# Properties

 Label 550.2.a.d.1.1 Level $550$ Weight $2$ Character 550.1 Self dual yes Analytic conductor $4.392$ 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] = [550,2,Mod(1,550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, 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("550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$550 = 2 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 550.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: $$4.39177211117$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 550.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} -1.00000 q^{19} -1.00000 q^{21} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} +5.00000 q^{27} +1.00000 q^{28} -9.00000 q^{29} +5.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -3.00000 q^{34} -2.00000 q^{36} -5.00000 q^{37} +1.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} +1.00000 q^{42} -8.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -3.00000 q^{51} -2.00000 q^{52} -9.00000 q^{53} -5.00000 q^{54} -1.00000 q^{56} +1.00000 q^{57} +9.00000 q^{58} +6.00000 q^{59} +5.00000 q^{61} -5.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{66} -8.00000 q^{67} +3.00000 q^{68} +6.00000 q^{69} -9.00000 q^{71} +2.00000 q^{72} +10.0000 q^{73} +5.00000 q^{74} -1.00000 q^{76} -1.00000 q^{77} -2.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -1.00000 q^{84} +8.00000 q^{86} +9.00000 q^{87} +1.00000 q^{88} -15.0000 q^{89} -2.00000 q^{91} -6.00000 q^{92} -5.00000 q^{93} +6.00000 q^{94} +1.00000 q^{96} -8.00000 q^{97} +6.00000 q^{98} +2.00000 q^{99} +O(q^{100})$$

## 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 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ −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 $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 2.00000 0.471405
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 1.00000 0.213201
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 5.00000 0.962250
$$28$$ 1.00000 0.188982
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 1.00000 0.174078
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ −5.00000 −0.821995 −0.410997 0.911636i $$-0.634819\pi$$
−0.410997 + 0.911636i $$0.634819\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ −2.00000 −0.277350
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ 1.00000 0.132453
$$58$$ 9.00000 1.18176
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ −5.00000 −0.635001
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −9.00000 −1.06810 −0.534052 0.845452i $$-0.679331\pi$$
−0.534052 + 0.845452i $$0.679331\pi$$
$$72$$ 2.00000 0.235702
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 5.00000 0.581238
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −1.00000 −0.113961
$$78$$ −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
$$82$$ 6.00000 0.662589
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 9.00000 0.964901
$$88$$ 1.00000 0.106600
$$89$$ −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
$$92$$ −6.00000 −0.625543
$$93$$ −5.00000 −0.518476
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 6.00000 0.606092
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 3.00000 0.297044
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 5.00000 0.481125
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 5.00000 0.474579
$$112$$ 1.00000 0.0944911
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ −9.00000 −0.835629
$$117$$ 4.00000 0.369800
$$118$$ −6.00000 −0.552345
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −5.00000 −0.452679
$$123$$ 6.00000 0.541002
$$124$$ 5.00000 0.449013
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ −1.00000 −0.0867110
