# Properties

 Label 2646.2.d.d.2645.5 Level $2646$ Weight $2$ Character 2646.2645 Analytic conductor $21.128$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.1284163748$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2645.5 Root $$-0.258819 - 0.965926i$$ of defining polynomial Character $$\chi$$ $$=$$ 2646.2645 Dual form 2646.2.d.d.2645.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{4} -2.44949 q^{5} -1.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{4} -2.44949 q^{5} -1.00000i q^{8} -2.44949i q^{10} -4.24264i q^{11} -0.717439i q^{13} +1.00000 q^{16} -2.44949 q^{17} -4.89898i q^{19} +2.44949 q^{20} +4.24264 q^{22} +6.00000i q^{23} +1.00000 q^{25} +0.717439 q^{26} -1.75736i q^{29} +9.08052i q^{31} +1.00000i q^{32} -2.44949i q^{34} +5.24264 q^{37} +4.89898 q^{38} +2.44949i q^{40} +2.44949 q^{41} +7.00000 q^{43} +4.24264i q^{44} -6.00000 q^{46} -12.8418 q^{47} +1.00000i q^{50} +0.717439i q^{52} +14.4853i q^{53} +10.3923i q^{55} +1.75736 q^{58} +2.44949 q^{59} +4.18154i q^{61} -9.08052 q^{62} -1.00000 q^{64} +1.75736i q^{65} +13.4853 q^{67} +2.44949 q^{68} -12.7279i q^{71} +5.49333i q^{73} +5.24264i q^{74} +4.89898i q^{76} +0.757359 q^{79} -2.44949 q^{80} +2.44949i q^{82} -15.2913 q^{83} +6.00000 q^{85} +7.00000i q^{86} -4.24264 q^{88} -3.04384 q^{89} -6.00000i q^{92} -12.8418i q^{94} +12.0000i q^{95} +3.16693i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{4}+O(q^{10})$$ 8 * q - 8 * q^4 $$8 q - 8 q^{4} + 8 q^{16} + 8 q^{25} + 8 q^{37} + 56 q^{43} - 48 q^{46} + 48 q^{58} - 8 q^{64} + 40 q^{67} + 40 q^{79} + 48 q^{85}+O(q^{100})$$ 8 * q - 8 * q^4 + 8 * q^16 + 8 * q^25 + 8 * q^37 + 56 * q^43 - 48 * q^46 + 48 * q^58 - 8 * q^64 + 40 * q^67 + 40 * q^79 + 48 * q^85

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ −2.44949 −1.09545 −0.547723 0.836660i $$-0.684505\pi$$
−0.547723 + 0.836660i $$0.684505\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ − 1.00000i − 0.353553i
$$9$$ 0 0
$$10$$ − 2.44949i − 0.774597i
$$11$$ − 4.24264i − 1.27920i −0.768706 0.639602i $$-0.779099\pi$$
0.768706 0.639602i $$-0.220901\pi$$
$$12$$ 0 0
$$13$$ − 0.717439i − 0.198982i −0.995038 0.0994909i $$-0.968279\pi$$
0.995038 0.0994909i $$-0.0317214\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.44949 −0.594089 −0.297044 0.954864i $$-0.596001\pi$$
−0.297044 + 0.954864i $$0.596001\pi$$
$$18$$ 0 0
$$19$$ − 4.89898i − 1.12390i −0.827170 0.561951i $$-0.810051\pi$$
0.827170 0.561951i $$-0.189949\pi$$
$$20$$ 2.44949 0.547723
$$21$$ 0 0
$$22$$ 4.24264 0.904534
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0.717439 0.140701
$$27$$ 0 0
$$28$$ 0 0
$$29$$ − 1.75736i − 0.326333i −0.986599 0.163167i $$-0.947829\pi$$
0.986599 0.163167i $$-0.0521708\pi$$
$$30$$ 0 0
$$31$$ 9.08052i 1.63091i 0.578821 + 0.815455i $$0.303513\pi$$
−0.578821 + 0.815455i $$0.696487\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ − 2.44949i − 0.420084i
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 5.24264 0.861885 0.430942 0.902379i $$-0.358181\pi$$
0.430942 + 0.902379i $$0.358181\pi$$
$$38$$ 4.89898 0.794719
$$39$$ 0 0
$$40$$ 2.44949i 0.387298i
$$41$$ 2.44949 0.382546 0.191273 0.981537i $$-0.438738\pi$$
0.191273 + 0.981537i $$0.438738\pi$$
$$42$$ 0 0
$$43$$ 7.00000 1.06749 0.533745 0.845645i $$-0.320784\pi$$
0.533745 + 0.845645i $$0.320784\pi$$
$$44$$ 4.24264i 0.639602i
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ −12.8418 −1.87317 −0.936584 0.350443i $$-0.886031\pi$$
−0.936584 + 0.350443i $$0.886031\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1.00000i 0.141421i
$$51$$ 0 0
$$52$$ 0.717439i 0.0994909i
$$53$$ 14.4853i 1.98971i 0.101327 + 0.994853i $$0.467691\pi$$
−0.101327 + 0.994853i $$0.532309\pi$$
$$54$$ 0 0
$$55$$ 10.3923i 1.40130i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 1.75736 0.230753
$$59$$ 2.44949 0.318896 0.159448 0.987206i $$-0.449029\pi$$
0.159448 + 0.987206i $$0.449029\pi$$
$$60$$ 0 0
$$61$$ 4.18154i 0.535391i 0.963504 + 0.267696i $$0.0862622\pi$$
−0.963504 + 0.267696i $$0.913738\pi$$
$$62$$ −9.08052 −1.15323
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 1.75736i 0.217974i
$$66$$ 0 0
$$67$$ 13.4853 1.64749 0.823745 0.566961i $$-0.191881\pi$$
0.823745 + 0.566961i $$0.191881\pi$$
$$68$$ 2.44949 0.297044
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 12.7279i − 1.51053i −0.655422 0.755263i $$-0.727509\pi$$
0.655422 0.755263i $$-0.272491\pi$$
$$72$$ 0 0
$$73$$ 5.49333i 0.642945i 0.946919 + 0.321473i $$0.104178\pi$$
−0.946919 + 0.321473i $$0.895822\pi$$
$$74$$ 5.24264i 0.609445i
$$75$$ 0 0
$$76$$ 4.89898i 0.561951i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0.757359 0.0852096 0.0426048 0.999092i $$-0.486434\pi$$
