Properties

 Label 7098.2.a.bk.1.1 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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Newspace parameters

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$56.6778153547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{129})$$ Defining polynomial: $$x^{2} - x - 32$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$-5.17891$$ of defining polynomial Character $$\chi$$ $$=$$ 7098.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -6.17891 q^{11} -1.00000 q^{12} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} +8.17891 q^{19} +2.00000 q^{20} -1.00000 q^{21} +6.17891 q^{22} +3.17891 q^{23} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} +8.17891 q^{29} +2.00000 q^{30} +7.17891 q^{31} -1.00000 q^{32} +6.17891 q^{33} -5.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -8.17891 q^{38} -2.00000 q^{40} -4.17891 q^{41} +1.00000 q^{42} +0.821092 q^{43} -6.17891 q^{44} +2.00000 q^{45} -3.17891 q^{46} -4.17891 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -5.00000 q^{51} -3.00000 q^{53} +1.00000 q^{54} -12.3578 q^{55} -1.00000 q^{56} -8.17891 q^{57} -8.17891 q^{58} +0.821092 q^{59} -2.00000 q^{60} -3.00000 q^{61} -7.17891 q^{62} +1.00000 q^{63} +1.00000 q^{64} -6.17891 q^{66} +15.1789 q^{67} +5.00000 q^{68} -3.17891 q^{69} -2.00000 q^{70} -9.17891 q^{71} -1.00000 q^{72} -4.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +8.17891 q^{76} -6.17891 q^{77} +1.82109 q^{79} +2.00000 q^{80} +1.00000 q^{81} +4.17891 q^{82} +5.17891 q^{83} -1.00000 q^{84} +10.0000 q^{85} -0.821092 q^{86} -8.17891 q^{87} +6.17891 q^{88} -9.00000 q^{89} -2.00000 q^{90} +3.17891 q^{92} -7.17891 q^{93} +4.17891 q^{94} +16.3578 q^{95} +1.00000 q^{96} -12.3578 q^{97} -1.00000 q^{98} -6.17891 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 4q^{10} - q^{11} - 2q^{12} - 2q^{14} - 4q^{15} + 2q^{16} + 10q^{17} - 2q^{18} + 5q^{19} + 4q^{20} - 2q^{21} + q^{22} - 5q^{23} + 2q^{24} - 2q^{25} - 2q^{27} + 2q^{28} + 5q^{29} + 4q^{30} + 3q^{31} - 2q^{32} + q^{33} - 10q^{34} + 4q^{35} + 2q^{36} - 4q^{37} - 5q^{38} - 4q^{40} + 3q^{41} + 2q^{42} + 13q^{43} - q^{44} + 4q^{45} + 5q^{46} + 3q^{47} - 2q^{48} + 2q^{49} + 2q^{50} - 10q^{51} - 6q^{53} + 2q^{54} - 2q^{55} - 2q^{56} - 5q^{57} - 5q^{58} + 13q^{59} - 4q^{60} - 6q^{61} - 3q^{62} + 2q^{63} + 2q^{64} - q^{66} + 19q^{67} + 10q^{68} + 5q^{69} - 4q^{70} - 7q^{71} - 2q^{72} - 8q^{73} + 4q^{74} + 2q^{75} + 5q^{76} - q^{77} + 15q^{79} + 4q^{80} + 2q^{81} - 3q^{82} - q^{83} - 2q^{84} + 20q^{85} - 13q^{86} - 5q^{87} + q^{88} - 18q^{89} - 4q^{90} - 5q^{92} - 3q^{93} - 3q^{94} + 10q^{95} + 2q^{96} - 2q^{97} - 2q^{98} - q^{99} + O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −2.00000 −0.632456
$$11$$ −6.17891 −1.86301 −0.931505 0.363727i $$-0.881504\pi$$
−0.931505 + 0.363727i $$0.881504\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −1.00000 −0.267261
$$15$$ −2.00000 −0.516398
$$16$$ 1.00000 0.250000
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 8.17891 1.87637 0.938185 0.346134i $$-0.112506\pi$$
0.938185 + 0.346134i $$0.112506\pi$$
$$20$$ 2.00000 0.447214
$$21$$ −1.00000 −0.218218
$$22$$ 6.17891 1.31735
$$23$$ 3.17891 0.662848 0.331424 0.943482i $$-0.392471\pi$$
0.331424 + 0.943482i $$0.392471\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ 8.17891 1.51879 0.759393 0.650633i $$-0.225496\pi$$
0.759393 + 0.650633i $$0.225496\pi$$
$$30$$ 2.00000 0.365148
$$31$$ 7.17891 1.28937 0.644685 0.764448i $$-0.276989\pi$$
0.644685 + 0.764448i $$0.276989\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 6.17891 1.07561
$$34$$ −5.00000 −0.857493
$$35$$ 2.00000 0.338062
$$36$$ 1.00000 0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −8.17891 −1.32679
$$39$$ 0 0
$$40$$ −2.00000 −0.316228
$$41$$ −4.17891 −0.652636 −0.326318 0.945260i $$-0.605808\pi$$
−0.326318 + 0.945260i $$0.605808\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 0.821092 0.125215 0.0626077 0.998038i $$-0.480058\pi$$
0.0626077 + 0.998038i $$0.480058\pi$$
$$44$$ −6.17891 −0.931505
$$45$$ 2.00000 0.298142
$$46$$ −3.17891 −0.468704
$$47$$ −4.17891 −0.609556 −0.304778 0.952423i $$-0.598582\pi$$
−0.304778 + 0.952423i $$0.598582\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ −5.00000 −0.700140
$$52$$ 0 0
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −12.3578 −1.66633
$$56$$ −1.00000 −0.133631
$$57$$ −8.17891 −1.08332
$$58$$ −8.17891 −1.07394
