# Properties

 Label 845.2.c.g.506.5 Level $845$ Weight $2$ Character 845.506 Analytic conductor $6.747$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 506.5 Root $$1.40994 - 0.109843i$$ of defining polynomial Character $$\chi$$ $$=$$ 845.506 Dual form 845.2.c.g.506.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.219687i q^{2} -1.60020 q^{3} +1.95174 q^{4} +1.00000i q^{5} -0.351542i q^{6} -0.332247i q^{7} +0.868145i q^{8} -0.439374 q^{9} +O(q^{10})$$ $$q+0.219687i q^{2} -1.60020 q^{3} +1.95174 q^{4} +1.00000i q^{5} -0.351542i q^{6} -0.332247i q^{7} +0.868145i q^{8} -0.439374 q^{9} -0.219687 q^{10} -5.37182i q^{11} -3.12316 q^{12} +0.0729902 q^{14} -1.60020i q^{15} +3.71276 q^{16} +5.06430 q^{17} -0.0965246i q^{18} -2.26795i q^{19} +1.95174i q^{20} +0.531659i q^{21} +1.18012 q^{22} +2.83918 q^{23} -1.38920i q^{24} -1.00000 q^{25} +5.50367 q^{27} -0.648458i q^{28} -2.90348 q^{29} +0.351542 q^{30} -5.46410i q^{31} +2.55193i q^{32} +8.59596i q^{33} +1.11256i q^{34} +0.332247 q^{35} -0.857542 q^{36} +5.97201i q^{37} +0.498239 q^{38} -0.868145 q^{40} -3.73205i q^{41} -0.116799 q^{42} +5.06430 q^{43} -10.4844i q^{44} -0.439374i q^{45} +0.623730i q^{46} +8.34285i q^{47} -5.94114 q^{48} +6.88961 q^{49} -0.219687i q^{50} -8.10387 q^{51} -1.56063 q^{53} +1.20908i q^{54} +5.37182 q^{55} +0.288438 q^{56} +3.62916i q^{57} -0.637855i q^{58} +2.70732i q^{59} -3.12316i q^{60} +14.1039 q^{61} +1.20039 q^{62} +0.145980i q^{63} +6.86488 q^{64} -1.88842 q^{66} -10.3322i q^{67} +9.88418 q^{68} -4.54324 q^{69} +0.0729902i q^{70} -12.7973i q^{71} -0.381440i q^{72} -9.68922i q^{73} -1.31197 q^{74} +1.60020 q^{75} -4.42644i q^{76} -1.78477 q^{77} +4.51851 q^{79} +3.71276i q^{80} -7.48883 q^{81} +0.819883 q^{82} +4.26371i q^{83} +1.03766i q^{84} +5.06430i q^{85} +1.11256i q^{86} +4.64613 q^{87} +4.66351 q^{88} +3.22584i q^{89} +0.0965246 q^{90} +5.54133 q^{92} +8.74363i q^{93} -1.83281 q^{94} +2.26795 q^{95} -4.08359i q^{96} +2.50791i q^{97} +1.51356i q^{98} +2.36023i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} - 4q^{4} + 8q^{9} + O(q^{10})$$ $$8q - 4q^{3} - 4q^{4} + 8q^{9} + 4q^{10} + 20q^{12} + 4q^{14} + 4q^{16} + 4q^{17} + 24q^{22} + 20q^{23} - 8q^{25} - 4q^{27} + 16q^{29} - 8q^{30} - 20q^{35} - 40q^{36} - 16q^{38} - 12q^{40} - 8q^{42} + 4q^{43} - 56q^{48} - 24q^{49} - 8q^{51} - 24q^{53} - 24q^{56} + 56q^{61} - 8q^{62} - 8q^{64} + 12q^{66} + 28q^{68} + 32q^{69} - 20q^{74} + 4q^{75} - 36q^{77} - 16q^{79} - 16q^{81} - 8q^{82} - 44q^{87} + 36q^{88} + 40q^{90} + 44q^{92} - 64q^{94} + 32q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.219687i 0.155342i 0.996979 + 0.0776710i $$0.0247484\pi$$
−0.996979 + 0.0776710i $$0.975252\pi$$
$$3$$ −1.60020 −0.923873 −0.461937 0.886913i $$-0.652845\pi$$
−0.461937 + 0.886913i $$0.652845\pi$$
$$4$$ 1.95174 0.975869
$$5$$ 1.00000i 0.447214i
$$6$$ − 0.351542i − 0.143516i
$$7$$ − 0.332247i − 0.125577i −0.998027 0.0627887i $$-0.980001\pi$$
0.998027 0.0627887i $$-0.0199994\pi$$
$$8$$ 0.868145i 0.306936i
$$9$$ −0.439374 −0.146458
$$10$$ −0.219687 −0.0694711
$$11$$ − 5.37182i − 1.61966i −0.586662 0.809832i $$-0.699558\pi$$
0.586662 0.809832i $$-0.300442\pi$$
$$12$$ −3.12316 −0.901579
$$13$$ 0 0
$$14$$ 0.0729902 0.0195074
$$15$$ − 1.60020i − 0.413169i
$$16$$ 3.71276 0.928189
$$17$$ 5.06430 1.22827 0.614136 0.789200i $$-0.289505\pi$$
0.614136 + 0.789200i $$0.289505\pi$$
$$18$$ − 0.0965246i − 0.0227511i
$$19$$ − 2.26795i − 0.520303i −0.965568 0.260152i $$-0.916227\pi$$
0.965568 0.260152i $$-0.0837725\pi$$
$$20$$ 1.95174i 0.436422i
$$21$$ 0.531659i 0.116018i
$$22$$ 1.18012 0.251602
$$23$$ 2.83918 0.592010 0.296005 0.955186i $$-0.404346\pi$$
0.296005 + 0.955186i $$0.404346\pi$$
$$24$$ − 1.38920i − 0.283570i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 5.50367 1.05918
$$28$$ − 0.648458i − 0.122547i
$$29$$ −2.90348 −0.539162 −0.269581 0.962978i $$-0.586885\pi$$
−0.269581 + 0.962978i $$0.586885\pi$$
$$30$$ 0.351542 0.0641825
$$31$$ − 5.46410i − 0.981382i −0.871334 0.490691i $$-0.836744\pi$$
0.871334 0.490691i $$-0.163256\pi$$
$$32$$ 2.55193i 0.451122i
$$33$$ 8.59596i 1.49636i
$$34$$ 1.11256i 0.190802i
$$35$$ 0.332247 0.0561599
$$36$$ −0.857542 −0.142924
$$37$$ 5.97201i 0.981793i 0.871218 + 0.490896i $$0.163331\pi$$
−0.871218 + 0.490896i $$0.836669\pi$$
$$38$$ 0.498239 0.0808250
$$39$$ 0 0
$$40$$ −0.868145 −0.137266
$$41$$ − 3.73205i − 0.582848i −0.956594 0.291424i $$-0.905871\pi$$
0.956594 0.291424i $$-0.0941291\pi$$
$$42$$ −0.116799 −0.0180224
$$43$$ 5.06430 0.772298 0.386149 0.922436i $$-0.373805\pi$$
0.386149 + 0.922436i $$0.373805\pi$$
$$44$$ − 10.4844i − 1.58058i
$$45$$ − 0.439374i − 0.0654980i
$$46$$ 0.623730i 0.0919640i
$$47$$ 8.34285i 1.21693i 0.793581 + 0.608465i $$0.208214\pi$$
−0.793581 + 0.608465i $$0.791786\pi$$
$$48$$ −5.94114 −0.857529
$$49$$ 6.88961 0.984230
$$50$$ − 0.219687i − 0.0310684i
$$51$$ −8.10387 −1.13477
$$52$$ 0 0
$$53$$ −1.56063 −0.214369 −0.107184 0.994239i $$-0.534183\pi$$
−0.107184 + 0.994239i $$0.534183\pi$$
$$54$$ 1.20908i 0.164536i
$$55$$ 5.37182 0.724336
$$56$$ 0.288438 0.0385442
$$57$$ 3.62916i 0.480694i
$$58$$ − 0.637855i − 0.0837545i
$$59$$ 2.70732i 0.352463i 0.984349 + 0.176232i $$0.0563908\pi$$
−0.984349 + 0.176232i $$0.943609\pi$$
$$60$$ − 3.12316i − 0.403199i
$$61$$ 14.1039 1.80582 0.902908 0.429835i $$-0.141428\pi$$
0.902908 + 0.429835i $$0.141428\pi$$
$$62$$ 1.20039 0.152450
$$63$$ 0.145980i 0.0183918i
$$64$$ 6.86488 0.858111
