# Properties

 Label 510.2.a.c.1.1 Level $510$ Weight $2$ Character 510.1 Self dual yes Analytic conductor $4.072$ 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] = [510,2,Mod(1,510)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(510, 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("510.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$510 = 2 \cdot 3 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 510.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$4.07237050309$$ 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 $$\chi$$ $$=$$ 510.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{21} -4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{42} -12.0000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -4.00000 q^{56} +4.00000 q^{57} +2.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +2.00000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} +1.00000 q^{68} +4.00000 q^{69} +4.00000 q^{70} -4.00000 q^{71} +1.00000 q^{72} -14.0000 q^{73} -6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +16.0000 q^{77} +2.00000 q^{78} -12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} +4.00000 q^{84} -1.00000 q^{85} -12.0000 q^{86} -2.00000 q^{87} -4.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} +8.00000 q^{91} -4.00000 q^{92} -4.00000 q^{93} +8.00000 q^{94} +4.00000 q^{95} -1.00000 q^{96} +18.0000 q^{97} +9.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −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
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −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
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 4.00000 0.872872
$$22$$ −4.00000 −0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ −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
$$33$$ 4.00000 0.696311
$$34$$ 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 $$0.664172\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 2.00000 0.320256
$$40$$ −1.00000 −0.158114
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 4.00000 0.617213
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ −1.00000 −0.149071
$$46$$ −4.00000 −0.589768
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 9.00000 1.28571
$$50$$ 1.00000 0.141421
$$51$$ −1.00000 −0.140028
$$52$$ −2.00000 −0.277350
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 4.00000 0.539360
$$56$$ −4.00000 −0.534522
$$57$$ 4.00000 0.529813
$$58$$ 2.00000 0.262613
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 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
$$66$$ 4.00000 0.492366
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 4.00000 0.481543
$$70$$ 4.00000 0.478091
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ −1.00000 −0.115470
$$76$$ −4.00000 −0.458831
$$77$$ 16.0000 1.82337
$$78$$ 2.00000 0.226455
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 4.00000 0.436436
$$85$$ −1.00000 −0.108465
$$86$$ −12.0000 −1.29399
$$87$$ −2.00000 −0.214423
$$88$$ −4.00000 −0.426401
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 8.00000 0.838628
$$92$$ −4.00000 −0.417029
$$93$$ −4.00000 −0.414781
$$94$$ 8.00000 0.825137
$$95$$ 4.00000 0.410391
$$96$$ −1.00000 −0.102062
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ 9.00000 0.909137
$$99$$ −4.00000 −0.402015
$$100$$ 1.00000 0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ −1.00000 −0.0990148
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ −4.00000 −0.390360
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 6.00000 0.569495
$$112$$ −4.00000 −0.377964
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 4.00000 0.373002
