# Properties

 Label 114.2.a.c.1.1 Level $114$ Weight $2$ Character 114.1 Self dual yes Analytic conductor $0.910$ 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] = [114,2,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, 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("114.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.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: $$0.910294583043$$ 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 $$\chi$$ $$=$$ 114.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -4.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -4.00000 q^{21} -6.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +6.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} -4.00000 q^{42} -4.00000 q^{43} -6.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -5.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{56} +1.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +14.0000 q^{61} +2.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +8.00000 q^{67} +6.00000 q^{68} -6.00000 q^{69} +1.00000 q^{72} +14.0000 q^{73} -4.00000 q^{74} -5.00000 q^{75} +1.00000 q^{76} -4.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -4.00000 q^{86} +6.00000 q^{87} -6.00000 q^{89} +16.0000 q^{91} -6.00000 q^{92} +2.00000 q^{93} +6.00000 q^{94} +1.00000 q^{96} -10.0000 q^{97} +9.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 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
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 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
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 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
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000 0.192450
$$28$$ −4.00000 −0.755929
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 1.00000 0.162221
$$39$$ −4.00000 −0.640513
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ −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
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 9.00000 1.28571
$$50$$ −5.00000 −0.707107
$$51$$ 6.00000 0.840168
$$52$$ −4.00000 −0.554700
$$53$$ 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
$$56$$ −4.00000 −0.534522
$$57$$ 1.00000 0.132453
$$58$$ 6.00000 0.787839
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 2.00000 0.254000
$$63$$ −4.00000 −0.503953
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 6.00000 0.727607
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ −5.00000 −0.577350
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ −4.00000 −0.452911
$$79$$ −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 $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 16.0000 1.67726
$$92$$ −6.00000 −0.625543
$$93$$ 2.00000 0.207390
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ −4.00000 −0.377964
$$113$$ −18.0000 −1.69330 −0.846649 0.532152i $$-0.821383\pi$$
−0.846649 + 0.532152i $$0.821383\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ −4.00000 −0.369800
$$118$$ −12.0000 −1.10469
$$119$$ −24.0000 −2.20008
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 14.0000 1.26750
$$123$$ 6.00000 0.541002
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 9.00000 0.742307
$$148$$ −4.00000 −0.328798
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ −5.00000 −0.408248
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 24.0000 1.89146
$$162$$ 1.00000 0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ −4.00000 −0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 20.0000 1.51186
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ −6.00000 −0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 16.0000 1.18600
$$183$$ 14.0000 1.03491
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 2.00000 0.146647
$$187$$ 0 0
$$188$$ 6.00000 0.437595
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ −5.00000 −0.353553
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ −24.0000 −1.68447
$$204$$ 6.00000 0.420084
$$205$$ 0 0
$$206$$ −10.0000 −0.696733
$$207$$ −6.00000 −0.417029
$$208$$ −4.00000 −0.277350
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −8.00000 −0.543075
$$218$$ −16.0000 −1.08366
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ −4.00000 −0.268462
$$223$$ −22.0000 −1.47323 −0.736614 0.676313i $$-0.763577\pi$$
−0.736614 + 0.676313i $$0.763577\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ −5.00000 −0.333333
$$226$$ −18.0000 −1.19734
