# Properties

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

## $q$-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 $$-0.975252\pi$$
0.996979 0.0776710i $$-0.0247484\pi$$
$$3$$ −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 $$0.0199994\pi$$
−0.998027 + 0.0627887i $$0.980001\pi$$
$$8$$ − 0.868145i − 0.306936i
$$9$$ −0.439374 −0.146458
$$10$$ −0.219687 −0.0694711
$$11$$ 5.37182i 1.61966i 0.586662 + 0.809832i $$0.300442\pi$$
−0.586662 + 0.809832i $$0.699558\pi$$
$$12$$ −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 $$0.0837725\pi$$
−0.965568 + 0.260152i $$0.916227\pi$$
$$20$$ − 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 $$0.163256\pi$$
−0.871334 + 0.490691i $$0.836744\pi$$
$$32$$ − 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 $$-0.836669\pi$$
0.871218 0.490896i $$-0.163331\pi$$
$$38$$ 0.498239 0.0808250
$$39$$ 0 0
$$40$$ −0.868145 −0.137266
$$41$$ 3.73205i 0.582848i 0.956594 + 0.291424i $$0.0941291\pi$$
−0.956594 + 0.291424i $$0.905871\pi$$
$$42$$ −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 $$-0.791786\pi$$
0.793581 0.608465i $$-0.208214\pi$$
$$48$$ −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 $$-0.943609\pi$$
0.984349 0.176232i $$-0.0563908\pi$$
$$60$$ 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 $$0.217414\pi$$
−0.775667 + 0.631142i $$0.782586\pi$$
$$68$$ 9.88418 1.19863
$$69$$ −4.54324 −0.546942
$$70$$ − 0.0729902i − 0.00872400i
$$71$$ 12.7973i 1.51876i 0.650645 + 0.759382i $$0.274498\pi$$
−0.650645 + 0.759382i $$0.725502\pi$$
$$72$$ 0.381440i 0.0449531i
$$73$$ 9.68922i 1.13404i 0.823705 + 0.567019i $$0.191903\pi$$
−0.823705 + 0.567019i $$0.808097\pi$$
$$74$$ −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 $$-0.924818\pi$$
0.972236 0.234001i $$-0.0751821\pi$$
$$84$$ − 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 $$-0.945310\pi$$
0.985276 0.170969i $$-0.0546898\pi$$
$$90$$ 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 $$-0.959363\pi$$
0.991862 0.127320i $$-0.0406375\pi$$
$$98$$ − 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 $$-0.819204\pi$$
0.842985 0.537937i $$-0.180796\pi$$
$$110$$ − 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 $$0.278843\pi$$
−0.640221 + 0.768190i $$0.721157\pi$$
$$138$$ 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 $$-0.955388\pi$$
0.990194 0.139696i $$-0.0446124\pi$$
$$150$$ − 0.351542i − 0.0287033i
$$151$$ − 7.96141i − 0.647890i −0.946076 0.323945i $$-0.894991\pi$$
0.946076 0.323945i $$-0.105009\pi$$
$$152$$ 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 $$-0.754347\pi$$
0.716697 0.697384i $$-0.245653\pi$$
$$164$$ 7.28398i 0.568784i
$$165$$ −8.59596 −0.669194
$$166$$ −0.936681 −0.0727006
$$167$$ 6.29366i 0.487018i 0.969899 + 0.243509i $$0.0782985\pi$$
−0.969899 + 0.243509i $$0.921702\pi$$
$$168$$ −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 $$-0.693222\pi$$
0.570427 0.821348i $$-0.306778\pi$$
$$194$$ −0.550955 −0.0395563
$$195$$ 0 0
$$196$$ 13.4467 0.960480
$$197$$ − 0.643026i − 0.0458137i −0.999738 0.0229068i $$-0.992708\pi$$
0.999738 0.0229068i $$-0.00729211\pi$$
$$198$$ −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 $$0.111443\pi$$
