# Properties

 Label 8470.2.a.l.1.1 Level $8470$ Weight $2$ Character 8470.1 Self dual yes Analytic conductor $67.633$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8470.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.6332905120$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 8470.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +7.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{18} -5.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} +9.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -7.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +1.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +5.00000 q^{38} +7.00000 q^{39} +1.00000 q^{40} -12.0000 q^{41} +1.00000 q^{42} +4.00000 q^{43} +2.00000 q^{45} -9.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +7.00000 q^{52} -6.00000 q^{53} +5.00000 q^{54} +1.00000 q^{56} -5.00000 q^{57} +9.00000 q^{59} -1.00000 q^{60} -14.0000 q^{61} -2.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -7.00000 q^{65} +2.00000 q^{67} +6.00000 q^{68} +9.00000 q^{69} -1.00000 q^{70} +2.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -5.00000 q^{76} -7.00000 q^{78} -11.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -9.00000 q^{83} -1.00000 q^{84} -6.00000 q^{85} -4.00000 q^{86} +12.0000 q^{89} -2.00000 q^{90} -7.00000 q^{91} +9.00000 q^{92} +2.00000 q^{93} -6.00000 q^{94} +5.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ −2.00000 −0.666667
$$10$$ 1.00000 0.316228
$$11$$ 0 0
$$12$$ 1.00000 0.288675
$$13$$ 7.00000 1.94145 0.970725 0.240192i $$-0.0772105\pi$$
0.970725 + 0.240192i $$0.0772105\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 2.00000 0.471405
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 9.00000 1.87663 0.938315 0.345782i $$-0.112386\pi$$
0.938315 + 0.345782i $$0.112386\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ −7.00000 −1.37281
$$27$$ −5.00000 −0.962250
$$28$$ −1.00000 −0.188982
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 1.00000 0.169031
$$36$$ −2.00000 −0.333333
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 5.00000 0.811107
$$39$$ 7.00000 1.12090
$$40$$ 1.00000 0.158114
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ −9.00000 −1.32698
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 6.00000 0.840168
$$52$$ 7.00000 0.970725
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 5.00000 0.680414
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ −5.00000 −0.662266
$$58$$ 0 0
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ −7.00000 −0.868243
$$66$$ 0 0
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 9.00000 1.08347
$$70$$ −1.00000 −0.119523
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 2.00000 0.235702
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 1.00000 0.115470
$$76$$ −5.00000 −0.573539
$$77$$ 0 0
$$78$$ −7.00000 −0.792594
$$79$$ −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
$$82$$ 12.0000 1.32518
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ −6.00000 −0.650791
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 12.0000 1.27200 0.635999 0.771690i $$-0.280588\pi$$
0.635999 + 0.771690i $$0.280588\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ −7.00000 −0.733799
$$92$$ 9.00000 0.938315
$$93$$ 2.00000 0.207390
$$94$$ −6.00000 −0.618853
$$95$$ 5.00000 0.512989
$$96$$ −1.00000 −0.102062
$$97$$ 8.00000 0.812277 0.406138 0.913812i $$-0.366875\pi$$
0.406138 + 0.913812i $$0.366875\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −7.00000 −0.686406
$$105$$ 1.00000 0.0975900
$$106$$ 6.00000 0.582772
$$107$$ 18.0000 1.74013 0.870063 0.492941i $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ −5.00000 −0.481125
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ −1.00000 −0.0944911
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 5.00000 0.468293
$$115$$ −9.00000 −0.839254
