# Properties

 Label 234.2.a.c.1.1 Level $234$ Weight $2$ Character 234.1 Self dual yes Analytic conductor $1.868$ 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] = [234,2,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 234.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$1.86849940730$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -8.00000 q^{19} -2.00000 q^{20} +4.00000 q^{22} -1.00000 q^{25} +1.00000 q^{26} +4.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -8.00000 q^{35} -2.00000 q^{37} -8.00000 q^{38} -2.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} +4.00000 q^{44} -8.00000 q^{47} +9.00000 q^{49} -1.00000 q^{50} +1.00000 q^{52} +10.0000 q^{53} -8.00000 q^{55} +4.00000 q^{56} -6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -16.0000 q^{67} -2.00000 q^{68} -8.00000 q^{70} +8.00000 q^{71} +2.00000 q^{73} -2.00000 q^{74} -8.00000 q^{76} +16.0000 q^{77} +8.00000 q^{79} -2.00000 q^{80} +10.0000 q^{82} -12.0000 q^{83} +4.00000 q^{85} +4.00000 q^{86} +4.00000 q^{88} -14.0000 q^{89} +4.00000 q^{91} -8.00000 q^{94} +16.0000 q^{95} +10.0000 q^{97} +9.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −2.00000 −0.632456
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 4.00000 0.755929
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ −8.00000 −1.35225
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −8.00000 −1.29777
$$39$$ 0 0
$$40$$ −2.00000 −0.316228
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ −8.00000 −1.07872
$$56$$ 4.00000 0.534522
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −16.0000 −1.95471 −0.977356 0.211604i $$-0.932131\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ −8.00000 −0.956183
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −8.00000 −0.917663
$$77$$ 16.0000 1.82337
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 0 0
$$82$$ 10.0000 1.10432
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 16.0000 1.64157
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −8.00000 −0.762770
$$111$$ 0 0
$$112$$ 4.00000 0.377964
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −2.00000 −0.175412
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ −32.0000 −2.77475
$$134$$ −16.0000 −1.38219
$$135$$ 0 0
$$136$$ −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
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ −8.00000 −0.676123
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 12.0000 0.996546
$$146$$ 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 $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ −8.00000 −0.648886
$$153$$ 0 0
$$154$$ 16.0000 1.28932
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ −2.00000 −0.158114
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 10.0000 0.760286 0.380143 0.924928i $$-0.375875\pi$$
0.380143 + 0.924928i $$0.375875\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ −14.0000 −1.04934
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 16.0000 1.16076
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ −18.0000 −1.28245 −0.641223 0.767354i $$-0.721573\pi$$
−0.641223 + 0.767354i $$0.721573\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ −24.0000 −1.68447
$$204$$ 0 0
$$205$$ −20.0000 −1.39686
$$206$$ 16.0000 1.11477
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −32.0000 −2.21349
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ −2.00000 −0.135457
$$219$$ 0 0
$$220$$ −8.00000 −0.539360
$$221$$ −2.00000 −0.134535
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 16.0000 1.04372
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ −8.00000 −0.518563
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ −18.0000 −1.14998
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ 12.0000 0.758947
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ −8.00000 −0.497096
$$260$$ −2.00000 −0.124035
$$261$$ 0 0
$$262$$ −4.00000 −0.247121
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ −20.0000 −1.22859
$$266$$ −32.0000 −1.96205
$$267$$ 0 0
$$268$$ −16.0000 −0.977356
$$269$$ 26.0000 1.58525 0.792624 0.609711i $$-0.208714\pi$$
