# Properties

 Label 5520.2.a.cc.1.4 Level $5520$ Weight $2$ Character 5520.1 Self dual yes Analytic conductor $44.077$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5520.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.0774219157$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.20087896.1 Defining polynomial: $$x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4$$ x^5 - x^4 - 21*x^3 + 5*x^2 + 84*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 2760) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.27399$$ of defining polynomial Character $$\chi$$ $$=$$ 5520.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} +3.55148 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} +3.55148 q^{7} +1.00000 q^{9} +2.79304 q^{11} +3.73151 q^{13} +1.00000 q^{15} -7.83097 q^{17} -6.27949 q^{19} +3.55148 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.75844 q^{29} +2.48995 q^{31} +2.79304 q^{33} +3.55148 q^{35} -1.55148 q^{37} +3.73151 q^{39} +5.78954 q^{41} +4.54798 q^{43} +1.00000 q^{45} +6.27949 q^{47} +5.61301 q^{49} -7.83097 q^{51} +8.89250 q^{53} +2.79304 q^{55} -6.27949 q^{57} -3.31342 q^{59} +11.2180 q^{61} +3.55148 q^{63} +3.73151 q^{65} -5.52105 q^{67} +1.00000 q^{69} -6.34452 q^{71} +15.3824 q^{73} +1.00000 q^{75} +9.91943 q^{77} +9.62051 q^{79} +1.00000 q^{81} -5.83097 q^{83} -7.83097 q^{85} +2.75844 q^{87} +0.390487 q^{89} +13.2524 q^{91} +2.48995 q^{93} -6.27949 q^{95} +0.961896 q^{97} +2.79304 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10})$$ 5 * q + 5 * q^3 + 5 * q^5 + 4 * q^7 + 5 * q^9 $$5 q + 5 q^{3} + 5 q^{5} + 4 q^{7} + 5 q^{9} + 4 q^{11} + 4 q^{13} + 5 q^{15} + 10 q^{17} + 4 q^{19} + 4 q^{21} + 5 q^{23} + 5 q^{25} + 5 q^{27} + 10 q^{29} - 6 q^{31} + 4 q^{33} + 4 q^{35} + 6 q^{37} + 4 q^{39} + 12 q^{41} + 2 q^{43} + 5 q^{45} - 4 q^{47} + 19 q^{49} + 10 q^{51} + 4 q^{55} + 4 q^{57} - 6 q^{59} + 16 q^{61} + 4 q^{63} + 4 q^{65} + 4 q^{67} + 5 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 16 q^{77} - 18 q^{79} + 5 q^{81} + 20 q^{83} + 10 q^{85} + 10 q^{87} + 18 q^{89} - 20 q^{91} - 6 q^{93} + 4 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100})$$ 5 * q + 5 * q^3 + 5 * q^5 + 4 * q^7 + 5 * q^9 + 4 * q^11 + 4 * q^13 + 5 * q^15 + 10 * q^17 + 4 * q^19 + 4 * q^21 + 5 * q^23 + 5 * q^25 + 5 * q^27 + 10 * q^29 - 6 * q^31 + 4 * q^33 + 4 * q^35 + 6 * q^37 + 4 * q^39 + 12 * q^41 + 2 * q^43 + 5 * q^45 - 4 * q^47 + 19 * q^49 + 10 * q^51 + 4 * q^55 + 4 * q^57 - 6 * q^59 + 16 * q^61 + 4 * q^63 + 4 * q^65 + 4 * q^67 + 5 * q^69 - 8 * q^71 + 14 * q^73 + 5 * q^75 + 16 * q^77 - 18 * q^79 + 5 * q^81 + 20 * q^83 + 10 * q^85 + 10 * q^87 + 18 * q^89 - 20 * q^91 - 6 * q^93 + 4 * q^95 + 4 * q^97 + 4 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 3.55148 1.34233 0.671167 0.741306i $$-0.265793\pi$$
0.671167 + 0.741306i $$0.265793\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.79304 0.842134 0.421067 0.907030i $$-0.361656\pi$$
0.421067 + 0.907030i $$0.361656\pi$$
$$12$$ 0 0
$$13$$ 3.73151 1.03493 0.517467 0.855703i $$-0.326875\pi$$
0.517467 + 0.855703i $$0.326875\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −7.83097 −1.89929 −0.949644 0.313330i $$-0.898555\pi$$
−0.949644 + 0.313330i $$0.898555\pi$$
$$18$$ 0 0
$$19$$ −6.27949 −1.44061 −0.720307 0.693656i $$-0.755999\pi$$
−0.720307 + 0.693656i $$0.755999\pi$$
$$20$$ 0 0
$$21$$ 3.55148 0.774997
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 2.75844 0.512229 0.256115 0.966646i $$-0.417558\pi$$
0.256115 + 0.966646i $$0.417558\pi$$
$$30$$ 0 0
$$31$$ 2.48995 0.447208 0.223604 0.974680i $$-0.428218\pi$$
0.223604 + 0.974680i $$0.428218\pi$$
$$32$$ 0 0
$$33$$ 2.79304 0.486206
$$34$$ 0 0
$$35$$ 3.55148 0.600310
$$36$$ 0 0
$$37$$ −1.55148 −0.255062 −0.127531 0.991835i $$-0.540705\pi$$
−0.127531 + 0.991835i $$0.540705\pi$$
$$38$$ 0 0
$$39$$ 3.73151 0.597519
$$40$$ 0 0
$$41$$ 5.78954 0.904174 0.452087 0.891974i $$-0.350680\pi$$
0.452087 + 0.891974i $$0.350680\pi$$
$$42$$ 0 0
$$43$$ 4.54798 0.693560 0.346780 0.937946i $$-0.387275\pi$$
0.346780 + 0.937946i $$0.387275\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 6.27949 0.915957 0.457979 0.888963i $$-0.348574\pi$$
0.457979 + 0.888963i $$0.348574\pi$$
$$48$$ 0 0
