# Properties

 Label 1176.2.a.o.1.2 Level $1176$ Weight $2$ Character 1176.1 Self dual yes Analytic conductor $9.390$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1176.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.39040727770$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1176.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +3.41421 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +3.41421 q^{5} +1.00000 q^{9} +4.82843 q^{11} +1.41421 q^{13} +3.41421 q^{15} -6.24264 q^{17} +1.17157 q^{19} -0.828427 q^{23} +6.65685 q^{25} +1.00000 q^{27} -8.48528 q^{29} +10.8284 q^{31} +4.82843 q^{33} -9.65685 q^{37} +1.41421 q^{39} -3.41421 q^{41} -8.00000 q^{43} +3.41421 q^{45} +1.17157 q^{47} -6.24264 q^{51} +9.31371 q^{53} +16.4853 q^{55} +1.17157 q^{57} +10.8284 q^{59} -5.89949 q^{61} +4.82843 q^{65} -8.00000 q^{67} -0.828427 q^{69} +4.82843 q^{71} +3.07107 q^{73} +6.65685 q^{75} -13.6569 q^{79} +1.00000 q^{81} -7.31371 q^{83} -21.3137 q^{85} -8.48528 q^{87} +14.7279 q^{89} +10.8284 q^{93} +4.00000 q^{95} -16.2426 q^{97} +4.82843 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} + 4q^{11} + 4q^{15} - 4q^{17} + 8q^{19} + 4q^{23} + 2q^{25} + 2q^{27} + 16q^{31} + 4q^{33} - 8q^{37} - 4q^{41} - 16q^{43} + 4q^{45} + 8q^{47} - 4q^{51} - 4q^{53} + 16q^{55} + 8q^{57} + 16q^{59} + 8q^{61} + 4q^{65} - 16q^{67} + 4q^{69} + 4q^{71} - 8q^{73} + 2q^{75} - 16q^{79} + 2q^{81} + 8q^{83} - 20q^{85} + 4q^{89} + 16q^{93} + 8q^{95} - 24q^{97} + 4q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 3.41421 1.52688 0.763441 0.645877i $$-0.223508\pi$$
0.763441 + 0.645877i $$0.223508\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.82843 1.45583 0.727913 0.685670i $$-0.240491\pi$$
0.727913 + 0.685670i $$0.240491\pi$$
$$12$$ 0 0
$$13$$ 1.41421 0.392232 0.196116 0.980581i $$-0.437167\pi$$
0.196116 + 0.980581i $$0.437167\pi$$
$$14$$ 0 0
$$15$$ 3.41421 0.881546
$$16$$ 0 0
$$17$$ −6.24264 −1.51406 −0.757031 0.653379i $$-0.773351\pi$$
−0.757031 + 0.653379i $$0.773351\pi$$
$$18$$ 0 0
$$19$$ 1.17157 0.268777 0.134389 0.990929i $$-0.457093\pi$$
0.134389 + 0.990929i $$0.457093\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.828427 −0.172739 −0.0863695 0.996263i $$-0.527527\pi$$
−0.0863695 + 0.996263i $$0.527527\pi$$
$$24$$ 0 0
$$25$$ 6.65685 1.33137
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −8.48528 −1.57568 −0.787839 0.615882i $$-0.788800\pi$$
−0.787839 + 0.615882i $$0.788800\pi$$
$$30$$ 0 0
$$31$$ 10.8284 1.94484 0.972421 0.233231i $$-0.0749297\pi$$
0.972421 + 0.233231i $$0.0749297\pi$$
$$32$$ 0 0
$$33$$ 4.82843 0.840521
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −9.65685 −1.58758 −0.793789 0.608194i $$-0.791894\pi$$
−0.793789 + 0.608194i $$0.791894\pi$$
$$38$$ 0 0
$$39$$ 1.41421 0.226455
$$40$$ 0 0
$$41$$ −3.41421 −0.533211 −0.266605 0.963806i $$-0.585902\pi$$
−0.266605 + 0.963806i $$0.585902\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 3.41421 0.508961
$$46$$ 0 0
$$47$$ 1.17157 0.170891 0.0854457 0.996343i $$-0.472769\pi$$
0.0854457 + 0.996343i $$0.472769\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −6.24264 −0.874145
$$52$$ 0 0
$$53$$ 9.31371 1.27934 0.639668 0.768651i $$-0.279072\pi$$
0.639668 + 0.768651i $$0.279072\pi$$
$$54$$ 0 0
$$55$$ 16.4853 2.22287
$$56$$ 0 0
$$57$$ 1.17157 0.155179
$$58$$ 0 0
$$59$$ 10.8284 1.40974 0.704871 0.709336i $$-0.251005\pi$$
0.704871 + 0.709336i $$0.251005\pi$$
$$60$$ 0 0
$$61$$ −5.89949 −0.755353 −0.377676 0.925938i $$-0.623277\pi$$
−0.377676 + 0.925938i $$0.623277\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 4.82843 0.598893
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 0 0
$$69$$ −0.828427 −0.0997309
$$70$$ 0 0
$$71$$ 4.82843 0.573029 0.286514 0.958076i $$-0.407503\pi$$
0.286514 + 0.958076i $$0.407503\pi$$
$$72$$ 0 0
$$73$$ 3.07107 0.359441 0.179721 0.983718i $$-0.442481\pi$$
0.179721 + 0.983718i $$0.442481\pi$$
$$74$$ 0 0
$$75$$ 6.65685 0.768667
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −13.6569 −1.53652 −0.768258 0.640140i $$-0.778876\pi$$
