# Properties

 Label 8016.2.a.w.1.1 Level $8016$ Weight $2$ Character 8016.1 Self dual yes Analytic conductor $64.008$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8016 = 2^{4} \cdot 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8016.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.0080822603$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4008) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.80982$$ of defining polynomial Character $$\chi$$ $$=$$ 8016.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -3.20729 q^{5} -3.68779 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -3.20729 q^{5} -3.68779 q^{7} +1.00000 q^{9} -2.41235 q^{11} -0.292389 q^{13} -3.20729 q^{15} +7.19061 q^{17} +2.96484 q^{19} -3.68779 q^{21} -5.50281 q^{23} +5.28672 q^{25} +1.00000 q^{27} +3.32252 q^{29} +3.32412 q^{31} -2.41235 q^{33} +11.8278 q^{35} -0.357486 q^{37} -0.292389 q^{39} +3.49808 q^{41} +11.1409 q^{43} -3.20729 q^{45} +0.406706 q^{47} +6.59982 q^{49} +7.19061 q^{51} -10.6914 q^{53} +7.73710 q^{55} +2.96484 q^{57} +3.65193 q^{59} +9.48361 q^{61} -3.68779 q^{63} +0.937777 q^{65} -4.18677 q^{67} -5.50281 q^{69} -2.68125 q^{71} -6.27452 q^{73} +5.28672 q^{75} +8.89624 q^{77} -6.66323 q^{79} +1.00000 q^{81} -10.2090 q^{83} -23.0624 q^{85} +3.32252 q^{87} -4.51995 q^{89} +1.07827 q^{91} +3.32412 q^{93} -9.50909 q^{95} +1.77054 q^{97} -2.41235 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} + O(q^{10})$$ $$7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} - q^{11} - 2q^{13} - 3q^{15} + 11q^{17} - 2q^{19} - 8q^{21} - 17q^{23} + 4q^{25} + 7q^{27} - 7q^{29} - 10q^{31} - q^{33} - 10q^{35} - 21q^{37} - 2q^{39} + 8q^{41} + 12q^{43} - 3q^{45} - 25q^{47} - 7q^{49} + 11q^{51} - 7q^{53} - 15q^{55} - 2q^{57} - 3q^{59} - 14q^{61} - 8q^{63} + 4q^{65} - 4q^{67} - 17q^{69} - 27q^{71} - 12q^{73} + 4q^{75} + 16q^{77} - 8q^{79} + 7q^{81} - 15q^{83} - 3q^{85} - 7q^{87} + 14q^{89} + 3q^{91} - 10q^{93} - 37q^{95} + 3q^{97} - q^{99} + O(q^{100})$$

## 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
$$5$$ −3.20729 −1.43434 −0.717172 0.696896i $$-0.754564\pi$$
−0.717172 + 0.696896i $$0.754564\pi$$
$$6$$ 0 0
$$7$$ −3.68779 −1.39385 −0.696927 0.717142i $$-0.745450\pi$$
−0.696927 + 0.717142i $$0.745450\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.41235 −0.727350 −0.363675 0.931526i $$-0.618478\pi$$
−0.363675 + 0.931526i $$0.618478\pi$$
$$12$$ 0 0
$$13$$ −0.292389 −0.0810942 −0.0405471 0.999178i $$-0.512910\pi$$
−0.0405471 + 0.999178i $$0.512910\pi$$
$$14$$ 0 0
$$15$$ −3.20729 −0.828119
$$16$$ 0 0
$$17$$ 7.19061 1.74398 0.871989 0.489525i $$-0.162830\pi$$
0.871989 + 0.489525i $$0.162830\pi$$
$$18$$ 0 0
$$19$$ 2.96484 0.680180 0.340090 0.940393i $$-0.389542\pi$$
0.340090 + 0.940393i $$0.389542\pi$$
$$20$$ 0 0
$$21$$ −3.68779 −0.804743
$$22$$ 0 0
$$23$$ −5.50281 −1.14742 −0.573708 0.819060i $$-0.694496\pi$$
−0.573708 + 0.819060i $$0.694496\pi$$
$$24$$ 0 0
$$25$$ 5.28672 1.05734
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 3.32252 0.616976 0.308488 0.951228i $$-0.400177\pi$$
0.308488 + 0.951228i $$0.400177\pi$$
$$30$$ 0 0
$$31$$ 3.32412 0.597029 0.298514 0.954405i $$-0.403509\pi$$
0.298514 + 0.954405i $$0.403509\pi$$
$$32$$ 0 0
$$33$$ −2.41235 −0.419936
$$34$$ 0 0
$$35$$ 11.8278 1.99927
$$36$$ 0 0
$$37$$ −0.357486 −0.0587703 −0.0293852 0.999568i $$-0.509355\pi$$
−0.0293852 + 0.999568i $$0.509355\pi$$
$$38$$ 0 0
$$39$$ −0.292389 −0.0468197
$$40$$ 0 0
$$41$$ 3.49808 0.546309 0.273154 0.961970i $$-0.411933\pi$$
0.273154 + 0.961970i $$0.411933\pi$$
$$42$$ 0 0
$$43$$ 11.1409 1.69898 0.849488 0.527608i $$-0.176911\pi$$
0.849488 + 0.527608i $$0.176911\pi$$
$$44$$ 0 0
$$45$$ −3.20729 −0.478115
$$46$$ 0 0
$$47$$ 0.406706 0.0593241 0.0296621 0.999560i $$-0.490557\pi$$
0.0296621 + 0.999560i $$0.490557\pi$$
$$48$$ 0 0
$$49$$ 6.59982 0.942832
$$50$$ 0 0
$$51$$ 7.19061 1.00689
$$52$$ 0 0
$$53$$ −10.6914 −1.46858 −0.734291 0.678834i $$-0.762485\pi$$
−0.734291 + 0.678834i $$0.762485\pi$$
$$54$$ 0 0
$$55$$ 7.73710 1.04327
$$56$$ 0 0
$$57$$ 2.96484 0.392702
$$58$$ 0 0
$$59$$ 3.65193 0.475441 0.237720 0.971334i $$-0.423600\pi$$
0.237720 + 0.971334i $$0.423600\pi$$
$$60$$ 0 0
