# Properties

 Label 4225.2.a.bq.1.6 Level $4225$ Weight $2$ Character 4225.1 Self dual yes Analytic conductor $33.737$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# 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: $$6$$ Coefficient field: 6.6.199374400.1 Defining polynomial: $$x^{6} - 8 x^{4} + 10 x^{2} - 1$$ 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 1.6 Root $$2.54574$$ of defining polynomial Character $$\chi$$ $$=$$ 4225.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.54574 q^{2} -2.15293 q^{3} +4.48079 q^{4} -5.48079 q^{6} +2.93855 q^{7} +6.31544 q^{8} +1.63509 q^{9} +O(q^{10})$$ $$q+2.54574 q^{2} -2.15293 q^{3} +4.48079 q^{4} -5.48079 q^{6} +2.93855 q^{7} +6.31544 q^{8} +1.63509 q^{9} +0.635089 q^{11} -9.64680 q^{12} +7.48079 q^{14} +7.11588 q^{16} -1.22396 q^{17} +4.16251 q^{18} -1.36491 q^{19} -6.32648 q^{21} +1.61677 q^{22} +2.15293 q^{23} -13.5967 q^{24} +2.93855 q^{27} +13.1670 q^{28} +3.00000 q^{29} +8.96157 q^{31} +5.48429 q^{32} -1.36730 q^{33} -3.11588 q^{34} +7.32648 q^{36} -1.22396 q^{37} -3.47471 q^{38} +9.96157 q^{41} -16.1056 q^{42} -1.36730 q^{43} +2.84570 q^{44} +5.48079 q^{46} +6.16379 q^{47} -15.3200 q^{48} +1.63509 q^{49} +2.63509 q^{51} +0.642285 q^{53} +7.48079 q^{54} +18.5582 q^{56} +2.93855 q^{57} +7.63722 q^{58} -7.59666 q^{59} -2.27018 q^{61} +22.8138 q^{62} +4.80479 q^{63} -0.270178 q^{64} -3.48079 q^{66} -8.03003 q^{67} -5.48429 q^{68} -4.63509 q^{69} -2.63509 q^{71} +10.3263 q^{72} +10.3263 q^{73} -3.11588 q^{74} -6.11588 q^{76} +1.86624 q^{77} +1.03843 q^{79} -11.2318 q^{81} +25.3596 q^{82} -11.8452 q^{83} -28.3476 q^{84} -3.48079 q^{86} -6.45878 q^{87} +4.01086 q^{88} +12.5582 q^{89} +9.64680 q^{92} -19.2936 q^{93} +15.6914 q^{94} -11.8073 q^{96} +14.7838 q^{97} +4.16251 q^{98} +1.03843 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 4q^{4} - 10q^{6} + 6q^{9} + O(q^{10})$$ $$6q + 4q^{4} - 10q^{6} + 6q^{9} + 22q^{14} + 16q^{16} - 12q^{19} + 4q^{21} - 32q^{24} + 18q^{29} + 8q^{31} + 8q^{34} + 2q^{36} + 14q^{41} - 2q^{44} + 10q^{46} + 6q^{49} + 12q^{51} + 22q^{54} + 16q^{56} + 4q^{59} - 6q^{61} + 6q^{64} + 2q^{66} - 24q^{69} - 12q^{71} + 8q^{74} - 10q^{76} + 52q^{79} - 14q^{81} - 90q^{84} + 2q^{86} - 20q^{89} + 56q^{94} - 6q^{96} + 52q^{99} + O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.54574 1.80011 0.900055 0.435777i $$-0.143526\pi$$
0.900055 + 0.435777i $$0.143526\pi$$
$$3$$ −2.15293 −1.24299 −0.621496 0.783417i $$-0.713475\pi$$
−0.621496 + 0.783417i $$0.713475\pi$$
$$4$$ 4.48079 2.24039
$$5$$ 0 0
$$6$$ −5.48079 −2.23752
$$7$$ 2.93855 1.11067 0.555334 0.831627i $$-0.312590\pi$$
0.555334 + 0.831627i $$0.312590\pi$$
$$8$$ 6.31544 2.23284
$$9$$ 1.63509 0.545030
$$10$$ 0 0
$$11$$ 0.635089 0.191487 0.0957433 0.995406i $$-0.469477\pi$$
0.0957433 + 0.995406i $$0.469477\pi$$
$$12$$ −9.64680 −2.78479
$$13$$ 0 0
$$14$$ 7.48079 1.99932
$$15$$ 0 0
$$16$$ 7.11588 1.77897
$$17$$ −1.22396 −0.296853 −0.148427 0.988923i $$-0.547421\pi$$
−0.148427 + 0.988923i $$0.547421\pi$$
$$18$$ 4.16251 0.981113
$$19$$ −1.36491 −0.313132 −0.156566 0.987667i $$-0.550042\pi$$
−0.156566 + 0.987667i $$0.550042\pi$$
$$20$$ 0 0
$$21$$ −6.32648 −1.38055
$$22$$ 1.61677 0.344697
$$23$$ 2.15293 0.448916 0.224458 0.974484i $$-0.427939\pi$$
0.224458 + 0.974484i $$0.427939\pi$$
$$24$$ −13.5967 −2.77541
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 2.93855 0.565525
$$28$$ 13.1670 2.48833
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 8.96157 1.60955 0.804773 0.593583i $$-0.202287\pi$$
0.804773 + 0.593583i $$0.202287\pi$$
$$32$$ 5.48429 0.969495
$$33$$ −1.36730 −0.238016
$$34$$ −3.11588 −0.534368
$$35$$ 0 0
$$36$$ 7.32648 1.22108
$$37$$ −1.22396 −0.201217 −0.100609 0.994926i $$-0.532079\pi$$
−0.100609 + 0.994926i $$0.532079\pi$$
$$38$$ −3.47471 −0.563672
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.96157 1.55574 0.777868 0.628427i $$-0.216301\pi$$
0.777868 + 0.628427i $$0.216301\pi$$
$$42$$ −16.1056 −2.48514
$$43$$ −1.36730 −0.208511 −0.104256 0.994551i $$-0.533246\pi$$
−0.104256 + 0.994551i $$0.533246\pi$$
$$44$$ 2.84570 0.429005
$$45$$ 0 0
$$46$$ 5.48079 0.808098
$$47$$ 6.16379 0.899081 0.449540 0.893260i $$-0.351588\pi$$
0.449540 + 0.893260i $$0.351588\pi$$
$$48$$ −15.3200 −2.21124
$$49$$ 1.63509 0.233584
$$50$$ 0 0
$$51$$ 2.63509 0.368986
$$52$$ 0 0
$$53$$ 0.642285 0.0882246 0.0441123 0.999027i $$-0.485954\pi$$
0.0441123 + 0.999027i $$0.485954\pi$$
$$54$$ 7.48079 1.01801
$$55$$ 0 0
$$56$$ 18.5582 2.47995
$$57$$ 2.93855 0.389221
$$58$$ 7.63722 1.00282
$$59$$ −7.59666 −0.989001 −0.494501 0.869177i $$-0.664649\pi$$
−0.494501 + 0.869177i $$0.664649\pi$$
$$60$$ 0 0
$$61$$ −2.27018 −0.290666 −0.145333 0.989383i $$-0.546425\pi$$
−0.145333 + 0.989383i $$0.546425\pi$$
$$62$$ 22.8138 2.89736
$$63$$ 4.80479 0.605347
$$64$$ −0.270178 −0.0337722
$$65$$ 0 0
$$66$$ −3.48079 −0.428455
$$67$$ −8.03003 −0.981024 −0.490512 0.871434i $$-0.663190\pi$$
−0.490512 + 0.871434i $$0.663190\pi$$
$$68$$ −5.48429 −0.665068
$$69$$ −4.63509 −0.557999
$$70$$ 0 0
