# Properties

 Label 4200.2.a.bp.1.2 Level $4200$ Weight $2$ Character 4200.1 Self dual yes Analytic conductor $33.537$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.5371688489$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 840) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 4200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -3.26180 q^{11} +0.340173 q^{13} +5.75872 q^{17} -6.49693 q^{19} +1.00000 q^{21} -8.49693 q^{23} +1.00000 q^{27} -2.00000 q^{29} -8.34017 q^{31} -3.26180 q^{33} -6.15676 q^{37} +0.340173 q^{39} +0.340173 q^{41} -8.68035 q^{43} +1.00000 q^{49} +5.75872 q^{51} -8.34017 q^{53} -6.49693 q^{57} +6.83710 q^{59} +15.3607 q^{61} +1.00000 q^{63} +14.8371 q^{67} -8.49693 q^{69} -15.9421 q^{71} -1.50307 q^{73} -3.26180 q^{77} +8.68035 q^{79} +1.00000 q^{81} +6.83710 q^{83} -2.00000 q^{87} -15.1773 q^{89} +0.340173 q^{91} -8.34017 q^{93} -6.49693 q^{97} -3.26180 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 2 q^{11} - 10 q^{13} - 8 q^{17} - 2 q^{19} + 3 q^{21} - 8 q^{23} + 3 q^{27} - 6 q^{29} - 14 q^{31} - 2 q^{33} - 12 q^{37} - 10 q^{39} - 10 q^{41} - 4 q^{43} + 3 q^{49} - 8 q^{51} - 14 q^{53} - 2 q^{57} - 8 q^{59} + 2 q^{61} + 3 q^{63} + 16 q^{67} - 8 q^{69} - 18 q^{71} - 22 q^{73} - 2 q^{77} + 4 q^{79} + 3 q^{81} - 8 q^{83} - 6 q^{87} - 6 q^{89} - 10 q^{91} - 14 q^{93} - 2 q^{97} - 2 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 - 2 * q^11 - 10 * q^13 - 8 * q^17 - 2 * q^19 + 3 * q^21 - 8 * q^23 + 3 * q^27 - 6 * q^29 - 14 * q^31 - 2 * q^33 - 12 * q^37 - 10 * q^39 - 10 * q^41 - 4 * q^43 + 3 * q^49 - 8 * q^51 - 14 * q^53 - 2 * q^57 - 8 * q^59 + 2 * q^61 + 3 * q^63 + 16 * q^67 - 8 * q^69 - 18 * q^71 - 22 * q^73 - 2 * q^77 + 4 * q^79 + 3 * q^81 - 8 * q^83 - 6 * q^87 - 6 * q^89 - 10 * q^91 - 14 * q^93 - 2 * q^97 - 2 * 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.26180 −0.983468 −0.491734 0.870745i $$-0.663637\pi$$
−0.491734 + 0.870745i $$0.663637\pi$$
$$12$$ 0 0
$$13$$ 0.340173 0.0943470 0.0471735 0.998887i $$-0.484979\pi$$
0.0471735 + 0.998887i $$0.484979\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.75872 1.39670 0.698348 0.715759i $$-0.253919\pi$$
0.698348 + 0.715759i $$0.253919\pi$$
$$18$$ 0 0
$$19$$ −6.49693 −1.49050 −0.745249 0.666786i $$-0.767669\pi$$
−0.745249 + 0.666786i $$0.767669\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 0 0
$$23$$ −8.49693 −1.77173 −0.885866 0.463941i $$-0.846435\pi$$
−0.885866 + 0.463941i $$0.846435\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −8.34017 −1.49794 −0.748970 0.662604i $$-0.769451\pi$$
−0.748970 + 0.662604i $$0.769451\pi$$
$$32$$ 0 0
$$33$$ −3.26180 −0.567806
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.15676 −1.01216 −0.506082 0.862485i $$-0.668907\pi$$
−0.506082 + 0.862485i $$0.668907\pi$$
$$38$$ 0 0
$$39$$ 0.340173 0.0544713
$$40$$ 0 0
$$41$$ 0.340173 0.0531261 0.0265630 0.999647i $$-0.491544\pi$$
0.0265630 + 0.999647i $$0.491544\pi$$
$$42$$ 0 0
$$43$$ −8.68035 −1.32374 −0.661870 0.749618i $$-0.730237\pi$$
−0.661870 + 0.749618i $$0.730237\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 5.75872 0.806383
$$52$$ 0 0
$$53$$ −8.34017 −1.14561 −0.572805 0.819691i $$-0.694145\pi$$
−0.572805 + 0.819691i $$0.694145\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −6.49693 −0.860539
$$58$$ 0 0
$$59$$ 6.83710 0.890115 0.445057 0.895502i $$-0.353183\pi$$
0.445057 + 0.895502i $$0.353183\pi$$
$$60$$ 0 0
$$61$$ 15.3607 1.96674 0.983368 0.181627i $$-0.0581363\pi$$
0.983368 + 0.181627i $$0.0581363\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 14.8371 1.81264 0.906320 0.422592i $$-0.138880\pi$$
0.906320 + 0.422592i $$0.138880\pi$$
$$68$$ 0 0
$$69$$ −8.49693 −1.02291
$$70$$ 0 0
$$71$$ −15.9421 −1.89198 −0.945992 0.324190i $$-0.894908\pi$$
−0.945992 + 0.324190i $$0.894908\pi$$
$$72$$ 0 0
$$73$$ −1.50307 −0.175921 −0.0879606 0.996124i $$-0.528035\pi$$
−0.0879606 + 0.996124i $$0.528035\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.26180 −0.371716
$$78$$ 0 0
$$79$$ 8.68035 0.976615 0.488308 0.872672i $$-0.337614\pi$$
