# Properties

 Label 390.2.b.c.181.2 Level $390$ Weight $2$ Character 390.181 Analytic conductor $3.114$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 181.2 Root $$2.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 390.181 Dual form 390.2.b.c.181.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +4.60555i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +4.60555i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +3.60555 q^{13} +4.60555 q^{14} +1.00000i q^{15} +1.00000 q^{16} +4.60555 q^{17} -1.00000i q^{18} +4.60555i q^{19} +1.00000i q^{20} -4.60555i q^{21} -1.39445 q^{23} -1.00000i q^{24} -1.00000 q^{25} -3.60555i q^{26} -1.00000 q^{27} -4.60555i q^{28} +4.60555 q^{29} +1.00000 q^{30} -6.00000i q^{31} -1.00000i q^{32} -4.60555i q^{34} +4.60555 q^{35} -1.00000 q^{36} +9.21110i q^{37} +4.60555 q^{38} -3.60555 q^{39} +1.00000 q^{40} -3.21110i q^{41} -4.60555 q^{42} +8.00000 q^{43} -1.00000i q^{45} +1.39445i q^{46} +9.21110i q^{47} -1.00000 q^{48} -14.2111 q^{49} +1.00000i q^{50} -4.60555 q^{51} -3.60555 q^{52} +6.00000 q^{53} +1.00000i q^{54} -4.60555 q^{56} -4.60555i q^{57} -4.60555i q^{58} +9.21110i q^{59} -1.00000i q^{60} -11.2111 q^{61} -6.00000 q^{62} +4.60555i q^{63} -1.00000 q^{64} -3.60555i q^{65} +3.21110i q^{67} -4.60555 q^{68} +1.39445 q^{69} -4.60555i q^{70} -9.21110i q^{71} +1.00000i q^{72} +1.39445i q^{73} +9.21110 q^{74} +1.00000 q^{75} -4.60555i q^{76} +3.60555i q^{78} -14.4222 q^{79} -1.00000i q^{80} +1.00000 q^{81} -3.21110 q^{82} -2.78890i q^{83} +4.60555i q^{84} -4.60555i q^{85} -8.00000i q^{86} -4.60555 q^{87} -15.2111i q^{89} -1.00000 q^{90} +16.6056i q^{91} +1.39445 q^{92} +6.00000i q^{93} +9.21110 q^{94} +4.60555 q^{95} +1.00000i q^{96} -1.39445i q^{97} +14.2111i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 4 q^{14} + 4 q^{16} + 4 q^{17} - 20 q^{23} - 4 q^{25} - 4 q^{27} + 4 q^{29} + 4 q^{30} + 4 q^{35} - 4 q^{36} + 4 q^{38} + 4 q^{40} - 4 q^{42} + 32 q^{43} - 4 q^{48} - 28 q^{49} - 4 q^{51} + 24 q^{53} - 4 q^{56} - 16 q^{61} - 24 q^{62} - 4 q^{64} - 4 q^{68} + 20 q^{69} + 8 q^{74} + 4 q^{75} + 4 q^{81} + 16 q^{82} - 4 q^{87} - 4 q^{90} + 20 q^{92} + 8 q^{94} + 4 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 + 4 * q^12 + 4 * q^14 + 4 * q^16 + 4 * q^17 - 20 * q^23 - 4 * q^25 - 4 * q^27 + 4 * q^29 + 4 * q^30 + 4 * q^35 - 4 * q^36 + 4 * q^38 + 4 * q^40 - 4 * q^42 + 32 * q^43 - 4 * q^48 - 28 * q^49 - 4 * q^51 + 24 * q^53 - 4 * q^56 - 16 * q^61 - 24 * q^62 - 4 * q^64 - 4 * q^68 + 20 * q^69 + 8 * q^74 + 4 * q^75 + 4 * q^81 + 16 * q^82 - 4 * q^87 - 4 * q^90 + 20 * q^92 + 8 * q^94 + 4 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 1.00000i − 0.447214i
$$6$$ 1.00000i 0.408248i
$$7$$ 4.60555i 1.74073i 0.492403 + 0.870367i $$0.336119\pi$$
−0.492403 + 0.870367i $$0.663881\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 3.60555 1.00000
$$14$$ 4.60555 1.23089
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 4.60555 1.11701 0.558505 0.829501i $$-0.311375\pi$$
0.558505 + 0.829501i $$0.311375\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 4.60555i 1.05659i 0.849062 + 0.528293i $$0.177168\pi$$
−0.849062 + 0.528293i $$0.822832\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ − 4.60555i − 1.00501i
$$22$$ 0 0
$$23$$ −1.39445 −0.290763 −0.145381 0.989376i $$-0.546441\pi$$
−0.145381 + 0.989376i $$0.546441\pi$$
$$24$$ − 1.00000i − 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ − 3.60555i − 0.707107i
$$27$$ −1.00000 −0.192450
$$28$$ − 4.60555i − 0.870367i
$$29$$ 4.60555 0.855229 0.427615 0.903961i $$-0.359354\pi$$
0.427615 + 0.903961i $$0.359354\pi$$
$$30$$ 1.00000 0.182574
$$31$$ − 6.00000i − 1.07763i −0.842424 0.538816i $$-0.818872\pi$$
0.842424 0.538816i $$-0.181128\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ − 4.60555i − 0.789846i
$$35$$ 4.60555 0.778480
$$36$$ −1.00000 −0.166667
$$37$$ 9.21110i 1.51430i 0.653243 + 0.757148i $$0.273408\pi$$
−0.653243 + 0.757148i $$0.726592\pi$$
$$38$$ 4.60555 0.747119
$$39$$ −3.60555 −0.577350
$$40$$ 1.00000 0.158114
$$41$$ − 3.21110i − 0.501490i −0.968053 0.250745i $$-0.919324\pi$$
0.968053 0.250745i $$-0.0806756\pi$$
$$42$$ −4.60555 −0.710652
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ − 1.00000i − 0.149071i
$$46$$ 1.39445i 0.205600i
$$47$$ 9.21110i 1.34358i 0.740743 + 0.671789i $$0.234474\pi$$
−0.740743 + 0.671789i $$0.765526\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −14.2111 −2.03016
$$50$$ 1.00000i 0.141421i
$$51$$ −4.60555 −0.644906
$$52$$ −3.60555 −0.500000
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 1.00000i 0.136083i
$$55$$ 0 0
$$56$$ −4.60555 −0.615443
$$57$$ − 4.60555i − 0.610020i
$$58$$ − 4.60555i − 0.604739i
$$59$$ 9.21110i 1.19918i 0.800306 + 0.599592i $$0.204670\pi$$
−0.800306 + 0.599592i $$0.795330\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ −11.2111 −1.43543 −0.717717 0.696335i $$-0.754813\pi$$
−0.717717 + 0.696335i $$0.754813\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ 4.60555i 0.580245i
$$64$$ −1.00000 −0.125000
$$65$$ − 3.60555i − 0.447214i
$$66$$ 0 0
$$67$$ 3.21110i 0.392299i 0.980574 + 0.196149i $$0.0628437\pi$$
