# Properties

 Label 1089.2.a.h.1.1 Level $1089$ Weight $2$ Character 1089.1 Self dual yes Analytic conductor $8.696$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1089.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.69570878012$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 99) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1089.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} -4.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} -4.00000 q^{10} +2.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{19} +4.00000 q^{20} +4.00000 q^{23} +11.0000 q^{25} +2.00000 q^{26} -2.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} +5.00000 q^{32} -2.00000 q^{34} -8.00000 q^{35} -6.00000 q^{37} +6.00000 q^{38} +12.0000 q^{40} +10.0000 q^{41} -6.00000 q^{43} +4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} +11.0000 q^{50} -2.00000 q^{52} -6.00000 q^{56} +6.00000 q^{58} +4.00000 q^{59} +6.00000 q^{61} +4.00000 q^{62} +7.00000 q^{64} -8.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} -8.00000 q^{70} +2.00000 q^{73} -6.00000 q^{74} -6.00000 q^{76} +10.0000 q^{79} +4.00000 q^{80} +10.0000 q^{82} -12.0000 q^{83} +8.00000 q^{85} -6.00000 q^{86} +4.00000 q^{91} -4.00000 q^{92} -8.00000 q^{94} -24.0000 q^{95} +2.00000 q^{97} -3.00000 q^{98} +O(q^{100})$$

## 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 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ −4.00000 −1.26491
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 4.00000 0.894427
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ −8.00000 −1.35225
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 6.00000 0.973329
$$39$$ 0 0
$$40$$ 12.0000 1.89737
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 11.0000 1.55563
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −6.00000 −0.801784
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ −8.00000 −0.992278
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ −8.00000 −0.956183
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 4.00000 0.447214
$$81$$ 0 0
$$82$$ 10.0000 1.10432
$$83$$ −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
$$86$$ −6.00000 −0.646997
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ −24.0000 −2.46235
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 0 0
$$100$$ −11.0000 −1.10000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 0 0
$$115$$ −16.0000 −1.49201
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 6.00000 0.543214
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ −24.0000 −2.14663
$$126$$ 0 0
$$127$$ 10.0000 0.887357 0.443678 0.896186i $$-0.353673\pi$$
0.443678 + 0.896186i $$0.353673\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ −8.00000 −0.701646
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 12.0000 1.04053
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 4.00000 0.341743 0.170872 0.985293i $$-0.445342\pi$$
0.170872 + 0.985293i $$0.445342\pi$$
$$138$$ 0 0
$$139$$ −10.0000 −0.848189 −0.424094 0.905618i $$-0.639408\pi$$
−0.424094 + 0.905618i $$0.639408\pi$$
$$140$$ 8.00000 0.676123
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −24.0000 −1.99309
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ 0 0
$$151$$ −14.0000 −1.13930 −0.569652 0.821886i $$-0.692922\pi$$
−0.569652 + 0.821886i $$0.692922\pi$$
$$152$$ −18.0000 −1.45999
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −16.0000 −1.28515
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 10.0000 0.795557
$$159$$ 0 0
$$160$$ −20.0000 −1.58114
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 8.00000 0.613572
$$171$$ 0 0
$$172$$ 6.00000 0.457496
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 22.0000 1.66304
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ 24.0000 1.76452
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ −24.0000 −1.74114
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ −33.0000 −2.33345
$$201$$ 0 0
$$202$$ 14.0000 0.985037
$$203$$ 12.0000 0.842235
$$204$$ 0 0
$$205$$ −40.0000 −2.79372
