# Properties

 Label 5070.2.a.n.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

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -4.00000 q^{22} +2.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} +8.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +4.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} +2.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +2.00000 q^{56} -2.00000 q^{57} +8.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +8.00000 q^{67} +4.00000 q^{68} -2.00000 q^{69} -2.00000 q^{70} +1.00000 q^{72} -6.00000 q^{74} -1.00000 q^{75} +2.00000 q^{76} -8.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000 q^{83} -2.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} -8.00000 q^{87} -4.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} +2.00000 q^{92} +4.00000 q^{93} -2.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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 $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 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
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −2.00000 −0.436436
$$22$$ −4.00000 −0.852803
$$23$$ 2.00000 0.417029 0.208514 0.978019i $$-0.433137\pi$$
0.208514 + 0.978019i $$0.433137\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000 0.377964
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 4.00000 0.696311
$$34$$ 4.00000 0.685994
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ −1.00000 −0.149071
$$46$$ 2.00000 0.294884
$$47$$ 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
$$51$$ −4.00000 −0.560112
$$52$$ 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
$$55$$ 4.00000 0.539360
$$56$$ 2.00000 0.267261
$$57$$ −2.00000 −0.264906
$$58$$ 8.00000 1.05045
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 4.00000 0.485071
$$69$$ −2.00000 −0.240772
$$70$$ −2.00000 −0.239046
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ −1.00000 −0.115470
$$76$$ 2.00000 0.229416
$$77$$ −8.00000 −0.911685
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −10.0000 −1.10432
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ −4.00000 −0.433861
$$86$$ 4.00000 0.431331
$$87$$ −8.00000 −0.857690
$$88$$ −4.00000 −0.426401
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 2.00000 0.208514
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ −1.00000 −0.102062
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ −4.00000 −0.402015
$$100$$ 1.00000 0.100000
$$101$$ 20.0000 1.99007 0.995037 0.0995037i $$-0.0317255\pi$$
0.995037 + 0.0995037i $$0.0317255\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 6.00000 0.582772
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 6.00000 0.569495
$$112$$ 2.00000 0.188982
$$113$$ −16.0000 −1.50515 −0.752577 0.658505i $$-0.771189\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ −2.00000 −0.186501
$$116$$ 8.00000 0.742781
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 8.00000 0.733359
$$120$$ 1.00000 0.0912871
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ 10.0000 0.901670
$$124$$ −4.00000 −0.359211
$$125$$ −1.00000 −0.0894427
$$126$$ 2.00000 0.178174
$$127$$ 20.0000 1.77471 0.887357 0.461084i $$-0.152539\pi$$
0.887357 + 0.461084i $$0.152539\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 4.00000 0.346844
$$134$$ 8.00000 0.691095
$$135$$ 1.00000 0.0860663
$$136$$ 4.00000 0.342997