0.488308 + 0.872672i $$0.337614\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.83710 0.750469 0.375235 0.926930i $$-0.377562\pi$$
0.375235 + 0.926930i $$0.377562\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ −15.1773 −1.60879 −0.804394 0.594096i $$-0.797510\pi$$
−0.804394 + 0.594096i $$0.797510\pi$$
$$90$$ 0 0
$$91$$ 0.340173 0.0356598
$$92$$ 0 0
$$93$$ −8.34017 −0.864836
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −6.49693 −0.659663 −0.329832 0.944040i $$-0.606992\pi$$
−0.329832 + 0.944040i $$0.606992\pi$$
$$98$$ 0 0
$$99$$ −3.26180 −0.327823
$$100$$ 0 0
$$101$$ −2.18342 −0.217258 −0.108629 0.994082i $$-0.534646\pi$$
−0.108629 + 0.994082i $$0.534646\pi$$
$$102$$ 0 0
$$103$$ −5.84324 −0.575752 −0.287876 0.957668i $$-0.592949\pi$$
−0.287876 + 0.957668i $$0.592949\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −8.18342 −0.791121 −0.395560 0.918440i $$-0.629450\pi$$
−0.395560 + 0.918440i $$0.629450\pi$$
$$108$$ 0 0
$$109$$ 16.8371 1.61270 0.806351 0.591437i $$-0.201439\pi$$
0.806351 + 0.591437i $$0.201439\pi$$
$$110$$ 0 0
$$111$$ −6.15676 −0.584373
$$112$$ 0 0
$$113$$ −13.0205 −1.22487 −0.612434 0.790522i $$-0.709809\pi$$
−0.612434 + 0.790522i $$0.709809\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0.340173 0.0314490
$$118$$ 0 0
$$119$$ 5.75872 0.527901
$$120$$ 0 0
$$121$$ −0.360692 −0.0327902
$$122$$ 0 0
$$123$$ 0.340173 0.0306724
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1.84324 −0.163562 −0.0817808 0.996650i $$-0.526061\pi$$
−0.0817808 + 0.996650i $$0.526061\pi$$
$$128$$ 0 0
$$129$$ −8.68035 −0.764262
$$130$$ 0 0
$$131$$ −0.313511 −0.0273916 −0.0136958 0.999906i $$-0.504360\pi$$
−0.0136958 + 0.999906i $$0.504360\pi$$
$$132$$ 0 0
$$133$$ −6.49693 −0.563355
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2.18342 −0.186542 −0.0932710 0.995641i $$-0.529732\pi$$
−0.0932710 + 0.995641i $$0.529732\pi$$
$$138$$ 0 0
$$139$$ −1.02052 −0.0865593 −0.0432796 0.999063i $$-0.513781\pi$$
−0.0432796 + 0.999063i $$0.513781\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −1.10957 −0.0927873
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1.00000 0.0824786
$$148$$ 0 0
$$149$$ 14.6803 1.20266 0.601330 0.799000i $$-0.294638\pi$$
0.601330 + 0.799000i $$0.294638\pi$$
$$150$$ 0 0
$$151$$ 2.15676 0.175514 0.0877571 0.996142i $$-0.472030\pi$$
0.0877571 + 0.996142i $$0.472030\pi$$
$$152$$ 0 0
$$153$$ 5.75872 0.465565
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.18342 −0.493490 −0.246745 0.969080i $$-0.579361\pi$$
−0.246745 + 0.969080i $$0.579361\pi$$
$$158$$ 0 0
$$159$$ −8.34017 −0.661419
$$160$$ 0 0
$$161$$ −8.49693 −0.669652
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.47641 0.114248 0.0571240 0.998367i $$-0.481807\pi$$
0.0571240 + 0.998367i $$0.481807\pi$$
$$168$$ 0 0
$$169$$ −12.8843 −0.991099
$$170$$ 0 0
$$171$$ −6.49693 −0.496833
$$172$$ 0 0
$$173$$ 1.75872 0.133713 0.0668566 0.997763i $$-0.478703\pi$$
0.0668566 + 0.997763i $$0.478703\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.83710 0.513908
$$178$$ 0 0
$$179$$ 0.424694 0.0317431 0.0158716 0.999874i $$-0.494948\pi$$
0.0158716 + 0.999874i $$0.494948\pi$$
$$180$$ 0 0
$$181$$ 10.3668 0.770561 0.385280 0.922800i $$-0.374105\pi$$
0.385280 + 0.922800i $$0.374105\pi$$
$$182$$ 0 0
$$183$$ 15.3607 1.13550
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −18.7838 −1.37361
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −21.7321 −1.57248 −0.786238 0.617923i $$-0.787974\pi$$
−0.786238 + 0.617923i $$0.787974\pi$$
$$192$$ 0 0
$$193$$ 8.36683 0.602258 0.301129 0.953583i $$-0.402637\pi$$
0.301129 + 0.953583i $$0.402637\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −13.3340 −0.950010 −0.475005 0.879983i $$-0.657554\pi$$
−0.475005 + 0.879983i $$0.657554\pi$$
$$198$$ 0 0
$$199$$ 6.49693 0.460555 0.230278 0.973125i $$-0.426037\pi$$
0.230278 + 0.973125i $$0.426037\pi$$
$$200$$ 0 0
$$201$$ 14.8371 1.04653
$$202$$ 0 0
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −8.49693 −0.590577