$$61$$ 9.48361 1.21425 0.607126 0.794606i $$-0.292322\pi$$
0.607126 + 0.794606i $$0.292322\pi$$
$$62$$ 0 0
$$63$$ −3.68779 −0.464618
$$64$$ 0 0
$$65$$ 0.937777 0.116317
$$66$$ 0 0
$$67$$ −4.18677 −0.511496 −0.255748 0.966744i $$-0.582322\pi$$
−0.255748 + 0.966744i $$0.582322\pi$$
$$68$$ 0 0
$$69$$ −5.50281 −0.662461
$$70$$ 0 0
$$71$$ −2.68125 −0.318206 −0.159103 0.987262i $$-0.550860\pi$$
−0.159103 + 0.987262i $$0.550860\pi$$
$$72$$ 0 0
$$73$$ −6.27452 −0.734377 −0.367189 0.930147i $$-0.619680\pi$$
−0.367189 + 0.930147i $$0.619680\pi$$
$$74$$ 0 0
$$75$$ 5.28672 0.610457
$$76$$ 0 0
$$77$$ 8.89624 1.01382
$$78$$ 0 0
$$79$$ −6.66323 −0.749672 −0.374836 0.927091i $$-0.622301\pi$$
−0.374836 + 0.927091i $$0.622301\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −10.2090 −1.12059 −0.560293 0.828295i $$-0.689311\pi$$
−0.560293 + 0.828295i $$0.689311\pi$$
$$84$$ 0 0
$$85$$ −23.0624 −2.50147
$$86$$ 0 0
$$87$$ 3.32252 0.356211
$$88$$ 0 0
$$89$$ −4.51995 −0.479114 −0.239557 0.970882i $$-0.577002\pi$$
−0.239557 + 0.970882i $$0.577002\pi$$
$$90$$ 0 0
$$91$$ 1.07827 0.113034
$$92$$ 0 0
$$93$$ 3.32412 0.344695
$$94$$ 0 0
$$95$$ −9.50909 −0.975613
$$96$$ 0 0
$$97$$ 1.77054 0.179771 0.0898853 0.995952i $$-0.471350\pi$$
0.0898853 + 0.995952i $$0.471350\pi$$
$$98$$ 0 0
$$99$$ −2.41235 −0.242450
$$100$$ 0 0
$$101$$ 18.8099 1.87165 0.935826 0.352463i $$-0.114656\pi$$
0.935826 + 0.352463i $$0.114656\pi$$
$$102$$ 0 0
$$103$$ −5.82075 −0.573536 −0.286768 0.958000i $$-0.592581\pi$$
−0.286768 + 0.958000i $$0.592581\pi$$
$$104$$ 0 0
$$105$$ 11.8278 1.15428
$$106$$ 0 0
$$107$$ −9.11502 −0.881182 −0.440591 0.897708i $$-0.645231\pi$$
−0.440591 + 0.897708i $$0.645231\pi$$
$$108$$ 0 0
$$109$$ −18.8744 −1.80784 −0.903922 0.427697i $$-0.859325\pi$$
−0.903922 + 0.427697i $$0.859325\pi$$
$$110$$ 0 0
$$111$$ −0.357486 −0.0339311
$$112$$ 0 0
$$113$$ 12.5304 1.17876 0.589381 0.807855i $$-0.299372\pi$$
0.589381 + 0.807855i $$0.299372\pi$$
$$114$$ 0 0
$$115$$ 17.6491 1.64579
$$116$$ 0 0
$$117$$ −0.292389 −0.0270314
$$118$$ 0 0
$$119$$ −26.5175 −2.43085
$$120$$ 0 0
$$121$$ −5.18058 −0.470962
$$122$$ 0 0
$$123$$ 3.49808 0.315412
$$124$$ 0 0
$$125$$ −0.919585 −0.0822502
$$126$$ 0 0
$$127$$ 1.13671 0.100867 0.0504333 0.998727i $$-0.483940\pi$$
0.0504333 + 0.998727i $$0.483940\pi$$
$$128$$ 0 0
$$129$$ 11.1409 0.980904
$$130$$ 0 0
$$131$$ −12.1630 −1.06269 −0.531343 0.847157i $$-0.678312\pi$$
−0.531343 + 0.847157i $$0.678312\pi$$
$$132$$ 0 0
$$133$$ −10.9337 −0.948072
$$134$$ 0 0
$$135$$ −3.20729 −0.276040
$$136$$ 0 0
$$137$$ −14.4346 −1.23323 −0.616616 0.787264i $$-0.711497\pi$$
−0.616616 + 0.787264i $$0.711497\pi$$
$$138$$ 0 0
$$139$$ 0.433633 0.0367803 0.0183902 0.999831i $$-0.494146\pi$$
0.0183902 + 0.999831i $$0.494146\pi$$
$$140$$ 0 0
$$141$$ 0.406706 0.0342508
$$142$$ 0 0
$$143$$ 0.705344 0.0589839
$$144$$ 0 0
$$145$$ −10.6563 −0.884956
$$146$$ 0 0
$$147$$ 6.59982 0.544344
$$148$$ 0 0
$$149$$ −19.0992 −1.56466 −0.782332 0.622862i $$-0.785970\pi$$
−0.782332 + 0.622862i $$0.785970\pi$$
$$150$$ 0 0
$$151$$ −8.70546 −0.708440 −0.354220 0.935162i $$-0.615254\pi$$
−0.354220 + 0.935162i $$0.615254\pi$$
$$152$$ 0 0
$$153$$ 7.19061 0.581326
$$154$$ 0 0
$$155$$ −10.6614 −0.856345
$$156$$ 0 0
$$157$$ 22.6106 1.80452 0.902261 0.431191i $$-0.141907\pi$$
0.902261 + 0.431191i $$0.141907\pi$$
$$158$$ 0 0
$$159$$ −10.6914 −0.847887
$$160$$ 0 0
$$161$$ 20.2932 1.59933
$$162$$ 0 0
$$163$$ 9.50323 0.744350 0.372175 0.928162i $$-0.378612\pi$$
0.372175 + 0.928162i $$0.378612\pi$$
$$164$$ 0 0
$$165$$ 7.73710 0.602332
$$166$$ 0 0
$$167$$ 1.00000 0.0773823
$$168$$ 0 0
$$169$$ −12.9145 −0.993424
$$170$$ 0 0
$$171$$ 2.96484 0.226727
$$172$$ 0 0
$$173$$ 3.88163 0.295115 0.147557 0.989053i $$-0.452859\pi$$
0.147557 + 0.989053i $$0.452859\pi$$
$$174$$ 0 0
$$175$$ −19.4963 −1.47378
$$176$$ 0 0
$$177$$ 3.65193 0.274496
$$178$$ 0 0
$$179$$ 7.65909 0.572467 0.286234 0.958160i $$-0.407597\pi$$
0.286234 + 0.958160i $$0.407597\pi$$
$$180$$ 0 0
$$181$$ 9.78926 0.727630 0.363815 0.931471i $$-0.381474\pi$$
0.363815 + 0.931471i $$0.381474\pi$$
$$182$$ 0 0
$$183$$ 9.48361 0.701048
$$184$$ 0 0
$$185$$ 1.14656 0.0842969
$$186$$ 0 0
$$187$$ −17.3462 −1.26848
$$188$$ 0 0