$$49$$ 5.61301 0.801859
$$50$$ 0 0
$$51$$ −7.83097 −1.09655
$$52$$ 0 0
$$53$$ 8.89250 1.22148 0.610739 0.791832i $$-0.290872\pi$$
0.610739 + 0.791832i $$0.290872\pi$$
$$54$$ 0 0
$$55$$ 2.79304 0.376614
$$56$$ 0 0
$$57$$ −6.27949 −0.831738
$$58$$ 0 0
$$59$$ −3.31342 −0.431371 −0.215685 0.976463i $$-0.569199\pi$$
−0.215685 + 0.976463i $$0.569199\pi$$
$$60$$ 0 0
$$61$$ 11.2180 1.43631 0.718156 0.695882i $$-0.244986\pi$$
0.718156 + 0.695882i $$0.244986\pi$$
$$62$$ 0 0
$$63$$ 3.55148 0.447444
$$64$$ 0 0
$$65$$ 3.73151 0.462837
$$66$$ 0 0
$$67$$ −5.52105 −0.674503 −0.337252 0.941415i $$-0.609497\pi$$
−0.337252 + 0.941415i $$0.609497\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −6.34452 −0.752956 −0.376478 0.926426i $$-0.622865\pi$$
−0.376478 + 0.926426i $$0.622865\pi$$
$$72$$ 0 0
$$73$$ 15.3824 1.80038 0.900190 0.435498i $$-0.143428\pi$$
0.900190 + 0.435498i $$0.143428\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 9.91943 1.13042
$$78$$ 0 0
$$79$$ 9.62051 1.08239 0.541196 0.840897i $$-0.317972\pi$$
0.541196 + 0.840897i $$0.317972\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −5.83097 −0.640032 −0.320016 0.947412i $$-0.603688\pi$$
−0.320016 + 0.947412i $$0.603688\pi$$
$$84$$ 0 0
$$85$$ −7.83097 −0.849388
$$86$$ 0 0
$$87$$ 2.75844 0.295736
$$88$$ 0 0
$$89$$ 0.390487 0.0413915 0.0206958 0.999786i $$-0.493412\pi$$
0.0206958 + 0.999786i $$0.493412\pi$$
$$90$$ 0 0
$$91$$ 13.2524 1.38923
$$92$$ 0 0
$$93$$ 2.48995 0.258195
$$94$$ 0 0
$$95$$ −6.27949 −0.644262
$$96$$ 0 0
$$97$$ 0.961896 0.0976657 0.0488329 0.998807i $$-0.484450\pi$$
0.0488329 + 0.998807i $$0.484450\pi$$
$$98$$ 0 0
$$99$$ 2.79304 0.280711
$$100$$ 0 0
$$101$$ −16.9035 −1.68196 −0.840980 0.541066i $$-0.818021\pi$$
−0.840980 + 0.541066i $$0.818021\pi$$
$$102$$ 0 0
$$103$$ 6.04143 0.595280 0.297640 0.954678i $$-0.403801\pi$$
0.297640 + 0.954678i $$0.403801\pi$$
$$104$$ 0 0
$$105$$ 3.55148 0.346589
$$106$$ 0 0
$$107$$ −12.3141 −1.19045 −0.595224 0.803560i $$-0.702937\pi$$
−0.595224 + 0.803560i $$0.702937\pi$$
$$108$$ 0 0
$$109$$ 14.8579 1.42313 0.711564 0.702621i $$-0.247987\pi$$
0.711564 + 0.702621i $$0.247987\pi$$
$$110$$ 0 0
$$111$$ −1.55148 −0.147260
$$112$$ 0 0
$$113$$ 3.21113 0.302077 0.151039 0.988528i $$-0.451738\pi$$
0.151039 + 0.988528i $$0.451738\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 3.73151 0.344978
$$118$$ 0 0
$$119$$ −27.8115 −2.54948
$$120$$ 0 0
$$121$$ −3.19892 −0.290811
$$122$$ 0 0
$$123$$ 5.78954 0.522025
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −0.145425 −0.0129043 −0.00645217 0.999979i $$-0.502054\pi$$
−0.00645217 + 0.999979i $$0.502054\pi$$
$$128$$ 0 0
$$129$$ 4.54798 0.400427
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −22.3015 −1.93378
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −6.16450 −0.526668 −0.263334 0.964705i $$-0.584822\pi$$
−0.263334 + 0.964705i $$0.584822\pi$$
$$138$$ 0 0
$$139$$ −14.0690 −1.19332 −0.596660 0.802494i $$-0.703506\pi$$
−0.596660 + 0.802494i $$0.703506\pi$$
$$140$$ 0 0
$$141$$ 6.27949 0.528828
$$142$$ 0 0
$$143$$ 10.4223 0.871553
$$144$$ 0 0
$$145$$ 2.75844 0.229076
$$146$$ 0 0
$$147$$ 5.61301 0.462954
$$148$$ 0 0
$$149$$ 13.7964 1.13024 0.565121 0.825008i $$-0.308829\pi$$
0.565121 + 0.825008i $$0.308829\pi$$
$$150$$ 0 0
$$151$$ −16.1989 −1.31825 −0.659125 0.752034i $$-0.729073\pi$$
−0.659125 + 0.752034i $$0.729073\pi$$
$$152$$ 0 0
$$153$$ −7.83097 −0.633096
$$154$$ 0 0
$$155$$ 2.48995 0.199997
$$156$$ 0 0
$$157$$ −14.1105 −1.12614 −0.563068 0.826410i $$-0.690379\pi$$
−0.563068 + 0.826410i $$0.690379\pi$$
$$158$$ 0 0
$$159$$ 8.89250 0.705221
$$160$$ 0 0
$$161$$ 3.55148 0.279896
$$162$$ 0 0
$$163$$ −21.2480 −1.66427 −0.832137 0.554571i $$-0.812882\pi$$
−0.832137 + 0.554571i $$0.812882\pi$$
$$164$$ 0 0
$$165$$ 2.79304 0.217438
$$166$$ 0 0
$$167$$ −19.3895 −1.50040 −0.750200 0.661211i $$-0.770043\pi$$
−0.750200 + 0.661211i $$0.770043\pi$$
$$168$$ 0 0
$$169$$ 0.924149 0.0710884
$$170$$ 0 0
$$171$$ −6.27949 −0.480204
$$172$$ 0 0
$$173$$ −13.1030 −0.996200 −0.498100 0.867120i $$-0.665969\pi$$
−0.498100 + 0.867120i $$0.665969\pi$$
$$174$$ 0 0
$$175$$ 3.55148 0.268467
$$176$$ 0 0
$$177$$ −3.31342 −0.249052
$$178$$ 0 0
$$179$$ 14.0759 1.05208 0.526039 0.850460i $$-0.323676\pi$$
