Properties

 Label 350.2.a.a.1.1 Level $350$ Weight $2$ Character 350.1 Self dual yes Analytic conductor $2.795$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,2,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.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: $$2.79476407074$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} -7.00000 q^{19} +1.00000 q^{21} -3.00000 q^{22} +1.00000 q^{24} +2.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} -8.00000 q^{37} +7.00000 q^{38} +2.00000 q^{39} -9.00000 q^{41} -1.00000 q^{42} -8.00000 q^{43} +3.00000 q^{44} +6.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} -2.00000 q^{52} +12.0000 q^{53} -5.00000 q^{54} +1.00000 q^{56} +7.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +7.00000 q^{67} -3.00000 q^{68} +6.00000 q^{71} +2.00000 q^{72} -5.00000 q^{73} +8.00000 q^{74} -7.00000 q^{76} -3.00000 q^{77} -2.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} +9.00000 q^{82} +9.00000 q^{83} +1.00000 q^{84} +8.00000 q^{86} +6.00000 q^{87} -3.00000 q^{88} -15.0000 q^{89} +2.00000 q^{91} +4.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} -6.00000 q^{99} +O(q^{100})$$

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 2.00000 0.471405
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ −3.00000 −0.639602
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 5.00000 0.962250
$$28$$ −1.00000 −0.188982
$$29$$ −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
$$33$$ −3.00000 −0.522233
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 7.00000 1.13555
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ −2.00000 −0.277350
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 7.00000 0.927173
$$58$$ 6.00000 0.787839
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 2.00000 0.235702
$$73$$ −5.00000 −0.585206 −0.292603 0.956234i $$-0.594521\pi$$
−0.292603 + 0.956234i $$0.594521\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ −7.00000 −0.802955
$$77$$ −3.00000 −0.341882
$$78$$ −2.00000 −0.226455
$$79$$ 14.0000 1.57512 0.787562 0.616236i $$-0.211343\pi$$
0.787562 + 0.616236i $$0.211343\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 9.00000 0.993884
$$83$$ 9.00000 0.987878 0.493939 0.869496i $$-0.335557\pi$$
0.493939 + 0.869496i $$0.335557\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 6.00000 0.643268
$$88$$ −3.00000 −0.319801
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ −3.00000 −0.297044
$$103$$ −20.0000 −1.97066 −0.985329 0.170664i $$-0.945409\pi$$
−0.985329 + 0.170664i $$0.945409\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 5.00000 0.481125
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ −1.00000 −0.0944911
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ −7.00000 −0.655610
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 4.00000 0.369800
$$118$$ −12.0000 −1.10469
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 10.0000 0.905357
$$123$$ 9.00000 0.811503
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ −3.00000 −0.261116
$$133$$ 7.00000 0.606977
$$134$$ −7.00000 −0.604708
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ −21.0000 −1.79415 −0.897076 0.441877i $$-0.854313\pi$$
−0.897076 + 0.441877i $$0.854313\pi$$
$$138$$ 0 0
$$139$$ −7.00000 −0.593732 −0.296866 0.954919i $$-0.595942\pi$$
−0.296866 + 0.954919i $$0.595942\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ −6.00000 −0.503509
$$143$$ −6.00000 −0.501745
