# Properties

 Label 5070.2.a.d.1.1 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} +5.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} +5.00000 q^{29} +1.00000 q^{30} -11.0000 q^{31} -1.00000 q^{32} +5.00000 q^{33} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +3.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} -2.00000 q^{42} -11.0000 q^{43} -5.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +6.00000 q^{53} +1.00000 q^{54} -5.00000 q^{55} +2.00000 q^{56} +2.00000 q^{57} -5.00000 q^{58} -15.0000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +11.0000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +16.0000 q^{67} -2.00000 q^{68} +1.00000 q^{69} +2.00000 q^{70} -1.00000 q^{72} -6.00000 q^{73} -3.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} +10.0000 q^{77} -11.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +6.00000 q^{83} +2.00000 q^{84} -2.00000 q^{85} +11.0000 q^{86} -5.00000 q^{87} +5.00000 q^{88} +2.00000 q^{89} -1.00000 q^{90} -1.00000 q^{92} +11.0000 q^{93} -9.00000 q^{94} -2.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} -5.00000 q^{99} +O(q^{100})$$

## 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
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.00000 0.436436
$$22$$ 5.00000 1.06600
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −2.00000 −0.377964
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −11.0000 −1.97566 −0.987829 0.155543i $$-0.950287\pi$$
−0.987829 + 0.155543i $$0.950287\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 5.00000 0.870388
$$34$$ 2.00000 0.342997
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −11.0000 −1.67748 −0.838742 0.544529i $$-0.816708\pi$$
−0.838742 + 0.544529i $$0.816708\pi$$
$$44$$ −5.00000 −0.753778
$$45$$ 1.00000 0.149071
$$46$$ 1.00000 0.147442
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −5.00000 −0.674200
$$56$$ 2.00000 0.267261
$$57$$ 2.00000 0.264906
$$58$$ −5.00000 −0.656532
$$59$$ −15.0000 −1.95283 −0.976417 0.215894i $$-0.930733\pi$$
−0.976417 + 0.215894i $$0.930733\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 11.0000 1.39700
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −5.00000 −0.615457
$$67$$ 16.0000 1.95471 0.977356 0.211604i $$-0.0678686\pi$$
0.977356 + 0.211604i $$0.0678686\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 1.00000 0.120386
$$70$$ 2.00000 0.239046
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −3.00000 −0.348743
$$75$$ −1.00000 −0.115470
$$76$$ −2.00000 −0.229416
$$77$$ 10.0000 1.13961
$$78$$ 0 0
$$79$$ −11.0000 −1.23760 −0.618798 0.785550i $$-0.712380\pi$$
−0.618798 + 0.785550i $$0.712380\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 2.00000 0.218218
$$85$$ −2.00000 −0.216930
$$86$$ 11.0000 1.18616
$$87$$ −5.00000 −0.536056
$$88$$ 5.00000 0.533002
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −1.00000 −0.104257
$$93$$ 11.0000 1.14065
$$94$$ −9.00000 −0.928279
$$95$$ −2.00000 −0.205196
$$96$$ 1.00000 0.102062
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 3.00000 0.303046
$$99$$ −5.00000 −0.502519
$$100$$ 1.00000 0.100000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ −6.00000 −0.582772
$$107$$ 10.0000 0.966736 0.483368 0.875417i $$-0.339413\pi$$
0.483368 + 0.875417i $$0.339413\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 5.00000 0.476731
$$111$$ −3.00000 −0.284747
$$112$$ −2.00000 −0.188982
$$113$$ 11.0000 1.03479 0.517396 0.855746i $$-0.326901\pi$$
0.517396 + 0.855746i $$0.326901\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ −1.00000 −0.0932505
$$116$$ 5.00000 0.464238
$$117$$ 0 0
$$118$$ 15.0000 1.38086
$$119$$ 4.00000 0.366679
$$120$$ 1.00000 0.0912871
$$121$$ 14.0000 1.27273
$$122$$ −10.0000 −0.905357
$$123$$ 2.00000 0.180334
$$124$$ −11.0000 −0.987829
$$125$$ 1.00000 0.0894427
$$126$$ 2.00000 0.178174
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 11.0000 0.968496
$$130$$ 0 0
$$131$$ −1.00000 −0.0873704 −0.0436852 0.999045i $$-0.513910\pi$$
−0.0436852 + 0.999045i $$0.513910\pi$$
$$132$$ 5.00000 0.435194
$$133$$ 4.00000 0.346844
$$134$$ −16.0000 −1.38219
$$135$$ −1.00000 −0.0860663
$$136$$ 2.00000 0.171499
