Properties

 Label 5610.2.a.bf.1.1 Level $5610$ Weight $2$ Character 5610.1 Self dual yes Analytic conductor $44.796$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5610.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: $$44.7960755339$$ 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 $$\chi$$ $$=$$ 5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} -4.00000 q^{21} +1.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} -1.00000 q^{45} -8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} +1.00000 q^{51} -2.00000 q^{52} -14.0000 q^{53} +1.00000 q^{54} -1.00000 q^{55} -4.00000 q^{56} +2.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +1.00000 q^{66} -16.0000 q^{67} +1.00000 q^{68} +4.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{77} -2.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} -4.00000 q^{84} -1.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} +1.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} +8.00000 q^{91} +4.00000 q^{93} -8.00000 q^{94} +1.00000 q^{96} +6.00000 q^{97} +9.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 1.00000 0.301511
$$12$$ 1.00000 0.288675
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536
$$18$$ 1.00000 0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −4.00000 −0.872872
$$22$$ 1.00000 0.213201
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ −4.00000 −0.755929
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.00000 0.174078
$$34$$ 1.00000 0.171499
$$35$$ 4.00000 0.676123
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ −1.00000 −0.158114
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ −4.00000 −0.617213
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 9.00000 1.28571
$$50$$ 1.00000 0.141421
$$51$$ 1.00000 0.140028
$$52$$ −2.00000 −0.277350
$$53$$ −14.0000 −1.92305 −0.961524 0.274721i $$-0.911414\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −1.00000 −0.134840
$$56$$ −4.00000 −0.534522
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −4.00000 −0.503953
$$64$$ 1.00000 0.125000
$$65$$ 2.00000 0.248069
$$66$$ 1.00000 0.123091
$$67$$ −16.0000 −1.95471 −0.977356 0.211604i $$-0.932131\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 0 0
$$70$$ 4.00000 0.478091
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −4.00000 −0.455842
$$78$$ −2.00000 −0.226455
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ −1.00000 −0.108465
$$86$$ 4.00000 0.431331
$$87$$ 2.00000 0.214423
$$88$$ 1.00000 0.106600
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 8.00000 0.838628
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 1.00000 0.100504
$$100$$ 1.00000 0.100000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 4.00000 0.390360
$$106$$ −14.0000 −1.35980
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ −6.00000 −0.569495
$$112$$ −4.00000 −0.377964
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ −2.00000 −0.184900
$$118$$ 4.00000 0.368230
$$119$$ −4.00000 −0.366679
$$120$$ −1.00000 −0.0912871
$$121$$ 1.00000 0.0909091
$$122$$ 2.00000 0.181071
$$123$$ −2.00000 −0.180334
$$124$$ 4.00000 0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ −4.00000 −0.356348
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 2.00000 0.175412
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ 0 0
$$134$$ −16.0000 −1.38219
$$135$$ −1.00000 −0.0860663
$$136$$ 1.00000 0.0857493
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 4.00000 0.338062
$$141$$ −8.00000 −0.673722
$$142$$ −12.0000 −1.00702
$$143$$ −2.00000 −0.167248
$$144$$ 1.00000 0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ −6.00000 −0.496564
$$147$$ 9.00000 0.742307
$$148$$ −6.00000 −0.493197
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 1.00000 0.0808452