$$116$$ 2.00000 0.185695
$$117$$ −2.00000 −0.184900
$$118$$ 12.0000 1.10469
$$119$$ −4.00000 −0.366679
$$120$$ 1.00000 0.0912871
$$121$$ 5.00000 0.454545
$$122$$ 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 $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 12.0000 1.05654
$$130$$ 2.00000 0.175412
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 16.0000 1.38738
$$134$$ 4.00000 0.345547
$$135$$ 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 $$0.582506\pi$$
$$138$$ 4.00000 0.340503
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 4.00000 0.338062
$$141$$ −8.00000 −0.673722
$$142$$ −4.00000 −0.335673
$$143$$ 8.00000 0.668994
$$144$$ 1.00000 0.0833333
$$145$$ −2.00000 −0.166091
$$146$$ −14.0000 −1.15865
$$147$$ −9.00000 −0.742307
$$148$$ −6.00000 −0.493197
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 1.00000 0.0808452
$$154$$ 16.0000 1.28932
$$155$$ −4.00000 −0.321288
$$156$$ 2.00000 0.160128
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ −12.0000 −0.954669
$$159$$ 2.00000 0.158610
$$160$$ −1.00000 −0.0790569
$$161$$ 16.0000 1.26098
$$162$$ 1.00000 0.0785674
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ 2.00000 0.156174
$$165$$ −4.00000 −0.311400
$$166$$ −4.00000 −0.310460
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 4.00000 0.308607
$$169$$ −9.00000 −0.692308
$$170$$ −1.00000 −0.0766965
$$171$$ −4.00000 −0.305888
$$172$$ −12.0000 −0.914991
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ −4.00000 −0.302372
$$176$$ −4.00000 −0.301511
$$177$$ −12.0000 −0.901975
$$178$$ 10.0000 0.749532
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 8.00000 0.592999
$$183$$ −2.00000 −0.147844
$$184$$ −4.00000 −0.294884
$$185$$ 6.00000 0.441129
$$186$$ −4.00000 −0.293294
$$187$$ −4.00000 −0.292509
$$188$$ 8.00000 0.583460
$$189$$ 4.00000 0.290957
$$190$$ 4.00000 0.290191
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ 18.0000 1.29232
$$195$$ −2.00000 −0.143223
$$196$$ 9.00000 0.642857
$$197$$ −14.0000 −0.997459 −0.498729 0.866758i $$-0.666200\pi$$
−0.498729 + 0.866758i $$0.666200\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −4.00000 −0.282138
$$202$$ −10.0000 −0.703598
$$203$$ −8.00000 −0.561490
$$204$$ −1.00000 −0.0700140
$$205$$ −2.00000 −0.139686
$$206$$ 0 0
$$207$$ −4.00000 −0.278019
$$208$$ −2.00000 −0.138675
$$209$$ 16.0000 1.10674
$$210$$ −4.00000 −0.276026
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 4.00000 0.274075
$$214$$ 12.0000 0.820303
$$215$$ 12.0000 0.818393
$$216$$ −1.00000 −0.0680414
$$217$$ −16.0000 −1.08615
$$218$$ 10.0000 0.677285
$$219$$ 14.0000 0.946032
$$220$$ 4.00000 0.269680
$$221$$ −2.00000 −0.134535
$$222$$ 6.00000 0.402694
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ 1.00000 0.0666667
$$226$$ −6.00000 −0.399114
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 4.00000 0.263752
$$231$$ −16.0000 −1.05272
$$232$$ 2.00000 0.131306
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ −8.00000 −0.521862
$$236$$ 12.0000 0.781133
$$237$$ 12.0000 0.779484
$$238$$ −4.00000 −0.259281
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ −9.00000 −0.574989
$$246$$ −2.00000 −0.127515
$$247$$ 8.00000 0.509028
$$248$$ 4.00000 0.254000
$$249$$ 4.00000 0.253490
$$250$$ −1.00000 −0.0632456
$$251$$ −28.0000 −1.76734 −0.883672 0.468106i $$-0.844936\pi$$
−0.883672 + 0.468106i $$0.844936\pi$$
$$252$$ −4.00000 −0.251976
$$253$$ 16.0000 1.00591
$$254$$ −16.0000 −1.00393
$$255$$ 1.00000 0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 24.0000 1.49129
$$260$$ 2.00000 0.124035
$$261$$ 2.00000 0.123797
$$262$$ 20.0000 1.23560
