# Properties

 Label 6422.2.a.h.1.1 Level $6422$ Weight $2$ Character 6422.1 Self dual yes Analytic conductor $51.280$ 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] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6422 = 2 \cdot 13^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6422.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: $$51.2799281781$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 6422.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +6.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} +1.00000 q^{21} +6.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -5.00000 q^{27} +1.00000 q^{28} +9.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} -2.00000 q^{37} -1.00000 q^{38} +1.00000 q^{42} +8.00000 q^{43} +6.00000 q^{44} +3.00000 q^{46} +1.00000 q^{48} -6.00000 q^{49} -5.00000 q^{50} +3.00000 q^{51} -3.00000 q^{53} -5.00000 q^{54} +1.00000 q^{56} -1.00000 q^{57} +9.00000 q^{58} -9.00000 q^{59} -10.0000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} -5.00000 q^{67} +3.00000 q^{68} +3.00000 q^{69} +6.00000 q^{71} -2.00000 q^{72} +7.00000 q^{73} -2.00000 q^{74} -5.00000 q^{75} -1.00000 q^{76} +6.00000 q^{77} -10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +1.00000 q^{84} +8.00000 q^{86} +9.00000 q^{87} +6.00000 q^{88} +12.0000 q^{89} +3.00000 q^{92} +4.00000 q^{93} +1.00000 q^{96} +10.0000 q^{97} -6.00000 q^{98} -12.0000 q^{99} +O(q^{100})$$

## 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 $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 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
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 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
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 6.00000 1.27920
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 1.00000 0.188982
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 6.00000 1.04447
$$34$$ 3.00000 0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.00000 −0.857143
$$50$$ −5.00000 −0.707107
$$51$$ 3.00000 0.420084
$$52$$ 0 0
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −1.00000 −0.132453
$$58$$ 9.00000 1.18176
$$59$$ −9.00000 −1.17170 −0.585850 0.810419i $$-0.699239\pi$$
−0.585850 + 0.810419i $$0.699239\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ −5.00000 −0.610847 −0.305424 0.952217i $$-0.598798\pi$$
−0.305424 + 0.952217i $$0.598798\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ −2.00000 −0.235702
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −5.00000 −0.577350
$$76$$ −1.00000 −0.114708
$$77$$ 6.00000 0.683763
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 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
$$88$$ 6.00000 0.639602
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 3.00000 0.312772
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −6.00000 −0.606092
$$99$$ −12.0000 −1.20605
$$100$$ −5.00000 −0.500000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 3.00000 0.297044
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −3.00000 −0.291386
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 1.00000 0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 9.00000 0.835629
$$117$$ 0 0
$$118$$ −9.00000 −0.828517
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ −10.0000 −0.905357
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 6.00000 0.522233
$$133$$ −1.00000 −0.0867110
$$134$$ −5.00000 −0.431934
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ 9.00000 0.768922 0.384461 0.923141i $$-0.374387\pi$$
