# Properties

 Label 1014.2.b.e.337.4 Level $1014$ Weight $2$ Character 1014.337 Analytic conductor $8.097$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1014.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.09683076496$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.4 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1014.337 Dual form 1014.2.b.e.337.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.73205i q^{5} +1.00000i q^{6} +2.73205i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.73205i q^{5} +1.00000i q^{6} +2.73205i q^{7} -1.00000i q^{8} +1.00000 q^{9} -3.73205 q^{10} -1.26795i q^{11} -1.00000 q^{12} -2.73205 q^{14} +3.73205i q^{15} +1.00000 q^{16} +5.73205 q^{17} +1.00000i q^{18} +4.73205i q^{19} -3.73205i q^{20} +2.73205i q^{21} +1.26795 q^{22} -4.19615 q^{23} -1.00000i q^{24} -8.92820 q^{25} +1.00000 q^{27} -2.73205i q^{28} -4.46410 q^{29} -3.73205 q^{30} +1.46410i q^{31} +1.00000i q^{32} -1.26795i q^{33} +5.73205i q^{34} -10.1962 q^{35} -1.00000 q^{36} -3.53590i q^{37} -4.73205 q^{38} +3.73205 q^{40} -9.39230i q^{41} -2.73205 q^{42} +9.66025 q^{43} +1.26795i q^{44} +3.73205i q^{45} -4.19615i q^{46} -2.19615i q^{47} +1.00000 q^{48} -0.464102 q^{49} -8.92820i q^{50} +5.73205 q^{51} -6.46410 q^{53} +1.00000i q^{54} +4.73205 q^{55} +2.73205 q^{56} +4.73205i q^{57} -4.46410i q^{58} -8.00000i q^{59} -3.73205i q^{60} -9.19615 q^{61} -1.46410 q^{62} +2.73205i q^{63} -1.00000 q^{64} +1.26795 q^{66} +13.1244i q^{67} -5.73205 q^{68} -4.19615 q^{69} -10.1962i q^{70} +4.73205i q^{71} -1.00000i q^{72} +6.26795i q^{73} +3.53590 q^{74} -8.92820 q^{75} -4.73205i q^{76} +3.46410 q^{77} -2.53590 q^{79} +3.73205i q^{80} +1.00000 q^{81} +9.39230 q^{82} -0.196152i q^{83} -2.73205i q^{84} +21.3923i q^{85} +9.66025i q^{86} -4.46410 q^{87} -1.26795 q^{88} +9.46410i q^{89} -3.73205 q^{90} +4.19615 q^{92} +1.46410i q^{93} +2.19615 q^{94} -17.6603 q^{95} +1.00000i q^{96} -6.00000i q^{97} -0.464102i q^{98} -1.26795i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 $$4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 8 q^{10} - 4 q^{12} - 4 q^{14} + 4 q^{16} + 16 q^{17} + 12 q^{22} + 4 q^{23} - 8 q^{25} + 4 q^{27} - 4 q^{29} - 8 q^{30} - 20 q^{35} - 4 q^{36} - 12 q^{38} + 8 q^{40} - 4 q^{42} + 4 q^{43} + 4 q^{48} + 12 q^{49} + 16 q^{51} - 12 q^{53} + 12 q^{55} + 4 q^{56} - 16 q^{61} + 8 q^{62} - 4 q^{64} + 12 q^{66} - 16 q^{68} + 4 q^{69} + 28 q^{74} - 8 q^{75} - 24 q^{79} + 4 q^{81} - 4 q^{82} - 4 q^{87} - 12 q^{88} - 8 q^{90} - 4 q^{92} - 12 q^{94} - 36 q^{95}+O(q^{100})$$ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 8 * q^10 - 4 * q^12 - 4 * q^14 + 4 * q^16 + 16 * q^17 + 12 * q^22 + 4 * q^23 - 8 * q^25 + 4 * q^27 - 4 * q^29 - 8 * q^30 - 20 * q^35 - 4 * q^36 - 12 * q^38 + 8 * q^40 - 4 * q^42 + 4 * q^43 + 4 * q^48 + 12 * q^49 + 16 * q^51 - 12 * q^53 + 12 * q^55 + 4 * q^56 - 16 * q^61 + 8 * q^62 - 4 * q^64 + 12 * q^66 - 16 * q^68 + 4 * q^69 + 28 * q^74 - 8 * q^75 - 24 * q^79 + 4 * q^81 - 4 * q^82 - 4 * q^87 - 12 * q^88 - 8 * q^90 - 4 * q^92 - 12 * q^94 - 36 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 3.73205i 1.66902i 0.550990 + 0.834512i $$0.314250\pi$$
−0.550990 + 0.834512i $$0.685750\pi$$
$$6$$ 1.00000i 0.408248i
$$7$$ 2.73205i 1.03262i 0.856402 + 0.516309i $$0.172694\pi$$
−0.856402 + 0.516309i $$0.827306\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −3.73205 −1.18018
$$11$$ − 1.26795i − 0.382301i −0.981561 0.191151i $$-0.938778\pi$$
0.981561 0.191151i $$-0.0612219\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −2.73205 −0.730171
$$15$$ 3.73205i 0.963611i
$$16$$ 1.00000 0.250000
$$17$$ 5.73205 1.39023 0.695113 0.718900i $$-0.255354\pi$$
0.695113 + 0.718900i $$0.255354\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 4.73205i 1.08561i 0.839860 + 0.542803i $$0.182637\pi$$
−0.839860 + 0.542803i $$0.817363\pi$$
$$20$$ − 3.73205i − 0.834512i
$$21$$ 2.73205i 0.596182i
$$22$$ 1.26795 0.270328
$$23$$ −4.19615 −0.874958 −0.437479 0.899229i $$-0.644129\pi$$
−0.437479 + 0.899229i $$0.644129\pi$$
$$24$$ − 1.00000i − 0.204124i
$$25$$ −8.92820 −1.78564
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ − 2.73205i − 0.516309i
$$29$$ −4.46410 −0.828963 −0.414481 0.910058i $$-0.636037\pi$$
−0.414481 + 0.910058i $$0.636037\pi$$
$$30$$ −3.73205 −0.681376
$$31$$ 1.46410i 0.262960i 0.991319 + 0.131480i $$0.0419730\pi$$
−0.991319 + 0.131480i $$0.958027\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 1.26795i − 0.220722i
$$34$$ 5.73205i 0.983039i
$$35$$ −10.1962 −1.72346
$$36$$ −1.00000 −0.166667
$$37$$ − 3.53590i − 0.581298i −0.956830 0.290649i $$-0.906129\pi$$
0.956830 0.290649i $$-0.0938712\pi$$
$$38$$ −4.73205 −0.767640
$$39$$ 0 0
$$40$$ 3.73205 0.590089
$$41$$ − 9.39230i − 1.46683i −0.679780 0.733416i $$-0.737925\pi$$
0.679780 0.733416i $$-0.262075\pi$$
$$42$$ −2.73205 −0.421565
$$43$$ 9.66025 1.47317 0.736587 0.676342i $$-0.236436\pi$$
0.736587 + 0.676342i $$0.236436\pi$$
$$44$$ 1.26795i 0.191151i
$$45$$ 3.73205i 0.556341i
$$46$$ − 4.19615i − 0.618689i
$$47$$ − 2.19615i − 0.320342i −0.987089 0.160171i $$-0.948795\pi$$
0.987089 0.160171i $$-0.0512045\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −0.464102 −0.0663002
$$50$$ − 8.92820i − 1.26264i
$$51$$ 5.73205 0.802648
$$52$$ 0 0
$$53$$ −6.46410 −0.887913 −0.443956 0.896048i $$-0.646425\pi$$
−0.443956 + 0.896048i $$0.646425\pi$$
$$54$$ 1.00000i 0.136083i
$$55$$ 4.73205 0.638070
$$56$$ 2.73205 0.365086
$$57$$ 4.73205i 0.626775i