0.792624 + 0.609711i $$0.208714\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 0 0
$$280$$ −8.00000 −0.478091
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 40.0000 2.36113
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 12.0000 0.704664
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ −26.0000 −1.51894 −0.759468 0.650545i $$-0.774541\pi$$
−0.759468 + 0.650545i $$0.774541\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 12.0000 0.690522
$$303$$ 0 0
$$304$$ −8.00000 −0.458831
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 0 0
$$310$$ 8.00000 0.454369
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ −24.0000 −1.34374
$$320$$ −2.00000 −0.111803
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 16.0000 0.890264
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ −16.0000 −0.886158
$$327$$ 0 0
$$328$$ 10.0000 0.552158
$$329$$ −32.0000 −1.76422
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 32.0000 1.74835
$$336$$ 0 0
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 10.0000 0.537603
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ −16.0000 −0.849192
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ −4.00000 −0.209370
$$366$$ 0 0
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 4.00000 0.207950
$$371$$ 40.0000 2.07670
$$372$$ 0 0
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ −6.00000 −0.309016
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 16.0000 0.820783
$$381$$ 0 0
$$382$$ 8.00000 0.409316
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ −32.0000 −1.63087
$$386$$ −14.0000 −0.712581
$$387$$ 0 0
$$388$$ 10.0000 0.507673
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 9.00000 0.454569
$$393$$ 0 0
$$394$$ −18.0000 −0.906827
$$395$$ −16.0000 −0.805047
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ −24.0000 −1.19110
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ −20.0000 −0.987730
$$411$$ 0 0
$$412$$ 16.0000 0.788263
$$413$$ −16.0000 −0.787309
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −32.0000 −1.56517
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ −8.00000 −0.387147
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ −30.0000 −1.44171 −0.720854 0.693087i $$-0.756250\pi$$
−0.720854 + 0.693087i $$0.756250\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ −8.00000 −0.381385
$$441$$ 0 0
$$442$$ −2.00000 −0.0951303
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 0 0
$$445$$ 28.0000 1.32733
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ 4.00000 0.188982
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 40.0000 1.88353
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ −8.00000 −0.375046
$$456$$ 0 0
$$457$$ −30.0000 −1.40334 −0.701670 0.712502i $$-0.747562\pi$$
−0.701670 + 0.712502i $$0.747562\pi$$
$$458$$ 22.0000 1.02799
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 4.00000 0.185098 0.0925490 0.995708i $$-0.470499\pi$$
0.0925490 + 0.995708i $$0.470499\pi$$
$$468$$ 0 0
$$469$$ −64.0000 −2.95525
$$470$$ 16.0000 0.738025
$$471$$ 0 0
$$472$$ −4.00000 −0.184115
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ 8.00000 0.367065
$$476$$ −8.00000 −0.366679
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −20.0000 −0.908153
$$486$$ 0 0
$$487$$ 4.00000 0.181257 0.0906287 0.995885i $$-0.471112\pi$$
0.0906287 + 0.995885i $$0.471112\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 0 0
$$490$$ −18.0000 −0.813157
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 32.0000 1.43540
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 0 0
$$502$$ −4.00000 −0.178529
$$503$$ −40.0000 −1.78351 −0.891756 0.452517i $$-0.850526\pi$$
−0.891756 + 0.452517i $$0.850526\pi$$
$$504$$ 0 0
$$505$$ −4.00000 −0.177998
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −42.0000 −1.86162 −0.930809 0.365507i $$-0.880896\pi$$
−0.930809 + 0.365507i $$0.880896\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ −32.0000 −1.41009
$$516$$ 0 0
$$517$$ −32.0000 −1.40736
$$518$$ −8.00000 −0.351500
$$519$$ 0 0
$$520$$ −2.00000 −0.0877058