0.384461 + 0.923141i $$0.374387\pi$$
$$138$$ 3.00000 0.255377
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 6.00000 0.503509
$$143$$ 0 0
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ 7.00000 0.579324
$$147$$ −6.00000 −0.494872
$$148$$ −2.00000 −0.164399
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ −5.00000 −0.408248
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ −6.00000 −0.485071
$$154$$ 6.00000 0.483494
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −22.0000 −1.75579 −0.877896 0.478852i $$-0.841053\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ −3.00000 −0.237915
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 1.00000 0.0785674
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 8.00000 0.609994
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 9.00000 0.682288
$$175$$ −5.00000 −0.377964
$$176$$ 6.00000 0.452267
$$177$$ −9.00000 −0.676481
$$178$$ 12.0000 0.899438
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 3.00000 0.221163
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ 18.0000 1.31629
$$188$$ 0 0
$$189$$ −5.00000 −0.363696
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ −12.0000 −0.852803
$$199$$ 11.0000 0.779769 0.389885 0.920864i $$-0.372515\pi$$
0.389885 + 0.920864i $$0.372515\pi$$
$$200$$ −5.00000 −0.353553
$$201$$ −5.00000 −0.352673
$$202$$ 18.0000 1.26648
$$203$$ 9.00000 0.631676
$$204$$ 3.00000 0.210042
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ −6.00000 −0.415029
$$210$$ 0 0
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ 6.00000 0.411113
$$214$$ −9.00000 −0.615227
$$215$$ 0 0
$$216$$ −5.00000 −0.340207
$$217$$ 4.00000 0.271538
$$218$$ −11.0000 −0.745014
$$219$$ 7.00000 0.473016
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ −26.0000 −1.74109 −0.870544 0.492090i $$-0.836233\pi$$
−0.870544 + 0.492090i $$0.836233\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 10.0000 0.666667
$$226$$ 6.00000 0.399114
$$227$$ 15.0000 0.995585 0.497792 0.867296i $$-0.334144\pi$$
0.497792 + 0.867296i $$0.334144\pi$$
$$228$$ −1.00000 −0.0662266
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 6.00000 0.394771
$$232$$ 9.00000 0.590879
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −9.00000 −0.585850
$$237$$ −10.0000 −0.649570
$$238$$ 3.00000 0.194461
$$239$$ 21.0000 1.35838 0.679189 0.733964i $$-0.262332\pi$$
0.679189 + 0.733964i $$0.262332\pi$$
$$240$$ 0 0
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 25.0000 1.60706
$$243$$ 16.0000 1.02640
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 18.0000 1.13165
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 8.00000 0.498058
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ −18.0000 −1.11417
$$262$$ 0 0
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 6.00000 0.369274
$$265$$ 0 0
$$266$$ −1.00000 −0.0613139
$$267$$ 12.0000 0.734388
$$268$$ −5.00000 −0.305424
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −11.0000 −0.668202 −0.334101 0.942537i $$-0.608433\pi$$
−0.334101 + 0.942537i $$0.608433\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 0 0
$$274$$ 9.00000 0.543710
$$275$$ −30.0000 −1.80907
$$276$$ 3.00000 0.180579
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ −22.0000 −1.30776 −0.653882 0.756596i $$-0.726861\pi$$
−0.653882 + 0.756596i $$0.726861\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −2.00000 −0.117851
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ 7.00000 0.409644
$$293$$ 21.0000 1.22683 0.613417 0.789760i $$-0.289795\pi$$
0.613417 + 0.789760i $$0.289795\pi$$
