# Properties

 Label 102.2.a.c.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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -10.0000 q^{29} -2.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} -2.00000 q^{40} +10.0000 q^{41} +12.0000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} +4.00000 q^{57} -10.0000 q^{58} +12.0000 q^{59} -2.00000 q^{60} -10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} +1.00000 q^{68} +1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} -8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000 q^{83} -2.00000 q^{85} +12.0000 q^{86} -10.0000 q^{87} -4.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} +8.00000 q^{93} -8.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −2.00000 −0.632456
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536
$$18$$ 1.00000 0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −1.00000 −0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −4.00000 −0.696311
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 4.00000 0.648886
$$39$$ −2.00000 −0.320256
$$40$$ −2.00000 −0.316228
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −7.00000 −1.00000
$$50$$ −1.00000 −0.141421
$$51$$ 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
$$55$$ 8.00000 1.07872
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ −10.0000 −1.31306
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 4.00000 0.496139
$$66$$ −4.00000 −0.492366
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −1.00000 −0.115470
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 12.0000 1.29399
$$87$$ −10.0000 −1.07211
$$88$$ −4.00000 −0.426401
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ −8.00000 −0.820783
$$96$$ 1.00000 0.102062
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ −4.00000 −0.402015
$$100$$ −1.00000 −0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 8.00000 0.762770
$$111$$ −2.00000 −0.189832
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −10.0000 −0.928477
$$117$$ −2.00000 −0.184900
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ −2.00000 −0.182574
$$121$$ 5.00000 0.454545
$$122$$ −10.0000 −0.905357
$$123$$ 10.0000 0.901670
$$124$$ 8.00000 0.718421
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 12.0000 1.05654
$$130$$ 4.00000 0.350823
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ 0 0
$$134$$ −12.0000 −1.03664
$$135$$ −2.00000 −0.172133
$$136$$ 1.00000 0.0857493
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ 1.00000 0.0833333
$$145$$ 20.0000 1.66091
$$146$$ 10.0000 0.827606
$$147$$ −7.00000 −0.577350
$$148$$ −2.00000 −0.164399
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 1.00000 0.0808452
$$154$$ 0 0
$$155$$ −16.0000 −1.28515
$$156$$ −2.00000 −0.160128
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 6.00000 0.475831
$$160$$ −2.00000 −0.158114
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 8.00000 0.622799
$$166$$ 4.00000 0.310460
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −2.00000 −0.153393
$$171$$ 4.00000 0.305888
$$172$$ 12.0000 0.914991
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −10.0000 −0.758098
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 12.0000 0.901975
$$178$$ −6.00000 −0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ −2.00000 −0.149071
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 0 0
$$185$$ 4.00000 0.294086
$$186$$ 8.00000 0.586588
$$187$$ −4.00000 −0.292509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ −8.00000 −0.580381
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 4.00000 0.286446
$$196$$ −7.00000 −0.500000
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −12.0000 −0.846415
$$202$$ −10.0000 −0.703598
$$203$$ 0 0
$$204$$ 1.00000 0.0700140
$$205$$ −20.0000 −1.39686
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ −24.0000 −1.63679
