# Properties

 Label 4225.2.a.bl.1.2 Level $4225$ Weight $2$ Character 4225.1 Self dual yes Analytic conductor $33.737$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4225 = 5^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.7367948540$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.4752.1 Defining polynomial: $$x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.219687$$ of defining polynomial Character $$\chi$$ $$=$$ 4225.1

## $q$-expansion

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

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.219687 −0.155342 −0.0776710 0.996979i $$-0.524748\pi$$
−0.0776710 + 0.996979i $$0.524748\pi$$
$$3$$ 1.60020 0.923873 0.461937 0.886913i $$-0.347155\pi$$
0.461937 + 0.886913i $$0.347155\pi$$
$$4$$ −1.95174 −0.975869
$$5$$ 0 0
$$6$$ −0.351542 −0.143516
$$7$$ −0.332247 −0.125577 −0.0627887 0.998027i $$-0.519999\pi$$
−0.0627887 + 0.998027i $$0.519999\pi$$
$$8$$ 0.868145 0.306936
$$9$$ −0.439374 −0.146458
$$10$$ 0 0
$$11$$ 5.37182 1.61966 0.809832 0.586662i $$-0.199558\pi$$
0.809832 + 0.586662i $$0.199558\pi$$
$$12$$ −3.12316 −0.901579
$$13$$ 0 0
$$14$$ 0.0729902 0.0195074
$$15$$ 0 0
$$16$$ 3.71276 0.928189
$$17$$ 5.06430 1.22827 0.614136 0.789200i $$-0.289505\pi$$
0.614136 + 0.789200i $$0.289505\pi$$
$$18$$ 0.0965246 0.0227511
$$19$$ −2.26795 −0.520303 −0.260152 0.965568i $$-0.583773\pi$$
−0.260152 + 0.965568i $$0.583773\pi$$
$$20$$ 0 0
$$21$$ −0.531659 −0.116018
$$22$$ −1.18012 −0.251602
$$23$$ 2.83918 0.592010 0.296005 0.955186i $$-0.404346\pi$$
0.296005 + 0.955186i $$0.404346\pi$$
$$24$$ 1.38920 0.283570
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.50367 −1.05918
$$28$$ 0.648458 0.122547
$$29$$ −2.90348 −0.539162 −0.269581 0.962978i $$-0.586885\pi$$
−0.269581 + 0.962978i $$0.586885\pi$$
$$30$$ 0 0
$$31$$ −5.46410 −0.981382 −0.490691 0.871334i $$-0.663256\pi$$
−0.490691 + 0.871334i $$0.663256\pi$$
$$32$$ −2.55193 −0.451122
$$33$$ 8.59596 1.49636
$$34$$ −1.11256 −0.190802
$$35$$ 0 0
$$36$$ 0.857542 0.142924
$$37$$ 5.97201 0.981793 0.490896 0.871218i $$-0.336669\pi$$
0.490896 + 0.871218i $$0.336669\pi$$
$$38$$ 0.498239 0.0808250
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −3.73205 −0.582848 −0.291424 0.956594i $$-0.594129\pi$$
−0.291424 + 0.956594i $$0.594129\pi$$
$$42$$ 0.116799 0.0180224
$$43$$ 5.06430 0.772298 0.386149 0.922436i $$-0.373805\pi$$
0.386149 + 0.922436i $$0.373805\pi$$
$$44$$ −10.4844 −1.58058
$$45$$ 0 0
$$46$$ −0.623730 −0.0919640
$$47$$ 8.34285 1.21693 0.608465 0.793581i $$-0.291786\pi$$
0.608465 + 0.793581i $$0.291786\pi$$
$$48$$ 5.94114 0.857529
$$49$$ −6.88961 −0.984230
$$50$$ 0 0
$$51$$ 8.10387 1.13477
$$52$$ 0 0
$$53$$ 1.56063 0.214369 0.107184 0.994239i $$-0.465817\pi$$
0.107184 + 0.994239i $$0.465817\pi$$
$$54$$ 1.20908 0.164536
$$55$$ 0 0
$$56$$ −0.288438 −0.0385442
$$57$$ −3.62916 −0.480694
$$58$$ 0.637855 0.0837545
$$59$$ −2.70732 −0.352463 −0.176232 0.984349i $$-0.556391\pi$$
−0.176232 + 0.984349i $$0.556391\pi$$
$$60$$ 0 0
$$61$$ 14.1039 1.80582 0.902908 0.429835i $$-0.141428\pi$$
0.902908 + 0.429835i $$0.141428\pi$$
$$62$$ 1.20039 0.152450
$$63$$ 0.145980 0.0183918
$$64$$ −6.86488 −0.858111
$$65$$ 0 0
$$66$$ −1.88842 −0.232448
$$67$$ 10.3322 1.26228 0.631142 0.775667i $$-0.282586\pi$$
0.631142 + 0.775667i $$0.282586\pi$$
$$68$$ −9.88418 −1.19863
$$69$$ 4.54324 0.546942
$$70$$ 0 0
$$71$$ −12.7973 −1.51876 −0.759382 0.650645i $$-0.774498\pi$$
−0.759382 + 0.650645i $$0.774498\pi$$
$$72$$ −0.381440 −0.0449531
$$73$$ −9.68922 −1.13404 −0.567019 0.823705i $$-0.691903\pi$$
−0.567019 + 0.823705i $$0.691903\pi$$
$$74$$ −1.31197 −0.152514
$$75$$ 0 0
$$76$$ 4.42644 0.507748
$$77$$ −1.78477 −0.203393
$$78$$ 0 0
$$79$$ 4.51851 0.508372 0.254186 0.967155i $$-0.418192\pi$$
0.254186 + 0.967155i $$0.418192\pi$$
$$80$$ 0 0
$$81$$ −7.48883 −0.832092
$$82$$ 0.819883 0.0905409
$$83$$ −4.26371 −0.468003 −0.234001 0.972236i $$-0.575182\pi$$
−0.234001 + 0.972236i $$0.575182\pi$$
$$84$$ 1.03766 0.113218
$$85$$ 0 0
$$86$$ −1.11256 −0.119970
$$87$$ −4.64613 −0.498117
$$88$$ 4.66351 0.497132
$$89$$ −3.22584 −0.341938 −0.170969 0.985276i $$-0.554690\pi$$
−0.170969 + 0.985276i $$0.554690\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −5.54133 −0.577724
$$93$$ −8.74363 −0.906672
$$94$$ −1.83281 −0.189040
$$95$$ 0 0
$$96$$ −4.08359 −0.416780
$$97$$ −2.50791 −0.254640 −0.127320 0.991862i $$-0.540637\pi$$
−0.127320 + 0.991862i $$0.540637\pi$$
$$98$$ 1.51356 0.152892
$$99$$ −2.36023 −0.237213
$$100$$ 0 0
$$101$$ 12.4467 1.23849 0.619247 0.785196i $$-0.287438\pi$$
0.619247 + 0.785196i $$0.287438\pi$$
$$102$$ −1.78031 −0.176277
$$103$$ 15.0247 1.48043 0.740215 0.672370i $$-0.234724\pi$$
