# Properties

 Label 1045.2.a.b.1.1 Level $1045$ Weight $2$ Character 1045.1 Self dual yes Analytic conductor $8.344$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.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: $$8.34436701122$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1045.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{16} -6.00000 q^{17} -3.00000 q^{18} +1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -6.00000 q^{29} +4.00000 q^{31} +5.00000 q^{32} -6.00000 q^{34} +3.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} -3.00000 q^{40} -10.0000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -3.00000 q^{45} -8.00000 q^{46} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{55} -6.00000 q^{58} -8.00000 q^{59} +14.0000 q^{61} +4.00000 q^{62} +7.00000 q^{64} +2.00000 q^{65} +8.00000 q^{67} +6.00000 q^{68} -4.00000 q^{71} +9.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} -1.00000 q^{76} -16.0000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -10.0000 q^{82} -4.00000 q^{83} -6.00000 q^{85} +4.00000 q^{86} +3.00000 q^{88} +10.0000 q^{89} -3.00000 q^{90} +8.00000 q^{92} +1.00000 q^{95} +10.0000 q^{97} -7.00000 q^{98} +3.00000 q^{99} +O(q^{100})$$

## 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 $$0.384973\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ −3.00000 −1.00000
$$10$$ 1.00000 0.316228
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ 1.00000 0.229416
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ −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
$$40$$ −3.00000 −0.474342
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −3.00000 −0.447214
$$46$$ −8.00000 −1.17954
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 9.00000 1.06066
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 9.00000 1.00000
$$82$$ −10.0000 −1.10432
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ −6.00000 −0.650791
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 3.00000 0.319801
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ −3.00000 −0.316228
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 3.00000 0.301511
$$100$$ −1.00000 −0.100000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 6.00000 0.557086
$$117$$ −6.00000 −0.554700
$$118$$ −8.00000 −0.736460
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 14.0000 1.26750
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 18.0000 1.54349
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −4.00000 −0.335673
$$143$$ −2.00000 −0.167248
$$144$$ 3.00000 0.250000
$$145$$ −6.00000 −0.498273
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ −3.00000 −0.243332
$$153$$ 18.0000 1.45521
$$154$$ 0 0
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ 0 0
$$160$$ 5.00000 0.395285
$$161$$ 0 0
$$162$$ 9.00000 0.707107
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ −6.00000 −0.460179
$$171$$ −3.00000 −0.229416
$$172$$ −4.00000 −0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ 16.0000 1.19590 0.597948 0.801535i $$-0.295983\pi$$
0.597948 + 0.801535i $$0.295983\pi$$
$$180$$ 3.00000 0.223607
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 24.0000 1.76930
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ 6.00000 0.438763
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 1.00000 0.0725476
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ 3.00000 0.213201
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ 14.0000 0.985037
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −10.0000 −0.698430
$$206$$ −4.00000 −0.278693
$$207$$ 24.0000 1.66812
$$208$$ −2.00000 −0.138675
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −6.00000 −0.406371
$$219$$ 0 0
$$220$$ 1.00000 0.0674200
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ −6.00000 −0.399114
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ 18.0000 1.18176
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 0 0
$$244$$ −14.0000 −0.896258
$$245$$ −7.00000 −0.447214
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ −12.0000 −0.762001
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 8.00000 0.502956
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2.00000 −0.124035
$$261$$ 18.0000 1.11417
$$262$$ 12.0000 0.741362
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −8.00000 −0.488678
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 12.0000 0.719712
$$279$$ −12.0000 −0.718421
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 4.00000 0.237356
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 0 0
$$288$$ −15.0000 −0.883883
$$289$$ 19.0000 1.11765
$$290$$ −6.00000 −0.352332
$$291$$ 0 0
$$292$$ −2.00000 −0.117041
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ 14.0000 0.810998
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −16.0000 −0.920697
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 14.0000 0.801638