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ −2.00000 −0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 6.00000 0.413057 0.206529 0.978441i $$-0.433783\pi$$
0.206529 + 0.978441i $$0.433783\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 24.0000 1.63679
$$216$$ 0 0
$$217$$ 8.00000 0.543075
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 10.0000 0.668153
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ −16.0000 −1.05501
$$231$$ 0 0
$$232$$ −18.0000 −1.18176
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 32.0000 2.08745
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ −4.00000 −0.259281
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −6.00000 −0.384111
$$245$$ 12.0000 0.766652
$$246$$ 0 0
$$247$$ 12.0000 0.763542
$$248$$ −12.0000 −0.762001
$$249$$ 0 0
$$250$$ −24.0000 −1.51789
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 10.0000 0.627456
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 8.00000 0.499026 0.249513 0.968371i $$-0.419729\pi$$
0.249513 + 0.968371i $$0.419729\pi$$
$$258$$ 0 0
$$259$$ −12.0000 −0.745644
$$260$$ 8.00000 0.496139
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ 32.0000 1.97320 0.986602 0.163144i $$-0.0521635\pi$$
0.986602 + 0.163144i $$0.0521635\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 12.0000 0.735767
$$267$$ 0 0
$$268$$ −8.00000 −0.488678
$$269$$ −16.0000 −0.975537 −0.487769 0.872973i $$-0.662189\pi$$
−0.487769 + 0.872973i $$0.662189\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 4.00000 0.241649
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −10.0000 −0.599760
$$279$$ 0 0
$$280$$ 24.0000 1.43427
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ 2.00000 0.118888 0.0594438 0.998232i $$-0.481067\pi$$
0.0594438 + 0.998232i $$0.481067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 20.0000 1.18056
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ −24.0000 −1.40933
$$291$$ 0 0
$$292$$ −2.00000 −0.117041
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ −16.0000 −0.931556
$$296$$ 18.0000 1.04623
$$297$$ 0 0
$$298$$ 2.00000 0.115857
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ −14.0000 −0.805609
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ −24.0000 −1.37424
$$306$$ 0 0
$$307$$ 22.0000 1.25561 0.627803 0.778372i $$-0.283954\pi$$
0.627803 + 0.778372i $$0.283954\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −16.0000 −0.908739
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −4.00000 −0.224662 −0.112331 0.993671i $$-0.535832\pi$$
−0.112331 + 0.993671i $$0.535832\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −28.0000 −1.56525
$$321$$ 0 0
$$322$$ 8.00000 0.445823
$$323$$ −12.0000 −0.667698
$$324$$ 0 0
$$325$$ 22.0000 1.22034
$$326$$ −20.0000 −1.10770
$$327$$ 0 0
$$328$$ −30.0000 −1.65647
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −32.0000 −1.74835
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ −8.00000 −0.433861
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 18.0000 0.970495
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 0 0
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 22.0000 1.17595
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 24.0000 1.27739 0.638696 0.769460i $$-0.279474\pi$$
0.638696 + 0.769460i $$0.279474\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −24.0000 −1.26844
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 10.0000 0.525588
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ −8.00000 −0.418739
$$366$$ 0 0
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 0 0
$$370$$ 24.0000 1.24770
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 24.0000 1.23771
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 24.0000 1.23117
$$381$$ 0 0
$$382$$ 16.0000 0.818631
$$383$$ −20.0000 −1.02195 −0.510976 0.859595i $$-0.670716\pi$$
−0.510976 + 0.859595i $$0.670716\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ 36.0000 1.82527 0.912636 0.408773i $$-0.134043\pi$$
0.912636 + 0.408773i $$0.134043\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 9.00000 0.454569
$$393$$ 0 0