$$59$$ 0.821092 0.106897 0.0534485 0.998571i $$-0.482979\pi$$
0.0534485 + 0.998571i $$0.482979\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ −3.00000 −0.384111 −0.192055 0.981384i $$-0.561515\pi$$
−0.192055 + 0.981384i $$0.561515\pi$$
$$62$$ −7.17891 −0.911722
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −6.17891 −0.760571
$$67$$ 15.1789 1.85440 0.927199 0.374568i $$-0.122209\pi$$
0.927199 + 0.374568i $$0.122209\pi$$
$$68$$ 5.00000 0.606339
$$69$$ −3.17891 −0.382696
$$70$$ −2.00000 −0.239046
$$71$$ −9.17891 −1.08934 −0.544668 0.838652i $$-0.683344\pi$$
−0.544668 + 0.838652i $$0.683344\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 1.00000 0.115470
$$76$$ 8.17891 0.938185
$$77$$ −6.17891 −0.704152
$$78$$ 0 0
$$79$$ 1.82109 0.204889 0.102444 0.994739i $$-0.467334\pi$$
0.102444 + 0.994739i $$0.467334\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ 4.17891 0.461483
$$83$$ 5.17891 0.568459 0.284230 0.958756i $$-0.408262\pi$$
0.284230 + 0.958756i $$0.408262\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 10.0000 1.08465
$$86$$ −0.821092 −0.0885406
$$87$$ −8.17891 −0.876871
$$88$$ 6.17891 0.658674
$$89$$ −9.00000 −0.953998 −0.476999 0.878904i $$-0.658275\pi$$
−0.476999 + 0.878904i $$0.658275\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ 0 0
$$92$$ 3.17891 0.331424
$$93$$ −7.17891 −0.744418
$$94$$ 4.17891 0.431021
$$95$$ 16.3578 1.67828
$$96$$ 1.00000 0.102062
$$97$$ −12.3578 −1.25475 −0.627373 0.778719i $$-0.715870\pi$$
−0.627373 + 0.778719i $$0.715870\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ −6.17891 −0.621004
$$100$$ −1.00000 −0.100000
$$101$$ −16.3578 −1.62766 −0.813832 0.581101i $$-0.802622\pi$$
−0.813832 + 0.581101i $$0.802622\pi$$
$$102$$ 5.00000 0.495074
$$103$$ 3.17891 0.313227 0.156614 0.987660i $$-0.449942\pi$$
0.156614 + 0.987660i $$0.449942\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 3.00000 0.291386
$$107$$ 7.82109 0.756093 0.378047 0.925787i $$-0.376596\pi$$
0.378047 + 0.925787i $$0.376596\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 12.3578 1.17827
$$111$$ 2.00000 0.189832
$$112$$ 1.00000 0.0944911
$$113$$ −4.35782 −0.409949 −0.204974 0.978767i $$-0.565711\pi$$
−0.204974 + 0.978767i $$0.565711\pi$$
$$114$$ 8.17891 0.766025
$$115$$ 6.35782 0.592869
$$116$$ 8.17891 0.759393
$$117$$ 0 0
$$118$$ −0.821092 −0.0755876
$$119$$ 5.00000 0.458349
$$120$$ 2.00000 0.182574
$$121$$ 27.1789 2.47081
$$122$$ 3.00000 0.271607
$$123$$ 4.17891 0.376799
$$124$$ 7.17891 0.644685
$$125$$ −12.0000 −1.07331
$$126$$ −1.00000 −0.0890871
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −0.821092 −0.0722931
$$130$$ 0 0
$$131$$ 19.5367 1.70693 0.853466 0.521149i $$-0.174496\pi$$
0.853466 + 0.521149i $$0.174496\pi$$
$$132$$ 6.17891 0.537805
$$133$$ 8.17891 0.709201
$$134$$ −15.1789 −1.31126
$$135$$ −2.00000 −0.172133
$$136$$ −5.00000 −0.428746
$$137$$ 16.3578 1.39754 0.698771 0.715345i $$-0.253731\pi$$
0.698771 + 0.715345i $$0.253731\pi$$
$$138$$ 3.17891 0.270607
$$139$$ −18.1789 −1.54191 −0.770957 0.636887i $$-0.780222\pi$$
−0.770957 + 0.636887i $$0.780222\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 4.17891 0.351928
$$142$$ 9.17891 0.770277
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 16.3578 1.35844
$$146$$ 4.00000 0.331042
$$147$$ −1.00000 −0.0824786
$$148$$ −2.00000 −0.164399
$$149$$ −4.82109 −0.394959 −0.197480 0.980307i $$-0.563276\pi$$
−0.197480 + 0.980307i $$0.563276\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 14.5367 1.18298 0.591491 0.806312i $$-0.298540\pi$$
0.591491 + 0.806312i $$0.298540\pi$$
$$152$$ −8.17891 −0.663397
$$153$$ 5.00000 0.404226
$$154$$ 6.17891 0.497911
$$155$$ 14.3578 1.15325
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ −1.82109 −0.144878
$$159$$ 3.00000 0.237915
$$160$$ −2.00000 −0.158114
$$161$$ 3.17891 0.250533
$$162$$ −1.00000 −0.0785674
$$163$$ 8.82109 0.690921 0.345461 0.938433i $$-0.387723\pi$$
0.345461 + 0.938433i $$0.387723\pi$$
$$164$$ −4.17891 −0.326318
$$165$$ 12.3578 0.962055
$$166$$ −5.17891 −0.401961
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 0 0
$$170$$ −10.0000 −0.766965
$$171$$ 8.17891 0.625457
$$172$$ 0.821092 0.0626077
$$173$$ 4.00000 0.304114 0.152057 0.988372i $$-0.451410\pi$$
0.152057 + 0.988372i $$0.451410\pi$$
$$174$$ 8.17891 0.620041
$$175$$ −1.00000 −0.0755929
$$176$$ −6.17891 −0.465753
$$177$$ −0.821092 −0.0617170
$$178$$ 9.00000 0.674579
