# Properties

 Label 5070.2.a.e.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

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} +4.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} -1.00000 q^{32} +4.00000 q^{33} -8.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -6.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -8.00000 q^{51} -10.0000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +2.00000 q^{56} -6.00000 q^{57} +4.00000 q^{58} -4.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} -2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -12.0000 q^{67} +8.00000 q^{68} -6.00000 q^{69} +2.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} +8.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} +6.00000 q^{76} +8.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} +8.00000 q^{85} +4.00000 q^{86} +4.00000 q^{87} +4.00000 q^{88} +14.0000 q^{89} -1.00000 q^{90} +6.00000 q^{92} +6.00000 q^{95} +1.00000 q^{96} +16.0000 q^{97} +3.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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
$$7$$ −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
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 8.00000 1.94029 0.970143 0.242536i $$-0.0779791\pi$$
0.970143 + 0.242536i $$0.0779791\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 2.00000 0.436436
$$22$$ 4.00000 0.852803
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −2.00000 −0.377964
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.00000 0.696311
$$34$$ −8.00000 −1.37199
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 1.00000 0.149071
$$46$$ −6.00000 −0.884652
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ −8.00000 −1.12022
$$52$$ 0 0
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −4.00000 −0.539360
$$56$$ 2.00000 0.267261
$$57$$ −6.00000 −0.794719
$$58$$ 4.00000 0.525226
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 8.00000 0.970143
$$69$$ −6.00000 −0.722315
$$70$$ 2.00000 0.239046
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 8.00000 0.936329 0.468165 0.883641i $$-0.344915\pi$$
0.468165 + 0.883641i $$0.344915\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −1.00000 −0.115470
$$76$$ 6.00000 0.688247
$$77$$ 8.00000 0.911685
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 8.00000 0.867722
$$86$$ 4.00000 0.431331
$$87$$ 4.00000 0.428845
$$88$$ 4.00000 0.426401
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 6.00000 0.615587
$$96$$ 1.00000 0.102062
$$97$$ 16.0000 1.62455 0.812277 0.583272i $$-0.198228\pi$$
0.812277 + 0.583272i $$0.198228\pi$$
$$98$$ 3.00000 0.303046
$$99$$ −4.00000 −0.402015
$$100$$ 1.00000 0.100000
$$101$$ −16.0000 −1.59206 −0.796030 0.605257i $$-0.793070\pi$$
−0.796030 + 0.605257i $$0.793070\pi$$
$$102$$ 8.00000 0.792118
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 10.0000 0.971286
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 4.00000 0.381385
$$111$$ −2.00000 −0.189832
$$112$$ −2.00000 −0.188982
$$113$$ 20.0000 1.88144 0.940721 0.339182i $$-0.110150\pi$$
0.940721 + 0.339182i $$0.110150\pi$$
$$114$$ 6.00000 0.561951
$$115$$ 6.00000 0.559503
$$116$$ −4.00000 −0.371391
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ −16.0000 −1.46672
$$120$$ 1.00000 0.0912871
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ −2.00000 −0.180334
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 2.00000 0.178174
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 10.0000 0.873704 0.436852 0.899533i $$-0.356093\pi$$
0.436852 + 0.899533i $$0.356093\pi$$