0.740215 + 0.672370i $$0.234724\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −0.342849 −0.0333004
$$107$$ 13.0643 1.26297 0.631487 0.775387i $$-0.282445\pi$$
0.631487 + 0.775387i $$0.282445\pi$$
$$108$$ 10.7417 1.03362
$$109$$ 11.2325 1.07587 0.537937 0.842985i $$-0.319204\pi$$
0.537937 + 0.842985i $$0.319204\pi$$
$$110$$ 0 0
$$111$$ 9.55639 0.907052
$$112$$ −1.23355 −0.116560
$$113$$ 18.3438 1.72564 0.862821 0.505509i $$-0.168695\pi$$
0.862821 + 0.505509i $$0.168695\pi$$
$$114$$ 0.797279 0.0746721
$$115$$ 0 0
$$116$$ 5.66682 0.526151
$$117$$ 0 0
$$118$$ 0.594763 0.0547524
$$119$$ −1.68260 −0.154243
$$120$$ 0 0
$$121$$ 17.8564 1.62331
$$122$$ −3.09843 −0.280519
$$123$$ −5.97201 −0.538478
$$124$$ 10.6645 0.957700
$$125$$ 0 0
$$126$$ −0.0320700 −0.00285702
$$127$$ −3.23996 −0.287500 −0.143750 0.989614i $$-0.545916\pi$$
−0.143750 + 0.989614i $$0.545916\pi$$
$$128$$ 6.61199 0.584423
$$129$$ 8.10387 0.713506
$$130$$ 0 0
$$131$$ 0.175664 0.0153478 0.00767390 0.999971i $$-0.497557\pi$$
0.00767390 + 0.999971i $$0.497557\pi$$
$$132$$ −16.7771 −1.46025
$$133$$ 0.753518 0.0653383
$$134$$ −2.26986 −0.196086
$$135$$ 0 0
$$136$$ 4.39654 0.377001
$$137$$ −17.9829 −1.53638 −0.768190 0.640221i $$-0.778843\pi$$
−0.768190 + 0.640221i $$0.778843\pi$$
$$138$$ −0.998090 −0.0849631
$$139$$ 11.9861 1.01665 0.508325 0.861165i $$-0.330265\pi$$
0.508325 + 0.861165i $$0.330265\pi$$
$$140$$ 0 0
$$141$$ 13.3502 1.12429
$$142$$ 2.81140 0.235928
$$143$$ 0 0
$$144$$ −1.63129 −0.135941
$$145$$ 0 0
$$146$$ 2.12859 0.176164
$$147$$ −11.0247 −0.909304
$$148$$ −11.6558 −0.958101
$$149$$ 3.41041 0.279391 0.139696 0.990194i $$-0.455388\pi$$
0.139696 + 0.990194i $$0.455388\pi$$
$$150$$ 0 0
$$151$$ −7.96141 −0.647890 −0.323945 0.946076i $$-0.605009\pi$$
−0.323945 + 0.946076i $$0.605009\pi$$
$$152$$ −1.96891 −0.159700
$$153$$ −2.22512 −0.179890
$$154$$ 0.392090 0.0315955
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 16.4329 1.31148 0.655742 0.754985i $$-0.272356\pi$$
0.655742 + 0.754985i $$0.272356\pi$$
$$158$$ −0.992658 −0.0789716
$$159$$ 2.49731 0.198049
$$160$$ 0 0
$$161$$ −0.943307 −0.0743430
$$162$$ 1.64520 0.129259
$$163$$ 17.8072 1.39477 0.697384 0.716697i $$-0.254347\pi$$
0.697384 + 0.716697i $$0.254347\pi$$
$$164$$ 7.28398 0.568784
$$165$$ 0 0
$$166$$ 0.936681 0.0727006
$$167$$ −6.29366 −0.487018 −0.243509 0.969899i $$-0.578298\pi$$
−0.243509 + 0.969899i $$0.578298\pi$$
$$168$$ −0.461557 −0.0356099
$$169$$ 0 0
$$170$$ 0 0
$$171$$ 0.996477 0.0762025
$$172$$ −9.88418 −0.753662
$$173$$ −15.9751 −1.21457 −0.607283 0.794486i $$-0.707740\pi$$
−0.607283 + 0.794486i $$0.707740\pi$$
$$174$$ 1.02069 0.0773786
$$175$$ 0 0
$$176$$ 19.9442 1.50335
$$177$$ −4.33225 −0.325632
$$178$$ 0.708674 0.0531173
$$179$$ 23.6174 1.76525 0.882625 0.470079i $$-0.155774\pi$$
0.882625 + 0.470079i $$0.155774\pi$$
$$180$$ 0 0
$$181$$ −2.62590 −0.195182 −0.0975909 0.995227i $$-0.531114\pi$$
−0.0975909 + 0.995227i $$0.531114\pi$$
$$182$$ 0 0
$$183$$ 22.5689 1.66834
$$184$$ 2.46482 0.181709
$$185$$ 0 0
$$186$$ 1.92086 0.140844
$$187$$ 27.2045 1.98939
$$188$$ −16.2831 −1.18756
$$189$$ 1.82858 0.133009
$$190$$ 0 0
$$191$$ −2.01582 −0.145860 −0.0729298 0.997337i $$-0.523235\pi$$
−0.0729298 + 0.997337i $$0.523235\pi$$
$$192$$ −10.9852 −0.792786
$$193$$ 22.8211 1.64270 0.821348 0.570427i $$-0.193222\pi$$
0.821348 + 0.570427i $$0.193222\pi$$
$$194$$ 0.550955 0.0395563
$$195$$ 0 0
$$196$$ 13.4467 0.960480
$$197$$ −0.643026 −0.0458137 −0.0229068 0.999738i $$-0.507292\pi$$
−0.0229068 + 0.999738i $$0.507292\pi$$
$$198$$ 0.518513 0.0368491
$$199$$ 3.06684 0.217403 0.108701 0.994074i $$-0.465331\pi$$
0.108701 + 0.994074i $$0.465331\pi$$
$$200$$ 0 0
$$201$$ 16.5336 1.16619
$$202$$ −2.73438 −0.192390
$$203$$ 0.964670 0.0677065
$$204$$ −15.8166 −1.10739
$$205$$ 0 0
$$206$$ −3.30074 −0.229973
$$207$$ −1.24746 −0.0867045
$$208$$ 0 0
$$209$$ −12.1830 −0.842716
$$210$$ 0 0
$$211$$ −8.20039 −0.564538 −0.282269 0.959335i $$-0.591087\pi$$
−0.282269 + 0.959335i $$0.591087\pi$$
$$212$$ −3.04593 −0.209196
$$213$$ −20.4782 −1.40314
$$214$$ −2.87005 −0.196193
$$215$$ 0 0
$$216$$ −4.77798 −0.325101
$$217$$ 1.81543 0.123239
$$218$$ −2.46762 −0.167129
$$219$$ −15.5046 −1.04771
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −2.09941 −0.140903
$$223$$ 10.2442 0.686002 0.343001 0.939335i $$-0.388557\pi$$
0.343001 + 0.939335i $$0.388557\pi$$
$$224$$ 0.847871 0.0566508
$$225$$ 0 0
$$226$$ −4.02990 −0.268065
$$227$$ 7.04381 0.467514 0.233757 0.972295i $$-0.424898\pi$$
0.233757 + 0.972295i $$0.424898\pi$$
$$228$$ 7.08317 0.469095
$$229$$ −1.32899 −0.0878219 −0.0439109 0.999035i $$-0.513982\pi$$
−0.0439109 + 0.999035i $$0.513982\pi$$
$$230$$ 0 0