$$179$$ −14.3578 −1.07315 −0.536577 0.843851i $$-0.680283\pi$$
−0.536577 + 0.843851i $$0.680283\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 7.82109 0.581337 0.290669 0.956824i $$-0.406122\pi$$
0.290669 + 0.956824i $$0.406122\pi$$
$$182$$ 0 0
$$183$$ 3.00000 0.221766
$$184$$ −3.17891 −0.234352
$$185$$ −4.00000 −0.294086
$$186$$ 7.17891 0.526383
$$187$$ −30.8945 −2.25923
$$188$$ −4.17891 −0.304778
$$189$$ −1.00000 −0.0727393
$$190$$ −16.3578 −1.18672
$$191$$ 27.5367 1.99249 0.996244 0.0865933i $$-0.0275981\pi$$
0.996244 + 0.0865933i $$0.0275981\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −20.5367 −1.47827 −0.739133 0.673560i $$-0.764764\pi$$
−0.739133 + 0.673560i $$0.764764\pi$$
$$194$$ 12.3578 0.887240
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −17.0000 −1.21120 −0.605600 0.795769i $$-0.707067\pi$$
−0.605600 + 0.795769i $$0.707067\pi$$
$$198$$ 6.17891 0.439116
$$199$$ 12.8211 0.908863 0.454432 0.890782i $$-0.349842\pi$$
0.454432 + 0.890782i $$0.349842\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −15.1789 −1.07064
$$202$$ 16.3578 1.15093
$$203$$ 8.17891 0.574047
$$204$$ −5.00000 −0.350070
$$205$$ −8.35782 −0.583735
$$206$$ −3.17891 −0.221485
$$207$$ 3.17891 0.220949
$$208$$ 0 0
$$209$$ −50.5367 −3.49570
$$210$$ 2.00000 0.138013
$$211$$ 16.3578 1.12612 0.563059 0.826417i $$-0.309624\pi$$
0.563059 + 0.826417i $$0.309624\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ 9.17891 0.628928
$$214$$ −7.82109 −0.534639
$$215$$ 1.64218 0.111996
$$216$$ 1.00000 0.0680414
$$217$$ 7.17891 0.487336
$$218$$ 4.00000 0.270914
$$219$$ 4.00000 0.270295
$$220$$ −12.3578 −0.833164
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ −9.17891 −0.614665 −0.307333 0.951602i $$-0.599436\pi$$
−0.307333 + 0.951602i $$0.599436\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −1.00000 −0.0666667
$$226$$ 4.35782 0.289878
$$227$$ −8.35782 −0.554728 −0.277364 0.960765i $$-0.589461\pi$$
−0.277364 + 0.960765i $$0.589461\pi$$
$$228$$ −8.17891 −0.541661
$$229$$ −1.00000 −0.0660819 −0.0330409 0.999454i $$-0.510519\pi$$
−0.0330409 + 0.999454i $$0.510519\pi$$
$$230$$ −6.35782 −0.419222
$$231$$ 6.17891 0.406542
$$232$$ −8.17891 −0.536972
$$233$$ −0.357817 −0.0234414 −0.0117207 0.999931i $$-0.503731\pi$$
−0.0117207 + 0.999931i $$0.503731\pi$$
$$234$$ 0 0
$$235$$ −8.35782 −0.545204
$$236$$ 0.821092 0.0534485
$$237$$ −1.82109 −0.118293
$$238$$ −5.00000 −0.324102
$$239$$ 1.17891 0.0762572 0.0381286 0.999273i $$-0.487860\pi$$
0.0381286 + 0.999273i $$0.487860\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ 18.3578 1.18253 0.591265 0.806477i $$-0.298629\pi$$
0.591265 + 0.806477i $$0.298629\pi$$
$$242$$ −27.1789 −1.74713
$$243$$ −1.00000 −0.0641500
$$244$$ −3.00000 −0.192055
$$245$$ 2.00000 0.127775
$$246$$ −4.17891 −0.266437
$$247$$ 0 0
$$248$$ −7.17891 −0.455861
$$249$$ −5.17891 −0.328200
$$250$$ 12.0000 0.758947
$$251$$ 10.8211 0.683021 0.341511 0.939878i $$-0.389061\pi$$
0.341511 + 0.939878i $$0.389061\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ −19.6422 −1.23489
$$254$$ −12.0000 −0.752947
$$255$$ −10.0000 −0.626224
$$256$$ 1.00000 0.0625000
$$257$$ 25.7156 1.60410 0.802049 0.597259i $$-0.203743\pi$$
0.802049 + 0.597259i $$0.203743\pi$$
$$258$$ 0.821092 0.0511189
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ 8.17891 0.506262
$$262$$ −19.5367 −1.20698
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ −6.17891 −0.380286
$$265$$ −6.00000 −0.368577
$$266$$ −8.17891 −0.501481
$$267$$ 9.00000 0.550791
$$268$$ 15.1789 0.927199
$$269$$ 18.3578 1.11930 0.559648 0.828730i $$-0.310936\pi$$
0.559648 + 0.828730i $$0.310936\pi$$
$$270$$ 2.00000 0.121716
$$271$$ −31.1789 −1.89398 −0.946992 0.321257i $$-0.895895\pi$$
−0.946992 + 0.321257i $$0.895895\pi$$
$$272$$ 5.00000 0.303170
$$273$$ 0 0
$$274$$ −16.3578 −0.988212
$$275$$ 6.17891 0.372602
$$276$$ −3.17891 −0.191348
$$277$$ 28.3578 1.70386 0.851928 0.523659i $$-0.175433\pi$$
0.851928 + 0.523659i $$0.175433\pi$$
$$278$$ 18.1789 1.09030
$$279$$ 7.17891 0.429790
$$280$$ −2.00000 −0.119523
$$281$$ −5.64218 −0.336584 −0.168292 0.985737i $$-0.553825\pi$$
−0.168292 + 0.985737i $$0.553825\pi$$
$$282$$ −4.17891 −0.248850
$$283$$ 30.3578 1.80458 0.902292 0.431125i $$-0.141883\pi$$
0.902292 + 0.431125i $$0.141883\pi$$
$$284$$ −9.17891 −0.544668
$$285$$ −16.3578 −0.968953
$$286$$ 0 0
$$287$$ −4.17891 −0.246673
$$288$$ −1.00000 −0.0589256
$$289$$ 8.00000 0.470588