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ −2.00000 −0.170251
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −8.00000 −0.664364
$$146$$ 0 0
$$147$$ 3.00000 0.247436
$$148$$ −6.00000 −0.493197
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 4.00000 0.323381
$$154$$ −8.00000 −0.644658
$$155$$ 4.00000 0.321288
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ −6.00000 −0.475831
$$160$$ −1.00000 −0.0790569
$$161$$ 4.00000 0.315244
$$162$$ 1.00000 0.0785674
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ −4.00000 −0.311400
$$166$$ 12.0000 0.931381
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 0 0
$$170$$ −4.00000 −0.306786
$$171$$ 2.00000 0.152944
$$172$$ 4.00000 0.304997
$$173$$ 14.0000 1.06440 0.532200 0.846619i $$-0.321365\pi$$
0.532200 + 0.846619i $$0.321365\pi$$
$$174$$ −8.00000 −0.606478
$$175$$ 2.00000 0.151186
$$176$$ −4.00000 −0.301511
$$177$$ −12.0000 −0.901975
$$178$$ 10.0000 0.749532
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 2.00000 0.147442
$$185$$ 6.00000 0.441129
$$186$$ 4.00000 0.293294
$$187$$ −16.0000 −1.17004
$$188$$ 0 0
$$189$$ −2.00000 −0.145479
$$190$$ −2.00000 −0.145095
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −12.0000 −0.863779 −0.431889 0.901927i $$-0.642153\pi$$
−0.431889 + 0.901927i $$0.642153\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −8.00000 −0.564276
$$202$$ 20.0000 1.40720
$$203$$ 16.0000 1.12298
$$204$$ −4.00000 −0.280056
$$205$$ 10.0000 0.698430
$$206$$ 4.00000 0.278693
$$207$$ 2.00000 0.139010
$$208$$ 0 0
$$209$$ −8.00000 −0.553372
$$210$$ 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
$$214$$ 4.00000 0.273434
$$215$$ −4.00000 −0.272798
$$216$$ −1.00000 −0.0680414
$$217$$ −8.00000 −0.543075
$$218$$ −4.00000 −0.270914
$$219$$ 0 0
$$220$$ 4.00000 0.269680
$$221$$ 0 0
$$222$$ 6.00000 0.402694
$$223$$ 22.0000 1.47323 0.736614 0.676313i $$-0.236423\pi$$
0.736614 + 0.676313i $$0.236423\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 1.00000 0.0666667
$$226$$ −16.0000 −1.06430
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −2.00000 −0.132453
$$229$$ −28.0000 −1.85029 −0.925146 0.379611i $$-0.876058\pi$$
−0.925146 + 0.379611i $$0.876058\pi$$
$$230$$ −2.00000 −0.131876
$$231$$ 8.00000 0.526361
$$232$$ 8.00000 0.525226
$$233$$ 28.0000 1.83434 0.917170 0.398495i $$-0.130467\pi$$
0.917170 + 0.398495i $$0.130467\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 8.00000 0.519656
$$238$$ 8.00000 0.518563
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 3.00000 0.191663
$$246$$ 10.0000 0.637577
$$247$$ 0 0
$$248$$ −4.00000 −0.254000
$$249$$ −12.0000 −0.760469
$$250$$ −1.00000 −0.0632456
$$251$$ 10.0000 0.631194 0.315597 0.948893i $$-0.397795\pi$$
0.315597 + 0.948893i $$0.397795\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −8.00000 −0.502956
$$254$$ 20.0000 1.25491
$$255$$ 4.00000 0.250490
$$256$$ 1.00000 0.0625000
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ −12.0000 −0.745644
$$260$$ 0 0
$$261$$ 8.00000 0.495188
$$262$$ 14.0000 0.864923
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 4.00000 0.246183
$$265$$ −6.00000 −0.368577
$$266$$ 4.00000 0.245256
$$267$$ −10.0000 −0.611990
$$268$$ 8.00000 0.488678
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 28.0000 1.70088 0.850439 0.526073i $$-0.176336\pi$$
0.850439 + 0.526073i $$0.176336\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ 2.00000 0.120824
