# Properties

 Label 1078.2.a.e.1.1 Level $1078$ Weight $2$ Character 1078.1 Self dual yes Analytic conductor $8.608$ 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] = [1078,2,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1078.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1078.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: $$8.60787333789$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1078.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} +2.00000 q^{18} +2.00000 q^{19} +1.00000 q^{22} -6.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} +1.00000 q^{26} -5.00000 q^{27} +9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -2.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +5.00000 q^{50} -6.00000 q^{51} -1.00000 q^{52} +5.00000 q^{54} +2.00000 q^{57} -9.00000 q^{58} -3.00000 q^{59} +11.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{66} +11.0000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +2.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} -5.00000 q^{75} +2.00000 q^{76} +1.00000 q^{78} +5.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -6.00000 q^{83} +4.00000 q^{86} +9.00000 q^{87} +1.00000 q^{88} -18.0000 q^{89} -6.00000 q^{92} -4.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} -13.0000 q^{97} +2.00000 q^{99} +O(q^{100})$$

## 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 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 1.00000 0.288675
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 2.00000 0.471405
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −5.00000 −1.00000
$$26$$ 1.00000 0.196116
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.00000 −0.174078
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ 5.00000 0.707107
$$51$$ −6.00000 −0.840168
$$52$$ −1.00000 −0.138675
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ −9.00000 −1.18176
$$59$$ −3.00000 −0.390567 −0.195283 0.980747i $$-0.562563\pi$$
−0.195283 + 0.980747i $$0.562563\pi$$
$$60$$ 0 0
$$61$$ 11.0000 1.40841 0.704203 0.709999i $$-0.251305\pi$$
0.704203 + 0.709999i $$0.251305\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.00000 0.123091
$$67$$ 11.0000 1.34386 0.671932 0.740613i $$-0.265465\pi$$
0.671932 + 0.740613i $$0.265465\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 2.00000 0.235702
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −5.00000 −0.577350
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ 1.00000 0.113228
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 9.00000 0.964901
$$88$$ 1.00000 0.106600
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −6.00000 −0.625543
$$93$$ −4.00000 −0.414781
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ −13.0000 −1.31995 −0.659975 0.751288i $$-0.729433\pi$$
−0.659975 + 0.751288i $$0.729433\pi$$
$$98$$ 0 0
$$99$$ 2.00000 0.201008
$$100$$ −5.00000 −0.500000
$$101$$ −15.0000 −1.49256 −0.746278 0.665635i $$-0.768161\pi$$
−0.746278 + 0.665635i $$0.768161\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ 9.00000 0.835629
$$117$$ 2.00000 0.184900
$$118$$ 3.00000 0.276172
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −11.0000 −0.995893
$$123$$ −6.00000 −0.541002
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 0 0
$$134$$ −11.0000 −0.950255
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ −9.00000 −0.768922 −0.384461 0.923141i $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ 6.00000 0.510754
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ 1.00000 0.0836242
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 5.00000 0.408248
$$151$$ −19.0000 −1.54620 −0.773099 0.634285i $$-0.781294\pi$$
−0.773099 + 0.634285i $$0.781294\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 12.0000 0.970143