$$290$$ −16.3578 −0.960564
$$291$$ 12.3578 0.724428
$$292$$ −4.00000 −0.234082
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 1.64218 0.0956116
$$296$$ 2.00000 0.116248
$$297$$ 6.17891 0.358537
$$298$$ 4.82109 0.279278
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 0.821092 0.0473269
$$302$$ −14.5367 −0.836495
$$303$$ 16.3578 0.939732
$$304$$ 8.17891 0.469093
$$305$$ −6.00000 −0.343559
$$306$$ −5.00000 −0.285831
$$307$$ −24.8945 −1.42081 −0.710403 0.703795i $$-0.751487\pi$$
−0.710403 + 0.703795i $$0.751487\pi$$
$$308$$ −6.17891 −0.352076
$$309$$ −3.17891 −0.180842
$$310$$ −14.3578 −0.815469
$$311$$ −8.53673 −0.484073 −0.242037 0.970267i $$-0.577815\pi$$
−0.242037 + 0.970267i $$0.577815\pi$$
$$312$$ 0 0
$$313$$ 4.00000 0.226093 0.113047 0.993590i $$-0.463939\pi$$
0.113047 + 0.993590i $$0.463939\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 2.00000 0.112687
$$316$$ 1.82109 0.102444
$$317$$ −7.17891 −0.403208 −0.201604 0.979467i $$-0.564615\pi$$
−0.201604 + 0.979467i $$0.564615\pi$$
$$318$$ −3.00000 −0.168232
$$319$$ −50.5367 −2.82951
$$320$$ 2.00000 0.111803
$$321$$ −7.82109 −0.436531
$$322$$ −3.17891 −0.177154
$$323$$ 40.8945 2.27543
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −8.82109 −0.488555
$$327$$ 4.00000 0.221201
$$328$$ 4.17891 0.230742
$$329$$ −4.17891 −0.230391
$$330$$ −12.3578 −0.680275
$$331$$ 24.3578 1.33883 0.669413 0.742890i $$-0.266546\pi$$
0.669413 + 0.742890i $$0.266546\pi$$
$$332$$ 5.17891 0.284230
$$333$$ −2.00000 −0.109599
$$334$$ −8.00000 −0.437741
$$335$$ 30.3578 1.65862
$$336$$ −1.00000 −0.0545545
$$337$$ −0.536725 −0.0292373 −0.0146186 0.999893i $$-0.504653\pi$$
−0.0146186 + 0.999893i $$0.504653\pi$$
$$338$$ 0 0
$$339$$ 4.35782 0.236684
$$340$$ 10.0000 0.542326
$$341$$ −44.3578 −2.40211
$$342$$ −8.17891 −0.442265
$$343$$ 1.00000 0.0539949
$$344$$ −0.821092 −0.0442703
$$345$$ −6.35782 −0.342293
$$346$$ −4.00000 −0.215041
$$347$$ −7.82109 −0.419858 −0.209929 0.977717i $$-0.567323\pi$$
−0.209929 + 0.977717i $$0.567323\pi$$
$$348$$ −8.17891 −0.438436
$$349$$ 3.17891 0.170163 0.0850815 0.996374i $$-0.472885\pi$$
0.0850815 + 0.996374i $$0.472885\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 0 0
$$352$$ 6.17891 0.329337
$$353$$ 7.17891 0.382095 0.191047 0.981581i $$-0.438812\pi$$
0.191047 + 0.981581i $$0.438812\pi$$
$$354$$ 0.821092 0.0436405
$$355$$ −18.3578 −0.974332
$$356$$ −9.00000 −0.476999
$$357$$ −5.00000 −0.264628
$$358$$ 14.3578 0.758834
$$359$$ −12.3578 −0.652221 −0.326110 0.945332i $$-0.605738\pi$$
−0.326110 + 0.945332i $$0.605738\pi$$
$$360$$ −2.00000 −0.105409
$$361$$ 47.8945 2.52077
$$362$$ −7.82109 −0.411067
$$363$$ −27.1789 −1.42652
$$364$$ 0 0
$$365$$ −8.00000 −0.418739
$$366$$ −3.00000 −0.156813
$$367$$ 3.17891 0.165938 0.0829688 0.996552i $$-0.473560\pi$$
0.0829688 + 0.996552i $$0.473560\pi$$
$$368$$ 3.17891 0.165712
$$369$$ −4.17891 −0.217545
$$370$$ 4.00000 0.207950
$$371$$ −3.00000 −0.155752
$$372$$ −7.17891 −0.372209
$$373$$ −20.0000 −1.03556 −0.517780 0.855514i $$-0.673242\pi$$
−0.517780 + 0.855514i $$0.673242\pi$$
$$374$$ 30.8945 1.59752
$$375$$ 12.0000 0.619677
$$376$$ 4.17891 0.215511
$$377$$ 0 0
$$378$$ 1.00000 0.0514344
$$379$$ −20.7156 −1.06409 −0.532045 0.846716i $$-0.678576\pi$$
−0.532045 + 0.846716i $$0.678576\pi$$
$$380$$ 16.3578 0.839138
$$381$$ −12.0000 −0.614779
$$382$$ −27.5367 −1.40890
$$383$$ −9.82109 −0.501834 −0.250917 0.968009i $$-0.580732\pi$$
−0.250917 + 0.968009i $$0.580732\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ −12.3578 −0.629813
$$386$$ 20.5367 1.04529
$$387$$ 0.821092 0.0417384
$$388$$ −12.3578 −0.627373
$$389$$ −0.821092 −0.0416310 −0.0208155 0.999783i $$-0.506626\pi$$
−0.0208155 + 0.999783i $$0.506626\pi$$
$$390$$ 0 0
$$391$$ 15.8945 0.803822
$$392$$ −1.00000 −0.0505076
$$393$$ −19.5367 −0.985497
$$394$$ 17.0000 0.856448
$$395$$ 3.64218 0.183258
$$396$$ −6.17891 −0.310502
$$397$$ −3.00000 −0.150566 −0.0752828 0.997162i $$-0.523986\pi$$
−0.0752828 + 0.997162i $$0.523986\pi$$
$$398$$ −12.8211 −0.642663
$$399$$ −8.17891 −0.409458
$$400$$ −1.00000 −0.0500000
$$401$$ −32.3578 −1.61587 −0.807936 0.589270i $$-0.799415\pi$$
−0.807936 + 0.589270i $$0.799415\pi$$
$$402$$ 15.1789 0.757055
$$403$$ 0 0
$$404$$ −16.3578 −0.813832
$$405$$ 2.00000 0.0993808
$$406$$ −8.17891 −0.405912
$$407$$ 12.3578 0.612554
$$408$$ 5.00000 0.247537
$$409$$ 13.6422 0.674563 0.337281 0.941404i $$-0.390493\pi$$
0.337281 + 0.941404i $$0.390493\pi$$