$$154$$ −4.00000 −0.322329
$$155$$ −4.00000 −0.321288
$$156$$ −2.00000 −0.160128
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −14.0000 −1.11027
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ −1.00000 −0.0778499
$$166$$ −12.0000 −0.931381
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −4.00000 −0.308607
$$169$$ −9.00000 −0.692308
$$170$$ −1.00000 −0.0766965
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 2.00000 0.151620
$$175$$ −4.00000 −0.302372
$$176$$ 1.00000 0.0753778
$$177$$ 4.00000 0.300658
$$178$$ −6.00000 −0.449719
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 8.00000 0.592999
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 4.00000 0.293294
$$187$$ 1.00000 0.0731272
$$188$$ −8.00000 −0.583460
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −22.0000 −1.58359 −0.791797 0.610784i $$-0.790854\pi$$
−0.791797 + 0.610784i $$0.790854\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 2.00000 0.143223
$$196$$ 9.00000 0.642857
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −16.0000 −1.12855
$$202$$ −14.0000 −0.985037
$$203$$ −8.00000 −0.561490
$$204$$ 1.00000 0.0700140
$$205$$ 2.00000 0.139686
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 4.00000 0.276026
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −14.0000 −0.961524
$$213$$ −12.0000 −0.822226
$$214$$ 4.00000 0.273434
$$215$$ −4.00000 −0.272798
$$216$$ 1.00000 0.0680414
$$217$$ −16.0000 −1.08615
$$218$$ −14.0000 −0.948200
$$219$$ −6.00000 −0.405442
$$220$$ −1.00000 −0.0674200
$$221$$ −2.00000 −0.134535
$$222$$ −6.00000 −0.402694
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ 1.00000 0.0666667
$$226$$ 6.00000 0.399114
$$227$$ 28.0000 1.85843 0.929213 0.369546i $$-0.120487\pi$$
0.929213 + 0.369546i $$0.120487\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 2.00000 0.131306
$$233$$ 26.0000 1.70332 0.851658 0.524097i $$-0.175597\pi$$
0.851658 + 0.524097i $$0.175597\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 8.00000 0.521862
$$236$$ 4.00000 0.260378
$$237$$ −8.00000 −0.519656
$$238$$ −4.00000 −0.259281
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 1.00000 0.0641500
$$244$$ 2.00000 0.128037
$$245$$ −9.00000 −0.574989
$$246$$ −2.00000 −0.127515
$$247$$ 0 0
$$248$$ 4.00000 0.254000
$$249$$ −12.0000 −0.760469
$$250$$ −1.00000 −0.0632456
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ 0 0
$$254$$ 16.0000 1.00393
$$255$$ −1.00000 −0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 24.0000 1.49129
$$260$$ 2.00000 0.124035
$$261$$ 2.00000 0.123797
$$262$$ −20.0000 −1.23560
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ 14.0000 0.860013
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ −16.0000 −0.977356
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ 8.00000 0.484182
$$274$$ −6.00000 −0.362473
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ 20.0000 1.19952
$$279$$ 4.00000 0.239474
$$280$$ 4.00000 0.239046
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 8.00000 0.472225
$$288$$ 1.00000 0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ −2.00000 −0.117444
$$291$$ 6.00000 0.351726
$$292$$ −6.00000 −0.351123
$$293$$ 22.0000 1.28525 0.642627 0.766179i $$-0.277845\pi$$
0.642627 + 0.766179i $$0.277845\pi$$
$$294$$ 9.00000 0.524891
$$295$$ −4.00000 −0.232889
$$296$$ −6.00000 −0.348743
$$297$$ 1.00000 0.0580259
$$298$$ 18.0000 1.04271
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ −16.0000 −0.922225
$$302$$ −16.0000 −0.920697
$$303$$ −14.0000 −0.804279
$$304$$ 0 0
$$305$$ −2.00000 −0.114520
$$306$$ 1.00000 0.0571662
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ −8.00000 −0.455104
$$310$$ −4.00000 −0.227185
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 4.00000 0.225374