$$58$$ − 4.46410i − 0.586165i
$$59$$ − 8.00000i − 1.04151i −0.853706 0.520756i $$-0.825650\pi$$
0.853706 0.520756i $$-0.174350\pi$$
$$60$$ − 3.73205i − 0.481806i
$$61$$ −9.19615 −1.17745 −0.588723 0.808335i $$-0.700369\pi$$
−0.588723 + 0.808335i $$0.700369\pi$$
$$62$$ −1.46410 −0.185941
$$63$$ 2.73205i 0.344206i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 1.26795 0.156074
$$67$$ 13.1244i 1.60340i 0.597730 + 0.801698i $$0.296070\pi$$
−0.597730 + 0.801698i $$0.703930\pi$$
$$68$$ −5.73205 −0.695113
$$69$$ −4.19615 −0.505157
$$70$$ − 10.1962i − 1.21867i
$$71$$ 4.73205i 0.561591i 0.959768 + 0.280796i $$0.0905983\pi$$
−0.959768 + 0.280796i $$0.909402\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 6.26795i 0.733608i 0.930298 + 0.366804i $$0.119548\pi$$
−0.930298 + 0.366804i $$0.880452\pi$$
$$74$$ 3.53590 0.411040
$$75$$ −8.92820 −1.03094
$$76$$ − 4.73205i − 0.542803i
$$77$$ 3.46410 0.394771
$$78$$ 0 0
$$79$$ −2.53590 −0.285311 −0.142655 0.989772i $$-0.545564\pi$$
−0.142655 + 0.989772i $$0.545564\pi$$
$$80$$ 3.73205i 0.417256i
$$81$$ 1.00000 0.111111
$$82$$ 9.39230 1.03721
$$83$$ − 0.196152i − 0.0215305i −0.999942 0.0107653i $$-0.996573\pi$$
0.999942 0.0107653i $$-0.00342676\pi$$
$$84$$ − 2.73205i − 0.298091i
$$85$$ 21.3923i 2.32032i
$$86$$ 9.66025i 1.04169i
$$87$$ −4.46410 −0.478602
$$88$$ −1.26795 −0.135164
$$89$$ 9.46410i 1.00319i 0.865102 + 0.501596i $$0.167254\pi$$
−0.865102 + 0.501596i $$0.832746\pi$$
$$90$$ −3.73205 −0.393393
$$91$$ 0 0
$$92$$ 4.19615 0.437479
$$93$$ 1.46410i 0.151820i
$$94$$ 2.19615 0.226516
$$95$$ −17.6603 −1.81190
$$96$$ 1.00000i 0.102062i
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ − 0.464102i − 0.0468813i
$$99$$ − 1.26795i − 0.127434i
$$100$$ 8.92820 0.892820
$$101$$ −1.92820 −0.191863 −0.0959317 0.995388i $$-0.530583\pi$$
−0.0959317 + 0.995388i $$0.530583\pi$$
$$102$$ 5.73205i 0.567558i
$$103$$ 15.2679 1.50440 0.752198 0.658937i $$-0.228994\pi$$
0.752198 + 0.658937i $$0.228994\pi$$
$$104$$ 0 0
$$105$$ −10.1962 −0.995043
$$106$$ − 6.46410i − 0.627849i
$$107$$ 10.1962 0.985699 0.492850 0.870114i $$-0.335955\pi$$
0.492850 + 0.870114i $$0.335955\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 1.46410i 0.140236i 0.997539 + 0.0701178i $$0.0223375\pi$$
−0.997539 + 0.0701178i $$0.977662\pi$$
$$110$$ 4.73205i 0.451183i
$$111$$ − 3.53590i − 0.335613i
$$112$$ 2.73205i 0.258155i
$$113$$ −1.33975 −0.126033 −0.0630163 0.998012i $$-0.520072\pi$$
−0.0630163 + 0.998012i $$0.520072\pi$$
$$114$$ −4.73205 −0.443197
$$115$$ − 15.6603i − 1.46033i
$$116$$ 4.46410 0.414481
$$117$$ 0 0
$$118$$ 8.00000 0.736460
$$119$$ 15.6603i 1.43557i
$$120$$ 3.73205 0.340688
$$121$$ 9.39230 0.853846
$$122$$ − 9.19615i − 0.832581i
$$123$$ − 9.39230i − 0.846876i
$$124$$ − 1.46410i − 0.131480i
$$125$$ − 14.6603i − 1.31125i
$$126$$ −2.73205 −0.243390
$$127$$ 9.85641 0.874615 0.437307 0.899312i $$-0.355932\pi$$
0.437307 + 0.899312i $$0.355932\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 9.66025 0.850538
$$130$$ 0 0
$$131$$ 6.53590 0.571044 0.285522 0.958372i $$-0.407833\pi$$
0.285522 + 0.958372i $$0.407833\pi$$
$$132$$ 1.26795i 0.110361i
$$133$$ −12.9282 −1.12102
$$134$$ −13.1244 −1.13377
$$135$$ 3.73205i 0.321204i
$$136$$ − 5.73205i − 0.491519i
$$137$$ − 11.9282i − 1.01910i −0.860442 0.509548i $$-0.829813\pi$$
0.860442 0.509548i $$-0.170187\pi$$
$$138$$ − 4.19615i − 0.357200i
$$139$$ 17.8564 1.51456 0.757280 0.653090i $$-0.226528\pi$$
0.757280 + 0.653090i $$0.226528\pi$$
$$140$$ 10.1962 0.861732
$$141$$ − 2.19615i − 0.184949i
$$142$$ −4.73205 −0.397105
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 16.6603i − 1.38356i
$$146$$ −6.26795 −0.518739
$$147$$ −0.464102 −0.0382785
$$148$$ 3.53590i 0.290649i
$$149$$ 13.1962i 1.08107i 0.841321 + 0.540535i $$0.181778\pi$$
−0.841321 + 0.540535i $$0.818222\pi$$
$$150$$ − 8.92820i − 0.728985i
$$151$$ − 6.73205i − 0.547847i −0.961752 0.273923i $$-0.911679\pi$$
0.961752 0.273923i $$-0.0883214\pi$$
$$152$$ 4.73205 0.383820
$$153$$ 5.73205 0.463409
$$154$$ 3.46410i 0.279145i
$$155$$ −5.46410 −0.438887
$$156$$ 0 0
$$157$$ 7.58846 0.605625 0.302812 0.953050i $$-0.402074\pi$$
0.302812 + 0.953050i $$0.402074\pi$$
$$158$$ − 2.53590i − 0.201745i
$$159$$ −6.46410 −0.512637
$$160$$ −3.73205 −0.295045
$$161$$ − 11.4641i − 0.903498i
$$162$$ 1.00000i 0.0785674i
$$163$$ 13.4641i 1.05459i 0.849682 + 0.527295i $$0.176794\pi$$
−0.849682 + 0.527295i $$0.823206\pi$$
$$164$$ 9.39230i 0.733416i
$$165$$ 4.73205 0.368390
$$166$$ 0.196152 0.0152244
$$167$$ 9.46410i 0.732354i 0.930545 + 0.366177i $$0.119334\pi$$
−0.930545 + 0.366177i $$0.880666\pi$$
$$168$$ 2.73205 0.210782
$$169$$ 0 0
$$170$$ −21.3923 −1.64071
$$171$$ 4.73205i 0.361869i
$$172$$ −9.66025 −0.736587
$$173$$ −4.39230 −0.333941 −0.166970 0.985962i $$-0.553398\pi$$
−0.166970 + 0.985962i $$0.553398\pi$$
$$174$$ − 4.46410i − 0.338423i
$$175$$ − 24.3923i − 1.84388i
$$176$$ − 1.26795i − 0.0955753i
$$177$$ − 8.00000i − 0.601317i
$$178$$ −9.46410 −0.709364
$$179$$ 16.0526 1.19982 0.599912 0.800066i $$-0.295202\pi$$
0.599912 + 0.800066i $$0.295202\pi$$
$$180$$ − 3.73205i − 0.278171i
$$181$$ −19.1962 −1.42684 −0.713419 0.700737i $$-0.752855\pi$$
−0.713419 + 0.700737i $$0.752855\pi$$
$$182$$ 0 0
$$183$$ −9.19615 −0.679799
$$184$$ 4.19615i 0.309344i
$$185$$ 13.1962 0.970200
$$186$$ −1.46410 −0.107353
$$187$$ − 7.26795i − 0.531485i
$$188$$ 2.19615i 0.160171i
$$189$$ 2.73205i 0.198727i
$$190$$ − 17.6603i − 1.28121i
$$191$$ −6.92820 −0.501307 −0.250654 0.968077i $$-0.580646\pi$$