$$116$$ 0 0
$$117$$ −14.0000 −1.29430
$$118$$ −9.00000 −0.828517
$$119$$ −6.00000 −0.550019
$$120$$ 1.00000 0.0912871
$$121$$ 0 0
$$122$$ 14.0000 1.26750
$$123$$ −12.0000 −1.08200
$$124$$ 2.00000 0.179605
$$125$$ −1.00000 −0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ −5.00000 −0.443678 −0.221839 0.975083i $$-0.571206\pi$$
−0.221839 + 0.975083i $$0.571206\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 7.00000 0.613941
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ −2.00000 −0.172774
$$135$$ 5.00000 0.430331
$$136$$ −6.00000 −0.514496
$$137$$ −3.00000 −0.256307 −0.128154 0.991754i $$-0.540905\pi$$
−0.128154 + 0.991754i $$0.540905\pi$$
$$138$$ −9.00000 −0.766131
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 1.00000 0.0845154
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −2.00000 −0.166667
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 1.00000 0.0824786
$$148$$ 2.00000 0.164399
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −5.00000 −0.406894 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$152$$ 5.00000 0.405554
$$153$$ −12.0000 −0.970143
$$154$$ 0 0
$$155$$ −2.00000 −0.160644
$$156$$ 7.00000 0.560449
$$157$$ 17.0000 1.35675 0.678374 0.734717i $$-0.262685\pi$$
0.678374 + 0.734717i $$0.262685\pi$$
$$158$$ 11.0000 0.875113
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ −9.00000 −0.709299
$$162$$ −1.00000 −0.0785674
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ 9.00000 0.698535
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 36.0000 2.76923
$$170$$ 6.00000 0.460179
$$171$$ 10.0000 0.764719
$$172$$ 4.00000 0.304997
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ 9.00000 0.676481
$$178$$ −12.0000 −0.899438
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 2.00000 0.149071
$$181$$ −13.0000 −0.966282 −0.483141 0.875542i $$-0.660504\pi$$
−0.483141 + 0.875542i $$0.660504\pi$$
$$182$$ 7.00000 0.518875
$$183$$ −14.0000 −1.03491
$$184$$ −9.00000 −0.663489
$$185$$ −2.00000 −0.147043
$$186$$ −2.00000 −0.146647
$$187$$ 0 0
$$188$$ 6.00000 0.437595
$$189$$ 5.00000 0.363696
$$190$$ −5.00000 −0.362738
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 13.0000 0.935760 0.467880 0.883792i $$-0.345018\pi$$
0.467880 + 0.883792i $$0.345018\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ −7.00000 −0.501280
$$196$$ 1.00000 0.0714286
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 2.00000 0.141069
$$202$$ 9.00000 0.633238
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 12.0000 0.838116
$$206$$ −8.00000 −0.557386
$$207$$ −18.0000 −1.25109
$$208$$ 7.00000 0.485363
$$209$$ 0 0
$$210$$ −1.00000 −0.0690066
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ −18.0000 −1.23045
$$215$$ −4.00000 −0.272798
$$216$$ 5.00000 0.340207
$$217$$ −2.00000 −0.135769
$$218$$ 2.00000 0.135457
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 42.0000 2.82523
$$222$$ −2.00000 −0.134231
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −2.00000 −0.133333
$$226$$ 15.0000 0.997785
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ −5.00000 −0.331133
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 9.00000 0.593442
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3.00000 −0.196537 −0.0982683 0.995160i $$-0.531330\pi$$
−0.0982683 + 0.995160i $$0.531330\pi$$
$$234$$ 14.0000 0.915209
$$235$$ −6.00000 −0.391397
$$236$$ 9.00000 0.585850
$$237$$ −11.0000 −0.714527
$$238$$ 6.00000 0.388922
$$239$$ 15.0000 0.970269 0.485135 0.874439i $$-0.338771\pi$$
0.485135 + 0.874439i $$0.338771\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −20.0000 −1.28831 −0.644157 0.764894i $$-0.722792\pi$$
−0.644157 + 0.764894i $$0.722792\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ −14.0000 −0.896258
$$245$$ −1.00000 −0.0638877
$$246$$ 12.0000 0.765092
$$247$$ −35.0000 −2.22700
$$248$$ −2.00000 −0.127000
$$249$$ −9.00000 −0.570352