$$306$$ 18.0000 1.02899
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 4.00000 0.227185
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 7.00000 0.391312
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6.00000 −0.333849
$$324$$ −9.00000 −0.500000
$$325$$ 2.00000 0.110940
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ 30.0000 1.65647
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −32.0000 −1.75888 −0.879440 0.476011i $$-0.842082\pi$$
−0.879440 + 0.476011i $$0.842082\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 6.00000 0.328798
$$334$$ 16.0000 0.875481
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ 6.00000 0.325396
$$341$$ −4.00000 −0.216612
$$342$$ −3.00000 −0.162221
$$343$$ 0 0
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5.00000 −0.266501
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ −4.00000 −0.212298
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 16.0000 0.845626
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 9.00000 0.474342
$$361$$ 1.00000 0.0526316
$$362$$ 14.0000 0.735824
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 30.0000 1.56174
$$370$$ −2.00000 −0.103975
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 32.0000 1.64373 0.821865 0.569683i $$-0.192934\pi$$
0.821865 + 0.569683i $$0.192934\pi$$
$$380$$ −1.00000 −0.0512989
$$381$$ 0 0
$$382$$ −16.0000 −0.818631
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ −12.0000 −0.609994
$$388$$ −10.0000 −0.507673
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ 21.0000 1.06066
$$393$$ 0 0
$$394$$ −10.0000 −0.503793
$$395$$ −16.0000 −0.805047
$$396$$ −3.00000 −0.150756
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ −14.0000 −0.696526
$$405$$ 9.00000 0.447214
$$406$$ 0 0
$$407$$ 2.00000 0.0991363
$$408$$ 0 0
$$409$$ 30.0000 1.48340 0.741702 0.670729i $$-0.234019\pi$$
0.741702 + 0.670729i $$0.234019\pi$$
$$410$$ −10.0000 −0.493865
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ 0 0
$$414$$ 24.0000 1.17954
$$415$$ −4.00000 −0.196352
$$416$$ 10.0000 0.490290
$$417$$ 0 0
$$418$$ −1.00000 −0.0489116
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ −8.00000 −0.382692
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 3.00000 0.143019
$$441$$ 21.0000 1.00000
$$442$$ −12.0000 −0.570782
$$443$$ −28.0000 −1.33032 −0.665160 0.746701i $$-0.731637\pi$$
−0.665160 + 0.746701i $$0.731637\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ −3.00000 −0.141421
$$451$$ 10.0000 0.470882
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 0 0
$$460$$ 8.00000 0.373002
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ 0 0
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 6.00000 0.277350
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 24.0000 1.10469
$$473$$ −4.00000 −0.183920
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ −16.0000 −0.731823
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 22.0000 1.00207
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 10.0000 0.454077
$$486$$ 0 0
$$487$$ 4.00000 0.181257 0.0906287 0.995885i $$-0.471112\pi$$
0.0906287 + 0.995885i $$0.471112\pi$$
$$488$$ −42.0000 −1.90125
$$489$$ 0 0
$$490$$ −7.00000 −0.316228
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ 36.0000 1.62136
$$494$$ 2.00000 0.0899843
$$495$$ 3.00000 0.134840
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 36.0000 1.61158 0.805791 0.592200i $$-0.201741\pi$$
0.805791 + 0.592200i $$0.201741\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 12.0000 0.535586
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ 14.0000 0.622992
$$506$$ 8.00000 0.355643
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −30.0000 −1.32324
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −6.00000 −0.263117
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 18.0000 0.787839
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ −24.0000 −1.04546
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ −2.00000 −0.0868744
$$531$$ 24.0000 1.04151
$$532$$ 0 0
$$533$$ −20.0000 −0.866296
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ −24.0000 −1.03664
$$537$$ 0 0
$$538$$ 6.00000 0.258678
$$539$$ 7.00000 0.301511
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ −30.0000 −1.28624
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ −36.0000 −1.53925 −0.769624 0.638497i $$-0.779557\pi$$
−0.769624 + 0.638497i $$0.779557\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −42.0000 −1.79252
$$550$$ −1.00000 −0.0426401
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ 22.0000 0.932170 0.466085 0.884740i $$-0.345664\pi$$
0.466085 + 0.884740i $$0.345664\pi$$
$$558$$ −12.0000 −0.508001
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 30.0000 1.26547
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ 12.0000 0.503509
