0.350823
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 2.00000 0.172133
$$136$$ −2.00000 −0.171499
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 4.00000 0.336861
$$142$$ 8.00000 0.671345
$$143$$ −2.00000 −0.167248
$$144$$ 1.00000 0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ 6.00000 0.496564
$$147$$ 1.00000 0.0824786
$$148$$ −2.00000 −0.164399
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ −1.00000 −0.0805823
$$155$$ 8.00000 0.642575
$$156$$ −2.00000 −0.160128
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −2.00000 −0.158610
$$160$$ 2.00000 0.158114
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 2.00000 0.155700
$$166$$ −8.00000 −0.620920
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ −4.00000 −0.306786
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 1.00000 0.0755929
$$176$$ 1.00000 0.0753778
$$177$$ −12.0000 −0.901975
$$178$$ −14.0000 −1.04934
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 2.00000 0.148250
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ −4.00000 −0.294086
$$186$$ 4.00000 0.293294
$$187$$ −2.00000 −0.146254
$$188$$ 4.00000 0.291730
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ −4.00000 −0.286446
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 12.0000 0.846415
$$202$$ −10.0000 −0.703598
$$203$$ 2.00000 0.140372
$$204$$ −2.00000 −0.140028
$$205$$ −20.0000 −1.39686
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ −2.00000 −0.138013
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 8.00000 0.548151
$$214$$ −12.0000 −0.820303
$$215$$ 8.00000 0.545595
$$216$$ 1.00000 0.0680414
$$217$$ −4.00000 −0.271538
$$218$$ 14.0000 0.948200
$$219$$ 6.00000 0.405442
$$220$$ 2.00000 0.134840
$$221$$ 4.00000 0.269069
$$222$$ −2.00000 −0.134231
$$223$$ 20.0000 1.33930 0.669650 0.742677i $$-0.266444\pi$$
0.669650 + 0.742677i $$0.266444\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −1.00000 −0.0666667
$$226$$ 18.0000 1.19734
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ −1.00000 −0.0657952
$$232$$ −2.00000 −0.131306
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 8.00000 0.521862
$$236$$ −12.0000 −0.781133
$$237$$ −8.00000 −0.519656
$$238$$ 2.00000 0.129641
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 2.00000 0.129099
$$241$$ 30.0000 1.93247 0.966235 0.257663i $$-0.0829523\pi$$
0.966235 + 0.257663i $$0.0829523\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 2.00000 0.127775
$$246$$ −10.0000 −0.637577
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ −8.00000 −0.506979
$$250$$ −12.0000 −0.758947
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ −4.00000 −0.250490
$$256$$ 1.00000 0.0625000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 2.00000 0.124274
$$260$$ −4.00000 −0.248069
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ 32.0000 1.97320 0.986602 0.163144i $$-0.0521635\pi$$
0.986602 + 0.163144i $$0.0521635\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ −4.00000 −0.245718
$$266$$ 0 0
$$267$$ −14.0000 −0.856786
$$268$$ 12.0000 0.733017
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 2.00000 0.121716
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 2.00000 0.121046
$$274$$ 10.0000 0.604122
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ −2.00000 −0.119523
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 4.00000 0.238197
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 10.0000 0.590281
$$288$$ 1.00000 0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ −4.00000 −0.234888
$$291$$ −14.0000 −0.820695
$$292$$ 6.00000 0.351123
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ −24.0000 −1.39733
$$296$$ −2.00000 −0.116248
$$297$$ 1.00000 0.0580259
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ −4.00000 −0.230556
$$302$$ 8.00000 0.460348