−0.250654 + 0.968077i $$0.580646\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 11.7321i 0.844491i 0.906481 + 0.422246i $$0.138758\pi$$
−0.906481 + 0.422246i $$0.861242\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ 0.464102 0.0331501
$$197$$ − 17.8564i − 1.27222i −0.771600 0.636108i $$-0.780543\pi$$
0.771600 0.636108i $$-0.219457\pi$$
$$198$$ 1.26795 0.0901092
$$199$$ 14.1962 1.00634 0.503169 0.864188i $$-0.332167\pi$$
0.503169 + 0.864188i $$0.332167\pi$$
$$200$$ 8.92820i 0.631319i
$$201$$ 13.1244i 0.925721i
$$202$$ − 1.92820i − 0.135668i
$$203$$ − 12.1962i − 0.856002i
$$204$$ −5.73205 −0.401324
$$205$$ 35.0526 2.44818
$$206$$ 15.2679i 1.06377i
$$207$$ −4.19615 −0.291653
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ − 10.1962i − 0.703601i
$$211$$ 16.3923 1.12849 0.564246 0.825606i $$-0.309167\pi$$
0.564246 + 0.825606i $$0.309167\pi$$
$$212$$ 6.46410 0.443956
$$213$$ 4.73205i 0.324235i
$$214$$ 10.1962i 0.696995i
$$215$$ 36.0526i 2.45876i
$$216$$ − 1.00000i − 0.0680414i
$$217$$ −4.00000 −0.271538
$$218$$ −1.46410 −0.0991615
$$219$$ 6.26795i 0.423549i
$$220$$ −4.73205 −0.319035
$$221$$ 0 0
$$222$$ 3.53590 0.237314
$$223$$ − 26.9282i − 1.80325i −0.432523 0.901623i $$-0.642377\pi$$
0.432523 0.901623i $$-0.357623\pi$$
$$224$$ −2.73205 −0.182543
$$225$$ −8.92820 −0.595214
$$226$$ − 1.33975i − 0.0891186i
$$227$$ − 12.1962i − 0.809487i −0.914430 0.404744i $$-0.867361\pi$$
0.914430 0.404744i $$-0.132639\pi$$
$$228$$ − 4.73205i − 0.313388i
$$229$$ 11.8564i 0.783493i 0.920073 + 0.391747i $$0.128129\pi$$
−0.920073 + 0.391747i $$0.871871\pi$$
$$230$$ 15.6603 1.03261
$$231$$ 3.46410 0.227921
$$232$$ 4.46410i 0.293083i
$$233$$ 7.85641 0.514690 0.257345 0.966320i $$-0.417152\pi$$
0.257345 + 0.966320i $$0.417152\pi$$
$$234$$ 0 0
$$235$$ 8.19615 0.534658
$$236$$ 8.00000i 0.520756i
$$237$$ −2.53590 −0.164724
$$238$$ −15.6603 −1.01510
$$239$$ 7.66025i 0.495501i 0.968824 + 0.247750i $$0.0796913\pi$$
−0.968824 + 0.247750i $$0.920309\pi$$
$$240$$ 3.73205i 0.240903i
$$241$$ − 13.5885i − 0.875309i −0.899143 0.437655i $$-0.855809\pi$$
0.899143 0.437655i $$-0.144191\pi$$
$$242$$ 9.39230i 0.603760i
$$243$$ 1.00000 0.0641500
$$244$$ 9.19615 0.588723
$$245$$ − 1.73205i − 0.110657i
$$246$$ 9.39230 0.598831
$$247$$ 0 0
$$248$$ 1.46410 0.0929705
$$249$$ − 0.196152i − 0.0124307i
$$250$$ 14.6603 0.927196
$$251$$ −13.4641 −0.849847 −0.424923 0.905229i $$-0.639699\pi$$
−0.424923 + 0.905229i $$0.639699\pi$$
$$252$$ − 2.73205i − 0.172103i
$$253$$ 5.32051i 0.334497i
$$254$$ 9.85641i 0.618446i
$$255$$ 21.3923i 1.33964i
$$256$$ 1.00000 0.0625000
$$257$$ −9.33975 −0.582597 −0.291299 0.956632i $$-0.594087\pi$$
−0.291299 + 0.956632i $$0.594087\pi$$
$$258$$ 9.66025i 0.601421i
$$259$$ 9.66025 0.600259
$$260$$ 0 0
$$261$$ −4.46410 −0.276321
$$262$$ 6.53590i 0.403789i
$$263$$ −10.0526 −0.619867 −0.309934 0.950758i $$-0.600307\pi$$
−0.309934 + 0.950758i $$0.600307\pi$$
$$264$$ −1.26795 −0.0780369
$$265$$ − 24.1244i − 1.48195i
$$266$$ − 12.9282i − 0.792679i
$$267$$ 9.46410i 0.579194i
$$268$$ − 13.1244i − 0.801698i
$$269$$ 5.46410 0.333152 0.166576 0.986029i $$-0.446729\pi$$
0.166576 + 0.986029i $$0.446729\pi$$
$$270$$ −3.73205 −0.227125
$$271$$ 21.8564i 1.32768i 0.747874 + 0.663841i $$0.231075\pi$$
−0.747874 + 0.663841i $$0.768925\pi$$
$$272$$ 5.73205 0.347557
$$273$$ 0 0
$$274$$ 11.9282 0.720609
$$275$$ 11.3205i 0.682652i
$$276$$ 4.19615 0.252579
$$277$$ 5.73205 0.344406 0.172203 0.985062i $$-0.444912\pi$$
0.172203 + 0.985062i $$0.444912\pi$$
$$278$$ 17.8564i 1.07096i
$$279$$ 1.46410i 0.0876535i
$$280$$ 10.1962i 0.609337i
$$281$$ − 12.3205i − 0.734980i −0.930027 0.367490i $$-0.880217\pi$$
0.930027 0.367490i $$-0.119783\pi$$
$$282$$ 2.19615 0.130779
$$283$$ −25.6603 −1.52534 −0.762672 0.646786i $$-0.776113\pi$$
−0.762672 + 0.646786i $$0.776113\pi$$
$$284$$ − 4.73205i − 0.280796i
$$285$$ −17.6603 −1.04610
$$286$$ 0 0
$$287$$ 25.6603 1.51468
$$288$$ 1.00000i 0.0589256i
$$289$$ 15.8564 0.932730
$$290$$ 16.6603 0.978324
$$291$$ − 6.00000i − 0.351726i
$$292$$ − 6.26795i − 0.366804i
$$293$$ − 30.5167i − 1.78280i −0.453215 0.891401i $$-0.649723\pi$$
0.453215 0.891401i $$-0.350277\pi$$
$$294$$ − 0.464102i − 0.0270670i
$$295$$ 29.8564 1.73831
$$296$$ −3.53590 −0.205520
$$297$$ − 1.26795i − 0.0735739i
$$298$$ −13.1962 −0.764433
$$299$$ 0 0
$$300$$ 8.92820 0.515470
$$301$$ 26.3923i 1.52123i
$$302$$ 6.73205 0.387386
$$303$$ −1.92820 −0.110772
$$304$$ 4.73205i 0.271402i
$$305$$ − 34.3205i − 1.96519i
$$306$$ 5.73205i 0.327680i
$$307$$ 22.5885i 1.28919i 0.764524 + 0.644596i $$0.222974\pi$$
−0.764524 + 0.644596i $$0.777026\pi$$
$$308$$ −3.46410 −0.197386
$$309$$ 15.2679 0.868563
$$310$$ − 5.46410i − 0.310340i
$$311$$ 1.66025 0.0941444 0.0470722 0.998891i $$-0.485011\pi$$
0.0470722 + 0.998891i $$0.485011\pi$$
$$312$$ 0 0
$$313$$ 6.53590 0.369431 0.184715 0.982792i $$-0.440864\pi$$
0.184715 + 0.982792i $$0.440864\pi$$
$$314$$ 7.58846i 0.428241i
$$315$$ −10.1962 −0.574488
$$316$$ 2.53590 0.142655
$$317$$ 20.6603i 1.16040i 0.814476 + 0.580198i $$0.197025\pi$$
−0.814476 + 0.580198i $$0.802975\pi$$
$$318$$ − 6.46410i − 0.362489i
$$319$$ 5.66025i 0.316913i
$$320$$ − 3.73205i − 0.208628i
$$321$$ 10.1962 0.569094
$$322$$ 11.4641 0.638869
$$323$$ 27.1244i 1.50924i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −13.4641 −0.745708
$$327$$ 1.46410i 0.0809650i
$$328$$ −9.39230 −0.518603
$$329$$ 6.00000 0.330791
$$330$$ 4.73205i 0.260491i