0.526039 + 0.850460i $$0.323676\pi$$
$$180$$ 0 0
$$181$$ 21.1374 1.57113 0.785565 0.618779i $$-0.212373\pi$$
0.785565 + 0.618779i $$0.212373\pi$$
$$182$$ 0 0
$$183$$ 11.2180 0.829255
$$184$$ 0 0
$$185$$ −1.55148 −0.114067
$$186$$ 0 0
$$187$$ −21.8722 −1.59945
$$188$$ 0 0
$$189$$ 3.55148 0.258332
$$190$$ 0 0
$$191$$ −2.81647 −0.203793 −0.101896 0.994795i $$-0.532491\pi$$
−0.101896 + 0.994795i $$0.532491\pi$$
$$192$$ 0 0
$$193$$ 15.4560 1.11255 0.556274 0.830999i $$-0.312230\pi$$
0.556274 + 0.830999i $$0.312230\pi$$
$$194$$ 0 0
$$195$$ 3.73151 0.267219
$$196$$ 0 0
$$197$$ 25.7311 1.83327 0.916634 0.399728i $$-0.130895\pi$$
0.916634 + 0.399728i $$0.130895\pi$$
$$198$$ 0 0
$$199$$ −5.95857 −0.422392 −0.211196 0.977444i $$-0.567736\pi$$
−0.211196 + 0.977444i $$0.567736\pi$$
$$200$$ 0 0
$$201$$ −5.52105 −0.389425
$$202$$ 0 0
$$203$$ 9.79654 0.687583
$$204$$ 0 0
$$205$$ 5.78954 0.404359
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ −17.5389 −1.21319
$$210$$ 0 0
$$211$$ 6.06903 0.417809 0.208905 0.977936i $$-0.433010\pi$$
0.208905 + 0.977936i $$0.433010\pi$$
$$212$$ 0 0
$$213$$ −6.34452 −0.434719
$$214$$ 0 0
$$215$$ 4.54798 0.310170
$$216$$ 0 0
$$217$$ 8.84300 0.600302
$$218$$ 0 0
$$219$$ 15.3824 1.03945
$$220$$ 0 0
$$221$$ −29.2213 −1.96564
$$222$$ 0 0
$$223$$ 1.57509 0.105476 0.0527379 0.998608i $$-0.483205\pi$$
0.0527379 + 0.998608i $$0.483205\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 21.7034 1.44050 0.720251 0.693713i $$-0.244026\pi$$
0.720251 + 0.693713i $$0.244026\pi$$
$$228$$ 0 0
$$229$$ 23.2136 1.53400 0.766999 0.641649i $$-0.221749\pi$$
0.766999 + 0.641649i $$0.221749\pi$$
$$230$$ 0 0
$$231$$ 9.91943 0.652651
$$232$$ 0 0
$$233$$ −18.1989 −1.19225 −0.596125 0.802891i $$-0.703294\pi$$
−0.596125 + 0.802891i $$0.703294\pi$$
$$234$$ 0 0
$$235$$ 6.27949 0.409629
$$236$$ 0 0
$$237$$ 9.62051 0.624919
$$238$$ 0 0
$$239$$ −21.9236 −1.41812 −0.709060 0.705148i $$-0.750880\pi$$
−0.709060 + 0.705148i $$0.750880\pi$$
$$240$$ 0 0
$$241$$ 17.7964 1.14636 0.573182 0.819428i $$-0.305709\pi$$
0.573182 + 0.819428i $$0.305709\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 5.61301 0.358602
$$246$$ 0 0
$$247$$ −23.4320 −1.49094
$$248$$ 0 0
$$249$$ −5.83097 −0.369523
$$250$$ 0 0
$$251$$ 23.2785 1.46932 0.734661 0.678434i $$-0.237341\pi$$
0.734661 + 0.678434i $$0.237341\pi$$
$$252$$ 0 0
$$253$$ 2.79304 0.175597
$$254$$ 0 0
$$255$$ −7.83097 −0.490394
$$256$$ 0 0
$$257$$ 25.5884 1.59616 0.798079 0.602552i $$-0.205850\pi$$
0.798079 + 0.602552i $$0.205850\pi$$
$$258$$ 0 0
$$259$$ −5.51005 −0.342378
$$260$$ 0 0
$$261$$ 2.75844 0.170743
$$262$$ 0 0
$$263$$ 8.57051 0.528481 0.264240 0.964457i $$-0.414879\pi$$
0.264240 + 0.964457i $$0.414879\pi$$
$$264$$ 0 0
$$265$$ 8.89250 0.546262
$$266$$ 0 0
$$267$$ 0.390487 0.0238974
$$268$$ 0 0
$$269$$ −13.4405 −0.819481 −0.409740 0.912202i $$-0.634381\pi$$
−0.409740 + 0.912202i $$0.634381\pi$$
$$270$$ 0 0
$$271$$ −3.38699 −0.205745 −0.102872 0.994695i $$-0.532803\pi$$
−0.102872 + 0.994695i $$0.532803\pi$$
$$272$$ 0 0
$$273$$ 13.2524 0.802070
$$274$$ 0 0
$$275$$ 2.79304 0.168427
$$276$$ 0 0
$$277$$ −18.7469 −1.12639 −0.563196 0.826323i $$-0.690428\pi$$
−0.563196 + 0.826323i $$0.690428\pi$$
$$278$$ 0 0
$$279$$ 2.48995 0.149069
$$280$$ 0 0
$$281$$ −5.21396 −0.311039 −0.155519 0.987833i $$-0.549705\pi$$
−0.155519 + 0.987833i $$0.549705\pi$$
$$282$$ 0 0
$$283$$ 18.9771 1.12807 0.564035 0.825751i $$-0.309248\pi$$
0.564035 + 0.825751i $$0.309248\pi$$
$$284$$ 0 0
$$285$$ −6.27949 −0.371965
$$286$$ 0 0
$$287$$ 20.5614 1.21370
$$288$$ 0 0
$$289$$ 44.3240 2.60730
$$290$$ 0 0
$$291$$ 0.961896 0.0563873
$$292$$ 0 0
$$293$$ −27.3976 −1.60059 −0.800293 0.599609i $$-0.795323\pi$$
−0.800293 + 0.599609i $$0.795323\pi$$
$$294$$ 0 0
$$295$$ −3.31342 −0.192915
$$296$$ 0 0
$$297$$ 2.79304 0.162069
$$298$$ 0 0
$$299$$ 3.73151 0.215799
$$300$$ 0 0
$$301$$ 16.1521 0.930989
$$302$$ 0 0
$$303$$ −16.9035 −0.971080
$$304$$ 0 0
$$305$$ 11.2180 0.642338
$$306$$ 0 0
$$307$$ −1.18353 −0.0675475 −0.0337738 0.999430i $$-0.510753\pi$$
−0.0337738 + 0.999430i $$0.510753\pi$$
$$308$$ 0 0
$$309$$ 6.04143 0.343685
$$310$$ 0 0
$$311$$ −18.5770 −1.05340 −0.526702 0.850050i $$-0.676572\pi$$
−0.526702 + 0.850050i $$0.676572\pi$$