$$132$$ 4.00000 0.348155
$$133$$ −12.0000 −1.04053
$$134$$ 12.0000 1.03664
$$135$$ −1.00000 −0.0860663
$$136$$ −8.00000 −0.685994
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 6.00000 0.510754
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ −8.00000 −0.662085
$$147$$ 3.00000 0.247436
$$148$$ 2.00000 0.164399
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ 8.00000 0.646762
$$154$$ −8.00000 −0.644658
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 22.0000 1.75579 0.877896 0.478852i $$-0.158947\pi$$
0.877896 + 0.478852i $$0.158947\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 10.0000 0.793052
$$160$$ −1.00000 −0.0790569
$$161$$ −12.0000 −0.945732
$$162$$ −1.00000 −0.0785674
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 4.00000 0.311400
$$166$$ 12.0000 0.931381
$$167$$ 4.00000 0.309529 0.154765 0.987951i $$-0.450538\pi$$
0.154765 + 0.987951i $$0.450538\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 0 0
$$170$$ −8.00000 −0.613572
$$171$$ 6.00000 0.458831
$$172$$ −4.00000 −0.304997
$$173$$ 22.0000 1.67263 0.836315 0.548250i $$-0.184706\pi$$
0.836315 + 0.548250i $$0.184706\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ −2.00000 −0.151186
$$176$$ −4.00000 −0.301511
$$177$$ 4.00000 0.300658
$$178$$ −14.0000 −1.04934
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ −6.00000 −0.442326
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ −32.0000 −2.34007
$$188$$ 0 0
$$189$$ 2.00000 0.145479
$$190$$ −6.00000 −0.435286
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ −16.0000 −1.14873
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 12.0000 0.846415
$$202$$ 16.0000 1.12576
$$203$$ 8.00000 0.561490
$$204$$ −8.00000 −0.560112
$$205$$ 2.00000 0.139686
$$206$$ 12.0000 0.836080
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ −24.0000 −1.66011
$$210$$ −2.00000 −0.138013
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ −10.0000 −0.686803
$$213$$ −8.00000 −0.548151
$$214$$ −12.0000 −0.820303
$$215$$ −4.00000 −0.272798
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 12.0000 0.812743
$$219$$ −8.00000 −0.540590
$$220$$ −4.00000 −0.269680
$$221$$ 0 0
$$222$$ 2.00000 0.134231
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 1.00000 0.0666667
$$226$$ −20.0000 −1.33038
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ −6.00000 −0.397360
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ −8.00000 −0.526361
$$232$$ 4.00000 0.262613
$$233$$ 24.0000 1.57229 0.786146 0.618041i $$-0.212073\pi$$
0.786146 + 0.618041i $$0.212073\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ −8.00000 −0.519656
$$238$$ 16.0000 1.03713
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ −3.00000 −0.191663
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ −1.00000 −0.0632456
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ −24.0000 −1.50887
$$254$$ −4.00000 −0.250982
$$255$$ −8.00000 −0.500979
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ −10.0000 −0.617802
$$263$$ 2.00000 0.123325 0.0616626 0.998097i $$-0.480360\pi$$
0.0616626 + 0.998097i $$0.480360\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ −10.0000 −0.614295
$$266$$ 12.0000 0.735767
$$267$$ −14.0000 −0.856786
$$268$$ −12.0000 −0.733017
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 8.00000 0.485071
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −4.00000 −0.241209
$$276$$ −6.00000 −0.361158
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 8.00000 0.479808
$$279$$ 0 0
$$280$$ 2.00000 0.119523
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 8.00000 0.474713