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ −5.00000 −0.397779
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 17.0000 1.33154 0.665771 0.746156i $$-0.268103\pi$$
0.665771 + 0.746156i $$0.268103\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ −4.00000 −0.304997
$$173$$ −21.0000 −1.59660 −0.798300 0.602260i $$-0.794267\pi$$
−0.798300 + 0.602260i $$0.794267\pi$$
$$174$$ −9.00000 −0.682288
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ −3.00000 −0.225494
$$178$$ 18.0000 1.34916
$$179$$ −15.0000 −1.12115 −0.560576 0.828103i $$-0.689420\pi$$
−0.560576 + 0.828103i $$0.689420\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 11.0000 0.813143
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ 6.00000 0.438763
$$188$$ −6.00000 −0.437595
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 13.0000 0.933346
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 11.0000 0.775880
$$202$$ 15.0000 1.05540
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 16.0000 1.11477
$$207$$ 12.0000 0.834058
$$208$$ −1.00000 −0.0693375
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ 0 0
$$218$$ −2.00000 −0.135457
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ −2.00000 −0.134231
$$223$$ 26.0000 1.74109 0.870544 0.492090i $$-0.163767\pi$$
0.870544 + 0.492090i $$0.163767\pi$$
$$224$$ 0 0
$$225$$ 10.0000 0.666667
$$226$$ −9.00000 −0.598671
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 2.00000 0.132453
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −9.00000 −0.590879
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −3.00000 −0.195283
$$237$$ 5.00000 0.324785
$$238$$ 0 0
$$239$$ −9.00000 −0.582162 −0.291081 0.956698i $$-0.594015\pi$$
−0.291081 + 0.956698i $$0.594015\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 16.0000 1.02640
$$244$$ 11.0000 0.704203
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ −2.00000 −0.127257
$$248$$ 4.00000 0.254000
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 6.00000 0.377217
$$254$$ 7.00000 0.439219
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 3.00000 0.187135 0.0935674 0.995613i $$-0.470173\pi$$
0.0935674 + 0.995613i $$0.470173\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ −18.0000 −1.11204
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −18.0000 −1.10158
$$268$$ 11.0000 0.671932
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 0 0
$$271$$ 29.0000 1.76162 0.880812 0.473466i $$-0.156997\pi$$
0.880812 + 0.473466i $$0.156997\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ 9.00000 0.543710
$$275$$ 5.00000 0.301511
$$276$$ −6.00000 −0.361158
$$277$$ 29.0000 1.74244 0.871221 0.490892i $$-0.163329\pi$$
0.871221 + 0.490892i $$0.163329\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 6.00000 0.357295
$$283$$ −10.0000 −0.594438 −0.297219 0.954809i $$-0.596059\pi$$
−0.297219 + 0.954809i $$0.596059\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −1.00000 −0.0591312
$$287$$ 0 0
$$288$$ 2.00000 0.117851
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −13.0000 −0.762073
$$292$$ 2.00000 0.117041
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 5.00000 0.290129
$$298$$ −6.00000 −0.347571
$$299$$ 6.00000 0.346989
$$300$$ −5.00000 −0.288675
$$301$$ 0 0
$$302$$ 19.0000 1.09333
$$303$$ −15.0000 −0.861727
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ −12.0000 −0.685994
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ 17.0000 0.960897 0.480448 0.877023i $$-0.340474\pi$$
0.480448 + 0.877023i $$0.340474\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ 5.00000 0.281272
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 0 0
$$319$$ −9.00000 −0.503903
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ −12.0000 −0.667698
$$324$$ 1.00000 0.0555556
$$325$$ 5.00000 0.277350