$$208$$ 0 0
$$209$$ 21.1917 1.46586
$$210$$ 0 0
$$211$$ 6.83710 0.470685 0.235343 0.971912i $$-0.424379\pi$$
0.235343 + 0.971912i $$0.424379\pi$$
$$212$$ 0 0
$$213$$ −15.9421 −1.09234
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −8.34017 −0.566168
$$218$$ 0 0
$$219$$ −1.50307 −0.101568
$$220$$ 0 0
$$221$$ 1.95896 0.131774
$$222$$ 0 0
$$223$$ −17.3607 −1.16256 −0.581279 0.813704i $$-0.697447\pi$$
−0.581279 + 0.813704i $$0.697447\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −10.1568 −0.674128 −0.337064 0.941482i $$-0.609434\pi$$
−0.337064 + 0.941482i $$0.609434\pi$$
$$228$$ 0 0
$$229$$ 14.9939 0.990822 0.495411 0.868659i $$-0.335017\pi$$
0.495411 + 0.868659i $$0.335017\pi$$
$$230$$ 0 0
$$231$$ −3.26180 −0.214610
$$232$$ 0 0
$$233$$ 11.5441 0.756280 0.378140 0.925748i $$-0.376564\pi$$
0.378140 + 0.925748i $$0.376564\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.68035 0.563849
$$238$$ 0 0
$$239$$ −10.8950 −0.704736 −0.352368 0.935861i $$-0.614624\pi$$
−0.352368 + 0.935861i $$0.614624\pi$$
$$240$$ 0 0
$$241$$ −6.68035 −0.430319 −0.215159 0.976579i $$-0.569027\pi$$
−0.215159 + 0.976579i $$0.569027\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.21008 −0.140624
$$248$$ 0 0
$$249$$ 6.83710 0.433284
$$250$$ 0 0
$$251$$ 24.1978 1.52735 0.763676 0.645600i $$-0.223393\pi$$
0.763676 + 0.645600i $$0.223393\pi$$
$$252$$ 0 0
$$253$$ 27.7152 1.74244
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −15.2351 −0.950342 −0.475171 0.879894i $$-0.657614\pi$$
−0.475171 + 0.879894i $$0.657614\pi$$
$$258$$ 0 0
$$259$$ −6.15676 −0.382562
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ −29.1773 −1.79915 −0.899574 0.436769i $$-0.856123\pi$$
−0.899574 + 0.436769i $$0.856123\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −15.1773 −0.928834
$$268$$ 0 0
$$269$$ −6.13009 −0.373758 −0.186879 0.982383i $$-0.559837\pi$$
−0.186879 + 0.982383i $$0.559837\pi$$
$$270$$ 0 0
$$271$$ 25.7009 1.56122 0.780608 0.625021i $$-0.214909\pi$$
0.780608 + 0.625021i $$0.214909\pi$$
$$272$$ 0 0
$$273$$ 0.340173 0.0205882
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −11.2039 −0.673179 −0.336590 0.941651i $$-0.609274\pi$$
−0.336590 + 0.941651i $$0.609274\pi$$
$$278$$ 0 0
$$279$$ −8.34017 −0.499313
$$280$$ 0 0
$$281$$ 24.3545 1.45287 0.726435 0.687235i $$-0.241176\pi$$
0.726435 + 0.687235i $$0.241176\pi$$
$$282$$ 0 0
$$283$$ 6.15676 0.365981 0.182991 0.983115i $$-0.441422\pi$$
0.182991 + 0.983115i $$0.441422\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0.340173 0.0200798
$$288$$ 0 0
$$289$$ 16.1629 0.950759
$$290$$ 0 0
$$291$$ −6.49693 −0.380857
$$292$$ 0 0
$$293$$ −1.75872 −0.102746 −0.0513729 0.998680i $$-0.516360\pi$$
−0.0513729 + 0.998680i $$0.516360\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.26180 −0.189269
$$298$$ 0 0
$$299$$ −2.89043 −0.167158
$$300$$ 0 0
$$301$$ −8.68035 −0.500327
$$302$$ 0 0
$$303$$ −2.18342 −0.125434
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 27.2039 1.55261 0.776305 0.630357i $$-0.217092\pi$$
0.776305 + 0.630357i $$0.217092\pi$$
$$308$$ 0 0
$$309$$ −5.84324 −0.332411
$$310$$ 0 0
$$311$$ −21.8432 −1.23862 −0.619308 0.785148i $$-0.712587\pi$$
−0.619308 + 0.785148i $$0.712587\pi$$
$$312$$ 0 0
$$313$$ −25.3874 −1.43498 −0.717489 0.696570i $$-0.754709\pi$$
−0.717489 + 0.696570i $$0.754709\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 22.6947 1.27466 0.637331 0.770590i $$-0.280038\pi$$
0.637331 + 0.770590i $$0.280038\pi$$
$$318$$ 0 0
$$319$$ 6.52359 0.365251
$$320$$ 0 0
$$321$$ −8.18342 −0.456754
$$322$$ 0 0
$$323$$ −37.4140 −2.08177
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 16.8371 0.931094
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 5.47641 0.301011 0.150505 0.988609i $$-0.451910\pi$$
0.150505 + 0.988609i $$0.451910\pi$$
$$332$$ 0 0
$$333$$ −6.15676 −0.337388
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 13.6742 0.744881 0.372441 0.928056i $$-0.378521\pi$$
0.372441 + 0.928056i $$0.378521\pi$$
$$338$$ 0 0
$$339$$ −13.0205 −0.707178
$$340$$ 0 0