0.0426048 + 0.999092i $$0.486434\pi$$
$$80$$ −2.44949 −0.273861
$$81$$ 0 0
$$82$$ 2.44949i 0.270501i
$$83$$ −15.2913 −1.67844 −0.839218 0.543795i $$-0.816987\pi$$
−0.839218 + 0.543795i $$0.816987\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ 7.00000i 0.754829i
$$87$$ 0 0
$$88$$ −4.24264 −0.452267
$$89$$ −3.04384 −0.322646 −0.161323 0.986902i $$-0.551576\pi$$
−0.161323 + 0.986902i $$0.551576\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 6.00000i − 0.625543i
$$93$$ 0 0
$$94$$ − 12.8418i − 1.32453i
$$95$$ 12.0000i 1.23117i
$$96$$ 0 0
$$97$$ 3.16693i 0.321553i 0.986991 + 0.160776i $$0.0513998\pi$$
−0.986991 + 0.160776i $$0.948600\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ −7.34847 −0.731200 −0.365600 0.930772i $$-0.619136\pi$$
−0.365600 + 0.930772i $$0.619136\pi$$
$$102$$ 0 0
$$103$$ 11.1097i 1.09468i 0.836912 + 0.547338i $$0.184359\pi$$
−0.836912 + 0.547338i $$0.815641\pi$$
$$104$$ −0.717439 −0.0703507
$$105$$ 0 0
$$106$$ −14.4853 −1.40693
$$107$$ − 2.48528i − 0.240261i −0.992758 0.120131i $$-0.961669\pi$$
0.992758 0.120131i $$-0.0383313\pi$$
$$108$$ 0 0
$$109$$ −17.7279 −1.69803 −0.849013 0.528371i $$-0.822803\pi$$
−0.849013 + 0.528371i $$0.822803\pi$$
$$110$$ −10.3923 −0.990867
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 10.2426i 0.963547i 0.876296 + 0.481773i $$0.160007\pi$$
−0.876296 + 0.481773i $$0.839993\pi$$
$$114$$ 0 0
$$115$$ − 14.6969i − 1.37050i
$$116$$ 1.75736i 0.163167i
$$117$$ 0 0
$$118$$ 2.44949i 0.225494i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −4.18154 −0.378579
$$123$$ 0 0
$$124$$ − 9.08052i − 0.815455i
$$125$$ 9.79796 0.876356
$$126$$ 0 0
$$127$$ −7.72792 −0.685742 −0.342871 0.939382i $$-0.611399\pi$$
−0.342871 + 0.939382i $$0.611399\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 0 0
$$130$$ −1.75736 −0.154131
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 13.4853i 1.16495i
$$135$$ 0 0
$$136$$ 2.44949i 0.210042i
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 8.06591i 0.684141i 0.939674 + 0.342071i $$0.111128\pi$$
−0.939674 + 0.342071i $$0.888872\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 12.7279 1.06810
$$143$$ −3.04384 −0.254538
$$144$$ 0 0
$$145$$ 4.30463i 0.357480i
$$146$$ −5.49333 −0.454631
$$147$$ 0 0
$$148$$ −5.24264 −0.430942
$$149$$ 16.2426i 1.33065i 0.746554 + 0.665324i $$0.231707\pi$$
−0.746554 + 0.665324i $$0.768293\pi$$
$$150$$ 0 0
$$151$$ 8.75736 0.712664 0.356332 0.934359i $$-0.384027\pi$$
0.356332 + 0.934359i $$0.384027\pi$$
$$152$$ −4.89898 −0.397360
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 22.2426i − 1.78657i
$$156$$ 0 0
$$157$$ 10.3923i 0.829396i 0.909959 + 0.414698i $$0.136113\pi$$
−0.909959 + 0.414698i $$0.863887\pi$$
$$158$$ 0.757359i 0.0602523i
$$159$$ 0 0
$$160$$ − 2.44949i − 0.193649i
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −9.48528 −0.742945 −0.371472 0.928444i $$-0.621147\pi$$
−0.371472 + 0.928444i $$0.621147\pi$$
$$164$$ −2.44949 −0.191273
$$165$$ 0 0
$$166$$ − 15.2913i − 1.18683i
$$167$$ 0.594346 0.0459919 0.0229959 0.999736i $$-0.492680\pi$$
0.0229959 + 0.999736i $$0.492680\pi$$
$$168$$ 0 0
$$169$$ 12.4853 0.960406
$$170$$ 6.00000i 0.460179i
$$171$$ 0 0
$$172$$ −7.00000 −0.533745
$$173$$ −20.7846 −1.58022 −0.790112 0.612962i $$-0.789978\pi$$
−0.790112 + 0.612962i $$0.789978\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ − 4.24264i − 0.319801i
$$177$$ 0 0
$$178$$ − 3.04384i − 0.228145i
$$179$$ − 6.72792i − 0.502869i −0.967874 0.251434i $$-0.919098\pi$$
0.967874 0.251434i $$-0.0809022\pi$$
$$180$$ 0 0
$$181$$ − 9.79796i − 0.728277i −0.931345 0.364138i $$-0.881364\pi$$
0.931345 0.364138i $$-0.118636\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 6.00000 0.442326
$$185$$ −12.8418 −0.944148
$$186$$ 0 0
$$187$$ 10.3923i 0.759961i
$$188$$ 12.8418 0.936584
$$189$$ 0 0
$$190$$ −12.0000 −0.870572
$$191$$ 21.2132i 1.53493i 0.641089 + 0.767467i $$0.278483\pi$$
−0.641089 + 0.767467i $$0.721517\pi$$
$$192$$ 0 0
$$193$$ −1.48528 −0.106913 −0.0534564 0.998570i $$-0.517024\pi$$
−0.0534564 + 0.998570i $$0.517024\pi$$
$$194$$ −3.16693 −0.227372
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.9706i 1.20910i 0.796566 + 0.604551i $$0.206648\pi$$
−0.796566 + 0.604551i $$0.793352\pi$$
$$198$$ 0 0
$$199$$ 20.9077i 1.48211i 0.671446 + 0.741054i $$0.265674\pi$$
−0.671446 + 0.741054i $$0.734326\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ 0 0
$$202$$ − 7.34847i − 0.517036i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −6.00000 −0.419058
$$206$$ −11.1097 −0.774053
$$207$$ 0 0
$$208$$ − 0.717439i − 0.0497454i
$$209$$ −20.7846 −1.43770
$$210$$ 0 0
$$211$$ 3.48528 0.239937 0.119968 0.992778i $$-0.461721\pi$$