$$394$$ −2.00000 −0.100759
$$395$$ −40.0000 −2.01262
$$396$$ 0 0
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 12.0000 0.601506
$$399$$ 0 0
$$400$$ −11.0000 −0.550000
$$401$$ 28.0000 1.39825 0.699127 0.714998i $$-0.253572\pi$$
0.699127 + 0.714998i $$0.253572\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ −14.0000 −0.696526
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ −40.0000 −1.97546
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 48.0000 2.35623
$$416$$ 10.0000 0.490290
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −32.0000 −1.56330 −0.781651 0.623716i $$-0.785622\pi$$
−0.781651 + 0.623716i $$0.785622\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 6.00000 0.292075
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −22.0000 −1.06716
$$426$$ 0 0
$$427$$ 12.0000 0.580721
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 24.0000 1.15738
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 8.00000 0.384012
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −4.00000 −0.190261
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 0 0
$$448$$ 14.0000 0.661438
$$449$$ −32.0000 −1.51017 −0.755087 0.655625i $$-0.772405\pi$$
−0.755087 + 0.655625i $$0.772405\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −12.0000 −0.564433
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ −16.0000 −0.750092
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 0 0
$$460$$ 16.0000 0.746004
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 32.0000 1.47605
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 66.0000 3.02829
$$476$$ 4.00000 0.183340
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 32.0000 1.46212 0.731059 0.682315i $$-0.239027\pi$$
0.731059 + 0.682315i $$0.239027\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 26.0000 1.18427
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −8.00000 −0.363261
$$486$$ 0 0
$$487$$ −4.00000 −0.181257 −0.0906287 0.995885i $$-0.528888\pi$$
−0.0906287 + 0.995885i $$0.528888\pi$$
$$488$$ −18.0000 −0.814822
$$489$$ 0 0
$$490$$ 12.0000 0.542105
$$491$$ 28.0000 1.26362 0.631811 0.775122i $$-0.282312\pi$$
0.631811 + 0.775122i $$0.282312\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 12.0000 0.539906
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −16.0000 −0.716258 −0.358129 0.933672i $$-0.616585\pi$$
−0.358129 + 0.933672i $$0.616585\pi$$
$$500$$ 24.0000 1.07331
$$501$$ 0 0
$$502$$ 8.00000 0.357057
$$503$$ 40.0000 1.78351 0.891756 0.452517i $$-0.149474\pi$$
0.891756 + 0.452517i $$0.149474\pi$$
$$504$$ 0 0
$$505$$ −56.0000 −2.49197
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −10.0000 −0.443678
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ 8.00000 0.352865
$$515$$ −32.0000 −1.41009
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −12.0000 −0.527250
$$519$$ 0 0
$$520$$ 24.0000 1.05247
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 0 0
$$523$$ −38.0000 −1.66162 −0.830812 0.556553i $$-0.812124\pi$$
−0.830812 + 0.556553i $$0.812124\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ −8.00000 −0.348485
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −12.0000 −0.520266
$$533$$ 20.0000 0.866296
$$534$$ 0 0
$$535$$ 48.0000 2.07522
$$536$$ −24.0000 −1.03664
$$537$$ 0 0
$$538$$ −16.0000 −0.689809
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ −2.00000 −0.0859074
$$543$$ 0 0
$$544$$ −10.0000 −0.428746
$$545$$ −8.00000 −0.342682
$$546$$ 0 0
$$547$$ −38.0000 −1.62476 −0.812381 0.583127i $$-0.801829\pi$$
−0.812381 + 0.583127i $$0.801829\pi$$
$$548$$ −4.00000 −0.170872
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 36.0000 1.53365
$$552$$ 0 0
$$553$$ 20.0000 0.850487
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ 10.0000 0.424094
$$557$$ −38.0000 −1.61011 −0.805056 0.593199i $$-0.797865\pi$$
−0.805056 + 0.593199i $$0.797865\pi$$
$$558$$ 0 0
$$559$$ −12.0000 −0.507546
$$560$$ 8.00000 0.338062
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ 28.0000 1.18006 0.590030 0.807382i $$-0.299116\pi$$
0.590030 + 0.807382i $$0.299116\pi$$
$$564$$ 0 0