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 2.00000 0.122859
$$266$$ 16.0000 0.981023
$$267$$ −10.0000 −0.611990
$$268$$ 4.00000 0.244339
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ −8.00000 −0.484182
$$274$$ −6.00000 −0.362473
$$275$$ −4.00000 −0.241209
$$276$$ 4.00000 0.240772
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 4.00000 0.239474
$$280$$ 4.00000 0.239046
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ −4.00000 −0.236940
$$286$$ 8.00000 0.473050
$$287$$ −8.00000 −0.472225
$$288$$ 1.00000 0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ −2.00000 −0.117444
$$291$$ −18.0000 −1.05518
$$292$$ −14.0000 −0.819288
$$293$$ −26.0000 −1.51894 −0.759468 0.650545i $$-0.774541\pi$$
−0.759468 + 0.650545i $$0.774541\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ −12.0000 −0.698667
$$296$$ −6.00000 −0.348743
$$297$$ 4.00000 0.232104
$$298$$ −2.00000 −0.115857
$$299$$ 8.00000 0.462652
$$300$$ −1.00000 −0.0577350
$$301$$ 48.0000 2.76667
$$302$$ −8.00000 −0.460348
$$303$$ 10.0000 0.574485
$$304$$ −4.00000 −0.229416
$$305$$ −2.00000 −0.114520
$$306$$ 1.00000 0.0571662
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ 2.00000 0.113228
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 4.00000 0.225374
$$316$$ −12.0000 −0.675053
$$317$$ 10.0000 0.561656 0.280828 0.959758i $$-0.409391\pi$$
0.280828 + 0.959758i $$0.409391\pi$$
$$318$$ 2.00000 0.112154
$$319$$ −8.00000 −0.447914
$$320$$ −1.00000 −0.0559017
$$321$$ −12.0000 −0.669775
$$322$$ 16.0000 0.891645
$$323$$ −4.00000 −0.222566
$$324$$ 1.00000 0.0555556
$$325$$ −2.00000 −0.110940
$$326$$ −20.0000 −1.10770
$$327$$ −10.0000 −0.553001
$$328$$ 2.00000 0.110432
$$329$$ −32.0000 −1.76422
$$330$$ −4.00000 −0.220193
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ −6.00000 −0.328798
$$334$$ 12.0000 0.656611
$$335$$ −4.00000 −0.218543
$$336$$ 4.00000 0.218218
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 6.00000 0.325875
$$340$$ −1.00000 −0.0542326
$$341$$ −16.0000 −0.866449
$$342$$ −4.00000 −0.216295
$$343$$ −8.00000 −0.431959
$$344$$ −12.0000 −0.646997
$$345$$ −4.00000 −0.215353
$$346$$ −14.0000 −0.752645
$$347$$ 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 $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ −4.00000 −0.213809
$$351$$ 2.00000 0.106752
$$352$$ −4.00000 −0.213201
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 4.00000 0.212298
$$356$$ 10.0000 0.529999
$$357$$ 4.00000 0.211702
$$358$$ 12.0000 0.634220
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ −22.0000 −1.15629
$$363$$ −5.00000 −0.262432
$$364$$ 8.00000 0.419314
$$365$$ 14.0000 0.732793
$$366$$ −2.00000 −0.104542
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 2.00000 0.104116
$$370$$ 6.00000 0.311925
$$371$$ 8.00000 0.415339
$$372$$ −4.00000 −0.207390
$$373$$ 38.0000 1.96757 0.983783 0.179364i $$-0.0574041\pi$$
0.983783 + 0.179364i $$0.0574041\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 1.00000 0.0516398
$$376$$ 8.00000 0.412568
$$377$$ −4.00000 −0.206010
$$378$$ 4.00000 0.205738
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 16.0000 0.819705
$$382$$ −8.00000 −0.409316
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −16.0000 −0.815436
$$386$$ 18.0000 0.916176
$$387$$ −12.0000 −0.609994
$$388$$ 18.0000 0.913812
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ −2.00000 −0.101274
$$391$$ −4.00000 −0.202289
$$392$$ 9.00000 0.454569
$$393$$ −20.0000 −1.00887
$$394$$ −14.0000 −0.705310
$$395$$ 12.0000 0.603786
$$396$$ −4.00000 −0.201008
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ 4.00000 0.200502
$$399$$ −16.0000 −0.801002