−0.768258 + 0.640140i $$0.778876\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −7.31371 −0.802784 −0.401392 0.915906i $$-0.631473\pi$$
−0.401392 + 0.915906i $$0.631473\pi$$
$$84$$ 0 0
$$85$$ −21.3137 −2.31180
$$86$$ 0 0
$$87$$ −8.48528 −0.909718
$$88$$ 0 0
$$89$$ 14.7279 1.56116 0.780578 0.625058i $$-0.214925\pi$$
0.780578 + 0.625058i $$0.214925\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 10.8284 1.12286
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ −16.2426 −1.64919 −0.824595 0.565723i $$-0.808597\pi$$
−0.824595 + 0.565723i $$0.808597\pi$$
$$98$$ 0 0
$$99$$ 4.82843 0.485275
$$100$$ 0 0
$$101$$ 0.585786 0.0582879 0.0291440 0.999575i $$-0.490722\pi$$
0.0291440 + 0.999575i $$0.490722\pi$$
$$102$$ 0 0
$$103$$ 5.17157 0.509570 0.254785 0.966998i $$-0.417995\pi$$
0.254785 + 0.966998i $$0.417995\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −2.48528 −0.240261 −0.120131 0.992758i $$-0.538331\pi$$
−0.120131 + 0.992758i $$0.538331\pi$$
$$108$$ 0 0
$$109$$ 11.3137 1.08366 0.541828 0.840489i $$-0.317732\pi$$
0.541828 + 0.840489i $$0.317732\pi$$
$$110$$ 0 0
$$111$$ −9.65685 −0.916588
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −2.82843 −0.263752
$$116$$ 0 0
$$117$$ 1.41421 0.130744
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 12.3137 1.11943
$$122$$ 0 0
$$123$$ −3.41421 −0.307849
$$124$$ 0 0
$$125$$ 5.65685 0.505964
$$126$$ 0 0
$$127$$ 7.31371 0.648987 0.324493 0.945888i $$-0.394806\pi$$
0.324493 + 0.945888i $$0.394806\pi$$
$$128$$ 0 0
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ 15.3137 1.33796 0.668982 0.743278i $$-0.266730\pi$$
0.668982 + 0.743278i $$0.266730\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 3.41421 0.293849
$$136$$ 0 0
$$137$$ −12.4853 −1.06669 −0.533345 0.845898i $$-0.679065\pi$$
−0.533345 + 0.845898i $$0.679065\pi$$
$$138$$ 0 0
$$139$$ 9.65685 0.819084 0.409542 0.912291i $$-0.365689\pi$$
0.409542 + 0.912291i $$0.365689\pi$$
$$140$$ 0 0
$$141$$ 1.17157 0.0986642
$$142$$ 0 0
$$143$$ 6.82843 0.571022
$$144$$ 0 0
$$145$$ −28.9706 −2.40587
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ −1.65685 −0.134833 −0.0674164 0.997725i $$-0.521476\pi$$
−0.0674164 + 0.997725i $$0.521476\pi$$
$$152$$ 0 0
$$153$$ −6.24264 −0.504688
$$154$$ 0 0
$$155$$ 36.9706 2.96955
$$156$$ 0 0
$$157$$ 5.89949 0.470831 0.235415 0.971895i $$-0.424355\pi$$
0.235415 + 0.971895i $$0.424355\pi$$
$$158$$ 0 0
$$159$$ 9.31371 0.738625
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −2.34315 −0.183529 −0.0917647 0.995781i $$-0.529251\pi$$
−0.0917647 + 0.995781i $$0.529251\pi$$
$$164$$ 0 0
$$165$$ 16.4853 1.28338
$$166$$ 0 0
$$167$$ 6.82843 0.528400 0.264200 0.964468i $$-0.414892\pi$$
0.264200 + 0.964468i $$0.414892\pi$$
$$168$$ 0 0
$$169$$ −11.0000 −0.846154
$$170$$ 0 0
$$171$$ 1.17157 0.0895924
$$172$$ 0 0
$$173$$ −0.585786 −0.0445365 −0.0222683 0.999752i $$-0.507089\pi$$
−0.0222683 + 0.999752i $$0.507089\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 10.8284 0.813914
$$178$$ 0 0
$$179$$ −21.7990 −1.62933 −0.814667 0.579930i $$-0.803080\pi$$
−0.814667 + 0.579930i $$0.803080\pi$$
$$180$$ 0 0
$$181$$ 9.89949 0.735824 0.367912 0.929861i $$-0.380073\pi$$
0.367912 + 0.929861i $$0.380073\pi$$
$$182$$ 0 0
$$183$$ −5.89949 −0.436103
$$184$$ 0 0
$$185$$ −32.9706 −2.42404
$$186$$ 0 0
$$187$$ −30.1421 −2.20421
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 20.8284 1.50709 0.753546 0.657395i $$-0.228342\pi$$
0.753546 + 0.657395i $$0.228342\pi$$
$$192$$ 0 0
$$193$$ −20.6274 −1.48479 −0.742397 0.669960i $$-0.766311\pi$$
−0.742397 + 0.669960i $$0.766311\pi$$
$$194$$ 0 0
$$195$$ 4.82843 0.345771
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ −5.65685 −0.401004 −0.200502 0.979693i $$-0.564257\pi$$
−0.200502 + 0.979693i $$0.564257\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −11.6569 −0.814150
$$206$$ 0 0
$$207$$ −0.828427 −0.0575797
$$208$$ 0 0
$$209$$ 5.65685 0.391293
$$210$$ 0 0
$$211$$ −25.6569 −1.76629 −0.883145 0.469099i $$-0.844579\pi$$
−0.883145 + 0.469099i $$0.844579\pi$$
$$212$$ 0 0
$$213$$ 4.82843 0.330838
$$214$$ 0 0
$$215$$ −27.3137 −1.86278
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 3.07107 0.207524