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ −1.00000 −0.0589256
$$289$$ 47.0000 2.76471
$$290$$ 4.00000 0.234888
$$291$$ −16.0000 −0.937937
$$292$$ 8.00000 0.468165
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ −4.00000 −0.232889
$$296$$ −2.00000 −0.116248
$$297$$ 4.00000 0.232104
$$298$$ −10.0000 −0.579284
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ 16.0000 0.919176
$$304$$ 6.00000 0.344124
$$305$$ −10.0000 −0.572598
$$306$$ −8.00000 −0.457330
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 8.00000 0.455842
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ −22.0000 −1.24153
$$315$$ −2.00000 −0.112687
$$316$$ 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
$$319$$ 16.0000 0.895828
$$320$$ 1.00000 0.0559017
$$321$$ −12.0000 −0.669775
$$322$$ 12.0000 0.668734
$$323$$ 48.0000 2.67079
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ 12.0000 0.663602
$$328$$ −2.00000 −0.110432
$$329$$ 0 0
$$330$$ −4.00000 −0.220193
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 2.00000 0.109599
$$334$$ −4.00000 −0.218870
$$335$$ −12.0000 −0.655630
$$336$$ 2.00000 0.109109
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 0 0
$$339$$ −20.0000 −1.08625
$$340$$ 8.00000 0.433861
$$341$$ 0 0
$$342$$ −6.00000 −0.324443
$$343$$ 20.0000 1.07990
$$344$$ 4.00000 0.215666
$$345$$ −6.00000 −0.323029
$$346$$ −22.0000 −1.18273
$$347$$ −24.0000 −1.28839 −0.644194 0.764862i $$-0.722807\pi$$
−0.644194 + 0.764862i $$0.722807\pi$$
$$348$$ 4.00000 0.214423
$$349$$ 28.0000 1.49881 0.749403 0.662114i $$-0.230341\pi$$
0.749403 + 0.662114i $$0.230341\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 8.00000 0.424596
$$356$$ 14.0000 0.741999
$$357$$ 16.0000 0.846810
$$358$$ 10.0000 0.528516
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 17.0000 0.894737
$$362$$ −10.0000 −0.525588
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ 8.00000 0.418739
$$366$$ −10.0000 −0.522708
$$367$$ −36.0000 −1.87918 −0.939592 0.342296i $$-0.888796\pi$$
−0.939592 + 0.342296i $$0.888796\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 2.00000 0.104116
$$370$$ −2.00000 −0.103975
$$371$$ 20.0000 1.03835
$$372$$ 0 0
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 32.0000 1.65468
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 18.0000 0.924598 0.462299 0.886724i $$-0.347025\pi$$
0.462299 + 0.886724i $$0.347025\pi$$
$$380$$ 6.00000 0.307794
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 8.00000 0.407718
$$386$$ −4.00000 −0.203595
$$387$$ −4.00000 −0.203331
$$388$$ 16.0000 0.812277
$$389$$ 8.00000 0.405616 0.202808 0.979219i $$-0.434993\pi$$
0.202808 + 0.979219i $$0.434993\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ 3.00000 0.151523
$$393$$ −10.0000 −0.504433
$$394$$ −18.0000 −0.906827
$$395$$ 8.00000 0.402524
$$396$$ −4.00000 −0.201008
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 0 0
$$399$$ 12.0000 0.600751
$$400$$ 1.00000 0.0500000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ −12.0000 −0.598506
$$403$$ 0 0
$$404$$ −16.0000 −0.796030
$$405$$ 1.00000 0.0496904
$$406$$ −8.00000 −0.397033
$$407$$ −8.00000 −0.396545
$$408$$ 8.00000 0.396059
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ −2.00000 −0.0987730
$$411$$ −6.00000 −0.295958
$$412$$ −12.0000 −0.591198
$$413$$ 8.00000 0.393654
$$414$$ −6.00000 −0.294884
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ 24.0000 1.17388
$$419$$ 10.0000 0.488532 0.244266 0.969708i $$-0.421453\pi$$
0.244266 + 0.969708i $$0.421453\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 0 0