$$65$$ 0 0
$$66$$ −1.88842 −0.232448
$$67$$ − 10.3322i − 1.26228i −0.775667 0.631142i $$-0.782586\pi$$
0.775667 0.631142i $$-0.217414\pi$$
$$68$$ 9.88418 1.19863
$$69$$ −4.54324 −0.546942
$$70$$ 0.0729902i 0.00872400i
$$71$$ − 12.7973i − 1.51876i −0.650645 0.759382i $$-0.725502\pi$$
0.650645 0.759382i $$-0.274498\pi$$
$$72$$ − 0.381440i − 0.0449531i
$$73$$ − 9.68922i − 1.13404i −0.823705 0.567019i $$-0.808097\pi$$
0.823705 0.567019i $$-0.191903\pi$$
$$74$$ −1.31197 −0.152514
$$75$$ 1.60020 0.184775
$$76$$ − 4.42644i − 0.507748i
$$77$$ −1.78477 −0.203393
$$78$$ 0 0
$$79$$ 4.51851 0.508372 0.254186 0.967155i $$-0.418192\pi$$
0.254186 + 0.967155i $$0.418192\pi$$
$$80$$ 3.71276i 0.415099i
$$81$$ −7.48883 −0.832092
$$82$$ 0.819883 0.0905409
$$83$$ 4.26371i 0.468003i 0.972236 + 0.234001i $$0.0751821\pi$$
−0.972236 + 0.234001i $$0.924818\pi$$
$$84$$ 1.03766i 0.113218i
$$85$$ 5.06430i 0.549300i
$$86$$ 1.11256i 0.119970i
$$87$$ 4.64613 0.498117
$$88$$ 4.66351 0.497132
$$89$$ 3.22584i 0.341938i 0.985276 + 0.170969i $$0.0546898\pi$$
−0.985276 + 0.170969i $$0.945310\pi$$
$$90$$ 0.0965246 0.0101746
$$91$$ 0 0
$$92$$ 5.54133 0.577724
$$93$$ 8.74363i 0.906672i
$$94$$ −1.83281 −0.189040
$$95$$ 2.26795 0.232687
$$96$$ − 4.08359i − 0.416780i
$$97$$ 2.50791i 0.254640i 0.991862 + 0.127320i $$0.0406375\pi$$
−0.991862 + 0.127320i $$0.959363\pi$$
$$98$$ 1.51356i 0.152892i
$$99$$ 2.36023i 0.237213i
$$100$$ −1.95174 −0.195174
$$101$$ −12.4467 −1.23849 −0.619247 0.785196i $$-0.712562\pi$$
−0.619247 + 0.785196i $$0.712562\pi$$
$$102$$ − 1.78031i − 0.176277i
$$103$$ 15.0247 1.48043 0.740215 0.672370i $$-0.234724\pi$$
0.740215 + 0.672370i $$0.234724\pi$$
$$104$$ 0 0
$$105$$ −0.531659 −0.0518846
$$106$$ − 0.342849i − 0.0333004i
$$107$$ −13.0643 −1.26297 −0.631487 0.775387i $$-0.717555\pi$$
−0.631487 + 0.775387i $$0.717555\pi$$
$$108$$ 10.7417 1.03362
$$109$$ 11.2325i 1.07587i 0.842985 + 0.537937i $$0.180796\pi$$
−0.842985 + 0.537937i $$0.819204\pi$$
$$110$$ 1.18012i 0.112520i
$$111$$ − 9.55639i − 0.907052i
$$112$$ − 1.23355i − 0.116560i
$$113$$ −18.3438 −1.72564 −0.862821 0.505509i $$-0.831305\pi$$
−0.862821 + 0.505509i $$0.831305\pi$$
$$114$$ −0.797279 −0.0746721
$$115$$ 2.83918i 0.264755i
$$116$$ −5.66682 −0.526151
$$117$$ 0 0
$$118$$ −0.594763 −0.0547524
$$119$$ − 1.68260i − 0.154243i
$$120$$ 1.38920 0.126816
$$121$$ −17.8564 −1.62331
$$122$$ 3.09843i 0.280519i
$$123$$ 5.97201i 0.538478i
$$124$$ − 10.6645i − 0.957700i
$$125$$ − 1.00000i − 0.0894427i
$$126$$ −0.0320700 −0.00285702
$$127$$ −3.23996 −0.287500 −0.143750 0.989614i $$-0.545916\pi$$
−0.143750 + 0.989614i $$0.545916\pi$$
$$128$$ 6.61199i 0.584423i
$$129$$ −8.10387 −0.713506
$$130$$ 0 0
$$131$$ 0.175664 0.0153478 0.00767390 0.999971i $$-0.497557\pi$$
0.00767390 + 0.999971i $$0.497557\pi$$
$$132$$ 16.7771i 1.46025i
$$133$$ −0.753518 −0.0653383
$$134$$ 2.26986 0.196086
$$135$$ 5.50367i 0.473681i
$$136$$ 4.39654i 0.377001i
$$137$$ − 17.9829i − 1.53638i −0.640221 0.768190i $$-0.721157\pi$$
0.640221 0.768190i $$-0.278843\pi$$
$$138$$ − 0.998090i − 0.0849631i
$$139$$ 11.9861 1.01665 0.508325 0.861165i $$-0.330265\pi$$
0.508325 + 0.861165i $$0.330265\pi$$
$$140$$ 0.648458 0.0548047
$$141$$ − 13.3502i − 1.12429i
$$142$$ 2.81140 0.235928
$$143$$ 0 0
$$144$$ −1.63129 −0.135941
$$145$$ − 2.90348i − 0.241121i
$$146$$ 2.12859 0.176164
$$147$$ −11.0247 −0.909304
$$148$$ 11.6558i 0.958101i
$$149$$ 3.41041i 0.279391i 0.990194 + 0.139696i $$0.0446124\pi$$
−0.990194 + 0.139696i $$0.955388\pi$$
$$150$$ 0.351542i 0.0287033i
$$151$$ 7.96141i 0.647890i 0.946076 + 0.323945i $$0.105009\pi$$
−0.946076 + 0.323945i $$0.894991\pi$$
$$152$$ 1.96891 0.159700
$$153$$ −2.22512 −0.179890
$$154$$ − 0.392090i − 0.0315955i
$$155$$ 5.46410 0.438887
$$156$$ 0 0
$$157$$ −16.4329 −1.31148 −0.655742 0.754985i $$-0.727644\pi$$
−0.655742 + 0.754985i $$0.727644\pi$$
$$158$$ 0.992658i 0.0789716i
$$159$$ 2.49731 0.198049
$$160$$ −2.55193 −0.201748
$$161$$ − 0.943307i − 0.0743430i
$$162$$ − 1.64520i − 0.129259i
$$163$$ 17.8072i 1.39477i 0.716697 + 0.697384i $$0.245653\pi$$
−0.716697 + 0.697384i $$0.754347\pi$$
$$164$$ − 7.28398i − 0.568784i
$$165$$ −8.59596 −0.669194
$$166$$ −0.936681 −0.0727006
$$167$$ − 6.29366i − 0.487018i −0.969899 0.243509i $$-0.921702\pi$$
0.969899 0.243509i $$-0.0782985\pi$$
$$168$$ −0.461557 −0.0356099
$$169$$ 0 0
$$170$$ −1.11256 −0.0853294
$$171$$ 0.996477i 0.0762025i
$$172$$ 9.88418 0.753662
$$173$$ −15.9751 −1.21457 −0.607283 0.794486i $$-0.707740\pi$$
−0.607283 + 0.794486i $$0.707740\pi$$
$$174$$ 1.02069i 0.0773786i
$$175$$ 0.332247i 0.0251155i
$$176$$ − 19.9442i − 1.50335i
$$177$$ − 4.33225i − 0.325632i
$$178$$ −0.708674 −0.0531173
$$179$$ −23.6174 −1.76525 −0.882625 0.470079i $$-0.844226\pi$$
−0.882625 + 0.470079i $$0.844226\pi$$
$$180$$ − 0.857542i − 0.0639174i
$$181$$ 2.62590 0.195182 0.0975909 0.995227i $$-0.468886\pi$$
0.0975909 + 0.995227i $$0.468886\pi$$
$$182$$ 0 0
$$183$$ −22.5689 −1.66834
$$184$$ 2.46482i 0.181709i
$$185$$ −5.97201 −0.439071
$$186$$ −1.92086 −0.140844
$$187$$ − 27.2045i − 1.98939i
$$188$$ 16.2831i 1.18756i
$$189$$ − 1.82858i − 0.133009i
$$190$$ 0.498239i 0.0361460i
$$191$$ −2.01582 −0.145860 −0.0729298 0.997337i $$-0.523235\pi$$
−0.0729298 + 0.997337i $$0.523235\pi$$
$$192$$ −10.9852 −0.792786
$$193$$ 22.8211i 1.64270i 0.570427 + 0.821348i $$0.306778\pi$$
−0.570427 + 0.821348i $$0.693222\pi$$
$$194$$ −0.550955 −0.0395563
$$195$$ 0 0