$$294$$ −6.00000 −0.349927
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ −30.0000 −1.74078
$$298$$ 0 0
$$299$$ 0 0
$$300$$ −5.00000 −0.288675
$$301$$ 8.00000 0.461112
$$302$$ 10.0000 0.575435
$$303$$ 18.0000 1.03407
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 6.00000 0.341882
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ −21.0000 −1.19080 −0.595400 0.803429i $$-0.703007\pi$$
−0.595400 + 0.803429i $$0.703007\pi$$
$$312$$ 0 0
$$313$$ −19.0000 −1.07394 −0.536972 0.843600i $$-0.680432\pi$$
−0.536972 + 0.843600i $$0.680432\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 9.00000 0.505490 0.252745 0.967533i $$-0.418667\pi$$
0.252745 + 0.967533i $$0.418667\pi$$
$$318$$ −3.00000 −0.168232
$$319$$ 54.0000 3.02342
$$320$$ 0 0
$$321$$ −9.00000 −0.502331
$$322$$ 3.00000 0.167183
$$323$$ −3.00000 −0.166924
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ −11.0000 −0.608301
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 1.00000 0.0549650 0.0274825 0.999622i $$-0.491251\pi$$
0.0274825 + 0.999622i $$0.491251\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 4.00000 0.219199
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 1.00000 0.0545545
$$337$$ −4.00000 −0.217894 −0.108947 0.994048i $$-0.534748\pi$$
−0.108947 + 0.994048i $$0.534748\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ 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 $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ 9.00000 0.482451
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ −5.00000 −0.267261
$$351$$ 0 0
$$352$$ 6.00000 0.319801
$$353$$ 15.0000 0.798369 0.399185 0.916871i $$-0.369293\pi$$
0.399185 + 0.916871i $$0.369293\pi$$
$$354$$ −9.00000 −0.478345
$$355$$ 0 0
$$356$$ 12.0000 0.635999
$$357$$ 3.00000 0.158777
$$358$$ 0 0
$$359$$ −21.0000 −1.10834 −0.554169 0.832404i $$-0.686964\pi$$
−0.554169 + 0.832404i $$0.686964\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 2.00000 0.105118
$$363$$ 25.0000 1.31216
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 3.00000 0.156386
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −3.00000 −0.155752
$$372$$ 4.00000 0.207390
$$373$$ 23.0000 1.19089 0.595447 0.803394i $$-0.296975\pi$$
0.595447 + 0.803394i $$0.296975\pi$$
$$374$$ 18.0000 0.930758
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −5.00000 −0.257172
$$379$$ 7.00000 0.359566 0.179783 0.983706i $$-0.442460\pi$$
0.179783 + 0.983706i $$0.442460\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ 3.00000 0.153493
$$383$$ −18.0000 −0.919757 −0.459879 0.887982i $$-0.652107\pi$$
−0.459879 + 0.887982i $$0.652107\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −16.0000 −0.813326
$$388$$ 10.0000 0.507673
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ 9.00000 0.455150
$$392$$ −6.00000 −0.303046
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ −12.0000 −0.603023
$$397$$ −20.0000 −1.00377 −0.501886 0.864934i $$-0.667360\pi$$
−0.501886 + 0.864934i $$0.667360\pi$$
$$398$$ 11.0000 0.551380
$$399$$ −1.00000 −0.0500626
$$400$$ −5.00000 −0.250000
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ −5.00000 −0.249377
$$403$$ 0 0
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 9.00000 0.446663
$$407$$ −12.0000 −0.594818
$$408$$ 3.00000 0.148522
$$409$$ −32.0000 −1.58230 −0.791149 0.611623i $$-0.790517\pi$$
−0.791149 + 0.611623i $$0.790517\pi$$
$$410$$ 0 0
$$411$$ 9.00000 0.443937
$$412$$ 14.0000 0.689730
$$413$$ −9.00000 −0.442861
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ −6.00000 −0.293470
