# Properties

 Label 435.2.a.d.1.1 Level $435$ Weight $2$ Character 435.1 Self dual yes Analytic conductor $3.473$ 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] = [435,2,Mod(1,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("435.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$435 = 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 435.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: $$3.47349248793$$ 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 $$\chi$$ $$=$$ 435.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{21} -4.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +1.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} +5.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +4.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} -3.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -6.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{55} -12.0000 q^{56} -4.00000 q^{57} +1.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} +6.00000 q^{65} -4.00000 q^{66} +8.00000 q^{67} -6.00000 q^{68} -4.00000 q^{69} +4.00000 q^{70} -8.00000 q^{71} -3.00000 q^{72} -2.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -16.0000 q^{77} +6.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +8.00000 q^{83} -4.00000 q^{84} +6.00000 q^{85} +4.00000 q^{86} +1.00000 q^{87} +12.0000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +24.0000 q^{91} +4.00000 q^{92} -8.00000 q^{93} -4.00000 q^{95} +5.00000 q^{96} -2.00000 q^{97} +9.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.00000 0.408248
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 1.00000 0.258199
$$16$$ −1.00000 −0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 4.00000 0.872872
$$22$$ −4.00000 −0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 1.00000 0.200000
$$26$$ 6.00000 1.17670
$$27$$ 1.00000 0.192450
$$28$$ −4.00000 −0.755929
$$29$$ 1.00000 0.185695
$$30$$ 1.00000 0.182574
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 5.00000 0.883883
$$33$$ −4.00000 −0.696311
$$34$$ 6.00000 1.02899
$$35$$ 4.00000 0.676123
$$36$$ −1.00000 −0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 6.00000 0.960769
$$40$$ −3.00000 −0.474342
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 4.00000 0.617213
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 1.00000 0.149071
$$46$$ −4.00000 −0.589768
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 9.00000 1.28571
$$50$$ 1.00000 0.141421
$$51$$ 6.00000 0.840168
$$52$$ −6.00000 −0.832050
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −4.00000 −0.539360
$$56$$ −12.0000 −1.60357
$$57$$ −4.00000 −0.529813
$$58$$ 1.00000 0.131306
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 4.00000 0.503953
$$64$$ 7.00000 0.875000
$$65$$ 6.00000 0.744208
$$66$$ −4.00000 −0.492366
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ −4.00000 −0.481543
$$70$$ 4.00000 0.478091
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 1.00000 0.115470
$$76$$ 4.00000 0.458831
$$77$$ −16.0000 −1.82337
$$78$$ 6.00000 0.679366
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 6.00000 0.650791
$$86$$ 4.00000 0.431331
$$87$$ 1.00000 0.107211
$$88$$ 12.0000 1.27920
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 24.0000 2.51588
$$92$$ 4.00000 0.417029
$$93$$ −8.00000 −0.829561
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 5.00000 0.510310
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 9.00000 0.909137
$$99$$ −4.00000 −0.402015
$$100$$ −1.00000 −0.100000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ −18.0000 −1.76505
$$105$$ 4.00000 0.390360
$$106$$ −10.0000 −0.971286
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 2.00000 0.189832