$$231$$ −2.85598 −0.187909
$$232$$ −2.52064 −0.165488
$$233$$ 1.24746 0.0817238 0.0408619 0.999165i $$-0.486990\pi$$
0.0408619 + 0.999165i $$0.486990\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 5.28398 0.343958
$$237$$ 7.23050 0.469672
$$238$$ 0.369644 0.0239605
$$239$$ 9.94207 0.643099 0.321549 0.946893i $$-0.395796\pi$$
0.321549 + 0.946893i $$0.395796\pi$$
$$240$$ 0 0
$$241$$ 22.5869 1.45495 0.727475 0.686134i $$-0.240694\pi$$
0.727475 + 0.686134i $$0.240694\pi$$
$$242$$ −3.92282 −0.252168
$$243$$ 4.52742 0.290434
$$244$$ −27.5270 −1.76224
$$245$$ 0 0
$$246$$ 1.31197 0.0836483
$$247$$ 0 0
$$248$$ −4.74363 −0.301221
$$249$$ −6.82277 −0.432376
$$250$$ 0 0
$$251$$ −6.76836 −0.427215 −0.213608 0.976920i $$-0.568521\pi$$
−0.213608 + 0.976920i $$0.568521\pi$$
$$252$$ −0.284915 −0.0179480
$$253$$ 15.2515 0.958856
$$254$$ 0.711777 0.0446609
$$255$$ 0 0
$$256$$ 12.2772 0.767325
$$257$$ 10.2538 0.639616 0.319808 0.947482i $$-0.396382\pi$$
0.319808 + 0.947482i $$0.396382\pi$$
$$258$$ −1.78031 −0.110837
$$259$$ −1.98418 −0.123291
$$260$$ 0 0
$$261$$ 1.27571 0.0789645
$$262$$ −0.0385910 −0.00238416
$$263$$ −18.6570 −1.15044 −0.575220 0.817999i $$-0.695083\pi$$
−0.575220 + 0.817999i $$0.695083\pi$$
$$264$$ 7.46254 0.459287
$$265$$ 0 0
$$266$$ −0.165538 −0.0101498
$$267$$ −5.16197 −0.315907
$$268$$ −20.1658 −1.23182
$$269$$ 17.9579 1.09491 0.547456 0.836835i $$-0.315596\pi$$
0.547456 + 0.836835i $$0.315596\pi$$
$$270$$ 0 0
$$271$$ −30.8977 −1.87690 −0.938450 0.345415i $$-0.887738\pi$$
−0.938450 + 0.345415i $$0.887738\pi$$
$$272$$ 18.8025 1.14007
$$273$$ 0 0
$$274$$ 3.95060 0.238665
$$275$$ 0 0
$$276$$ −8.86721 −0.533744
$$277$$ −26.5045 −1.59250 −0.796250 0.604967i $$-0.793186\pi$$
−0.796250 + 0.604967i $$0.793186\pi$$
$$278$$ −2.63320 −0.157929
$$279$$ 2.40078 0.143731
$$280$$ 0 0
$$281$$ −4.97766 −0.296942 −0.148471 0.988917i $$-0.547435\pi$$
−0.148471 + 0.988917i $$0.547435\pi$$
$$282$$ −2.93286 −0.174649
$$283$$ 12.5863 0.748180 0.374090 0.927392i $$-0.377955\pi$$
0.374090 + 0.927392i $$0.377955\pi$$
$$284$$ 24.9770 1.48211
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1.23996 0.0731926
$$288$$ 1.12125 0.0660704
$$289$$ 8.64711 0.508653
$$290$$ 0 0
$$291$$ −4.01315 −0.235255
$$292$$ 18.9108 1.10667
$$293$$ 16.9176 0.988337 0.494168 0.869366i $$-0.335473\pi$$
0.494168 + 0.869366i $$0.335473\pi$$
$$294$$ 2.42199 0.141253
$$295$$ 0 0
$$296$$ 5.18457 0.301347
$$297$$ −29.5647 −1.71552
$$298$$ −0.749222 −0.0434012
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −1.68260 −0.0969832
$$302$$ 1.74902 0.100645
$$303$$ 19.9172 1.14421
$$304$$ −8.42034 −0.482940
$$305$$ 0 0
$$306$$ 0.488829 0.0279445
$$307$$ −4.30426 −0.245657 −0.122828 0.992428i $$-0.539197\pi$$
−0.122828 + 0.992428i $$0.539197\pi$$
$$308$$ 3.48340 0.198485
$$309$$ 24.0425 1.36773
$$310$$ 0 0
$$311$$ −2.22512 −0.126175 −0.0630875 0.998008i $$-0.520095\pi$$
−0.0630875 + 0.998008i $$0.520095\pi$$
$$312$$ 0 0
$$313$$ −7.20887 −0.407469 −0.203735 0.979026i $$-0.565308\pi$$
−0.203735 + 0.979026i $$0.565308\pi$$
$$314$$ −3.61008 −0.203729
$$315$$ 0 0
$$316$$ −8.81895 −0.496105
$$317$$ −0.321644 −0.0180653 −0.00903266 0.999959i $$-0.502875\pi$$
−0.00903266 + 0.999959i $$0.502875\pi$$
$$318$$ −0.548626 −0.0307654
$$319$$ −15.5969 −0.873261
$$320$$ 0 0
$$321$$ 20.9054 1.16683
$$322$$ 0.207232 0.0115486
$$323$$ −11.4856 −0.639074
$$324$$ 14.6162 0.812013
$$325$$ 0 0
$$326$$ −3.91201 −0.216666
$$327$$ 17.9741 0.993972
$$328$$ −3.23996 −0.178897
$$329$$ −2.77188 −0.152819
$$330$$ 0 0
$$331$$ −16.6320 −0.914178 −0.457089 0.889421i $$-0.651108\pi$$
−0.457089 + 0.889421i $$0.651108\pi$$
$$332$$ 8.32164 0.456710
$$333$$ −2.62395 −0.143791
$$334$$ 1.38263 0.0756543
$$335$$ 0 0
$$336$$ −1.97392 −0.107686
$$337$$ −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
$$340$$ 0 0
$$341$$ −29.3521 −1.58951
$$342$$ −0.218913 −0.0118375
$$343$$ 4.61478 0.249174
$$344$$ 4.39654 0.237046
$$345$$ 0 0
$$346$$ 3.50952 0.188673
$$347$$ 6.27360 0.336784 0.168392 0.985720i $$-0.446143\pi$$
0.168392 + 0.985720i $$0.446143\pi$$
$$348$$ 9.06802 0.486097
$$349$$ −7.06994 −0.378445 −0.189223 0.981934i $$-0.560597\pi$$
−0.189223 + 0.981934i $$0.560597\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −13.7085 −0.730666
$$353$$ −21.7898 −1.15976 −0.579878 0.814704i $$-0.696900\pi$$
−0.579878 + 0.814704i $$0.696900\pi$$
$$354$$ 0.951738 0.0505843
$$355$$ 0 0
$$356$$ 6.29598 0.333687
$$357$$ −2.69248 −0.142501
$$358$$ −5.18844 −0.274217
$$359$$ −23.9737 −1.26528 −0.632642 0.774444i $$-0.718029\pi$$
−0.632642 + 0.774444i $$0.718029\pi$$
$$360$$ 0 0
$$361$$ −13.8564 −0.729285
$$362$$ 0.576876 0.0303199
$$363$$ 28.5737 1.49973
