# Properties

 Label 385.2.a.f.1.1 Level $385$ Weight $2$ Character 385.1 Self dual yes Analytic conductor $3.074$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$385 = 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 385.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.07424047782$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 385.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.67513 q^{2} -2.48119 q^{3} +5.15633 q^{4} -1.00000 q^{5} +6.63752 q^{6} -1.00000 q^{7} -8.44358 q^{8} +3.15633 q^{9} +O(q^{10})$$ $$q-2.67513 q^{2} -2.48119 q^{3} +5.15633 q^{4} -1.00000 q^{5} +6.63752 q^{6} -1.00000 q^{7} -8.44358 q^{8} +3.15633 q^{9} +2.67513 q^{10} -1.00000 q^{11} -12.7938 q^{12} +5.83146 q^{13} +2.67513 q^{14} +2.48119 q^{15} +12.2750 q^{16} +5.44358 q^{17} -8.44358 q^{18} -1.35026 q^{19} -5.15633 q^{20} +2.48119 q^{21} +2.67513 q^{22} -3.19394 q^{23} +20.9502 q^{24} +1.00000 q^{25} -15.5999 q^{26} -0.387873 q^{27} -5.15633 q^{28} -3.61213 q^{29} -6.63752 q^{30} -5.28726 q^{31} -15.9502 q^{32} +2.48119 q^{33} -14.5623 q^{34} +1.00000 q^{35} +16.2750 q^{36} -8.54420 q^{37} +3.61213 q^{38} -14.4690 q^{39} +8.44358 q^{40} -5.02539 q^{41} -6.63752 q^{42} +5.89446 q^{43} -5.15633 q^{44} -3.15633 q^{45} +8.54420 q^{46} -11.8315 q^{47} -30.4568 q^{48} +1.00000 q^{49} -2.67513 q^{50} -13.5066 q^{51} +30.0689 q^{52} -0.231548 q^{53} +1.03761 q^{54} +1.00000 q^{55} +8.44358 q^{56} +3.35026 q^{57} +9.66291 q^{58} +13.5999 q^{59} +12.7938 q^{60} -1.41327 q^{61} +14.1441 q^{62} -3.15633 q^{63} +18.1187 q^{64} -5.83146 q^{65} -6.63752 q^{66} -10.8568 q^{67} +28.0689 q^{68} +7.92478 q^{69} -2.67513 q^{70} -15.5369 q^{71} -26.6507 q^{72} -11.3684 q^{73} +22.8568 q^{74} -2.48119 q^{75} -6.96239 q^{76} +1.00000 q^{77} +38.7064 q^{78} +1.96968 q^{79} -12.2750 q^{80} -8.50659 q^{81} +13.4436 q^{82} +10.6253 q^{83} +12.7938 q^{84} -5.44358 q^{85} -15.7685 q^{86} +8.96239 q^{87} +8.44358 q^{88} +7.22425 q^{89} +8.44358 q^{90} -5.83146 q^{91} -16.4690 q^{92} +13.1187 q^{93} +31.6507 q^{94} +1.35026 q^{95} +39.5755 q^{96} -0.836381 q^{97} -2.67513 q^{98} -3.15633 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 3 q^{7} - 9 q^{8} - q^{9} + O(q^{10})$$ $$3 q - 3 q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 3 q^{7} - 9 q^{8} - q^{9} + 3 q^{10} - 3 q^{11} - 12 q^{12} + 2 q^{13} + 3 q^{14} + 2 q^{15} + 5 q^{16} - 9 q^{18} + 6 q^{19} - 5 q^{20} + 2 q^{21} + 3 q^{22} - 10 q^{23} + 26 q^{24} + 3 q^{25} - 20 q^{26} - 2 q^{27} - 5 q^{28} - 10 q^{29} - 4 q^{30} - 10 q^{31} - 11 q^{32} + 2 q^{33} - 6 q^{34} + 3 q^{35} + 17 q^{36} - 16 q^{37} + 10 q^{38} - 12 q^{39} + 9 q^{40} - 4 q^{42} - 2 q^{43} - 5 q^{44} + q^{45} + 16 q^{46} - 20 q^{47} - 34 q^{48} + 3 q^{49} - 3 q^{50} - 20 q^{51} + 32 q^{52} - 12 q^{53} + 14 q^{54} + 3 q^{55} + 9 q^{56} - 2 q^{58} + 14 q^{59} + 12 q^{60} + 10 q^{61} + 6 q^{62} + q^{63} + 33 q^{64} - 2 q^{65} - 4 q^{66} - 2 q^{67} + 26 q^{68} + 2 q^{69} - 3 q^{70} - 24 q^{71} - 23 q^{72} + 4 q^{73} + 38 q^{74} - 2 q^{75} - 10 q^{76} + 3 q^{77} + 42 q^{78} + 8 q^{79} - 5 q^{80} - 5 q^{81} + 24 q^{82} - 10 q^{83} + 12 q^{84} - 36 q^{86} + 16 q^{87} + 9 q^{88} + 20 q^{89} + 9 q^{90} - 2 q^{91} - 18 q^{92} + 18 q^{93} + 38 q^{94} - 6 q^{95} + 40 q^{96} - 3 q^{98} + q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.67513 −1.89160 −0.945802 0.324745i $$-0.894721\pi$$
−0.945802 + 0.324745i $$0.894721\pi$$
$$3$$ −2.48119 −1.43252 −0.716259 0.697834i $$-0.754147\pi$$
−0.716259 + 0.697834i $$0.754147\pi$$
$$4$$ 5.15633 2.57816
$$5$$ −1.00000 −0.447214
$$6$$ 6.63752 2.70976
$$7$$ −1.00000 −0.377964
$$8$$ −8.44358 −2.98526
$$9$$ 3.15633 1.05211
$$10$$ 2.67513 0.845951
$$11$$ −1.00000 −0.301511
$$12$$ −12.7938 −3.69326
$$13$$ 5.83146 1.61735 0.808677 0.588252i $$-0.200184\pi$$
0.808677 + 0.588252i $$0.200184\pi$$
$$14$$ 2.67513 0.714959
$$15$$ 2.48119 0.640642
$$16$$ 12.2750 3.06876
$$17$$ 5.44358 1.32026 0.660131 0.751150i $$-0.270501\pi$$
0.660131 + 0.751150i $$0.270501\pi$$
$$18$$ −8.44358 −1.99017
$$19$$ −1.35026 −0.309771 −0.154886 0.987932i $$-0.549501\pi$$
−0.154886 + 0.987932i $$0.549501\pi$$
$$20$$ −5.15633 −1.15299
$$21$$ 2.48119 0.541441
$$22$$ 2.67513 0.570340
$$23$$ −3.19394 −0.665982 −0.332991 0.942930i $$-0.608058\pi$$
−0.332991 + 0.942930i $$0.608058\pi$$
$$24$$ 20.9502 4.27644
$$25$$ 1.00000 0.200000
$$26$$ −15.5999 −3.05939
$$27$$ −0.387873 −0.0746462
$$28$$ −5.15633 −0.974454
$$29$$ −3.61213 −0.670755 −0.335378 0.942084i $$-0.608864\pi$$
−0.335378 + 0.942084i $$0.608864\pi$$
$$30$$ −6.63752 −1.21184
$$31$$ −5.28726 −0.949620 −0.474810 0.880088i $$-0.657483\pi$$
−0.474810 + 0.880088i $$0.657483\pi$$
$$32$$ −15.9502 −2.81962
$$33$$ 2.48119 0.431920
$$34$$ −14.5623 −2.49741
$$35$$ 1.00000 0.169031
$$36$$ 16.2750 2.71251
$$37$$ −8.54420 −1.40466 −0.702329 0.711853i $$-0.747856\pi$$
−0.702329 + 0.711853i $$0.747856\pi$$
$$38$$ 3.61213 0.585964
$$39$$ −14.4690 −2.31689
$$40$$ 8.44358 1.33505
$$41$$ −5.02539 −0.784834 −0.392417 0.919787i $$-0.628361\pi$$
−0.392417 + 0.919787i $$0.628361\pi$$
$$42$$ −6.63752 −1.02419
$$43$$ 5.89446 0.898897 0.449448 0.893306i $$-0.351621\pi$$
0.449448 + 0.893306i $$0.351621\pi$$
$$44$$ −5.15633 −0.777345
$$45$$ −3.15633 −0.470517
$$46$$ 8.54420 1.25977