$$71$$ −2.63509 −0.312728 −0.156364 0.987700i $$-0.549977\pi$$
−0.156364 + 0.987700i $$0.549977\pi$$
$$72$$ 10.3263 1.21697
$$73$$ 10.3263 1.20860 0.604301 0.796756i $$-0.293453\pi$$
0.604301 + 0.796756i $$0.293453\pi$$
$$74$$ −3.11588 −0.362213
$$75$$ 0 0
$$76$$ −6.11588 −0.701539
$$77$$ 1.86624 0.212678
$$78$$ 0 0
$$79$$ 1.03843 0.116832 0.0584161 0.998292i $$-0.481395\pi$$
0.0584161 + 0.998292i $$0.481395\pi$$
$$80$$ 0 0
$$81$$ −11.2318 −1.24797
$$82$$ 25.3596 2.80050
$$83$$ −11.8452 −1.30018 −0.650092 0.759855i $$-0.725270\pi$$
−0.650092 + 0.759855i $$0.725270\pi$$
$$84$$ −28.3476 −3.09298
$$85$$ 0 0
$$86$$ −3.48079 −0.375343
$$87$$ −6.45878 −0.692454
$$88$$ 4.01086 0.427559
$$89$$ 12.5582 1.33117 0.665585 0.746322i $$-0.268182\pi$$
0.665585 + 0.746322i $$0.268182\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 9.64680 1.00575
$$93$$ −19.2936 −2.00065
$$94$$ 15.6914 1.61844
$$95$$ 0 0
$$96$$ −11.8073 −1.20507
$$97$$ 14.7838 1.50107 0.750534 0.660832i $$-0.229797\pi$$
0.750534 + 0.660832i $$0.229797\pi$$
$$98$$ 4.16251 0.420477
$$99$$ 1.03843 0.104366
$$100$$ 0 0
$$101$$ 13.2318 1.31661 0.658304 0.752752i $$-0.271274\pi$$
0.658304 + 0.752752i $$0.271274\pi$$
$$102$$ 6.70825 0.664216
$$103$$ 10.9686 1.08077 0.540383 0.841419i $$-0.318279\pi$$
0.540383 + 0.841419i $$0.318279\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 1.63509 0.158814
$$107$$ −10.6736 −1.03186 −0.515928 0.856632i $$-0.672553\pi$$
−0.515928 + 0.856632i $$0.672553\pi$$
$$108$$ 13.1670 1.26700
$$109$$ 3.27018 0.313226 0.156613 0.987660i $$-0.449942\pi$$
0.156613 + 0.987660i $$0.449942\pi$$
$$110$$ 0 0
$$111$$ 2.63509 0.250112
$$112$$ 20.9104 1.97584
$$113$$ −5.52981 −0.520201 −0.260100 0.965582i $$-0.583756\pi$$
−0.260100 + 0.965582i $$0.583756\pi$$
$$114$$ 7.48079 0.700640
$$115$$ 0 0
$$116$$ 13.4424 1.24809
$$117$$ 0 0
$$118$$ −19.3391 −1.78031
$$119$$ −3.59666 −0.329705
$$120$$ 0 0
$$121$$ −10.5967 −0.963333
$$122$$ −5.77928 −0.523231
$$123$$ −21.4465 −1.93377
$$124$$ 40.1549 3.60602
$$125$$ 0 0
$$126$$ 12.2318 1.08969
$$127$$ −17.2317 −1.52907 −0.764534 0.644584i $$-0.777031\pi$$
−0.764534 + 0.644584i $$0.777031\pi$$
$$128$$ −11.6564 −1.03029
$$129$$ 2.94369 0.259178
$$130$$ 0 0
$$131$$ 10.0000 0.873704 0.436852 0.899533i $$-0.356093\pi$$
0.436852 + 0.899533i $$0.356093\pi$$
$$132$$ −6.12658 −0.533250
$$133$$ −4.01086 −0.347786
$$134$$ −20.4424 −1.76595
$$135$$ 0 0
$$136$$ −7.72982 −0.662827
$$137$$ 8.67231 0.740926 0.370463 0.928847i $$-0.379199\pi$$
0.370463 + 0.928847i $$0.379199\pi$$
$$138$$ −11.7997 −1.00446
$$139$$ 14.3265 1.21516 0.607578 0.794260i $$-0.292141\pi$$
0.607578 + 0.794260i $$0.292141\pi$$
$$140$$ 0 0
$$141$$ −13.2702 −1.11755
$$142$$ −6.70825 −0.562944
$$143$$ 0 0
$$144$$ 11.6351 0.969591
$$145$$ 0 0
$$146$$ 26.2881 2.17562
$$147$$ −3.52022 −0.290343
$$148$$ −5.48429 −0.450806
$$149$$ 17.1549 1.40538 0.702692 0.711494i $$-0.251981\pi$$
0.702692 + 0.711494i $$0.251981\pi$$
$$150$$ 0 0
$$151$$ 21.3828 1.74011 0.870053 0.492957i $$-0.164084\pi$$
0.870053 + 0.492957i $$0.164084\pi$$
$$152$$ −8.62001 −0.699175
$$153$$ −2.00128 −0.161794
$$154$$ 4.75096 0.382844
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −18.3646 −1.46566 −0.732829 0.680413i $$-0.761800\pi$$
−0.732829 + 0.680413i $$0.761800\pi$$
$$158$$ 2.64356 0.210311
$$159$$ −1.38279 −0.109662
$$160$$ 0 0
$$161$$ 6.32648 0.498597
$$162$$ −28.5931 −2.24649
$$163$$ 4.01086 0.314155 0.157078 0.987586i $$-0.449793\pi$$
0.157078 + 0.987586i $$0.449793\pi$$
$$164$$ 44.6357 3.48546
$$165$$ 0 0
$$166$$ −30.1549 −2.34047
$$167$$ −2.93855 −0.227392 −0.113696 0.993516i $$-0.536269\pi$$
−0.113696 + 0.993516i $$0.536269\pi$$
$$168$$ −39.9545 −3.08256
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −2.23175 −0.170666
$$172$$ −6.12658 −0.467147
$$173$$ 1.36730 0.103954 0.0519769 0.998648i $$-0.483448\pi$$
0.0519769 + 0.998648i $$0.483448\pi$$
$$174$$ −16.4424 −1.24649
$$175$$ 0 0
$$176$$ 4.51921 0.340649
$$177$$ 16.3550 1.22932
$$178$$ 31.9700 2.39625
$$179$$ 7.78613 0.581963 0.290981 0.956729i $$-0.406018\pi$$
0.290981 + 0.956729i $$0.406018\pi$$
$$180$$ 0 0
$$181$$ −3.86684 −0.287420 −0.143710 0.989620i $$-0.545903\pi$$
−0.143710 + 0.989620i $$0.545903\pi$$
$$182$$ 0 0
$$183$$ 4.88752 0.361296
$$184$$ 13.5967 1.00236
$$185$$ 0 0
$$186$$ −49.1165 −3.60139
$$187$$ −0.777322 −0.0568434
$$188$$ 27.6186 2.01429
$$189$$ 8.63509 0.628110
$$190$$ 0 0
$$191$$ 4.94369 0.357713 0.178857 0.983875i $$-0.442760\pi$$
0.178857 + 0.983875i $$0.442760\pi$$
$$192$$ 0.581673 0.0419786
$$193$$ −4.95644 −0.356772 −0.178386 0.983961i $$-0.557088\pi$$
−0.178386 + 0.983961i $$0.557088\pi$$
$$194$$ 37.6357 2.70208
$$195$$ 0 0
$$196$$ 7.32648 0.523320
$$197$$ −6.74546 −0.480594 −0.240297 0.970699i $$-0.577245\pi$$
−0.240297 + 0.970699i $$0.577245\pi$$
$$198$$ 2.64356 0.187870
$$199$$ 5.17544 0.366878 0.183439 0.983031i $$-0.441277\pi$$
0.183439 + 0.983031i $$0.441277\pi$$
$$200$$ 0 0
$$201$$ 17.2881 1.21941
$$202$$ 33.6846 2.37004
$$203$$ 8.81566 0.618738