$$112$$ −4.00000 −0.377964
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ −4.00000 −0.373002
$$116$$ −1.00000 −0.0928477
$$117$$ 6.00000 0.554700
$$118$$ −12.0000 −1.10469
$$119$$ 24.0000 2.20008
$$120$$ −3.00000 −0.273861
$$121$$ 5.00000 0.454545
$$122$$ −10.0000 −0.905357
$$123$$ −6.00000 −0.541002
$$124$$ 8.00000 0.718421
$$125$$ 1.00000 0.0894427
$$126$$ 4.00000 0.356348
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 4.00000 0.352180
$$130$$ 6.00000 0.526235
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 4.00000 0.348155
$$133$$ −16.0000 −1.38738
$$134$$ 8.00000 0.691095
$$135$$ 1.00000 0.0860663
$$136$$ −18.0000 −1.54349
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ −24.0000 −2.00698
$$144$$ −1.00000 −0.0833333
$$145$$ 1.00000 0.0830455
$$146$$ −2.00000 −0.165521
$$147$$ 9.00000 0.742307
$$148$$ −2.00000 −0.164399
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 12.0000 0.973329
$$153$$ 6.00000 0.485071
$$154$$ −16.0000 −1.28932
$$155$$ −8.00000 −0.642575
$$156$$ −6.00000 −0.480384
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 0 0
$$159$$ −10.0000 −0.793052
$$160$$ 5.00000 0.395285
$$161$$ −16.0000 −1.26098
$$162$$ 1.00000 0.0785674
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ 6.00000 0.468521
$$165$$ −4.00000 −0.311400
$$166$$ 8.00000 0.620920
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −12.0000 −0.925820
$$169$$ 23.0000 1.76923
$$170$$ 6.00000 0.460179
$$171$$ −4.00000 −0.305888
$$172$$ −4.00000 −0.304997
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 1.00000 0.0758098
$$175$$ 4.00000 0.302372
$$176$$ 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 $$0.648028\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 24.0000 1.77900
$$183$$ −10.0000 −0.739221
$$184$$ 12.0000 0.884652
$$185$$ 2.00000 0.147043
$$186$$ −8.00000 −0.586588
$$187$$ −24.0000 −1.75505
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ −4.00000 −0.290191
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 7.00000 0.505181
$$193$$ 22.0000 1.58359 0.791797 0.610784i $$-0.209146\pi$$
0.791797 + 0.610784i $$0.209146\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 6.00000 0.429669
$$196$$ −9.00000 −0.642857
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 8.00000 0.564276
$$202$$ −2.00000 −0.140720
$$203$$ 4.00000 0.280745
$$204$$ −6.00000 −0.420084
$$205$$ −6.00000 −0.419058
$$206$$ −12.0000 −0.836080
$$207$$ −4.00000 −0.278019
$$208$$ −6.00000 −0.416025
$$209$$ 16.0000 1.10674
$$210$$ 4.00000 0.276026
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 10.0000 0.686803
$$213$$ −8.00000 −0.548151
$$214$$ 8.00000 0.546869
$$215$$ 4.00000 0.272798
$$216$$ −3.00000 −0.204124
$$217$$ −32.0000 −2.17230
$$218$$ 14.0000 0.948200
$$219$$ −2.00000 −0.135147
$$220$$ 4.00000 0.269680
$$221$$ 36.0000 2.42162
$$222$$ 2.00000 0.134231
$$223$$ 28.0000 1.87502 0.937509 0.347960i $$-0.113126\pi$$
0.937509 + 0.347960i $$0.113126\pi$$
$$224$$ 20.0000 1.33631
$$225$$ 1.00000 0.0666667
$$226$$ −2.00000 −0.133038
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ −16.0000 −1.05272
$$232$$ −3.00000 −0.196960
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 24.0000 1.55569
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 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
$$245$$ 9.00000 0.574989
$$246$$ −6.00000 −0.382546
$$247$$ −24.0000 −1.52708
$$248$$ 24.0000 1.52400
$$249$$ 8.00000 0.506979
$$250$$ 1.00000 0.0632456
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ 16.0000 1.00591
$$254$$ −16.0000 −1.00393
$$255$$ 6.00000 0.375735
$$256$$ −17.0000 −1.06250
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 8.00000 0.497096
$$260$$ −6.00000 −0.372104
$$261$$ 1.00000 0.0618984