$$189$$ −3.68779 −0.268248
$$190$$ 0 0
$$191$$ −14.9580 −1.08232 −0.541162 0.840918i $$-0.682015\pi$$
−0.541162 + 0.840918i $$0.682015\pi$$
$$192$$ 0 0
$$193$$ −0.569004 −0.0409578 −0.0204789 0.999790i $$-0.506519\pi$$
−0.0204789 + 0.999790i $$0.506519\pi$$
$$194$$ 0 0
$$195$$ 0.937777 0.0671556
$$196$$ 0 0
$$197$$ 24.1326 1.71938 0.859688 0.510820i $$-0.170658\pi$$
0.859688 + 0.510820i $$0.170658\pi$$
$$198$$ 0 0
$$199$$ −10.6151 −0.752482 −0.376241 0.926522i $$-0.622784\pi$$
−0.376241 + 0.926522i $$0.622784\pi$$
$$200$$ 0 0
$$201$$ −4.18677 −0.295312
$$202$$ 0 0
$$203$$ −12.2528 −0.859975
$$204$$ 0 0
$$205$$ −11.2194 −0.783595
$$206$$ 0 0
$$207$$ −5.50281 −0.382472
$$208$$ 0 0
$$209$$ −7.15222 −0.494729
$$210$$ 0 0
$$211$$ −14.9203 −1.02716 −0.513579 0.858042i $$-0.671681\pi$$
−0.513579 + 0.858042i $$0.671681\pi$$
$$212$$ 0 0
$$213$$ −2.68125 −0.183716
$$214$$ 0 0
$$215$$ −35.7322 −2.43692
$$216$$ 0 0
$$217$$ −12.2587 −0.832171
$$218$$ 0 0
$$219$$ −6.27452 −0.423993
$$220$$ 0 0
$$221$$ −2.10246 −0.141427
$$222$$ 0 0
$$223$$ 14.5962 0.977431 0.488715 0.872443i $$-0.337466\pi$$
0.488715 + 0.872443i $$0.337466\pi$$
$$224$$ 0 0
$$225$$ 5.28672 0.352448
$$226$$ 0 0
$$227$$ −7.67692 −0.509535 −0.254767 0.967002i $$-0.581999\pi$$
−0.254767 + 0.967002i $$0.581999\pi$$
$$228$$ 0 0
$$229$$ −7.74042 −0.511501 −0.255751 0.966743i $$-0.582323\pi$$
−0.255751 + 0.966743i $$0.582323\pi$$
$$230$$ 0 0
$$231$$ 8.89624 0.585330
$$232$$ 0 0
$$233$$ −15.6688 −1.02650 −0.513250 0.858239i $$-0.671558\pi$$
−0.513250 + 0.858239i $$0.671558\pi$$
$$234$$ 0 0
$$235$$ −1.30442 −0.0850912
$$236$$ 0 0
$$237$$ −6.66323 −0.432823
$$238$$ 0 0
$$239$$ −20.3027 −1.31327 −0.656636 0.754208i $$-0.728021\pi$$
−0.656636 + 0.754208i $$0.728021\pi$$
$$240$$ 0 0
$$241$$ 7.43764 0.479100 0.239550 0.970884i $$-0.423000\pi$$
0.239550 + 0.970884i $$0.423000\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −21.1675 −1.35235
$$246$$ 0 0
$$247$$ −0.866886 −0.0551586
$$248$$ 0 0
$$249$$ −10.2090 −0.646970
$$250$$ 0 0
$$251$$ −28.3028 −1.78646 −0.893228 0.449604i $$-0.851565\pi$$
−0.893228 + 0.449604i $$0.851565\pi$$
$$252$$ 0 0
$$253$$ 13.2747 0.834573
$$254$$ 0 0
$$255$$ −23.0624 −1.44422
$$256$$ 0 0
$$257$$ −7.82120 −0.487873 −0.243936 0.969791i $$-0.578439\pi$$
−0.243936 + 0.969791i $$0.578439\pi$$
$$258$$ 0 0
$$259$$ 1.31833 0.0819173
$$260$$ 0 0
$$261$$ 3.32252 0.205659
$$262$$ 0 0
$$263$$ 6.44120 0.397182 0.198591 0.980082i $$-0.436364\pi$$
0.198591 + 0.980082i $$0.436364\pi$$
$$264$$ 0 0
$$265$$ 34.2906 2.10645
$$266$$ 0 0
$$267$$ −4.51995 −0.276617
$$268$$ 0 0
$$269$$ 21.7914 1.32865 0.664323 0.747446i $$-0.268720\pi$$
0.664323 + 0.747446i $$0.268720\pi$$
$$270$$ 0 0
$$271$$ 10.1627 0.617340 0.308670 0.951169i $$-0.400116\pi$$
0.308670 + 0.951169i $$0.400116\pi$$
$$272$$ 0 0
$$273$$ 1.07827 0.0652599
$$274$$ 0 0
$$275$$ −12.7534 −0.769059
$$276$$ 0 0
$$277$$ 5.38295 0.323430 0.161715 0.986838i $$-0.448297\pi$$
0.161715 + 0.986838i $$0.448297\pi$$
$$278$$ 0 0
$$279$$ 3.32412 0.199010
$$280$$ 0 0
$$281$$ −17.2566 −1.02945 −0.514723 0.857357i $$-0.672105\pi$$
−0.514723 + 0.857357i $$0.672105\pi$$
$$282$$ 0 0
$$283$$ 6.13015 0.364399 0.182200 0.983262i $$-0.441678\pi$$
0.182200 + 0.983262i $$0.441678\pi$$
$$284$$ 0 0
$$285$$ −9.50909 −0.563270
$$286$$ 0 0
$$287$$ −12.9002 −0.761475
$$288$$ 0 0
$$289$$ 34.7048 2.04146
$$290$$ 0 0
$$291$$ 1.77054 0.103791
$$292$$ 0 0
$$293$$ 17.6848 1.03316 0.516578 0.856240i $$-0.327206\pi$$
0.516578 + 0.856240i $$0.327206\pi$$
$$294$$ 0 0
$$295$$ −11.7128 −0.681945
$$296$$ 0 0
$$297$$ −2.41235 −0.139979
$$298$$ 0 0
$$299$$ 1.60896 0.0930488
$$300$$ 0 0
$$301$$ −41.0854 −2.36813
$$302$$ 0 0
$$303$$ 18.8099 1.08060
$$304$$ 0 0
$$305$$ −30.4167 −1.74165
$$306$$ 0 0
$$307$$ −11.1101 −0.634085 −0.317042 0.948411i $$-0.602690\pi$$
−0.317042 + 0.948411i $$0.602690\pi$$
$$308$$ 0 0
$$309$$ −5.82075 −0.331131
$$310$$ 0 0
$$311$$ −28.3481 −1.60747 −0.803737 0.594985i $$-0.797158\pi$$
−0.803737 + 0.594985i $$0.797158\pi$$
$$312$$ 0 0
$$313$$ 31.4209 1.77601 0.888006 0.459831i $$-0.152090\pi$$
0.888006 + 0.459831i $$0.152090\pi$$
$$314$$ 0 0
$$315$$ 11.8278 0.666423
$$316$$ 0 0