$$204$$ 11.8073 0.826674
$$205$$ 0 0
$$206$$ 27.9231 1.94550
$$207$$ 3.52022 0.244673
$$208$$ 0 0
$$209$$ −0.866840 −0.0599606
$$210$$ 0 0
$$211$$ −14.0179 −0.965031 −0.482515 0.875888i $$-0.660277\pi$$
−0.482515 + 0.875888i $$0.660277\pi$$
$$212$$ 2.87794 0.197658
$$213$$ 5.67315 0.388718
$$214$$ −27.1722 −1.85745
$$215$$ 0 0
$$216$$ 18.5582 1.26273
$$217$$ 26.3341 1.78767
$$218$$ 8.32502 0.563841
$$219$$ −22.2318 −1.50228
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 6.70825 0.450228
$$223$$ 0.00830491 0.000556138 0 0.000278069 1.00000i $$-0.499911\pi$$
0.000278069 1.00000i $$0.499911\pi$$
$$224$$ 16.1159 1.07679
$$225$$ 0 0
$$226$$ −14.0774 −0.936418
$$227$$ −11.2636 −0.747589 −0.373795 0.927511i $$-0.621944\pi$$
−0.373795 + 0.927511i $$0.621944\pi$$
$$228$$ 13.1670 0.872008
$$229$$ 16.5404 1.09302 0.546509 0.837453i $$-0.315957\pi$$
0.546509 + 0.837453i $$0.315957\pi$$
$$230$$ 0 0
$$231$$ −4.01788 −0.264357
$$232$$ 18.9463 1.24389
$$233$$ 6.94941 0.455271 0.227636 0.973746i $$-0.426900\pi$$
0.227636 + 0.973746i $$0.426900\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −34.0390 −2.21575
$$237$$ −2.23566 −0.145221
$$238$$ −9.15616 −0.593506
$$239$$ −4.00000 −0.258738 −0.129369 0.991596i $$-0.541295\pi$$
−0.129369 + 0.991596i $$0.541295\pi$$
$$240$$ 0 0
$$241$$ −19.7721 −1.27363 −0.636817 0.771015i $$-0.719749\pi$$
−0.636817 + 0.771015i $$0.719749\pi$$
$$242$$ −26.9763 −1.73410
$$243$$ 15.3655 0.985695
$$244$$ −10.1722 −0.651207
$$245$$ 0 0
$$246$$ −54.5973 −3.48099
$$247$$ 0 0
$$248$$ 56.5962 3.59386
$$249$$ 25.5019 1.61612
$$250$$ 0 0
$$251$$ 3.67352 0.231870 0.115935 0.993257i $$-0.463014\pi$$
0.115935 + 0.993257i $$0.463014\pi$$
$$252$$ 21.5293 1.35622
$$253$$ 1.36730 0.0859614
$$254$$ −43.8674 −2.75249
$$255$$ 0 0
$$256$$ −29.1338 −1.82086
$$257$$ −13.2648 −0.827439 −0.413719 0.910404i $$-0.635771\pi$$
−0.413719 + 0.910404i $$0.635771\pi$$
$$258$$ 7.49387 0.466548
$$259$$ −3.59666 −0.223486
$$260$$ 0 0
$$261$$ 4.90527 0.303628
$$262$$ 25.4574 1.57276
$$263$$ −30.2705 −1.86656 −0.933279 0.359152i $$-0.883066\pi$$
−0.933279 + 0.359152i $$0.883066\pi$$
$$264$$ −8.63509 −0.531453
$$265$$ 0 0
$$266$$ −10.2106 −0.626053
$$267$$ −27.0369 −1.65463
$$268$$ −35.9809 −2.19788
$$269$$ −22.2496 −1.35658 −0.678292 0.734792i $$-0.737280\pi$$
−0.678292 + 0.734792i $$0.737280\pi$$
$$270$$ 0 0
$$271$$ 11.8284 0.718525 0.359262 0.933237i $$-0.383028\pi$$
0.359262 + 0.933237i $$0.383028\pi$$
$$272$$ −8.70953 −0.528093
$$273$$ 0 0
$$274$$ 22.0774 1.33375
$$275$$ 0 0
$$276$$ −20.7688 −1.25014
$$277$$ −16.7851 −1.00852 −0.504259 0.863553i $$-0.668234\pi$$
−0.504259 + 0.863553i $$0.668234\pi$$
$$278$$ 36.4715 2.18741
$$279$$ 14.6530 0.877250
$$280$$ 0 0
$$281$$ 10.5967 0.632144 0.316072 0.948735i $$-0.397636\pi$$
0.316072 + 0.948735i $$0.397636\pi$$
$$282$$ −33.7824 −2.01171
$$283$$ 8.81566 0.524036 0.262018 0.965063i $$-0.415612\pi$$
0.262018 + 0.965063i $$0.415612\pi$$
$$284$$ −11.8073 −0.700633
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 29.2726 1.72791
$$288$$ 8.96730 0.528403
$$289$$ −15.5019 −0.911878
$$290$$ 0 0
$$291$$ −31.8284 −1.86581
$$292$$ 46.2699 2.70774
$$293$$ −28.2526 −1.65053 −0.825267 0.564742i $$-0.808976\pi$$
−0.825267 + 0.564742i $$0.808976\pi$$
$$294$$ −8.96157 −0.522650
$$295$$ 0 0
$$296$$ −7.72982 −0.449287
$$297$$ 1.86624 0.108290
$$298$$ 43.6719 2.52984
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −4.01788 −0.231587
$$302$$ 54.4350 3.13238
$$303$$ −28.4870 −1.63653
$$304$$ −9.71254 −0.557052
$$305$$ 0 0
$$306$$ −5.09473 −0.291247
$$307$$ 12.7219 0.726077 0.363039 0.931774i $$-0.381739\pi$$
0.363039 + 0.931774i $$0.381739\pi$$
$$308$$ 8.36223 0.476482
$$309$$ −23.6145 −1.34338
$$310$$ 0 0
$$311$$ 27.9231 1.58338 0.791688 0.610925i $$-0.209202\pi$$
0.791688 + 0.610925i $$0.209202\pi$$
$$312$$ 0 0
$$313$$ −24.5807 −1.38938 −0.694692 0.719307i $$-0.744460\pi$$
−0.694692 + 0.719307i $$0.744460\pi$$
$$314$$ −46.7516 −2.63834
$$315$$ 0 0
$$316$$ 4.65297 0.261750
$$317$$ −0.234377 −0.0131639 −0.00658196 0.999978i $$-0.502095\pi$$
−0.00658196 + 0.999978i $$0.502095\pi$$
$$318$$ −3.52022 −0.197404
$$319$$ 1.90527 0.106674
$$320$$ 0 0
$$321$$ 22.9795 1.28259
$$322$$ 16.1056 0.897529
$$323$$ 1.67059 0.0929543
$$324$$ −50.3271 −2.79595
$$325$$ 0 0
$$326$$ 10.2106 0.565513
$$327$$ −7.04045 −0.389338
$$328$$ 62.9117 3.47372
$$329$$ 18.1126 0.998581
$$330$$ 0 0
$$331$$ 18.3265 1.00731 0.503657 0.863904i $$-0.331987\pi$$
0.503657 + 0.863904i $$0.331987\pi$$
$$332$$ −53.0760 −2.91292
$$333$$ −2.00128 −0.109669
$$334$$ −7.48079 −0.409330
$$335$$ 0 0
$$336$$ −45.0185 −2.45596
$$337$$ 21.2949 1.16001 0.580003 0.814614i $$-0.303051\pi$$
0.580003 + 0.814614i $$0.303051\pi$$
$$338$$ 0 0
$$339$$ 11.9053 0.646605
$$340$$ 0 0
$$341$$ 5.69140 0.308206
$$342$$ −5.68146 −0.307218
$$343$$ −15.7651 −0.851234
$$344$$ −8.63509 −0.465573
$$345$$ 0 0
$$346$$ 3.48079 0.187128
$$347$$ 3.81521 0.204811 0.102406 0.994743i $$-0.467346\pi$$
0.102406 + 0.994743i $$0.467346\pi$$
$$348$$ −28.9404 −1.55137