$$220$$ 0 0
$$221$$ −8.82843 −0.593864
$$222$$ 0 0
$$223$$ −2.34315 −0.156909 −0.0784543 0.996918i $$-0.524998\pi$$
−0.0784543 + 0.996918i $$0.524998\pi$$
$$224$$ 0 0
$$225$$ 6.65685 0.443790
$$226$$ 0 0
$$227$$ 19.7990 1.31411 0.657053 0.753845i $$-0.271803\pi$$
0.657053 + 0.753845i $$0.271803\pi$$
$$228$$ 0 0
$$229$$ 0.928932 0.0613856 0.0306928 0.999529i $$-0.490229\pi$$
0.0306928 + 0.999529i $$0.490229\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −11.5147 −0.754354 −0.377177 0.926141i $$-0.623105\pi$$
−0.377177 + 0.926141i $$0.623105\pi$$
$$234$$ 0 0
$$235$$ 4.00000 0.260931
$$236$$ 0 0
$$237$$ −13.6569 −0.887108
$$238$$ 0 0
$$239$$ −8.82843 −0.571063 −0.285532 0.958369i $$-0.592170\pi$$
−0.285532 + 0.958369i $$0.592170\pi$$
$$240$$ 0 0
$$241$$ −2.10051 −0.135305 −0.0676527 0.997709i $$-0.521551\pi$$
−0.0676527 + 0.997709i $$0.521551\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.65685 0.105423
$$248$$ 0 0
$$249$$ −7.31371 −0.463487
$$250$$ 0 0
$$251$$ −8.48528 −0.535586 −0.267793 0.963476i $$-0.586294\pi$$
−0.267793 + 0.963476i $$0.586294\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 0 0
$$255$$ −21.3137 −1.33472
$$256$$ 0 0
$$257$$ −21.7574 −1.35719 −0.678593 0.734514i $$-0.737410\pi$$
−0.678593 + 0.734514i $$0.737410\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −8.48528 −0.525226
$$262$$ 0 0
$$263$$ 19.1716 1.18217 0.591085 0.806609i $$-0.298700\pi$$
0.591085 + 0.806609i $$0.298700\pi$$
$$264$$ 0 0
$$265$$ 31.7990 1.95340
$$266$$ 0 0
$$267$$ 14.7279 0.901334
$$268$$ 0 0
$$269$$ −18.0416 −1.10002 −0.550009 0.835159i $$-0.685376\pi$$
−0.550009 + 0.835159i $$0.685376\pi$$
$$270$$ 0 0
$$271$$ −18.8284 −1.14375 −0.571873 0.820342i $$-0.693783\pi$$
−0.571873 + 0.820342i $$0.693783\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 32.1421 1.93824
$$276$$ 0 0
$$277$$ −6.00000 −0.360505 −0.180253 0.983620i $$-0.557691\pi$$
−0.180253 + 0.983620i $$0.557691\pi$$
$$278$$ 0 0
$$279$$ 10.8284 0.648281
$$280$$ 0 0
$$281$$ 4.48528 0.267569 0.133785 0.991010i $$-0.457287\pi$$
0.133785 + 0.991010i $$0.457287\pi$$
$$282$$ 0 0
$$283$$ −9.17157 −0.545193 −0.272597 0.962128i $$-0.587882\pi$$
−0.272597 + 0.962128i $$0.587882\pi$$
$$284$$ 0 0
$$285$$ 4.00000 0.236940
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 21.9706 1.29239
$$290$$ 0 0
$$291$$ −16.2426 −0.952160
$$292$$ 0 0
$$293$$ −13.0711 −0.763620 −0.381810 0.924241i $$-0.624699\pi$$
−0.381810 + 0.924241i $$0.624699\pi$$
$$294$$ 0 0
$$295$$ 36.9706 2.15251
$$296$$ 0 0
$$297$$ 4.82843 0.280174
$$298$$ 0 0
$$299$$ −1.17157 −0.0677538
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0.585786 0.0336526
$$304$$ 0 0
$$305$$ −20.1421 −1.15334
$$306$$ 0 0
$$307$$ −28.4853 −1.62574 −0.812870 0.582445i $$-0.802096\pi$$
−0.812870 + 0.582445i $$0.802096\pi$$
$$308$$ 0 0
$$309$$ 5.17157 0.294201
$$310$$ 0 0
$$311$$ 2.14214 0.121469 0.0607347 0.998154i $$-0.480656\pi$$
0.0607347 + 0.998154i $$0.480656\pi$$
$$312$$ 0 0
$$313$$ −14.5858 −0.824437 −0.412219 0.911085i $$-0.635246\pi$$
−0.412219 + 0.911085i $$0.635246\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1.31371 −0.0737852 −0.0368926 0.999319i $$-0.511746\pi$$
−0.0368926 + 0.999319i $$0.511746\pi$$
$$318$$ 0 0
$$319$$ −40.9706 −2.29391
$$320$$ 0 0
$$321$$ −2.48528 −0.138715
$$322$$ 0 0
$$323$$ −7.31371 −0.406946
$$324$$ 0 0
$$325$$ 9.41421 0.522207
$$326$$ 0 0
$$327$$ 11.3137 0.625650
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −31.3137 −1.72116 −0.860579 0.509318i $$-0.829898\pi$$
−0.860579 + 0.509318i $$0.829898\pi$$
$$332$$ 0 0
$$333$$ −9.65685 −0.529192
$$334$$ 0 0
$$335$$ −27.3137 −1.49231
$$336$$ 0 0
$$337$$ 16.9706 0.924445 0.462223 0.886764i $$-0.347052\pi$$
0.462223 + 0.886764i $$0.347052\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 52.2843 2.83135
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −2.82843 −0.152277
$$346$$ 0 0
$$347$$ 24.1421 1.29602 0.648009 0.761633i $$-0.275602\pi$$
0.648009 + 0.761633i $$0.275602\pi$$
$$348$$ 0 0
$$349$$ −6.38478 −0.341769 −0.170885 0.985291i $$-0.554663\pi$$
−0.170885 + 0.985291i $$0.554663\pi$$
