# Properties

 Label 1170.2.a.e.1.1 Level $1170$ Weight $2$ Character 1170.1 Self dual yes Analytic conductor $9.342$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1170.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.34249703649$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 1170.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} -6.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} +2.00000 q^{28} +4.00000 q^{29} -1.00000 q^{32} +8.00000 q^{34} +2.00000 q^{35} -2.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} +6.00000 q^{46} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} +10.0000 q^{53} -4.00000 q^{55} -2.00000 q^{56} -4.00000 q^{58} -4.00000 q^{59} -10.0000 q^{61} +1.00000 q^{64} -1.00000 q^{65} +12.0000 q^{67} -8.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} -8.00000 q^{73} +2.00000 q^{74} -6.00000 q^{76} -8.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} -8.00000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +14.0000 q^{89} -2.00000 q^{91} -6.00000 q^{92} -6.00000 q^{95} -16.0000 q^{97} +3.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −8.00000 −1.94029 −0.970143 0.242536i $$-0.922021\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 8.00000 1.37199
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 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 $$0.450084\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ −4.00000 −0.525226
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −8.00000 −0.970143
$$69$$ 0 0
$$70$$ −2.00000 −0.239046
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 0 0
$$73$$ −8.00000 −0.936329 −0.468165 0.883641i $$-0.655085\pi$$
−0.468165 + 0.883641i $$0.655085\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ −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
$$81$$ 0 0
$$82$$ −2.00000 −0.220863
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ −8.00000 −0.867722
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ −6.00000 −0.625543
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 16.0000 1.59206 0.796030 0.605257i $$-0.206930\pi$$
0.796030 + 0.605257i $$0.206930\pi$$
$$102$$ 0 0
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 12.0000 1.14939 0.574696 0.818367i $$-0.305120\pi$$
0.574696 + 0.818367i $$0.305120\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 0 0
$$112$$ 2.00000 0.188982
$$113$$ −20.0000 −1.88144 −0.940721 0.339182i $$-0.889850\pi$$
−0.940721 + 0.339182i $$0.889850\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ −16.0000 −1.46672
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 1.00000 0.0877058
$$131$$ −10.0000 −0.873704 −0.436852 0.899533i $$-0.643907\pi$$
−0.436852 + 0.899533i $$0.643907\pi$$
$$132$$ 0 0
$$133$$ −12.0000 −1.04053
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 8.00000 0.685994
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −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
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ 8.00000 0.662085
$$147$$ 0 0
$$148$$ −2.00000 −0.164399
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 6.00000 0.486664
$$153$$ 0 0
$$154$$ 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
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ −12.0000 −0.945732
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 4.00000 0.309529 0.154765 0.987951i $$-0.450538\pi$$
0.154765 + 0.987951i $$0.450538\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 8.00000 0.613572
$$171$$ 0 0
$$172$$ −4.00000 −0.304997
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ −14.0000 −1.04934
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 2.00000 0.148250
$$183$$ 0 0
$$184$$ 6.00000 0.442326
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ 32.0000 2.34007
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 6.00000 0.435286
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 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 $$0.278427\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −16.0000 −1.12576
$$203$$ 8.00000 0.561490
$$204$$ 0 0
$$205$$ 2.00000 0.139686
$$206$$ 12.0000 0.836080
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 24.0000 1.66011
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −12.0000 −0.812743
$$219$$ 0 0
$$220$$ −4.00000 −0.269680
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 20.0000 1.33038