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ 10.0000 0.675737
$$220$$ 8.00000 0.539360
$$221$$ −2.00000 −0.134535
$$222$$ −2.00000 −0.134231
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 2.00000 0.133038
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −10.0000 −0.656532
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 1.00000 0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 14.0000 0.894427
$$246$$ 10.0000 0.637577
$$247$$ −8.00000 −0.509028
$$248$$ 8.00000 0.508001
$$249$$ 4.00000 0.253490
$$250$$ 12.0000 0.758947
$$251$$ 28.0000 1.76734 0.883672 0.468106i $$-0.155064\pi$$
0.883672 + 0.468106i $$0.155064\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −2.00000 −0.125245
$$256$$ 1.00000 0.0625000
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 0 0
$$260$$ 4.00000 0.248069
$$261$$ −10.0000 −0.618984
$$262$$ −12.0000 −0.741362
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ −12.0000 −0.737154
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ −12.0000 −0.733017
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ 30.0000 1.80253 0.901263 0.433273i $$-0.142641\pi$$
0.901263 + 0.433273i $$0.142641\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 0 0
$$285$$ −8.00000 −0.473879
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 20.0000 1.17444
$$291$$ −14.0000 −0.820695
$$292$$ 10.0000 0.585206
$$293$$ −26.0000 −1.51894 −0.759468 0.650545i $$-0.774541\pi$$
−0.759468 + 0.650545i $$0.774541\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ −24.0000 −1.39733
$$296$$ −2.00000 −0.116248
$$297$$ −4.00000 −0.232104
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 24.0000 1.38104
$$303$$ −10.0000 −0.574485
$$304$$ 4.00000 0.229416
$$305$$ 20.0000 1.14520
$$306$$ 1.00000 0.0571662
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ −16.0000 −0.908739
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 40.0000 2.23957
$$320$$ −2.00000 −0.111803
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 4.00000 0.222566
$$324$$ 1.00000 0.0555556
$$325$$ 2.00000 0.110940
$$326$$ 4.00000 0.221540
$$327$$ −10.0000 −0.553001
$$328$$ 10.0000 0.552158
$$329$$ 0 0
$$330$$ 8.00000 0.440386
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 4.00000 0.219529
$$333$$ −2.00000 −0.109599
$$334$$ 16.0000 0.875481
$$335$$ 24.0000 1.31126
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 2.00000 0.108625
$$340$$ −2.00000 −0.108465
$$341$$ −32.0000 −1.73290
$$342$$ 4.00000 0.216295
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ −10.0000 −0.536056
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ −4.00000 −0.213201
$$353$$ −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
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ −2.00000 −0.105409
$$361$$ −3.00000 −0.157895
$$362$$ 14.0000 0.735824
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ −20.0000 −1.04685
$$366$$ −10.0000 −0.522708
$$367$$ 24.0000 1.25279 0.626395 0.779506i $$-0.284530\pi$$
0.626395 + 0.779506i $$0.284530\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 4.00000 0.207950
$$371$$ 0 0
$$372$$ 8.00000 0.414781
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ 20.0000 1.03005
$$378$$ 0 0
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ −8.00000 −0.410391
$$381$$ 0 0
$$382$$ −16.0000 −0.818631
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 18.0000 0.916176
$$387$$ 12.0000 0.609994
$$388$$ −14.0000 −0.710742
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 4.00000 0.202548
$$391$$ 0 0
$$392$$ −7.00000 −0.353553
$$393$$ −12.0000 −0.605320
$$394$$ 14.0000 0.705310
$$395$$ 16.0000 0.805047
$$396$$ −4.00000 −0.201008
$$397$$ −26.0000 −1.30490 −0.652451 0.757831i $$-0.726259\pi$$
−0.652451 + 0.757831i $$0.726259\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ −12.0000 −0.598506