$$303$$ −10.0000 −0.574485
$$304$$ 0 0
$$305$$ −4.00000 −0.229039
$$306$$ −2.00000 −0.114332
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ −1.00000 −0.0569803
$$309$$ 4.00000 0.227552
$$310$$ 8.00000 0.454369
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ −30.0000 −1.69570 −0.847850 0.530236i $$-0.822103\pi$$
−0.847850 + 0.530236i $$0.822103\pi$$
$$314$$ 18.0000 1.01580
$$315$$ −2.00000 −0.112687
$$316$$ −8.00000 −0.450035
$$317$$ 22.0000 1.23564 0.617822 0.786318i $$-0.288015\pi$$
0.617822 + 0.786318i $$0.288015\pi$$
$$318$$ −2.00000 −0.112154
$$319$$ −2.00000 −0.111979
$$320$$ 2.00000 0.111803
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 2.00000 0.110940
$$326$$ 4.00000 0.221540
$$327$$ 14.0000 0.774202
$$328$$ −10.0000 −0.552158
$$329$$ −4.00000 −0.220527
$$330$$ 2.00000 0.110096
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ 24.0000 1.31126
$$336$$ −1.00000 −0.0545545
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 18.0000 0.977626
$$340$$ −4.00000 −0.216930
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −2.00000 −0.106752
$$352$$ 1.00000 0.0533002
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 16.0000 0.849192
$$356$$ −14.0000 −0.741999
$$357$$ 2.00000 0.105851
$$358$$ 12.0000 0.634220
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 2.00000 0.105409
$$361$$ −19.0000 −1.00000
$$362$$ 2.00000 0.105118
$$363$$ 1.00000 0.0524864
$$364$$ 2.00000 0.104828
$$365$$ 12.0000 0.628109
$$366$$ −2.00000 −0.104542
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ −4.00000 −0.207950
$$371$$ 2.00000 0.103835
$$372$$ 4.00000 0.207390
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ −12.0000 −0.619677
$$376$$ 4.00000 0.206284
$$377$$ 4.00000 0.206010
$$378$$ −1.00000 −0.0514344
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ −8.00000 −0.409316
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ −2.00000 −0.101929
$$386$$ 2.00000 0.101797
$$387$$ 4.00000 0.203331
$$388$$ −14.0000 −0.710742
$$389$$ 22.0000 1.11544 0.557722 0.830028i $$-0.311675\pi$$
0.557722 + 0.830028i $$0.311675\pi$$
$$390$$ −4.00000 −0.202548
$$391$$ 0 0
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ −16.0000 −0.805047
$$396$$ 1.00000 0.0502519
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ 20.0000 1.00251
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 12.0000 0.598506
$$403$$ −8.00000 −0.398508
$$404$$ −10.0000 −0.497519
$$405$$ 2.00000 0.0993808
$$406$$ 2.00000 0.0992583
$$407$$ −2.00000 −0.0991363
$$408$$ −2.00000 −0.0990148
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ −20.0000 −0.987730
$$411$$ 10.0000 0.493264
$$412$$ 4.00000 0.197066
$$413$$ 12.0000 0.590481
$$414$$ 0 0
$$415$$ −16.0000 −0.785409
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 4.00000 0.194487
$$424$$ −2.00000 −0.0971286
$$425$$ 2.00000 0.0970143
$$426$$ 8.00000 0.387601
$$427$$ 2.00000 0.0967868
$$428$$ −12.0000 −0.580042
$$429$$ −2.00000 −0.0965609
$$430$$ 8.00000 0.385794
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 18.0000 0.865025 0.432512 0.901628i $$-0.357627\pi$$
0.432512 + 0.901628i $$0.357627\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ −4.00000 −0.191785
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ 6.00000 0.286691
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 2.00000 0.0953463
$$441$$ 1.00000 0.0476190
$$442$$ 4.00000 0.190261
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ −28.0000 −1.32733
$$446$$ 20.0000 0.947027
$$447$$ −10.0000 −0.472984
$$448$$ −1.00000 −0.0472456
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −10.0000 −0.470882
$$452$$ 18.0000 0.846649
$$453$$ 8.00000 0.375873
$$454$$ 0 0
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ −38.0000 −1.77757 −0.888783 0.458329i $$-0.848448\pi$$