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −8.00000 −0.333623
$$576$$ −21.0000 −0.875000
$$577$$ −46.0000 −1.91501 −0.957503 0.288425i $$-0.906868\pi$$
−0.957503 + 0.288425i $$0.906868\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 0 0
$$580$$ 6.00000 0.249136
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 2.00000 0.0828315
$$584$$ −6.00000 −0.248282
$$585$$ −6.00000 −0.248069
$$586$$ −14.0000 −0.578335
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ −8.00000 −0.329355
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ 34.0000 1.39621 0.698106 0.715994i $$-0.254026\pi$$
0.698106 + 0.715994i $$0.254026\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −14.0000 −0.573462
$$597$$ 0 0
$$598$$ −16.0000 −0.654289
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ −24.0000 −0.977356
$$604$$ 16.0000 0.651031
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 0 0
$$610$$ 14.0000 0.566843
$$611$$ 0 0
$$612$$ −18.0000 −0.727607
$$613$$ 46.0000 1.85792 0.928961 0.370177i $$-0.120703\pi$$
0.928961 + 0.370177i $$0.120703\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −22.0000 −0.885687 −0.442843 0.896599i $$-0.646030\pi$$
−0.442843 + 0.896599i $$0.646030\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 48.0000 1.90934
$$633$$ 0 0
$$634$$ 30.0000 1.19145
$$635$$ −8.00000 −0.317470
$$636$$ 0 0
$$637$$ −14.0000 −0.554700
$$638$$ 6.00000 0.237542
$$639$$ 12.0000 0.474713
$$640$$ −3.00000 −0.118585
$$641$$ −14.0000 −0.552967 −0.276483 0.961019i $$-0.589169\pi$$
−0.276483 + 0.961019i $$0.589169\pi$$
$$642$$ 0 0
$$643$$ 20.0000 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −6.00000 −0.236067
$$647$$ −16.0000 −0.629025 −0.314512 0.949253i $$-0.601841\pi$$
−0.314512 + 0.949253i $$0.601841\pi$$
$$648$$ −27.0000 −1.06066
$$649$$ 8.00000 0.314027
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ 0 0
$$655$$ 12.0000 0.468879
$$656$$ 10.0000 0.390434
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −32.0000 −1.24372
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 48.0000 1.85857
$$668$$ −16.0000 −0.619059
$$669$$ 0 0
$$670$$ 8.00000 0.309067
$$671$$ −14.0000 −0.540464
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ −18.0000 −0.693334
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 34.0000 1.30673 0.653363 0.757045i $$-0.273358\pi$$
0.653363 + 0.757045i $$0.273358\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 18.0000 0.690268
$$681$$ 0 0
$$682$$ −4.00000 −0.153168
$$683$$ −40.0000 −1.53056 −0.765279 0.643699i $$-0.777399\pi$$
−0.765279 + 0.643699i $$0.777399\pi$$
$$684$$ 3.00000 0.114708
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ 60.0000 2.27266
$$698$$ −10.0000 −0.378506
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −26.0000 −0.982006 −0.491003 0.871158i $$-0.663370\pi$$
−0.491003 + 0.871158i $$0.663370\pi$$
$$702$$ 0 0
$$703$$ −2.00000 −0.0754314
$$704$$ −7.00000 −0.263822
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ −4.00000 −0.150117
$$711$$ 48.0000 1.80014
$$712$$ −30.0000 −1.12430
$$713$$ −32.0000 −1.19841
$$714$$ 0 0
$$715$$ −2.00000 −0.0747958
$$716$$ −16.0000 −0.597948
$$717$$ 0 0
$$718$$ −8.00000 −0.298557
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 3.00000 0.111803
$$721$$ 0 0
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ −14.0000 −0.520306
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 2.00000 0.0740233
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ 30.0000 1.10808 0.554038 0.832492i $$-0.313086\pi$$
0.554038 + 0.832492i $$0.313086\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ −40.0000 −1.47442
$$737$$ −8.00000 −0.294684
$$738$$ 30.0000 1.10432
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ 0 0
$$745$$ 14.0000 0.512920
$$746$$ −6.00000 −0.219676
$$747$$ 12.0000 0.439057
$$748$$ −6.00000 −0.219382
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ −42.0000 −1.52652 −0.763258 0.646094i $$-0.776401\pi$$
−0.763258 + 0.646094i $$0.776401\pi$$
$$758$$ 32.0000 1.16229
$$759$$ 0 0
$$760$$ −3.00000 −0.108821
$$761$$ 34.0000 1.23250 0.616250 0.787551i $$-0.288651\pi$$
0.616250 + 0.787551i $$0.288651\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 16.0000 0.578860
$$765$$ 18.0000 0.650791
$$766$$ −36.0000 −1.30073
$$767$$ −16.0000 −0.577727
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 4.00000 0.143684
$$776$$ −30.0000 −1.07694
$$777$$ 0 0
$$778$$ 6.00000 0.215110
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ 48.0000 1.71648
$$783$$ 0 0
$$784$$ 7.00000 0.250000
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 10.0000 0.356235
$$789$$ 0 0
$$790$$ −16.0000 −0.569254
$$791$$ 0 0
$$792$$ −9.00000 −0.319801
$$793$$ 28.0000 0.994309
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 5.00000 0.176777
$$801$$ −30.0000 −1.06000
$$802$$ −22.0000 −0.776847
$$803$$ −2.00000 −0.0705785