$$262$$ 12.0000 0.741362
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 12.0000 0.738549
$$265$$ −10.0000 −0.614295
$$266$$ −16.0000 −0.981023
$$267$$ −6.00000 −0.367194
$$268$$ −8.00000 −0.488678
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 24.0000 1.45255
$$274$$ −2.00000 −0.120824
$$275$$ −4.00000 −0.241209
$$276$$ 4.00000 0.240772
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ 4.00000 0.239904
$$279$$ −8.00000 −0.478947
$$280$$ −12.0000 −0.717137
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 8.00000 0.475551 0.237775 0.971320i $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ 8.00000 0.474713
$$285$$ −4.00000 −0.236940
$$286$$ −24.0000 −1.41915
$$287$$ −24.0000 −1.41668
$$288$$ 5.00000 0.294628
$$289$$ 19.0000 1.11765
$$290$$ 1.00000 0.0587220
$$291$$ −2.00000 −0.117242
$$292$$ 2.00000 0.117041
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 9.00000 0.524891
$$295$$ −12.0000 −0.698667
$$296$$ −6.00000 −0.348743
$$297$$ −4.00000 −0.232104
$$298$$ 6.00000 0.347571
$$299$$ −24.0000 −1.38796
$$300$$ −1.00000 −0.0577350
$$301$$ 16.0000 0.922225
$$302$$ 16.0000 0.920697
$$303$$ −2.00000 −0.114897
$$304$$ 4.00000 0.229416
$$305$$ −10.0000 −0.572598
$$306$$ 6.00000 0.342997
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 16.0000 0.911685
$$309$$ −12.0000 −0.682656
$$310$$ −8.00000 −0.454369
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ −18.0000 −1.01905
$$313$$ −30.0000 −1.69570 −0.847850 0.530236i $$-0.822103\pi$$
−0.847850 + 0.530236i $$0.822103\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 4.00000 0.225374
$$316$$ 0 0
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ −4.00000 −0.223957
$$320$$ 7.00000 0.391312
$$321$$ 8.00000 0.446516
$$322$$ −16.0000 −0.891645
$$323$$ −24.0000 −1.33540
$$324$$ −1.00000 −0.0555556
$$325$$ 6.00000 0.332820
$$326$$ −12.0000 −0.664619
$$327$$ 14.0000 0.774202
$$328$$ 18.0000 0.993884
$$329$$ 0 0
$$330$$ −4.00000 −0.220193
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ −8.00000 −0.439057
$$333$$ 2.00000 0.109599
$$334$$ 12.0000 0.656611
$$335$$ 8.00000 0.437087
$$336$$ −4.00000 −0.218218
$$337$$ 30.0000 1.63420 0.817102 0.576493i $$-0.195579\pi$$
0.817102 + 0.576493i $$0.195579\pi$$
$$338$$ 23.0000 1.25104
$$339$$ −2.00000 −0.108625
$$340$$ −6.00000 −0.325396
$$341$$ 32.0000 1.73290
$$342$$ −4.00000 −0.216295
$$343$$ 8.00000 0.431959
$$344$$ −12.0000 −0.646997
$$345$$ −4.00000 −0.215353
$$346$$ 6.00000 0.322562
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ −1.00000 −0.0536056
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 6.00000 0.320256
$$352$$ −20.0000 −1.06600
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ −8.00000 −0.424596
$$356$$ 6.00000 0.317999
$$357$$ 24.0000 1.27021
$$358$$ −12.0000 −0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ −3.00000 −0.158114
$$361$$ −3.00000 −0.157895
$$362$$ 22.0000 1.15629
$$363$$ 5.00000 0.262432
$$364$$ −24.0000 −1.25794
$$365$$ −2.00000 −0.104685
$$366$$ −10.0000 −0.522708
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −6.00000 −0.312348
$$370$$ 2.00000 0.103975
$$371$$ −40.0000 −2.07670
$$372$$ 8.00000 0.414781
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 6.00000 0.309016
$$378$$ 4.00000 0.205738
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 4.00000 0.205196
$$381$$ −16.0000 −0.819705
$$382$$ −16.0000 −0.818631
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ −16.0000 −0.815436
$$386$$ 22.0000 1.11977
$$387$$ 4.00000 0.203331
$$388$$ 2.00000 0.101535
$$389$$ −2.00000 −0.101404 −0.0507020 0.998714i $$-0.516146\pi$$
−0.0507020 + 0.998714i $$0.516146\pi$$
$$390$$ 6.00000 0.303822
$$391$$ −24.0000 −1.21373
$$392$$ −27.0000 −1.36371