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −4.95810 −0.259164
$$367$$ −6.39133 −0.333625 −0.166812 0.985989i $$-0.553347\pi$$
−0.166812 + 0.985989i $$0.553347\pi$$
$$368$$ 10.5412 0.549497
$$369$$ 1.63977 0.0853628
$$370$$ 0 0
$$371$$ −0.518513 −0.0269198
$$372$$ 17.0653 0.884793
$$373$$ 20.0801 1.03971 0.519855 0.854255i $$-0.325986\pi$$
0.519855 + 0.854255i $$0.325986\pi$$
$$374$$ −5.97647 −0.309036
$$375$$ 0 0
$$376$$ 7.24280 0.373519
$$377$$ 0 0
$$378$$ −0.401714 −0.0206619
$$379$$ 5.46182 0.280555 0.140277 0.990112i $$-0.455201\pi$$
0.140277 + 0.990112i $$0.455201\pi$$
$$380$$ 0 0
$$381$$ −5.18457 −0.265614
$$382$$ 0.442849 0.0226581
$$383$$ −5.66775 −0.289609 −0.144804 0.989460i $$-0.546255\pi$$
−0.144804 + 0.989460i $$0.546255\pi$$
$$384$$ 10.5805 0.539933
$$385$$ 0 0
$$386$$ −5.01349 −0.255180
$$387$$ −2.22512 −0.113109
$$388$$ 4.89478 0.248495
$$389$$ 10.6174 0.538325 0.269162 0.963095i $$-0.413253\pi$$
0.269162 + 0.963095i $$0.413253\pi$$
$$390$$ 0 0
$$391$$ 14.3784 0.727149
$$392$$ −5.98118 −0.302095
$$393$$ 0.281096 0.0141794
$$394$$ 0.141264 0.00711679
$$395$$ 0 0
$$396$$ 4.60656 0.231488
$$397$$ 28.0338 1.40697 0.703487 0.710708i $$-0.251625\pi$$
0.703487 + 0.710708i $$0.251625\pi$$
$$398$$ −0.673745 −0.0337718
$$399$$ 1.20578 0.0603643
$$400$$ 0 0
$$401$$ 22.5143 1.12431 0.562155 0.827032i $$-0.309973\pi$$
0.562155 + 0.827032i $$0.309973\pi$$
$$402$$ −3.63222 −0.181159
$$403$$ 0 0
$$404$$ −24.2927 −1.20861
$$405$$ 0 0
$$406$$ −0.211925 −0.0105177
$$407$$ 32.0805 1.59017
$$408$$ 7.03533 0.348301
$$409$$ −4.28772 −0.212014 −0.106007 0.994365i $$-0.533807\pi$$
−0.106007 + 0.994365i $$0.533807\pi$$
$$410$$ 0 0
$$411$$ −28.7761 −1.41942
$$412$$ −29.3243 −1.44471
$$413$$ 0.899499 0.0442614
$$414$$ 0.274051 0.0134689
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 19.1802 0.939257
$$418$$ 2.67645 0.130909
$$419$$ 17.7116 0.865266 0.432633 0.901570i $$-0.357585\pi$$
0.432633 + 0.901570i $$0.357585\pi$$
$$420$$ 0 0
$$421$$ −12.8787 −0.627672 −0.313836 0.949477i $$-0.601614\pi$$
−0.313836 + 0.949477i $$0.601614\pi$$
$$422$$ 1.80152 0.0876965
$$423$$ −3.66563 −0.178229
$$424$$ 1.35485 0.0657973
$$425$$ 0 0
$$426$$ 4.49880 0.217967
$$427$$ −4.68596 −0.226770
$$428$$ −25.4981 −1.23250
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −9.49845 −0.457524 −0.228762 0.973482i $$-0.573468\pi$$
−0.228762 + 0.973482i $$0.573468\pi$$
$$432$$ −20.4338 −0.983121
$$433$$ 1.39628 0.0671010 0.0335505 0.999437i $$-0.489319\pi$$
0.0335505 + 0.999437i $$0.489319\pi$$
$$434$$ −0.398826 −0.0191443
$$435$$ 0 0
$$436$$ −21.9228 −1.04991
$$437$$ −6.43911 −0.308024
$$438$$ 3.40617 0.162753
$$439$$ 4.16180 0.198632 0.0993159 0.995056i $$-0.468335\pi$$
0.0993159 + 0.995056i $$0.468335\pi$$
$$440$$ 0 0
$$441$$ 3.02711 0.144148
$$442$$ 0 0
$$443$$ −9.54563 −0.453526 −0.226763 0.973950i $$-0.572814\pi$$
−0.226763 + 0.973950i $$0.572814\pi$$
$$444$$ −18.6516 −0.885164
$$445$$ 0 0
$$446$$ −2.25052 −0.106565
$$447$$ 5.45732 0.258122
$$448$$ 2.28083 0.107759
$$449$$ −21.7171 −1.02489 −0.512446 0.858720i $$-0.671260\pi$$
−0.512446 + 0.858720i $$0.671260\pi$$
$$450$$ 0 0
$$451$$ −20.0479 −0.944018
$$452$$ −35.8023 −1.68400
$$453$$ −12.7398 −0.598569
$$454$$ −1.54743 −0.0726246
$$455$$ 0 0
$$456$$ −3.15064 −0.147542
$$457$$ 4.72259 0.220914 0.110457 0.993881i $$-0.464769\pi$$
0.110457 + 0.993881i $$0.464769\pi$$
$$458$$ 0.291961 0.0136424
$$459$$ −27.8722 −1.30096
$$460$$ 0 0
$$461$$ 1.78151 0.0829730 0.0414865 0.999139i $$-0.486791\pi$$
0.0414865 + 0.999139i $$0.486791\pi$$
$$462$$ 0.627421 0.0291902
$$463$$ 6.80200 0.316116 0.158058 0.987430i $$-0.449477\pi$$
0.158058 + 0.987430i $$0.449477\pi$$
$$464$$ −10.7799 −0.500444
$$465$$ 0 0
$$466$$ −0.274051 −0.0126952
$$467$$ −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
$$470$$ 0 0
$$471$$ 26.2958 1.21165
$$472$$ −2.35035 −0.108184
$$473$$ 27.2045 1.25086
$$474$$ −1.58845 −0.0729598
$$475$$ 0 0
$$476$$ 3.28398 0.150521
$$477$$ −0.685698 −0.0313960
$$478$$ −2.18414 −0.0999003
$$479$$ 35.1807 1.60745 0.803724 0.595002i $$-0.202849\pi$$
0.803724 + 0.595002i $$0.202849\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −4.96204 −0.226015
$$483$$ −1.50948 −0.0686835
$$484$$ −34.8510 −1.58414
$$485$$ 0 0
$$486$$ −0.994615 −0.0451166
$$487$$ −10.3040 −0.466919 −0.233459 0.972367i $$-0.575005\pi$$
−0.233459 + 0.972367i $$0.575005\pi$$
$$488$$ 12.2442 0.554269
$$489$$ 28.4950 1.28859
$$490$$ 0 0
$$491$$ 9.33198 0.421147 0.210573 0.977578i $$-0.432467\pi$$
0.210573 + 0.977578i $$0.432467\pi$$
$$492$$ 11.6558 0.525484
$$493$$ −14.7041 −0.662238
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −20.2869 −0.910907
$$497$$ 4.25187 0.190722