$$326$$ −17.0000 −0.941543
$$327$$ 2.00000 0.110600
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 35.0000 1.92377 0.961887 0.273447i $$-0.0881639\pi$$
0.961887 + 0.273447i $$0.0881639\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ −4.00000 −0.219199
$$334$$ −3.00000 −0.164153
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 9.00000 0.488813
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ 4.00000 0.216295
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 21.0000 1.12897
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 9.00000 0.482451
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 5.00000 0.266880
$$352$$ 1.00000 0.0533002
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 3.00000 0.159448
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ 15.0000 0.792775
$$359$$ −3.00000 −0.158334 −0.0791670 0.996861i $$-0.525226\pi$$
−0.0791670 + 0.996861i $$0.525226\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −2.00000 −0.105118
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −11.0000 −0.574979
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ 0 0
$$372$$ −4.00000 −0.207390
$$373$$ −31.0000 −1.60512 −0.802560 0.596572i $$-0.796529\pi$$
−0.802560 + 0.596572i $$0.796529\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ −9.00000 −0.463524
$$378$$ 0 0
$$379$$ 23.0000 1.18143 0.590715 0.806880i $$-0.298846\pi$$
0.590715 + 0.806880i $$0.298846\pi$$
$$380$$ 0 0
$$381$$ −7.00000 −0.358621
$$382$$ −6.00000 −0.306987
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 8.00000 0.406663
$$388$$ −13.0000 −0.659975
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ 36.0000 1.82060
$$392$$ 0 0
$$393$$ 18.0000 0.907980
$$394$$ 3.00000 0.151138
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ −14.0000 −0.701757
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ −33.0000 −1.64794 −0.823971 0.566632i $$-0.808246\pi$$
−0.823971 + 0.566632i $$0.808246\pi$$
$$402$$ −11.0000 −0.548630
$$403$$ 4.00000 0.199254
$$404$$ −15.0000 −0.746278
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.00000 −0.0991363
$$408$$ 6.00000 0.297044
$$409$$ −4.00000 −0.197787 −0.0988936 0.995098i $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 0 0
$$411$$ −9.00000 −0.443937
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ −12.0000 −0.589768
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 8.00000 0.391762
$$418$$ 2.00000 0.0978232
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ 10.0000 0.486792
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ 30.0000 1.45521
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 1.00000 0.0482805
$$430$$ 0 0
$$431$$ −3.00000 −0.144505 −0.0722525 0.997386i $$-0.523019\pi$$
−0.0722525 + 0.997386i $$0.523019\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ −12.0000 −0.574038
$$438$$ −2.00000 −0.0955637
$$439$$ −1.00000 −0.0477274 −0.0238637 0.999715i $$-0.507597\pi$$
−0.0238637 + 0.999715i $$0.507597\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −6.00000 −0.285391
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ 0 0
$$446$$ −26.0000 −1.23114
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ −10.0000 −0.471405
$$451$$ 6.00000 0.282529
$$452$$ 9.00000 0.423324
$$453$$ −19.0000 −0.892698
$$454$$ −18.0000 −0.844782
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 16.0000 0.747631
$$459$$ 30.0000 1.40028
$$460$$ 0 0
$$461$$ 21.0000 0.978068 0.489034 0.872265i $$-0.337349\pi$$
0.489034 + 0.872265i $$0.337349\pi$$
$$462$$ 0 0
$$463$$ −22.0000 −1.02243 −0.511213 0.859454i $$-0.670804\pi$$
−0.511213 + 0.859454i $$0.670804\pi$$
$$464$$ 9.00000 0.417815
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −4.00000 −0.184310
$$472$$ 3.00000 0.138086
$$473$$ 4.00000 0.183920
$$474$$ −5.00000 −0.229658