$$331$$ 20.0000i 1.09930i 0.835395 + 0.549650i $$0.185239\pi$$
−0.835395 + 0.549650i $$0.814761\pi$$
$$332$$ 0.196152i 0.0107653i
$$333$$ − 3.53590i − 0.193766i
$$334$$ −9.46410 −0.517853
$$335$$ −48.9808 −2.67610
$$336$$ 2.73205i 0.149046i
$$337$$ 20.8564 1.13612 0.568060 0.822987i $$-0.307694\pi$$
0.568060 + 0.822987i $$0.307694\pi$$
$$338$$ 0 0
$$339$$ −1.33975 −0.0727650
$$340$$ − 21.3923i − 1.16016i
$$341$$ 1.85641 0.100530
$$342$$ −4.73205 −0.255880
$$343$$ 17.8564i 0.964155i
$$344$$ − 9.66025i − 0.520846i
$$345$$ − 15.6603i − 0.843120i
$$346$$ − 4.39230i − 0.236132i
$$347$$ 33.1244 1.77821 0.889104 0.457705i $$-0.151328\pi$$
0.889104 + 0.457705i $$0.151328\pi$$
$$348$$ 4.46410 0.239301
$$349$$ − 15.3205i − 0.820088i −0.912066 0.410044i $$-0.865513\pi$$
0.912066 0.410044i $$-0.134487\pi$$
$$350$$ 24.3923 1.30382
$$351$$ 0 0
$$352$$ 1.26795 0.0675819
$$353$$ − 21.7846i − 1.15948i −0.814802 0.579739i $$-0.803155\pi$$
0.814802 0.579739i $$-0.196845\pi$$
$$354$$ 8.00000 0.425195
$$355$$ −17.6603 −0.937309
$$356$$ − 9.46410i − 0.501596i
$$357$$ 15.6603i 0.828829i
$$358$$ 16.0526i 0.848404i
$$359$$ 1.12436i 0.0593412i 0.999560 + 0.0296706i $$0.00944584\pi$$
−0.999560 + 0.0296706i $$0.990554\pi$$
$$360$$ 3.73205 0.196696
$$361$$ −3.39230 −0.178542
$$362$$ − 19.1962i − 1.00893i
$$363$$ 9.39230 0.492968
$$364$$ 0 0
$$365$$ −23.3923 −1.22441
$$366$$ − 9.19615i − 0.480691i
$$367$$ −11.2679 −0.588182 −0.294091 0.955777i $$-0.595017\pi$$
−0.294091 + 0.955777i $$0.595017\pi$$
$$368$$ −4.19615 −0.218740
$$369$$ − 9.39230i − 0.488944i
$$370$$ 13.1962i 0.686035i
$$371$$ − 17.6603i − 0.916875i
$$372$$ − 1.46410i − 0.0759101i
$$373$$ −13.7321 −0.711019 −0.355509 0.934673i $$-0.615693\pi$$
−0.355509 + 0.934673i $$0.615693\pi$$
$$374$$ 7.26795 0.375817
$$375$$ − 14.6603i − 0.757052i
$$376$$ −2.19615 −0.113258
$$377$$ 0 0
$$378$$ −2.73205 −0.140522
$$379$$ − 5.46410i − 0.280672i −0.990104 0.140336i $$-0.955182\pi$$
0.990104 0.140336i $$-0.0448183\pi$$
$$380$$ 17.6603 0.905952
$$381$$ 9.85641 0.504959
$$382$$ − 6.92820i − 0.354478i
$$383$$ − 1.46410i − 0.0748121i −0.999300 0.0374060i $$-0.988091\pi$$
0.999300 0.0374060i $$-0.0119095\pi$$
$$384$$ − 1.00000i − 0.0510310i
$$385$$ 12.9282i 0.658882i
$$386$$ −11.7321 −0.597146
$$387$$ 9.66025 0.491058
$$388$$ 6.00000i 0.304604i
$$389$$ −11.7846 −0.597503 −0.298752 0.954331i $$-0.596570\pi$$
−0.298752 + 0.954331i $$0.596570\pi$$
$$390$$ 0 0
$$391$$ −24.0526 −1.21639
$$392$$ 0.464102i 0.0234407i
$$393$$ 6.53590 0.329692
$$394$$ 17.8564 0.899593
$$395$$ − 9.46410i − 0.476191i
$$396$$ 1.26795i 0.0637168i
$$397$$ − 20.3923i − 1.02346i −0.859146 0.511730i $$-0.829005\pi$$
0.859146 0.511730i $$-0.170995\pi$$
$$398$$ 14.1962i 0.711589i
$$399$$ −12.9282 −0.647220
$$400$$ −8.92820 −0.446410
$$401$$ − 8.07180i − 0.403086i −0.979480 0.201543i $$-0.935404\pi$$
0.979480 0.201543i $$-0.0645956\pi$$
$$402$$ −13.1244 −0.654583
$$403$$ 0 0
$$404$$ 1.92820 0.0959317
$$405$$ 3.73205i 0.185447i
$$406$$ 12.1962 0.605285
$$407$$ −4.48334 −0.222231
$$408$$ − 5.73205i − 0.283779i
$$409$$ 17.7321i 0.876793i 0.898782 + 0.438397i $$0.144454\pi$$
−0.898782 + 0.438397i $$0.855546\pi$$
$$410$$ 35.0526i 1.73112i
$$411$$ − 11.9282i − 0.588375i
$$412$$ −15.2679 −0.752198
$$413$$ 21.8564 1.07548
$$414$$ − 4.19615i − 0.206230i
$$415$$ 0.732051 0.0359350
$$416$$ 0 0
$$417$$ 17.8564 0.874432
$$418$$ 6.00000i 0.293470i
$$419$$ −17.4641 −0.853177 −0.426589 0.904446i $$-0.640285\pi$$
−0.426589 + 0.904446i $$0.640285\pi$$
$$420$$ 10.1962 0.497521
$$421$$ − 22.7128i − 1.10695i −0.832864 0.553477i $$-0.813301\pi$$
0.832864 0.553477i $$-0.186699\pi$$
$$422$$ 16.3923i 0.797965i
$$423$$ − 2.19615i − 0.106781i
$$424$$ 6.46410i 0.313925i
$$425$$ −51.1769 −2.48244
$$426$$ −4.73205 −0.229269
$$427$$ − 25.1244i − 1.21585i
$$428$$ −10.1962 −0.492850
$$429$$ 0 0
$$430$$ −36.0526 −1.73861
$$431$$ 13.1244i 0.632178i 0.948730 + 0.316089i $$0.102370\pi$$
−0.948730 + 0.316089i $$0.897630\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −12.8564 −0.617839 −0.308920 0.951088i $$-0.599967\pi$$
−0.308920 + 0.951088i $$0.599967\pi$$
$$434$$ − 4.00000i − 0.192006i
$$435$$ − 16.6603i − 0.798798i
$$436$$ − 1.46410i − 0.0701178i
$$437$$ − 19.8564i − 0.949861i
$$438$$ −6.26795 −0.299494
$$439$$ 0.339746 0.0162152 0.00810760 0.999967i $$-0.497419\pi$$
0.00810760 + 0.999967i $$0.497419\pi$$
$$440$$ − 4.73205i − 0.225592i
$$441$$ −0.464102 −0.0221001
$$442$$ 0 0
$$443$$ 15.6077 0.741544 0.370772 0.928724i $$-0.379093\pi$$
0.370772 + 0.928724i $$0.379093\pi$$
$$444$$ 3.53590i 0.167806i
$$445$$ −35.3205 −1.67435
$$446$$ 26.9282 1.27509
$$447$$ 13.1962i 0.624157i
$$448$$ − 2.73205i − 0.129077i
$$449$$ − 11.3205i − 0.534248i −0.963662 0.267124i $$-0.913927\pi$$
0.963662 0.267124i $$-0.0860733\pi$$
$$450$$ − 8.92820i − 0.420880i
$$451$$ −11.9090 −0.560771
$$452$$ 1.33975 0.0630163
$$453$$ − 6.73205i − 0.316299i
$$454$$ 12.1962 0.572394
$$455$$ 0 0
$$456$$ 4.73205 0.221599
$$457$$ 1.33975i 0.0626707i 0.999509 + 0.0313353i $$0.00997598\pi$$
−0.999509 + 0.0313353i $$0.990024\pi$$
$$458$$ −11.8564 −0.554013
$$459$$ 5.73205 0.267549
$$460$$ 15.6603i 0.730163i
$$461$$ 22.2679i 1.03712i 0.855041 + 0.518561i $$0.173532\pi$$
−0.855041 + 0.518561i $$0.826468\pi$$
$$462$$ 3.46410i 0.161165i
$$463$$ − 10.0526i − 0.467182i −0.972335 0.233591i $$-0.924952\pi$$
0.972335 0.233591i $$-0.0750477\pi$$
$$464$$ −4.46410 −0.207241
$$465$$ −5.46410 −0.253392
$$466$$ 7.85641i 0.363941i
$$467$$ −18.5885 −0.860171 −0.430086 0.902788i $$-0.641517\pi$$