0.119968 + 0.992778i $$0.461721\pi$$
$$212$$ − 14.4853i − 0.994853i
$$213$$ 0 0
$$214$$ 2.48528 0.169890
$$215$$ −17.1464 −1.16938
$$216$$ 0 0
$$217$$ 0 0
$$218$$ − 17.7279i − 1.20069i
$$219$$ 0 0
$$220$$ − 10.3923i − 0.700649i
$$221$$ 1.75736i 0.118213i
$$222$$ 0 0
$$223$$ − 10.3923i − 0.695920i −0.937509 0.347960i $$-0.886874\pi$$
0.937509 0.347960i $$-0.113126\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −10.2426 −0.681330
$$227$$ 25.0892 1.66523 0.832616 0.553851i $$-0.186842\pi$$
0.832616 + 0.553851i $$0.186842\pi$$
$$228$$ 0 0
$$229$$ − 5.61642i − 0.371143i −0.982631 0.185572i $$-0.940586\pi$$
0.982631 0.185572i $$-0.0594137\pi$$
$$230$$ 14.6969 0.969087
$$231$$ 0 0
$$232$$ −1.75736 −0.115376
$$233$$ 20.4853i 1.34204i 0.741441 + 0.671018i $$0.234143\pi$$
−0.741441 + 0.671018i $$0.765857\pi$$
$$234$$ 0 0
$$235$$ 31.4558 2.05195
$$236$$ −2.44949 −0.159448
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.2426i 1.05065i 0.850902 + 0.525325i $$0.176056\pi$$
−0.850902 + 0.525325i $$0.823944\pi$$
$$240$$ 0 0
$$241$$ 1.13770i 0.0732860i 0.999328 + 0.0366430i $$0.0116664\pi$$
−0.999328 + 0.0366430i $$0.988334\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ 0 0
$$244$$ − 4.18154i − 0.267696i
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.51472 −0.223636
$$248$$ 9.08052 0.576614
$$249$$ 0 0
$$250$$ 9.79796i 0.619677i
$$251$$ 15.2913 0.965177 0.482589 0.875847i $$-0.339697\pi$$
0.482589 + 0.875847i $$0.339697\pi$$
$$252$$ 0 0
$$253$$ 25.4558 1.60040
$$254$$ − 7.72792i − 0.484893i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 25.0892 1.56502 0.782512 0.622636i $$-0.213938\pi$$
0.782512 + 0.622636i $$0.213938\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ − 1.75736i − 0.108987i
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 6.72792i 0.414861i 0.978250 + 0.207431i $$0.0665102\pi$$
−0.978250 + 0.207431i $$0.933490\pi$$
$$264$$ 0 0
$$265$$ − 35.4815i − 2.17961i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −13.4853 −0.823745
$$269$$ 9.79796 0.597392 0.298696 0.954348i $$-0.403448\pi$$
0.298696 + 0.954348i $$0.403448\pi$$
$$270$$ 0 0
$$271$$ − 18.8785i − 1.14679i −0.819280 0.573393i $$-0.805627\pi$$
0.819280 0.573393i $$-0.194373\pi$$
$$272$$ −2.44949 −0.148522
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ − 4.24264i − 0.255841i
$$276$$ 0 0
$$277$$ 23.7279 1.42567 0.712836 0.701330i $$-0.247410\pi$$
0.712836 + 0.701330i $$0.247410\pi$$
$$278$$ −8.06591 −0.483761
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 13.7574i − 0.820695i −0.911929 0.410348i $$-0.865407\pi$$
0.911929 0.410348i $$-0.134593\pi$$
$$282$$ 0 0
$$283$$ − 6.03668i − 0.358844i −0.983772 0.179422i $$-0.942577\pi$$
0.983772 0.179422i $$-0.0574227\pi$$
$$284$$ 12.7279i 0.755263i
$$285$$ 0 0
$$286$$ − 3.04384i − 0.179986i
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −11.0000 −0.647059
$$290$$ −4.30463 −0.252777
$$291$$ 0 0
$$292$$ − 5.49333i − 0.321473i
$$293$$ 12.8418 0.750226 0.375113 0.926979i $$-0.377604\pi$$
0.375113 + 0.926979i $$0.377604\pi$$
$$294$$ 0 0
$$295$$ −6.00000 −0.349334
$$296$$ − 5.24264i − 0.304722i
$$297$$ 0 0
$$298$$ −16.2426 −0.940911
$$299$$ 4.30463 0.248943
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8.75736i 0.503929i
$$303$$ 0 0
$$304$$ − 4.89898i − 0.280976i
$$305$$ − 10.2426i − 0.586492i
$$306$$ 0 0
$$307$$ 26.8213i 1.53077i 0.643571 + 0.765386i $$0.277452\pi$$
−0.643571 + 0.765386i $$0.722548\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 22.2426 1.26330
$$311$$ −17.1464 −0.972285 −0.486142 0.873880i $$-0.661596\pi$$
−0.486142 + 0.873880i $$0.661596\pi$$
$$312$$ 0 0
$$313$$ − 20.1903i − 1.14122i −0.821221 0.570611i $$-0.806707\pi$$
0.821221 0.570611i $$-0.193293\pi$$
$$314$$ −10.3923 −0.586472
$$315$$ 0 0
$$316$$ −0.757359 −0.0426048
$$317$$ − 22.2426i − 1.24927i −0.780916 0.624636i $$-0.785248\pi$$
0.780916 0.624636i $$-0.214752\pi$$
$$318$$ 0 0
$$319$$ −7.45584 −0.417447
$$320$$ 2.44949 0.136931
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 12.0000i 0.667698i
$$324$$ 0 0
$$325$$ − 0.717439i − 0.0397964i
$$326$$ − 9.48528i − 0.525341i
$$327$$ 0 0
$$328$$ − 2.44949i − 0.135250i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 10.0000 0.549650 0.274825 0.961494i $$-0.411380\pi$$
0.274825 + 0.961494i $$0.411380\pi$$
$$332$$ 15.2913 0.839218
$$333$$ 0 0
$$334$$ 0.594346i 0.0325212i
$$335$$ −33.0321 −1.80473
$$336$$ 0 0
$$337$$ −21.4558 −1.16877 −0.584387 0.811475i $$-0.698665\pi$$
−0.584387 + 0.811475i $$0.698665\pi$$
$$338$$ 12.4853i 0.679110i
$$339$$ 0 0
$$340$$ −6.00000 −0.325396
$$341$$ 38.5254 2.08627
$$342$$ 0 0
$$343$$ 0 0
$$344$$ − 7.00000i − 0.377415i