$$475$$ −10.0000 −0.458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 9.00000 0.411650
$$479$$ 15.0000 0.685367 0.342684 0.939451i $$-0.388664\pi$$
0.342684 + 0.939451i $$0.388664\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −16.0000 −0.725775
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ −11.0000 −0.497947
$$489$$ 17.0000 0.768767
$$490$$ 0 0
$$491$$ 42.0000 1.89543 0.947717 0.319113i $$-0.103385\pi$$
0.947717 + 0.319113i $$0.103385\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −54.0000 −2.43204
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 6.00000 0.268866
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ 0 0
$$501$$ 3.00000 0.134030
$$502$$ 12.0000 0.535586
$$503$$ −33.0000 −1.47140 −0.735699 0.677309i $$-0.763146\pi$$
−0.735699 + 0.677309i $$0.763146\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −6.00000 −0.266733
$$507$$ −12.0000 −0.532939
$$508$$ −7.00000 −0.310575
$$509$$ −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
$$513$$ −10.0000 −0.441511
$$514$$ −3.00000 −0.132324
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 6.00000 0.263880
$$518$$ 0 0
$$519$$ −21.0000 −0.921798
$$520$$ 0 0
$$521$$ −42.0000 −1.84005 −0.920027 0.391856i $$-0.871833\pi$$
−0.920027 + 0.391856i $$0.871833\pi$$
$$522$$ 18.0000 0.787839
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ 9.00000 0.392419
$$527$$ 24.0000 1.04546
$$528$$ −1.00000 −0.0435194
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ 6.00000 0.259889
$$534$$ 18.0000 0.778936
$$535$$ 0 0
$$536$$ −11.0000 −0.475128
$$537$$ −15.0000 −0.647298
$$538$$ −12.0000 −0.517357
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −25.0000 −1.07483 −0.537417 0.843317i $$-0.680600\pi$$
−0.537417 + 0.843317i $$0.680600\pi$$
$$542$$ −29.0000 −1.24566
$$543$$ 2.00000 0.0858282
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −9.00000 −0.384461
$$549$$ −22.0000 −0.938937
$$550$$ −5.00000 −0.213201
$$551$$ 18.0000 0.766826
$$552$$ 6.00000 0.255377
$$553$$ 0 0
$$554$$ −29.0000 −1.23209
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 6.00000 0.253320
$$562$$ −18.0000 −0.759284
$$563$$ −18.0000 −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ 10.0000 0.420331
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −36.0000 −1.50920 −0.754599 0.656186i $$-0.772169\pi$$
−0.754599 + 0.656186i $$0.772169\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 1.00000 0.0418121
$$573$$ 6.00000 0.250654
$$574$$ 0 0
$$575$$ 30.0000 1.25109
$$576$$ −2.00000 −0.0833333
$$577$$ −7.00000 −0.291414 −0.145707 0.989328i $$-0.546546\pi$$
−0.145707 + 0.989328i $$0.546546\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 13.0000 0.538867
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ 9.00000 0.371470 0.185735 0.982600i $$-0.440533\pi$$
0.185735 + 0.982600i $$0.440533\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ −3.00000 −0.123404
$$592$$ 2.00000 0.0821995
$$593$$ 36.0000 1.47834 0.739171 0.673517i $$-0.235217\pi$$
0.739171 + 0.673517i $$0.235217\pi$$
$$594$$ −5.00000 −0.205152
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 14.0000 0.572982
$$598$$ −6.00000 −0.245358
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 5.00000 0.204124
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ −22.0000 −0.895909
$$604$$ −19.0000 −0.773099
$$605$$ 0 0
$$606$$ 15.0000 0.609333
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6.00000 0.242734
$$612$$ 12.0000 0.485071
$$613$$ −10.0000 −0.403896 −0.201948 0.979396i $$-0.564727\pi$$
−0.201948 + 0.979396i $$0.564727\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 21.0000 0.845428 0.422714 0.906263i $$-0.361077\pi$$
0.422714 + 0.906263i $$0.361077\pi$$
$$618$$ 16.0000 0.643614
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 30.0000 1.20386