$$403$$ −16.0000 −0.797017
$$404$$ −10.0000 −0.497519
$$405$$ −2.00000 −0.0993808
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ 1.00000 0.0495074
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ −20.0000 −0.987730
$$411$$ 10.0000 0.493264
$$412$$ −8.00000 −0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ −2.00000 −0.0980581
$$417$$ −4.00000 −0.195881
$$418$$ −16.0000 −0.782586
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ −28.0000 −1.36302
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ −1.00000 −0.0485071
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −4.00000 −0.193347
$$429$$ 8.00000 0.386244
$$430$$ −24.0000 −1.15738
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ 20.0000 0.958927
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ 10.0000 0.477818
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 8.00000 0.381385
$$441$$ −7.00000 −0.333333
$$442$$ −2.00000 −0.0951303
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 12.0000 0.568855
$$446$$ 16.0000 0.757622
$$447$$ −10.0000 −0.472984
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −40.0000 −1.88353
$$452$$ 2.00000 0.0940721
$$453$$ 24.0000 1.12762
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −26.0000 −1.21490
$$459$$ 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
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ −10.0000 −0.464238
$$465$$ −16.0000 −0.741982
$$466$$ 26.0000 1.20443
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 12.0000 0.552345
$$473$$ −48.0000 −2.20704
$$474$$ −8.00000 −0.367452
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ 4.00000 0.182384
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 28.0000 1.27141
$$486$$ 1.00000 0.0453609
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 4.00000 0.180886
$$490$$ 14.0000 0.632456
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 10.0000 0.450835
$$493$$ −10.0000 −0.450377
$$494$$ −8.00000 −0.359937
$$495$$ 8.00000 0.359573
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 4.00000 0.179244
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 16.0000 0.714827
$$502$$ 28.0000 1.24970
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 20.0000 0.889988
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ −2.00000 −0.0885615
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 4.00000 0.176604
$$514$$ 2.00000 0.0882162
$$515$$ 16.0000 0.705044
$$516$$ 12.0000 0.528271
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 4.00000 0.175412
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ −10.0000 −0.437688
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 8.00000 0.348485
$$528$$ −4.00000 −0.174078
$$529$$ −23.0000 −1.00000
$$530$$ −12.0000 −0.521247
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ −20.0000 −0.866296
$$534$$ −6.00000 −0.259645
$$535$$ 8.00000 0.345870
$$536$$ −12.0000 −0.518321
$$537$$ −12.0000 −0.517838
$$538$$ 6.00000 0.258678
$$539$$ 28.0000 1.20605
$$540$$ −2.00000 −0.0860663
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 14.0000 0.600798
$$544$$ 1.00000 0.0428746
$$545$$ 20.0000 0.856706
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 10.0000 0.427179
$$549$$ −10.0000 −0.426790
$$550$$ 4.00000 0.170561
$$551$$ −40.0000 −1.70406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 30.0000 1.27458
$$555$$ 4.00000 0.169791
$$556$$ −4.00000 −0.169638
$$557$$ −34.0000 −1.44063 −0.720313 0.693649i $$-0.756002\pi$$
−0.720313 + 0.693649i $$0.756002\pi$$
$$558$$ 8.00000 0.338667
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ −6.00000 −0.253095
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 0 0
$$565$$ −4.00000 −0.168281
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ −8.00000 −0.335083
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ 8.00000 0.334497