−0.980574 + 0.196149i $$0.937156\pi$$
$$68$$ −4.60555 −0.558505
$$69$$ 1.39445 0.167872
$$70$$ − 4.60555i − 0.550469i
$$71$$ − 9.21110i − 1.09316i −0.837408 0.546578i $$-0.815930\pi$$
0.837408 0.546578i $$-0.184070\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 1.39445i 0.163208i 0.996665 + 0.0816039i $$0.0260043\pi$$
−0.996665 + 0.0816039i $$0.973996\pi$$
$$74$$ 9.21110 1.07077
$$75$$ 1.00000 0.115470
$$76$$ − 4.60555i − 0.528293i
$$77$$ 0 0
$$78$$ 3.60555i 0.408248i
$$79$$ −14.4222 −1.62262 −0.811312 0.584613i $$-0.801246\pi$$
−0.811312 + 0.584613i $$0.801246\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ −3.21110 −0.354607
$$83$$ − 2.78890i − 0.306121i −0.988217 0.153061i $$-0.951087\pi$$
0.988217 0.153061i $$-0.0489130\pi$$
$$84$$ 4.60555i 0.502507i
$$85$$ − 4.60555i − 0.499542i
$$86$$ − 8.00000i − 0.862662i
$$87$$ −4.60555 −0.493767
$$88$$ 0 0
$$89$$ − 15.2111i − 1.61237i −0.591661 0.806187i $$-0.701528\pi$$
0.591661 0.806187i $$-0.298472\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 16.6056i 1.74073i
$$92$$ 1.39445 0.145381
$$93$$ 6.00000i 0.622171i
$$94$$ 9.21110 0.950053
$$95$$ 4.60555 0.472520
$$96$$ 1.00000i 0.102062i
$$97$$ − 1.39445i − 0.141585i −0.997491 0.0707924i $$-0.977447\pi$$
0.997491 0.0707924i $$-0.0225528\pi$$
$$98$$ 14.2111i 1.43554i
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −7.39445 −0.735775 −0.367888 0.929870i $$-0.619919\pi$$
−0.367888 + 0.929870i $$0.619919\pi$$
$$102$$ 4.60555i 0.456018i
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 3.60555i 0.353553i
$$105$$ −4.60555 −0.449456
$$106$$ − 6.00000i − 0.582772i
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ − 1.39445i − 0.133564i −0.997768 0.0667820i $$-0.978727\pi$$
0.997768 0.0667820i $$-0.0212732\pi$$
$$110$$ 0 0
$$111$$ − 9.21110i − 0.874279i
$$112$$ 4.60555i 0.435184i
$$113$$ −13.8167 −1.29976 −0.649881 0.760036i $$-0.725181\pi$$
−0.649881 + 0.760036i $$0.725181\pi$$
$$114$$ −4.60555 −0.431349
$$115$$ 1.39445i 0.130033i
$$116$$ −4.60555 −0.427615
$$117$$ 3.60555 0.333333
$$118$$ 9.21110 0.847951
$$119$$ 21.2111i 1.94442i
$$120$$ −1.00000 −0.0912871
$$121$$ 11.0000 1.00000
$$122$$ 11.2111i 1.01501i
$$123$$ 3.21110i 0.289535i
$$124$$ 6.00000i 0.538816i
$$125$$ 1.00000i 0.0894427i
$$126$$ 4.60555 0.410295
$$127$$ 1.21110 0.107468 0.0537340 0.998555i $$-0.482888\pi$$
0.0537340 + 0.998555i $$0.482888\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ −3.60555 −0.316228
$$131$$ 22.6056 1.97506 0.987528 0.157443i $$-0.0503250\pi$$
0.987528 + 0.157443i $$0.0503250\pi$$
$$132$$ 0 0
$$133$$ −21.2111 −1.83924
$$134$$ 3.21110 0.277397
$$135$$ 1.00000i 0.0860663i
$$136$$ 4.60555i 0.394923i
$$137$$ − 3.21110i − 0.274343i −0.990547 0.137172i $$-0.956199\pi$$
0.990547 0.137172i $$-0.0438011\pi$$
$$138$$ − 1.39445i − 0.118703i
$$139$$ 17.2111 1.45983 0.729913 0.683540i $$-0.239560\pi$$
0.729913 + 0.683540i $$0.239560\pi$$
$$140$$ −4.60555 −0.389240
$$141$$ − 9.21110i − 0.775715i
$$142$$ −9.21110 −0.772979
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 4.60555i − 0.382470i
$$146$$ 1.39445 0.115405
$$147$$ 14.2111 1.17211
$$148$$ − 9.21110i − 0.757148i
$$149$$ − 15.2111i − 1.24614i −0.782165 0.623071i $$-0.785885\pi$$
0.782165 0.623071i $$-0.214115\pi$$
$$150$$ − 1.00000i − 0.0816497i
$$151$$ − 6.00000i − 0.488273i −0.969741 0.244137i $$-0.921495\pi$$
0.969741 0.244137i $$-0.0785045\pi$$
$$152$$ −4.60555 −0.373560
$$153$$ 4.60555 0.372337
$$154$$ 0 0
$$155$$ −6.00000 −0.481932
$$156$$ 3.60555 0.288675
$$157$$ 20.4222 1.62987 0.814935 0.579553i $$-0.196773\pi$$
0.814935 + 0.579553i $$0.196773\pi$$
$$158$$ 14.4222i 1.14737i
$$159$$ −6.00000 −0.475831
$$160$$ −1.00000 −0.0790569
$$161$$ − 6.42221i − 0.506141i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 24.4222i − 1.91289i −0.291905 0.956447i $$-0.594289\pi$$
0.291905 0.956447i $$-0.405711\pi$$
$$164$$ 3.21110i 0.250745i
$$165$$ 0 0
$$166$$ −2.78890 −0.216460
$$167$$ − 9.21110i − 0.712777i −0.934338 0.356388i $$-0.884008\pi$$
0.934338 0.356388i $$-0.115992\pi$$
$$168$$ 4.60555 0.355326
$$169$$ 13.0000 1.00000
$$170$$ −4.60555 −0.353230
$$171$$ 4.60555i 0.352195i
$$172$$ −8.00000 −0.609994
$$173$$ −12.4222 −0.944443 −0.472221 0.881480i $$-0.656548\pi$$
−0.472221 + 0.881480i $$0.656548\pi$$
$$174$$ 4.60555i 0.349146i
$$175$$ − 4.60555i − 0.348147i
$$176$$ 0 0
$$177$$ − 9.21110i − 0.692349i
$$178$$ −15.2111 −1.14012
$$179$$ −19.8167 −1.48117 −0.740583 0.671965i $$-0.765451\pi$$
−0.740583 + 0.671965i $$0.765451\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ 8.42221 0.626018 0.313009 0.949750i $$-0.398663\pi$$
0.313009 + 0.949750i $$0.398663\pi$$
$$182$$ 16.6056 1.23089
$$183$$ 11.2111 0.828749
$$184$$ − 1.39445i − 0.102800i
$$185$$ 9.21110 0.677214
$$186$$ 6.00000 0.439941
$$187$$ 0 0
$$188$$ − 9.21110i − 0.671789i
$$189$$ − 4.60555i − 0.335005i
$$190$$ − 4.60555i − 0.334122i
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ − 7.81665i − 0.562655i −0.959612 0.281328i $$-0.909225\pi$$
0.959612 0.281328i $$-0.0907747\pi$$
$$194$$ −1.39445 −0.100116
$$195$$ 3.60555i 0.258199i
$$196$$ 14.2111 1.01508
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 22.4222 1.58947 0.794734 0.606958i $$-0.207610\pi$$
