# Properties

 Label 4410.2.a.b.1.1 Level $4410$ Weight $2$ Character 4410.1 Self dual yes Analytic conductor $35.214$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4410.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} -4.00000 q^{11} +6.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -6.00000 q^{26} -6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} -10.0000 q^{37} +1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} -4.00000 q^{44} +8.00000 q^{47} -1.00000 q^{50} +6.00000 q^{52} +2.00000 q^{53} +4.00000 q^{55} +6.00000 q^{58} -8.00000 q^{59} +14.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} -12.0000 q^{67} +2.00000 q^{68} +16.0000 q^{71} -2.00000 q^{73} +10.0000 q^{74} -8.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} +8.00000 q^{83} -2.00000 q^{85} -4.00000 q^{86} +4.00000 q^{88} +10.0000 q^{89} -8.00000 q^{94} -2.00000 q^{97} +O(q^{100})$$

## 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
$$7$$ 0 0
$$8$$ −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
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −6.00000 −1.17670
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ 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 $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 6.00000 0.832050
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ −2.00000 −0.220863
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 8.00000 0.736460
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −14.0000 −1.26750
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 6.00000 0.526235
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −16.0000 −1.34269
$$143$$ −24.0000 −2.00698
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ −10.0000 −0.821995
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 2.00000 0.153393
$$171$$ 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
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ −10.0000 −0.749532
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 10.0000 0.735215
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −14.0000 −0.997459 −0.498729 0.866758i $$-0.666200\pi$$
−0.498729 + 0.866758i $$0.666200\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 16.0000 1.11477
$$207$$ 0 0
$$208$$ 6.00000 0.416025
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −6.00000 −0.406371
$$219$$ 0 0
$$220$$ 4.00000 0.269680
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ −8.00000 −0.520756
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 8.00000 0.508001
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −6.00000 −0.372104
$$261$$ 0 0
$$262$$ 16.0000 0.988483
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 16.0000 0.959616
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ −32.0000 −1.90220 −0.951101 0.308879i $$-0.900046\pi$$
−0.951101 + 0.308879i $$0.900046\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 24.0000 1.41915
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ −6.00000 −0.352332
$$291$$ 0 0
$$292$$ −2.00000 −0.117041
$$293$$ 10.0000 0.584206 0.292103 0.956387i $$-0.405645\pi$$
0.292103 + 0.956387i $$0.405645\pi$$
$$294$$ 0 0
$$295$$ 8.00000 0.465778
$$296$$ 10.0000 0.581238
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −14.0000 −0.801638
$$306$$ 0 0
$$307$$ 8.00000 0.456584 0.228292 0.973593i $$-0.426686\pi$$
0.228292 + 0.973593i $$0.426686\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −8.00000 −0.454369
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 10.0000 0.564333
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 6.00000 0.332820
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ −2.00000 −0.110432
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ 0 0
$$340$$ −2.00000 −0.108465
$$341$$ 32.0000 1.73290
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ −16.0000 −0.849192
$$356$$ 10.0000 0.529999
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ −14.0000 −0.735824
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 0 0
$$367$$ −16.0000 −0.835193 −0.417597 0.908633i $$-0.637127\pi$$
−0.417597 + 0.908633i $$0.637127\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ −10.0000 −0.519875
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ −8.00000 −0.412568
