# Properties

 Label 1050.2.a.q.1.1 Level $1050$ Weight $2$ Character 1050.1 Self dual yes Analytic conductor $8.384$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1050.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +1.00000 q^{21} -4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} +10.0000 q^{29} -8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +2.00000 q^{39} -2.00000 q^{41} +1.00000 q^{42} -8.00000 q^{43} -4.00000 q^{44} +8.00000 q^{46} -4.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +6.00000 q^{51} +2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} +1.00000 q^{56} +10.0000 q^{58} +4.00000 q^{59} -6.00000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} +6.00000 q^{68} +8.00000 q^{69} -12.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} -4.00000 q^{77} +2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} -8.00000 q^{86} +10.0000 q^{87} -4.00000 q^{88} +14.0000 q^{89} +2.00000 q^{91} +8.00000 q^{92} -8.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −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 $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ −4.00000 −0.852803
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 1.00000 0.192450
$$28$$ 1.00000 0.188982
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −4.00000 −0.696311
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 2.00000 0.277350
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 10.0000 1.31306
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ −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 $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 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 $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 10.0000 1.07211
$$88$$ −4.00000 −0.426401
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 8.00000 0.834058
$$93$$ −8.00000 −0.829561
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 6.00000 0.594089
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ 1.00000 0.0944911
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 10.0000 0.928477
$$117$$ 2.00000 0.184900
$$118$$ 4.00000 0.368230
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −6.00000 −0.543214
$$123$$ −2.00000 −0.180334
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 8.00000 0.681005
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ −12.0000 −1.00702
$$143$$ −8.00000 −0.668994
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 1.00000 0.0824786
$$148$$ −2.00000 −0.164399
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 6.00000 0.485071
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 1.00000 0.0785674
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 4.00000 0.300658
$$178$$ 14.0000 1.04934
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 2.00000 0.148250
$$183$$ −6.00000 −0.443533
$$184$$ 8.00000 0.589768
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ −24.0000 −1.75505
$$188$$ −4.00000 −0.291730
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 22.0000 1.56744 0.783718 0.621117i $$-0.213321\pi$$
0.783718 + 0.621117i $$0.213321\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −6.00000 −0.422159
$$203$$ 10.0000 0.701862
$$204$$ 6.00000 0.420084
$$205$$ 0 0
$$206$$ 16.0000 1.11477
$$207$$ 8.00000 0.556038
$$208$$ 2.00000 0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ −12.0000 −0.822226
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −8.00000 −0.543075
$$218$$ −18.0000 −1.21911
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ −2.00000 −0.134231
$$223$$ 24.0000 1.60716 0.803579 0.595198i $$-0.202926\pi$$
0.803579 + 0.595198i $$0.202926\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 10.0000 0.656532
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ −8.00000 −0.519656
$$238$$ 6.00000 0.388922
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 1.00000 0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ 0 0
$$248$$ −8.00000 −0.508001
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ −32.0000 −2.01182
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 4.00000 0.247121
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 14.0000 0.856786
$$268$$ 0 0
$$269$$ 2.00000 0.121942 0.0609711 0.998140i $$-0.480580\pi$$
0.0609711 + 0.998140i $$0.480580\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 2.00000 0.121046
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ −4.00000 −0.238197
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ −2.00000 −0.118056
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 6.00000 0.351123
$$293$$ 30.0000 1.75262 0.876309 0.481749i $$-0.159998\pi$$
0.876309 + 0.481749i $$0.159998\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ −4.00000 −0.232104
$$298$$ −6.00000 −0.347571
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ −6.00000 −0.344691
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 2.00000 0.113228
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ −40.0000 −2.23957
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 8.00000 0.445823
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ −18.0000 −0.995402
