# Properties

 Label 5070.2.b.y.1351.3 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.3 Root $$-0.445042i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.y.1351.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -0.307979i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -0.307979i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -0.335126i q^{11} -1.00000 q^{12} -0.307979 q^{14} +1.00000i q^{15} +1.00000 q^{16} +6.85086 q^{17} -1.00000i q^{18} +5.80194i q^{19} -1.00000i q^{20} -0.307979i q^{21} -0.335126 q^{22} -2.35690 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +0.307979i q^{28} -5.91185 q^{29} +1.00000 q^{30} +0.0609989i q^{31} -1.00000i q^{32} -0.335126i q^{33} -6.85086i q^{34} +0.307979 q^{35} -1.00000 q^{36} -7.07606i q^{37} +5.80194 q^{38} -1.00000 q^{40} +3.40581i q^{41} -0.307979 q^{42} +0.0489173 q^{43} +0.335126i q^{44} +1.00000i q^{45} +2.35690i q^{46} -7.00000i q^{47} +1.00000 q^{48} +6.90515 q^{49} +1.00000i q^{50} +6.85086 q^{51} +12.1739 q^{53} -1.00000i q^{54} +0.335126 q^{55} +0.307979 q^{56} +5.80194i q^{57} +5.91185i q^{58} +13.1347i q^{59} -1.00000i q^{60} +2.14675 q^{61} +0.0609989 q^{62} -0.307979i q^{63} -1.00000 q^{64} -0.335126 q^{66} +11.9608i q^{67} -6.85086 q^{68} -2.35690 q^{69} -0.307979i q^{70} +9.56465i q^{71} +1.00000i q^{72} -2.31336i q^{73} -7.07606 q^{74} -1.00000 q^{75} -5.80194i q^{76} -0.103211 q^{77} -0.0760644 q^{79} +1.00000i q^{80} +1.00000 q^{81} +3.40581 q^{82} +3.84117i q^{83} +0.307979i q^{84} +6.85086i q^{85} -0.0489173i q^{86} -5.91185 q^{87} +0.335126 q^{88} +4.41119i q^{89} +1.00000 q^{90} +2.35690 q^{92} +0.0609989i q^{93} -7.00000 q^{94} -5.80194 q^{95} -1.00000i q^{96} +6.13169i q^{97} -6.90515i q^{98} -0.335126i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10})$$ $$6q + 6q^{3} - 6q^{4} + 6q^{9} + 6q^{10} - 6q^{12} - 12q^{14} + 6q^{16} + 14q^{17} - 6q^{23} - 6q^{25} + 6q^{27} - 28q^{29} + 6q^{30} + 12q^{35} - 6q^{36} + 26q^{38} - 6q^{40} - 12q^{42} - 18q^{43} + 6q^{48} - 10q^{49} + 14q^{51} + 6q^{53} + 12q^{56} - 42q^{61} + 20q^{62} - 6q^{64} - 14q^{68} - 6q^{69} - 12q^{74} - 6q^{75} + 42q^{77} + 30q^{79} + 6q^{81} - 6q^{82} - 28q^{87} + 6q^{90} + 6q^{92} - 42q^{94} - 26q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000i 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ − 0.307979i − 0.116405i −0.998305 0.0582025i $$-0.981463\pi$$
0.998305 0.0582025i $$-0.0185369\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ − 0.335126i − 0.101044i −0.998723 0.0505221i $$-0.983911\pi$$
0.998723 0.0505221i $$-0.0160885\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −0.307979 −0.0823107
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 6.85086 1.66158 0.830788 0.556589i $$-0.187890\pi$$
0.830788 + 0.556589i $$0.187890\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 5.80194i 1.33106i 0.746373 + 0.665528i $$0.231794\pi$$
−0.746373 + 0.665528i $$0.768206\pi$$
$$20$$ − 1.00000i − 0.223607i
$$21$$ − 0.307979i − 0.0672064i
$$22$$ −0.335126 −0.0714490
$$23$$ −2.35690 −0.491447 −0.245723 0.969340i $$-0.579025\pi$$
−0.245723 + 0.969340i $$0.579025\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0.307979i 0.0582025i
$$29$$ −5.91185 −1.09780 −0.548902 0.835887i $$-0.684954\pi$$
−0.548902 + 0.835887i $$0.684954\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 0.0609989i 0.0109557i 0.999985 + 0.00547787i $$0.00174367\pi$$
−0.999985 + 0.00547787i $$0.998256\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 0.335126i − 0.0583379i
$$34$$ − 6.85086i − 1.17491i
$$35$$ 0.307979 0.0520579
$$36$$ −1.00000 −0.166667
$$37$$ − 7.07606i − 1.16330i −0.813440 0.581649i $$-0.802408\pi$$
0.813440 0.581649i $$-0.197592\pi$$
$$38$$ 5.80194 0.941199
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 3.40581i 0.531899i 0.963987 + 0.265949i $$0.0856854\pi$$
−0.963987 + 0.265949i $$0.914315\pi$$
$$42$$ −0.307979 −0.0475221
$$43$$ 0.0489173 0.00745982 0.00372991 0.999993i $$-0.498813\pi$$
0.00372991 + 0.999993i $$0.498813\pi$$
$$44$$ 0.335126i 0.0505221i
$$45$$ 1.00000i 0.149071i
$$46$$ 2.35690i 0.347505i
$$47$$ − 7.00000i − 1.02105i −0.859861 0.510527i $$-0.829450\pi$$
0.859861 0.510527i $$-0.170550\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 6.90515 0.986450
$$50$$ 1.00000i 0.141421i
$$51$$ 6.85086 0.959312
$$52$$ 0 0
$$53$$ 12.1739 1.67221 0.836107 0.548567i $$-0.184826\pi$$
0.836107 + 0.548567i $$0.184826\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 0.335126 0.0451883
$$56$$ 0.307979 0.0411554
$$57$$ 5.80194i 0.768485i
$$58$$ 5.91185i 0.776264i
$$59$$ 13.1347i 1.70999i 0.518638 + 0.854994i $$0.326439\pi$$
−0.518638 + 0.854994i $$0.673561\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ 2.14675 0.274863 0.137432 0.990511i $$-0.456115\pi$$
0.137432 + 0.990511i $$0.456115\pi$$
$$62$$ 0.0609989 0.00774687
$$63$$ − 0.307979i − 0.0388016i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −0.335126 −0.0412511
$$67$$ 11.9608i 1.46124i 0.682784 + 0.730620i $$0.260769\pi$$
−0.682784 + 0.730620i $$0.739231\pi$$
$$68$$ −6.85086 −0.830788
$$69$$ −2.35690 −0.283737
$$70$$ − 0.307979i − 0.0368105i
$$71$$ 9.56465i 1.13511i 0.823334 + 0.567557i $$0.192111\pi$$