$$573$$ −16.0000 −0.668410
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 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
$$579$$ 18.0000 0.748054
$$580$$ 20.0000 0.830455
$$581$$ 0 0
$$582$$ −14.0000 −0.580319
$$583$$ −24.0000 −0.993978
$$584$$ 10.0000 0.413803
$$585$$ 4.00000 0.165380
$$586$$ −26.0000 −1.07405
$$587$$ −20.0000 −0.825488 −0.412744 0.910847i $$-0.635430\pi$$
−0.412744 + 0.910847i $$0.635430\pi$$
$$588$$ −7.00000 −0.288675
$$589$$ 32.0000 1.31854
$$590$$ −24.0000 −0.988064
$$591$$ 14.0000 0.575883
$$592$$ −2.00000 −0.0821995
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 0 0
$$603$$ −12.0000 −0.488678
$$604$$ 24.0000 0.976546
$$605$$ −10.0000 −0.406558
$$606$$ −10.0000 −0.406222
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ 20.0000 0.809776
$$611$$ 0 0
$$612$$ 1.00000 0.0404226
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ −20.0000 −0.806478
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ −16.0000 −0.642575
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ −2.00000 −0.0800641
$$625$$ −19.0000 −0.760000
$$626$$ 10.0000 0.399680
$$627$$ −16.0000 −0.638978
$$628$$ −2.00000 −0.0798087
$$629$$ −2.00000 −0.0797452
$$630$$ 0 0
$$631$$ −24.0000 −0.955425 −0.477712 0.878516i $$-0.658534\pi$$
−0.477712 + 0.878516i $$0.658534\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ −28.0000 −1.11290
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ 14.0000 0.554700
$$638$$ 40.0000 1.58362
$$639$$ 0 0
$$640$$ −2.00000 −0.0790569
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ −4.00000 −0.157867
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ 0 0
$$645$$ −24.0000 −0.944999
$$646$$ 4.00000 0.157378
$$647$$ −40.0000 −1.57256 −0.786281 0.617869i $$-0.787996\pi$$
−0.786281 + 0.617869i $$0.787996\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −48.0000 −1.88416
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 24.0000 0.937758
$$656$$ 10.0000 0.390434
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 8.00000 0.311400
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ −2.00000 −0.0776736
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 16.0000 0.619059
$$669$$ 16.0000 0.618596
$$670$$ 24.0000 0.927201
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ −46.0000 −1.77317 −0.886585 0.462566i $$-0.846929\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ −1.00000 −0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ 46.0000 1.76792 0.883962 0.467559i $$-0.154866\pi$$
0.883962 + 0.467559i $$0.154866\pi$$
$$678$$ 2.00000 0.0768095
$$679$$ 0 0
$$680$$ −2.00000 −0.0766965
$$681$$ 4.00000 0.153280
$$682$$ −32.0000 −1.22534
$$683$$ −20.0000 −0.765279 −0.382639 0.923898i $$-0.624985\pi$$
−0.382639 + 0.923898i $$0.624985\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −20.0000 −0.764161
$$686$$ 0 0
$$687$$ −26.0000 −0.991962
$$688$$ 12.0000 0.457496
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ 8.00000 0.303457
$$696$$ −10.0000 −0.379049
$$697$$ 10.0000 0.378777
$$698$$ 14.0000 0.529908
$$699$$ 26.0000 0.983410
$$700$$ 0 0
$$701$$ 46.0000 1.73740 0.868698 0.495342i $$-0.164957\pi$$
0.868698 + 0.495342i $$0.164957\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −8.00000 −0.301726
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ 0 0
$$708$$ 12.0000 0.450988
$$709$$ 46.0000 1.72757 0.863783 0.503864i $$-0.168089\pi$$
0.863783 + 0.503864i $$0.168089\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ −6.00000 −0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −16.0000 −0.598366
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ −2.00000 −0.0745356
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ 2.00000 0.0743808
$$724$$ 14.0000 0.520306
$$725$$ 10.0000 0.371391
$$726$$ 5.00000 0.185567