$$393$$ 12.0000 0.605320
$$394$$ 14.0000 0.705310
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ −10.0000 −0.501886 −0.250943 0.968002i $$-0.580741\pi$$
−0.250943 + 0.968002i $$0.580741\pi$$
$$398$$ −24.0000 −1.20301
$$399$$ −16.0000 −0.801002
$$400$$ −1.00000 −0.0500000
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 8.00000 0.399004
$$403$$ −48.0000 −2.39105
$$404$$ 2.00000 0.0995037
$$405$$ 1.00000 0.0496904
$$406$$ 4.00000 0.198517
$$407$$ −8.00000 −0.396545
$$408$$ −18.0000 −0.891133
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ −6.00000 −0.296319
$$411$$ −2.00000 −0.0986527
$$412$$ 12.0000 0.591198
$$413$$ −48.0000 −2.36193
$$414$$ −4.00000 −0.196589
$$415$$ 8.00000 0.392705
$$416$$ 30.0000 1.47087
$$417$$ 4.00000 0.195881
$$418$$ 16.0000 0.782586
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ −4.00000 −0.195180
$$421$$ −18.0000 −0.877266 −0.438633 0.898666i $$-0.644537\pi$$
−0.438633 + 0.898666i $$0.644537\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 0 0
$$424$$ 30.0000 1.45693
$$425$$ 6.00000 0.291043
$$426$$ −8.00000 −0.387601
$$427$$ −40.0000 −1.93574
$$428$$ −8.00000 −0.386695
$$429$$ −24.0000 −1.15873
$$430$$ 4.00000 0.192897
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ −32.0000 −1.53605
$$435$$ 1.00000 0.0479463
$$436$$ −14.0000 −0.670478
$$437$$ 16.0000 0.765384
$$438$$ −2.00000 −0.0955637
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 12.0000 0.572078
$$441$$ 9.00000 0.428571
$$442$$ 36.0000 1.71235
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ −6.00000 −0.284427
$$446$$ 28.0000 1.32584
$$447$$ 6.00000 0.283790
$$448$$ 28.0000 1.32288
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 24.0000 1.13012
$$452$$ 2.00000 0.0940721
$$453$$ 16.0000 0.751746
$$454$$ 24.0000 1.12638
$$455$$ 24.0000 1.12514
$$456$$ 12.0000 0.561951
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ 6.00000 0.280056
$$460$$ 4.00000 0.186501
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ −16.0000 −0.744387
$$463$$ 36.0000 1.67306 0.836531 0.547920i $$-0.184580\pi$$
0.836531 + 0.547920i $$0.184580\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ −8.00000 −0.370991
$$466$$ −22.0000 −1.01913
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 36.0000 1.65703
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ −24.0000 −1.10004
$$477$$ −10.0000 −0.457869
$$478$$ −8.00000 −0.365911
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 5.00000 0.228218
$$481$$ 12.0000 0.547153
$$482$$ 2.00000 0.0910975
$$483$$ −16.0000 −0.728025
$$484$$ −5.00000 −0.227273
$$485$$ −2.00000 −0.0908153
$$486$$ 1.00000 0.0453609
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ 30.0000 1.35804
$$489$$ −12.0000 −0.542659
$$490$$ 9.00000 0.406579
$$491$$ −44.0000 −1.98569 −0.992846 0.119401i $$-0.961903\pi$$
−0.992846 + 0.119401i $$0.961903\pi$$
$$492$$ 6.00000 0.270501
$$493$$ 6.00000 0.270226
$$494$$ −24.0000 −1.07981
$$495$$ −4.00000 −0.179787
$$496$$ 8.00000 0.359211
$$497$$ −32.0000 −1.43540
$$498$$ 8.00000 0.358489
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 12.0000 0.536120
$$502$$ −20.0000 −0.892644
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ −12.0000 −0.534522
$$505$$ −2.00000 −0.0889988
$$506$$ 16.0000 0.711287
$$507$$ 23.0000 1.02147
$$508$$ 16.0000 0.709885
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 6.00000 0.265684
$$511$$ −8.00000 −0.353899
$$512$$ −11.0000 −0.486136
$$513$$ −4.00000 −0.176604
$$514$$ −22.0000 −0.970378
$$515$$ −12.0000 −0.528783
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 8.00000 0.351500
$$519$$ 6.00000 0.263371
$$520$$ −18.0000 −0.789352
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 1.00000 0.0437688