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 6.00000 0.395628
$$231$$ 0 0
$$232$$ −4.00000 −0.262613
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 16.0000 1.03713
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 6.00000 0.381771
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ 0 0
$$253$$ 24.0000 1.50887
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ −1.00000 −0.0620174
$$261$$ 0 0
$$262$$ 10.0000 0.617802
$$263$$ −2.00000 −0.123325 −0.0616626 0.998097i $$-0.519640\pi$$
−0.0616626 + 0.998097i $$0.519640\pi$$
$$264$$ 0 0
$$265$$ 10.0000 0.614295
$$266$$ 12.0000 0.735767
$$267$$ 0 0
$$268$$ 12.0000 0.733017
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −8.00000 −0.485071
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −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
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 4.00000 0.236113
$$288$$ 0 0
$$289$$ 47.0000 2.76471
$$290$$ −4.00000 −0.234888
$$291$$ 0 0
$$292$$ −8.00000 −0.468165
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ −10.0000 −0.572598
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ −8.00000 −0.455842
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 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
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 0 0
$$319$$ −16.0000 −0.895828
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 12.0000 0.668734
$$323$$ 48.0000 2.67079
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ −2.00000 −0.110432
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ −4.00000 −0.218870
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ −8.00000 −0.433861
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ 24.0000 1.28839 0.644194 0.764862i $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ 0 0
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ −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 $$0.378476\pi$$
$$354$$ 0 0
$$355$$ 8.00000 0.424596
$$356$$ 14.0000 0.741999
$$357$$ 0 0
$$358$$ −10.0000 −0.528516
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ −8.00000 −0.418739
$$366$$ 0 0
$$367$$ −36.0000 −1.87918 −0.939592 0.342296i $$-0.888796\pi$$
−0.939592 + 0.342296i $$0.888796\pi$$
$$368$$ −6.00000 −0.312772
$$369$$ 0 0
$$370$$ 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
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −18.0000 −0.924598 −0.462299 0.886724i $$-0.652975\pi$$
−0.462299 + 0.886724i $$0.652975\pi$$
$$380$$ −6.00000 −0.307794
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ −8.00000 −0.407718
$$386$$ 4.00000 0.203595
$$387$$ 0 0
$$388$$ −16.0000 −0.812277
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ 3.00000 0.151523
$$393$$ 0 0
$$394$$ −18.0000 −0.906827
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 16.0000 0.796030
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ −2.00000 −0.0987730
$$411$$ 0 0
$$412$$ −12.0000 −0.591198
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −24.0000 −1.17388
$$419$$ −10.0000 −0.488532 −0.244266 0.969708i $$-0.578547\pi$$
−0.244266 + 0.969708i $$0.578547\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 0 0
$$424$$ −10.0000 −0.485643
$$425$$ −8.00000 −0.388057
$$426$$ 0 0
$$427$$ −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 $$0.5\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 12.0000 0.574696
$$437$$ 36.0000 1.72211
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 4.00000 0.190693
$$441$$ 0 0
$$442$$ −8.00000 −0.380521
$$443$$ −16.0000 −0.760183 −0.380091 0.924949i $$-0.624107\pi$$
−0.380091 + 0.924949i $$0.624107\pi$$
$$444$$ 0 0
$$445$$ 14.0000 0.663664
$$446$$ 2.00000 0.0947027
$$447$$ 0 0
$$448$$ 2.00000 0.0944911
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ −20.0000 −0.940721
$$453$$ 0 0
$$454$$ −4.00000 −0.187729
$$455$$ −2.00000 −0.0937614
$$456$$ 0 0
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ −4.00000 −0.186908
$$459$$ 0 0
$$460$$ −6.00000 −0.279751
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ −26.0000 −1.20832 −0.604161 0.796862i $$-0.706492\pi$$
−0.604161 + 0.796862i $$0.706492\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ 0 0
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.00000 0.184115
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ −6.00000 −0.275299
$$476$$ −16.0000 −0.733359
$$477$$ 0 0
$$478$$ 16.0000 0.731823