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −20.0000 −0.740233
$$731$$ 12.0000 0.443836
$$732$$ −10.0000 −0.369611
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ 24.0000 0.885856
$$735$$ 14.0000 0.516398
$$736$$ 0 0
$$737$$ 48.0000 1.76810
$$738$$ 10.0000 0.368105
$$739$$ 52.0000 1.91285 0.956425 0.291977i $$-0.0943129\pi$$
0.956425 + 0.291977i $$0.0943129\pi$$
$$740$$ 4.00000 0.147043
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 20.0000 0.732743
$$746$$ 6.00000 0.219676
$$747$$ 4.00000 0.146352
$$748$$ −4.00000 −0.146254
$$749$$ 0 0
$$750$$ 12.0000 0.438178
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 28.0000 1.02038
$$754$$ 20.0000 0.728357
$$755$$ −48.0000 −1.74690
$$756$$ 0 0
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 0 0
$$760$$ −8.00000 −0.290191
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ −2.00000 −0.0723102
$$766$$ −16.0000 −0.578103
$$767$$ −24.0000 −0.866590
$$768$$ 1.00000 0.0360844
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 2.00000 0.0720282
$$772$$ 18.0000 0.647834
$$773$$ 38.0000 1.36677 0.683383 0.730061i $$-0.260508\pi$$
0.683383 + 0.730061i $$0.260508\pi$$
$$774$$ 12.0000 0.431331
$$775$$ −8.00000 −0.287368
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ −26.0000 −0.932145
$$779$$ 40.0000 1.43315
$$780$$ 4.00000 0.143223
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −10.0000 −0.357371
$$784$$ −7.00000 −0.250000
$$785$$ 4.00000 0.142766
$$786$$ −12.0000 −0.428026
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 14.0000 0.498729
$$789$$ −8.00000 −0.284808
$$790$$ 16.0000 0.569254
$$791$$ 0 0
$$792$$ −4.00000 −0.142134
$$793$$ 20.0000 0.710221
$$794$$ −26.0000 −0.922705
$$795$$ −12.0000 −0.425596
$$796$$ 0 0
$$797$$ 14.0000 0.495905 0.247953 0.968772i $$-0.420242\pi$$
0.247953 + 0.968772i $$0.420242\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ −6.00000 −0.212000
$$802$$ −14.0000 −0.494357
$$803$$ −40.0000 −1.41157
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 6.00000 0.211210
$$808$$ −10.0000 −0.351799
$$809$$ −22.0000 −0.773479 −0.386739 0.922189i $$-0.626399\pi$$
−0.386739 + 0.922189i $$0.626399\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ 16.0000 0.561144
$$814$$ 8.00000 0.280400
$$815$$ −8.00000 −0.280228
$$816$$ 1.00000 0.0350070
$$817$$ 48.0000 1.67931
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ −20.0000 −0.698430
$$821$$ −34.0000 −1.18661 −0.593304 0.804978i $$-0.702177\pi$$
−0.593304 + 0.804978i $$0.702177\pi$$
$$822$$ 10.0000 0.348790
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ −20.0000 −0.695468 −0.347734 0.937593i $$-0.613049\pi$$
−0.347734 + 0.937593i $$0.613049\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ −8.00000 −0.277684
$$831$$ 30.0000 1.04069
$$832$$ −2.00000 −0.0693375
$$833$$ −7.00000 −0.242536
$$834$$ −4.00000 −0.138509
$$835$$ −32.0000 −1.10741
$$836$$ −16.0000 −0.553372
$$837$$ 8.00000 0.276520
$$838$$ 4.00000 0.138178
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 22.0000 0.758170
$$843$$ −6.00000 −0.206651
$$844$$ −28.0000 −0.963800
$$845$$ 18.0000 0.619219
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ 12.0000 0.411839
$$850$$ −1.00000 −0.0342997
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −2.00000 −0.0684787 −0.0342393 0.999414i $$-0.510901\pi$$
−0.0342393 + 0.999414i $$0.510901\pi$$
$$854$$ 0 0
$$855$$ −8.00000 −0.273594
$$856$$ −4.00000 −0.136717
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 8.00000 0.273115
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ −24.0000 −0.818393
$$861$$ 0 0
$$862$$ −8.00000 −0.272481
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −12.0000 −0.408012
$$866$$ −14.0000 −0.475739
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 20.0000 0.678064
$$871$$ 24.0000 0.813209
$$872$$ −10.0000 −0.338643