$$400$$ 1.00000 0.0500000
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ −8.00000 −0.398508
$$404$$ −10.0000 −0.497519
$$405$$ −1.00000 −0.0496904
$$406$$ −8.00000 −0.397033
$$407$$ 24.0000 1.18964
$$408$$ −1.00000 −0.0495074
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ −2.00000 −0.0987730
$$411$$ 6.00000 0.295958
$$412$$ 0 0
$$413$$ −48.0000 −2.36193
$$414$$ −4.00000 −0.196589
$$415$$ 4.00000 0.196352
$$416$$ −2.00000 −0.0980581
$$417$$ 20.0000 0.979404
$$418$$ 16.0000 0.782586
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ −4.00000 −0.195180
$$421$$ −18.0000 −0.877266 −0.438633 0.898666i $$-0.644537\pi$$
−0.438633 + 0.898666i $$0.644537\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 8.00000 0.388973
$$424$$ −2.00000 −0.0971286
$$425$$ 1.00000 0.0485071
$$426$$ 4.00000 0.193801
$$427$$ −8.00000 −0.387147
$$428$$ 12.0000 0.580042
$$429$$ −8.00000 −0.386244
$$430$$ 12.0000 0.578691
$$431$$ −4.00000 −0.192673 −0.0963366 0.995349i $$-0.530713\pi$$
−0.0963366 + 0.995349i $$0.530713\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 2.00000 0.0958927
$$436$$ 10.0000 0.478913
$$437$$ 16.0000 0.765384
$$438$$ 14.0000 0.668946
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ 4.00000 0.190693
$$441$$ 9.00000 0.428571
$$442$$ −2.00000 −0.0951303
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 6.00000 0.284747
$$445$$ −10.0000 −0.474045
$$446$$ −8.00000 −0.378811
$$447$$ 2.00000 0.0945968
$$448$$ −4.00000 −0.188982
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −8.00000 −0.376705
$$452$$ −6.00000 −0.282216
$$453$$ 8.00000 0.375873
$$454$$ −20.0000 −0.938647
$$455$$ −8.00000 −0.375046
$$456$$ 4.00000 0.187317
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 6.00000 0.280362
$$459$$ −1.00000 −0.0466760
$$460$$ 4.00000 0.186501
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ −16.0000 −0.744387
$$463$$ 40.0000 1.85896 0.929479 0.368875i $$-0.120257\pi$$
0.929479 + 0.368875i $$0.120257\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 4.00000 0.185496
$$466$$ 18.0000 0.833834
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ −16.0000 −0.738811
$$470$$ −8.00000 −0.369012
$$471$$ 10.0000 0.460776
$$472$$ 12.0000 0.552345
$$473$$ 48.0000 2.20704
$$474$$ 12.0000 0.551178
$$475$$ −4.00000 −0.183533
$$476$$ −4.00000 −0.183340
$$477$$ −2.00000 −0.0915737
$$478$$ −24.0000 −1.09773
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 12.0000 0.547153
$$482$$ 10.0000 0.455488
$$483$$ −16.0000 −0.728025
$$484$$ 5.00000 0.227273
$$485$$ −18.0000 −0.817338
$$486$$ −1.00000 −0.0453609
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ 20.0000 0.904431
$$490$$ −9.00000 −0.406579
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ 2.00000 0.0900755
$$494$$ 8.00000 0.359937
$$495$$ 4.00000 0.179787
$$496$$ 4.00000 0.179605
$$497$$ 16.0000 0.717698
$$498$$ 4.00000 0.179244
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −12.0000 −0.536120
$$502$$ −28.0000 −1.24970
$$503$$ 44.0000 1.96186 0.980932 0.194354i $$-0.0622609\pi$$
0.980932 + 0.194354i $$0.0622609\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 10.0000 0.444994
$$506$$ 16.0000 0.711287
$$507$$ 9.00000 0.399704
$$508$$ −16.0000 −0.709885
$$509$$ 38.0000 1.68432 0.842160 0.539227i $$-0.181284\pi$$
0.842160 + 0.539227i $$0.181284\pi$$
$$510$$ 1.00000 0.0442807
$$511$$ 56.0000 2.47729
$$512$$ 1.00000 0.0441942
$$513$$ 4.00000 0.176604
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 12.0000 0.528271
$$517$$ −32.0000 −1.40736
$$518$$ 24.0000 1.05450
$$519$$ 14.0000 0.614532
$$520$$ 2.00000 0.0877058
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 20.0000 0.873704