$$350$$ 0 0
$$351$$ 1.41421 0.0754851
$$352$$ 0 0
$$353$$ 3.89949 0.207549 0.103775 0.994601i $$-0.466908\pi$$
0.103775 + 0.994601i $$0.466908\pi$$
$$354$$ 0 0
$$355$$ 16.4853 0.874948
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 15.4558 0.815728 0.407864 0.913043i $$-0.366274\pi$$
0.407864 + 0.913043i $$0.366274\pi$$
$$360$$ 0 0
$$361$$ −17.6274 −0.927759
$$362$$ 0 0
$$363$$ 12.3137 0.646302
$$364$$ 0 0
$$365$$ 10.4853 0.548825
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ −3.41421 −0.177737
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 37.3137 1.93203 0.966015 0.258485i $$-0.0832232\pi$$
0.966015 + 0.258485i $$0.0832232\pi$$
$$374$$ 0 0
$$375$$ 5.65685 0.292119
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ −23.3137 −1.19754 −0.598772 0.800919i $$-0.704345\pi$$
−0.598772 + 0.800919i $$0.704345\pi$$
$$380$$ 0 0
$$381$$ 7.31371 0.374693
$$382$$ 0 0
$$383$$ −8.97056 −0.458374 −0.229187 0.973382i $$-0.573607\pi$$
−0.229187 + 0.973382i $$0.573607\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.00000 −0.406663
$$388$$ 0 0
$$389$$ 14.1421 0.717035 0.358517 0.933523i $$-0.383282\pi$$
0.358517 + 0.933523i $$0.383282\pi$$
$$390$$ 0 0
$$391$$ 5.17157 0.261538
$$392$$ 0 0
$$393$$ 15.3137 0.772474
$$394$$ 0 0
$$395$$ −46.6274 −2.34608
$$396$$ 0 0
$$397$$ 32.7279 1.64257 0.821284 0.570520i $$-0.193258\pi$$
0.821284 + 0.570520i $$0.193258\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4.48528 0.223984 0.111992 0.993709i $$-0.464277\pi$$
0.111992 + 0.993709i $$0.464277\pi$$
$$402$$ 0 0
$$403$$ 15.3137 0.762830
$$404$$ 0 0
$$405$$ 3.41421 0.169654
$$406$$ 0 0
$$407$$ −46.6274 −2.31124
$$408$$ 0 0
$$409$$ 7.75736 0.383577 0.191788 0.981436i $$-0.438571\pi$$
0.191788 + 0.981436i $$0.438571\pi$$
$$410$$ 0 0
$$411$$ −12.4853 −0.615854
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −24.9706 −1.22576
$$416$$ 0 0
$$417$$ 9.65685 0.472898
$$418$$ 0 0
$$419$$ 5.17157 0.252648 0.126324 0.991989i $$-0.459682\pi$$
0.126324 + 0.991989i $$0.459682\pi$$
$$420$$ 0 0
$$421$$ 29.3137 1.42866 0.714331 0.699808i $$-0.246731\pi$$
0.714331 + 0.699808i $$0.246731\pi$$
$$422$$ 0 0
$$423$$ 1.17157 0.0569638
$$424$$ 0 0
$$425$$ −41.5563 −2.01578
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 6.82843 0.329680
$$430$$ 0 0
$$431$$ −13.5147 −0.650981 −0.325491 0.945545i $$-0.605529\pi$$
−0.325491 + 0.945545i $$0.605529\pi$$
$$432$$ 0 0
$$433$$ −10.3848 −0.499061 −0.249530 0.968367i $$-0.580276\pi$$
−0.249530 + 0.968367i $$0.580276\pi$$
$$434$$ 0 0
$$435$$ −28.9706 −1.38903
$$436$$ 0 0
$$437$$ −0.970563 −0.0464283
$$438$$ 0 0
$$439$$ 19.3137 0.921793 0.460897 0.887454i $$-0.347528\pi$$
0.460897 + 0.887454i $$0.347528\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 21.5147 1.02220 0.511098 0.859523i $$-0.329239\pi$$
0.511098 + 0.859523i $$0.329239\pi$$
$$444$$ 0 0
$$445$$ 50.2843 2.38370
$$446$$ 0 0
$$447$$ −10.0000 −0.472984
$$448$$ 0 0
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −16.4853 −0.776262
$$452$$ 0 0
$$453$$ −1.65685 −0.0778458
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −24.6274 −1.15202 −0.576011 0.817442i $$-0.695391\pi$$
−0.576011 + 0.817442i $$0.695391\pi$$
$$458$$ 0 0
$$459$$ −6.24264 −0.291382
$$460$$ 0 0
$$461$$ 9.75736 0.454446 0.227223 0.973843i $$-0.427035\pi$$
0.227223 + 0.973843i $$0.427035\pi$$
$$462$$ 0 0
$$463$$ −12.9706 −0.602793 −0.301397 0.953499i $$-0.597453\pi$$
−0.301397 + 0.953499i $$0.597453\pi$$
$$464$$ 0 0
$$465$$ 36.9706 1.71447
$$466$$ 0 0
$$467$$ −5.17157 −0.239312 −0.119656 0.992815i $$-0.538179\pi$$
−0.119656 + 0.992815i $$0.538179\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 5.89949 0.271834
$$472$$ 0 0
$$473$$ −38.6274 −1.77609
$$474$$ 0 0
$$475$$ 7.79899 0.357842
$$476$$ 0 0
$$477$$ 9.31371 0.426445
$$478$$ 0 0
$$479$$ 19.1127 0.873281 0.436641 0.899636i $$-0.356168\pi$$
0.436641 + 0.899636i $$0.356168\pi$$
$$480$$ 0 0
$$481$$ −13.6569 −0.622699
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −55.4558 −2.51812
$$486$$ 0 0
$$487$$ 12.9706 0.587752 0.293876 0.955844i $$-0.405055\pi$$
0.293876 + 0.955844i $$0.405055\pi$$