$$424$$ 10.0000 0.485643
$$425$$ 8.00000 0.388057
$$426$$ 8.00000 0.387601
$$427$$ 20.0000 0.967868
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 4.00000 0.191785
$$436$$ −12.0000 −0.574696
$$437$$ 36.0000 1.72211
$$438$$ 8.00000 0.382255
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 4.00000 0.190693
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 16.0000 0.760183 0.380091 0.924949i $$-0.375893\pi$$
0.380091 + 0.924949i $$0.375893\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 14.0000 0.663664
$$446$$ −2.00000 −0.0947027
$$447$$ −10.0000 −0.472984
$$448$$ −2.00000 −0.0944911
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −8.00000 −0.376705
$$452$$ 20.0000 0.940721
$$453$$ 0 0
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 6.00000 0.280976
$$457$$ −8.00000 −0.374224 −0.187112 0.982339i $$-0.559913\pi$$
−0.187112 + 0.982339i $$0.559913\pi$$
$$458$$ 4.00000 0.186908
$$459$$ −8.00000 −0.373408
$$460$$ 6.00000 0.279751
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 8.00000 0.372194
$$463$$ 26.0000 1.20832 0.604161 0.796862i $$-0.293508\pi$$
0.604161 + 0.796862i $$0.293508\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ −24.0000 −1.11178
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ 0 0
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ −22.0000 −1.01371
$$472$$ 4.00000 0.184115
$$473$$ 16.0000 0.735681
$$474$$ 8.00000 0.367452
$$475$$ 6.00000 0.275299
$$476$$ −16.0000 −0.733359
$$477$$ −10.0000 −0.457869
$$478$$ 16.0000 0.731823
$$479$$ −32.0000 −1.46212 −0.731059 0.682315i $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ 2.00000 0.0910975
$$483$$ 12.0000 0.546019
$$484$$ 5.00000 0.227273
$$485$$ 16.0000 0.726523
$$486$$ 1.00000 0.0453609
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ 10.0000 0.452679
$$489$$ −16.0000 −0.723545
$$490$$ 3.00000 0.135526
$$491$$ 42.0000 1.89543 0.947717 0.319113i $$-0.103385\pi$$
0.947717 + 0.319113i $$0.103385\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ −32.0000 −1.44121
$$494$$ 0 0
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ −16.0000 −0.717698
$$498$$ −12.0000 −0.537733
$$499$$ −38.0000 −1.70111 −0.850557 0.525883i $$-0.823735\pi$$
−0.850557 + 0.525883i $$0.823735\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −4.00000 −0.178707
$$502$$ −6.00000 −0.267793
$$503$$ 10.0000 0.445878 0.222939 0.974832i $$-0.428435\pi$$
0.222939 + 0.974832i $$0.428435\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ −16.0000 −0.711991
$$506$$ 24.0000 1.06693
$$507$$ 0 0
$$508$$ 4.00000 0.177471
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 8.00000 0.354246
$$511$$ −16.0000 −0.707798
$$512$$ −1.00000 −0.0441942
$$513$$ −6.00000 −0.264906
$$514$$ −12.0000 −0.529297
$$515$$ −12.0000 −0.528783
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 4.00000 0.175750
$$519$$ −22.0000 −0.965693
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ 4.00000 0.175075
$$523$$ 36.0000 1.57417 0.787085 0.616844i $$-0.211589\pi$$
0.787085 + 0.616844i $$0.211589\pi$$
$$524$$ 10.0000 0.436852
$$525$$ 2.00000 0.0872872
$$526$$ −2.00000 −0.0872041
$$527$$ 0 0
$$528$$ 4.00000 0.174078
$$529$$ 13.0000 0.565217
$$530$$ 10.0000 0.434372
$$531$$ −4.00000 −0.173585
$$532$$ −12.0000 −0.520266
$$533$$ 0 0
$$534$$ 14.0000 0.605839
$$535$$ 12.0000 0.518805
$$536$$ 12.0000 0.518321
$$537$$ 10.0000 0.431532
$$538$$ 24.0000 1.03471
$$539$$ 12.0000 0.516877
$$540$$ −1.00000 −0.0430331
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ 16.0000 0.687259
$$543$$ −10.0000 −0.429141