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 4.00000 0.174574
$$526$$ 0 0
$$527$$ −48.0000 −2.09091
$$528$$ 4.00000 0.174078
$$529$$ −7.00000 −0.304348
$$530$$ −10.0000 −0.434372
$$531$$ −12.0000 −0.520756
$$532$$ 16.0000 0.693688
$$533$$ −36.0000 −1.55933
$$534$$ −6.00000 −0.259645
$$535$$ 8.00000 0.345870
$$536$$ −24.0000 −1.03664
$$537$$ −12.0000 −0.517838
$$538$$ 6.00000 0.258678
$$539$$ −36.0000 −1.55063
$$540$$ −1.00000 −0.0430331
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 22.0000 0.944110
$$544$$ 30.0000 1.28624
$$545$$ 14.0000 0.599694
$$546$$ 24.0000 1.02711
$$547$$ 40.0000 1.71028 0.855138 0.518400i $$-0.173472\pi$$
0.855138 + 0.518400i $$0.173472\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ −10.0000 −0.426790
$$550$$ −4.00000 −0.170561
$$551$$ −4.00000 −0.170406
$$552$$ 12.0000 0.510754
$$553$$ 0 0
$$554$$ −26.0000 −1.10463
$$555$$ 2.00000 0.0848953
$$556$$ −4.00000 −0.169638
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 24.0000 1.01509
$$560$$ −4.00000 −0.169031
$$561$$ −24.0000 −1.01328
$$562$$ 10.0000 0.421825
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 0 0
$$565$$ −2.00000 −0.0841406
$$566$$ 8.00000 0.336265
$$567$$ 4.00000 0.167984
$$568$$ 24.0000 1.00702
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 24.0000 1.00349
$$573$$ −16.0000 −0.668410
$$574$$ −24.0000 −1.00174
$$575$$ −4.00000 −0.166812
$$576$$ 7.00000 0.291667
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 22.0000 0.914289
$$580$$ −1.00000 −0.0415227
$$581$$ 32.0000 1.32758
$$582$$ −2.00000 −0.0829027
$$583$$ 40.0000 1.65663
$$584$$ 6.00000 0.248282
$$585$$ 6.00000 0.248069
$$586$$ 18.0000 0.743573
$$587$$ −16.0000 −0.660391 −0.330195 0.943913i $$-0.607115\pi$$
−0.330195 + 0.943913i $$0.607115\pi$$
$$588$$ −9.00000 −0.371154
$$589$$ 32.0000 1.31854
$$590$$ −12.0000 −0.494032
$$591$$ 14.0000 0.575883
$$592$$ −2.00000 −0.0821995
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 24.0000 0.983904
$$596$$ −6.00000 −0.245770
$$597$$ −24.0000 −0.982255
$$598$$ −24.0000 −0.981433
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ −14.0000 −0.571072 −0.285536 0.958368i $$-0.592172\pi$$
−0.285536 + 0.958368i $$0.592172\pi$$
$$602$$ 16.0000 0.652111
$$603$$ 8.00000 0.325785
$$604$$ −16.0000 −0.651031
$$605$$ 5.00000 0.203279
$$606$$ −2.00000 −0.0812444
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ −20.0000 −0.811107
$$609$$ 4.00000 0.162088
$$610$$ −10.0000 −0.404888
$$611$$ 0 0
$$612$$ −6.00000 −0.242536
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ 12.0000 0.484281
$$615$$ −6.00000 −0.241943
$$616$$ 48.0000 1.93398
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ −12.0000 −0.482711
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 8.00000 0.321288
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ −24.0000 −0.961540
$$624$$ −6.00000 −0.240192
$$625$$ 1.00000 0.0400000
$$626$$ −30.0000 −1.19904
$$627$$ 16.0000 0.638978
$$628$$ −18.0000 −0.718278
$$629$$ 12.0000 0.478471
$$630$$ 4.00000 0.159364
$$631$$ 24.0000 0.955425 0.477712 0.878516i $$-0.341466\pi$$
0.477712 + 0.878516i $$0.341466\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 2.00000 0.0794301
$$635$$ −16.0000 −0.634941
$$636$$ 10.0000 0.396526
$$637$$ 54.0000 2.13956
$$638$$ −4.00000 −0.158362
$$639$$ −8.00000 −0.316475
$$640$$ −3.00000 −0.118585
$$641$$ 34.0000 1.34292 0.671460 0.741041i $$-0.265668\pi$$
0.671460 + 0.741041i $$0.265668\pi$$
$$642$$ 8.00000 0.315735
$$643$$ 16.0000 0.630978 0.315489 0.948929i $$-0.397831\pi$$
0.315489 + 0.948929i $$0.397831\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 4.00000 0.157500
$$646$$ −24.0000 −0.944267
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ 48.0000 1.88416