$$312$$ 0 0
$$313$$ −8.50094 −0.480502 −0.240251 0.970711i $$-0.577230\pi$$
−0.240251 + 0.970711i $$0.577230\pi$$
$$314$$ 0 0
$$315$$ 3.55148 0.200103
$$316$$ 0 0
$$317$$ 21.5884 1.21252 0.606262 0.795265i $$-0.292668\pi$$
0.606262 + 0.795265i $$0.292668\pi$$
$$318$$ 0 0
$$319$$ 7.70443 0.431366
$$320$$ 0 0
$$321$$ −12.3141 −0.687305
$$322$$ 0 0
$$323$$ 49.1745 2.73614
$$324$$ 0 0
$$325$$ 3.73151 0.206987
$$326$$ 0 0
$$327$$ 14.8579 0.821644
$$328$$ 0 0
$$329$$ 22.3015 1.22952
$$330$$ 0 0
$$331$$ −24.0288 −1.32074 −0.660372 0.750939i $$-0.729601\pi$$
−0.660372 + 0.750939i $$0.729601\pi$$
$$332$$ 0 0
$$333$$ −1.55148 −0.0850206
$$334$$ 0 0
$$335$$ −5.52105 −0.301647
$$336$$ 0 0
$$337$$ −19.7960 −1.07836 −0.539178 0.842192i $$-0.681265\pi$$
−0.539178 + 0.842192i $$0.681265\pi$$
$$338$$ 0 0
$$339$$ 3.21113 0.174405
$$340$$ 0 0
$$341$$ 6.95452 0.376609
$$342$$ 0 0
$$343$$ −4.92585 −0.265971
$$344$$ 0 0
$$345$$ 1.00000 0.0538382
$$346$$ 0 0
$$347$$ −3.52388 −0.189172 −0.0945859 0.995517i $$-0.530153\pi$$
−0.0945859 + 0.995517i $$0.530153\pi$$
$$348$$ 0 0
$$349$$ 9.10979 0.487636 0.243818 0.969821i $$-0.421600\pi$$
0.243818 + 0.969821i $$0.421600\pi$$
$$350$$ 0 0
$$351$$ 3.73151 0.199173
$$352$$ 0 0
$$353$$ −24.7224 −1.31584 −0.657920 0.753088i $$-0.728564\pi$$
−0.657920 + 0.753088i $$0.728564\pi$$
$$354$$ 0 0
$$355$$ −6.34452 −0.336732
$$356$$ 0 0
$$357$$ −27.8115 −1.47194
$$358$$ 0 0
$$359$$ −11.3895 −0.601112 −0.300556 0.953764i $$-0.597172\pi$$
−0.300556 + 0.953764i $$0.597172\pi$$
$$360$$ 0 0
$$361$$ 20.4320 1.07537
$$362$$ 0 0
$$363$$ −3.19892 −0.167900
$$364$$ 0 0
$$365$$ 15.3824 0.805154
$$366$$ 0 0
$$367$$ −11.1306 −0.581011 −0.290505 0.956873i $$-0.593823\pi$$
−0.290505 + 0.956873i $$0.593823\pi$$
$$368$$ 0 0
$$369$$ 5.78954 0.301391
$$370$$ 0 0
$$371$$ 31.5815 1.63963
$$372$$ 0 0
$$373$$ 8.51755 0.441022 0.220511 0.975385i $$-0.429228\pi$$
0.220511 + 0.975385i $$0.429228\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 10.2931 0.530123
$$378$$ 0 0
$$379$$ 21.1320 1.08548 0.542738 0.839902i $$-0.317388\pi$$
0.542738 + 0.839902i $$0.317388\pi$$
$$380$$ 0 0
$$381$$ −0.145425 −0.00745033
$$382$$ 0 0
$$383$$ −9.37562 −0.479072 −0.239536 0.970887i $$-0.576995\pi$$
−0.239536 + 0.970887i $$0.576995\pi$$
$$384$$ 0 0
$$385$$ 9.91943 0.505541
$$386$$ 0 0
$$387$$ 4.54798 0.231187
$$388$$ 0 0
$$389$$ −16.6352 −0.843437 −0.421719 0.906727i $$-0.638573\pi$$
−0.421719 + 0.906727i $$0.638573\pi$$
$$390$$ 0 0
$$391$$ −7.83097 −0.396029
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 9.62051 0.484060
$$396$$ 0 0
$$397$$ 23.9261 1.20081 0.600407 0.799694i $$-0.295005\pi$$
0.600407 + 0.799694i $$0.295005\pi$$
$$398$$ 0 0
$$399$$ −22.3015 −1.11647
$$400$$ 0 0
$$401$$ −12.4457 −0.621508 −0.310754 0.950490i $$-0.600582\pi$$
−0.310754 + 0.950490i $$0.600582\pi$$
$$402$$ 0 0
$$403$$ 9.29125 0.462830
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ −4.33335 −0.214796
$$408$$ 0 0
$$409$$ 27.0419 1.33714 0.668568 0.743651i $$-0.266907\pi$$
0.668568 + 0.743651i $$0.266907\pi$$
$$410$$ 0 0
$$411$$ −6.16450 −0.304072
$$412$$ 0 0
$$413$$ −11.7676 −0.579043
$$414$$ 0 0
$$415$$ −5.83097 −0.286231
$$416$$ 0 0
$$417$$ −14.0690 −0.688963
$$418$$ 0 0
$$419$$ −10.0949 −0.493169 −0.246585 0.969121i $$-0.579308\pi$$
−0.246585 + 0.969121i $$0.579308\pi$$
$$420$$ 0 0
$$421$$ −28.5268 −1.39031 −0.695156 0.718858i $$-0.744665\pi$$
−0.695156 + 0.718858i $$0.744665\pi$$
$$422$$ 0 0
$$423$$ 6.27949 0.305319
$$424$$ 0 0
$$425$$ −7.83097 −0.379858
$$426$$ 0 0
$$427$$ 39.8403 1.92801
$$428$$ 0 0
$$429$$ 10.4223 0.503191
$$430$$ 0 0
$$431$$ −21.6081 −1.04082 −0.520412 0.853915i $$-0.674222\pi$$
−0.520412 + 0.853915i $$0.674222\pi$$
$$432$$ 0 0
$$433$$ −8.21009 −0.394552 −0.197276 0.980348i $$-0.563209\pi$$
−0.197276 + 0.980348i $$0.563209\pi$$
$$434$$ 0 0
$$435$$ 2.75844 0.132257
$$436$$ 0 0
$$437$$ −6.27949 −0.300389
$$438$$ 0 0
$$439$$ −24.4429 −1.16660 −0.583298 0.812258i $$-0.698238\pi$$
−0.583298 + 0.812258i $$0.698238\pi$$
$$440$$ 0 0
$$441$$ 5.61301 0.267286
$$442$$ 0 0
$$443$$ −32.5074 −1.54447 −0.772237 0.635335i $$-0.780862\pi$$
−0.772237 + 0.635335i $$0.780862\pi$$
$$444$$ 0 0
$$445$$ 0.390487 0.0185109
$$446$$ 0 0