$$316$$ −8.00000 −0.450035
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ −14.0000 −0.785081
$$319$$ 2.00000 0.111979
$$320$$ −1.00000 −0.0559017
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ 4.00000 0.221540
$$327$$ −14.0000 −0.774202
$$328$$ −2.00000 −0.110432
$$329$$ 32.0000 1.76422
$$330$$ −1.00000 −0.0550482
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ −6.00000 −0.328798
$$334$$ −12.0000 −0.656611
$$335$$ 16.0000 0.874173
$$336$$ −4.00000 −0.218218
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 6.00000 0.325875
$$340$$ −1.00000 −0.0542326
$$341$$ 4.00000 0.216612
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 2.00000 0.107211
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ −2.00000 −0.106752
$$352$$ 1.00000 0.0533002
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 12.0000 0.636894
$$356$$ −6.00000 −0.317999
$$357$$ −4.00000 −0.211702
$$358$$ 20.0000 1.05703
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −19.0000 −1.00000
$$362$$ 18.0000 0.946059
$$363$$ 1.00000 0.0524864
$$364$$ 8.00000 0.419314
$$365$$ 6.00000 0.314054
$$366$$ 2.00000 0.104542
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 0 0
$$369$$ −2.00000 −0.104116
$$370$$ 6.00000 0.311925
$$371$$ 56.0000 2.90738
$$372$$ 4.00000 0.207390
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 1.00000 0.0517088
$$375$$ −1.00000 −0.0516398
$$376$$ −8.00000 −0.412568
$$377$$ −4.00000 −0.206010
$$378$$ −4.00000 −0.205738
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 8.00000 0.409316
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 4.00000 0.203859
$$386$$ −22.0000 −1.11977
$$387$$ 4.00000 0.203331
$$388$$ 6.00000 0.304604
$$389$$ 38.0000 1.92668 0.963338 0.268290i $$-0.0864585\pi$$
0.963338 + 0.268290i $$0.0864585\pi$$
$$390$$ 2.00000 0.101274
$$391$$ 0 0
$$392$$ 9.00000 0.454569
$$393$$ −20.0000 −1.00887
$$394$$ 2.00000 0.100759
$$395$$ 8.00000 0.402524
$$396$$ 1.00000 0.0502519
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 20.0000 1.00251
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 10.0000 0.499376 0.249688 0.968326i $$-0.419672\pi$$
0.249688 + 0.968326i $$0.419672\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ −8.00000 −0.398508
$$404$$ −14.0000 −0.696526
$$405$$ −1.00000 −0.0496904
$$406$$ −8.00000 −0.397033
$$407$$ −6.00000 −0.297409
$$408$$ 1.00000 0.0495074
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ −6.00000 −0.295958
$$412$$ −8.00000 −0.394132
$$413$$ −16.0000 −0.787309
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ −2.00000 −0.0980581
$$417$$ 20.0000 0.979404
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 4.00000 0.195180
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 4.00000 0.194717
$$423$$ −8.00000 −0.388973
$$424$$ −14.0000 −0.679900
$$425$$ 1.00000 0.0485071
$$426$$ −12.0000 −0.581402
$$427$$ −8.00000 −0.387147
$$428$$ 4.00000 0.193347
$$429$$ −2.00000 −0.0965609
$$430$$ −4.00000 −0.192897
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ −2.00000 −0.0958927
$$436$$ −14.0000 −0.670478
$$437$$ 0 0
$$438$$ −6.00000 −0.286691
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ −1.00000 −0.0476731
$$441$$ 9.00000 0.428571
$$442$$ −2.00000 −0.0951303
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 6.00000 0.284427
$$446$$ −24.0000 −1.13643
$$447$$ 18.0000 0.851371
$$448$$ −4.00000 −0.188982
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −2.00000 −0.0941763
$$452$$ 6.00000 0.282216
$$453$$ −16.0000 −0.751746
$$454$$ 28.0000 1.31411
$$455$$ −8.00000 −0.375046
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ −4.00000 −0.185496
$$466$$ 26.0000 1.20443
$$467$$ 24.0000 1.11059 0.555294 0.831654i $$-0.312606\pi$$