$$377$$ −36.0000 −1.85409
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 24.0000 1.22795
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 14.0000 0.705310
$$395$$ 8.00000 0.402524
$$396$$ 0 0
$$397$$ −10.0000 −0.501886 −0.250943 0.968002i $$-0.580741\pi$$
−0.250943 + 0.968002i $$0.580741\pi$$
$$398$$ −16.0000 −0.802008
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ 0 0
$$403$$ −48.0000 −2.39105
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 40.0000 1.98273
$$408$$ 0 0
$$409$$ 30.0000 1.48340 0.741702 0.670729i $$-0.234019\pi$$
0.741702 + 0.670729i $$0.234019\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ 0 0
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8.00000 −0.392705
$$416$$ −6.00000 −0.294174
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 24.0000 1.17248 0.586238 0.810139i $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 0 0
$$424$$ −2.00000 −0.0971286
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −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
$$436$$ 6.00000 0.287348
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ 0 0
$$442$$ −12.0000 −0.570782
$$443$$ 20.0000 0.950229 0.475114 0.879924i $$-0.342407\pi$$
0.475114 + 0.879924i $$0.342407\pi$$
$$444$$ 0 0
$$445$$ −10.0000 −0.474045
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −8.00000 −0.376705
$$452$$ −2.00000 −0.0940721
$$453$$ 0 0
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ −16.0000 −0.743583 −0.371792 0.928316i $$-0.621256\pi$$
−0.371792 + 0.928316i $$0.621256\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ −40.0000 −1.85098 −0.925490 0.378773i $$-0.876346\pi$$
−0.925490 + 0.378773i $$0.876346\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 8.00000 0.369012
$$471$$ 0 0
$$472$$ 8.00000 0.368230
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 16.0000 0.731823
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 2.00000 0.0908153
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ −14.0000 −0.633750
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −12.0000 −0.537194 −0.268597 0.963253i $$-0.586560\pi$$
−0.268597 + 0.963253i $$0.586560\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ 6.00000 0.266996
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ −32.0000 −1.40736
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 6.00000 0.263117
$$521$$ 2.00000 0.0876216 0.0438108 0.999040i $$-0.486050\pi$$
0.0438108 + 0.999040i $$0.486050\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ −16.0000 −0.696971
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 2.00000 0.0868744
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 6.00000 0.258678
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ −6.00000 −0.257012
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 18.0000 0.764747
$$555$$ 0 0
$$556$$ −16.0000 −0.678551
$$557$$ −22.0000 −0.932170 −0.466085 0.884740i $$-0.654336\pi$$
−0.466085 + 0.884740i $$0.654336\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 26.0000 1.09674
$$563$$ −16.0000 −0.674320 −0.337160 0.941447i $$-0.609466\pi$$
−0.337160 + 0.941447i $$0.609466\pi$$
$$564$$ 0 0
$$565$$ 2.00000 0.0841406
$$566$$ 32.0000 1.34506
$$567$$ 0 0
$$568$$ −16.0000 −0.671345
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ −24.0000 −1.00349
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 0 0
$$580$$ 6.00000 0.249136
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −8.00000 −0.331326
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −10.0000 −0.413096
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −8.00000 −0.329355
$$591$$ 0 0
$$592$$ −10.0000 −0.410997
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 8.00000 0.325515
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 14.0000 0.566843
$$611$$ 48.0000 1.94187
$$612$$ 0 0
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ 24.0000 0.962312
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ −10.0000 −0.399043
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 0 0
$$634$$ 22.0000 0.873732
$$635$$ 8.00000 0.317470
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −24.0000 −0.950169
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ 32.0000 1.26196 0.630978 0.775800i $$-0.282654\pi$$