−0.823334 + 0.567557i $$0.807889\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 2.31336i − 0.270758i −0.990794 0.135379i $$-0.956775\pi$$
0.990794 0.135379i $$-0.0432252\pi$$
$$74$$ −7.07606 −0.822576
$$75$$ −1.00000 −0.115470
$$76$$ − 5.80194i − 0.665528i
$$77$$ −0.103211 −0.0117620
$$78$$ 0 0
$$79$$ −0.0760644 −0.00855792 −0.00427896 0.999991i $$-0.501362\pi$$
−0.00427896 + 0.999991i $$0.501362\pi$$
$$80$$ 1.00000i 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 3.40581 0.376109
$$83$$ 3.84117i 0.421623i 0.977527 + 0.210811i $$0.0676106\pi$$
−0.977527 + 0.210811i $$0.932389\pi$$
$$84$$ 0.307979i 0.0336032i
$$85$$ 6.85086i 0.743080i
$$86$$ − 0.0489173i − 0.00527489i
$$87$$ −5.91185 −0.633817
$$88$$ 0.335126 0.0357245
$$89$$ 4.41119i 0.467585i 0.972286 + 0.233793i $$0.0751137\pi$$
−0.972286 + 0.233793i $$0.924886\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 2.35690 0.245723
$$93$$ 0.0609989i 0.00632529i
$$94$$ −7.00000 −0.721995
$$95$$ −5.80194 −0.595266
$$96$$ − 1.00000i − 0.102062i
$$97$$ 6.13169i 0.622578i 0.950315 + 0.311289i $$0.100761\pi$$
−0.950315 + 0.311289i $$0.899239\pi$$
$$98$$ − 6.90515i − 0.697525i
$$99$$ − 0.335126i − 0.0336814i
$$100$$ 1.00000 0.100000
$$101$$ −1.32975 −0.132315 −0.0661575 0.997809i $$-0.521074\pi$$
−0.0661575 + 0.997809i $$0.521074\pi$$
$$102$$ − 6.85086i − 0.678336i
$$103$$ −8.92154 −0.879066 −0.439533 0.898227i $$-0.644856\pi$$
−0.439533 + 0.898227i $$0.644856\pi$$
$$104$$ 0 0
$$105$$ 0.307979 0.0300556
$$106$$ − 12.1739i − 1.18243i
$$107$$ 14.4644 1.39833 0.699164 0.714961i $$-0.253556\pi$$
0.699164 + 0.714961i $$0.253556\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 7.13706i 0.683607i 0.939772 + 0.341803i $$0.111038\pi$$
−0.939772 + 0.341803i $$0.888962\pi$$
$$110$$ − 0.335126i − 0.0319530i
$$111$$ − 7.07606i − 0.671630i
$$112$$ − 0.307979i − 0.0291012i
$$113$$ −16.3424 −1.53737 −0.768683 0.639630i $$-0.779088\pi$$
−0.768683 + 0.639630i $$0.779088\pi$$
$$114$$ 5.80194 0.543401
$$115$$ − 2.35690i − 0.219782i
$$116$$ 5.91185 0.548902
$$117$$ 0 0
$$118$$ 13.1347 1.20914
$$119$$ − 2.10992i − 0.193416i
$$120$$ −1.00000 −0.0912871
$$121$$ 10.8877 0.989790
$$122$$ − 2.14675i − 0.194358i
$$123$$ 3.40581i 0.307092i
$$124$$ − 0.0609989i − 0.00547787i
$$125$$ − 1.00000i − 0.0894427i
$$126$$ −0.307979 −0.0274369
$$127$$ 0.911854 0.0809140 0.0404570 0.999181i $$-0.487119\pi$$
0.0404570 + 0.999181i $$0.487119\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 0.0489173 0.00430693
$$130$$ 0 0
$$131$$ 13.0368 1.13903 0.569517 0.821980i $$-0.307130\pi$$
0.569517 + 0.821980i $$0.307130\pi$$
$$132$$ 0.335126i 0.0291689i
$$133$$ 1.78687 0.154941
$$134$$ 11.9608 1.03325
$$135$$ 1.00000i 0.0860663i
$$136$$ 6.85086i 0.587456i
$$137$$ − 12.8877i − 1.10107i −0.834812 0.550535i $$-0.814424\pi$$
0.834812 0.550535i $$-0.185576\pi$$
$$138$$ 2.35690i 0.200632i
$$139$$ −2.01208 −0.170663 −0.0853313 0.996353i $$-0.527195\pi$$
−0.0853313 + 0.996353i $$0.527195\pi$$
$$140$$ −0.307979 −0.0260289
$$141$$ − 7.00000i − 0.589506i
$$142$$ 9.56465 0.802647
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 5.91185i − 0.490953i
$$146$$ −2.31336 −0.191455
$$147$$ 6.90515 0.569527
$$148$$ 7.07606i 0.581649i
$$149$$ 3.44935i 0.282582i 0.989968 + 0.141291i $$0.0451253\pi$$
−0.989968 + 0.141291i $$0.954875\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ − 2.84117i − 0.231211i −0.993295 0.115605i $$-0.963119\pi$$
0.993295 0.115605i $$-0.0368808\pi$$
$$152$$ −5.80194 −0.470599
$$153$$ 6.85086 0.553859
$$154$$ 0.103211i 0.00831702i
$$155$$ −0.0609989 −0.00489955
$$156$$ 0 0
$$157$$ 16.2228 1.29472 0.647361 0.762184i $$-0.275873\pi$$
0.647361 + 0.762184i $$0.275873\pi$$
$$158$$ 0.0760644i 0.00605136i
$$159$$ 12.1739 0.965453
$$160$$ 1.00000 0.0790569
$$161$$ 0.725873i 0.0572068i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 0.0924579i 0.00724186i 0.999993 + 0.00362093i $$0.00115258\pi$$
−0.999993 + 0.00362093i $$0.998847\pi$$
$$164$$ − 3.40581i − 0.265949i
$$165$$ 0.335126 0.0260895
$$166$$ 3.84117 0.298132
$$167$$ 9.64071i 0.746021i 0.927827 + 0.373010i $$0.121674\pi$$
−0.927827 + 0.373010i $$0.878326\pi$$
$$168$$ 0.307979 0.0237611
$$169$$ 0 0
$$170$$ 6.85086 0.525437
$$171$$ 5.80194i 0.443685i
$$172$$ −0.0489173 −0.00372991
$$173$$ −0.719169 −0.0546774 −0.0273387 0.999626i $$-0.508703\pi$$
−0.0273387 + 0.999626i $$0.508703\pi$$
$$174$$ 5.91185i 0.448176i
$$175$$ 0.307979i 0.0232810i
$$176$$ − 0.335126i − 0.0252610i
$$177$$ 13.1347i 0.987262i
$$178$$ 4.41119 0.330633
$$179$$ 2.01208 0.150390 0.0751950 0.997169i $$-0.476042\pi$$
0.0751950 + 0.997169i $$0.476042\pi$$
$$180$$ − 1.00000i − 0.0745356i
$$181$$ 2.57135 0.191127 0.0955635 0.995423i $$-0.469535\pi$$
0.0955635 + 0.995423i $$0.469535\pi$$
$$182$$ 0 0
$$183$$ 2.14675 0.158692
$$184$$ − 2.35690i − 0.173753i
$$185$$ 7.07606 0.520243
$$186$$ 0.0609989 0.00447266
$$187$$ − 2.29590i − 0.167893i
$$188$$ 7.00000i 0.510527i
$$189$$ − 0.307979i − 0.0224021i
$$190$$ 5.80194i 0.420917i
$$191$$ 3.80194 0.275099 0.137549 0.990495i $$-0.456077\pi$$
0.137549 + 0.990495i $$0.456077\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 8.86294i 0.637968i 0.947760 + 0.318984i $$0.103342\pi$$
−0.947760 + 0.318984i $$0.896658\pi$$
$$194$$ 6.13169 0.440229
$$195$$ 0 0
$$196$$ −6.90515 −0.493225