$$345$$ 0 0
$$346$$ − 20.7846i − 1.11739i
$$347$$ 4.97056i 0.266834i 0.991060 + 0.133417i $$0.0425949\pi$$
−0.991060 + 0.133417i $$0.957405\pi$$
$$348$$ 0 0
$$349$$ − 0.123093i − 0.00658902i −0.999995 0.00329451i $$-0.998951\pi$$
0.999995 0.00329451i $$-0.00104868\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.24264 0.226134
$$353$$ 10.9867 0.584760 0.292380 0.956302i $$-0.405553\pi$$
0.292380 + 0.956302i $$0.405553\pi$$
$$354$$ 0 0
$$355$$ 31.1769i 1.65470i
$$356$$ 3.04384 0.161323
$$357$$ 0 0
$$358$$ 6.72792 0.355582
$$359$$ − 14.4853i − 0.764504i −0.924058 0.382252i $$-0.875149\pi$$
0.924058 0.382252i $$-0.124851\pi$$
$$360$$ 0 0
$$361$$ −5.00000 −0.263158
$$362$$ 9.79796 0.514969
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 13.4558i − 0.704311i
$$366$$ 0 0
$$367$$ 29.9882i 1.56537i 0.622416 + 0.782686i $$0.286151\pi$$
−0.622416 + 0.782686i $$0.713849\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ 0 0
$$370$$ − 12.8418i − 0.667613i
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ −10.3923 −0.537373
$$375$$ 0 0
$$376$$ 12.8418i 0.662265i
$$377$$ −1.26080 −0.0649344
$$378$$ 0 0
$$379$$ −7.48528 −0.384493 −0.192247 0.981347i $$-0.561577\pi$$
−0.192247 + 0.981347i $$0.561577\pi$$
$$380$$ − 12.0000i − 0.615587i
$$381$$ 0 0
$$382$$ −21.2132 −1.08536
$$383$$ 5.49333 0.280696 0.140348 0.990102i $$-0.455178\pi$$
0.140348 + 0.990102i $$0.455178\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ − 1.48528i − 0.0755988i
$$387$$ 0 0
$$388$$ − 3.16693i − 0.160776i
$$389$$ − 15.5147i − 0.786627i −0.919404 0.393314i $$-0.871329\pi$$
0.919404 0.393314i $$-0.128671\pi$$
$$390$$ 0 0
$$391$$ − 14.6969i − 0.743256i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −16.9706 −0.854965
$$395$$ −1.85514 −0.0933424
$$396$$ 0 0
$$397$$ − 15.1682i − 0.761270i −0.924725 0.380635i $$-0.875705\pi$$
0.924725 0.380635i $$-0.124295\pi$$
$$398$$ −20.9077 −1.04801
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ − 19.7574i − 0.986635i −0.869849 0.493318i $$-0.835784\pi$$
0.869849 0.493318i $$-0.164216\pi$$
$$402$$ 0 0
$$403$$ 6.51472 0.324521
$$404$$ 7.34847 0.365600
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 22.2426i − 1.10253i
$$408$$ 0 0
$$409$$ 3.76127i 0.185983i 0.995667 + 0.0929915i $$0.0296430\pi$$
−0.995667 + 0.0929915i $$0.970357\pi$$
$$410$$ − 6.00000i − 0.296319i
$$411$$ 0 0
$$412$$ − 11.1097i − 0.547338i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 37.4558 1.83864
$$416$$ 0.717439 0.0351753
$$417$$ 0 0
$$418$$ − 20.7846i − 1.01661i
$$419$$ 7.94282 0.388032 0.194016 0.980998i $$-0.437849\pi$$
0.194016 + 0.980998i $$0.437849\pi$$
$$420$$ 0 0
$$421$$ −23.4558 −1.14317 −0.571584 0.820544i $$-0.693671\pi$$
−0.571584 + 0.820544i $$0.693671\pi$$
$$422$$ 3.48528i 0.169661i
$$423$$ 0 0
$$424$$ 14.4853 0.703467
$$425$$ −2.44949 −0.118818
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 2.48528i 0.120131i
$$429$$ 0 0
$$430$$ − 17.1464i − 0.826874i
$$431$$ − 1.75736i − 0.0846490i −0.999104 0.0423245i $$-0.986524\pi$$
0.999104 0.0423245i $$-0.0134763\pi$$
$$432$$ 0 0
$$433$$ 2.57258i 0.123630i 0.998088 + 0.0618152i $$0.0196889\pi$$
−0.998088 + 0.0618152i $$0.980311\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 17.7279 0.849013
$$437$$ 29.3939 1.40610
$$438$$ 0 0
$$439$$ 4.30463i 0.205449i 0.994710 + 0.102724i $$0.0327560\pi$$
−0.994710 + 0.102724i $$0.967244\pi$$
$$440$$ 10.3923 0.495434
$$441$$ 0 0
$$442$$ −1.75736 −0.0835891
$$443$$ − 25.4558i − 1.20944i −0.796437 0.604722i $$-0.793284\pi$$
0.796437 0.604722i $$-0.206716\pi$$
$$444$$ 0 0
$$445$$ 7.45584 0.353441
$$446$$ 10.3923 0.492090
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 5.27208i 0.248805i 0.992232 + 0.124402i $$0.0397014\pi$$
−0.992232 + 0.124402i $$0.960299\pi$$
$$450$$ 0 0
$$451$$ − 10.3923i − 0.489355i
$$452$$ − 10.2426i − 0.481773i
$$453$$ 0 0
$$454$$ 25.0892i 1.17750i
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −23.0000 −1.07589 −0.537947 0.842978i $$-0.680800\pi$$
−0.537947 + 0.842978i $$0.680800\pi$$
$$458$$ 5.61642 0.262438
$$459$$ 0 0
$$460$$ 14.6969i 0.685248i
$$461$$ 21.4511 0.999076 0.499538 0.866292i $$-0.333503\pi$$
0.499538 + 0.866292i $$0.333503\pi$$
$$462$$ 0 0
$$463$$ −22.0000 −1.02243 −0.511213 0.859454i $$-0.670804\pi$$
−0.511213 + 0.859454i $$0.670804\pi$$
$$464$$ − 1.75736i − 0.0815834i
$$465$$ 0 0
$$466$$ −20.4853 −0.948962
$$467$$ 17.7408 0.820945 0.410473 0.911873i $$-0.365364\pi$$
0.410473 + 0.911873i $$0.365364\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 31.4558i 1.45095i
$$471$$ 0 0
$$472$$ − 2.44949i − 0.112747i
$$473$$ − 29.6985i − 1.36554i
$$474$$ 0 0