$$349$$ −24.3265 −1.30217 −0.651083 0.759006i $$-0.725685\pi$$
−0.651083 + 0.759006i $$0.725685\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 3.48301 0.185645
$$353$$ −27.0591 −1.44021 −0.720104 0.693866i $$-0.755906\pi$$
−0.720104 + 0.693866i $$0.755906\pi$$
$$354$$ 41.6357 2.21291
$$355$$ 0 0
$$356$$ 56.2708 2.98235
$$357$$ 7.74335 0.409821
$$358$$ 19.8215 1.04760
$$359$$ −27.0039 −1.42521 −0.712605 0.701566i $$-0.752485\pi$$
−0.712605 + 0.701566i $$0.752485\pi$$
$$360$$ 0 0
$$361$$ −17.1370 −0.901948
$$362$$ −9.84396 −0.517387
$$363$$ 22.8138 1.19742
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 12.4424 0.650373
$$367$$ 6.94111 0.362323 0.181161 0.983453i $$-0.442014\pi$$
0.181161 + 0.983453i $$0.442014\pi$$
$$368$$ 15.3200 0.798608
$$369$$ 16.2881 0.847922
$$370$$ 0 0
$$371$$ 1.88739 0.0979882
$$372$$ −86.4505 −4.48225
$$373$$ −2.31288 −0.119756 −0.0598781 0.998206i $$-0.519071\pi$$
−0.0598781 + 0.998206i $$0.519071\pi$$
$$374$$ −1.97886 −0.102324
$$375$$ 0 0
$$376$$ 38.9270 2.00751
$$377$$ 0 0
$$378$$ 21.9827 1.13067
$$379$$ −5.17544 −0.265845 −0.132922 0.991126i $$-0.542436\pi$$
−0.132922 + 0.991126i $$0.542436\pi$$
$$380$$ 0 0
$$381$$ 37.0986 1.90062
$$382$$ 12.5854 0.643923
$$383$$ −20.6609 −1.05572 −0.527861 0.849331i $$-0.677006\pi$$
−0.527861 + 0.849331i $$0.677006\pi$$
$$384$$ 25.0953 1.28064
$$385$$ 0 0
$$386$$ −12.6178 −0.642229
$$387$$ −2.23566 −0.113645
$$388$$ 66.2430 3.36298
$$389$$ 19.7477 1.00125 0.500624 0.865665i $$-0.333104\pi$$
0.500624 + 0.865665i $$0.333104\pi$$
$$390$$ 0 0
$$391$$ −2.63509 −0.133262
$$392$$ 10.3263 0.521557
$$393$$ −21.5293 −1.08601
$$394$$ −17.1722 −0.865122
$$395$$ 0 0
$$396$$ 4.65297 0.233820
$$397$$ −9.38902 −0.471222 −0.235611 0.971847i $$-0.575709\pi$$
−0.235611 + 0.971847i $$0.575709\pi$$
$$398$$ 13.1753 0.660420
$$399$$ 8.63509 0.432295
$$400$$ 0 0
$$401$$ −24.5019 −1.22357 −0.611784 0.791025i $$-0.709548\pi$$
−0.611784 + 0.791025i $$0.709548\pi$$
$$402$$ 44.0109 2.19506
$$403$$ 0 0
$$404$$ 59.2887 2.94972
$$405$$ 0 0
$$406$$ 22.4424 1.11380
$$407$$ −0.777322 −0.0385304
$$408$$ 16.6417 0.823889
$$409$$ −36.1165 −1.78584 −0.892922 0.450211i $$-0.851349\pi$$
−0.892922 + 0.450211i $$0.851349\pi$$
$$410$$ 0 0
$$411$$ −18.6708 −0.920965
$$412$$ 49.1479 2.42134
$$413$$ −22.3232 −1.09845
$$414$$ 8.96157 0.440437
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −30.8439 −1.51043
$$418$$ −2.20675 −0.107936
$$419$$ −6.86684 −0.335467 −0.167734 0.985832i $$-0.553645\pi$$
−0.167734 + 0.985832i $$0.553645\pi$$
$$420$$ 0 0
$$421$$ −33.9795 −1.65606 −0.828029 0.560686i $$-0.810538\pi$$
−0.828029 + 0.560686i $$0.810538\pi$$
$$422$$ −35.6859 −1.73716
$$423$$ 10.0783 0.490026
$$424$$ 4.05631 0.196992
$$425$$ 0 0
$$426$$ 14.4424 0.699735
$$427$$ −6.67104 −0.322834
$$428$$ −47.8261 −2.31176
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 16.2496 0.782717 0.391359 0.920238i $$-0.372005\pi$$
0.391359 + 0.920238i $$0.372005\pi$$
$$432$$ 20.9104 1.00605
$$433$$ −0.256261 −0.0123151 −0.00615756 0.999981i $$-0.501960\pi$$
−0.00615756 + 0.999981i $$0.501960\pi$$
$$434$$ 67.0396 3.21800
$$435$$ 0 0
$$436$$ 14.6530 0.701750
$$437$$ −2.93855 −0.140570
$$438$$ −56.5962 −2.70427
$$439$$ −7.59666 −0.362569 −0.181284 0.983431i $$-0.558025\pi$$
−0.181284 + 0.983431i $$0.558025\pi$$
$$440$$ 0 0
$$441$$ 2.67352 0.127310
$$442$$ 0 0
$$443$$ −4.32246 −0.205366 −0.102683 0.994714i $$-0.532743\pi$$
−0.102683 + 0.994714i $$0.532743\pi$$
$$444$$ 11.8073 0.560348
$$445$$ 0 0
$$446$$ 0.0211421 0.00100111
$$447$$ −36.9332 −1.74688
$$448$$ −0.793931 −0.0375097
$$449$$ −3.28806 −0.155173 −0.0775865 0.996986i $$-0.524721\pi$$
−0.0775865 + 0.996986i $$0.524721\pi$$
$$450$$ 0 0
$$451$$ 6.32648 0.297903
$$452$$ −24.7779 −1.16545
$$453$$ −46.0356 −2.16294
$$454$$ −28.6741 −1.34574
$$455$$ 0 0
$$456$$ 18.5582 0.869069
$$457$$ −15.4261 −0.721602 −0.360801 0.932643i $$-0.617497\pi$$
−0.360801 + 0.932643i $$0.617497\pi$$
$$458$$ 42.1074 1.96755
$$459$$ −3.59666 −0.167878
$$460$$ 0 0
$$461$$ −25.8847 −1.20557 −0.602786 0.797903i $$-0.705943\pi$$
−0.602786 + 0.797903i $$0.705943\pi$$
$$462$$ −10.2285 −0.475872
$$463$$ 7.04045 0.327197 0.163599 0.986527i $$-0.447690\pi$$
0.163599 + 0.986527i $$0.447690\pi$$
$$464$$ 21.3476 0.991039
$$465$$ 0 0
$$466$$ 17.6914 0.819538
$$467$$ −18.8113 −0.870482 −0.435241 0.900314i $$-0.643337\pi$$
−0.435241 + 0.900314i $$0.643337\pi$$
$$468$$ 0 0
$$469$$ −23.5967 −1.08959
$$470$$ 0 0
$$471$$ 39.5377 1.82180
$$472$$ −47.9762 −2.20828
$$473$$ −0.868356 −0.0399271
$$474$$ −5.69140 −0.261414
$$475$$ 0 0
$$476$$ −16.1159 −0.738670
$$477$$ 1.05019 0.0480850
$$478$$ −10.1830 −0.465758
$$479$$ 19.4775 0.889951 0.444975 0.895543i $$-0.353212\pi$$
0.444975 + 0.895543i $$0.353212\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −50.3346 −2.29268
$$483$$ −13.6205 −0.619752
$$484$$ −47.4814 −2.15824
$$485$$ 0 0
$$486$$ 39.1165 1.77436
$$487$$ −32.3241 −1.46474 −0.732372 0.680905i $$-0.761587\pi$$
−0.732372 + 0.680905i $$0.761587\pi$$
$$488$$ −14.3372 −0.649013