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ −42.0000 −1.47755
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ 9.00000 0.316228
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 2.00000 0.0701000
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 30.0000 1.04893
$$819$$ 0 0
$$820$$ 10.0000 0.349215
$$821$$ 30.0000 1.04701 0.523504 0.852023i $$-0.324625\pi$$
0.523504 + 0.852023i $$0.324625\pi$$
$$822$$ 0 0
$$823$$ 48.0000 1.67317 0.836587 0.547833i $$-0.184547\pi$$
0.836587 + 0.547833i $$0.184547\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ −24.0000 −0.834058
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ 0 0
$$832$$ 14.0000 0.485363
$$833$$ 42.0000 1.45521
$$834$$ 0 0
$$835$$ 16.0000 0.553703
$$836$$ 1.00000 0.0345857
$$837$$ 0 0
$$838$$ −28.0000 −0.967244
$$839$$ 4.00000 0.138095 0.0690477 0.997613i $$-0.478004\pi$$
0.0690477 + 0.997613i $$0.478004\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 6.00000 0.206774
$$843$$ 0 0
$$844$$ −4.00000 −0.137686
$$845$$ −9.00000 −0.309609
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ −6.00000 −0.205798
$$851$$ 16.0000 0.548473
$$852$$ 0 0
$$853$$ 38.0000 1.30110 0.650548 0.759465i $$-0.274539\pi$$
0.650548 + 0.759465i $$0.274539\pi$$
$$854$$ 0 0
$$855$$ −3.00000 −0.102598
$$856$$ 36.0000 1.23045
$$857$$ 30.0000 1.02478 0.512390 0.858753i $$-0.328760\pi$$
0.512390 + 0.858753i $$0.328760\pi$$
$$858$$ 0 0
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 4.00000 0.136162 0.0680808 0.997680i $$-0.478312\pi$$
0.0680808 + 0.997680i $$0.478312\pi$$
$$864$$ 0 0
$$865$$ 18.0000 0.612018
$$866$$ −22.0000 −0.747590
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ 18.0000 0.609557
$$873$$ −30.0000 −1.01535
$$874$$ −8.00000 −0.270604
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ 0 0
$$880$$ 1.00000 0.0337100
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 21.0000 0.707107
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −28.0000 −0.940678
$$887$$ −40.0000 −1.34307 −0.671534 0.740973i $$-0.734364\pi$$
−0.671534 + 0.740973i $$0.734364\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 10.0000 0.335201
$$891$$ −9.00000 −0.301511
$$892$$ 4.00000 0.133930
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 16.0000 0.534821
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 2.00000 0.0667409
$$899$$ −24.0000 −0.800445
$$900$$ 3.00000 0.100000
$$901$$ 12.0000 0.399778
$$902$$ 10.0000 0.332964
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 14.0000 0.465376
$$906$$ 0 0
$$907$$ −40.0000 −1.32818 −0.664089 0.747653i $$-0.731180\pi$$
−0.664089 + 0.747653i $$0.731180\pi$$
$$908$$ −4.00000 −0.132745
$$909$$ −42.0000 −1.39305
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ 0 0
$$913$$ 4.00000 0.132381
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 24.0000 0.791257
$$921$$ 0 0
$$922$$ −10.0000 −0.329332
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ −24.0000 −0.788689
$$927$$ 12.0000 0.394132
$$928$$ −30.0000 −0.984798
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ −7.00000 −0.229416
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 6.00000 0.196221
$$936$$ 18.0000 0.588348
$$937$$ −30.0000 −0.980057 −0.490029 0.871706i $$-0.663014\pi$$
−0.490029 + 0.871706i $$0.663014\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 0 0
$$943$$ 80.0000 2.60516
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ 4.00000 0.129845
$$950$$ 1.00000 0.0324443
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 6.00000 0.194257
$$955$$ −16.0000 −0.517748
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −4.00000 −0.128965
$$963$$ 36.0000 1.16008
$$964$$ −22.0000 −0.708572
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ −3.00000 −0.0964237
$$969$$ 0 0
$$970$$ 10.0000 0.321081
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 4.00000 0.128168
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ 58.0000 1.85558 0.927792 0.373097i $$-0.121704\pi$$
0.927792 + 0.373097i $$0.121704\pi$$
$$978$$ 0 0
$$979$$ −10.0000 −0.319601
$$980$$ 7.00000 0.223607
$$981$$ 18.0000 0.574696
$$982$$ −28.0000 −0.893516
$$983$$ −4.00000 −0.127580 −0.0637901 0.997963i $$-0.520319\pi$$
−0.0637901 + 0.997963i $$0.520319\pi$$
$$984$$ 0 0
$$985$$ −10.0000 −0.318626
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ −32.0000 −1.01754
$$990$$ 3.00000 0.0953463
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ 20.0000 0.635001
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 14.0000 0.443384 0.221692 0.975117i $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ 36.0000 1.13956
$$999$$ 0 0
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 1045.2.a.b.1.1 1
3.2 odd 2 9405.2.a.d.1.1 1
5.4 even 2 5225.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.b.1.1 1 1.1 even 1 trivial
5225.2.a.a.1.1 1 5.4 even 2
9405.2.a.d.1.1 1 3.2 odd 2