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 9.00000 0.755263
$$143$$ 2.00000 0.167248
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 6.00000 0.494872
$$148$$ −5.00000 −0.410997
$$149$$ 21.0000 1.72039 0.860194 0.509968i $$-0.170343\pi$$
0.860194 + 0.509968i $$0.170343\pi$$
$$150$$ 0 0
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ −6.00000 −0.485071
$$154$$ 1.00000 0.0805823
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ −5.00000 −0.399043 −0.199522 0.979893i $$-0.563939\pi$$
−0.199522 + 0.979893i $$0.563939\pi$$
$$158$$ −14.0000 −1.11378
$$159$$ 9.00000 0.713746
$$160$$ 0 0
$$161$$ −6.00000 −0.472866
$$162$$ −1.00000 −0.0785674
$$163$$ −5.00000 −0.391630 −0.195815 0.980641i $$-0.562735\pi$$
−0.195815 + 0.980641i $$0.562735\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 9.00000 0.696441 0.348220 0.937413i $$-0.386786\pi$$
0.348220 + 0.937413i $$0.386786\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ −8.00000 −0.609994
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ −9.00000 −0.682288
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ −6.00000 −0.450988
$$178$$ 15.0000 1.12430
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 2.00000 0.148250
$$183$$ −5.00000 −0.369611
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 5.00000 0.366618
$$187$$ −3.00000 −0.219382
$$188$$ −6.00000 −0.437595
$$189$$ 5.00000 0.363696
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 7.00000 0.503871 0.251936 0.967744i $$-0.418933\pi$$
0.251936 + 0.967744i $$0.418933\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ −25.0000 −1.77220 −0.886102 0.463491i $$-0.846597\pi$$
−0.886102 + 0.463491i $$0.846597\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 18.0000 1.26648
$$203$$ −9.00000 −0.631676
$$204$$ −3.00000 −0.210042
$$205$$ 0 0
$$206$$ −16.0000 −1.11477
$$207$$ 12.0000 0.834058
$$208$$ −2.00000 −0.138675
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −13.0000 −0.894957 −0.447478 0.894295i $$-0.647678\pi$$
−0.447478 + 0.894295i $$0.647678\pi$$
$$212$$ −9.00000 −0.618123
$$213$$ 9.00000 0.616670
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −5.00000 −0.340207
$$217$$ 5.00000 0.339422
$$218$$ −2.00000 −0.135457
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ −5.00000 −0.335578
$$223$$ 10.0000 0.669650 0.334825 0.942280i $$-0.391323\pi$$
0.334825 + 0.942280i $$0.391323\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 1.00000 0.0657952
$$232$$ 9.00000 0.590879
$$233$$ 27.0000 1.76883 0.884414 0.466702i $$-0.154558\pi$$
0.884414 + 0.466702i $$0.154558\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ −14.0000 −0.909398
$$238$$ −3.00000 −0.194461
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ −16.0000 −1.02640
$$244$$ 5.00000 0.320092
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 2.00000 0.127257
$$248$$ −5.00000 −0.317500
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 6.00000 0.377217
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ −5.00000 −0.310685
$$260$$ 0 0
$$261$$ 18.0000 1.11417
$$262$$ −3.00000 −0.185341
$$263$$ −21.0000 −1.29492 −0.647458 0.762101i $$-0.724168\pi$$
−0.647458 + 0.762101i $$0.724168\pi$$
$$264$$ −1.00000 −0.0615457
$$265$$ 0 0
$$266$$ 1.00000 0.0613139
$$267$$ 15.0000 0.917985
$$268$$ −8.00000 −0.488678
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 2.00000 0.121046
$$274$$ 12.0000 0.724947
$$275$$ 0 0
$$276$$ 6.00000 0.361158
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ 4.00000 0.239904
$$279$$ −10.0000 −0.598684
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ 22.0000 1.30776 0.653882 0.756596i $$-0.273139\pi$$
0.653882 + 0.756596i $$0.273139\pi$$
$$284$$ −9.00000 −0.534052
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ −6.00000 −0.354169