$$47$$ −11.8315 −1.72580 −0.862898 0.505379i $$-0.831353\pi$$
−0.862898 + 0.505379i $$0.831353\pi$$
$$48$$ −30.4568 −4.39605
$$49$$ 1.00000 0.142857
$$50$$ −2.67513 −0.378321
$$51$$ −13.5066 −1.89130
$$52$$ 30.0689 4.16980
$$53$$ −0.231548 −0.0318056 −0.0159028 0.999874i $$-0.505062\pi$$
−0.0159028 + 0.999874i $$0.505062\pi$$
$$54$$ 1.03761 0.141201
$$55$$ 1.00000 0.134840
$$56$$ 8.44358 1.12832
$$57$$ 3.35026 0.443753
$$58$$ 9.66291 1.26880
$$59$$ 13.5999 1.77056 0.885279 0.465061i $$-0.153968\pi$$
0.885279 + 0.465061i $$0.153968\pi$$
$$60$$ 12.7938 1.65168
$$61$$ −1.41327 −0.180950 −0.0904751 0.995899i $$-0.528839\pi$$
−0.0904751 + 0.995899i $$0.528839\pi$$
$$62$$ 14.1441 1.79630
$$63$$ −3.15633 −0.397660
$$64$$ 18.1187 2.26484
$$65$$ −5.83146 −0.723303
$$66$$ −6.63752 −0.817022
$$67$$ −10.8568 −1.32638 −0.663188 0.748453i $$-0.730797\pi$$
−0.663188 + 0.748453i $$0.730797\pi$$
$$68$$ 28.0689 3.40385
$$69$$ 7.92478 0.954031
$$70$$ −2.67513 −0.319739
$$71$$ −15.5369 −1.84389 −0.921946 0.387319i $$-0.873401\pi$$
−0.921946 + 0.387319i $$0.873401\pi$$
$$72$$ −26.6507 −3.14081
$$73$$ −11.3684 −1.33057 −0.665283 0.746591i $$-0.731689\pi$$
−0.665283 + 0.746591i $$0.731689\pi$$
$$74$$ 22.8568 2.65705
$$75$$ −2.48119 −0.286504
$$76$$ −6.96239 −0.798641
$$77$$ 1.00000 0.113961
$$78$$ 38.7064 4.38264
$$79$$ 1.96968 0.221607 0.110803 0.993842i $$-0.464658\pi$$
0.110803 + 0.993842i $$0.464658\pi$$
$$80$$ −12.2750 −1.37239
$$81$$ −8.50659 −0.945176
$$82$$ 13.4436 1.48460
$$83$$ 10.6253 1.16628 0.583139 0.812372i $$-0.301824\pi$$
0.583139 + 0.812372i $$0.301824\pi$$
$$84$$ 12.7938 1.39592
$$85$$ −5.44358 −0.590439
$$86$$ −15.7685 −1.70036
$$87$$ 8.96239 0.960869
$$88$$ 8.44358 0.900089
$$89$$ 7.22425 0.765769 0.382885 0.923796i $$-0.374931\pi$$
0.382885 + 0.923796i $$0.374931\pi$$
$$90$$ 8.44358 0.890032
$$91$$ −5.83146 −0.611303
$$92$$ −16.4690 −1.71701
$$93$$ 13.1187 1.36035
$$94$$ 31.6507 3.26452
$$95$$ 1.35026 0.138534
$$96$$ 39.5755 4.03915
$$97$$ −0.836381 −0.0849216 −0.0424608 0.999098i $$-0.513520\pi$$
−0.0424608 + 0.999098i $$0.513520\pi$$
$$98$$ −2.67513 −0.270229
$$99$$ −3.15633 −0.317223
$$100$$ 5.15633 0.515633
$$101$$ 7.41327 0.737648 0.368824 0.929499i $$-0.379761\pi$$
0.368824 + 0.929499i $$0.379761\pi$$
$$102$$ 36.1319 3.57759
$$103$$ 4.21933 0.415743 0.207871 0.978156i $$-0.433346\pi$$
0.207871 + 0.978156i $$0.433346\pi$$
$$104$$ −49.2384 −4.82822
$$105$$ −2.48119 −0.242140
$$106$$ 0.619421 0.0601635
$$107$$ −11.5369 −1.11531 −0.557657 0.830071i $$-0.688300\pi$$
−0.557657 + 0.830071i $$0.688300\pi$$
$$108$$ −2.00000 −0.192450
$$109$$ −2.18664 −0.209442 −0.104721 0.994502i $$-0.533395\pi$$
−0.104721 + 0.994502i $$0.533395\pi$$
$$110$$ −2.67513 −0.255064
$$111$$ 21.1998 2.01220
$$112$$ −12.2750 −1.15988
$$113$$ −9.35026 −0.879599 −0.439799 0.898096i $$-0.644950\pi$$
−0.439799 + 0.898096i $$0.644950\pi$$
$$114$$ −8.96239 −0.839405
$$115$$ 3.19394 0.297836
$$116$$ −18.6253 −1.72932
$$117$$ 18.4060 1.70163
$$118$$ −36.3815 −3.34919
$$119$$ −5.44358 −0.499012
$$120$$ −20.9502 −1.91248
$$121$$ 1.00000 0.0909091
$$122$$ 3.78067 0.342286
$$123$$ 12.4690 1.12429
$$124$$ −27.2628 −2.44827
$$125$$ −1.00000 −0.0894427
$$126$$ 8.44358 0.752214
$$127$$ −16.9624 −1.50517 −0.752584 0.658496i $$-0.771193\pi$$
−0.752584 + 0.658496i $$0.771193\pi$$
$$128$$ −16.5696 −1.46456
$$129$$ −14.6253 −1.28769
$$130$$ 15.5999 1.36820
$$131$$ −9.92478 −0.867132 −0.433566 0.901122i $$-0.642745\pi$$
−0.433566 + 0.901122i $$0.642745\pi$$
$$132$$ 12.7938 1.11356
$$133$$ 1.35026 0.117083
$$134$$ 29.0435 2.50898
$$135$$ 0.387873 0.0333828
$$136$$ −45.9633 −3.94132
$$137$$ −10.9927 −0.939170 −0.469585 0.882887i $$-0.655596\pi$$
−0.469585 + 0.882887i $$0.655596\pi$$
$$138$$ −21.1998 −1.80465
$$139$$ 6.88717 0.584162 0.292081 0.956394i $$-0.405652\pi$$
0.292081 + 0.956394i $$0.405652\pi$$
$$140$$ 5.15633 0.435789
$$141$$ 29.3561 2.47223
$$142$$ 41.5633 3.48791
$$143$$ −5.83146 −0.487651
$$144$$ 38.7440 3.22867
$$145$$ 3.61213 0.299971
$$146$$ 30.4119 2.51690
$$147$$ −2.48119 −0.204645
$$148$$ −44.0567 −3.62144
$$149$$ 22.8119 1.86883 0.934414 0.356190i $$-0.115924\pi$$
0.934414 + 0.356190i $$0.115924\pi$$
$$150$$ 6.63752 0.541951
$$151$$ −3.24472 −0.264052 −0.132026 0.991246i $$-0.542148\pi$$
−0.132026 + 0.991246i $$0.542148\pi$$
$$152$$ 11.4010 0.924747
$$153$$ 17.1817 1.38906
$$154$$ −2.67513 −0.215568
$$155$$ 5.28726 0.424683
$$156$$ −74.6067 −5.97332
$$157$$ −5.42548 −0.433001 −0.216500 0.976283i $$-0.569464\pi$$
−0.216500 + 0.976283i $$0.569464\pi$$
$$158$$ −5.26916 −0.419192
$$159$$ 0.574515 0.0455620
$$160$$ 15.9502 1.26097
$$161$$ 3.19394 0.251717
$$162$$ 22.7562 1.78790
$$163$$ 3.38058 0.264787 0.132394 0.991197i $$-0.457734\pi$$
0.132394 + 0.991197i $$0.457734\pi$$
$$164$$ −25.9126 −2.02343
$$165$$ −2.48119 −0.193161
$$166$$ −28.4241 −2.20614
$$167$$ 11.2750 0.872489 0.436244 0.899828i $$-0.356308\pi$$
0.436244 + 0.899828i $$0.356308\pi$$
$$168$$ −20.9502 −1.61634
$$169$$ 21.0059 1.61584
$$170$$ 14.5623 1.11688
$$171$$ −4.26187 −0.325913
$$172$$ 30.3938 2.31750
$$173$$ −8.98049 −0.682774 −0.341387 0.939923i $$-0.610897\pi$$
−0.341387 + 0.939923i $$0.610897\pi$$
$$174$$ −23.9756 −1.81758
$$175$$ −1.00000 −0.0755929