$$521$$ 14.0000 0.613351 0.306676 0.951814i $$-0.400783\pi$$
0.306676 + 0.951814i $$0.400783\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 8.00000 0.348485
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ −20.0000 −0.868744
$$531$$ 0 0
$$532$$ −32.0000 −1.38738
$$533$$ 10.0000 0.433148
$$534$$ 0 0
$$535$$ 24.0000 1.03761
$$536$$ −16.0000 −0.691095
$$537$$ 0 0
$$538$$ 26.0000 1.12094
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ −4.00000 −0.171815
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ 4.00000 0.171341
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 10.0000 0.427179
$$549$$ 0 0
$$550$$ −4.00000 −0.170561
$$551$$ 48.0000 2.04487
$$552$$ 0 0
$$553$$ 32.0000 1.36078
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ −8.00000 −0.338062
$$561$$ 0 0
$$562$$ 26.0000 1.09674
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ −12.0000 −0.504844
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ 8.00000 0.335673
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ 40.0000 1.66957
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 12.0000 0.498273
$$581$$ −48.0000 −1.99138
$$582$$ 0 0
$$583$$ 40.0000 1.65663
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ −4.00000 −0.165098 −0.0825488 0.996587i $$-0.526306\pi$$
−0.0825488 + 0.996587i $$0.526306\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 8.00000 0.329355
$$591$$ 0 0
$$592$$ −2.00000 −0.0821995
$$593$$ 42.0000 1.72473 0.862367 0.506284i $$-0.168981\pi$$
0.862367 + 0.506284i $$0.168981\pi$$
$$594$$ 0 0
$$595$$ 16.0000 0.655936
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 16.0000 0.652111
$$603$$ 0 0
$$604$$ 12.0000 0.488273
$$605$$ −10.0000 −0.406558
$$606$$ 0 0
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ −8.00000 −0.324443
$$609$$ 0 0
$$610$$ 4.00000 0.161955
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ 16.0000 0.644658
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 32.0000 1.28619 0.643094 0.765787i $$-0.277650\pi$$
0.643094 + 0.765787i $$0.277650\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −56.0000 −2.24359
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −6.00000 −0.239808
$$627$$ 0 0
$$628$$ 14.0000 0.558661
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ −36.0000 −1.43314 −0.716569 0.697517i $$-0.754288\pi$$
−0.716569 + 0.697517i $$0.754288\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 0 0
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 9.00000 0.356593
$$638$$ −24.0000 −0.950169
$$639$$ 0 0
$$640$$ −2.00000 −0.0790569
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 16.0000 0.629512
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ −1.00000 −0.0392232
$$651$$ 0 0
$$652$$ −16.0000 −0.626608
$$653$$ 10.0000 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$654$$ 0 0
$$655$$ 8.00000 0.312586
$$656$$ 10.0000 0.390434
$$657$$ 0 0
$$658$$ −32.0000 −1.24749
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ 8.00000 0.310929
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 64.0000 2.48181
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 32.0000 1.23627
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −38.0000 −1.46046 −0.730229 0.683202i $$-0.760587\pi$$
−0.730229 + 0.683202i $$0.760587\pi$$
$$678$$ 0 0
$$679$$ 40.0000 1.53506
$$680$$ 4.00000 0.153393
$$681$$ 0 0
$$682$$ −16.0000 −0.612672
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 0 0
$$685$$ −20.0000 −0.764161
$$686$$ 8.00000 0.305441
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 10.0000 0.380970
$$690$$ 0 0
$$691$$ −32.0000 −1.21734 −0.608669 0.793424i $$-0.708296\pi$$
−0.608669 + 0.793424i $$0.708296\pi$$
$$692$$ 10.0000 0.380143
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ −24.0000 −0.910372
$$696$$ 0 0
$$697$$ −20.0000 −0.757554
$$698$$ 6.00000 0.227103
$$699$$ 0 0
$$700$$ −4.00000 −0.151186
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ 0 0
$$703$$ 16.0000 0.603451
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ 8.00000 0.300871
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ 0 0