$$275$$ −4.00000 −0.241209
$$276$$ −2.00000 −0.120386
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ −4.00000 −0.239474
$$280$$ −2.00000 −0.119523
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ −20.0000 −1.18056
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ −8.00000 −0.469776
$$291$$ −8.00000 −0.468968
$$292$$ 0 0
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 3.00000 0.174964
$$295$$ −12.0000 −0.698667
$$296$$ −6.00000 −0.348743
$$297$$ 4.00000 0.232104
$$298$$ 22.0000 1.27443
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 8.00000 0.461112
$$302$$ 4.00000 0.230174
$$303$$ −20.0000 −1.14897
$$304$$ 2.00000 0.114708
$$305$$ 2.00000 0.114520
$$306$$ 4.00000 0.228665
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ −8.00000 −0.455842
$$309$$ −4.00000 −0.227552
$$310$$ 4.00000 0.227185
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ −2.00000 −0.112687
$$316$$ −8.00000 −0.450035
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ −32.0000 −1.79166
$$320$$ −1.00000 −0.0559017
$$321$$ −4.00000 −0.223258
$$322$$ 4.00000 0.222911
$$323$$ 8.00000 0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 4.00000 0.221201
$$328$$ −10.0000 −0.552158
$$329$$ 0 0
$$330$$ −4.00000 −0.220193
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −6.00000 −0.328798
$$334$$ 12.0000 0.656611
$$335$$ −8.00000 −0.437087
$$336$$ −2.00000 −0.109109
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 0 0
$$339$$ 16.0000 0.869001
$$340$$ −4.00000 −0.216930
$$341$$ 16.0000 0.866449
$$342$$ 2.00000 0.108148
$$343$$ −20.0000 −1.07990
$$344$$ 4.00000 0.215666
$$345$$ 2.00000 0.107676
$$346$$ 14.0000 0.752645
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ −8.00000 −0.428845
$$349$$ 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
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ −8.00000 −0.423405
$$358$$ 2.00000 0.105703
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −15.0000 −0.789474
$$362$$ −22.0000 −1.15629
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ −4.00000 −0.208798 −0.104399 0.994535i $$-0.533292\pi$$
−0.104399 + 0.994535i $$0.533292\pi$$
$$368$$ 2.00000 0.104257
$$369$$ −10.0000 −0.520579
$$370$$ 6.00000 0.311925
$$371$$ 12.0000 0.623009
$$372$$ 4.00000 0.207390
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ −16.0000 −0.827340
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 14.0000 0.719132 0.359566 0.933120i $$-0.382925\pi$$
0.359566 + 0.933120i $$0.382925\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ −20.0000 −1.02463
$$382$$ −8.00000 −0.409316
$$383$$ 20.0000 1.02195 0.510976 0.859595i $$-0.329284\pi$$
0.510976 + 0.859595i $$0.329284\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 8.00000 0.407718
$$386$$ −12.0000 −0.610784
$$387$$ 4.00000 0.203331
$$388$$ 8.00000 0.406138
$$389$$ 20.0000 1.01404 0.507020 0.861934i $$-0.330747\pi$$
0.507020 + 0.861934i $$0.330747\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ −3.00000 −0.151523
$$393$$ −14.0000 −0.706207
$$394$$ −10.0000 −0.503793
$$395$$ 8.00000 0.402524
$$396$$ −4.00000 −0.201008
$$397$$ −6.00000 −0.301131 −0.150566 0.988600i $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ 24.0000 1.20301
$$399$$ −4.00000 −0.200250
$$400$$ 1.00000 0.0500000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 0 0
$$404$$ 20.0000 0.995037