$$341$$ 27.2039 1.47318
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −0.183417 −0.00984637 −0.00492318 0.999988i $$-0.501567\pi$$
−0.00492318 + 0.999988i $$0.501567\pi$$
$$348$$ 0 0
$$349$$ −7.67420 −0.410791 −0.205395 0.978679i $$-0.565848\pi$$
−0.205395 + 0.978679i $$0.565848\pi$$
$$350$$ 0 0
$$351$$ 0.340173 0.0181571
$$352$$ 0 0
$$353$$ −19.4329 −1.03431 −0.517155 0.855892i $$-0.673009\pi$$
−0.517155 + 0.855892i $$0.673009\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 5.75872 0.304784
$$358$$ 0 0
$$359$$ −25.4186 −1.34154 −0.670770 0.741666i $$-0.734036\pi$$
−0.670770 + 0.741666i $$0.734036\pi$$
$$360$$ 0 0
$$361$$ 23.2101 1.22158
$$362$$ 0 0
$$363$$ −0.360692 −0.0189314
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −9.36069 −0.488624 −0.244312 0.969697i $$-0.578562\pi$$
−0.244312 + 0.969697i $$0.578562\pi$$
$$368$$ 0 0
$$369$$ 0.340173 0.0177087
$$370$$ 0 0
$$371$$ −8.34017 −0.433000
$$372$$ 0 0
$$373$$ −27.5174 −1.42480 −0.712400 0.701774i $$-0.752392\pi$$
−0.712400 + 0.701774i $$0.752392\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −0.680346 −0.0350396
$$378$$ 0 0
$$379$$ 9.84324 0.505614 0.252807 0.967517i $$-0.418646\pi$$
0.252807 + 0.967517i $$0.418646\pi$$
$$380$$ 0 0
$$381$$ −1.84324 −0.0944323
$$382$$ 0 0
$$383$$ 27.8310 1.42210 0.711048 0.703144i $$-0.248221\pi$$
0.711048 + 0.703144i $$0.248221\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.68035 −0.441247
$$388$$ 0 0
$$389$$ −3.67420 −0.186289 −0.0931447 0.995653i $$-0.529692\pi$$
−0.0931447 + 0.995653i $$0.529692\pi$$
$$390$$ 0 0
$$391$$ −48.9315 −2.47457
$$392$$ 0 0
$$393$$ −0.313511 −0.0158145
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −22.8638 −1.14750 −0.573750 0.819031i $$-0.694512\pi$$
−0.573750 + 0.819031i $$0.694512\pi$$
$$398$$ 0 0
$$399$$ −6.49693 −0.325253
$$400$$ 0 0
$$401$$ 5.31965 0.265651 0.132825 0.991139i $$-0.457595\pi$$
0.132825 + 0.991139i $$0.457595\pi$$
$$402$$ 0 0
$$403$$ −2.83710 −0.141326
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 20.0821 0.995432
$$408$$ 0 0
$$409$$ 27.7275 1.37104 0.685519 0.728055i $$-0.259575\pi$$
0.685519 + 0.728055i $$0.259575\pi$$
$$410$$ 0 0
$$411$$ −2.18342 −0.107700
$$412$$ 0 0
$$413$$ 6.83710 0.336432
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −1.02052 −0.0499750
$$418$$ 0 0
$$419$$ −23.5174 −1.14890 −0.574451 0.818539i $$-0.694785\pi$$
−0.574451 + 0.818539i $$0.694785\pi$$
$$420$$ 0 0
$$421$$ −1.15061 −0.0560774 −0.0280387 0.999607i $$-0.508926\pi$$
−0.0280387 + 0.999607i $$0.508926\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 15.3607 0.743356
$$428$$ 0 0
$$429$$ −1.10957 −0.0535708
$$430$$ 0 0
$$431$$ 17.4186 0.839022 0.419511 0.907750i $$-0.362202\pi$$
0.419511 + 0.907750i $$0.362202\pi$$
$$432$$ 0 0
$$433$$ −26.0144 −1.25017 −0.625086 0.780556i $$-0.714936\pi$$
−0.625086 + 0.780556i $$0.714936\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 55.2039 2.64076
$$438$$ 0 0
$$439$$ 5.13624 0.245139 0.122570 0.992460i $$-0.460887\pi$$
0.122570 + 0.992460i $$0.460887\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 18.5380 0.880765 0.440383 0.897810i $$-0.354843\pi$$
0.440383 + 0.897810i $$0.354843\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 14.6803 0.694357
$$448$$ 0 0
$$449$$ −10.3135 −0.486725 −0.243362 0.969935i $$-0.578250\pi$$
−0.243362 + 0.969935i $$0.578250\pi$$
$$450$$ 0 0
$$451$$ −1.10957 −0.0522478
$$452$$ 0 0
$$453$$ 2.15676 0.101333
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.36683 0.204272 0.102136 0.994770i $$-0.467432\pi$$
0.102136 + 0.994770i $$0.467432\pi$$
$$458$$ 0 0
$$459$$ 5.75872 0.268794
$$460$$ 0 0
$$461$$ −33.7009 −1.56961 −0.784803 0.619745i $$-0.787236\pi$$
−0.784803 + 0.619745i $$0.787236\pi$$
$$462$$ 0 0
$$463$$ −10.4703 −0.486595 −0.243297 0.969952i $$-0.578229\pi$$
−0.243297 + 0.969952i $$0.578229\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −3.51745 −0.162768 −0.0813840 0.996683i $$-0.525934\pi$$
−0.0813840 + 0.996683i $$0.525934\pi$$
$$468$$ 0 0
$$469$$ 14.8371 0.685114