$$176$$ −12.2750 −0.925266
$$177$$ −33.7440 −2.53636
$$178$$ −19.3258 −1.44853
$$179$$ −26.2374 −1.96108 −0.980539 0.196326i $$-0.937099\pi$$
−0.980539 + 0.196326i $$0.937099\pi$$
$$180$$ −16.2750 −1.21307
$$181$$ −11.1998 −0.832476 −0.416238 0.909256i $$-0.636652\pi$$
−0.416238 + 0.909256i $$0.636652\pi$$
$$182$$ 15.5999 1.15634
$$183$$ 3.50659 0.259214
$$184$$ 26.9683 1.98813
$$185$$ 8.54420 0.628182
$$186$$ −35.0943 −2.57324
$$187$$ −5.44358 −0.398074
$$188$$ −61.0068 −4.44938
$$189$$ 0.387873 0.0282136
$$190$$ −3.61213 −0.262051
$$191$$ 11.1998 0.810390 0.405195 0.914230i $$-0.367204\pi$$
0.405195 + 0.914230i $$0.367204\pi$$
$$192$$ −44.9560 −3.24442
$$193$$ 0.604833 0.0435368 0.0217684 0.999763i $$-0.493070\pi$$
0.0217684 + 0.999763i $$0.493070\pi$$
$$194$$ 2.23743 0.160638
$$195$$ 14.4690 1.03614
$$196$$ 5.15633 0.368309
$$197$$ 15.3054 1.09046 0.545231 0.838286i $$-0.316442\pi$$
0.545231 + 0.838286i $$0.316442\pi$$
$$198$$ 8.44358 0.600059
$$199$$ −12.5623 −0.890518 −0.445259 0.895402i $$-0.646888\pi$$
−0.445259 + 0.895402i $$0.646888\pi$$
$$200$$ −8.44358 −0.597051
$$201$$ 26.9380 1.90006
$$202$$ −19.8315 −1.39534
$$203$$ 3.61213 0.253522
$$204$$ −69.6444 −4.87608
$$205$$ 5.02539 0.350989
$$206$$ −11.2873 −0.786421
$$207$$ −10.0811 −0.700685
$$208$$ 71.5814 4.96327
$$209$$ 1.35026 0.0933996
$$210$$ 6.63752 0.458032
$$211$$ −4.43866 −0.305570 −0.152785 0.988259i $$-0.548824\pi$$
−0.152785 + 0.988259i $$0.548824\pi$$
$$212$$ −1.19394 −0.0819999
$$213$$ 38.5501 2.64141
$$214$$ 30.8627 2.10973
$$215$$ −5.89446 −0.401999
$$216$$ 3.27504 0.222838
$$217$$ 5.28726 0.358922
$$218$$ 5.84955 0.396182
$$219$$ 28.2071 1.90606
$$220$$ 5.15633 0.347639
$$221$$ 31.7440 2.13533
$$222$$ −56.7123 −3.80628
$$223$$ −7.78067 −0.521032 −0.260516 0.965469i $$-0.583893\pi$$
−0.260516 + 0.965469i $$0.583893\pi$$
$$224$$ 15.9502 1.06572
$$225$$ 3.15633 0.210422
$$226$$ 25.0132 1.66385
$$227$$ −10.4485 −0.693492 −0.346746 0.937959i $$-0.612713\pi$$
−0.346746 + 0.937959i $$0.612713\pi$$
$$228$$ 17.2750 1.14407
$$229$$ −29.4518 −1.94623 −0.973116 0.230316i $$-0.926024\pi$$
−0.973116 + 0.230316i $$0.926024\pi$$
$$230$$ −8.54420 −0.563388
$$231$$ −2.48119 −0.163251
$$232$$ 30.4993 2.00238
$$233$$ −8.73084 −0.571976 −0.285988 0.958233i $$-0.592322\pi$$
−0.285988 + 0.958233i $$0.592322\pi$$
$$234$$ −49.2384 −3.21881
$$235$$ 11.8315 0.771799
$$236$$ 70.1255 4.56478
$$237$$ −4.88717 −0.317456
$$238$$ 14.5623 0.943933
$$239$$ −21.2144 −1.37225 −0.686123 0.727486i $$-0.740689\pi$$
−0.686123 + 0.727486i $$0.740689\pi$$
$$240$$ 30.4568 1.96598
$$241$$ −9.33804 −0.601516 −0.300758 0.953700i $$-0.597240\pi$$
−0.300758 + 0.953700i $$0.597240\pi$$
$$242$$ −2.67513 −0.171964
$$243$$ 22.2701 1.42863
$$244$$ −7.28726 −0.466519
$$245$$ −1.00000 −0.0638877
$$246$$ −33.3561 −2.12671
$$247$$ −7.87399 −0.501010
$$248$$ 44.6434 2.83486
$$249$$ −26.3634 −1.67071
$$250$$ 2.67513 0.169190
$$251$$ 1.87636 0.118435 0.0592174 0.998245i $$-0.481139\pi$$
0.0592174 + 0.998245i $$0.481139\pi$$
$$252$$ −16.2750 −1.02523
$$253$$ 3.19394 0.200801
$$254$$ 45.3766 2.84718
$$255$$ 13.5066 0.845815
$$256$$ 8.08840 0.505525
$$257$$ 27.1392 1.69290 0.846448 0.532472i $$-0.178737\pi$$
0.846448 + 0.532472i $$0.178737\pi$$
$$258$$ 39.1246 2.43579
$$259$$ 8.54420 0.530911
$$260$$ −30.0689 −1.86479
$$261$$ −11.4010 −0.705707
$$262$$ 26.5501 1.64027
$$263$$ 12.8119 0.790018 0.395009 0.918677i $$-0.370741\pi$$
0.395009 + 0.918677i $$0.370741\pi$$
$$264$$ −20.9502 −1.28939
$$265$$ 0.231548 0.0142239
$$266$$ −3.61213 −0.221474
$$267$$ −17.9248 −1.09698
$$268$$ −55.9814 −3.41961
$$269$$ −6.26187 −0.381793 −0.190896 0.981610i $$-0.561139\pi$$
−0.190896 + 0.981610i $$0.561139\pi$$
$$270$$ −1.03761 −0.0631470
$$271$$ −5.73813 −0.348567 −0.174283 0.984696i $$-0.555761\pi$$
−0.174283 + 0.984696i $$0.555761\pi$$
$$272$$ 66.8202 4.05157
$$273$$ 14.4690 0.875702
$$274$$ 29.4069 1.77654
$$275$$ −1.00000 −0.0603023
$$276$$ 40.8627 2.45965
$$277$$ −8.35756 −0.502157 −0.251078 0.967967i $$-0.580785\pi$$
−0.251078 + 0.967967i $$0.580785\pi$$
$$278$$ −18.4241 −1.10500
$$279$$ −16.6883 −0.999103
$$280$$ −8.44358 −0.504601
$$281$$ −8.44851 −0.503996 −0.251998 0.967728i $$-0.581088\pi$$
−0.251998 + 0.967728i $$0.581088\pi$$
$$282$$ −78.5315 −4.67648
$$283$$ 0.836381 0.0497177 0.0248588 0.999691i $$-0.492086\pi$$
0.0248588 + 0.999691i $$0.492086\pi$$
$$284$$ −80.1133 −4.75385
$$285$$ −3.35026 −0.198452
$$286$$ 15.5999 0.922442
$$287$$ 5.02539 0.296640
$$288$$ −50.3439 −2.96654
$$289$$ 12.6326 0.743094
$$290$$ −9.66291 −0.567426
$$291$$ 2.07522 0.121652
$$292$$ −58.6190 −3.43042
$$293$$ −2.71862 −0.158824 −0.0794118 0.996842i $$-0.525304\pi$$
−0.0794118 + 0.996842i $$0.525304\pi$$
$$294$$ 6.63752 0.387108
$$295$$ −13.5999 −0.791817
$$296$$ 72.1436 4.19326
$$297$$ 0.387873 0.0225067
$$298$$ −61.0249 −3.53508
$$299$$ −18.6253 −1.07713
$$300$$ −12.7938 −0.738653
$$301$$ −5.89446 −0.339751
$$302$$ 8.68006 0.499481
$$303$$ −18.3938 −1.05669
$$304$$ −16.5745 −0.950614
$$305$$ 1.41327 0.0809234
$$306$$ −45.9633 −2.62755
$$307$$ −8.36344 −0.477326 −0.238663 0.971102i $$-0.576709\pi$$
−0.238663 + 0.971102i $$0.576709\pi$$
$$308$$ 5.15633 0.293809
$$309$$ −10.4690 −0.595559
$$310$$ −14.1441 −0.803331