0.555294 + 0.831654i $$0.312606\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 64.0000 2.95525
$$470$$ 8.00000 0.369012
$$471$$ −14.0000 −0.645086
$$472$$ 4.00000 0.184115
$$473$$ 4.00000 0.183920
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ −4.00000 −0.183340
$$477$$ −14.0000 −0.641016
$$478$$ 0 0
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 12.0000 0.547153
$$482$$ 14.0000 0.637683
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −6.00000 −0.272446
$$486$$ 1.00000 0.0453609
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 4.00000 0.180886
$$490$$ −9.00000 −0.406579
$$491$$ −40.0000 −1.80517 −0.902587 0.430507i $$-0.858335\pi$$
−0.902587 + 0.430507i $$0.858335\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ 2.00000 0.0900755
$$494$$ 0 0
$$495$$ −1.00000 −0.0449467
$$496$$ 4.00000 0.179605
$$497$$ 48.0000 2.15309
$$498$$ −12.0000 −0.537733
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −12.0000 −0.536120
$$502$$ −12.0000 −0.535586
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 14.0000 0.622992
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 16.0000 0.709885
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ −1.00000 −0.0442807
$$511$$ 24.0000 1.06170
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −30.0000 −1.32324
$$515$$ 8.00000 0.352522
$$516$$ 4.00000 0.176090
$$517$$ −8.00000 −0.351840
$$518$$ 24.0000 1.05450
$$519$$ 18.0000 0.790112
$$520$$ 2.00000 0.0877058
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ −4.00000 −0.174574
$$526$$ −8.00000 −0.348817
$$527$$ 4.00000 0.174243
$$528$$ 1.00000 0.0435194
$$529$$ −23.0000 −1.00000
$$530$$ 14.0000 0.608121
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ 4.00000 0.173259
$$534$$ −6.00000 −0.259645
$$535$$ −4.00000 −0.172935
$$536$$ −16.0000 −0.691095
$$537$$ 20.0000 0.863064
$$538$$ −30.0000 −1.29339
$$539$$ 9.00000 0.387657
$$540$$ −1.00000 −0.0430331
$$541$$ −46.0000 −1.97769 −0.988847 0.148933i $$-0.952416\pi$$
−0.988847 + 0.148933i $$0.952416\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 18.0000 0.772454
$$544$$ 1.00000 0.0428746
$$545$$ 14.0000 0.599694
$$546$$ 8.00000 0.342368
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 2.00000 0.0853579
$$550$$ 1.00000 0.0426401
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 32.0000 1.36078
$$554$$ −22.0000 −0.934690
$$555$$ 6.00000 0.254686
$$556$$ 20.0000 0.848189
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −8.00000 −0.338364
$$560$$ 4.00000 0.169031
$$561$$ 1.00000 0.0422200
$$562$$ 18.0000 0.759284
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ −6.00000 −0.252422
$$566$$ 4.00000 0.168133
$$567$$ −4.00000 −0.167984
$$568$$ −12.0000 −0.503509
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ −2.00000 −0.0836242
$$573$$ 8.00000 0.334205
$$574$$ 8.00000 0.333914
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ −22.0000 −0.914289
$$580$$ −2.00000 −0.0830455
$$581$$ 48.0000 1.99138
$$582$$ 6.00000 0.248708
$$583$$ −14.0000 −0.579821
$$584$$ −6.00000 −0.248282
$$585$$ 2.00000 0.0826898
$$586$$ 22.0000 0.908812
$$587$$ −8.00000 −0.330195 −0.165098 0.986277i $$-0.552794\pi$$
−0.165098 + 0.986277i $$0.552794\pi$$
$$588$$ 9.00000 0.371154
$$589$$ 0 0
$$590$$ −4.00000 −0.164677
$$591$$ 2.00000 0.0822690
$$592$$ −6.00000 −0.246598
$$593$$ 2.00000 0.0821302 0.0410651 0.999156i $$-0.486925\pi$$
0.0410651 + 0.999156i $$0.486925\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 4.00000 0.163984
$$596$$ 18.0000 0.737309
$$597$$ 20.0000 0.818546
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ −16.0000 −0.652111
$$603$$ −16.0000 −0.651570
$$604$$ −16.0000 −0.651031
$$605$$ −1.00000 −0.0406558
$$606$$ −14.0000 −0.568711
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 0 0