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ 5.00000 0.413803
$$147$$ −1.00000 −0.0824786
$$148$$ −8.00000 −0.657596
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 7.00000 0.567775
$$153$$ 6.00000 0.485071
$$154$$ 3.00000 0.241747
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −20.0000 −1.59617 −0.798087 0.602542i $$-0.794154\pi$$
−0.798087 + 0.602542i $$0.794154\pi$$
$$158$$ −14.0000 −1.11378
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −5.00000 −0.391630 −0.195815 0.980641i $$-0.562735\pi$$
−0.195815 + 0.980641i $$0.562735\pi$$
$$164$$ −9.00000 −0.702782
$$165$$ 0 0
$$166$$ −9.00000 −0.698535
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 14.0000 1.07061
$$172$$ −8.00000 −0.609994
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ 3.00000 0.226134
$$177$$ −12.0000 −0.901975
$$178$$ 15.0000 1.12430
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 10.0000 0.739221
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ −9.00000 −0.658145
$$188$$ 6.00000 0.437595
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ −6.00000 −0.434145 −0.217072 0.976156i $$-0.569651\pi$$
−0.217072 + 0.976156i $$0.569651\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −5.00000 −0.359908 −0.179954 0.983675i $$-0.557595\pi$$
−0.179954 + 0.983675i $$0.557595\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 6.00000 0.426401
$$199$$ 14.0000 0.992434 0.496217 0.868199i $$-0.334722\pi$$
0.496217 + 0.868199i $$0.334722\pi$$
$$200$$ 0 0
$$201$$ −7.00000 −0.493742
$$202$$ 0 0
$$203$$ 6.00000 0.421117
$$204$$ 3.00000 0.210042
$$205$$ 0 0
$$206$$ 20.0000 1.39347
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ −21.0000 −1.45260
$$210$$ 0 0
$$211$$ 17.0000 1.17033 0.585164 0.810915i $$-0.301030\pi$$
0.585164 + 0.810915i $$0.301030\pi$$
$$212$$ 12.0000 0.824163
$$213$$ −6.00000 −0.411113
$$214$$ 3.00000 0.205076
$$215$$ 0 0
$$216$$ −5.00000 −0.340207
$$217$$ 4.00000 0.271538
$$218$$ −14.0000 −0.948200
$$219$$ 5.00000 0.337869
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ −8.00000 −0.536925
$$223$$ −14.0000 −0.937509 −0.468755 0.883328i $$-0.655297\pi$$
−0.468755 + 0.883328i $$0.655297\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −9.00000 −0.598671
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 7.00000 0.463586
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 3.00000 0.197386
$$232$$ 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
$$237$$ −14.0000 −0.909398
$$238$$ −3.00000 −0.194461
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −25.0000 −1.61039 −0.805196 0.593009i $$-0.797940\pi$$
−0.805196 + 0.593009i $$0.797940\pi$$
$$242$$ 2.00000 0.128565
$$243$$ −16.0000 −1.02640
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −9.00000 −0.573819
$$247$$ 14.0000 0.890799
$$248$$ 4.00000 0.254000
$$249$$ −9.00000 −0.570352
$$250$$ 0 0
$$251$$ 15.0000 0.946792 0.473396 0.880850i $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 0 0
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 30.0000 1.87135 0.935674 0.352865i $$-0.114792\pi$$
0.935674 + 0.352865i $$0.114792\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 12.0000 0.742781
$$262$$ 0 0
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 0 0
$$266$$ −7.00000 −0.429198
$$267$$ 15.0000 0.917985
$$268$$ 7.00000 0.427593
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ −2.00000 −0.121046
$$274$$ 21.0000 1.26866
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 7.00000 0.419832
$$279$$ 8.00000 0.478947
