# Properties

 Label 165.2.a.c.1.1 Level $165$ Weight $2$ Character 165.1 Self dual yes Analytic conductor $1.318$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.31753163335$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 165.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} +1.00000 q^{5} -2.70928 q^{6} +1.07838 q^{7} -9.04945 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-2.70928 q^{2} +1.00000 q^{3} +5.34017 q^{4} +1.00000 q^{5} -2.70928 q^{6} +1.07838 q^{7} -9.04945 q^{8} +1.00000 q^{9} -2.70928 q^{10} +1.00000 q^{11} +5.34017 q^{12} -4.34017 q^{13} -2.92162 q^{14} +1.00000 q^{15} +13.8371 q^{16} +7.75872 q^{17} -2.70928 q^{18} +5.26180 q^{19} +5.34017 q^{20} +1.07838 q^{21} -2.70928 q^{22} -2.15676 q^{23} -9.04945 q^{24} +1.00000 q^{25} +11.7587 q^{26} +1.00000 q^{27} +5.75872 q^{28} +1.41855 q^{29} -2.70928 q^{30} -4.68035 q^{31} -19.3896 q^{32} +1.00000 q^{33} -21.0205 q^{34} +1.07838 q^{35} +5.34017 q^{36} -2.00000 q^{37} -14.2557 q^{38} -4.34017 q^{39} -9.04945 q^{40} -9.41855 q^{41} -2.92162 q^{42} +7.60197 q^{43} +5.34017 q^{44} +1.00000 q^{45} +5.84324 q^{46} +4.68035 q^{47} +13.8371 q^{48} -5.83710 q^{49} -2.70928 q^{50} +7.75872 q^{51} -23.1773 q^{52} +0.156755 q^{53} -2.70928 q^{54} +1.00000 q^{55} -9.75872 q^{56} +5.26180 q^{57} -3.84324 q^{58} +6.15676 q^{59} +5.34017 q^{60} -4.15676 q^{61} +12.6803 q^{62} +1.07838 q^{63} +24.8576 q^{64} -4.34017 q^{65} -2.70928 q^{66} -8.68035 q^{67} +41.4329 q^{68} -2.15676 q^{69} -2.92162 q^{70} -4.68035 q^{71} -9.04945 q^{72} -10.4969 q^{73} +5.41855 q^{74} +1.00000 q^{75} +28.0989 q^{76} +1.07838 q^{77} +11.7587 q^{78} -8.09890 q^{79} +13.8371 q^{80} +1.00000 q^{81} +25.5174 q^{82} -11.0205 q^{83} +5.75872 q^{84} +7.75872 q^{85} -20.5958 q^{86} +1.41855 q^{87} -9.04945 q^{88} -12.8371 q^{89} -2.70928 q^{90} -4.68035 q^{91} -11.5174 q^{92} -4.68035 q^{93} -12.6803 q^{94} +5.26180 q^{95} -19.3896 q^{96} +14.6803 q^{97} +15.8143 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 - 9 * q^8 + 3 * q^9 $$3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 9 q^{8} + 3 q^{9} - q^{10} + 3 q^{11} + 5 q^{12} - 2 q^{13} - 12 q^{14} + 3 q^{15} + 13 q^{16} - 2 q^{17} - q^{18} + 8 q^{19} + 5 q^{20} - q^{22} - 9 q^{24} + 3 q^{25} + 10 q^{26} + 3 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} + 8 q^{31} - 29 q^{32} + 3 q^{33} - 30 q^{34} + 5 q^{36} - 6 q^{37} - 2 q^{39} - 9 q^{40} - 14 q^{41} - 12 q^{42} + 4 q^{43} + 5 q^{44} + 3 q^{45} + 24 q^{46} - 8 q^{47} + 13 q^{48} + 11 q^{49} - q^{50} - 2 q^{51} - 30 q^{52} - 6 q^{53} - q^{54} + 3 q^{55} - 4 q^{56} + 8 q^{57} - 18 q^{58} + 12 q^{59} + 5 q^{60} - 6 q^{61} + 16 q^{62} + 13 q^{64} - 2 q^{65} - q^{66} - 4 q^{67} + 42 q^{68} - 12 q^{70} + 8 q^{71} - 9 q^{72} - 14 q^{73} + 2 q^{74} + 3 q^{75} + 48 q^{76} + 10 q^{78} + 12 q^{79} + 13 q^{80} + 3 q^{81} + 26 q^{82} - 8 q^{84} - 2 q^{85} - 8 q^{86} - 10 q^{87} - 9 q^{88} - 10 q^{89} - q^{90} + 8 q^{91} + 16 q^{92} + 8 q^{93} - 16 q^{94} + 8 q^{95} - 29 q^{96} + 22 q^{97} + 39 q^{98} + 3 q^{99}+O(q^{100})$$ 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 - 9 * q^8 + 3 * q^9 - q^10 + 3 * q^11 + 5 * q^12 - 2 * q^13 - 12 * q^14 + 3 * q^15 + 13 * q^16 - 2 * q^17 - q^18 + 8 * q^19 + 5 * q^20 - q^22 - 9 * q^24 + 3 * q^25 + 10 * q^26 + 3 * q^27 - 8 * q^28 - 10 * q^29 - q^30 + 8 * q^31 - 29 * q^32 + 3 * q^33 - 30 * q^34 + 5 * q^36 - 6 * q^37 - 2 * q^39 - 9 * q^40 - 14 * q^41 - 12 * q^42 + 4 * q^43 + 5 * q^44 + 3 * q^45 + 24 * q^46 - 8 * q^47 + 13 * q^48 + 11 * q^49 - q^50 - 2 * q^51 - 30 * q^52 - 6 * q^53 - q^54 + 3 * q^55 - 4 * q^56 + 8 * q^57 - 18 * q^58 + 12 * q^59 + 5 * q^60 - 6 * q^61 + 16 * q^62 + 13 * q^64 - 2 * q^65 - q^66 - 4 * q^67 + 42 * q^68 - 12 * q^70 + 8 * q^71 - 9 * q^72 - 14 * q^73 + 2 * q^74 + 3 * q^75 + 48 * q^76 + 10 * q^78 + 12 * q^79 + 13 * q^80 + 3 * q^81 + 26 * q^82 - 8 * q^84 - 2 * q^85 - 8 * q^86 - 10 * q^87 - 9 * q^88 - 10 * q^89 - q^90 + 8 * q^91 + 16 * q^92 + 8 * q^93 - 16 * q^94 + 8 * q^95 - 29 * q^96 + 22 * q^97 + 39 * q^98 + 3 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.70928 −1.91575 −0.957873 0.287190i $$-0.907279\pi$$
−0.957873 + 0.287190i $$0.907279\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 5.34017 2.67009
$$5$$ 1.00000 0.447214
$$6$$ −2.70928 −1.10606
$$7$$ 1.07838 0.407588 0.203794 0.979014i $$-0.434673\pi$$
0.203794 + 0.979014i $$0.434673\pi$$
$$8$$ −9.04945 −3.19946
$$9$$ 1.00000 0.333333
$$10$$ −2.70928 −0.856748
$$11$$ 1.00000 0.301511
$$12$$ 5.34017 1.54158
$$13$$ −4.34017 −1.20375 −0.601874 0.798591i $$-0.705579\pi$$
−0.601874 + 0.798591i $$0.705579\pi$$
$$14$$ −2.92162 −0.780836
$$15$$ 1.00000 0.258199
$$16$$ 13.8371 3.45928
$$17$$ 7.75872 1.88177 0.940883 0.338730i $$-0.109997\pi$$
0.940883 + 0.338730i $$0.109997\pi$$
$$18$$ −2.70928 −0.638582
$$19$$ 5.26180 1.20714 0.603569 0.797311i $$-0.293745\pi$$
0.603569 + 0.797311i $$0.293745\pi$$
$$20$$ 5.34017 1.19410
$$21$$ 1.07838 0.235321
$$22$$ −2.70928 −0.577619
$$23$$ −2.15676 −0.449715 −0.224857 0.974392i $$-0.572192\pi$$
−0.224857 + 0.974392i $$0.572192\pi$$
$$24$$ −9.04945 −1.84721
$$25$$ 1.00000 0.200000
$$26$$ 11.7587 2.30608
$$27$$ 1.00000 0.192450
$$28$$ 5.75872 1.08830
$$29$$ 1.41855 0.263418 0.131709 0.991288i $$-0.457954\pi$$
0.131709 + 0.991288i $$0.457954\pi$$
$$30$$ −2.70928 −0.494644
$$31$$ −4.68035 −0.840615 −0.420307 0.907382i $$-0.638078\pi$$