$$525$$ 4.00000 0.174574
$$526$$ −16.0000 −0.697633
$$527$$ 4.00000 0.174243
$$528$$ 4.00000 0.174078
$$529$$ −7.00000 −0.304348
$$530$$ 2.00000 0.0868744
$$531$$ 12.0000 0.520756
$$532$$ 16.0000 0.693688
$$533$$ −4.00000 −0.173259
$$534$$ −10.0000 −0.432742
$$535$$ −12.0000 −0.518805
$$536$$ 4.00000 0.172774
$$537$$ −12.0000 −0.517838
$$538$$ −30.0000 −1.29339
$$539$$ −36.0000 −1.55063
$$540$$ 1.00000 0.0430331
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ 0 0
$$543$$ 22.0000 0.944110
$$544$$ 1.00000 0.0428746
$$545$$ −10.0000 −0.428353
$$546$$ −8.00000 −0.342368
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 2.00000 0.0853579
$$550$$ −4.00000 −0.170561
$$551$$ −8.00000 −0.340811
$$552$$ 4.00000 0.170251
$$553$$ 48.0000 2.04117
$$554$$ 2.00000 0.0849719
$$555$$ −6.00000 −0.254686
$$556$$ −20.0000 −0.848189
$$557$$ −26.0000 −1.10166 −0.550828 0.834619i $$-0.685688\pi$$
−0.550828 + 0.834619i $$0.685688\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 24.0000 1.01509
$$560$$ 4.00000 0.169031
$$561$$ 4.00000 0.168880
$$562$$ 10.0000 0.421825
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 6.00000 0.252422
$$566$$ −20.0000 −0.840663
$$567$$ −4.00000 −0.167984
$$568$$ −4.00000 −0.167836
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ −4.00000 −0.167542
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ 8.00000 0.334497
$$573$$ 8.00000 0.334205
$$574$$ −8.00000 −0.333914
$$575$$ −4.00000 −0.166812
$$576$$ 1.00000 0.0416667
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ −18.0000 −0.748054
$$580$$ −2.00000 −0.0830455
$$581$$ 16.0000 0.663792
$$582$$ −18.0000 −0.746124
$$583$$ 8.00000 0.331326
$$584$$ −14.0000 −0.579324
$$585$$ 2.00000 0.0826898
$$586$$ −26.0000 −1.07405
$$587$$ 44.0000 1.81607 0.908037 0.418890i $$-0.137581\pi$$
0.908037 + 0.418890i $$0.137581\pi$$
$$588$$ −9.00000 −0.371154
$$589$$ −16.0000 −0.659269
$$590$$ −12.0000 −0.494032
$$591$$ 14.0000 0.575883
$$592$$ −6.00000 −0.246598
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 4.00000 0.163984
$$596$$ −2.00000 −0.0819232
$$597$$ −4.00000 −0.163709
$$598$$ 8.00000 0.327144
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 48.0000 1.95633
$$603$$ 4.00000 0.162893
$$604$$ −8.00000 −0.325515
$$605$$ −5.00000 −0.203279
$$606$$ 10.0000 0.406222
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 8.00000 0.324176
$$610$$ −2.00000 −0.0809776
$$611$$ −16.0000 −0.647291
$$612$$ 1.00000 0.0404226
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 2.00000 0.0806478
$$616$$ 16.0000 0.644658
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 4.00000 0.160514
$$622$$ 20.0000 0.801927
$$623$$ −40.0000 −1.60257
$$624$$ 2.00000 0.0800641
$$625$$ 1.00000 0.0400000
$$626$$ 10.0000 0.399680
$$627$$ −16.0000 −0.638978
$$628$$ −10.0000 −0.399043
$$629$$ −6.00000 −0.239236
$$630$$ 4.00000 0.159364
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −12.0000 −0.477334
$$633$$ −4.00000 −0.158986
$$634$$ 10.0000 0.397151
$$635$$ 16.0000 0.634941
$$636$$ 2.00000 0.0793052
$$637$$ −18.0000 −0.713186
$$638$$ −8.00000 −0.316723
$$639$$ −4.00000 −0.158238
$$640$$ −1.00000 −0.0395285
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ 28.0000 1.10421 0.552106 0.833774i $$-0.313824\pi$$
0.552106 + 0.833774i $$0.313824\pi$$
$$644$$ 16.0000 0.630488
$$645$$ −12.0000 −0.472500
$$646$$ −4.00000 −0.157378
$$647$$ −16.0000 −0.629025 −0.314512 0.949253i $$-0.601841\pi$$
−0.314512 + 0.949253i $$0.601841\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −48.0000 −1.88416
$$650$$ −2.00000 −0.0784465
$$651$$ 16.0000 0.627089
$$652$$ −20.0000 −0.783260