$$250$$ 1.00000 0.0632456
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 0 0
$$254$$ 5.00000 0.313728
$$255$$ −6.00000 −0.375735
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ −2.00000 −0.124274
$$260$$ −7.00000 −0.434122
$$261$$ 0 0
$$262$$ −3.00000 −0.185341
$$263$$ 21.0000 1.29492 0.647458 0.762101i $$-0.275832\pi$$
0.647458 + 0.762101i $$0.275832\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ −5.00000 −0.306570
$$267$$ 12.0000 0.734388
$$268$$ 2.00000 0.122169
$$269$$ 15.0000 0.914566 0.457283 0.889321i $$-0.348823\pi$$
0.457283 + 0.889321i $$0.348823\pi$$
$$270$$ −5.00000 −0.304290
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 6.00000 0.363803
$$273$$ −7.00000 −0.423659
$$274$$ 3.00000 0.181237
$$275$$ 0 0
$$276$$ 9.00000 0.541736
$$277$$ 28.0000 1.68236 0.841178 0.540758i $$-0.181862\pi$$
0.841178 + 0.540758i $$0.181862\pi$$
$$278$$ 5.00000 0.299880
$$279$$ −4.00000 −0.239474
$$280$$ −1.00000 −0.0597614
$$281$$ 15.0000 0.894825 0.447412 0.894328i $$-0.352346\pi$$
0.447412 + 0.894328i $$0.352346\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ −5.00000 −0.297219 −0.148610 0.988896i $$-0.547480\pi$$
−0.148610 + 0.988896i $$0.547480\pi$$
$$284$$ 0 0
$$285$$ 5.00000 0.296174
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ 2.00000 0.117851
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 10.0000 0.585206
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ −9.00000 −0.524000
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 63.0000 3.64338
$$300$$ 1.00000 0.0577350
$$301$$ −4.00000 −0.230556
$$302$$ 5.00000 0.287718
$$303$$ −9.00000 −0.517036
$$304$$ −5.00000 −0.286770
$$305$$ 14.0000 0.801638
$$306$$ 12.0000 0.685994
$$307$$ 28.0000 1.59804 0.799022 0.601302i $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 2.00000 0.113592
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ −7.00000 −0.396297
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ −17.0000 −0.959366
$$315$$ −2.00000 −0.112687
$$316$$ −11.0000 −0.618798
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 18.0000 1.00466
$$322$$ 9.00000 0.501550
$$323$$ −30.0000 −1.66924
$$324$$ 1.00000 0.0555556
$$325$$ 7.00000 0.388290
$$326$$ −2.00000 −0.110770
$$327$$ −2.00000 −0.110600
$$328$$ 12.0000 0.662589
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −9.00000 −0.493939
$$333$$ −4.00000 −0.219199
$$334$$ 12.0000 0.656611
$$335$$ −2.00000 −0.109272
$$336$$ −1.00000 −0.0545545
$$337$$ −23.0000 −1.25289 −0.626445 0.779466i $$-0.715491\pi$$
−0.626445 + 0.779466i $$0.715491\pi$$
$$338$$ −36.0000 −1.95814
$$339$$ −15.0000 −0.814688
$$340$$ −6.00000 −0.325396
$$341$$ 0 0
$$342$$ −10.0000 −0.540738
$$343$$ −1.00000 −0.0539949
$$344$$ −4.00000 −0.215666
$$345$$ −9.00000 −0.484544
$$346$$ 6.00000 0.322562
$$347$$ −18.0000 −0.966291 −0.483145 0.875540i $$-0.660506\pi$$
−0.483145 + 0.875540i $$0.660506\pi$$
$$348$$ 0 0
$$349$$ 25.0000 1.33822 0.669110 0.743164i $$-0.266676\pi$$
0.669110 + 0.743164i $$0.266676\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −35.0000 −1.86816
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ −9.00000 −0.478345
$$355$$ 0 0
$$356$$ 12.0000 0.635999
$$357$$ −6.00000 −0.317554
$$358$$ 12.0000 0.634220
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ −2.00000 −0.105409
$$361$$ 6.00000 0.315789
$$362$$ 13.0000 0.683265
$$363$$ 0 0
$$364$$ −7.00000 −0.366900
$$365$$ −10.0000 −0.523424
$$366$$ 14.0000 0.731792
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 9.00000 0.469157
$$369$$ 24.0000 1.24939
$$370$$ 2.00000 0.103975
$$371$$ 6.00000 0.311504
$$372$$ 2.00000 0.103695
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ −6.00000 −0.309426
$$377$$ 0 0
$$378$$ −5.00000 −0.257172
$$379$$ −10.0000 −0.513665 −0.256833 0.966456i $$-0.582679\pi$$
−0.256833 + 0.966456i $$0.582679\pi$$
$$380$$ 5.00000 0.256495
$$381$$ −5.00000 −0.256158