$$405$$ −1.00000 −0.0496904
$$406$$ 16.0000 0.794067
$$407$$ 24.0000 1.18964
$$408$$ −4.00000 −0.198030
$$409$$ 18.0000 0.890043 0.445021 0.895520i $$-0.353196\pi$$
0.445021 + 0.895520i $$0.353196\pi$$
$$410$$ 10.0000 0.493865
$$411$$ −2.00000 −0.0986527
$$412$$ 4.00000 0.197066
$$413$$ 24.0000 1.18096
$$414$$ 2.00000 0.0982946
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ 16.0000 0.783523
$$418$$ −8.00000 −0.391293
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ 16.0000 0.779792 0.389896 0.920859i $$-0.372511\pi$$
0.389896 + 0.920859i $$0.372511\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 8.00000 0.383571
$$436$$ −4.00000 −0.191565
$$437$$ 4.00000 0.191346
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 4.00000 0.190693
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −16.0000 −0.760183 −0.380091 0.924949i $$-0.624107\pi$$
−0.380091 + 0.924949i $$0.624107\pi$$
$$444$$ 6.00000 0.284747
$$445$$ −10.0000 −0.474045
$$446$$ 22.0000 1.04173
$$447$$ −22.0000 −1.04056
$$448$$ 2.00000 0.0944911
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 40.0000 1.88353
$$452$$ −16.0000 −0.752577
$$453$$ −4.00000 −0.187936
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 16.0000 0.748448 0.374224 0.927338i $$-0.377909\pi$$
0.374224 + 0.927338i $$0.377909\pi$$
$$458$$ −28.0000 −1.30835
$$459$$ −4.00000 −0.186704
$$460$$ −2.00000 −0.0932505
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 8.00000 0.372194
$$463$$ −2.00000 −0.0929479 −0.0464739 0.998920i $$-0.514798\pi$$
−0.0464739 + 0.998920i $$0.514798\pi$$
$$464$$ 8.00000 0.371391
$$465$$ −4.00000 −0.185496
$$466$$ 28.0000 1.29707
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ 12.0000 0.552345
$$473$$ −16.0000 −0.735681
$$474$$ 8.00000 0.367452
$$475$$ 2.00000 0.0917663
$$476$$ 8.00000 0.366679
$$477$$ 6.00000 0.274721
$$478$$ 8.00000 0.365911
$$479$$ −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
$$482$$ 22.0000 1.00207
$$483$$ −4.00000 −0.182006
$$484$$ 5.00000 0.227273
$$485$$ −8.00000 −0.363261
$$486$$ −1.00000 −0.0453609
$$487$$ −38.0000 −1.72194 −0.860972 0.508652i $$-0.830144\pi$$
−0.860972 + 0.508652i $$0.830144\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 12.0000 0.542659
$$490$$ 3.00000 0.135526
$$491$$ 38.0000 1.71492 0.857458 0.514554i $$-0.172042\pi$$
0.857458 + 0.514554i $$0.172042\pi$$
$$492$$ 10.0000 0.450835
$$493$$ 32.0000 1.44121
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ 14.0000 0.626726 0.313363 0.949633i $$-0.398544\pi$$
0.313363 + 0.949633i $$0.398544\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −12.0000 −0.536120
$$502$$ 10.0000 0.446322
$$503$$ −42.0000 −1.87269 −0.936344 0.351085i $$-0.885813\pi$$
−0.936344 + 0.351085i $$0.885813\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ −20.0000 −0.889988
$$506$$ −8.00000 −0.355643
$$507$$ 0 0
$$508$$ 20.0000 0.887357
$$509$$ −38.0000 −1.68432 −0.842160 0.539227i $$-0.818716\pi$$
−0.842160 + 0.539227i $$0.818716\pi$$
$$510$$ 4.00000 0.177123
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −2.00000 −0.0883022
$$514$$ −8.00000 −0.352865
$$515$$ −4.00000 −0.176261
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ −12.0000 −0.527250
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 8.00000 0.350150
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 14.0000 0.611593
$$525$$ −2.00000 −0.0872872