−0.888783 + 0.458329i $$0.848448\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ −1.00000 −0.0465242
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 8.00000 0.370991
$$466$$ −6.00000 −0.277945
$$467$$ 4.00000 0.185098 0.0925490 0.995708i $$-0.470499\pi$$
0.0925490 + 0.995708i $$0.470499\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ −12.0000 −0.554109
$$470$$ 8.00000 0.369012
$$471$$ 18.0000 0.829396
$$472$$ −12.0000 −0.552345
$$473$$ 4.00000 0.183920
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 2.00000 0.0916698
$$477$$ −2.00000 −0.0915737
$$478$$ 16.0000 0.731823
$$479$$ −32.0000 −1.46212 −0.731059 0.682315i $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ 4.00000 0.182384
$$482$$ 30.0000 1.36646
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −28.0000 −1.27141
$$486$$ 1.00000 0.0453609
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 4.00000 0.180886
$$490$$ 2.00000 0.0903508
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ 2.00000 0.0898933
$$496$$ 4.00000 0.179605
$$497$$ −8.00000 −0.358849
$$498$$ −8.00000 −0.358489
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ 4.00000 0.178529
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ −20.0000 −0.889988
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 8.00000 0.354943
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ −6.00000 −0.265424
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −6.00000 −0.264649
$$515$$ 8.00000 0.352522
$$516$$ 4.00000 0.176090
$$517$$ 4.00000 0.175920
$$518$$ 2.00000 0.0878750
$$519$$ 6.00000 0.263371
$$520$$ −4.00000 −0.175412
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ −2.00000 −0.0875376
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 0 0
$$525$$ 1.00000 0.0436436
$$526$$ 32.0000 1.39527
$$527$$ −8.00000 −0.348485
$$528$$ 1.00000 0.0435194
$$529$$ −23.0000 −1.00000
$$530$$ −4.00000 −0.173749
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 20.0000 0.866296
$$534$$ −14.0000 −0.605839
$$535$$ −24.0000 −1.03761
$$536$$ 12.0000 0.518321
$$537$$ 12.0000 0.517838
$$538$$ 10.0000 0.431131
$$539$$ 1.00000 0.0430730
$$540$$ 2.00000 0.0860663
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ 2.00000 0.0858282
$$544$$ −2.00000 −0.0857493
$$545$$ 28.0000 1.19939
$$546$$ 2.00000 0.0855921
$$547$$ −44.0000 −1.88130 −0.940652 0.339372i $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ 10.0000 0.427179
$$549$$ −2.00000 −0.0853579
$$550$$ −1.00000 −0.0426401
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ −10.0000 −0.424859
$$555$$ −4.00000 −0.169791
$$556$$ 0 0
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −8.00000 −0.338364
$$560$$ −2.00000 −0.0845154
$$561$$ −2.00000 −0.0844401
$$562$$ 10.0000 0.421825
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 36.0000 1.51453
$$566$$ 16.0000 0.672530
$$567$$ −1.00000 −0.0419961
$$568$$ 8.00000 0.335673
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ −8.00000 −0.334205
$$574$$ 10.0000 0.417392
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 2.00000 0.0831172
$$580$$ −4.00000 −0.166091
$$581$$ 8.00000 0.331896
$$582$$ −14.0000 −0.580319
$$583$$ −2.00000 −0.0828315
$$584$$ 6.00000 0.248282
$$585$$ −4.00000 −0.165380
$$586$$ 14.0000 0.578335
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 0 0
$$590$$ −24.0000 −0.988064
$$591$$ 6.00000 0.246807
$$592$$ −2.00000 −0.0821995
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 4.00000 0.163984
$$596$$ −10.0000 −0.409616
$$597$$ 20.0000 0.818546
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −18.0000 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$602$$ −4.00000 −0.163028
$$603$$ 12.0000 0.488678
$$604$$ 8.00000 0.325515
$$605$$ 2.00000 0.0813116
$$606$$ −10.0000 −0.406222
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ 2.00000 0.0810441