$$328$$ −2.00000 −0.110432
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 4.00000 0.219529
$$333$$ −2.00000 −0.109599
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 1.00000 0.0545545
$$337$$ 6.00000 0.326841 0.163420 0.986557i $$-0.447747\pi$$
0.163420 + 0.986557i $$0.447747\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 32.0000 1.73290
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 10.0000 0.536056
$$349$$ 18.0000 0.963518 0.481759 0.876304i $$-0.339998\pi$$
0.481759 + 0.876304i $$0.339998\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ −4.00000 −0.213201
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ 6.00000 0.317554
$$358$$ −4.00000 −0.211407
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 18.0000 0.946059
$$363$$ 5.00000 0.262432
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ −6.00000 −0.313625
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ 8.00000 0.417029
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ −10.0000 −0.519174
$$372$$ −8.00000 −0.414781
$$373$$ 30.0000 1.55334 0.776671 0.629907i $$-0.216907\pi$$
0.776671 + 0.629907i $$0.216907\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ 20.0000 1.03005
$$378$$ 1.00000 0.0514344
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 12.0000 0.613973
$$383$$ 28.0000 1.43073 0.715367 0.698749i $$-0.246260\pi$$
0.715367 + 0.698749i $$0.246260\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ −8.00000 −0.406663
$$388$$ −2.00000 −0.101535
$$389$$ 10.0000 0.507020 0.253510 0.967333i $$-0.418415\pi$$
0.253510 + 0.967333i $$0.418415\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ 1.00000 0.0505076
$$393$$ 4.00000 0.201773
$$394$$ 22.0000 1.10834
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 24.0000 1.20301
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 34.0000 1.69788 0.848939 0.528490i $$-0.177242\pi$$
0.848939 + 0.528490i $$0.177242\pi$$
$$402$$ 0 0
$$403$$ −16.0000 −0.797017
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 10.0000 0.496292
$$407$$ 8.00000 0.396545
$$408$$ 6.00000 0.297044
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 16.0000 0.788263
$$413$$ 4.00000 0.196827
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ −16.0000 −0.783523
$$418$$ 0 0
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ −4.00000 −0.194487
$$424$$ −10.0000 −0.485643
$$425$$ 0 0
$$426$$ −12.0000 −0.581402
$$427$$ −6.00000 −0.290360
$$428$$ 4.00000 0.193347
$$429$$ −8.00000 −0.386244
$$430$$ 0 0
$$431$$ −36.0000 −1.73406 −0.867029 0.498257i $$-0.833974\pi$$
−0.867029 + 0.498257i $$0.833974\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ −18.0000 −0.862044
$$437$$ 0 0
$$438$$ 6.00000 0.286691
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 12.0000 0.570782
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ 24.0000 1.13643
$$447$$ −6.00000 −0.283790
$$448$$ 1.00000 0.0472456
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ −22.0000 −1.02799
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ 18.0000 0.833834
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ 4.00000 0.184115
$$473$$ 32.0000 1.47136
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 6.00000 0.275010
$$477$$ −10.0000 −0.457869
$$478$$ −12.0000 −0.548867
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ −14.0000 −0.637683
$$483$$ 8.00000 0.364013
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ 60.0000 2.70226
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ −12.0000 −0.538274
$$498$$ 4.00000 0.179244
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ −20.0000 −0.892644
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ −32.0000 −1.42257
$$507$$ −9.00000 −0.399704
$$508$$ −16.0000 −0.709885
$$509$$ 2.00000 0.0886484 0.0443242 0.999017i $$-0.485887\pi$$
0.0443242 + 0.999017i $$0.485887\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ 16.0000 0.703679
$$518$$ −2.00000 −0.0878750
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 10.0000 0.437688
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −48.0000 −2.09091
$$528$$ −4.00000 −0.174078
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ −4.00000 −0.173259
$$534$$ 14.0000 0.605839
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −4.00000 −0.172613
$$538$$ 2.00000 0.0862261
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 18.0000 0.772454
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ 32.0000 1.36822 0.684111 0.729378i $$-0.260191\pi$$
0.684111 + 0.729378i $$0.260191\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 8.00000 0.340503
$$553$$ −8.00000 −0.340195
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ −24.0000 −1.01328
$$562$$ −22.0000 −0.928014
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ −4.00000 −0.168430
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 1.00000 0.0419961
$$568$$ −12.0000 −0.503509
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 12.0000 0.501307
$$574$$ −2.00000 −0.0834784
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 19.0000 0.790296
$$579$$ −10.0000 −0.415586