$$280$$ 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
$$283$$ 1.00000 0.0594438 0.0297219 0.999558i $$-0.490538\pi$$
0.0297219 + 0.999558i $$0.490538\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ 9.00000 0.531253
$$288$$ 2.00000 0.117851
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ −5.00000 −0.292603
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ 15.0000 0.870388
$$298$$ −12.0000 −0.695141
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ −7.00000 −0.401478
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 7.00000 0.399511 0.199756 0.979846i $$-0.435985\pi$$
0.199756 + 0.979846i $$0.435985\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 20.0000 1.13776
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 20.0000 1.12867
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 12.0000 0.672927
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ 3.00000 0.167444
$$322$$ 0 0
$$323$$ 21.0000 1.16847
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 5.00000 0.276924
$$327$$ −14.0000 −0.774202
$$328$$ 9.00000 0.496942
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ 9.00000 0.493939
$$333$$ 16.0000 0.876795
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 1.00000 0.0545545
$$337$$ 13.0000 0.708155 0.354078 0.935216i $$-0.384795\pi$$
0.354078 + 0.935216i $$0.384795\pi$$
$$338$$ 9.00000 0.489535
$$339$$ −9.00000 −0.488813
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ −14.0000 −0.757033
$$343$$ −1.00000 −0.0539949
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 21.0000 1.12734 0.563670 0.826000i $$-0.309389\pi$$
0.563670 + 0.826000i $$0.309389\pi$$
$$348$$ 6.00000 0.321634
$$349$$ 8.00000 0.428230 0.214115 0.976808i $$-0.431313\pi$$
0.214115 + 0.976808i $$0.431313\pi$$
$$350$$ 0 0
$$351$$ −10.0000 −0.533761
$$352$$ −3.00000 −0.159901
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ −15.0000 −0.794998
$$357$$ −3.00000 −0.158777
$$358$$ −3.00000 −0.158555
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ −2.00000 −0.105118
$$363$$ 2.00000 0.104973
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ 18.0000 0.937043
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 4.00000 0.207390
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 9.00000 0.465379
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ 12.0000 0.618031
$$378$$ 5.00000 0.257172
$$379$$ 17.0000 0.873231 0.436616 0.899648i $$-0.356177\pi$$
0.436616 + 0.899648i $$0.356177\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ 6.00000 0.306987
$$383$$ 30.0000 1.53293 0.766464 0.642287i $$-0.222014\pi$$
0.766464 + 0.642287i $$0.222014\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 5.00000 0.254493
$$387$$ 16.0000 0.813326
$$388$$ 10.0000 0.507673
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −1.00000 −0.0505076
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ −14.0000 −0.701757
$$399$$ −7.00000 −0.350438
$$400$$ 0 0
$$401$$ −27.0000 −1.34832 −0.674158 0.738587i $$-0.735493\pi$$
−0.674158 + 0.738587i $$0.735493\pi$$
$$402$$ 7.00000 0.349128
$$403$$ 8.00000 0.398508
$$404$$ 0 0
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ −24.0000 −1.18964
$$408$$ −3.00000 −0.148522
$$409$$ −25.0000 −1.23617 −0.618085 0.786111i $$-0.712091\pi$$
−0.618085 + 0.786111i $$0.712091\pi$$
$$410$$ 0 0
$$411$$ 21.0000 1.03585
$$412$$ −20.0000 −0.985329
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 7.00000 0.342791
$$418$$ 21.0000 1.02714
$$419$$ −3.00000 −0.146560 −0.0732798 0.997311i $$-0.523347\pi$$