$$137$$ −11.0000 −0.939793 −0.469897 0.882721i $$-0.655709\pi$$
−0.469897 + 0.882721i $$0.655709\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ 2.00000 0.169638 0.0848189 0.996396i $$-0.472969\pi$$
0.0848189 + 0.996396i $$0.472969\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −9.00000 −0.757937
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 5.00000 0.415227
$$146$$ 6.00000 0.496564
$$147$$ 3.00000 0.247436
$$148$$ 3.00000 0.246598
$$149$$ 17.0000 1.39269 0.696347 0.717705i $$-0.254807\pi$$
0.696347 + 0.717705i $$0.254807\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 2.00000 0.162221
$$153$$ −2.00000 −0.161690
$$154$$ −10.0000 −0.805823
$$155$$ −11.0000 −0.883541
$$156$$ 0 0
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ 11.0000 0.875113
$$159$$ −6.00000 −0.475831
$$160$$ −1.00000 −0.0790569
$$161$$ 2.00000 0.157622
$$162$$ −1.00000 −0.0785674
$$163$$ −15.0000 −1.17489 −0.587445 0.809264i $$-0.699866\pi$$
−0.587445 + 0.809264i $$0.699866\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 5.00000 0.389249
$$166$$ −6.00000 −0.465690
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 0 0
$$170$$ 2.00000 0.153393
$$171$$ −2.00000 −0.152944
$$172$$ −11.0000 −0.838742
$$173$$ 20.0000 1.52057 0.760286 0.649589i $$-0.225059\pi$$
0.760286 + 0.649589i $$0.225059\pi$$
$$174$$ 5.00000 0.379049
$$175$$ −2.00000 −0.151186
$$176$$ −5.00000 −0.376889
$$177$$ 15.0000 1.12747
$$178$$ −2.00000 −0.149906
$$179$$ −13.0000 −0.971666 −0.485833 0.874052i $$-0.661484\pi$$
−0.485833 + 0.874052i $$0.661484\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ 1.00000 0.0737210
$$185$$ 3.00000 0.220564
$$186$$ −11.0000 −0.806559
$$187$$ 10.0000 0.731272
$$188$$ 9.00000 0.656392
$$189$$ 2.00000 0.145479
$$190$$ 2.00000 0.145095
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −24.0000 −1.72756 −0.863779 0.503871i $$-0.831909\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −8.00000 −0.569976 −0.284988 0.958531i $$-0.591990\pi$$
−0.284988 + 0.958531i $$0.591990\pi$$
$$198$$ 5.00000 0.355335
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −16.0000 −1.12855
$$202$$ −2.00000 −0.140720
$$203$$ −10.0000 −0.701862
$$204$$ 2.00000 0.140028
$$205$$ −2.00000 −0.139686
$$206$$ −10.0000 −0.696733
$$207$$ −1.00000 −0.0695048
$$208$$ 0 0
$$209$$ 10.0000 0.691714
$$210$$ −2.00000 −0.138013
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ −10.0000 −0.683586
$$215$$ −11.0000 −0.750194
$$216$$ 1.00000 0.0680414
$$217$$ 22.0000 1.49346
$$218$$ 2.00000 0.135457
$$219$$ 6.00000 0.405442
$$220$$ −5.00000 −0.337100
$$221$$ 0 0
$$222$$ 3.00000 0.201347
$$223$$ 26.0000 1.74109 0.870544 0.492090i $$-0.163767\pi$$
0.870544 + 0.492090i $$0.163767\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 1.00000 0.0666667
$$226$$ −11.0000 −0.731709
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 2.00000 0.132453
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ −10.0000 −0.657952
$$232$$ −5.00000 −0.328266
$$233$$ 1.00000 0.0655122 0.0327561 0.999463i $$-0.489572\pi$$
0.0327561 + 0.999463i $$0.489572\pi$$
$$234$$ 0 0
$$235$$ 9.00000 0.587095
$$236$$ −15.0000 −0.976417
$$237$$ 11.0000 0.714527
$$238$$ −4.00000 −0.259281
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −7.00000 −0.450910 −0.225455 0.974254i $$-0.572387\pi$$
−0.225455 + 0.974254i $$0.572387\pi$$
$$242$$ −14.0000 −0.899954
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ −3.00000 −0.191663
$$246$$ −2.00000 −0.127515
$$247$$ 0 0
$$248$$ 11.0000 0.698501
$$249$$ −6.00000 −0.380235
$$250$$ −1.00000 −0.0632456
$$251$$ 25.0000 1.57799 0.788993 0.614402i $$-0.210603\pi$$
0.788993 + 0.614402i $$0.210603\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 5.00000 0.314347
$$254$$ −2.00000 −0.125491
$$255$$ 2.00000 0.125245
$$256$$ 1.00000 0.0625000
$$257$$ −17.0000 −1.06043 −0.530215 0.847863i $$-0.677889\pi$$
−0.530215 + 0.847863i $$0.677889\pi$$
$$258$$ −11.0000 −0.684830
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 5.00000 0.309492
$$262$$ 1.00000 0.0617802
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ −5.00000 −0.307729
$$265$$ 6.00000 0.368577
$$266$$ −4.00000 −0.245256