$$382$$ −3.00000 −0.153493
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −13.0000 −0.661683
$$387$$ −8.00000 −0.406663
$$388$$ 8.00000 0.406138
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 7.00000 0.354459
$$391$$ 54.0000 2.73090
$$392$$ −1.00000 −0.0505076
$$393$$ 3.00000 0.151330
$$394$$ −12.0000 −0.604551
$$395$$ 11.0000 0.553470
$$396$$ 0 0
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 5.00000 0.250313
$$400$$ 1.00000 0.0500000
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ −2.00000 −0.0997509
$$403$$ 14.0000 0.697390
$$404$$ −9.00000 −0.447767
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ −6.00000 −0.297044
$$409$$ 4.00000 0.197787 0.0988936 0.995098i $$-0.468470\pi$$
0.0988936 + 0.995098i $$0.468470\pi$$
$$410$$ −12.0000 −0.592638
$$411$$ −3.00000 −0.147979
$$412$$ 8.00000 0.394132
$$413$$ −9.00000 −0.442861
$$414$$ 18.0000 0.884652
$$415$$ 9.00000 0.441793
$$416$$ −7.00000 −0.343203
$$417$$ −5.00000 −0.244851
$$418$$ 0 0
$$419$$ −3.00000 −0.146560 −0.0732798 0.997311i $$-0.523347\pi$$
−0.0732798 + 0.997311i $$0.523347\pi$$
$$420$$ 1.00000 0.0487950
$$421$$ 8.00000 0.389896 0.194948 0.980814i $$-0.437546\pi$$
0.194948 + 0.980814i $$0.437546\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ −12.0000 −0.583460
$$424$$ 6.00000 0.291386
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 14.0000 0.677507
$$428$$ 18.0000 0.870063
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ −33.0000 −1.58955 −0.794777 0.606902i $$-0.792412\pi$$
−0.794777 + 0.606902i $$0.792412\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ 8.00000 0.384455 0.192228 0.981350i $$-0.438429\pi$$
0.192228 + 0.981350i $$0.438429\pi$$
$$434$$ 2.00000 0.0960031
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ −45.0000 −2.15264
$$438$$ −10.0000 −0.477818
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ −2.00000 −0.0952381
$$442$$ −42.0000 −1.99774
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ −12.0000 −0.568855
$$446$$ −2.00000 −0.0947027
$$447$$ 6.00000 0.283790
$$448$$ −1.00000 −0.0472456
$$449$$ −27.0000 −1.27421 −0.637104 0.770778i $$-0.719868\pi$$
−0.637104 + 0.770778i $$0.719868\pi$$
$$450$$ 2.00000 0.0942809
$$451$$ 0 0
$$452$$ −15.0000 −0.705541
$$453$$ −5.00000 −0.234920
$$454$$ −24.0000 −1.12638
$$455$$ 7.00000 0.328165
$$456$$ 5.00000 0.234146
$$457$$ 1.00000 0.0467780 0.0233890 0.999726i $$-0.492554\pi$$
0.0233890 + 0.999726i $$0.492554\pi$$
$$458$$ −26.0000 −1.21490
$$459$$ −30.0000 −1.40028
$$460$$ −9.00000 −0.419627
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 5.00000 0.232370 0.116185 0.993228i $$-0.462933\pi$$
0.116185 + 0.993228i $$0.462933\pi$$
$$464$$ 0 0
$$465$$ −2.00000 −0.0927478
$$466$$ 3.00000 0.138972
$$467$$ 21.0000 0.971764 0.485882 0.874024i $$-0.338498\pi$$
0.485882 + 0.874024i $$0.338498\pi$$
$$468$$ −14.0000 −0.647150
$$469$$ −2.00000 −0.0923514
$$470$$ 6.00000 0.276759
$$471$$ 17.0000 0.783319
$$472$$ −9.00000 −0.414259
$$473$$ 0 0
$$474$$ 11.0000 0.505247
$$475$$ −5.00000 −0.229416
$$476$$ −6.00000 −0.275010
$$477$$ 12.0000 0.549442
$$478$$ −15.0000 −0.686084
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 14.0000 0.638345
$$482$$ 20.0000 0.910975
$$483$$ −9.00000 −0.409514
$$484$$ 0 0
$$485$$ −8.00000 −0.363261
$$486$$ −16.0000 −0.725775
$$487$$ −25.0000 −1.13286 −0.566429 0.824110i $$-0.691675\pi$$
−0.566429 + 0.824110i $$0.691675\pi$$
$$488$$ 14.0000 0.633750
$$489$$ 2.00000 0.0904431
$$490$$ 1.00000 0.0451754
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ −12.0000 −0.541002
$$493$$ 0 0
$$494$$ 35.0000 1.57472
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ 0 0
$$498$$ 9.00000 0.403300
$$499$$ 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
$$502$$ −24.0000 −1.07117
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 9.00000 0.400495
$$506$$ 0 0
$$507$$ 36.0000 1.59882
