# Properties

 Label 102.2.a.b.1.1 Level $102$ Weight $2$ Character 102.1 Self dual yes Analytic conductor $0.814$ 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] = [102,2,Mod(1,102)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(102, 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("102.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$102 = 2 \cdot 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 102.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: $$0.814474100617$$ 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 $$\chi$$ $$=$$ 102.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} -6.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -10.0000 q^{31} -1.00000 q^{32} +1.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} +6.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +5.00000 q^{50} -1.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -2.00000 q^{56} -4.00000 q^{57} -12.0000 q^{59} +8.00000 q^{61} +10.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} -1.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -8.00000 q^{74} -5.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -18.0000 q^{89} +4.00000 q^{91} -6.00000 q^{92} -10.0000 q^{93} -12.0000 q^{94} -1.00000 q^{96} +14.0000 q^{97} +3.00000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ −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
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 5.00000 0.707107
$$51$$ −1.00000 −0.140028
$$52$$ 2.00000 0.277350
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 10.0000 1.27000
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ −5.00000 −0.577350
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −18.0000 −1.90800 −0.953998 0.299813i $$-0.903076\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ −6.00000 −0.625543
$$93$$ −10.0000 −1.03695
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 20.0000 1.91565 0.957826 0.287348i $$-0.0927736\pi$$
0.957826 + 0.287348i $$0.0927736\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ 2.00000 0.188982
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 12.0000 1.10469
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ −8.00000 −0.724286
$$123$$ 6.00000 0.541002
$$124$$ −10.0000 −0.898027
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 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
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −8.00000 −0.693688
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 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
$$141$$ 12.0000 1.01058
$$142$$ −6.00000 −0.503509
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ −3.00000 −0.247436
$$148$$ 8.00000 0.657596
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 5.00000 0.408248
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 4.00000 0.324443
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 10.0000 0.795557
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ −1.00000 −0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ −4.00000 −0.304997
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ −10.0000 −0.755929
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 18.0000 1.34916
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 8.00000 0.594635 0.297318 0.954779i $$-0.403908\pi$$
0.297318 + 0.954779i $$0.403908\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 8.00000 0.591377
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 10.0000 0.733236
$$187$$ 0 0
$$188$$ 12.0000 0.875190
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 5.00000 0.353553
$$201$$ −4.00000 −0.282138
$$202$$ −6.00000 −0.422159
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ −6.00000 −0.417029
$$208$$ 2.00000 0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 6.00000 0.411113
$$214$$ 0 0
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ −20.0000 −1.35769
$$218$$ −20.0000 −1.35457
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ −8.00000 −0.536925
$$223$$ −28.0000 −1.87502 −0.937509 0.347960i $$-0.886874\pi$$
−0.937509 + 0.347960i $$0.886874\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ −5.00000 −0.333333