$$447$$ 13.7964 0.652546
$$448$$ 0 0
$$449$$ −24.5635 −1.15922 −0.579612 0.814893i $$-0.696796\pi$$
−0.579612 + 0.814893i $$0.696796\pi$$
$$450$$ 0 0
$$451$$ 16.1704 0.761436
$$452$$ 0 0
$$453$$ −16.1989 −0.761092
$$454$$ 0 0
$$455$$ 13.2524 0.621281
$$456$$ 0 0
$$457$$ −29.6798 −1.38836 −0.694180 0.719801i $$-0.744233\pi$$
−0.694180 + 0.719801i $$0.744233\pi$$
$$458$$ 0 0
$$459$$ −7.83097 −0.365518
$$460$$ 0 0
$$461$$ −34.2239 −1.59397 −0.796983 0.604001i $$-0.793572\pi$$
−0.796983 + 0.604001i $$0.793572\pi$$
$$462$$ 0 0
$$463$$ 27.8074 1.29232 0.646159 0.763203i $$-0.276374\pi$$
0.646159 + 0.763203i $$0.276374\pi$$
$$464$$ 0 0
$$465$$ 2.48995 0.115469
$$466$$ 0 0
$$467$$ −22.3207 −1.03288 −0.516440 0.856323i $$-0.672743\pi$$
−0.516440 + 0.856323i $$0.672743\pi$$
$$468$$ 0 0
$$469$$ −19.6079 −0.905408
$$470$$ 0 0
$$471$$ −14.1105 −0.650175
$$472$$ 0 0
$$473$$ 12.7027 0.584070
$$474$$ 0 0
$$475$$ −6.27949 −0.288123
$$476$$ 0 0
$$477$$ 8.89250 0.407160
$$478$$ 0 0
$$479$$ 32.7535 1.49655 0.748274 0.663390i $$-0.230883\pi$$
0.748274 + 0.663390i $$0.230883\pi$$
$$480$$ 0 0
$$481$$ −5.78936 −0.263972
$$482$$ 0 0
$$483$$ 3.55148 0.161598
$$484$$ 0 0
$$485$$ 0.961896 0.0436774
$$486$$ 0 0
$$487$$ 25.9996 1.17816 0.589078 0.808076i $$-0.299491\pi$$
0.589078 + 0.808076i $$0.299491\pi$$
$$488$$ 0 0
$$489$$ −21.2480 −0.960869
$$490$$ 0 0
$$491$$ 7.67348 0.346299 0.173150 0.984896i $$-0.444606\pi$$
0.173150 + 0.984896i $$0.444606\pi$$
$$492$$ 0 0
$$493$$ −21.6012 −0.972871
$$494$$ 0 0
$$495$$ 2.79304 0.125538
$$496$$ 0 0
$$497$$ −22.5324 −1.01072
$$498$$ 0 0
$$499$$ −15.2951 −0.684701 −0.342350 0.939572i $$-0.611223\pi$$
−0.342350 + 0.939572i $$0.611223\pi$$
$$500$$ 0 0
$$501$$ −19.3895 −0.866257
$$502$$ 0 0
$$503$$ 9.79654 0.436806 0.218403 0.975859i $$-0.429915\pi$$
0.218403 + 0.975859i $$0.429915\pi$$
$$504$$ 0 0
$$505$$ −16.9035 −0.752196
$$506$$ 0 0
$$507$$ 0.924149 0.0410429
$$508$$ 0 0
$$509$$ 6.25713 0.277342 0.138671 0.990338i $$-0.455717\pi$$
0.138671 + 0.990338i $$0.455717\pi$$
$$510$$ 0 0
$$511$$ 54.6305 2.41671
$$512$$ 0 0
$$513$$ −6.27949 −0.277246
$$514$$ 0 0
$$515$$ 6.04143 0.266217
$$516$$ 0 0
$$517$$ 17.5389 0.771358
$$518$$ 0 0
$$519$$ −13.1030 −0.575156
$$520$$ 0 0
$$521$$ −8.72384 −0.382198 −0.191099 0.981571i $$-0.561205\pi$$
−0.191099 + 0.981571i $$0.561205\pi$$
$$522$$ 0 0
$$523$$ 8.66404 0.378852 0.189426 0.981895i $$-0.439337\pi$$
0.189426 + 0.981895i $$0.439337\pi$$
$$524$$ 0 0
$$525$$ 3.55148 0.154999
$$526$$ 0 0
$$527$$ −19.4987 −0.849376
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −3.31342 −0.143790
$$532$$ 0 0
$$533$$ 21.6037 0.935761
$$534$$ 0 0
$$535$$ −12.3141 −0.532384
$$536$$ 0 0
$$537$$ 14.0759 0.607418
$$538$$ 0 0
$$539$$ 15.6774 0.675273
$$540$$ 0 0
$$541$$ −36.5660 −1.57209 −0.786047 0.618167i $$-0.787876\pi$$
−0.786047 + 0.618167i $$0.787876\pi$$
$$542$$ 0 0
$$543$$ 21.1374 0.907092
$$544$$ 0 0
$$545$$ 14.8579 0.636442
$$546$$ 0 0
$$547$$ 0.827834 0.0353956 0.0176978 0.999843i $$-0.494366\pi$$
0.0176978 + 0.999843i $$0.494366\pi$$
$$548$$ 0 0
$$549$$ 11.2180 0.478771
$$550$$ 0 0
$$551$$ −17.3216 −0.737924
$$552$$ 0 0
$$553$$ 34.1670 1.45293
$$554$$ 0 0
$$555$$ −1.55148 −0.0658567
$$556$$ 0 0
$$557$$ 0.125533 0.00531902 0.00265951 0.999996i $$-0.499153\pi$$
0.00265951 + 0.999996i $$0.499153\pi$$
$$558$$ 0 0
$$559$$ 16.9708 0.717789
$$560$$ 0 0
$$561$$ −21.8722 −0.923446
$$562$$ 0 0
$$563$$ 38.3207 1.61503 0.807513 0.589849i $$-0.200813\pi$$
0.807513 + 0.589849i $$0.200813\pi$$
$$564$$ 0 0
$$565$$ 3.21113 0.135093
$$566$$ 0 0
$$567$$ 3.55148 0.149148
$$568$$ 0 0
$$569$$ −28.8715 −1.21036 −0.605179 0.796090i $$-0.706898\pi$$
−0.605179 + 0.796090i $$0.706898\pi$$
$$570$$ 0 0
$$571$$ −15.0514 −0.629881 −0.314940 0.949111i $$-0.601985\pi$$
−0.314940 + 0.949111i $$0.601985\pi$$
$$572$$ 0 0
$$573$$ −2.81647 −0.117660
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −21.0561 −0.876577 −0.438288 0.898834i $$-0.644415\pi$$
−0.438288 + 0.898834i $$0.644415\pi$$
$$578$$ 0 0
$$579$$ 15.4560 0.642330
$$580$$ 0 0
$$581$$ −20.7086 −0.859136
$$582$$ 0 0
$$583$$ 24.8371 1.02865
$$584$$ 0 0
$$585$$ 3.73151 0.154279
$$586$$ 0 0
$$587$$ −22.2751 −0.919393 −0.459696 0.888076i $$-0.652042\pi$$