$$565$$ −48.0000 −2.01938
$$566$$ 2.00000 0.0840663
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 34.0000 1.42286 0.711428 0.702759i $$-0.248049\pi$$
0.711428 + 0.702759i $$0.248049\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 20.0000 0.834784
$$575$$ 44.0000 1.83493
$$576$$ 0 0
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ 0 0
$$580$$ 24.0000 0.996546
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ −4.00000 −0.165098 −0.0825488 0.996587i $$-0.526306\pi$$
−0.0825488 + 0.996587i $$0.526306\pi$$
$$588$$ 0 0
$$589$$ 24.0000 0.988903
$$590$$ −16.0000 −0.658710
$$591$$ 0 0
$$592$$ 6.00000 0.246598
$$593$$ 26.0000 1.06769 0.533846 0.845582i $$-0.320746\pi$$
0.533846 + 0.845582i $$0.320746\pi$$
$$594$$ 0 0
$$595$$ 16.0000 0.655936
$$596$$ −2.00000 −0.0819232
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ −28.0000 −1.14405 −0.572024 0.820237i $$-0.693842\pi$$
−0.572024 + 0.820237i $$0.693842\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ −12.0000 −0.489083
$$603$$ 0 0
$$604$$ 14.0000 0.569652
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 10.0000 0.405887 0.202944 0.979190i $$-0.434949\pi$$
0.202944 + 0.979190i $$0.434949\pi$$
$$608$$ 30.0000 1.21666
$$609$$ 0 0
$$610$$ −24.0000 −0.971732
$$611$$ −16.0000 −0.647291
$$612$$ 0 0
$$613$$ 46.0000 1.85792 0.928961 0.370177i $$-0.120703\pi$$
0.928961 + 0.370177i $$0.120703\pi$$
$$614$$ 22.0000 0.887848
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.0000 −0.483102 −0.241551 0.970388i $$-0.577656\pi$$
−0.241551 + 0.970388i $$0.577656\pi$$
$$618$$ 0 0
$$619$$ 44.0000 1.76851 0.884255 0.467005i $$-0.154667\pi$$
0.884255 + 0.467005i $$0.154667\pi$$
$$620$$ 16.0000 0.642575
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ −34.0000 −1.35891
$$627$$ 0 0
$$628$$ 10.0000 0.399043
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 4.00000 0.159237 0.0796187 0.996825i $$-0.474630\pi$$
0.0796187 + 0.996825i $$0.474630\pi$$
$$632$$ −30.0000 −1.19334
$$633$$ 0 0
$$634$$ −4.00000 −0.158860
$$635$$ −40.0000 −1.58735
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 12.0000 0.474342
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ 28.0000 1.10079 0.550397 0.834903i $$-0.314476\pi$$
0.550397 + 0.834903i $$0.314476\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 22.0000 0.862911
$$651$$ 0 0
$$652$$ 20.0000 0.783260
$$653$$ −16.0000 −0.626128 −0.313064 0.949732i $$-0.601356\pi$$
−0.313064 + 0.949732i $$0.601356\pi$$
$$654$$ 0 0
$$655$$ −48.0000 −1.87552
$$656$$ −10.0000 −0.390434
$$657$$ 0 0
$$658$$ −16.0000 −0.623745
$$659$$ −44.0000 −1.71400 −0.856998 0.515319i $$-0.827673\pi$$
−0.856998 + 0.515319i $$0.827673\pi$$
$$660$$ 0 0
$$661$$ −50.0000 −1.94477 −0.972387 0.233373i $$-0.925024\pi$$
−0.972387 + 0.233373i $$0.925024\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ 36.0000 1.39707
$$665$$ −48.0000 −1.86136
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 0 0
$$669$$ 0 0
$$670$$ −32.0000 −1.23627
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ −24.0000 −0.920358
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −32.0000 −1.22445 −0.612223 0.790685i $$-0.709725\pi$$
−0.612223 + 0.790685i $$0.709725\pi$$
$$684$$ 0 0
$$685$$ −16.0000 −0.611329
$$686$$ −20.0000 −0.763604
$$687$$ 0 0
$$688$$ 6.00000 0.228748
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ 0 0
$$694$$ −20.0000 −0.759190
$$695$$ 40.0000 1.51729
$$696$$ 0 0
$$697$$ −20.0000 −0.757554
$$698$$ −18.0000 −0.681310
$$699$$ 0 0
$$700$$ −22.0000 −0.831522
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ −36.0000 −1.35777
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ 28.0000 1.05305
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 0 0
$$718$$ −8.00000 −0.298557
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 17.0000 0.632674
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ 66.0000 2.45118
$$726$$ 0 0
$$727$$ 12.0000 0.445055 0.222528 0.974926i $$-0.428569\pi$$
0.222528 + 0.974926i $$0.428569\pi$$
$$728$$ −12.0000 −0.444750
$$729$$ 0 0
$$730$$ −8.00000 −0.296093