$$226$$ 6.00000 0.399114
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −30.0000 −1.96537 −0.982683 0.185296i $$-0.940675\pi$$
−0.982683 + 0.185296i $$0.940675\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ −10.0000 −0.649570
$$238$$ 2.00000 0.129641
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 1.00000 0.0641500
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ −8.00000 −0.509028
$$248$$ 10.0000 0.635001
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 16.0000 0.994192
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ −18.0000 −1.10158
$$268$$ −4.00000 −0.244339
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 4.00000 0.242091
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ −4.00000 −0.240337 −0.120168 0.992754i $$-0.538343\pi$$
−0.120168 + 0.992754i $$0.538343\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ −12.0000 −0.714590
$$283$$ −16.0000 −0.951101 −0.475551 0.879688i $$-0.657751\pi$$
−0.475551 + 0.879688i $$0.657751\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ −1.00000 −0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 2.00000 0.117041
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ −12.0000 −0.693978
$$300$$ −5.00000 −0.288675
$$301$$ −8.00000 −0.461112
$$302$$ −8.00000 −0.460348
$$303$$ 6.00000 0.344691
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 12.0000 0.668734
$$323$$ 4.00000 0.222566
$$324$$ 1.00000 0.0555556
$$325$$ −10.0000 −0.554700
$$326$$ −20.0000 −1.10770
$$327$$ 20.0000 1.10600
$$328$$ −6.00000 −0.331295
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 8.00000 0.438397
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000 0.216295
$$343$$ −20.0000 −1.07990
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 10.0000 0.534522
$$351$$ 2.00000 0.106752
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ −2.00000 −0.105851
$$358$$ 12.0000 0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −8.00000 −0.420471
$$363$$ −11.0000 −0.577350
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ −8.00000 −0.418167
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ −10.0000 −0.518476
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 12.0000 0.613973
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −4.00000 −0.203331
$$388$$ 14.0000 0.710742
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ 3.00000 0.151523
$$393$$ 0 0
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −16.0000 −0.803017 −0.401508 0.915855i $$-0.631514\pi$$
−0.401508 + 0.915855i $$0.631514\pi$$
$$398$$ −2.00000 −0.100251
$$399$$ −8.00000 −0.400501
$$400$$ −5.00000 −0.250000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 4.00000 0.199502
$$403$$ −20.0000 −0.996271
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 1.00000 0.0495074
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ −4.00000 −0.197066
$$413$$ −24.0000 −1.18096
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ −22.0000 −1.07221 −0.536107 0.844150i $$-0.680106\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ 12.0000 0.583460
$$424$$ −6.00000 −0.291386
$$425$$ 5.00000 0.242536
$$426$$ −6.00000 −0.290701
$$427$$ 16.0000 0.774294
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 14.0000 0.672797 0.336399 0.941720i $$-0.390791\pi$$
0.336399 + 0.941720i $$0.390791\pi$$
$$434$$ 20.0000 0.960031
$$435$$ 0 0
$$436$$ 20.0000 0.957826
$$437$$ 24.0000 1.14808
$$438$$ −2.00000 −0.0955637
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 2.00000 0.0951303
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 0 0
$$446$$ 28.0000 1.32584
$$447$$ −6.00000 −0.283790
$$448$$ 2.00000 0.0944911
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 5.00000 0.235702
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ 8.00000 0.375873
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ 20.0000 0.929479 0.464739 0.885448i $$-0.346148\pi$$
0.464739 + 0.885448i $$0.346148\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 30.0000 1.38972
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ 12.0000 0.552345
$$473$$ 0 0
$$474$$ 10.0000 0.459315
$$475$$ 20.0000 0.917663
$$476$$ −2.00000 −0.0916698
$$477$$ 6.00000 0.274721
$$478$$ 12.0000 0.548867