−0.420307 + 0.907382i $$0.638078\pi$$
$$32$$ −19.3896 −3.42763
$$33$$ 1.00000 0.174078
$$34$$ −21.0205 −3.60499
$$35$$ 1.07838 0.182279
$$36$$ 5.34017 0.890029
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −14.2557 −2.31257
$$39$$ −4.34017 −0.694984
$$40$$ −9.04945 −1.43084
$$41$$ −9.41855 −1.47093 −0.735465 0.677562i $$-0.763036\pi$$
−0.735465 + 0.677562i $$0.763036\pi$$
$$42$$ −2.92162 −0.450816
$$43$$ 7.60197 1.15929 0.579645 0.814869i $$-0.303191\pi$$
0.579645 + 0.814869i $$0.303191\pi$$
$$44$$ 5.34017 0.805061
$$45$$ 1.00000 0.149071
$$46$$ 5.84324 0.861539
$$47$$ 4.68035 0.682699 0.341349 0.939937i $$-0.389116\pi$$
0.341349 + 0.939937i $$0.389116\pi$$
$$48$$ 13.8371 1.99721
$$49$$ −5.83710 −0.833872
$$50$$ −2.70928 −0.383149
$$51$$ 7.75872 1.08644
$$52$$ −23.1773 −3.21411
$$53$$ 0.156755 0.0215320 0.0107660 0.999942i $$-0.496573\pi$$
0.0107660 + 0.999942i $$0.496573\pi$$
$$54$$ −2.70928 −0.368686
$$55$$ 1.00000 0.134840
$$56$$ −9.75872 −1.30406
$$57$$ 5.26180 0.696942
$$58$$ −3.84324 −0.504643
$$59$$ 6.15676 0.801541 0.400771 0.916178i $$-0.368742\pi$$
0.400771 + 0.916178i $$0.368742\pi$$
$$60$$ 5.34017 0.689413
$$61$$ −4.15676 −0.532218 −0.266109 0.963943i $$-0.585738\pi$$
−0.266109 + 0.963943i $$0.585738\pi$$
$$62$$ 12.6803 1.61041
$$63$$ 1.07838 0.135863
$$64$$ 24.8576 3.10720
$$65$$ −4.34017 −0.538332
$$66$$ −2.70928 −0.333489
$$67$$ −8.68035 −1.06047 −0.530237 0.847850i $$-0.677897\pi$$
−0.530237 + 0.847850i $$0.677897\pi$$
$$68$$ 41.4329 5.02448
$$69$$ −2.15676 −0.259643
$$70$$ −2.92162 −0.349201
$$71$$ −4.68035 −0.555455 −0.277727 0.960660i $$-0.589581\pi$$
−0.277727 + 0.960660i $$0.589581\pi$$
$$72$$ −9.04945 −1.06649
$$73$$ −10.4969 −1.22857 −0.614286 0.789083i $$-0.710556\pi$$
−0.614286 + 0.789083i $$0.710556\pi$$
$$74$$ 5.41855 0.629894
$$75$$ 1.00000 0.115470
$$76$$ 28.0989 3.22316
$$77$$ 1.07838 0.122893
$$78$$ 11.7587 1.33141
$$79$$ −8.09890 −0.911197 −0.455599 0.890185i $$-0.650575\pi$$
−0.455599 + 0.890185i $$0.650575\pi$$
$$80$$ 13.8371 1.54703
$$81$$ 1.00000 0.111111
$$82$$ 25.5174 2.81793
$$83$$ −11.0205 −1.20966 −0.604830 0.796355i $$-0.706759\pi$$
−0.604830 + 0.796355i $$0.706759\pi$$
$$84$$ 5.75872 0.628328
$$85$$ 7.75872 0.841552
$$86$$ −20.5958 −2.22090
$$87$$ 1.41855 0.152085
$$88$$ −9.04945 −0.964674
$$89$$ −12.8371 −1.36073 −0.680365 0.732873i $$-0.738179\pi$$
−0.680365 + 0.732873i $$0.738179\pi$$
$$90$$ −2.70928 −0.285583
$$91$$ −4.68035 −0.490634
$$92$$ −11.5174 −1.20078
$$93$$ −4.68035 −0.485329
$$94$$ −12.6803 −1.30788
$$95$$ 5.26180 0.539849
$$96$$ −19.3896 −1.97894
$$97$$ 14.6803 1.49056 0.745282 0.666750i $$-0.232315\pi$$
0.745282 + 0.666750i $$0.232315\pi$$
$$98$$ 15.8143 1.59749
$$99$$ 1.00000 0.100504
$$100$$ 5.34017 0.534017
$$101$$ −15.5753 −1.54980 −0.774900 0.632083i $$-0.782200\pi$$
−0.774900 + 0.632083i $$0.782200\pi$$
$$102$$ −21.0205 −2.08134
$$103$$ 6.83710 0.673680 0.336840 0.941562i $$-0.390642\pi$$
0.336840 + 0.941562i $$0.390642\pi$$
$$104$$ 39.2762 3.85135
$$105$$ 1.07838 0.105239
$$106$$ −0.424694 −0.0412499
$$107$$ 6.34017 0.612928 0.306464 0.951882i $$-0.400854\pi$$
0.306464 + 0.951882i $$0.400854\pi$$
$$108$$ 5.34017 0.513858
$$109$$ 2.31351 0.221594 0.110797 0.993843i $$-0.464660\pi$$
0.110797 + 0.993843i $$0.464660\pi$$
$$110$$ −2.70928 −0.258319
$$111$$ −2.00000 −0.189832
$$112$$ 14.9216 1.40996
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −14.2557 −1.33516
$$115$$ −2.15676 −0.201118
$$116$$ 7.57531 0.703350
$$117$$ −4.34017 −0.401249
$$118$$ −16.6803 −1.53555
$$119$$ 8.36683 0.766987
$$120$$ −9.04945 −0.826098
$$121$$ 1.00000 0.0909091
$$122$$ 11.2618 1.01960
$$123$$ −9.41855 −0.849242
$$124$$ −24.9939 −2.24451
$$125$$ 1.00000 0.0894427
$$126$$ −2.92162 −0.260279
$$127$$ −2.24128 −0.198881 −0.0994406 0.995044i $$-0.531705\pi$$
−0.0994406 + 0.995044i $$0.531705\pi$$
$$128$$ −28.5669 −2.52498
$$129$$ 7.60197 0.669316
$$130$$ 11.7587 1.03131
$$131$$ 8.68035 0.758405 0.379203 0.925314i $$-0.376198\pi$$
0.379203 + 0.925314i $$0.376198\pi$$
$$132$$ 5.34017 0.464802
$$133$$ 5.67420 0.492016
$$134$$ 23.5174 2.03160
$$135$$ 1.00000 0.0860663
$$136$$ −70.2122 −6.02064
$$137$$ −15.3607 −1.31235 −0.656176 0.754608i $$-0.727827\pi$$
−0.656176 + 0.754608i $$0.727827\pi$$
$$138$$ 5.84324 0.497410
$$139$$ 8.58145 0.727869 0.363935 0.931425i $$-0.381433\pi$$
0.363935 + 0.931425i $$0.381433\pi$$
$$140$$ 5.75872 0.486701
$$141$$ 4.68035 0.394156
$$142$$ 12.6803 1.06411
$$143$$ −4.34017 −0.362943
$$144$$ 13.8371 1.15309
$$145$$ 1.41855 0.117804
$$146$$ 28.4391 2.35363
$$147$$ −5.83710 −0.481436
$$148$$ −10.6803 −0.877919
$$149$$ −18.0989 −1.48272 −0.741360 0.671108i $$-0.765819\pi$$
−0.741360 + 0.671108i $$0.765819\pi$$
$$150$$ −2.70928 −0.221211
$$151$$ 22.9360 1.86651 0.933253 0.359221i $$-0.116958\pi$$
0.933253 + 0.359221i $$0.116958\pi$$
$$152$$ −47.6163 −3.86220
$$153$$ 7.75872 0.627256
$$154$$ −2.92162 −0.235431
$$155$$ −4.68035 −0.375934
$$156$$ −23.1773 −1.85567
$$157$$ −10.9939 −0.877405 −0.438703 0.898632i $$-0.644562\pi$$
−0.438703 + 0.898632i $$0.644562\pi$$
$$158$$ 21.9421 1.74562
$$159$$ 0.156755 0.0124315
$$160$$ −19.3896 −1.53288
$$161$$ −2.32580 −0.183298
$$162$$ −2.70928 −0.212861
$$163$$ −6.52359 −0.510967 −0.255484 0.966813i $$-0.582235\pi$$
−0.255484 + 0.966813i $$0.582235\pi$$
$$164$$ −50.2967 −3.92751