$$317$$ −7.57882 −0.425669 −0.212834 0.977088i $$-0.568269\pi$$
−0.212834 + 0.977088i $$0.568269\pi$$
$$318$$ 0 0
$$319$$ −8.01507 −0.448758
$$320$$ 0 0
$$321$$ −9.11502 −0.508751
$$322$$ 0 0
$$323$$ 21.3190 1.18622
$$324$$ 0 0
$$325$$ −1.54578 −0.0857444
$$326$$ 0 0
$$327$$ −18.8744 −1.04376
$$328$$ 0 0
$$329$$ −1.49985 −0.0826892
$$330$$ 0 0
$$331$$ −29.3509 −1.61327 −0.806636 0.591049i $$-0.798714\pi$$
−0.806636 + 0.591049i $$0.798714\pi$$
$$332$$ 0 0
$$333$$ −0.357486 −0.0195901
$$334$$ 0 0
$$335$$ 13.4282 0.733661
$$336$$ 0 0
$$337$$ −24.2626 −1.32167 −0.660833 0.750533i $$-0.729797\pi$$
−0.660833 + 0.750533i $$0.729797\pi$$
$$338$$ 0 0
$$339$$ 12.5304 0.680559
$$340$$ 0 0
$$341$$ −8.01892 −0.434249
$$342$$ 0 0
$$343$$ 1.47578 0.0796844
$$344$$ 0 0
$$345$$ 17.6491 0.950197
$$346$$ 0 0
$$347$$ −16.9314 −0.908924 −0.454462 0.890766i $$-0.650169\pi$$
−0.454462 + 0.890766i $$0.650169\pi$$
$$348$$ 0 0
$$349$$ 2.41024 0.129017 0.0645087 0.997917i $$-0.479452\pi$$
0.0645087 + 0.997917i $$0.479452\pi$$
$$350$$ 0 0
$$351$$ −0.292389 −0.0156066
$$352$$ 0 0
$$353$$ −12.2607 −0.652570 −0.326285 0.945271i $$-0.605797\pi$$
−0.326285 + 0.945271i $$0.605797\pi$$
$$354$$ 0 0
$$355$$ 8.59956 0.456417
$$356$$ 0 0
$$357$$ −26.5175 −1.40345
$$358$$ 0 0
$$359$$ −12.1673 −0.642166 −0.321083 0.947051i $$-0.604047\pi$$
−0.321083 + 0.947051i $$0.604047\pi$$
$$360$$ 0 0
$$361$$ −10.2097 −0.537355
$$362$$ 0 0
$$363$$ −5.18058 −0.271910
$$364$$ 0 0
$$365$$ 20.1242 1.05335
$$366$$ 0 0
$$367$$ −7.82364 −0.408391 −0.204195 0.978930i $$-0.565458\pi$$
−0.204195 + 0.978930i $$0.565458\pi$$
$$368$$ 0 0
$$369$$ 3.49808 0.182103
$$370$$ 0 0
$$371$$ 39.4278 2.04699
$$372$$ 0 0
$$373$$ −18.1180 −0.938114 −0.469057 0.883168i $$-0.655406\pi$$
−0.469057 + 0.883168i $$0.655406\pi$$
$$374$$ 0 0
$$375$$ −0.919585 −0.0474872
$$376$$ 0 0
$$377$$ −0.971469 −0.0500332
$$378$$ 0 0
$$379$$ −0.116915 −0.00600553 −0.00300276 0.999995i $$-0.500956\pi$$
−0.00300276 + 0.999995i $$0.500956\pi$$
$$380$$ 0 0
$$381$$ 1.13671 0.0582354
$$382$$ 0 0
$$383$$ 0.723537 0.0369710 0.0184855 0.999829i $$-0.494116\pi$$
0.0184855 + 0.999829i $$0.494116\pi$$
$$384$$ 0 0
$$385$$ −28.5328 −1.45417
$$386$$ 0 0
$$387$$ 11.1409 0.566325
$$388$$ 0 0
$$389$$ −24.8520 −1.26004 −0.630022 0.776577i $$-0.716954\pi$$
−0.630022 + 0.776577i $$0.716954\pi$$
$$390$$ 0 0
$$391$$ −39.5686 −2.00107
$$392$$ 0 0
$$393$$ −12.1630 −0.613542
$$394$$ 0 0
$$395$$ 21.3709 1.07529
$$396$$ 0 0
$$397$$ 14.6468 0.735102 0.367551 0.930003i $$-0.380196\pi$$
0.367551 + 0.930003i $$0.380196\pi$$
$$398$$ 0 0
$$399$$ −10.9337 −0.547370
$$400$$ 0 0
$$401$$ −5.34823 −0.267078 −0.133539 0.991044i $$-0.542634\pi$$
−0.133539 + 0.991044i $$0.542634\pi$$
$$402$$ 0 0
$$403$$ −0.971935 −0.0484156
$$404$$ 0 0
$$405$$ −3.20729 −0.159372
$$406$$ 0 0
$$407$$ 0.862380 0.0427466
$$408$$ 0 0
$$409$$ 16.7435 0.827913 0.413957 0.910297i $$-0.364146\pi$$
0.413957 + 0.910297i $$0.364146\pi$$
$$410$$ 0 0
$$411$$ −14.4346 −0.712007
$$412$$ 0 0
$$413$$ −13.4676 −0.662695
$$414$$ 0 0
$$415$$ 32.7433 1.60731
$$416$$ 0 0
$$417$$ 0.433633 0.0212351
$$418$$ 0 0
$$419$$ −5.99429 −0.292840 −0.146420 0.989222i $$-0.546775\pi$$
−0.146420 + 0.989222i $$0.546775\pi$$
$$420$$ 0 0
$$421$$ 2.60581 0.126999 0.0634996 0.997982i $$-0.479774\pi$$
0.0634996 + 0.997982i $$0.479774\pi$$
$$422$$ 0 0
$$423$$ 0.406706 0.0197747
$$424$$ 0 0
$$425$$ 38.0147 1.84398
$$426$$ 0 0
$$427$$ −34.9736 −1.69249
$$428$$ 0 0
$$429$$ 0.705344 0.0340543
$$430$$ 0 0
$$431$$ −3.60689 −0.173738 −0.0868690 0.996220i $$-0.527686\pi$$
−0.0868690 + 0.996220i $$0.527686\pi$$
$$432$$ 0 0
$$433$$ 11.1030 0.533576 0.266788 0.963755i $$-0.414038\pi$$
0.266788 + 0.963755i $$0.414038\pi$$
$$434$$ 0 0
$$435$$ −10.6563 −0.510930
$$436$$ 0 0
$$437$$ −16.3149 −0.780450
$$438$$ 0 0
$$439$$ 37.7654 1.80245 0.901223 0.433356i $$-0.142671\pi$$
0.901223 + 0.433356i $$0.142671\pi$$
$$440$$ 0 0
$$441$$ 6.59982 0.314277
$$442$$ 0 0
$$443$$ 21.3294 1.01339 0.506695 0.862125i $$-0.330867\pi$$
0.506695 + 0.862125i $$0.330867\pi$$
$$444$$ 0 0
$$445$$ 14.4968 0.687215
$$446$$ 0 0
$$447$$ −19.0992 −0.903359
$$448$$ 0 0
$$449$$ 28.7910 1.35873 0.679365 0.733800i $$-0.262255\pi$$
0.679365 + 0.733800i $$0.262255\pi$$