−0.939335 + 0.343001i $$0.888557\pi$$
$$224$$ 0.847871 0.0566508
$$225$$ 0.439374 0.0292916
$$226$$ 4.02990i 0.268065i
$$227$$ 7.04381i 0.467514i 0.972295 + 0.233757i $$0.0751020\pi$$
−0.972295 + 0.233757i $$0.924898\pi$$
$$228$$ − 7.08317i − 0.469095i
$$229$$ − 1.32899i − 0.0878219i −0.999035 0.0439109i $$-0.986018\pi$$
0.999035 0.0439109i $$-0.0139818\pi$$
$$230$$ −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 $$-0.895796\pi$$
0.946893 0.321549i $$-0.104204\pi$$
$$240$$ 5.94114i 0.383499i
$$241$$ 22.5869i 1.45495i 0.686134 + 0.727475i $$0.259306\pi$$
−0.686134 + 0.727475i $$0.740694\pi$$
$$242$$ 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 $$-0.612262\pi$$
0.345415 0.938450i $$-0.387738\pi$$
$$272$$ 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 $$-0.952565\pi$$
0.988917 0.148471i $$-0.0474352\pi$$
$$282$$ 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 $$-0.835473\pi$$
0.869366 0.494168i $$-0.164527\pi$$
$$294$$ 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 $$0.0391965\pi$$
−0.992428 + 0.122828i $$0.960803\pi$$
$$308$$ −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 $$-0.997125\pi$$
0.999959 0.00903266i $$-0.00287522\pi$$
$$318$$ − 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 $$0.151108\pi$$
−0.889421 + 0.457089i $$0.848892\pi$$
$$332$$ − 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 $$-0.939403\pi$$
0.981934 0.189223i $$-0.0605968\pi$$
$$350$$ −0.0729902 −0.00390149
$$351$$ 0 0
$$352$$ 13.7085 0.730666
$$353$$ − 21.7898i − 1.15976i −0.814704 0.579878i $$-0.803100\pi$$
0.814704 0.579878i $$-0.196900\pi$$
$$354$$ 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 $$-0.781971\pi$$
0.774444 0.632642i $$-0.218029\pi$$
$$360$$ 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 $$-0.955201\pi$$
0.990112 0.140277i $$-0.0447994\pi$$
$$380$$ 4.42644 0.227072
$$381$$ 5.18457 0.265614
$$382$$ 0.442849i 0.0226581i
$$383$$ − 5.66775i − 0.289609i −0.989460 0.144804i $$-0.953745\pi$$
0.989460 0.144804i $$-0.0462553\pi$$
$$384$$ 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 $$-0.751625\pi$$
0.710708 0.703487i $$-0.248375\pi$$
$$398$$ 0.673745i 0.0337718i
$$399$$ 1.20578 0.0603643
$$400$$ −3.71276 −0.185638
$$401$$ 22.5143i 1.12431i 0.827032 + 0.562155i $$0.190027\pi$$
−0.827032 + 0.562155i $$0.809973\pi$$
$$402$$ −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 $$0.0338066\pi$$
−0.994365 + 0.106007i $$0.966193\pi$$
$$410$$ − 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 $$0.101614\pi$$
−0.949477 + 0.313836i $$0.898386\pi$$
$$422$$ 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 $$0.0734678\pi$$
−0.973482 + 0.228762i $$0.926532\pi$$
$$432$$ 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 $$-0.828740\pi$$
0.858720 0.512446i $$-0.171260\pi$$
$$450$$ − 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 $$0.0352314\pi$$
−0.993881 + 0.110457i $$0.964769\pi$$
$$458$$ −0.291961 −0.0136424
$$459$$ 27.8722 1.30096
$$460$$ − 5.54133i − 0.258366i
$$461$$ − 1.78151i − 0.0829730i −0.999139 0.0414865i $$-0.986791\pi$$
0.999139 0.0414865i $$-0.0132094\pi$$
$$462$$ − 0.627421i − 0.0291902i
$$463$$ − 6.80200i − 0.316116i −0.987430 0.158058i $$-0.949477\pi$$