$$267$$ −2.00000 −0.122398
$$268$$ 16.0000 0.977356
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −13.0000 −0.789694 −0.394847 0.918747i $$-0.629202\pi$$
−0.394847 + 0.918747i $$0.629202\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ 11.0000 0.664534
$$275$$ −5.00000 −0.301511
$$276$$ 1.00000 0.0601929
$$277$$ −11.0000 −0.660926 −0.330463 0.943819i $$-0.607205\pi$$
−0.330463 + 0.943819i $$0.607205\pi$$
$$278$$ −2.00000 −0.119952
$$279$$ −11.0000 −0.658553
$$280$$ 2.00000 0.119523
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 9.00000 0.535942
$$283$$ 19.0000 1.12943 0.564716 0.825285i $$-0.308986\pi$$
0.564716 + 0.825285i $$0.308986\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 4.00000 0.236113
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ −5.00000 −0.293610
$$291$$ 2.00000 0.117242
$$292$$ −6.00000 −0.351123
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ −15.0000 −0.873334
$$296$$ −3.00000 −0.174371
$$297$$ 5.00000 0.290129
$$298$$ −17.0000 −0.984784
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 22.0000 1.26806
$$302$$ −8.00000 −0.460348
$$303$$ −2.00000 −0.114897
$$304$$ −2.00000 −0.114708
$$305$$ 10.0000 0.572598
$$306$$ 2.00000 0.114332
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 10.0000 0.569803
$$309$$ −10.0000 −0.568880
$$310$$ 11.0000 0.624758
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ 0 0
$$313$$ 20.0000 1.13047 0.565233 0.824931i $$-0.308786\pi$$
0.565233 + 0.824931i $$0.308786\pi$$
$$314$$ 7.00000 0.395033
$$315$$ −2.00000 −0.112687
$$316$$ −11.0000 −0.618798
$$317$$ 16.0000 0.898650 0.449325 0.893368i $$-0.351665\pi$$
0.449325 + 0.893368i $$0.351665\pi$$
$$318$$ 6.00000 0.336463
$$319$$ −25.0000 −1.39973
$$320$$ 1.00000 0.0559017
$$321$$ −10.0000 −0.558146
$$322$$ −2.00000 −0.111456
$$323$$ 4.00000 0.222566
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 15.0000 0.830773
$$327$$ 2.00000 0.110600
$$328$$ 2.00000 0.110432
$$329$$ −18.0000 −0.992372
$$330$$ −5.00000 −0.275241
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 3.00000 0.164399
$$334$$ −3.00000 −0.164153
$$335$$ 16.0000 0.874173
$$336$$ 2.00000 0.109109
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 0 0
$$339$$ −11.0000 −0.597438
$$340$$ −2.00000 −0.108465
$$341$$ 55.0000 2.97842
$$342$$ 2.00000 0.108148
$$343$$ 20.0000 1.07990
$$344$$ 11.0000 0.593080
$$345$$ 1.00000 0.0538382
$$346$$ −20.0000 −1.07521
$$347$$ 34.0000 1.82522 0.912608 0.408836i $$-0.134065\pi$$
0.912608 + 0.408836i $$0.134065\pi$$
$$348$$ −5.00000 −0.268028
$$349$$ 20.0000 1.07058 0.535288 0.844670i $$-0.320203\pi$$
0.535288 + 0.844670i $$0.320203\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 5.00000 0.266501
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ −15.0000 −0.797241
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ −4.00000 −0.211702
$$358$$ 13.0000 0.687071
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −15.0000 −0.789474
$$362$$ 16.0000 0.840941
$$363$$ −14.0000 −0.734809
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 10.0000 0.522708
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ −2.00000 −0.104116
$$370$$ −3.00000 −0.155963
$$371$$ −12.0000 −0.623009
$$372$$ 11.0000 0.570323
$$373$$ −19.0000 −0.983783 −0.491891 0.870657i $$-0.663694\pi$$
−0.491891 + 0.870657i $$0.663694\pi$$
$$374$$ −10.0000 −0.517088
$$375$$ −1.00000 −0.0516398
$$376$$ −9.00000 −0.464140
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ −2.00000 −0.102733 −0.0513665 0.998680i $$-0.516358\pi$$
−0.0513665 + 0.998680i $$0.516358\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ −2.00000 −0.102463
$$382$$ −4.00000 −0.204658
$$383$$ 31.0000 1.58403 0.792013 0.610504i $$-0.209033\pi$$
0.792013 + 0.610504i $$0.209033\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 10.0000 0.509647
$$386$$ 24.0000 1.22157
$$387$$ −11.0000 −0.559161
$$388$$ −2.00000 −0.101535
$$389$$ 5.00000 0.253510 0.126755 0.991934i $$-0.459544\pi$$
0.126755 + 0.991934i $$0.459544\pi$$
$$390$$ 0 0
$$391$$ 2.00000 0.101144
$$392$$ 3.00000 0.151523
$$393$$ 1.00000 0.0504433
$$394$$ 8.00000 0.403034
$$395$$ −11.0000 −0.553470
$$396$$ −5.00000 −0.251259