$$450$$ 0 0
$$451$$ −8.43859 −0.397358
$$452$$ 0 0
$$453$$ −8.70546 −0.409018
$$454$$ 0 0
$$455$$ −3.45833 −0.162129
$$456$$ 0 0
$$457$$ 18.7766 0.878332 0.439166 0.898406i $$-0.355274\pi$$
0.439166 + 0.898406i $$0.355274\pi$$
$$458$$ 0 0
$$459$$ 7.19061 0.335629
$$460$$ 0 0
$$461$$ −12.0426 −0.560878 −0.280439 0.959872i $$-0.590480\pi$$
−0.280439 + 0.959872i $$0.590480\pi$$
$$462$$ 0 0
$$463$$ −21.0030 −0.976094 −0.488047 0.872817i $$-0.662291\pi$$
−0.488047 + 0.872817i $$0.662291\pi$$
$$464$$ 0 0
$$465$$ −10.6614 −0.494411
$$466$$ 0 0
$$467$$ 36.3609 1.68258 0.841291 0.540582i $$-0.181796\pi$$
0.841291 + 0.540582i $$0.181796\pi$$
$$468$$ 0 0
$$469$$ 15.4400 0.712951
$$470$$ 0 0
$$471$$ 22.6106 1.04184
$$472$$ 0 0
$$473$$ −26.8758 −1.23575
$$474$$ 0 0
$$475$$ 15.6743 0.719184
$$476$$ 0 0
$$477$$ −10.6914 −0.489528
$$478$$ 0 0
$$479$$ −10.2203 −0.466976 −0.233488 0.972360i $$-0.575014\pi$$
−0.233488 + 0.972360i $$0.575014\pi$$
$$480$$ 0 0
$$481$$ 0.104525 0.00476593
$$482$$ 0 0
$$483$$ 20.2932 0.923375
$$484$$ 0 0
$$485$$ −5.67862 −0.257853
$$486$$ 0 0
$$487$$ 27.8144 1.26039 0.630195 0.776437i $$-0.282975\pi$$
0.630195 + 0.776437i $$0.282975\pi$$
$$488$$ 0 0
$$489$$ 9.50323 0.429751
$$490$$ 0 0
$$491$$ −6.01324 −0.271374 −0.135687 0.990752i $$-0.543324\pi$$
−0.135687 + 0.990752i $$0.543324\pi$$
$$492$$ 0 0
$$493$$ 23.8909 1.07599
$$494$$ 0 0
$$495$$ 7.73710 0.347757
$$496$$ 0 0
$$497$$ 9.88791 0.443533
$$498$$ 0 0
$$499$$ −6.63868 −0.297188 −0.148594 0.988898i $$-0.547475\pi$$
−0.148594 + 0.988898i $$0.547475\pi$$
$$500$$ 0 0
$$501$$ 1.00000 0.0446767
$$502$$ 0 0
$$503$$ −42.5316 −1.89639 −0.948195 0.317689i $$-0.897093\pi$$
−0.948195 + 0.317689i $$0.897093\pi$$
$$504$$ 0 0
$$505$$ −60.3287 −2.68459
$$506$$ 0 0
$$507$$ −12.9145 −0.573553
$$508$$ 0 0
$$509$$ −35.1257 −1.55692 −0.778461 0.627693i $$-0.783999\pi$$
−0.778461 + 0.627693i $$0.783999\pi$$
$$510$$ 0 0
$$511$$ 23.1391 1.02362
$$512$$ 0 0
$$513$$ 2.96484 0.130901
$$514$$ 0 0
$$515$$ 18.6689 0.822648
$$516$$ 0 0
$$517$$ −0.981115 −0.0431494
$$518$$ 0 0
$$519$$ 3.88163 0.170385
$$520$$ 0 0
$$521$$ −37.8196 −1.65691 −0.828454 0.560057i $$-0.810779\pi$$
−0.828454 + 0.560057i $$0.810779\pi$$
$$522$$ 0 0
$$523$$ 2.84372 0.124347 0.0621737 0.998065i $$-0.480197\pi$$
0.0621737 + 0.998065i $$0.480197\pi$$
$$524$$ 0 0
$$525$$ −19.4963 −0.850889
$$526$$ 0 0
$$527$$ 23.9024 1.04121
$$528$$ 0 0
$$529$$ 7.28097 0.316564
$$530$$ 0 0
$$531$$ 3.65193 0.158480
$$532$$ 0 0
$$533$$ −1.02280 −0.0443025
$$534$$ 0 0
$$535$$ 29.2345 1.26392
$$536$$ 0 0
$$537$$ 7.65909 0.330514
$$538$$ 0 0
$$539$$ −15.9211 −0.685769
$$540$$ 0 0
$$541$$ 37.0033 1.59090 0.795448 0.606022i $$-0.207236\pi$$
0.795448 + 0.606022i $$0.207236\pi$$
$$542$$ 0 0
$$543$$ 9.78926 0.420097
$$544$$ 0 0
$$545$$ 60.5358 2.59307
$$546$$ 0 0
$$547$$ 33.2638 1.42226 0.711128 0.703063i $$-0.248185\pi$$
0.711128 + 0.703063i $$0.248185\pi$$
$$548$$ 0 0
$$549$$ 9.48361 0.404750
$$550$$ 0 0
$$551$$ 9.85072 0.419655
$$552$$ 0 0
$$553$$ 24.5726 1.04493
$$554$$ 0 0
$$555$$ 1.14656 0.0486688
$$556$$ 0 0
$$557$$ −38.8570 −1.64642 −0.823212 0.567734i $$-0.807820\pi$$
−0.823212 + 0.567734i $$0.807820\pi$$
$$558$$ 0 0
$$559$$ −3.25749 −0.137777
$$560$$ 0 0
$$561$$ −17.3462 −0.732359
$$562$$ 0 0
$$563$$ −15.4700 −0.651984 −0.325992 0.945373i $$-0.605698\pi$$
−0.325992 + 0.945373i $$0.605698\pi$$
$$564$$ 0 0
$$565$$ −40.1887 −1.69075
$$566$$ 0 0
$$567$$ −3.68779 −0.154873
$$568$$ 0 0
$$569$$ −21.7094 −0.910107 −0.455053 0.890464i $$-0.650380\pi$$
−0.455053 + 0.890464i $$0.650380\pi$$
$$570$$ 0 0
$$571$$ −22.9576 −0.960745 −0.480372 0.877065i $$-0.659498\pi$$
−0.480372 + 0.877065i $$0.659498\pi$$
$$572$$ 0 0
$$573$$ −14.9580 −0.624880
$$574$$ 0 0
$$575$$ −29.0918 −1.21321
$$576$$ 0 0
$$577$$ −23.8858 −0.994381 −0.497190 0.867641i $$-0.665635\pi$$
−0.497190 + 0.867641i $$0.665635\pi$$
$$578$$ 0 0
$$579$$ −0.569004 −0.0236470
$$580$$ 0 0
$$581$$ 37.6488 1.56193
$$582$$ 0 0
$$583$$ 25.7915 1.06817
$$584$$ 0 0
$$585$$ 0.937777 0.0387723
$$586$$ 0 0
$$587$$ 18.2904 0.754926 0.377463 0.926025i $$-0.376797\pi$$
0.377463 + 0.926025i $$0.376797\pi$$
$$588$$ 0 0
$$589$$ 9.85546 0.406087
$$590$$ 0 0