$$410$$ 8.35782 0.412763
$$411$$ −16.3578 −0.806872
$$412$$ 3.17891 0.156614
$$413$$ 0.821092 0.0404033
$$414$$ −3.17891 −0.156235
$$415$$ 10.3578 0.508445
$$416$$ 0 0
$$417$$ 18.1789 0.890225
$$418$$ 50.5367 2.47183
$$419$$ 29.8945 1.46044 0.730222 0.683210i $$-0.239417\pi$$
0.730222 + 0.683210i $$0.239417\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 34.7156 1.69194 0.845968 0.533233i $$-0.179023\pi$$
0.845968 + 0.533233i $$0.179023\pi$$
$$422$$ −16.3578 −0.796286
$$423$$ −4.17891 −0.203185
$$424$$ 3.00000 0.145693
$$425$$ −5.00000 −0.242536
$$426$$ −9.17891 −0.444720
$$427$$ −3.00000 −0.145180
$$428$$ 7.82109 0.378047
$$429$$ 0 0
$$430$$ −1.64218 −0.0791931
$$431$$ 1.17891 0.0567860 0.0283930 0.999597i $$-0.490961\pi$$
0.0283930 + 0.999597i $$0.490961\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 11.6422 0.559488 0.279744 0.960075i $$-0.409750\pi$$
0.279744 + 0.960075i $$0.409750\pi$$
$$434$$ −7.17891 −0.344599
$$435$$ −16.3578 −0.784297
$$436$$ −4.00000 −0.191565
$$437$$ 26.0000 1.24375
$$438$$ −4.00000 −0.191127
$$439$$ −4.71563 −0.225065 −0.112532 0.993648i $$-0.535896\pi$$
−0.112532 + 0.993648i $$0.535896\pi$$
$$440$$ 12.3578 0.589136
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −8.17891 −0.388592 −0.194296 0.980943i $$-0.562242\pi$$
−0.194296 + 0.980943i $$0.562242\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ −18.0000 −0.853282
$$446$$ 9.17891 0.434634
$$447$$ 4.82109 0.228030
$$448$$ 1.00000 0.0472456
$$449$$ 14.0000 0.660701 0.330350 0.943858i $$-0.392833\pi$$
0.330350 + 0.943858i $$0.392833\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 25.8211 1.21587
$$452$$ −4.35782 −0.204974
$$453$$ −14.5367 −0.682995
$$454$$ 8.35782 0.392252
$$455$$ 0 0
$$456$$ 8.17891 0.383012
$$457$$ 31.8945 1.49196 0.745982 0.665966i $$-0.231981\pi$$
0.745982 + 0.665966i $$0.231981\pi$$
$$458$$ 1.00000 0.0467269
$$459$$ −5.00000 −0.233380
$$460$$ 6.35782 0.296435
$$461$$ 24.7156 1.15112 0.575561 0.817759i $$-0.304784\pi$$
0.575561 + 0.817759i $$0.304784\pi$$
$$462$$ −6.17891 −0.287469
$$463$$ 28.8945 1.34284 0.671422 0.741076i $$-0.265684\pi$$
0.671422 + 0.741076i $$0.265684\pi$$
$$464$$ 8.17891 0.379696
$$465$$ −14.3578 −0.665828
$$466$$ 0.357817 0.0165755
$$467$$ 33.8945 1.56845 0.784226 0.620475i $$-0.213060\pi$$
0.784226 + 0.620475i $$0.213060\pi$$
$$468$$ 0 0
$$469$$ 15.1789 0.700897
$$470$$ 8.35782 0.385517
$$471$$ −2.00000 −0.0921551
$$472$$ −0.821092 −0.0377938
$$473$$ −5.07345 −0.233277
$$474$$ 1.82109 0.0836455
$$475$$ −8.17891 −0.375274
$$476$$ 5.00000 0.229175
$$477$$ −3.00000 −0.137361
$$478$$ −1.17891 −0.0539220
$$479$$ 29.8211 1.36256 0.681280 0.732023i $$-0.261424\pi$$
0.681280 + 0.732023i $$0.261424\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ 0 0
$$482$$ −18.3578 −0.836176
$$483$$ −3.17891 −0.144645
$$484$$ 27.1789 1.23540
$$485$$ −24.7156 −1.12228
$$486$$ 1.00000 0.0453609
$$487$$ 38.1789 1.73005 0.865026 0.501727i $$-0.167302\pi$$
0.865026 + 0.501727i $$0.167302\pi$$
$$488$$ 3.00000 0.135804
$$489$$ −8.82109 −0.398904
$$490$$ −2.00000 −0.0903508
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 4.17891 0.188400
$$493$$ 40.8945 1.84180
$$494$$ 0 0
$$495$$ −12.3578 −0.555443
$$496$$ 7.17891 0.322343
$$497$$ −9.17891 −0.411730
$$498$$ 5.17891 0.232072
$$499$$ 3.17891 0.142307 0.0711537 0.997465i $$-0.477332\pi$$
0.0711537 + 0.997465i $$0.477332\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ −8.00000 −0.357414
$$502$$ −10.8211 −0.482969
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ −32.7156 −1.45583
$$506$$ 19.6422 0.873202
$$507$$ 0 0
$$508$$ 12.0000 0.532414
$$509$$ −28.7156 −1.27280 −0.636399 0.771360i $$-0.719577\pi$$
−0.636399 + 0.771360i $$0.719577\pi$$
$$510$$ 10.0000 0.442807
$$511$$ −4.00000 −0.176950
$$512$$ −1.00000 −0.0441942
$$513$$ −8.17891 −0.361108
$$514$$ −25.7156 −1.13427
$$515$$ 6.35782 0.280159
$$516$$ −0.821092 −0.0361465
$$517$$ 25.8211 1.13561
$$518$$ 2.00000 0.0878750
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ −4.17891 −0.183081 −0.0915406 0.995801i $$-0.529179\pi$$
−0.0915406 + 0.995801i $$0.529179\pi$$
$$522$$ −8.17891 −0.357981
$$523$$ −22.5367 −0.985462 −0.492731 0.870182i $$-0.664001\pi$$
−0.492731 + 0.870182i $$0.664001\pi$$
$$524$$ 19.5367 0.853466
$$525$$ 1.00000 0.0436436
$$526$$ 12.0000 0.523225
$$527$$ 35.8945 1.56359
$$528$$ 6.17891 0.268902
$$529$$ −12.8945 −0.560632
$$530$$ 6.00000 0.260623