$$311$$ 4.43629 0.251559 0.125779 0.992058i $$-0.459857\pi$$
0.125779 + 0.992058i $$0.459857\pi$$
$$312$$ 122.170 6.91651
$$313$$ 29.7889 1.68377 0.841885 0.539658i $$-0.181446\pi$$
0.841885 + 0.539658i $$0.181446\pi$$
$$314$$ 14.5139 0.819066
$$315$$ 3.15633 0.177839
$$316$$ 10.1563 0.571338
$$317$$ 15.4010 0.865009 0.432504 0.901632i $$-0.357630\pi$$
0.432504 + 0.901632i $$0.357630\pi$$
$$318$$ −1.53690 −0.0861853
$$319$$ 3.61213 0.202240
$$320$$ −18.1187 −1.01287
$$321$$ 28.6253 1.59771
$$322$$ −8.54420 −0.476150
$$323$$ −7.35026 −0.408980
$$324$$ −43.8627 −2.43682
$$325$$ 5.83146 0.323471
$$326$$ −9.04349 −0.500873
$$327$$ 5.42548 0.300030
$$328$$ 42.4323 2.34293
$$329$$ 11.8315 0.652289
$$330$$ 6.63752 0.365383
$$331$$ 6.26187 0.344183 0.172092 0.985081i $$-0.444947\pi$$
0.172092 + 0.985081i $$0.444947\pi$$
$$332$$ 54.7875 3.00685
$$333$$ −26.9683 −1.47785
$$334$$ −30.1622 −1.65040
$$335$$ 10.8568 0.593173
$$336$$ 30.4568 1.66155
$$337$$ −15.8700 −0.864495 −0.432248 0.901755i $$-0.642279\pi$$
−0.432248 + 0.901755i $$0.642279\pi$$
$$338$$ −56.1935 −3.05652
$$339$$ 23.1998 1.26004
$$340$$ −28.0689 −1.52225
$$341$$ 5.28726 0.286321
$$342$$ 11.4010 0.616498
$$343$$ −1.00000 −0.0539949
$$344$$ −49.7704 −2.68344
$$345$$ −7.92478 −0.426656
$$346$$ 24.0240 1.29154
$$347$$ 6.79147 0.364585 0.182293 0.983244i $$-0.441648\pi$$
0.182293 + 0.983244i $$0.441648\pi$$
$$348$$ 46.2130 2.47728
$$349$$ 26.7489 1.43184 0.715919 0.698183i $$-0.246008\pi$$
0.715919 + 0.698183i $$0.246008\pi$$
$$350$$ 2.67513 0.142992
$$351$$ −2.26187 −0.120729
$$352$$ 15.9502 0.850147
$$353$$ −16.8627 −0.897512 −0.448756 0.893654i $$-0.648133\pi$$
−0.448756 + 0.893654i $$0.648133\pi$$
$$354$$ 90.2697 4.79778
$$355$$ 15.5369 0.824613
$$356$$ 37.2506 1.97428
$$357$$ 13.5066 0.714844
$$358$$ 70.1886 3.70958
$$359$$ −3.79289 −0.200181 −0.100091 0.994978i $$-0.531913\pi$$
−0.100091 + 0.994978i $$0.531913\pi$$
$$360$$ 26.6507 1.40461
$$361$$ −17.1768 −0.904042
$$362$$ 29.9610 1.57471
$$363$$ −2.48119 −0.130229
$$364$$ −30.0689 −1.57604
$$365$$ 11.3684 0.595047
$$366$$ −9.38058 −0.490331
$$367$$ −6.36977 −0.332500 −0.166250 0.986084i $$-0.553166\pi$$
−0.166250 + 0.986084i $$0.553166\pi$$
$$368$$ −39.2057 −2.04374
$$369$$ −15.8618 −0.825731
$$370$$ −22.8568 −1.18827
$$371$$ 0.231548 0.0120214
$$372$$ 67.6444 3.50720
$$373$$ −21.3317 −1.10451 −0.552257 0.833674i $$-0.686233\pi$$
−0.552257 + 0.833674i $$0.686233\pi$$
$$374$$ 14.5623 0.752998
$$375$$ 2.48119 0.128128
$$376$$ 99.8999 5.15194
$$377$$ −21.0640 −1.08485
$$378$$ −1.03761 −0.0533690
$$379$$ 24.7875 1.27325 0.636624 0.771174i $$-0.280330\pi$$
0.636624 + 0.771174i $$0.280330\pi$$
$$380$$ 6.96239 0.357163
$$381$$ 42.0870 2.15618
$$382$$ −29.9610 −1.53294
$$383$$ −5.45817 −0.278900 −0.139450 0.990229i $$-0.544533\pi$$
−0.139450 + 0.990229i $$0.544533\pi$$
$$384$$ 41.1124 2.09801
$$385$$ −1.00000 −0.0509647
$$386$$ −1.61801 −0.0823544
$$387$$ 18.6048 0.945737
$$388$$ −4.31265 −0.218942
$$389$$ −13.7235 −0.695811 −0.347906 0.937530i $$-0.613107\pi$$
−0.347906 + 0.937530i $$0.613107\pi$$
$$390$$ −38.7064 −1.95997
$$391$$ −17.3865 −0.879271
$$392$$ −8.44358 −0.426465
$$393$$ 24.6253 1.24218
$$394$$ −40.9438 −2.06272
$$395$$ −1.96968 −0.0991055
$$396$$ −16.2750 −0.817851
$$397$$ 2.11142 0.105969 0.0529846 0.998595i $$-0.483127\pi$$
0.0529846 + 0.998595i $$0.483127\pi$$
$$398$$ 33.6058 1.68451
$$399$$ −3.35026 −0.167723
$$400$$ 12.2750 0.613752
$$401$$ 19.1490 0.956257 0.478128 0.878290i $$-0.341315\pi$$
0.478128 + 0.878290i $$0.341315\pi$$
$$402$$ −72.0625 −3.59415
$$403$$ −30.8324 −1.53587
$$404$$ 38.2252 1.90178
$$405$$ 8.50659 0.422696
$$406$$ −9.66291 −0.479562
$$407$$ 8.54420 0.423520
$$408$$ 114.044 5.64602
$$409$$ −18.6883 −0.924077 −0.462039 0.886860i $$-0.652882\pi$$
−0.462039 + 0.886860i $$0.652882\pi$$
$$410$$ −13.4436 −0.663931
$$411$$ 27.2750 1.34538
$$412$$ 21.7562 1.07185
$$413$$ −13.5999 −0.669208
$$414$$ 26.9683 1.32542
$$415$$ −10.6253 −0.521575
$$416$$ −93.0127 −4.56032
$$417$$ −17.0884 −0.836822
$$418$$ −3.61213 −0.176675
$$419$$ −0.773377 −0.0377819 −0.0188910 0.999822i $$-0.506014\pi$$
−0.0188910 + 0.999822i $$0.506014\pi$$
$$420$$ −12.7938 −0.624276
$$421$$ −10.5198 −0.512702 −0.256351 0.966584i $$-0.582520\pi$$
−0.256351 + 0.966584i $$0.582520\pi$$
$$422$$ 11.8740 0.578017
$$423$$ −37.3439 −1.81572
$$424$$ 1.95509 0.0949478
$$425$$ 5.44358 0.264053
$$426$$ −103.127 −4.99650
$$427$$ 1.41327 0.0683927
$$428$$ −59.4880 −2.87546
$$429$$ 14.4690 0.698569
$$430$$ 15.7685 0.760422
$$431$$ −24.7308 −1.19124 −0.595621 0.803265i $$-0.703094\pi$$
−0.595621 + 0.803265i $$0.703094\pi$$
$$432$$ −4.76116 −0.229071
$$433$$ −18.5599 −0.891933 −0.445967 0.895050i $$-0.647140\pi$$
−0.445967 + 0.895050i $$0.647140\pi$$
$$434$$ −14.1441 −0.678939
$$435$$ −8.96239 −0.429714
$$436$$ −11.2750 −0.539976
$$437$$ 4.31265 0.206302
$$438$$ −75.4577 −3.60551
$$439$$ −1.42548 −0.0680347 −0.0340173 0.999421i $$-0.510830\pi$$
−0.0340173 + 0.999421i $$0.510830\pi$$
$$440$$ −8.44358 −0.402532
$$441$$ 3.15633 0.150301
$$442$$ −84.9194 −4.03920
$$443$$ 40.1925 1.90960 0.954802 0.297242i $$-0.0960668\pi$$
0.954802 + 0.297242i $$0.0960668\pi$$
$$444$$ 109.313 5.18777
$$445$$ −7.22425 −0.342462
$$446$$ 20.8143 0.985586