$$591$$ 24.1326 0.992682
$$592$$ 0 0
$$593$$ −33.7008 −1.38393 −0.691964 0.721932i $$-0.743254\pi$$
−0.691964 + 0.721932i $$0.743254\pi$$
$$594$$ 0 0
$$595$$ 85.0493 3.48668
$$596$$ 0 0
$$597$$ −10.6151 −0.434446
$$598$$ 0 0
$$599$$ 22.1340 0.904372 0.452186 0.891924i $$-0.350644\pi$$
0.452186 + 0.891924i $$0.350644\pi$$
$$600$$ 0 0
$$601$$ 6.70740 0.273601 0.136800 0.990599i $$-0.456318\pi$$
0.136800 + 0.990599i $$0.456318\pi$$
$$602$$ 0 0
$$603$$ −4.18677 −0.170499
$$604$$ 0 0
$$605$$ 16.6156 0.675521
$$606$$ 0 0
$$607$$ −30.2730 −1.22874 −0.614371 0.789017i $$-0.710590\pi$$
−0.614371 + 0.789017i $$0.710590\pi$$
$$608$$ 0 0
$$609$$ −12.2528 −0.496507
$$610$$ 0 0
$$611$$ −0.118916 −0.00481084
$$612$$ 0 0
$$613$$ 2.52415 0.101949 0.0509747 0.998700i $$-0.483767\pi$$
0.0509747 + 0.998700i $$0.483767\pi$$
$$614$$ 0 0
$$615$$ −11.2194 −0.452409
$$616$$ 0 0
$$617$$ 12.6084 0.507593 0.253797 0.967258i $$-0.418321\pi$$
0.253797 + 0.967258i $$0.418321\pi$$
$$618$$ 0 0
$$619$$ 3.97122 0.159617 0.0798085 0.996810i $$-0.474569\pi$$
0.0798085 + 0.996810i $$0.474569\pi$$
$$620$$ 0 0
$$621$$ −5.50281 −0.220820
$$622$$ 0 0
$$623$$ 16.6687 0.667816
$$624$$ 0 0
$$625$$ −23.4842 −0.939368
$$626$$ 0 0
$$627$$ −7.15222 −0.285632
$$628$$ 0 0
$$629$$ −2.57054 −0.102494
$$630$$ 0 0
$$631$$ 48.8417 1.94436 0.972179 0.234238i $$-0.0752594\pi$$
0.972179 + 0.234238i $$0.0752594\pi$$
$$632$$ 0 0
$$633$$ −14.9203 −0.593030
$$634$$ 0 0
$$635$$ −3.64576 −0.144677
$$636$$ 0 0
$$637$$ −1.92972 −0.0764582
$$638$$ 0 0
$$639$$ −2.68125 −0.106069
$$640$$ 0 0
$$641$$ −11.7538 −0.464246 −0.232123 0.972686i $$-0.574567\pi$$
−0.232123 + 0.972686i $$0.574567\pi$$
$$642$$ 0 0
$$643$$ 18.9663 0.747958 0.373979 0.927437i $$-0.377993\pi$$
0.373979 + 0.927437i $$0.377993\pi$$
$$644$$ 0 0
$$645$$ −35.7322 −1.40695
$$646$$ 0 0
$$647$$ −37.2745 −1.46541 −0.732705 0.680546i $$-0.761743\pi$$
−0.732705 + 0.680546i $$0.761743\pi$$
$$648$$ 0 0
$$649$$ −8.80972 −0.345812
$$650$$ 0 0
$$651$$ −12.2587 −0.480454
$$652$$ 0 0
$$653$$ −9.70587 −0.379820 −0.189910 0.981802i $$-0.560820\pi$$
−0.189910 + 0.981802i $$0.560820\pi$$
$$654$$ 0 0
$$655$$ 39.0103 1.52426
$$656$$ 0 0
$$657$$ −6.27452 −0.244792
$$658$$ 0 0
$$659$$ −21.9798 −0.856211 −0.428105 0.903729i $$-0.640819\pi$$
−0.428105 + 0.903729i $$0.640819\pi$$
$$660$$ 0 0
$$661$$ −24.2779 −0.944300 −0.472150 0.881518i $$-0.656522\pi$$
−0.472150 + 0.881518i $$0.656522\pi$$
$$662$$ 0 0
$$663$$ −2.10246 −0.0816526
$$664$$ 0 0
$$665$$ 35.0676 1.35986
$$666$$ 0 0
$$667$$ −18.2832 −0.707928
$$668$$ 0 0
$$669$$ 14.5962 0.564320
$$670$$ 0 0
$$671$$ −22.8778 −0.883186
$$672$$ 0 0
$$673$$ 27.1769 1.04759 0.523795 0.851844i $$-0.324516\pi$$
0.523795 + 0.851844i $$0.324516\pi$$
$$674$$ 0 0
$$675$$ 5.28672 0.203486
$$676$$ 0 0
$$677$$ 31.2103 1.19951 0.599755 0.800184i $$-0.295265\pi$$
0.599755 + 0.800184i $$0.295265\pi$$
$$678$$ 0 0
$$679$$ −6.52937 −0.250574
$$680$$ 0 0
$$681$$ −7.67692 −0.294180
$$682$$ 0 0
$$683$$ −8.68394 −0.332282 −0.166141 0.986102i $$-0.553131\pi$$
−0.166141 + 0.986102i $$0.553131\pi$$
$$684$$ 0 0
$$685$$ 46.2960 1.76888
$$686$$ 0 0
$$687$$ −7.74042 −0.295316
$$688$$ 0 0
$$689$$ 3.12606 0.119094
$$690$$ 0 0
$$691$$ 10.3984 0.395573 0.197787 0.980245i $$-0.436625\pi$$
0.197787 + 0.980245i $$0.436625\pi$$
$$692$$ 0 0
$$693$$ 8.89624 0.337940
$$694$$ 0 0
$$695$$ −1.39079 −0.0527556
$$696$$ 0 0
$$697$$ 25.1533 0.952751
$$698$$ 0 0
$$699$$ −15.6688 −0.592650
$$700$$ 0 0
$$701$$ 33.3918 1.26119 0.630596 0.776111i $$-0.282811\pi$$
0.630596 + 0.776111i $$0.282811\pi$$
$$702$$ 0 0
$$703$$ −1.05989 −0.0399744
$$704$$ 0 0
$$705$$ −1.30442 −0.0491274
$$706$$ 0 0
$$707$$ −69.3669 −2.60881
$$708$$ 0 0
$$709$$ −5.92848 −0.222649 −0.111324 0.993784i $$-0.535509\pi$$
−0.111324 + 0.993784i $$0.535509\pi$$
$$710$$ 0 0
$$711$$ −6.66323 −0.249891
$$712$$ 0 0
$$713$$ −18.2920 −0.685040
$$714$$ 0 0
$$715$$ −2.26224 −0.0846032
$$716$$ 0 0
$$717$$ −20.3027 −0.758218
$$718$$ 0 0
$$719$$ 4.67728 0.174433 0.0872166 0.996189i $$-0.472203\pi$$
0.0872166 + 0.996189i $$0.472203\pi$$
$$720$$ 0 0
$$721$$ 21.4657 0.799426
$$722$$ 0 0
$$723$$ 7.43764 0.276609
$$724$$ 0 0
$$725$$ 17.5652 0.652356
$$726$$ 0 0