$$397$$ 3.00000 0.150566 0.0752828 0.997162i $$-0.476014\pi$$
0.0752828 + 0.997162i $$0.476014\pi$$
$$398$$ −24.0000 −1.20301
$$399$$ −4.00000 −0.200250
$$400$$ 1.00000 0.0500000
$$401$$ 36.0000 1.79775 0.898877 0.438201i $$-0.144384\pi$$
0.898877 + 0.438201i $$0.144384\pi$$
$$402$$ 16.0000 0.798007
$$403$$ 0 0
$$404$$ 2.00000 0.0995037
$$405$$ 1.00000 0.0496904
$$406$$ 10.0000 0.496292
$$407$$ −15.0000 −0.743522
$$408$$ −2.00000 −0.0990148
$$409$$ 30.0000 1.48340 0.741702 0.670729i $$-0.234019\pi$$
0.741702 + 0.670729i $$0.234019\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ 11.0000 0.542590
$$412$$ 10.0000 0.492665
$$413$$ 30.0000 1.47620
$$414$$ 1.00000 0.0491473
$$415$$ 6.00000 0.294528
$$416$$ 0 0
$$417$$ −2.00000 −0.0979404
$$418$$ −10.0000 −0.489116
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ −16.0000 −0.779792 −0.389896 0.920859i $$-0.627489\pi$$
−0.389896 + 0.920859i $$0.627489\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 9.00000 0.437595
$$424$$ −6.00000 −0.291386
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ −20.0000 −0.967868
$$428$$ 10.0000 0.483368
$$429$$ 0 0
$$430$$ 11.0000 0.530467
$$431$$ 40.0000 1.92673 0.963366 0.268190i $$-0.0864254\pi$$
0.963366 + 0.268190i $$0.0864254\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 28.0000 1.34559 0.672797 0.739827i $$-0.265093\pi$$
0.672797 + 0.739827i $$0.265093\pi$$
$$434$$ −22.0000 −1.05603
$$435$$ −5.00000 −0.239732
$$436$$ −2.00000 −0.0957826
$$437$$ 2.00000 0.0956730
$$438$$ −6.00000 −0.286691
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 5.00000 0.238366
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ −3.00000 −0.142374
$$445$$ 2.00000 0.0948091
$$446$$ −26.0000 −1.23114
$$447$$ −17.0000 −0.804072
$$448$$ −2.00000 −0.0944911
$$449$$ 12.0000 0.566315 0.283158 0.959073i $$-0.408618\pi$$
0.283158 + 0.959073i $$0.408618\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 10.0000 0.470882
$$452$$ 11.0000 0.517396
$$453$$ −8.00000 −0.375873
$$454$$ 0 0
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 38.0000 1.77757 0.888783 0.458329i $$-0.151552\pi$$
0.888783 + 0.458329i $$0.151552\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 2.00000 0.0933520
$$460$$ −1.00000 −0.0466252
$$461$$ −39.0000 −1.81641 −0.908206 0.418524i $$-0.862547\pi$$
−0.908206 + 0.418524i $$0.862547\pi$$
$$462$$ 10.0000 0.465242
$$463$$ 14.0000 0.650635 0.325318 0.945605i $$-0.394529\pi$$
0.325318 + 0.945605i $$0.394529\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 11.0000 0.510113
$$466$$ −1.00000 −0.0463241
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ 0 0
$$469$$ −32.0000 −1.47762
$$470$$ −9.00000 −0.415139
$$471$$ 7.00000 0.322543
$$472$$ 15.0000 0.690431
$$473$$ 55.0000 2.52890
$$474$$ −11.0000 −0.505247
$$475$$ −2.00000 −0.0917663
$$476$$ 4.00000 0.183340
$$477$$ 6.00000 0.274721
$$478$$ 20.0000 0.914779
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ 7.00000 0.318841
$$483$$ −2.00000 −0.0910032
$$484$$ 14.0000 0.636364
$$485$$ −2.00000 −0.0908153
$$486$$ 1.00000 0.0453609
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 15.0000 0.678323
$$490$$ 3.00000 0.135526
$$491$$ −16.0000 −0.722070 −0.361035 0.932552i $$-0.617576\pi$$
−0.361035 + 0.932552i $$0.617576\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ −10.0000 −0.450377
$$494$$ 0 0
$$495$$ −5.00000 −0.224733
$$496$$ −11.0000 −0.493915
$$497$$ 0 0
$$498$$ 6.00000 0.268866
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −3.00000 −0.134030
$$502$$ −25.0000 −1.11580
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 2.00000 0.0889988
$$506$$ −5.00000 −0.222277
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ 29.0000 1.28540 0.642701 0.766117i $$-0.277814\pi$$
0.642701 + 0.766117i $$0.277814\pi$$
$$510$$ −2.00000 −0.0885615
$$511$$ 12.0000 0.530849
$$512$$ −1.00000 −0.0441942
$$513$$ 2.00000 0.0883022
$$514$$ 17.0000 0.749838
$$515$$ 10.0000 0.440653
$$516$$ 11.0000 0.484248
$$517$$ −45.0000 −1.97910
$$518$$ 6.00000 0.263625
$$519$$ −20.0000 −0.877903
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ −5.00000 −0.218844
$$523$$ −31.0000 −1.35554 −0.677768 0.735276i $$-0.737052\pi$$