$$475$$ − 4.89898i − 0.224781i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −16.2426 −0.742921
$$479$$ 2.44949 0.111920 0.0559600 0.998433i $$-0.482178\pi$$
0.0559600 + 0.998433i $$0.482178\pi$$
$$480$$ 0 0
$$481$$ − 3.76127i − 0.171499i
$$482$$ −1.13770 −0.0518210
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ − 7.75736i − 0.352244i
$$486$$ 0 0
$$487$$ 22.0000 0.996915 0.498458 0.866914i $$-0.333900\pi$$
0.498458 + 0.866914i $$0.333900\pi$$
$$488$$ 4.18154 0.189289
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 34.9706i 1.57820i 0.614265 + 0.789100i $$0.289453\pi$$
−0.614265 + 0.789100i $$0.710547\pi$$
$$492$$ 0 0
$$493$$ 4.30463i 0.193871i
$$494$$ − 3.51472i − 0.158135i
$$495$$ 0 0
$$496$$ 9.08052i 0.407727i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −24.4558 −1.09479 −0.547397 0.836873i $$-0.684381\pi$$
−0.547397 + 0.836873i $$0.684381\pi$$
$$500$$ −9.79796 −0.438178
$$501$$ 0 0
$$502$$ 15.2913i 0.682483i
$$503$$ −0.594346 −0.0265006 −0.0132503 0.999912i $$-0.504218\pi$$
−0.0132503 + 0.999912i $$0.504218\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 25.4558i 1.13165i
$$507$$ 0 0
$$508$$ 7.72792 0.342871
$$509$$ −9.20361 −0.407943 −0.203971 0.978977i $$-0.565385\pi$$
−0.203971 + 0.978977i $$0.565385\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ 25.0892i 1.10664i
$$515$$ − 27.2132i − 1.19916i
$$516$$ 0 0
$$517$$ 54.4831i 2.39616i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 1.75736 0.0770653
$$521$$ −29.9882 −1.31381 −0.656904 0.753974i $$-0.728134\pi$$
−0.656904 + 0.753974i $$0.728134\pi$$
$$522$$ 0 0
$$523$$ 27.4156i 1.19880i 0.800449 + 0.599401i $$0.204595\pi$$
−0.800449 + 0.599401i $$0.795405\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −6.72792 −0.293351
$$527$$ − 22.2426i − 0.968905i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 35.4815 1.54122
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 1.75736i − 0.0761197i
$$534$$ 0 0
$$535$$ 6.08767i 0.263193i
$$536$$ − 13.4853i − 0.582475i
$$537$$ 0 0
$$538$$ 9.79796i 0.422420i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −5.45584 −0.234565 −0.117283 0.993099i $$-0.537418\pi$$
−0.117283 + 0.993099i $$0.537418\pi$$
$$542$$ 18.8785 0.810900
$$543$$ 0 0
$$544$$ − 2.44949i − 0.105021i
$$545$$ 43.4244 1.86010
$$546$$ 0 0
$$547$$ 39.9706 1.70902 0.854509 0.519437i $$-0.173858\pi$$
0.854509 + 0.519437i $$0.173858\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 0 0
$$550$$ 4.24264 0.180907
$$551$$ −8.60927 −0.366767
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 23.7279i 1.00810i
$$555$$ 0 0
$$556$$ − 8.06591i − 0.342071i
$$557$$ 21.2132i 0.898832i 0.893323 + 0.449416i $$0.148368\pi$$
−0.893323 + 0.449416i $$0.851632\pi$$
$$558$$ 0 0
$$559$$ − 5.02207i − 0.212411i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 13.7574 0.580319
$$563$$ −45.8739 −1.93335 −0.966676 0.256002i $$-0.917595\pi$$
−0.966676 + 0.256002i $$0.917595\pi$$
$$564$$ 0 0
$$565$$ − 25.0892i − 1.05551i
$$566$$ 6.03668 0.253741
$$567$$ 0 0
$$568$$ −12.7279 −0.534052
$$569$$ − 10.2426i − 0.429394i −0.976681 0.214697i $$-0.931124\pi$$
0.976681 0.214697i $$-0.0688764\pi$$
$$570$$ 0 0
$$571$$ −22.0000 −0.920671 −0.460336 0.887745i $$-0.652271\pi$$
−0.460336 + 0.887745i $$0.652271\pi$$
$$572$$ 3.04384 0.127269
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6.00000i 0.250217i
$$576$$ 0 0
$$577$$ − 27.4156i − 1.14133i −0.821184 0.570664i $$-0.806686\pi$$
0.821184 0.570664i $$-0.193314\pi$$
$$578$$ − 11.0000i − 0.457540i
$$579$$ 0 0
$$580$$ − 4.30463i − 0.178740i
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 61.4558 2.54524
$$584$$ 5.49333 0.227315
$$585$$ 0 0
$$586$$ 12.8418i 0.530490i
$$587$$ 32.4377 1.33885 0.669424 0.742881i $$-0.266541\pi$$
0.669424 + 0.742881i $$0.266541\pi$$
$$588$$ 0 0
$$589$$ 44.4853 1.83298
$$590$$ − 6.00000i − 0.247016i
$$591$$ 0 0
$$592$$ 5.24264 0.215471
$$593$$ −1.85514 −0.0761816 −0.0380908 0.999274i $$-0.512128\pi$$
−0.0380908 + 0.999274i $$0.512128\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ − 16.2426i − 0.665324i
$$597$$ 0 0
$$598$$ 4.30463i 0.176030i
$$599$$ − 3.51472i − 0.143608i −0.997419 0.0718038i $$-0.977124\pi$$
0.997419 0.0718038i $$-0.0228755\pi$$
$$600$$ 0 0
$$601$$ 41.5182i 1.69356i 0.531940 + 0.846782i $$0.321463\pi$$
−0.531940 + 0.846782i $$0.678537\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −8.75736 −0.356332
$$605$$ 17.1464 0.697101
$$606$$ 0 0
$$607$$ 36.0759i 1.46428i 0.681157 + 0.732138i $$0.261477\pi$$
−0.681157 + 0.732138i $$0.738523\pi$$
$$608$$ 4.89898 0.198680
$$609$$ 0 0
$$610$$ 10.2426 0.414712
$$611$$ 9.21320i 0.372726i
$$612$$ 0 0
$$613$$ −28.2132 −1.13952 −0.569760 0.821811i $$-0.692964\pi$$