$$609$$ −8.00000 −0.324176
$$610$$ −2.00000 −0.0809776
$$611$$ 16.0000 0.647291
$$612$$ 1.00000 0.0404226
$$613$$ −10.0000 −0.403896 −0.201948 0.979396i $$-0.564727\pi$$
−0.201948 + 0.979396i $$0.564727\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 2.00000 0.0806478
$$616$$ −4.00000 −0.161165
$$617$$ 46.0000 1.85189 0.925945 0.377658i $$-0.123271\pi$$
0.925945 + 0.377658i $$0.123271\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ −12.0000 −0.481156
$$623$$ 24.0000 0.961540
$$624$$ −2.00000 −0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ 14.0000 0.559553
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ −6.00000 −0.239236
$$630$$ 4.00000 0.159364
$$631$$ −24.0000 −0.955425 −0.477712 0.878516i $$-0.658534\pi$$
−0.477712 + 0.878516i $$0.658534\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 4.00000 0.158986
$$634$$ −30.0000 −1.19145
$$635$$ −16.0000 −0.634941
$$636$$ −14.0000 −0.555136
$$637$$ −18.0000 −0.713186
$$638$$ 2.00000 0.0791808
$$639$$ −12.0000 −0.474713
$$640$$ −1.00000 −0.0395285
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 4.00000 0.157867
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 0 0
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 4.00000 0.157014
$$650$$ −2.00000 −0.0784465
$$651$$ −16.0000 −0.627089
$$652$$ 4.00000 0.156652
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ 20.0000 0.781465
$$656$$ −2.00000 −0.0780869
$$657$$ −6.00000 −0.234082
$$658$$ 32.0000 1.24749
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ −1.00000 −0.0389249
$$661$$ 46.0000 1.78919 0.894596 0.446875i $$-0.147463\pi$$
0.894596 + 0.446875i $$0.147463\pi$$
$$662$$ −12.0000 −0.466393
$$663$$ −2.00000 −0.0776736
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ −24.0000 −0.927894
$$670$$ 16.0000 0.618134
$$671$$ 2.00000 0.0772091
$$672$$ −4.00000 −0.154303
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 1.00000 0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 6.00000 0.230429
$$679$$ −24.0000 −0.921035
$$680$$ −1.00000 −0.0383482
$$681$$ 28.0000 1.07296
$$682$$ 4.00000 0.153168
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 0 0
$$685$$ 6.00000 0.229248
$$686$$ −8.00000 −0.305441
$$687$$ 14.0000 0.534133
$$688$$ 4.00000 0.152499
$$689$$ 28.0000 1.06672
$$690$$ 0 0
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ 18.0000 0.684257
$$693$$ −4.00000 −0.151947
$$694$$ 20.0000 0.759190
$$695$$ −20.0000 −0.758643
$$696$$ 2.00000 0.0758098
$$697$$ −2.00000 −0.0757554
$$698$$ −22.0000 −0.832712
$$699$$ 26.0000 0.983410
$$700$$ −4.00000 −0.151186
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ 0 0
$$704$$ 1.00000 0.0376889
$$705$$ 8.00000 0.301297
$$706$$ 2.00000 0.0752710
$$707$$ 56.0000 2.10610
$$708$$ 4.00000 0.150329
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 12.0000 0.450352
$$711$$ −8.00000 −0.300023
$$712$$ −6.00000 −0.224860
$$713$$ 0 0
$$714$$ −4.00000 −0.149696
$$715$$ 2.00000 0.0747958
$$716$$ 20.0000 0.747435
$$717$$ 0 0
$$718$$ 24.0000 0.895672
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 32.0000 1.19174
$$722$$ −19.0000 −0.707107
$$723$$ 14.0000 0.520666
$$724$$ 18.0000 0.668965
$$725$$ 2.00000 0.0742781
$$726$$ 1.00000 0.0371135
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 8.00000 0.296500
$$729$$ 1.00000 0.0370370
$$730$$ 6.00000 0.222070
$$731$$ 4.00000 0.147945
$$732$$ 2.00000 0.0739221
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ 8.00000 0.295285
$$735$$ −9.00000 −0.331970
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ −2.00000 −0.0736210
$$739$$ 8.00000 0.294285 0.147142 0.989115i $$-0.452992\pi$$
0.147142 + 0.989115i $$0.452992\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 0 0
$$742$$ 56.0000 2.05582