−0.0732798 + 0.997311i $$0.523347\pi$$
$$420$$ 0 0
$$421$$ 20.0000 0.974740 0.487370 0.873195i $$-0.337956\pi$$
0.487370 + 0.873195i $$0.337956\pi$$
$$422$$ −17.0000 −0.827547
$$423$$ −12.0000 −0.583460
$$424$$ −12.0000 −0.582772
$$425$$ 0 0
$$426$$ 6.00000 0.290701
$$427$$ 10.0000 0.483934
$$428$$ −3.00000 −0.145010
$$429$$ 6.00000 0.289683
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 5.00000 0.240563
$$433$$ −11.0000 −0.528626 −0.264313 0.964437i $$-0.585145\pi$$
−0.264313 + 0.964437i $$0.585145\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ −5.00000 −0.238909
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ −6.00000 −0.285391
$$443$$ −21.0000 −0.997740 −0.498870 0.866677i $$-0.666252\pi$$
−0.498870 + 0.866677i $$0.666252\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ −12.0000 −0.567581
$$448$$ −1.00000 −0.0472456
$$449$$ −15.0000 −0.707894 −0.353947 0.935266i $$-0.615161\pi$$
−0.353947 + 0.935266i $$0.615161\pi$$
$$450$$ 0 0
$$451$$ −27.0000 −1.27138
$$452$$ 9.00000 0.423324
$$453$$ −8.00000 −0.375873
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −7.00000 −0.327805
$$457$$ −17.0000 −0.795226 −0.397613 0.917553i $$-0.630161\pi$$
−0.397613 + 0.917553i $$0.630161\pi$$
$$458$$ −26.0000 −1.21490
$$459$$ −15.0000 −0.700140
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ −3.00000 −0.139573
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ −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
$$469$$ −7.00000 −0.323230
$$470$$ 0 0
$$471$$ 20.0000 0.921551
$$472$$ −12.0000 −0.552345
$$473$$ −24.0000 −1.10352
$$474$$ 14.0000 0.643041
$$475$$ 0 0
$$476$$ 3.00000 0.137505
$$477$$ −24.0000 −1.09888
$$478$$ 12.0000 0.548867
$$479$$ 18.0000 0.822441 0.411220 0.911536i $$-0.365103\pi$$
0.411220 + 0.911536i $$0.365103\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 25.0000 1.13872
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ 16.0000 0.725775
$$487$$ 34.0000 1.54069 0.770344 0.637629i $$-0.220085\pi$$
0.770344 + 0.637629i $$0.220085\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 5.00000 0.226108
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 9.00000 0.405751
$$493$$ 18.0000 0.810679
$$494$$ −14.0000 −0.629890
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ −6.00000 −0.269137
$$498$$ 9.00000 0.403300
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ −15.0000 −0.669483
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9.00000 0.399704
$$508$$ −2.00000 −0.0887357
$$509$$ 42.0000 1.86162 0.930809 0.365507i $$-0.119104\pi$$
0.930809 + 0.365507i $$0.119104\pi$$
$$510$$ 0 0
$$511$$ 5.00000 0.221187
$$512$$ −1.00000 −0.0441942
$$513$$ −35.0000 −1.54529
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 18.0000 0.791639
$$518$$ −8.00000 −0.351500
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ 39.0000 1.70862 0.854311 0.519763i $$-0.173980\pi$$
0.854311 + 0.519763i $$0.173980\pi$$
$$522$$ −12.0000 −0.525226
$$523$$ 7.00000 0.306089 0.153044 0.988219i $$-0.451092\pi$$
0.153044 + 0.988219i $$0.451092\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 6.00000 0.261612
$$527$$ 12.0000 0.522728
$$528$$ −3.00000 −0.130558
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −24.0000 −1.04151
$$532$$ 7.00000 0.303488
$$533$$ 18.0000 0.779667
$$534$$ −15.0000 −0.649113
$$535$$ 0 0
$$536$$ −7.00000 −0.302354
$$537$$ −3.00000 −0.129460
$$538$$ 6.00000 0.258678
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ −2.00000 −0.0859074