−0.677768 + 0.735276i $$0.737052\pi$$
$$524$$ −1.00000 −0.0436852
$$525$$ 2.00000 0.0872872
$$526$$ −21.0000 −0.915644
$$527$$ 22.0000 0.958335
$$528$$ 5.00000 0.217597
$$529$$ −22.0000 −0.956522
$$530$$ −6.00000 −0.260623
$$531$$ −15.0000 −0.650945
$$532$$ 4.00000 0.173422
$$533$$ 0 0
$$534$$ 2.00000 0.0865485
$$535$$ 10.0000 0.432338
$$536$$ −16.0000 −0.691095
$$537$$ 13.0000 0.560991
$$538$$ −14.0000 −0.603583
$$539$$ 15.0000 0.646096
$$540$$ −1.00000 −0.0430331
$$541$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$542$$ 13.0000 0.558398
$$543$$ 16.0000 0.686626
$$544$$ 2.00000 0.0857493
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ −11.0000 −0.469897
$$549$$ 10.0000 0.426790
$$550$$ 5.00000 0.213201
$$551$$ −10.0000 −0.426014
$$552$$ −1.00000 −0.0425628
$$553$$ 22.0000 0.935535
$$554$$ 11.0000 0.467345
$$555$$ −3.00000 −0.127343
$$556$$ 2.00000 0.0848189
$$557$$ 26.0000 1.10166 0.550828 0.834619i $$-0.314312\pi$$
0.550828 + 0.834619i $$0.314312\pi$$
$$558$$ 11.0000 0.465667
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ −10.0000 −0.422200
$$562$$ 10.0000 0.421825
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ −9.00000 −0.378968
$$565$$ 11.0000 0.462773
$$566$$ −19.0000 −0.798630
$$567$$ −2.00000 −0.0839921
$$568$$ 0 0
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ −2.00000 −0.0837708
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 0 0
$$573$$ −4.00000 −0.167102
$$574$$ −4.00000 −0.166957
$$575$$ −1.00000 −0.0417029
$$576$$ 1.00000 0.0416667
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 24.0000 0.997406
$$580$$ 5.00000 0.207614
$$581$$ −12.0000 −0.497844
$$582$$ −2.00000 −0.0829027
$$583$$ −30.0000 −1.24247
$$584$$ 6.00000 0.248282
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ −18.0000 −0.742940 −0.371470 0.928445i $$-0.621146\pi$$
−0.371470 + 0.928445i $$0.621146\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 22.0000 0.906494
$$590$$ 15.0000 0.617540
$$591$$ 8.00000 0.329076
$$592$$ 3.00000 0.123299
$$593$$ 31.0000 1.27302 0.636509 0.771270i $$-0.280378\pi$$
0.636509 + 0.771270i $$0.280378\pi$$
$$594$$ −5.00000 −0.205152
$$595$$ 4.00000 0.163984
$$596$$ 17.0000 0.696347
$$597$$ −24.0000 −0.982255
$$598$$ 0 0
$$599$$ 42.0000 1.71607 0.858037 0.513588i $$-0.171684\pi$$
0.858037 + 0.513588i $$0.171684\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −3.00000 −0.122373 −0.0611863 0.998126i $$-0.519488\pi$$
−0.0611863 + 0.998126i $$0.519488\pi$$
$$602$$ −22.0000 −0.896653
$$603$$ 16.0000 0.651570
$$604$$ 8.00000 0.325515
$$605$$ 14.0000 0.569181
$$606$$ 2.00000 0.0812444
$$607$$ 18.0000 0.730597 0.365299 0.930890i $$-0.380967\pi$$
0.365299 + 0.930890i $$0.380967\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 10.0000 0.405220
$$610$$ −10.0000 −0.404888
$$611$$ 0 0
$$612$$ −2.00000 −0.0808452
$$613$$ −29.0000 −1.17130 −0.585649 0.810564i $$-0.699160\pi$$
−0.585649 + 0.810564i $$0.699160\pi$$
$$614$$ 0 0
$$615$$ 2.00000 0.0806478
$$616$$ −10.0000 −0.402911
$$617$$ 41.0000 1.65060 0.825299 0.564696i $$-0.191007\pi$$
0.825299 + 0.564696i $$0.191007\pi$$
$$618$$ 10.0000 0.402259
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ −11.0000 −0.441771
$$621$$ 1.00000 0.0401286
$$622$$ −20.0000 −0.801927
$$623$$ −4.00000 −0.160257
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −20.0000 −0.799361
$$627$$ −10.0000 −0.399362
$$628$$ −7.00000 −0.279330
$$629$$ −6.00000 −0.239236
$$630$$ 2.00000 0.0796819
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 11.0000 0.437557
$$633$$ 4.00000 0.158986
$$634$$ −16.0000 −0.635441
$$635$$ 2.00000 0.0793676
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 25.0000 0.989759
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 10.0000 0.394669
$$643$$ 8.00000 0.315489 0.157745 0.987480i $$-0.449578\pi$$
0.157745 + 0.987480i $$0.449578\pi$$
$$644$$ 2.00000 0.0788110
$$645$$ 11.0000 0.433125
$$646$$ −4.00000 −0.157378
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 75.0000 2.94401
$$650$$ 0 0
$$651$$ −22.0000 −0.862248
$$652$$ −15.0000 −0.587445
$$653$$ 34.0000 1.33052 0.665261 0.746611i $$-0.268320\pi$$