$$508$$ −5.00000 −0.221839
$$509$$ −9.00000 −0.398918 −0.199459 0.979906i $$-0.563918\pi$$
−0.199459 + 0.979906i $$0.563918\pi$$
$$510$$ 6.00000 0.265684
$$511$$ −10.0000 −0.442374
$$512$$ −1.00000 −0.0441942
$$513$$ 25.0000 1.10378
$$514$$ −18.0000 −0.793946
$$515$$ −8.00000 −0.352522
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 2.00000 0.0878750
$$519$$ −6.00000 −0.263371
$$520$$ 7.00000 0.306970
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ 0 0
$$523$$ 7.00000 0.306089 0.153044 0.988219i $$-0.451092\pi$$
0.153044 + 0.988219i $$0.451092\pi$$
$$524$$ 3.00000 0.131056
$$525$$ −1.00000 −0.0436436
$$526$$ −21.0000 −0.915644
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ 58.0000 2.52174
$$530$$ −6.00000 −0.260623
$$531$$ −18.0000 −0.781133
$$532$$ 5.00000 0.216777
$$533$$ −84.0000 −3.63844
$$534$$ −12.0000 −0.519291
$$535$$ −18.0000 −0.778208
$$536$$ −2.00000 −0.0863868
$$537$$ −12.0000 −0.517838
$$538$$ −15.0000 −0.646696
$$539$$ 0 0
$$540$$ 5.00000 0.215166
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 20.0000 0.859074
$$543$$ −13.0000 −0.557883
$$544$$ −6.00000 −0.257248
$$545$$ 2.00000 0.0856706
$$546$$ 7.00000 0.299572
$$547$$ −44.0000 −1.88130 −0.940652 0.339372i $$-0.889785\pi$$
−0.940652 + 0.339372i $$0.889785\pi$$
$$548$$ −3.00000 −0.128154
$$549$$ 28.0000 1.19501
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −9.00000 −0.383065
$$553$$ 11.0000 0.467768
$$554$$ −28.0000 −1.18961
$$555$$ −2.00000 −0.0848953
$$556$$ −5.00000 −0.212047
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 28.0000 1.18427
$$560$$ 1.00000 0.0422577
$$561$$ 0 0
$$562$$ −15.0000 −0.632737
$$563$$ 39.0000 1.64365 0.821827 0.569737i $$-0.192955\pi$$
0.821827 + 0.569737i $$0.192955\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 15.0000 0.631055
$$566$$ 5.00000 0.210166
$$567$$ −1.00000 −0.0419961
$$568$$ 0 0
$$569$$ 21.0000 0.880366 0.440183 0.897908i $$-0.354914\pi$$
0.440183 + 0.897908i $$0.354914\pi$$
$$570$$ −5.00000 −0.209427
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 0 0
$$573$$ 3.00000 0.125327
$$574$$ −12.0000 −0.500870
$$575$$ 9.00000 0.375326
$$576$$ −2.00000 −0.0833333
$$577$$ 20.0000 0.832611 0.416305 0.909225i $$-0.363325\pi$$
0.416305 + 0.909225i $$0.363325\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 13.0000 0.540262
$$580$$ 0 0
$$581$$ 9.00000 0.373383
$$582$$ −8.00000 −0.331611
$$583$$ 0 0
$$584$$ −10.0000 −0.413803
$$585$$ 14.0000 0.578829
$$586$$ −9.00000 −0.371787
$$587$$ 15.0000 0.619116 0.309558 0.950881i $$-0.399819\pi$$
0.309558 + 0.950881i $$0.399819\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ −10.0000 −0.412043
$$590$$ 9.00000 0.370524
$$591$$ 12.0000 0.493614
$$592$$ 2.00000 0.0821995
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 0 0
$$595$$ 6.00000 0.245976
$$596$$ 6.00000 0.245770
$$597$$ −16.0000 −0.654836
$$598$$ −63.0000 −2.57626
$$599$$ 15.0000 0.612883 0.306442 0.951889i $$-0.400862\pi$$
0.306442 + 0.951889i $$0.400862\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 4.00000 0.163028
$$603$$ −4.00000 −0.162893
$$604$$ −5.00000 −0.203447
$$605$$ 0 0
$$606$$ 9.00000 0.365600
$$607$$ −14.0000 −0.568242 −0.284121 0.958788i $$-0.591702\pi$$
−0.284121 + 0.958788i $$0.591702\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 0 0
$$610$$ −14.0000 −0.566843
$$611$$ 42.0000 1.69914
$$612$$ −12.0000 −0.485071
$$613$$ −20.0000 −0.807792 −0.403896 0.914805i $$-0.632344\pi$$
−0.403896 + 0.914805i $$0.632344\pi$$
$$614$$ −28.0000 −1.12999
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ −25.0000 −1.00483 −0.502417 0.864625i $$-0.667556\pi$$
−0.502417 + 0.864625i $$0.667556\pi$$
$$620$$ −2.00000 −0.0803219
$$621$$ −45.0000 −1.80579
$$622$$ 12.0000 0.481156
$$623$$ −12.0000 −0.480770
$$624$$ 7.00000 0.280224
$$625$$ 1.00000 0.0400000
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ 17.0000 0.678374