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ −10.0000 −0.649570
$$238$$ −24.0000 −1.55569
$$239$$ −18.0000 −1.16432 −0.582162 0.813073i $$-0.697793\pi$$
−0.582162 + 0.813073i $$0.697793\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 1.00000 0.0641500
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ −4.00000 −0.254514
$$248$$ 2.00000 0.127000
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ 0 0
$$254$$ 2.00000 0.125491
$$255$$ 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
$$261$$ 6.00000 0.371391
$$262$$ 12.0000 0.741362
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ −6.00000 −0.367194
$$268$$ 8.00000 0.488678
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 16.0000 0.968364
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 6.00000 0.357295
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −24.0000 −1.41668
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 14.0000 0.819288
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ −4.00000 −0.232495
$$297$$ 0 0
$$298$$ 12.0000 0.695141
$$299$$ 24.0000 1.38796
$$300$$ −5.00000 −0.288675
$$301$$ 16.0000 0.922225
$$302$$ −10.0000 −0.575435
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ −10.0000 −0.568880
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 24.0000 1.33747
$$323$$ 6.00000 0.333849
$$324$$ 1.00000 0.0555556
$$325$$ 20.0000 1.10940
$$326$$ −4.00000 −0.221540
$$327$$ −16.0000 −0.884802
$$328$$ 6.00000 0.331295
$$329$$ −24.0000 −1.32316
$$330$$ 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
$$333$$ −4.00000 −0.219199
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 3.00000 0.163178
$$339$$ −18.0000 −0.977626
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 1.00000 0.0540738
$$343$$ −8.00000 −0.431959
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 24.0000 1.28839 0.644194 0.764862i $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ 6.00000 0.321634
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 20.0000 1.06904
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ −24.0000 −1.27021
$$358$$ 12.0000 0.634220
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −16.0000 −0.840941
$$363$$ −11.0000 −0.577350
$$364$$ 16.0000 0.838628
$$365$$ 0 0
$$366$$ 14.0000 0.731792
$$367$$ −4.00000 −0.208798 −0.104399 0.994535i $$-0.533292\pi$$
−0.104399 + 0.994535i $$0.533292\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ 2.00000 0.103695
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ −24.0000 −1.23606
$$378$$ −4.00000 −0.205738
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ 18.0000 0.920960
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ −4.00000 −0.203331
$$388$$ −10.0000 −0.507673
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 9.00000 0.454569
$$393$$ 12.0000 0.605320
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 26.0000 1.30490 0.652451 0.757831i $$-0.273741\pi$$
0.652451 + 0.757831i $$0.273741\pi$$
$$398$$ 20.0000 1.00251
$$399$$ −4.00000 −0.200250
$$400$$ −5.00000 −0.250000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 8.00000 0.399004
$$403$$ −8.00000 −0.398508
$$404$$ 0 0
$$405$$ 0 0
$$406$$ −24.0000 −1.19110
$$407$$ 0 0
$$408$$ 6.00000 0.297044
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ −10.0000 −0.492665
$$413$$ 48.0000 2.36193
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ 20.0000 0.979404
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 6.00000 0.291730
$$424$$ 6.00000 0.291386
$$425$$ −30.0000 −1.45521
$$426$$ 0 0
$$427$$ −56.0000 −2.71003
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −10.0000 −0.480569 −0.240285 0.970702i $$-0.577241\pi$$
−0.240285 + 0.970702i $$0.577241\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ −6.00000 −0.287019
$$438$$ 14.0000 0.668946
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ −24.0000 −1.14156
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ −22.0000 −1.04173
$$447$$ 12.0000 0.567581
$$448$$ −4.00000 −0.188982
$$449$$ −30.0000 −1.41579 −0.707894 0.706319i $$-0.750354\pi$$
−0.707894 + 0.706319i $$0.750354\pi$$
$$450$$ −5.00000 −0.235702
$$451$$ 0 0
$$452$$ −18.0000 −0.846649
$$453$$ −10.0000 −0.469841
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ −4.00000 −0.184900
$$469$$ −32.0000 −1.47762
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ −12.0000 −0.552345
$$473$$ 0 0
$$474$$ −10.0000 −0.459315
$$475$$ −5.00000 −0.229416