$$196$$ 13.4467 0.960480
$$197$$ 0.643026i 0.0458137i 0.999738 + 0.0229068i $$0.00729211\pi$$
−0.999738 + 0.0229068i $$0.992708\pi$$
$$198$$ −0.518513 −0.0368491
$$199$$ −3.06684 −0.217403 −0.108701 0.994074i $$-0.534669\pi$$
−0.108701 + 0.994074i $$0.534669\pi$$
$$200$$ − 0.868145i − 0.0613871i
$$201$$ 16.5336i 1.16619i
$$202$$ − 2.73438i − 0.192390i
$$203$$ 0.964670i 0.0677065i
$$204$$ −15.8166 −1.10739
$$205$$ 3.73205 0.260658
$$206$$ 3.30074i 0.229973i
$$207$$ −1.24746 −0.0867045
$$208$$ 0 0
$$209$$ −12.1830 −0.842716
$$210$$ − 0.116799i − 0.00805987i
$$211$$ −8.20039 −0.564538 −0.282269 0.959335i $$-0.591087\pi$$
−0.282269 + 0.959335i $$0.591087\pi$$
$$212$$ −3.04593 −0.209196
$$213$$ 20.4782i 1.40314i
$$214$$ − 2.87005i − 0.196193i
$$215$$ 5.06430i 0.345382i
$$216$$ 4.77798i 0.325101i
$$217$$ −1.81543 −0.123239
$$218$$ −2.46762 −0.167129
$$219$$ 15.5046i 1.04771i
$$220$$ 10.4844 0.706856
$$221$$ 0 0
$$222$$ 2.09941 0.140903
$$223$$ − 10.2442i − 0.686002i −0.939335 0.343001i $$-0.888557\pi$$
0.939335 0.343001i $$-0.111443\pi$$
$$224$$ 0.847871 0.0566508
$$225$$ 0.439374 0.0292916
$$226$$ − 4.02990i − 0.268065i
$$227$$ − 7.04381i − 0.467514i −0.972295 0.233757i $$-0.924898\pi$$
0.972295 0.233757i $$-0.0751020\pi$$
$$228$$ 7.08317i 0.469095i
$$229$$ 1.32899i 0.0878219i 0.999035 + 0.0439109i $$0.0139818\pi$$
−0.999035 + 0.0439109i $$0.986018\pi$$
$$230$$ −0.623730 −0.0411275
$$231$$ 2.85598 0.187909
$$232$$ − 2.52064i − 0.165488i
$$233$$ 1.24746 0.0817238 0.0408619 0.999165i $$-0.486990\pi$$
0.0408619 + 0.999165i $$0.486990\pi$$
$$234$$ 0 0
$$235$$ −8.34285 −0.544227
$$236$$ 5.28398i 0.343958i
$$237$$ −7.23050 −0.469672
$$238$$ 0.369644 0.0239605
$$239$$ 9.94207i 0.643099i 0.946893 + 0.321549i $$0.104204\pi$$
−0.946893 + 0.321549i $$0.895796\pi$$
$$240$$ − 5.94114i − 0.383499i
$$241$$ − 22.5869i − 1.45495i −0.686134 0.727475i $$-0.740694\pi$$
0.686134 0.727475i $$-0.259306\pi$$
$$242$$ − 3.92282i − 0.252168i
$$243$$ −4.52742 −0.290434
$$244$$ 27.5270 1.76224
$$245$$ 6.88961i 0.440161i
$$246$$ −1.31197 −0.0836483
$$247$$ 0 0
$$248$$ 4.74363 0.301221
$$249$$ − 6.82277i − 0.432376i
$$250$$ 0.219687 0.0138942
$$251$$ 6.76836 0.427215 0.213608 0.976920i $$-0.431479\pi$$
0.213608 + 0.976920i $$0.431479\pi$$
$$252$$ 0.284915i 0.0179480i
$$253$$ − 15.2515i − 0.958856i
$$254$$ − 0.711777i − 0.0446609i
$$255$$ − 8.10387i − 0.507484i
$$256$$ 12.2772 0.767325
$$257$$ 10.2538 0.639616 0.319808 0.947482i $$-0.396382\pi$$
0.319808 + 0.947482i $$0.396382\pi$$
$$258$$ − 1.78031i − 0.110837i
$$259$$ 1.98418 0.123291
$$260$$ 0 0
$$261$$ 1.27571 0.0789645
$$262$$ 0.0385910i 0.00238416i
$$263$$ 18.6570 1.15044 0.575220 0.817999i $$-0.304917\pi$$
0.575220 + 0.817999i $$0.304917\pi$$
$$264$$ −7.46254 −0.459287
$$265$$ − 1.56063i − 0.0958685i
$$266$$ − 0.165538i − 0.0101498i
$$267$$ − 5.16197i − 0.315907i
$$268$$ − 20.1658i − 1.23182i
$$269$$ 17.9579 1.09491 0.547456 0.836835i $$-0.315596\pi$$
0.547456 + 0.836835i $$0.315596\pi$$
$$270$$ −1.20908 −0.0735825
$$271$$ 30.8977i 1.87690i 0.345415 + 0.938450i $$0.387738\pi$$
−0.345415 + 0.938450i $$0.612262\pi$$
$$272$$ 18.8025 1.14007
$$273$$ 0 0
$$274$$ 3.95060 0.238665
$$275$$ 5.37182i 0.323933i
$$276$$ −8.86721 −0.533744
$$277$$ −26.5045 −1.59250 −0.796250 0.604967i $$-0.793186\pi$$
−0.796250 + 0.604967i $$0.793186\pi$$
$$278$$ 2.63320i 0.157929i
$$279$$ 2.40078i 0.143731i
$$280$$ 0.288438i 0.0172375i
$$281$$ 4.97766i 0.296942i 0.988917 + 0.148471i $$0.0474352\pi$$
−0.988917 + 0.148471i $$0.952565\pi$$
$$282$$ 2.93286 0.174649
$$283$$ 12.5863 0.748180 0.374090 0.927392i $$-0.377955\pi$$
0.374090 + 0.927392i $$0.377955\pi$$
$$284$$ − 24.9770i − 1.48211i
$$285$$ −3.62916 −0.214973
$$286$$ 0 0
$$287$$ −1.23996 −0.0731926
$$288$$ − 1.12125i − 0.0660704i
$$289$$ 8.64711 0.508653
$$290$$ 0.637855 0.0374562
$$291$$ − 4.01315i − 0.235255i
$$292$$ − 18.9108i − 1.10667i
$$293$$ 16.9176i 0.988337i 0.869366 + 0.494168i $$0.164527\pi$$
−0.869366 + 0.494168i $$0.835473\pi$$
$$294$$ − 2.42199i − 0.141253i
$$295$$ −2.70732 −0.157626
$$296$$ −5.18457 −0.301347
$$297$$ − 29.5647i − 1.71552i
$$298$$ −0.749222 −0.0434012
$$299$$ 0 0
$$300$$ 3.12316 0.180316
$$301$$ − 1.68260i − 0.0969832i
$$302$$ −1.74902 −0.100645
$$303$$ 19.9172 1.14421
$$304$$ − 8.42034i − 0.482940i
$$305$$ 14.1039i 0.807585i
$$306$$ − 0.488829i − 0.0279445i
$$307$$ − 4.30426i − 0.245657i −0.992428 0.122828i $$-0.960803\pi$$
0.992428 0.122828i $$-0.0391965\pi$$
$$308$$ −3.48340 −0.198485
$$309$$ −24.0425 −1.36773
$$310$$ 1.20039i 0.0681776i
$$311$$ 2.22512 0.126175 0.0630875 0.998008i $$-0.479905\pi$$
0.0630875 + 0.998008i $$0.479905\pi$$
$$312$$ 0 0
$$313$$ 7.20887 0.407469 0.203735 0.979026i $$-0.434692\pi$$
0.203735 + 0.979026i $$0.434692\pi$$
$$314$$ − 3.61008i − 0.203729i
$$315$$ −0.145980 −0.00822506
$$316$$ 8.81895 0.496105
$$317$$ 0.321644i 0.0180653i 0.999959 + 0.00903266i $$0.00287522\pi$$
−0.999959 + 0.00903266i $$0.997125\pi$$
$$318$$ 0.548626i 0.0307654i
$$319$$ 15.5969i 0.873261i
$$320$$ 6.86488i 0.383759i
$$321$$ 20.9054 1.16683
$$322$$ 0.207232 0.0115486
$$323$$ − 11.4856i − 0.639074i
$$324$$ −14.6162 −0.812013
$$325$$ 0 0
$$326$$ −3.91201 −0.216666
$$327$$ − 17.9741i − 0.993972i
$$328$$ 3.23996 0.178897
$$329$$ 2.77188 0.152819
$$330$$ − 1.88842i − 0.103954i
$$331$$ − 16.6320i − 0.914178i −0.889421 0.457089i $$-0.848892\pi$$
0.889421 0.457089i $$-0.151108\pi$$
$$332$$ 8.32164i 0.456710i
$$333$$ − 2.62395i − 0.143791i