$$288$$ 2.00000 0.117851
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 10.0000 0.585206
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ −6.00000 −0.349927
$$295$$ 0 0
$$296$$ 5.00000 0.290619
$$297$$ −5.00000 −0.290129
$$298$$ −21.0000 −1.21650
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ −2.00000 −0.115087
$$303$$ 18.0000 1.03407
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ −2.00000 −0.114146 −0.0570730 0.998370i $$-0.518177\pi$$
−0.0570730 + 0.998370i $$0.518177\pi$$
$$308$$ −1.00000 −0.0569803
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ −3.00000 −0.170114 −0.0850572 0.996376i $$-0.527107\pi$$
−0.0850572 + 0.996376i $$0.527107\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 5.00000 0.282166
$$315$$ 0 0
$$316$$ 14.0000 0.787562
$$317$$ 9.00000 0.505490 0.252745 0.967533i $$-0.418667\pi$$
0.252745 + 0.967533i $$0.418667\pi$$
$$318$$ −9.00000 −0.504695
$$319$$ 9.00000 0.503903
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 6.00000 0.334367
$$323$$ −3.00000 −0.166924
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 5.00000 0.276924
$$327$$ −2.00000 −0.110600
$$328$$ 6.00000 0.331295
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ 32.0000 1.75888 0.879440 0.476011i $$-0.157918\pi$$
0.879440 + 0.476011i $$0.157918\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 10.0000 0.547997
$$334$$ −9.00000 −0.492458
$$335$$ 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
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ −5.00000 −0.270765
$$342$$ −2.00000 −0.108148
$$343$$ −13.0000 −0.701934
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ −18.0000 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$348$$ 9.00000 0.482451
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ −10.0000 −0.533761
$$352$$ 1.00000 0.0533002
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 6.00000 0.318896
$$355$$ 0 0
$$356$$ −15.0000 −0.794998
$$357$$ −3.00000 −0.158777
$$358$$ −24.0000 −1.26844
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ −14.0000 −0.735824
$$363$$ −1.00000 −0.0524864
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 5.00000 0.261354
$$367$$ 4.00000 0.208798 0.104399 0.994535i $$-0.466708\pi$$
0.104399 + 0.994535i $$0.466708\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ 12.0000 0.624695
$$370$$ 0 0
$$371$$ −9.00000 −0.467257
$$372$$ −5.00000 −0.259238
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 3.00000 0.155126
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ 18.0000 0.927047
$$378$$ −5.00000 −0.257172
$$379$$ −10.0000 −0.513665 −0.256833 0.966456i $$-0.582679\pi$$
−0.256833 + 0.966456i $$0.582679\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ −12.0000 −0.613973
$$383$$ −30.0000 −1.53293 −0.766464 0.642287i $$-0.777986\pi$$
−0.766464 + 0.642287i $$0.777986\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −7.00000 −0.356291
$$387$$ 16.0000 0.813326
$$388$$ −8.00000 −0.406138
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 6.00000 0.303046
$$393$$ −3.00000 −0.151330
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 25.0000 1.25314
$$399$$ 1.00000 0.0500626
$$400$$ 0 0
$$401$$ 3.00000 0.149813 0.0749064 0.997191i $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ −10.0000 −0.498135
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 9.00000 0.446663
$$407$$ 5.00000 0.247841
$$408$$ 3.00000 0.148522
$$409$$ 32.0000 1.58230 0.791149 0.611623i $$-0.209483\pi$$
0.791149 + 0.611623i $$0.209483\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ 16.0000 0.788263
$$413$$ 6.00000 0.295241
$$414$$ −12.0000 −0.589768
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 4.00000 0.195881