$$544$$ −8.00000 −0.342997
$$545$$ −12.0000 −0.514024
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −10.0000 −0.426790
$$550$$ 4.00000 0.170561
$$551$$ −24.0000 −1.02243
$$552$$ 6.00000 0.255377
$$553$$ −16.0000 −0.680389
$$554$$ 10.0000 0.424859
$$555$$ −2.00000 −0.0848953
$$556$$ −8.00000 −0.339276
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ 32.0000 1.35104
$$562$$ −6.00000 −0.253095
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 0 0
$$565$$ 20.0000 0.841406
$$566$$ 4.00000 0.168133
$$567$$ −2.00000 −0.0839921
$$568$$ −8.00000 −0.335673
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 6.00000 0.251312
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 4.00000 0.166957
$$575$$ 6.00000 0.250217
$$576$$ 1.00000 0.0416667
$$577$$ −12.0000 −0.499567 −0.249783 0.968302i $$-0.580359\pi$$
−0.249783 + 0.968302i $$0.580359\pi$$
$$578$$ −47.0000 −1.95494
$$579$$ −4.00000 −0.166234
$$580$$ −4.00000 −0.166091
$$581$$ 24.0000 0.995688
$$582$$ 16.0000 0.663221
$$583$$ 40.0000 1.65663
$$584$$ −8.00000 −0.331042
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 0 0
$$590$$ 4.00000 0.164677
$$591$$ −18.0000 −0.740421
$$592$$ 2.00000 0.0821995
$$593$$ −22.0000 −0.903432 −0.451716 0.892162i $$-0.649188\pi$$
−0.451716 + 0.892162i $$0.649188\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ −16.0000 −0.655936
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ −12.0000 −0.488678
$$604$$ 0 0
$$605$$ 5.00000 0.203279
$$606$$ −16.0000 −0.649956
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ −8.00000 −0.324176
$$610$$ 10.0000 0.404888
$$611$$ 0 0
$$612$$ 8.00000 0.323381
$$613$$ 10.0000 0.403896 0.201948 0.979396i $$-0.435273\pi$$
0.201948 + 0.979396i $$0.435273\pi$$
$$614$$ 12.0000 0.484281
$$615$$ −2.00000 −0.0806478
$$616$$ −8.00000 −0.322329
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ −12.0000 −0.482711
$$619$$ 10.0000 0.401934 0.200967 0.979598i $$-0.435592\pi$$
0.200967 + 0.979598i $$0.435592\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ −24.0000 −0.962312
$$623$$ −28.0000 −1.12180
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −6.00000 −0.239808
$$627$$ 24.0000 0.958468
$$628$$ 22.0000 0.877896
$$629$$ 16.0000 0.637962
$$630$$ 2.00000 0.0796819
$$631$$ −12.0000 −0.477712 −0.238856 0.971055i $$-0.576772\pi$$
−0.238856 + 0.971055i $$0.576772\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 20.0000 0.794929
$$634$$ −30.0000 −1.19145
$$635$$ 4.00000 0.158735
$$636$$ 10.0000 0.396526
$$637$$ 0 0
$$638$$ −16.0000 −0.633446
$$639$$ 8.00000 0.316475
$$640$$ −1.00000 −0.0395285
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 12.0000 0.473602
$$643$$ −44.0000 −1.73519 −0.867595 0.497271i $$-0.834335\pi$$
−0.867595 + 0.497271i $$0.834335\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 4.00000 0.157500
$$646$$ −48.0000 −1.88853
$$647$$ −30.0000 −1.17942 −0.589711 0.807614i $$-0.700758\pi$$
−0.589711 + 0.807614i $$0.700758\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16.0000 0.626608
$$653$$ 42.0000 1.64359 0.821794 0.569785i $$-0.192974\pi$$
0.821794 + 0.569785i $$0.192974\pi$$
$$654$$ −12.0000 −0.469237
$$655$$ 10.0000 0.390732
$$656$$ 2.00000 0.0780869
$$657$$ 8.00000 0.312110
$$658$$ 0 0
$$659$$ 2.00000 0.0779089 0.0389545 0.999241i $$-0.487597\pi$$
0.0389545 + 0.999241i $$0.487597\pi$$
$$660$$ 4.00000 0.155700
$$661$$ −48.0000 −1.86698 −0.933492 0.358599i $$-0.883255\pi$$
−0.933492 + 0.358599i $$0.883255\pi$$
$$662$$ 10.0000 0.388661
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ −12.0000 −0.465340