$$498$$ 1.49887 0.0671661
$$499$$ −23.9421 −1.07179 −0.535897 0.844283i $$-0.680026\pi$$
−0.535897 + 0.844283i $$0.680026\pi$$
$$500$$ 0 0
$$501$$ −10.0711 −0.449943
$$502$$ 1.48692 0.0663645
$$503$$ −42.1443 −1.87912 −0.939560 0.342385i $$-0.888765\pi$$
−0.939560 + 0.342385i $$0.888765\pi$$
$$504$$ 0.126732 0.00564510
$$505$$ 0 0
$$506$$ −3.35056 −0.148951
$$507$$ 0 0
$$508$$ 6.32355 0.280562
$$509$$ 33.5602 1.48753 0.743765 0.668441i $$-0.233038\pi$$
0.743765 + 0.668441i $$0.233038\pi$$
$$510$$ 0 0
$$511$$ 3.21921 0.142409
$$512$$ −15.9211 −0.703621
$$513$$ 12.4820 0.551096
$$514$$ −2.25263 −0.0993593
$$515$$ 0 0
$$516$$ −15.8166 −0.696288
$$517$$ 44.8162 1.97102
$$518$$ 0.435898 0.0191523
$$519$$ −25.5633 −1.12210
$$520$$ 0 0
$$521$$ 12.4649 0.546098 0.273049 0.962000i $$-0.411968\pi$$
0.273049 + 0.962000i $$0.411968\pi$$
$$522$$ −0.280257 −0.0122665
$$523$$ 5.65956 0.247475 0.123738 0.992315i $$-0.460512\pi$$
0.123738 + 0.992315i $$0.460512\pi$$
$$524$$ −0.342849 −0.0149774
$$525$$ 0 0
$$526$$ 4.09870 0.178712
$$527$$ −27.6718 −1.20540
$$528$$ 31.9147 1.38891
$$529$$ −14.9391 −0.649525
$$530$$ 0 0
$$531$$ 1.18953 0.0516211
$$532$$ −1.47067 −0.0637616
$$533$$ 0 0
$$534$$ 1.13402 0.0490737
$$535$$ 0 0
$$536$$ 8.96989 0.387440
$$537$$ 37.7925 1.63087
$$538$$ −3.94511 −0.170086
$$539$$ −37.0097 −1.59412
$$540$$ 0 0
$$541$$ 15.4750 0.665321 0.332660 0.943047i $$-0.392054\pi$$
0.332660 + 0.943047i $$0.392054\pi$$
$$542$$ 6.78781 0.291562
$$543$$ −4.20196 −0.180323
$$544$$ −12.9237 −0.554101
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −25.1765 −1.07647 −0.538234 0.842795i $$-0.680908\pi$$
−0.538234 + 0.842795i $$0.680908\pi$$
$$548$$ 35.0979 1.49931
$$549$$ −6.19687 −0.264476
$$550$$ 0 0
$$551$$ 6.58493 0.280528
$$552$$ 3.94419 0.167876
$$553$$ −1.50126 −0.0638401
$$554$$ 5.82269 0.247382
$$555$$ 0 0
$$556$$ −23.3938 −0.992118
$$557$$ 42.3489 1.79438 0.897190 0.441645i $$-0.145605\pi$$
0.897190 + 0.441645i $$0.145605\pi$$
$$558$$ −0.527420 −0.0223275
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 43.5325 1.83794
$$562$$ 1.09353 0.0461276
$$563$$ −23.7905 −1.00265 −0.501326 0.865259i $$-0.667154\pi$$
−0.501326 + 0.865259i $$0.667154\pi$$
$$564$$ −26.0561 −1.09716
$$565$$ 0 0
$$566$$ −2.76505 −0.116224
$$567$$ 2.48814 0.104492
$$568$$ −11.1099 −0.466162
$$569$$ −26.7421 −1.12109 −0.560543 0.828125i $$-0.689407\pi$$
−0.560543 + 0.828125i $$0.689407\pi$$
$$570$$ 0 0
$$571$$ −16.7159 −0.699539 −0.349769 0.936836i $$-0.613740\pi$$
−0.349769 + 0.936836i $$0.613740\pi$$
$$572$$ 0 0
$$573$$ −3.22571 −0.134756
$$574$$ −0.272403 −0.0113699
$$575$$ 0 0
$$576$$ 3.01625 0.125677
$$577$$ −20.6768 −0.860786 −0.430393 0.902642i $$-0.641625\pi$$
−0.430393 + 0.902642i $$0.641625\pi$$
$$578$$ −1.89966 −0.0790153
$$579$$ 36.5182 1.51764
$$580$$ 0 0
$$581$$ 1.41660 0.0587706
$$582$$ 0.881636 0.0365450
$$583$$ 8.38340 0.347205
$$584$$ −8.41165 −0.348076
$$585$$ 0 0
$$586$$ −3.71657 −0.153530
$$587$$ 20.7972 0.858391 0.429196 0.903212i $$-0.358797\pi$$
0.429196 + 0.903212i $$0.358797\pi$$
$$588$$ 21.5174 0.887362
$$589$$ 12.3923 0.510616
$$590$$ 0 0
$$591$$ −1.02897 −0.0423260
$$592$$ 22.1726 0.911289
$$593$$ −21.8475 −0.897169 −0.448585 0.893740i $$-0.648072\pi$$
−0.448585 + 0.893740i $$0.648072\pi$$
$$594$$ 6.49498 0.266492
$$595$$ 0 0
$$596$$ −6.65622 −0.272649
$$597$$ 4.90755 0.200853
$$598$$ 0 0
$$599$$ −3.58040 −0.146291 −0.0731456 0.997321i $$-0.523304\pi$$
−0.0731456 + 0.997321i $$0.523304\pi$$
$$600$$ 0 0
$$601$$ 21.3486 0.870829 0.435414 0.900230i $$-0.356602\pi$$
0.435414 + 0.900230i $$0.356602\pi$$
$$602$$ 0.369644 0.0150656
$$603$$ −4.53972 −0.184872
$$604$$ 15.5386 0.632256
$$605$$ 0 0
$$606$$ −4.37554 −0.177744
$$607$$ 3.29976 0.133933 0.0669665 0.997755i $$-0.478668\pi$$
0.0669665 + 0.997755i $$0.478668\pi$$
$$608$$ 5.78766 0.234720
$$609$$ 1.54366 0.0625523
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 4.34285 0.175549
$$613$$ 9.88635 0.399306 0.199653 0.979867i $$-0.436019\pi$$
0.199653 + 0.979867i $$0.436019\pi$$
$$614$$ 0.945589 0.0381609
$$615$$ 0 0
$$616$$ −1.54944 −0.0624286
$$617$$ −45.7169 −1.84049 −0.920246 0.391339i $$-0.872012\pi$$
−0.920246 + 0.391339i $$0.872012\pi$$
$$618$$ −5.28182 −0.212466
$$619$$ −19.9143 −0.800425 −0.400212 0.916422i $$-0.631064\pi$$
−0.400212 + 0.916422i $$0.631064\pi$$
$$620$$ 0 0
$$621$$ −15.6259 −0.627046
$$622$$ 0.488829 0.0196003
$$623$$ 1.07177 0.0429397
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1.58369 0.0632971
$$627$$ −19.4952 −0.778563
$$628$$ −32.0726 −1.27984
$$629$$ 30.2440 1.20591
$$630$$ 0 0
$$631$$ −14.5958 −0.581050 −0.290525 0.956867i $$-0.593830\pi$$
−0.290525 + 0.956867i $$0.593830\pi$$