$$526$$ −18.0000 −0.784837
$$527$$ −16.0000 −0.696971
$$528$$ 4.00000 0.174078
$$529$$ −19.0000 −0.826087
$$530$$ −6.00000 −0.260623
$$531$$ 12.0000 0.520756
$$532$$ 4.00000 0.173422
$$533$$ 0 0
$$534$$ −10.0000 −0.432742
$$535$$ −4.00000 −0.172935
$$536$$ 8.00000 0.345547
$$537$$ −2.00000 −0.0863064
$$538$$ −4.00000 −0.172452
$$539$$ 12.0000 0.516877
$$540$$ 1.00000 0.0430331
$$541$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$542$$ 28.0000 1.20270
$$543$$ 22.0000 0.944110
$$544$$ 4.00000 0.171499
$$545$$ 4.00000 0.171341
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ −2.00000 −0.0853579
$$550$$ −4.00000 −0.170561
$$551$$ 16.0000 0.681623
$$552$$ −2.00000 −0.0851257
$$553$$ −16.0000 −0.680389
$$554$$ 22.0000 0.934690
$$555$$ −6.00000 −0.254686
$$556$$ −16.0000 −0.678551
$$557$$ 10.0000 0.423714 0.211857 0.977301i $$-0.432049\pi$$
0.211857 + 0.977301i $$0.432049\pi$$
$$558$$ −4.00000 −0.169334
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ 16.0000 0.675521
$$562$$ 10.0000 0.421825
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ 16.0000 0.673125
$$566$$ 28.0000 1.17693
$$567$$ 2.00000 0.0839921
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 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
$$573$$ 8.00000 0.334205
$$574$$ −20.0000 −0.834784
$$575$$ 2.00000 0.0834058
$$576$$ 1.00000 0.0416667
$$577$$ 4.00000 0.166522 0.0832611 0.996528i $$-0.473466\pi$$
0.0832611 + 0.996528i $$0.473466\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 12.0000 0.498703
$$580$$ −8.00000 −0.332182
$$581$$ 24.0000 0.995688
$$582$$ −8.00000 −0.331611
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 3.00000 0.123718
$$589$$ −8.00000 −0.329634
$$590$$ −12.0000 −0.494032
$$591$$ 10.0000 0.411345
$$592$$ −6.00000 −0.246598
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 4.00000 0.164122
$$595$$ −8.00000 −0.327968
$$596$$ 22.0000 0.901155
$$597$$ −24.0000 −0.982255
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ 8.00000 0.326056
$$603$$ 8.00000 0.325785
$$604$$ 4.00000 0.162758
$$605$$ −5.00000 −0.203279
$$606$$ −20.0000 −0.812444
$$607$$ −12.0000 −0.487065 −0.243532 0.969893i $$-0.578306\pi$$
−0.243532 + 0.969893i $$0.578306\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ −16.0000 −0.648353
$$610$$ 2.00000 0.0809776
$$611$$ 0 0
$$612$$ 4.00000 0.161690
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ 0 0
$$615$$ −10.0000 −0.403239
$$616$$ −8.00000 −0.322329
$$617$$ −38.0000 −1.52982 −0.764911 0.644136i $$-0.777217\pi$$
−0.764911 + 0.644136i $$0.777217\pi$$
$$618$$ −4.00000 −0.160904
$$619$$ −26.0000 −1.04503 −0.522514 0.852631i $$-0.675006\pi$$
−0.522514 + 0.852631i $$0.675006\pi$$
$$620$$ 4.00000 0.160644
$$621$$ −2.00000 −0.0802572
$$622$$ −16.0000 −0.641542
$$623$$ 20.0000 0.801283
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −10.0000 −0.399680
$$627$$ 8.00000 0.319489
$$628$$ −10.0000 −0.399043
$$629$$ −24.0000 −0.956943
$$630$$ −2.00000 −0.0796819
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 4.00000 0.158986
$$634$$ 2.00000 0.0794301
$$635$$ −20.0000 −0.793676
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ −32.0000 −1.26689
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ −4.00000 −0.157867
$$643$$ 40.0000 1.57745 0.788723 0.614749i $$-0.210743\pi$$
0.788723 + 0.614749i $$0.210743\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 4.00000 0.157500