$$610$$ −4.00000 −0.161955
$$611$$ −8.00000 −0.323645
$$612$$ −2.00000 −0.0808452
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ 8.00000 0.322854
$$615$$ −20.0000 −0.806478
$$616$$ −1.00000 −0.0402911
$$617$$ 26.0000 1.04672 0.523360 0.852111i $$-0.324678\pi$$
0.523360 + 0.852111i $$0.324678\pi$$
$$618$$ 4.00000 0.160904
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ 20.0000 0.801927
$$623$$ 14.0000 0.560898
$$624$$ −2.00000 −0.0800641
$$625$$ −19.0000 −0.760000
$$626$$ −30.0000 −1.19904
$$627$$ 0 0
$$628$$ 18.0000 0.718278
$$629$$ 4.00000 0.159490
$$630$$ −2.00000 −0.0796819
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 12.0000 0.476957
$$634$$ 22.0000 0.873732
$$635$$ 16.0000 0.634941
$$636$$ −2.00000 −0.0793052
$$637$$ −2.00000 −0.0792429
$$638$$ −2.00000 −0.0791808
$$639$$ 8.00000 0.316475
$$640$$ 2.00000 0.0790569
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 44.0000 1.73519 0.867595 0.497271i $$-0.165665\pi$$
0.867595 + 0.497271i $$0.165665\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ 0 0
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −12.0000 −0.471041
$$650$$ 2.00000 0.0784465
$$651$$ −4.00000 −0.156772
$$652$$ 4.00000 0.156652
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ 6.00000 0.234082
$$658$$ −4.00000 −0.155936
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 2.00000 0.0778499
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 4.00000 0.155347
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 20.0000 0.773245
$$670$$ 24.0000 0.927201
$$671$$ −2.00000 −0.0772091
$$672$$ −1.00000 −0.0385758
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ −1.00000 −0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ 14.0000 0.538064 0.269032 0.963131i $$-0.413296\pi$$
0.269032 + 0.963131i $$0.413296\pi$$
$$678$$ 18.0000 0.691286
$$679$$ 14.0000 0.537271
$$680$$ −4.00000 −0.153393
$$681$$ 0 0
$$682$$ 4.00000 0.153168
$$683$$ −28.0000 −1.07139 −0.535695 0.844411i $$-0.679950\pi$$
−0.535695 + 0.844411i $$0.679950\pi$$
$$684$$ 0 0
$$685$$ 20.0000 0.764161
$$686$$ −1.00000 −0.0381802
$$687$$ −14.0000 −0.534133
$$688$$ 4.00000 0.152499
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 6.00000 0.228086
$$693$$ −1.00000 −0.0379869
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ −2.00000 −0.0758098
$$697$$ 20.0000 0.757554
$$698$$ −26.0000 −0.984115
$$699$$ −6.00000 −0.226941
$$700$$ 1.00000 0.0377964
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 0 0
$$704$$ 1.00000 0.0376889
$$705$$ 8.00000 0.301297
$$706$$ −6.00000 −0.225813
$$707$$ 10.0000 0.376089
$$708$$ −12.0000 −0.450988
$$709$$ −2.00000 −0.0751116 −0.0375558 0.999295i $$-0.511957\pi$$
−0.0375558 + 0.999295i $$0.511957\pi$$
$$710$$ 16.0000 0.600469
$$711$$ −8.00000 −0.300023
$$712$$ −14.0000 −0.524672
$$713$$ 0 0
$$714$$ 2.00000 0.0748481
$$715$$ −4.00000 −0.149592
$$716$$ 12.0000 0.448461
$$717$$ 16.0000 0.597531
$$718$$ −32.0000 −1.19423
$$719$$ 28.0000 1.04422 0.522112 0.852877i $$-0.325144\pi$$
0.522112 + 0.852877i $$0.325144\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ −4.00000 −0.148968
$$722$$ −19.0000 −0.707107
$$723$$ 30.0000 1.11571
$$724$$ 2.00000 0.0743294
$$725$$ 2.00000 0.0742781
$$726$$ 1.00000 0.0371135
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 12.0000 0.444140
$$731$$ −8.00000 −0.295891
$$732$$ −2.00000 −0.0739221
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ 12.0000 0.442026
$$738$$ −10.0000 −0.368105
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 0 0
$$742$$ 2.00000 0.0734223
$$743$$ −40.0000 −1.46746 −0.733729 0.679442i $$-0.762222\pi$$
−0.733729 + 0.679442i $$0.762222\pi$$
$$744$$ 4.00000 0.146647
$$745$$ −20.0000 −0.732743
$$746$$ 22.0000 0.805477
$$747$$ −8.00000 −0.292705
$$748$$ −2.00000 −0.0731272
$$749$$ 12.0000 0.438470