$$334$$ 1.38263 0.0756543
$$335$$ 10.3322 0.564511
$$336$$ 1.97392i 0.107686i
$$337$$ −24.2186 −1.31927 −0.659636 0.751586i $$-0.729289\pi$$
−0.659636 + 0.751586i $$0.729289\pi$$
$$338$$ 0 0
$$339$$ 29.3537 1.59427
$$340$$ 9.88418i 0.536045i
$$341$$ −29.3521 −1.58951
$$342$$ −0.218913 −0.0118375
$$343$$ − 4.61478i − 0.249174i
$$344$$ 4.39654i 0.237046i
$$345$$ − 4.54324i − 0.244600i
$$346$$ − 3.50952i − 0.188673i
$$347$$ −6.27360 −0.336784 −0.168392 0.985720i $$-0.553857\pi$$
−0.168392 + 0.985720i $$0.553857\pi$$
$$348$$ 9.06802 0.486097
$$349$$ 7.06994i 0.378445i 0.981934 + 0.189223i $$0.0605968\pi$$
−0.981934 + 0.189223i $$0.939403\pi$$
$$350$$ −0.0729902 −0.00390149
$$351$$ 0 0
$$352$$ 13.7085 0.730666
$$353$$ 21.7898i 1.15976i 0.814704 + 0.579878i $$0.196900\pi$$
−0.814704 + 0.579878i $$0.803100\pi$$
$$354$$ 0.951738 0.0505843
$$355$$ 12.7973 0.679212
$$356$$ 6.29598i 0.333687i
$$357$$ 2.69248i 0.142501i
$$358$$ − 5.18844i − 0.274217i
$$359$$ 23.9737i 1.26528i 0.774444 + 0.632642i $$0.218029\pi$$
−0.774444 + 0.632642i $$0.781971\pi$$
$$360$$ 0.381440 0.0201037
$$361$$ 13.8564 0.729285
$$362$$ 0.576876i 0.0303199i
$$363$$ 28.5737 1.49973
$$364$$ 0 0
$$365$$ 9.68922 0.507157
$$366$$ − 4.95810i − 0.259164i
$$367$$ 6.39133 0.333625 0.166812 0.985989i $$-0.446653\pi$$
0.166812 + 0.985989i $$0.446653\pi$$
$$368$$ 10.5412 0.549497
$$369$$ 1.63977i 0.0853628i
$$370$$ − 1.31197i − 0.0682062i
$$371$$ 0.518513i 0.0269198i
$$372$$ 17.0653i 0.884793i
$$373$$ −20.0801 −1.03971 −0.519855 0.854255i $$-0.674014\pi$$
−0.519855 + 0.854255i $$0.674014\pi$$
$$374$$ 5.97647 0.309036
$$375$$ 1.60020i 0.0826338i
$$376$$ −7.24280 −0.373519
$$377$$ 0 0
$$378$$ 0.401714 0.0206619
$$379$$ 5.46182i 0.280555i 0.990112 + 0.140277i $$0.0447994\pi$$
−0.990112 + 0.140277i $$0.955201\pi$$
$$380$$ 4.42644 0.227072
$$381$$ 5.18457 0.265614
$$382$$ − 0.442849i − 0.0226581i
$$383$$ 5.66775i 0.289609i 0.989460 + 0.144804i $$0.0462553\pi$$
−0.989460 + 0.144804i $$0.953745\pi$$
$$384$$ − 10.5805i − 0.539933i
$$385$$ − 1.78477i − 0.0909602i
$$386$$ −5.01349 −0.255180
$$387$$ −2.22512 −0.113109
$$388$$ 4.89478i 0.248495i
$$389$$ −10.6174 −0.538325 −0.269162 0.963095i $$-0.586747\pi$$
−0.269162 + 0.963095i $$0.586747\pi$$
$$390$$ 0 0
$$391$$ 14.3784 0.727149
$$392$$ 5.98118i 0.302095i
$$393$$ −0.281096 −0.0141794
$$394$$ −0.141264 −0.00711679
$$395$$ 4.51851i 0.227351i
$$396$$ 4.60656i 0.231488i
$$397$$ 28.0338i 1.40697i 0.710708 + 0.703487i $$0.248375\pi$$
−0.710708 + 0.703487i $$0.751625\pi$$
$$398$$ − 0.673745i − 0.0337718i
$$399$$ 1.20578 0.0603643
$$400$$ −3.71276 −0.185638
$$401$$ − 22.5143i − 1.12431i −0.827032 0.562155i $$-0.809973\pi$$
0.827032 0.562155i $$-0.190027\pi$$
$$402$$ −3.63222 −0.181159
$$403$$ 0 0
$$404$$ −24.2927 −1.20861
$$405$$ − 7.48883i − 0.372123i
$$406$$ −0.211925 −0.0105177
$$407$$ 32.0805 1.59017
$$408$$ − 7.03533i − 0.348301i
$$409$$ − 4.28772i − 0.212014i −0.994365 0.106007i $$-0.966193\pi$$
0.994365 0.106007i $$-0.0338066\pi$$
$$410$$ 0.819883i 0.0404911i
$$411$$ 28.7761i 1.41942i
$$412$$ 29.3243 1.44471
$$413$$ 0.899499 0.0442614
$$414$$ − 0.274051i − 0.0134689i
$$415$$ −4.26371 −0.209297
$$416$$ 0 0
$$417$$ −19.1802 −0.939257
$$418$$ − 2.67645i − 0.130909i
$$419$$ 17.7116 0.865266 0.432633 0.901570i $$-0.357585\pi$$
0.432633 + 0.901570i $$0.357585\pi$$
$$420$$ −1.03766 −0.0506326
$$421$$ − 12.8787i − 0.627672i −0.949477 0.313836i $$-0.898386\pi$$
0.949477 0.313836i $$-0.101614\pi$$
$$422$$ − 1.80152i − 0.0876965i
$$423$$ − 3.66563i − 0.178229i
$$424$$ − 1.35485i − 0.0657973i
$$425$$ −5.06430 −0.245655
$$426$$ −4.49880 −0.217967
$$427$$ − 4.68596i − 0.226770i
$$428$$ −25.4981 −1.23250
$$429$$ 0 0
$$430$$ −1.11256 −0.0536524
$$431$$ − 9.49845i − 0.457524i −0.973482 0.228762i $$-0.926532\pi$$
0.973482 0.228762i $$-0.0734678\pi$$
$$432$$ 20.4338 0.983121
$$433$$ 1.39628 0.0671010 0.0335505 0.999437i $$-0.489319\pi$$
0.0335505 + 0.999437i $$0.489319\pi$$
$$434$$ − 0.398826i − 0.0191443i
$$435$$ 4.64613i 0.222765i
$$436$$ 21.9228i 1.04991i
$$437$$ − 6.43911i − 0.308024i
$$438$$ −3.40617 −0.162753
$$439$$ −4.16180 −0.198632 −0.0993159 0.995056i $$-0.531665\pi$$
−0.0993159 + 0.995056i $$0.531665\pi$$
$$440$$ 4.66351i 0.222324i
$$441$$ −3.02711 −0.144148
$$442$$ 0 0
$$443$$ 9.54563 0.453526 0.226763 0.973950i $$-0.427186\pi$$
0.226763 + 0.973950i $$0.427186\pi$$
$$444$$ − 18.6516i − 0.885164i
$$445$$ −3.22584 −0.152919
$$446$$ 2.25052 0.106565
$$447$$ − 5.45732i − 0.258122i
$$448$$ − 2.28083i − 0.107759i
$$449$$ 21.7171i 1.02489i 0.858720 + 0.512446i $$0.171260\pi$$
−0.858720 + 0.512446i $$0.828740\pi$$
$$450$$ 0.0965246i 0.00455022i
$$451$$ −20.0479 −0.944018
$$452$$ −35.8023 −1.68400
$$453$$ − 12.7398i − 0.598569i
$$454$$ 1.54743 0.0726246
$$455$$ 0 0
$$456$$ −3.15064 −0.147542
$$457$$ − 4.72259i − 0.220914i −0.993881 0.110457i $$-0.964769\pi$$
0.993881 0.110457i $$-0.0352314\pi$$
$$458$$ −0.291961 −0.0136424
$$459$$ 27.8722 1.30096
$$460$$ 5.54133i 0.258366i
$$461$$ 1.78151i 0.0829730i 0.999139 + 0.0414865i $$0.0132094\pi$$
−0.999139 + 0.0414865i $$0.986791\pi$$
$$462$$ 0.627421i 0.0291902i
$$463$$ 6.80200i 0.316116i 0.987430 + 0.158058i $$0.0505232\pi$$
−0.987430 + 0.158058i $$0.949477\pi$$
$$464$$ −10.7799 −0.500444
$$465$$ −8.74363 −0.405476
$$466$$ 0.274051i 0.0126952i
$$467$$ −18.2374 −0.843927 −0.421963 0.906613i $$-0.638659\pi$$
−0.421963 + 0.906613i $$0.638659\pi$$
$$468$$ 0 0