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ −2.00000 −0.0829027
$$583$$ 40.0000 1.65663
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 22.0000 0.904959
$$592$$ −2.00000 −0.0821995
$$593$$ 38.0000 1.56047 0.780236 0.625485i $$-0.215099\pi$$
0.780236 + 0.625485i $$0.215099\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 24.0000 0.982255
$$598$$ 16.0000 0.654289
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 0 0
$$609$$ 10.0000 0.405220
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 6.00000 0.242536
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 16.0000 0.643614
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 0 0
$$623$$ 14.0000 0.560898
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ 6.00000 0.239808
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ −4.00000 −0.158986
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ −10.0000 −0.396526
$$637$$ 2.00000 0.0792429
$$638$$ −40.0000 −1.58362
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 4.00000 0.157867
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ −16.0000 −0.626608
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ −18.0000 −0.703856
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ 6.00000 0.234082
$$658$$ −4.00000 −0.155936
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 12.0000 0.466041
$$664$$ 4.00000 0.155230
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 80.0000 3.09761
$$668$$ 12.0000 0.464294
$$669$$ 24.0000 0.927894
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 1.00000 0.0385758
$$673$$ −10.0000 −0.385472 −0.192736 0.981251i $$-0.561736\pi$$
−0.192736 + 0.981251i $$0.561736\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 22.0000 0.845529 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 32.0000 1.22534
$$683$$ −28.0000 −1.07139 −0.535695 0.844411i $$-0.679950\pi$$
−0.535695 + 0.844411i $$0.679950\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ −22.0000 −0.839352
$$688$$ −8.00000 −0.304997
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ −4.00000 −0.151947
$$694$$ −20.0000 −0.759190
$$695$$ 0 0
$$696$$ 10.0000 0.379049
$$697$$ −12.0000 −0.454532
$$698$$ 18.0000 0.681310
$$699$$ 18.0000 0.680823
$$700$$ 0 0
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ 0 0
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ −6.00000 −0.225653
$$708$$ 4.00000 0.150329
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 14.0000 0.524672
$$713$$ −64.0000 −2.39682
$$714$$ 6.00000 0.224544
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ −12.0000 −0.448148
$$718$$ −20.0000 −0.746393
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ −19.0000 −0.707107
$$723$$ −14.0000 −0.520666
$$724$$ 18.0000 0.668965
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −48.0000 −1.77534
$$732$$ −6.00000 −0.221766
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ 0 0
$$738$$ −2.00000 −0.0736210
$$739$$ −4.00000 −0.147142 −0.0735712 0.997290i $$-0.523440\pi$$
−0.0735712 + 0.997290i $$0.523440\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −10.0000 −0.367112
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ 30.0000 1.09838
$$747$$ 4.00000 0.146352
$$748$$ −24.0000 −0.877527
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ −4.00000 −0.145865
$$753$$ −20.0000 −0.728841
$$754$$ 20.0000 0.728357
$$755$$ 0 0
$$756$$ 1.00000 0.0363696
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ −28.0000 −1.01701
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ −16.0000 −0.579619
$$763$$ −18.0000 −0.651644
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 28.0000 1.01168
$$767$$ 8.00000 0.288863
$$768$$ 1.00000 0.0360844
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ 14.0000 0.504198
$$772$$ −10.0000 −0.359908
$$773$$ −18.0000 −0.647415 −0.323708 0.946157i $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ −8.00000 −0.287554
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ −2.00000 −0.0717496
$$778$$ 10.0000 0.358517
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ 48.0000 1.71648
$$783$$ 10.0000 0.357371
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 4.00000 0.142675
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ 22.0000 0.783718
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ −4.00000 −0.142134
$$793$$ −12.0000 −0.426132
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ 24.0000 0.850657
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 14.0000 0.494666
$$802$$ 34.0000 1.20058
$$803$$ −24.0000 −0.846942
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 2.00000 0.0704033
$$808$$ −6.00000 −0.211079
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 10.0000 0.350931
$$813$$ 8.00000 0.280572
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 0 0
$$818$$ 26.0000 0.909069
$$819$$ 2.00000 0.0698857
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 4.00000 0.139178
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ 22.0000 0.763172
$$832$$ 2.00000 0.0693375
$$833$$ 6.00000 0.207888
$$834$$ −16.0000 −0.554035
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −8.00000 −0.276520