$$712$$ −14.0000 −0.524672
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −8.00000 −0.299183
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 64.0000 2.38348
$$722$$ 45.0000 1.67473
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ −40.0000 −1.48352 −0.741759 0.670667i $$-0.766008\pi$$
−0.741759 + 0.670667i $$0.766008\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 0 0
$$730$$ −4.00000 −0.148047
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −64.0000 −2.35747
$$738$$ 0 0
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ 40.0000 1.46845
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ −12.0000 −0.439646
$$746$$ 6.00000 0.219676
$$747$$ 0 0
$$748$$ −8.00000 −0.292509
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ −6.00000 −0.218507
$$755$$ −24.0000 −0.873449
$$756$$ 0 0
$$757$$ 54.0000 1.96266 0.981332 0.192323i $$-0.0616021\pi$$
0.981332 + 0.192323i $$0.0616021\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 16.0000 0.580381
$$761$$ 26.0000 0.942499 0.471250 0.882000i $$-0.343803\pi$$
0.471250 + 0.882000i $$0.343803\pi$$
$$762$$ 0 0
$$763$$ −8.00000 −0.289619
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ −4.00000 −0.144432
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ −32.0000 −1.15320
$$771$$ 0 0
$$772$$ −14.0000 −0.503871
$$773$$ 54.0000 1.94225 0.971123 0.238581i $$-0.0766824\pi$$
0.971123 + 0.238581i $$0.0766824\pi$$
$$774$$ 0 0
$$775$$ 4.00000 0.143684
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 26.0000 0.932145
$$779$$ −80.0000 −2.86630
$$780$$ 0 0
$$781$$ 32.0000 1.14505
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 9.00000 0.321429
$$785$$ −28.0000 −0.999363
$$786$$ 0 0
$$787$$ 40.0000 1.42585 0.712923 0.701242i $$-0.247371\pi$$
0.712923 + 0.701242i $$0.247371\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ 0 0
$$790$$ −16.0000 −0.569254
$$791$$ 24.0000 0.853342
$$792$$ 0 0
$$793$$ −2.00000 −0.0710221
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −6.00000 −0.211867
$$803$$ 8.00000 0.282314
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 2.00000 0.0703598
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ −24.0000 −0.842235
$$813$$ 0 0
$$814$$ −8.00000 −0.280400
$$815$$ 32.0000 1.12091
$$816$$ 0 0
$$817$$ −32.0000 −1.11954
$$818$$ 2.00000 0.0699284
$$819$$ 0 0
$$820$$ −20.0000 −0.698430
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 0 0
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 24.0000 0.833052
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −32.0000 −1.10674
$$837$$ 0 0
$$838$$ −4.00000 −0.138178
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 22.0000 0.758170
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ −2.00000 −0.0688021
$$846$$ 0 0
$$847$$ 20.0000 0.687208
$$848$$ 10.0000 0.343401
$$849$$ 0 0
$$850$$ 2.00000 0.0685994
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −2.00000 −0.0684787 −0.0342393 0.999414i $$-0.510901\pi$$
−0.0342393 + 0.999414i $$0.510901\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ 8.00000 0.272481
$$863$$ −40.0000 −1.36162 −0.680808 0.732462i $$-0.738371\pi$$
−0.680808 + 0.732462i $$0.738371\pi$$
$$864$$ 0 0
$$865$$ −20.0000 −0.680020
$$866$$ −30.0000 −1.01944
$$867$$ 0 0
$$868$$ −16.0000 −0.543075
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ −2.00000 −0.0677285
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 48.0000 1.62270
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 16.0000 0.539974
$$879$$ 0 0
$$880$$ −8.00000 −0.269680
$$881$$ −26.0000 −0.875962 −0.437981 0.898984i $$-0.644306\pi$$
−0.437981 + 0.898984i $$0.644306\pi$$
$$882$$ 0 0
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 28.0000 0.938562
$$891$$ 0 0
$$892$$ −4.00000 −0.133930
$$893$$ 64.0000 2.14168
$$894$$ 0 0
$$895$$ −24.0000 −0.802232
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ 40.0000 1.33185
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 20.0000 0.664822
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ 0 0
$$910$$ −8.00000 −0.265197
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ −30.0000 −0.992312