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 4.00000 0.146647
$$745$$ −18.0000 −0.659469
$$746$$ −2.00000 −0.0732252
$$747$$ −12.0000 −0.439057
$$748$$ 1.00000 0.0365636
$$749$$ −16.0000 −0.584627
$$750$$ −1.00000 −0.0365148
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −12.0000 −0.437304
$$754$$ −4.00000 −0.145671
$$755$$ 16.0000 0.582300
$$756$$ −4.00000 −0.145479
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −54.0000 −1.95750 −0.978749 0.205061i $$-0.934261\pi$$
−0.978749 + 0.205061i $$0.934261\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 56.0000 2.02734
$$764$$ 8.00000 0.289430
$$765$$ −1.00000 −0.0361551
$$766$$ 24.0000 0.867155
$$767$$ −8.00000 −0.288863
$$768$$ 1.00000 0.0360844
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 4.00000 0.144150
$$771$$ −30.0000 −1.08042
$$772$$ −22.0000 −0.791797
$$773$$ −30.0000 −1.07903 −0.539513 0.841978i $$-0.681391\pi$$
−0.539513 + 0.841978i $$0.681391\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 4.00000 0.143684
$$776$$ 6.00000 0.215387
$$777$$ 24.0000 0.860995
$$778$$ 38.0000 1.36237
$$779$$ 0 0
$$780$$ 2.00000 0.0716115
$$781$$ −12.0000 −0.429394
$$782$$ 0 0
$$783$$ 2.00000 0.0714742
$$784$$ 9.00000 0.321429
$$785$$ 14.0000 0.499681
$$786$$ −20.0000 −0.713376
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ −8.00000 −0.284808
$$790$$ 8.00000 0.284627
$$791$$ −24.0000 −0.853342
$$792$$ 1.00000 0.0355335
$$793$$ −4.00000 −0.142044
$$794$$ 2.00000 0.0709773
$$795$$ 14.0000 0.496529
$$796$$ 20.0000 0.708881
$$797$$ −14.0000 −0.495905 −0.247953 0.968772i $$-0.579758\pi$$
−0.247953 + 0.968772i $$0.579758\pi$$
$$798$$ 0 0
$$799$$ −8.00000 −0.283020
$$800$$ 1.00000 0.0353553
$$801$$ −6.00000 −0.212000
$$802$$ 10.0000 0.353112
$$803$$ −6.00000 −0.211735
$$804$$ −16.0000 −0.564276
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ −30.0000 −1.05605
$$808$$ −14.0000 −0.492518
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 16.0000 0.561144
$$814$$ −6.00000 −0.210300
$$815$$ −4.00000 −0.140114
$$816$$ 1.00000 0.0350070
$$817$$ 0 0
$$818$$ −14.0000 −0.489499
$$819$$ 8.00000 0.279543
$$820$$ 2.00000 0.0698430
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 1.00000 0.0348155
$$826$$ −16.0000 −0.556711
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 12.0000 0.416526
$$831$$ −22.0000 −0.763172
$$832$$ −2.00000 −0.0693375
$$833$$ 9.00000 0.311832
$$834$$ 20.0000 0.692543
$$835$$ 12.0000 0.415277
$$836$$ 0 0
$$837$$ 4.00000 0.138260
$$838$$ −12.0000 −0.414533
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 4.00000 0.138013
$$841$$ −25.0000 −0.862069
$$842$$ −2.00000 −0.0689246
$$843$$ 18.0000 0.619953
$$844$$ 4.00000 0.137686
$$845$$ 9.00000 0.309609
$$846$$ −8.00000 −0.275046
$$847$$ −4.00000 −0.137442
$$848$$ −14.0000 −0.480762
$$849$$ 4.00000 0.137280
$$850$$ 1.00000 0.0342997
$$851$$ 0 0
$$852$$ −12.0000 −0.411113
$$853$$ −6.00000 −0.205436 −0.102718 0.994711i $$-0.532754\pi$$
−0.102718 + 0.994711i $$0.532754\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ 58.0000 1.98124 0.990621 0.136637i $$-0.0436295\pi$$
0.990621 + 0.136637i $$0.0436295\pi$$
$$858$$ −2.00000 −0.0682789
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 8.00000 0.272639
$$862$$ 8.00000 0.272481
$$863$$ −56.0000 −1.90626 −0.953131 0.302558i $$-0.902160\pi$$
−0.953131 + 0.302558i $$0.902160\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −18.0000 −0.612018
$$866$$ −22.0000 −0.747590
$$867$$ 1.00000 0.0339618
$$868$$ −16.0000 −0.543075
$$869$$ −8.00000 −0.271381
$$870$$ −2.00000 −0.0678064
$$871$$ 32.0000 1.08428
$$872$$ −14.0000 −0.474100
$$873$$ 6.00000 0.203069
$$874$$ 0 0
$$875$$ 4.00000 0.135225
$$876$$ −6.00000 −0.202721