$$622$$ −24.0000 −0.962312
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 25.0000 1.00000
$$626$$ −17.0000 −0.679457
$$627$$ −2.00000 −0.0798723
$$628$$ −4.00000 −0.159617
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ −5.00000 −0.198889
$$633$$ −10.0000 −0.397464
$$634$$ −12.0000 −0.476581
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 9.00000 0.356313
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 9.00000 0.355479 0.177739 0.984078i $$-0.443122\pi$$
0.177739 + 0.984078i $$0.443122\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 5.00000 0.197181 0.0985904 0.995128i $$-0.468567\pi$$
0.0985904 + 0.995128i $$0.468567\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ −12.0000 −0.471769 −0.235884 0.971781i $$-0.575799\pi$$
−0.235884 + 0.971781i $$0.575799\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 3.00000 0.117760
$$650$$ −5.00000 −0.196116
$$651$$ 0 0
$$652$$ 17.0000 0.665771
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ −4.00000 −0.156055
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ −35.0000 −1.36031
$$663$$ 6.00000 0.233021
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ −54.0000 −2.09089
$$668$$ 3.00000 0.116073
$$669$$ 26.0000 1.00522
$$670$$ 0 0
$$671$$ −11.0000 −0.424650
$$672$$ 0 0
$$673$$ −28.0000 −1.07932 −0.539660 0.841883i $$-0.681447\pi$$
−0.539660 + 0.841883i $$0.681447\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 25.0000 0.962250
$$676$$ −12.0000 −0.461538
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ −9.00000 −0.345643
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ −4.00000 −0.153168
$$683$$ −21.0000 −0.803543 −0.401771 0.915740i $$-0.631605\pi$$
−0.401771 + 0.915740i $$0.631605\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −16.0000 −0.610438
$$688$$ −4.00000 −0.152499
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 11.0000 0.418460 0.209230 0.977866i $$-0.432904\pi$$
0.209230 + 0.977866i $$0.432904\pi$$
$$692$$ −21.0000 −0.798300
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ −9.00000 −0.341144
$$697$$ 36.0000 1.36360
$$698$$ −2.00000 −0.0757011
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 39.0000 1.47301 0.736505 0.676432i $$-0.236475\pi$$
0.736505 + 0.676432i $$0.236475\pi$$
$$702$$ −5.00000 −0.188713
$$703$$ 4.00000 0.150863
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ −3.00000 −0.112747
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ 18.0000 0.674579
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −15.0000 −0.560576
$$717$$ −9.00000 −0.336111
$$718$$ 3.00000 0.111959
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 15.0000 0.558242
$$723$$ 26.0000 0.966950
$$724$$ 2.00000 0.0743294
$$725$$ −45.0000 −1.67126
$$726$$ −1.00000 −0.0371135
$$727$$ 14.0000 0.519231 0.259616 0.965712i $$-0.416404\pi$$
0.259616 + 0.965712i $$0.416404\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 11.0000 0.406572
$$733$$ −25.0000 −0.923396 −0.461698 0.887037i $$-0.652760\pi$$
−0.461698 + 0.887037i $$0.652760\pi$$
$$734$$ 10.0000 0.369107
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ −11.0000 −0.405190
$$738$$ −12.0000 −0.441726
$$739$$ 50.0000 1.83928 0.919640 0.392763i $$-0.128481\pi$$
0.919640 + 0.392763i $$0.128481\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 0 0
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ 31.0000 1.13499
$$747$$ 12.0000 0.439057
$$748$$ 6.00000 0.219382
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ −12.0000 −0.437304
$$754$$ 9.00000 0.327761
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −46.0000 −1.67190 −0.835949 0.548807i $$-0.815082\pi$$
−0.835949 + 0.548807i $$0.815082\pi$$
$$758$$ −23.0000 −0.835398
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 7.00000 0.253583