$$489$$ −8.63509 −0.390492
$$490$$ 0 0
$$491$$ 28.6708 1.29390 0.646949 0.762534i $$-0.276045\pi$$
0.646949 + 0.762534i $$0.276045\pi$$
$$492$$ −96.0973 −4.33240
$$493$$ −3.67187 −0.165373
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 63.7694 2.86333
$$497$$ −7.74335 −0.347337
$$498$$ 64.9213 2.90919
$$499$$ −28.9616 −1.29650 −0.648249 0.761428i $$-0.724498\pi$$
−0.648249 + 0.761428i $$0.724498\pi$$
$$500$$ 0 0
$$501$$ 6.32648 0.282646
$$502$$ 9.35181 0.417392
$$503$$ 28.1093 1.25333 0.626665 0.779289i $$-0.284419\pi$$
0.626665 + 0.779289i $$0.284419\pi$$
$$504$$ 30.3444 1.35165
$$505$$ 0 0
$$506$$ 3.48079 0.154740
$$507$$ 0 0
$$508$$ −77.2116 −3.42571
$$509$$ −21.1126 −0.935800 −0.467900 0.883781i $$-0.654989\pi$$
−0.467900 + 0.883781i $$0.654989\pi$$
$$510$$ 0 0
$$511$$ 30.3444 1.34236
$$512$$ −50.8542 −2.24746
$$513$$ −4.01086 −0.177084
$$514$$ −33.7688 −1.48948
$$515$$ 0 0
$$516$$ 13.1901 0.580660
$$517$$ 3.91455 0.172162
$$518$$ −9.15616 −0.402299
$$519$$ −2.94369 −0.129214
$$520$$ 0 0
$$521$$ 0.673516 0.0295073 0.0147536 0.999891i $$-0.495304\pi$$
0.0147536 + 0.999891i $$0.495304\pi$$
$$522$$ 12.4875 0.546564
$$523$$ 29.8626 1.30580 0.652900 0.757444i $$-0.273552\pi$$
0.652900 + 0.757444i $$0.273552\pi$$
$$524$$ 44.8079 1.95744
$$525$$ 0 0
$$526$$ −77.0608 −3.36001
$$527$$ −10.9686 −0.477799
$$528$$ −9.72953 −0.423423
$$529$$ −18.3649 −0.798474
$$530$$ 0 0
$$531$$ −12.4212 −0.539035
$$532$$ −17.9718 −0.779177
$$533$$ 0 0
$$534$$ −68.8290 −2.97852
$$535$$ 0 0
$$536$$ −50.7131 −2.19047
$$537$$ −16.7630 −0.723375
$$538$$ −56.6418 −2.44200
$$539$$ 1.03843 0.0447282
$$540$$ 0 0
$$541$$ −6.28806 −0.270345 −0.135172 0.990822i $$-0.543159\pi$$
−0.135172 + 0.990822i $$0.543159\pi$$
$$542$$ 30.1121 1.29342
$$543$$ 8.32502 0.357261
$$544$$ −6.71254 −0.287798
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 3.03789 0.129891 0.0649454 0.997889i $$-0.479313\pi$$
0.0649454 + 0.997889i $$0.479313\pi$$
$$548$$ 38.8588 1.65997
$$549$$ −3.71194 −0.158422
$$550$$ 0 0
$$551$$ −4.09473 −0.174442
$$552$$ −29.2726 −1.24592
$$553$$ 3.05147 0.129762
$$554$$ −42.7304 −1.81544
$$555$$ 0 0
$$556$$ 64.1939 2.72243
$$557$$ 20.6996 0.877071 0.438536 0.898714i $$-0.355497\pi$$
0.438536 + 0.898714i $$0.355497\pi$$
$$558$$ 37.3026 1.57915
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 1.67352 0.0706559
$$562$$ 26.9763 1.13793
$$563$$ −10.9603 −0.461921 −0.230960 0.972963i $$-0.574187\pi$$
−0.230960 + 0.972963i $$0.574187\pi$$
$$564$$ −59.4608 −2.50375
$$565$$ 0 0
$$566$$ 22.4424 0.943323
$$567$$ −33.0051 −1.38608
$$568$$ −16.6417 −0.698272
$$569$$ 42.7131 1.79063 0.895314 0.445436i $$-0.146951\pi$$
0.895314 + 0.445436i $$0.146951\pi$$
$$570$$ 0 0
$$571$$ −23.6145 −0.988238 −0.494119 0.869394i $$-0.664509\pi$$
−0.494119 + 0.869394i $$0.664509\pi$$
$$572$$ 0 0
$$573$$ −10.6434 −0.444635
$$574$$ 74.5204 3.11042
$$575$$ 0 0
$$576$$ −0.441765 −0.0184069
$$577$$ −18.3646 −0.764530 −0.382265 0.924053i $$-0.624856\pi$$
−0.382265 + 0.924053i $$0.624856\pi$$
$$578$$ −39.4639 −1.64148
$$579$$ 10.6708 0.443465
$$580$$ 0 0
$$581$$ −34.8079 −1.44407
$$582$$ −81.0268 −3.35867
$$583$$ 0.407908 0.0168938
$$584$$ 65.2151 2.69862
$$585$$ 0 0
$$586$$ −71.9237 −2.97114
$$587$$ −0.702897 −0.0290116 −0.0145058 0.999895i $$-0.504618\pi$$
−0.0145058 + 0.999895i $$0.504618\pi$$
$$588$$ −15.7734 −0.650483
$$589$$ −12.2318 −0.504001
$$590$$ 0 0
$$591$$ 14.5225 0.597375
$$592$$ −8.70953 −0.357959
$$593$$ −37.1593 −1.52595 −0.762975 0.646428i $$-0.776262\pi$$
−0.762975 + 0.646428i $$0.776262\pi$$
$$594$$ 4.75096 0.194934
$$595$$ 0 0
$$596$$ 76.8674 3.14861
$$597$$ −11.1423 −0.456026
$$598$$ 0 0
$$599$$ −15.6914 −0.641133 −0.320567 0.947226i $$-0.603873\pi$$
−0.320567 + 0.947226i $$0.603873\pi$$
$$600$$ 0 0
$$601$$ 12.0039 0.489648 0.244824 0.969568i $$-0.421270\pi$$
0.244824 + 0.969568i $$0.421270\pi$$
$$602$$ −10.2285 −0.416881
$$603$$ −13.1298 −0.534687
$$604$$ 95.8117 3.89852
$$605$$ 0 0
$$606$$ −72.5204 −2.94594
$$607$$ −38.6865 −1.57024 −0.785119 0.619345i $$-0.787398\pi$$
−0.785119 + 0.619345i $$0.787398\pi$$
$$608$$ −7.48557 −0.303580
$$609$$ −18.9795 −0.769086
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −8.96730 −0.362482
$$613$$ −17.2840 −0.698095 −0.349047 0.937105i $$-0.613495\pi$$
−0.349047 + 0.937105i $$0.613495\pi$$
$$614$$ 32.3866 1.30702
$$615$$ 0 0
$$616$$ 11.7861 0.474877
$$617$$ 26.4691 1.06561 0.532803 0.846240i $$-0.321139\pi$$
0.532803 + 0.846240i $$0.321139\pi$$
$$618$$ −60.1165 −2.41824
$$619$$ 31.0039 1.24615 0.623075 0.782162i $$-0.285883\pi$$
0.623075 + 0.782162i $$0.285883\pi$$
$$620$$ 0 0
$$621$$ 6.32648 0.253873
$$622$$ 71.0850 2.85025
$$623$$ 36.9030 1.47849
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −62.5761 −2.50104
$$627$$ 1.86624 0.0745305
$$628$$ −82.2880 −3.28365
$$629$$ 1.49807 0.0597320
$$630$$ 0 0
$$631$$ 20.7131 0.824577 0.412288 0.911053i $$-0.364730\pi$$
0.412288 + 0.911053i $$0.364730\pi$$
$$632$$ 6.55812 0.260868
$$633$$ 30.1795 1.19953
$$634$$ −0.596662 −0.0236965