$$666$$ −2.00000 −0.0774984
$$667$$ −24.0000 −0.929284
$$668$$ 4.00000 0.154765
$$669$$ −2.00000 −0.0773245
$$670$$ 12.0000 0.463600
$$671$$ 40.0000 1.54418
$$672$$ −2.00000 −0.0771517
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −30.0000 −1.15299 −0.576497 0.817099i $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ 20.0000 0.768095
$$679$$ −32.0000 −1.22805
$$680$$ −8.00000 −0.306786
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ 44.0000 1.68361 0.841807 0.539779i $$-0.181492\pi$$
0.841807 + 0.539779i $$0.181492\pi$$
$$684$$ 6.00000 0.229416
$$685$$ 6.00000 0.229248
$$686$$ −20.0000 −0.763604
$$687$$ 4.00000 0.152610
$$688$$ −4.00000 −0.152499
$$689$$ 0 0
$$690$$ 6.00000 0.228416
$$691$$ 14.0000 0.532585 0.266293 0.963892i $$-0.414201\pi$$
0.266293 + 0.963892i $$0.414201\pi$$
$$692$$ 22.0000 0.836315
$$693$$ 8.00000 0.303895
$$694$$ 24.0000 0.911028
$$695$$ −8.00000 −0.303457
$$696$$ −4.00000 −0.151620
$$697$$ 16.0000 0.606043
$$698$$ −28.0000 −1.05982
$$699$$ −24.0000 −0.907763
$$700$$ −2.00000 −0.0755929
$$701$$ 32.0000 1.20862 0.604312 0.796748i $$-0.293448\pi$$
0.604312 + 0.796748i $$0.293448\pi$$
$$702$$ 0 0
$$703$$ 12.0000 0.452589
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ 32.0000 1.20348
$$708$$ 4.00000 0.150329
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 8.00000 0.300023
$$712$$ −14.0000 −0.524672
$$713$$ 0 0
$$714$$ −16.0000 −0.598785
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ 16.0000 0.597531
$$718$$ −24.0000 −0.895672
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 24.0000 0.893807
$$722$$ −17.0000 −0.632674
$$723$$ 2.00000 0.0743808
$$724$$ 10.0000 0.371647
$$725$$ −4.00000 −0.148556
$$726$$ 5.00000 0.185567
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −8.00000 −0.296093
$$731$$ −32.0000 −1.18356
$$732$$ 10.0000 0.369611
$$733$$ −38.0000 −1.40356 −0.701781 0.712393i $$-0.747612\pi$$
−0.701781 + 0.712393i $$0.747612\pi$$
$$734$$ 36.0000 1.32878
$$735$$ 3.00000 0.110657
$$736$$ −6.00000 −0.221163
$$737$$ 48.0000 1.76810
$$738$$ −2.00000 −0.0736210
$$739$$ −30.0000 −1.10357 −0.551784 0.833987i $$-0.686053\pi$$
−0.551784 + 0.833987i $$0.686053\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 0 0
$$742$$ −20.0000 −0.734223
$$743$$ −12.0000 −0.440237 −0.220119 0.975473i $$-0.570644\pi$$
−0.220119 + 0.975473i $$0.570644\pi$$
$$744$$ 0 0
$$745$$ 10.0000 0.366372
$$746$$ 2.00000 0.0732252
$$747$$ −12.0000 −0.439057
$$748$$ −32.0000 −1.17004
$$749$$ −24.0000 −0.876941
$$750$$ 1.00000 0.0365148
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 0 0
$$753$$ −6.00000 −0.218652
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ −6.00000 −0.218074 −0.109037 0.994038i $$-0.534777\pi$$
−0.109037 + 0.994038i $$0.534777\pi$$
$$758$$ −18.0000 −0.653789
$$759$$ 24.0000 0.871145
$$760$$ −6.00000 −0.217643
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 4.00000 0.144905
$$763$$ 24.0000 0.868858
$$764$$ 0 0
$$765$$ 8.00000 0.289241
$$766$$ −12.0000 −0.433578
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ −8.00000 −0.288300
$$771$$ −12.0000 −0.432169
$$772$$ 4.00000 0.143963
$$773$$ −38.0000 −1.36677 −0.683383 0.730061i $$-0.739492\pi$$
−0.683383 + 0.730061i $$0.739492\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −16.0000 −0.574367
$$777$$ 4.00000 0.143499
$$778$$ −8.00000 −0.286814
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ −48.0000 −1.71648
$$783$$ 4.00000 0.142948
$$784$$ −3.00000 −0.107143
$$785$$ 22.0000 0.785214
$$786$$ 10.0000 0.356688
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 18.0000 0.641223