$$629$$ 12.0000 0.478471
$$630$$ 2.00000 0.0796819
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ 11.0000 0.437557
$$633$$ 4.00000 0.158986
$$634$$ −18.0000 −0.714871
$$635$$ 5.00000 0.198419
$$636$$ −6.00000 −0.237915
$$637$$ 7.00000 0.277350
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 27.0000 1.06644 0.533218 0.845978i $$-0.320983\pi$$
0.533218 + 0.845978i $$0.320983\pi$$
$$642$$ −18.0000 −0.710403
$$643$$ 8.00000 0.315489 0.157745 0.987480i $$-0.449578\pi$$
0.157745 + 0.987480i $$0.449578\pi$$
$$644$$ −9.00000 −0.354650
$$645$$ −4.00000 −0.157500
$$646$$ 30.0000 1.18033
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ −7.00000 −0.274563
$$651$$ −2.00000 −0.0783862
$$652$$ 2.00000 0.0783260
$$653$$ 36.0000 1.40879 0.704394 0.709809i $$-0.251219\pi$$
0.704394 + 0.709809i $$0.251219\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ −3.00000 −0.117220
$$656$$ −12.0000 −0.468521
$$657$$ −20.0000 −0.780274
$$658$$ 6.00000 0.233904
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ −7.00000 −0.272268 −0.136134 0.990690i $$-0.543468\pi$$
−0.136134 + 0.990690i $$0.543468\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 42.0000 1.63114
$$664$$ 9.00000 0.349268
$$665$$ −5.00000 −0.193892
$$666$$ 4.00000 0.154997
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ 2.00000 0.0773245
$$670$$ 2.00000 0.0772667
$$671$$ 0 0
$$672$$ 1.00000 0.0385758
$$673$$ −41.0000 −1.58043 −0.790217 0.612827i $$-0.790032\pi$$
−0.790217 + 0.612827i $$0.790032\pi$$
$$674$$ 23.0000 0.885927
$$675$$ −5.00000 −0.192450
$$676$$ 36.0000 1.38462
$$677$$ −33.0000 −1.26829 −0.634147 0.773213i $$-0.718648\pi$$
−0.634147 + 0.773213i $$0.718648\pi$$
$$678$$ 15.0000 0.576072
$$679$$ −8.00000 −0.307012
$$680$$ 6.00000 0.230089
$$681$$ 24.0000 0.919682
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 10.0000 0.382360
$$685$$ 3.00000 0.114624
$$686$$ 1.00000 0.0381802
$$687$$ 26.0000 0.991962
$$688$$ 4.00000 0.152499
$$689$$ −42.0000 −1.60007
$$690$$ 9.00000 0.342624
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ 18.0000 0.683271
$$695$$ 5.00000 0.189661
$$696$$ 0 0
$$697$$ −72.0000 −2.72719
$$698$$ −25.0000 −0.946264
$$699$$ −3.00000 −0.113470
$$700$$ −1.00000 −0.0377964
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 35.0000 1.32099
$$703$$ −10.0000 −0.377157
$$704$$ 0 0
$$705$$ −6.00000 −0.225973
$$706$$ −18.0000 −0.677439
$$707$$ 9.00000 0.338480
$$708$$ 9.00000 0.338241
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 0 0
$$711$$ 22.0000 0.825064
$$712$$ −12.0000 −0.449719
$$713$$ 18.0000 0.674105
$$714$$ 6.00000 0.224544
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 15.0000 0.560185
$$718$$ −24.0000 −0.895672
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ −8.00000 −0.297936
$$722$$ −6.00000 −0.223297
$$723$$ −20.0000 −0.743808
$$724$$ −13.0000 −0.483141
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 20.0000 0.741759 0.370879 0.928681i $$-0.379056\pi$$
0.370879 + 0.928681i $$0.379056\pi$$
$$728$$ 7.00000 0.259437
$$729$$ 13.0000 0.481481
$$730$$ 10.0000 0.370117
$$731$$ 24.0000 0.887672
$$732$$ −14.0000 −0.517455
$$733$$ 49.0000 1.80986 0.904928 0.425564i $$-0.139924\pi$$
0.904928 + 0.425564i $$0.139924\pi$$
$$734$$ 28.0000 1.03350
$$735$$ −1.00000 −0.0368856
$$736$$ −9.00000 −0.331744
$$737$$ 0 0
$$738$$ −24.0000 −0.883452
$$739$$ −38.0000 −1.39785 −0.698926 0.715194i $$-0.746338\pi$$
−0.698926 + 0.715194i $$0.746338\pi$$
$$740$$ −2.00000 −0.0735215
$$741$$ −35.0000 −1.28576
$$742$$ −6.00000 −0.220267
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ −2.00000 −0.0733236
$$745$$ −6.00000 −0.219823
$$746$$ −22.0000 −0.805477
$$747$$ 18.0000 0.658586
$$748$$ 0 0
$$749$$ −18.0000 −0.657706
$$750$$ 1.00000 0.0365148
$$751$$ −25.0000 −0.912263 −0.456131 0.889912i $$-0.650765\pi$$
−0.456131 + 0.889912i $$0.650765\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 24.0000 0.874609