0.794734 + 0.606958i $$0.207610\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ − 3.21110i − 0.226494i
$$202$$ 7.39445i 0.520272i
$$203$$ 21.2111i 1.48873i
$$204$$ 4.60555 0.322453
$$205$$ −3.21110 −0.224273
$$206$$ 4.00000i 0.278693i
$$207$$ −1.39445 −0.0969209
$$208$$ 3.60555 0.250000
$$209$$ 0 0
$$210$$ 4.60555i 0.317813i
$$211$$ −17.2111 −1.18486 −0.592431 0.805622i $$-0.701832\pi$$
−0.592431 + 0.805622i $$0.701832\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 9.21110i 0.631134i
$$214$$ 0 0
$$215$$ − 8.00000i − 0.545595i
$$216$$ − 1.00000i − 0.0680414i
$$217$$ 27.6333 1.87587
$$218$$ −1.39445 −0.0944440
$$219$$ − 1.39445i − 0.0942281i
$$220$$ 0 0
$$221$$ 16.6056 1.11701
$$222$$ −9.21110 −0.618209
$$223$$ − 1.81665i − 0.121652i −0.998148 0.0608261i $$-0.980627\pi$$
0.998148 0.0608261i $$-0.0193735\pi$$
$$224$$ 4.60555 0.307721
$$225$$ −1.00000 −0.0666667
$$226$$ 13.8167i 0.919070i
$$227$$ − 24.0000i − 1.59294i −0.604681 0.796468i $$-0.706699\pi$$
0.604681 0.796468i $$-0.293301\pi$$
$$228$$ 4.60555i 0.305010i
$$229$$ 19.8167i 1.30952i 0.755836 + 0.654761i $$0.227231\pi$$
−0.755836 + 0.654761i $$0.772769\pi$$
$$230$$ 1.39445 0.0919472
$$231$$ 0 0
$$232$$ 4.60555i 0.302369i
$$233$$ −1.81665 −0.119013 −0.0595065 0.998228i $$-0.518953\pi$$
−0.0595065 + 0.998228i $$0.518953\pi$$
$$234$$ − 3.60555i − 0.235702i
$$235$$ 9.21110 0.600866
$$236$$ − 9.21110i − 0.599592i
$$237$$ 14.4222 0.936823
$$238$$ 21.2111 1.37491
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ 6.42221i 0.413691i 0.978374 + 0.206845i $$0.0663197\pi$$
−0.978374 + 0.206845i $$0.933680\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ −1.00000 −0.0641500
$$244$$ 11.2111 0.717717
$$245$$ 14.2111i 0.907914i
$$246$$ 3.21110 0.204732
$$247$$ 16.6056i 1.05659i
$$248$$ 6.00000 0.381000
$$249$$ 2.78890i 0.176739i
$$250$$ 1.00000 0.0632456
$$251$$ −13.3944 −0.845450 −0.422725 0.906258i $$-0.638926\pi$$
−0.422725 + 0.906258i $$0.638926\pi$$
$$252$$ − 4.60555i − 0.290122i
$$253$$ 0 0
$$254$$ − 1.21110i − 0.0759913i
$$255$$ 4.60555i 0.288411i
$$256$$ 1.00000 0.0625000
$$257$$ −28.6056 −1.78437 −0.892183 0.451675i $$-0.850827\pi$$
−0.892183 + 0.451675i $$0.850827\pi$$
$$258$$ 8.00000i 0.498058i
$$259$$ −42.4222 −2.63599
$$260$$ 3.60555i 0.223607i
$$261$$ 4.60555 0.285076
$$262$$ − 22.6056i − 1.39658i
$$263$$ −7.81665 −0.481996 −0.240998 0.970526i $$-0.577475\pi$$
−0.240998 + 0.970526i $$0.577475\pi$$
$$264$$ 0 0
$$265$$ − 6.00000i − 0.368577i
$$266$$ 21.2111i 1.30054i
$$267$$ 15.2111i 0.930904i
$$268$$ − 3.21110i − 0.196149i
$$269$$ 25.8167 1.57407 0.787035 0.616909i $$-0.211615\pi$$
0.787035 + 0.616909i $$0.211615\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 0.422205i 0.0256471i 0.999918 + 0.0128236i $$0.00408198\pi$$
−0.999918 + 0.0128236i $$0.995918\pi$$
$$272$$ 4.60555 0.279253
$$273$$ − 16.6056i − 1.00501i
$$274$$ −3.21110 −0.193990
$$275$$ 0 0
$$276$$ −1.39445 −0.0839359
$$277$$ −16.4222 −0.986715 −0.493357 0.869827i $$-0.664230\pi$$
−0.493357 + 0.869827i $$0.664230\pi$$
$$278$$ − 17.2111i − 1.03225i
$$279$$ − 6.00000i − 0.359211i
$$280$$ 4.60555i 0.275234i
$$281$$ − 27.2111i − 1.62328i −0.584159 0.811639i $$-0.698576\pi$$
0.584159 0.811639i $$-0.301424\pi$$
$$282$$ −9.21110 −0.548513
$$283$$ 10.4222 0.619536 0.309768 0.950812i $$-0.399749\pi$$
0.309768 + 0.950812i $$0.399749\pi$$
$$284$$ 9.21110i 0.546578i
$$285$$ −4.60555 −0.272809
$$286$$ 0 0
$$287$$ 14.7889 0.872961
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 4.21110 0.247712
$$290$$ −4.60555 −0.270447
$$291$$ 1.39445i 0.0817440i
$$292$$ − 1.39445i − 0.0816039i
$$293$$ − 18.0000i − 1.05157i −0.850617 0.525786i $$-0.823771\pi$$
0.850617 0.525786i $$-0.176229\pi$$
$$294$$ − 14.2111i − 0.828808i
$$295$$ 9.21110 0.536291
$$296$$ −9.21110 −0.535384
$$297$$ 0 0
$$298$$ −15.2111 −0.881156
$$299$$ −5.02776 −0.290763
$$300$$ −1.00000 −0.0577350
$$301$$ 36.8444i 2.12368i
$$302$$ −6.00000 −0.345261
$$303$$ 7.39445 0.424800
$$304$$ 4.60555i 0.264146i
$$305$$ 11.2111i 0.641946i
$$306$$ − 4.60555i − 0.263282i
$$307$$ 8.78890i 0.501609i 0.968038 + 0.250804i $$0.0806951\pi$$
−0.968038 + 0.250804i $$0.919305\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 6.00000i 0.340777i
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ − 3.60555i − 0.204124i
$$313$$ 3.57779 0.202229 0.101114 0.994875i $$-0.467759\pi$$
0.101114 + 0.994875i $$0.467759\pi$$
$$314$$ − 20.4222i − 1.15249i
$$315$$ 4.60555 0.259493
$$316$$ 14.4222 0.811312
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 1.00000i 0.0559017i
$$321$$ 0 0
$$322$$ −6.42221 −0.357895
$$323$$ 21.2111i 1.18022i
$$324$$ −1.00000 −0.0555556
$$325$$ −3.60555 −0.200000
$$326$$ −24.4222 −1.35262
$$327$$ 1.39445i 0.0771132i
$$328$$ 3.21110 0.177303
$$329$$ −42.4222 −2.33881
$$330$$ 0 0
$$331$$ 16.6056i 0.912724i 0.889794 + 0.456362i $$0.150848\pi$$
−0.889794 + 0.456362i $$0.849152\pi$$
$$332$$ 2.78890i 0.153061i
$$333$$ 9.21110i 0.504765i
$$334$$ −9.21110 −0.504009
$$335$$ 3.21110 0.175441
$$336$$ − 4.60555i − 0.251253i
$$337$$ 13.6333 0.742654 0.371327 0.928502i $$-0.378903\pi$$