$$531$$ 0.821092 0.0356323
$$532$$ 8.17891 0.354601
$$533$$ 0 0
$$534$$ −9.00000 −0.389468
$$535$$ 15.6422 0.676271
$$536$$ −15.1789 −0.655629
$$537$$ 14.3578 0.619586
$$538$$ −18.3578 −0.791462
$$539$$ −6.17891 −0.266144
$$540$$ −2.00000 −0.0860663
$$541$$ −2.35782 −0.101370 −0.0506852 0.998715i $$-0.516141\pi$$
−0.0506852 + 0.998715i $$0.516141\pi$$
$$542$$ 31.1789 1.33925
$$543$$ −7.82109 −0.335635
$$544$$ −5.00000 −0.214373
$$545$$ −8.00000 −0.342682
$$546$$ 0 0
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ 16.3578 0.698771
$$549$$ −3.00000 −0.128037
$$550$$ −6.17891 −0.263470
$$551$$ 66.8945 2.84980
$$552$$ 3.17891 0.135303
$$553$$ 1.82109 0.0774407
$$554$$ −28.3578 −1.20481
$$555$$ 4.00000 0.169791
$$556$$ −18.1789 −0.770957
$$557$$ 21.7156 0.920121 0.460060 0.887888i $$-0.347828\pi$$
0.460060 + 0.887888i $$0.347828\pi$$
$$558$$ −7.17891 −0.303907
$$559$$ 0 0
$$560$$ 2.00000 0.0845154
$$561$$ 30.8945 1.30437
$$562$$ 5.64218 0.238001
$$563$$ 4.35782 0.183660 0.0918300 0.995775i $$-0.470728\pi$$
0.0918300 + 0.995775i $$0.470728\pi$$
$$564$$ 4.17891 0.175964
$$565$$ −8.71563 −0.366669
$$566$$ −30.3578 −1.27603
$$567$$ 1.00000 0.0419961
$$568$$ 9.17891 0.385138
$$569$$ −26.3578 −1.10498 −0.552489 0.833520i $$-0.686322\pi$$
−0.552489 + 0.833520i $$0.686322\pi$$
$$570$$ 16.3578 0.685154
$$571$$ −22.8211 −0.955033 −0.477516 0.878623i $$-0.658463\pi$$
−0.477516 + 0.878623i $$0.658463\pi$$
$$572$$ 0 0
$$573$$ −27.5367 −1.15036
$$574$$ 4.17891 0.174424
$$575$$ −3.17891 −0.132570
$$576$$ 1.00000 0.0416667
$$577$$ 5.64218 0.234887 0.117444 0.993080i $$-0.462530\pi$$
0.117444 + 0.993080i $$0.462530\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 20.5367 0.853477
$$580$$ 16.3578 0.679221
$$581$$ 5.17891 0.214857
$$582$$ −12.3578 −0.512248
$$583$$ 18.5367 0.767713
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 12.0000 0.495715
$$587$$ 11.1789 0.461403 0.230701 0.973025i $$-0.425898\pi$$
0.230701 + 0.973025i $$0.425898\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 58.7156 2.41934
$$590$$ −1.64218 −0.0676076
$$591$$ 17.0000 0.699287
$$592$$ −2.00000 −0.0821995
$$593$$ 33.0000 1.35515 0.677574 0.735455i $$-0.263031\pi$$
0.677574 + 0.735455i $$0.263031\pi$$
$$594$$ −6.17891 −0.253524
$$595$$ 10.0000 0.409960
$$596$$ −4.82109 −0.197480
$$597$$ −12.8211 −0.524732
$$598$$ 0 0
$$599$$ 28.8211 1.17760 0.588799 0.808280i $$-0.299601\pi$$
0.588799 + 0.808280i $$0.299601\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ −0.821092 −0.0334652
$$603$$ 15.1789 0.618133
$$604$$ 14.5367 0.591491
$$605$$ 54.3578 2.20996
$$606$$ −16.3578 −0.664491
$$607$$ 33.5367 1.36121 0.680607 0.732649i $$-0.261716\pi$$
0.680607 + 0.732649i $$0.261716\pi$$
$$608$$ −8.17891 −0.331699
$$609$$ −8.17891 −0.331426
$$610$$ 6.00000 0.242933
$$611$$ 0 0
$$612$$ 5.00000 0.202113
$$613$$ −38.3578 −1.54926 −0.774629 0.632416i $$-0.782063\pi$$
−0.774629 + 0.632416i $$0.782063\pi$$
$$614$$ 24.8945 1.00466
$$615$$ 8.35782 0.337020
$$616$$ 6.17891 0.248955
$$617$$ −32.0000 −1.28827 −0.644136 0.764911i $$-0.722783\pi$$
−0.644136 + 0.764911i $$0.722783\pi$$
$$618$$ 3.17891 0.127874
$$619$$ 8.53673 0.343120 0.171560 0.985174i $$-0.445119\pi$$
0.171560 + 0.985174i $$0.445119\pi$$
$$620$$ 14.3578 0.576624
$$621$$ −3.17891 −0.127565
$$622$$ 8.53673 0.342291
$$623$$ −9.00000 −0.360577
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −4.00000 −0.159872
$$627$$ 50.5367 2.01824
$$628$$ 2.00000 0.0798087
$$629$$ −10.0000 −0.398726
$$630$$ −2.00000 −0.0796819
$$631$$ 30.8945 1.22989 0.614946 0.788569i $$-0.289178\pi$$
0.614946 + 0.788569i $$0.289178\pi$$
$$632$$ −1.82109 −0.0724391
$$633$$ −16.3578 −0.650165
$$634$$ 7.17891 0.285111
$$635$$ 24.0000 0.952411
$$636$$ 3.00000 0.118958
$$637$$ 0 0
$$638$$ 50.5367 2.00077
$$639$$ −9.17891 −0.363112
$$640$$ −2.00000 −0.0790569
$$641$$ −41.0735 −1.62230 −0.811152 0.584836i $$-0.801159\pi$$
−0.811152 + 0.584836i $$0.801159\pi$$
$$642$$ 7.82109 0.308674
$$643$$ 20.1789 0.795778 0.397889 0.917433i $$-0.369743\pi$$
0.397889 + 0.917433i $$0.369743\pi$$
$$644$$ 3.17891 0.125267
$$645$$ −1.64218 −0.0646609
$$646$$ −40.8945 −1.60897
$$647$$ −8.89454 −0.349681 −0.174840 0.984597i $$-0.555941\pi$$
−0.174840 + 0.984597i $$0.555941\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −5.07345 −0.199150
$$650$$ 0 0
$$651$$ −7.17891 −0.281364
$$652$$ 8.82109 0.345461
$$653$$ 21.0000 0.821794 0.410897 0.911682i $$-0.365216\pi$$