$$838$$ 28.0000 0.967244
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −26.0000 −0.896019
$$843$$ −22.0000 −0.757720
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ 5.00000 0.171802
$$848$$ −10.0000 −0.343401
$$849$$ −28.0000 −0.960958
$$850$$ 0 0
$$851$$ −16.0000 −0.548473
$$852$$ −12.0000 −0.411113
$$853$$ −22.0000 −0.753266 −0.376633 0.926363i $$-0.622918\pi$$
−0.376633 + 0.926363i $$0.622918\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ 38.0000 1.29806 0.649028 0.760765i $$-0.275176\pi$$
0.649028 + 0.760765i $$0.275176\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ 24.0000 0.818869 0.409435 0.912339i $$-0.365726\pi$$
0.409435 + 0.912339i $$0.365726\pi$$
$$860$$ 0 0
$$861$$ −2.00000 −0.0681598
$$862$$ −36.0000 −1.22616
$$863$$ 8.00000 0.272323 0.136162 0.990687i $$-0.456523\pi$$
0.136162 + 0.990687i $$0.456523\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ 19.0000 0.645274
$$868$$ −8.00000 −0.271538
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −18.0000 −0.609557
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 32.0000 1.07995
$$879$$ 30.0000 1.01187
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ −48.0000 −1.61533 −0.807664 0.589643i $$-0.799269\pi$$
−0.807664 + 0.589643i $$0.799269\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ −20.0000 −0.671534 −0.335767 0.941945i $$-0.608996\pi$$
−0.335767 + 0.941945i $$0.608996\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 24.0000 0.803579
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 16.0000 0.534224
$$898$$ 26.0000 0.867631
$$899$$ −80.0000 −2.66815
$$900$$ 0 0
$$901$$ −60.0000 −1.99889
$$902$$ 8.00000 0.266371
$$903$$ −8.00000 −0.266223
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −8.00000 −0.265636 −0.132818 0.991140i $$-0.542403\pi$$
−0.132818 + 0.991140i $$0.542403\pi$$
$$908$$ −20.0000 −0.663723
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 4.00000 0.132092
$$918$$ 6.00000 0.198030
$$919$$ −24.0000 −0.791687 −0.395843 0.918318i $$-0.629548\pi$$
−0.395843 + 0.918318i $$0.629548\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 18.0000 0.592798
$$923$$ −24.0000 −0.789970
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ −8.00000 −0.262896
$$927$$ 16.0000 0.525509
$$928$$ 10.0000 0.328266
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 18.0000 0.589610
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 0 0
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 10.0000 0.325818
$$943$$ −16.0000 −0.521032
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 32.0000 1.04041
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 6.00000 0.194461
$$953$$ 50.0000 1.61966 0.809829 0.586665i $$-0.199560\pi$$
0.809829 + 0.586665i $$0.199560\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 0 0
$$956$$ −12.0000 −0.388108
$$957$$ −40.0000 −1.29302
$$958$$ −8.00000 −0.258468
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −4.00000 −0.128965
$$963$$ 4.00000 0.128898
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −16.0000 −0.512936
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ −6.00000 −0.191957 −0.0959785 0.995383i $$-0.530598\pi$$
−0.0959785 + 0.995383i $$0.530598\pi$$
$$978$$ −16.0000 −0.511624
$$979$$ −56.0000 −1.78977
$$980$$ 0 0
$$981$$ −18.0000 −0.574696
$$982$$ 20.0000 0.638226
$$983$$ −28.0000 −0.893061 −0.446531 0.894768i $$-0.647341\pi$$
−0.446531 + 0.894768i $$0.647341\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ 0 0
$$986$$ 60.0000 1.91079
$$987$$ −4.00000 −0.127321
$$988$$ 0 0
$$989$$ −64.0000 −2.03508
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ 20.0000 0.634681
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ 10.0000 0.316703 0.158352 0.987383i $$-0.449382\pi$$
0.158352 + 0.987383i $$0.449382\pi$$
$$998$$ 28.0000 0.886325
$$999$$ −2.00000 −0.0632772
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1050.2.a.q.1.1 1
3.2 odd 2 3150.2.a.t.1.1 1
4.3 odd 2 8400.2.a.m.1.1 1
5.2 odd 4 1050.2.g.f.799.2 2
5.3 odd 4 1050.2.g.f.799.1 2
5.4 even 2 210.2.a.a.1.1 1
7.6 odd 2 7350.2.a.bo.1.1 1
15.2 even 4 3150.2.g.t.2899.1 2
15.8 even 4 3150.2.g.t.2899.2 2
15.14 odd 2 630.2.a.i.1.1 1
20.19 odd 2 1680.2.a.o.1.1 1
35.4 even 6 1470.2.i.t.961.1 2
35.9 even 6 1470.2.i.t.361.1 2
35.19 odd 6 1470.2.i.n.361.1 2
35.24 odd 6 1470.2.i.n.961.1 2
35.34 odd 2 1470.2.a.g.1.1 1
40.19 odd 2 6720.2.a.z.1.1 1
40.29 even 2 6720.2.a.cg.1.1 1
60.59 even 2 5040.2.a.bg.1.1 1
105.104 even 2 4410.2.a.bc.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.a.a.1.1 1 5.4 even 2
630.2.a.i.1.1 1 15.14 odd 2
1050.2.a.q.1.1 1 1.1 even 1 trivial
1050.2.g.f.799.1 2 5.3 odd 4
1050.2.g.f.799.2 2 5.2 odd 4
1470.2.a.g.1.1 1 35.34 odd 2
1470.2.i.n.361.1 2 35.19 odd 6
1470.2.i.n.961.1 2 35.24 odd 6
1470.2.i.t.361.1 2 35.9 even 6
1470.2.i.t.961.1 2 35.4 even 6
1680.2.a.o.1.1 1 20.19 odd 2
3150.2.a.t.1.1 1 3.2 odd 2
3150.2.g.t.2899.1 2 15.2 even 4
3150.2.g.t.2899.2 2 15.8 even 4
4410.2.a.bc.1.1 1 105.104 even 2
5040.2.a.bg.1.1 1 60.59 even 2
6720.2.a.z.1.1 1 40.19 odd 2
6720.2.a.cg.1.1 1 40.29 even 2
7350.2.a.bo.1.1 1 7.6 odd 2
8400.2.a.m.1.1 1 4.3 odd 2