$$635$$ 0 0
$$636$$ −6.19599 −0.245687
$$637$$ 0 0
$$638$$ 4.85031 0.192026
$$639$$ −4.30860 −0.170446
$$640$$ 0 0
$$641$$ 21.1895 0.836934 0.418467 0.908232i $$-0.362568\pi$$
0.418467 + 0.908232i $$0.362568\pi$$
$$642$$ 58.4997 2.30880
$$643$$ −11.5336 −0.454843 −0.227421 0.973796i $$-0.573029\pi$$
−0.227421 + 0.973796i $$0.573029\pi$$
$$644$$ 28.3476 1.11705
$$645$$ 0 0
$$646$$ 4.25289 0.167328
$$647$$ 34.8464 1.36995 0.684977 0.728565i $$-0.259812\pi$$
0.684977 + 0.728565i $$0.259812\pi$$
$$648$$ −70.9334 −2.78653
$$649$$ −4.82456 −0.189380
$$650$$ 0 0
$$651$$ −56.6953 −2.22206
$$652$$ 17.9718 0.703831
$$653$$ −22.3232 −0.873574 −0.436787 0.899565i $$-0.643884\pi$$
−0.436787 + 0.899565i $$0.643884\pi$$
$$654$$ −17.9231 −0.700850
$$655$$ 0 0
$$656$$ 70.8853 2.76761
$$657$$ 16.8844 0.658724
$$658$$ 46.1100 1.79755
$$659$$ −0.866840 −0.0337673 −0.0168836 0.999857i $$-0.505374\pi$$
−0.0168836 + 0.999857i $$0.505374\pi$$
$$660$$ 0 0
$$661$$ −13.3086 −0.517645 −0.258822 0.965925i $$-0.583334\pi$$
−0.258822 + 0.965925i $$0.583334\pi$$
$$662$$ 46.6544 1.81328
$$663$$ 0 0
$$664$$ −74.8079 −2.90311
$$665$$ 0 0
$$666$$ −5.09473 −0.197417
$$667$$ 6.45878 0.250085
$$668$$ −13.1670 −0.509448
$$669$$ −0.0178799 −0.000691275 0
$$670$$ 0 0
$$671$$ −1.44176 −0.0556587
$$672$$ −34.6963 −1.33844
$$673$$ 5.51320 0.212518 0.106259 0.994338i $$-0.466113\pi$$
0.106259 + 0.994338i $$0.466113\pi$$
$$674$$ 54.2112 2.08814
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4.80479 0.184663 0.0923316 0.995728i $$-0.470568\pi$$
0.0923316 + 0.995728i $$0.470568\pi$$
$$678$$ 30.3077 1.16396
$$679$$ 43.4430 1.66719
$$680$$ 0 0
$$681$$ 24.2496 0.929248
$$682$$ 14.4888 0.554805
$$683$$ 11.7625 0.450080 0.225040 0.974350i $$-0.427749\pi$$
0.225040 + 0.974350i $$0.427749\pi$$
$$684$$ −10.0000 −0.382360
$$685$$ 0 0
$$686$$ −40.1338 −1.53231
$$687$$ −35.6102 −1.35861
$$688$$ −9.72953 −0.370935
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −4.86684 −0.185143 −0.0925717 0.995706i $$-0.529509\pi$$
−0.0925717 + 0.995706i $$0.529509\pi$$
$$692$$ 6.12658 0.232897
$$693$$ 3.05147 0.115916
$$694$$ 9.71254 0.368683
$$695$$ 0 0
$$696$$ −40.7900 −1.54614
$$697$$ −12.1925 −0.461825
$$698$$ −61.9289 −2.34404
$$699$$ −14.9616 −0.565899
$$700$$ 0 0
$$701$$ 21.3828 0.807617 0.403808 0.914844i $$-0.367686\pi$$
0.403808 + 0.914844i $$0.367686\pi$$
$$702$$ 0 0
$$703$$ 1.67059 0.0630076
$$704$$ −0.171587 −0.00646692
$$705$$ 0 0
$$706$$ −68.8853 −2.59253
$$707$$ 38.8822 1.46232
$$708$$ 73.2835 2.75416
$$709$$ 26.1165 0.980825 0.490412 0.871491i $$-0.336846\pi$$
0.490412 + 0.871491i $$0.336846\pi$$
$$710$$ 0 0
$$711$$ 1.69792 0.0636770
$$712$$ 79.3107 2.97229
$$713$$ 19.2936 0.722551
$$714$$ 19.7125 0.737723
$$715$$ 0 0
$$716$$ 34.8880 1.30383
$$717$$ 8.61170 0.321610
$$718$$ −68.7448 −2.56553
$$719$$ 36.6774 1.36784 0.683918 0.729559i $$-0.260275\pi$$
0.683918 + 0.729559i $$0.260275\pi$$
$$720$$ 0 0
$$721$$ 32.2318 1.20037
$$722$$ −43.6264 −1.62361
$$723$$ 42.5679 1.58312
$$724$$ −17.3265 −0.643934
$$725$$ 0 0
$$726$$ 58.0780 2.15548
$$727$$ 26.2596 0.973916 0.486958 0.873425i $$-0.338107\pi$$
0.486958 + 0.873425i $$0.338107\pi$$
$$728$$ 0 0
$$729$$ 0.614542 0.0227608
$$730$$ 0 0
$$731$$ 1.67352 0.0618972
$$732$$ 21.9000 0.809446
$$733$$ 31.7811 1.17386 0.586931 0.809637i $$-0.300336\pi$$
0.586931 + 0.809637i $$0.300336\pi$$
$$734$$ 17.6703 0.652221
$$735$$ 0 0
$$736$$ 11.8073 0.435222
$$737$$ −5.09978 −0.187853
$$738$$ 41.4651 1.52635
$$739$$ 34.1370 1.25575 0.627875 0.778314i $$-0.283925\pi$$
0.627875 + 0.778314i $$0.283925\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 4.80479 0.176390
$$743$$ 3.12062 0.114485 0.0572423 0.998360i $$-0.481769\pi$$
0.0572423 + 0.998360i $$0.481769\pi$$
$$744$$ −121.847 −4.46715
$$745$$ 0 0
$$746$$ −5.88798 −0.215574
$$747$$ −19.3680 −0.708639
$$748$$ −3.48301 −0.127352
$$749$$ −31.3649 −1.14605
$$750$$ 0 0
$$751$$ 1.48405 0.0541537 0.0270769 0.999633i $$-0.491380\pi$$
0.0270769 + 0.999633i $$0.491380\pi$$
$$752$$ 43.8607 1.59944
$$753$$ −7.90881 −0.288213
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 38.6920 1.40721
$$757$$ −5.09978 −0.185355 −0.0926774 0.995696i $$-0.529543\pi$$
−0.0926774 + 0.995696i $$0.529543\pi$$
$$758$$ −13.1753 −0.478550
$$759$$ −2.94369 −0.106849
$$760$$ 0 0
$$761$$ 29.7861 1.07975 0.539873 0.841746i $$-0.318472\pi$$
0.539873 + 0.841746i $$0.318472\pi$$
$$762$$ 94.4433 3.42132
$$763$$ 9.60959 0.347890
$$764$$ 22.1516 0.801418
$$765$$ 0 0
$$766$$ −52.5973 −1.90042
$$767$$ 0 0
$$768$$ 62.7228 2.26331
$$769$$ −19.0986 −0.688713 −0.344356 0.938839i $$-0.611903\pi$$
−0.344356 + 0.938839i $$0.611903\pi$$
$$770$$ 0 0
$$771$$ 28.5582 1.02850
$$772$$ −22.2088 −0.799311
$$773$$ 49.2306 1.77070 0.885351 0.464923i $$-0.153918\pi$$
0.885351 + 0.464923i $$0.153918\pi$$
$$774$$ −5.69140 −0.204573
$$775$$ 0 0
$$776$$ 93.3661 3.35165
$$777$$ 7.74335 0.277791
$$778$$ 50.2725 1.80236
$$779$$ −13.5967 −0.487151
$$780$$ 0 0
$$781$$ −1.67352 −0.0598831
$$782$$ −6.70825 −0.239886