$$488$$ 0 0
$$489$$ −2.34315 −0.105961
$$490$$ 0 0
$$491$$ −4.14214 −0.186932 −0.0934660 0.995622i $$-0.529795\pi$$
−0.0934660 + 0.995622i $$0.529795\pi$$
$$492$$ 0 0
$$493$$ 52.9706 2.38567
$$494$$ 0 0
$$495$$ 16.4853 0.740958
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 28.2843 1.26618 0.633089 0.774079i $$-0.281787\pi$$
0.633089 + 0.774079i $$0.281787\pi$$
$$500$$ 0 0
$$501$$ 6.82843 0.305072
$$502$$ 0 0
$$503$$ 30.6274 1.36561 0.682805 0.730601i $$-0.260760\pi$$
0.682805 + 0.730601i $$0.260760\pi$$
$$504$$ 0 0
$$505$$ 2.00000 0.0889988
$$506$$ 0 0
$$507$$ −11.0000 −0.488527
$$508$$ 0 0
$$509$$ 22.9289 1.01631 0.508154 0.861267i $$-0.330328\pi$$
0.508154 + 0.861267i $$0.330328\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 1.17157 0.0517262
$$514$$ 0 0
$$515$$ 17.6569 0.778054
$$516$$ 0 0
$$517$$ 5.65685 0.248788
$$518$$ 0 0
$$519$$ −0.585786 −0.0257132
$$520$$ 0 0
$$521$$ 5.27208 0.230974 0.115487 0.993309i $$-0.463157\pi$$
0.115487 + 0.993309i $$0.463157\pi$$
$$522$$ 0 0
$$523$$ −1.65685 −0.0724492 −0.0362246 0.999344i $$-0.511533\pi$$
−0.0362246 + 0.999344i $$0.511533\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −67.5980 −2.94461
$$528$$ 0 0
$$529$$ −22.3137 −0.970161
$$530$$ 0 0
$$531$$ 10.8284 0.469914
$$532$$ 0 0
$$533$$ −4.82843 −0.209142
$$534$$ 0 0
$$535$$ −8.48528 −0.366851
$$536$$ 0 0
$$537$$ −21.7990 −0.940696
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −8.62742 −0.370922 −0.185461 0.982652i $$-0.559378\pi$$
−0.185461 + 0.982652i $$0.559378\pi$$
$$542$$ 0 0
$$543$$ 9.89949 0.424828
$$544$$ 0 0
$$545$$ 38.6274 1.65462
$$546$$ 0 0
$$547$$ −4.97056 −0.212526 −0.106263 0.994338i $$-0.533889\pi$$
−0.106263 + 0.994338i $$0.533889\pi$$
$$548$$ 0 0
$$549$$ −5.89949 −0.251784
$$550$$ 0 0
$$551$$ −9.94113 −0.423506
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −32.9706 −1.39952
$$556$$ 0 0
$$557$$ 43.9411 1.86185 0.930923 0.365216i $$-0.119005\pi$$
0.930923 + 0.365216i $$0.119005\pi$$
$$558$$ 0 0
$$559$$ −11.3137 −0.478519
$$560$$ 0 0
$$561$$ −30.1421 −1.27260
$$562$$ 0 0
$$563$$ 26.8284 1.13068 0.565342 0.824857i $$-0.308744\pi$$
0.565342 + 0.824857i $$0.308744\pi$$
$$564$$ 0 0
$$565$$ 20.4853 0.861822
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.82843 0.286263 0.143131 0.989704i $$-0.454283\pi$$
0.143131 + 0.989704i $$0.454283\pi$$
$$570$$ 0 0
$$571$$ 40.2843 1.68584 0.842922 0.538036i $$-0.180833\pi$$
0.842922 + 0.538036i $$0.180833\pi$$
$$572$$ 0 0
$$573$$ 20.8284 0.870120
$$574$$ 0 0
$$575$$ −5.51472 −0.229980
$$576$$ 0 0
$$577$$ −9.41421 −0.391919 −0.195959 0.980612i $$-0.562782\pi$$
−0.195959 + 0.980612i $$0.562782\pi$$
$$578$$ 0 0
$$579$$ −20.6274 −0.857246
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 44.9706 1.86249
$$584$$ 0 0
$$585$$ 4.82843 0.199631
$$586$$ 0 0
$$587$$ −26.8284 −1.10733 −0.553664 0.832740i $$-0.686771\pi$$
−0.553664 + 0.832740i $$0.686771\pi$$
$$588$$ 0 0
$$589$$ 12.6863 0.522730
$$590$$ 0 0
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ 29.0711 1.19381 0.596903 0.802314i $$-0.296398\pi$$
0.596903 + 0.802314i $$0.296398\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −5.65685 −0.231520
$$598$$ 0 0
$$599$$ 16.1421 0.659550 0.329775 0.944060i $$-0.393027\pi$$
0.329775 + 0.944060i $$0.393027\pi$$
$$600$$ 0 0
$$601$$ −12.2426 −0.499388 −0.249694 0.968325i $$-0.580330\pi$$
−0.249694 + 0.968325i $$0.580330\pi$$
$$602$$ 0 0
$$603$$ −8.00000 −0.325785
$$604$$ 0 0
$$605$$ 42.0416 1.70924
$$606$$ 0 0
$$607$$ 31.5980 1.28252 0.641261 0.767323i $$-0.278412\pi$$
0.641261 + 0.767323i $$0.278412\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.65685 0.0670291
$$612$$ 0 0
$$613$$ −2.34315 −0.0946388 −0.0473194 0.998880i $$-0.515068\pi$$
−0.0473194 + 0.998880i $$0.515068\pi$$
$$614$$ 0 0
$$615$$ −11.6569 −0.470050
$$616$$ 0 0
$$617$$ 21.4558 0.863780 0.431890 0.901926i $$-0.357847\pi$$
0.431890 + 0.901926i $$0.357847\pi$$
$$618$$ 0 0
$$619$$ −40.2843 −1.61916 −0.809581 0.587008i $$-0.800306\pi$$
−0.809581 + 0.587008i $$0.800306\pi$$
$$620$$ 0 0
$$621$$ −0.828427 −0.0332436