$$418$$ −1.00000 −0.0489116
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −28.0000 −1.36464 −0.682318 0.731055i $$-0.739028\pi$$
−0.682318 + 0.731055i $$0.739028\pi$$
$$422$$ 13.0000 0.632830
$$423$$ 12.0000 0.583460
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ −9.00000 −0.436051
$$427$$ 5.00000 0.241967
$$428$$ 12.0000 0.580042
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 5.00000 0.240563
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ −5.00000 −0.240008
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 6.00000 0.287019
$$438$$ 10.0000 0.477818
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 12.0000 0.571429
$$442$$ 6.00000 0.285391
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 5.00000 0.237289
$$445$$ 0 0
$$446$$ −10.0000 −0.473514
$$447$$ −21.0000 −0.993266
$$448$$ 1.00000 0.0472456
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 6.00000 0.282529
$$452$$ 12.0000 0.564433
$$453$$ −2.00000 −0.0939682
$$454$$ 18.0000 0.844782
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 1.00000 0.0467780 0.0233890 0.999726i $$-0.492554\pi$$
0.0233890 + 0.999726i $$0.492554\pi$$
$$458$$ 22.0000 1.02799
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ −15.0000 −0.698620 −0.349310 0.937007i $$-0.613584\pi$$
−0.349310 + 0.937007i $$0.613584\pi$$
$$462$$ −1.00000 −0.0465242
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ −9.00000 −0.417815
$$465$$ 0 0
$$466$$ −27.0000 −1.25075
$$467$$ 15.0000 0.694117 0.347059 0.937843i $$-0.387180\pi$$
0.347059 + 0.937843i $$0.387180\pi$$
$$468$$ 4.00000 0.184900
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 5.00000 0.230388
$$472$$ −6.00000 −0.276172
$$473$$ 8.00000 0.367840
$$474$$ 14.0000 0.643041
$$475$$ 0 0
$$476$$ 3.00000 0.137505
$$477$$ 18.0000 0.824163
$$478$$ 6.00000 0.274434
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 10.0000 0.455961
$$482$$ −14.0000 −0.637683
$$483$$ 6.00000 0.273009
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 16.0000 0.725775
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ −5.00000 −0.226339
$$489$$ 5.00000 0.226108
$$490$$ 0 0
$$491$$ −9.00000 −0.406164 −0.203082 0.979162i $$-0.565096\pi$$
−0.203082 + 0.979162i $$0.565096\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −27.0000 −1.21602
$$494$$ −2.00000 −0.0899843
$$495$$ 0 0
$$496$$ 5.00000 0.224507
$$497$$ −9.00000 −0.403705
$$498$$ 6.00000 0.268866
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ 0 0
$$501$$ −9.00000 −0.402090
$$502$$ 18.0000 0.803379
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ −6.00000 −0.266733
$$507$$ 9.00000 0.399704
$$508$$ 16.0000 0.709885
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 10.0000 0.442374
$$512$$ −1.00000 −0.0441942
$$513$$ −5.00000 −0.220755
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 6.00000 0.263880
$$518$$ 5.00000 0.219687
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ −18.0000 −0.787839
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 3.00000 0.131056
$$525$$ 0 0
$$526$$ 21.0000 0.915644
$$527$$ 15.0000 0.653410
$$528$$ 1.00000 0.0435194
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ −1.00000 −0.0433555
$$533$$ 12.0000 0.519778
$$534$$ −15.0000 −0.649113
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ −24.0000 −1.03568
$$538$$ 12.0000 0.517357
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −13.0000 −0.558914 −0.279457 0.960158i $$-0.590154\pi$$
−0.279457 + 0.960158i $$0.590154\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ −14.0000 −0.600798
$$544$$ −3.00000 −0.128624
$$545$$ 0 0
$$546$$ −2.00000 −0.0855921
$$547$$ −44.0000 −1.88130 −0.940652 0.339372i $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 9.00000 0.383413