−0.569760 + 0.821811i $$0.692964\pi$$
$$614$$ −26.8213 −1.08242
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4.24264i 0.170802i 0.996347 + 0.0854011i $$0.0272172\pi$$
−0.996347 + 0.0854011i $$0.972783\pi$$
$$618$$ 0 0
$$619$$ 12.7187i 0.511208i 0.966782 + 0.255604i $$0.0822743\pi$$
−0.966782 + 0.255604i $$0.917726\pi$$
$$620$$ 22.2426i 0.893286i
$$621$$ 0 0
$$622$$ − 17.1464i − 0.687509i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 20.1903 0.806965
$$627$$ 0 0
$$628$$ − 10.3923i − 0.414698i
$$629$$ −12.8418 −0.512036
$$630$$ 0 0
$$631$$ −14.7574 −0.587481 −0.293741 0.955885i $$-0.594900\pi$$
−0.293741 + 0.955885i $$0.594900\pi$$
$$632$$ − 0.757359i − 0.0301261i
$$633$$ 0 0
$$634$$ 22.2426 0.883368
$$635$$ 18.9295 0.751193
$$636$$ 0 0
$$637$$ 0 0
$$638$$ − 7.45584i − 0.295180i
$$639$$ 0 0
$$640$$ 2.44949i 0.0968246i
$$641$$ − 33.2132i − 1.31184i −0.754829 0.655921i $$-0.772280\pi$$
0.754829 0.655921i $$-0.227720\pi$$
$$642$$ 0 0
$$643$$ − 1.73205i − 0.0683054i −0.999417 0.0341527i $$-0.989127\pi$$
0.999417 0.0341527i $$-0.0108733\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ 20.7846 0.817127 0.408564 0.912730i $$-0.366030\pi$$
0.408564 + 0.912730i $$0.366030\pi$$
$$648$$ 0 0
$$649$$ − 10.3923i − 0.407934i
$$650$$ 0.717439 0.0281403
$$651$$ 0 0
$$652$$ 9.48528 0.371472
$$653$$ − 2.48528i − 0.0972566i −0.998817 0.0486283i $$-0.984515\pi$$
0.998817 0.0486283i $$-0.0154850\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 2.44949 0.0956365
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 22.2426i 0.866450i 0.901286 + 0.433225i $$0.142624\pi$$
−0.901286 + 0.433225i $$0.857376\pi$$
$$660$$ 0 0
$$661$$ 4.89898i 0.190548i 0.995451 + 0.0952741i $$0.0303728\pi$$
−0.995451 + 0.0952741i $$0.969627\pi$$
$$662$$ 10.0000i 0.388661i
$$663$$ 0 0
$$664$$ 15.2913i 0.593417i
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10.5442 0.408271
$$668$$ −0.594346 −0.0229959
$$669$$ 0 0
$$670$$ − 33.0321i − 1.27614i
$$671$$ 17.7408 0.684875
$$672$$ 0 0
$$673$$ −45.4558 −1.75219 −0.876097 0.482135i $$-0.839862\pi$$
−0.876097 + 0.482135i $$0.839862\pi$$
$$674$$ − 21.4558i − 0.826448i
$$675$$ 0 0
$$676$$ −12.4853 −0.480203
$$677$$ −14.6969 −0.564849 −0.282425 0.959289i $$-0.591139\pi$$
−0.282425 + 0.959289i $$0.591139\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ − 6.00000i − 0.230089i
$$681$$ 0 0
$$682$$ 38.5254i 1.47521i
$$683$$ − 10.2426i − 0.391924i −0.980612 0.195962i $$-0.937217\pi$$
0.980612 0.195962i $$-0.0627829\pi$$
$$684$$ 0 0
$$685$$ − 14.6969i − 0.561541i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 7.00000 0.266872
$$689$$ 10.3923 0.395915
$$690$$ 0 0
$$691$$ 2.57258i 0.0978657i 0.998802 + 0.0489328i $$0.0155820\pi$$
−0.998802 + 0.0489328i $$0.984418\pi$$
$$692$$ 20.7846 0.790112
$$693$$ 0 0
$$694$$ −4.97056 −0.188680
$$695$$ − 19.7574i − 0.749439i
$$696$$ 0 0
$$697$$ −6.00000 −0.227266
$$698$$ 0.123093 0.00465914
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 20.4853i 0.773718i 0.922139 + 0.386859i $$0.126440\pi$$
−0.922139 + 0.386859i $$0.873560\pi$$
$$702$$ 0 0
$$703$$ − 25.6836i − 0.968675i
$$704$$ 4.24264i 0.159901i
$$705$$ 0 0
$$706$$ 10.9867i 0.413488i
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 16.2132 0.608900 0.304450 0.952528i $$-0.401527\pi$$
0.304450 + 0.952528i $$0.401527\pi$$
$$710$$ −31.1769 −1.17005
$$711$$ 0 0
$$712$$ 3.04384i 0.114073i
$$713$$ −54.4831 −2.04041
$$714$$ 0 0
$$715$$ 7.45584 0.278833
$$716$$ 6.72792i 0.251434i
$$717$$ 0 0
$$718$$ 14.4853 0.540586
$$719$$ 52.6280 1.96269 0.981346 0.192249i $$-0.0615781\pi$$
0.981346 + 0.192249i $$0.0615781\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 5.00000i − 0.186081i
$$723$$ 0 0
$$724$$ 9.79796i 0.364138i
$$725$$ − 1.75736i − 0.0652667i
$$726$$ 0 0
$$727$$ − 28.0821i − 1.04151i −0.853707 0.520754i $$-0.825651\pi$$
0.853707 0.520754i $$-0.174349\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 13.4558 0.498023
$$731$$ −17.1464 −0.634184
$$732$$ 0 0
$$733$$ 1.31178i 0.0484519i 0.999707 + 0.0242259i $$0.00771211\pi$$
−0.999707 + 0.0242259i $$0.992288\pi$$
$$734$$ −29.9882 −1.10689
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ − 57.2132i − 2.10748i
$$738$$ 0 0
$$739$$ −8.45584 −0.311053 −0.155527 0.987832i $$-0.549707\pi$$
−0.155527 + 0.987832i $$0.549707\pi$$
$$740$$ 12.8418 0.472074
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 18.7279i 0.687061i 0.939142 + 0.343530i $$0.111623\pi$$
−0.939142 + 0.343530i $$0.888377\pi$$
$$744$$ 0 0
$$745$$ − 39.7862i − 1.45765i
$$746$$ − 22.0000i − 0.805477i
$$747$$ 0 0
$$748$$ − 10.3923i − 0.379980i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 53.4558 1.95063 0.975316 0.220815i $$-0.0708717\pi$$