$$646$$ 8.00000 0.314756
$$647$$ −18.0000 −0.707653 −0.353827 0.935311i $$-0.615120\pi$$
−0.353827 + 0.935311i $$0.615120\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 8.00000 0.313545
$$652$$ −12.0000 −0.469956
$$653$$ −38.0000 −1.48705 −0.743527 0.668705i $$-0.766849\pi$$
−0.743527 + 0.668705i $$0.766849\pi$$
$$654$$ 4.00000 0.156412
$$655$$ −14.0000 −0.547025
$$656$$ −10.0000 −0.390434
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ −4.00000 −0.155700
$$661$$ 16.0000 0.622328 0.311164 0.950356i $$-0.399281\pi$$
0.311164 + 0.950356i $$0.399281\pi$$
$$662$$ 10.0000 0.388661
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ −4.00000 −0.155113
$$666$$ −6.00000 −0.232495
$$667$$ 16.0000 0.619522
$$668$$ 12.0000 0.464294
$$669$$ −22.0000 −0.850569
$$670$$ −8.00000 −0.309067
$$671$$ 8.00000 0.308837
$$672$$ −2.00000 −0.0771517
$$673$$ −46.0000 −1.77317 −0.886585 0.462566i $$-0.846929\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 16.0000 0.614476
$$679$$ 16.0000 0.614024
$$680$$ −4.00000 −0.153393
$$681$$ 12.0000 0.459841
$$682$$ 16.0000 0.612672
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ −2.00000 −0.0764161
$$686$$ −20.0000 −0.763604
$$687$$ 28.0000 1.06827
$$688$$ 4.00000 0.152499
$$689$$ 0 0
$$690$$ 2.00000 0.0761387
$$691$$ 18.0000 0.684752 0.342376 0.939563i $$-0.388768\pi$$
0.342376 + 0.939563i $$0.388768\pi$$
$$692$$ 14.0000 0.532200
$$693$$ −8.00000 −0.303895
$$694$$ 16.0000 0.607352
$$695$$ 16.0000 0.606915
$$696$$ −8.00000 −0.303239
$$697$$ −40.0000 −1.51511
$$698$$ 28.0000 1.05982
$$699$$ −28.0000 −1.05906
$$700$$ 2.00000 0.0755929
$$701$$ −20.0000 −0.755390 −0.377695 0.925930i $$-0.623283\pi$$
−0.377695 + 0.925930i $$0.623283\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 40.0000 1.50435
$$708$$ −12.0000 −0.450988
$$709$$ −4.00000 −0.150223 −0.0751116 0.997175i $$-0.523931\pi$$
−0.0751116 + 0.997175i $$0.523931\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 10.0000 0.374766
$$713$$ −8.00000 −0.299602
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ 2.00000 0.0747435
$$717$$ −8.00000 −0.298765
$$718$$ −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
$$721$$ 8.00000 0.297936
$$722$$ −15.0000 −0.558242
$$723$$ −22.0000 −0.818189
$$724$$ −22.0000 −0.817624
$$725$$ 8.00000 0.297113
$$726$$ −5.00000 −0.185567
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ 2.00000 0.0739221
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ −4.00000 −0.147643
$$735$$ −3.00000 −0.110657
$$736$$ 2.00000 0.0737210
$$737$$ −32.0000 −1.17874
$$738$$ −10.0000 −0.368105
$$739$$ −50.0000 −1.83928 −0.919640 0.392763i $$-0.871519\pi$$
−0.919640 + 0.392763i $$0.871519\pi$$
$$740$$ 6.00000 0.220564
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 4.00000 0.146647
$$745$$ −22.0000 −0.806018
$$746$$ 14.0000 0.512576
$$747$$ 12.0000 0.439057
$$748$$ −16.0000 −0.585018
$$749$$ 8.00000 0.292314
$$750$$ 1.00000 0.0365148
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ −10.0000 −0.364420
$$754$$ 0 0
$$755$$ −4.00000 −0.145575
$$756$$ −2.00000 −0.0727393
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ 14.0000 0.508503
$$759$$ 8.00000 0.290382
$$760$$ −2.00000 −0.0725476
$$761$$ 46.0000 1.66750 0.833749 0.552143i $$-0.186190\pi$$
0.833749 + 0.552143i $$0.186190\pi$$
$$762$$ −20.0000 −0.724524
$$763$$ −8.00000 −0.289619
$$764$$ −8.00000 −0.289430