$$543$$ −2.00000 −0.0858282
$$544$$ 3.00000 0.128624
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ −35.0000 −1.49649 −0.748246 0.663421i $$-0.769104\pi$$
−0.748246 + 0.663421i $$0.769104\pi$$
$$548$$ −21.0000 −0.897076
$$549$$ 20.0000 0.853579
$$550$$ 0 0
$$551$$ 42.0000 1.78926
$$552$$ 0 0
$$553$$ −14.0000 −0.595341
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −7.00000 −0.296866
$$557$$ −36.0000 −1.52537 −0.762684 0.646771i $$-0.776119\pi$$
−0.762684 + 0.646771i $$0.776119\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 9.00000 0.379980
$$562$$ 18.0000 0.759284
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ −1.00000 −0.0420331
$$567$$ −1.00000 −0.0419961
$$568$$ −6.00000 −0.251754
$$569$$ −27.0000 −1.13190 −0.565949 0.824440i $$-0.691490\pi$$
−0.565949 + 0.824440i $$0.691490\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −6.00000 −0.250873
$$573$$ 6.00000 0.250654
$$574$$ −9.00000 −0.375653
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ 7.00000 0.291414 0.145707 0.989328i $$-0.453454\pi$$
0.145707 + 0.989328i $$0.453454\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 5.00000 0.207793
$$580$$ 0 0
$$581$$ −9.00000 −0.373383
$$582$$ 10.0000 0.414513
$$583$$ 36.0000 1.49097
$$584$$ 5.00000 0.206901
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ −39.0000 −1.60970 −0.804851 0.593477i $$-0.797755\pi$$
−0.804851 + 0.593477i $$0.797755\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 28.0000 1.15372
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ −8.00000 −0.328798
$$593$$ 27.0000 1.10876 0.554379 0.832265i $$-0.312956\pi$$
0.554379 + 0.832265i $$0.312956\pi$$
$$594$$ −15.0000 −0.615457
$$595$$ 0 0
$$596$$ 12.0000 0.491539
$$597$$ −14.0000 −0.572982
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ −14.0000 −0.570124
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −44.0000 −1.78590 −0.892952 0.450151i $$-0.851370\pi$$
−0.892952 + 0.450151i $$0.851370\pi$$
$$608$$ 7.00000 0.283887
$$609$$ −6.00000 −0.243132
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ 6.00000 0.242536
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −7.00000 −0.282497
$$615$$ 0 0
$$616$$ 3.00000 0.120873
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ −20.0000 −0.804518
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 18.0000 0.721734
$$623$$ 15.0000 0.600962
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 21.0000 0.838659
$$628$$ −20.0000 −0.798087
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ −14.0000 −0.556890
$$633$$ −17.0000 −0.675689
$$634$$ −12.0000 −0.476581
$$635$$ 0 0
$$636$$ −12.0000 −0.475831
$$637$$ −2.00000 −0.0792429
$$638$$ 18.0000 0.712627
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ −3.00000 −0.118401
$$643$$ 40.0000 1.57745 0.788723 0.614749i $$-0.210743\pi$$
0.788723 + 0.614749i $$0.210743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −21.0000 −0.826234
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ −4.00000 −0.156772
$$652$$ −5.00000 −0.195815
$$653$$ −36.0000 −1.40879 −0.704394 0.709809i $$-0.748781\pi$$
−0.704394 + 0.709809i $$0.748781\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 0 0
$$656$$ −9.00000 −0.351391
$$657$$ 10.0000 0.390137
$$658$$ 6.00000 0.233904
$$659$$ 9.00000 0.350590 0.175295 0.984516i $$-0.443912\pi$$
0.175295 + 0.984516i $$0.443912\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 25.0000 0.971653
$$663$$ −6.00000 −0.233021
$$664$$ −9.00000 −0.349268