$$447$$ −56.6009 −2.67713
$$448$$ −18.1187 −0.856029
$$449$$ 12.6556 0.597256 0.298628 0.954370i $$-0.403471\pi$$
0.298628 + 0.954370i $$0.403471\pi$$
$$450$$ −8.44358 −0.398034
$$451$$ 5.02539 0.236636
$$452$$ −48.2130 −2.26775
$$453$$ 8.05079 0.378259
$$454$$ 27.9511 1.31181
$$455$$ 5.83146 0.273383
$$456$$ −28.2882 −1.32472
$$457$$ −0.544198 −0.0254565 −0.0127283 0.999919i $$-0.504052\pi$$
−0.0127283 + 0.999919i $$0.504052\pi$$
$$458$$ 78.7875 3.68150
$$459$$ −2.11142 −0.0985526
$$460$$ 16.4690 0.767870
$$461$$ −11.5755 −0.539123 −0.269562 0.962983i $$-0.586879\pi$$
−0.269562 + 0.962983i $$0.586879\pi$$
$$462$$ 6.63752 0.308805
$$463$$ 23.7948 1.10584 0.552919 0.833235i $$-0.313514\pi$$
0.552919 + 0.833235i $$0.313514\pi$$
$$464$$ −44.3390 −2.05839
$$465$$ −13.1187 −0.608366
$$466$$ 23.3561 1.08195
$$467$$ −2.66784 −0.123453 −0.0617264 0.998093i $$-0.519661\pi$$
−0.0617264 + 0.998093i $$0.519661\pi$$
$$468$$ 94.9072 4.38709
$$469$$ 10.8568 0.501323
$$470$$ −31.6507 −1.45994
$$471$$ 13.4617 0.620282
$$472$$ −114.832 −5.28557
$$473$$ −5.89446 −0.271028
$$474$$ 13.0738 0.600500
$$475$$ −1.35026 −0.0619543
$$476$$ −28.0689 −1.28654
$$477$$ −0.730841 −0.0334629
$$478$$ 56.7513 2.59574
$$479$$ −10.7104 −0.489369 −0.244685 0.969603i $$-0.578684\pi$$
−0.244685 + 0.969603i $$0.578684\pi$$
$$480$$ −39.5755 −1.80636
$$481$$ −49.8251 −2.27183
$$482$$ 24.9805 1.13783
$$483$$ −7.92478 −0.360590
$$484$$ 5.15633 0.234378
$$485$$ 0.836381 0.0379781
$$486$$ −59.5755 −2.70240
$$487$$ 17.4314 0.789891 0.394945 0.918705i $$-0.370764\pi$$
0.394945 + 0.918705i $$0.370764\pi$$
$$488$$ 11.9330 0.540183
$$489$$ −8.38787 −0.379313
$$490$$ 2.67513 0.120850
$$491$$ −28.3693 −1.28029 −0.640145 0.768254i $$-0.721126\pi$$
−0.640145 + 0.768254i $$0.721126\pi$$
$$492$$ 64.2941 2.89860
$$493$$ −19.6629 −0.885573
$$494$$ 21.0640 0.947712
$$495$$ 3.15633 0.141866
$$496$$ −64.9013 −2.91415
$$497$$ 15.5369 0.696925
$$498$$ 70.5256 3.16033
$$499$$ 27.4763 1.23001 0.615003 0.788524i $$-0.289155\pi$$
0.615003 + 0.788524i $$0.289155\pi$$
$$500$$ −5.15633 −0.230598
$$501$$ −27.9756 −1.24986
$$502$$ −5.01951 −0.224032
$$503$$ 20.2981 0.905046 0.452523 0.891753i $$-0.350524\pi$$
0.452523 + 0.891753i $$0.350524\pi$$
$$504$$ 26.6507 1.18712
$$505$$ −7.41327 −0.329886
$$506$$ −8.54420 −0.379836
$$507$$ −52.1197 −2.31472
$$508$$ −87.4636 −3.88057
$$509$$ −24.2619 −1.07539 −0.537694 0.843140i $$-0.680704\pi$$
−0.537694 + 0.843140i $$0.680704\pi$$
$$510$$ −36.1319 −1.59995
$$511$$ 11.3684 0.502907
$$512$$ 11.5017 0.508306
$$513$$ 0.523730 0.0231233
$$514$$ −72.6009 −3.20229
$$515$$ −4.21933 −0.185926
$$516$$ −75.4128 −3.31986
$$517$$ 11.8315 0.520347
$$518$$ −22.8568 −1.00427
$$519$$ 22.2823 0.978086
$$520$$ 49.2384 2.15925
$$521$$ 2.20123 0.0964377 0.0482188 0.998837i $$-0.484646\pi$$
0.0482188 + 0.998837i $$0.484646\pi$$
$$522$$ 30.4993 1.33492
$$523$$ −22.1378 −0.968017 −0.484008 0.875063i $$-0.660820\pi$$
−0.484008 + 0.875063i $$0.660820\pi$$
$$524$$ −51.1754 −2.23561
$$525$$ 2.48119 0.108288
$$526$$ −34.2736 −1.49440
$$527$$ −28.7816 −1.25375
$$528$$ 30.4568 1.32546
$$529$$ −12.7988 −0.556468
$$530$$ −0.619421 −0.0269059
$$531$$ 42.9257 1.86282
$$532$$ 6.96239 0.301858
$$533$$ −29.3054 −1.26936
$$534$$ 47.9511 2.07505
$$535$$ 11.5369 0.498784
$$536$$ 91.6707 3.95957
$$537$$ 65.1002 2.80928
$$538$$ 16.7513 0.722200
$$539$$ −1.00000 −0.0430730
$$540$$ 2.00000 0.0860663
$$541$$ 23.0640 0.991597 0.495799 0.868438i $$-0.334875\pi$$
0.495799 + 0.868438i $$0.334875\pi$$
$$542$$ 15.3503 0.659350
$$543$$ 27.7889 1.19254
$$544$$ −86.8261 −3.72264
$$545$$ 2.18664 0.0936655
$$546$$ −38.7064 −1.65648
$$547$$ 21.3766 0.913998 0.456999 0.889467i $$-0.348924\pi$$
0.456999 + 0.889467i $$0.348924\pi$$
$$548$$ −56.6820 −2.42133
$$549$$ −4.46073 −0.190379
$$550$$ 2.67513 0.114068
$$551$$ 4.87732 0.207781
$$552$$ −66.9135 −2.84803
$$553$$ −1.96968 −0.0837594
$$554$$ 22.3576 0.949882
$$555$$ −21.1998 −0.899882
$$556$$ 35.5125 1.50606
$$557$$ −9.19394 −0.389560 −0.194780 0.980847i $$-0.562399\pi$$
−0.194780 + 0.980847i $$0.562399\pi$$
$$558$$ 44.6434 1.88991
$$559$$ 34.3733 1.45384
$$560$$ 12.2750 0.518715
$$561$$ 13.5066 0.570249
$$562$$ 22.6009 0.953360
$$563$$ −9.79877 −0.412969 −0.206484 0.978450i $$-0.566202\pi$$
−0.206484 + 0.978450i $$0.566202\pi$$
$$564$$ 151.370 6.37382
$$565$$ 9.35026 0.393368
$$566$$ −2.23743 −0.0940461
$$567$$ 8.50659 0.357243
$$568$$ 131.187 5.50449
$$569$$ −33.5125 −1.40492 −0.702458 0.711725i $$-0.747914\pi$$
−0.702458 + 0.711725i $$0.747914\pi$$
$$570$$ 8.96239 0.375393
$$571$$ 43.1392 1.80532 0.902659 0.430356i $$-0.141612\pi$$
0.902659 + 0.430356i $$0.141612\pi$$
$$572$$ −30.0689 −1.25724
$$573$$ −27.7889 −1.16090
$$574$$ −13.4436 −0.561124
$$575$$ −3.19394 −0.133196
$$576$$ 57.1886 2.38286
$$577$$ −14.8510 −0.618254 −0.309127 0.951021i $$-0.600037\pi$$
−0.309127 + 0.951021i $$0.600037\pi$$
$$578$$ −33.7938 −1.40564
$$579$$ −1.50071 −0.0623673
$$580$$ 18.6253 0.773374
$$581$$ −10.6253 −0.440812
$$582$$ −5.55149 −0.230117
$$583$$ 0.231548 0.00958974
$$584$$ 95.9897 3.97208
$$585$$ −18.4060 −0.760993
$$586$$ 7.27267 0.300431
$$587$$ 14.7938 0.610607 0.305304 0.952255i $$-0.401242\pi$$
0.305304 + 0.952255i $$0.401242\pi$$
$$588$$ −12.7938 −0.527609