$$789$$ −2.00000 −0.0712019
$$790$$ −8.00000 −0.284627
$$791$$ −40.0000 −1.42224
$$792$$ 4.00000 0.142134
$$793$$ 0 0
$$794$$ 14.0000 0.496841
$$795$$ 10.0000 0.354663
$$796$$ 0 0
$$797$$ −10.0000 −0.354218 −0.177109 0.984191i $$-0.556675\pi$$
−0.177109 + 0.984191i $$0.556675\pi$$
$$798$$ −12.0000 −0.424795
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 14.0000 0.494666
$$802$$ 30.0000 1.05934
$$803$$ −32.0000 −1.12926
$$804$$ 12.0000 0.423207
$$805$$ −12.0000 −0.422944
$$806$$ 0 0
$$807$$ 24.0000 0.844840
$$808$$ 16.0000 0.562878
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 38.0000 1.33436 0.667180 0.744896i $$-0.267501\pi$$
0.667180 + 0.744896i $$0.267501\pi$$
$$812$$ 8.00000 0.280745
$$813$$ 16.0000 0.561144
$$814$$ 8.00000 0.280400
$$815$$ 16.0000 0.560456
$$816$$ −8.00000 −0.280056
$$817$$ −24.0000 −0.839654
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ 2.00000 0.0698430
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 6.00000 0.209274
$$823$$ −52.0000 −1.81261 −0.906303 0.422628i $$-0.861108\pi$$
−0.906303 + 0.422628i $$0.861108\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 4.00000 0.139262
$$826$$ −8.00000 −0.278356
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 6.00000 0.208514
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 12.0000 0.416526
$$831$$ 10.0000 0.346896
$$832$$ 0 0
$$833$$ −24.0000 −0.831551
$$834$$ −8.00000 −0.277017
$$835$$ 4.00000 0.138426
$$836$$ −24.0000 −0.830057
$$837$$ 0 0
$$838$$ −10.0000 −0.345444
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ −2.00000 −0.0690066
$$841$$ −13.0000 −0.448276
$$842$$ −8.00000 −0.275698
$$843$$ −6.00000 −0.206651
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −10.0000 −0.343604
$$848$$ −10.0000 −0.343401
$$849$$ 4.00000 0.137280
$$850$$ −8.00000 −0.274398
$$851$$ 12.0000 0.411355
$$852$$ −8.00000 −0.274075
$$853$$ −46.0000 −1.57501 −0.787505 0.616308i $$-0.788628\pi$$
−0.787505 + 0.616308i $$0.788628\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ 6.00000 0.205196
$$856$$ −12.0000 −0.410152
$$857$$ 4.00000 0.136637 0.0683187 0.997664i $$-0.478237\pi$$
0.0683187 + 0.997664i $$0.478237\pi$$
$$858$$ 0 0
$$859$$ 36.0000 1.22830 0.614152 0.789188i $$-0.289498\pi$$
0.614152 + 0.789188i $$0.289498\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 4.00000 0.136320
$$862$$ 0 0
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 22.0000 0.748022
$$866$$ 2.00000 0.0679628
$$867$$ −47.0000 −1.59620
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ −4.00000 −0.135613
$$871$$ 0 0
$$872$$ 12.0000 0.406371
$$873$$ 16.0000 0.541518
$$874$$ −36.0000 −1.21772
$$875$$ −2.00000 −0.0676123
$$876$$ −8.00000 −0.270295
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ 32.0000 1.07995
$$879$$ −26.0000 −0.876958
$$880$$ −4.00000 −0.134840
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ 0 0
$$885$$ 4.00000 0.134459
$$886$$ −16.0000 −0.537531
$$887$$ −2.00000 −0.0671534 −0.0335767 0.999436i $$-0.510690\pi$$
−0.0335767 + 0.999436i $$0.510690\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ −8.00000 −0.268311
$$890$$ −14.0000 −0.469281
$$891$$ −4.00000 −0.134005
$$892$$ 2.00000 0.0669650
$$893$$ 0 0
$$894$$ 10.0000 0.334450
$$895$$ −10.0000 −0.334263
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ −80.0000 −2.66519
$$902$$ 8.00000 0.266371
$$903$$ −8.00000 −0.266223
$$904$$ −20.0000 −0.665190
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ 52.0000 1.72663 0.863316 0.504664i $$-0.168384\pi$$
0.863316 + 0.504664i $$0.168384\pi$$