0.630978 + 0.775800i $$0.282654\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 16.0000 0.629025 0.314512 0.949253i $$-0.398159\pi$$
0.314512 + 0.949253i $$0.398159\pi$$
$$648$$ 0 0
$$649$$ 32.0000 1.25611
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 0 0
$$655$$ 16.0000 0.625172
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −28.0000 −1.09073 −0.545363 0.838200i $$-0.683608\pi$$
−0.545363 + 0.838200i $$0.683608\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ −12.0000 −0.463600
$$671$$ −56.0000 −2.16186
$$672$$ 0 0
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 2.00000 0.0766965
$$681$$ 0 0
$$682$$ −32.0000 −1.22534
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ −22.0000 −0.836315
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 16.0000 0.606915
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 10.0000 0.378506
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −2.00000 −0.0751116 −0.0375558 0.999295i $$-0.511957\pi$$
−0.0375558 + 0.999295i $$0.511957\pi$$
$$710$$ 16.0000 0.600469
$$711$$ 0 0
$$712$$ −10.0000 −0.374766
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 24.0000 0.897549
$$716$$ 12.0000 0.448461
$$717$$ 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
$$721$$ 0 0
$$722$$ 19.0000 0.707107
$$723$$ 0 0
$$724$$ 14.0000 0.520306
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −2.00000 −0.0740233
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 48.0000 1.76810
$$738$$ 0 0
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 10.0000 0.367607
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −48.0000 −1.76095 −0.880475 0.474093i $$-0.842776\pi$$
−0.880475 + 0.474093i $$0.842776\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ −14.0000 −0.512576
$$747$$ 0 0
$$748$$ −8.00000 −0.292509
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ 36.0000 1.31104
$$755$$ −8.00000 −0.291150
$$756$$ 0 0
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ 12.0000 0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ −48.0000 −1.73318
$$768$$ 0 0
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ 14.0000 0.501924
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −64.0000 −2.29010
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 10.0000 0.356915
$$786$$ 0 0
$$787$$ 8.00000 0.285169 0.142585 0.989783i $$-0.454459\pi$$
0.142585 + 0.989783i $$0.454459\pi$$
$$788$$ −14.0000 −0.498729
$$789$$ 0 0
$$790$$ −8.00000 −0.284627
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 84.0000 2.98293
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −14.0000 −0.494357
$$803$$ 8.00000 0.282314
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 48.0000 1.69073
$$807$$ 0 0
$$808$$ 6.00000 0.211079
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −8.00000 −0.280918 −0.140459 0.990086i $$-0.544858\pi$$
−0.140459 + 0.990086i $$0.544858\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −40.0000 −1.40200
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −30.0000 −1.04893
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 0 0
$$823$$ −8.00000 −0.278862 −0.139431 0.990232i $$-0.544527\pi$$
−0.139431 + 0.990232i $$0.544527\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −44.0000 −1.53003 −0.765015 0.644013i $$-0.777268\pi$$
−0.765015 + 0.644013i $$0.777268\pi$$
$$828$$ 0 0
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 8.00000 0.277684
$$831$$ 0 0
$$832$$ 6.00000 0.208013
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −24.0000 −0.829066
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 10.0000 0.344623
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ −23.0000 −0.791224
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ −2.00000 −0.0685994
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 46.0000 1.57501 0.787505 0.616308i $$-0.211372\pi$$
0.787505 + 0.616308i $$0.211372\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −54.0000 −1.84460 −0.922302 0.386469i $$-0.873695\pi$$
−0.922302 + 0.386469i $$0.873695\pi$$
$$858$$ 0 0
$$859$$ 48.0000 1.63774 0.818869 0.573980i $$-0.194601\pi$$
0.818869 + 0.573980i $$0.194601\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 32.0000 1.08929 0.544646 0.838666i $$-0.316664\pi$$
0.544646 + 0.838666i $$0.316664\pi$$