$$589$$ 7.13918 0.294165
$$590$$ 36.3815 1.49780
$$591$$ −37.9756 −1.56211
$$592$$ −104.880 −4.31056
$$593$$ −27.4191 −1.12597 −0.562985 0.826467i $$-0.690347\pi$$
−0.562985 + 0.826467i $$0.690347\pi$$
$$594$$ −1.03761 −0.0425737
$$595$$ 5.44358 0.223165
$$596$$ 117.626 4.81814
$$597$$ 31.1695 1.27568
$$598$$ 49.8251 2.03750
$$599$$ 11.3258 0.462761 0.231380 0.972863i $$-0.425676\pi$$
0.231380 + 0.972863i $$0.425676\pi$$
$$600$$ 20.9502 0.855287
$$601$$ 15.5393 0.633860 0.316930 0.948449i $$-0.397348\pi$$
0.316930 + 0.948449i $$0.397348\pi$$
$$602$$ 15.7685 0.642674
$$603$$ −34.2677 −1.39549
$$604$$ −16.7308 −0.680768
$$605$$ −1.00000 −0.0406558
$$606$$ 49.2057 1.99884
$$607$$ 17.7235 0.719377 0.359688 0.933073i $$-0.382883\pi$$
0.359688 + 0.933073i $$0.382883\pi$$
$$608$$ 21.5369 0.873437
$$609$$ −8.96239 −0.363174
$$610$$ −3.78067 −0.153075
$$611$$ −68.9946 −2.79122
$$612$$ 88.5945 3.58122
$$613$$ 22.2941 0.900450 0.450225 0.892915i $$-0.351344\pi$$
0.450225 + 0.892915i $$0.351344\pi$$
$$614$$ 22.3733 0.902912
$$615$$ −12.4690 −0.502798
$$616$$ −8.44358 −0.340202
$$617$$ −30.9438 −1.24575 −0.622876 0.782321i $$-0.714036\pi$$
−0.622876 + 0.782321i $$0.714036\pi$$
$$618$$ 28.0059 1.12656
$$619$$ 32.4119 1.30274 0.651371 0.758759i $$-0.274194\pi$$
0.651371 + 0.758759i $$0.274194\pi$$
$$620$$ 27.2628 1.09490
$$621$$ 1.23884 0.0497130
$$622$$ −11.8677 −0.475850
$$623$$ −7.22425 −0.289434
$$624$$ −177.607 −7.10998
$$625$$ 1.00000 0.0400000
$$626$$ −79.6893 −3.18502
$$627$$ −3.35026 −0.133797
$$628$$ −27.9756 −1.11635
$$629$$ −46.5111 −1.85452
$$630$$ −8.44358 −0.336400
$$631$$ −27.3258 −1.08782 −0.543912 0.839142i $$-0.683057\pi$$
−0.543912 + 0.839142i $$0.683057\pi$$
$$632$$ −16.6312 −0.661553
$$633$$ 11.0132 0.437734
$$634$$ −41.1998 −1.63625
$$635$$ 16.9624 0.673132
$$636$$ 2.96239 0.117466
$$637$$ 5.83146 0.231051
$$638$$ −9.66291 −0.382558
$$639$$ −49.0395 −1.93997
$$640$$ 16.5696 0.654971
$$641$$ 19.4460 0.768069 0.384034 0.923319i $$-0.374534\pi$$
0.384034 + 0.923319i $$0.374534\pi$$
$$642$$ −76.5764 −3.02223
$$643$$ 5.29314 0.208741 0.104370 0.994538i $$-0.466717\pi$$
0.104370 + 0.994538i $$0.466717\pi$$
$$644$$ 16.4690 0.648969
$$645$$ 14.6253 0.575871
$$646$$ 19.6629 0.773627
$$647$$ 35.0966 1.37979 0.689896 0.723909i $$-0.257656\pi$$
0.689896 + 0.723909i $$0.257656\pi$$
$$648$$ 71.8261 2.82159
$$649$$ −13.5999 −0.533843
$$650$$ −15.5999 −0.611879
$$651$$ −13.1187 −0.514163
$$652$$ 17.4314 0.682665
$$653$$ 27.7988 1.08785 0.543925 0.839134i $$-0.316938\pi$$
0.543925 + 0.839134i $$0.316938\pi$$
$$654$$ −14.5139 −0.567538
$$655$$ 9.92478 0.387793
$$656$$ −61.6869 −2.40847
$$657$$ −35.8822 −1.39990
$$658$$ −31.6507 −1.23387
$$659$$ 19.6180 0.764209 0.382105 0.924119i $$-0.375199\pi$$
0.382105 + 0.924119i $$0.375199\pi$$
$$660$$ −12.7938 −0.498000
$$661$$ 21.5633 0.838713 0.419357 0.907822i $$-0.362256\pi$$
0.419357 + 0.907822i $$0.362256\pi$$
$$662$$ −16.7513 −0.651058
$$663$$ −78.7631 −3.05890
$$664$$ −89.7156 −3.48164
$$665$$ −1.35026 −0.0523609
$$666$$ 72.1436 2.79551
$$667$$ 11.5369 0.446711
$$668$$ 58.1378 2.24942
$$669$$ 19.3054 0.746388
$$670$$ −29.0435 −1.12205
$$671$$ 1.41327 0.0545585
$$672$$ −39.5755 −1.52666
$$673$$ −21.0679 −0.812109 −0.406054 0.913849i $$-0.633096\pi$$
−0.406054 + 0.913849i $$0.633096\pi$$
$$674$$ 42.4544 1.63528
$$675$$ −0.387873 −0.0149292
$$676$$ 108.313 4.16589
$$677$$ 34.5174 1.32661 0.663306 0.748349i $$-0.269153\pi$$
0.663306 + 0.748349i $$0.269153\pi$$
$$678$$ −62.0625 −2.38350
$$679$$ 0.836381 0.0320973
$$680$$ 45.9633 1.76261
$$681$$ 25.9248 0.993440
$$682$$ −14.1441 −0.541606
$$683$$ 33.7802 1.29256 0.646282 0.763099i $$-0.276323\pi$$
0.646282 + 0.763099i $$0.276323\pi$$
$$684$$ −21.9756 −0.840257
$$685$$ 10.9927 0.420010
$$686$$ 2.67513 0.102137
$$687$$ 73.0757 2.78801
$$688$$ 72.3547 2.75850
$$689$$ −1.35026 −0.0514409
$$690$$ 21.1998 0.807063
$$691$$ −13.8618 −0.527327 −0.263663 0.964615i $$-0.584931\pi$$
−0.263663 + 0.964615i $$0.584931\pi$$
$$692$$ −46.3063 −1.76030
$$693$$ 3.15633 0.119899
$$694$$ −18.1681 −0.689651
$$695$$ −6.88717 −0.261245
$$696$$ −75.6747 −2.86844
$$697$$ −27.3561 −1.03619
$$698$$ −71.5569 −2.70847
$$699$$ 21.6629 0.819367
$$700$$ −5.15633 −0.194891
$$701$$ 40.5256 1.53063 0.765316 0.643655i $$-0.222583\pi$$
0.765316 + 0.643655i $$0.222583\pi$$
$$702$$ 6.05079 0.228372
$$703$$ 11.5369 0.435123
$$704$$ −18.1187 −0.682875
$$705$$ −29.3561 −1.10562
$$706$$ 45.1100 1.69774
$$707$$ −7.41327 −0.278805
$$708$$ −173.995 −6.53914
$$709$$ −0.850969 −0.0319588 −0.0159794 0.999872i $$-0.505087\pi$$
−0.0159794 + 0.999872i $$0.505087\pi$$
$$710$$ −41.5633 −1.55984
$$711$$ 6.21696 0.233154
$$712$$ −60.9986 −2.28602
$$713$$ 16.8872 0.632429
$$714$$ −36.1319 −1.35220
$$715$$ 5.83146 0.218084
$$716$$ −135.289 −5.05598
$$717$$ 52.6371 1.96577
$$718$$ 10.1465 0.378663
$$719$$ 22.5769 0.841976 0.420988 0.907066i $$-0.361683\pi$$
0.420988 + 0.907066i $$0.361683\pi$$
$$720$$ −38.7440 −1.44390
$$721$$ −4.21933 −0.157136
$$722$$ 45.9502 1.71009
$$723$$ 23.1695 0.861683
$$724$$ −57.7499 −2.14626
$$725$$ −3.61213 −0.134151
$$726$$ 6.63752 0.246341
$$727$$ 12.5174 0.464244 0.232122 0.972687i $$-0.425433\pi$$
0.232122 + 0.972687i $$0.425433\pi$$
$$728$$ 49.2384 1.82490