$$552$$ −6.00000 −0.255377
$$553$$ 14.0000 0.595341
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 10.0000 0.423334
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ 3.00000 0.126660
$$562$$ −18.0000 −0.759284
$$563$$ −6.00000 −0.252870 −0.126435 0.991975i $$-0.540353\pi$$
−0.126435 + 0.991975i $$0.540353\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 0 0
$$566$$ −22.0000 −0.924729
$$567$$ 1.00000 0.0419961
$$568$$ 9.00000 0.377632
$$569$$ −12.0000 −0.503066 −0.251533 0.967849i $$-0.580935\pi$$
−0.251533 + 0.967849i $$0.580935\pi$$
$$570$$ 0 0
$$571$$ −31.0000 −1.29731 −0.648655 0.761083i $$-0.724668\pi$$
−0.648655 + 0.761083i $$0.724668\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ −12.0000 −0.501307
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ −26.0000 −1.08239 −0.541197 0.840896i $$-0.682029\pi$$
−0.541197 + 0.840896i $$0.682029\pi$$
$$578$$ 8.00000 0.332756
$$579$$ −7.00000 −0.290910
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ −8.00000 −0.331611
$$583$$ 9.00000 0.372742
$$584$$ −10.0000 −0.413803
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ −27.0000 −1.11441 −0.557205 0.830375i $$-0.688126\pi$$
−0.557205 + 0.830375i $$0.688126\pi$$
$$588$$ 6.00000 0.247436
$$589$$ −5.00000 −0.206021
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ −5.00000 −0.205499
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 5.00000 0.205152
$$595$$ 0 0
$$596$$ 21.0000 0.860194
$$597$$ 25.0000 1.02318
$$598$$ −12.0000 −0.490716
$$599$$ −3.00000 −0.122577 −0.0612883 0.998120i $$-0.519521\pi$$
−0.0612883 + 0.998120i $$0.519521\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 8.00000 0.326056
$$603$$ 16.0000 0.651570
$$604$$ 2.00000 0.0813788
$$605$$ 0 0
$$606$$ −18.0000 −0.731200
$$607$$ −5.00000 −0.202944 −0.101472 0.994838i $$-0.532355\pi$$
−0.101472 + 0.994838i $$0.532355\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 9.00000 0.364698
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ −6.00000 −0.242536
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ 2.00000 0.0807134
$$615$$ 0 0
$$616$$ 1.00000 0.0402911
$$617$$ 36.0000 1.44931 0.724653 0.689114i $$-0.242000\pi$$
0.724653 + 0.689114i $$0.242000\pi$$
$$618$$ 16.0000 0.643614
$$619$$ −16.0000 −0.643094 −0.321547 0.946894i $$-0.604203\pi$$
−0.321547 + 0.946894i $$0.604203\pi$$
$$620$$ 0 0
$$621$$ −30.0000 −1.20386
$$622$$ 3.00000 0.120289
$$623$$ −15.0000 −0.600962
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ 14.0000 0.559553
$$627$$ −1.00000 −0.0399362
$$628$$ −5.00000 −0.199522
$$629$$ −15.0000 −0.598089
$$630$$ 0 0
$$631$$ −1.00000 −0.0398094 −0.0199047 0.999802i $$-0.506336\pi$$
−0.0199047 + 0.999802i $$0.506336\pi$$
$$632$$ −14.0000 −0.556890
$$633$$ 13.0000 0.516704
$$634$$ −9.00000 −0.357436
$$635$$ 0 0
$$636$$ 9.00000 0.356873
$$637$$ 12.0000 0.475457
$$638$$ −9.00000 −0.356313
$$639$$ 18.0000 0.712069
$$640$$ 0 0
$$641$$ −9.00000 −0.355479 −0.177739 0.984078i $$-0.556878\pi$$
−0.177739 + 0.984078i $$0.556878\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −5.00000 −0.197181 −0.0985904 0.995128i $$-0.531433\pi$$
−0.0985904 + 0.995128i $$0.531433\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 0 0
$$646$$ 3.00000 0.118033
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −6.00000 −0.235521
$$650$$ 0 0
$$651$$ −5.00000 −0.195965
$$652$$ −5.00000 −0.195815
$$653$$ 9.00000 0.352197 0.176099 0.984373i $$-0.443652\pi$$
0.176099 + 0.984373i $$0.443652\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ −20.0000 −0.780274
$$658$$ 6.00000 0.233904
$$659$$ −27.0000 −1.05177 −0.525885 0.850555i $$-0.676266\pi$$
−0.525885 + 0.850555i $$0.676266\pi$$
$$660$$ 0 0
$$661$$ −34.0000 −1.32245 −0.661223 0.750189i $$-0.729962\pi$$