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −17.0000 −0.828529 −0.414265 0.910156i $$-0.635961\pi$$
−0.414265 + 0.910156i $$0.635961\pi$$
$$422$$ 5.00000 0.243396
$$423$$ 0 0
$$424$$ −3.00000 −0.145693
$$425$$ −15.0000 −0.727607
$$426$$ 6.00000 0.290701
$$427$$ −10.0000 −0.483934
$$428$$ −9.00000 −0.435031
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −6.00000 −0.289010 −0.144505 0.989504i $$-0.546159\pi$$
−0.144505 + 0.989504i $$0.546159\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ −11.0000 −0.526804
$$437$$ −3.00000 −0.143509
$$438$$ 7.00000 0.334473
$$439$$ −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
$$442$$ 0 0
$$443$$ −18.0000 −0.855206 −0.427603 0.903967i $$-0.640642\pi$$
−0.427603 + 0.903967i $$0.640642\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ −26.0000 −1.23114
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 10.0000 0.471405
$$451$$ 0 0
$$452$$ 6.00000 0.282216
$$453$$ 10.0000 0.469841
$$454$$ 15.0000 0.703985
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ −17.0000 −0.795226 −0.397613 0.917553i $$-0.630161\pi$$
−0.397613 + 0.917553i $$0.630161\pi$$
$$458$$ 22.0000 1.02799
$$459$$ −15.0000 −0.700140
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 6.00000 0.279145
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 9.00000 0.417815
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 18.0000 0.832941 0.416470 0.909149i $$-0.363267\pi$$
0.416470 + 0.909149i $$0.363267\pi$$
$$468$$ 0 0
$$469$$ −5.00000 −0.230879
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ −9.00000 −0.414259
$$473$$ 48.0000 2.20704
$$474$$ −10.0000 −0.459315
$$475$$ 5.00000 0.229416
$$476$$ 3.00000 0.137505
$$477$$ 6.00000 0.274721
$$478$$ 21.0000 0.960518
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −8.00000 −0.364390
$$483$$ 3.00000 0.136505
$$484$$ 25.0000 1.13636
$$485$$ 0 0
$$486$$ 16.0000 0.725775
$$487$$ −2.00000 −0.0906287 −0.0453143 0.998973i $$-0.514429\pi$$
−0.0453143 + 0.998973i $$0.514429\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 0 0
$$493$$ 27.0000 1.21602
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 6.00000 0.269137
$$498$$ 6.00000 0.268866
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 6.00000 0.267793
$$503$$ −21.0000 −0.936344 −0.468172 0.883637i $$-0.655087\pi$$
−0.468172 + 0.883637i $$0.655087\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 18.0000 0.800198
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 7.00000 0.309662
$$512$$ 1.00000 0.0441942
$$513$$ 5.00000 0.220755
$$514$$ 12.0000 0.529297
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ −2.00000 −0.0878750
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −36.0000 −1.57719 −0.788594 0.614914i $$-0.789191\pi$$
−0.788594 + 0.614914i $$0.789191\pi$$
$$522$$ −18.0000 −0.787839
$$523$$ 11.0000 0.480996 0.240498 0.970650i $$-0.422689\pi$$
0.240498 + 0.970650i $$0.422689\pi$$
$$524$$ 0 0
$$525$$ −5.00000 −0.218218
$$526$$ 24.0000 1.04645
$$527$$ 12.0000 0.522728
$$528$$ 6.00000 0.261116
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 18.0000 0.781133
$$532$$ −1.00000 −0.0433555
$$533$$ 0 0
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ −5.00000 −0.215967
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ −11.0000 −0.472490
$$543$$ 2.00000 0.0858282
$$544$$ 3.00000 0.128624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ 9.00000 0.384461
$$549$$ 20.0000 0.853579
$$550$$ −30.0000 −1.27920
$$551$$ −9.00000 −0.383413
$$552$$ 3.00000 0.127688
$$553$$ −10.0000 −0.425243