$$729$$ −29.7367 −1.10136
$$730$$ −30.4119 −1.12559
$$731$$ 32.0870 1.18678
$$732$$ 18.0811 0.668297
$$733$$ 16.6678 0.615641 0.307820 0.951445i $$-0.400400\pi$$
0.307820 + 0.951445i $$0.400400\pi$$
$$734$$ 17.0400 0.628957
$$735$$ 2.48119 0.0915202
$$736$$ 50.9438 1.87781
$$737$$ 10.8568 0.399917
$$738$$ 42.4323 1.56196
$$739$$ 42.7005 1.57076 0.785382 0.619011i $$-0.212467\pi$$
0.785382 + 0.619011i $$0.212467\pi$$
$$740$$ 44.0567 1.61956
$$741$$ 19.5369 0.717706
$$742$$ −0.619421 −0.0227397
$$743$$ −19.6873 −0.722259 −0.361129 0.932516i $$-0.617609\pi$$
−0.361129 + 0.932516i $$0.617609\pi$$
$$744$$ −110.769 −4.06099
$$745$$ −22.8119 −0.835765
$$746$$ 57.0651 2.08930
$$747$$ 33.5369 1.22705
$$748$$ −28.0689 −1.02630
$$749$$ 11.5369 0.421549
$$750$$ −6.63752 −0.242368
$$751$$ −5.85940 −0.213813 −0.106906 0.994269i $$-0.534094\pi$$
−0.106906 + 0.994269i $$0.534094\pi$$
$$752$$ −145.232 −5.29605
$$753$$ −4.65562 −0.169660
$$754$$ 56.3488 2.05210
$$755$$ 3.24472 0.118088
$$756$$ 2.00000 0.0727393
$$757$$ −40.5863 −1.47513 −0.737567 0.675274i $$-0.764025\pi$$
−0.737567 + 0.675274i $$0.764025\pi$$
$$758$$ −66.3098 −2.40848
$$759$$ −7.92478 −0.287651
$$760$$ −11.4010 −0.413559
$$761$$ 21.8472 0.791960 0.395980 0.918259i $$-0.370405\pi$$
0.395980 + 0.918259i $$0.370405\pi$$
$$762$$ −112.588 −4.07864
$$763$$ 2.18664 0.0791618
$$764$$ 57.7499 2.08932
$$765$$ −17.1817 −0.621206
$$766$$ 14.6013 0.527567
$$767$$ 79.3073 2.86362
$$768$$ −20.0689 −0.724173
$$769$$ 45.2892 1.63317 0.816585 0.577226i $$-0.195865\pi$$
0.816585 + 0.577226i $$0.195865\pi$$
$$770$$ 2.67513 0.0964050
$$771$$ −67.3376 −2.42510
$$772$$ 3.11871 0.112245
$$773$$ −33.8153 −1.21625 −0.608125 0.793841i $$-0.708078\pi$$
−0.608125 + 0.793841i $$0.708078\pi$$
$$774$$ −49.7704 −1.78896
$$775$$ −5.28726 −0.189924
$$776$$ 7.06205 0.253513
$$777$$ −21.1998 −0.760539
$$778$$ 36.7123 1.31620
$$779$$ 6.78560 0.243119
$$780$$ 74.6067 2.67135
$$781$$ 15.5369 0.555954
$$782$$ 46.5111 1.66323
$$783$$ 1.40105 0.0500693
$$784$$ 12.2750 0.438394
$$785$$ 5.42548 0.193644
$$786$$ −65.8759 −2.34972
$$787$$ 1.27504 0.0454502 0.0227251 0.999742i $$-0.492766\pi$$
0.0227251 + 0.999742i $$0.492766\pi$$
$$788$$ 78.9194 2.81139
$$789$$ −31.7889 −1.13172
$$790$$ 5.26916 0.187468
$$791$$ 9.35026 0.332457
$$792$$ 26.6507 0.946991
$$793$$ −8.24140 −0.292661
$$794$$ −5.64832 −0.200452
$$795$$ −0.574515 −0.0203760
$$796$$ −64.7753 −2.29590
$$797$$ −42.5256 −1.50634 −0.753168 0.657829i $$-0.771475\pi$$
−0.753168 + 0.657829i $$0.771475\pi$$
$$798$$ 8.96239 0.317265
$$799$$ −64.4055 −2.27850
$$800$$ −15.9502 −0.563924
$$801$$ 22.8021 0.805672
$$802$$ −51.2262 −1.80886
$$803$$ 11.3684 0.401181
$$804$$ 138.901 4.89865
$$805$$ −3.19394 −0.112571
$$806$$ 82.4807 2.90526
$$807$$ 15.5369 0.546925
$$808$$ −62.5945 −2.20207
$$809$$ −14.7151 −0.517356 −0.258678 0.965964i $$-0.583287\pi$$
−0.258678 + 0.965964i $$0.583287\pi$$
$$810$$ −22.7562 −0.799573
$$811$$ 51.7743 1.81804 0.909021 0.416750i $$-0.136831\pi$$
0.909021 + 0.416750i $$0.136831\pi$$
$$812$$ 18.6253 0.653620
$$813$$ 14.2374 0.499328
$$814$$ −22.8568 −0.801132
$$815$$ −3.38058 −0.118417
$$816$$ −165.794 −5.80395
$$817$$ −7.95906 −0.278452
$$818$$ 49.9937 1.74799
$$819$$ −18.4060 −0.643157
$$820$$ 25.9126 0.904906
$$821$$ 2.64974 0.0924765 0.0462383 0.998930i $$-0.485277\pi$$
0.0462383 + 0.998930i $$0.485277\pi$$
$$822$$ −72.9643 −2.54492
$$823$$ 5.76845 0.201076 0.100538 0.994933i $$-0.467944\pi$$
0.100538 + 0.994933i $$0.467944\pi$$
$$824$$ −35.6263 −1.24110
$$825$$ 2.48119 0.0863841
$$826$$ 36.3815 1.26588
$$827$$ −13.4920 −0.469163 −0.234581 0.972096i $$-0.575372\pi$$
−0.234581 + 0.972096i $$0.575372\pi$$
$$828$$ −51.9814 −1.80648
$$829$$ −4.70052 −0.163256 −0.0816280 0.996663i $$-0.526012\pi$$
−0.0816280 + 0.996663i $$0.526012\pi$$
$$830$$ 28.4241 0.986614
$$831$$ 20.7367 0.719349
$$832$$ 105.658 3.66305
$$833$$ 5.44358 0.188609
$$834$$ 45.7137 1.58294
$$835$$ −11.2750 −0.390189
$$836$$ 6.96239 0.240799
$$837$$ 2.05079 0.0708855
$$838$$ 2.06888 0.0714684
$$839$$ −38.8045 −1.33968 −0.669839 0.742506i $$-0.733637\pi$$
−0.669839 + 0.742506i $$0.733637\pi$$
$$840$$ 20.9502 0.722850
$$841$$ −15.9525 −0.550088
$$842$$ 28.1417 0.969828
$$843$$ 20.9624 0.721983
$$844$$ −22.8872 −0.787809
$$845$$ −21.0059 −0.722624
$$846$$ 99.8999 3.43463
$$847$$ −1.00000 −0.0343604
$$848$$ −2.84226 −0.0976036
$$849$$ −2.07522 −0.0712215
$$850$$ −14.5623 −0.499483
$$851$$ 27.2896 0.935476
$$852$$ 198.777 6.80998
$$853$$ 20.6824 0.708153 0.354076 0.935217i $$-0.384795\pi$$
0.354076 + 0.935217i $$0.384795\pi$$
$$854$$ −3.78067 −0.129372
$$855$$ 4.26187 0.145753
$$856$$ 97.4128 3.32950
$$857$$ 26.3453 0.899940 0.449970 0.893044i $$-0.351435\pi$$
0.449970 + 0.893044i $$0.351435\pi$$
$$858$$ −38.7064 −1.32141
$$859$$ −8.51151 −0.290409 −0.145205 0.989402i $$-0.546384\pi$$
−0.145205 + 0.989402i $$0.546384\pi$$
$$860$$ −30.3938 −1.03642
$$861$$ −12.4690 −0.424942
$$862$$ 66.1582 2.25336
$$863$$ 7.56722 0.257591 0.128796 0.991671i $$-0.458889\pi$$
0.128796 + 0.991671i $$0.458889\pi$$
$$864$$ 6.18664 0.210474
$$865$$ 8.98049 0.305346
$$866$$ 49.6502 1.68718
$$867$$ −31.3439 −1.06450
$$868$$ 27.2628 0.925360
$$869$$ −1.96968 −0.0668169
$$870$$ 23.9756 0.812848