−0.459696 + 0.888076i $$0.652042\pi$$
$$588$$ 0 0
$$589$$ −15.6356 −0.644253
$$590$$ 0 0
$$591$$ 25.7311 1.05844
$$592$$ 0 0
$$593$$ −8.45642 −0.347263 −0.173632 0.984811i $$-0.555550\pi$$
−0.173632 + 0.984811i $$0.555550\pi$$
$$594$$ 0 0
$$595$$ −27.8115 −1.14016
$$596$$ 0 0
$$597$$ −5.95857 −0.243868
$$598$$ 0 0
$$599$$ 10.0040 0.408752 0.204376 0.978892i $$-0.434483\pi$$
0.204376 + 0.978892i $$0.434483\pi$$
$$600$$ 0 0
$$601$$ 14.8119 0.604191 0.302096 0.953278i $$-0.402314\pi$$
0.302096 + 0.953278i $$0.402314\pi$$
$$602$$ 0 0
$$603$$ −5.52105 −0.224834
$$604$$ 0 0
$$605$$ −3.19892 −0.130055
$$606$$ 0 0
$$607$$ −10.8883 −0.441944 −0.220972 0.975280i $$-0.570923\pi$$
−0.220972 + 0.975280i $$0.570923\pi$$
$$608$$ 0 0
$$609$$ 9.79654 0.396976
$$610$$ 0 0
$$611$$ 23.4320 0.947955
$$612$$ 0 0
$$613$$ 3.35238 0.135401 0.0677007 0.997706i $$-0.478434\pi$$
0.0677007 + 0.997706i $$0.478434\pi$$
$$614$$ 0 0
$$615$$ 5.78954 0.233457
$$616$$ 0 0
$$617$$ −21.8848 −0.881050 −0.440525 0.897740i $$-0.645208\pi$$
−0.440525 + 0.897740i $$0.645208\pi$$
$$618$$ 0 0
$$619$$ 10.0762 0.404997 0.202498 0.979283i $$-0.435094\pi$$
0.202498 + 0.979283i $$0.435094\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 1.38681 0.0555613
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −17.5389 −0.700435
$$628$$ 0 0
$$629$$ 12.1496 0.484436
$$630$$ 0 0
$$631$$ −6.50485 −0.258954 −0.129477 0.991582i $$-0.541330\pi$$
−0.129477 + 0.991582i $$0.541330\pi$$
$$632$$ 0 0
$$633$$ 6.06903 0.241222
$$634$$ 0 0
$$635$$ −0.145425 −0.00577100
$$636$$ 0 0
$$637$$ 20.9450 0.829871
$$638$$ 0 0
$$639$$ −6.34452 −0.250985
$$640$$ 0 0
$$641$$ −1.60723 −0.0634816 −0.0317408 0.999496i $$-0.510105\pi$$
−0.0317408 + 0.999496i $$0.510105\pi$$
$$642$$ 0 0
$$643$$ 7.73605 0.305080 0.152540 0.988297i $$-0.451255\pi$$
0.152540 + 0.988297i $$0.451255\pi$$
$$644$$ 0 0
$$645$$ 4.54798 0.179076
$$646$$ 0 0
$$647$$ 8.89232 0.349593 0.174797 0.984605i $$-0.444073\pi$$
0.174797 + 0.984605i $$0.444073\pi$$
$$648$$ 0 0
$$649$$ −9.25452 −0.363272
$$650$$ 0 0
$$651$$ 8.84300 0.346584
$$652$$ 0 0
$$653$$ 31.2055 1.22117 0.610583 0.791952i $$-0.290935\pi$$
0.610583 + 0.791952i $$0.290935\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 15.3824 0.600126
$$658$$ 0 0
$$659$$ 21.7051 0.845509 0.422755 0.906244i $$-0.361063\pi$$
0.422755 + 0.906244i $$0.361063\pi$$
$$660$$ 0 0
$$661$$ 30.0564 1.16906 0.584529 0.811372i $$-0.301279\pi$$
0.584529 + 0.811372i $$0.301279\pi$$
$$662$$ 0 0
$$663$$ −29.2213 −1.13486
$$664$$ 0 0
$$665$$ −22.3015 −0.864814
$$666$$ 0 0
$$667$$ 2.75844 0.106807
$$668$$ 0 0
$$669$$ 1.57509 0.0608965
$$670$$ 0 0
$$671$$ 31.3322 1.20957
$$672$$ 0 0
$$673$$ 16.2795 0.627528 0.313764 0.949501i $$-0.398410\pi$$
0.313764 + 0.949501i $$0.398410\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −4.70723 −0.180914 −0.0904568 0.995900i $$-0.528833\pi$$
−0.0904568 + 0.995900i $$0.528833\pi$$
$$678$$ 0 0
$$679$$ 3.41616 0.131100
$$680$$ 0 0
$$681$$ 21.7034 0.831675
$$682$$ 0 0
$$683$$ −9.82536 −0.375957 −0.187978 0.982173i $$-0.560193\pi$$
−0.187978 + 0.982173i $$0.560193\pi$$
$$684$$ 0 0
$$685$$ −6.16450 −0.235533
$$686$$ 0 0
$$687$$ 23.2136 0.885654
$$688$$ 0 0
$$689$$ 33.1824 1.26415
$$690$$ 0 0
$$691$$ 43.1718 1.64233 0.821167 0.570689i $$-0.193324\pi$$
0.821167 + 0.570689i $$0.193324\pi$$
$$692$$ 0 0
$$693$$ 9.91943 0.376808
$$694$$ 0 0
$$695$$ −14.0690 −0.533669
$$696$$ 0 0
$$697$$ −45.3377 −1.71729
$$698$$ 0 0
$$699$$ −18.1989 −0.688346
$$700$$ 0 0
$$701$$ −1.44539 −0.0545915 −0.0272957 0.999627i $$-0.508690\pi$$
−0.0272957 + 0.999627i $$0.508690\pi$$
$$702$$ 0 0
$$703$$ 9.74250 0.367445
$$704$$ 0 0
$$705$$ 6.27949 0.236499
$$706$$ 0 0
$$707$$ −60.0324 −2.25775
$$708$$ 0 0
$$709$$ −28.9697 −1.08798 −0.543991 0.839091i $$-0.683087\pi$$
−0.543991 + 0.839091i $$0.683087\pi$$
$$710$$ 0 0
$$711$$ 9.62051 0.360797
$$712$$ 0 0
$$713$$ 2.48995 0.0932492
$$714$$ 0 0
$$715$$ 10.4223 0.389770
$$716$$ 0 0
$$717$$ −21.9236 −0.818752
$$718$$ 0 0
$$719$$ −42.4742 −1.58402 −0.792011 0.610507i $$-0.790966\pi$$
−0.792011 + 0.610507i $$0.790966\pi$$
$$720$$ 0 0
$$721$$ 21.4560 0.799064
$$722$$ 0 0
$$723$$ 17.7964 0.661854
$$724$$ 0 0
$$725$$ 2.75844 0.102446
$$726$$ 0 0