$$479$$ −32.0000 −1.46212 −0.731059 0.682315i $$-0.760973\pi$$
−0.731059 + 0.682315i $$0.760973\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −16.0000 −0.726523
$$486$$ 0 0
$$487$$ 26.0000 1.17817 0.589086 0.808070i $$-0.299488\pi$$
0.589086 + 0.808070i $$0.299488\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 0 0
$$490$$ 3.00000 0.135526
$$491$$ −42.0000 −1.89543 −0.947717 0.319113i $$-0.896615\pi$$
−0.947717 + 0.319113i $$0.896615\pi$$
$$492$$ 0 0
$$493$$ −32.0000 −1.44121
$$494$$ −6.00000 −0.269953
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 16.0000 0.717698
$$498$$ 0 0
$$499$$ 38.0000 1.70111 0.850557 0.525883i $$-0.176265\pi$$
0.850557 + 0.525883i $$0.176265\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ 6.00000 0.267793
$$503$$ −10.0000 −0.445878 −0.222939 0.974832i $$-0.571565\pi$$
−0.222939 + 0.974832i $$0.571565\pi$$
$$504$$ 0 0
$$505$$ 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 $$0.630614\pi$$
$$510$$ 0 0
$$511$$ −16.0000 −0.707798
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 12.0000 0.529297
$$515$$ −12.0000 −0.528783
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 4.00000 0.175750
$$519$$ 0 0
$$520$$ 1.00000 0.0438529
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ 36.0000 1.57417 0.787085 0.616844i $$-0.211589\pi$$
0.787085 + 0.616844i $$0.211589\pi$$
$$524$$ −10.0000 −0.436852
$$525$$ 0 0
$$526$$ 2.00000 0.0872041
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −10.0000 −0.434372
$$531$$ 0 0
$$532$$ −12.0000 −0.520266
$$533$$ −2.00000 −0.0866296
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ −24.0000 −1.03471
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ −8.00000 −0.343947 −0.171973 0.985102i $$-0.555014\pi$$
−0.171973 + 0.985102i $$0.555014\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ 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
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −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
$$559$$ 4.00000 0.169182
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ −20.0000 −0.841406
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ −4.00000 −0.166957
$$575$$ −6.00000 −0.250217
$$576$$ 0 0
$$577$$ 12.0000 0.499567 0.249783 0.968302i $$-0.419641\pi$$
0.249783 + 0.968302i $$0.419641\pi$$
$$578$$ −47.0000 −1.95494
$$579$$ 0 0
$$580$$ 4.00000 0.166091
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ −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 $$0.766569\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 4.00000 0.164677
$$591$$ 0 0
$$592$$ −2.00000 −0.0821995
$$593$$ −22.0000 −0.903432 −0.451716 0.892162i $$-0.649188\pi$$
−0.451716 + 0.892162i $$0.649188\pi$$
$$594$$ 0 0
$$595$$ −16.0000 −0.655936
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ −6.00000 −0.245358
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 8.00000 0.326056
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 5.00000 0.203279
$$606$$ 0 0
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ 6.00000 0.243332
$$609$$ 0 0
$$610$$ 10.0000 0.404888
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −10.0000 −0.403896 −0.201948 0.979396i $$-0.564727\pi$$
−0.201948 + 0.979396i $$0.564727\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 8.00000 0.322329
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 24.0000 0.962312
$$623$$ 28.0000 1.12180
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −6.00000 −0.239808
$$627$$ 0 0
$$628$$ 22.0000 0.877896
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ 12.0000 0.477712 0.238856 0.971055i $$-0.423228\pi$$
0.238856 + 0.971055i $$0.423228\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 0 0
$$634$$ −30.0000 −1.19145
$$635$$ 4.00000 0.158735
$$636$$ 0 0
$$637$$ 3.00000 0.118864
$$638$$ 16.0000 0.633446
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ 44.0000 1.73519 0.867595 0.497271i $$-0.165665\pi$$
0.867595 + 0.497271i $$0.165665\pi$$
$$644$$ −12.0000 −0.472866
$$645$$ 0 0
$$646$$ −48.0000 −1.88853
$$647$$ 30.0000 1.17942 0.589711 0.807614i $$-0.299242\pi$$
0.589711 + 0.807614i $$0.299242\pi$$
$$648$$ 0 0
$$649$$ 16.0000 0.628055
$$650$$ 1.00000 0.0392232
$$651$$ 0 0
$$652$$ −16.0000 −0.626608
$$653$$ −42.0000 −1.64359 −0.821794 0.569785i $$-0.807026\pi$$
−0.821794 + 0.569785i $$0.807026\pi$$
$$654$$ 0 0
$$655$$ −10.0000 −0.390732
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −2.00000 −0.0779089 −0.0389545 0.999241i $$-0.512403\pi$$