$$750$$ −12.0000 −0.438178
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 4.00000 0.145865
$$753$$ 4.00000 0.145768
$$754$$ 4.00000 0.145671
$$755$$ 16.0000 0.582300
$$756$$ −1.00000 −0.0363696
$$757$$ −18.0000 −0.654221 −0.327111 0.944986i $$-0.606075\pi$$
−0.327111 + 0.944986i $$0.606075\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −2.00000 −0.0724999 −0.0362500 0.999343i $$-0.511541\pi$$
−0.0362500 + 0.999343i $$0.511541\pi$$
$$762$$ 8.00000 0.289809
$$763$$ −14.0000 −0.506834
$$764$$ −8.00000 −0.289430
$$765$$ −4.00000 −0.144620
$$766$$ 12.0000 0.433578
$$767$$ 24.0000 0.866590
$$768$$ 1.00000 0.0360844
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ −2.00000 −0.0720750
$$771$$ −6.00000 −0.216085
$$772$$ 2.00000 0.0719816
$$773$$ 10.0000 0.359675 0.179838 0.983696i $$-0.442443\pi$$
0.179838 + 0.983696i $$0.442443\pi$$
$$774$$ 4.00000 0.143777
$$775$$ −4.00000 −0.143684
$$776$$ −14.0000 −0.502571
$$777$$ 2.00000 0.0717496
$$778$$ 22.0000 0.788738
$$779$$ 0 0
$$780$$ −4.00000 −0.143223
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 1.00000 0.0357143
$$785$$ 36.0000 1.28490
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 32.0000 1.13923
$$790$$ −16.0000 −0.569254
$$791$$ −18.0000 −0.640006
$$792$$ 1.00000 0.0355335
$$793$$ 4.00000 0.142044
$$794$$ −30.0000 −1.06466
$$795$$ −4.00000 −0.141865
$$796$$ 20.0000 0.708881
$$797$$ 34.0000 1.20434 0.602171 0.798367i $$-0.294303\pi$$
0.602171 + 0.798367i $$0.294303\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ −1.00000 −0.0353553
$$801$$ −14.0000 −0.494666
$$802$$ 18.0000 0.635602
$$803$$ 6.00000 0.211735
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 10.0000 0.352017
$$808$$ −10.0000 −0.351799
$$809$$ −14.0000 −0.492214 −0.246107 0.969243i $$-0.579151\pi$$
−0.246107 + 0.969243i $$0.579151\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ −8.00000 −0.280572
$$814$$ −2.00000 −0.0701000
$$815$$ 8.00000 0.280228
$$816$$ −2.00000 −0.0700140
$$817$$ 0 0
$$818$$ 14.0000 0.489499
$$819$$ 2.00000 0.0698857
$$820$$ −20.0000 −0.698430
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ 10.0000 0.348790
$$823$$ −48.0000 −1.67317 −0.836587 0.547833i $$-0.815453\pi$$
−0.836587 + 0.547833i $$0.815453\pi$$
$$824$$ 4.00000 0.139347
$$825$$ −1.00000 −0.0348155
$$826$$ 12.0000 0.417533
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −6.00000 −0.208389 −0.104194 0.994557i $$-0.533226\pi$$
−0.104194 + 0.994557i $$0.533226\pi$$
$$830$$ −16.0000 −0.555368
$$831$$ −10.0000 −0.346896
$$832$$ −2.00000 −0.0693375
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 4.00000 0.138260
$$838$$ −12.0000 −0.414533
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ −2.00000 −0.0690066
$$841$$ −25.0000 −0.862069
$$842$$ 6.00000 0.206774
$$843$$ 10.0000 0.344418
$$844$$ 12.0000 0.413057
$$845$$ −18.0000 −0.619219
$$846$$ 4.00000 0.137523
$$847$$ −1.00000 −0.0343604
$$848$$ −2.00000 −0.0686803
$$849$$ 16.0000 0.549119
$$850$$ 2.00000 0.0685994
$$851$$ 0 0
$$852$$ 8.00000 0.274075
$$853$$ 6.00000 0.205436 0.102718 0.994711i $$-0.467246\pi$$
0.102718 + 0.994711i $$0.467246\pi$$
$$854$$ 2.00000 0.0684386
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −34.0000 −1.16142 −0.580709 0.814111i $$-0.697225\pi$$
−0.580709 + 0.814111i $$0.697225\pi$$
$$858$$ −2.00000 −0.0682789
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 10.0000 0.340799
$$862$$ −40.0000 −1.36241
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 12.0000 0.408012
$$866$$ 18.0000 0.611665
$$867$$ −13.0000 −0.441503
$$868$$ −4.00000 −0.135769
$$869$$ −8.00000 −0.271381
$$870$$ −4.00000 −0.135613
$$871$$ −24.0000 −0.813209
$$872$$ 14.0000 0.474100
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ 12.0000 0.405674
$$876$$ 6.00000 0.202721