$$665$$ 0 0
$$666$$ −16.0000 −0.619987
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ −30.0000 −1.15814
$$672$$ −1.00000 −0.0385758
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ −13.0000 −0.500741
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 9.00000 0.345643
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 12.0000 0.459504
$$683$$ −3.00000 −0.114792 −0.0573959 0.998351i $$-0.518280\pi$$
−0.0573959 + 0.998351i $$0.518280\pi$$
$$684$$ 14.0000 0.535303
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ −26.0000 −0.991962
$$688$$ −8.00000 −0.304997
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ −19.0000 −0.722794 −0.361397 0.932412i $$-0.617700\pi$$
−0.361397 + 0.932412i $$0.617700\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 6.00000 0.227921
$$694$$ −21.0000 −0.797149
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 27.0000 1.02270
$$698$$ −8.00000 −0.302804
$$699$$ 6.00000 0.226941
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 10.0000 0.377426
$$703$$ 56.0000 2.11208
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 0 0
$$708$$ −12.0000 −0.450988
$$709$$ −28.0000 −1.05156 −0.525781 0.850620i $$-0.676227\pi$$
−0.525781 + 0.850620i $$0.676227\pi$$
$$710$$ 0 0
$$711$$ −28.0000 −1.05008
$$712$$ 15.0000 0.562149
$$713$$ 0 0
$$714$$ 3.00000 0.112272
$$715$$ 0 0
$$716$$ 3.00000 0.112115
$$717$$ 12.0000 0.448148
$$718$$ 6.00000 0.223918
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 20.0000 0.744839
$$722$$ −30.0000 −1.11648
$$723$$ 25.0000 0.929760
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ 34.0000 1.26099 0.630495 0.776193i $$-0.282852\pi$$
0.630495 + 0.776193i $$0.282852\pi$$
$$728$$ −2.00000 −0.0741249
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 10.0000 0.369611
$$733$$ 40.0000 1.47743 0.738717 0.674016i $$-0.235432\pi$$
0.738717 + 0.674016i $$0.235432\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 21.0000 0.773545
$$738$$ −18.0000 −0.662589
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ −14.0000 −0.514303
$$742$$ 12.0000 0.440534
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 0 0
$$746$$ −4.00000 −0.146450
$$747$$ −18.0000 −0.658586
$$748$$ −9.00000 −0.329073
$$749$$ 3.00000 0.109618
$$750$$ 0 0
$$751$$ −46.0000 −1.67856 −0.839282 0.543696i $$-0.817024\pi$$
−0.839282 + 0.543696i $$0.817024\pi$$
$$752$$ 6.00000 0.218797
$$753$$ −15.0000 −0.546630
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ −5.00000 −0.181848
$$757$$ 34.0000 1.23575 0.617876 0.786276i $$-0.287994\pi$$
0.617876 + 0.786276i $$0.287994\pi$$
$$758$$ −17.0000 −0.617468
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −9.00000 −0.326250 −0.163125 0.986605i $$-0.552157\pi$$
−0.163125 + 0.986605i $$0.552157\pi$$
$$762$$ −2.00000 −0.0724524
$$763$$ −14.0000 −0.506834
$$764$$ −6.00000 −0.217072
$$765$$ 0 0
$$766$$ −30.0000 −1.08394
$$767$$ −24.0000 −0.866590
$$768$$ −1.00000 −0.0360844
$$769$$ 23.0000 0.829401 0.414701 0.909958i $$-0.363886\pi$$
0.414701 + 0.909958i $$0.363886\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ −5.00000 −0.179954
$$773$$ 12.0000 0.431610 0.215805 0.976436i $$-0.430762\pi$$
0.215805 + 0.976436i $$0.430762\pi$$
$$774$$ −16.0000 −0.575108
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ −8.00000 −0.286998
$$778$$ 24.0000 0.860442
$$779$$ 63.0000 2.25721
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ 0 0
$$783$$ −30.0000 −1.07211
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ −9.00000 −0.320003