−0.661223 + 0.750189i $$0.729962\pi$$
$$662$$ −32.0000 −1.24372
$$663$$ 6.00000 0.233021
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 54.0000 2.09089
$$668$$ 9.00000 0.348220
$$669$$ −10.0000 −0.386622
$$670$$ 0 0
$$671$$ −5.00000 −0.193023
$$672$$ 1.00000 0.0385758
$$673$$ −41.0000 −1.58043 −0.790217 0.612827i $$-0.790032\pi$$
−0.790217 + 0.612827i $$0.790032\pi$$
$$674$$ −13.0000 −0.500741
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 12.0000 0.460857
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 5.00000 0.191460
$$683$$ 9.00000 0.344375 0.172188 0.985064i $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ 13.0000 0.496342
$$687$$ 22.0000 0.839352
$$688$$ −8.00000 −0.304997
$$689$$ 18.0000 0.685745
$$690$$ 0 0
$$691$$ −10.0000 −0.380418 −0.190209 0.981744i $$-0.560917\pi$$
−0.190209 + 0.981744i $$0.560917\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 2.00000 0.0759737
$$694$$ 18.0000 0.683271
$$695$$ 0 0
$$696$$ −9.00000 −0.341144
$$697$$ −18.0000 −0.681799
$$698$$ −2.00000 −0.0757011
$$699$$ −27.0000 −1.02123
$$700$$ 0 0
$$701$$ −3.00000 −0.113308 −0.0566542 0.998394i $$-0.518043\pi$$
−0.0566542 + 0.998394i $$0.518043\pi$$
$$702$$ 10.0000 0.377426
$$703$$ 5.00000 0.188579
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ −18.0000 −0.676960
$$708$$ −6.00000 −0.225494
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ 0 0
$$711$$ −28.0000 −1.05008
$$712$$ 15.0000 0.562149
$$713$$ −30.0000 −1.12351
$$714$$ 3.00000 0.112272
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 6.00000 0.224074
$$718$$ −24.0000 −0.895672
$$719$$ −15.0000 −0.559406 −0.279703 0.960087i $$-0.590236\pi$$
−0.279703 + 0.960087i $$0.590236\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 18.0000 0.669891
$$723$$ −14.0000 −0.520666
$$724$$ 14.0000 0.520306
$$725$$ 0 0
$$726$$ 1.00000 0.0371135
$$727$$ −26.0000 −0.964287 −0.482143 0.876092i $$-0.660142\pi$$
−0.482143 + 0.876092i $$0.660142\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ −5.00000 −0.184805
$$733$$ 4.00000 0.147743 0.0738717 0.997268i $$-0.476464\pi$$
0.0738717 + 0.997268i $$0.476464\pi$$
$$734$$ −4.00000 −0.147643
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ 8.00000 0.294684
$$738$$ −12.0000 −0.441726
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 9.00000 0.330400
$$743$$ 9.00000 0.330178 0.165089 0.986279i $$-0.447209\pi$$
0.165089 + 0.986279i $$0.447209\pi$$
$$744$$ 5.00000 0.183309
$$745$$ 0 0
$$746$$ 14.0000 0.512576
$$747$$ −12.0000 −0.439057
$$748$$ −3.00000 −0.109691
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ −7.00000 −0.255434 −0.127717 0.991811i $$-0.540765\pi$$
−0.127717 + 0.991811i $$0.540765\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ 18.0000 0.655956
$$754$$ −18.0000 −0.655521
$$755$$ 0 0
$$756$$ 5.00000 0.181848
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 10.0000 0.363216
$$759$$ −6.00000 −0.217786
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 2.00000 0.0724049
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 30.0000 1.08394
$$767$$ −12.0000 −0.433295
$$768$$ −1.00000 −0.0360844
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 7.00000 0.251936
$$773$$ −45.0000 −1.61854 −0.809269 0.587439i $$-0.800136\pi$$
−0.809269 + 0.587439i $$0.800136\pi$$
$$774$$ −16.0000 −0.575108
$$775$$ 0 0
$$776$$ 8.00000 0.287183
$$777$$ 5.00000 0.179374
$$778$$ −6.00000 −0.215110
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 9.00000 0.322045
$$782$$ 18.0000 0.643679
$$783$$ −45.0000 −1.60817
$$784$$ −6.00000 −0.214286
$$785$$ 0 0
$$786$$ 3.00000 0.107006
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ 21.0000 0.747620
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ −2.00000 −0.0710669