$$469$$ −3.43285 −0.158514
$$470$$ − 1.83281i − 0.0845414i
$$471$$ 26.2958 1.21165
$$472$$ −2.35035 −0.108184
$$473$$ − 27.2045i − 1.25086i
$$474$$ − 1.58845i − 0.0729598i
$$475$$ 2.26795i 0.104061i
$$476$$ − 3.28398i − 0.150521i
$$477$$ 0.685698 0.0313960
$$478$$ −2.18414 −0.0999003
$$479$$ − 35.1807i − 1.60745i −0.595002 0.803724i $$-0.702849\pi$$
0.595002 0.803724i $$-0.297151\pi$$
$$480$$ 4.08359 0.186390
$$481$$ 0 0
$$482$$ 4.96204 0.226015
$$483$$ 1.50948i 0.0686835i
$$484$$ −34.8510 −1.58414
$$485$$ −2.50791 −0.113878
$$486$$ − 0.994615i − 0.0451166i
$$487$$ 10.3040i 0.466919i 0.972367 + 0.233459i $$0.0750046\pi$$
−0.972367 + 0.233459i $$0.924995\pi$$
$$488$$ 12.2442i 0.554269i
$$489$$ − 28.4950i − 1.28859i
$$490$$ −1.51356 −0.0683756
$$491$$ −9.33198 −0.421147 −0.210573 0.977578i $$-0.567533\pi$$
−0.210573 + 0.977578i $$0.567533\pi$$
$$492$$ 11.6558i 0.525484i
$$493$$ −14.7041 −0.662238
$$494$$ 0 0
$$495$$ −2.36023 −0.106085
$$496$$ − 20.2869i − 0.910907i
$$497$$ −4.25187 −0.190722
$$498$$ 1.49887 0.0671661
$$499$$ − 23.9421i − 1.07179i −0.844283 0.535897i $$-0.819974\pi$$
0.844283 0.535897i $$-0.180026\pi$$
$$500$$ − 1.95174i − 0.0872844i
$$501$$ 10.0711i 0.449943i
$$502$$ 1.48692i 0.0663645i
$$503$$ 42.1443 1.87912 0.939560 0.342385i $$-0.111235\pi$$
0.939560 + 0.342385i $$0.111235\pi$$
$$504$$ −0.126732 −0.00564510
$$505$$ − 12.4467i − 0.553872i
$$506$$ 3.35056 0.148951
$$507$$ 0 0
$$508$$ −6.32355 −0.280562
$$509$$ 33.5602i 1.48753i 0.668441 + 0.743765i $$0.266962\pi$$
−0.668441 + 0.743765i $$0.733038\pi$$
$$510$$ 1.78031 0.0788336
$$511$$ −3.21921 −0.142409
$$512$$ 15.9211i 0.703621i
$$513$$ − 12.4820i − 0.551096i
$$514$$ 2.25263i 0.0993593i
$$515$$ 15.0247i 0.662069i
$$516$$ −15.8166 −0.696288
$$517$$ 44.8162 1.97102
$$518$$ 0.435898i 0.0191523i
$$519$$ 25.5633 1.12210
$$520$$ 0 0
$$521$$ 12.4649 0.546098 0.273049 0.962000i $$-0.411968\pi$$
0.273049 + 0.962000i $$0.411968\pi$$
$$522$$ 0.280257i 0.0122665i
$$523$$ −5.65956 −0.247475 −0.123738 0.992315i $$-0.539488\pi$$
−0.123738 + 0.992315i $$0.539488\pi$$
$$524$$ 0.342849 0.0149774
$$525$$ − 0.531659i − 0.0232035i
$$526$$ 4.09870i 0.178712i
$$527$$ − 27.6718i − 1.20540i
$$528$$ 31.9147i 1.38891i
$$529$$ −14.9391 −0.649525
$$530$$ 0.342849 0.0148924
$$531$$ − 1.18953i − 0.0516211i
$$532$$ −1.47067 −0.0637616
$$533$$ 0 0
$$534$$ 1.13402 0.0490737
$$535$$ − 13.0643i − 0.564819i
$$536$$ 8.96989 0.387440
$$537$$ 37.7925 1.63087
$$538$$ 3.94511i 0.170086i
$$539$$ − 37.0097i − 1.59412i
$$540$$ 10.7417i 0.462250i
$$541$$ − 15.4750i − 0.665321i −0.943047 0.332660i $$-0.892054\pi$$
0.943047 0.332660i $$-0.107946\pi$$
$$542$$ −6.78781 −0.291562
$$543$$ −4.20196 −0.180323
$$544$$ 12.9237i 0.554101i
$$545$$ −11.2325 −0.481146
$$546$$ 0 0
$$547$$ 25.1765 1.07647 0.538234 0.842795i $$-0.319092\pi$$
0.538234 + 0.842795i $$0.319092\pi$$
$$548$$ − 35.0979i − 1.49931i
$$549$$ −6.19687 −0.264476
$$550$$ −1.18012 −0.0503204
$$551$$ 6.58493i 0.280528i
$$552$$ − 3.94419i − 0.167876i
$$553$$ − 1.50126i − 0.0638401i
$$554$$ − 5.82269i − 0.247382i
$$555$$ 9.55639 0.405646
$$556$$ 23.3938 0.992118
$$557$$ 42.3489i 1.79438i 0.441645 + 0.897190i $$0.354395\pi$$
−0.441645 + 0.897190i $$0.645605\pi$$
$$558$$ −0.527420 −0.0223275
$$559$$ 0 0
$$560$$ 1.23355 0.0521270
$$561$$ 43.5325i 1.83794i
$$562$$ −1.09353 −0.0461276
$$563$$ −23.7905 −1.00265 −0.501326 0.865259i $$-0.667154\pi$$
−0.501326 + 0.865259i $$0.667154\pi$$
$$564$$ − 26.0561i − 1.09716i
$$565$$ − 18.3438i − 0.771731i
$$566$$ 2.76505i 0.116224i
$$567$$ 2.48814i 0.104492i
$$568$$ 11.1099 0.466162
$$569$$ 26.7421 1.12109 0.560543 0.828125i $$-0.310593\pi$$
0.560543 + 0.828125i $$0.310593\pi$$
$$570$$ − 0.797279i − 0.0333944i
$$571$$ 16.7159 0.699539 0.349769 0.936836i $$-0.386260\pi$$
0.349769 + 0.936836i $$0.386260\pi$$
$$572$$ 0 0
$$573$$ 3.22571 0.134756
$$574$$ − 0.272403i − 0.0113699i
$$575$$ −2.83918 −0.118402
$$576$$ −3.01625 −0.125677
$$577$$ 20.6768i 0.860786i 0.902642 + 0.430393i $$0.141625\pi$$
−0.902642 + 0.430393i $$0.858375\pi$$
$$578$$ 1.89966i 0.0790153i
$$579$$ − 36.5182i − 1.51764i
$$580$$ − 5.66682i − 0.235302i
$$581$$ 1.41660 0.0587706
$$582$$ 0.881636 0.0365450
$$583$$ 8.38340i 0.347205i
$$584$$ 8.41165 0.348076
$$585$$ 0 0
$$586$$ −3.71657 −0.153530
$$587$$ − 20.7972i − 0.858391i −0.903212 0.429196i $$-0.858797\pi$$
0.903212 0.429196i $$-0.141203\pi$$
$$588$$ −21.5174 −0.887362
$$589$$ −12.3923 −0.510616
$$590$$ − 0.594763i − 0.0244860i
$$591$$ − 1.02897i − 0.0423260i
$$592$$ 22.1726i 0.911289i
$$593$$ − 21.8475i − 0.897169i −0.893740 0.448585i $$-0.851928\pi$$
0.893740 0.448585i $$-0.148072\pi$$
$$594$$ 6.49498 0.266492
$$595$$ 1.68260 0.0689797
$$596$$ 6.65622i 0.272649i
$$597$$ 4.90755 0.200853
$$598$$ 0 0
$$599$$ −3.58040 −0.146291 −0.0731456 0.997321i $$-0.523304\pi$$
−0.0731456 + 0.997321i $$0.523304\pi$$
$$600$$ 1.38920i 0.0567139i
$$601$$ 21.3486 0.870829 0.435414 0.900230i $$-0.356602\pi$$
0.435414 + 0.900230i $$0.356602\pi$$
$$602$$ 0.369644 0.0150656
$$603$$ 4.53972i 0.184872i
$$604$$ 15.5386i 0.632256i
$$605$$ − 17.8564i − 0.725966i
$$606$$ 4.37554i 0.177744i
$$607$$ −3.29976 −0.133933 −0.0669665 0.997755i $$-0.521332\pi$$
−0.0669665 + 0.997755i $$0.521332\pi$$
$$608$$ 5.78766 0.234720
$$609$$ − 1.54366i − 0.0625523i
$$610$$ −3.09843 −0.125452
$$611$$ 0 0
$$612$$ −4.34285 −0.175549
$$613$$ − 9.88635i − 0.399306i −0.979867 0.199653i $$-0.936019\pi$$
0.979867 0.199653i $$-0.0639815\pi$$