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ 22.0000 0.742042
$$880$$ −1.00000 −0.0337100
$$881$$ −38.0000 −1.28025 −0.640126 0.768270i $$-0.721118\pi$$
−0.640126 + 0.768270i $$0.721118\pi$$
$$882$$ 9.00000 0.303046
$$883$$ −32.0000 −1.07689 −0.538443 0.842662i $$-0.680987\pi$$
−0.538443 + 0.842662i $$0.680987\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ −4.00000 −0.134459
$$886$$ 24.0000 0.806296
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ −64.0000 −2.14649
$$890$$ 6.00000 0.201120
$$891$$ 1.00000 0.0335013
$$892$$ −24.0000 −0.803579
$$893$$ 0 0
$$894$$ 18.0000 0.602010
$$895$$ −20.0000 −0.668526
$$896$$ −4.00000 −0.133631
$$897$$ 0 0
$$898$$ −14.0000 −0.467186
$$899$$ 8.00000 0.266815
$$900$$ 1.00000 0.0333333
$$901$$ −14.0000 −0.466408
$$902$$ −2.00000 −0.0665927
$$903$$ −16.0000 −0.532447
$$904$$ 6.00000 0.199557
$$905$$ −18.0000 −0.598340
$$906$$ −16.0000 −0.531564
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 28.0000 0.929213
$$909$$ −14.0000 −0.464351
$$910$$ −8.00000 −0.265197
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ 0 0
$$913$$ −12.0000 −0.397142
$$914$$ −22.0000 −0.727695
$$915$$ −2.00000 −0.0661180
$$916$$ 14.0000 0.462573
$$917$$ 80.0000 2.64183
$$918$$ 1.00000 0.0330049
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ −6.00000 −0.197599
$$923$$ 24.0000 0.789970
$$924$$ −4.00000 −0.131590
$$925$$ −6.00000 −0.197279
$$926$$ −24.0000 −0.788689
$$927$$ −8.00000 −0.262754
$$928$$ 2.00000 0.0656532
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ 0 0
$$932$$ 26.0000 0.851658
$$933$$ −12.0000 −0.392862
$$934$$ 24.0000 0.785304
$$935$$ −1.00000 −0.0327035
$$936$$ −2.00000 −0.0653720
$$937$$ −38.0000 −1.24141 −0.620703 0.784046i $$-0.713153\pi$$
−0.620703 + 0.784046i $$0.713153\pi$$
$$938$$ 64.0000 2.08967
$$939$$ 14.0000 0.456873
$$940$$ 8.00000 0.260931
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ −14.0000 −0.456145
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 4.00000 0.130120
$$946$$ 4.00000 0.130051
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ −30.0000 −0.972817
$$952$$ −4.00000 −0.129641
$$953$$ 10.0000 0.323932 0.161966 0.986796i $$-0.448217\pi$$
0.161966 + 0.986796i $$0.448217\pi$$
$$954$$ −14.0000 −0.453267
$$955$$ −8.00000 −0.258874
$$956$$ 0 0
$$957$$ 2.00000 0.0646508
$$958$$ 32.0000 1.03387
$$959$$ 24.0000 0.775000
$$960$$ −1.00000 −0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ 12.0000 0.386896
$$963$$ 4.00000 0.128898
$$964$$ 14.0000 0.450910
$$965$$ 22.0000 0.708205
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ 0 0
$$970$$ −6.00000 −0.192648
$$971$$ −4.00000 −0.128366 −0.0641831 0.997938i $$-0.520444\pi$$
−0.0641831 + 0.997938i $$0.520444\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −80.0000 −2.56468
$$974$$ −8.00000 −0.256337
$$975$$ −2.00000 −0.0640513
$$976$$ 2.00000 0.0640184
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 4.00000 0.127906
$$979$$ −6.00000 −0.191761
$$980$$ −9.00000 −0.287494
$$981$$ −14.0000 −0.446986
$$982$$ −40.0000 −1.27645
$$983$$ 48.0000 1.53096 0.765481 0.643458i $$-0.222501\pi$$
0.765481 + 0.643458i $$0.222501\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ −2.00000 −0.0637253
$$986$$ 2.00000 0.0636930
$$987$$ 32.0000 1.01857
$$988$$ 0 0
$$989$$ 0 0
$$990$$ −1.00000 −0.0317821
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −12.0000 −0.380808
$$994$$ 48.0000 1.52247
$$995$$ −20.0000 −0.634043
$$996$$ −12.0000 −0.380235
$$997$$ 18.0000 0.570066 0.285033 0.958518i $$-0.407995\pi$$
0.285033 + 0.958518i $$0.407995\pi$$
$$998$$ 28.0000 0.886325
$$999$$ −6.00000 −0.189832
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 5610.2.a.bf.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.bf.1.1 1 1.1 even 1 trivial