0.371327 + 0.928502i $$0.378903\pi$$
$$338$$ − 13.0000i − 0.707107i
$$339$$ 13.8167 0.750418
$$340$$ 4.60555i 0.249771i
$$341$$ 0 0
$$342$$ 4.60555 0.249040
$$343$$ − 33.2111i − 1.79323i
$$344$$ 8.00000i 0.431331i
$$345$$ − 1.39445i − 0.0750746i
$$346$$ 12.4222i 0.667822i
$$347$$ −27.6333 −1.48343 −0.741717 0.670713i $$-0.765988\pi$$
−0.741717 + 0.670713i $$0.765988\pi$$
$$348$$ 4.60555 0.246883
$$349$$ 7.81665i 0.418416i 0.977871 + 0.209208i $$0.0670886\pi$$
−0.977871 + 0.209208i $$0.932911\pi$$
$$350$$ −4.60555 −0.246177
$$351$$ −3.60555 −0.192450
$$352$$ 0 0
$$353$$ 8.78890i 0.467786i 0.972262 + 0.233893i $$0.0751465\pi$$
−0.972262 + 0.233893i $$0.924853\pi$$
$$354$$ −9.21110 −0.489565
$$355$$ −9.21110 −0.488875
$$356$$ 15.2111i 0.806187i
$$357$$ − 21.2111i − 1.12261i
$$358$$ 19.8167i 1.04734i
$$359$$ − 15.6333i − 0.825094i −0.910936 0.412547i $$-0.864639\pi$$
0.910936 0.412547i $$-0.135361\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −2.21110 −0.116374
$$362$$ − 8.42221i − 0.442661i
$$363$$ −11.0000 −0.577350
$$364$$ − 16.6056i − 0.870367i
$$365$$ 1.39445 0.0729888
$$366$$ − 11.2111i − 0.586014i
$$367$$ −19.6333 −1.02485 −0.512425 0.858732i $$-0.671253\pi$$
−0.512425 + 0.858732i $$0.671253\pi$$
$$368$$ −1.39445 −0.0726907
$$369$$ − 3.21110i − 0.167163i
$$370$$ − 9.21110i − 0.478862i
$$371$$ 27.6333i 1.43465i
$$372$$ − 6.00000i − 0.311086i
$$373$$ −20.4222 −1.05742 −0.528711 0.848802i $$-0.677324\pi$$
−0.528711 + 0.848802i $$0.677324\pi$$
$$374$$ 0 0
$$375$$ − 1.00000i − 0.0516398i
$$376$$ −9.21110 −0.475026
$$377$$ 16.6056 0.855229
$$378$$ −4.60555 −0.236884
$$379$$ − 35.0278i − 1.79925i −0.436658 0.899627i $$-0.643838\pi$$
0.436658 0.899627i $$-0.356162\pi$$
$$380$$ −4.60555 −0.236260
$$381$$ −1.21110 −0.0620467
$$382$$ 12.0000i 0.613973i
$$383$$ 27.6333i 1.41200i 0.708214 + 0.705998i $$0.249501\pi$$
−0.708214 + 0.705998i $$0.750499\pi$$
$$384$$ − 1.00000i − 0.0510310i
$$385$$ 0 0
$$386$$ −7.81665 −0.397857
$$387$$ 8.00000 0.406663
$$388$$ 1.39445i 0.0707924i
$$389$$ 4.60555 0.233511 0.116755 0.993161i $$-0.462751\pi$$
0.116755 + 0.993161i $$0.462751\pi$$
$$390$$ 3.60555 0.182574
$$391$$ −6.42221 −0.324785
$$392$$ − 14.2111i − 0.717769i
$$393$$ −22.6056 −1.14030
$$394$$ 6.00000 0.302276
$$395$$ 14.4222i 0.725660i
$$396$$ 0 0
$$397$$ 3.63331i 0.182350i 0.995835 + 0.0911752i $$0.0290623\pi$$
−0.995835 + 0.0911752i $$0.970938\pi$$
$$398$$ − 22.4222i − 1.12392i
$$399$$ 21.2111 1.06188
$$400$$ −1.00000 −0.0500000
$$401$$ − 8.78890i − 0.438897i −0.975624 0.219448i $$-0.929574\pi$$
0.975624 0.219448i $$-0.0704257\pi$$
$$402$$ −3.21110 −0.160155
$$403$$ − 21.6333i − 1.07763i
$$404$$ 7.39445 0.367888
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 21.2111 1.05269
$$407$$ 0 0
$$408$$ − 4.60555i − 0.228009i
$$409$$ 14.7889i 0.731264i 0.930760 + 0.365632i $$0.119147\pi$$
−0.930760 + 0.365632i $$0.880853\pi$$
$$410$$ 3.21110i 0.158585i
$$411$$ 3.21110i 0.158392i
$$412$$ 4.00000 0.197066
$$413$$ −42.4222 −2.08746
$$414$$ 1.39445i 0.0685334i
$$415$$ −2.78890 −0.136902
$$416$$ − 3.60555i − 0.176777i
$$417$$ −17.2111 −0.842831
$$418$$ 0 0
$$419$$ 4.18335 0.204370 0.102185 0.994765i $$-0.467417\pi$$
0.102185 + 0.994765i $$0.467417\pi$$
$$420$$ 4.60555 0.224728
$$421$$ 19.8167i 0.965805i 0.875674 + 0.482902i $$0.160417\pi$$
−0.875674 + 0.482902i $$0.839583\pi$$
$$422$$ 17.2111i 0.837823i
$$423$$ 9.21110i 0.447859i
$$424$$ 6.00000i 0.291386i
$$425$$ −4.60555 −0.223402
$$426$$ 9.21110 0.446279
$$427$$ − 51.6333i − 2.49871i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ − 12.0000i − 0.578020i −0.957326 0.289010i $$-0.906674\pi$$
0.957326 0.289010i $$-0.0933260\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 19.2111 0.923227 0.461613 0.887081i $$-0.347271\pi$$
0.461613 + 0.887081i $$0.347271\pi$$
$$434$$ − 27.6333i − 1.32644i
$$435$$ 4.60555i 0.220819i
$$436$$ 1.39445i 0.0667820i
$$437$$ − 6.42221i − 0.307216i
$$438$$ −1.39445 −0.0666293
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −14.2111 −0.676719
$$442$$ − 16.6056i − 0.789846i
$$443$$ 15.6333 0.742761 0.371380 0.928481i $$-0.378885\pi$$
0.371380 + 0.928481i $$0.378885\pi$$
$$444$$ 9.21110i 0.437140i
$$445$$ −15.2111 −0.721075
$$446$$ −1.81665 −0.0860211
$$447$$ 15.2111i 0.719460i
$$448$$ − 4.60555i − 0.217592i
$$449$$ − 33.6333i − 1.58725i −0.608405 0.793627i $$-0.708190\pi$$
0.608405 0.793627i $$-0.291810\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ 0 0
$$452$$ 13.8167 0.649881
$$453$$ 6.00000i 0.281905i
$$454$$ −24.0000 −1.12638
$$455$$ 16.6056 0.778480
$$456$$ 4.60555 0.215675
$$457$$ − 38.2389i − 1.78874i −0.447330 0.894369i $$-0.647625\pi$$
0.447330 0.894369i $$-0.352375\pi$$
$$458$$ 19.8167 0.925971
$$459$$ −4.60555 −0.214969
$$460$$ − 1.39445i − 0.0650165i
$$461$$ 33.6333i 1.56646i 0.621733 + 0.783230i $$0.286429\pi$$
−0.621733 + 0.783230i $$0.713571\pi$$
$$462$$ 0 0
$$463$$ 31.3944i 1.45902i 0.683968 + 0.729512i $$0.260253\pi$$
−0.683968 + 0.729512i $$0.739747\pi$$
$$464$$ 4.60555 0.213807
$$465$$ 6.00000 0.278243
$$466$$ 1.81665i 0.0841549i
$$467$$ −30.4222 −1.40777 −0.703886 0.710313i $$-0.748553\pi$$
−0.703886 + 0.710313i $$0.748553\pi$$