−0.430086 + 0.902788i $$0.641517\pi$$
$$468$$ 0 0
$$469$$ −35.8564 −1.65570
$$470$$ 8.19615i 0.378060i
$$471$$ 7.58846 0.349658
$$472$$ −8.00000 −0.368230
$$473$$ − 12.2487i − 0.563196i
$$474$$ − 2.53590i − 0.116478i
$$475$$ − 42.2487i − 1.93850i
$$476$$ − 15.6603i − 0.717787i
$$477$$ −6.46410 −0.295971
$$478$$ −7.66025 −0.350372
$$479$$ − 33.4641i − 1.52901i −0.644616 0.764507i $$-0.722983\pi$$
0.644616 0.764507i $$-0.277017\pi$$
$$480$$ −3.73205 −0.170344
$$481$$ 0 0
$$482$$ 13.5885 0.618937
$$483$$ − 11.4641i − 0.521635i
$$484$$ −9.39230 −0.426923
$$485$$ 22.3923 1.01678
$$486$$ 1.00000i 0.0453609i
$$487$$ − 3.12436i − 0.141578i −0.997491 0.0707890i $$-0.977448\pi$$
0.997491 0.0707890i $$-0.0225517\pi$$
$$488$$ 9.19615i 0.416290i
$$489$$ 13.4641i 0.608868i
$$490$$ 1.73205 0.0782461
$$491$$ −8.73205 −0.394072 −0.197036 0.980396i $$-0.563132\pi$$
−0.197036 + 0.980396i $$0.563132\pi$$
$$492$$ 9.39230i 0.423438i
$$493$$ −25.5885 −1.15245
$$494$$ 0 0
$$495$$ 4.73205 0.212690
$$496$$ 1.46410i 0.0657401i
$$497$$ −12.9282 −0.579909
$$498$$ 0.196152 0.00878980
$$499$$ 32.0000i 1.43252i 0.697835 + 0.716258i $$0.254147\pi$$
−0.697835 + 0.716258i $$0.745853\pi$$
$$500$$ 14.6603i 0.655626i
$$501$$ 9.46410i 0.422825i
$$502$$ − 13.4641i − 0.600932i
$$503$$ 40.9808 1.82724 0.913621 0.406567i $$-0.133274\pi$$
0.913621 + 0.406567i $$0.133274\pi$$
$$504$$ 2.73205 0.121695
$$505$$ − 7.19615i − 0.320225i
$$506$$ −5.32051 −0.236525
$$507$$ 0 0
$$508$$ −9.85641 −0.437307
$$509$$ 13.7321i 0.608662i 0.952566 + 0.304331i $$0.0984330\pi$$
−0.952566 + 0.304331i $$0.901567\pi$$
$$510$$ −21.3923 −0.947267
$$511$$ −17.1244 −0.757537
$$512$$ 1.00000i 0.0441942i
$$513$$ 4.73205i 0.208925i
$$514$$ − 9.33975i − 0.411959i
$$515$$ 56.9808i 2.51087i
$$516$$ −9.66025 −0.425269
$$517$$ −2.78461 −0.122467
$$518$$ 9.66025i 0.424447i
$$519$$ −4.39230 −0.192801
$$520$$ 0 0
$$521$$ 41.4449 1.81573 0.907866 0.419260i $$-0.137710\pi$$
0.907866 + 0.419260i $$0.137710\pi$$
$$522$$ − 4.46410i − 0.195388i
$$523$$ −22.4449 −0.981445 −0.490723 0.871316i $$-0.663267\pi$$
−0.490723 + 0.871316i $$0.663267\pi$$
$$524$$ −6.53590 −0.285522
$$525$$ − 24.3923i − 1.06457i
$$526$$ − 10.0526i − 0.438312i
$$527$$ 8.39230i 0.365575i
$$528$$ − 1.26795i − 0.0551804i
$$529$$ −5.39230 −0.234448
$$530$$ 24.1244 1.04790
$$531$$ − 8.00000i − 0.347170i
$$532$$ 12.9282 0.560509
$$533$$ 0 0
$$534$$ −9.46410 −0.409552
$$535$$ 38.0526i 1.64516i
$$536$$ 13.1244 0.566886
$$537$$ 16.0526 0.692719
$$538$$ 5.46410i 0.235574i
$$539$$ 0.588457i 0.0253466i
$$540$$ − 3.73205i − 0.160602i
$$541$$ − 5.67949i − 0.244180i −0.992519 0.122090i $$-0.961040\pi$$
0.992519 0.122090i $$-0.0389597\pi$$
$$542$$ −21.8564 −0.938813
$$543$$ −19.1962 −0.823786
$$544$$ 5.73205i 0.245760i
$$545$$ −5.46410 −0.234056
$$546$$ 0 0
$$547$$ −4.19615 −0.179415 −0.0897073 0.995968i $$-0.528593\pi$$
−0.0897073 + 0.995968i $$0.528593\pi$$
$$548$$ 11.9282i 0.509548i
$$549$$ −9.19615 −0.392482
$$550$$ −11.3205 −0.482708
$$551$$ − 21.1244i − 0.899928i
$$552$$ 4.19615i 0.178600i
$$553$$ − 6.92820i − 0.294617i
$$554$$ 5.73205i 0.243532i
$$555$$ 13.1962 0.560145
$$556$$ −17.8564 −0.757280
$$557$$ − 42.3731i − 1.79540i −0.440603 0.897702i $$-0.645235\pi$$
0.440603 0.897702i $$-0.354765\pi$$
$$558$$ −1.46410 −0.0619804
$$559$$ 0 0
$$560$$ −10.1962 −0.430866
$$561$$ − 7.26795i − 0.306853i
$$562$$ 12.3205 0.519709
$$563$$ −34.9282 −1.47205 −0.736024 0.676955i $$-0.763299\pi$$
−0.736024 + 0.676955i $$0.763299\pi$$
$$564$$ 2.19615i 0.0924747i
$$565$$ − 5.00000i − 0.210352i
$$566$$ − 25.6603i − 1.07858i
$$567$$ 2.73205i 0.114735i
$$568$$ 4.73205 0.198552
$$569$$ −30.6410 −1.28454 −0.642269 0.766479i $$-0.722007\pi$$
−0.642269 + 0.766479i $$0.722007\pi$$
$$570$$ − 17.6603i − 0.739707i
$$571$$ −14.0526 −0.588081 −0.294041 0.955793i $$-0.595000\pi$$
−0.294041 + 0.955793i $$0.595000\pi$$
$$572$$ 0 0
$$573$$ −6.92820 −0.289430
$$574$$ 25.6603i 1.07104i
$$575$$ 37.4641 1.56236
$$576$$ −1.00000 −0.0416667
$$577$$ 3.73205i 0.155367i 0.996978 + 0.0776837i $$0.0247524\pi$$
−0.996978 + 0.0776837i $$0.975248\pi$$
$$578$$ 15.8564i 0.659540i
$$579$$ 11.7321i 0.487567i
$$580$$ 16.6603i 0.691779i
$$581$$ 0.535898 0.0222328
$$582$$ 6.00000 0.248708
$$583$$ 8.19615i 0.339450i
$$584$$ 6.26795 0.259370
$$585$$ 0 0
$$586$$ 30.5167 1.26063
$$587$$ 16.0000i 0.660391i 0.943913 + 0.330195i $$0.107115\pi$$
−0.943913 + 0.330195i $$0.892885\pi$$
$$588$$ 0.464102 0.0191392
$$589$$ −6.92820 −0.285472
$$590$$ 29.8564i 1.22917i
$$591$$ − 17.8564i − 0.734514i
$$592$$ − 3.53590i − 0.145325i
$$593$$ 9.14359i 0.375482i 0.982219 + 0.187741i $$0.0601166\pi$$
−0.982219 + 0.187741i $$0.939883\pi$$
$$594$$ 1.26795 0.0520246
$$595$$ −58.4449 −2.39601
$$596$$ − 13.1962i − 0.540535i
$$597$$ 14.1962 0.581010
$$598$$ 0 0
$$599$$ −2.53590 −0.103614 −0.0518070 0.998657i $$-0.516498\pi$$
−0.0518070 + 0.998657i $$0.516498\pi$$
$$600$$ 8.92820i 0.364492i
$$601$$ 7.92820 0.323398 0.161699 0.986840i $$-0.448303\pi$$
0.161699 + 0.986840i $$0.448303\pi$$
$$602$$ −26.3923 −1.07567
$$603$$ 13.1244i 0.534465i
$$604$$ 6.73205i 0.273923i
$$605$$ 35.0526i 1.42509i
$$606$$ − 1.92820i − 0.0783279i
$$607$$ −40.7846 −1.65540 −0.827698 0.561174i $$-0.810350\pi$$
−0.827698 + 0.561174i $$0.810350\pi$$
$$608$$ −4.73205 −0.191910
$$609$$ − 12.1962i − 0.494213i
$$610$$ 34.3205 1.38960
$$611$$ 0 0
$$612$$ −5.73205 −0.231704
$$613$$ − 9.39230i − 0.379352i −0.981847 0.189676i $$-0.939256\pi$$
0.981847 0.189676i $$-0.0607437\pi$$