$$754$$ 0 0
$$755$$ 5.00000 0.181969
$$756$$ 5.00000 0.181848
$$757$$ −16.0000 −0.581530 −0.290765 0.956795i $$-0.593910\pi$$
−0.290765 + 0.956795i $$0.593910\pi$$
$$758$$ 10.0000 0.363216
$$759$$ 0 0
$$760$$ −5.00000 −0.181369
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 5.00000 0.181131
$$763$$ 2.00000 0.0724049
$$764$$ 3.00000 0.108536
$$765$$ 12.0000 0.433861
$$766$$ −24.0000 −0.867155
$$767$$ 63.0000 2.27480
$$768$$ 1.00000 0.0360844
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 13.0000 0.467880
$$773$$ 27.0000 0.971123 0.485561 0.874203i $$-0.338615\pi$$
0.485561 + 0.874203i $$0.338615\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 2.00000 0.0718421
$$776$$ −8.00000 −0.287183
$$777$$ −2.00000 −0.0717496
$$778$$ −18.0000 −0.645331
$$779$$ 60.0000 2.14972
$$780$$ −7.00000 −0.250640
$$781$$ 0 0
$$782$$ −54.0000 −1.93104
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ −17.0000 −0.606756
$$786$$ −3.00000 −0.107006
$$787$$ 52.0000 1.85360 0.926800 0.375555i $$-0.122548\pi$$
0.926800 + 0.375555i $$0.122548\pi$$
$$788$$ 12.0000 0.427482
$$789$$ 21.0000 0.747620
$$790$$ −11.0000 −0.391362
$$791$$ 15.0000 0.533339
$$792$$ 0 0
$$793$$ −98.0000 −3.48008
$$794$$ 34.0000 1.20661
$$795$$ 6.00000 0.212798
$$796$$ −16.0000 −0.567105
$$797$$ −27.0000 −0.956389 −0.478195 0.878254i $$-0.658709\pi$$
−0.478195 + 0.878254i $$0.658709\pi$$
$$798$$ −5.00000 −0.176998
$$799$$ 36.0000 1.27359
$$800$$ −1.00000 −0.0353553
$$801$$ −24.0000 −0.847998
$$802$$ −18.0000 −0.635602
$$803$$ 0 0
$$804$$ 2.00000 0.0705346
$$805$$ 9.00000 0.317208
$$806$$ −14.0000 −0.493129
$$807$$ 15.0000 0.528025
$$808$$ 9.00000 0.316619
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 16.0000 0.561836 0.280918 0.959732i $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ 0 0
$$813$$ −20.0000 −0.701431
$$814$$ 0 0
$$815$$ −2.00000 −0.0700569
$$816$$ 6.00000 0.210042
$$817$$ −20.0000 −0.699711
$$818$$ −4.00000 −0.139857
$$819$$ 14.0000 0.489200
$$820$$ 12.0000 0.419058
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 3.00000 0.104637
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ 9.00000 0.313150
$$827$$ 18.0000 0.625921 0.312961 0.949766i $$-0.398679\pi$$
0.312961 + 0.949766i $$0.398679\pi$$
$$828$$ −18.0000 −0.625543
$$829$$ −19.0000 −0.659897 −0.329949 0.943999i $$-0.607031\pi$$
−0.329949 + 0.943999i $$0.607031\pi$$
$$830$$ −9.00000 −0.312395
$$831$$ 28.0000 0.971309
$$832$$ 7.00000 0.242681
$$833$$ 6.00000 0.207888
$$834$$ 5.00000 0.173136
$$835$$ 12.0000 0.415277
$$836$$ 0 0
$$837$$ −10.0000 −0.345651
$$838$$ 3.00000 0.103633
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ −1.00000 −0.0345033
$$841$$ −29.0000 −1.00000
$$842$$ −8.00000 −0.275698
$$843$$ 15.0000 0.516627
$$844$$ 4.00000 0.137686
$$845$$ −36.0000 −1.23844
$$846$$ 12.0000 0.412568
$$847$$ 0 0
$$848$$ −6.00000 −0.206041
$$849$$ −5.00000 −0.171600
$$850$$ −6.00000 −0.205798
$$851$$ 18.0000 0.617032
$$852$$ 0 0
$$853$$ 7.00000 0.239675 0.119838 0.992793i $$-0.461763\pi$$
0.119838 + 0.992793i $$0.461763\pi$$
$$854$$ −14.0000 −0.479070
$$855$$ −10.0000 −0.341993
$$856$$ −18.0000 −0.615227
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 12.0000 0.408959
$$862$$ 33.0000 1.12398
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 6.00000 0.204006
$$866$$ −8.00000 −0.271851
$$867$$ 19.0000 0.645274
$$868$$ −2.00000 −0.0678844
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 14.0000 0.474372
$$872$$ 2.00000 0.0677285
$$873$$ −16.0000 −0.541518
$$874$$ 45.0000 1.52215
$$875$$ 1.00000 0.0338062
$$876$$ 10.0000 0.337869
$$877$$ 52.0000 1.75592 0.877958 0.478738i $$-0.158906\pi$$
0.877958 + 0.478738i $$0.158906\pi$$
$$878$$ −28.0000 −0.944954
$$879$$ 9.00000 0.303562
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 2.00000 0.0673435
$$883$$ −34.0000 −1.14419 −0.572096 0.820187i $$-0.693869\pi$$