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ −24.0000 −1.01691 −0.508456 0.861088i $$-0.669784\pi$$
−0.508456 + 0.861088i $$0.669784\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 18.0000 0.759961
$$562$$ 0 0
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −22.0000 −0.924729
$$567$$ 1.00000 0.0419961
$$568$$ 6.00000 0.251754
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 3.00000 0.125327
$$574$$ 0 0
$$575$$ −15.0000 −0.625543
$$576$$ −2.00000 −0.0833333
$$577$$ −11.0000 −0.457936 −0.228968 0.973434i $$-0.573535\pi$$
−0.228968 + 0.973434i $$0.573535\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 6.00000 0.248922
$$582$$ 10.0000 0.414513
$$583$$ −18.0000 −0.745484
$$584$$ 7.00000 0.289662
$$585$$ 0 0
$$586$$ 21.0000 0.867502
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −2.00000 −0.0821995
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ −30.0000 −1.23091
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 11.0000 0.450200
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ −5.00000 −0.204124
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ 8.00000 0.326056
$$603$$ 10.0000 0.407231
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 18.0000 0.731200
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 9.00000 0.364698
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −6.00000 −0.242536
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 6.00000 0.241747
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 14.0000 0.563163
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 0 0
$$621$$ −15.0000 −0.601929
$$622$$ −21.0000 −0.842023
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ −19.0000 −0.759393
$$627$$ −6.00000 −0.239617
$$628$$ −22.0000 −0.877896
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ 5.00000 0.198732
$$634$$ 9.00000 0.357436
$$635$$ 0 0
$$636$$ −3.00000 −0.118958
$$637$$ 0 0
$$638$$ 54.0000 2.13788
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ −9.00000 −0.355202
$$643$$ 22.0000 0.867595 0.433798 0.901010i $$-0.357173\pi$$
0.433798 + 0.901010i $$0.357173\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 0 0
$$646$$ −3.00000 −0.118033
$$647$$ 27.0000 1.06148 0.530740 0.847535i $$-0.321914\pi$$
0.530740 + 0.847535i $$0.321914\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −54.0000 −2.11969
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ −20.0000 −0.783260
$$653$$ 24.0000 0.939193 0.469596 0.882881i $$-0.344399\pi$$
0.469596 + 0.882881i $$0.344399\pi$$
$$654$$ −11.0000 −0.430134
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −14.0000 −0.546192
$$658$$ 0 0
$$659$$ −45.0000 −1.75295 −0.876476 0.481446i $$-0.840112\pi$$
−0.876476 + 0.481446i $$0.840112\pi$$
$$660$$ 0 0
$$661$$ 13.0000 0.505641 0.252821 0.967513i $$-0.418642\pi$$
0.252821 + 0.967513i $$0.418642\pi$$
$$662$$ 1.00000 0.0388661
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ 27.0000 1.04544
$$668$$ −12.0000 −0.464294
$$669$$ −26.0000 −1.00522
$$670$$ 0 0
$$671$$ −60.0000 −2.31627
$$672$$ 1.00000 0.0385758
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ −4.00000 −0.154074
$$675$$ 25.0000 0.962250
$$676$$ 0 0
$$677$$ −33.0000 −1.26829 −0.634147 0.773213i $$-0.718648\pi$$
−0.634147 + 0.773213i $$0.718648\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 15.0000 0.574801
$$682$$ 24.0000 0.919007
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ −13.0000 −0.496342
$$687$$ 22.0000 0.839352
$$688$$ 8.00000 0.304997
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ 6.00000 0.228086
$$693$$ −12.0000 −0.455842
$$694$$ 18.0000 0.683271