$$908$$ 4.00000 0.132745
$$909$$ −16.0000 −0.530687
$$910$$ 0 0
$$911$$ −20.0000 −0.662630 −0.331315 0.943520i $$-0.607492\pi$$
−0.331315 + 0.943520i $$0.607492\pi$$
$$912$$ −6.00000 −0.198680
$$913$$ 48.0000 1.58857
$$914$$ 8.00000 0.264616
$$915$$ 10.0000 0.330590
$$916$$ −4.00000 −0.132164
$$917$$ −20.0000 −0.660458
$$918$$ 8.00000 0.264039
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ −6.00000 −0.197814
$$921$$ 12.0000 0.395413
$$922$$ −6.00000 −0.197599
$$923$$ 0 0
$$924$$ −8.00000 −0.263181
$$925$$ 2.00000 0.0657596
$$926$$ −26.0000 −0.854413
$$927$$ −12.0000 −0.394132
$$928$$ 4.00000 0.131306
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ 24.0000 0.786146
$$933$$ −24.0000 −0.785725
$$934$$ −28.0000 −0.916188
$$935$$ −32.0000 −1.04651
$$936$$ 0 0
$$937$$ 18.0000 0.588034 0.294017 0.955800i $$-0.405008\pi$$
0.294017 + 0.955800i $$0.405008\pi$$
$$938$$ −24.0000 −0.783628
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ 34.0000 1.10837 0.554184 0.832394i $$-0.313030\pi$$
0.554184 + 0.832394i $$0.313030\pi$$
$$942$$ 22.0000 0.716799
$$943$$ 12.0000 0.390774
$$944$$ −4.00000 −0.130189
$$945$$ 2.00000 0.0650600
$$946$$ −16.0000 −0.520205
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ −6.00000 −0.194666
$$951$$ −30.0000 −0.972817
$$952$$ 16.0000 0.518563
$$953$$ 24.0000 0.777436 0.388718 0.921357i $$-0.372918\pi$$
0.388718 + 0.921357i $$0.372918\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ −16.0000 −0.517207
$$958$$ 32.0000 1.03387
$$959$$ −12.0000 −0.387500
$$960$$ −1.00000 −0.0322749
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 12.0000 0.386695
$$964$$ −2.00000 −0.0644157
$$965$$ 4.00000 0.128765
$$966$$ −12.0000 −0.386094
$$967$$ 14.0000 0.450210 0.225105 0.974335i $$-0.427728\pi$$
0.225105 + 0.974335i $$0.427728\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ −48.0000 −1.54198
$$970$$ −16.0000 −0.513729
$$971$$ 38.0000 1.21948 0.609739 0.792602i $$-0.291274\pi$$
0.609739 + 0.792602i $$0.291274\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 16.0000 0.512936
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 16.0000 0.511624
$$979$$ −56.0000 −1.78977
$$980$$ −3.00000 −0.0958315
$$981$$ −12.0000 −0.383131
$$982$$ −42.0000 −1.34027
$$983$$ 52.0000 1.65854 0.829271 0.558846i $$-0.188756\pi$$
0.829271 + 0.558846i $$0.188756\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 18.0000 0.573528
$$986$$ 32.0000 1.01909
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 4.00000 0.127128
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ 10.0000 0.317340
$$994$$ 16.0000 0.507489
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 38.0000 1.20287
$$999$$ −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 5070.2.a.e.1.1 1
13.5 odd 4 5070.2.b.e.1351.2 2
13.8 odd 4 5070.2.b.e.1351.1 2
13.12 even 2 390.2.a.e.1.1 1
39.38 odd 2 1170.2.a.e.1.1 1
52.51 odd 2 3120.2.a.o.1.1 1
65.12 odd 4 1950.2.e.f.1249.2 2
65.38 odd 4 1950.2.e.f.1249.1 2
65.64 even 2 1950.2.a.h.1.1 1
156.155 even 2 9360.2.a.bh.1.1 1
195.38 even 4 5850.2.e.i.5149.2 2
195.77 even 4 5850.2.e.i.5149.1 2
195.194 odd 2 5850.2.a.bi.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.e.1.1 1 13.12 even 2
1170.2.a.e.1.1 1 39.38 odd 2
1950.2.a.h.1.1 1 65.64 even 2
1950.2.e.f.1249.1 2 65.38 odd 4
1950.2.e.f.1249.2 2 65.12 odd 4
3120.2.a.o.1.1 1 52.51 odd 2
5070.2.a.e.1.1 1 1.1 even 1 trivial
5070.2.b.e.1351.1 2 13.8 odd 4
5070.2.b.e.1351.2 2 13.5 odd 4
5850.2.a.bi.1.1 1 195.194 odd 2
5850.2.e.i.5149.1 2 195.77 even 4
5850.2.e.i.5149.2 2 195.38 even 4
9360.2.a.bh.1.1 1 156.155 even 2