$$470$$ 0 0
$$471$$ −6.18342 −0.284917
$$472$$ 0 0
$$473$$ 28.3135 1.30186
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −8.34017 −0.381870
$$478$$ 0 0
$$479$$ −21.8432 −0.998043 −0.499022 0.866590i $$-0.666307\pi$$
−0.499022 + 0.866590i $$0.666307\pi$$
$$480$$ 0 0
$$481$$ −2.09436 −0.0954947
$$482$$ 0 0
$$483$$ −8.49693 −0.386624
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 7.88428 0.357271 0.178635 0.983915i $$-0.442832\pi$$
0.178635 + 0.983915i $$0.442832\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 19.7731 0.892347 0.446174 0.894946i $$-0.352786\pi$$
0.446174 + 0.894946i $$0.352786\pi$$
$$492$$ 0 0
$$493$$ −11.5174 −0.518720
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −15.9421 −0.715103
$$498$$ 0 0
$$499$$ 39.5174 1.76904 0.884522 0.466499i $$-0.154485\pi$$
0.884522 + 0.466499i $$0.154485\pi$$
$$500$$ 0 0
$$501$$ 1.47641 0.0659611
$$502$$ 0 0
$$503$$ −11.2039 −0.499559 −0.249779 0.968303i $$-0.580358\pi$$
−0.249779 + 0.968303i $$0.580358\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −12.8843 −0.572211
$$508$$ 0 0
$$509$$ −23.5441 −1.04357 −0.521787 0.853076i $$-0.674734\pi$$
−0.521787 + 0.853076i $$0.674734\pi$$
$$510$$ 0 0
$$511$$ −1.50307 −0.0664920
$$512$$ 0 0
$$513$$ −6.49693 −0.286846
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 1.75872 0.0771994
$$520$$ 0 0
$$521$$ 24.6537 1.08010 0.540049 0.841634i $$-0.318406\pi$$
0.540049 + 0.841634i $$0.318406\pi$$
$$522$$ 0 0
$$523$$ −2.63931 −0.115409 −0.0577044 0.998334i $$-0.518378\pi$$
−0.0577044 + 0.998334i $$0.518378\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −48.0288 −2.09217
$$528$$ 0 0
$$529$$ 49.1978 2.13903
$$530$$ 0 0
$$531$$ 6.83710 0.296705
$$532$$ 0 0
$$533$$ 0.115718 0.00501229
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0.424694 0.0183269
$$538$$ 0 0
$$539$$ −3.26180 −0.140495
$$540$$ 0 0
$$541$$ 25.1506 1.08131 0.540655 0.841245i $$-0.318177\pi$$
0.540655 + 0.841245i $$0.318177\pi$$
$$542$$ 0 0
$$543$$ 10.3668 0.444883
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −35.1506 −1.50293 −0.751466 0.659772i $$-0.770653\pi$$
−0.751466 + 0.659772i $$0.770653\pi$$
$$548$$ 0 0
$$549$$ 15.3607 0.655578
$$550$$ 0 0
$$551$$ 12.9939 0.553557
$$552$$ 0 0
$$553$$ 8.68035 0.369126
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 26.3812 1.11781 0.558904 0.829232i $$-0.311222\pi$$
0.558904 + 0.829232i $$0.311222\pi$$
$$558$$ 0 0
$$559$$ −2.95282 −0.124891
$$560$$ 0 0
$$561$$ −18.7838 −0.793052
$$562$$ 0 0
$$563$$ 12.9939 0.547626 0.273813 0.961783i $$-0.411715\pi$$
0.273813 + 0.961783i $$0.411715\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −6.36683 −0.266912 −0.133456 0.991055i $$-0.542607\pi$$
−0.133456 + 0.991055i $$0.542607\pi$$
$$570$$ 0 0
$$571$$ 2.63931 0.110452 0.0552258 0.998474i $$-0.482412\pi$$
0.0552258 + 0.998474i $$0.482412\pi$$
$$572$$ 0 0
$$573$$ −21.7321 −0.907870
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 33.3340 1.38771 0.693857 0.720113i $$-0.255910\pi$$
0.693857 + 0.720113i $$0.255910\pi$$
$$578$$ 0 0
$$579$$ 8.36683 0.347714
$$580$$ 0 0
$$581$$ 6.83710 0.283651
$$582$$ 0 0
$$583$$ 27.2039 1.12667
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −17.3074 −0.714352 −0.357176 0.934037i $$-0.616260\pi$$
−0.357176 + 0.934037i $$0.616260\pi$$
$$588$$ 0 0
$$589$$ 54.1855 2.23267
$$590$$ 0 0
$$591$$ −13.3340 −0.548489
$$592$$ 0 0
$$593$$ 47.1194 1.93496 0.967481 0.252943i $$-0.0813984\pi$$
0.967481 + 0.252943i $$0.0813984\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 6.49693 0.265902
$$598$$ 0 0
$$599$$ −13.1050 −0.535457 −0.267729 0.963494i $$-0.586273\pi$$
−0.267729 + 0.963494i $$0.586273\pi$$
$$600$$ 0 0
$$601$$ 29.0349 1.18436 0.592179 0.805806i $$-0.298268\pi$$
0.592179 + 0.805806i $$0.298268\pi$$
$$602$$ 0 0
$$603$$ 14.8371 0.604213
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 25.9877 1.05481 0.527404 0.849614i $$-0.323165\pi$$
0.527404 + 0.849614i $$0.323165\pi$$
$$608$$ 0 0
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 20.1978 0.815781 0.407891 0.913031i $$-0.366264\pi$$