$$165$$ 1.00000 0.0778499
$$166$$ 29.8576 2.31740
$$167$$ 1.97334 0.152701 0.0763507 0.997081i $$-0.475673\pi$$
0.0763507 + 0.997081i $$0.475673\pi$$
$$168$$ −9.75872 −0.752902
$$169$$ 5.83710 0.449008
$$170$$ −21.0205 −1.61220
$$171$$ 5.26180 0.402380
$$172$$ 40.5958 3.09540
$$173$$ 3.75872 0.285770 0.142885 0.989739i $$-0.454362\pi$$
0.142885 + 0.989739i $$0.454362\pi$$
$$174$$ −3.84324 −0.291356
$$175$$ 1.07838 0.0815177
$$176$$ 13.8371 1.04301
$$177$$ 6.15676 0.462770
$$178$$ 34.7792 2.60681
$$179$$ 15.1506 1.13241 0.566205 0.824264i $$-0.308411\pi$$
0.566205 + 0.824264i $$0.308411\pi$$
$$180$$ 5.34017 0.398033
$$181$$ 4.83710 0.359539 0.179769 0.983709i $$-0.442465\pi$$
0.179769 + 0.983709i $$0.442465\pi$$
$$182$$ 12.6803 0.939930
$$183$$ −4.15676 −0.307276
$$184$$ 19.5174 1.43885
$$185$$ −2.00000 −0.147043
$$186$$ 12.6803 0.929768
$$187$$ 7.75872 0.567374
$$188$$ 24.9939 1.82286
$$189$$ 1.07838 0.0784404
$$190$$ −14.2557 −1.03421
$$191$$ 2.52359 0.182601 0.0913003 0.995823i $$-0.470898\pi$$
0.0913003 + 0.995823i $$0.470898\pi$$
$$192$$ 24.8576 1.79394
$$193$$ 0.0266620 0.00191917 0.000959586 1.00000i $$-0.499695\pi$$
0.000959586 1.00000i $$0.499695\pi$$
$$194$$ −39.7731 −2.85554
$$195$$ −4.34017 −0.310806
$$196$$ −31.1711 −2.22651
$$197$$ 21.1194 1.50470 0.752348 0.658766i $$-0.228921\pi$$
0.752348 + 0.658766i $$0.228921\pi$$
$$198$$ −2.70928 −0.192540
$$199$$ 10.5236 0.745998 0.372999 0.927832i $$-0.378330\pi$$
0.372999 + 0.927832i $$0.378330\pi$$
$$200$$ −9.04945 −0.639893
$$201$$ −8.68035 −0.612264
$$202$$ 42.1978 2.96903
$$203$$ 1.52973 0.107366
$$204$$ 41.4329 2.90089
$$205$$ −9.41855 −0.657820
$$206$$ −18.5236 −1.29060
$$207$$ −2.15676 −0.149905
$$208$$ −60.0554 −4.16409
$$209$$ 5.26180 0.363966
$$210$$ −2.92162 −0.201611
$$211$$ 9.57531 0.659191 0.329596 0.944122i $$-0.393088\pi$$
0.329596 + 0.944122i $$0.393088\pi$$
$$212$$ 0.837101 0.0574924
$$213$$ −4.68035 −0.320692
$$214$$ −17.1773 −1.17421
$$215$$ 7.60197 0.518450
$$216$$ −9.04945 −0.615737
$$217$$ −5.04718 −0.342625
$$218$$ −6.26794 −0.424518
$$219$$ −10.4969 −0.709317
$$220$$ 5.34017 0.360034
$$221$$ −33.6742 −2.26517
$$222$$ 5.41855 0.363669
$$223$$ −2.15676 −0.144427 −0.0722135 0.997389i $$-0.523006\pi$$
−0.0722135 + 0.997389i $$0.523006\pi$$
$$224$$ −20.9093 −1.39706
$$225$$ 1.00000 0.0666667
$$226$$ 16.2557 1.08131
$$227$$ 9.65983 0.641145 0.320573 0.947224i $$-0.396125\pi$$
0.320573 + 0.947224i $$0.396125\pi$$
$$228$$ 28.0989 1.86089
$$229$$ −3.36069 −0.222081 −0.111040 0.993816i $$-0.535418\pi$$
−0.111040 + 0.993816i $$0.535418\pi$$
$$230$$ 5.84324 0.385292
$$231$$ 1.07838 0.0709520
$$232$$ −12.8371 −0.842797
$$233$$ −2.39803 −0.157100 −0.0785501 0.996910i $$-0.525029\pi$$
−0.0785501 + 0.996910i $$0.525029\pi$$
$$234$$ 11.7587 0.768692
$$235$$ 4.68035 0.305312
$$236$$ 32.8781 2.14018
$$237$$ −8.09890 −0.526080
$$238$$ −22.6681 −1.46935
$$239$$ −7.20394 −0.465984 −0.232992 0.972479i $$-0.574852\pi$$
−0.232992 + 0.972479i $$0.574852\pi$$
$$240$$ 13.8371 0.893181
$$241$$ −5.20394 −0.335215 −0.167608 0.985854i $$-0.553604\pi$$
−0.167608 + 0.985854i $$0.553604\pi$$
$$242$$ −2.70928 −0.174159
$$243$$ 1.00000 0.0641500
$$244$$ −22.1978 −1.42107
$$245$$ −5.83710 −0.372919
$$246$$ 25.5174 1.62693
$$247$$ −22.8371 −1.45309
$$248$$ 42.3545 2.68952
$$249$$ −11.0205 −0.698397
$$250$$ −2.70928 −0.171350
$$251$$ 15.3197 0.966968 0.483484 0.875353i $$-0.339371\pi$$
0.483484 + 0.875353i $$0.339371\pi$$
$$252$$ 5.75872 0.362765
$$253$$ −2.15676 −0.135594
$$254$$ 6.07223 0.381006
$$255$$ 7.75872 0.485870
$$256$$ 27.6803 1.73002
$$257$$ 4.15676 0.259291 0.129646 0.991560i $$-0.458616\pi$$
0.129646 + 0.991560i $$0.458616\pi$$
$$258$$ −20.5958 −1.28224
$$259$$ −2.15676 −0.134014
$$260$$ −23.1773 −1.43739
$$261$$ 1.41855 0.0878061
$$262$$ −23.5174 −1.45291
$$263$$ −18.7070 −1.15352 −0.576762 0.816912i $$-0.695684\pi$$
−0.576762 + 0.816912i $$0.695684\pi$$
$$264$$ −9.04945 −0.556955
$$265$$ 0.156755 0.00962941
$$266$$ −15.3730 −0.942578
$$267$$ −12.8371 −0.785618
$$268$$ −46.3545 −2.83155
$$269$$ 23.3607 1.42433 0.712163 0.702014i $$-0.247716\pi$$
0.712163 + 0.702014i $$0.247716\pi$$
$$270$$ −2.70928 −0.164881
$$271$$ −5.57531 −0.338676 −0.169338 0.985558i $$-0.554163\pi$$
−0.169338 + 0.985558i $$0.554163\pi$$
$$272$$ 107.358 6.50955
$$273$$ −4.68035 −0.283267
$$274$$ 41.6163 2.51414
$$275$$ 1.00000 0.0603023
$$276$$ −11.5174 −0.693269
$$277$$ −26.0144 −1.56305 −0.781526 0.623872i $$-0.785558\pi$$
−0.781526 + 0.623872i $$0.785558\pi$$
$$278$$ −23.2495 −1.39441
$$279$$ −4.68035 −0.280205
$$280$$ −9.75872 −0.583195
$$281$$ −9.41855 −0.561864 −0.280932 0.959728i $$-0.590643\pi$$
−0.280932 + 0.959728i $$0.590643\pi$$
$$282$$ −12.6803 −0.755104
$$283$$ 14.2413 0.846556 0.423278 0.906000i $$-0.360879\pi$$
0.423278 + 0.906000i $$0.360879\pi$$
$$284$$ −24.9939 −1.48311
$$285$$ 5.26180 0.311682
$$286$$ 11.7587 0.695308
$$287$$ −10.1568 −0.599534
$$288$$ −19.3896 −1.14254
$$289$$ 43.1978 2.54105
$$290$$ −3.84324 −0.225683
$$291$$ 14.6803 0.860577
$$292$$ −56.0554 −3.28039
$$293$$ −15.7587 −0.920634 −0.460317 0.887754i $$-0.652264\pi$$
−0.460317 + 0.887754i $$0.652264\pi$$
$$294$$ 15.8143 0.922310
$$295$$ 6.15676 0.358460
$$296$$ 18.0989 1.05198
$$297$$ 1.00000 0.0580259
$$298$$ 49.0349 2.84052
$$299$$ 9.36069 0.541343
$$300$$ 5.34017 0.308315
$$301$$ 8.19779 0.472513
$$302$$ −62.1399 −3.57575