$$614$$ −22.5885 −0.911596
$$615$$ 35.0526 1.41346
$$616$$ − 3.46410i − 0.139573i
$$617$$ − 13.2487i − 0.533373i −0.963783 0.266687i $$-0.914071\pi$$
0.963783 0.266687i $$-0.0859288\pi$$
$$618$$ 15.2679i 0.614167i
$$619$$ − 17.4641i − 0.701942i −0.936386 0.350971i $$-0.885852\pi$$
0.936386 0.350971i $$-0.114148\pi$$
$$620$$ 5.46410 0.219444
$$621$$ −4.19615 −0.168386
$$622$$ 1.66025i 0.0665701i
$$623$$ −25.8564 −1.03592
$$624$$ 0 0
$$625$$ 10.0718 0.402872
$$626$$ 6.53590i 0.261227i
$$627$$ 6.00000 0.239617
$$628$$ −7.58846 −0.302812
$$629$$ − 20.2679i − 0.808136i
$$630$$ − 10.1962i − 0.406224i
$$631$$ 7.71281i 0.307042i 0.988145 + 0.153521i $$0.0490613\pi$$
−0.988145 + 0.153521i $$0.950939\pi$$
$$632$$ 2.53590i 0.100873i
$$633$$ 16.3923 0.651536
$$634$$ −20.6603 −0.820524
$$635$$ 36.7846i 1.45975i
$$636$$ 6.46410 0.256318
$$637$$ 0 0
$$638$$ −5.66025 −0.224092
$$639$$ 4.73205i 0.187197i
$$640$$ 3.73205 0.147522
$$641$$ 25.9808 1.02618 0.513089 0.858335i $$-0.328501\pi$$
0.513089 + 0.858335i $$0.328501\pi$$
$$642$$ 10.1962i 0.402410i
$$643$$ − 13.8564i − 0.546443i −0.961951 0.273222i $$-0.911911\pi$$
0.961951 0.273222i $$-0.0880892\pi$$
$$644$$ 11.4641i 0.451749i
$$645$$ 36.0526i 1.41957i
$$646$$ −27.1244 −1.06719
$$647$$ 22.2487 0.874687 0.437344 0.899295i $$-0.355919\pi$$
0.437344 + 0.899295i $$0.355919\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −10.1436 −0.398171
$$650$$ 0 0
$$651$$ −4.00000 −0.156772
$$652$$ − 13.4641i − 0.527295i
$$653$$ −17.4641 −0.683423 −0.341712 0.939805i $$-0.611007\pi$$
−0.341712 + 0.939805i $$0.611007\pi$$
$$654$$ −1.46410 −0.0572509
$$655$$ 24.3923i 0.953086i
$$656$$ − 9.39230i − 0.366708i
$$657$$ 6.26795i 0.244536i
$$658$$ 6.00000i 0.233904i
$$659$$ −10.2487 −0.399233 −0.199617 0.979874i $$-0.563970\pi$$
−0.199617 + 0.979874i $$0.563970\pi$$
$$660$$ −4.73205 −0.184195
$$661$$ − 11.3923i − 0.443109i −0.975148 0.221555i $$-0.928887\pi$$
0.975148 0.221555i $$-0.0711131\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ −0.196152 −0.00761219
$$665$$ − 48.2487i − 1.87100i
$$666$$ 3.53590 0.137013
$$667$$ 18.7321 0.725308
$$668$$ − 9.46410i − 0.366177i
$$669$$ − 26.9282i − 1.04110i
$$670$$ − 48.9808i − 1.89229i
$$671$$ 11.6603i 0.450139i
$$672$$ −2.73205 −0.105391
$$673$$ −27.9282 −1.07655 −0.538277 0.842768i $$-0.680924\pi$$
−0.538277 + 0.842768i $$0.680924\pi$$
$$674$$ 20.8564i 0.803359i
$$675$$ −8.92820 −0.343647
$$676$$ 0 0
$$677$$ −45.4641 −1.74733 −0.873664 0.486530i $$-0.838262\pi$$
−0.873664 + 0.486530i $$0.838262\pi$$
$$678$$ − 1.33975i − 0.0514526i
$$679$$ 16.3923 0.629079
$$680$$ 21.3923 0.820357
$$681$$ − 12.1962i − 0.467358i
$$682$$ 1.85641i 0.0710855i
$$683$$ 10.1436i 0.388134i 0.980988 + 0.194067i $$0.0621679\pi$$
−0.980988 + 0.194067i $$0.937832\pi$$
$$684$$ − 4.73205i − 0.180934i
$$685$$ 44.5167 1.70089
$$686$$ −17.8564 −0.681761
$$687$$ 11.8564i 0.452350i
$$688$$ 9.66025 0.368294
$$689$$ 0 0
$$690$$ 15.6603 0.596176
$$691$$ 43.6603i 1.66091i 0.557082 + 0.830457i $$0.311921\pi$$
−0.557082 + 0.830457i $$0.688079\pi$$
$$692$$ 4.39230 0.166970
$$693$$ 3.46410 0.131590
$$694$$ 33.1244i 1.25738i
$$695$$ 66.6410i 2.52784i
$$696$$ 4.46410i 0.169211i
$$697$$ − 53.8372i − 2.03923i
$$698$$ 15.3205 0.579890
$$699$$ 7.85641 0.297157
$$700$$ 24.3923i 0.921942i
$$701$$ 3.32051 0.125414 0.0627069 0.998032i $$-0.480027\pi$$
0.0627069 + 0.998032i $$0.480027\pi$$
$$702$$ 0 0
$$703$$ 16.7321 0.631061
$$704$$ 1.26795i 0.0477876i
$$705$$ 8.19615 0.308685
$$706$$ 21.7846 0.819875
$$707$$ − 5.26795i − 0.198122i
$$708$$ 8.00000i 0.300658i
$$709$$ − 13.1436i − 0.493618i −0.969064 0.246809i $$-0.920618\pi$$
0.969064 0.246809i $$-0.0793820\pi$$
$$710$$ − 17.6603i − 0.662778i
$$711$$ −2.53590 −0.0951036
$$712$$ 9.46410 0.354682
$$713$$ − 6.14359i − 0.230079i
$$714$$ −15.6603 −0.586070
$$715$$ 0 0
$$716$$ −16.0526 −0.599912
$$717$$ 7.66025i 0.286077i
$$718$$ −1.12436 −0.0419606
$$719$$ 29.4641 1.09883 0.549413 0.835551i $$-0.314851\pi$$
0.549413 + 0.835551i $$0.314851\pi$$
$$720$$ 3.73205i 0.139085i
$$721$$ 41.7128i 1.55347i
$$722$$ − 3.39230i − 0.126249i
$$723$$ − 13.5885i − 0.505360i
$$724$$ 19.1962 0.713419
$$725$$ 39.8564 1.48023
$$726$$ 9.39230i 0.348581i
$$727$$ 30.9808 1.14901 0.574506 0.818500i $$-0.305194\pi$$
0.574506 + 0.818500i $$0.305194\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ − 23.3923i − 0.865788i
$$731$$ 55.3731 2.04805
$$732$$ 9.19615 0.339900
$$733$$ 19.0000i 0.701781i 0.936416 + 0.350891i $$0.114121\pi$$
−0.936416 + 0.350891i $$0.885879\pi$$
$$734$$ − 11.2679i − 0.415908i
$$735$$ − 1.73205i − 0.0638877i
$$736$$ − 4.19615i − 0.154672i
$$737$$ 16.6410 0.612980
$$738$$ 9.39230 0.345736
$$739$$ − 2.92820i − 0.107716i −0.998549 0.0538578i $$-0.982848\pi$$
0.998549 0.0538578i $$-0.0171518\pi$$
$$740$$ −13.1962 −0.485100
$$741$$ 0 0
$$742$$ 17.6603 0.648328
$$743$$ − 48.3923i − 1.77534i −0.460479 0.887671i $$-0.652322\pi$$
0.460479 0.887671i $$-0.347678\pi$$
$$744$$ 1.46410 0.0536766
$$745$$ −49.2487 −1.80433
$$746$$ − 13.7321i − 0.502766i
$$747$$ − 0.196152i − 0.00717684i
$$748$$ 7.26795i 0.265743i
$$749$$ 27.8564i 1.01785i
$$750$$ 14.6603 0.535317
$$751$$ 49.9090 1.82120 0.910602 0.413284i $$-0.135618\pi$$
0.910602 + 0.413284i $$0.135618\pi$$
$$752$$ − 2.19615i − 0.0800854i
$$753$$ −13.4641 −0.490659
$$754$$ 0 0
$$755$$ 25.1244 0.914369
$$756$$ − 2.73205i − 0.0993637i
$$757$$ 20.9282 0.760648 0.380324 0.924853i $$-0.375812\pi$$
0.380324 + 0.924853i $$0.375812\pi$$