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −13.9706 −0.558823
$$626$$ 0 0
$$627$$ 5.65685 0.225913
$$628$$ 0 0
$$629$$ 60.2843 2.40369
$$630$$ 0 0
$$631$$ −8.28427 −0.329792 −0.164896 0.986311i $$-0.552729\pi$$
−0.164896 + 0.986311i $$0.552729\pi$$
$$632$$ 0 0
$$633$$ −25.6569 −1.01977
$$634$$ 0 0
$$635$$ 24.9706 0.990927
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 4.82843 0.191010
$$640$$ 0 0
$$641$$ 25.1716 0.994217 0.497109 0.867688i $$-0.334395\pi$$
0.497109 + 0.867688i $$0.334395\pi$$
$$642$$ 0 0
$$643$$ 19.5147 0.769585 0.384793 0.923003i $$-0.374273\pi$$
0.384793 + 0.923003i $$0.374273\pi$$
$$644$$ 0 0
$$645$$ −27.3137 −1.07548
$$646$$ 0 0
$$647$$ 14.8284 0.582966 0.291483 0.956576i $$-0.405851\pi$$
0.291483 + 0.956576i $$0.405851\pi$$
$$648$$ 0 0
$$649$$ 52.2843 2.05234
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.82843 0.110685 0.0553425 0.998467i $$-0.482375\pi$$
0.0553425 + 0.998467i $$0.482375\pi$$
$$654$$ 0 0
$$655$$ 52.2843 2.04292
$$656$$ 0 0
$$657$$ 3.07107 0.119814
$$658$$ 0 0
$$659$$ 25.5147 0.993912 0.496956 0.867776i $$-0.334451\pi$$
0.496956 + 0.867776i $$0.334451\pi$$
$$660$$ 0 0
$$661$$ −23.3553 −0.908417 −0.454209 0.890895i $$-0.650078\pi$$
−0.454209 + 0.890895i $$0.650078\pi$$
$$662$$ 0 0
$$663$$ −8.82843 −0.342868
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 7.02944 0.272181
$$668$$ 0 0
$$669$$ −2.34315 −0.0905912
$$670$$ 0 0
$$671$$ −28.4853 −1.09966
$$672$$ 0 0
$$673$$ 15.3137 0.590300 0.295150 0.955451i $$-0.404630\pi$$
0.295150 + 0.955451i $$0.404630\pi$$
$$674$$ 0 0
$$675$$ 6.65685 0.256222
$$676$$ 0 0
$$677$$ −1.55635 −0.0598154 −0.0299077 0.999553i $$-0.509521\pi$$
−0.0299077 + 0.999553i $$0.509521\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 19.7990 0.758699
$$682$$ 0 0
$$683$$ 12.1421 0.464606 0.232303 0.972643i $$-0.425374\pi$$
0.232303 + 0.972643i $$0.425374\pi$$
$$684$$ 0 0
$$685$$ −42.6274 −1.62871
$$686$$ 0 0
$$687$$ 0.928932 0.0354410
$$688$$ 0 0
$$689$$ 13.1716 0.501797
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 32.9706 1.25064
$$696$$ 0 0
$$697$$ 21.3137 0.807314
$$698$$ 0 0
$$699$$ −11.5147 −0.435527
$$700$$ 0 0
$$701$$ −10.8284 −0.408984 −0.204492 0.978868i $$-0.565554\pi$$
−0.204492 + 0.978868i $$0.565554\pi$$
$$702$$ 0 0
$$703$$ −11.3137 −0.426705
$$704$$ 0 0
$$705$$ 4.00000 0.150649
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −8.00000 −0.300446 −0.150223 0.988652i $$-0.547999\pi$$
−0.150223 + 0.988652i $$0.547999\pi$$
$$710$$ 0 0
$$711$$ −13.6569 −0.512172
$$712$$ 0 0
$$713$$ −8.97056 −0.335950
$$714$$ 0 0
$$715$$ 23.3137 0.871883
$$716$$ 0 0
$$717$$ −8.82843 −0.329704
$$718$$ 0 0
$$719$$ −27.3137 −1.01863 −0.509315 0.860580i $$-0.670101\pi$$
−0.509315 + 0.860580i $$0.670101\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −2.10051 −0.0781186
$$724$$ 0 0
$$725$$ −56.4853 −2.09781
$$726$$ 0 0
$$727$$ −25.4558 −0.944105 −0.472052 0.881570i $$-0.656487\pi$$
−0.472052 + 0.881570i $$0.656487\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 49.9411 1.84714
$$732$$ 0 0
$$733$$ 20.2426 0.747679 0.373839 0.927493i $$-0.378041\pi$$
0.373839 + 0.927493i $$0.378041\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −38.6274 −1.42286
$$738$$ 0 0
$$739$$ −36.2843 −1.33474 −0.667369 0.744727i $$-0.732580\pi$$
−0.667369 + 0.744727i $$0.732580\pi$$
$$740$$ 0 0
$$741$$ 1.65685 0.0608661
$$742$$ 0 0
$$743$$ −16.8284 −0.617375 −0.308688 0.951163i $$-0.599890\pi$$
−0.308688 + 0.951163i $$0.599890\pi$$
$$744$$ 0 0
$$745$$ −34.1421 −1.25087
$$746$$ 0 0
$$747$$ −7.31371 −0.267595
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 18.3431 0.669351 0.334675 0.942333i $$-0.391373\pi$$
0.334675 + 0.942333i $$0.391373\pi$$
$$752$$ 0 0
$$753$$ −8.48528 −0.309221
$$754$$ 0 0
$$755$$ −5.65685 −0.205874
$$756$$ 0 0
$$757$$ 11.3137 0.411204 0.205602 0.978636i $$-0.434085\pi$$
0.205602 + 0.978636i $$0.434085\pi$$
$$758$$ 0 0
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ −18.2426 −0.661295 −0.330648 0.943754i $$-0.607267\pi$$
−0.330648 + 0.943754i $$0.607267\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −21.3137 −0.770599
$$766$$ 0 0