$$695$$ 0 0
$$696$$ 9.00000 0.341144
$$697$$ 0 0
$$698$$ 10.0000 0.378506
$$699$$ −6.00000 −0.226941
$$700$$ −5.00000 −0.188982
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ 6.00000 0.226134
$$705$$ 0 0
$$706$$ 15.0000 0.564532
$$707$$ 18.0000 0.676960
$$708$$ −9.00000 −0.338241
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 20.0000 0.750059
$$712$$ 12.0000 0.449719
$$713$$ 12.0000 0.449404
$$714$$ 3.00000 0.112272
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 21.0000 0.784259
$$718$$ −21.0000 −0.783713
$$719$$ 39.0000 1.45445 0.727227 0.686397i $$-0.240809\pi$$
0.727227 + 0.686397i $$0.240809\pi$$
$$720$$ 0 0
$$721$$ 14.0000 0.521387
$$722$$ 1.00000 0.0372161
$$723$$ −8.00000 −0.297523
$$724$$ 2.00000 0.0743294
$$725$$ −45.0000 −1.67126
$$726$$ 25.0000 0.927837
$$727$$ −37.0000 −1.37225 −0.686127 0.727482i $$-0.740691\pi$$
−0.686127 + 0.727482i $$0.740691\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ −10.0000 −0.369611
$$733$$ −32.0000 −1.18195 −0.590973 0.806691i $$-0.701256\pi$$
−0.590973 + 0.806691i $$0.701256\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ 3.00000 0.110581
$$737$$ −30.0000 −1.10506
$$738$$ 0 0
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −3.00000 −0.110133
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 0 0
$$746$$ 23.0000 0.842090
$$747$$ −12.0000 −0.439057
$$748$$ 18.0000 0.658145
$$749$$ −9.00000 −0.328853
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ 6.00000 0.218652
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −5.00000 −0.181848
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 7.00000 0.254251
$$759$$ 18.0000 0.653359
$$760$$ 0 0
$$761$$ 21.0000 0.761249 0.380625 0.924730i $$-0.375709\pi$$
0.380625 + 0.924730i $$0.375709\pi$$
$$762$$ 2.00000 0.0724524
$$763$$ −11.0000 −0.398227
$$764$$ 3.00000 0.108536
$$765$$ 0 0
$$766$$ −18.0000 −0.650366
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −5.00000 −0.180305 −0.0901523 0.995928i $$-0.528735\pi$$
−0.0901523 + 0.995928i $$0.528735\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ −14.0000 −0.503871
$$773$$ −51.0000 −1.83434 −0.917171 0.398493i $$-0.869533\pi$$
−0.917171 + 0.398493i $$0.869533\pi$$
$$774$$ −16.0000 −0.575108
$$775$$ −20.0000 −0.718421
$$776$$ 10.0000 0.358979
$$777$$ −2.00000 −0.0717496
$$778$$ 18.0000 0.645331
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 9.00000 0.321839
$$783$$ −45.0000 −1.60817
$$784$$ −6.00000 −0.214286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 31.0000 1.10503 0.552515 0.833503i $$-0.313668\pi$$
0.552515 + 0.833503i $$0.313668\pi$$
$$788$$ 0 0
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ −12.0000 −0.426401
$$793$$ 0 0
$$794$$ −20.0000 −0.709773
$$795$$ 0 0
$$796$$ 11.0000 0.389885
$$797$$ −39.0000 −1.38145 −0.690725 0.723117i $$-0.742709\pi$$
−0.690725 + 0.723117i $$0.742709\pi$$
$$798$$ −1.00000 −0.0353996
$$799$$ 0 0
$$800$$ −5.00000 −0.176777
$$801$$ −24.0000 −0.847998
$$802$$ 0 0
$$803$$ 42.0000 1.48215
$$804$$ −5.00000 −0.176336
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.00000 −0.211210
$$808$$ 18.0000 0.633238
$$809$$ 9.00000 0.316423 0.158212 0.987405i $$-0.449427\pi$$
0.158212 + 0.987405i $$0.449427\pi$$
$$810$$ 0 0
$$811$$ −11.0000 −0.386262 −0.193131 0.981173i $$-0.561864\pi$$
−0.193131 + 0.981173i $$0.561864\pi$$
$$812$$ 9.00000 0.315838
$$813$$ −11.0000 −0.385787
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ 3.00000 0.105021
$$817$$ −8.00000 −0.279885
$$818$$ −32.0000 −1.11885
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 9.00000 0.313911