$$758$$ 5.46410 0.198465
$$759$$ 5.32051i 0.193122i
$$760$$ 17.6603i 0.640605i
$$761$$ − 11.3205i − 0.410368i −0.978723 0.205184i $$-0.934221\pi$$
0.978723 0.205184i $$-0.0657793\pi$$
$$762$$ 9.85641i 0.357060i
$$763$$ −4.00000 −0.144810
$$764$$ 6.92820 0.250654
$$765$$ 21.3923i 0.773440i
$$766$$ 1.46410 0.0529001
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ − 43.8564i − 1.58150i −0.612138 0.790751i $$-0.709690\pi$$
0.612138 0.790751i $$-0.290310\pi$$
$$770$$ −12.9282 −0.465900
$$771$$ −9.33975 −0.336363
$$772$$ − 11.7321i − 0.422246i
$$773$$ − 48.9282i − 1.75983i −0.475136 0.879913i $$-0.657601\pi$$
0.475136 0.879913i $$-0.342399\pi$$
$$774$$ 9.66025i 0.347231i
$$775$$ − 13.0718i − 0.469553i
$$776$$ −6.00000 −0.215387
$$777$$ 9.66025 0.346560
$$778$$ − 11.7846i − 0.422499i
$$779$$ 44.4449 1.59240
$$780$$ 0 0
$$781$$ 6.00000 0.214697
$$782$$ − 24.0526i − 0.860118i
$$783$$ −4.46410 −0.159534
$$784$$ −0.464102 −0.0165751
$$785$$ 28.3205i 1.01080i
$$786$$ 6.53590i 0.233128i
$$787$$ − 4.67949i − 0.166806i −0.996516 0.0834029i $$-0.973421\pi$$
0.996516 0.0834029i $$-0.0265789\pi$$
$$788$$ 17.8564i 0.636108i
$$789$$ −10.0526 −0.357881
$$790$$ 9.46410 0.336718
$$791$$ − 3.66025i − 0.130144i
$$792$$ −1.26795 −0.0450546
$$793$$ 0 0
$$794$$ 20.3923 0.723696
$$795$$ − 24.1244i − 0.855603i
$$796$$ −14.1962 −0.503169
$$797$$ 34.0000 1.20434 0.602171 0.798367i $$-0.294303\pi$$
0.602171 + 0.798367i $$0.294303\pi$$
$$798$$ − 12.9282i − 0.457653i
$$799$$ − 12.5885i − 0.445348i
$$800$$ − 8.92820i − 0.315660i
$$801$$ 9.46410i 0.334398i
$$802$$ 8.07180 0.285025
$$803$$ 7.94744 0.280459
$$804$$ − 13.1244i − 0.462860i
$$805$$ 42.7846 1.50796
$$806$$ 0 0
$$807$$ 5.46410 0.192345
$$808$$ 1.92820i 0.0678340i
$$809$$ −53.5885 −1.88407 −0.942035 0.335515i $$-0.891090\pi$$
−0.942035 + 0.335515i $$0.891090\pi$$
$$810$$ −3.73205 −0.131131
$$811$$ 17.1769i 0.603163i 0.953440 + 0.301582i $$0.0975145\pi$$
−0.953440 + 0.301582i $$0.902485\pi$$
$$812$$ 12.1962i 0.428001i
$$813$$ 21.8564i 0.766538i
$$814$$ − 4.48334i − 0.157141i
$$815$$ −50.2487 −1.76014
$$816$$ 5.73205 0.200662
$$817$$ 45.7128i 1.59929i
$$818$$ −17.7321 −0.619987
$$819$$ 0 0
$$820$$ −35.0526 −1.22409
$$821$$ − 0.928203i − 0.0323945i −0.999869 0.0161973i $$-0.994844\pi$$
0.999869 0.0161973i $$-0.00515597\pi$$
$$822$$ 11.9282 0.416044
$$823$$ −41.5692 −1.44901 −0.724506 0.689269i $$-0.757932\pi$$
−0.724506 + 0.689269i $$0.757932\pi$$
$$824$$ − 15.2679i − 0.531884i
$$825$$ 11.3205i 0.394130i
$$826$$ 21.8564i 0.760482i
$$827$$ − 26.5359i − 0.922744i −0.887207 0.461372i $$-0.847357\pi$$
0.887207 0.461372i $$-0.152643\pi$$
$$828$$ 4.19615 0.145826
$$829$$ 12.1244 0.421096 0.210548 0.977583i $$-0.432475\pi$$
0.210548 + 0.977583i $$0.432475\pi$$
$$830$$ 0.732051i 0.0254099i
$$831$$ 5.73205 0.198843
$$832$$ 0 0
$$833$$ −2.66025 −0.0921723
$$834$$ 17.8564i 0.618317i
$$835$$ −35.3205 −1.22232
$$836$$ −6.00000 −0.207514
$$837$$ 1.46410i 0.0506068i
$$838$$ − 17.4641i − 0.603287i
$$839$$ − 41.8564i − 1.44504i −0.691348 0.722522i $$-0.742983\pi$$
0.691348 0.722522i $$-0.257017\pi$$
$$840$$ 10.1962i 0.351801i
$$841$$ −9.07180 −0.312821
$$842$$ 22.7128 0.782735
$$843$$ − 12.3205i − 0.424341i
$$844$$ −16.3923 −0.564246
$$845$$ 0 0
$$846$$ 2.19615 0.0755053
$$847$$ 25.6603i 0.881697i
$$848$$ −6.46410 −0.221978
$$849$$ −25.6603 −0.880658
$$850$$ − 51.1769i − 1.75535i
$$851$$ 14.8372i 0.508612i
$$852$$ − 4.73205i − 0.162117i
$$853$$ 54.1769i 1.85498i 0.373845 + 0.927491i $$0.378039\pi$$
−0.373845 + 0.927491i $$0.621961\pi$$
$$854$$ 25.1244 0.859738
$$855$$ −17.6603 −0.603968
$$856$$ − 10.1962i − 0.348497i
$$857$$ −39.4449 −1.34741 −0.673705 0.739000i $$-0.735298\pi$$
−0.673705 + 0.739000i $$0.735298\pi$$
$$858$$ 0 0
$$859$$ −47.1244 −1.60786 −0.803931 0.594722i $$-0.797262\pi$$
−0.803931 + 0.594722i $$0.797262\pi$$
$$860$$ − 36.0526i − 1.22938i
$$861$$ 25.6603 0.874499
$$862$$ −13.1244 −0.447017
$$863$$ − 17.1244i − 0.582920i −0.956583 0.291460i $$-0.905859\pi$$
0.956583 0.291460i $$-0.0941410\pi$$
$$864$$ 1.00000i 0.0340207i
$$865$$ − 16.3923i − 0.557355i
$$866$$ − 12.8564i − 0.436878i
$$867$$ 15.8564 0.538512
$$868$$ 4.00000 0.135769
$$869$$ 3.21539i 0.109075i
$$870$$ 16.6603 0.564836
$$871$$ 0 0
$$872$$ 1.46410 0.0495807
$$873$$ − 6.00000i − 0.203069i
$$874$$ 19.8564 0.671653
$$875$$ 40.0526 1.35402
$$876$$ − 6.26795i − 0.211774i
$$877$$ 23.9282i 0.807998i 0.914759 + 0.403999i $$0.132380\pi$$
−0.914759 + 0.403999i $$0.867620\pi$$
$$878$$ 0.339746i 0.0114659i
$$879$$ − 30.5167i − 1.02930i
$$880$$ 4.73205 0.159517
$$881$$ 27.8372 0.937858 0.468929 0.883236i $$-0.344640\pi$$
0.468929 + 0.883236i $$0.344640\pi$$
$$882$$ − 0.464102i − 0.0156271i
$$883$$ −42.9282 −1.44465 −0.722325 0.691554i $$-0.756926\pi$$
−0.722325 + 0.691554i $$0.756926\pi$$
$$884$$ 0 0
$$885$$ 29.8564 1.00361
$$886$$ 15.6077i 0.524351i
$$887$$ 37.8564 1.27109 0.635547 0.772062i $$-0.280775\pi$$
0.635547 + 0.772062i $$0.280775\pi$$
$$888$$ −3.53590 −0.118657
$$889$$ 26.9282i 0.903143i
$$890$$ − 35.3205i − 1.18395i
$$891$$ − 1.26795i − 0.0424779i
$$892$$ 26.9282i 0.901623i
$$893$$ 10.3923 0.347765
$$894$$ −13.1962 −0.441345
$$895$$ 59.9090i 2.00254i
$$896$$ 2.73205 0.0912714
$$897$$ 0 0
$$898$$ 11.3205 0.377770
$$899$$ − 6.53590i − 0.217984i
$$900$$ 8.92820 0.297607
$$901$$ −37.0526 −1.23440
$$902$$ − 11.9090i − 0.396525i
$$903$$ 26.3923i 0.878281i
$$904$$ 1.33975i 0.0445593i
$$905$$ − 71.6410i − 2.38143i
$$906$$ 6.73205 0.223657