$$727$$ −27.8553 −1.03310 −0.516548 0.856258i $$-0.672783\pi$$
−0.516548 + 0.856258i $$0.672783\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 80.1101 2.96298
$$732$$ 0 0
$$733$$ −23.6613 −0.873949 −0.436974 0.899474i $$-0.643950\pi$$
−0.436974 + 0.899474i $$0.643950\pi$$
$$734$$ 0 0
$$735$$ −21.1675 −0.780777
$$736$$ 0 0
$$737$$ 10.0999 0.372036
$$738$$ 0 0
$$739$$ −2.47093 −0.0908945 −0.0454473 0.998967i $$-0.514471\pi$$
−0.0454473 + 0.998967i $$0.514471\pi$$
$$740$$ 0 0
$$741$$ −0.866886 −0.0318459
$$742$$ 0 0
$$743$$ −18.8973 −0.693276 −0.346638 0.937999i $$-0.612677\pi$$
−0.346638 + 0.937999i $$0.612677\pi$$
$$744$$ 0 0
$$745$$ 61.2565 2.24427
$$746$$ 0 0
$$747$$ −10.2090 −0.373529
$$748$$ 0 0
$$749$$ 33.6143 1.22824
$$750$$ 0 0
$$751$$ −39.2273 −1.43142 −0.715712 0.698396i $$-0.753898\pi$$
−0.715712 + 0.698396i $$0.753898\pi$$
$$752$$ 0 0
$$753$$ −28.3028 −1.03141
$$754$$ 0 0
$$755$$ 27.9210 1.01615
$$756$$ 0 0
$$757$$ 34.1963 1.24288 0.621442 0.783460i $$-0.286547\pi$$
0.621442 + 0.783460i $$0.286547\pi$$
$$758$$ 0 0
$$759$$ 13.2747 0.481841
$$760$$ 0 0
$$761$$ −11.0455 −0.400398 −0.200199 0.979755i $$-0.564159\pi$$
−0.200199 + 0.979755i $$0.564159\pi$$
$$762$$ 0 0
$$763$$ 69.6051 2.51987
$$764$$ 0 0
$$765$$ −23.0624 −0.833822
$$766$$ 0 0
$$767$$ −1.06778 −0.0385555
$$768$$ 0 0
$$769$$ 8.17138 0.294668 0.147334 0.989087i $$-0.452931\pi$$
0.147334 + 0.989087i $$0.452931\pi$$
$$770$$ 0 0
$$771$$ −7.82120 −0.281673
$$772$$ 0 0
$$773$$ −47.9579 −1.72493 −0.862463 0.506120i $$-0.831079\pi$$
−0.862463 + 0.506120i $$0.831079\pi$$
$$774$$ 0 0
$$775$$ 17.5737 0.631264
$$776$$ 0 0
$$777$$ 1.31833 0.0472950
$$778$$ 0 0
$$779$$ 10.3712 0.371588
$$780$$ 0 0
$$781$$ 6.46812 0.231447
$$782$$ 0 0
$$783$$ 3.32252 0.118737
$$784$$ 0 0
$$785$$ −72.5187 −2.58830
$$786$$ 0 0
$$787$$ −53.9997 −1.92488 −0.962440 0.271495i $$-0.912482\pi$$
−0.962440 + 0.271495i $$0.912482\pi$$
$$788$$ 0 0
$$789$$ 6.44120 0.229313
$$790$$ 0 0
$$791$$ −46.2096 −1.64302
$$792$$ 0 0
$$793$$ −2.77290 −0.0984687
$$794$$ 0 0
$$795$$ 34.2906 1.21616
$$796$$ 0 0
$$797$$ 5.87813 0.208214 0.104107 0.994566i $$-0.466802\pi$$
0.104107 + 0.994566i $$0.466802\pi$$
$$798$$ 0 0
$$799$$ 2.92446 0.103460
$$800$$ 0 0
$$801$$ −4.51995 −0.159705
$$802$$ 0 0
$$803$$ 15.1363 0.534149
$$804$$ 0 0
$$805$$ −65.0863 −2.29399
$$806$$ 0 0
$$807$$ 21.7914 0.767094
$$808$$ 0 0
$$809$$ −3.91670 −0.137704 −0.0688519 0.997627i $$-0.521934\pi$$
−0.0688519 + 0.997627i $$0.521934\pi$$
$$810$$ 0 0
$$811$$ −13.4761 −0.473211 −0.236605 0.971606i $$-0.576035\pi$$
−0.236605 + 0.971606i $$0.576035\pi$$
$$812$$ 0 0
$$813$$ 10.1627 0.356421
$$814$$ 0 0
$$815$$ −30.4796 −1.06765
$$816$$ 0 0
$$817$$ 33.0310 1.15561
$$818$$ 0 0
$$819$$ 1.07827 0.0376778
$$820$$ 0 0
$$821$$ 22.0639 0.770035 0.385018 0.922909i $$-0.374195\pi$$
0.385018 + 0.922909i $$0.374195\pi$$
$$822$$ 0 0
$$823$$ 12.8677 0.448540 0.224270 0.974527i $$-0.428000\pi$$
0.224270 + 0.974527i $$0.428000\pi$$
$$824$$ 0 0
$$825$$ −12.7534 −0.444016
$$826$$ 0 0
$$827$$ 23.6784 0.823380 0.411690 0.911324i $$-0.364939\pi$$
0.411690 + 0.911324i $$0.364939\pi$$
$$828$$ 0 0
$$829$$ 32.1173 1.11548 0.557741 0.830015i $$-0.311668\pi$$
0.557741 + 0.830015i $$0.311668\pi$$
$$830$$ 0 0
$$831$$ 5.38295 0.186732
$$832$$ 0 0
$$833$$ 47.4567 1.64428
$$834$$ 0 0
$$835$$ −3.20729 −0.110993
$$836$$ 0 0
$$837$$ 3.32412 0.114898
$$838$$ 0 0
$$839$$ −4.63931 −0.160167 −0.0800835 0.996788i $$-0.525519\pi$$
−0.0800835 + 0.996788i $$0.525519\pi$$
$$840$$ 0 0
$$841$$ −17.9609 −0.619340
$$842$$ 0 0
$$843$$ −17.2566 −0.594351
$$844$$ 0 0
$$845$$ 41.4206 1.42491
$$846$$ 0 0
$$847$$ 19.1049 0.656452
$$848$$ 0 0
$$849$$ 6.13015 0.210386
$$850$$ 0 0
$$851$$ 1.96718 0.0674340
$$852$$ 0 0
$$853$$ −4.45984 −0.152702 −0.0763510 0.997081i $$-0.524327\pi$$
−0.0763510 + 0.997081i $$0.524327\pi$$
$$854$$ 0 0
$$855$$ −9.50909 −0.325204
$$856$$ 0 0
$$857$$ 48.4705 1.65572 0.827860 0.560935i $$-0.189558\pi$$
0.827860 + 0.560935i $$0.189558\pi$$
$$858$$ 0 0
$$859$$ 11.3642 0.387741 0.193871 0.981027i $$-0.437896\pi$$
0.193871 + 0.981027i $$0.437896\pi$$
$$860$$ 0 0
$$861$$ −12.9002 −0.439638
$$862$$ 0 0
$$863$$ −31.2403 −1.06343 −0.531717 0.846922i $$-0.678453\pi$$