0.407891 + 0.913031i $$0.366264\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 47.8576 1.92668 0.963338 0.268292i $$-0.0864592\pi$$
0.963338 + 0.268292i $$0.0864592\pi$$
$$618$$ 0 0
$$619$$ −18.3812 −0.738803 −0.369402 0.929270i $$-0.620437\pi$$
−0.369402 + 0.929270i $$0.620437\pi$$
$$620$$ 0 0
$$621$$ −8.49693 −0.340970
$$622$$ 0 0
$$623$$ −15.1773 −0.608065
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 21.1917 0.846313
$$628$$ 0 0
$$629$$ −35.4551 −1.41369
$$630$$ 0 0
$$631$$ −7.94668 −0.316352 −0.158176 0.987411i $$-0.550561\pi$$
−0.158176 + 0.987411i $$0.550561\pi$$
$$632$$ 0 0
$$633$$ 6.83710 0.271750
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0.340173 0.0134781
$$638$$ 0 0
$$639$$ −15.9421 −0.630661
$$640$$ 0 0
$$641$$ 15.7275 0.621200 0.310600 0.950541i $$-0.399470\pi$$
0.310600 + 0.950541i $$0.399470\pi$$
$$642$$ 0 0
$$643$$ 44.5646 1.75746 0.878729 0.477322i $$-0.158392\pi$$
0.878729 + 0.477322i $$0.158392\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −5.16290 −0.202974 −0.101487 0.994837i $$-0.532360\pi$$
−0.101487 + 0.994837i $$0.532360\pi$$
$$648$$ 0 0
$$649$$ −22.3012 −0.875400
$$650$$ 0 0
$$651$$ −8.34017 −0.326877
$$652$$ 0 0
$$653$$ −7.49079 −0.293137 −0.146569 0.989201i $$-0.546823\pi$$
−0.146569 + 0.989201i $$0.546823\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −1.50307 −0.0586404
$$658$$ 0 0
$$659$$ −22.7259 −0.885276 −0.442638 0.896700i $$-0.645957\pi$$
−0.442638 + 0.896700i $$0.645957\pi$$
$$660$$ 0 0
$$661$$ −12.3258 −0.479418 −0.239709 0.970845i $$-0.577052\pi$$
−0.239709 + 0.970845i $$0.577052\pi$$
$$662$$ 0 0
$$663$$ 1.95896 0.0760798
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.9939 0.658005
$$668$$ 0 0
$$669$$ −17.3607 −0.671203
$$670$$ 0 0
$$671$$ −50.1034 −1.93422
$$672$$ 0 0
$$673$$ 25.6742 0.989668 0.494834 0.868988i $$-0.335229\pi$$
0.494834 + 0.868988i $$0.335229\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −29.2762 −1.12517 −0.562587 0.826738i $$-0.690194\pi$$
−0.562587 + 0.826738i $$0.690194\pi$$
$$678$$ 0 0
$$679$$ −6.49693 −0.249329
$$680$$ 0 0
$$681$$ −10.1568 −0.389208
$$682$$ 0 0
$$683$$ −25.5441 −0.977418 −0.488709 0.872447i $$-0.662532\pi$$
−0.488709 + 0.872447i $$0.662532\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 14.9939 0.572051
$$688$$ 0 0
$$689$$ −2.83710 −0.108085
$$690$$ 0 0
$$691$$ 26.3812 1.00359 0.501794 0.864987i $$-0.332673\pi$$
0.501794 + 0.864987i $$0.332673\pi$$
$$692$$ 0 0
$$693$$ −3.26180 −0.123905
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 1.95896 0.0742010
$$698$$ 0 0
$$699$$ 11.5441 0.436638
$$700$$ 0 0
$$701$$ −23.6742 −0.894162 −0.447081 0.894493i $$-0.647536\pi$$
−0.447081 + 0.894493i $$0.647536\pi$$
$$702$$ 0 0
$$703$$ 40.0000 1.50863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2.18342 −0.0821159
$$708$$ 0 0
$$709$$ −40.7214 −1.52932 −0.764662 0.644432i $$-0.777094\pi$$
−0.764662 + 0.644432i $$0.777094\pi$$
$$710$$ 0 0
$$711$$ 8.68035 0.325538
$$712$$ 0 0
$$713$$ 70.8659 2.65395
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −10.8950 −0.406880
$$718$$ 0 0
$$719$$ 22.3545 0.833684 0.416842 0.908979i $$-0.363137\pi$$
0.416842 + 0.908979i $$0.363137\pi$$
$$720$$ 0 0
$$721$$ −5.84324 −0.217614
$$722$$ 0 0
$$723$$ −6.68035 −0.248445
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 5.10957 0.189504 0.0947518 0.995501i $$-0.469794\pi$$
0.0947518 + 0.995501i $$0.469794\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −49.9877 −1.84886
$$732$$ 0 0
$$733$$ −41.6475 −1.53829 −0.769144 0.639076i $$-0.779317\pi$$
−0.769144 + 0.639076i $$0.779317\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −48.3956 −1.78267
$$738$$ 0 0
$$739$$ 32.3135 1.18867 0.594336 0.804217i $$-0.297415\pi$$
0.594336 + 0.804217i $$0.297415\pi$$
$$740$$ 0 0
$$741$$ −2.21008 −0.0811893
$$742$$ 0 0
$$743$$ −30.1711 −1.10687 −0.553436 0.832892i $$-0.686684\pi$$
−0.553436 + 0.832892i $$0.686684\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.83710 0.250156