$$871$$ −63.3112 −2.14522
$$872$$ 18.4631 0.625239
$$873$$ −2.63989 −0.0893467
$$874$$ −11.5369 −0.390242
$$875$$ 1.00000 0.0338062
$$876$$ 145.445 4.91413
$$877$$ 17.2955 0.584028 0.292014 0.956414i $$-0.405675\pi$$
0.292014 + 0.956414i $$0.405675\pi$$
$$878$$ 3.81336 0.128695
$$879$$ 6.74543 0.227518
$$880$$ 12.2750 0.413791
$$881$$ 20.4504 0.688992 0.344496 0.938788i $$-0.388050\pi$$
0.344496 + 0.938788i $$0.388050\pi$$
$$882$$ −8.44358 −0.284310
$$883$$ −49.6589 −1.67116 −0.835578 0.549371i $$-0.814867\pi$$
−0.835578 + 0.549371i $$0.814867\pi$$
$$884$$ 163.682 5.50524
$$885$$ 33.7440 1.13429
$$886$$ −107.520 −3.61221
$$887$$ −47.1100 −1.58180 −0.790900 0.611946i $$-0.790387\pi$$
−0.790900 + 0.611946i $$0.790387\pi$$
$$888$$ −179.002 −6.00693
$$889$$ 16.9624 0.568900
$$890$$ 19.3258 0.647803
$$891$$ 8.50659 0.284981
$$892$$ −40.1197 −1.34331
$$893$$ 15.9756 0.534602
$$894$$ 151.415 5.06407
$$895$$ 26.2374 0.877020
$$896$$ 16.5696 0.553551
$$897$$ 46.2130 1.54301
$$898$$ −33.8554 −1.12977
$$899$$ 19.0982 0.636962
$$900$$ 16.2750 0.542501
$$901$$ −1.26045 −0.0419917
$$902$$ −13.4436 −0.447622
$$903$$ 14.6253 0.486700
$$904$$ 78.9497 2.62583
$$905$$ 11.1998 0.372294
$$906$$ −21.5369 −0.715516
$$907$$ 14.4591 0.480107 0.240054 0.970760i $$-0.422835\pi$$
0.240054 + 0.970760i $$0.422835\pi$$
$$908$$ −53.8759 −1.78793
$$909$$ 23.3987 0.776085
$$910$$ −15.5999 −0.517132
$$911$$ −31.5369 −1.04486 −0.522432 0.852681i $$-0.674975\pi$$
−0.522432 + 0.852681i $$0.674975\pi$$
$$912$$ 41.1246 1.36177
$$913$$ −10.6253 −0.351646
$$914$$ 1.45580 0.0481536
$$915$$ −3.50659 −0.115924
$$916$$ −151.863 −5.01770
$$917$$ 9.92478 0.327745
$$918$$ 5.64832 0.186422
$$919$$ 5.26328 0.173620 0.0868098 0.996225i $$-0.472333\pi$$
0.0868098 + 0.996225i $$0.472333\pi$$
$$920$$ −26.9683 −0.889117
$$921$$ 20.7513 0.683779
$$922$$ 30.9659 1.01981
$$923$$ −90.6028 −2.98223
$$924$$ −12.7938 −0.420887
$$925$$ −8.54420 −0.280932
$$926$$ −63.6542 −2.09181
$$927$$ 13.3176 0.437407
$$928$$ 57.6140 1.89127
$$929$$ 26.0508 0.854699 0.427349 0.904087i $$-0.359447\pi$$
0.427349 + 0.904087i $$0.359447\pi$$
$$930$$ 35.0943 1.15079
$$931$$ −1.35026 −0.0442530
$$932$$ −45.0191 −1.47465
$$933$$ −11.0073 −0.360363
$$934$$ 7.13681 0.233524
$$935$$ 5.44358 0.178024
$$936$$ −155.412 −5.07981
$$937$$ −29.3439 −0.958624 −0.479312 0.877645i $$-0.659114\pi$$
−0.479312 + 0.877645i $$0.659114\pi$$
$$938$$ −29.0435 −0.948304
$$939$$ −73.9121 −2.41203
$$940$$ 61.0068 1.98982
$$941$$ −28.6375 −0.933556 −0.466778 0.884374i $$-0.654585\pi$$
−0.466778 + 0.884374i $$0.654585\pi$$
$$942$$ −36.0118 −1.17333
$$943$$ 16.0508 0.522685
$$944$$ 166.939 5.43341
$$945$$ −0.387873 −0.0126175
$$946$$ 15.7685 0.512677
$$947$$ −52.8178 −1.71635 −0.858174 0.513358i $$-0.828401\pi$$
−0.858174 + 0.513358i $$0.828401\pi$$
$$948$$ −25.1998 −0.818452
$$949$$ −66.2941 −2.15200
$$950$$ 3.61213 0.117193
$$951$$ −38.2130 −1.23914
$$952$$ 45.9633 1.48968
$$953$$ 37.1939 1.20483 0.602415 0.798183i $$-0.294205\pi$$
0.602415 + 0.798183i $$0.294205\pi$$
$$954$$ 1.95509 0.0632985
$$955$$ −11.1998 −0.362418
$$956$$ −109.388 −3.53787
$$957$$ −8.96239 −0.289713
$$958$$ 28.6516 0.925693
$$959$$ 10.9927 0.354973
$$960$$ 44.9560 1.45095
$$961$$ −3.04491 −0.0982228
$$962$$ 133.289 4.29740
$$963$$ −36.4142 −1.17343
$$964$$ −48.1500 −1.55081
$$965$$ −0.604833 −0.0194703
$$966$$ 21.1998 0.682093
$$967$$ 4.07125 0.130923 0.0654613 0.997855i $$-0.479148\pi$$
0.0654613 + 0.997855i $$0.479148\pi$$
$$968$$ −8.44358 −0.271387
$$969$$ 18.2374 0.585871
$$970$$ −2.23743 −0.0718395
$$971$$ 0.773377 0.0248188 0.0124094 0.999923i $$-0.496050\pi$$
0.0124094 + 0.999923i $$0.496050\pi$$
$$972$$ 114.832 3.68324
$$973$$ −6.88717 −0.220792
$$974$$ −46.6312 −1.49416
$$975$$ −14.4690 −0.463378
$$976$$ −17.3479 −0.555292
$$977$$ −37.8740 −1.21170 −0.605848 0.795580i $$-0.707166\pi$$
−0.605848 + 0.795580i $$0.707166\pi$$
$$978$$ 22.4387 0.717509
$$979$$ −7.22425 −0.230888
$$980$$ −5.15633 −0.164713
$$981$$ −6.90175 −0.220356
$$982$$ 75.8916 2.42180
$$983$$ −15.5794 −0.496907 −0.248453 0.968644i $$-0.579922\pi$$
−0.248453 + 0.968644i $$0.579922\pi$$
$$984$$ −105.283 −3.35629
$$985$$ −15.3054 −0.487669
$$986$$ 52.6009 1.67515
$$987$$ −29.3561 −0.934416
$$988$$ −40.6009 −1.29169
$$989$$ −18.8265 −0.598649
$$990$$ −8.44358 −0.268355
$$991$$ 27.0982 0.860804 0.430402 0.902637i $$-0.358372\pi$$
0.430402 + 0.902637i $$0.358372\pi$$
$$992$$ 84.3327 2.67756
$$993$$ −15.5369 −0.493049
$$994$$ −41.5633 −1.31831
$$995$$ 12.5623 0.398252
$$996$$ −135.938 −4.30737
$$997$$ 50.4060 1.59637 0.798187 0.602410i $$-0.205793\pi$$
0.798187 + 0.602410i $$0.205793\pi$$
$$998$$ −73.5026 −2.32668
$$999$$ 3.31406 0.104852
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 385.2.a.f.1.1 3
3.2 odd 2 3465.2.a.bh.1.3 3
4.3 odd 2 6160.2.a.bn.1.3 3
5.2 odd 4 1925.2.b.n.1849.1 6
5.3 odd 4 1925.2.b.n.1849.6 6
5.4 even 2 1925.2.a.v.1.3 3
7.6 odd 2 2695.2.a.g.1.1 3
11.10 odd 2 4235.2.a.q.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.f.1.1 3 1.1 even 1 trivial
1925.2.a.v.1.3 3 5.4 even 2
1925.2.b.n.1849.1 6 5.2 odd 4
1925.2.b.n.1849.6 6 5.3 odd 4
2695.2.a.g.1.1 3 7.6 odd 2
3465.2.a.bh.1.3 3 3.2 odd 2
4235.2.a.q.1.3 3 11.10 odd 2
6160.2.a.bn.1.3 3 4.3 odd 2