$$793$$ −10.0000 −0.355110
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −25.0000 −0.886102
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ −1.00000 −0.0353996
$$799$$ −18.0000 −0.636794
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ −3.00000 −0.105934
$$803$$ −10.0000 −0.352892
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ 10.0000 0.352235
$$807$$ 12.0000 0.422420
$$808$$ 18.0000 0.633238
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ −43.0000 −1.50993 −0.754967 0.655763i $$-0.772347\pi$$
−0.754967 + 0.655763i $$0.772347\pi$$
$$812$$ −9.00000 −0.315838
$$813$$ −20.0000 −0.701431
$$814$$ −5.00000 −0.175250
$$815$$ 0 0
$$816$$ −3.00000 −0.105021
$$817$$ 8.00000 0.279885
$$818$$ −32.0000 −1.11885
$$819$$ 4.00000 0.139771
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ −12.0000 −0.418548
$$823$$ 40.0000 1.39431 0.697156 0.716919i $$-0.254448\pi$$
0.697156 + 0.716919i $$0.254448\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ −6.00000 −0.208767
$$827$$ 48.0000 1.66912 0.834562 0.550914i $$-0.185721\pi$$
0.834562 + 0.550914i $$0.185721\pi$$
$$828$$ 12.0000 0.417029
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ −2.00000 −0.0693375
$$833$$ −18.0000 −0.623663
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 1.00000 0.0345857
$$837$$ 25.0000 0.864126
$$838$$ −12.0000 −0.414533
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 28.0000 0.964944
$$843$$ −18.0000 −0.619953
$$844$$ −13.0000 −0.447478
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 1.00000 0.0343604
$$848$$ −9.00000 −0.309061
$$849$$ −22.0000 −0.755038
$$850$$ 0 0
$$851$$ 30.0000 1.02839
$$852$$ 9.00000 0.308335
$$853$$ −38.0000 −1.30110 −0.650548 0.759465i $$-0.725461\pi$$
−0.650548 + 0.759465i $$0.725461\pi$$
$$854$$ −5.00000 −0.171096
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 2.00000 0.0682789
$$859$$ 26.0000 0.887109 0.443554 0.896248i $$-0.353717\pi$$
0.443554 + 0.896248i $$0.353717\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ 0 0
$$863$$ 6.00000 0.204242 0.102121 0.994772i $$-0.467437\pi$$
0.102121 + 0.994772i $$0.467437\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ 8.00000 0.271694
$$868$$ 5.00000 0.169711
$$869$$ −14.0000 −0.474917
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ −2.00000 −0.0677285
$$873$$ 16.0000 0.541518
$$874$$ −6.00000 −0.202953
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ 34.0000 1.14810 0.574049 0.818821i $$-0.305372\pi$$
0.574049 + 0.818821i $$0.305372\pi$$
$$878$$ 28.0000 0.944954
$$879$$ 30.0000 1.01187
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ −12.0000 −0.404061
$$883$$ 25.0000 0.841317 0.420658 0.907219i $$-0.361799\pi$$
0.420658 + 0.907219i $$0.361799\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ −5.00000 −0.167789
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 10.0000 0.334825
$$893$$ 6.00000 0.200782
$$894$$ 21.0000 0.702345
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ −12.0000 −0.400668
$$898$$ 6.00000 0.200223
$$899$$ −45.0000 −1.50083
$$900$$ 0 0
$$901$$ −27.0000 −0.899500
$$902$$ −6.00000 −0.199778
$$903$$ 8.00000 0.266223
$$904$$ −12.0000 −0.399114
$$905$$ 0 0
$$906$$ 2.00000 0.0664455
$$907$$ 7.00000 0.232431 0.116216 0.993224i $$-0.462924\pi$$
0.116216 + 0.993224i $$0.462924\pi$$
$$908$$ −18.0000 −0.597351
$$909$$ 36.0000 1.19404
$$910$$ 0 0
$$911$$ 3.00000 0.0993944 0.0496972 0.998764i $$-0.484174\pi$$
0.0496972 + 0.998764i $$0.484174\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ −6.00000 −0.198571
$$914$$ −1.00000 −0.0330771
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 3.00000 0.0990687
$$918$$ −15.0000 −0.495074
$$919$$ −52.0000 −1.71532 −0.857661 0.514216i $$-0.828083\pi$$