$$792$$ 6.00000 0.213201
$$793$$ 20.0000 0.710221
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ 14.0000 0.496217
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 7.00000 0.247797
$$799$$ −18.0000 −0.636794
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 27.0000 0.953403
$$803$$ −15.0000 −0.529339
$$804$$ −7.00000 −0.246871
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 6.00000 0.211210
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 6.00000 0.210559
$$813$$ −2.00000 −0.0701431
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ 3.00000 0.105021
$$817$$ 56.0000 1.95919
$$818$$ 25.0000 0.874105
$$819$$ −4.00000 −0.139771
$$820$$ 0 0
$$821$$ 24.0000 0.837606 0.418803 0.908077i $$-0.362450\pi$$
0.418803 + 0.908077i $$0.362450\pi$$
$$822$$ −21.0000 −0.732459
$$823$$ −26.0000 −0.906303 −0.453152 0.891434i $$-0.649700\pi$$
−0.453152 + 0.891434i $$0.649700\pi$$
$$824$$ 20.0000 0.696733
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ −9.00000 −0.312961 −0.156480 0.987681i $$-0.550015\pi$$
−0.156480 + 0.987681i $$0.550015\pi$$
$$828$$ 0 0
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ −2.00000 −0.0693375
$$833$$ −3.00000 −0.103944
$$834$$ −7.00000 −0.242390
$$835$$ 0 0
$$836$$ −21.0000 −0.726300
$$837$$ −20.0000 −0.691301
$$838$$ 3.00000 0.103633
$$839$$ −6.00000 −0.207143 −0.103572 0.994622i $$-0.533027\pi$$
−0.103572 + 0.994622i $$0.533027\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −20.0000 −0.689246
$$843$$ 18.0000 0.619953
$$844$$ 17.0000 0.585164
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ 2.00000 0.0687208
$$848$$ 12.0000 0.412082
$$849$$ −1.00000 −0.0343199
$$850$$ 0 0
$$851$$ 0 0
$$852$$ −6.00000 −0.205557
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ −10.0000 −0.342193
$$855$$ 0 0
$$856$$ 3.00000 0.102538
$$857$$ −15.0000 −0.512390 −0.256195 0.966625i $$-0.582469\pi$$
−0.256195 + 0.966625i $$0.582469\pi$$
$$858$$ −6.00000 −0.204837
$$859$$ −31.0000 −1.05771 −0.528853 0.848713i $$-0.677378\pi$$
−0.528853 + 0.848713i $$0.677378\pi$$
$$860$$ 0 0
$$861$$ −9.00000 −0.306719
$$862$$ 36.0000 1.22616
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ 11.0000 0.373795
$$867$$ 8.00000 0.271694
$$868$$ 4.00000 0.135769
$$869$$ 42.0000 1.42475
$$870$$ 0 0
$$871$$ −14.0000 −0.474372
$$872$$ −14.0000 −0.474100
$$873$$ −20.0000 −0.676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 5.00000 0.168934
$$877$$ −32.0000 −1.08056 −0.540282 0.841484i $$-0.681682\pi$$
−0.540282 + 0.841484i $$0.681682\pi$$
$$878$$ 4.00000 0.134993
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 2.00000 0.0673435
$$883$$ −47.0000 −1.58168 −0.790838 0.612026i $$-0.790355\pi$$
−0.790838 + 0.612026i $$0.790355\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 21.0000 0.705509
$$887$$ −6.00000 −0.201460 −0.100730 0.994914i $$-0.532118\pi$$
−0.100730 + 0.994914i $$0.532118\pi$$
$$888$$ −8.00000 −0.268462
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ −14.0000 −0.468755
$$893$$ −42.0000 −1.40548
$$894$$ 12.0000 0.401340
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 15.0000 0.500556
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 27.0000 0.899002
$$903$$ −8.00000 −0.266223
$$904$$ −9.00000 −0.299336
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 7.00000 0.231793
$$913$$ 27.0000 0.893570
$$914$$ 17.0000 0.562310
$$915$$ 0 0
$$916$$ 26.0000 0.859064
$$917$$ 0 0
$$918$$ 15.0000 0.495074