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 10.0000 0.455488
$$483$$ −12.0000 −0.546019
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −10.0000 −0.453143 −0.226572 0.973995i $$-0.572752\pi$$
−0.226572 + 0.973995i $$0.572752\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 0 0
$$494$$ 8.00000 0.359937
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ 12.0000 0.538274
$$498$$ −12.0000 −0.537733
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ 0 0
$$501$$ 18.0000 0.804181
$$502$$ −12.0000 −0.535586
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 8.00000 0.354943
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ −16.0000 −0.703000
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ −10.0000 −0.436436
$$526$$ −12.0000 −0.523225
$$527$$ 10.0000 0.435607
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ −8.00000 −0.346844
$$533$$ 12.0000 0.519778
$$534$$ 18.0000 0.778936
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ −12.0000 −0.517838
$$538$$ −24.0000 −1.03471
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16.0000 −0.687894 −0.343947 0.938989i $$-0.611764\pi$$
−0.343947 + 0.938989i $$0.611764\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 8.00000 0.343313
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ −4.00000 −0.171184
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 8.00000 0.341432
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 6.00000 0.255377
$$553$$ −20.0000 −0.850487
$$554$$ 4.00000 0.169944
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 10.0000 0.423334
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −30.0000 −1.26547
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ 2.00000 0.0839921
$$568$$ −6.00000 −0.251754
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ −12.0000 −0.500870
$$575$$ 30.0000 1.25109
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ −14.0000 −0.580319
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ 40.0000 1.64817
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 8.00000 0.328798
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 2.00000 0.0818546
$$598$$ 12.0000 0.490716
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 5.00000 0.204124
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 8.00000 0.326056
$$603$$ −4.00000 −0.162893
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ −10.0000 −0.405887 −0.202944 0.979190i $$-0.565051\pi$$
−0.202944 + 0.979190i $$0.565051\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ −1.00000 −0.0404226
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 4.00000 0.160904
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ −18.0000 −0.721734
$$623$$ −36.0000 −1.44231
$$624$$ 2.00000 0.0800641
$$625$$ 25.0000 1.00000
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ −10.0000 −0.399043
$$629$$ −8.00000 −0.318981
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ 10.0000 0.397779
$$633$$ 8.00000 0.317971
$$634$$ −12.0000 −0.476581
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 10.0000 0.392232
$$651$$ −20.0000 −0.783862
$$652$$ 20.0000 0.783260
$$653$$ −24.0000 −0.939193 −0.469596 0.882881i $$-0.655601\pi$$
−0.469596 + 0.882881i $$0.655601\pi$$
$$654$$ −20.0000 −0.782062
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 2.00000 0.0780274
$$658$$ −24.0000 −0.935617
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ −2.00000 −0.0776736
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ 0 0
$$668$$ 18.0000 0.696441
$$669$$ −28.0000 −1.08254
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −2.00000 −0.0771517
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ 22.0000 0.847408
$$675$$ −5.00000 −0.192450
$$676$$ −9.00000 −0.346154
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 28.0000 1.07454
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 14.0000 0.534133
$$688$$ −4.00000 −0.152499
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −6.00000 −0.227266
$$698$$ −26.0000 −0.984115
$$699$$ −30.0000 −1.13470
$$700$$ −10.0000 −0.377964
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −32.0000 −1.20690
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 12.0000 0.451306