$$614$$ 0.945589 0.0381609
$$615$$ −5.97201 −0.240815
$$616$$ − 1.54944i − 0.0624286i
$$617$$ 45.7169i 1.84049i 0.391339 + 0.920246i $$0.372012\pi$$
−0.391339 + 0.920246i $$0.627988\pi$$
$$618$$ − 5.28182i − 0.212466i
$$619$$ 19.9143i 0.800425i 0.916422 + 0.400212i $$0.131064\pi$$
−0.916422 + 0.400212i $$0.868936\pi$$
$$620$$ 10.6645 0.428296
$$621$$ 15.6259 0.627046
$$622$$ 0.488829i 0.0196003i
$$623$$ 1.07177 0.0429397
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 1.58369i 0.0632971i
$$627$$ 19.4952 0.778563
$$628$$ −32.0726 −1.27984
$$629$$ 30.2440i 1.20591i
$$630$$ − 0.0320700i − 0.00127770i
$$631$$ 14.5958i 0.581050i 0.956867 + 0.290525i $$0.0938299\pi$$
−0.956867 + 0.290525i $$0.906170\pi$$
$$632$$ 3.92272i 0.156038i
$$633$$ 13.1222 0.521562
$$634$$ −0.0706609 −0.00280630
$$635$$ − 3.23996i − 0.128574i
$$636$$ 4.87409 0.193270
$$637$$ 0 0
$$638$$ −3.42644 −0.135654
$$639$$ 5.62281i 0.222435i
$$640$$ −6.61199 −0.261362
$$641$$ 14.1637 0.559431 0.279716 0.960083i $$-0.409760\pi$$
0.279716 + 0.960083i $$0.409760\pi$$
$$642$$ 4.59265i 0.181257i
$$643$$ − 16.7716i − 0.661408i −0.943735 0.330704i $$-0.892714\pi$$
0.943735 0.330704i $$-0.107286\pi$$
$$644$$ − 1.84109i − 0.0725490i
$$645$$ − 8.10387i − 0.319089i
$$646$$ 2.52323 0.0992751
$$647$$ 2.99168 0.117615 0.0588075 0.998269i $$-0.481270\pi$$
0.0588075 + 0.998269i $$0.481270\pi$$
$$648$$ − 6.50139i − 0.255399i
$$649$$ 14.5432 0.570872
$$650$$ 0 0
$$651$$ 2.90504 0.113858
$$652$$ 34.7550i 1.36111i
$$653$$ −11.6643 −0.456461 −0.228230 0.973607i $$-0.573294\pi$$
−0.228230 + 0.973607i $$0.573294\pi$$
$$654$$ 3.94868 0.154406
$$655$$ 0.175664i 0.00686374i
$$656$$ − 13.8562i − 0.540993i
$$657$$ 4.25719i 0.166089i
$$658$$ 0.608946i 0.0237392i
$$659$$ −1.81047 −0.0705260 −0.0352630 0.999378i $$-0.511227\pi$$
−0.0352630 + 0.999378i $$0.511227\pi$$
$$660$$ −16.7771 −0.653046
$$661$$ − 12.3406i − 0.479992i −0.970774 0.239996i $$-0.922854\pi$$
0.970774 0.239996i $$-0.0771462\pi$$
$$662$$ 3.65383 0.142010
$$663$$ 0 0
$$664$$ −3.70152 −0.143647
$$665$$ − 0.753518i − 0.0292202i
$$666$$ 0.576446 0.0223368
$$667$$ −8.24348 −0.319189
$$668$$ − 12.2836i − 0.475265i
$$669$$ 16.3927i 0.633779i
$$670$$ 2.26986i 0.0876923i
$$671$$ − 75.7634i − 2.92481i
$$672$$ −1.35676 −0.0523381
$$673$$ −9.26625 −0.357188 −0.178594 0.983923i $$-0.557155\pi$$
−0.178594 + 0.983923i $$0.557155\pi$$
$$674$$ − 5.32051i − 0.204938i
$$675$$ −5.50367 −0.211836
$$676$$ 0 0
$$677$$ 13.8984 0.534158 0.267079 0.963675i $$-0.413941\pi$$
0.267079 + 0.963675i $$0.413941\pi$$
$$678$$ 6.44863i 0.247658i
$$679$$ 0.833244 0.0319770
$$680$$ −4.39654 −0.168600
$$681$$ 11.2715i 0.431924i
$$682$$ − 6.44828i − 0.246917i
$$683$$ − 37.7512i − 1.44451i −0.691626 0.722255i $$-0.743105\pi$$
0.691626 0.722255i $$-0.256895\pi$$
$$684$$ 1.94486i 0.0743637i
$$685$$ 17.9829 0.687090
$$686$$ 1.01381 0.0387073
$$687$$ − 2.12664i − 0.0811363i
$$688$$ 18.8025 0.716838
$$689$$ 0 0
$$690$$ 0.998090 0.0379966
$$691$$ 1.65291i 0.0628797i 0.999506 + 0.0314399i $$0.0100093\pi$$
−0.999506 + 0.0314399i $$0.989991\pi$$
$$692$$ −31.1792 −1.18526
$$693$$ 0.784180 0.0297885
$$694$$ − 1.37823i − 0.0523168i
$$695$$ 11.9861i 0.454660i
$$696$$ 4.03351i 0.152890i
$$697$$ − 18.9002i − 0.715897i
$$698$$ −1.55317 −0.0587885
$$699$$ −1.99618 −0.0755025
$$700$$ 0.648458i 0.0245094i
$$701$$ −20.4819 −0.773590 −0.386795 0.922166i $$-0.626418\pi$$
−0.386795 + 0.922166i $$0.626418\pi$$
$$702$$ 0 0
$$703$$ 13.5442 0.510830
$$704$$ − 36.8769i − 1.38985i
$$705$$ 13.3502 0.502797
$$706$$ −4.78694 −0.180159
$$707$$ 4.13538i 0.155527i
$$708$$ − 8.45541i − 0.317774i
$$709$$ 21.9417i 0.824039i 0.911175 + 0.412020i $$0.135177\pi$$
−0.911175 + 0.412020i $$0.864823\pi$$
$$710$$ 2.81140i 0.105510i
$$711$$ −1.98532 −0.0744552
$$712$$ −2.80049 −0.104953
$$713$$ − 15.5136i − 0.580987i
$$714$$ −0.591503 −0.0221364
$$715$$ 0 0
$$716$$ −46.0950 −1.72265
$$717$$ − 15.9093i − 0.594142i
$$718$$ −5.26671 −0.196552
$$719$$ −38.8475 −1.44877 −0.724384 0.689397i $$-0.757876\pi$$
−0.724384 + 0.689397i $$0.757876\pi$$
$$720$$ − 1.63129i − 0.0607945i
$$721$$ − 4.99191i − 0.185909i
$$722$$ 3.04407i 0.113289i
$$723$$ 36.1434i 1.34419i
$$724$$ 5.12507 0.190472
$$725$$ 2.90348 0.107832
$$726$$ 6.27728i 0.232972i
$$727$$ 30.6598 1.13711 0.568555 0.822645i $$-0.307503\pi$$
0.568555 + 0.822645i $$0.307503\pi$$
$$728$$ 0 0
$$729$$ 29.7112 1.10042
$$730$$ 2.12859i 0.0787828i
$$731$$ 25.6471 0.948593
$$732$$ −44.0487 −1.62809
$$733$$ − 24.3858i − 0.900709i −0.892850 0.450355i $$-0.851298\pi$$
0.892850 0.450355i $$-0.148702\pi$$
$$734$$ 1.40409i 0.0518259i
$$735$$ − 11.0247i − 0.406653i
$$736$$ 7.24539i 0.267069i
$$737$$ −55.5029 −2.04448
$$738$$ −0.360235 −0.0132604
$$739$$ 38.2788i 1.40811i 0.710146 + 0.704054i $$0.248629\pi$$
−0.710146 + 0.704054i $$0.751371\pi$$
$$740$$ −11.6558 −0.428476
$$741$$ 0 0
$$742$$ −0.113910 −0.00418178
$$743$$ 40.0079i 1.46775i 0.679286 + 0.733874i $$0.262290\pi$$
−0.679286 + 0.733874i $$0.737710\pi$$
$$744$$ −7.59074 −0.278290
$$745$$ −3.41041 −0.124948
$$746$$ − 4.41134i − 0.161511i
$$747$$ − 1.87336i − 0.0685427i
$$748$$ − 53.0960i − 1.94138i
$$749$$ 4.34057i 0.158601i
$$750$$ −0.351542 −0.0128365
$$751$$ −25.6020 −0.934230 −0.467115 0.884197i $$-0.654707\pi$$
−0.467115 + 0.884197i $$0.654707\pi$$
$$752$$ 30.9750i 1.12954i
$$753$$ −10.8307 −0.394693
$$754$$ 0 0
$$755$$ −7.96141 −0.289745
$$756$$ − 3.56890i − 0.129800i