$$763$$ 0 0
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ 36.0000 1.30073
$$767$$ 3.00000 0.108324
$$768$$ 1.00000 0.0360844
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 3.00000 0.108042
$$772$$ 14.0000 0.503871
$$773$$ 24.0000 0.863220 0.431610 0.902060i $$-0.357946\pi$$
0.431610 + 0.902060i $$0.357946\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 20.0000 0.718421
$$776$$ 13.0000 0.466673
$$777$$ 0 0
$$778$$ −12.0000 −0.430221
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −36.0000 −1.28736
$$783$$ −45.0000 −1.60817
$$784$$ 0 0
$$785$$ 0 0
$$786$$ −18.0000 −0.642039
$$787$$ 14.0000 0.499046 0.249523 0.968369i $$-0.419726\pi$$
0.249523 + 0.968369i $$0.419726\pi$$
$$788$$ −3.00000 −0.106871
$$789$$ −9.00000 −0.320408
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −2.00000 −0.0710669
$$793$$ −11.0000 −0.390621
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ 14.0000 0.496217
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ 36.0000 1.27359
$$800$$ 5.00000 0.176777
$$801$$ 36.0000 1.27200
$$802$$ 33.0000 1.16527
$$803$$ −2.00000 −0.0705785
$$804$$ 11.0000 0.387940
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 12.0000 0.422420
$$808$$ 15.0000 0.527698
$$809$$ −48.0000 −1.68759 −0.843795 0.536666i $$-0.819684\pi$$
−0.843795 + 0.536666i $$0.819684\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ 29.0000 1.01707
$$814$$ 2.00000 0.0701000
$$815$$ 0 0
$$816$$ −6.00000 −0.210042
$$817$$ −8.00000 −0.279885
$$818$$ 4.00000 0.139857
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −15.0000 −0.523504 −0.261752 0.965135i $$-0.584300\pi$$
−0.261752 + 0.965135i $$0.584300\pi$$
$$822$$ 9.00000 0.313911
$$823$$ −10.0000 −0.348578 −0.174289 0.984695i $$-0.555763\pi$$
−0.174289 + 0.984695i $$0.555763\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 5.00000 0.174078
$$826$$ 0 0
$$827$$ 30.0000 1.04320 0.521601 0.853189i $$-0.325335\pi$$
0.521601 + 0.853189i $$0.325335\pi$$
$$828$$ 12.0000 0.417029
$$829$$ −52.0000 −1.80603 −0.903017 0.429604i $$-0.858653\pi$$
−0.903017 + 0.429604i $$0.858653\pi$$
$$830$$ 0 0
$$831$$ 29.0000 1.00600
$$832$$ −1.00000 −0.0346688
$$833$$ 0 0
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ 20.0000 0.691301
$$838$$ 12.0000 0.414533
$$839$$ 6.00000 0.207143 0.103572 0.994622i $$-0.466973\pi$$
0.103572 + 0.994622i $$0.466973\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 28.0000 0.964944
$$843$$ 18.0000 0.619953
$$844$$ −10.0000 −0.344214
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −10.0000 −0.343199
$$850$$ −30.0000 −1.02899
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 36.0000 1.22974 0.614868 0.788630i $$-0.289209\pi$$
0.614868 + 0.788630i $$0.289209\pi$$
$$858$$ −1.00000 −0.0341394
$$859$$ −37.0000 −1.26242 −0.631212 0.775610i $$-0.717442\pi$$
−0.631212 + 0.775610i $$0.717442\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 3.00000 0.102180
$$863$$ 30.0000 1.02121 0.510606 0.859815i $$-0.329421\pi$$
0.510606 + 0.859815i $$0.329421\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 19.0000 0.645274
$$868$$ 0 0
$$869$$ −5.00000 −0.169613
$$870$$ 0 0
$$871$$ −11.0000 −0.372721
$$872$$ −2.00000 −0.0677285
$$873$$ 26.0000 0.879967
$$874$$ 12.0000 0.405906
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ 17.0000 0.574049 0.287025 0.957923i $$-0.407334\pi$$
0.287025 + 0.957923i $$0.407334\pi$$
$$878$$ 1.00000 0.0337484
$$879$$ −30.0000 −1.01187
$$880$$ 0 0
$$881$$ 33.0000 1.11180 0.555899 0.831250i $$-0.312374\pi$$
0.555899 + 0.831250i $$0.312374\pi$$
$$882$$ 0 0
$$883$$ 29.0000 0.975928 0.487964 0.872864i $$-0.337740\pi$$
0.487964 + 0.872864i $$0.337740\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 3.00000 0.100730 0.0503651 0.998731i $$-0.483962\pi$$