$$476$$ −24.0000 −1.10004
$$477$$ 6.00000 0.274721
$$478$$ −18.0000 −0.823301
$$479$$ −6.00000 −0.274147 −0.137073 0.990561i $$-0.543770\pi$$
−0.137073 + 0.990561i $$0.543770\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 14.0000 0.637683
$$483$$ 24.0000 1.09204
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ 14.0000 0.633750
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 36.0000 1.62466 0.812329 0.583200i $$-0.198200\pi$$
0.812329 + 0.583200i $$0.198200\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 36.0000 1.62136
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 12.0000 0.535586
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3.00000 0.133235
$$508$$ 2.00000 0.0887357
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −56.0000 −2.47729
$$512$$ 1.00000 0.0441942
$$513$$ 1.00000 0.0441511
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 16.0000 0.703000
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 8.00000 0.349816 0.174908 0.984585i $$-0.444037\pi$$
0.174908 + 0.984585i $$0.444037\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 20.0000 0.872872
$$526$$ −18.0000 −0.784837
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ −4.00000 −0.173422
$$533$$ −24.0000 −1.03956
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 12.0000 0.517838
$$538$$ −6.00000 −0.258678
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 20.0000 0.859074
$$543$$ −16.0000 −0.686626
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 16.0000 0.684737
$$547$$ 32.0000 1.36822 0.684111 0.729378i $$-0.260191\pi$$
0.684111 + 0.729378i $$0.260191\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ −6.00000 −0.255377
$$553$$ 40.0000 1.70097
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ 12.0000 0.508456 0.254228 0.967144i $$-0.418179\pi$$
0.254228 + 0.967144i $$0.418179\pi$$
$$558$$ 2.00000 0.0846668
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −18.0000 −0.759284
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ −4.00000 −0.167984
$$568$$ 0 0
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 18.0000 0.751961
$$574$$ −24.0000 −1.00174
$$575$$ 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 $$0.486745\pi$$
$$578$$ 19.0000 0.790296
$$579$$ −10.0000 −0.415586
$$580$$ 0 0
$$581$$ 48.0000 1.99138
$$582$$ −10.0000 −0.414513
$$583$$ 0 0
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 9.00000 0.371154
$$589$$ 2.00000 0.0824086
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ −4.00000 −0.164399
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 12.0000 0.491539
$$597$$ 20.0000 0.818546
$$598$$ 24.0000 0.981433
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ −5.00000 −0.204124
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 16.0000 0.652111
$$603$$ 8.00000 0.325785
$$604$$ −10.0000 −0.406894
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −34.0000 −1.38002 −0.690009 0.723801i $$-0.742393\pi$$
−0.690009 + 0.723801i $$0.742393\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 6.00000 0.242536
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ −10.0000 −0.402259
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 30.0000 1.20289
$$623$$ 24.0000 0.961540
$$624$$ −4.00000 −0.160128
$$625$$ 25.0000 1.00000
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ −4.00000 −0.158986
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ −36.0000 −1.42637
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 24.0000 0.945732
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ −6.00000 −0.235884 −0.117942 0.993020i $$-0.537630\pi$$
−0.117942 + 0.993020i $$0.537630\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 20.0000 0.784465
$$651$$ −8.00000 −0.313545
$$652$$ −4.00000 −0.156652
$$653$$ 36.0000 1.40879 0.704394 0.709809i $$-0.251219\pi$$
0.704394 + 0.709809i $$0.251219\pi$$
$$654$$ −16.0000 −0.625650
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 14.0000 0.546192
$$658$$ −24.0000 −0.935617
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ 8.00000 0.310929
$$663$$ −24.0000 −0.932083
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −4.00000 −0.154997
$$667$$ −36.0000 −1.39393
$$668$$ −12.0000 −0.464294
$$669$$ −22.0000 −0.850569
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −4.00000 −0.154303
$$673$$ −10.0000 −0.385472 −0.192736 0.981251i $$-0.561736\pi$$
−0.192736 + 0.981251i $$0.561736\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ −5.00000 −0.192450