$$783$$ 8.81566 0.315046
$$784$$ 11.6351 0.415539
$$785$$ 0 0
$$786$$ −54.8079 −1.95493
$$787$$ −9.78335 −0.348739 −0.174369 0.984680i $$-0.555789\pi$$
−0.174369 + 0.984680i $$0.555789\pi$$
$$788$$ −30.2250 −1.07672
$$789$$ 65.1701 2.32012
$$790$$ 0 0
$$791$$ −16.2496 −0.577770
$$792$$ 6.55812 0.233033
$$793$$ 0 0
$$794$$ −23.9020 −0.848250
$$795$$ 0 0
$$796$$ 23.1901 0.821950
$$797$$ −16.5371 −0.585775 −0.292887 0.956147i $$-0.594616\pi$$
−0.292887 + 0.956147i $$0.594616\pi$$
$$798$$ 21.9827 0.778178
$$799$$ −7.54421 −0.266895
$$800$$ 0 0
$$801$$ 20.5338 0.725527
$$802$$ −62.3755 −2.20256
$$803$$ 6.55812 0.231431
$$804$$ 77.4641 2.73195
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 47.9018 1.68622
$$808$$ 83.5643 2.93978
$$809$$ −31.8424 −1.11952 −0.559760 0.828655i $$-0.689107\pi$$
−0.559760 + 0.828655i $$0.689107\pi$$
$$810$$ 0 0
$$811$$ −13.3470 −0.468678 −0.234339 0.972155i $$-0.575293\pi$$
−0.234339 + 0.972155i $$0.575293\pi$$
$$812$$ 39.5011 1.38622
$$813$$ −25.4657 −0.893121
$$814$$ −1.97886 −0.0693589
$$815$$ 0 0
$$816$$ 18.7510 0.656415
$$817$$ 1.86624 0.0652915
$$818$$ −91.9431 −3.21472
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −11.6735 −0.407409 −0.203704 0.979032i $$-0.565298\pi$$
−0.203704 + 0.979032i $$0.565298\pi$$
$$822$$ −47.5311 −1.65784
$$823$$ −32.4317 −1.13050 −0.565249 0.824920i $$-0.691220\pi$$
−0.565249 + 0.824920i $$0.691220\pi$$
$$824$$ 69.2714 2.41318
$$825$$ 0 0
$$826$$ −56.8290 −1.97733
$$827$$ 27.3319 0.950425 0.475212 0.879871i $$-0.342371\pi$$
0.475212 + 0.879871i $$0.342371\pi$$
$$828$$ 15.7734 0.548163
$$829$$ 3.54036 0.122962 0.0614808 0.998108i $$-0.480418\pi$$
0.0614808 + 0.998108i $$0.480418\pi$$
$$830$$ 0 0
$$831$$ 36.1370 1.25358
$$832$$ 0 0
$$833$$ −2.00128 −0.0693402
$$834$$ −78.5204 −2.71894
$$835$$ 0 0
$$836$$ −3.88412 −0.134335
$$837$$ 26.3341 0.910238
$$838$$ −17.4812 −0.603877
$$839$$ −44.7900 −1.54632 −0.773161 0.634210i $$-0.781326\pi$$
−0.773161 + 0.634210i $$0.781326\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ −86.5028 −2.98108
$$843$$ −22.8138 −0.785750
$$844$$ −62.8111 −2.16205
$$845$$ 0 0
$$846$$ 25.6568 0.882100
$$847$$ −31.1388 −1.06994
$$848$$ 4.57042 0.156949
$$849$$ −18.9795 −0.651373
$$850$$ 0 0
$$851$$ −2.63509 −0.0903297
$$852$$ 25.4202 0.870881
$$853$$ −31.3732 −1.07420 −0.537099 0.843519i $$-0.680480\pi$$
−0.537099 + 0.843519i $$0.680480\pi$$
$$854$$ −16.9827 −0.581137
$$855$$ 0 0
$$856$$ −67.4084 −2.30397
$$857$$ 21.2813 0.726955 0.363478 0.931603i $$-0.381589\pi$$
0.363478 + 0.931603i $$0.381589\pi$$
$$858$$ 0 0
$$859$$ 56.8502 1.93970 0.969851 0.243698i $$-0.0783607\pi$$
0.969851 + 0.243698i $$0.0783607\pi$$
$$860$$ 0 0
$$861$$ −63.0217 −2.14778
$$862$$ 41.3673 1.40898
$$863$$ 32.8011 1.11656 0.558282 0.829651i $$-0.311461\pi$$
0.558282 + 0.829651i $$0.311461\pi$$
$$864$$ 16.1159 0.548273
$$865$$ 0 0
$$866$$ −0.652374 −0.0221686
$$867$$ 33.3745 1.13346
$$868$$ 117.997 4.00509
$$869$$ 0.659493 0.0223718
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 20.6526 0.699385
$$873$$ 24.1728 0.818126
$$874$$ −7.48079 −0.253041
$$875$$ 0 0
$$876$$ −99.6157 −3.36570
$$877$$ −36.0651 −1.21783 −0.608916 0.793235i $$-0.708395\pi$$
−0.608916 + 0.793235i $$0.708395\pi$$
$$878$$ −19.3391 −0.652664
$$879$$ 60.8257 2.05160
$$880$$ 0 0
$$881$$ −46.0396 −1.55111 −0.775557 0.631277i $$-0.782531\pi$$
−0.775557 + 0.631277i $$0.782531\pi$$
$$882$$ 6.80607 0.229172
$$883$$ −0.802236 −0.0269974 −0.0134987 0.999909i $$-0.504297\pi$$
−0.0134987 + 0.999909i $$0.504297\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −11.0039 −0.369682
$$887$$ −8.22568 −0.276191 −0.138096 0.990419i $$-0.544098\pi$$
−0.138096 + 0.990419i $$0.544098\pi$$
$$888$$ 16.6417 0.558460
$$889$$ −50.6363 −1.69829
$$890$$ 0 0
$$891$$ −7.13316 −0.238970
$$892$$ 0.0372125 0.00124597
$$893$$ −8.41302 −0.281531
$$894$$ −94.0223 −3.14458
$$895$$ 0 0
$$896$$ −34.2529 −1.14431
$$897$$ 0 0
$$898$$ −8.37054 −0.279328
$$899$$ 26.8847 0.896656
$$900$$ 0 0
$$901$$ −0.786129 −0.0261897
$$902$$ 16.1056 0.536257
$$903$$ 8.65020 0.287861
$$904$$ −34.9231 −1.16153
$$905$$ 0 0
$$906$$ −117.195 −3.89353
$$907$$ 30.4359 1.01061 0.505305 0.862941i $$-0.331380\pi$$
0.505305 + 0.862941i $$0.331380\pi$$
$$908$$ −50.4697 −1.67489
$$909$$ 21.6351 0.717591
$$910$$ 0 0
$$911$$ 43.6145 1.44501 0.722507 0.691363i $$-0.242990\pi$$
0.722507 + 0.691363i $$0.242990\pi$$
$$912$$ 20.9104 0.692412
$$913$$ −7.52278 −0.248968
$$914$$ −39.2708 −1.29896
$$915$$ 0 0
$$916$$ 74.1138 2.44879
$$917$$ 29.3855 0.970395
$$918$$ −9.15616 −0.302198
$$919$$ −37.0217 −1.22123 −0.610617 0.791926i $$-0.709079\pi$$
−0.610617 + 0.791926i $$0.709079\pi$$
$$920$$ 0 0
$$921$$ −27.3893 −0.902509
$$922$$ −65.8957 −2.17016
$$923$$ 0 0
$$924$$ −18.0033 −0.592264
$$925$$ 0 0
$$926$$ 17.9231 0.588991
$$927$$ 17.9346 0.589050
$$928$$ 16.4529 0.540092
$$929$$ 4.76825 0.156441 0.0782206 0.996936i $$-0.475076\pi$$
0.0782206 + 0.996936i $$0.475076\pi$$
$$930$$ 0 0
$$931$$ −2.23175 −0.0731427
$$932$$ 31.1388 1.01999