$$823$$ 41.0000 1.42917 0.714585 0.699549i $$-0.246616\pi$$
0.714585 + 0.699549i $$0.246616\pi$$
$$824$$ 14.0000 0.487713
$$825$$ −30.0000 −1.04447
$$826$$ −9.00000 −0.313150
$$827$$ −33.0000 −1.14752 −0.573761 0.819023i $$-0.694516\pi$$
−0.573761 + 0.819023i $$0.694516\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 11.0000 0.382046 0.191023 0.981586i $$-0.438820\pi$$
0.191023 + 0.981586i $$0.438820\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ −6.00000 −0.207514
$$837$$ −20.0000 −0.691301
$$838$$ −12.0000 −0.414533
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ −17.0000 −0.585859
$$843$$ 0 0
$$844$$ 5.00000 0.172107
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 25.0000 0.859010
$$848$$ −3.00000 −0.103020
$$849$$ −22.0000 −0.755038
$$850$$ −15.0000 −0.514496
$$851$$ −6.00000 −0.205677
$$852$$ 6.00000 0.205557
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ −10.0000 −0.342193
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ −12.0000 −0.409912 −0.204956 0.978771i $$-0.565705\pi$$
−0.204956 + 0.978771i $$0.565705\pi$$
$$858$$ 0 0
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −6.00000 −0.204361
$$863$$ 18.0000 0.612727 0.306364 0.951915i $$-0.400888\pi$$
0.306364 + 0.951915i $$0.400888\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ −8.00000 −0.271694
$$868$$ 4.00000 0.135769
$$869$$ −60.0000 −2.03536
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −11.0000 −0.372507
$$873$$ −20.0000 −0.676897
$$874$$ −3.00000 −0.101477
$$875$$ 0 0
$$876$$ 7.00000 0.236508
$$877$$ −23.0000 −0.776655 −0.388327 0.921521i $$-0.626947\pi$$
−0.388327 + 0.921521i $$0.626947\pi$$
$$878$$ −28.0000 −0.944954
$$879$$ 21.0000 0.708312
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 12.0000 0.404061
$$883$$ −34.0000 −1.14419 −0.572096 0.820187i $$-0.693869\pi$$
−0.572096 + 0.820187i $$0.693869\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −18.0000 −0.604722
$$887$$ −42.0000 −1.41022 −0.705111 0.709097i $$-0.749103\pi$$
−0.705111 + 0.709097i $$0.749103\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ 6.00000 0.201008
$$892$$ −26.0000 −0.870544
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ 36.0000 1.20067
$$900$$ 10.0000 0.333333
$$901$$ −9.00000 −0.299833
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 10.0000 0.332228
$$907$$ −37.0000 −1.22856 −0.614282 0.789086i $$-0.710554\pi$$
−0.614282 + 0.789086i $$0.710554\pi$$
$$908$$ 15.0000 0.497792
$$909$$ −36.0000 −1.19404
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ −1.00000 −0.0331133
$$913$$ 36.0000 1.19143
$$914$$ −17.0000 −0.562310
$$915$$ 0 0
$$916$$ 22.0000 0.726900
$$917$$ 0 0
$$918$$ −15.0000 −0.495074
$$919$$ −7.00000 −0.230909 −0.115454 0.993313i $$-0.536832\pi$$
−0.115454 + 0.993313i $$0.536832\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 12.0000 0.395199
$$923$$ 0 0
$$924$$ 6.00000 0.197386
$$925$$ 10.0000 0.328798
$$926$$ 4.00000 0.131448
$$927$$ −28.0000 −0.919641
$$928$$ 9.00000 0.295439
$$929$$ −33.0000 −1.08269 −0.541347 0.840799i $$-0.682086\pi$$
−0.541347 + 0.840799i $$0.682086\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ −6.00000 −0.196537
$$933$$ −21.0000 −0.687509
$$934$$ 18.0000 0.588978
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −7.00000 −0.228680 −0.114340 0.993442i $$-0.536475\pi$$
−0.114340 + 0.993442i $$0.536475\pi$$
$$938$$ −5.00000 −0.163256
$$939$$ −19.0000 −0.620042
$$940$$ 0 0
$$941$$ −21.0000 −0.684580 −0.342290 0.939594i $$-0.611203\pi$$
−0.342290 + 0.939594i $$0.611203\pi$$