$$727$$ 3.10857 0.115291 0.0576453 0.998337i $$-0.481641\pi$$
0.0576453 + 0.998337i $$0.481641\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −35.6151 −1.31727
$$732$$ 0 0
$$733$$ 6.20453 0.229170 0.114585 0.993413i $$-0.463446\pi$$
0.114585 + 0.993413i $$0.463446\pi$$
$$734$$ 0 0
$$735$$ 5.61301 0.207039
$$736$$ 0 0
$$737$$ −15.4205 −0.568022
$$738$$ 0 0
$$739$$ 21.5010 0.790926 0.395463 0.918482i $$-0.370584\pi$$
0.395463 + 0.918482i $$0.370584\pi$$
$$740$$ 0 0
$$741$$ −23.4320 −0.860794
$$742$$ 0 0
$$743$$ 25.4469 0.933558 0.466779 0.884374i $$-0.345414\pi$$
0.466779 + 0.884374i $$0.345414\pi$$
$$744$$ 0 0
$$745$$ 13.7964 0.505460
$$746$$ 0 0
$$747$$ −5.83097 −0.213344
$$748$$ 0 0
$$749$$ −43.7332 −1.59798
$$750$$ 0 0
$$751$$ −49.9066 −1.82112 −0.910560 0.413378i $$-0.864349\pi$$
−0.910560 + 0.413378i $$0.864349\pi$$
$$752$$ 0 0
$$753$$ 23.2785 0.848314
$$754$$ 0 0
$$755$$ −16.1989 −0.589539
$$756$$ 0 0
$$757$$ 31.8486 1.15756 0.578778 0.815485i $$-0.303530\pi$$
0.578778 + 0.815485i $$0.303530\pi$$
$$758$$ 0 0
$$759$$ 2.79304 0.101381
$$760$$ 0 0
$$761$$ 7.63541 0.276783 0.138392 0.990378i $$-0.455807\pi$$
0.138392 + 0.990378i $$0.455807\pi$$
$$762$$ 0 0
$$763$$ 52.7675 1.91031
$$764$$ 0 0
$$765$$ −7.83097 −0.283129
$$766$$ 0 0
$$767$$ −12.3641 −0.446440
$$768$$ 0 0
$$769$$ −3.36880 −0.121482 −0.0607410 0.998154i $$-0.519346\pi$$
−0.0607410 + 0.998154i $$0.519346\pi$$
$$770$$ 0 0
$$771$$ 25.5884 0.921543
$$772$$ 0 0
$$773$$ 5.77800 0.207820 0.103910 0.994587i $$-0.466865\pi$$
0.103910 + 0.994587i $$0.466865\pi$$
$$774$$ 0 0
$$775$$ 2.48995 0.0894415
$$776$$ 0 0
$$777$$ −5.51005 −0.197672
$$778$$ 0 0
$$779$$ −36.3553 −1.30257
$$780$$ 0 0
$$781$$ −17.7205 −0.634090
$$782$$ 0 0
$$783$$ 2.75844 0.0985786
$$784$$ 0 0
$$785$$ −14.1105 −0.503624
$$786$$ 0 0
$$787$$ −6.46988 −0.230626 −0.115313 0.993329i $$-0.536787\pi$$
−0.115313 + 0.993329i $$0.536787\pi$$
$$788$$ 0 0
$$789$$ 8.57051 0.305118
$$790$$ 0 0
$$791$$ 11.4043 0.405489
$$792$$ 0 0
$$793$$ 41.8599 1.48649
$$794$$ 0 0
$$795$$ 8.89250 0.315385
$$796$$ 0 0
$$797$$ −21.9636 −0.777990 −0.388995 0.921240i $$-0.627178\pi$$
−0.388995 + 0.921240i $$0.627178\pi$$
$$798$$ 0 0
$$799$$ −49.1745 −1.73967
$$800$$ 0 0
$$801$$ 0.390487 0.0137972
$$802$$ 0 0
$$803$$ 42.9638 1.51616
$$804$$ 0 0
$$805$$ 3.55148 0.125173
$$806$$ 0 0
$$807$$ −13.4405 −0.473127
$$808$$ 0 0
$$809$$ −42.7222 −1.50203 −0.751017 0.660283i $$-0.770436\pi$$
−0.751017 + 0.660283i $$0.770436\pi$$
$$810$$ 0 0
$$811$$ 30.2347 1.06169 0.530843 0.847470i $$-0.321876\pi$$
0.530843 + 0.847470i $$0.321876\pi$$
$$812$$ 0 0
$$813$$ −3.38699 −0.118787
$$814$$ 0 0
$$815$$ −21.2480 −0.744286
$$816$$ 0 0
$$817$$ −28.5590 −0.999152
$$818$$ 0 0
$$819$$ 13.2524 0.463076
$$820$$ 0 0
$$821$$ −18.3260 −0.639581 −0.319791 0.947488i $$-0.603613\pi$$
−0.319791 + 0.947488i $$0.603613\pi$$
$$822$$ 0 0
$$823$$ 14.5081 0.505721 0.252861 0.967503i $$-0.418629\pi$$
0.252861 + 0.967503i $$0.418629\pi$$
$$824$$ 0 0
$$825$$ 2.79304 0.0972412
$$826$$ 0 0
$$827$$ 28.0452 0.975228 0.487614 0.873059i $$-0.337867\pi$$
0.487614 + 0.873059i $$0.337867\pi$$
$$828$$ 0 0
$$829$$ −26.2281 −0.910939 −0.455470 0.890251i $$-0.650529\pi$$
−0.455470 + 0.890251i $$0.650529\pi$$
$$830$$ 0 0
$$831$$ −18.7469 −0.650323
$$832$$ 0 0
$$833$$ −43.9553 −1.52296
$$834$$ 0 0
$$835$$ −19.3895 −0.671000
$$836$$ 0 0
$$837$$ 2.48995 0.0860651
$$838$$ 0 0
$$839$$ −12.2129 −0.421637 −0.210819 0.977525i $$-0.567613\pi$$
−0.210819 + 0.977525i $$0.567613\pi$$
$$840$$ 0 0
$$841$$ −21.3910 −0.737621
$$842$$ 0 0
$$843$$ −5.21396 −0.179578
$$844$$ 0 0
$$845$$ 0.924149 0.0317917
$$846$$ 0 0
$$847$$ −11.3609 −0.390365
$$848$$ 0 0
$$849$$ 18.9771 0.651291
$$850$$ 0 0
$$851$$ −1.55148 −0.0531841
$$852$$ 0 0
$$853$$ 54.2928 1.85895 0.929474 0.368886i $$-0.120261\pi$$
0.929474 + 0.368886i $$0.120261\pi$$
$$854$$ 0 0
$$855$$ −6.27949 −0.214754
$$856$$ 0 0
$$857$$ 15.0491 0.514067 0.257034 0.966402i $$-0.417255\pi$$
0.257034 + 0.966402i $$0.417255\pi$$
$$858$$ 0 0
$$859$$ −34.2351 −1.16809 −0.584043 0.811723i $$-0.698530\pi$$
−0.584043 + 0.811723i $$0.698530\pi$$
$$860$$ 0 0
$$861$$ 20.5614 0.700732
$$862$$ 0 0
$$863$$ −14.6488 −0.498652 −0.249326 0.968420i $$-0.580209\pi$$
−0.249326 + 0.968420i $$0.580209\pi$$