0.975316 + 0.220815i $$0.0708717\pi$$
$$752$$ −12.8418 −0.468292
$$753$$ 0 0
$$754$$ − 1.26080i − 0.0459156i
$$755$$ −21.4511 −0.780684
$$756$$ 0 0
$$757$$ 32.7574 1.19059 0.595293 0.803509i $$-0.297036\pi$$
0.595293 + 0.803509i $$0.297036\pi$$
$$758$$ − 7.48528i − 0.271878i
$$759$$ 0 0
$$760$$ 12.0000 0.435286
$$761$$ 24.4949 0.887939 0.443970 0.896042i $$-0.353570\pi$$
0.443970 + 0.896042i $$0.353570\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ − 21.2132i − 0.767467i
$$765$$ 0 0
$$766$$ 5.49333i 0.198482i
$$767$$ − 1.75736i − 0.0634546i
$$768$$ 0 0
$$769$$ − 40.3805i − 1.45616i −0.685493 0.728080i $$-0.740413\pi$$
0.685493 0.728080i $$-0.259587\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.48528 0.0534564
$$773$$ −9.79796 −0.352408 −0.176204 0.984354i $$-0.556382\pi$$
−0.176204 + 0.984354i $$0.556382\pi$$
$$774$$ 0 0
$$775$$ 9.08052i 0.326182i
$$776$$ 3.16693 0.113686
$$777$$ 0 0
$$778$$ 15.5147 0.556230
$$779$$ − 12.0000i − 0.429945i
$$780$$ 0 0
$$781$$ −54.0000 −1.93227
$$782$$ 14.6969 0.525561
$$783$$ 0 0
$$784$$ 0 0
$$785$$ − 25.4558i − 0.908558i
$$786$$ 0 0
$$787$$ 28.2562i 1.00722i 0.863930 + 0.503612i $$0.167996\pi$$
−0.863930 + 0.503612i $$0.832004\pi$$
$$788$$ − 16.9706i − 0.604551i
$$789$$ 0 0
$$790$$ − 1.85514i − 0.0660031i
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 3.00000 0.106533
$$794$$ 15.1682 0.538299
$$795$$ 0 0
$$796$$ − 20.9077i − 0.741054i
$$797$$ −17.7408 −0.628410 −0.314205 0.949355i $$-0.601738\pi$$
−0.314205 + 0.949355i $$0.601738\pi$$
$$798$$ 0 0
$$799$$ 31.4558 1.11283
$$800$$ 1.00000i 0.0353553i
$$801$$ 0 0
$$802$$ 19.7574 0.697657
$$803$$ 23.3062 0.822458
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 6.51472i 0.229471i
$$807$$ 0 0
$$808$$ 7.34847i 0.258518i
$$809$$ − 14.4853i − 0.509275i −0.967037 0.254638i $$-0.918044\pi$$
0.967037 0.254638i $$-0.0819562\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 22.2426 0.779604
$$815$$ 23.2341 0.813855
$$816$$ 0 0
$$817$$ − 34.2929i − 1.19976i
$$818$$ −3.76127 −0.131510
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ 22.9706i 0.801678i 0.916148 + 0.400839i $$0.131281\pi$$
−0.916148 + 0.400839i $$0.868719\pi$$
$$822$$ 0 0
$$823$$ −2.75736 −0.0961155 −0.0480578 0.998845i $$-0.515303\pi$$
−0.0480578 + 0.998845i $$0.515303\pi$$
$$824$$ 11.1097 0.387026
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 3.51472i − 0.122219i −0.998131 0.0611094i $$-0.980536\pi$$
0.998131 0.0611094i $$-0.0194638\pi$$
$$828$$ 0 0
$$829$$ − 9.79796i − 0.340297i −0.985418 0.170149i $$-0.945575\pi$$
0.985418 0.170149i $$-0.0544248\pi$$
$$830$$ 37.4558i 1.30011i
$$831$$ 0 0
$$832$$ 0.717439i 0.0248727i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −1.45584 −0.0503816
$$836$$ 20.7846 0.718851
$$837$$ 0 0
$$838$$ 7.94282i 0.274380i
$$839$$ 15.2913 0.527914 0.263957 0.964534i $$-0.414972\pi$$
0.263957 + 0.964534i $$0.414972\pi$$
$$840$$ 0 0
$$841$$ 25.9117 0.893506
$$842$$ − 23.4558i − 0.808342i
$$843$$ 0 0
$$844$$ −3.48528 −0.119968
$$845$$ −30.5826 −1.05207
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 14.4853i 0.497427i
$$849$$ 0 0
$$850$$ − 2.44949i − 0.0840168i
$$851$$ 31.4558i 1.07829i
$$852$$ 0 0
$$853$$ 31.7713i 1.08783i 0.839141 + 0.543914i $$0.183058\pi$$
−0.839141 + 0.543914i $$0.816942\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −2.48528 −0.0849452
$$857$$ −27.5387 −0.940705 −0.470353 0.882479i $$-0.655873\pi$$
−0.470353 + 0.882479i $$0.655873\pi$$
$$858$$ 0 0
$$859$$ 28.2562i 0.964088i 0.876147 + 0.482044i $$0.160105\pi$$
−0.876147 + 0.482044i $$0.839895\pi$$
$$860$$ 17.1464 0.584688
$$861$$ 0 0
$$862$$ 1.75736 0.0598559
$$863$$ 37.4558i 1.27501i 0.770445 + 0.637506i $$0.220034\pi$$
−0.770445 + 0.637506i $$0.779966\pi$$
$$864$$ 0 0
$$865$$ 50.9117 1.73105
$$866$$ −2.57258 −0.0874199
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 3.21320i − 0.109000i
$$870$$ 0 0
$$871$$ − 9.67487i − 0.327820i
$$872$$ 17.7279i 0.600343i
$$873$$ 0 0
$$874$$ 29.3939i 0.994263i
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −46.2132 −1.56051 −0.780254 0.625462i $$-0.784910\pi$$
−0.780254 + 0.625462i $$0.784910\pi$$
$$878$$ −4.30463 −0.145274
$$879$$ 0 0
$$880$$ 10.3923i 0.350325i
$$881$$ −25.0892 −0.845278 −0.422639 0.906298i $$-0.638896\pi$$
−0.422639 + 0.906298i $$0.638896\pi$$
$$882$$ 0 0
$$883$$ −2.00000 −0.0673054 −0.0336527 0.999434i $$-0.510714\pi$$
−0.0336527 + 0.999434i $$0.510714\pi$$
$$884$$ − 1.75736i − 0.0591064i
$$885$$ 0 0
$$886$$ 25.4558 0.855206
$$887$$ 9.20361 0.309027 0.154514 0.987991i $$-0.450619\pi$$
0.154514 + 0.987991i $$0.450619\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 7.45584i 0.249920i