$$676$$ 3.00000 0.115385
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ −18.0000 −0.691286
$$679$$ 40.0000 1.53506
$$680$$ 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 $$0.573736\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ −10.0000 −0.381524
$$688$$ −4.00000 −0.152499
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 36.0000 1.36360
$$698$$ 2.00000 0.0757011
$$699$$ −6.00000 −0.226941
$$700$$ 20.0000 0.755929
$$701$$ −48.0000 −1.81293 −0.906467 0.422276i $$-0.861231\pi$$
−0.906467 + 0.422276i $$0.861231\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ −4.00000 −0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ −12.0000 −0.450988
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ −6.00000 −0.224860
$$713$$ −12.0000 −0.449404
$$714$$ −24.0000 −0.898177
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ −18.0000 −0.672222
$$718$$ −6.00000 −0.223918
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 0 0
$$721$$ 40.0000 1.48968
$$722$$ 1.00000 0.0372161
$$723$$ 14.0000 0.520666
$$724$$ −16.0000 −0.594635
$$725$$ −30.0000 −1.11417
$$726$$ −11.0000 −0.408248
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 16.0000 0.592999
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 14.0000 0.517455
$$733$$ 50.0000 1.84679 0.923396 0.383849i $$-0.125402\pi$$
0.923396 + 0.383849i $$0.125402\pi$$
$$734$$ −4.00000 −0.147643
$$735$$ 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
$$741$$ −4.00000 −0.146944
$$742$$ −24.0000 −0.881068
$$743$$ −12.0000 −0.440237 −0.220119 0.975473i $$-0.570644\pi$$
−0.220119 + 0.975473i $$0.570644\pi$$
$$744$$ 2.00000 0.0733236
$$745$$ 0 0
$$746$$ −4.00000 −0.146450
$$747$$ −12.0000 −0.439057
$$748$$ 0 0
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ −10.0000 −0.364905 −0.182453 0.983215i $$-0.558404\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 12.0000 0.437304
$$754$$ −24.0000 −0.874028
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ −46.0000 −1.67190 −0.835949 0.548807i $$-0.815082\pi$$
−0.835949 + 0.548807i $$0.815082\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 2.00000 0.0724524
$$763$$ 64.0000 2.31696
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ 48.0000 1.73318
$$768$$ 1.00000 0.0360844
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ −10.0000 −0.359908
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ −10.0000 −0.359211
$$776$$ −10.0000 −0.358979
$$777$$ 16.0000 0.573997
$$778$$ 24.0000 0.860442
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −36.0000 −1.28736
$$783$$ 6.00000 0.214423
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 20.0000 0.712923 0.356462 0.934310i $$-0.383983\pi$$
0.356462 + 0.934310i $$0.383983\pi$$
$$788$$ 12.0000 0.427482
$$789$$ −18.0000 −0.640817
$$790$$ 0 0
$$791$$ 72.0000 2.56003
$$792$$ 0 0
$$793$$ −56.0000 −1.98862
$$794$$ 26.0000 0.922705
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ 36.0000 1.27359
$$800$$ −5.00000 −0.176777
$$801$$ −6.00000 −0.212000
$$802$$ 30.0000 1.05934
$$803$$ 0 0
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ −6.00000 −0.211210
$$808$$ 0 0
$$809$$ −42.0000 −1.47664 −0.738321 0.674450i $$-0.764381\pi$$
−0.738321 + 0.674450i $$0.764381\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ −24.0000 −0.842235
$$813$$ 20.0000 0.701431
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ −4.00000 −0.139942
$$818$$ 14.0000 0.489499
$$819$$ 16.0000 0.559085
$$820$$ 0 0
$$821$$ −24.0000 −0.837606 −0.418803 0.908077i $$-0.637550\pi$$
−0.418803 + 0.908077i $$0.637550\pi$$
$$822$$ −18.0000 −0.627822
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ −10.0000 −0.348367
$$825$$ 0 0
$$826$$ 48.0000 1.67013
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ −16.0000 −0.555703 −0.277851 0.960624i $$-0.589622\pi$$
−0.277851 + 0.960624i $$0.589622\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ −4.00000 −0.138675
$$833$$ 54.0000 1.87099
$$834$$ 20.0000 0.692543
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 2.00000 0.0691301
$$838$$ 0 0
$$839$$ 36.0000 1.24286 0.621429 0.783470i $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −28.0000 −0.964944
$$843$$ −18.0000 −0.619953
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ 44.0000 1.51186
$$848$$ 6.00000 0.206041
$$849$$ −4.00000 −0.137280
$$850$$ −30.0000 −1.02899
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ 38.0000 1.30110 0.650548 0.759465i $$-0.274539\pi$$