−0.0389545 + 0.999241i $$0.512403\pi$$
$$660$$ 0 0
$$661$$ 48.0000 1.86698 0.933492 0.358599i $$-0.116745\pi$$
0.933492 + 0.358599i $$0.116745\pi$$
$$662$$ −10.0000 −0.388661
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ −12.0000 −0.465340
$$666$$ 0 0
$$667$$ −24.0000 −0.929284
$$668$$ 4.00000 0.154765
$$669$$ 0 0
$$670$$ −12.0000 −0.463600
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ 26.0000 1.00223 0.501113 0.865382i $$-0.332924\pi$$
0.501113 + 0.865382i $$0.332924\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 30.0000 1.15299 0.576497 0.817099i $$-0.304419\pi$$
0.576497 + 0.817099i $$0.304419\pi$$
$$678$$ 0 0
$$679$$ −32.0000 −1.22805
$$680$$ 8.00000 0.306786
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 44.0000 1.68361 0.841807 0.539779i $$-0.181492\pi$$
0.841807 + 0.539779i $$0.181492\pi$$
$$684$$ 0 0
$$685$$ 6.00000 0.229248
$$686$$ 20.0000 0.763604
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −10.0000 −0.380970
$$690$$ 0 0
$$691$$ −14.0000 −0.532585 −0.266293 0.963892i $$-0.585799\pi$$
−0.266293 + 0.963892i $$0.585799\pi$$
$$692$$ −22.0000 −0.836315
$$693$$ 0 0
$$694$$ −24.0000 −0.911028
$$695$$ −8.00000 −0.303457
$$696$$ 0 0
$$697$$ −16.0000 −0.606043
$$698$$ 28.0000 1.05982
$$699$$ 0 0
$$700$$ 2.00000 0.0755929
$$701$$ −32.0000 −1.20862 −0.604312 0.796748i $$-0.706552\pi$$
−0.604312 + 0.796748i $$0.706552\pi$$
$$702$$ 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
$$708$$ 0 0
$$709$$ −4.00000 −0.150223 −0.0751116 0.997175i $$-0.523931\pi$$
−0.0751116 + 0.997175i $$0.523931\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 0 0
$$712$$ −14.0000 −0.524672
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ 10.0000 0.373718
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ −17.0000 −0.632674
$$723$$ 0 0
$$724$$ 10.0000 0.371647
$$725$$ 4.00000 0.148556
$$726$$ 0 0
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ 0 0
$$730$$ 8.00000 0.296093
$$731$$ 32.0000 1.18356
$$732$$ 0 0
$$733$$ 38.0000 1.40356 0.701781 0.712393i $$-0.252388\pi$$
0.701781 + 0.712393i $$0.252388\pi$$
$$734$$ 36.0000 1.32878
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ −48.0000 −1.76810
$$738$$ 0 0
$$739$$ 30.0000 1.10357 0.551784 0.833987i $$-0.313947\pi$$
0.551784 + 0.833987i $$0.313947\pi$$
$$740$$ −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
$$747$$ 0 0
$$748$$ 32.0000 1.17004
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −6.00000 −0.218074 −0.109037 0.994038i $$-0.534777\pi$$
−0.109037 + 0.994038i $$0.534777\pi$$
$$758$$ 18.0000 0.653789
$$759$$ 0 0
$$760$$ 6.00000 0.217643
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ 24.0000 0.868858
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 4.00000 0.144432
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 8.00000 0.288300
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ −38.0000 −1.36677 −0.683383 0.730061i $$-0.739492\pi$$
−0.683383 + 0.730061i $$0.739492\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 16.0000 0.574367
$$777$$ 0 0
$$778$$ 8.00000 0.286814
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ −48.0000 −1.71648
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 22.0000 0.785214
$$786$$ 0 0
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ 18.0000 0.641223
$$789$$ 0 0
$$790$$ −8.00000 −0.284627
$$791$$ −40.0000 −1.42224
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 10.0000 0.354218 0.177109 0.984191i $$-0.443325\pi$$
0.177109 + 0.984191i $$0.443325\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 30.0000 1.05934
$$803$$ 32.0000 1.12926
$$804$$ 0 0
$$805$$ −12.0000 −0.422944
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −16.0000 −0.562878
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ −38.0000 −1.33436 −0.667180 0.744896i $$-0.732499\pi$$
−0.667180 + 0.744896i $$0.732499\pi$$
$$812$$ 8.00000 0.280745
$$813$$ 0 0
$$814$$ −8.00000 −0.280400
$$815$$ −16.0000 −0.560456
$$816$$ 0 0
$$817$$ 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 $$0.444169\pi$$
$$822$$ 0 0
$$823$$ −52.0000 −1.81261 −0.906303 0.422628i $$-0.861108\pi$$
−0.906303 + 0.422628i $$0.861108\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 12.0000 0.416526
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ 24.0000 0.831551
$$834$$ 0 0