$$891$$ 0 0
$$892$$ 10.3923i 0.347960i
$$893$$ 62.9117i 2.10526i
$$894$$ 0 0
$$895$$ 16.4800i 0.550865i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −5.27208 −0.175932
$$899$$ 15.9577 0.532220
$$900$$ 0 0
$$901$$ − 35.4815i − 1.18206i
$$902$$ 10.3923 0.346026
$$903$$ 0 0
$$904$$ 10.2426 0.340665
$$905$$ 24.0000i 0.797787i
$$906$$ 0 0
$$907$$ −13.4853 −0.447771 −0.223886 0.974615i $$-0.571874\pi$$
−0.223886 + 0.974615i $$0.571874\pi$$
$$908$$ −25.0892 −0.832616
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 10.2426i − 0.339354i −0.985500 0.169677i $$-0.945728\pi$$
0.985500 0.169677i $$-0.0542724\pi$$
$$912$$ 0 0
$$913$$ 64.8754i 2.14706i
$$914$$ − 23.0000i − 0.760772i
$$915$$ 0 0
$$916$$ 5.61642i 0.185572i
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 25.7279 0.848686 0.424343 0.905502i $$-0.360505\pi$$
0.424343 + 0.905502i $$0.360505\pi$$
$$920$$ −14.6969 −0.484544
$$921$$ 0 0
$$922$$ 21.4511i 0.706453i
$$923$$ −9.13151 −0.300567
$$924$$ 0 0
$$925$$ 5.24264 0.172377
$$926$$ − 22.0000i − 0.722965i
$$927$$ 0 0
$$928$$ 1.75736 0.0576881
$$929$$ 28.7274 0.942516 0.471258 0.881995i $$-0.343800\pi$$
0.471258 + 0.881995i $$0.343800\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 20.4853i − 0.671018i
$$933$$ 0 0
$$934$$ 17.7408i 0.580496i
$$935$$ − 25.4558i − 0.832495i
$$936$$ 0 0
$$937$$ − 33.5033i − 1.09451i −0.836967 0.547253i $$-0.815674\pi$$
0.836967 0.547253i $$-0.184326\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −31.4558 −1.02598
$$941$$ 42.2357 1.37684 0.688422 0.725311i $$-0.258304\pi$$
0.688422 + 0.725311i $$0.258304\pi$$
$$942$$ 0 0
$$943$$ 14.6969i 0.478598i
$$944$$ 2.44949 0.0797241
$$945$$ 0 0
$$946$$ 29.6985 0.965581
$$947$$ − 16.2426i − 0.527815i −0.964548 0.263907i $$-0.914989\pi$$
0.964548 0.263907i $$-0.0850114\pi$$
$$948$$ 0 0
$$949$$ 3.94113 0.127934
$$950$$ 4.89898 0.158944
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 1.02944i − 0.0333467i −0.999861 0.0166734i $$-0.994692\pi$$
0.999861 0.0166734i $$-0.00530755\pi$$
$$954$$ 0 0
$$955$$ − 51.9615i − 1.68144i
$$956$$ − 16.2426i − 0.525325i
$$957$$ 0 0
$$958$$ 2.44949i 0.0791394i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −51.4558 −1.65987
$$962$$ 3.76127 0.121268
$$963$$ 0 0
$$964$$ − 1.13770i − 0.0366430i
$$965$$ 3.63818 0.117117
$$966$$ 0 0
$$967$$ −42.6985 −1.37309 −0.686545 0.727087i $$-0.740874\pi$$
−0.686545 + 0.727087i $$0.740874\pi$$
$$968$$ 7.00000i 0.224989i
$$969$$ 0 0
$$970$$ 7.75736 0.249074
$$971$$ −40.3805 −1.29587 −0.647936 0.761694i $$-0.724368\pi$$
−0.647936 + 0.761694i $$0.724368\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 22.0000i 0.704925i
$$975$$ 0 0
$$976$$ 4.18154i 0.133848i
$$977$$ 6.00000i 0.191957i 0.995383 + 0.0959785i $$0.0305980\pi$$
−0.995383 + 0.0959785i $$0.969402\pi$$
$$978$$ 0 0
$$979$$ 12.9139i 0.412730i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −34.9706 −1.11596
$$983$$ 23.2341 0.741053 0.370526 0.928822i $$-0.379177\pi$$
0.370526 + 0.928822i $$0.379177\pi$$
$$984$$ 0 0
$$985$$ − 41.5692i − 1.32451i
$$986$$ −4.30463 −0.137087
$$987$$ 0 0
$$988$$ 3.51472 0.111818
$$989$$ 42.0000i 1.33552i
$$990$$ 0 0
$$991$$ −20.2132 −0.642094 −0.321047 0.947063i $$-0.604035\pi$$
−0.321047 + 0.947063i $$0.604035\pi$$
$$992$$ −9.08052 −0.288307
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 51.2132i − 1.62357i
$$996$$ 0 0
$$997$$ − 19.4728i − 0.616711i −0.951271 0.308355i $$-0.900221\pi$$
0.951271 0.308355i $$-0.0997786\pi$$
$$998$$ − 24.4558i − 0.774136i
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2646.2.d.d.2645.5 8
3.2 odd 2 inner 2646.2.d.d.2645.4 8
7.2 even 3 378.2.k.d.269.4 yes 8
7.3 odd 6 378.2.k.d.215.1 8
7.6 odd 2 inner 2646.2.d.d.2645.7 8
21.2 odd 6 378.2.k.d.269.1 yes 8
21.17 even 6 378.2.k.d.215.4 yes 8
21.20 even 2 inner 2646.2.d.d.2645.2 8
63.2 odd 6 1134.2.t.f.1025.4 8
63.16 even 3 1134.2.t.f.1025.1 8
63.23 odd 6 1134.2.l.e.269.1 8
63.31 odd 6 1134.2.t.f.593.4 8
63.38 even 6 1134.2.l.e.215.2 8
63.52 odd 6 1134.2.l.e.215.3 8
63.58 even 3 1134.2.l.e.269.4 8
63.59 even 6 1134.2.t.f.593.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.d.215.1 8 7.3 odd 6
378.2.k.d.215.4 yes 8 21.17 even 6
378.2.k.d.269.1 yes 8 21.2 odd 6
378.2.k.d.269.4 yes 8 7.2 even 3
1134.2.l.e.215.2 8 63.38 even 6
1134.2.l.e.215.3 8 63.52 odd 6
1134.2.l.e.269.1 8 63.23 odd 6
1134.2.l.e.269.4 8 63.58 even 3
1134.2.t.f.593.1 8 63.59 even 6
1134.2.t.f.593.4 8 63.31 odd 6
1134.2.t.f.1025.1 8 63.16 even 3
1134.2.t.f.1025.4 8 63.2 odd 6
2646.2.d.d.2645.2 8 21.20 even 2 inner
2646.2.d.d.2645.4 8 3.2 odd 2 inner
2646.2.d.d.2645.5 8 1.1 even 1 trivial
2646.2.d.d.2645.7 8 7.6 odd 2 inner