$$757$$ −1.84848 −0.0671840 −0.0335920 0.999436i $$-0.510695\pi$$
−0.0335920 + 0.999436i $$0.510695\pi$$
$$758$$ −1.19989 −0.0435820
$$759$$ 24.4055i 0.885862i
$$760$$ 1.96891i 0.0714198i
$$761$$ 26.2124i 0.950199i 0.879932 + 0.475099i $$0.157588\pi$$
−0.879932 + 0.475099i $$0.842412\pi$$
$$762$$ 1.13898i 0.0412610i
$$763$$ 3.73195 0.135106
$$764$$ −3.93435 −0.142340
$$765$$ − 2.22512i − 0.0804494i
$$766$$ −1.24513 −0.0449884
$$767$$ 0 0
$$768$$ −19.6459 −0.708911
$$769$$ 44.3495i 1.59928i 0.600478 + 0.799641i $$0.294977\pi$$
−0.600478 + 0.799641i $$0.705023\pi$$
$$770$$ 0.392090 0.0141299
$$771$$ −16.4081 −0.590924
$$772$$ 44.5408i 1.60306i
$$773$$ 23.2638i 0.836742i 0.908276 + 0.418371i $$0.137399\pi$$
−0.908276 + 0.418371i $$0.862601\pi$$
$$774$$ − 0.488829i − 0.0175706i
$$775$$ 5.46410i 0.196276i
$$776$$ −2.17723 −0.0781580
$$777$$ −3.17508 −0.113905
$$778$$ − 2.33251i − 0.0836245i
$$779$$ −8.46410 −0.303258
$$780$$ 0 0
$$781$$ −68.7449 −2.45989
$$782$$ 3.15875i 0.112957i
$$783$$ −15.9798 −0.571071
$$784$$ 25.5794 0.913552
$$785$$ − 16.4329i − 0.586514i
$$786$$ − 0.0617531i − 0.00220266i
$$787$$ − 47.9133i − 1.70793i −0.520334 0.853963i $$-0.674192\pi$$
0.520334 0.853963i $$-0.325808\pi$$
$$788$$ 1.25502i 0.0447081i
$$789$$ −29.8548 −1.06286
$$790$$ −0.992658 −0.0353172
$$791$$ 6.09467i 0.216702i
$$792$$ −2.04903 −0.0728090
$$793$$ 0 0
$$794$$ −6.15865 −0.218562
$$795$$ 2.49731i 0.0885704i
$$796$$ −5.98567 −0.212156
$$797$$ 20.6952 0.733060 0.366530 0.930406i $$-0.380546\pi$$
0.366530 + 0.930406i $$0.380546\pi$$
$$798$$ 0.264893i 0.00937712i
$$799$$ 42.2507i 1.49472i
$$800$$ − 2.55193i − 0.0902245i
$$801$$ − 1.41735i − 0.0500795i
$$802$$ 4.94609 0.174653
$$803$$ −52.0487 −1.83676
$$804$$ 32.2693i 1.13805i
$$805$$ 0.943307 0.0332472
$$806$$ 0 0
$$807$$ −28.7361 −1.01156
$$808$$ − 10.8056i − 0.380138i
$$809$$ 15.8915 0.558714 0.279357 0.960187i $$-0.409879\pi$$
0.279357 + 0.960187i $$0.409879\pi$$
$$810$$ 1.64520 0.0578063
$$811$$ 23.8796i 0.838525i 0.907865 + 0.419263i $$0.137711\pi$$
−0.907865 + 0.419263i $$0.862289\pi$$
$$812$$ 1.88278i 0.0660727i
$$813$$ − 49.4423i − 1.73402i
$$814$$ 7.04768i 0.247021i
$$815$$ −17.8072 −0.623759
$$816$$ −30.0877 −1.05328
$$817$$ − 11.4856i − 0.401829i
$$818$$ 0.941956 0.0329347
$$819$$ 0 0
$$820$$ 7.28398 0.254368
$$821$$ − 15.9097i − 0.555251i −0.960689 0.277626i $$-0.910453\pi$$
0.960689 0.277626i $$-0.0895475\pi$$
$$822$$ −6.32174 −0.220496
$$823$$ 14.8115 0.516295 0.258147 0.966106i $$-0.416888\pi$$
0.258147 + 0.966106i $$0.416888\pi$$
$$824$$ 13.0436i 0.454397i
$$825$$ − 8.59596i − 0.299273i
$$826$$ 0.197608i 0.00687566i
$$827$$ 33.9498i 1.18055i 0.807202 + 0.590275i $$0.200981\pi$$
−0.807202 + 0.590275i $$0.799019\pi$$
$$828$$ −2.43472 −0.0846122
$$829$$ −23.3146 −0.809749 −0.404875 0.914372i $$-0.632685\pi$$
−0.404875 + 0.914372i $$0.632685\pi$$
$$830$$ − 0.936681i − 0.0325127i
$$831$$ 42.4124 1.47127
$$832$$ 0 0
$$833$$ 34.8910 1.20890
$$834$$ − 4.21363i − 0.145906i
$$835$$ 6.29366 0.217801
$$836$$ −23.7780 −0.822380
$$837$$ − 30.0726i − 1.03946i
$$838$$ 3.89100i 0.134412i
$$839$$ − 14.7930i − 0.510710i −0.966847 0.255355i $$-0.917808\pi$$
0.966847 0.255355i $$-0.0821924\pi$$
$$840$$ − 0.461557i − 0.0159252i
$$841$$ −20.5698 −0.709305
$$842$$ 2.82929 0.0975038
$$843$$ − 7.96523i − 0.274337i
$$844$$ −16.0050 −0.550915
$$845$$ 0 0
$$846$$ 0.805291 0.0276865
$$847$$ 5.93273i 0.203851i
$$848$$ −5.79422 −0.198974
$$849$$ −20.1406 −0.691223
$$850$$ − 1.11256i − 0.0381605i
$$851$$ 16.9556i 0.581231i
$$852$$ 39.9681i 1.36929i
$$853$$ − 16.3452i − 0.559650i −0.960051 0.279825i $$-0.909724\pi$$
0.960051 0.279825i $$-0.0902765\pi$$
$$854$$ 1.02944 0.0352268
$$855$$ −0.996477 −0.0340788
$$856$$ − 11.3417i − 0.387651i
$$857$$ 34.1418 1.16626 0.583132 0.812378i $$-0.301827\pi$$
0.583132 + 0.812378i $$0.301827\pi$$
$$858$$ 0 0
$$859$$ −45.1996 −1.54219 −0.771096 0.636719i $$-0.780291\pi$$
−0.771096 + 0.636719i $$0.780291\pi$$
$$860$$ 9.88418i 0.337048i
$$861$$ 1.98418 0.0676207
$$862$$ 2.08669 0.0710728
$$863$$ − 4.75058i − 0.161712i −0.996726 0.0808559i $$-0.974235\pi$$
0.996726 0.0808559i $$-0.0257653\pi$$
$$864$$ 14.0450i 0.477821i
$$865$$ − 15.9751i − 0.543170i
$$866$$ 0.306745i 0.0104236i
$$867$$ −13.8371 −0.469931
$$868$$ −3.54324 −0.120265
$$869$$ − 24.2726i − 0.823392i
$$870$$ −1.02069 −0.0346048
$$871$$ 0 0
$$872$$ −9.75140 −0.330224
$$873$$ − 1.10191i − 0.0372940i
$$874$$ 1.41459 0.0478492
$$875$$ −0.332247 −0.0112320
$$876$$ 30.2610i 1.02242i
$$877$$ − 2.25506i − 0.0761481i −0.999275 0.0380741i $$-0.987878\pi$$
0.999275 0.0380741i $$-0.0121223\pi$$
$$878$$ − 0.914293i − 0.0308559i
$$879$$ − 27.0715i − 0.913098i
$$880$$ 19.9442 0.672320
$$881$$ 2.98304 0.100501 0.0502507 0.998737i $$-0.483998\pi$$
0.0502507 + 0.998737i $$0.483998\pi$$
$$882$$ − 0.665017i − 0.0223923i
$$883$$ −28.2874 −0.951947 −0.475973 0.879460i $$-0.657904\pi$$
−0.475973 + 0.879460i $$0.657904\pi$$
$$884$$ 0 0
$$885$$ 4.33225 0.145627
$$886$$ 2.09705i 0.0704517i
$$887$$ −27.9816 −0.939531 −0.469766 0.882791i $$-0.655662\pi$$
−0.469766 + 0.882791i $$0.655662\pi$$
$$888$$ 8.29633 0.278407
$$889$$ 1.07647i 0.0361035i
$$890$$ − 0.708674i − 0.0237548i
$$891$$ 40.2286i 1.34771i
$$892$$ − 19.9940i − 0.669448i
$$893$$ 18.9212 0.633172
$$894$$ 1.19890 0.0400973
$$895$$ − 23.6174i − 0.789443i
$$896$$ 2.19681 0.0733903
$$897$$ 0 0
$$898$$ −4.77095 −0.159209
$$899$$ 15.8649i 0.529124i