$$197$$ − 22.6504i − 1.61377i −0.590706 0.806887i $$-0.701151\pi$$
0.590706 0.806887i $$-0.298849\pi$$
$$198$$ −0.335126 −0.0238163
$$199$$ 5.46681 0.387532 0.193766 0.981048i $$-0.437930\pi$$
0.193766 + 0.981048i $$0.437930\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ 11.9608i 0.843648i
$$202$$ 1.32975i 0.0935608i
$$203$$ 1.82072i 0.127790i
$$204$$ −6.85086 −0.479656
$$205$$ −3.40581 −0.237872
$$206$$ 8.92154i 0.621593i
$$207$$ −2.35690 −0.163816
$$208$$ 0 0
$$209$$ 1.94438 0.134495
$$210$$ − 0.307979i − 0.0212525i
$$211$$ 11.0422 0.760177 0.380089 0.924950i $$-0.375893\pi$$
0.380089 + 0.924950i $$0.375893\pi$$
$$212$$ −12.1739 −0.836107
$$213$$ 9.56465i 0.655359i
$$214$$ − 14.4644i − 0.988767i
$$215$$ 0.0489173i 0.00333613i
$$216$$ 1.00000i 0.0680414i
$$217$$ 0.0187864 0.00127530
$$218$$ 7.13706 0.483383
$$219$$ − 2.31336i − 0.156322i
$$220$$ −0.335126 −0.0225942
$$221$$ 0 0
$$222$$ −7.07606 −0.474914
$$223$$ 18.3884i 1.23138i 0.787990 + 0.615688i $$0.211122\pi$$
−0.787990 + 0.615688i $$0.788878\pi$$
$$224$$ −0.307979 −0.0205777
$$225$$ −1.00000 −0.0666667
$$226$$ 16.3424i 1.08708i
$$227$$ 12.0073i 0.796952i 0.917179 + 0.398476i $$0.130461\pi$$
−0.917179 + 0.398476i $$0.869539\pi$$
$$228$$ − 5.80194i − 0.384243i
$$229$$ − 25.4601i − 1.68245i −0.540684 0.841226i $$-0.681835\pi$$
0.540684 0.841226i $$-0.318165\pi$$
$$230$$ −2.35690 −0.155409
$$231$$ −0.103211 −0.00679082
$$232$$ − 5.91185i − 0.388132i
$$233$$ 2.10992 0.138225 0.0691126 0.997609i $$-0.477983\pi$$
0.0691126 + 0.997609i $$0.477983\pi$$
$$234$$ 0 0
$$235$$ 7.00000 0.456630
$$236$$ − 13.1347i − 0.854994i
$$237$$ −0.0760644 −0.00494091
$$238$$ −2.10992 −0.136766
$$239$$ 3.95838i 0.256046i 0.991771 + 0.128023i $$0.0408632\pi$$
−0.991771 + 0.128023i $$0.959137\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ 23.1564i 1.49164i 0.666149 + 0.745819i $$0.267942\pi$$
−0.666149 + 0.745819i $$0.732058\pi$$
$$242$$ − 10.8877i − 0.699887i
$$243$$ 1.00000 0.0641500
$$244$$ −2.14675 −0.137432
$$245$$ 6.90515i 0.441154i
$$246$$ 3.40581 0.217147
$$247$$ 0 0
$$248$$ −0.0609989 −0.00387344
$$249$$ 3.84117i 0.243424i
$$250$$ −1.00000 −0.0632456
$$251$$ 5.54048 0.349712 0.174856 0.984594i $$-0.444054\pi$$
0.174856 + 0.984594i $$0.444054\pi$$
$$252$$ 0.307979i 0.0194008i
$$253$$ 0.789856i 0.0496578i
$$254$$ − 0.911854i − 0.0572148i
$$255$$ 6.85086i 0.429017i
$$256$$ 1.00000 0.0625000
$$257$$ −6.60819 −0.412207 −0.206104 0.978530i $$-0.566078\pi$$
−0.206104 + 0.978530i $$0.566078\pi$$
$$258$$ − 0.0489173i − 0.00304546i
$$259$$ −2.17928 −0.135414
$$260$$ 0 0
$$261$$ −5.91185 −0.365935
$$262$$ − 13.0368i − 0.805418i
$$263$$ −0.660563 −0.0407320 −0.0203660 0.999793i $$-0.506483\pi$$
−0.0203660 + 0.999793i $$0.506483\pi$$
$$264$$ 0.335126 0.0206256
$$265$$ 12.1739i 0.747837i
$$266$$ − 1.78687i − 0.109560i
$$267$$ 4.41119i 0.269960i
$$268$$ − 11.9608i − 0.730620i
$$269$$ −26.7928 −1.63359 −0.816794 0.576929i $$-0.804251\pi$$
−0.816794 + 0.576929i $$0.804251\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ − 14.0151i − 0.851355i −0.904875 0.425677i $$-0.860036\pi$$
0.904875 0.425677i $$-0.139964\pi$$
$$272$$ 6.85086 0.415394
$$273$$ 0 0
$$274$$ −12.8877 −0.778574
$$275$$ 0.335126i 0.0202088i
$$276$$ 2.35690 0.141868
$$277$$ 23.4252 1.40748 0.703742 0.710456i $$-0.251511\pi$$
0.703742 + 0.710456i $$0.251511\pi$$
$$278$$ 2.01208i 0.120677i
$$279$$ 0.0609989i 0.00365191i
$$280$$ 0.307979i 0.0184052i
$$281$$ − 8.40044i − 0.501128i −0.968100 0.250564i $$-0.919384\pi$$
0.968100 0.250564i $$-0.0806161\pi$$
$$282$$ −7.00000 −0.416844
$$283$$ 31.6142 1.87927 0.939633 0.342183i $$-0.111166\pi$$
0.939633 + 0.342183i $$0.111166\pi$$
$$284$$ − 9.56465i − 0.567557i
$$285$$ −5.80194 −0.343677
$$286$$ 0 0
$$287$$ 1.04892 0.0619156
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 29.9342 1.76084
$$290$$ −5.91185 −0.347156
$$291$$ 6.13169i 0.359446i
$$292$$ 2.31336i 0.135379i
$$293$$ 17.5254i 1.02385i 0.859031 + 0.511923i $$0.171067\pi$$
−0.859031 + 0.511923i $$0.828933\pi$$
$$294$$ − 6.90515i − 0.402716i
$$295$$ −13.1347 −0.764730
$$296$$ 7.07606 0.411288
$$297$$ − 0.335126i − 0.0194460i
$$298$$ 3.44935 0.199816
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ − 0.0150655i 0 0.000868360i
$$302$$ −2.84117 −0.163491
$$303$$ −1.32975 −0.0763921
$$304$$ 5.80194i 0.332764i
$$305$$ 2.14675i 0.122923i
$$306$$ − 6.85086i − 0.391637i
$$307$$ − 20.6732i − 1.17988i −0.807446 0.589942i $$-0.799151\pi$$
0.807446 0.589942i $$-0.200849\pi$$
$$308$$ 0.103211 0.00588102
$$309$$ −8.92154 −0.507529
$$310$$ 0.0609989i 0.00346451i
$$311$$ 22.0248 1.24891 0.624455 0.781061i $$-0.285321\pi$$
0.624455 + 0.781061i $$0.285321\pi$$
$$312$$ 0 0
$$313$$ 21.7192 1.22764 0.613820 0.789446i $$-0.289632\pi$$
0.613820 + 0.789446i $$0.289632\pi$$
$$314$$ − 16.2228i − 0.915506i
$$315$$ 0.307979 0.0173526
$$316$$ 0.0760644 0.00427896
$$317$$ − 16.4644i − 0.924734i −0.886689 0.462367i $$-0.847000\pi$$
0.886689 0.462367i $$-0.153000\pi$$
$$318$$ − 12.1739i − 0.682678i
$$319$$ 1.98121i 0.110927i
$$320$$ − 1.00000i − 0.0559017i
$$321$$ 14.4644 0.807325
$$322$$ 0.725873 0.0404513
$$323$$ 39.7482i 2.21165i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 0.0924579 0.00512077
$$327$$ 7.13706i 0.394681i
$$328$$ −3.40581 −0.188055
$$329$$ −2.15585 −0.118856