0.410897 + 0.911682i $$0.365216\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ 39.0735 1.52673
$$656$$ −4.17891 −0.163159
$$657$$ −4.00000 −0.156055
$$658$$ 4.17891 0.162911
$$659$$ −16.5367 −0.644179 −0.322090 0.946709i $$-0.604385\pi$$
−0.322090 + 0.946709i $$0.604385\pi$$
$$660$$ 12.3578 0.481027
$$661$$ 17.5367 0.682100 0.341050 0.940045i $$-0.389218\pi$$
0.341050 + 0.940045i $$0.389218\pi$$
$$662$$ −24.3578 −0.946693
$$663$$ 0 0
$$664$$ −5.17891 −0.200981
$$665$$ 16.3578 0.634329
$$666$$ 2.00000 0.0774984
$$667$$ 26.0000 1.00672
$$668$$ 8.00000 0.309529
$$669$$ 9.17891 0.354877
$$670$$ −30.3578 −1.17282
$$671$$ 18.5367 0.715602
$$672$$ 1.00000 0.0385758
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ 0.536725 0.0206739
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ −4.35782 −0.167361
$$679$$ −12.3578 −0.474249
$$680$$ −10.0000 −0.383482
$$681$$ 8.35782 0.320272
$$682$$ 44.3578 1.69855
$$683$$ 14.3578 0.549387 0.274693 0.961532i $$-0.411424\pi$$
0.274693 + 0.961532i $$0.411424\pi$$
$$684$$ 8.17891 0.312728
$$685$$ 32.7156 1.25000
$$686$$ −1.00000 −0.0381802
$$687$$ 1.00000 0.0381524
$$688$$ 0.821092 0.0313038
$$689$$ 0 0
$$690$$ 6.35782 0.242038
$$691$$ 40.0000 1.52167 0.760836 0.648944i $$-0.224789\pi$$
0.760836 + 0.648944i $$0.224789\pi$$
$$692$$ 4.00000 0.152057
$$693$$ −6.17891 −0.234717
$$694$$ 7.82109 0.296885
$$695$$ −36.3578 −1.37913
$$696$$ 8.17891 0.310021
$$697$$ −20.8945 −0.791437
$$698$$ −3.17891 −0.120323
$$699$$ 0.357817 0.0135339
$$700$$ −1.00000 −0.0377964
$$701$$ 23.7156 0.895727 0.447864 0.894102i $$-0.352185\pi$$
0.447864 + 0.894102i $$0.352185\pi$$
$$702$$ 0 0
$$703$$ −16.3578 −0.616947
$$704$$ −6.17891 −0.232876
$$705$$ 8.35782 0.314774
$$706$$ −7.17891 −0.270182
$$707$$ −16.3578 −0.615199
$$708$$ −0.821092 −0.0308585
$$709$$ 16.0000 0.600893 0.300446 0.953799i $$-0.402864\pi$$
0.300446 + 0.953799i $$0.402864\pi$$
$$710$$ 18.3578 0.688957
$$711$$ 1.82109 0.0682963
$$712$$ 9.00000 0.337289
$$713$$ 22.8211 0.854657
$$714$$ 5.00000 0.187120
$$715$$ 0 0
$$716$$ −14.3578 −0.536577
$$717$$ −1.17891 −0.0440271
$$718$$ 12.3578 0.461190
$$719$$ −25.8211 −0.962964 −0.481482 0.876456i $$-0.659901\pi$$
−0.481482 + 0.876456i $$0.659901\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ 3.17891 0.118389
$$722$$ −47.8945 −1.78245
$$723$$ −18.3578 −0.682735
$$724$$ 7.82109 0.290669
$$725$$ −8.17891 −0.303757
$$726$$ 27.1789 1.00870
$$727$$ 0.463275 0.0171819 0.00859096 0.999963i $$-0.497265\pi$$
0.00859096 + 0.999963i $$0.497265\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 8.00000 0.296093
$$731$$ 4.10546 0.151846
$$732$$ 3.00000 0.110883
$$733$$ 25.7156 0.949829 0.474914 0.880032i $$-0.342479\pi$$
0.474914 + 0.880032i $$0.342479\pi$$
$$734$$ −3.17891 −0.117336
$$735$$ −2.00000 −0.0737711
$$736$$ −3.17891 −0.117176
$$737$$ −93.7891 −3.45477
$$738$$ 4.17891 0.153828
$$739$$ 4.82109 0.177347 0.0886734 0.996061i $$-0.471737\pi$$
0.0886734 + 0.996061i $$0.471737\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 0 0
$$742$$ 3.00000 0.110133
$$743$$ −43.1789 −1.58408 −0.792040 0.610469i $$-0.790981\pi$$
−0.792040 + 0.610469i $$0.790981\pi$$
$$744$$ 7.17891 0.263192
$$745$$ −9.64218 −0.353262
$$746$$ 20.0000 0.732252
$$747$$ 5.17891 0.189486
$$748$$ −30.8945 −1.12962
$$749$$ 7.82109 0.285776
$$750$$ −12.0000 −0.438178
$$751$$ 20.8945 0.762453 0.381226 0.924482i $$-0.375502\pi$$
0.381226 + 0.924482i $$0.375502\pi$$
$$752$$ −4.17891 −0.152389
$$753$$ −10.8211 −0.394343
$$754$$ 0 0
$$755$$ 29.0735 1.05809
$$756$$ −1.00000 −0.0363696
$$757$$ 2.35782 0.0856963 0.0428482 0.999082i $$-0.486357\pi$$
0.0428482 + 0.999082i $$0.486357\pi$$
$$758$$ 20.7156 0.752426
$$759$$ 19.6422 0.712966
$$760$$ −16.3578 −0.593360
$$761$$ −47.4313 −1.71938 −0.859691 0.510814i $$-0.829344\pi$$
−0.859691 + 0.510814i $$0.829344\pi$$
$$762$$ 12.0000 0.434714
$$763$$ −4.00000 −0.144810
$$764$$ 27.5367 0.996244
$$765$$ 10.0000 0.361551
$$766$$ 9.82109 0.354850
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 4.35782 0.157147 0.0785734 0.996908i $$-0.474963\pi$$
0.0785734 + 0.996908i $$0.474963\pi$$
$$770$$ 12.3578 0.445345
$$771$$ −25.7156 −0.926126
$$772$$ −20.5367 −0.739133
$$773$$ 30.7156 1.10476 0.552382 0.833591i $$-0.313719\pi$$
0.552382 + 0.833591i $$0.313719\pi$$
$$774$$ −0.821092 −0.0295135
$$775$$ −7.17891 −0.257874
$$776$$ 12.3578 0.443620
$$777$$ 2.00000 0.0717496
$$778$$ 0.821092 0.0294376