0.650548 + 0.759465i $$0.274539\pi$$
$$854$$ −56.0000 −1.91628
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ −12.0000 −0.408722
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −10.0000 −0.339814
$$867$$ 19.0000 0.645274
$$868$$ −8.00000 −0.271538
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −32.0000 −1.08428
$$872$$ −16.0000 −0.541828
$$873$$ −10.0000 −0.338449
$$874$$ −6.00000 −0.202953
$$875$$ 0 0
$$876$$ 14.0000 0.473016
$$877$$ 44.0000 1.48577 0.742887 0.669417i $$-0.233456\pi$$
0.742887 + 0.669417i $$0.233456\pi$$
$$878$$ 26.0000 0.877457
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 9.00000 0.303046
$$883$$ 20.0000 0.673054 0.336527 0.941674i $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ −24.0000 −0.807207
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ −4.00000 −0.134231
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −22.0000 −0.736614
$$893$$ 6.00000 0.200782
$$894$$ 12.0000 0.401340
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ 24.0000 0.801337
$$898$$ −30.0000 −1.00111
$$899$$ 12.0000 0.400222
$$900$$ −5.00000 −0.166667
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 16.0000 0.532447
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ −10.0000 −0.332228
$$907$$ 8.00000 0.265636 0.132818 0.991140i $$-0.457597\pi$$
0.132818 + 0.991140i $$0.457597\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ 0 0
$$914$$ 2.00000 0.0661541
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ −48.0000 −1.58510
$$918$$ 6.00000 0.198030
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ −16.0000 −0.527218
$$922$$ 12.0000 0.395199
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 20.0000 0.657596
$$926$$ −4.00000 −0.131448
$$927$$ −10.0000 −0.328443
$$928$$ 6.00000 0.196960
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 9.00000 0.294963
$$932$$ −6.00000 −0.196537
$$933$$ 30.0000 0.982156
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ −58.0000 −1.89478 −0.947389 0.320085i $$-0.896288\pi$$
−0.947389 + 0.320085i $$0.896288\pi$$
$$938$$ −32.0000 −1.04484
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ −36.0000 −1.17232
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ −56.0000 −1.81784
$$950$$ −5.00000 −0.162221
$$951$$ 18.0000 0.583690
$$952$$ −24.0000 −0.777844
$$953$$ 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
$$956$$ −18.0000 −0.582162
$$957$$ 0 0
$$958$$ −6.00000 −0.193851
$$959$$ 72.0000 2.32500
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 16.0000 0.515861
$$963$$ 12.0000 0.386695
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ 24.0000 0.772187
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ −11.0000 −0.353553
$$969$$ 6.00000 0.192748
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −80.0000 −2.56468
$$974$$ 38.0000 1.21760
$$975$$ 20.0000 0.640513
$$976$$ 14.0000 0.448129
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 36.0000 1.14881
$$983$$ −48.0000 −1.53096 −0.765481 0.643458i $$-0.777499\pi$$
−0.765481 + 0.643458i $$0.777499\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 36.0000 1.14647
$$987$$ −24.0000 −0.763928
$$988$$ −4.00000 −0.127257
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 26.0000 0.825917 0.412959 0.910750i $$-0.364495\pi$$
0.412959 + 0.910750i $$0.364495\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ 8.00000 0.253872
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ 26.0000 0.823428 0.411714 0.911313i $$-0.364930\pi$$
0.411714 + 0.911313i $$0.364930\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ −4.00000 −0.126554
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 114.2.a.c.1.1 1
3.2 odd 2 342.2.a.c.1.1 1
4.3 odd 2 912.2.a.c.1.1 1
5.2 odd 4 2850.2.d.p.799.2 2
5.3 odd 4 2850.2.d.p.799.1 2
5.4 even 2 2850.2.a.g.1.1 1
7.6 odd 2 5586.2.a.u.1.1 1
8.3 odd 2 3648.2.a.bc.1.1 1
8.5 even 2 3648.2.a.i.1.1 1
12.11 even 2 2736.2.a.o.1.1 1
15.14 odd 2 8550.2.a.bj.1.1 1
19.18 odd 2 2166.2.a.a.1.1 1
57.56 even 2 6498.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.c.1.1 1 1.1 even 1 trivial
342.2.a.c.1.1 1 3.2 odd 2
912.2.a.c.1.1 1 4.3 odd 2
2166.2.a.a.1.1 1 19.18 odd 2
2736.2.a.o.1.1 1 12.11 even 2
2850.2.a.g.1.1 1 5.4 even 2
2850.2.d.p.799.1 2 5.3 odd 4
2850.2.d.p.799.2 2 5.2 odd 4
3648.2.a.i.1.1 1 8.5 even 2
3648.2.a.bc.1.1 1 8.3 odd 2
5586.2.a.u.1.1 1 7.6 odd 2
6498.2.a.t.1.1 1 57.56 even 2
8550.2.a.bj.1.1 1 15.14 odd 2