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 16.0000 0.539974
$$879$$ 14.0000 0.472208
$$880$$ 2.00000 0.0674200
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ 4.00000 0.134535
$$885$$ −24.0000 −0.806751
$$886$$ −12.0000 −0.403148
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ −8.00000 −0.268311
$$890$$ −28.0000 −0.938562
$$891$$ 1.00000 0.0335013
$$892$$ 20.0000 0.669650
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 24.0000 0.802232
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ −8.00000 −0.266815
$$900$$ −1.00000 −0.0333333
$$901$$ 4.00000 0.133259
$$902$$ −10.0000 −0.332964
$$903$$ −4.00000 −0.133112
$$904$$ 18.0000 0.598671
$$905$$ 4.00000 0.132964
$$906$$ 8.00000 0.265782
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ 0 0
$$909$$ −10.0000 −0.331679
$$910$$ 4.00000 0.132599
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 0 0
$$913$$ −8.00000 −0.264761
$$914$$ −38.0000 −1.25693
$$915$$ −4.00000 −0.132236
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ −2.00000 −0.0660098
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 8.00000 0.263609
$$922$$ −18.0000 −0.592798
$$923$$ −16.0000 −0.526646
$$924$$ −1.00000 −0.0328976
$$925$$ 2.00000 0.0657596
$$926$$ 16.0000 0.525793
$$927$$ 4.00000 0.131377
$$928$$ −2.00000 −0.0656532
$$929$$ −54.0000 −1.77168 −0.885841 0.463988i $$-0.846418\pi$$
−0.885841 + 0.463988i $$0.846418\pi$$
$$930$$ 8.00000 0.262330
$$931$$ 0 0
$$932$$ −6.00000 −0.196537
$$933$$ 20.0000 0.654771
$$934$$ 4.00000 0.130884
$$935$$ −4.00000 −0.130814
$$936$$ −2.00000 −0.0653720
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ −12.0000 −0.391814
$$939$$ −30.0000 −0.979013
$$940$$ 8.00000 0.260931
$$941$$ 6.00000 0.195594 0.0977972 0.995206i $$-0.468820\pi$$
0.0977972 + 0.995206i $$0.468820\pi$$
$$942$$ 18.0000 0.586472
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ −2.00000 −0.0650600
$$946$$ 4.00000 0.130051
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 22.0000 0.713399
$$952$$ 2.00000 0.0648204
$$953$$ 18.0000 0.583077 0.291539 0.956559i $$-0.405833\pi$$
0.291539 + 0.956559i $$0.405833\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ −16.0000 −0.517748
$$956$$ 16.0000 0.517477
$$957$$ −2.00000 −0.0646508
$$958$$ −32.0000 −1.03387
$$959$$ −10.0000 −0.322917
$$960$$ 2.00000 0.0645497
$$961$$ −15.0000 −0.483871
$$962$$ 4.00000 0.128965
$$963$$ −12.0000 −0.386695
$$964$$ 30.0000 0.966235
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ −28.0000 −0.899026
$$971$$ −52.0000 −1.66876 −0.834380 0.551190i $$-0.814174\pi$$
−0.834380 + 0.551190i $$0.814174\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 2.00000 0.0640513
$$976$$ −2.00000 −0.0640184
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ 4.00000 0.127906
$$979$$ −14.0000 −0.447442
$$980$$ 2.00000 0.0638877
$$981$$ 14.0000 0.446986
$$982$$ −36.0000 −1.14881
$$983$$ −4.00000 −0.127580 −0.0637901 0.997963i $$-0.520319\pi$$
−0.0637901 + 0.997963i $$0.520319\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 12.0000 0.382352
$$986$$ 4.00000 0.127386
$$987$$ −4.00000 −0.127321
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 2.00000 0.0635642
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 20.0000 0.634681
$$994$$ −8.00000 −0.253745
$$995$$ 40.0000 1.26809
$$996$$ −8.00000 −0.253490
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 28.0000 0.886325
$$999$$ −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 462.2.a.g.1.1 1
3.2 odd 2 1386.2.a.a.1.1 1
4.3 odd 2 3696.2.a.m.1.1 1
7.6 odd 2 3234.2.a.p.1.1 1
11.10 odd 2 5082.2.a.n.1.1 1
21.20 even 2 9702.2.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.g.1.1 1 1.1 even 1 trivial
1386.2.a.a.1.1 1 3.2 odd 2
3234.2.a.p.1.1 1 7.6 odd 2
3696.2.a.m.1.1 1 4.3 odd 2
5082.2.a.n.1.1 1 11.10 odd 2
9702.2.a.r.1.1 1 21.20 even 2