$$650$$ 6.00000 0.235339
$$651$$ −32.0000 −1.25418
$$652$$ 12.0000 0.469956
$$653$$ 50.0000 1.95665 0.978326 0.207072i $$-0.0663936\pi$$
0.978326 + 0.207072i $$0.0663936\pi$$
$$654$$ 14.0000 0.547443
$$655$$ 12.0000 0.468879
$$656$$ 6.00000 0.234261
$$657$$ −2.00000 −0.0780274
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 4.00000 0.155700
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 36.0000 1.39812
$$664$$ −24.0000 −0.931381
$$665$$ −16.0000 −0.620453
$$666$$ 2.00000 0.0774984
$$667$$ −4.00000 −0.154881
$$668$$ −12.0000 −0.464294
$$669$$ 28.0000 1.08254
$$670$$ 8.00000 0.309067
$$671$$ 40.0000 1.54418
$$672$$ 20.0000 0.771517
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 30.0000 1.15556
$$675$$ 1.00000 0.0384900
$$676$$ −23.0000 −0.884615
$$677$$ 50.0000 1.92166 0.960828 0.277145i $$-0.0893883\pi$$
0.960828 + 0.277145i $$0.0893883\pi$$
$$678$$ −2.00000 −0.0768095
$$679$$ −8.00000 −0.307012
$$680$$ −18.0000 −0.690268
$$681$$ 24.0000 0.919682
$$682$$ 32.0000 1.22534
$$683$$ 40.0000 1.53056 0.765279 0.643699i $$-0.222601\pi$$
0.765279 + 0.643699i $$0.222601\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −2.00000 −0.0764161
$$686$$ 8.00000 0.305441
$$687$$ −2.00000 −0.0763048
$$688$$ −4.00000 −0.152499
$$689$$ −60.0000 −2.28582
$$690$$ −4.00000 −0.152277
$$691$$ 44.0000 1.67384 0.836919 0.547326i $$-0.184354\pi$$
0.836919 + 0.547326i $$0.184354\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ −16.0000 −0.607790
$$694$$ 16.0000 0.607352
$$695$$ 4.00000 0.151729
$$696$$ −3.00000 −0.113715
$$697$$ −36.0000 −1.36360
$$698$$ −34.0000 −1.28692
$$699$$ −22.0000 −0.832116
$$700$$ −4.00000 −0.151186
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 6.00000 0.226455
$$703$$ −8.00000 −0.301726
$$704$$ −28.0000 −1.05529
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ −8.00000 −0.300871
$$708$$ 12.0000 0.450988
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 0 0
$$712$$ 18.0000 0.674579
$$713$$ 32.0000 1.19841
$$714$$ 24.0000 0.898177
$$715$$ −24.0000 −0.897549
$$716$$ 12.0000 0.448461
$$717$$ −8.00000 −0.298765
$$718$$ 0 0
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ −48.0000 −1.78761
$$722$$ −3.00000 −0.111648
$$723$$ 2.00000 0.0743808
$$724$$ −22.0000 −0.817624
$$725$$ 1.00000 0.0371391
$$726$$ 5.00000 0.185567
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ −72.0000 −2.66850
$$729$$ 1.00000 0.0370370
$$730$$ −2.00000 −0.0740233
$$731$$ 24.0000 0.887672
$$732$$ 10.0000 0.369611
$$733$$ 2.00000 0.0738717 0.0369358 0.999318i $$-0.488240\pi$$
0.0369358 + 0.999318i $$0.488240\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 9.00000 0.331970
$$736$$ −20.0000 −0.737210
$$737$$ −32.0000 −1.17874
$$738$$ −6.00000 −0.220863
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ −2.00000 −0.0735215
$$741$$ −24.0000 −0.881662
$$742$$ −40.0000 −1.46845
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ 24.0000 0.879883
$$745$$ 6.00000 0.219823
$$746$$ −2.00000 −0.0732252
$$747$$ 8.00000 0.292705
$$748$$ 24.0000 0.877527
$$749$$ 32.0000 1.16925
$$750$$ 1.00000 0.0365148
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 0 0
$$753$$ −20.0000 −0.728841
$$754$$ 6.00000 0.218507
$$755$$ 16.0000 0.582300
$$756$$ −4.00000 −0.145479
$$757$$ −14.0000 −0.508839 −0.254419 0.967094i $$-0.581884\pi$$
−0.254419 + 0.967094i $$0.581884\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ 16.0000 0.580763
$$760$$ 12.0000 0.435286
$$761$$ 26.0000 0.942499 0.471250 0.882000i $$-0.343803\pi$$
0.471250 + 0.882000i $$0.343803\pi$$
$$762$$ −16.0000 −0.579619
$$763$$ 56.0000 2.02734
$$764$$ 16.0000 0.578860
$$765$$ 6.00000 0.216930
$$766$$ 20.0000 0.722629
$$767$$ −72.0000 −2.59977
$$768$$ −17.0000 −0.613435