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ −20.0000 −0.781465
$$656$$ 2.00000 0.0780869
$$657$$ −14.0000 −0.546192
$$658$$ −32.0000 −1.24749
$$659$$ −28.0000 −1.09073 −0.545363 0.838200i $$-0.683608\pi$$
−0.545363 + 0.838200i $$0.683608\pi$$
$$660$$ −4.00000 −0.155700
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 2.00000 0.0776736
$$664$$ −4.00000 −0.155230
$$665$$ −16.0000 −0.620453
$$666$$ −6.00000 −0.232495
$$667$$ −8.00000 −0.309761
$$668$$ 12.0000 0.464294
$$669$$ 8.00000 0.309298
$$670$$ −4.00000 −0.154533
$$671$$ −8.00000 −0.308837
$$672$$ 4.00000 0.154303
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ −1.00000 −0.0384900
$$676$$ −9.00000 −0.346154
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 6.00000 0.230429
$$679$$ −72.0000 −2.76311
$$680$$ −1.00000 −0.0383482
$$681$$ 20.0000 0.766402
$$682$$ −16.0000 −0.612672
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 6.00000 0.229248
$$686$$ −8.00000 −0.305441
$$687$$ −6.00000 −0.228914
$$688$$ −12.0000 −0.457496
$$689$$ 4.00000 0.152388
$$690$$ −4.00000 −0.152277
$$691$$ −44.0000 −1.67384 −0.836919 0.547326i $$-0.815646\pi$$
−0.836919 + 0.547326i $$0.815646\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 16.0000 0.607790
$$694$$ 20.0000 0.759190
$$695$$ 20.0000 0.758643
$$696$$ −2.00000 −0.0758098
$$697$$ 2.00000 0.0757554
$$698$$ 22.0000 0.832712
$$699$$ −18.0000 −0.680823
$$700$$ −4.00000 −0.151186
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ 24.0000 0.905177
$$704$$ −4.00000 −0.150756
$$705$$ 8.00000 0.301297
$$706$$ 2.00000 0.0752710
$$707$$ 40.0000 1.50435
$$708$$ −12.0000 −0.450988
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 4.00000 0.150117
$$711$$ −12.0000 −0.450035
$$712$$ 10.0000 0.374766
$$713$$ −16.0000 −0.599205
$$714$$ 4.00000 0.149696
$$715$$ −8.00000 −0.299183
$$716$$ 12.0000 0.448461
$$717$$ 24.0000 0.896296
$$718$$ −24.0000 −0.895672
$$719$$ −44.0000 −1.64092 −0.820462 0.571702i $$-0.806283\pi$$
−0.820462 + 0.571702i $$0.806283\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ −10.0000 −0.371904
$$724$$ −22.0000 −0.817624
$$725$$ 2.00000 0.0742781
$$726$$ −5.00000 −0.185567
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 8.00000 0.296500
$$729$$ 1.00000 0.0370370
$$730$$ 14.0000 0.518163
$$731$$ −12.0000 −0.443836
$$732$$ −2.00000 −0.0739221
$$733$$ −18.0000 −0.664845 −0.332423 0.943131i $$-0.607866\pi$$
−0.332423 + 0.943131i $$0.607866\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 9.00000 0.331970
$$736$$ −4.00000 −0.147442
$$737$$ −16.0000 −0.589368
$$738$$ 2.00000 0.0736210
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 6.00000 0.220564
$$741$$ −8.00000 −0.293887
$$742$$ 8.00000 0.293689
$$743$$ 28.0000 1.02722 0.513610 0.858024i $$-0.328308\pi$$
0.513610 + 0.858024i $$0.328308\pi$$
$$744$$ −4.00000 −0.146647
$$745$$ 2.00000 0.0732743
$$746$$ 38.0000 1.39128
$$747$$ −4.00000 −0.146352
$$748$$ −4.00000 −0.146254
$$749$$ −48.0000 −1.75388
$$750$$ 1.00000 0.0365148
$$751$$ 12.0000 0.437886 0.218943 0.975738i $$-0.429739\pi$$
0.218943 + 0.975738i $$0.429739\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 28.0000 1.02038
$$754$$ −4.00000 −0.145671
$$755$$ 8.00000 0.291150
$$756$$ 4.00000 0.145479
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ 20.0000 0.726433
$$759$$ −16.0000 −0.580763
$$760$$ 4.00000 0.145095
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 16.0000 0.579619
$$763$$ −40.0000 −1.44810
$$764$$ −8.00000 −0.289430
$$765$$ −1.00000 −0.0361551
$$766$$ 8.00000 0.289052
$$767$$ −24.0000 −0.866590
$$768$$ −1.00000 −0.0360844