$$864$$ 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
$$871$$ −72.0000 −2.43963
$$872$$ −6.00000 −0.203186
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ 0 0
$$880$$ 4.00000 0.134840
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ −12.0000 −0.403832 −0.201916 0.979403i $$-0.564717\pi$$
−0.201916 + 0.979403i $$0.564717\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ −16.0000 −0.537227 −0.268614 0.963248i $$-0.586566\pi$$
−0.268614 + 0.963248i $$0.586566\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 10.0000 0.335201
$$891$$ 0 0
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −30.0000 −1.00111
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 8.00000 0.266371
$$903$$ 0 0
$$904$$ 2.00000 0.0665190
$$905$$ −14.0000 −0.465376
$$906$$ 0 0
$$907$$ −52.0000 −1.72663 −0.863316 0.504664i $$-0.831616\pi$$
−0.863316 + 0.504664i $$0.831616\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ −32.0000 −1.05905
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −10.0000 −0.329332
$$923$$ 96.0000 3.15988
$$924$$ 0 0
$$925$$ −10.0000 −0.328798
$$926$$ 16.0000 0.525793
$$927$$ 0 0
$$928$$ 6.00000 0.196960
$$929$$ 58.0000 1.90292 0.951459 0.307775i $$-0.0995844\pi$$
0.951459 + 0.307775i $$0.0995844\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ 40.0000 1.30884
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ −50.0000 −1.63343 −0.816714 0.577042i $$-0.804207\pi$$
−0.816714 + 0.577042i $$0.804207\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −8.00000 −0.260931
$$941$$ 2.00000 0.0651981 0.0325991 0.999469i $$-0.489622\pi$$
0.0325991 + 0.999469i $$0.489622\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ −44.0000 −1.42981 −0.714904 0.699223i $$-0.753530\pi$$
−0.714904 + 0.699223i $$0.753530\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 54.0000 1.74923 0.874616 0.484817i $$-0.161114\pi$$
0.874616 + 0.484817i $$0.161114\pi$$
$$954$$ 0 0
$$955$$ 24.0000 0.776622
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ −24.0000 −0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 60.0000 1.93448
$$963$$ 0 0
$$964$$ −10.0000 −0.322078
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 0 0
$$970$$ −2.00000 −0.0642161
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12.0000 0.382935
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 0 0
$$985$$ 14.0000 0.446077
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −16.0000 −0.507234
$$996$$ 0 0
$$997$$ 22.0000 0.696747 0.348373 0.937356i $$-0.386734\pi$$
0.348373 + 0.937356i $$0.386734\pi$$
$$998$$ 12.0000 0.379853
$$999$$ 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 4410.2.a.b.1.1 1
3.2 odd 2 490.2.a.h.1.1 1
7.6 odd 2 630.2.a.d.1.1 1
12.11 even 2 3920.2.a.t.1.1 1
15.2 even 4 2450.2.c.k.99.2 2
15.8 even 4 2450.2.c.k.99.1 2
15.14 odd 2 2450.2.a.l.1.1 1
21.2 odd 6 490.2.e.c.361.1 2
21.5 even 6 490.2.e.d.361.1 2
21.11 odd 6 490.2.e.c.471.1 2
21.17 even 6 490.2.e.d.471.1 2
21.20 even 2 70.2.a.a.1.1 1
28.27 even 2 5040.2.a.bm.1.1 1
35.13 even 4 3150.2.g.c.2899.2 2
35.27 even 4 3150.2.g.c.2899.1 2
35.34 odd 2 3150.2.a.bj.1.1 1
84.83 odd 2 560.2.a.d.1.1 1
105.62 odd 4 350.2.c.b.99.2 2
105.83 odd 4 350.2.c.b.99.1 2
105.104 even 2 350.2.a.b.1.1 1
168.83 odd 2 2240.2.a.q.1.1 1
168.125 even 2 2240.2.a.n.1.1 1
231.230 odd 2 8470.2.a.j.1.1 1
420.83 even 4 2800.2.g.n.449.1 2
420.167 even 4 2800.2.g.n.449.2 2
420.419 odd 2 2800.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.a.a.1.1 1 21.20 even 2
350.2.a.b.1.1 1 105.104 even 2
350.2.c.b.99.1 2 105.83 odd 4
350.2.c.b.99.2 2 105.62 odd 4
490.2.a.h.1.1 1 3.2 odd 2
490.2.e.c.361.1 2 21.2 odd 6
490.2.e.c.471.1 2 21.11 odd 6
490.2.e.d.361.1 2 21.5 even 6
490.2.e.d.471.1 2 21.17 even 6
560.2.a.d.1.1 1 84.83 odd 2
630.2.a.d.1.1 1 7.6 odd 2
2240.2.a.n.1.1 1 168.125 even 2
2240.2.a.q.1.1 1 168.83 odd 2
2450.2.a.l.1.1 1 15.14 odd 2
2450.2.c.k.99.1 2 15.8 even 4
2450.2.c.k.99.2 2 15.2 even 4
2800.2.a.m.1.1 1 420.419 odd 2
2800.2.g.n.449.1 2 420.83 even 4
2800.2.g.n.449.2 2 420.167 even 4
3150.2.a.bj.1.1 1 35.34 odd 2
3150.2.g.c.2899.1 2 35.27 even 4
3150.2.g.c.2899.2 2 35.13 even 4
3920.2.a.t.1.1 1 12.11 even 2
4410.2.a.b.1.1 1 1.1 even 1 trivial
5040.2.a.bm.1.1 1 28.27 even 2
8470.2.a.j.1.1 1 231.230 odd 2