$$330$$ − 0.335126i − 0.0184481i
$$331$$ 26.4373i 1.45312i 0.687101 + 0.726562i $$0.258883\pi$$
−0.687101 + 0.726562i $$0.741117\pi$$
$$332$$ − 3.84117i − 0.210811i
$$333$$ − 7.07606i − 0.387766i
$$334$$ 9.64071 0.527516
$$335$$ −11.9608 −0.653487
$$336$$ − 0.307979i − 0.0168016i
$$337$$ −27.4698 −1.49638 −0.748188 0.663487i $$-0.769076\pi$$
−0.748188 + 0.663487i $$0.769076\pi$$
$$338$$ 0 0
$$339$$ −16.3424 −0.887598
$$340$$ − 6.85086i − 0.371540i
$$341$$ 0.0204423 0.00110701
$$342$$ 5.80194 0.313733
$$343$$ − 4.28249i − 0.231233i
$$344$$ 0.0489173i 0.00263745i
$$345$$ − 2.35690i − 0.126891i
$$346$$ 0.719169i 0.0386627i
$$347$$ −33.1323 −1.77863 −0.889317 0.457291i $$-0.848820\pi$$
−0.889317 + 0.457291i $$0.848820\pi$$
$$348$$ 5.91185 0.316909
$$349$$ − 26.3739i − 1.41176i −0.708331 0.705881i $$-0.750551\pi$$
0.708331 0.705881i $$-0.249449\pi$$
$$350$$ 0.307979 0.0164621
$$351$$ 0 0
$$352$$ −0.335126 −0.0178623
$$353$$ 21.0291i 1.11926i 0.828741 + 0.559632i $$0.189058\pi$$
−0.828741 + 0.559632i $$0.810942\pi$$
$$354$$ 13.1347 0.698100
$$355$$ −9.56465 −0.507639
$$356$$ − 4.41119i − 0.233793i
$$357$$ − 2.10992i − 0.111669i
$$358$$ − 2.01208i − 0.106342i
$$359$$ 30.8877i 1.63019i 0.579327 + 0.815095i $$0.303315\pi$$
−0.579327 + 0.815095i $$0.696685\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −14.6625 −0.771710
$$362$$ − 2.57135i − 0.135147i
$$363$$ 10.8877 0.571456
$$364$$ 0 0
$$365$$ 2.31336 0.121087
$$366$$ − 2.14675i − 0.112213i
$$367$$ 11.9584 0.624222 0.312111 0.950046i $$-0.398964\pi$$
0.312111 + 0.950046i $$0.398964\pi$$
$$368$$ −2.35690 −0.122862
$$369$$ 3.40581i 0.177300i
$$370$$ − 7.07606i − 0.367867i
$$371$$ − 3.74930i − 0.194654i
$$372$$ − 0.0609989i − 0.00316265i
$$373$$ 25.0315 1.29608 0.648040 0.761606i $$-0.275589\pi$$
0.648040 + 0.761606i $$0.275589\pi$$
$$374$$ −2.29590 −0.118718
$$375$$ − 1.00000i − 0.0516398i
$$376$$ 7.00000 0.360997
$$377$$ 0 0
$$378$$ −0.307979 −0.0158407
$$379$$ − 34.4989i − 1.77209i −0.463602 0.886044i $$-0.653443\pi$$
0.463602 0.886044i $$-0.346557\pi$$
$$380$$ 5.80194 0.297633
$$381$$ 0.911854 0.0467157
$$382$$ − 3.80194i − 0.194524i
$$383$$ 10.4644i 0.534707i 0.963599 + 0.267353i $$0.0861491\pi$$
−0.963599 + 0.267353i $$0.913851\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ − 0.103211i − 0.00526014i
$$386$$ 8.86294 0.451112
$$387$$ 0.0489173 0.00248661
$$388$$ − 6.13169i − 0.311289i
$$389$$ −25.2218 −1.27879 −0.639397 0.768877i $$-0.720816\pi$$
−0.639397 + 0.768877i $$0.720816\pi$$
$$390$$ 0 0
$$391$$ −16.1468 −0.816576
$$392$$ 6.90515i 0.348763i
$$393$$ 13.0368 0.657621
$$394$$ −22.6504 −1.14111
$$395$$ − 0.0760644i − 0.00382722i
$$396$$ 0.335126i 0.0168407i
$$397$$ − 21.3860i − 1.07333i −0.843795 0.536665i $$-0.819684\pi$$
0.843795 0.536665i $$-0.180316\pi$$
$$398$$ − 5.46681i − 0.274027i
$$399$$ 1.78687 0.0894555
$$400$$ −1.00000 −0.0500000
$$401$$ 11.8062i 0.589576i 0.955563 + 0.294788i $$0.0952490\pi$$
−0.955563 + 0.294788i $$0.904751\pi$$
$$402$$ 11.9608 0.596549
$$403$$ 0 0
$$404$$ 1.32975 0.0661575
$$405$$ 1.00000i 0.0496904i
$$406$$ 1.82072 0.0903610
$$407$$ −2.37137 −0.117544
$$408$$ 6.85086i 0.339168i
$$409$$ − 5.65950i − 0.279844i −0.990163 0.139922i $$-0.955315\pi$$
0.990163 0.139922i $$-0.0446852\pi$$
$$410$$ 3.40581i 0.168201i
$$411$$ − 12.8877i − 0.635703i
$$412$$ 8.92154 0.439533
$$413$$ 4.04520 0.199051
$$414$$ 2.35690i 0.115835i
$$415$$ −3.84117 −0.188555
$$416$$ 0 0
$$417$$ −2.01208 −0.0985321
$$418$$ − 1.94438i − 0.0951026i
$$419$$ 17.0084 0.830913 0.415456 0.909613i $$-0.363622\pi$$
0.415456 + 0.909613i $$0.363622\pi$$
$$420$$ −0.307979 −0.0150278
$$421$$ − 10.2851i − 0.501267i −0.968082 0.250634i $$-0.919361\pi$$
0.968082 0.250634i $$-0.0806389\pi$$
$$422$$ − 11.0422i − 0.537526i
$$423$$ − 7.00000i − 0.340352i
$$424$$ 12.1739i 0.591217i
$$425$$ −6.85086 −0.332315
$$426$$ 9.56465 0.463409
$$427$$ − 0.661154i − 0.0319955i
$$428$$ −14.4644 −0.699164
$$429$$ 0 0
$$430$$ 0.0489173 0.00235900
$$431$$ − 35.9657i − 1.73241i −0.499693 0.866203i $$-0.666554\pi$$
0.499693 0.866203i $$-0.333446\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −7.63342 −0.366839 −0.183419 0.983035i $$-0.558717\pi$$
−0.183419 + 0.983035i $$0.558717\pi$$
$$434$$ − 0.0187864i 0 0.000901774i
$$435$$ − 5.91185i − 0.283452i
$$436$$ − 7.13706i − 0.341803i
$$437$$ − 13.6746i − 0.654143i
$$438$$ −2.31336 −0.110536
$$439$$ 34.4373 1.64360 0.821801 0.569775i $$-0.192970\pi$$
0.821801 + 0.569775i $$0.192970\pi$$
$$440$$ 0.335126i 0.0159765i
$$441$$ 6.90515 0.328817
$$442$$ 0 0
$$443$$ 33.6431 1.59843 0.799216 0.601044i $$-0.205248\pi$$
0.799216 + 0.601044i $$0.205248\pi$$
$$444$$ 7.07606i 0.335815i
$$445$$ −4.41119 −0.209110
$$446$$ 18.3884 0.870714
$$447$$ 3.44935i 0.163149i
$$448$$ 0.307979i 0.0145506i
$$449$$ 27.9729i 1.32012i 0.751213 + 0.660060i $$0.229469\pi$$
−0.751213 + 0.660060i $$0.770531\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ 1.14138 0.0537453
$$452$$ 16.3424 0.768683
$$453$$ − 2.84117i − 0.133490i
$$454$$ 12.0073 0.563530
$$455$$ 0 0
$$456$$ −5.80194 −0.271701
$$457$$ − 6.18060i − 0.289116i −0.989496 0.144558i $$-0.953824\pi$$
0.989496 0.144558i $$-0.0461761\pi$$
$$458$$ −25.4601 −1.18967
$$459$$ 6.85086 0.319771
$$460$$ 2.35690i 0.109891i