$$767$$ 15.3137 0.552946
$$768$$ 0 0
$$769$$ 0.928932 0.0334982 0.0167491 0.999860i $$-0.494668\pi$$
0.0167491 + 0.999860i $$0.494668\pi$$
$$770$$ 0 0
$$771$$ −21.7574 −0.783572
$$772$$ 0 0
$$773$$ −0.585786 −0.0210693 −0.0105346 0.999945i $$-0.503353\pi$$
−0.0105346 + 0.999945i $$0.503353\pi$$
$$774$$ 0 0
$$775$$ 72.0833 2.58931
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −4.00000 −0.143315
$$780$$ 0 0
$$781$$ 23.3137 0.834230
$$782$$ 0 0
$$783$$ −8.48528 −0.303239
$$784$$ 0 0
$$785$$ 20.1421 0.718904
$$786$$ 0 0
$$787$$ 34.6274 1.23433 0.617167 0.786832i $$-0.288280\pi$$
0.617167 + 0.786832i $$0.288280\pi$$
$$788$$ 0 0
$$789$$ 19.1716 0.682526
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −8.34315 −0.296274
$$794$$ 0 0
$$795$$ 31.7990 1.12779
$$796$$ 0 0
$$797$$ −0.786797 −0.0278698 −0.0139349 0.999903i $$-0.504436\pi$$
−0.0139349 + 0.999903i $$0.504436\pi$$
$$798$$ 0 0
$$799$$ −7.31371 −0.258740
$$800$$ 0 0
$$801$$ 14.7279 0.520386
$$802$$ 0 0
$$803$$ 14.8284 0.523284
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −18.0416 −0.635095
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 12.9706 0.455458 0.227729 0.973725i $$-0.426870\pi$$
0.227729 + 0.973725i $$0.426870\pi$$
$$812$$ 0 0
$$813$$ −18.8284 −0.660342
$$814$$ 0 0
$$815$$ −8.00000 −0.280228
$$816$$ 0 0
$$817$$ −9.37258 −0.327905
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ −13.6569 −0.476048 −0.238024 0.971259i $$-0.576500\pi$$
−0.238024 + 0.971259i $$0.576500\pi$$
$$824$$ 0 0
$$825$$ 32.1421 1.11905
$$826$$ 0 0
$$827$$ 31.4558 1.09383 0.546913 0.837189i $$-0.315803\pi$$
0.546913 + 0.837189i $$0.315803\pi$$
$$828$$ 0 0
$$829$$ 28.7279 0.997762 0.498881 0.866671i $$-0.333744\pi$$
0.498881 + 0.866671i $$0.333744\pi$$
$$830$$ 0 0
$$831$$ −6.00000 −0.208138
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 23.3137 0.806804
$$836$$ 0 0
$$837$$ 10.8284 0.374285
$$838$$ 0 0
$$839$$ 54.8284 1.89289 0.946444 0.322869i $$-0.104647\pi$$
0.946444 + 0.322869i $$0.104647\pi$$
$$840$$ 0 0
$$841$$ 43.0000 1.48276
$$842$$ 0 0
$$843$$ 4.48528 0.154481
$$844$$ 0 0
$$845$$ −37.5563 −1.29198
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −9.17157 −0.314768
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ 0 0
$$853$$ −12.0416 −0.412298 −0.206149 0.978521i $$-0.566093\pi$$
−0.206149 + 0.978521i $$0.566093\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 0 0
$$857$$ 15.6985 0.536250 0.268125 0.963384i $$-0.413596\pi$$
0.268125 + 0.963384i $$0.413596\pi$$
$$858$$ 0 0
$$859$$ −33.1716 −1.13180 −0.565900 0.824474i $$-0.691471\pi$$
−0.565900 + 0.824474i $$0.691471\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −46.4853 −1.58238 −0.791189 0.611572i $$-0.790537\pi$$
−0.791189 + 0.611572i $$0.790537\pi$$
$$864$$ 0 0
$$865$$ −2.00000 −0.0680020
$$866$$ 0 0
$$867$$ 21.9706 0.746159
$$868$$ 0 0
$$869$$ −65.9411 −2.23690
$$870$$ 0 0
$$871$$ −11.3137 −0.383350
$$872$$ 0 0
$$873$$ −16.2426 −0.549730
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −36.2843 −1.22523 −0.612616 0.790380i $$-0.709883\pi$$
−0.612616 + 0.790380i $$0.709883\pi$$
$$878$$ 0 0
$$879$$ −13.0711 −0.440876
$$880$$ 0 0
$$881$$ −10.9289 −0.368205 −0.184103 0.982907i $$-0.558938\pi$$
−0.184103 + 0.982907i $$0.558938\pi$$
$$882$$ 0 0
$$883$$ 29.6569 0.998033 0.499016 0.866593i $$-0.333695\pi$$
0.499016 + 0.866593i $$0.333695\pi$$
$$884$$ 0 0
$$885$$ 36.9706 1.24275
$$886$$ 0 0
$$887$$ −29.4558 −0.989030 −0.494515 0.869169i $$-0.664654\pi$$
−0.494515 + 0.869169i $$0.664654\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.82843 0.161758
$$892$$ 0 0
$$893$$ 1.37258 0.0459317
$$894$$ 0 0
$$895$$ −74.4264 −2.48780
$$896$$ 0 0
$$897$$ −1.17157 −0.0391177
$$898$$ 0 0
$$899$$ −91.8823 −3.06444
$$900$$ 0 0
$$901$$ −58.1421 −1.93700
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 33.7990 1.12352
$$906$$ 0 0
$$907$$ −40.9706 −1.36041 −0.680203 0.733024i $$-0.738108\pi$$
−0.680203 + 0.733024i $$0.738108\pi$$
$$908$$ 0 0
$$909$$ 0.585786 0.0194293
$$910$$ 0 0
$$911$$ −37.5147 −1.24292 −0.621459 0.783447i $$-0.713460\pi$$
−0.621459 + 0.783447i $$0.713460\pi$$