$$765$$ −4.00000 −0.144620
$$766$$ 20.0000 0.722629
$$767$$ 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
$$771$$ 8.00000 0.288113
$$772$$ −12.0000 −0.431889
$$773$$ −26.0000 −0.935155 −0.467578 0.883952i $$-0.654873\pi$$
−0.467578 + 0.883952i $$0.654873\pi$$
$$774$$ 4.00000 0.143777
$$775$$ −4.00000 −0.143684
$$776$$ 8.00000 0.287183
$$777$$ 12.0000 0.430498
$$778$$ 20.0000 0.717035
$$779$$ −20.0000 −0.716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 8.00000 0.286079
$$783$$ −8.00000 −0.285897
$$784$$ −3.00000 −0.107143
$$785$$ 10.0000 0.356915
$$786$$ −14.0000 −0.499363
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ −10.0000 −0.356235
$$789$$ 18.0000 0.640817
$$790$$ 8.00000 0.284627
$$791$$ −32.0000 −1.13779
$$792$$ −4.00000 −0.142134
$$793$$ 0 0
$$794$$ −6.00000 −0.212932
$$795$$ 6.00000 0.212798
$$796$$ 24.0000 0.850657
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ 10.0000 0.353333
$$802$$ 30.0000 1.05934
$$803$$ 0 0
$$804$$ −8.00000 −0.282138
$$805$$ −4.00000 −0.140981
$$806$$ 0 0
$$807$$ 4.00000 0.140807
$$808$$ 20.0000 0.703598
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 34.0000 1.19390 0.596951 0.802278i $$-0.296379\pi$$
0.596951 + 0.802278i $$0.296379\pi$$
$$812$$ 16.0000 0.561490
$$813$$ −28.0000 −0.982003
$$814$$ 24.0000 0.841200
$$815$$ 12.0000 0.420342
$$816$$ −4.00000 −0.140028
$$817$$ 8.00000 0.279885
$$818$$ 18.0000 0.629355
$$819$$ 0 0
$$820$$ 10.0000 0.349215
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ −20.0000 −0.697156 −0.348578 0.937280i $$-0.613335\pi$$
−0.348578 + 0.937280i $$0.613335\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 4.00000 0.139262
$$826$$ 24.0000 0.835067
$$827$$ 44.0000 1.53003 0.765015 0.644013i $$-0.222732\pi$$
0.765015 + 0.644013i $$0.222732\pi$$
$$828$$ 2.00000 0.0695048
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ −22.0000 −0.763172
$$832$$ 0 0
$$833$$ −12.0000 −0.415775
$$834$$ 16.0000 0.554035
$$835$$ −12.0000 −0.415277
$$836$$ −8.00000 −0.276686
$$837$$ 4.00000 0.138260
$$838$$ 14.0000 0.483622
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ 35.0000 1.20690
$$842$$ 16.0000 0.551396
$$843$$ −10.0000 −0.344418
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 6.00000 0.206041
$$849$$ −28.0000 −0.960958
$$850$$ 4.00000 0.137199
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ −2.00000 −0.0683986
$$856$$ 4.00000 0.136717
$$857$$ −16.0000 −0.546550 −0.273275 0.961936i $$-0.588107\pi$$
−0.273275 + 0.961936i $$0.588107\pi$$
$$858$$ 0 0
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 20.0000 0.681598
$$862$$ −16.0000 −0.544962
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −14.0000 −0.476014
$$866$$ −26.0000 −0.883516
$$867$$ 1.00000 0.0339618
$$868$$ −8.00000 −0.271538
$$869$$ 32.0000 1.08553
$$870$$ 8.00000 0.271225
$$871$$ 0 0
$$872$$ −4.00000 −0.135457
$$873$$ 8.00000 0.270759
$$874$$ 4.00000 0.135302
$$875$$ −2.00000 −0.0676123
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 8.00000 0.269987
$$879$$ −6.00000 −0.202375
$$880$$ 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
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ 0 0
$$885$$ 12.0000 0.403376
$$886$$ −16.0000 −0.537531
$$887$$ −6.00000 −0.201460 −0.100730 0.994914i $$-0.532118\pi$$
−0.100730 + 0.994914i $$0.532118\pi$$
$$888$$ 6.00000 0.201347
$$889$$ 40.0000 1.34156
$$890$$ −10.0000 −0.335201