$$632$$ 3.92272 0.156038
$$633$$ −13.1222 −0.521562
$$634$$ 0.0706609 0.00280630
$$635$$ 0 0
$$636$$ −4.87409 −0.193270
$$637$$ 0 0
$$638$$ 3.42644 0.135654
$$639$$ 5.62281 0.222435
$$640$$ 0 0
$$641$$ −14.1637 −0.559431 −0.279716 0.960083i $$-0.590240\pi$$
−0.279716 + 0.960083i $$0.590240\pi$$
$$642$$ −4.59265 −0.181257
$$643$$ 16.7716 0.661408 0.330704 0.943735i $$-0.392714\pi$$
0.330704 + 0.943735i $$0.392714\pi$$
$$644$$ 1.84109 0.0725490
$$645$$ 0 0
$$646$$ 2.52323 0.0992751
$$647$$ 2.99168 0.117615 0.0588075 0.998269i $$-0.481270\pi$$
0.0588075 + 0.998269i $$0.481270\pi$$
$$648$$ −6.50139 −0.255399
$$649$$ −14.5432 −0.570872
$$650$$ 0 0
$$651$$ 2.90504 0.113858
$$652$$ −34.7550 −1.36111
$$653$$ 11.6643 0.456461 0.228230 0.973607i $$-0.426706\pi$$
0.228230 + 0.973607i $$0.426706\pi$$
$$654$$ −3.94868 −0.154406
$$655$$ 0 0
$$656$$ −13.8562 −0.540993
$$657$$ 4.25719 0.166089
$$658$$ 0.608946 0.0237392
$$659$$ −1.81047 −0.0705260 −0.0352630 0.999378i $$-0.511227\pi$$
−0.0352630 + 0.999378i $$0.511227\pi$$
$$660$$ 0 0
$$661$$ 12.3406 0.479992 0.239996 0.970774i $$-0.422854\pi$$
0.239996 + 0.970774i $$0.422854\pi$$
$$662$$ 3.65383 0.142010
$$663$$ 0 0
$$664$$ −3.70152 −0.143647
$$665$$ 0 0
$$666$$ 0.576446 0.0223368
$$667$$ −8.24348 −0.319189
$$668$$ 12.2836 0.475265
$$669$$ 16.3927 0.633779
$$670$$ 0 0
$$671$$ 75.7634 2.92481
$$672$$ 1.35676 0.0523381
$$673$$ −9.26625 −0.357188 −0.178594 0.983923i $$-0.557155\pi$$
−0.178594 + 0.983923i $$0.557155\pi$$
$$674$$ 5.32051 0.204938
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −13.8984 −0.534158 −0.267079 0.963675i $$-0.586059\pi$$
−0.267079 + 0.963675i $$0.586059\pi$$
$$678$$ −6.44863 −0.247658
$$679$$ 0.833244 0.0319770
$$680$$ 0 0
$$681$$ 11.2715 0.431924
$$682$$ 6.44828 0.246917
$$683$$ −37.7512 −1.44451 −0.722255 0.691626i $$-0.756895\pi$$
−0.722255 + 0.691626i $$0.756895\pi$$
$$684$$ −1.94486 −0.0743637
$$685$$ 0 0
$$686$$ −1.01381 −0.0387073
$$687$$ −2.12664 −0.0811363
$$688$$ 18.8025 0.716838
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1.65291 0.0628797 0.0314399 0.999506i $$-0.489991\pi$$
0.0314399 + 0.999506i $$0.489991\pi$$
$$692$$ 31.1792 1.18526
$$693$$ 0.784180 0.0297885
$$694$$ −1.37823 −0.0523168
$$695$$ 0 0
$$696$$ −4.03351 −0.152890
$$697$$ −18.9002 −0.715897
$$698$$ 1.55317 0.0587885
$$699$$ 1.99618 0.0755025
$$700$$ 0 0
$$701$$ 20.4819 0.773590 0.386795 0.922166i $$-0.373582\pi$$
0.386795 + 0.922166i $$0.373582\pi$$
$$702$$ 0 0
$$703$$ −13.5442 −0.510830
$$704$$ −36.8769 −1.38985
$$705$$ 0 0
$$706$$ 4.78694 0.180159
$$707$$ −4.13538 −0.155527
$$708$$ 8.45541 0.317774
$$709$$ −21.9417 −0.824039 −0.412020 0.911175i $$-0.635177\pi$$
−0.412020 + 0.911175i $$0.635177\pi$$
$$710$$ 0 0
$$711$$ −1.98532 −0.0744552
$$712$$ −2.80049 −0.104953
$$713$$ −15.5136 −0.580987
$$714$$ 0.591503 0.0221364
$$715$$ 0 0
$$716$$ −46.0950 −1.72265
$$717$$ 15.9093 0.594142
$$718$$ 5.26671 0.196552
$$719$$ 38.8475 1.44877 0.724384 0.689397i $$-0.242124\pi$$
0.724384 + 0.689397i $$0.242124\pi$$
$$720$$ 0 0
$$721$$ −4.99191 −0.185909
$$722$$ 3.04407 0.113289
$$723$$ 36.1434 1.34419
$$724$$ 5.12507 0.190472
$$725$$ 0 0
$$726$$ −6.27728 −0.232972
$$727$$ 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
$$730$$ 0 0
$$731$$ 25.6471 0.948593
$$732$$ −44.0487 −1.62809
$$733$$ 24.3858 0.900709 0.450355 0.892850i $$-0.351298\pi$$
0.450355 + 0.892850i $$0.351298\pi$$
$$734$$ 1.40409 0.0518259
$$735$$ 0 0
$$736$$ −7.24539 −0.267069
$$737$$ 55.5029 2.04448
$$738$$ −0.360235 −0.0132604
$$739$$ −38.2788 −1.40811 −0.704054 0.710146i $$-0.748629\pi$$
−0.704054 + 0.710146i $$0.748629\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0.113910 0.00418178
$$743$$ −40.0079 −1.46775 −0.733874 0.679286i $$-0.762290\pi$$
−0.733874 + 0.679286i $$0.762290\pi$$
$$744$$ −7.59074 −0.278290
$$745$$ 0 0
$$746$$ −4.41134 −0.161511
$$747$$ 1.87336 0.0685427
$$748$$ −53.0960 −1.94138
$$749$$ −4.34057 −0.158601
$$750$$ 0 0
$$751$$ 25.6020 0.934230 0.467115 0.884197i $$-0.345293\pi$$
0.467115 + 0.884197i $$0.345293\pi$$
$$752$$ 30.9750 1.12954
$$753$$ −10.8307 −0.394693
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −3.56890 −0.129800
$$757$$ 1.84848 0.0671840 0.0335920 0.999436i $$-0.489305\pi$$
0.0335920 + 0.999436i $$0.489305\pi$$
$$758$$ −1.19989 −0.0435820
$$759$$ 24.4055 0.885862
$$760$$ 0 0
$$761$$ −26.2124 −0.950199 −0.475099 0.879932i $$-0.657588\pi$$
−0.475099 + 0.879932i $$0.657588\pi$$
$$762$$ 1.13898 0.0412610
$$763$$ −3.73195 −0.135106
$$764$$ 3.93435 0.142340
$$765$$ 0 0
$$766$$ 1.24513 0.0449884
$$767$$ 0 0
$$768$$ 19.6459 0.708911
$$769$$ 44.3495 1.59928 0.799641 0.600478i $$-0.205023\pi$$