$$303$$ −15.5753 −0.894778
$$304$$ 72.8080 4.17582
$$305$$ −4.15676 −0.238015
$$306$$ −21.0205 −1.20166
$$307$$ −18.9216 −1.07991 −0.539957 0.841693i $$-0.681560\pi$$
−0.539957 + 0.841693i $$0.681560\pi$$
$$308$$ 5.75872 0.328134
$$309$$ 6.83710 0.388949
$$310$$ 12.6803 0.720195
$$311$$ −20.8781 −1.18389 −0.591945 0.805978i $$-0.701640\pi$$
−0.591945 + 0.805978i $$0.701640\pi$$
$$312$$ 39.2762 2.22358
$$313$$ 6.31351 0.356861 0.178430 0.983953i $$-0.442898\pi$$
0.178430 + 0.983953i $$0.442898\pi$$
$$314$$ 29.7854 1.68089
$$315$$ 1.07838 0.0607597
$$316$$ −43.2495 −2.43297
$$317$$ 31.3607 1.76139 0.880696 0.473682i $$-0.157075\pi$$
0.880696 + 0.473682i $$0.157075\pi$$
$$318$$ −0.424694 −0.0238156
$$319$$ 1.41855 0.0794236
$$320$$ 24.8576 1.38958
$$321$$ 6.34017 0.353874
$$322$$ 6.30122 0.351154
$$323$$ 40.8248 2.27155
$$324$$ 5.34017 0.296676
$$325$$ −4.34017 −0.240749
$$326$$ 17.6742 0.978884
$$327$$ 2.31351 0.127937
$$328$$ 85.2327 4.70619
$$329$$ 5.04718 0.278260
$$330$$ −2.70928 −0.149141
$$331$$ 19.2039 1.05554 0.527772 0.849386i $$-0.323028\pi$$
0.527772 + 0.849386i $$0.323028\pi$$
$$332$$ −58.8515 −3.22989
$$333$$ −2.00000 −0.109599
$$334$$ −5.34632 −0.292537
$$335$$ −8.68035 −0.474258
$$336$$ 14.9216 0.814041
$$337$$ 13.5031 0.735559 0.367780 0.929913i $$-0.380118\pi$$
0.367780 + 0.929913i $$0.380118\pi$$
$$338$$ −15.8143 −0.860185
$$339$$ −6.00000 −0.325875
$$340$$ 41.4329 2.24702
$$341$$ −4.68035 −0.253455
$$342$$ −14.2557 −0.770857
$$343$$ −13.8432 −0.747465
$$344$$ −68.7936 −3.70910
$$345$$ −2.15676 −0.116116
$$346$$ −10.1834 −0.547464
$$347$$ 6.34017 0.340358 0.170179 0.985413i $$-0.445565\pi$$
0.170179 + 0.985413i $$0.445565\pi$$
$$348$$ 7.57531 0.406079
$$349$$ 16.1568 0.864851 0.432426 0.901670i $$-0.357658\pi$$
0.432426 + 0.901670i $$0.357658\pi$$
$$350$$ −2.92162 −0.156167
$$351$$ −4.34017 −0.231661
$$352$$ −19.3896 −1.03347
$$353$$ −13.2039 −0.702775 −0.351387 0.936230i $$-0.614290\pi$$
−0.351387 + 0.936230i $$0.614290\pi$$
$$354$$ −16.6803 −0.886550
$$355$$ −4.68035 −0.248407
$$356$$ −68.5523 −3.63327
$$357$$ 8.36683 0.442820
$$358$$ −41.0472 −2.16941
$$359$$ 3.31965 0.175205 0.0876023 0.996156i $$-0.472080\pi$$
0.0876023 + 0.996156i $$0.472080\pi$$
$$360$$ −9.04945 −0.476948
$$361$$ 8.68649 0.457184
$$362$$ −13.1050 −0.688786
$$363$$ 1.00000 0.0524864
$$364$$ −24.9939 −1.31003
$$365$$ −10.4969 −0.549434
$$366$$ 11.2618 0.588663
$$367$$ −36.1445 −1.88673 −0.943363 0.331762i $$-0.892357\pi$$
−0.943363 + 0.331762i $$0.892357\pi$$
$$368$$ −29.8432 −1.55569
$$369$$ −9.41855 −0.490310
$$370$$ 5.41855 0.281697
$$371$$ 0.169042 0.00877620
$$372$$ −24.9939 −1.29587
$$373$$ −2.81044 −0.145519 −0.0727595 0.997350i $$-0.523181\pi$$
−0.0727595 + 0.997350i $$0.523181\pi$$
$$374$$ −21.0205 −1.08695
$$375$$ 1.00000 0.0516398
$$376$$ −42.3545 −2.18427
$$377$$ −6.15676 −0.317089
$$378$$ −2.92162 −0.150272
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 28.0989 1.44144
$$381$$ −2.24128 −0.114824
$$382$$ −6.83710 −0.349817
$$383$$ 33.5585 1.71476 0.857379 0.514685i $$-0.172091\pi$$
0.857379 + 0.514685i $$0.172091\pi$$
$$384$$ −28.5669 −1.45780
$$385$$ 1.07838 0.0549592
$$386$$ −0.0722347 −0.00367665
$$387$$ 7.60197 0.386430
$$388$$ 78.3956 3.97993
$$389$$ 12.8371 0.650867 0.325433 0.945565i $$-0.394490\pi$$
0.325433 + 0.945565i $$0.394490\pi$$
$$390$$ 11.7587 0.595426
$$391$$ −16.7337 −0.846258
$$392$$ 52.8225 2.66794
$$393$$ 8.68035 0.437866
$$394$$ −57.2183 −2.88262
$$395$$ −8.09890 −0.407500
$$396$$ 5.34017 0.268354
$$397$$ −5.31965 −0.266986 −0.133493 0.991050i $$-0.542619\pi$$
−0.133493 + 0.991050i $$0.542619\pi$$
$$398$$ −28.5113 −1.42914
$$399$$ 5.67420 0.284065
$$400$$ 13.8371 0.691855
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 23.5174 1.17294
$$403$$ 20.3135 1.01189
$$404$$ −83.1748 −4.13810
$$405$$ 1.00000 0.0496904
$$406$$ −4.14447 −0.205687
$$407$$ −2.00000 −0.0991363
$$408$$ −70.2122 −3.47602
$$409$$ 26.1978 1.29540 0.647699 0.761897i $$-0.275732\pi$$
0.647699 + 0.761897i $$0.275732\pi$$
$$410$$ 25.5174 1.26022
$$411$$ −15.3607 −0.757687
$$412$$ 36.5113 1.79878
$$413$$ 6.63931 0.326699
$$414$$ 5.84324 0.287180
$$415$$ −11.0205 −0.540976
$$416$$ 84.1543 4.12600
$$417$$ 8.58145 0.420235
$$418$$ −14.2557 −0.697267
$$419$$ −2.83710 −0.138601 −0.0693007 0.997596i $$-0.522077\pi$$
−0.0693007 + 0.997596i $$0.522077\pi$$
$$420$$ 5.75872 0.280997
$$421$$ 11.4764 0.559326 0.279663 0.960098i $$-0.409777\pi$$
0.279663 + 0.960098i $$0.409777\pi$$
$$422$$ −25.9421 −1.26284
$$423$$ 4.68035 0.227566
$$424$$ −1.41855 −0.0688909
$$425$$ 7.75872 0.376353
$$426$$ 12.6803 0.614365
$$427$$ −4.48255 −0.216926
$$428$$ 33.8576 1.63657
$$429$$ −4.34017 −0.209546
$$430$$ −20.5958 −0.993219
$$431$$ 23.5708 1.13536 0.567682 0.823248i $$-0.307840\pi$$
0.567682 + 0.823248i $$0.307840\pi$$
$$432$$ 13.8371 0.665738
$$433$$ −14.9939 −0.720559 −0.360279 0.932844i $$-0.617319\pi$$
−0.360279 + 0.932844i $$0.617319\pi$$
$$434$$ 13.6742 0.656383
$$435$$ 1.41855 0.0680143
$$436$$ 12.3545 0.591676
$$437$$ −11.3484 −0.542868
$$438$$ 28.4391 1.35887
$$439$$ 4.77924 0.228101 0.114050 0.993475i $$-0.463617\pi$$
0.114050 + 0.993475i $$0.463617\pi$$
$$440$$ −9.04945 −0.431416
$$441$$ −5.83710 −0.277957
$$442$$ 91.2327 4.33950
$$443$$ 20.1978 0.959626 0.479813 0.877371i $$-0.340704\pi$$
0.479813 + 0.877371i $$0.340704\pi$$