0.665261 + 0.746611i $$0.268320\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ −1.00000 −0.0390732
$$656$$ −2.00000 −0.0780869
$$657$$ −6.00000 −0.234082
$$658$$ 18.0000 0.701713
$$659$$ −39.0000 −1.51922 −0.759612 0.650376i $$-0.774611\pi$$
−0.759612 + 0.650376i $$0.774611\pi$$
$$660$$ 5.00000 0.194625
$$661$$ 32.0000 1.24466 0.622328 0.782757i $$-0.286187\pi$$
0.622328 + 0.782757i $$0.286187\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 4.00000 0.155113
$$666$$ −3.00000 −0.116248
$$667$$ −5.00000 −0.193601
$$668$$ 3.00000 0.116073
$$669$$ −26.0000 −1.00522
$$670$$ −16.0000 −0.618134
$$671$$ −50.0000 −1.93023
$$672$$ −2.00000 −0.0771517
$$673$$ 8.00000 0.308377 0.154189 0.988041i $$-0.450724\pi$$
0.154189 + 0.988041i $$0.450724\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −4.00000 −0.153732 −0.0768662 0.997041i $$-0.524491\pi$$
−0.0768662 + 0.997041i $$0.524491\pi$$
$$678$$ 11.0000 0.422452
$$679$$ 4.00000 0.153506
$$680$$ 2.00000 0.0766965
$$681$$ 0 0
$$682$$ −55.0000 −2.10606
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ −11.0000 −0.420288
$$686$$ −20.0000 −0.763604
$$687$$ −10.0000 −0.381524
$$688$$ −11.0000 −0.419371
$$689$$ 0 0
$$690$$ −1.00000 −0.0380693
$$691$$ −30.0000 −1.14125 −0.570627 0.821209i $$-0.693300\pi$$
−0.570627 + 0.821209i $$0.693300\pi$$
$$692$$ 20.0000 0.760286
$$693$$ 10.0000 0.379869
$$694$$ −34.0000 −1.29062
$$695$$ 2.00000 0.0758643
$$696$$ 5.00000 0.189525
$$697$$ 4.00000 0.151511
$$698$$ −20.0000 −0.757011
$$699$$ −1.00000 −0.0378235
$$700$$ −2.00000 −0.0755929
$$701$$ 13.0000 0.491003 0.245502 0.969396i $$-0.421047\pi$$
0.245502 + 0.969396i $$0.421047\pi$$
$$702$$ 0 0
$$703$$ −6.00000 −0.226294
$$704$$ −5.00000 −0.188445
$$705$$ −9.00000 −0.338960
$$706$$ 18.0000 0.677439
$$707$$ −4.00000 −0.150435
$$708$$ 15.0000 0.563735
$$709$$ −8.00000 −0.300446 −0.150223 0.988652i $$-0.547999\pi$$
−0.150223 + 0.988652i $$0.547999\pi$$
$$710$$ 0 0
$$711$$ −11.0000 −0.412532
$$712$$ −2.00000 −0.0749532
$$713$$ 11.0000 0.411953
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ −13.0000 −0.485833
$$717$$ 20.0000 0.746914
$$718$$ −12.0000 −0.447836
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ −20.0000 −0.744839
$$722$$ 15.0000 0.558242
$$723$$ 7.00000 0.260333
$$724$$ −16.0000 −0.594635
$$725$$ 5.00000 0.185695
$$726$$ 14.0000 0.519589
$$727$$ −40.0000 −1.48352 −0.741759 0.670667i $$-0.766008\pi$$
−0.741759 + 0.670667i $$0.766008\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 6.00000 0.222070
$$731$$ 22.0000 0.813699
$$732$$ −10.0000 −0.369611
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 3.00000 0.110657
$$736$$ 1.00000 0.0368605
$$737$$ −80.0000 −2.94684
$$738$$ 2.00000 0.0736210
$$739$$ 44.0000 1.61857 0.809283 0.587419i $$-0.199856\pi$$
0.809283 + 0.587419i $$0.199856\pi$$
$$740$$ 3.00000 0.110282
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ 51.0000 1.87101 0.935504 0.353315i $$-0.114946\pi$$
0.935504 + 0.353315i $$0.114946\pi$$
$$744$$ −11.0000 −0.403280
$$745$$ 17.0000 0.622832
$$746$$ 19.0000 0.695639
$$747$$ 6.00000 0.219529
$$748$$ 10.0000 0.365636
$$749$$ −20.0000 −0.730784
$$750$$ 1.00000 0.0365148
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ 9.00000 0.328196
$$753$$ −25.0000 −0.911051
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ 2.00000 0.0727393
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 2.00000 0.0726433
$$759$$ −5.00000 −0.181489
$$760$$ 2.00000 0.0725476
$$761$$ 20.0000 0.724999 0.362500 0.931984i $$-0.381923\pi$$
0.362500 + 0.931984i $$0.381923\pi$$
$$762$$ 2.00000 0.0724524
$$763$$ 4.00000 0.144810
$$764$$ 4.00000 0.144715
$$765$$ −2.00000 −0.0723102
$$766$$ −31.0000 −1.12008
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 43.0000 1.55062 0.775310 0.631581i $$-0.217594\pi$$
0.775310 + 0.631581i $$0.217594\pi$$
$$770$$ −10.0000 −0.360375
$$771$$ 17.0000 0.612240
$$772$$ −24.0000 −0.863779
$$773$$ 32.0000 1.15096 0.575480 0.817816i $$-0.304815\pi$$
0.575480 + 0.817816i $$0.304815\pi$$
$$774$$ 11.0000 0.395387
$$775$$ −11.0000 −0.395132
$$776$$ 2.00000 0.0717958
$$777$$ 6.00000 0.215249
$$778$$ −5.00000 −0.179259
$$779$$ 4.00000 0.143315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −2.00000 −0.0715199