−0.857661 + 0.514216i $$0.828083\pi$$
$$920$$ 0 0
$$921$$ 2.00000 0.0659022
$$922$$ 15.0000 0.493999
$$923$$ 18.0000 0.592477
$$924$$ 1.00000 0.0328976
$$925$$ 0 0
$$926$$ 14.0000 0.460069
$$927$$ −32.0000 −1.05102
$$928$$ 9.00000 0.295439
$$929$$ −21.0000 −0.688988 −0.344494 0.938789i $$-0.611949\pi$$
−0.344494 + 0.938789i $$0.611949\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 27.0000 0.884414
$$933$$ 3.00000 0.0982156
$$934$$ −15.0000 −0.490815
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ 8.00000 0.261209
$$939$$ 14.0000 0.456873
$$940$$ 0 0
$$941$$ 27.0000 0.880175 0.440087 0.897955i $$-0.354947\pi$$
0.440087 + 0.897955i $$0.354947\pi$$
$$942$$ −5.00000 −0.162909
$$943$$ 36.0000 1.17232
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ −3.00000 −0.0974869 −0.0487435 0.998811i $$-0.515522\pi$$
−0.0487435 + 0.998811i $$0.515522\pi$$
$$948$$ −14.0000 −0.454699
$$949$$ −20.0000 −0.649227
$$950$$ 0 0
$$951$$ −9.00000 −0.291845
$$952$$ −3.00000 −0.0972306
$$953$$ 21.0000 0.680257 0.340128 0.940379i $$-0.389529\pi$$
0.340128 + 0.940379i $$0.389529\pi$$
$$954$$ −18.0000 −0.582772
$$955$$ 0 0
$$956$$ −6.00000 −0.194054
$$957$$ −9.00000 −0.290929
$$958$$ 0 0
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ −10.0000 −0.322413
$$963$$ −24.0000 −0.773389
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ −6.00000 −0.193047
$$967$$ −17.0000 −0.546683 −0.273342 0.961917i $$-0.588129\pi$$
−0.273342 + 0.961917i $$0.588129\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ 3.00000 0.0963739
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ −16.0000 −0.513200
$$973$$ −4.00000 −0.128234
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ 5.00000 0.160046
$$977$$ 48.0000 1.53566 0.767828 0.640656i $$-0.221338\pi$$
0.767828 + 0.640656i $$0.221338\pi$$
$$978$$ −5.00000 −0.159882
$$979$$ 15.0000 0.479402
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ 9.00000 0.287202
$$983$$ 42.0000 1.33959 0.669796 0.742545i $$-0.266382\pi$$
0.669796 + 0.742545i $$0.266382\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 27.0000 0.859855
$$987$$ 6.00000 0.190982
$$988$$ 2.00000 0.0636285
$$989$$ 48.0000 1.52631
$$990$$ 0 0
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ −5.00000 −0.158750
$$993$$ −32.0000 −1.01549
$$994$$ 9.00000 0.285463
$$995$$ 0 0
$$996$$ −6.00000 −0.190117
$$997$$ 28.0000 0.886769 0.443384 0.896332i $$-0.353778\pi$$
0.443384 + 0.896332i $$0.353778\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ −25.0000 −0.790965
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 550.2.a.d.1.1 1
3.2 odd 2 4950.2.a.bm.1.1 1
4.3 odd 2 4400.2.a.t.1.1 1
5.2 odd 4 550.2.b.c.199.1 2
5.3 odd 4 550.2.b.c.199.2 2
5.4 even 2 110.2.a.c.1.1 1
11.10 odd 2 6050.2.a.bc.1.1 1
15.2 even 4 4950.2.c.s.199.2 2
15.8 even 4 4950.2.c.s.199.1 2
15.14 odd 2 990.2.a.f.1.1 1
20.3 even 4 4400.2.b.j.4049.2 2
20.7 even 4 4400.2.b.j.4049.1 2
20.19 odd 2 880.2.a.d.1.1 1
35.34 odd 2 5390.2.a.x.1.1 1
40.19 odd 2 3520.2.a.ba.1.1 1
40.29 even 2 3520.2.a.k.1.1 1
55.54 odd 2 1210.2.a.e.1.1 1
60.59 even 2 7920.2.a.bc.1.1 1
220.219 even 2 9680.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.c.1.1 1 5.4 even 2
550.2.a.d.1.1 1 1.1 even 1 trivial
550.2.b.c.199.1 2 5.2 odd 4
550.2.b.c.199.2 2 5.3 odd 4
880.2.a.d.1.1 1 20.19 odd 2
990.2.a.f.1.1 1 15.14 odd 2
1210.2.a.e.1.1 1 55.54 odd 2
3520.2.a.k.1.1 1 40.29 even 2
3520.2.a.ba.1.1 1 40.19 odd 2
4400.2.a.t.1.1 1 4.3 odd 2
4400.2.b.j.4049.1 2 20.7 even 4
4400.2.b.j.4049.2 2 20.3 even 4
4950.2.a.bm.1.1 1 3.2 odd 2
4950.2.c.s.199.1 2 15.8 even 4
4950.2.c.s.199.2 2 15.2 even 4
5390.2.a.x.1.1 1 35.34 odd 2
6050.2.a.bc.1.1 1 11.10 odd 2
7920.2.a.bc.1.1 1 60.59 even 2
9680.2.a.g.1.1 1 220.219 even 2