$$891$$ −4.00000 −0.134005
$$892$$ 22.0000 0.736614
$$893$$ 0 0
$$894$$ −22.0000 −0.735790
$$895$$ −2.00000 −0.0668526
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ −32.0000 −1.06726
$$900$$ 1.00000 0.0333333
$$901$$ 24.0000 0.799556
$$902$$ 40.0000 1.33185
$$903$$ −8.00000 −0.266223
$$904$$ −16.0000 −0.532152
$$905$$ 22.0000 0.731305
$$906$$ −4.00000 −0.132891
$$907$$ −44.0000 −1.46100 −0.730498 0.682915i $$-0.760712\pi$$
−0.730498 + 0.682915i $$0.760712\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 20.0000 0.663358
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ −2.00000 −0.0662266
$$913$$ −48.0000 −1.58857
$$914$$ 16.0000 0.529233
$$915$$ −2.00000 −0.0661180
$$916$$ −28.0000 −0.925146
$$917$$ 28.0000 0.924641
$$918$$ −4.00000 −0.132020
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ −2.00000 −0.0659380
$$921$$ 0 0
$$922$$ −30.0000 −0.987997
$$923$$ 0 0
$$924$$ 8.00000 0.263181
$$925$$ −6.00000 −0.197279
$$926$$ −2.00000 −0.0657241
$$927$$ 4.00000 0.131377
$$928$$ 8.00000 0.262613
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ −4.00000 −0.131165
$$931$$ −6.00000 −0.196642
$$932$$ 28.0000 0.917170
$$933$$ 16.0000 0.523816
$$934$$ 12.0000 0.392652
$$935$$ 16.0000 0.523256
$$936$$ 0 0
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 38.0000 1.23876 0.619382 0.785090i $$-0.287383\pi$$
0.619382 + 0.785090i $$0.287383\pi$$
$$942$$ 10.0000 0.325818
$$943$$ −20.0000 −0.651290
$$944$$ 12.0000 0.390567
$$945$$ 2.00000 0.0650600
$$946$$ −16.0000 −0.520205
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 0 0
$$950$$ 2.00000 0.0648886
$$951$$ −2.00000 −0.0648544
$$952$$ 8.00000 0.259281
$$953$$ −36.0000 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 8.00000 0.258874
$$956$$ 8.00000 0.258738
$$957$$ 32.0000 1.03441
$$958$$ −24.0000 −0.775405
$$959$$ 4.00000 0.129167
$$960$$ 1.00000 0.0322749
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 4.00000 0.128898
$$964$$ 22.0000 0.708572
$$965$$ 12.0000 0.386294
$$966$$ −4.00000 −0.128698
$$967$$ −14.0000 −0.450210 −0.225105 0.974335i $$-0.572272\pi$$
−0.225105 + 0.974335i $$0.572272\pi$$
$$968$$ 5.00000 0.160706
$$969$$ −8.00000 −0.256997
$$970$$ −8.00000 −0.256865
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −32.0000 −1.02587
$$974$$ −38.0000 −1.21760
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ 12.0000 0.383718
$$979$$ −40.0000 −1.27841
$$980$$ 3.00000 0.0958315
$$981$$ −4.00000 −0.127710
$$982$$ 38.0000 1.21263
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 10.0000 0.318626
$$986$$ 32.0000 1.01909
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 4.00000 0.127128
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ −10.0000 −0.317340
$$994$$ 0 0
$$995$$ −24.0000 −0.760851
$$996$$ −12.0000 −0.380235
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 14.0000 0.443162
$$999$$ 6.00000 0.189832
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5070.2.a.n.1.1 1
13.5 odd 4 5070.2.b.f.1351.1 2
13.8 odd 4 5070.2.b.f.1351.2 2
13.12 even 2 390.2.a.b.1.1 1
39.38 odd 2 1170.2.a.j.1.1 1
52.51 odd 2 3120.2.a.y.1.1 1
65.12 odd 4 1950.2.e.m.1249.1 2
65.38 odd 4 1950.2.e.m.1249.2 2
65.64 even 2 1950.2.a.ba.1.1 1
156.155 even 2 9360.2.a.v.1.1 1
195.38 even 4 5850.2.e.h.5149.1 2
195.77 even 4 5850.2.e.h.5149.2 2
195.194 odd 2 5850.2.a.s.1.1 1

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