$$468$$ −3.60555 −0.166667
$$469$$ −14.7889 −0.682888
$$470$$ − 9.21110i − 0.424876i
$$471$$ −20.4222 −0.941006
$$472$$ −9.21110 −0.423975
$$473$$ 0 0
$$474$$ − 14.4222i − 0.662434i
$$475$$ − 4.60555i − 0.211317i
$$476$$ − 21.2111i − 0.972209i
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ − 5.57779i − 0.254856i −0.991848 0.127428i $$-0.959328\pi$$
0.991848 0.127428i $$-0.0406722\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 33.2111i 1.51430i
$$482$$ 6.42221 0.292523
$$483$$ 6.42221i 0.292220i
$$484$$ −11.0000 −0.500000
$$485$$ −1.39445 −0.0633187
$$486$$ 1.00000i 0.0453609i
$$487$$ 0.972244i 0.0440566i 0.999757 + 0.0220283i $$0.00701239\pi$$
−0.999757 + 0.0220283i $$0.992988\pi$$
$$488$$ − 11.2111i − 0.507503i
$$489$$ 24.4222i 1.10441i
$$490$$ 14.2111 0.641992
$$491$$ 7.81665 0.352761 0.176380 0.984322i $$-0.443561\pi$$
0.176380 + 0.984322i $$0.443561\pi$$
$$492$$ − 3.21110i − 0.144768i
$$493$$ 21.2111 0.955300
$$494$$ 16.6056 0.747119
$$495$$ 0 0
$$496$$ − 6.00000i − 0.269408i
$$497$$ 42.4222 1.90290
$$498$$ 2.78890 0.124973
$$499$$ − 23.0278i − 1.03086i −0.856930 0.515432i $$-0.827631\pi$$
0.856930 0.515432i $$-0.172369\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ 9.21110i 0.411522i
$$502$$ 13.3944i 0.597824i
$$503$$ 23.4500 1.04558 0.522791 0.852461i $$-0.324891\pi$$
0.522791 + 0.852461i $$0.324891\pi$$
$$504$$ −4.60555 −0.205148
$$505$$ 7.39445i 0.329049i
$$506$$ 0 0
$$507$$ −13.0000 −0.577350
$$508$$ −1.21110 −0.0537340
$$509$$ 33.6333i 1.49077i 0.666634 + 0.745385i $$0.267734\pi$$
−0.666634 + 0.745385i $$0.732266\pi$$
$$510$$ 4.60555 0.203937
$$511$$ −6.42221 −0.284102
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 4.60555i − 0.203340i
$$514$$ 28.6056i 1.26174i
$$515$$ 4.00000i 0.176261i
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ 42.4222i 1.86392i
$$519$$ 12.4222 0.545274
$$520$$ 3.60555 0.158114
$$521$$ 21.6333 0.947772 0.473886 0.880586i $$-0.342851\pi$$
0.473886 + 0.880586i $$0.342851\pi$$
$$522$$ − 4.60555i − 0.201580i
$$523$$ 32.8444 1.43619 0.718093 0.695947i $$-0.245015\pi$$
0.718093 + 0.695947i $$0.245015\pi$$
$$524$$ −22.6056 −0.987528
$$525$$ 4.60555i 0.201003i
$$526$$ 7.81665i 0.340822i
$$527$$ − 27.6333i − 1.20373i
$$528$$ 0 0
$$529$$ −21.0555 −0.915457
$$530$$ −6.00000 −0.260623
$$531$$ 9.21110i 0.399728i
$$532$$ 21.2111 0.919618
$$533$$ − 11.5778i − 0.501490i
$$534$$ 15.2111 0.658249
$$535$$ 0 0
$$536$$ −3.21110 −0.138699
$$537$$ 19.8167 0.855152
$$538$$ − 25.8167i − 1.11303i
$$539$$ 0 0
$$540$$ − 1.00000i − 0.0430331i
$$541$$ − 6.97224i − 0.299760i −0.988704 0.149880i $$-0.952111\pi$$
0.988704 0.149880i $$-0.0478888\pi$$
$$542$$ 0.422205 0.0181353
$$543$$ −8.42221 −0.361431
$$544$$ − 4.60555i − 0.197461i
$$545$$ −1.39445 −0.0597316
$$546$$ −16.6056 −0.710652
$$547$$ 14.4222 0.616649 0.308324 0.951281i $$-0.400232\pi$$
0.308324 + 0.951281i $$0.400232\pi$$
$$548$$ 3.21110i 0.137172i
$$549$$ −11.2111 −0.478478
$$550$$ 0 0
$$551$$ 21.2111i 0.903623i
$$552$$ 1.39445i 0.0593517i
$$553$$ − 66.4222i − 2.82456i
$$554$$ 16.4222i 0.697713i
$$555$$ −9.21110 −0.390990
$$556$$ −17.2111 −0.729913
$$557$$ − 11.5778i − 0.490567i −0.969451 0.245283i $$-0.921119\pi$$
0.969451 0.245283i $$-0.0788810\pi$$
$$558$$ −6.00000 −0.254000
$$559$$ 28.8444 1.21999
$$560$$ 4.60555 0.194620
$$561$$ 0 0
$$562$$ −27.2111 −1.14783
$$563$$ 34.0555 1.43527 0.717634 0.696420i $$-0.245225\pi$$
0.717634 + 0.696420i $$0.245225\pi$$
$$564$$ 9.21110i 0.387857i
$$565$$ 13.8167i 0.581271i
$$566$$ − 10.4222i − 0.438078i
$$567$$ 4.60555i 0.193415i
$$568$$ 9.21110 0.386489
$$569$$ 33.6333 1.40998 0.704991 0.709216i $$-0.250951\pi$$
0.704991 + 0.709216i $$0.250951\pi$$
$$570$$ 4.60555i 0.192905i
$$571$$ −30.0555 −1.25778 −0.628892 0.777493i $$-0.716491\pi$$
−0.628892 + 0.777493i $$0.716491\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ − 14.7889i − 0.617277i
$$575$$ 1.39445 0.0581525
$$576$$ −1.00000 −0.0416667
$$577$$ 37.3944i 1.55675i 0.627799 + 0.778376i $$0.283956\pi$$
−0.627799 + 0.778376i $$0.716044\pi$$
$$578$$ − 4.21110i − 0.175159i
$$579$$ 7.81665i 0.324849i
$$580$$ 4.60555i 0.191235i
$$581$$ 12.8444 0.532876
$$582$$ 1.39445 0.0578018
$$583$$ 0 0
$$584$$ −1.39445 −0.0577027
$$585$$ − 3.60555i − 0.149071i
$$586$$ −18.0000 −0.743573
$$587$$ 6.42221i 0.265073i 0.991178 + 0.132536i $$0.0423121\pi$$
−0.991178 + 0.132536i $$0.957688\pi$$
$$588$$ −14.2111 −0.586056
$$589$$ 27.6333 1.13861
$$590$$ − 9.21110i − 0.379215i
$$591$$ − 6.00000i − 0.246807i
$$592$$ 9.21110i 0.378574i
$$593$$ 24.4222i 1.00290i 0.865187 + 0.501450i $$0.167200\pi$$
−0.865187 + 0.501450i $$0.832800\pi$$
$$594$$ 0 0
$$595$$ 21.2111 0.869570
$$596$$ 15.2111i 0.623071i
$$597$$ −22.4222 −0.917680
$$598$$ 5.02776i 0.205600i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 1.00000i 0.0408248i
$$601$$ 1.63331 0.0666240 0.0333120 0.999445i $$-0.489394\pi$$
0.0333120 + 0.999445i $$0.489394\pi$$
$$602$$ 36.8444 1.50167
$$603$$ 3.21110i 0.130766i
$$604$$ 6.00000i 0.244137i
$$605$$ − 11.0000i − 0.447214i
$$606$$ − 7.39445i − 0.300379i
$$607$$ 17.2111 0.698577 0.349289 0.937015i $$-0.386423\pi$$
0.349289 + 0.937015i $$0.386423\pi$$