$$919$$ −34.0000 −1.12156 −0.560778 0.827966i $$-0.689498\pi$$
−0.560778 + 0.827966i $$0.689498\pi$$
$$920$$ 0 0
$$921$$ −7.00000 −0.230658
$$922$$ −18.0000 −0.592798
$$923$$ −12.0000 −0.394985
$$924$$ 3.00000 0.0986928
$$925$$ 0 0
$$926$$ 8.00000 0.262896
$$927$$ 40.0000 1.31377
$$928$$ 6.00000 0.196960
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ −7.00000 −0.229416
$$932$$ −6.00000 −0.196537
$$933$$ 18.0000 0.589294
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ −29.0000 −0.947389 −0.473694 0.880689i $$-0.657080\pi$$
−0.473694 + 0.880689i $$0.657080\pi$$
$$938$$ 7.00000 0.228558
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ −20.0000 −0.651635
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 24.0000 0.780307
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ −14.0000 −0.454699
$$949$$ 10.0000 0.324614
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ −3.00000 −0.0972306
$$953$$ −57.0000 −1.84641 −0.923206 0.384307i $$-0.874441\pi$$
−0.923206 + 0.384307i $$0.874441\pi$$
$$954$$ 24.0000 0.777029
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ 18.0000 0.581857
$$958$$ −18.0000 −0.581554
$$959$$ 21.0000 0.678125
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −16.0000 −0.515861
$$963$$ 6.00000 0.193347
$$964$$ −25.0000 −0.805196
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 34.0000 1.09337 0.546683 0.837340i $$-0.315890\pi$$
0.546683 + 0.837340i $$0.315890\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ −21.0000 −0.674617
$$970$$ 0 0
$$971$$ 9.00000 0.288824 0.144412 0.989518i $$-0.453871\pi$$
0.144412 + 0.989518i $$0.453871\pi$$
$$972$$ −16.0000 −0.513200
$$973$$ 7.00000 0.224410
$$974$$ −34.0000 −1.08943
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 3.00000 0.0959785 0.0479893 0.998848i $$-0.484719\pi$$
0.0479893 + 0.998848i $$0.484719\pi$$
$$978$$ −5.00000 −0.159882
$$979$$ −45.0000 −1.43821
$$980$$ 0 0
$$981$$ −28.0000 −0.893971
$$982$$ 12.0000 0.382935
$$983$$ −12.0000 −0.382741 −0.191370 0.981518i $$-0.561293\pi$$
−0.191370 + 0.981518i $$0.561293\pi$$
$$984$$ −9.00000 −0.286910
$$985$$ 0 0
$$986$$ −18.0000 −0.573237
$$987$$ 6.00000 0.190982
$$988$$ 14.0000 0.445399
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 25.0000 0.793351
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ −9.00000 −0.285176
$$997$$ −62.0000 −1.96356 −0.981780 0.190022i $$-0.939144\pi$$
−0.981780 + 0.190022i $$0.939144\pi$$
$$998$$ 28.0000 0.886325
$$999$$ −40.0000 −1.26554
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 350.2.a.a.1.1 1
3.2 odd 2 3150.2.a.x.1.1 1
4.3 odd 2 2800.2.a.x.1.1 1
5.2 odd 4 350.2.c.c.99.1 2
5.3 odd 4 350.2.c.c.99.2 2
5.4 even 2 350.2.a.e.1.1 yes 1
7.6 odd 2 2450.2.a.m.1.1 1
15.2 even 4 3150.2.g.f.2899.2 2
15.8 even 4 3150.2.g.f.2899.1 2
15.14 odd 2 3150.2.a.m.1.1 1
20.3 even 4 2800.2.g.i.449.2 2
20.7 even 4 2800.2.g.i.449.1 2
20.19 odd 2 2800.2.a.h.1.1 1
35.13 even 4 2450.2.c.h.99.2 2
35.27 even 4 2450.2.c.h.99.1 2
35.34 odd 2 2450.2.a.x.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.a.1.1 1 1.1 even 1 trivial
350.2.a.e.1.1 yes 1 5.4 even 2
350.2.c.c.99.1 2 5.2 odd 4
350.2.c.c.99.2 2 5.3 odd 4
2450.2.a.m.1.1 1 7.6 odd 2
2450.2.a.x.1.1 1 35.34 odd 2
2450.2.c.h.99.1 2 35.27 even 4
2450.2.c.h.99.2 2 35.13 even 4
2800.2.a.h.1.1 1 20.19 odd 2
2800.2.a.x.1.1 1 4.3 odd 2
2800.2.g.i.449.1 2 20.7 even 4
2800.2.g.i.449.2 2 20.3 even 4
3150.2.a.m.1.1 1 15.14 odd 2
3150.2.a.x.1.1 1 3.2 odd 2
3150.2.g.f.2899.1 2 15.8 even 4
3150.2.g.f.2899.2 2 15.2 even 4