$$461$$ 38.1062i 1.77478i 0.461017 + 0.887391i $$0.347485\pi$$
−0.461017 + 0.887391i $$0.652515\pi$$
$$462$$ 0.103211i 0.00480183i
$$463$$ 2.06829i 0.0961218i 0.998844 + 0.0480609i $$0.0153042\pi$$
−0.998844 + 0.0480609i $$0.984696\pi$$
$$464$$ −5.91185 −0.274451
$$465$$ −0.0609989 −0.00282876
$$466$$ − 2.10992i − 0.0977400i
$$467$$ 4.27114 0.197645 0.0988225 0.995105i $$-0.468492\pi$$
0.0988225 + 0.995105i $$0.468492\pi$$
$$468$$ 0 0
$$469$$ 3.68366 0.170096
$$470$$ − 7.00000i − 0.322886i
$$471$$ 16.2228 0.747508
$$472$$ −13.1347 −0.604572
$$473$$ − 0.0163935i 0 0.000753772i
$$474$$ 0.0760644i 0.00349375i
$$475$$ − 5.80194i − 0.266211i
$$476$$ 2.10992i 0.0967079i
$$477$$ 12.1739 0.557405
$$478$$ 3.95838 0.181052
$$479$$ 38.7560i 1.77081i 0.464823 + 0.885404i $$0.346118\pi$$
−0.464823 + 0.885404i $$0.653882\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ 23.1564 1.05475
$$483$$ 0.725873i 0.0330284i
$$484$$ −10.8877 −0.494895
$$485$$ −6.13169 −0.278426
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 33.4620i − 1.51631i −0.652075 0.758155i $$-0.726101\pi$$
0.652075 0.758155i $$-0.273899\pi$$
$$488$$ 2.14675i 0.0971789i
$$489$$ 0.0924579i 0.00418109i
$$490$$ 6.90515 0.311943
$$491$$ −35.1487 −1.58624 −0.793119 0.609067i $$-0.791544\pi$$
−0.793119 + 0.609067i $$0.791544\pi$$
$$492$$ − 3.40581i − 0.153546i
$$493$$ −40.5013 −1.82408
$$494$$ 0 0
$$495$$ 0.335126 0.0150628
$$496$$ 0.0609989i 0.00273893i
$$497$$ 2.94571 0.132133
$$498$$ 3.84117 0.172127
$$499$$ − 9.61463i − 0.430410i −0.976569 0.215205i $$-0.930958\pi$$
0.976569 0.215205i $$-0.0690420\pi$$
$$500$$ 1.00000i 0.0447214i
$$501$$ 9.64071i 0.430715i
$$502$$ − 5.54048i − 0.247284i
$$503$$ 18.5265 0.826055 0.413028 0.910719i $$-0.364471\pi$$
0.413028 + 0.910719i $$0.364471\pi$$
$$504$$ 0.307979 0.0137185
$$505$$ − 1.32975i − 0.0591730i
$$506$$ 0.789856 0.0351134
$$507$$ 0 0
$$508$$ −0.911854 −0.0404570
$$509$$ 5.43296i 0.240812i 0.992725 + 0.120406i $$0.0384196\pi$$
−0.992725 + 0.120406i $$0.961580\pi$$
$$510$$ 6.85086 0.303361
$$511$$ −0.712464 −0.0315175
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 5.80194i 0.256162i
$$514$$ 6.60819i 0.291475i
$$515$$ − 8.92154i − 0.393130i
$$516$$ −0.0489173 −0.00215347
$$517$$ −2.34588 −0.103172
$$518$$ 2.17928i 0.0957519i
$$519$$ −0.719169 −0.0315680
$$520$$ 0 0
$$521$$ −38.8001 −1.69986 −0.849932 0.526892i $$-0.823357\pi$$
−0.849932 + 0.526892i $$0.823357\pi$$
$$522$$ 5.91185i 0.258755i
$$523$$ 32.0629 1.40201 0.701007 0.713155i $$-0.252734\pi$$
0.701007 + 0.713155i $$0.252734\pi$$
$$524$$ −13.0368 −0.569517
$$525$$ 0.307979i 0.0134413i
$$526$$ 0.660563i 0.0288019i
$$527$$ 0.417895i 0.0182038i
$$528$$ − 0.335126i − 0.0145845i
$$529$$ −17.4450 −0.758480
$$530$$ 12.1739 0.528800
$$531$$ 13.1347i 0.569996i
$$532$$ −1.78687 −0.0774707
$$533$$ 0 0
$$534$$ 4.41119 0.190891
$$535$$ 14.4644i 0.625351i
$$536$$ −11.9608 −0.516627
$$537$$ 2.01208 0.0868277
$$538$$ 26.7928i 1.15512i
$$539$$ − 2.31409i − 0.0996750i
$$540$$ − 1.00000i − 0.0430331i
$$541$$ − 31.8334i − 1.36862i −0.729189 0.684312i $$-0.760103\pi$$
0.729189 0.684312i $$-0.239897\pi$$
$$542$$ −14.0151 −0.601999
$$543$$ 2.57135 0.110347
$$544$$ − 6.85086i − 0.293728i
$$545$$ −7.13706 −0.305718
$$546$$ 0 0
$$547$$ −44.4010 −1.89845 −0.949225 0.314597i $$-0.898131\pi$$
−0.949225 + 0.314597i $$0.898131\pi$$
$$548$$ 12.8877i 0.550535i
$$549$$ 2.14675 0.0916211
$$550$$ 0.335126 0.0142898
$$551$$ − 34.3002i − 1.46124i
$$552$$ − 2.35690i − 0.100316i
$$553$$ 0.0234262i 0 0.000996184i
$$554$$ − 23.4252i − 0.995241i
$$555$$ 7.07606 0.300362
$$556$$ 2.01208 0.0853313
$$557$$ 7.73423i 0.327710i 0.986484 + 0.163855i $$0.0523929\pi$$
−0.986484 + 0.163855i $$0.947607\pi$$
$$558$$ 0.0609989 0.00258229
$$559$$ 0 0
$$560$$ 0.307979 0.0130145
$$561$$ − 2.29590i − 0.0969328i
$$562$$ −8.40044 −0.354351
$$563$$ −44.5803 −1.87884 −0.939418 0.342774i $$-0.888633\pi$$
−0.939418 + 0.342774i $$0.888633\pi$$
$$564$$ 7.00000i 0.294753i
$$565$$ − 16.3424i − 0.687531i
$$566$$ − 31.6142i − 1.32884i
$$567$$ − 0.307979i − 0.0129339i
$$568$$ −9.56465 −0.401324
$$569$$ −24.9065 −1.04413 −0.522067 0.852905i $$-0.674839\pi$$
−0.522067 + 0.852905i $$0.674839\pi$$
$$570$$ 5.80194i 0.243016i
$$571$$ 0.873690 0.0365628 0.0182814 0.999833i $$-0.494181\pi$$
0.0182814 + 0.999833i $$0.494181\pi$$
$$572$$ 0 0
$$573$$ 3.80194 0.158828
$$574$$ − 1.04892i − 0.0437810i
$$575$$ 2.35690 0.0982894
$$576$$ −1.00000 −0.0416667
$$577$$ − 20.0871i − 0.836236i −0.908393 0.418118i $$-0.862690\pi$$
0.908393 0.418118i $$-0.137310\pi$$
$$578$$ − 29.9342i − 1.24510i
$$579$$ 8.86294i 0.368331i
$$580$$ 5.91185i 0.245476i
$$581$$ 1.18300 0.0490790
$$582$$ 6.13169 0.254167
$$583$$ − 4.07979i − 0.168967i
$$584$$ 2.31336 0.0957273
$$585$$ 0 0
$$586$$ 17.5254 0.723968
$$587$$ − 16.0901i − 0.664108i −0.943260 0.332054i $$-0.892258\pi$$
0.943260 0.332054i $$-0.107742\pi$$
$$588$$ −6.90515 −0.284764
$$589$$ −0.353912 −0.0145827
$$590$$ 13.1347i 0.540746i
$$591$$ − 22.6504i − 0.931713i
$$592$$ − 7.07606i − 0.290824i
$$593$$ − 4.85517i − 0.199378i −0.995019 0.0996889i $$-0.968215\pi$$
0.995019 0.0996889i $$-0.0317848\pi$$
$$594$$ −0.335126 −0.0137504
$$595$$ 2.10992 0.0864981
$$596$$ − 3.44935i − 0.141291i
$$597$$ 5.46681 0.223742
$$598$$ 0 0