$$864$$ 0 0
$$865$$ −13.1030 −0.445514
$$866$$ 0 0
$$867$$ 44.3240 1.50532
$$868$$ 0 0
$$869$$ 26.8705 0.911518
$$870$$ 0 0
$$871$$ −20.6018 −0.698066
$$872$$ 0 0
$$873$$ 0.961896 0.0325552
$$874$$ 0 0
$$875$$ 3.55148 0.120062
$$876$$ 0 0
$$877$$ −54.2319 −1.83128 −0.915641 0.401998i $$-0.868316\pi$$
−0.915641 + 0.401998i $$0.868316\pi$$
$$878$$ 0 0
$$879$$ −27.3976 −0.924098
$$880$$ 0 0
$$881$$ 44.3162 1.49305 0.746525 0.665357i $$-0.231721\pi$$
0.746525 + 0.665357i $$0.231721\pi$$
$$882$$ 0 0
$$883$$ −38.2704 −1.28790 −0.643951 0.765067i $$-0.722706\pi$$
−0.643951 + 0.765067i $$0.722706\pi$$
$$884$$ 0 0
$$885$$ −3.31342 −0.111379
$$886$$ 0 0
$$887$$ 5.80699 0.194980 0.0974898 0.995237i $$-0.468919\pi$$
0.0974898 + 0.995237i $$0.468919\pi$$
$$888$$ 0 0
$$889$$ −0.516473 −0.0173219
$$890$$ 0 0
$$891$$ 2.79304 0.0935704
$$892$$ 0 0
$$893$$ −39.4320 −1.31954
$$894$$ 0 0
$$895$$ 14.0759 0.470504
$$896$$ 0 0
$$897$$ 3.73151 0.124591
$$898$$ 0 0
$$899$$ 6.86837 0.229073
$$900$$ 0 0
$$901$$ −69.6369 −2.31994
$$902$$ 0 0
$$903$$ 16.1521 0.537507
$$904$$ 0 0
$$905$$ 21.1374 0.702630
$$906$$ 0 0
$$907$$ 12.3419 0.409805 0.204903 0.978782i $$-0.434312\pi$$
0.204903 + 0.978782i $$0.434312\pi$$
$$908$$ 0 0
$$909$$ −16.9035 −0.560654
$$910$$ 0 0
$$911$$ −27.2190 −0.901807 −0.450903 0.892573i $$-0.648898\pi$$
−0.450903 + 0.892573i $$0.648898\pi$$
$$912$$ 0 0
$$913$$ −16.2861 −0.538992
$$914$$ 0 0
$$915$$ 11.2180 0.370854
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −38.6836 −1.27605 −0.638027 0.770014i $$-0.720249\pi$$
−0.638027 + 0.770014i $$0.720249\pi$$
$$920$$ 0 0
$$921$$ −1.18353 −0.0389986
$$922$$ 0 0
$$923$$ −23.6746 −0.779260
$$924$$ 0 0
$$925$$ −1.55148 −0.0510124
$$926$$ 0 0
$$927$$ 6.04143 0.198427
$$928$$ 0 0
$$929$$ −17.1365 −0.562230 −0.281115 0.959674i $$-0.590704\pi$$
−0.281115 + 0.959674i $$0.590704\pi$$
$$930$$ 0 0
$$931$$ −35.2468 −1.15517
$$932$$ 0 0
$$933$$ −18.5770 −0.608183
$$934$$ 0 0
$$935$$ −21.8722 −0.715298
$$936$$ 0 0
$$937$$ 44.4382 1.45173 0.725866 0.687836i $$-0.241439\pi$$
0.725866 + 0.687836i $$0.241439\pi$$
$$938$$ 0 0
$$939$$ −8.50094 −0.277418
$$940$$ 0 0
$$941$$ −19.1745 −0.625069 −0.312535 0.949906i $$-0.601178\pi$$
−0.312535 + 0.949906i $$0.601178\pi$$
$$942$$ 0 0
$$943$$ 5.78954 0.188533
$$944$$ 0 0
$$945$$ 3.55148 0.115530
$$946$$ 0 0
$$947$$ −56.7662 −1.84465 −0.922327 0.386410i $$-0.873715\pi$$
−0.922327 + 0.386410i $$0.873715\pi$$
$$948$$ 0 0
$$949$$ 57.3997 1.86327
$$950$$ 0 0
$$951$$ 21.5884 0.700051
$$952$$ 0 0
$$953$$ 37.8953 1.22755 0.613774 0.789482i $$-0.289651\pi$$
0.613774 + 0.789482i $$0.289651\pi$$
$$954$$ 0 0
$$955$$ −2.81647 −0.0911389
$$956$$ 0 0
$$957$$ 7.70443 0.249049
$$958$$ 0 0
$$959$$ −21.8931 −0.706965
$$960$$ 0 0
$$961$$ −24.8002 −0.800005
$$962$$ 0 0
$$963$$ −12.3141 −0.396816
$$964$$ 0 0
$$965$$ 15.4560 0.497547
$$966$$ 0 0
$$967$$ −58.9924 −1.89707 −0.948534 0.316674i $$-0.897434\pi$$
−0.948534 + 0.316674i $$0.897434\pi$$
$$968$$ 0 0
$$969$$ 49.1745 1.57971
$$970$$ 0 0
$$971$$ −19.9675 −0.640787 −0.320394 0.947284i $$-0.603815\pi$$
−0.320394 + 0.947284i $$0.603815\pi$$
$$972$$ 0 0
$$973$$ −49.9659 −1.60183
$$974$$ 0 0
$$975$$ 3.73151 0.119504
$$976$$ 0 0
$$977$$ −5.54712 −0.177468 −0.0887341 0.996055i $$-0.528282\pi$$
−0.0887341 + 0.996055i $$0.528282\pi$$
$$978$$ 0 0
$$979$$ 1.09065 0.0348572
$$980$$ 0 0
$$981$$ 14.8579 0.474376
$$982$$ 0 0
$$983$$ −12.1636 −0.387959 −0.193979 0.981006i $$-0.562139\pi$$
−0.193979 + 0.981006i $$0.562139\pi$$
$$984$$ 0 0
$$985$$ 25.7311 0.819862
$$986$$ 0 0
$$987$$ 22.3015 0.709864
$$988$$ 0 0
$$989$$ 4.54798 0.144617
$$990$$ 0 0
$$991$$ −18.1991 −0.578113 −0.289057 0.957312i $$-0.593342\pi$$
−0.289057 + 0.957312i $$0.593342\pi$$
$$992$$ 0 0
$$993$$ −24.0288 −0.762531
$$994$$ 0 0
$$995$$ −5.95857 −0.188899
$$996$$ 0 0
$$997$$ 49.4097 1.56482 0.782411 0.622762i $$-0.213990\pi$$
0.782411 + 0.622762i $$0.213990\pi$$
$$998$$ 0 0
$$999$$ −1.55148 −0.0490867
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 5520.2.a.cc.1.4 5
4.3 odd 2 2760.2.a.w.1.2 5
12.11 even 2 8280.2.a.br.1.2 5

By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.w.1.2 5 4.3 odd 2
5520.2.a.cc.1.4 5 1.1 even 1 trivial
8280.2.a.br.1.2 5 12.11 even 2