$$444$$ −10.6803 −0.506867
$$445$$ −12.8371 −0.608537
$$446$$ 5.84324 0.276686
$$447$$ −18.0989 −0.856048
$$448$$ 26.8059 1.26646
$$449$$ −21.5708 −1.01799 −0.508994 0.860770i $$-0.669982\pi$$
−0.508994 + 0.860770i $$0.669982\pi$$
$$450$$ −2.70928 −0.127716
$$451$$ −9.41855 −0.443502
$$452$$ −32.0410 −1.50708
$$453$$ 22.9360 1.07763
$$454$$ −26.1711 −1.22827
$$455$$ −4.68035 −0.219418
$$456$$ −47.6163 −2.22984
$$457$$ 28.1711 1.31779 0.658895 0.752235i $$-0.271024\pi$$
0.658895 + 0.752235i $$0.271024\pi$$
$$458$$ 9.10504 0.425451
$$459$$ 7.75872 0.362146
$$460$$ −11.5174 −0.537004
$$461$$ −1.47187 −0.0685520 −0.0342760 0.999412i $$-0.510913\pi$$
−0.0342760 + 0.999412i $$0.510913\pi$$
$$462$$ −2.92162 −0.135926
$$463$$ −23.2039 −1.07838 −0.539189 0.842185i $$-0.681269\pi$$
−0.539189 + 0.842185i $$0.681269\pi$$
$$464$$ 19.6286 0.911236
$$465$$ −4.68035 −0.217046
$$466$$ 6.49693 0.300964
$$467$$ 14.1568 0.655097 0.327548 0.944834i $$-0.393778\pi$$
0.327548 + 0.944834i $$0.393778\pi$$
$$468$$ −23.1773 −1.07137
$$469$$ −9.36069 −0.432237
$$470$$ −12.6803 −0.584901
$$471$$ −10.9939 −0.506570
$$472$$ −55.7152 −2.56450
$$473$$ 7.60197 0.349539
$$474$$ 21.9421 1.00784
$$475$$ 5.26180 0.241428
$$476$$ 44.6803 2.04792
$$477$$ 0.156755 0.00717734
$$478$$ 19.5174 0.892707
$$479$$ −13.8432 −0.632514 −0.316257 0.948674i $$-0.602426\pi$$
−0.316257 + 0.948674i $$0.602426\pi$$
$$480$$ −19.3896 −0.885011
$$481$$ 8.68035 0.395790
$$482$$ 14.0989 0.642187
$$483$$ −2.32580 −0.105827
$$484$$ 5.34017 0.242735
$$485$$ 14.6803 0.666600
$$486$$ −2.70928 −0.122895
$$487$$ −40.9939 −1.85761 −0.928804 0.370570i $$-0.879162\pi$$
−0.928804 + 0.370570i $$0.879162\pi$$
$$488$$ 37.6163 1.70281
$$489$$ −6.52359 −0.295007
$$490$$ 15.8143 0.714418
$$491$$ 34.8371 1.57218 0.786088 0.618114i $$-0.212103\pi$$
0.786088 + 0.618114i $$0.212103\pi$$
$$492$$ −50.2967 −2.26755
$$493$$ 11.0061 0.495692
$$494$$ 61.8720 2.78375
$$495$$ 1.00000 0.0449467
$$496$$ −64.7624 −2.90792
$$497$$ −5.04718 −0.226397
$$498$$ 29.8576 1.33795
$$499$$ 15.1506 0.678235 0.339117 0.940744i $$-0.389872\pi$$
0.339117 + 0.940744i $$0.389872\pi$$
$$500$$ 5.34017 0.238820
$$501$$ 1.97334 0.0881622
$$502$$ −41.5052 −1.85247
$$503$$ −6.65368 −0.296673 −0.148337 0.988937i $$-0.547392\pi$$
−0.148337 + 0.988937i $$0.547392\pi$$
$$504$$ −9.75872 −0.434688
$$505$$ −15.5753 −0.693092
$$506$$ 5.84324 0.259764
$$507$$ 5.83710 0.259235
$$508$$ −11.9688 −0.531030
$$509$$ 41.3484 1.83274 0.916368 0.400337i $$-0.131107\pi$$
0.916368 + 0.400337i $$0.131107\pi$$
$$510$$ −21.0205 −0.930804
$$511$$ −11.3197 −0.500752
$$512$$ −17.8599 −0.789303
$$513$$ 5.26180 0.232314
$$514$$ −11.2618 −0.496736
$$515$$ 6.83710 0.301279
$$516$$ 40.5958 1.78713
$$517$$ 4.68035 0.205841
$$518$$ 5.84324 0.256737
$$519$$ 3.75872 0.164990
$$520$$ 39.2762 1.72237
$$521$$ 7.67420 0.336213 0.168106 0.985769i $$-0.446235\pi$$
0.168106 + 0.985769i $$0.446235\pi$$
$$522$$ −3.84324 −0.168214
$$523$$ 23.2351 1.01600 0.508001 0.861357i $$-0.330385\pi$$
0.508001 + 0.861357i $$0.330385\pi$$
$$524$$ 46.3545 2.02501
$$525$$ 1.07838 0.0470643
$$526$$ 50.6824 2.20986
$$527$$ −36.3135 −1.58184
$$528$$ 13.8371 0.602183
$$529$$ −18.3484 −0.797757
$$530$$ −0.424694 −0.0184475
$$531$$ 6.15676 0.267180
$$532$$ 30.3012 1.31372
$$533$$ 40.8781 1.77063
$$534$$ 34.7792 1.50505
$$535$$ 6.34017 0.274110
$$536$$ 78.5523 3.39294
$$537$$ 15.1506 0.653797
$$538$$ −63.2905 −2.72865
$$539$$ −5.83710 −0.251422
$$540$$ 5.34017 0.229804
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 15.1050 0.648817
$$543$$ 4.83710 0.207580
$$544$$ −150.439 −6.45001
$$545$$ 2.31351 0.0990999
$$546$$ 12.6803 0.542669
$$547$$ −23.0661 −0.986235 −0.493117 0.869963i $$-0.664143\pi$$
−0.493117 + 0.869963i $$0.664143\pi$$
$$548$$ −82.0288 −3.50409
$$549$$ −4.15676 −0.177406
$$550$$ −2.70928 −0.115524
$$551$$ 7.46412 0.317982
$$552$$ 19.5174 0.830718
$$553$$ −8.73367 −0.371393
$$554$$ 70.4801 2.99441
$$555$$ −2.00000 −0.0848953
$$556$$ 45.8264 1.94347
$$557$$ 10.5958 0.448960 0.224480 0.974479i $$-0.427932\pi$$
0.224480 + 0.974479i $$0.427932\pi$$
$$558$$ 12.6803 0.536802
$$559$$ −32.9939 −1.39549
$$560$$ 14.9216 0.630554
$$561$$ 7.75872 0.327574
$$562$$ 25.5174 1.07639
$$563$$ −36.2122 −1.52616 −0.763080 0.646303i $$-0.776314\pi$$
−0.763080 + 0.646303i $$0.776314\pi$$
$$564$$ 24.9939 1.05243
$$565$$ −6.00000 −0.252422
$$566$$ −38.5835 −1.62179
$$567$$ 1.07838 0.0452876
$$568$$ 42.3545 1.77716
$$569$$ −27.5753 −1.15602 −0.578008 0.816031i $$-0.696170\pi$$
−0.578008 + 0.816031i $$0.696170\pi$$
$$570$$ −14.2557 −0.597104
$$571$$ −27.9299 −1.16883 −0.584414 0.811456i $$-0.698676\pi$$
−0.584414 + 0.811456i $$0.698676\pi$$
$$572$$ −23.1773 −0.969091
$$573$$ 2.52359 0.105425
$$574$$ 27.5174 1.14856
$$575$$ −2.15676 −0.0899429
$$576$$ 24.8576 1.03573
$$577$$ 41.4017 1.72358 0.861788 0.507268i $$-0.169345\pi$$
0.861788 + 0.507268i $$0.169345\pi$$
$$578$$ −117.035 −4.86800
$$579$$ 0.0266620 0.00110803
$$580$$ 7.57531 0.314547
$$581$$ −11.8843 −0.493043
$$582$$ −39.7731 −1.64865
$$583$$ 0.156755 0.00649215
$$584$$ 94.9914 3.93077
$$585$$ −4.34017 −0.179444
$$586$$ 42.6947 1.76370
$$587$$ 8.48255 0.350112 0.175056 0.984558i $$-0.443989\pi$$
0.175056 + 0.984558i $$0.443989\pi$$
$$588$$ −31.1711 −1.28548
$$589$$ −24.6270 −1.01474
$$590$$ −16.6803 −0.686719
$$591$$ 21.1194 0.868737
$$592$$ −27.6742 −1.13740