$$748$$ 0 0
$$749$$ −8.18342 −0.299016
$$750$$ 0 0
$$751$$ −46.1855 −1.68533 −0.842667 0.538436i $$-0.819015\pi$$
−0.842667 + 0.538436i $$0.819015\pi$$
$$752$$ 0 0
$$753$$ 24.1978 0.881817
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −50.5523 −1.83736 −0.918678 0.395007i $$-0.870742\pi$$
−0.918678 + 0.395007i $$0.870742\pi$$
$$758$$ 0 0
$$759$$ 27.7152 1.00600
$$760$$ 0 0
$$761$$ −12.1711 −0.441203 −0.220602 0.975364i $$-0.570802\pi$$
−0.220602 + 0.975364i $$0.570802\pi$$
$$762$$ 0 0
$$763$$ 16.8371 0.609544
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 2.32580 0.0839797
$$768$$ 0 0
$$769$$ 7.36069 0.265433 0.132717 0.991154i $$-0.457630\pi$$
0.132717 + 0.991154i $$0.457630\pi$$
$$770$$ 0 0
$$771$$ −15.2351 −0.548680
$$772$$ 0 0
$$773$$ 10.5548 0.379629 0.189815 0.981820i $$-0.439211\pi$$
0.189815 + 0.981820i $$0.439211\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −6.15676 −0.220872
$$778$$ 0 0
$$779$$ −2.21008 −0.0791843
$$780$$ 0 0
$$781$$ 52.0000 1.86071
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −9.04718 −0.322497 −0.161249 0.986914i $$-0.551552\pi$$
−0.161249 + 0.986914i $$0.551552\pi$$
$$788$$ 0 0
$$789$$ −29.1773 −1.03874
$$790$$ 0 0
$$791$$ −13.0205 −0.462956
$$792$$ 0 0
$$793$$ 5.22529 0.185556
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −16.9627 −0.600848 −0.300424 0.953806i $$-0.597128\pi$$
−0.300424 + 0.953806i $$0.597128\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −15.1773 −0.536263
$$802$$ 0 0
$$803$$ 4.90271 0.173013
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.13009 −0.215790
$$808$$ 0 0
$$809$$ 15.9877 0.562098 0.281049 0.959693i $$-0.409318\pi$$
0.281049 + 0.959693i $$0.409318\pi$$
$$810$$ 0 0
$$811$$ −8.39350 −0.294736 −0.147368 0.989082i $$-0.547080\pi$$
−0.147368 + 0.989082i $$0.547080\pi$$
$$812$$ 0 0
$$813$$ 25.7009 0.901369
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 56.3956 1.97303
$$818$$ 0 0
$$819$$ 0.340173 0.0118866
$$820$$ 0 0
$$821$$ 38.0288 1.32721 0.663606 0.748082i $$-0.269025\pi$$
0.663606 + 0.748082i $$0.269025\pi$$
$$822$$ 0 0
$$823$$ 22.1568 0.772336 0.386168 0.922428i $$-0.373799\pi$$
0.386168 + 0.922428i $$0.373799\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −14.8227 −0.515437 −0.257718 0.966220i $$-0.582971\pi$$
−0.257718 + 0.966220i $$0.582971\pi$$
$$828$$ 0 0
$$829$$ 13.9467 0.484388 0.242194 0.970228i $$-0.422133\pi$$
0.242194 + 0.970228i $$0.422133\pi$$
$$830$$ 0 0
$$831$$ −11.2039 −0.388660
$$832$$ 0 0
$$833$$ 5.75872 0.199528
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −8.34017 −0.288279
$$838$$ 0 0
$$839$$ −25.4764 −0.879543 −0.439772 0.898110i $$-0.644941\pi$$
−0.439772 + 0.898110i $$0.644941\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 24.3545 0.838815
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −0.360692 −0.0123935
$$848$$ 0 0
$$849$$ 6.15676 0.211299
$$850$$ 0 0
$$851$$ 52.3135 1.79328
$$852$$ 0 0
$$853$$ 58.1588 1.99132 0.995660 0.0930604i $$-0.0296650\pi$$
0.995660 + 0.0930604i $$0.0296650\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −28.9093 −0.987524 −0.493762 0.869597i $$-0.664379\pi$$
−0.493762 + 0.869597i $$0.664379\pi$$
$$858$$ 0 0
$$859$$ 50.1834 1.71224 0.856118 0.516780i $$-0.172870\pi$$
0.856118 + 0.516780i $$0.172870\pi$$
$$860$$ 0 0
$$861$$ 0.340173 0.0115931
$$862$$ 0 0
$$863$$ 8.13009 0.276752 0.138376 0.990380i $$-0.455812\pi$$
0.138376 + 0.990380i $$0.455812\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 16.1629 0.548921
$$868$$ 0 0
$$869$$ −28.3135 −0.960470
$$870$$ 0 0
$$871$$ 5.04718 0.171017
$$872$$ 0 0
$$873$$ −6.49693 −0.219888
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −26.8371 −0.906225 −0.453112 0.891453i $$-0.649686\pi$$
−0.453112 + 0.891453i $$0.649686\pi$$
$$878$$ 0 0
$$879$$ −1.75872 −0.0593203
$$880$$ 0 0
$$881$$ −44.4846 −1.49873 −0.749363 0.662160i $$-0.769640\pi$$
−0.749363 + 0.662160i $$0.769640\pi$$
$$882$$ 0 0
$$883$$ −20.8781 −0.702605 −0.351303 0.936262i $$-0.614261\pi$$