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ 20.0000 0.737210
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 34.0000 1.25071 0.625355 0.780340i $$-0.284954\pi$$
0.625355 + 0.780340i $$0.284954\pi$$
$$740$$ −24.0000 −0.882258
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ 0 0
$$745$$ −8.00000 −0.293097
$$746$$ −34.0000 −1.24483
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 56.0000 2.03805
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 4.00000 0.145287
$$759$$ 0 0
$$760$$ 72.0000 2.61171
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 0 0
$$763$$ 4.00000 0.144810
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −14.0000 −0.503871
$$773$$ −24.0000 −0.863220 −0.431610 0.902060i $$-0.642054\pi$$
−0.431610 + 0.902060i $$0.642054\pi$$
$$774$$ 0 0
$$775$$ 44.0000 1.58053
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ 36.0000 1.29066
$$779$$ 60.0000 2.14972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −8.00000 −0.286079
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 40.0000 1.42766
$$786$$ 0 0
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 0 0
$$790$$ −40.0000 −1.42314
$$791$$ 24.0000 0.853342
$$792$$ 0 0
$$793$$ 12.0000 0.426132
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ 28.0000 0.991811 0.495905 0.868377i $$-0.334836\pi$$
0.495905 + 0.868377i $$0.334836\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 55.0000 1.94454
$$801$$ 0 0
$$802$$ 28.0000 0.988714
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −32.0000 −1.12785
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ −42.0000 −1.47755
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ −34.0000 −1.19390 −0.596951 0.802278i $$-0.703621\pi$$
−0.596951 + 0.802278i $$0.703621\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 80.0000 2.80228
$$816$$ 0 0
$$817$$ −36.0000 −1.25948
$$818$$ −6.00000 −0.209785
$$819$$ 0 0
$$820$$ 40.0000 1.39686
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ 0 0
$$823$$ −48.0000 −1.67317 −0.836587 0.547833i $$-0.815453\pi$$
−0.836587 + 0.547833i $$0.815453\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 48.0000 1.66610
$$831$$ 0 0
$$832$$ 14.0000 0.485363
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −32.0000 −1.10542
$$839$$ 20.0000 0.690477 0.345238 0.938515i $$-0.387798\pi$$
0.345238 + 0.938515i $$0.387798\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 34.0000 1.17172
$$843$$ 0 0
$$844$$ −6.00000 −0.206529
$$845$$ 36.0000 1.23844
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ −22.0000 −0.754594
$$851$$ −24.0000 −0.822709
$$852$$ 0 0
$$853$$ −50.0000 −1.71197 −0.855984 0.517003i $$-0.827048\pi$$
−0.855984 + 0.517003i $$0.827048\pi$$
$$854$$ 12.0000 0.410632
$$855$$ 0 0
$$856$$ 36.0000 1.23045
$$857$$ 14.0000 0.478231 0.239115 0.970991i $$-0.423143\pi$$
0.239115 + 0.970991i $$0.423143\pi$$
$$858$$ 0 0
$$859$$ −36.0000 −1.22830 −0.614152 0.789188i $$-0.710502\pi$$
−0.614152 + 0.789188i $$0.710502\pi$$
$$860$$ −24.0000 −0.818393
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 36.0000 1.22545 0.612727 0.790295i $$-0.290072\pi$$
0.612727 + 0.790295i $$0.290072\pi$$
$$864$$ 0 0
$$865$$ −24.0000 −0.816024
$$866$$ −2.00000 −0.0679628
$$867$$ 0 0
$$868$$ −8.00000 −0.271538
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 16.0000 0.542139
$$872$$ −6.00000 −0.203186
$$873$$ 0 0
$$874$$ 24.0000 0.811812
$$875$$ −48.0000 −1.62270
$$876$$ 0 0
$$877$$ −42.0000 −1.41824 −0.709120 0.705088i $$-0.750907\pi$$
−0.709120 + 0.705088i $$0.750907\pi$$
$$878$$ −10.0000 −0.337484
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −20.0000 −0.673817 −0.336909 0.941537i $$-0.609381\pi$$
−0.336909 + 0.941537i $$0.609381\pi$$
$$882$$ 0 0
$$883$$ −32.0000 −1.07689 −0.538443 0.842662i $$-0.680987\pi$$
−0.538443 + 0.842662i $$0.680987\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ −48.0000 −1.60626
$$894$$ 0 0
$$895$$ 96.0000 3.20893
$$896$$ −6.00000 −0.200446
$$897$$ 0 0
$$898$$ −32.0000 −1.06785
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −36.0000 −1.19734
$$905$$ −40.0000 −1.32964