$$593$$ 7.56093 0.310490 0.155245 0.987876i $$-0.450383\pi$$
0.155245 + 0.987876i $$0.450383\pi$$
$$594$$ −2.70928 −0.111163
$$595$$ 8.36683 0.343007
$$596$$ −96.6512 −3.95899
$$597$$ 10.5236 0.430702
$$598$$ −25.3607 −1.03708
$$599$$ 5.67420 0.231842 0.115921 0.993258i $$-0.463018\pi$$
0.115921 + 0.993258i $$0.463018\pi$$
$$600$$ −9.04945 −0.369442
$$601$$ −1.31965 −0.0538298 −0.0269149 0.999638i $$-0.508568\pi$$
−0.0269149 + 0.999638i $$0.508568\pi$$
$$602$$ −22.2101 −0.905215
$$603$$ −8.68035 −0.353491
$$604$$ 122.482 4.98373
$$605$$ 1.00000 0.0406558
$$606$$ 42.1978 1.71417
$$607$$ −2.24128 −0.0909706 −0.0454853 0.998965i $$-0.514483\pi$$
−0.0454853 + 0.998965i $$0.514483\pi$$
$$608$$ −102.024 −4.13763
$$609$$ 1.52973 0.0619879
$$610$$ 11.2618 0.455977
$$611$$ −20.3135 −0.821797
$$612$$ 41.4329 1.67483
$$613$$ 42.8638 1.73125 0.865626 0.500692i $$-0.166921\pi$$
0.865626 + 0.500692i $$0.166921\pi$$
$$614$$ 51.2639 2.06884
$$615$$ −9.41855 −0.379793
$$616$$ −9.75872 −0.393190
$$617$$ 11.3607 0.457364 0.228682 0.973501i $$-0.426558\pi$$
0.228682 + 0.973501i $$0.426558\pi$$
$$618$$ −18.5236 −0.745128
$$619$$ −45.1917 −1.81641 −0.908203 0.418530i $$-0.862545\pi$$
−0.908203 + 0.418530i $$0.862545\pi$$
$$620$$ −24.9939 −1.00378
$$621$$ −2.15676 −0.0865476
$$622$$ 56.5646 2.26803
$$623$$ −13.8432 −0.554618
$$624$$ −60.0554 −2.40414
$$625$$ 1.00000 0.0400000
$$626$$ −17.1050 −0.683655
$$627$$ 5.26180 0.210136
$$628$$ −58.7091 −2.34275
$$629$$ −15.5174 −0.618721
$$630$$ −2.92162 −0.116400
$$631$$ −9.78992 −0.389731 −0.194865 0.980830i $$-0.562427\pi$$
−0.194865 + 0.980830i $$0.562427\pi$$
$$632$$ 73.2905 2.91534
$$633$$ 9.57531 0.380584
$$634$$ −84.9647 −3.37438
$$635$$ −2.24128 −0.0889423
$$636$$ 0.837101 0.0331932
$$637$$ 25.3340 1.00377
$$638$$ −3.84324 −0.152156
$$639$$ −4.68035 −0.185152
$$640$$ −28.5669 −1.12921
$$641$$ 0.210079 0.00829764 0.00414882 0.999991i $$-0.498679\pi$$
0.00414882 + 0.999991i $$0.498679\pi$$
$$642$$ −17.1773 −0.677933
$$643$$ 14.5236 0.572754 0.286377 0.958117i $$-0.407549\pi$$
0.286377 + 0.958117i $$0.407549\pi$$
$$644$$ −12.4202 −0.489423
$$645$$ 7.60197 0.299327
$$646$$ −110.606 −4.35172
$$647$$ 15.4641 0.607957 0.303979 0.952679i $$-0.401685\pi$$
0.303979 + 0.952679i $$0.401685\pi$$
$$648$$ −9.04945 −0.355496
$$649$$ 6.15676 0.241674
$$650$$ 11.7587 0.461215
$$651$$ −5.04718 −0.197815
$$652$$ −34.8371 −1.36433
$$653$$ −17.8310 −0.697779 −0.348890 0.937164i $$-0.613441\pi$$
−0.348890 + 0.937164i $$0.613441\pi$$
$$654$$ −6.26794 −0.245096
$$655$$ 8.68035 0.339169
$$656$$ −130.325 −5.08835
$$657$$ −10.4969 −0.409524
$$658$$ −13.6742 −0.533076
$$659$$ −32.3135 −1.25876 −0.629378 0.777099i $$-0.716690\pi$$
−0.629378 + 0.777099i $$0.716690\pi$$
$$660$$ 5.34017 0.207866
$$661$$ −5.68649 −0.221179 −0.110589 0.993866i $$-0.535274\pi$$
−0.110589 + 0.993866i $$0.535274\pi$$
$$662$$ −52.0288 −2.02215
$$663$$ −33.6742 −1.30780
$$664$$ 99.7296 3.87026
$$665$$ 5.67420 0.220036
$$666$$ 5.41855 0.209965
$$667$$ −3.05947 −0.118463
$$668$$ 10.5380 0.407726
$$669$$ −2.15676 −0.0833850
$$670$$ 23.5174 0.908558
$$671$$ −4.15676 −0.160470
$$672$$ −20.9093 −0.806595
$$673$$ −21.0205 −0.810281 −0.405141 0.914254i $$-0.632777\pi$$
−0.405141 + 0.914254i $$0.632777\pi$$
$$674$$ −36.5835 −1.40915
$$675$$ 1.00000 0.0384900
$$676$$ 31.1711 1.19889
$$677$$ 36.7526 1.41252 0.706258 0.707954i $$-0.250382\pi$$
0.706258 + 0.707954i $$0.250382\pi$$
$$678$$ 16.2557 0.624295
$$679$$ 15.8310 0.607536
$$680$$ −70.2122 −2.69251
$$681$$ 9.65983 0.370165
$$682$$ 12.6803 0.485556
$$683$$ −17.3074 −0.662248 −0.331124 0.943587i $$-0.607428\pi$$
−0.331124 + 0.943587i $$0.607428\pi$$
$$684$$ 28.0989 1.07439
$$685$$ −15.3607 −0.586902
$$686$$ 37.5052 1.43195
$$687$$ −3.36069 −0.128218
$$688$$ 105.189 4.01030
$$689$$ −0.680346 −0.0259191
$$690$$ 5.84324 0.222449
$$691$$ 17.6742 0.672358 0.336179 0.941798i $$-0.390865\pi$$
0.336179 + 0.941798i $$0.390865\pi$$
$$692$$ 20.0722 0.763032
$$693$$ 1.07838 0.0409642
$$694$$ −17.1773 −0.652040
$$695$$ 8.58145 0.325513
$$696$$ −12.8371 −0.486589
$$697$$ −73.0759 −2.76795
$$698$$ −43.7731 −1.65684
$$699$$ −2.39803 −0.0907019
$$700$$ 5.75872 0.217659
$$701$$ −17.1050 −0.646048 −0.323024 0.946391i $$-0.604700\pi$$
−0.323024 + 0.946391i $$0.604700\pi$$
$$702$$ 11.7587 0.443804
$$703$$ −10.5236 −0.396905
$$704$$ 24.8576 0.936857
$$705$$ 4.68035 0.176272
$$706$$ 35.7731 1.34634
$$707$$ −16.7961 −0.631681
$$708$$ 32.8781 1.23564
$$709$$ 25.1506 0.944551 0.472276 0.881451i $$-0.343433\pi$$
0.472276 + 0.881451i $$0.343433\pi$$
$$710$$ 12.6803 0.475885
$$711$$ −8.09890 −0.303732
$$712$$ 116.169 4.35361
$$713$$ 10.0944 0.378037
$$714$$ −22.6681 −0.848331
$$715$$ −4.34017 −0.162313
$$716$$ 80.9069 3.02363
$$717$$ −7.20394 −0.269036
$$718$$ −8.99386 −0.335648
$$719$$ −1.78992 −0.0667528 −0.0333764 0.999443i $$-0.510626\pi$$
−0.0333764 + 0.999443i $$0.510626\pi$$
$$720$$ 13.8371 0.515678
$$721$$ 7.37298 0.274584
$$722$$ −23.5341 −0.875848
$$723$$ −5.20394 −0.193536
$$724$$ 25.8310 0.960000
$$725$$ 1.41855 0.0526837
$$726$$ −2.70928 −0.100551
$$727$$ 25.9877 0.963831 0.481915 0.876218i $$-0.339941\pi$$
0.481915 + 0.876218i $$0.339941\pi$$
$$728$$ 42.3545 1.56976
$$729$$ 1.00000 0.0370370
$$730$$ 28.4391 1.05258
$$731$$ 58.9816 2.18151
$$732$$ −22.1978 −0.820454
$$733$$ −41.0205 −1.51513 −0.757564 0.652761i $$-0.773610\pi$$