$$873$$ −14.0000 −0.473828
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 10.0000 0.337869
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 16.0000 0.539974
$$879$$ −26.0000 −0.876958
$$880$$ 8.00000 0.269680
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ −24.0000 −0.806751
$$886$$ −4.00000 −0.134383
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 0 0
$$890$$ 12.0000 0.402241
$$891$$ −4.00000 −0.134005
$$892$$ 16.0000 0.535720
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 2.00000 0.0667409
$$899$$ −80.0000 −2.66815
$$900$$ −1.00000 −0.0333333
$$901$$ 6.00000 0.199889
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ −28.0000 −0.930751
$$906$$ 24.0000 0.797347
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 4.00000 0.132745
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −16.0000 −0.529523
$$914$$ 10.0000 0.330771
$$915$$ 20.0000 0.661180
$$916$$ −26.0000 −0.859064
$$917$$ 0 0
$$918$$ 1.00000 0.0330049
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ 32.0000 1.05159
$$927$$ −8.00000 −0.262754
$$928$$ −10.0000 −0.328266
$$929$$ 2.00000 0.0656179 0.0328089 0.999462i $$-0.489555\pi$$
0.0328089 + 0.999462i $$0.489555\pi$$
$$930$$ −16.0000 −0.524661
$$931$$ −28.0000 −0.917663
$$932$$ 26.0000 0.851658
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 8.00000 0.261628
$$936$$ −2.00000 −0.0653720
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −10.0000 −0.325991 −0.162995 0.986627i $$-0.552116\pi$$
−0.162995 + 0.986627i $$0.552116\pi$$
$$942$$ −2.00000 −0.0651635
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −48.0000 −1.56061
$$947$$ −28.0000 −0.909878 −0.454939 0.890523i $$-0.650339\pi$$
−0.454939 + 0.890523i $$0.650339\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ −20.0000 −0.649227
$$950$$ −4.00000 −0.129777
$$951$$ 6.00000 0.194563
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 32.0000 1.03550
$$956$$ 0 0
$$957$$ 40.0000 1.29302
$$958$$ 24.0000 0.775405
$$959$$ 0 0
$$960$$ −2.00000 −0.0645497
$$961$$ 33.0000 1.06452
$$962$$ 4.00000 0.128965
$$963$$ −4.00000 −0.128898
$$964$$ 2.00000 0.0644157
$$965$$ −36.0000 −1.15888
$$966$$ 0 0
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 4.00000 0.128499
$$970$$ 28.0000 0.899026
$$971$$ −4.00000 −0.128366 −0.0641831 0.997938i $$-0.520444\pi$$
−0.0641831 + 0.997938i $$0.520444\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 2.00000 0.0640513
$$976$$ −10.0000 −0.320092
$$977$$ −46.0000 −1.47167 −0.735835 0.677161i $$-0.763210\pi$$
−0.735835 + 0.677161i $$0.763210\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 24.0000 0.767043
$$980$$ 14.0000 0.447214
$$981$$ −10.0000 −0.319275
$$982$$ 12.0000 0.382935
$$983$$ 48.0000 1.53096 0.765481 0.643458i $$-0.222501\pi$$
0.765481 + 0.643458i $$0.222501\pi$$
$$984$$ 10.0000 0.318788
$$985$$ −28.0000 −0.892154
$$986$$ −10.0000 −0.318465
$$987$$ 0 0
$$988$$ −8.00000 −0.254514
$$989$$ 0 0
$$990$$ 8.00000 0.254257
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ 8.00000 0.254000
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ 4.00000 0.126618
$$999$$ −2.00000 −0.0632772
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.c.1.1 1
3.2 odd 2 306.2.a.b.1.1 1
4.3 odd 2 816.2.a.b.1.1 1
5.2 odd 4 2550.2.d.m.2449.2 2
5.3 odd 4 2550.2.d.m.2449.1 2
5.4 even 2 2550.2.a.c.1.1 1
7.6 odd 2 4998.2.a.be.1.1 1
8.3 odd 2 3264.2.a.bc.1.1 1
8.5 even 2 3264.2.a.m.1.1 1
12.11 even 2 2448.2.a.p.1.1 1
15.14 odd 2 7650.2.a.ca.1.1 1
17.2 even 8 1734.2.f.e.1483.2 4
17.4 even 4 1734.2.b.b.577.1 2
17.8 even 8 1734.2.f.e.829.1 4
17.9 even 8 1734.2.f.e.829.2 4
17.13 even 4 1734.2.b.b.577.2 2
17.15 even 8 1734.2.f.e.1483.1 4
17.16 even 2 1734.2.a.j.1.1 1
24.5 odd 2 9792.2.a.k.1.1 1
24.11 even 2 9792.2.a.l.1.1 1
51.50 odd 2 5202.2.a.c.1.1 1

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