$$835$$ 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 $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 8.00000 0.275698
$$843$$ 0 0
$$844$$ −20.0000 −0.688428
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 10.0000 0.343401
$$849$$ 0 0
$$850$$ 8.00000 0.274398
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ 20.0000 0.684386
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −4.00000 −0.136637 −0.0683187 0.997664i $$-0.521763\pi$$
−0.0683187 + 0.997664i $$0.521763\pi$$
$$858$$ 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
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 12.0000 0.408485 0.204242 0.978920i $$-0.434527\pi$$
0.204242 + 0.978920i $$0.434527\pi$$
$$864$$ 0 0
$$865$$ −22.0000 −0.748022
$$866$$ 2.00000 0.0679628
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −32.0000 −1.08553
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ −12.0000 −0.406371
$$873$$ 0 0
$$874$$ −36.0000 −1.21772
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ 32.0000 1.07995
$$879$$ 0 0
$$880$$ −4.00000 −0.134840
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 0 0
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ 16.0000 0.537531
$$887$$ 2.00000 0.0671534 0.0335767 0.999436i $$-0.489310\pi$$
0.0335767 + 0.999436i $$0.489310\pi$$
$$888$$ 0 0
$$889$$ 8.00000 0.268311
$$890$$ −14.0000 −0.469281
$$891$$ 0 0
$$892$$ −2.00000 −0.0669650
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 10.0000 0.334263
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −80.0000 −2.66519
$$902$$ 8.00000 0.266371
$$903$$ 0 0
$$904$$ 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
$$909$$ 0 0
$$910$$ 2.00000 0.0662994
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ 0 0
$$913$$ 48.0000 1.58857
$$914$$ −8.00000 −0.264616
$$915$$ 0 0
$$916$$ 4.00000 0.132164
$$917$$ −20.0000 −0.660458
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 6.00000 0.197814
$$921$$ 0 0
$$922$$ −6.00000 −0.197599
$$923$$ −8.00000 −0.263323
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 26.0000 0.854413
$$927$$ 0 0
$$928$$ −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
$$933$$ 0 0
$$934$$ 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
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 34.0000 1.10837 0.554184 0.832394i $$-0.313030\pi$$
0.554184 + 0.832394i $$0.313030\pi$$
$$942$$ 0 0
$$943$$ −12.0000 −0.390774
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ 6.00000 0.194666
$$951$$ 0 0
$$952$$ 16.0000 0.518563
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 32.0000 1.03387
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −2.00000 −0.0644826
$$963$$ 0 0
$$964$$ 2.00000 0.0644157
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ −14.0000 −0.450210 −0.225105 0.974335i $$-0.572272\pi$$
−0.225105 + 0.974335i $$0.572272\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 0 0
$$970$$ 16.0000 0.513729
$$971$$ −38.0000 −1.21948 −0.609739 0.792602i $$-0.708726\pi$$
−0.609739 + 0.792602i $$0.708726\pi$$
$$972$$ 0 0
$$973$$ −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 $$0.659324\pi$$
$$978$$ 0 0
$$979$$ −56.0000 −1.78977
$$980$$ −3.00000 −0.0958315
$$981$$ 0 0
$$982$$ 42.0000 1.34027
$$983$$ 52.0000 1.65854 0.829271 0.558846i $$-0.188756\pi$$
0.829271 + 0.558846i $$0.188756\pi$$
$$984$$ 0 0
$$985$$ 18.0000 0.573528
$$986$$ 32.0000 1.01909
$$987$$ 0 0
$$988$$ 6.00000 0.190885
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −16.0000 −0.507489
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ −38.0000 −1.20287
$$999$$ 0 0
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 1170.2.a.e.1.1 1
3.2 odd 2 390.2.a.e.1.1 1
4.3 odd 2 9360.2.a.bh.1.1 1
5.2 odd 4 5850.2.e.i.5149.1 2
5.3 odd 4 5850.2.e.i.5149.2 2
5.4 even 2 5850.2.a.bi.1.1 1
12.11 even 2 3120.2.a.o.1.1 1
15.2 even 4 1950.2.e.f.1249.2 2
15.8 even 4 1950.2.e.f.1249.1 2
15.14 odd 2 1950.2.a.h.1.1 1
39.5 even 4 5070.2.b.e.1351.1 2
39.8 even 4 5070.2.b.e.1351.2 2
39.38 odd 2 5070.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.e.1.1 1 3.2 odd 2
1170.2.a.e.1.1 1 1.1 even 1 trivial
1950.2.a.h.1.1 1 15.14 odd 2
1950.2.e.f.1249.1 2 15.8 even 4
1950.2.e.f.1249.2 2 15.2 even 4
3120.2.a.o.1.1 1 12.11 even 2
5070.2.a.e.1.1 1 39.38 odd 2
5070.2.b.e.1351.1 2 39.5 even 4
5070.2.b.e.1351.2 2 39.8 even 4
5850.2.a.bi.1.1 1 5.4 even 2
5850.2.e.i.5149.1 2 5.2 odd 4
5850.2.e.i.5149.2 2 5.3 odd 4
9360.2.a.bh.1.1 1 4.3 odd 2