$$608$$ 4.60555 0.186780
$$609$$ − 21.2111i − 0.859517i
$$610$$ 11.2111 0.453924
$$611$$ 33.2111i 1.34358i
$$612$$ −4.60555 −0.186168
$$613$$ 33.2111i 1.34138i 0.741736 + 0.670692i $$0.234003\pi$$
−0.741736 + 0.670692i $$0.765997\pi$$
$$614$$ 8.78890 0.354691
$$615$$ 3.21110 0.129484
$$616$$ 0 0
$$617$$ − 12.4222i − 0.500099i −0.968233 0.250050i $$-0.919553\pi$$
0.968233 0.250050i $$-0.0804469\pi$$
$$618$$ − 4.00000i − 0.160904i
$$619$$ − 25.8167i − 1.03766i −0.854878 0.518829i $$-0.826368\pi$$
0.854878 0.518829i $$-0.173632\pi$$
$$620$$ 6.00000 0.240966
$$621$$ 1.39445 0.0559573
$$622$$ − 12.0000i − 0.481156i
$$623$$ 70.0555 2.80671
$$624$$ −3.60555 −0.144338
$$625$$ 1.00000 0.0400000
$$626$$ − 3.57779i − 0.142997i
$$627$$ 0 0
$$628$$ −20.4222 −0.814935
$$629$$ 42.4222i 1.69148i
$$630$$ − 4.60555i − 0.183490i
$$631$$ − 3.21110i − 0.127832i −0.997955 0.0639160i $$-0.979641\pi$$
0.997955 0.0639160i $$-0.0203590\pi$$
$$632$$ − 14.4222i − 0.573685i
$$633$$ 17.2111 0.684080
$$634$$ 18.0000 0.714871
$$635$$ − 1.21110i − 0.0480611i
$$636$$ 6.00000 0.237915
$$637$$ −51.2389 −2.03016
$$638$$ 0 0
$$639$$ − 9.21110i − 0.364386i
$$640$$ 1.00000 0.0395285
$$641$$ 0.422205 0.0166761 0.00833805 0.999965i $$-0.497346\pi$$
0.00833805 + 0.999965i $$0.497346\pi$$
$$642$$ 0 0
$$643$$ − 9.63331i − 0.379901i −0.981794 0.189950i $$-0.939167\pi$$
0.981794 0.189950i $$-0.0608327\pi$$
$$644$$ 6.42221i 0.253070i
$$645$$ 8.00000i 0.315000i
$$646$$ 21.2111 0.834540
$$647$$ −34.6056 −1.36048 −0.680242 0.732987i $$-0.738125\pi$$
−0.680242 + 0.732987i $$0.738125\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 3.60555i 0.141421i
$$651$$ −27.6333 −1.08303
$$652$$ 24.4222i 0.956447i
$$653$$ 39.2111 1.53445 0.767225 0.641379i $$-0.221637\pi$$
0.767225 + 0.641379i $$0.221637\pi$$
$$654$$ 1.39445 0.0545273
$$655$$ − 22.6056i − 0.883272i
$$656$$ − 3.21110i − 0.125372i
$$657$$ 1.39445i 0.0544026i
$$658$$ 42.4222i 1.65379i
$$659$$ −26.2389 −1.02212 −0.511060 0.859545i $$-0.670747\pi$$
−0.511060 + 0.859545i $$0.670747\pi$$
$$660$$ 0 0
$$661$$ − 50.2389i − 1.95407i −0.213090 0.977033i $$-0.568353\pi$$
0.213090 0.977033i $$-0.431647\pi$$
$$662$$ 16.6056 0.645393
$$663$$ −16.6056 −0.644906
$$664$$ 2.78890 0.108230
$$665$$ 21.2111i 0.822531i
$$666$$ 9.21110 0.356923
$$667$$ −6.42221 −0.248669
$$668$$ 9.21110i 0.356388i
$$669$$ 1.81665i 0.0702359i
$$670$$ − 3.21110i − 0.124056i
$$671$$ 0 0
$$672$$ −4.60555 −0.177663
$$673$$ −37.6333 −1.45066 −0.725329 0.688403i $$-0.758312\pi$$
−0.725329 + 0.688403i $$0.758312\pi$$
$$674$$ − 13.6333i − 0.525135i
$$675$$ 1.00000 0.0384900
$$676$$ −13.0000 −0.500000
$$677$$ −28.0555 −1.07826 −0.539130 0.842222i $$-0.681247\pi$$
−0.539130 + 0.842222i $$0.681247\pi$$
$$678$$ − 13.8167i − 0.530625i
$$679$$ 6.42221 0.246462
$$680$$ 4.60555 0.176615
$$681$$ 24.0000i 0.919682i
$$682$$ 0 0
$$683$$ − 9.21110i − 0.352453i −0.984350 0.176227i $$-0.943611\pi$$
0.984350 0.176227i $$-0.0563891\pi$$
$$684$$ − 4.60555i − 0.176098i
$$685$$ −3.21110 −0.122690
$$686$$ −33.2111 −1.26801
$$687$$ − 19.8167i − 0.756053i
$$688$$ 8.00000 0.304997
$$689$$ 21.6333 0.824163
$$690$$ −1.39445 −0.0530858
$$691$$ − 20.2389i − 0.769922i −0.922933 0.384961i $$-0.874215\pi$$
0.922933 0.384961i $$-0.125785\pi$$
$$692$$ 12.4222 0.472221
$$693$$ 0 0
$$694$$ 27.6333i 1.04895i
$$695$$ − 17.2111i − 0.652854i
$$696$$ − 4.60555i − 0.174573i
$$697$$ − 14.7889i − 0.560169i
$$698$$ 7.81665 0.295865
$$699$$ 1.81665 0.0687122
$$700$$ 4.60555i 0.174073i
$$701$$ 47.0278 1.77621 0.888107 0.459637i $$-0.152020\pi$$
0.888107 + 0.459637i $$0.152020\pi$$
$$702$$ 3.60555i 0.136083i
$$703$$ −42.4222 −1.59998
$$704$$ 0 0
$$705$$ −9.21110 −0.346910
$$706$$ 8.78890 0.330775
$$707$$ − 34.0555i − 1.28079i
$$708$$ 9.21110i 0.346174i
$$709$$ 1.39445i 0.0523696i 0.999657 + 0.0261848i $$0.00833584\pi$$
−0.999657 + 0.0261848i $$0.991664\pi$$
$$710$$ 9.21110i 0.345687i
$$711$$ −14.4222 −0.540875
$$712$$ 15.2111 0.570060
$$713$$ 8.36669i 0.313335i
$$714$$ −21.2111 −0.793806
$$715$$ 0 0
$$716$$ 19.8167 0.740583
$$717$$ 0 0
$$718$$ −15.6333 −0.583430
$$719$$ −51.6333 −1.92560 −0.962799 0.270220i $$-0.912904\pi$$
−0.962799 + 0.270220i $$0.912904\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ − 18.4222i − 0.686079i
$$722$$ 2.21110i 0.0822887i
$$723$$ − 6.42221i − 0.238844i
$$724$$ −8.42221 −0.313009
$$725$$ −4.60555 −0.171046
$$726$$ 11.0000i 0.408248i
$$727$$ −14.4222 −0.534890 −0.267445 0.963573i $$-0.586179\pi$$
−0.267445 + 0.963573i $$0.586179\pi$$
$$728$$ −16.6056 −0.615443
$$729$$ 1.00000 0.0370370
$$730$$ − 1.39445i − 0.0516109i
$$731$$ 36.8444 1.36274
$$732$$ −11.2111 −0.414374
$$733$$ 34.0555i 1.25787i 0.777458 + 0.628935i $$0.216509\pi$$
−0.777458 + 0.628935i $$0.783491\pi$$
$$734$$ 19.6333i 0.724679i
$$735$$ − 14.2111i − 0.524184i
$$736$$ 1.39445i 0.0514001i
$$737$$ 0 0
$$738$$ −3.21110 −0.118202
$$739$$ − 20.2389i − 0.744498i −0.928133 0.372249i $$-0.878587\pi$$
0.928133 0.372249i $$-0.121413\pi$$
$$740$$ −9.21110 −0.338607
$$741$$ − 16.6056i − 0.610020i
$$742$$ 27.6333 1.01445
$$743$$ − 36.8444i − 1.35169i −0.737044 0.675845i $$-0.763779\pi$$