$$942$$ −22.0000 −0.716799
$$943$$ 0 0
$$944$$ −9.00000 −0.292925
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ 48.0000 1.55979 0.779895 0.625910i $$-0.215272\pi$$
0.779895 + 0.625910i $$0.215272\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ 0 0
$$950$$ 5.00000 0.162221
$$951$$ 9.00000 0.291845
$$952$$ 3.00000 0.0972306
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 21.0000 0.679189
$$957$$ 54.0000 1.74557
$$958$$ −36.0000 −1.16311
$$959$$ 9.00000 0.290625
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 18.0000 0.580042
$$964$$ −8.00000 −0.257663
$$965$$ 0 0
$$966$$ 3.00000 0.0965234
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 25.0000 0.803530
$$969$$ −3.00000 −0.0963739
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 16.0000 0.513200
$$973$$ −4.00000 −0.128234
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 12.0000 0.383914 0.191957 0.981403i $$-0.438517\pi$$
0.191957 + 0.981403i $$0.438517\pi$$
$$978$$ −20.0000 −0.639529
$$979$$ 72.0000 2.30113
$$980$$ 0 0
$$981$$ 22.0000 0.702406
$$982$$ −36.0000 −1.14881
$$983$$ 30.0000 0.956851 0.478426 0.878128i $$-0.341208\pi$$
0.478426 + 0.878128i $$0.341208\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 27.0000 0.859855
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 1.00000 0.0317340
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 10.0000 0.316386
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 6422.2.a.h.1.1 1
13.12 even 2 38.2.a.a.1.1 1
39.38 odd 2 342.2.a.e.1.1 1
52.51 odd 2 304.2.a.c.1.1 1
65.12 odd 4 950.2.b.b.799.1 2
65.38 odd 4 950.2.b.b.799.2 2
65.64 even 2 950.2.a.d.1.1 1
91.90 odd 2 1862.2.a.b.1.1 1
104.51 odd 2 1216.2.a.m.1.1 1
104.77 even 2 1216.2.a.e.1.1 1
143.142 odd 2 4598.2.a.p.1.1 1
156.155 even 2 2736.2.a.n.1.1 1
195.194 odd 2 8550.2.a.m.1.1 1
247.12 odd 6 722.2.c.c.429.1 2
247.25 even 18 722.2.e.f.245.1 6
247.51 odd 18 722.2.e.e.245.1 6
247.64 even 6 722.2.c.e.429.1 2
247.90 odd 18 722.2.e.e.595.1 6
247.103 odd 6 722.2.c.c.653.1 2
247.116 odd 18 722.2.e.e.99.1 6
247.129 odd 18 722.2.e.e.415.1 6
247.142 even 18 722.2.e.f.423.1 6
247.155 odd 18 722.2.e.e.389.1 6
247.168 even 18 722.2.e.f.389.1 6
247.181 odd 18 722.2.e.e.423.1 6
247.194 even 18 722.2.e.f.415.1 6
247.207 even 18 722.2.e.f.99.1 6
247.220 even 6 722.2.c.e.653.1 2
247.233 even 18 722.2.e.f.595.1 6
247.246 odd 2 722.2.a.e.1.1 1
260.259 odd 2 7600.2.a.n.1.1 1
741.740 even 2 6498.2.a.f.1.1 1
988.987 even 2 5776.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.a.1.1 1 13.12 even 2
304.2.a.c.1.1 1 52.51 odd 2
342.2.a.e.1.1 1 39.38 odd 2
722.2.a.e.1.1 1 247.246 odd 2
722.2.c.c.429.1 2 247.12 odd 6
722.2.c.c.653.1 2 247.103 odd 6
722.2.c.e.429.1 2 247.64 even 6
722.2.c.e.653.1 2 247.220 even 6
722.2.e.e.99.1 6 247.116 odd 18
722.2.e.e.245.1 6 247.51 odd 18
722.2.e.e.389.1 6 247.155 odd 18
722.2.e.e.415.1 6 247.129 odd 18
722.2.e.e.423.1 6 247.181 odd 18
722.2.e.e.595.1 6 247.90 odd 18
722.2.e.f.99.1 6 247.207 even 18
722.2.e.f.245.1 6 247.25 even 18
722.2.e.f.389.1 6 247.168 even 18
722.2.e.f.415.1 6 247.194 even 18
722.2.e.f.423.1 6 247.142 even 18
722.2.e.f.595.1 6 247.233 even 18
950.2.a.d.1.1 1 65.64 even 2
950.2.b.b.799.1 2 65.12 odd 4
950.2.b.b.799.2 2 65.38 odd 4
1216.2.a.e.1.1 1 104.77 even 2
1216.2.a.m.1.1 1 104.51 odd 2
1862.2.a.b.1.1 1 91.90 odd 2
2736.2.a.n.1.1 1 156.155 even 2
4598.2.a.p.1.1 1 143.142 odd 2
5776.2.a.m.1.1 1 988.987 even 2
6422.2.a.h.1.1 1 1.1 even 1 trivial
6498.2.a.f.1.1 1 741.740 even 2
7600.2.a.n.1.1 1 260.259 odd 2
8550.2.a.m.1.1 1 195.194 odd 2