0.0503651 + 0.998731i $$0.483962\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 26.0000 0.870544
$$893$$ −12.0000 −0.401565
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ 18.0000 0.600668
$$899$$ −36.0000 −1.20067
$$900$$ 10.0000 0.333333
$$901$$ 0 0
$$902$$ −6.00000 −0.199778
$$903$$ 0 0
$$904$$ −9.00000 −0.299336
$$905$$ 0 0
$$906$$ 19.0000 0.631233
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 18.0000 0.597351
$$909$$ 30.0000 0.995037
$$910$$ 0 0
$$911$$ −6.00000 −0.198789 −0.0993944 0.995048i $$-0.531691\pi$$
−0.0993944 + 0.995048i $$0.531691\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ 6.00000 0.198571
$$914$$ 22.0000 0.727695
$$915$$ 0 0
$$916$$ −16.0000 −0.528655
$$917$$ 0 0
$$918$$ −30.0000 −0.990148
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ −21.0000 −0.691598
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ 22.0000 0.722965
$$927$$ 32.0000 1.05102
$$928$$ −9.00000 −0.295439
$$929$$ 39.0000 1.27955 0.639774 0.768563i $$-0.279028\pi$$
0.639774 + 0.768563i $$0.279028\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −6.00000 −0.196537
$$933$$ 24.0000 0.785725
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ 8.00000 0.261349 0.130674 0.991425i $$-0.458286\pi$$
0.130674 + 0.991425i $$0.458286\pi$$
$$938$$ 0 0
$$939$$ 17.0000 0.554774
$$940$$ 0 0
$$941$$ 15.0000 0.488986 0.244493 0.969651i $$-0.421378\pi$$
0.244493 + 0.969651i $$0.421378\pi$$
$$942$$ 4.00000 0.130327
$$943$$ 36.0000 1.17232
$$944$$ −3.00000 −0.0976417
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ 24.0000 0.779895 0.389948 0.920837i $$-0.372493\pi$$
0.389948 + 0.920837i $$0.372493\pi$$
$$948$$ 5.00000 0.162392
$$949$$ −2.00000 −0.0649227
$$950$$ 10.0000 0.324443
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ −36.0000 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −9.00000 −0.291081
$$957$$ −9.00000 −0.290929
$$958$$ −15.0000 −0.484628
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 2.00000 0.0644826
$$963$$ −24.0000 −0.773389
$$964$$ 26.0000 0.837404
$$965$$ 0 0
$$966$$ 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
$$969$$ −12.0000 −0.385496
$$970$$ 0 0
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 16.0000 0.513200
$$973$$ 0 0
$$974$$ −20.0000 −0.640841
$$975$$ 5.00000 0.160128
$$976$$ 11.0000 0.352101
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ −17.0000 −0.543600
$$979$$ 18.0000 0.575282
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ −42.0000 −1.34027
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 54.0000 1.71971
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 35.0000 1.11069
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −6.00000 −0.190117
$$997$$ −34.0000 −1.07679 −0.538395 0.842692i $$-0.680969\pi$$
−0.538395 + 0.842692i $$0.680969\pi$$
$$998$$ 40.0000 1.26618
$$999$$ −10.0000 −0.316386
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 1078.2.a.e.1.1 1
3.2 odd 2 9702.2.a.br.1.1 1
4.3 odd 2 8624.2.a.k.1.1 1
7.2 even 3 154.2.e.c.67.1 yes 2
7.3 odd 6 1078.2.e.k.177.1 2
7.4 even 3 154.2.e.c.23.1 2
7.5 odd 6 1078.2.e.k.67.1 2
7.6 odd 2 1078.2.a.c.1.1 1
21.2 odd 6 1386.2.k.e.991.1 2
21.11 odd 6 1386.2.k.e.793.1 2
21.20 even 2 9702.2.a.bs.1.1 1
28.11 odd 6 1232.2.q.d.177.1 2
28.23 odd 6 1232.2.q.d.529.1 2
28.27 even 2 8624.2.a.u.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.c.23.1 2 7.4 even 3
154.2.e.c.67.1 yes 2 7.2 even 3
1078.2.a.c.1.1 1 7.6 odd 2
1078.2.a.e.1.1 1 1.1 even 1 trivial
1078.2.e.k.67.1 2 7.5 odd 6
1078.2.e.k.177.1 2 7.3 odd 6
1232.2.q.d.177.1 2 28.11 odd 6
1232.2.q.d.529.1 2 28.23 odd 6
1386.2.k.e.793.1 2 21.11 odd 6
1386.2.k.e.991.1 2 21.2 odd 6
8624.2.a.k.1.1 1 4.3 odd 2
8624.2.a.u.1.1 1 28.27 even 2
9702.2.a.br.1.1 1 3.2 odd 2
9702.2.a.bs.1.1 1 21.20 even 2