−0.757564 + 0.652761i $$0.773610\pi$$
$$734$$ 97.9253 3.61449
$$735$$ −5.83710 −0.215305
$$736$$ 41.8187 1.54146
$$737$$ −8.68035 −0.319745
$$738$$ 25.5174 0.939310
$$739$$ −47.6163 −1.75160 −0.875798 0.482678i $$-0.839664\pi$$
−0.875798 + 0.482678i $$0.839664\pi$$
$$740$$ −10.6803 −0.392617
$$741$$ −22.8371 −0.838942
$$742$$ −0.457980 −0.0168130
$$743$$ 0.550252 0.0201868 0.0100934 0.999949i $$-0.496787\pi$$
0.0100934 + 0.999949i $$0.496787\pi$$
$$744$$ 42.3545 1.55279
$$745$$ −18.0989 −0.663092
$$746$$ 7.61425 0.278778
$$747$$ −11.0205 −0.403220
$$748$$ 41.4329 1.51494
$$749$$ 6.83710 0.249822
$$750$$ −2.70928 −0.0989287
$$751$$ 41.5585 1.51649 0.758245 0.651969i $$-0.226057\pi$$
0.758245 + 0.651969i $$0.226057\pi$$
$$752$$ 64.7624 2.36164
$$753$$ 15.3197 0.558279
$$754$$ 16.6803 0.607462
$$755$$ 22.9360 0.834726
$$756$$ 5.75872 0.209443
$$757$$ 1.31965 0.0479636 0.0239818 0.999712i $$-0.492366\pi$$
0.0239818 + 0.999712i $$0.492366\pi$$
$$758$$ 54.1855 1.96811
$$759$$ −2.15676 −0.0782853
$$760$$ −47.6163 −1.72723
$$761$$ −2.21461 −0.0802797 −0.0401399 0.999194i $$-0.512780\pi$$
−0.0401399 + 0.999194i $$0.512780\pi$$
$$762$$ 6.07223 0.219974
$$763$$ 2.49484 0.0903192
$$764$$ 13.4764 0.487559
$$765$$ 7.75872 0.280517
$$766$$ −90.9192 −3.28504
$$767$$ −26.7214 −0.964853
$$768$$ 27.6803 0.998828
$$769$$ −14.3668 −0.518081 −0.259041 0.965866i $$-0.583406\pi$$
−0.259041 + 0.965866i $$0.583406\pi$$
$$770$$ −2.92162 −0.105288
$$771$$ 4.15676 0.149702
$$772$$ 0.142380 0.00512436
$$773$$ 40.1568 1.44434 0.722169 0.691717i $$-0.243145\pi$$
0.722169 + 0.691717i $$0.243145\pi$$
$$774$$ −20.5958 −0.740302
$$775$$ −4.68035 −0.168123
$$776$$ −132.849 −4.76900
$$777$$ −2.15676 −0.0773732
$$778$$ −34.7792 −1.24690
$$779$$ −49.5585 −1.77562
$$780$$ −23.1773 −0.829880
$$781$$ −4.68035 −0.167476
$$782$$ 45.3361 1.62122
$$783$$ 1.41855 0.0506949
$$784$$ −80.7686 −2.88459
$$785$$ −10.9939 −0.392388
$$786$$ −23.5174 −0.838840
$$787$$ 49.5897 1.76768 0.883841 0.467788i $$-0.154949\pi$$
0.883841 + 0.467788i $$0.154949\pi$$
$$788$$ 112.781 4.01767
$$789$$ −18.7070 −0.665987
$$790$$ 21.9421 0.780666
$$791$$ −6.47027 −0.230056
$$792$$ −9.04945 −0.321558
$$793$$ 18.0410 0.640656
$$794$$ 14.4124 0.511477
$$795$$ 0.156755 0.00555954
$$796$$ 56.1978 1.99188
$$797$$ −46.7091 −1.65452 −0.827261 0.561818i $$-0.810102\pi$$
−0.827261 + 0.561818i $$0.810102\pi$$
$$798$$ −15.3730 −0.544198
$$799$$ 36.3135 1.28468
$$800$$ −19.3896 −0.685527
$$801$$ −12.8371 −0.453577
$$802$$ −5.41855 −0.191336
$$803$$ −10.4969 −0.370429
$$804$$ −46.3545 −1.63480
$$805$$ −2.32580 −0.0819736
$$806$$ −55.0349 −1.93852
$$807$$ 23.3607 0.822335
$$808$$ 140.948 4.95853
$$809$$ −18.5814 −0.653289 −0.326644 0.945147i $$-0.605918\pi$$
−0.326644 + 0.945147i $$0.605918\pi$$
$$810$$ −2.70928 −0.0951942
$$811$$ 27.3028 0.958732 0.479366 0.877615i $$-0.340867\pi$$
0.479366 + 0.877615i $$0.340867\pi$$
$$812$$ 8.16904 0.286677
$$813$$ −5.57531 −0.195535
$$814$$ 5.41855 0.189920
$$815$$ −6.52359 −0.228511
$$816$$ 107.358 3.75829
$$817$$ 40.0000 1.39942
$$818$$ −70.9770 −2.48165
$$819$$ −4.68035 −0.163545
$$820$$ −50.2967 −1.75644
$$821$$ −31.2085 −1.08918 −0.544592 0.838701i $$-0.683315\pi$$
−0.544592 + 0.838701i $$0.683315\pi$$
$$822$$ 41.6163 1.45154
$$823$$ −50.1855 −1.74936 −0.874678 0.484704i $$-0.838927\pi$$
−0.874678 + 0.484704i $$0.838927\pi$$
$$824$$ −61.8720 −2.15541
$$825$$ 1.00000 0.0348155
$$826$$ −17.9877 −0.625873
$$827$$ 27.3874 0.952352 0.476176 0.879350i $$-0.342023\pi$$
0.476176 + 0.879350i $$0.342023\pi$$
$$828$$ −11.5174 −0.400259
$$829$$ −26.1978 −0.909887 −0.454943 0.890520i $$-0.650341\pi$$
−0.454943 + 0.890520i $$0.650341\pi$$
$$830$$ 29.8576 1.03637
$$831$$ −26.0144 −0.902429
$$832$$ −107.886 −3.74029
$$833$$ −45.2885 −1.56915
$$834$$ −23.2495 −0.805065
$$835$$ 1.97334 0.0682902
$$836$$ 28.0989 0.971821
$$837$$ −4.68035 −0.161776
$$838$$ 7.68649 0.265525
$$839$$ 7.20394 0.248708 0.124354 0.992238i $$-0.460314\pi$$
0.124354 + 0.992238i $$0.460314\pi$$
$$840$$ −9.75872 −0.336708
$$841$$ −26.9877 −0.930611
$$842$$ −31.0928 −1.07153
$$843$$ −9.41855 −0.324392
$$844$$ 51.1338 1.76010
$$845$$ 5.83710 0.200802
$$846$$ −12.6803 −0.435959
$$847$$ 1.07838 0.0370535
$$848$$ 2.16904 0.0744852
$$849$$ 14.2413 0.488759
$$850$$ −21.0205 −0.720998
$$851$$ 4.31351 0.147865
$$852$$ −24.9939 −0.856275
$$853$$ 39.8043 1.36287 0.681437 0.731877i $$-0.261356\pi$$
0.681437 + 0.731877i $$0.261356\pi$$
$$854$$ 12.1445 0.415575
$$855$$ 5.26180 0.179950
$$856$$ −57.3751 −1.96104
$$857$$ −36.9504 −1.26220 −0.631100 0.775701i $$-0.717396\pi$$
−0.631100 + 0.775701i $$0.717396\pi$$
$$858$$ 11.7587 0.401436
$$859$$ 57.5052 1.96205 0.981025 0.193879i $$-0.0621070\pi$$
0.981025 + 0.193879i $$0.0621070\pi$$
$$860$$ 40.5958 1.38431
$$861$$ −10.1568 −0.346141
$$862$$ −63.8597 −2.17507
$$863$$ −1.89657 −0.0645599 −0.0322800 0.999479i $$-0.510277\pi$$
−0.0322800 + 0.999479i $$0.510277\pi$$
$$864$$ −19.3896 −0.659648
$$865$$ 3.75872 0.127800
$$866$$ 40.6225 1.38041
$$867$$ 43.1978 1.46707
$$868$$ −26.9528 −0.914838
$$869$$ −8.09890 −0.274736
$$870$$ −3.84324 −0.130298
$$871$$ 37.6742 1.27654
$$872$$ −20.9360 −0.708982
$$873$$ 14.6803 0.496854
$$874$$ 30.7460 1.04000
$$875$$ 1.07838 0.0364558
$$876$$ −56.0554 −1.89394
$$877$$ 32.5380 1.09873 0.549365 0.835583i $$-0.314870\pi$$