$$769$$ 42.0000 1.51456 0.757279 0.653091i $$-0.226528\pi$$
0.757279 + 0.653091i $$0.226528\pi$$
$$770$$ −16.0000 −0.576600
$$771$$ −22.0000 −0.792311
$$772$$ −22.0000 −0.791797
$$773$$ −38.0000 −1.36677 −0.683383 0.730061i $$-0.739492\pi$$
−0.683383 + 0.730061i $$0.739492\pi$$
$$774$$ 4.00000 0.143777
$$775$$ −8.00000 −0.287368
$$776$$ 6.00000 0.215387
$$777$$ 8.00000 0.286998
$$778$$ −2.00000 −0.0717035
$$779$$ 24.0000 0.859889
$$780$$ −6.00000 −0.214834
$$781$$ 32.0000 1.14505
$$782$$ −24.0000 −0.858238
$$783$$ 1.00000 0.0357371
$$784$$ −9.00000 −0.321429
$$785$$ 18.0000 0.642448
$$786$$ 12.0000 0.428026
$$787$$ −48.0000 −1.71102 −0.855508 0.517790i $$-0.826755\pi$$
−0.855508 + 0.517790i $$0.826755\pi$$
$$788$$ −14.0000 −0.498729
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −8.00000 −0.284447
$$792$$ 12.0000 0.426401
$$793$$ −60.0000 −2.13066
$$794$$ −10.0000 −0.354887
$$795$$ −10.0000 −0.354663
$$796$$ 24.0000 0.850657
$$797$$ 50.0000 1.77109 0.885545 0.464553i $$-0.153785\pi$$
0.885545 + 0.464553i $$0.153785\pi$$
$$798$$ −16.0000 −0.566394
$$799$$ 0 0
$$800$$ 5.00000 0.176777
$$801$$ −6.00000 −0.212000
$$802$$ 2.00000 0.0706225
$$803$$ 8.00000 0.282314
$$804$$ −8.00000 −0.282138
$$805$$ −16.0000 −0.563926
$$806$$ −48.0000 −1.69073
$$807$$ 6.00000 0.211210
$$808$$ 6.00000 0.211079
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ −4.00000 −0.140372
$$813$$ 16.0000 0.561144
$$814$$ −8.00000 −0.280400
$$815$$ −12.0000 −0.420342
$$816$$ −6.00000 −0.210042
$$817$$ −16.0000 −0.559769
$$818$$ −22.0000 −0.769212
$$819$$ 24.0000 0.838628
$$820$$ 6.00000 0.209529
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 36.0000 1.25412
$$825$$ −4.00000 −0.139262
$$826$$ −48.0000 −1.67013
$$827$$ 20.0000 0.695468 0.347734 0.937593i $$-0.386951\pi$$
0.347734 + 0.937593i $$0.386951\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 8.00000 0.277684
$$831$$ −26.0000 −0.901930
$$832$$ 42.0000 1.45609
$$833$$ 54.0000 1.87099
$$834$$ 4.00000 0.138509
$$835$$ 12.0000 0.415277
$$836$$ −16.0000 −0.553372
$$837$$ −8.00000 −0.276520
$$838$$ −4.00000 −0.138178
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ −12.0000 −0.414039
$$841$$ 1.00000 0.0344828
$$842$$ −18.0000 −0.620321
$$843$$ 10.0000 0.344418
$$844$$ −20.0000 −0.688428
$$845$$ 23.0000 0.791224
$$846$$ 0 0
$$847$$ 20.0000 0.687208
$$848$$ 10.0000 0.343401
$$849$$ 8.00000 0.274559
$$850$$ 6.00000 0.205798
$$851$$ −8.00000 −0.274236
$$852$$ 8.00000 0.274075
$$853$$ −38.0000 −1.30110 −0.650548 0.759465i $$-0.725461\pi$$
−0.650548 + 0.759465i $$0.725461\pi$$
$$854$$ −40.0000 −1.36877
$$855$$ −4.00000 −0.136797
$$856$$ −24.0000 −0.820303
$$857$$ 2.00000 0.0683187 0.0341593 0.999416i $$-0.489125\pi$$
0.0341593 + 0.999416i $$0.489125\pi$$
$$858$$ −24.0000 −0.819346
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ −24.0000 −0.817918
$$862$$ 24.0000 0.817443
$$863$$ 4.00000 0.136162 0.0680808 0.997680i $$-0.478312\pi$$
0.0680808 + 0.997680i $$0.478312\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 6.00000 0.204006
$$866$$ −34.0000 −1.15537
$$867$$ 19.0000 0.645274
$$868$$ 32.0000 1.08615
$$869$$ 0 0
$$870$$ 1.00000 0.0339032
$$871$$ 48.0000 1.62642
$$872$$ −42.0000 −1.42230
$$873$$ −2.00000 −0.0676897
$$874$$ 16.0000 0.541208
$$875$$ 4.00000 0.135225
$$876$$ 2.00000 0.0675737
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ 18.0000 0.607125
$$880$$ 4.00000 0.134840
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 9.00000 0.303046
$$883$$ −56.0000 −1.88455 −0.942275 0.334840i $$-0.891318\pi$$
−0.942275 + 0.334840i $$0.891318\pi$$
$$884$$ −36.0000 −1.21081
$$885$$ −12.0000 −0.403376
$$886$$ 20.0000 0.671913
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ −64.0000 −2.14649