$$708$$ −12.0000 −0.450988
$$709$$ 20.0000 0.751116 0.375558 0.926799i $$-0.377451\pi$$
0.375558 + 0.926799i $$0.377451\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ 18.0000 0.674579
$$713$$ 60.0000 2.24702
$$714$$ 2.00000 0.0748481
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ −12.0000 −0.448148
$$718$$ 0 0
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 3.00000 0.111648
$$723$$ −10.0000 −0.371904
$$724$$ 8.00000 0.297318
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ −4.00000 −0.148250
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 8.00000 0.295689
$$733$$ 26.0000 0.960332 0.480166 0.877178i $$-0.340576\pi$$
0.480166 + 0.877178i $$0.340576\pi$$
$$734$$ 10.0000 0.369107
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ 0 0
$$738$$ −6.00000 −0.220863
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ −12.0000 −0.440534
$$743$$ 42.0000 1.54083 0.770415 0.637542i $$-0.220049\pi$$
0.770415 + 0.637542i $$0.220049\pi$$
$$744$$ 10.0000 0.366618
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2.00000 0.0729810 0.0364905 0.999334i $$-0.488382\pi$$
0.0364905 + 0.999334i $$0.488382\pi$$
$$752$$ 12.0000 0.437595
$$753$$ 12.0000 0.437304
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −54.0000 −1.95750 −0.978749 0.205061i $$-0.934261\pi$$
−0.978749 + 0.205061i $$0.934261\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 40.0000 1.44810
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ −24.0000 −0.866590
$$768$$ 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
$$771$$ 18.0000 0.648254
$$772$$ 14.0000 0.503871
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 50.0000 1.79605
$$776$$ −14.0000 −0.502571
$$777$$ 16.0000 0.573997
$$778$$ 30.0000 1.07555
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −6.00000 −0.214560
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 16.0000 0.568177
$$794$$ 16.0000 0.567819
$$795$$ 0 0
$$796$$ 2.00000 0.0708881
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 8.00000 0.283197
$$799$$ −12.0000 −0.424529
$$800$$ 5.00000 0.176777
$$801$$ −18.0000 −0.635999
$$802$$ 18.0000 0.635602
$$803$$ 0 0
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 20.0000 0.704470
$$807$$ 24.0000 0.844840
$$808$$ −6.00000 −0.211079
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ 0 0
$$813$$ −16.0000 −0.561144
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ 16.0000 0.559769
$$818$$ −38.0000 −1.32864
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ −36.0000 −1.25641 −0.628204 0.778048i $$-0.716210\pi$$
−0.628204 + 0.778048i $$0.716210\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −34.0000 −1.18517 −0.592583 0.805510i $$-0.701892\pi$$
−0.592583 + 0.805510i $$0.701892\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 48.0000 1.66912 0.834562 0.550914i $$-0.185721\pi$$
0.834562 + 0.550914i $$0.185721\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 0 0
$$831$$ −4.00000 −0.138758
$$832$$ 2.00000 0.0693375
$$833$$ 3.00000 0.103944
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −10.0000 −0.345651
$$838$$ 36.0000 1.24360
$$839$$ −18.0000 −0.621429 −0.310715 0.950503i $$-0.600568\pi$$
−0.310715 + 0.950503i $$0.600568\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 22.0000 0.758170
$$843$$ 30.0000 1.03325
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ −22.0000 −0.755929
$$848$$ 6.00000 0.206041
$$849$$ −16.0000 −0.549119
$$850$$ −5.00000 −0.171499
$$851$$ −48.0000 −1.64542
$$852$$ 6.00000 0.205557
$$853$$ −28.0000 −0.958702 −0.479351 0.877623i $$-0.659128\pi$$
−0.479351 + 0.877623i $$0.659128\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ 12.0000 0.408959
$$862$$ 30.0000 1.02180
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 1.00000 0.0339618
$$868$$ −20.0000 −0.678844
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ −20.0000 −0.677285
$$873$$ 14.0000 0.473828
$$874$$ −24.0000 −0.811812
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ −4.00000 −0.135070 −0.0675352 0.997717i $$-0.521513\pi$$
−0.0675352 + 0.997717i $$0.521513\pi$$
$$878$$ −26.0000 −0.877457
$$879$$ 18.0000 0.607125
$$880$$ 0 0