$$915$$ 0 0
$$916$$ 22.0000 0.726900
$$917$$ −16.0000 −0.528367
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 6.00000 0.197599
$$923$$ 8.00000 0.263323
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ −20.0000 −0.657241
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ −46.0000 −1.50921 −0.754606 0.656179i $$-0.772172\pi$$
−0.754606 + 0.656179i $$0.772172\pi$$
$$930$$ 0 0
$$931$$ −72.0000 −2.35970
$$932$$ −18.0000 −0.589610
$$933$$ 0 0
$$934$$ 4.00000 0.130884
$$935$$ 16.0000 0.523256
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ −64.0000 −2.08967
$$939$$ 0 0
$$940$$ 16.0000 0.521862
$$941$$ 46.0000 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 0 0
$$949$$ 2.00000 0.0649227
$$950$$ 8.00000 0.259554
$$951$$ 0 0
$$952$$ −8.00000 −0.259281
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 0 0
$$955$$ −16.0000 −0.517748
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 16.0000 0.516937
$$959$$ 40.0000 1.29167
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −2.00000 −0.0644826
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 28.0000 0.901352
$$966$$ 0 0
$$967$$ 4.00000 0.128631 0.0643157 0.997930i $$-0.479514\pi$$
0.0643157 + 0.997930i $$0.479514\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ −20.0000 −0.642161
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ 0 0
$$973$$ 48.0000 1.53881
$$974$$ 4.00000 0.128168
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ 0 0
$$979$$ −56.0000 −1.78977
$$980$$ −18.0000 −0.574989
$$981$$ 0 0
$$982$$ −36.0000 −1.14881
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ 36.0000 1.14706
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −48.0000 −1.52477 −0.762385 0.647124i $$-0.775972\pi$$
−0.762385 + 0.647124i $$0.775972\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ 32.0000 1.01498
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 234.2.a.c.1.1 1
3.2 odd 2 78.2.a.a.1.1 1
4.3 odd 2 1872.2.a.c.1.1 1
5.2 odd 4 5850.2.e.bb.5149.2 2
5.3 odd 4 5850.2.e.bb.5149.1 2
5.4 even 2 5850.2.a.d.1.1 1
8.3 odd 2 7488.2.a.bk.1.1 1
8.5 even 2 7488.2.a.bz.1.1 1
9.2 odd 6 2106.2.e.q.1405.1 2
9.4 even 3 2106.2.e.j.703.1 2
9.5 odd 6 2106.2.e.q.703.1 2
9.7 even 3 2106.2.e.j.1405.1 2
12.11 even 2 624.2.a.h.1.1 1
13.5 odd 4 3042.2.b.g.1351.1 2
13.8 odd 4 3042.2.b.g.1351.2 2
13.12 even 2 3042.2.a.f.1.1 1
15.2 even 4 1950.2.e.i.1249.1 2
15.8 even 4 1950.2.e.i.1249.2 2
15.14 odd 2 1950.2.a.w.1.1 1
21.20 even 2 3822.2.a.j.1.1 1
24.5 odd 2 2496.2.a.t.1.1 1
24.11 even 2 2496.2.a.b.1.1 1
33.32 even 2 9438.2.a.t.1.1 1
39.2 even 12 1014.2.i.d.823.1 4
39.5 even 4 1014.2.b.b.337.2 2
39.8 even 4 1014.2.b.b.337.1 2
39.11 even 12 1014.2.i.d.823.2 4
39.17 odd 6 1014.2.e.c.991.1 2
39.20 even 12 1014.2.i.d.361.1 4
39.23 odd 6 1014.2.e.c.529.1 2
39.29 odd 6 1014.2.e.f.529.1 2
39.32 even 12 1014.2.i.d.361.2 4
39.35 odd 6 1014.2.e.f.991.1 2
39.38 odd 2 1014.2.a.d.1.1 1
156.155 even 2 8112.2.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.a.a.1.1 1 3.2 odd 2
234.2.a.c.1.1 1 1.1 even 1 trivial
624.2.a.h.1.1 1 12.11 even 2
1014.2.a.d.1.1 1 39.38 odd 2
1014.2.b.b.337.1 2 39.8 even 4
1014.2.b.b.337.2 2 39.5 even 4
1014.2.e.c.529.1 2 39.23 odd 6
1014.2.e.c.991.1 2 39.17 odd 6
1014.2.e.f.529.1 2 39.29 odd 6
1014.2.e.f.991.1 2 39.35 odd 6
1014.2.i.d.361.1 4 39.20 even 12
1014.2.i.d.361.2 4 39.32 even 12
1014.2.i.d.823.1 4 39.2 even 12
1014.2.i.d.823.2 4 39.11 even 12
1872.2.a.c.1.1 1 4.3 odd 2
1950.2.a.w.1.1 1 15.14 odd 2
1950.2.e.i.1249.1 2 15.2 even 4
1950.2.e.i.1249.2 2 15.8 even 4
2106.2.e.j.703.1 2 9.4 even 3
2106.2.e.j.1405.1 2 9.7 even 3
2106.2.e.q.703.1 2 9.5 odd 6
2106.2.e.q.1405.1 2 9.2 odd 6
2496.2.a.b.1.1 1 24.11 even 2
2496.2.a.t.1.1 1 24.5 odd 2
3042.2.a.f.1.1 1 13.12 even 2
3042.2.b.g.1351.1 2 13.5 odd 4
3042.2.b.g.1351.2 2 13.8 odd 4
3822.2.a.j.1.1 1 21.20 even 2
5850.2.a.d.1.1 1 5.4 even 2
5850.2.e.bb.5149.1 2 5.3 odd 4
5850.2.e.bb.5149.2 2 5.2 odd 4
7488.2.a.bk.1.1 1 8.3 odd 2
7488.2.a.bz.1.1 1 8.5 even 2
8112.2.a.v.1.1 1 156.155 even 2
9438.2.a.t.1.1 1 33.32 even 2