$$599$$ −34.4319 −1.40685 −0.703425 0.710770i $$-0.748347\pi$$
−0.703425 + 0.710770i $$0.748347\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ −24.5244 −1.00037 −0.500185 0.865919i $$-0.666735\pi$$
−0.500185 + 0.865919i $$0.666735\pi$$
$$602$$ −0.0150655 −0.000614024 0
$$603$$ 11.9608i 0.487080i
$$604$$ 2.84117i 0.115605i
$$605$$ 10.8877i 0.442648i
$$606$$ 1.32975i 0.0540174i
$$607$$ 17.6420 0.716068 0.358034 0.933708i $$-0.383447\pi$$
0.358034 + 0.933708i $$0.383447\pi$$
$$608$$ 5.80194 0.235300
$$609$$ 1.82072i 0.0737795i
$$610$$ 2.14675 0.0869194
$$611$$ 0 0
$$612$$ −6.85086 −0.276929
$$613$$ 44.5018i 1.79741i 0.438551 + 0.898706i $$0.355492\pi$$
−0.438551 + 0.898706i $$0.644508\pi$$
$$614$$ −20.6732 −0.834304
$$615$$ −3.40581 −0.137336
$$616$$ − 0.103211i − 0.00415851i
$$617$$ 32.1618i 1.29479i 0.762156 + 0.647393i $$0.224141\pi$$
−0.762156 + 0.647393i $$0.775859\pi$$
$$618$$ 8.92154i 0.358877i
$$619$$ − 24.8340i − 0.998162i −0.866555 0.499081i $$-0.833671\pi$$
0.866555 0.499081i $$-0.166329\pi$$
$$620$$ 0.0609989 0.00244978
$$621$$ −2.35690 −0.0945790
$$622$$ − 22.0248i − 0.883112i
$$623$$ 1.35855 0.0544292
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 21.7192i − 0.868073i
$$627$$ 1.94438 0.0776510
$$628$$ −16.2228 −0.647361
$$629$$ − 48.4771i − 1.93291i
$$630$$ − 0.307979i − 0.0122702i
$$631$$ − 10.5418i − 0.419663i −0.977738 0.209831i $$-0.932708\pi$$
0.977738 0.209831i $$-0.0672915\pi$$
$$632$$ − 0.0760644i − 0.00302568i
$$633$$ 11.0422 0.438889
$$634$$ −16.4644 −0.653886
$$635$$ 0.911854i 0.0361858i
$$636$$ −12.1739 −0.482726
$$637$$ 0 0
$$638$$ 1.98121 0.0784370
$$639$$ 9.56465i 0.378372i
$$640$$ −1.00000 −0.0395285
$$641$$ −33.5663 −1.32579 −0.662895 0.748713i $$-0.730672\pi$$
−0.662895 + 0.748713i $$0.730672\pi$$
$$642$$ − 14.4644i − 0.570865i
$$643$$ 8.24160i 0.325017i 0.986707 + 0.162509i $$0.0519585\pi$$
−0.986707 + 0.162509i $$0.948041\pi$$
$$644$$ − 0.725873i − 0.0286034i
$$645$$ 0.0489173i 0.00192612i
$$646$$ 39.7482 1.56387
$$647$$ 3.66056 0.143912 0.0719558 0.997408i $$-0.477076\pi$$
0.0719558 + 0.997408i $$0.477076\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 4.40176 0.172784
$$650$$ 0 0
$$651$$ 0.0187864 0.000736295 0
$$652$$ − 0.0924579i − 0.00362093i
$$653$$ −44.6064 −1.74558 −0.872791 0.488093i $$-0.837693\pi$$
−0.872791 + 0.488093i $$0.837693\pi$$
$$654$$ 7.13706 0.279081
$$655$$ 13.0368i 0.509391i
$$656$$ 3.40581i 0.132975i
$$657$$ − 2.31336i − 0.0902526i
$$658$$ 2.15585i 0.0840438i
$$659$$ −2.51035 −0.0977895 −0.0488947 0.998804i $$-0.515570\pi$$
−0.0488947 + 0.998804i $$0.515570\pi$$
$$660$$ −0.335126 −0.0130447
$$661$$ − 40.3096i − 1.56786i −0.620847 0.783932i $$-0.713211\pi$$
0.620847 0.783932i $$-0.286789\pi$$
$$662$$ 26.4373 1.02751
$$663$$ 0 0
$$664$$ −3.84117 −0.149066
$$665$$ 1.78687i 0.0692919i
$$666$$ −7.07606 −0.274192
$$667$$ 13.9336 0.539512
$$668$$ − 9.64071i − 0.373010i
$$669$$ 18.3884i 0.710935i
$$670$$ 11.9608i 0.462085i
$$671$$ − 0.719432i − 0.0277733i
$$672$$ −0.307979 −0.0118805
$$673$$ −42.6340 −1.64342 −0.821710 0.569906i $$-0.806980\pi$$
−0.821710 + 0.569906i $$0.806980\pi$$
$$674$$ 27.4698i 1.05810i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 18.5254 0.711990 0.355995 0.934488i $$-0.384142\pi$$
0.355995 + 0.934488i $$0.384142\pi$$
$$678$$ 16.3424i 0.627627i
$$679$$ 1.88843 0.0724712
$$680$$ −6.85086 −0.262718
$$681$$ 12.0073i 0.460121i
$$682$$ − 0.0204423i 0 0.000782776i
$$683$$ − 48.1769i − 1.84344i −0.387859 0.921719i $$-0.626785\pi$$
0.387859 0.921719i $$-0.373215\pi$$
$$684$$ − 5.80194i − 0.221843i
$$685$$ 12.8877 0.492413
$$686$$ −4.28249 −0.163506
$$687$$ − 25.4601i − 0.971364i
$$688$$ 0.0489173 0.00186496
$$689$$ 0 0
$$690$$ −2.35690 −0.0897255
$$691$$ 37.0084i 1.40786i 0.710267 + 0.703932i $$0.248574\pi$$
−0.710267 + 0.703932i $$0.751426\pi$$
$$692$$ 0.719169 0.0273387
$$693$$ −0.103211 −0.00392068
$$694$$ 33.1323i 1.25768i
$$695$$ − 2.01208i − 0.0763226i
$$696$$ − 5.91185i − 0.224088i
$$697$$ 23.3327i 0.883790i
$$698$$ −26.3739 −0.998266
$$699$$ 2.10992 0.0798044
$$700$$ − 0.307979i − 0.0116405i
$$701$$ −9.25188 −0.349439 −0.174719 0.984618i $$-0.555902\pi$$
−0.174719 + 0.984618i $$0.555902\pi$$
$$702$$ 0 0
$$703$$ 41.0549 1.54841
$$704$$ 0.335126i 0.0126305i
$$705$$ 7.00000 0.263635
$$706$$ 21.0291 0.791439
$$707$$ 0.409534i 0.0154021i
$$708$$ − 13.1347i − 0.493631i
$$709$$ 15.0573i 0.565488i 0.959195 + 0.282744i $$0.0912447\pi$$
−0.959195 + 0.282744i $$0.908755\pi$$
$$710$$ 9.56465i 0.358955i
$$711$$ −0.0760644 −0.00285264
$$712$$ −4.41119 −0.165316
$$713$$ − 0.143768i − 0.00538416i
$$714$$ −2.10992 −0.0789616
$$715$$ 0 0
$$716$$ −2.01208 −0.0751950
$$717$$ 3.95838i 0.147828i
$$718$$ 30.8877 1.15272
$$719$$ −24.8364 −0.926241 −0.463120 0.886295i $$-0.653270\pi$$
−0.463120 + 0.886295i $$0.653270\pi$$
$$720$$ 1.00000i 0.0372678i
$$721$$ 2.74764i 0.102328i
$$722$$ 14.6625i 0.545681i
$$723$$ 23.1564i 0.861197i
$$724$$ −2.57135 −0.0955635
$$725$$ 5.91185 0.219561
$$726$$ − 10.8877i − 0.404080i
$$727$$ 24.7590 0.918260 0.459130 0.888369i $$-0.348161\pi$$
0.459130 + 0.888369i $$0.348161\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ − 2.31336i − 0.0856211i
$$731$$ 0.335126 0.0123951
$$732$$ −2.14675 −0.0793462
$$733$$ − 15.4359i − 0.570140i −0.958507 0.285070i $$-0.907983\pi$$