0.987430 0.158058i $$-0.0505232\pi$$
$$464$$ −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 $$0.297151\pi$$
−0.595002 + 0.803724i $$0.702849\pi$$
$$480$$ 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 $$-0.924995\pi$$
0.972367 0.233459i $$-0.0750046\pi$$
$$488$$ − 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 $$0.180026\pi$$
−0.844283 + 0.535897i $$0.819974\pi$$
$$500$$ 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 $$-0.733038\pi$$
0.668441 0.743765i $$-0.266962\pi$$
$$510$$ 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 $$0.107946\pi$$
−0.943047 + 0.332660i $$0.892054\pi$$
$$542$$ −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 $$-0.645605\pi$$
0.441645 0.897190i $$-0.354395\pi$$
$$558$$ −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 $$-0.858375\pi$$
0.902642 0.430393i $$-0.141625\pi$$
$$578$$ − 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 $$0.141203\pi$$
−0.903212 + 0.429196i $$0.858797\pi$$
$$588$$ −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 $$0.148072\pi$$
−0.893740 + 0.448585i $$0.851928\pi$$
$$594$$ 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 $$0.0639815\pi$$
−0.979867 + 0.199653i $$0.936019\pi$$
$$614$$ 0.945589 0.0381609
$$615$$ −5.97201 −0.240815
$$616$$ 1.54944i 0.0624286i
$$617$$ − 45.7169i − 1.84049i −0.391339 0.920246i $$-0.627988\pi$$
0.391339 0.920246i $$-0.372012\pi$$
$$618$$ 5.28182i 0.212466i
$$619$$ − 19.9143i − 0.800425i −0.916422 0.400212i $$-0.868936\pi$$
0.916422 0.400212i $$-0.131064\pi$$
$$620$$ 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 $$-0.906170\pi$$
0.956867 0.290525i $$-0.0938299\pi$$
$$632$$ − 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 $$0.107286\pi$$
−0.943735 + 0.330704i $$0.892714\pi$$
$$644$$ 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 $$0.0771462\pi$$
−0.970774 + 0.239996i $$0.922854\pi$$
$$662$$ 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 $$0.256895\pi$$
−0.691626 + 0.722255i $$0.743105\pi$$
$$684$$ − 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 $$-0.989991\pi$$
0.999506 0.0314399i $$-0.0100093\pi$$
$$692$$ −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 $$-0.864823\pi$$
0.911175 0.412020i $$-0.135177\pi$$
$$710$$ − 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 $$0.148702\pi$$
−0.892850 + 0.450355i $$0.851298\pi$$
$$734$$ − 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 $$-0.751371\pi$$
0.710146 0.704054i $$-0.248629\pi$$
$$740$$ −11.6558 −0.428476
$$741$$ 0 0
$$742$$ −0.113910 −0.00418178
$$743$$ − 40.0079i − 1.46775i −0.679286 0.733874i $$-0.737710\pi$$
0.679286 0.733874i $$-0.262290\pi$$
$$744$$ −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 $$-0.842412\pi$$
0.879932 0.475099i $$-0.157588\pi$$
$$762$$ − 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 $$-0.705023\pi$$
0.600478 0.799641i $$-0.294977\pi$$
$$770$$ 0.392090 0.0141299
$$771$$ −16.4081 −0.590924
$$772$$ − 44.5408i − 1.60306i
$$773$$ − 23.2638i − 0.836742i −0.908276 0.418371i $$-0.862601\pi$$
0.908276 0.418371i $$-0.137399\pi$$
$$774$$ 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 $$0.325808\pi$$
−0.520334 + 0.853963i $$0.674192\pi$$