$$907$$ −36.3923 −1.20839 −0.604193 0.796838i $$-0.706505\pi$$
−0.604193 + 0.796838i $$0.706505\pi$$
$$908$$ 12.1962i 0.404744i
$$909$$ −1.92820 −0.0639545
$$910$$ 0 0
$$911$$ −2.53590 −0.0840181 −0.0420090 0.999117i $$-0.513376\pi$$
−0.0420090 + 0.999117i $$0.513376\pi$$
$$912$$ 4.73205i 0.156694i
$$913$$ −0.248711 −0.00823114
$$914$$ −1.33975 −0.0443149
$$915$$ − 34.3205i − 1.13460i
$$916$$ − 11.8564i − 0.391747i
$$917$$ 17.8564i 0.589670i
$$918$$ 5.73205i 0.189186i
$$919$$ 45.9615 1.51613 0.758065 0.652179i $$-0.226145\pi$$
0.758065 + 0.652179i $$0.226145\pi$$
$$920$$ −15.6603 −0.516303
$$921$$ 22.5885i 0.744315i
$$922$$ −22.2679 −0.733356
$$923$$ 0 0
$$924$$ −3.46410 −0.113961
$$925$$ 31.5692i 1.03799i
$$926$$ 10.0526 0.330348
$$927$$ 15.2679 0.501465
$$928$$ − 4.46410i − 0.146541i
$$929$$ − 39.2487i − 1.28771i −0.765148 0.643854i $$-0.777334\pi$$
0.765148 0.643854i $$-0.222666\pi$$
$$930$$ − 5.46410i − 0.179175i
$$931$$ − 2.19615i − 0.0719760i
$$932$$ −7.85641 −0.257345
$$933$$ 1.66025 0.0543543
$$934$$ − 18.5885i − 0.608233i
$$935$$ 27.1244 0.887061
$$936$$ 0 0
$$937$$ −5.24871 −0.171468 −0.0857340 0.996318i $$-0.527324\pi$$
−0.0857340 + 0.996318i $$0.527324\pi$$
$$938$$ − 35.8564i − 1.17075i
$$939$$ 6.53590 0.213291
$$940$$ −8.19615 −0.267329
$$941$$ 12.6410i 0.412085i 0.978543 + 0.206043i $$0.0660586\pi$$
−0.978543 + 0.206043i $$0.933941\pi$$
$$942$$ 7.58846i 0.247245i
$$943$$ 39.4115i 1.28342i
$$944$$ − 8.00000i − 0.260378i
$$945$$ −10.1962 −0.331681
$$946$$ 12.2487 0.398240
$$947$$ − 21.0718i − 0.684741i −0.939565 0.342371i $$-0.888770\pi$$
0.939565 0.342371i $$-0.111230\pi$$
$$948$$ 2.53590 0.0823622
$$949$$ 0 0
$$950$$ 42.2487 1.37073
$$951$$ 20.6603i 0.669955i
$$952$$ 15.6603 0.507552
$$953$$ −41.5692 −1.34656 −0.673280 0.739388i $$-0.735115\pi$$
−0.673280 + 0.739388i $$0.735115\pi$$
$$954$$ − 6.46410i − 0.209283i
$$955$$ − 25.8564i − 0.836694i
$$956$$ − 7.66025i − 0.247750i
$$957$$ 5.66025i 0.182970i
$$958$$ 33.4641 1.08118
$$959$$ 32.5885 1.05234
$$960$$ − 3.73205i − 0.120451i
$$961$$ 28.8564 0.930852
$$962$$ 0 0
$$963$$ 10.1962 0.328566
$$964$$ 13.5885i 0.437655i
$$965$$ −43.7846 −1.40948
$$966$$ 11.4641 0.368851
$$967$$ − 43.1244i − 1.38679i −0.720560 0.693393i $$-0.756115\pi$$
0.720560 0.693393i $$-0.243885\pi$$
$$968$$ − 9.39230i − 0.301880i
$$969$$ 27.1244i 0.871360i
$$970$$ 22.3923i 0.718974i
$$971$$ 30.2487 0.970727 0.485364 0.874312i $$-0.338687\pi$$
0.485364 + 0.874312i $$0.338687\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 48.7846i 1.56396i
$$974$$ 3.12436 0.100111
$$975$$ 0 0
$$976$$ −9.19615 −0.294362
$$977$$ 45.9282i 1.46937i 0.678407 + 0.734687i $$0.262671\pi$$
−0.678407 + 0.734687i $$0.737329\pi$$
$$978$$ −13.4641 −0.430534
$$979$$ 12.0000 0.383522
$$980$$ 1.73205i 0.0553283i
$$981$$ 1.46410i 0.0467452i
$$982$$ − 8.73205i − 0.278651i
$$983$$ − 20.7846i − 0.662926i −0.943468 0.331463i $$-0.892458\pi$$
0.943468 0.331463i $$-0.107542\pi$$
$$984$$ −9.39230 −0.299416
$$985$$ 66.6410 2.12336
$$986$$ − 25.5885i − 0.814902i
$$987$$ 6.00000 0.190982
$$988$$ 0 0
$$989$$ −40.5359 −1.28897
$$990$$ 4.73205i 0.150394i
$$991$$ 22.5885 0.717546 0.358773 0.933425i $$-0.383195\pi$$
0.358773 + 0.933425i $$0.383195\pi$$
$$992$$ −1.46410 −0.0464853
$$993$$ 20.0000i 0.634681i
$$994$$ − 12.9282i − 0.410058i
$$995$$ 52.9808i 1.67960i
$$996$$ 0.196152i 0.00621533i
$$997$$ 21.3397 0.675837 0.337918 0.941175i $$-0.390277\pi$$
0.337918 + 0.941175i $$0.390277\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ − 3.53590i − 0.111871i
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 1014.2.b.e.337.4 4
3.2 odd 2 3042.2.b.i.1351.1 4
13.2 odd 12 1014.2.e.g.529.2 4
13.3 even 3 78.2.i.a.43.1 4
13.4 even 6 78.2.i.a.49.1 yes 4
13.5 odd 4 1014.2.a.k.1.2 2
13.6 odd 12 1014.2.e.g.991.2 4
13.7 odd 12 1014.2.e.i.991.1 4
13.8 odd 4 1014.2.a.i.1.1 2
13.9 even 3 1014.2.i.a.361.2 4
13.10 even 6 1014.2.i.a.823.2 4
13.11 odd 12 1014.2.e.i.529.1 4
13.12 even 2 inner 1014.2.b.e.337.1 4
39.5 even 4 3042.2.a.p.1.1 2
39.8 even 4 3042.2.a.y.1.2 2
39.17 odd 6 234.2.l.c.127.2 4
39.29 odd 6 234.2.l.c.199.2 4
39.38 odd 2 3042.2.b.i.1351.4 4
52.3 odd 6 624.2.bv.e.433.2 4
52.31 even 4 8112.2.a.bp.1.2 2
52.43 odd 6 624.2.bv.e.49.1 4
52.47 even 4 8112.2.a.bj.1.1 2
65.3 odd 12 1950.2.y.b.199.1 4
65.4 even 6 1950.2.bc.d.751.2 4
65.17 odd 12 1950.2.y.b.49.1 4
65.29 even 6 1950.2.bc.d.901.2 4
65.42 odd 12 1950.2.y.g.199.2 4
65.43 odd 12 1950.2.y.g.49.2 4
156.95 even 6 1872.2.by.h.1297.2 4
156.107 even 6 1872.2.by.h.433.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.1 4 13.3 even 3
78.2.i.a.49.1 yes 4 13.4 even 6
234.2.l.c.127.2 4 39.17 odd 6
234.2.l.c.199.2 4 39.29 odd 6
624.2.bv.e.49.1 4 52.43 odd 6
624.2.bv.e.433.2 4 52.3 odd 6
1014.2.a.i.1.1 2 13.8 odd 4
1014.2.a.k.1.2 2 13.5 odd 4
1014.2.b.e.337.1 4 13.12 even 2 inner
1014.2.b.e.337.4 4 1.1 even 1 trivial
1014.2.e.g.529.2 4 13.2 odd 12
1014.2.e.g.991.2 4 13.6 odd 12
1014.2.e.i.529.1 4 13.11 odd 12
1014.2.e.i.991.1 4 13.7 odd 12
1014.2.i.a.361.2 4 13.9 even 3
1014.2.i.a.823.2 4 13.10 even 6
1872.2.by.h.433.1 4 156.107 even 6
1872.2.by.h.1297.2 4 156.95 even 6
1950.2.y.b.49.1 4 65.17 odd 12
1950.2.y.b.199.1 4 65.3 odd 12
1950.2.y.g.49.2 4 65.43 odd 12
1950.2.y.g.199.2 4 65.42 odd 12
1950.2.bc.d.751.2 4 65.4 even 6
1950.2.bc.d.901.2 4 65.29 even 6
3042.2.a.p.1.1 2 39.5 even 4
3042.2.a.y.1.2 2 39.8 even 4
3042.2.b.i.1351.1 4 3.2 odd 2
3042.2.b.i.1351.4 4 39.38 odd 2
8112.2.a.bj.1.1 2 52.47 even 4
8112.2.a.bp.1.2 2 52.31 even 4