$$783$$ −5.00000 −0.178685
$$784$$ −3.00000 −0.107143
$$785$$ −7.00000 −0.249841
$$786$$ −1.00000 −0.0356688
$$787$$ −31.0000 −1.10503 −0.552515 0.833503i $$-0.686332\pi$$
−0.552515 + 0.833503i $$0.686332\pi$$
$$788$$ −8.00000 −0.284988
$$789$$ −21.0000 −0.747620
$$790$$ 11.0000 0.391362
$$791$$ −22.0000 −0.782230
$$792$$ 5.00000 0.177667
$$793$$ 0 0
$$794$$ −3.00000 −0.106466
$$795$$ −6.00000 −0.212798
$$796$$ 24.0000 0.850657
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 4.00000 0.141598
$$799$$ −18.0000 −0.636794
$$800$$ −1.00000 −0.0353553
$$801$$ 2.00000 0.0706665
$$802$$ −36.0000 −1.27120
$$803$$ 30.0000 1.05868
$$804$$ −16.0000 −0.564276
$$805$$ 2.00000 0.0704907
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ −2.00000 −0.0703598
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −10.0000 −0.350931
$$813$$ 13.0000 0.455930
$$814$$ 15.0000 0.525750
$$815$$ −15.0000 −0.525427
$$816$$ 2.00000 0.0700140
$$817$$ 22.0000 0.769683
$$818$$ −30.0000 −1.04893
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ −51.0000 −1.77991 −0.889956 0.456046i $$-0.849265\pi$$
−0.889956 + 0.456046i $$0.849265\pi$$
$$822$$ −11.0000 −0.383669
$$823$$ −2.00000 −0.0697156 −0.0348578 0.999392i $$-0.511098\pi$$
−0.0348578 + 0.999392i $$0.511098\pi$$
$$824$$ −10.0000 −0.348367
$$825$$ 5.00000 0.174078
$$826$$ −30.0000 −1.04383
$$827$$ −50.0000 −1.73867 −0.869335 0.494223i $$-0.835453\pi$$
−0.869335 + 0.494223i $$0.835453\pi$$
$$828$$ −1.00000 −0.0347524
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ −6.00000 −0.208263
$$831$$ 11.0000 0.381586
$$832$$ 0 0
$$833$$ 6.00000 0.207888
$$834$$ 2.00000 0.0692543
$$835$$ 3.00000 0.103819
$$836$$ 10.0000 0.345857
$$837$$ 11.0000 0.380216
$$838$$ 4.00000 0.138178
$$839$$ −54.0000 −1.86429 −0.932144 0.362089i $$-0.882064\pi$$
−0.932144 + 0.362089i $$0.882064\pi$$
$$840$$ −2.00000 −0.0690066
$$841$$ −4.00000 −0.137931
$$842$$ 16.0000 0.551396
$$843$$ 10.0000 0.344418
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ −9.00000 −0.309426
$$847$$ −28.0000 −0.962091
$$848$$ 6.00000 0.206041
$$849$$ −19.0000 −0.652078
$$850$$ 2.00000 0.0685994
$$851$$ −3.00000 −0.102839
$$852$$ 0 0
$$853$$ −49.0000 −1.67773 −0.838864 0.544341i $$-0.816780\pi$$
−0.838864 + 0.544341i $$0.816780\pi$$
$$854$$ 20.0000 0.684386
$$855$$ −2.00000 −0.0683986
$$856$$ −10.0000 −0.341793
$$857$$ −25.0000 −0.853984 −0.426992 0.904255i $$-0.640427\pi$$
−0.426992 + 0.904255i $$0.640427\pi$$
$$858$$ 0 0
$$859$$ −50.0000 −1.70598 −0.852989 0.521929i $$-0.825213\pi$$
−0.852989 + 0.521929i $$0.825213\pi$$
$$860$$ −11.0000 −0.375097
$$861$$ −4.00000 −0.136320
$$862$$ −40.0000 −1.36241
$$863$$ 39.0000 1.32758 0.663788 0.747921i $$-0.268948\pi$$
0.663788 + 0.747921i $$0.268948\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 20.0000 0.680020
$$866$$ −28.0000 −0.951479
$$867$$ 13.0000 0.441503
$$868$$ 22.0000 0.746729
$$869$$ 55.0000 1.86575
$$870$$ 5.00000 0.169516
$$871$$ 0 0
$$872$$ 2.00000 0.0677285
$$873$$ −2.00000 −0.0676897
$$874$$ −2.00000 −0.0676510
$$875$$ −2.00000 −0.0676123
$$876$$ 6.00000 0.202721
$$877$$ −19.0000 −0.641584 −0.320792 0.947150i $$-0.603949\pi$$
−0.320792 + 0.947150i $$0.603949\pi$$
$$878$$ 4.00000 0.134993
$$879$$ −6.00000 −0.202375
$$880$$ −5.00000 −0.168550
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −1.00000 −0.0336527 −0.0168263 0.999858i $$-0.505356\pi$$
−0.0168263 + 0.999858i $$0.505356\pi$$
$$884$$ 0 0
$$885$$ 15.0000 0.504219
$$886$$ −20.0000 −0.671913
$$887$$ 3.00000 0.100730 0.0503651 0.998731i $$-0.483962\pi$$
0.0503651 + 0.998731i $$0.483962\pi$$
$$888$$ 3.00000 0.100673
$$889$$ −4.00000 −0.134156
$$890$$ −2.00000 −0.0670402
$$891$$ −5.00000 −0.167506
$$892$$ 26.0000 0.870544
$$893$$ −18.0000 −0.602347
$$894$$ 17.0000 0.568565
$$895$$ −13.0000 −0.434542
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −12.0000 −0.400445
$$899$$ −55.0000 −1.83435
$$900$$ 1.00000 0.0333333
$$901$$ −12.0000 −0.399778
$$902$$ −10.0000 −0.332964
$$903$$ −22.0000 −0.732114
$$904$$ −11.0000 −0.365855
$$905$$ −16.0000 −0.531858
$$906$$ 8.00000 0.265782
$$907$$ 19.0000 0.630885 0.315442 0.948945i $$-0.397847\pi$$