$$890$$ −6.00000 −0.201120
$$891$$ −4.00000 −0.134005
$$892$$ −28.0000 −0.937509
$$893$$ 0 0
$$894$$ 6.00000 0.200670
$$895$$ −12.0000 −0.401116
$$896$$ −12.0000 −0.400892
$$897$$ −24.0000 −0.801337
$$898$$ 26.0000 0.867631
$$899$$ −8.00000 −0.266815
$$900$$ −1.00000 −0.0333333
$$901$$ −60.0000 −1.99889
$$902$$ 24.0000 0.799113
$$903$$ 16.0000 0.532447
$$904$$ 6.00000 0.199557
$$905$$ 22.0000 0.731305
$$906$$ 16.0000 0.531564
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ −2.00000 −0.0663358
$$910$$ 24.0000 0.795592
$$911$$ −56.0000 −1.85536 −0.927681 0.373373i $$-0.878201\pi$$
−0.927681 + 0.373373i $$0.878201\pi$$
$$912$$ 4.00000 0.132453
$$913$$ −32.0000 −1.05905
$$914$$ 10.0000 0.330771
$$915$$ −10.0000 −0.330590
$$916$$ 2.00000 0.0660819
$$917$$ 48.0000 1.58510
$$918$$ 6.00000 0.198030
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 12.0000 0.395628
$$921$$ 12.0000 0.395413
$$922$$ 6.00000 0.197599
$$923$$ −48.0000 −1.57994
$$924$$ 16.0000 0.526361
$$925$$ 2.00000 0.0657596
$$926$$ 36.0000 1.18303
$$927$$ −12.0000 −0.394132
$$928$$ 5.00000 0.164133
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ −8.00000 −0.262330
$$931$$ −36.0000 −1.17985
$$932$$ 22.0000 0.720634
$$933$$ 0 0
$$934$$ −20.0000 −0.654420
$$935$$ −24.0000 −0.784884
$$936$$ −18.0000 −0.588348
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ 32.0000 1.04484
$$939$$ −30.0000 −0.979013
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 18.0000 0.586472
$$943$$ 24.0000 0.781548
$$944$$ 12.0000 0.390567
$$945$$ 4.00000 0.130120
$$946$$ −16.0000 −0.520205
$$947$$ 28.0000 0.909878 0.454939 0.890523i $$-0.349661\pi$$
0.454939 + 0.890523i $$0.349661\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ −4.00000 −0.129777
$$951$$ 2.00000 0.0648544
$$952$$ −72.0000 −2.33353
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ −16.0000 −0.517748
$$956$$ 8.00000 0.258738
$$957$$ −4.00000 −0.129302
$$958$$ −16.0000 −0.516937
$$959$$ −8.00000 −0.258333
$$960$$ 7.00000 0.225924
$$961$$ 33.0000 1.06452
$$962$$ 12.0000 0.386896
$$963$$ 8.00000 0.257796
$$964$$ −2.00000 −0.0644157
$$965$$ 22.0000 0.708205
$$966$$ −16.0000 −0.514792
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ −15.0000 −0.482118
$$969$$ −24.0000 −0.770991
$$970$$ −2.00000 −0.0642161
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 16.0000 0.512936
$$974$$ −28.0000 −0.897178
$$975$$ 6.00000 0.192154
$$976$$ 10.0000 0.320092
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ −12.0000 −0.383718
$$979$$ 24.0000 0.767043
$$980$$ −9.00000 −0.287494
$$981$$ 14.0000 0.446986
$$982$$ −44.0000 −1.40410
$$983$$ 40.0000 1.27580 0.637901 0.770118i $$-0.279803\pi$$
0.637901 + 0.770118i $$0.279803\pi$$
$$984$$ 18.0000 0.573819
$$985$$ 14.0000 0.446077
$$986$$ 6.00000 0.191079
$$987$$ 0 0
$$988$$ 24.0000 0.763542
$$989$$ −16.0000 −0.508770
$$990$$ −4.00000 −0.127128
$$991$$ −40.0000 −1.27064 −0.635321 0.772248i $$-0.719132\pi$$
−0.635321 + 0.772248i $$0.719132\pi$$
$$992$$ −40.0000 −1.27000
$$993$$ 4.00000 0.126936
$$994$$ −32.0000 −1.01498
$$995$$ −24.0000 −0.760851
$$996$$ −8.00000 −0.253490
$$997$$ 50.0000 1.58352 0.791758 0.610835i $$-0.209166\pi$$
0.791758 + 0.610835i $$0.209166\pi$$
$$998$$ −36.0000 −1.13956
$$999$$ 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 435.2.a.d.1.1 1
3.2 odd 2 1305.2.a.b.1.1 1
4.3 odd 2 6960.2.a.l.1.1 1
5.2 odd 4 2175.2.c.b.349.2 2
5.3 odd 4 2175.2.c.b.349.1 2
5.4 even 2 2175.2.a.b.1.1 1
15.14 odd 2 6525.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.d.1.1 1 1.1 even 1 trivial
1305.2.a.b.1.1 1 3.2 odd 2
2175.2.a.b.1.1 1 5.4 even 2
2175.2.c.b.349.1 2 5.3 odd 4
2175.2.c.b.349.2 2 5.2 odd 4
6525.2.a.j.1.1 1 15.14 odd 2
6960.2.a.l.1.1 1 4.3 odd 2