0.737044 0.675845i $$-0.236221\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ −15.2111 −0.557292
$$746$$ 20.4222i 0.747710i
$$747$$ − 2.78890i − 0.102040i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ −10.4222 −0.380312 −0.190156 0.981754i $$-0.560899\pi$$
−0.190156 + 0.981754i $$0.560899\pi$$
$$752$$ 9.21110i 0.335894i
$$753$$ 13.3944 0.488121
$$754$$ − 16.6056i − 0.604739i
$$755$$ −6.00000 −0.218362
$$756$$ 4.60555i 0.167502i
$$757$$ 12.7889 0.464820 0.232410 0.972618i $$-0.425339\pi$$
0.232410 + 0.972618i $$0.425339\pi$$
$$758$$ −35.0278 −1.27227
$$759$$ 0 0
$$760$$ 4.60555i 0.167061i
$$761$$ − 33.6333i − 1.21921i −0.792707 0.609603i $$-0.791329\pi$$
0.792707 0.609603i $$-0.208671\pi$$
$$762$$ 1.21110i 0.0438736i
$$763$$ 6.42221 0.232499
$$764$$ 12.0000 0.434145
$$765$$ − 4.60555i − 0.166514i
$$766$$ 27.6333 0.998432
$$767$$ 33.2111i 1.19918i
$$768$$ −1.00000 −0.0360844
$$769$$ 12.8444i 0.463181i 0.972813 + 0.231591i $$0.0743930\pi$$
−0.972813 + 0.231591i $$0.925607\pi$$
$$770$$ 0 0
$$771$$ 28.6056 1.03020
$$772$$ 7.81665i 0.281328i
$$773$$ − 30.0000i − 1.07903i −0.841978 0.539513i $$-0.818609\pi$$
0.841978 0.539513i $$-0.181391\pi$$
$$774$$ − 8.00000i − 0.287554i
$$775$$ 6.00000i 0.215526i
$$776$$ 1.39445 0.0500578
$$777$$ 42.4222 1.52189
$$778$$ − 4.60555i − 0.165117i
$$779$$ 14.7889 0.529867
$$780$$ − 3.60555i − 0.129099i
$$781$$ 0 0
$$782$$ 6.42221i 0.229658i
$$783$$ −4.60555 −0.164589
$$784$$ −14.2111 −0.507539
$$785$$ − 20.4222i − 0.728900i
$$786$$ 22.6056i 0.806313i
$$787$$ − 49.2666i − 1.75617i −0.478509 0.878083i $$-0.658823\pi$$
0.478509 0.878083i $$-0.341177\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 7.81665 0.278280
$$790$$ 14.4222 0.513119
$$791$$ − 63.6333i − 2.26254i
$$792$$ 0 0
$$793$$ −40.4222 −1.43543
$$794$$ 3.63331 0.128941
$$795$$ 6.00000i 0.212798i
$$796$$ −22.4222 −0.794734
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ − 21.2111i − 0.750865i
$$799$$ 42.4222i 1.50079i
$$800$$ 1.00000i 0.0353553i
$$801$$ − 15.2111i − 0.537458i
$$802$$ −8.78890 −0.310347
$$803$$ 0 0
$$804$$ 3.21110i 0.113247i
$$805$$ −6.42221 −0.226353
$$806$$ −21.6333 −0.762001
$$807$$ −25.8167 −0.908789
$$808$$ − 7.39445i − 0.260136i
$$809$$ −6.84441 −0.240637 −0.120318 0.992735i $$-0.538392\pi$$
−0.120318 + 0.992735i $$0.538392\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ − 32.2389i − 1.13206i −0.824385 0.566030i $$-0.808479\pi$$
0.824385 0.566030i $$-0.191521\pi$$
$$812$$ − 21.2111i − 0.744364i
$$813$$ − 0.422205i − 0.0148074i
$$814$$ 0 0
$$815$$ −24.4222 −0.855473
$$816$$ −4.60555 −0.161227
$$817$$ 36.8444i 1.28902i
$$818$$ 14.7889 0.517082
$$819$$ 16.6056i 0.580245i
$$820$$ 3.21110 0.112137
$$821$$ 3.21110i 0.112068i 0.998429 + 0.0560341i $$0.0178456\pi$$
−0.998429 + 0.0560341i $$0.982154\pi$$
$$822$$ 3.21110 0.112000
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ − 4.00000i − 0.139347i
$$825$$ 0 0
$$826$$ 42.4222i 1.47606i
$$827$$ 27.6333i 0.960904i 0.877021 + 0.480452i $$0.159527\pi$$
−0.877021 + 0.480452i $$0.840473\pi$$
$$828$$ 1.39445 0.0484604
$$829$$ 46.8444 1.62697 0.813487 0.581583i $$-0.197567\pi$$
0.813487 + 0.581583i $$0.197567\pi$$
$$830$$ 2.78890i 0.0968040i
$$831$$ 16.4222 0.569680
$$832$$ −3.60555 −0.125000
$$833$$ −65.4500 −2.26771
$$834$$ 17.2111i 0.595972i
$$835$$ −9.21110 −0.318763
$$836$$ 0 0
$$837$$ 6.00000i 0.207390i
$$838$$ − 4.18335i − 0.144511i
$$839$$ 18.4222i 0.636005i 0.948090 + 0.318003i $$0.103012\pi$$
−0.948090 + 0.318003i $$0.896988\pi$$
$$840$$ − 4.60555i − 0.158907i
$$841$$ −7.78890 −0.268583
$$842$$ 19.8167 0.682927
$$843$$ 27.2111i 0.937200i
$$844$$ 17.2111 0.592431
$$845$$ − 13.0000i − 0.447214i
$$846$$ 9.21110 0.316684
$$847$$ 50.6611i 1.74073i
$$848$$ 6.00000 0.206041
$$849$$ −10.4222 −0.357689
$$850$$ 4.60555i 0.157969i
$$851$$ − 12.8444i − 0.440301i
$$852$$ − 9.21110i − 0.315567i
$$853$$ 14.7889i 0.506362i 0.967419 + 0.253181i $$0.0814769\pi$$
−0.967419 + 0.253181i $$0.918523\pi$$
$$854$$ −51.6333 −1.76686
$$855$$ 4.60555 0.157507
$$856$$ 0 0
$$857$$ −23.0278 −0.786613 −0.393307 0.919407i $$-0.628669\pi$$
−0.393307 + 0.919407i $$0.628669\pi$$
$$858$$ 0 0
$$859$$ 25.2111 0.860192 0.430096 0.902783i $$-0.358480\pi$$
0.430096 + 0.902783i $$0.358480\pi$$
$$860$$ 8.00000i 0.272798i
$$861$$ −14.7889 −0.504004
$$862$$ −12.0000 −0.408722
$$863$$ − 51.6333i − 1.75762i −0.477173 0.878809i $$-0.658339\pi$$
0.477173 0.878809i $$-0.341661\pi$$
$$864$$ 1.00000i 0.0340207i
$$865$$ 12.4222i 0.422368i
$$866$$ − 19.2111i − 0.652820i
$$867$$ −4.21110 −0.143017
$$868$$ −27.6333 −0.937936
$$869$$ 0 0
$$870$$ 4.60555 0.156143
$$871$$ 11.5778i 0.392299i
$$872$$ 1.39445 0.0472220
$$873$$ − 1.39445i − 0.0471949i
$$874$$ −6.42221 −0.217234
$$875$$ −4.60555 −0.155696
$$876$$ 1.39445i 0.0471141i
$$877$$ 24.8444i 0.838936i 0.907770 + 0.419468i $$0.137783\pi$$
−0.907770 + 0.419468i $$0.862217\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ 18.0000i 0.607125i
$$880$$ 0 0
$$881$$ 39.2111 1.32106 0.660528 0.750802i $$-0.270333\pi$$
0.660528 + 0.750802i $$0.270333\pi$$
$$882$$ 14.2111i 0.478513i
$$883$$ 9.57779 0.322318 0.161159 0.986928i $$-0.448477\pi$$
0.161159 + 0.986928i $$0.448477\pi$$