−0.572096 + 0.820187i $$0.693869\pi$$
$$884$$ 42.0000 1.41261
$$885$$ −9.00000 −0.302532
$$886$$ 24.0000 0.806296
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 5.00000 0.167695
$$890$$ 12.0000 0.402241
$$891$$ 0 0
$$892$$ 2.00000 0.0669650
$$893$$ −30.0000 −1.00391
$$894$$ −6.00000 −0.200670
$$895$$ 12.0000 0.401116
$$896$$ 1.00000 0.0334077
$$897$$ 63.0000 2.10351
$$898$$ 27.0000 0.901002
$$899$$ 0 0
$$900$$ −2.00000 −0.0666667
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ −4.00000 −0.133112
$$904$$ 15.0000 0.498893
$$905$$ 13.0000 0.432135
$$906$$ 5.00000 0.166114
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 24.0000 0.796468
$$909$$ 18.0000 0.597022
$$910$$ −7.00000 −0.232048
$$911$$ −15.0000 −0.496972 −0.248486 0.968635i $$-0.579933\pi$$
−0.248486 + 0.968635i $$0.579933\pi$$
$$912$$ −5.00000 −0.165567
$$913$$ 0 0
$$914$$ −1.00000 −0.0330771
$$915$$ 14.0000 0.462826
$$916$$ 26.0000 0.859064
$$917$$ −3.00000 −0.0990687
$$918$$ 30.0000 0.990148
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 9.00000 0.296721
$$921$$ 28.0000 0.922631
$$922$$ −18.0000 −0.592798
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.00000 0.0657596
$$926$$ −5.00000 −0.164310
$$927$$ −16.0000 −0.525509
$$928$$ 0 0
$$929$$ −12.0000 −0.393707 −0.196854 0.980433i $$-0.563072\pi$$
−0.196854 + 0.980433i $$0.563072\pi$$
$$930$$ 2.00000 0.0655826
$$931$$ −5.00000 −0.163868
$$932$$ −3.00000 −0.0982683
$$933$$ −12.0000 −0.392862
$$934$$ −21.0000 −0.687141
$$935$$ 0 0
$$936$$ 14.0000 0.457604
$$937$$ −20.0000 −0.653372 −0.326686 0.945133i $$-0.605932\pi$$
−0.326686 + 0.945133i $$0.605932\pi$$
$$938$$ 2.00000 0.0653023
$$939$$ −10.0000 −0.326338
$$940$$ −6.00000 −0.195698
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ −17.0000 −0.553890
$$943$$ −108.000 −3.51696
$$944$$ 9.00000 0.292925
$$945$$ −5.00000 −0.162650
$$946$$ 0 0
$$947$$ 18.0000 0.584921 0.292461 0.956278i $$-0.405526\pi$$
0.292461 + 0.956278i $$0.405526\pi$$
$$948$$ −11.0000 −0.357263
$$949$$ 70.0000 2.27230
$$950$$ 5.00000 0.162221
$$951$$ 18.0000 0.583690
$$952$$ 6.00000 0.194461
$$953$$ 33.0000 1.06897 0.534487 0.845176i $$-0.320505\pi$$
0.534487 + 0.845176i $$0.320505\pi$$
$$954$$ −12.0000 −0.388514
$$955$$ −3.00000 −0.0970777
$$956$$ 15.0000 0.485135
$$957$$ 0 0
$$958$$ −24.0000 −0.775405
$$959$$ 3.00000 0.0968751
$$960$$ −1.00000 −0.0322749
$$961$$ −27.0000 −0.870968
$$962$$ −14.0000 −0.451378
$$963$$ −36.0000 −1.16008
$$964$$ −20.0000 −0.644157
$$965$$ −13.0000 −0.418485
$$966$$ 9.00000 0.289570
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ −30.0000 −0.963739
$$970$$ 8.00000 0.256865
$$971$$ 15.0000 0.481373 0.240686 0.970603i $$-0.422627\pi$$
0.240686 + 0.970603i $$0.422627\pi$$
$$972$$ 16.0000 0.513200
$$973$$ 5.00000 0.160293
$$974$$ 25.0000 0.801052
$$975$$ 7.00000 0.224179
$$976$$ −14.0000 −0.448129
$$977$$ 45.0000 1.43968 0.719839 0.694141i $$-0.244216\pi$$
0.719839 + 0.694141i $$0.244216\pi$$
$$978$$ −2.00000 −0.0639529
$$979$$ 0 0
$$980$$ −1.00000 −0.0319438
$$981$$ 4.00000 0.127710
$$982$$ 6.00000 0.191468
$$983$$ −42.0000 −1.33959 −0.669796 0.742545i $$-0.733618\pi$$
−0.669796 + 0.742545i $$0.733618\pi$$
$$984$$ 12.0000 0.382546
$$985$$ −12.0000 −0.382352
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ −35.0000 −1.11350
$$989$$ 36.0000 1.14473
$$990$$ 0 0
$$991$$ 23.0000 0.730619 0.365310 0.930886i $$-0.380963\pi$$
0.365310 + 0.930886i $$0.380963\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ −9.00000 −0.285176
$$997$$ −35.0000 −1.10846 −0.554231 0.832363i $$-0.686987\pi$$
−0.554231 + 0.832363i $$0.686987\pi$$
$$998$$ −14.0000 −0.443162
$$999$$ −10.0000 −0.316386
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 8470.2.a.l.1.1 1
11.10 odd 2 8470.2.a.bc.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.l.1.1 1 1.1 even 1 trivial
8470.2.a.bc.1.1 yes 1 11.10 odd 2