$$933$$ −60.1165 −1.96812
$$934$$ −47.8886 −1.56696
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −43.6264 −1.42521 −0.712606 0.701565i $$-0.752485\pi$$
−0.712606 + 0.701565i $$0.752485\pi$$
$$938$$ −60.0709 −1.96139
$$939$$ 52.9205 1.72699
$$940$$ 0 0
$$941$$ −18.2675 −0.595504 −0.297752 0.954643i $$-0.596237\pi$$
−0.297752 + 0.954643i $$0.596237\pi$$
$$942$$ 100.653 3.27944
$$943$$ 21.4465 0.698395
$$944$$ −54.0569 −1.75940
$$945$$ 0 0
$$946$$ −2.21061 −0.0718731
$$947$$ −19.9829 −0.649358 −0.324679 0.945824i $$-0.605256\pi$$
−0.324679 + 0.945824i $$0.605256\pi$$
$$948$$ −10.0175 −0.325353
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0.504596 0.0163626
$$952$$ −22.7145 −0.736181
$$953$$ 39.8635 1.29130 0.645652 0.763632i $$-0.276586\pi$$
0.645652 + 0.763632i $$0.276586\pi$$
$$954$$ 2.67352 0.0865583
$$955$$ 0 0
$$956$$ −17.9231 −0.579676
$$957$$ −4.10190 −0.132596
$$958$$ 49.5847 1.60201
$$959$$ 25.4840 0.822923
$$960$$ 0 0
$$961$$ 49.3098 1.59064
$$962$$ 0 0
$$963$$ −17.4523 −0.562392
$$964$$ −88.5946 −2.85344
$$965$$ 0 0
$$966$$ −34.6741 −1.11562
$$967$$ −43.8607 −1.41047 −0.705233 0.708975i $$-0.749158\pi$$
−0.705233 + 0.708975i $$0.749158\pi$$
$$968$$ −66.9225 −2.15097
$$969$$ −3.59666 −0.115541
$$970$$ 0 0
$$971$$ 60.9795 1.95692 0.978462 0.206428i $$-0.0661838\pi$$
0.978462 + 0.206428i $$0.0661838\pi$$
$$972$$ 68.8494 2.20835
$$973$$ 42.0991 1.34964
$$974$$ −82.2887 −2.63670
$$975$$ 0 0
$$976$$ −16.1543 −0.517087
$$977$$ 51.3697 1.64346 0.821731 0.569875i $$-0.193008\pi$$
0.821731 + 0.569875i $$0.193008\pi$$
$$978$$ −21.9827 −0.702929
$$979$$ 7.97560 0.254901
$$980$$ 0 0
$$981$$ 5.34703 0.170718
$$982$$ 72.9885 2.32916
$$983$$ 37.3026 1.18977 0.594885 0.803811i $$-0.297198\pi$$
0.594885 + 0.803811i $$0.297198\pi$$
$$984$$ −135.444 −4.31780
$$985$$ 0 0
$$986$$ −9.34763 −0.297689
$$987$$ −38.9951 −1.24123
$$988$$ 0 0
$$989$$ −2.94369 −0.0936040
$$990$$ 0 0
$$991$$ −51.5621 −1.63792 −0.818962 0.573848i $$-0.805450\pi$$
−0.818962 + 0.573848i $$0.805450\pi$$
$$992$$ 49.1479 1.56045
$$993$$ −39.4556 −1.25208
$$994$$ −19.7125 −0.625244
$$995$$ 0 0
$$996$$ 114.269 3.62074
$$997$$ −22.9489 −0.726798 −0.363399 0.931634i $$-0.618384\pi$$
−0.363399 + 0.931634i $$0.618384\pi$$
$$998$$ −73.7286 −2.33384
$$999$$ −3.59666 −0.113793
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.bq.1.6 6
5.2 odd 4 845.2.b.e.339.6 6
5.3 odd 4 845.2.b.e.339.1 6
5.4 even 2 inner 4225.2.a.bq.1.1 6
13.4 even 6 325.2.e.e.276.6 12
13.10 even 6 325.2.e.e.126.6 12
13.12 even 2 4225.2.a.br.1.1 6
65.2 even 12 845.2.l.f.654.1 24
65.3 odd 12 845.2.n.e.529.6 12
65.4 even 6 325.2.e.e.276.1 12
65.7 even 12 845.2.l.f.699.12 24
65.8 even 4 845.2.d.d.844.11 12
65.12 odd 4 845.2.b.d.339.1 6
65.17 odd 12 65.2.n.a.29.1 yes 12
65.18 even 4 845.2.d.d.844.1 12
65.22 odd 12 845.2.n.e.484.6 12
65.23 odd 12 65.2.n.a.9.1 12
65.28 even 12 845.2.l.f.654.12 24
65.32 even 12 845.2.l.f.699.2 24
65.33 even 12 845.2.l.f.699.1 24
65.37 even 12 845.2.l.f.654.11 24
65.38 odd 4 845.2.b.d.339.6 6
65.42 odd 12 845.2.n.e.529.1 12
65.43 odd 12 65.2.n.a.29.6 yes 12
65.47 even 4 845.2.d.d.844.2 12
65.48 odd 12 845.2.n.e.484.1 12
65.49 even 6 325.2.e.e.126.1 12
65.57 even 4 845.2.d.d.844.12 12
65.58 even 12 845.2.l.f.699.11 24
65.62 odd 12 65.2.n.a.9.6 yes 12
65.63 even 12 845.2.l.f.654.2 24
65.64 even 2 4225.2.a.br.1.6 6
195.17 even 12 585.2.bs.a.289.6 12
195.23 even 12 585.2.bs.a.334.6 12
195.62 even 12 585.2.bs.a.334.1 12
195.173 even 12 585.2.bs.a.289.1 12
260.23 even 12 1040.2.dh.a.529.2 12
260.43 even 12 1040.2.dh.a.289.5 12
260.127 even 12 1040.2.dh.a.529.5 12
260.147 even 12 1040.2.dh.a.289.2 12

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.n.a.9.1 12 65.23 odd 12
65.2.n.a.9.6 yes 12 65.62 odd 12
65.2.n.a.29.1 yes 12 65.17 odd 12
65.2.n.a.29.6 yes 12 65.43 odd 12
325.2.e.e.126.1 12 65.49 even 6
325.2.e.e.126.6 12 13.10 even 6
325.2.e.e.276.1 12 65.4 even 6
325.2.e.e.276.6 12 13.4 even 6
585.2.bs.a.289.1 12 195.173 even 12
585.2.bs.a.289.6 12 195.17 even 12
585.2.bs.a.334.1 12 195.62 even 12
585.2.bs.a.334.6 12 195.23 even 12
845.2.b.d.339.1 6 65.12 odd 4
845.2.b.d.339.6 6 65.38 odd 4
845.2.b.e.339.1 6 5.3 odd 4
845.2.b.e.339.6 6 5.2 odd 4
845.2.d.d.844.1 12 65.18 even 4
845.2.d.d.844.2 12 65.47 even 4
845.2.d.d.844.11 12 65.8 even 4
845.2.d.d.844.12 12 65.57 even 4
845.2.l.f.654.1 24 65.2 even 12
845.2.l.f.654.2 24 65.63 even 12
845.2.l.f.654.11 24 65.37 even 12
845.2.l.f.654.12 24 65.28 even 12
845.2.l.f.699.1 24 65.33 even 12
845.2.l.f.699.2 24 65.32 even 12
845.2.l.f.699.11 24 65.58 even 12
845.2.l.f.699.12 24 65.7 even 12
845.2.n.e.484.1 12 65.48 odd 12
845.2.n.e.484.6 12 65.22 odd 12
845.2.n.e.529.1 12 65.42 odd 12
845.2.n.e.529.6 12 65.3 odd 12
1040.2.dh.a.289.2 12 260.147 even 12
1040.2.dh.a.289.5 12 260.43 even 12
1040.2.dh.a.529.2 12 260.23 even 12
1040.2.dh.a.529.5 12 260.127 even 12
4225.2.a.bq.1.1 6 5.4 even 2 inner
4225.2.a.bq.1.6 6 1.1 even 1 trivial
4225.2.a.br.1.1 6 13.12 even 2
4225.2.a.br.1.6 6 65.64 even 2