0.799641 + 0.600478i $$0.205023\pi$$
$$770$$ 0 0
$$771$$ 16.4081 0.590924
$$772$$ −44.5408 −1.60306
$$773$$ −23.2638 −0.836742 −0.418371 0.908276i $$-0.637399\pi$$
−0.418371 + 0.908276i $$0.637399\pi$$
$$774$$ 0.488829 0.0175706
$$775$$ 0 0
$$776$$ −2.17723 −0.0781580
$$777$$ −3.17508 −0.113905
$$778$$ −2.33251 −0.0836245
$$779$$ 8.46410 0.303258
$$780$$ 0 0
$$781$$ −68.7449 −2.45989
$$782$$ −3.15875 −0.112957
$$783$$ 15.9798 0.571071
$$784$$ −25.5794 −0.913552
$$785$$ 0 0
$$786$$ −0.0617531 −0.00220266
$$787$$ −47.9133 −1.70793 −0.853963 0.520334i $$-0.825808\pi$$
−0.853963 + 0.520334i $$0.825808\pi$$
$$788$$ 1.25502 0.0447081
$$789$$ −29.8548 −1.06286
$$790$$ 0 0
$$791$$ −6.09467 −0.216702
$$792$$ −2.04903 −0.0728090
$$793$$ 0 0
$$794$$ −6.15865 −0.218562
$$795$$ 0 0
$$796$$ −5.98567 −0.212156
$$797$$ 20.6952 0.733060 0.366530 0.930406i $$-0.380546\pi$$
0.366530 + 0.930406i $$0.380546\pi$$
$$798$$ −0.264893 −0.00937712
$$799$$ 42.2507 1.49472
$$800$$ 0 0
$$801$$ 1.41735 0.0500795
$$802$$ −4.94609 −0.174653
$$803$$ −52.0487 −1.83676
$$804$$ −32.2693 −1.13805
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 28.7361 1.01156
$$808$$ 10.8056 0.380138
$$809$$ 15.8915 0.558714 0.279357 0.960187i $$-0.409879\pi$$
0.279357 + 0.960187i $$0.409879\pi$$
$$810$$ 0 0
$$811$$ 23.8796 0.838525 0.419263 0.907865i $$-0.362289\pi$$
0.419263 + 0.907865i $$0.362289\pi$$
$$812$$ −1.88278 −0.0660727
$$813$$ −49.4423 −1.73402
$$814$$ −7.04768 −0.247021
$$815$$ 0 0
$$816$$ 30.0877 1.05328
$$817$$ −11.4856 −0.401829
$$818$$ 0.941956 0.0329347
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −15.9097 −0.555251 −0.277626 0.960689i $$-0.589547\pi$$
−0.277626 + 0.960689i $$0.589547\pi$$
$$822$$ 6.32174 0.220496
$$823$$ 14.8115 0.516295 0.258147 0.966106i $$-0.416888\pi$$
0.258147 + 0.966106i $$0.416888\pi$$
$$824$$ 13.0436 0.454397
$$825$$ 0 0
$$826$$ −0.197608 −0.00687566
$$827$$ 33.9498 1.18055 0.590275 0.807202i $$-0.299019\pi$$
0.590275 + 0.807202i $$0.299019\pi$$
$$828$$ 2.43472 0.0846122
$$829$$ 23.3146 0.809749 0.404875 0.914372i $$-0.367315\pi$$
0.404875 + 0.914372i $$0.367315\pi$$
$$830$$ 0 0
$$831$$ −42.4124 −1.47127
$$832$$ 0 0
$$833$$ −34.8910 −1.20890
$$834$$ −4.21363 −0.145906
$$835$$ 0 0
$$836$$ 23.7780 0.822380
$$837$$ 30.0726 1.03946
$$838$$ −3.89100 −0.134412
$$839$$ 14.7930 0.510710 0.255355 0.966847i $$-0.417808\pi$$
0.255355 + 0.966847i $$0.417808\pi$$
$$840$$ 0 0
$$841$$ −20.5698 −0.709305
$$842$$ 2.82929 0.0975038
$$843$$ −7.96523 −0.274337
$$844$$ 16.0050 0.550915
$$845$$ 0 0
$$846$$ 0.805291 0.0276865
$$847$$ −5.93273 −0.203851
$$848$$ 5.79422 0.198974
$$849$$ 20.1406 0.691223
$$850$$ 0 0
$$851$$ 16.9556 0.581231
$$852$$ 39.9681 1.36929
$$853$$ −16.3452 −0.559650 −0.279825 0.960051i $$-0.590276\pi$$
−0.279825 + 0.960051i $$0.590276\pi$$
$$854$$ 1.02944 0.0352268
$$855$$ 0 0
$$856$$ 11.3417 0.387651
$$857$$ 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 $$0.780291\pi$$
$$860$$ 0 0
$$861$$ 1.98418 0.0676207
$$862$$ 2.08669 0.0710728
$$863$$ 4.75058 0.161712 0.0808559 0.996726i $$-0.474235\pi$$
0.0808559 + 0.996726i $$0.474235\pi$$
$$864$$ 14.0450 0.477821
$$865$$ 0 0
$$866$$ −0.306745 −0.0104236
$$867$$ 13.8371 0.469931
$$868$$ −3.54324 −0.120265
$$869$$ 24.2726 0.823392
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 9.75140 0.330224
$$873$$ 1.10191 0.0372940
$$874$$ 1.41459 0.0478492
$$875$$ 0 0
$$876$$ 30.2610 1.02242
$$877$$ 2.25506 0.0761481 0.0380741 0.999275i $$-0.487878\pi$$
0.0380741 + 0.999275i $$0.487878\pi$$
$$878$$ −0.914293 −0.0308559
$$879$$ 27.0715 0.913098
$$880$$ 0 0
$$881$$ −2.98304 −0.100501 −0.0502507 0.998737i $$-0.516002\pi$$
−0.0502507 + 0.998737i $$0.516002\pi$$
$$882$$ −0.665017 −0.0223923
$$883$$ −28.2874 −0.951947 −0.475973 0.879460i $$-0.657904\pi$$
−0.475973 + 0.879460i $$0.657904\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 2.09705 0.0704517
$$887$$ 27.9816 0.939531 0.469766 0.882791i $$-0.344338\pi$$
0.469766 + 0.882791i $$0.344338\pi$$
$$888$$ 8.29633 0.278407
$$889$$ 1.07647 0.0361035
$$890$$ 0 0
$$891$$ −40.2286 −1.34771
$$892$$ −19.9940 −0.669448
$$893$$ −18.9212 −0.633172
$$894$$ −1.19890 −0.0400973
$$895$$ 0 0
$$896$$ −2.19681 −0.0733903
$$897$$ 0 0
$$898$$ 4.77095 0.159209
$$899$$ 15.8649 0.529124
$$900$$ 0 0
$$901$$ 7.90348 0.263303
$$902$$ 4.40426 0.146646
$$903$$ −2.69248 −0.0896002
$$904$$ 15.9251 0.529661
$$905$$ 0 0
$$906$$ 2.79877 0.0929829
$$907$$ −16.5520 −0.549600 −0.274800 0.961501i $$-0.588612\pi$$
−0.274800 + 0.961501i $$0.588612\pi$$
$$908$$ −13.7477 −0.456232
$$909$$ −5.46876 −0.181387
$$910$$ 0 0
$$911$$ 7.04863 0.233532 0.116766 0.993159i $$-0.462747\pi$$
0.116766 + 0.993159i $$0.462747\pi$$