−0.531717 + 0.846922i $$0.678453\pi$$
$$864$$ 0 0
$$865$$ −12.4495 −0.423296
$$866$$ 0 0
$$867$$ 34.7048 1.17864
$$868$$ 0 0
$$869$$ 16.0740 0.545274
$$870$$ 0 0
$$871$$ 1.22417 0.0414793
$$872$$ 0 0
$$873$$ 1.77054 0.0599236
$$874$$ 0 0
$$875$$ 3.39124 0.114645
$$876$$ 0 0
$$877$$ −3.45334 −0.116611 −0.0583055 0.998299i $$-0.518570\pi$$
−0.0583055 + 0.998299i $$0.518570\pi$$
$$878$$ 0 0
$$879$$ 17.6848 0.596493
$$880$$ 0 0
$$881$$ −18.0397 −0.607772 −0.303886 0.952708i $$-0.598284\pi$$
−0.303886 + 0.952708i $$0.598284\pi$$
$$882$$ 0 0
$$883$$ 19.7645 0.665129 0.332565 0.943081i $$-0.392086\pi$$
0.332565 + 0.943081i $$0.392086\pi$$
$$884$$ 0 0
$$885$$ −11.7128 −0.393721
$$886$$ 0 0
$$887$$ −14.0636 −0.472209 −0.236105 0.971728i $$-0.575871\pi$$
−0.236105 + 0.971728i $$0.575871\pi$$
$$888$$ 0 0
$$889$$ −4.19195 −0.140593
$$890$$ 0 0
$$891$$ −2.41235 −0.0808167
$$892$$ 0 0
$$893$$ 1.20582 0.0403511
$$894$$ 0 0
$$895$$ −24.5649 −0.821115
$$896$$ 0 0
$$897$$ 1.60896 0.0537217
$$898$$ 0 0
$$899$$ 11.0444 0.368353
$$900$$ 0 0
$$901$$ −76.8780 −2.56118
$$902$$ 0 0
$$903$$ −41.0854 −1.36724
$$904$$ 0 0
$$905$$ −31.3970 −1.04367
$$906$$ 0 0
$$907$$ 13.8587 0.460170 0.230085 0.973171i $$-0.426100\pi$$
0.230085 + 0.973171i $$0.426100\pi$$
$$908$$ 0 0
$$909$$ 18.8099 0.623884
$$910$$ 0 0
$$911$$ 37.7036 1.24918 0.624588 0.780954i $$-0.285267\pi$$
0.624588 + 0.780954i $$0.285267\pi$$
$$912$$ 0 0
$$913$$ 24.6277 0.815058
$$914$$ 0 0
$$915$$ −30.4167 −1.00554
$$916$$ 0 0
$$917$$ 44.8546 1.48123
$$918$$ 0 0
$$919$$ 24.6184 0.812086 0.406043 0.913854i $$-0.366908\pi$$
0.406043 + 0.913854i $$0.366908\pi$$
$$920$$ 0 0
$$921$$ −11.1101 −0.366089
$$922$$ 0 0
$$923$$ 0.783970 0.0258047
$$924$$ 0 0
$$925$$ −1.88993 −0.0621404
$$926$$ 0 0
$$927$$ −5.82075 −0.191179
$$928$$ 0 0
$$929$$ 28.1432 0.923349 0.461675 0.887049i $$-0.347249\pi$$
0.461675 + 0.887049i $$0.347249\pi$$
$$930$$ 0 0
$$931$$ 19.5674 0.641295
$$932$$ 0 0
$$933$$ −28.3481 −0.928076
$$934$$ 0 0
$$935$$ 55.6345 1.81944
$$936$$ 0 0
$$937$$ 14.4087 0.470713 0.235357 0.971909i $$-0.424374\pi$$
0.235357 + 0.971909i $$0.424374\pi$$
$$938$$ 0 0
$$939$$ 31.4209 1.02538
$$940$$ 0 0
$$941$$ 31.8231 1.03740 0.518702 0.854955i $$-0.326415\pi$$
0.518702 + 0.854955i $$0.326415\pi$$
$$942$$ 0 0
$$943$$ −19.2493 −0.626844
$$944$$ 0 0
$$945$$ 11.8278 0.384759
$$946$$ 0 0
$$947$$ −7.53847 −0.244967 −0.122484 0.992471i $$-0.539086\pi$$
−0.122484 + 0.992471i $$0.539086\pi$$
$$948$$ 0 0
$$949$$ 1.83460 0.0595537
$$950$$ 0 0
$$951$$ −7.57882 −0.245760
$$952$$ 0 0
$$953$$ 9.31282 0.301672 0.150836 0.988559i $$-0.451803\pi$$
0.150836 + 0.988559i $$0.451803\pi$$
$$954$$ 0 0
$$955$$ 47.9747 1.55242
$$956$$ 0 0
$$957$$ −8.01507 −0.259090
$$958$$ 0 0
$$959$$ 53.2319 1.71895
$$960$$ 0 0
$$961$$ −19.9503 −0.643557
$$962$$ 0 0
$$963$$ −9.11502 −0.293727
$$964$$ 0 0
$$965$$ 1.82496 0.0587476
$$966$$ 0 0
$$967$$ −8.56361 −0.275387 −0.137693 0.990475i $$-0.543969\pi$$
−0.137693 + 0.990475i $$0.543969\pi$$
$$968$$ 0 0
$$969$$ 21.3190 0.684864
$$970$$ 0 0
$$971$$ 17.1699 0.551010 0.275505 0.961300i $$-0.411155\pi$$
0.275505 + 0.961300i $$0.411155\pi$$
$$972$$ 0 0
$$973$$ −1.59915 −0.0512664
$$974$$ 0 0
$$975$$ −1.54578 −0.0495045
$$976$$ 0 0
$$977$$ 16.8510 0.539112 0.269556 0.962985i $$-0.413123\pi$$
0.269556 + 0.962985i $$0.413123\pi$$
$$978$$ 0 0
$$979$$ 10.9037 0.348484
$$980$$ 0 0
$$981$$ −18.8744 −0.602615
$$982$$ 0 0
$$983$$ −23.9668 −0.764424 −0.382212 0.924075i $$-0.624838\pi$$
−0.382212 + 0.924075i $$0.624838\pi$$
$$984$$ 0 0
$$985$$ −77.4002 −2.46618
$$986$$ 0 0
$$987$$ −1.49985 −0.0477406
$$988$$ 0 0
$$989$$ −61.3065 −1.94943
$$990$$ 0 0
$$991$$ 17.7821 0.564867 0.282434 0.959287i $$-0.408858\pi$$
0.282434 + 0.959287i $$0.408858\pi$$
$$992$$ 0 0
$$993$$ −29.3509 −0.931423
$$994$$ 0 0
$$995$$ 34.0456 1.07932
$$996$$ 0 0
$$997$$ −34.0948 −1.07979 −0.539896 0.841732i $$-0.681536\pi$$
−0.539896 + 0.841732i $$0.681536\pi$$
$$998$$ 0 0
$$999$$ −0.357486 −0.0113104
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 8016.2.a.w.1.1 7
4.3 odd 2 4008.2.a.g.1.1 7

By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.g.1.1 7 4.3 odd 2
8016.2.a.w.1.1 7 1.1 even 1 trivial