−0.351303 + 0.936262i $$0.614261\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −22.4079 −0.752383 −0.376191 0.926542i $$-0.622766\pi$$
−0.376191 + 0.926542i $$0.622766\pi$$
$$888$$ 0 0
$$889$$ −1.84324 −0.0618204
$$890$$ 0 0
$$891$$ −3.26180 −0.109274
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −2.89043 −0.0965085
$$898$$ 0 0
$$899$$ 16.6803 0.556321
$$900$$ 0 0
$$901$$ −48.0288 −1.60007
$$902$$ 0 0
$$903$$ −8.68035 −0.288864
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −20.8781 −0.693247 −0.346624 0.938004i $$-0.612672\pi$$
−0.346624 + 0.938004i $$0.612672\pi$$
$$908$$ 0 0
$$909$$ −2.18342 −0.0724194
$$910$$ 0 0
$$911$$ 3.52198 0.116688 0.0583442 0.998297i $$-0.481418\pi$$
0.0583442 + 0.998297i $$0.481418\pi$$
$$912$$ 0 0
$$913$$ −22.3012 −0.738063
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −0.313511 −0.0103530
$$918$$ 0 0
$$919$$ 12.1978 0.402368 0.201184 0.979553i $$-0.435521\pi$$
0.201184 + 0.979553i $$0.435521\pi$$
$$920$$ 0 0
$$921$$ 27.2039 0.896400
$$922$$ 0 0
$$923$$ −5.42309 −0.178503
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −5.84324 −0.191917
$$928$$ 0 0
$$929$$ 2.86376 0.0939570 0.0469785 0.998896i $$-0.485041\pi$$
0.0469785 + 0.998896i $$0.485041\pi$$
$$930$$ 0 0
$$931$$ −6.49693 −0.212928
$$932$$ 0 0
$$933$$ −21.8432 −0.715116
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.7480 0.873821 0.436910 0.899505i $$-0.356073\pi$$
0.436910 + 0.899505i $$0.356073\pi$$
$$938$$ 0 0
$$939$$ −25.3874 −0.828485
$$940$$ 0 0
$$941$$ 10.0144 0.326459 0.163230 0.986588i $$-0.447809\pi$$
0.163230 + 0.986588i $$0.447809\pi$$
$$942$$ 0 0
$$943$$ −2.89043 −0.0941252
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 47.2183 1.53439 0.767194 0.641415i $$-0.221652\pi$$
0.767194 + 0.641415i $$0.221652\pi$$
$$948$$ 0 0
$$949$$ −0.511304 −0.0165976
$$950$$ 0 0
$$951$$ 22.6947 0.735927
$$952$$ 0 0
$$953$$ 14.6660 0.475077 0.237539 0.971378i $$-0.423659\pi$$
0.237539 + 0.971378i $$0.423659\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6.52359 0.210878
$$958$$ 0 0
$$959$$ −2.18342 −0.0705062
$$960$$ 0 0
$$961$$ 38.5585 1.24382
$$962$$ 0 0
$$963$$ −8.18342 −0.263707
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −57.3894 −1.84552 −0.922760 0.385375i $$-0.874072\pi$$
−0.922760 + 0.385375i $$0.874072\pi$$
$$968$$ 0 0
$$969$$ −37.4140 −1.20191
$$970$$ 0 0
$$971$$ −23.4017 −0.750997 −0.375499 0.926823i $$-0.622529\pi$$
−0.375499 + 0.926823i $$0.622529\pi$$
$$972$$ 0 0
$$973$$ −1.02052 −0.0327163
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 3.54411 0.113386 0.0566931 0.998392i $$-0.481944\pi$$
0.0566931 + 0.998392i $$0.481944\pi$$
$$978$$ 0 0
$$979$$ 49.5052 1.58219
$$980$$ 0 0
$$981$$ 16.8371 0.537567
$$982$$ 0 0
$$983$$ 11.7152 0.373658 0.186829 0.982392i $$-0.440179\pi$$
0.186829 + 0.982392i $$0.440179\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 73.7563 2.34531
$$990$$ 0 0
$$991$$ 56.5113 1.79514 0.897570 0.440871i $$-0.145330\pi$$
0.897570 + 0.440871i $$0.145330\pi$$
$$992$$ 0 0
$$993$$ 5.47641 0.173789
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 40.9048 1.29547 0.647734 0.761867i $$-0.275717\pi$$
0.647734 + 0.761867i $$0.275717\pi$$
$$998$$ 0 0
$$999$$ −6.15676 −0.194791
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 4200.2.a.bp.1.2 3
4.3 odd 2 8400.2.a.di.1.2 3
5.2 odd 4 840.2.t.d.169.1 6
5.3 odd 4 840.2.t.d.169.4 yes 6
5.4 even 2 4200.2.a.bn.1.2 3
15.2 even 4 2520.2.t.k.1009.6 6
15.8 even 4 2520.2.t.k.1009.5 6
20.3 even 4 1680.2.t.j.1009.1 6
20.7 even 4 1680.2.t.j.1009.4 6
20.19 odd 2 8400.2.a.dl.1.2 3
60.23 odd 4 5040.2.t.z.1009.5 6
60.47 odd 4 5040.2.t.z.1009.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.t.d.169.1 6 5.2 odd 4
840.2.t.d.169.4 yes 6 5.3 odd 4
1680.2.t.j.1009.1 6 20.3 even 4
1680.2.t.j.1009.4 6 20.7 even 4
2520.2.t.k.1009.5 6 15.8 even 4
2520.2.t.k.1009.6 6 15.2 even 4
4200.2.a.bn.1.2 3 5.4 even 2
4200.2.a.bp.1.2 3 1.1 even 1 trivial
5040.2.t.z.1009.5 6 60.23 odd 4
5040.2.t.z.1009.6 6 60.47 odd 4
8400.2.a.di.1.2 3 4.3 odd 2
8400.2.a.dl.1.2 3 20.19 odd 2