$$912$$ −13.4742 −0.446175
$$913$$ −22.9039 −0.758007
$$914$$ −1.03749 −0.0343172
$$915$$ 0 0
$$916$$ 2.59383 0.0857026
$$917$$ −0.0583636 −0.00192734
$$918$$ 6.12316 0.202094
$$919$$ 16.5438 0.545728 0.272864 0.962053i $$-0.412029\pi$$
0.272864 + 0.962053i $$0.412029\pi$$
$$920$$ 0 0
$$921$$ −6.88766 −0.226956
$$922$$ −0.391374 −0.0128892
$$923$$ 0 0
$$924$$ 5.57412 0.183375
$$925$$ 0 0
$$926$$ −1.49431 −0.0491060
$$927$$ −6.60147 −0.216821
$$928$$ 7.40948 0.243228
$$929$$ −33.8367 −1.11015 −0.555074 0.831801i $$-0.687310\pi$$
−0.555074 + 0.831801i $$0.687310\pi$$
$$930$$ 0 0
$$931$$ 15.6253 0.512098
$$932$$ −2.43472 −0.0797517
$$933$$ −3.56063 −0.116570
$$934$$ 4.00652 0.131097
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 30.4606 0.995104 0.497552 0.867434i $$-0.334232\pi$$
0.497552 + 0.867434i $$0.334232\pi$$
$$938$$ 0.754153 0.0246240
$$939$$ −11.5356 −0.376450
$$940$$ 0 0
$$941$$ 38.2101 1.24561 0.622807 0.782375i $$-0.285992\pi$$
0.622807 + 0.782375i $$0.285992\pi$$
$$942$$ −5.77684 −0.188220
$$943$$ −10.5960 −0.345052
$$944$$ −10.0516 −0.327153
$$945$$ 0 0
$$946$$ −5.97647 −0.194312
$$947$$ 52.4482 1.70434 0.852169 0.523266i $$-0.175287\pi$$
0.852169 + 0.523266i $$0.175287\pi$$
$$948$$ −14.1120 −0.458338
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −0.514693 −0.0166901
$$952$$ −1.46074 −0.0473427
$$953$$ −39.7500 −1.28763 −0.643814 0.765182i $$-0.722649\pi$$
−0.643814 + 0.765182i $$0.722649\pi$$
$$954$$ 0.150639 0.00487711
$$955$$ 0 0
$$956$$ −19.4043 −0.627580
$$957$$ −24.9581 −0.806782
$$958$$ −7.72874 −0.249704
$$959$$ 5.97475 0.192935
$$960$$ 0 0
$$961$$ −1.14359 −0.0368901
$$962$$ 0 0
$$963$$ −5.74011 −0.184972
$$964$$ −44.0837 −1.41984
$$965$$ 0 0
$$966$$ 0.331612 0.0106694
$$967$$ 25.7857 0.829214 0.414607 0.910001i $$-0.363919\pi$$
0.414607 + 0.910001i $$0.363919\pi$$
$$968$$ 15.5019 0.498251
$$969$$ −18.3792 −0.590424
$$970$$ 0 0
$$971$$ −55.5252 −1.78189 −0.890945 0.454111i $$-0.849957\pi$$
−0.890945 + 0.454111i $$0.849957\pi$$
$$972$$ −8.83634 −0.283426
$$973$$ −3.98235 −0.127668
$$974$$ 2.26365 0.0725321
$$975$$ 0 0
$$976$$ 52.3642 1.67614
$$977$$ −40.5161 −1.29622 −0.648112 0.761545i $$-0.724441\pi$$
−0.648112 + 0.761545i $$0.724441\pi$$
$$978$$ −6.25998 −0.200172
$$979$$ −17.3286 −0.553824
$$980$$ 0 0
$$981$$ −4.93525 −0.157570
$$982$$ −2.05011 −0.0654218
$$983$$ 34.8059 1.11014 0.555068 0.831805i $$-0.312692\pi$$
0.555068 + 0.831805i $$0.312692\pi$$
$$984$$ −5.18457 −0.165278
$$985$$ 0 0
$$986$$ 3.23029 0.102873
$$987$$ −4.43555 −0.141185
$$988$$ 0 0
$$989$$ 14.3784 0.457208
$$990$$ 0 0
$$991$$ 43.8855 1.39407 0.697034 0.717038i $$-0.254503\pi$$
0.697034 + 0.717038i $$0.254503\pi$$
$$992$$ 13.9440 0.442723
$$993$$ −26.6145 −0.844584
$$994$$ −0.934079 −0.0296272
$$995$$ 0 0
$$996$$ 13.3163 0.421942
$$997$$ 5.49137 0.173914 0.0869568 0.996212i $$-0.472286\pi$$
0.0869568 + 0.996212i $$0.472286\pi$$
$$998$$ 5.25976 0.166495
$$999$$ −32.8680 −1.03990
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 4225.2.a.bl.1.2 4
5.4 even 2 845.2.a.l.1.3 4
13.6 odd 12 325.2.n.d.101.3 8
13.11 odd 12 325.2.n.d.251.3 8
13.12 even 2 4225.2.a.bi.1.3 4
15.14 odd 2 7605.2.a.cj.1.2 4
65.4 even 6 845.2.e.m.146.3 8
65.9 even 6 845.2.e.n.146.2 8
65.19 odd 12 65.2.m.a.36.2 8
65.24 odd 12 65.2.m.a.56.2 yes 8
65.29 even 6 845.2.e.n.191.2 8
65.32 even 12 325.2.m.b.49.3 8
65.34 odd 4 845.2.c.g.506.5 8
65.37 even 12 325.2.m.c.199.2 8
65.44 odd 4 845.2.c.g.506.4 8
65.49 even 6 845.2.e.m.191.3 8
65.54 odd 12 845.2.m.g.316.3 8
65.58 even 12 325.2.m.c.49.2 8
65.59 odd 12 845.2.m.g.361.3 8
65.63 even 12 325.2.m.b.199.3 8
65.64 even 2 845.2.a.m.1.2 4
195.89 even 12 585.2.bu.c.316.3 8
195.149 even 12 585.2.bu.c.361.3 8
195.194 odd 2 7605.2.a.cf.1.3 4
260.19 even 12 1040.2.da.b.881.2 8
260.219 even 12 1040.2.da.b.641.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.2 8 65.19 odd 12
65.2.m.a.56.2 yes 8 65.24 odd 12
325.2.m.b.49.3 8 65.32 even 12
325.2.m.b.199.3 8 65.63 even 12
325.2.m.c.49.2 8 65.58 even 12
325.2.m.c.199.2 8 65.37 even 12
325.2.n.d.101.3 8 13.6 odd 12
325.2.n.d.251.3 8 13.11 odd 12
585.2.bu.c.316.3 8 195.89 even 12
585.2.bu.c.361.3 8 195.149 even 12
845.2.a.l.1.3 4 5.4 even 2
845.2.a.m.1.2 4 65.64 even 2
845.2.c.g.506.4 8 65.44 odd 4
845.2.c.g.506.5 8 65.34 odd 4
845.2.e.m.146.3 8 65.4 even 6
845.2.e.m.191.3 8 65.49 even 6
845.2.e.n.146.2 8 65.9 even 6
845.2.e.n.191.2 8 65.29 even 6
845.2.m.g.316.3 8 65.54 odd 12
845.2.m.g.361.3 8 65.59 odd 12
1040.2.da.b.641.2 8 260.219 even 12
1040.2.da.b.881.2 8 260.19 even 12
4225.2.a.bi.1.3 4 13.12 even 2
4225.2.a.bl.1.2 4 1.1 even 1 trivial
7605.2.a.cf.1.3 4 195.194 odd 2
7605.2.a.cj.1.2 4 15.14 odd 2