$$906$$ 0 0
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ −16.0000 −0.530395
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 24.0000 0.792550
$$918$$ 0 0
$$919$$ 14.0000 0.461817 0.230909 0.972975i $$-0.425830\pi$$
0.230909 + 0.972975i $$0.425830\pi$$
$$920$$ 48.0000 1.58251
$$921$$ 0 0
$$922$$ −6.00000 −0.197599
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −66.0000 −2.17007
$$926$$ 16.0000 0.525793
$$927$$ 0 0
$$928$$ 30.0000 0.984798
$$929$$ −12.0000 −0.393707 −0.196854 0.980433i $$-0.563072\pi$$
−0.196854 + 0.980433i $$0.563072\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −2.00000 −0.0653372 −0.0326686 0.999466i $$-0.510401\pi$$
−0.0326686 + 0.999466i $$0.510401\pi$$
$$938$$ 16.0000 0.522419
$$939$$ 0 0
$$940$$ −32.0000 −1.04372
$$941$$ 30.0000 0.977972 0.488986 0.872292i $$-0.337367\pi$$
0.488986 + 0.872292i $$0.337367\pi$$
$$942$$ 0 0
$$943$$ 40.0000 1.30258
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −48.0000 −1.55979 −0.779895 0.625910i $$-0.784728\pi$$
−0.779895 + 0.625910i $$0.784728\pi$$
$$948$$ 0 0
$$949$$ 4.00000 0.129845
$$950$$ 66.0000 2.14132
$$951$$ 0 0
$$952$$ 12.0000 0.388922
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ 0 0
$$955$$ −64.0000 −2.07099
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 32.0000 1.03387
$$959$$ 8.00000 0.258333
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −12.0000 −0.386896
$$963$$ 0 0
$$964$$ −26.0000 −0.837404
$$965$$ −56.0000 −1.80270
$$966$$ 0 0
$$967$$ 50.0000 1.60789 0.803946 0.594703i $$-0.202730\pi$$
0.803946 + 0.594703i $$0.202730\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −8.00000 −0.256865
$$971$$ 4.00000 0.128366 0.0641831 0.997938i $$-0.479556\pi$$
0.0641831 + 0.997938i $$0.479556\pi$$
$$972$$ 0 0
$$973$$ −20.0000 −0.641171
$$974$$ −4.00000 −0.128168
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ −36.0000 −1.15174 −0.575871 0.817541i $$-0.695337\pi$$
−0.575871 + 0.817541i $$0.695337\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −12.0000 −0.383326
$$981$$ 0 0
$$982$$ 28.0000 0.893516
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 0 0
$$985$$ 8.00000 0.254901
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ −12.0000 −0.381771
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −44.0000 −1.39771 −0.698853 0.715265i $$-0.746306\pi$$
−0.698853 + 0.715265i $$0.746306\pi$$
$$992$$ 20.0000 0.635001
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −48.0000 −1.52170
$$996$$ 0 0
$$997$$ −50.0000 −1.58352 −0.791758 0.610835i $$-0.790834\pi$$
−0.791758 + 0.610835i $$0.790834\pi$$
$$998$$ −16.0000 −0.506471
$$999$$ 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 1089.2.a.h.1.1 1
3.2 odd 2 1089.2.a.d.1.1 1
11.10 odd 2 99.2.a.a.1.1 1
33.32 even 2 99.2.a.c.1.1 yes 1
44.43 even 2 1584.2.a.b.1.1 1
55.32 even 4 2475.2.c.b.199.1 2
55.43 even 4 2475.2.c.b.199.2 2
55.54 odd 2 2475.2.a.j.1.1 1
77.76 even 2 4851.2.a.g.1.1 1
88.21 odd 2 6336.2.a.cl.1.1 1
88.43 even 2 6336.2.a.cm.1.1 1
99.32 even 6 891.2.e.c.298.1 2
99.43 odd 6 891.2.e.j.595.1 2
99.65 even 6 891.2.e.c.595.1 2
99.76 odd 6 891.2.e.j.298.1 2
132.131 odd 2 1584.2.a.r.1.1 1
165.32 odd 4 2475.2.c.g.199.2 2
165.98 odd 4 2475.2.c.g.199.1 2
165.164 even 2 2475.2.a.c.1.1 1
231.230 odd 2 4851.2.a.o.1.1 1
264.131 odd 2 6336.2.a.f.1.1 1
264.197 even 2 6336.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.a.a.1.1 1 11.10 odd 2
99.2.a.c.1.1 yes 1 33.32 even 2
891.2.e.c.298.1 2 99.32 even 6
891.2.e.c.595.1 2 99.65 even 6
891.2.e.j.298.1 2 99.76 odd 6
891.2.e.j.595.1 2 99.43 odd 6
1089.2.a.d.1.1 1 3.2 odd 2
1089.2.a.h.1.1 1 1.1 even 1 trivial
1584.2.a.b.1.1 1 44.43 even 2
1584.2.a.r.1.1 1 132.131 odd 2
2475.2.a.c.1.1 1 165.164 even 2
2475.2.a.j.1.1 1 55.54 odd 2
2475.2.c.b.199.1 2 55.32 even 4
2475.2.c.b.199.2 2 55.43 even 4
2475.2.c.g.199.1 2 165.98 odd 4
2475.2.c.g.199.2 2 165.32 odd 4
4851.2.a.g.1.1 1 77.76 even 2
4851.2.a.o.1.1 1 231.230 odd 2
6336.2.a.b.1.1 1 264.197 even 2
6336.2.a.f.1.1 1 264.131 odd 2
6336.2.a.cl.1.1 1 88.21 odd 2
6336.2.a.cm.1.1 1 88.43 even 2