$$912$$ 0 0
$$913$$ −35.3137 −1.16871
$$914$$ 0 0
$$915$$ −20.1421 −0.665878
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 41.2548 1.36087 0.680436 0.732808i $$-0.261791\pi$$
0.680436 + 0.732808i $$0.261791\pi$$
$$920$$ 0 0
$$921$$ −28.4853 −0.938622
$$922$$ 0 0
$$923$$ 6.82843 0.224760
$$924$$ 0 0
$$925$$ −64.2843 −2.11365
$$926$$ 0 0
$$927$$ 5.17157 0.169857
$$928$$ 0 0
$$929$$ 7.12994 0.233926 0.116963 0.993136i $$-0.462684\pi$$
0.116963 + 0.993136i $$0.462684\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 2.14214 0.0701304
$$934$$ 0 0
$$935$$ −102.912 −3.36557
$$936$$ 0 0
$$937$$ −25.8995 −0.846100 −0.423050 0.906106i $$-0.639040\pi$$
−0.423050 + 0.906106i $$0.639040\pi$$
$$938$$ 0 0
$$939$$ −14.5858 −0.475989
$$940$$ 0 0
$$941$$ −24.1005 −0.785654 −0.392827 0.919612i $$-0.628503\pi$$
−0.392827 + 0.919612i $$0.628503\pi$$
$$942$$ 0 0
$$943$$ 2.82843 0.0921063
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8.14214 −0.264584 −0.132292 0.991211i $$-0.542234\pi$$
−0.132292 + 0.991211i $$0.542234\pi$$
$$948$$ 0 0
$$949$$ 4.34315 0.140984
$$950$$ 0 0
$$951$$ −1.31371 −0.0425999
$$952$$ 0 0
$$953$$ −52.6274 −1.70477 −0.852385 0.522915i $$-0.824844\pi$$
−0.852385 + 0.522915i $$0.824844\pi$$
$$954$$ 0 0
$$955$$ 71.1127 2.30115
$$956$$ 0 0
$$957$$ −40.9706 −1.32439
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 86.2548 2.78241
$$962$$ 0 0
$$963$$ −2.48528 −0.0800871
$$964$$ 0 0
$$965$$ −70.4264 −2.26711
$$966$$ 0 0
$$967$$ 10.6274 0.341755 0.170877 0.985292i $$-0.445340\pi$$
0.170877 + 0.985292i $$0.445340\pi$$
$$968$$ 0 0
$$969$$ −7.31371 −0.234950
$$970$$ 0 0
$$971$$ 33.2548 1.06720 0.533599 0.845737i $$-0.320839\pi$$
0.533599 + 0.845737i $$0.320839\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 9.41421 0.301496
$$976$$ 0 0
$$977$$ 50.1421 1.60419 0.802095 0.597197i $$-0.203719\pi$$
0.802095 + 0.597197i $$0.203719\pi$$
$$978$$ 0 0
$$979$$ 71.1127 2.27277
$$980$$ 0 0
$$981$$ 11.3137 0.361219
$$982$$ 0 0
$$983$$ 30.6274 0.976863 0.488431 0.872602i $$-0.337569\pi$$
0.488431 + 0.872602i $$0.337569\pi$$
$$984$$ 0 0
$$985$$ 6.82843 0.217572
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 6.62742 0.210740
$$990$$ 0 0
$$991$$ 0.686292 0.0218008 0.0109004 0.999941i $$-0.496530\pi$$
0.0109004 + 0.999941i $$0.496530\pi$$
$$992$$ 0 0
$$993$$ −31.3137 −0.993711
$$994$$ 0 0
$$995$$ −19.3137 −0.612286
$$996$$ 0 0
$$997$$ −37.4142 −1.18492 −0.592460 0.805600i $$-0.701843\pi$$
−0.592460 + 0.805600i $$0.701843\pi$$
$$998$$ 0 0
$$999$$ −9.65685 −0.305529
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 1176.2.a.o.1.2 yes 2
3.2 odd 2 3528.2.a.bb.1.1 2
4.3 odd 2 2352.2.a.bb.1.2 2
7.2 even 3 1176.2.q.k.361.1 4
7.3 odd 6 1176.2.q.o.961.2 4
7.4 even 3 1176.2.q.k.961.1 4
7.5 odd 6 1176.2.q.o.361.2 4
7.6 odd 2 1176.2.a.j.1.1 2
8.3 odd 2 9408.2.a.du.1.1 2
8.5 even 2 9408.2.a.dg.1.1 2
12.11 even 2 7056.2.a.cg.1.1 2
21.2 odd 6 3528.2.s.bm.361.2 4
21.5 even 6 3528.2.s.bd.361.1 4
21.11 odd 6 3528.2.s.bm.3313.2 4
21.17 even 6 3528.2.s.bd.3313.1 4
21.20 even 2 3528.2.a.bl.1.2 2
28.3 even 6 2352.2.q.bc.961.2 4
28.11 odd 6 2352.2.q.be.961.1 4
28.19 even 6 2352.2.q.bc.1537.2 4
28.23 odd 6 2352.2.q.be.1537.1 4
28.27 even 2 2352.2.a.bd.1.1 2
56.13 odd 2 9408.2.a.ee.1.2 2
56.27 even 2 9408.2.a.ds.1.2 2
84.83 odd 2 7056.2.a.cx.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.j.1.1 2 7.6 odd 2
1176.2.a.o.1.2 yes 2 1.1 even 1 trivial
1176.2.q.k.361.1 4 7.2 even 3
1176.2.q.k.961.1 4 7.4 even 3
1176.2.q.o.361.2 4 7.5 odd 6
1176.2.q.o.961.2 4 7.3 odd 6
2352.2.a.bb.1.2 2 4.3 odd 2
2352.2.a.bd.1.1 2 28.27 even 2
2352.2.q.bc.961.2 4 28.3 even 6
2352.2.q.bc.1537.2 4 28.19 even 6
2352.2.q.be.961.1 4 28.11 odd 6
2352.2.q.be.1537.1 4 28.23 odd 6
3528.2.a.bb.1.1 2 3.2 odd 2
3528.2.a.bl.1.2 2 21.20 even 2
3528.2.s.bd.361.1 4 21.5 even 6
3528.2.s.bd.3313.1 4 21.17 even 6
3528.2.s.bm.361.2 4 21.2 odd 6
3528.2.s.bm.3313.2 4 21.11 odd 6
7056.2.a.cg.1.1 2 12.11 even 2
7056.2.a.cx.1.2 2 84.83 odd 2
9408.2.a.dg.1.1 2 8.5 even 2
9408.2.a.ds.1.2 2 56.27 even 2
9408.2.a.du.1.1 2 8.3 odd 2
9408.2.a.ee.1.2 2 56.13 odd 2