$$900$$ 0.857542 0.0285847
$$901$$ −7.90348 −0.263303
$$902$$ − 4.40426i − 0.146646i
$$903$$ 2.69248i 0.0896002i
$$904$$ − 15.9251i − 0.529661i
$$905$$ 2.62590i 0.0872879i
$$906$$ 2.79877 0.0929829
$$907$$ −16.5520 −0.549600 −0.274800 0.961501i $$-0.588612\pi$$
−0.274800 + 0.961501i $$0.588612\pi$$
$$908$$ − 13.7477i − 0.456232i
$$909$$ 5.46876 0.181387
$$910$$ 0 0
$$911$$ 7.04863 0.233532 0.116766 0.993159i $$-0.462747\pi$$
0.116766 + 0.993159i $$0.462747\pi$$
$$912$$ 13.4742i 0.446175i
$$913$$ 22.9039 0.758007
$$914$$ 1.03749 0.0343172
$$915$$ − 22.5689i − 0.746106i
$$916$$ 2.59383i 0.0857026i
$$917$$ − 0.0583636i − 0.00192734i
$$918$$ 6.12316i 0.202094i
$$919$$ 16.5438 0.545728 0.272864 0.962053i $$-0.412029\pi$$
0.272864 + 0.962053i $$0.412029\pi$$
$$920$$ −2.46482 −0.0812626
$$921$$ 6.88766i 0.226956i
$$922$$ −0.391374 −0.0128892
$$923$$ 0 0
$$924$$ 5.57412 0.183375
$$925$$ − 5.97201i − 0.196359i
$$926$$ −1.49431 −0.0491060
$$927$$ −6.60147 −0.216821
$$928$$ − 7.40948i − 0.243228i
$$929$$ − 33.8367i − 1.11015i −0.831801 0.555074i $$-0.812690\pi$$
0.831801 0.555074i $$-0.187310\pi$$
$$930$$ − 1.92086i − 0.0629875i
$$931$$ − 15.6253i − 0.512098i
$$932$$ 2.43472 0.0797517
$$933$$ −3.56063 −0.116570
$$934$$ − 4.00652i − 0.131097i
$$935$$ 27.2045 0.889681
$$936$$ 0 0
$$937$$ −30.4606 −0.995104 −0.497552 0.867434i $$-0.665768\pi$$
−0.497552 + 0.867434i $$0.665768\pi$$
$$938$$ − 0.754153i − 0.0246240i
$$939$$ −11.5356 −0.376450
$$940$$ −16.2831 −0.531095
$$941$$ 38.2101i 1.24561i 0.782375 + 0.622807i $$0.214008\pi$$
−0.782375 + 0.622807i $$0.785992\pi$$
$$942$$ 5.77684i 0.188220i
$$943$$ − 10.5960i − 0.345052i
$$944$$ 10.0516i 0.327153i
$$945$$ 1.82858 0.0594836
$$946$$ 5.97647 0.194312
$$947$$ 52.4482i 1.70434i 0.523266 + 0.852169i $$0.324713\pi$$
−0.523266 + 0.852169i $$0.675287\pi$$
$$948$$ −14.1120 −0.458338
$$949$$ 0 0
$$950$$ −0.498239 −0.0161650
$$951$$ − 0.514693i − 0.0166901i
$$952$$ 1.46074 0.0473427
$$953$$ −39.7500 −1.28763 −0.643814 0.765182i $$-0.722649\pi$$
−0.643814 + 0.765182i $$0.722649\pi$$
$$954$$ 0.150639i 0.00487711i
$$955$$ − 2.01582i − 0.0652304i
$$956$$ 19.4043i 0.627580i
$$957$$ − 24.9581i − 0.806782i
$$958$$ 7.72874 0.249704
$$959$$ −5.97475 −0.192935
$$960$$ − 10.9852i − 0.354544i
$$961$$ 1.14359 0.0368901
$$962$$ 0 0
$$963$$ 5.74011 0.184972
$$964$$ − 44.0837i − 1.41984i
$$965$$ −22.8211 −0.734636
$$966$$ −0.331612 −0.0106694
$$967$$ − 25.7857i − 0.829214i −0.910001 0.414607i $$-0.863919\pi$$
0.910001 0.414607i $$-0.136081\pi$$
$$968$$ − 15.5019i − 0.498251i
$$969$$ 18.3792i 0.590424i
$$970$$ − 0.550955i − 0.0176901i
$$971$$ −55.5252 −1.78189 −0.890945 0.454111i $$-0.849957\pi$$
−0.890945 + 0.454111i $$0.849957\pi$$
$$972$$ −8.83634 −0.283426
$$973$$ − 3.98235i − 0.127668i
$$974$$ −2.26365 −0.0725321
$$975$$ 0 0
$$976$$ 52.3642 1.67614
$$977$$ 40.5161i 1.29622i 0.761545 + 0.648112i $$0.224441\pi$$
−0.761545 + 0.648112i $$0.775559\pi$$
$$978$$ 6.25998 0.200172
$$979$$ 17.3286 0.553824
$$980$$ 13.4467i 0.429540i
$$981$$ − 4.93525i − 0.157570i
$$982$$ − 2.05011i − 0.0654218i
$$983$$ 34.8059i 1.11014i 0.831805 + 0.555068i $$0.187308\pi$$
−0.831805 + 0.555068i $$0.812692\pi$$
$$984$$ −5.18457 −0.165278
$$985$$ −0.643026 −0.0204885
$$986$$ − 3.23029i − 0.102873i
$$987$$ −4.43555 −0.141185
$$988$$ 0 0
$$989$$ 14.3784 0.457208
$$990$$ − 0.518513i − 0.0164794i
$$991$$ 43.8855 1.39407 0.697034 0.717038i $$-0.254503\pi$$
0.697034 + 0.717038i $$0.254503\pi$$
$$992$$ 13.9440 0.442723
$$993$$ 26.6145i 0.844584i
$$994$$ − 0.934079i − 0.0296272i
$$995$$ − 3.06684i − 0.0972254i
$$996$$ − 13.3163i − 0.421942i
$$997$$ −5.49137 −0.173914 −0.0869568 0.996212i $$-0.527714\pi$$
−0.0869568 + 0.996212i $$0.527714\pi$$
$$998$$ 5.25976 0.166495
$$999$$ 32.8680i 1.03990i
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 845.2.c.g.506.5 8
13.2 odd 12 845.2.e.n.191.2 8
13.3 even 3 65.2.m.a.56.2 yes 8
13.4 even 6 65.2.m.a.36.2 8
13.5 odd 4 845.2.a.l.1.3 4
13.6 odd 12 845.2.e.n.146.2 8
13.7 odd 12 845.2.e.m.146.3 8
13.8 odd 4 845.2.a.m.1.2 4
13.9 even 3 845.2.m.g.361.3 8
13.10 even 6 845.2.m.g.316.3 8
13.11 odd 12 845.2.e.m.191.3 8
13.12 even 2 inner 845.2.c.g.506.4 8
39.5 even 4 7605.2.a.cj.1.2 4
39.8 even 4 7605.2.a.cf.1.3 4
39.17 odd 6 585.2.bu.c.361.3 8
39.29 odd 6 585.2.bu.c.316.3 8
52.3 odd 6 1040.2.da.b.641.2 8
52.43 odd 6 1040.2.da.b.881.2 8
65.3 odd 12 325.2.m.c.199.2 8
65.4 even 6 325.2.n.d.101.3 8
65.17 odd 12 325.2.m.c.49.2 8
65.29 even 6 325.2.n.d.251.3 8
65.34 odd 4 4225.2.a.bi.1.3 4
65.42 odd 12 325.2.m.b.199.3 8
65.43 odd 12 325.2.m.b.49.3 8
65.44 odd 4 4225.2.a.bl.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.2 8 13.4 even 6
65.2.m.a.56.2 yes 8 13.3 even 3
325.2.m.b.49.3 8 65.43 odd 12
325.2.m.b.199.3 8 65.42 odd 12
325.2.m.c.49.2 8 65.17 odd 12
325.2.m.c.199.2 8 65.3 odd 12
325.2.n.d.101.3 8 65.4 even 6
325.2.n.d.251.3 8 65.29 even 6
585.2.bu.c.316.3 8 39.29 odd 6
585.2.bu.c.361.3 8 39.17 odd 6
845.2.a.l.1.3 4 13.5 odd 4
845.2.a.m.1.2 4 13.8 odd 4
845.2.c.g.506.4 8 13.12 even 2 inner
845.2.c.g.506.5 8 1.1 even 1 trivial
845.2.e.m.146.3 8 13.7 odd 12
845.2.e.m.191.3 8 13.11 odd 12
845.2.e.n.146.2 8 13.6 odd 12
845.2.e.n.191.2 8 13.2 odd 12
845.2.m.g.316.3 8 13.10 even 6
845.2.m.g.361.3 8 13.9 even 3
1040.2.da.b.641.2 8 52.3 odd 6
1040.2.da.b.881.2 8 52.43 odd 6
4225.2.a.bi.1.3 4 65.34 odd 4
4225.2.a.bl.1.2 4 65.44 odd 4
7605.2.a.cf.1.3 4 39.8 even 4
7605.2.a.cj.1.2 4 39.5 even 4