0.958507 0.285070i $$-0.0920168\pi$$
$$734$$ − 11.9584i − 0.441392i
$$735$$ 6.90515i 0.254700i
$$736$$ 2.35690i 0.0868763i
$$737$$ 4.00836 0.147650
$$738$$ 3.40581 0.125370
$$739$$ 20.3327i 0.747952i 0.927438 + 0.373976i $$0.122006\pi$$
−0.927438 + 0.373976i $$0.877994\pi$$
$$740$$ −7.07606 −0.260121
$$741$$ 0 0
$$742$$ −3.74930 −0.137641
$$743$$ 17.8243i 0.653910i 0.945040 + 0.326955i $$0.106023\pi$$
−0.945040 + 0.326955i $$0.893977\pi$$
$$744$$ −0.0609989 −0.00223633
$$745$$ −3.44935 −0.126375
$$746$$ − 25.0315i − 0.916467i
$$747$$ 3.84117i 0.140541i
$$748$$ 2.29590i 0.0839463i
$$749$$ − 4.45473i − 0.162772i
$$750$$ −1.00000 −0.0365148
$$751$$ −48.0616 −1.75379 −0.876896 0.480680i $$-0.840390\pi$$
−0.876896 + 0.480680i $$0.840390\pi$$
$$752$$ − 7.00000i − 0.255264i
$$753$$ 5.54048 0.201906
$$754$$ 0 0
$$755$$ 2.84117 0.103401
$$756$$ 0.307979i 0.0112011i
$$757$$ 32.9748 1.19849 0.599244 0.800566i $$-0.295468\pi$$
0.599244 + 0.800566i $$0.295468\pi$$
$$758$$ −34.4989 −1.25306
$$759$$ 0.789856i 0.0286700i
$$760$$ − 5.80194i − 0.210458i
$$761$$ − 11.9946i − 0.434805i −0.976082 0.217402i $$-0.930242\pi$$
0.976082 0.217402i $$-0.0697584\pi$$
$$762$$ − 0.911854i − 0.0330330i
$$763$$ 2.19806 0.0795752
$$764$$ −3.80194 −0.137549
$$765$$ 6.85086i 0.247693i
$$766$$ 10.4644 0.378095
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ − 18.5036i − 0.667259i −0.942704 0.333629i $$-0.891727\pi$$
0.942704 0.333629i $$-0.108273\pi$$
$$770$$ −0.103211 −0.00371948
$$771$$ −6.60819 −0.237988
$$772$$ − 8.86294i − 0.318984i
$$773$$ − 23.4946i − 0.845040i −0.906354 0.422520i $$-0.861146\pi$$
0.906354 0.422520i $$-0.138854\pi$$
$$774$$ − 0.0489173i − 0.00175830i
$$775$$ − 0.0609989i − 0.00219115i
$$776$$ −6.13169 −0.220115
$$777$$ −2.17928 −0.0781811
$$778$$ 25.2218i 0.904244i
$$779$$ −19.7603 −0.707987
$$780$$ 0 0
$$781$$ 3.20536 0.114697
$$782$$ 16.1468i 0.577407i
$$783$$ −5.91185 −0.211272
$$784$$ 6.90515 0.246612
$$785$$ 16.2228i 0.579017i
$$786$$ − 13.0368i − 0.465009i
$$787$$ − 7.23682i − 0.257965i −0.991647 0.128982i $$-0.958829\pi$$
0.991647 0.128982i $$-0.0411711\pi$$
$$788$$ 22.6504i 0.806887i
$$789$$ −0.660563 −0.0235166
$$790$$ −0.0760644 −0.00270625
$$791$$ 5.03311i 0.178957i
$$792$$ 0.335126 0.0119082
$$793$$ 0 0
$$794$$ −21.3860 −0.758959
$$795$$ 12.1739i 0.431764i
$$796$$ −5.46681 −0.193766
$$797$$ 46.7079 1.65448 0.827240 0.561849i $$-0.189910\pi$$
0.827240 + 0.561849i $$0.189910\pi$$
$$798$$ − 1.78687i − 0.0632546i
$$799$$ − 47.9560i − 1.69656i
$$800$$ 1.00000i 0.0353553i
$$801$$ 4.41119i 0.155862i
$$802$$ 11.8062 0.416893
$$803$$ −0.775265 −0.0273585
$$804$$ − 11.9608i − 0.421824i
$$805$$ −0.725873 −0.0255837
$$806$$ 0 0
$$807$$ −26.7928 −0.943153
$$808$$ − 1.32975i − 0.0467804i
$$809$$ −9.95838 −0.350118 −0.175059 0.984558i $$-0.556012\pi$$
−0.175059 + 0.984558i $$0.556012\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ − 36.2452i − 1.27274i −0.771384 0.636370i $$-0.780435\pi$$
0.771384 0.636370i $$-0.219565\pi$$
$$812$$ − 1.82072i − 0.0638949i
$$813$$ − 14.0151i − 0.491530i
$$814$$ 2.37137i 0.0831165i
$$815$$ −0.0924579 −0.00323866
$$816$$ 6.85086 0.239828
$$817$$ 0.283815i 0.00992944i
$$818$$ −5.65950 −0.197880
$$819$$ 0 0
$$820$$ 3.40581 0.118936
$$821$$ − 1.37867i − 0.0481158i −0.999711 0.0240579i $$-0.992341\pi$$
0.999711 0.0240579i $$-0.00765860\pi$$
$$822$$ −12.8877 −0.449510
$$823$$ 23.8713 0.832101 0.416051 0.909341i $$-0.363414\pi$$
0.416051 + 0.909341i $$0.363414\pi$$
$$824$$ − 8.92154i − 0.310797i
$$825$$ 0.335126i 0.0116676i
$$826$$ − 4.04520i − 0.140750i
$$827$$ 21.9571i 0.763521i 0.924261 + 0.381761i $$0.124682\pi$$
−0.924261 + 0.381761i $$0.875318\pi$$
$$828$$ 2.35690 0.0819078
$$829$$ 27.4862 0.954635 0.477317 0.878731i $$-0.341609\pi$$
0.477317 + 0.878731i $$0.341609\pi$$
$$830$$ 3.84117i 0.133329i
$$831$$ 23.4252 0.812611
$$832$$ 0 0
$$833$$ 47.3062 1.63906
$$834$$ 2.01208i 0.0696727i
$$835$$ −9.64071 −0.333631
$$836$$ −1.94438 −0.0672477
$$837$$ 0.0609989i 0.00210843i
$$838$$ − 17.0084i − 0.587544i
$$839$$ − 20.5749i − 0.710325i −0.934805 0.355163i $$-0.884425\pi$$
0.934805 0.355163i $$-0.115575\pi$$
$$840$$ 0.307979i 0.0106263i
$$841$$ 5.95002 0.205173
$$842$$ −10.2851 −0.354449
$$843$$ − 8.40044i − 0.289326i
$$844$$ −11.0422 −0.380089
$$845$$ 0 0
$$846$$ −7.00000 −0.240665
$$847$$ − 3.35317i − 0.115216i
$$848$$ 12.1739 0.418053
$$849$$ 31.6142 1.08499
$$850$$ 6.85086i 0.234982i
$$851$$ 16.6775i 0.571699i
$$852$$ − 9.56465i − 0.327679i
$$853$$ 50.3812i 1.72502i 0.506041 + 0.862509i $$0.331108\pi$$
−0.506041 + 0.862509i $$0.668892\pi$$
$$854$$ −0.661154 −0.0226242
$$855$$ −5.80194 −0.198422
$$856$$ 14.4644i 0.494384i
$$857$$ −21.6045 −0.737995 −0.368997 0.929430i $$-0.620299\pi$$
−0.368997 + 0.929430i $$0.620299\pi$$
$$858$$ 0 0
$$859$$ −38.2959 −1.30664 −0.653320 0.757082i $$-0.726624\pi$$
−0.653320 + 0.757082i $$0.726624\pi$$
$$860$$ − 0.0489173i − 0.00166807i
$$861$$ 1.04892 0.0357470
$$862$$ −35.9657 −1.22500
$$863$$ 8.61894i 0.293392i 0.989182 + 0.146696i $$0.0468639\pi$$
−0.989182 + 0.146696i $$0.953136\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ − 0.719169i − 0.0244525i
$$866$$ 7.63342i 0.259394i
$$867$$ 29.9342 1.01662
$$868$$ −0.0187864 −0.000637651 0