$$884$$ −16.6056 −0.558505
$$885$$ −9.21110 −0.309628
$$886$$ − 15.6333i − 0.525211i
$$887$$ 6.97224 0.234105 0.117053 0.993126i $$-0.462655\pi$$
0.117053 + 0.993126i $$0.462655\pi$$
$$888$$ 9.21110 0.309104
$$889$$ 5.57779i 0.187073i
$$890$$ 15.2111i 0.509877i
$$891$$ 0 0
$$892$$ 1.81665i 0.0608261i
$$893$$ −42.4222 −1.41960
$$894$$ 15.2111 0.508735
$$895$$ 19.8167i 0.662398i
$$896$$ −4.60555 −0.153861
$$897$$ 5.02776 0.167872
$$898$$ −33.6333 −1.12236
$$899$$ − 27.6333i − 0.921622i
$$900$$ 1.00000 0.0333333
$$901$$ 27.6333 0.920599
$$902$$ 0 0
$$903$$ − 36.8444i − 1.22611i
$$904$$ − 13.8167i − 0.459535i
$$905$$ − 8.42221i − 0.279964i
$$906$$ 6.00000 0.199337
$$907$$ 21.5778 0.716479 0.358239 0.933630i $$-0.383377\pi$$
0.358239 + 0.933630i $$0.383377\pi$$
$$908$$ 24.0000i 0.796468i
$$909$$ −7.39445 −0.245258
$$910$$ − 16.6056i − 0.550469i
$$911$$ −27.6333 −0.915532 −0.457766 0.889073i $$-0.651350\pi$$
−0.457766 + 0.889073i $$0.651350\pi$$
$$912$$ − 4.60555i − 0.152505i
$$913$$ 0 0
$$914$$ −38.2389 −1.26483
$$915$$ − 11.2111i − 0.370628i
$$916$$ − 19.8167i − 0.654761i
$$917$$ 104.111i 3.43805i
$$918$$ 4.60555i 0.152006i
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ −1.39445 −0.0459736
$$921$$ − 8.78890i − 0.289604i
$$922$$ 33.6333 1.10765
$$923$$ − 33.2111i − 1.09316i
$$924$$ 0 0
$$925$$ − 9.21110i − 0.302859i
$$926$$ 31.3944 1.03169
$$927$$ −4.00000 −0.131377
$$928$$ − 4.60555i − 0.151185i
$$929$$ − 39.2111i − 1.28647i −0.765667 0.643237i $$-0.777591\pi$$
0.765667 0.643237i $$-0.222409\pi$$
$$930$$ − 6.00000i − 0.196748i
$$931$$ − 65.4500i − 2.14504i
$$932$$ 1.81665 0.0595065
$$933$$ −12.0000 −0.392862
$$934$$ 30.4222i 0.995445i
$$935$$ 0 0
$$936$$ 3.60555i 0.117851i
$$937$$ −10.3667 −0.338665 −0.169333 0.985559i $$-0.554161\pi$$
−0.169333 + 0.985559i $$0.554161\pi$$
$$938$$ 14.7889i 0.482875i
$$939$$ −3.57779 −0.116757
$$940$$ −9.21110 −0.300433
$$941$$ 54.0000i 1.76035i 0.474650 + 0.880175i $$0.342575\pi$$
−0.474650 + 0.880175i $$0.657425\pi$$
$$942$$ 20.4222i 0.665391i
$$943$$ 4.47772i 0.145815i
$$944$$ 9.21110i 0.299796i
$$945$$ −4.60555 −0.149819
$$946$$ 0 0
$$947$$ 15.6333i 0.508014i 0.967202 + 0.254007i $$0.0817487\pi$$
−0.967202 + 0.254007i $$0.918251\pi$$
$$948$$ −14.4222 −0.468411
$$949$$ 5.02776i 0.163208i
$$950$$ −4.60555 −0.149424
$$951$$ − 18.0000i − 0.583690i
$$952$$ −21.2111 −0.687456
$$953$$ −20.2389 −0.655601 −0.327800 0.944747i $$-0.606307\pi$$
−0.327800 + 0.944747i $$0.606307\pi$$
$$954$$ − 6.00000i − 0.194257i
$$955$$ 12.0000i 0.388311i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −5.57779 −0.180210
$$959$$ 14.7889 0.477558
$$960$$ − 1.00000i − 0.0322749i
$$961$$ −5.00000 −0.161290
$$962$$ 33.2111 1.07077
$$963$$ 0 0
$$964$$ − 6.42221i − 0.206845i
$$965$$ −7.81665 −0.251627
$$966$$ 6.42221 0.206631
$$967$$ 8.23886i 0.264944i 0.991187 + 0.132472i $$0.0422914\pi$$
−0.991187 + 0.132472i $$0.957709\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ − 21.2111i − 0.681399i
$$970$$ 1.39445i 0.0447731i
$$971$$ −53.0278 −1.70174 −0.850871 0.525375i $$-0.823925\pi$$
−0.850871 + 0.525375i $$0.823925\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 79.2666i 2.54117i
$$974$$ 0.972244 0.0311527
$$975$$ 3.60555 0.115470
$$976$$ −11.2111 −0.358859
$$977$$ − 18.8444i − 0.602886i −0.953484 0.301443i $$-0.902532\pi$$
0.953484 0.301443i $$-0.0974683\pi$$
$$978$$ 24.4222 0.780936
$$979$$ 0 0
$$980$$ − 14.2111i − 0.453957i
$$981$$ − 1.39445i − 0.0445213i
$$982$$ − 7.81665i − 0.249439i
$$983$$ 42.4222i 1.35306i 0.736416 + 0.676529i $$0.236517\pi$$
−0.736416 + 0.676529i $$0.763483\pi$$
$$984$$ −3.21110 −0.102366
$$985$$ 6.00000 0.191176
$$986$$ − 21.2111i − 0.675499i
$$987$$ 42.4222 1.35031
$$988$$ − 16.6056i − 0.528293i
$$989$$ −11.1556 −0.354727
$$990$$ 0 0
$$991$$ −22.4222 −0.712265 −0.356132 0.934436i $$-0.615905\pi$$
−0.356132 + 0.934436i $$0.615905\pi$$
$$992$$ −6.00000 −0.190500
$$993$$ − 16.6056i − 0.526961i
$$994$$ − 42.4222i − 1.34555i
$$995$$ − 22.4222i − 0.710832i
$$996$$ − 2.78890i − 0.0883696i
$$997$$ −16.4222 −0.520096 −0.260048 0.965596i $$-0.583738\pi$$
−0.260048 + 0.965596i $$0.583738\pi$$
$$998$$ −23.0278 −0.728931
$$999$$ − 9.21110i − 0.291426i
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 390.2.b.c.181.2 4
3.2 odd 2 1170.2.b.d.181.4 4
4.3 odd 2 3120.2.g.q.961.1 4
5.2 odd 4 1950.2.f.n.649.3 4
5.3 odd 4 1950.2.f.m.649.2 4
5.4 even 2 1950.2.b.k.1351.3 4
13.5 odd 4 5070.2.a.z.1.1 2
13.8 odd 4 5070.2.a.bf.1.2 2
13.12 even 2 inner 390.2.b.c.181.3 yes 4
39.38 odd 2 1170.2.b.d.181.1 4
52.51 odd 2 3120.2.g.q.961.4 4
65.12 odd 4 1950.2.f.m.649.4 4
65.38 odd 4 1950.2.f.n.649.1 4
65.64 even 2 1950.2.b.k.1351.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 1.1 even 1 trivial
390.2.b.c.181.3 yes 4 13.12 even 2 inner
1170.2.b.d.181.1 4 39.38 odd 2
1170.2.b.d.181.4 4 3.2 odd 2
1950.2.b.k.1351.2 4 65.64 even 2
1950.2.b.k.1351.3 4 5.4 even 2
1950.2.f.m.649.2 4 5.3 odd 4
1950.2.f.m.649.4 4 65.12 odd 4
1950.2.f.n.649.1 4 65.38 odd 4
1950.2.f.n.649.3 4 5.2 odd 4
3120.2.g.q.961.1 4 4.3 odd 2
3120.2.g.q.961.4 4 52.51 odd 2
5070.2.a.z.1.1 2 13.5 odd 4
5070.2.a.bf.1.2 2 13.8 odd 4