$$788$$ − 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 $$-0.862289\pi$$
0.907865 0.419263i $$-0.137711\pi$$
$$812$$ − 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 $$0.0895475\pi$$
−0.960689 + 0.277626i $$0.910453\pi$$
$$822$$ −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 $$-0.799019\pi$$
0.807202 0.590275i $$-0.200981\pi$$
$$828$$ −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 $$0.0821924\pi$$
−0.966847 + 0.255355i $$0.917808\pi$$
$$840$$ 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 $$0.0902765\pi$$
−0.960051 + 0.279825i $$0.909724\pi$$
$$854$$ 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 $$0.0257653\pi$$
−0.996726 + 0.0808559i $$0.974235\pi$$
$$864$$ − 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 $$0.0121223\pi$$
−0.999275 + 0.0380741i $$0.987878\pi$$
$$878$$ 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 $$0.187310\pi$$
−0.831801 + 0.555074i $$0.812690\pi$$
$$930$$ 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 $$-0.785992\pi$$
0.782375 0.622807i $$-0.214008\pi$$
$$942$$ − 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 $$-0.675287\pi$$
0.523266 0.852169i $$-0.324713\pi$$
$$948$$ −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 $$0.136081\pi$$
−0.910001 + 0.414607i $$0.863919\pi$$
$$968$$ 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 $$-0.775559\pi$$
0.761545 0.648112i $$-0.224441\pi$$
$$978$$ 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 $$-0.812692\pi$$
0.831805 0.555068i $$-0.187308\pi$$
$$984$$ −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.506.4 8
13.2 odd 12 845.2.e.m.191.3 8
13.3 even 3 845.2.m.g.316.3 8
13.4 even 6 845.2.m.g.361.3 8
13.5 odd 4 845.2.a.m.1.2 4
13.6 odd 12 845.2.e.m.146.3 8
13.7 odd 12 845.2.e.n.146.2 8
13.8 odd 4 845.2.a.l.1.3 4
13.9 even 3 65.2.m.a.36.2 8
13.10 even 6 65.2.m.a.56.2 yes 8
13.11 odd 12 845.2.e.n.191.2 8
13.12 even 2 inner 845.2.c.g.506.5 8
39.5 even 4 7605.2.a.cf.1.3 4
39.8 even 4 7605.2.a.cj.1.2 4
39.23 odd 6 585.2.bu.c.316.3 8
39.35 odd 6 585.2.bu.c.361.3 8
52.23 odd 6 1040.2.da.b.641.2 8
52.35 odd 6 1040.2.da.b.881.2 8
65.9 even 6 325.2.n.d.101.3 8
65.22 odd 12 325.2.m.c.49.2 8
65.23 odd 12 325.2.m.c.199.2 8
65.34 odd 4 4225.2.a.bl.1.2 4
65.44 odd 4 4225.2.a.bi.1.3 4
65.48 odd 12 325.2.m.b.49.3 8
65.49 even 6 325.2.n.d.251.3 8
65.62 odd 12 325.2.m.b.199.3 8

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.2 8 13.9 even 3
65.2.m.a.56.2 yes 8 13.10 even 6
325.2.m.b.49.3 8 65.48 odd 12
325.2.m.b.199.3 8 65.62 odd 12
325.2.m.c.49.2 8 65.22 odd 12
325.2.m.c.199.2 8 65.23 odd 12
325.2.n.d.101.3 8 65.9 even 6
325.2.n.d.251.3 8 65.49 even 6
585.2.bu.c.316.3 8 39.23 odd 6
585.2.bu.c.361.3 8 39.35 odd 6
845.2.a.l.1.3 4 13.8 odd 4
845.2.a.m.1.2 4 13.5 odd 4
845.2.c.g.506.4 8 1.1 even 1 trivial
845.2.c.g.506.5 8 13.12 even 2 inner
845.2.e.m.146.3 8 13.6 odd 12
845.2.e.m.191.3 8 13.2 odd 12
845.2.e.n.146.2 8 13.7 odd 12
845.2.e.n.191.2 8 13.11 odd 12
845.2.m.g.316.3 8 13.3 even 3
845.2.m.g.361.3 8 13.4 even 6
1040.2.da.b.641.2 8 52.23 odd 6
1040.2.da.b.881.2 8 52.35 odd 6
4225.2.a.bi.1.3 4 65.44 odd 4
4225.2.a.bl.1.2 4 65.34 odd 4
7605.2.a.cf.1.3 4 39.5 even 4
7605.2.a.cj.1.2 4 39.8 even 4