$$869$$ 0.0254911i 0 0.000864727i
$$870$$ −5.91185 −0.200431
$$871$$ 0 0
$$872$$ −7.13706 −0.241691
$$873$$ 6.13169i 0.207526i
$$874$$ −13.6746 −0.462549
$$875$$ −0.307979 −0.0104116
$$876$$ 2.31336i 0.0781610i
$$877$$ − 33.1021i − 1.11778i −0.829242 0.558890i $$-0.811227\pi$$
0.829242 0.558890i $$-0.188773\pi$$
$$878$$ − 34.4373i − 1.16220i
$$879$$ 17.5254i 0.591118i
$$880$$ 0.335126 0.0112971
$$881$$ −20.8267 −0.701669 −0.350835 0.936437i $$-0.614102\pi$$
−0.350835 + 0.936437i $$0.614102\pi$$
$$882$$ − 6.90515i − 0.232508i
$$883$$ −0.785807 −0.0264445 −0.0132223 0.999913i $$-0.504209\pi$$
−0.0132223 + 0.999913i $$0.504209\pi$$
$$884$$ 0 0
$$885$$ −13.1347 −0.441517
$$886$$ − 33.6431i − 1.13026i
$$887$$ −44.2392 −1.48541 −0.742704 0.669620i $$-0.766457\pi$$
−0.742704 + 0.669620i $$0.766457\pi$$
$$888$$ 7.07606 0.237457
$$889$$ − 0.280831i − 0.00941878i
$$890$$ 4.41119i 0.147863i
$$891$$ − 0.335126i − 0.0112271i
$$892$$ − 18.3884i − 0.615688i
$$893$$ 40.6136 1.35908
$$894$$ 3.44935 0.115364
$$895$$ 2.01208i 0.0672565i
$$896$$ 0.307979 0.0102888
$$897$$ 0 0
$$898$$ 27.9729 0.933466
$$899$$ − 0.360617i − 0.0120272i
$$900$$ 1.00000 0.0333333
$$901$$ 83.4016 2.77851
$$902$$ − 1.14138i − 0.0380036i
$$903$$ − 0.0150655i 0 0.000501348i
$$904$$ − 16.3424i − 0.543541i
$$905$$ 2.57135i 0.0854746i
$$906$$ −2.84117 −0.0943914
$$907$$ −8.44371 −0.280369 −0.140184 0.990125i $$-0.544770\pi$$
−0.140184 + 0.990125i $$0.544770\pi$$
$$908$$ − 12.0073i − 0.398476i
$$909$$ −1.32975 −0.0441050
$$910$$ 0 0
$$911$$ −47.2801 −1.56646 −0.783230 0.621732i $$-0.786429\pi$$
−0.783230 + 0.621732i $$0.786429\pi$$
$$912$$ 5.80194i 0.192121i
$$913$$ 1.28727 0.0426025
$$914$$ −6.18060 −0.204436
$$915$$ 2.14675i 0.0709694i
$$916$$ 25.4601i 0.841226i
$$917$$ − 4.01507i − 0.132589i
$$918$$ − 6.85086i − 0.226112i
$$919$$ −11.1927 −0.369213 −0.184606 0.982813i $$-0.559101\pi$$
−0.184606 + 0.982813i $$0.559101\pi$$
$$920$$ 2.35690 0.0777046
$$921$$ − 20.6732i − 0.681206i
$$922$$ 38.1062 1.25496
$$923$$ 0 0
$$924$$ 0.103211 0.00339541
$$925$$ 7.07606i 0.232660i
$$926$$ 2.06829 0.0679684
$$927$$ −8.92154 −0.293022
$$928$$ 5.91185i 0.194066i
$$929$$ 1.11828i 0.0366895i 0.999832 + 0.0183447i $$0.00583964\pi$$
−0.999832 + 0.0183447i $$0.994160\pi$$
$$930$$ 0.0609989i 0.00200023i
$$931$$ 40.0632i 1.31302i
$$932$$ −2.10992 −0.0691126
$$933$$ 22.0248 0.721058
$$934$$ − 4.27114i − 0.139756i
$$935$$ 2.29590 0.0750839
$$936$$ 0 0
$$937$$ −29.9051 −0.976959 −0.488479 0.872575i $$-0.662448\pi$$
−0.488479 + 0.872575i $$0.662448\pi$$
$$938$$ − 3.68366i − 0.120276i
$$939$$ 21.7192 0.708778
$$940$$ −7.00000 −0.228315
$$941$$ − 32.7646i − 1.06810i −0.845454 0.534048i $$-0.820670\pi$$
0.845454 0.534048i $$-0.179330\pi$$
$$942$$ − 16.2228i − 0.528568i
$$943$$ − 8.02715i − 0.261400i
$$944$$ 13.1347i 0.427497i
$$945$$ 0.307979 0.0100185
$$946$$ −0.0163935 −0.000532997 0
$$947$$ 6.81535i 0.221469i 0.993850 + 0.110735i $$0.0353203\pi$$
−0.993850 + 0.110735i $$0.964680\pi$$
$$948$$ 0.0760644 0.00247046
$$949$$ 0 0
$$950$$ −5.80194 −0.188240
$$951$$ − 16.4644i − 0.533895i
$$952$$ 2.10992 0.0683828
$$953$$ −23.6886 −0.767348 −0.383674 0.923469i $$-0.625341\pi$$
−0.383674 + 0.923469i $$0.625341\pi$$
$$954$$ − 12.1739i − 0.394145i
$$955$$ 3.80194i 0.123028i
$$956$$ − 3.95838i − 0.128023i
$$957$$ 1.98121i 0.0640435i
$$958$$ 38.7560 1.25215
$$959$$ −3.96913 −0.128170
$$960$$ − 1.00000i − 0.0322749i
$$961$$ 30.9963 0.999880
$$962$$ 0 0
$$963$$ 14.4644 0.466109
$$964$$ − 23.1564i − 0.745819i
$$965$$ −8.86294 −0.285308
$$966$$ 0.725873 0.0233546
$$967$$ − 28.9075i − 0.929604i −0.885415 0.464802i $$-0.846126\pi$$
0.885415 0.464802i $$-0.153874\pi$$
$$968$$ 10.8877i 0.349944i
$$969$$ 39.7482i 1.27690i
$$970$$ 6.13169i 0.196877i
$$971$$ −47.2717 −1.51702 −0.758511 0.651660i $$-0.774073\pi$$
−0.758511 + 0.651660i $$0.774073\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0.619678i 0.0198660i
$$974$$ −33.4620 −1.07219
$$975$$ 0 0
$$976$$ 2.14675 0.0687159
$$977$$ − 33.9527i − 1.08624i −0.839654 0.543122i $$-0.817242\pi$$
0.839654 0.543122i $$-0.182758\pi$$
$$978$$ 0.0924579 0.00295648
$$979$$ 1.47830 0.0472468
$$980$$ − 6.90515i − 0.220577i
$$981$$ 7.13706i 0.227869i
$$982$$ 35.1487i 1.12164i
$$983$$ 22.9366i 0.731564i 0.930701 + 0.365782i $$0.119198\pi$$
−0.930701 + 0.365782i $$0.880802\pi$$
$$984$$ −3.40581 −0.108573
$$985$$ 22.6504 0.721702
$$986$$ 40.5013i 1.28982i
$$987$$ −2.15585 −0.0686215
$$988$$ 0 0
$$989$$ −0.115293 −0.00366611
$$990$$ − 0.335126i − 0.0106510i
$$991$$ −5.75781 −0.182903 −0.0914514 0.995810i $$-0.529151\pi$$
−0.0914514 + 0.995810i $$0.529151\pi$$
$$992$$ 0.0609989 0.00193672
$$993$$ 26.4373i 0.838961i
$$994$$ − 2.94571i − 0.0934321i
$$995$$ 5.46681i 0.173310i
$$996$$ − 3.84117i − 0.121712i
$$997$$ 3.86353 0.122359 0.0611796 0.998127i $$-0.480514\pi$$
0.0611796 + 0.998127i $$0.480514\pi$$
$$998$$ −9.61463 −0.304346
$$999$$ − 7.07606i − 0.223877i
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 5070.2.b.y.1351.3 6
13.5 odd 4 5070.2.a.bq.1.1 3
13.8 odd 4 5070.2.a.bv.1.3 yes 3
13.12 even 2 inner 5070.2.b.y.1351.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bq.1.1 3 13.5 odd 4
5070.2.a.bv.1.3 yes 3 13.8 odd 4
5070.2.b.y.1351.3 6 1.1 even 1 trivial
5070.2.b.y.1351.4 6 13.12 even 2 inner