0.549365 + 0.835583i $$0.314870\pi$$
$$878$$ −12.9483 −0.436983
$$879$$ −15.7587 −0.531529
$$880$$ 13.8371 0.466449
$$881$$ 18.1978 0.613099 0.306550 0.951855i $$-0.400825\pi$$
0.306550 + 0.951855i $$0.400825\pi$$
$$882$$ 15.8143 0.532496
$$883$$ 36.3956 1.22481 0.612405 0.790545i $$-0.290202\pi$$
0.612405 + 0.790545i $$0.290202\pi$$
$$884$$ −179.826 −6.04821
$$885$$ 6.15676 0.206957
$$886$$ −54.7214 −1.83840
$$887$$ 27.8699 0.935780 0.467890 0.883787i $$-0.345014\pi$$
0.467890 + 0.883787i $$0.345014\pi$$
$$888$$ 18.0989 0.607359
$$889$$ −2.41694 −0.0810616
$$890$$ 34.7792 1.16580
$$891$$ 1.00000 0.0335013
$$892$$ −11.5174 −0.385633
$$893$$ 24.6270 0.824112
$$894$$ 49.0349 1.63997
$$895$$ 15.1506 0.506429
$$896$$ −30.8059 −1.02915
$$897$$ 9.36069 0.312544
$$898$$ 58.4412 1.95021
$$899$$ −6.63931 −0.221433
$$900$$ 5.34017 0.178006
$$901$$ 1.21622 0.0405182
$$902$$ 25.5174 0.849638
$$903$$ 8.19779 0.272805
$$904$$ 54.2967 1.80588
$$905$$ 4.83710 0.160791
$$906$$ −62.1399 −2.06446
$$907$$ −27.9376 −0.927653 −0.463826 0.885926i $$-0.653524\pi$$
−0.463826 + 0.885926i $$0.653524\pi$$
$$908$$ 51.5851 1.71191
$$909$$ −15.5753 −0.516600
$$910$$ 12.6803 0.420349
$$911$$ −11.8843 −0.393744 −0.196872 0.980429i $$-0.563078\pi$$
−0.196872 + 0.980429i $$0.563078\pi$$
$$912$$ 72.8080 2.41091
$$913$$ −11.0205 −0.364726
$$914$$ −76.3234 −2.52455
$$915$$ −4.15676 −0.137418
$$916$$ −17.9467 −0.592975
$$917$$ 9.36069 0.309117
$$918$$ −21.0205 −0.693781
$$919$$ −45.6041 −1.50434 −0.752170 0.658970i $$-0.770993\pi$$
−0.752170 + 0.658970i $$0.770993\pi$$
$$920$$ 19.5174 0.643471
$$921$$ −18.9216 −0.623489
$$922$$ 3.98771 0.131328
$$923$$ 20.3135 0.668627
$$924$$ 5.75872 0.189448
$$925$$ −2.00000 −0.0657596
$$926$$ 62.8659 2.06590
$$927$$ 6.83710 0.224560
$$928$$ −27.5052 −0.902901
$$929$$ −25.1506 −0.825165 −0.412582 0.910920i $$-0.635373\pi$$
−0.412582 + 0.910920i $$0.635373\pi$$
$$930$$ 12.6803 0.415805
$$931$$ −30.7136 −1.00660
$$932$$ −12.8059 −0.419471
$$933$$ −20.8781 −0.683520
$$934$$ −38.3545 −1.25500
$$935$$ 7.75872 0.253737
$$936$$ 39.2762 1.28378
$$937$$ 5.33403 0.174255 0.0871276 0.996197i $$-0.472231\pi$$
0.0871276 + 0.996197i $$0.472231\pi$$
$$938$$ 25.3607 0.828056
$$939$$ 6.31351 0.206034
$$940$$ 24.9939 0.815210
$$941$$ 56.8203 1.85229 0.926144 0.377170i $$-0.123103\pi$$
0.926144 + 0.377170i $$0.123103\pi$$
$$942$$ 29.7854 0.970460
$$943$$ 20.3135 0.661499
$$944$$ 85.1917 2.77275
$$945$$ 1.07838 0.0350796
$$946$$ −20.5958 −0.669628
$$947$$ −20.9939 −0.682209 −0.341104 0.940025i $$-0.610801\pi$$
−0.341104 + 0.940025i $$0.610801\pi$$
$$948$$ −43.2495 −1.40468
$$949$$ 45.5585 1.47889
$$950$$ −14.2557 −0.462514
$$951$$ 31.3607 1.01694
$$952$$ −75.7152 −2.45395
$$953$$ −25.2351 −0.817446 −0.408723 0.912658i $$-0.634026\pi$$
−0.408723 + 0.912658i $$0.634026\pi$$
$$954$$ −0.424694 −0.0137500
$$955$$ 2.52359 0.0816615
$$956$$ −38.4703 −1.24422
$$957$$ 1.41855 0.0458552
$$958$$ 37.5052 1.21174
$$959$$ −16.5646 −0.534900
$$960$$ 24.8576 0.802276
$$961$$ −9.09436 −0.293367
$$962$$ −23.5174 −0.758233
$$963$$ 6.34017 0.204309
$$964$$ −27.7899 −0.895053
$$965$$ 0.0266620 0.000858280 0
$$966$$ 6.30122 0.202739
$$967$$ 13.1317 0.422287 0.211144 0.977455i $$-0.432281\pi$$
0.211144 + 0.977455i $$0.432281\pi$$
$$968$$ −9.04945 −0.290860
$$969$$ 40.8248 1.31148
$$970$$ −39.7731 −1.27704
$$971$$ 8.94053 0.286915 0.143458 0.989656i $$-0.454178\pi$$
0.143458 + 0.989656i $$0.454178\pi$$
$$972$$ 5.34017 0.171286
$$973$$ 9.25404 0.296671
$$974$$ 111.064 3.55871
$$975$$ −4.34017 −0.138997
$$976$$ −57.5174 −1.84109
$$977$$ 50.3956 1.61230 0.806149 0.591713i $$-0.201548\pi$$
0.806149 + 0.591713i $$0.201548\pi$$
$$978$$ 17.6742 0.565159
$$979$$ −12.8371 −0.410276
$$980$$ −31.1711 −0.995725
$$981$$ 2.31351 0.0738647
$$982$$ −94.3833 −3.01189
$$983$$ −32.1978 −1.02695 −0.513475 0.858105i $$-0.671642\pi$$
−0.513475 + 0.858105i $$0.671642\pi$$
$$984$$ 85.2327 2.71712
$$985$$ 21.1194 0.672921
$$986$$ −29.8187 −0.949620
$$987$$ 5.04718 0.160654
$$988$$ −121.954 −3.87988
$$989$$ −16.3956 −0.521349
$$990$$ −2.70928 −0.0861064
$$991$$ −46.7747 −1.48585 −0.742924 0.669376i $$-0.766562\pi$$
−0.742924 + 0.669376i $$0.766562\pi$$
$$992$$ 90.7501 2.88132
$$993$$ 19.2039 0.609419
$$994$$ 13.6742 0.433719
$$995$$ 10.5236 0.333620
$$996$$ −58.8515 −1.86478
$$997$$ 38.2122 1.21019 0.605096 0.796153i $$-0.293135\pi$$
0.605096 + 0.796153i $$0.293135\pi$$
$$998$$ −41.0472 −1.29933
$$999$$ −2.00000 −0.0632772
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 165.2.a.c.1.1 3
3.2 odd 2 495.2.a.e.1.3 3
4.3 odd 2 2640.2.a.be.1.2 3
5.2 odd 4 825.2.c.g.199.1 6
5.3 odd 4 825.2.c.g.199.6 6
5.4 even 2 825.2.a.k.1.3 3
7.6 odd 2 8085.2.a.bk.1.1 3
11.10 odd 2 1815.2.a.m.1.3 3
12.11 even 2 7920.2.a.cj.1.2 3
15.2 even 4 2475.2.c.r.199.6 6
15.8 even 4 2475.2.c.r.199.1 6
15.14 odd 2 2475.2.a.bb.1.1 3
33.32 even 2 5445.2.a.z.1.1 3
55.54 odd 2 9075.2.a.cf.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.1 3 1.1 even 1 trivial
495.2.a.e.1.3 3 3.2 odd 2
825.2.a.k.1.3 3 5.4 even 2
825.2.c.g.199.1 6 5.2 odd 4
825.2.c.g.199.6 6 5.3 odd 4
1815.2.a.m.1.3 3 11.10 odd 2
2475.2.a.bb.1.1 3 15.14 odd 2
2475.2.c.r.199.1 6 15.8 even 4
2475.2.c.r.199.6 6 15.2 even 4
2640.2.a.be.1.2 3 4.3 odd 2
5445.2.a.z.1.1 3 33.32 even 2
7920.2.a.cj.1.2 3 12.11 even 2
8085.2.a.bk.1.1 3 7.6 odd 2
9075.2.a.cf.1.1 3 55.54 odd 2