$$881$$ −54.0000 −1.81931 −0.909653 0.415369i $$-0.863653\pi$$
−0.909653 + 0.415369i $$0.863653\pi$$
$$882$$ 3.00000 0.101015
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ −18.0000 −0.604381 −0.302190 0.953248i $$-0.597718\pi$$
−0.302190 + 0.953248i $$0.597718\pi$$
$$888$$ −8.00000 −0.268462
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −28.0000 −0.937509
$$893$$ −48.0000 −1.60626
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ −12.0000 −0.400668
$$898$$ −6.00000 −0.200223
$$899$$ 0 0
$$900$$ −5.00000 −0.166667
$$901$$ −6.00000 −0.199889
$$902$$ 0 0
$$903$$ −8.00000 −0.266223
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ −40.0000 −1.32818 −0.664089 0.747653i $$-0.731180\pi$$
−0.664089 + 0.747653i $$0.731180\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 6.00000 0.198789 0.0993944 0.995048i $$-0.468309\pi$$
0.0993944 + 0.995048i $$0.468309\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 0 0
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ 1.00000 0.0330049
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 20.0000 0.659022
$$922$$ −30.0000 −0.987997
$$923$$ 12.0000 0.394985
$$924$$ 0 0
$$925$$ −40.0000 −1.31519
$$926$$ −20.0000 −0.657241
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ −30.0000 −0.982683
$$933$$ 18.0000 0.589294
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 8.00000 0.261209
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ 12.0000 0.391189 0.195594 0.980685i $$-0.437336\pi$$
0.195594 + 0.980685i $$0.437336\pi$$
$$942$$ 10.0000 0.325818
$$943$$ −36.0000 −1.17232
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 24.0000 0.779895 0.389948 0.920837i $$-0.372493\pi$$
0.389948 + 0.920837i $$0.372493\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ 4.00000 0.129845
$$950$$ −20.0000 −0.648886
$$951$$ 12.0000 0.389127
$$952$$ 2.00000 0.0648204
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ −6.00000 −0.193851
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 69.0000 2.22581
$$962$$ −16.0000 −0.515861
$$963$$ 0 0
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 12.0000 0.386094
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 4.00000 0.128499
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 16.0000 0.512936
$$974$$ 10.0000 0.320421
$$975$$ −10.0000 −0.320256
$$976$$ 8.00000 0.256074
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ −20.0000 −0.639529
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 20.0000 0.638551
$$982$$ 36.0000 1.14881
$$983$$ 6.00000 0.191370 0.0956851 0.995412i $$-0.469496\pi$$
0.0956851 + 0.995412i $$0.469496\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 24.0000 0.763928
$$988$$ −8.00000 −0.254514
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ 10.0000 0.317500
$$993$$ 20.0000 0.634681
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 8.00000 0.253109
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 102.2.a.b.1.1 1
3.2 odd 2 306.2.a.c.1.1 1
4.3 odd 2 816.2.a.d.1.1 1
5.2 odd 4 2550.2.d.g.2449.1 2
5.3 odd 4 2550.2.d.g.2449.2 2
5.4 even 2 2550.2.a.u.1.1 1
7.6 odd 2 4998.2.a.d.1.1 1
8.3 odd 2 3264.2.a.w.1.1 1
8.5 even 2 3264.2.a.i.1.1 1
12.11 even 2 2448.2.a.i.1.1 1
15.14 odd 2 7650.2.a.j.1.1 1
17.2 even 8 1734.2.f.b.1483.2 4
17.4 even 4 1734.2.b.f.577.1 2
17.8 even 8 1734.2.f.b.829.1 4
17.9 even 8 1734.2.f.b.829.2 4
17.13 even 4 1734.2.b.f.577.2 2
17.15 even 8 1734.2.f.b.1483.1 4
17.16 even 2 1734.2.a.b.1.1 1
24.5 odd 2 9792.2.a.bg.1.1 1
24.11 even 2 9792.2.a.ba.1.1 1
51.50 odd 2 5202.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.b.1.1 1 1.1 even 1 trivial
306.2.a.c.1.1 1 3.2 odd 2
816.2.a.d.1.1 1 4.3 odd 2
1734.2.a.b.1.1 1 17.16 even 2
1734.2.b.f.577.1 2 17.4 even 4
1734.2.b.f.577.2 2 17.13 even 4
1734.2.f.b.829.1 4 17.8 even 8
1734.2.f.b.829.2 4 17.9 even 8
1734.2.f.b.1483.1 4 17.15 even 8
1734.2.f.b.1483.2 4 17.2 even 8
2448.2.a.i.1.1 1 12.11 even 2
2550.2.a.u.1.1 1 5.4 even 2
2550.2.d.g.2449.1 2 5.2 odd 4
2550.2.d.g.2449.2 2 5.3 odd 4
3264.2.a.i.1.1 1 8.5 even 2
3264.2.a.w.1.1 1 8.3 odd 2
4998.2.a.d.1.1 1 7.6 odd 2
5202.2.a.j.1.1 1 51.50 odd 2
7650.2.a.j.1.1 1 15.14 odd 2
9792.2.a.ba.1.1 1 24.11 even 2
9792.2.a.bg.1.1 1 24.5 odd 2