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ −16.0000 −0.576600
$$771$$ −18.0000 −0.648254
$$772$$ 18.0000 0.647834
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 4.00000 0.143684
$$776$$ 18.0000 0.646162
$$777$$ −24.0000 −0.860995
$$778$$ −18.0000 −0.645331
$$779$$ −8.00000 −0.286630
$$780$$ −2.00000 −0.0716115
$$781$$ 16.0000 0.572525
$$782$$ −4.00000 −0.143040
$$783$$ −2.00000 −0.0714742
$$784$$ 9.00000 0.321429
$$785$$ 10.0000 0.356915
$$786$$ −20.0000 −0.713376
$$787$$ 36.0000 1.28326 0.641631 0.767014i $$-0.278258\pi$$
0.641631 + 0.767014i $$0.278258\pi$$
$$788$$ −14.0000 −0.498729
$$789$$ 16.0000 0.569615
$$790$$ 12.0000 0.426941
$$791$$ 24.0000 0.853342
$$792$$ −4.00000 −0.142134
$$793$$ −4.00000 −0.142044
$$794$$ −30.0000 −1.06466
$$795$$ −2.00000 −0.0709327
$$796$$ 4.00000 0.141776
$$797$$ −26.0000 −0.920967 −0.460484 0.887668i $$-0.652324\pi$$
−0.460484 + 0.887668i $$0.652324\pi$$
$$798$$ −16.0000 −0.566394
$$799$$ 8.00000 0.283020
$$800$$ 1.00000 0.0353553
$$801$$ 10.0000 0.353333
$$802$$ −6.00000 −0.211867
$$803$$ 56.0000 1.97620
$$804$$ −4.00000 −0.141069
$$805$$ −16.0000 −0.563926
$$806$$ −8.00000 −0.281788
$$807$$ 30.0000 1.05605
$$808$$ −10.0000 −0.351799
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ 20.0000 0.700569
$$816$$ −1.00000 −0.0350070
$$817$$ 48.0000 1.67931
$$818$$ 10.0000 0.349642
$$819$$ 8.00000 0.279543
$$820$$ −2.00000 −0.0698430
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −28.0000 −0.976019 −0.488009 0.872838i $$-0.662277\pi$$
−0.488009 + 0.872838i $$0.662277\pi$$
$$824$$ 0 0
$$825$$ 4.00000 0.139262
$$826$$ −48.0000 −1.67013
$$827$$ 44.0000 1.53003 0.765015 0.644013i $$-0.222732\pi$$
0.765015 + 0.644013i $$0.222732\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 4.00000 0.138842
$$831$$ −2.00000 −0.0693792
$$832$$ −2.00000 −0.0693375
$$833$$ 9.00000 0.311832
$$834$$ 20.0000 0.692543
$$835$$ −12.0000 −0.415277
$$836$$ 16.0000 0.553372
$$837$$ −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
$$842$$ −18.0000 −0.620321
$$843$$ −10.0000 −0.344418
$$844$$ 4.00000 0.137686
$$845$$ 9.00000 0.309609
$$846$$ 8.00000 0.275046
$$847$$ −20.0000 −0.687208
$$848$$ −2.00000 −0.0686803
$$849$$ 20.0000 0.686398
$$850$$ 1.00000 0.0342997
$$851$$ 24.0000 0.822709
$$852$$ 4.00000 0.137038
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 4.00000 0.136797
$$856$$ 12.0000 0.410152
$$857$$ −22.0000 −0.751506 −0.375753 0.926720i $$-0.622616\pi$$
−0.375753 + 0.926720i $$0.622616\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 12.0000 0.409197
$$861$$ 8.00000 0.272639
$$862$$ −4.00000 −0.136241
$$863$$ 32.0000 1.08929 0.544646 0.838666i $$-0.316664\pi$$
0.544646 + 0.838666i $$0.316664\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 14.0000 0.476014
$$866$$ −14.0000 −0.475739
$$867$$ −1.00000 −0.0339618
$$868$$ −16.0000 −0.543075
$$869$$ 48.0000 1.62829
$$870$$ 2.00000 0.0678064
$$871$$ −8.00000 −0.271070
$$872$$ 10.0000 0.338643
$$873$$ 18.0000 0.609208
$$874$$ 16.0000 0.541208
$$875$$ 4.00000 0.135225
$$876$$ 14.0000 0.473016
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ −12.0000 −0.404980
$$879$$ 26.0000 0.876958
$$880$$ 4.00000 0.134840
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 9.00000 0.303046
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 12.0000 0.403376
$$886$$ −36.0000 −1.20944
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ 6.00000 0.201347
$$889$$ 64.0000 2.14649
$$890$$ −10.0000 −0.335201
$$891$$ −4.00000 −0.134005
$$892$$ −8.00000 −0.267860
$$893$$ −32.0000 −1.07084
$$894$$ 2.00000 0.0668900
$$895$$ −12.0000 −0.401116
$$896$$ −4.00000 −0.133631
$$897$$ −8.00000 −0.267112