0.315442 + 0.948945i $$0.397847\pi$$
$$908$$ 0 0
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ −18.0000 −0.596367 −0.298183 0.954509i $$-0.596381\pi$$
−0.298183 + 0.954509i $$0.596381\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ −30.0000 −0.992855
$$914$$ −38.0000 −1.25693
$$915$$ −10.0000 −0.330590
$$916$$ 10.0000 0.330409
$$917$$ 2.00000 0.0660458
$$918$$ −2.00000 −0.0660098
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 1.00000 0.0329690
$$921$$ 0 0
$$922$$ 39.0000 1.28440
$$923$$ 0 0
$$924$$ −10.0000 −0.328976
$$925$$ 3.00000 0.0986394
$$926$$ −14.0000 −0.460069
$$927$$ 10.0000 0.328443
$$928$$ −5.00000 −0.164133
$$929$$ −16.0000 −0.524943 −0.262471 0.964940i $$-0.584538\pi$$
−0.262471 + 0.964940i $$0.584538\pi$$
$$930$$ −11.0000 −0.360704
$$931$$ 6.00000 0.196642
$$932$$ 1.00000 0.0327561
$$933$$ −20.0000 −0.654771
$$934$$ −6.00000 −0.196326
$$935$$ 10.0000 0.327035
$$936$$ 0 0
$$937$$ −50.0000 −1.63343 −0.816714 0.577042i $$-0.804207\pi$$
−0.816714 + 0.577042i $$0.804207\pi$$
$$938$$ 32.0000 1.04484
$$939$$ −20.0000 −0.652675
$$940$$ 9.00000 0.293548
$$941$$ −50.0000 −1.62995 −0.814977 0.579494i $$-0.803250\pi$$
−0.814977 + 0.579494i $$0.803250\pi$$
$$942$$ −7.00000 −0.228072
$$943$$ 2.00000 0.0651290
$$944$$ −15.0000 −0.488208
$$945$$ 2.00000 0.0650600
$$946$$ −55.0000 −1.78820
$$947$$ 8.00000 0.259965 0.129983 0.991516i $$-0.458508\pi$$
0.129983 + 0.991516i $$0.458508\pi$$
$$948$$ 11.0000 0.357263
$$949$$ 0 0
$$950$$ 2.00000 0.0648886
$$951$$ −16.0000 −0.518836
$$952$$ −4.00000 −0.129641
$$953$$ −51.0000 −1.65205 −0.826026 0.563632i $$-0.809404\pi$$
−0.826026 + 0.563632i $$0.809404\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 4.00000 0.129437
$$956$$ −20.0000 −0.646846
$$957$$ 25.0000 0.808135
$$958$$ 24.0000 0.775405
$$959$$ 22.0000 0.710417
$$960$$ −1.00000 −0.0322749
$$961$$ 90.0000 2.90323
$$962$$ 0 0
$$963$$ 10.0000 0.322245
$$964$$ −7.00000 −0.225455
$$965$$ −24.0000 −0.772587
$$966$$ 2.00000 0.0643489
$$967$$ 20.0000 0.643157 0.321578 0.946883i $$-0.395787\pi$$
0.321578 + 0.946883i $$0.395787\pi$$
$$968$$ −14.0000 −0.449977
$$969$$ −4.00000 −0.128499
$$970$$ 2.00000 0.0642161
$$971$$ 48.0000 1.54039 0.770197 0.637806i $$-0.220158\pi$$
0.770197 + 0.637806i $$0.220158\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −4.00000 −0.128234
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −21.0000 −0.671850 −0.335925 0.941889i $$-0.609049\pi$$
−0.335925 + 0.941889i $$0.609049\pi$$
$$978$$ −15.0000 −0.479647
$$979$$ −10.0000 −0.319601
$$980$$ −3.00000 −0.0958315
$$981$$ −2.00000 −0.0638551
$$982$$ 16.0000 0.510581
$$983$$ −3.00000 −0.0956851 −0.0478426 0.998855i $$-0.515235\pi$$
−0.0478426 + 0.998855i $$0.515235\pi$$
$$984$$ −2.00000 −0.0637577
$$985$$ −8.00000 −0.254901
$$986$$ 10.0000 0.318465
$$987$$ 18.0000 0.572946
$$988$$ 0 0
$$989$$ 11.0000 0.349780
$$990$$ 5.00000 0.158910
$$991$$ −17.0000 −0.540023 −0.270011 0.962857i $$-0.587027\pi$$
−0.270011 + 0.962857i $$0.587027\pi$$
$$992$$ 11.0000 0.349250
$$993$$ 28.0000 0.888553
$$994$$ 0 0
$$995$$ 24.0000 0.760851
$$996$$ −6.00000 −0.190117
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ −3.00000 −0.0949158
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 5070.2.a.d.1.1 1
13.3 even 3 390.2.i.f.61.1 2
13.5 odd 4 5070.2.b.h.1351.2 2
13.8 odd 4 5070.2.b.h.1351.1 2
13.9 even 3 390.2.i.f.211.1 yes 2
13.12 even 2 5070.2.a.p.1.1 1
39.29 odd 6 1170.2.i.a.451.1 2
39.35 odd 6 1170.2.i.a.991.1 2
65.3 odd 12 1950.2.z.e.1699.1 4
65.9 even 6 1950.2.i.d.601.1 2
65.22 odd 12 1950.2.z.e.1849.1 4
65.29 even 6 1950.2.i.d.451.1 2
65.42 odd 12 1950.2.z.e.1699.2 4
65.48 odd 12 1950.2.z.e.1849.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.f.61.1 2 13.3 even 3
390.2.i.f.211.1 yes 2 13.9 even 3
1170.2.i.a.451.1 2 39.29 odd 6
1170.2.i.a.991.1 2 39.35 odd 6
1950.2.i.d.451.1 2 65.29 even 6
1950.2.i.d.601.1 2 65.9 even 6
1950.2.z.e.1699.1 4 65.3 odd 12
1950.2.z.e.1699.2 4 65.42 odd 12
1950.2.z.e.1849.1 4 65.22 odd 12
1950.2.z.e.1849.2 4 65.48 odd 12
5070.2.a.d.1.1 1 1.1 even 1 trivial
5070.2.a.p.1.1 1 13.12 even 2
5070.2.b.h.1351.1 2 13.8 odd 4
5070.2.b.h.1351.2 2 13.5 odd 4