$$898$$ −6.00000 −0.200223
$$899$$ 8.00000 0.266815
$$900$$ 1.00000 0.0333333
$$901$$ −2.00000 −0.0666297
$$902$$ −8.00000 −0.266371
$$903$$ −48.0000 −1.59734
$$904$$ −6.00000 −0.199557
$$905$$ 22.0000 0.731305
$$906$$ 8.00000 0.265782
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ −10.0000 −0.331679
$$910$$ −8.00000 −0.265197
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 16.0000 0.529523
$$914$$ 10.0000 0.330771
$$915$$ 2.00000 0.0661180
$$916$$ 6.00000 0.198246
$$917$$ −80.0000 −2.64183
$$918$$ −1.00000 −0.0330049
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ 4.00000 0.131876
$$921$$ 4.00000 0.131804
$$922$$ 14.0000 0.461065
$$923$$ 8.00000 0.263323
$$924$$ −16.0000 −0.526361
$$925$$ −6.00000 −0.197279
$$926$$ 40.0000 1.31448
$$927$$ 0 0
$$928$$ 2.00000 0.0656532
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 4.00000 0.131165
$$931$$ −36.0000 −1.17985
$$932$$ 18.0000 0.589610
$$933$$ −20.0000 −0.654771
$$934$$ 36.0000 1.17796
$$935$$ 4.00000 0.130814
$$936$$ −2.00000 −0.0653720
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ −10.0000 −0.326338
$$940$$ −8.00000 −0.260931
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 10.0000 0.325818
$$943$$ −8.00000 −0.260516
$$944$$ 12.0000 0.390567
$$945$$ −4.00000 −0.130120
$$946$$ 48.0000 1.56061
$$947$$ −28.0000 −0.909878 −0.454939 0.890523i $$-0.650339\pi$$
−0.454939 + 0.890523i $$0.650339\pi$$
$$948$$ 12.0000 0.389742
$$949$$ 28.0000 0.908918
$$950$$ −4.00000 −0.129777
$$951$$ −10.0000 −0.324272
$$952$$ −4.00000 −0.129641
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 8.00000 0.258874
$$956$$ −24.0000 −0.776215
$$957$$ 8.00000 0.258603
$$958$$ 20.0000 0.646171
$$959$$ 24.0000 0.775000
$$960$$ 1.00000 0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ 12.0000 0.386896
$$963$$ 12.0000 0.386695
$$964$$ 10.0000 0.322078
$$965$$ −18.0000 −0.579441
$$966$$ −16.0000 −0.514792
$$967$$ −16.0000 −0.514525 −0.257263 0.966342i $$-0.582821\pi$$
−0.257263 + 0.966342i $$0.582821\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 4.00000 0.128499
$$970$$ −18.0000 −0.577945
$$971$$ 28.0000 0.898563 0.449281 0.893390i $$-0.351680\pi$$
0.449281 + 0.893390i $$0.351680\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 80.0000 2.56468
$$974$$ 20.0000 0.640841
$$975$$ 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 $$0.659324\pi$$
$$978$$ 20.0000 0.639529
$$979$$ −40.0000 −1.27841
$$980$$ −9.00000 −0.287494
$$981$$ 10.0000 0.319275
$$982$$ −12.0000 −0.382935
$$983$$ −28.0000 −0.893061 −0.446531 0.894768i $$-0.647341\pi$$
−0.446531 + 0.894768i $$0.647341\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 14.0000 0.446077
$$986$$ 2.00000 0.0636930
$$987$$ 32.0000 1.01857
$$988$$ 8.00000 0.254514
$$989$$ 48.0000 1.52631
$$990$$ 4.00000 0.127128
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ 4.00000 0.127000
$$993$$ −28.0000 −0.888553
$$994$$ 16.0000 0.507489
$$995$$ −4.00000 −0.126809
$$996$$ 4.00000 0.126745
$$997$$ −38.0000 −1.20347 −0.601736 0.798695i $$-0.705524\pi$$
−0.601736 + 0.798695i $$0.705524\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 510.2.a.c.1.1 1
3.2 odd 2 1530.2.a.d.1.1 1
4.3 odd 2 4080.2.a.x.1.1 1
5.2 odd 4 2550.2.d.b.2449.2 2
5.3 odd 4 2550.2.d.b.2449.1 2
5.4 even 2 2550.2.a.n.1.1 1
15.14 odd 2 7650.2.a.cn.1.1 1
17.16 even 2 8670.2.a.bb.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.c.1.1 1 1.1 even 1 trivial
1530.2.a.d.1.1 1 3.2 odd 2
2550.2.a.n.1.1 1 5.4 even 2
2550.2.d.b.2449.1 2 5.3 odd 4
2550.2.d.b.2449.2 2 5.2 odd 4
4080.2.a.x.1.1 1 4.3 odd 2
7650.2.a.cn.1.1 1 15.14 odd 2
8670.2.a.bb.1.1 1 17.16 even 2