$$779$$ −34.1789 −1.22459
$$780$$ 0 0
$$781$$ 56.7156 2.02944
$$782$$ −15.8945 −0.568388
$$783$$ −8.17891 −0.292290
$$784$$ 1.00000 0.0357143
$$785$$ 4.00000 0.142766
$$786$$ 19.5367 0.696852
$$787$$ 22.1789 0.790593 0.395296 0.918554i $$-0.370642\pi$$
0.395296 + 0.918554i $$0.370642\pi$$
$$788$$ −17.0000 −0.605600
$$789$$ 12.0000 0.427211
$$790$$ −3.64218 −0.129583
$$791$$ −4.35782 −0.154946
$$792$$ 6.17891 0.219558
$$793$$ 0 0
$$794$$ 3.00000 0.106466
$$795$$ 6.00000 0.212798
$$796$$ 12.8211 0.454432
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 8.17891 0.289530
$$799$$ −20.8945 −0.739196
$$800$$ 1.00000 0.0353553
$$801$$ −9.00000 −0.317999
$$802$$ 32.3578 1.14259
$$803$$ 24.7156 0.872196
$$804$$ −15.1789 −0.535319
$$805$$ 6.35782 0.224084
$$806$$ 0 0
$$807$$ −18.3578 −0.646226
$$808$$ 16.3578 0.575466
$$809$$ −20.0000 −0.703163 −0.351581 0.936157i $$-0.614356\pi$$
−0.351581 + 0.936157i $$0.614356\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ −32.7156 −1.14880 −0.574401 0.818574i $$-0.694765\pi$$
−0.574401 + 0.818574i $$0.694765\pi$$
$$812$$ 8.17891 0.287023
$$813$$ 31.1789 1.09349
$$814$$ −12.3578 −0.433141
$$815$$ 17.6422 0.617979
$$816$$ −5.00000 −0.175035
$$817$$ 6.71563 0.234950
$$818$$ −13.6422 −0.476988
$$819$$ 0 0
$$820$$ −8.35782 −0.291868
$$821$$ −45.7156 −1.59549 −0.797743 0.602997i $$-0.793973\pi$$
−0.797743 + 0.602997i $$0.793973\pi$$
$$822$$ 16.3578 0.570544
$$823$$ −12.7156 −0.443239 −0.221620 0.975133i $$-0.571134\pi$$
−0.221620 + 0.975133i $$0.571134\pi$$
$$824$$ −3.17891 −0.110743
$$825$$ −6.17891 −0.215122
$$826$$ −0.821092 −0.0285694
$$827$$ −4.71563 −0.163979 −0.0819893 0.996633i $$-0.526127\pi$$
−0.0819893 + 0.996633i $$0.526127\pi$$
$$828$$ 3.17891 0.110475
$$829$$ 8.17891 0.284065 0.142033 0.989862i $$-0.454636\pi$$
0.142033 + 0.989862i $$0.454636\pi$$
$$830$$ −10.3578 −0.359525
$$831$$ −28.3578 −0.983722
$$832$$ 0 0
$$833$$ 5.00000 0.173240
$$834$$ −18.1789 −0.629484
$$835$$ 16.0000 0.553703
$$836$$ −50.5367 −1.74785
$$837$$ −7.17891 −0.248139
$$838$$ −29.8945 −1.03269
$$839$$ −34.3578 −1.18616 −0.593082 0.805142i $$-0.702089\pi$$
−0.593082 + 0.805142i $$0.702089\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ 37.8945 1.30671
$$842$$ −34.7156 −1.19638
$$843$$ 5.64218 0.194327
$$844$$ 16.3578 0.563059
$$845$$ 0 0
$$846$$ 4.17891 0.143674
$$847$$ 27.1789 0.933878
$$848$$ −3.00000 −0.103020
$$849$$ −30.3578 −1.04188
$$850$$ 5.00000 0.171499
$$851$$ −6.35782 −0.217943
$$852$$ 9.17891 0.314464
$$853$$ 17.7156 0.606572 0.303286 0.952900i $$-0.401916\pi$$
0.303286 + 0.952900i $$0.401916\pi$$
$$854$$ 3.00000 0.102658
$$855$$ 16.3578 0.559426
$$856$$ −7.82109 −0.267319
$$857$$ 34.7156 1.18586 0.592932 0.805253i $$-0.297970\pi$$
0.592932 + 0.805253i $$0.297970\pi$$
$$858$$ 0 0
$$859$$ −36.8945 −1.25883 −0.629413 0.777071i $$-0.716705\pi$$
−0.629413 + 0.777071i $$0.716705\pi$$
$$860$$ 1.64218 0.0559980
$$861$$ 4.17891 0.142417
$$862$$ −1.17891 −0.0401538
$$863$$ −36.3578 −1.23763 −0.618817 0.785535i $$-0.712388\pi$$
−0.618817 + 0.785535i $$0.712388\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 8.00000 0.272008
$$866$$ −11.6422 −0.395617
$$867$$ −8.00000 −0.271694
$$868$$ 7.17891 0.243668
$$869$$ −11.2524 −0.381710
$$870$$ 16.3578 0.554582
$$871$$ 0 0
$$872$$ 4.00000 0.135457
$$873$$ −12.3578 −0.418249
$$874$$ −26.0000 −0.879463
$$875$$ −12.0000 −0.405674
$$876$$ 4.00000 0.135147
$$877$$ 17.6422 0.595734 0.297867 0.954607i $$-0.403725\pi$$
0.297867 + 0.954607i $$0.403725\pi$$
$$878$$ 4.71563 0.159145
$$879$$ 12.0000 0.404750
$$880$$ −12.3578 −0.416582
$$881$$ 45.5367 1.53417 0.767086 0.641545i $$-0.221706\pi$$
0.767086 + 0.641545i $$0.221706\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ −37.5367 −1.26321 −0.631606 0.775290i $$-0.717604\pi$$
−0.631606 + 0.775290i $$0.717604\pi$$
$$884$$ 0 0
$$885$$ −1.64218 −0.0552014
$$886$$ 8.17891 0.274776
$$887$$ −11.4633 −0.384899 −0.192450 0.981307i $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 12.0000 0.402467
$$890$$ 18.0000 0.603361
$$891$$ −6.17891 −0.207001
$$892$$ −9.17891 −0.307333
$$893$$ −34.1789 −1.14375
$$894$$ −4.82109 −0.161241
$$895$$ −28.7156 −0.959858
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −14.0000 −0.467186
$$899$$ 58.7156 1.95828
$$900$$ −1.00000 −0.0333333
$$901$$ −15.0000 −0.499722
$$902$$ −25.8211 −0.859748
$$903$$ −0.821092 −0.0273242
$$904$$ 4.35782 0.144939
$$905$$ 15.6422 0.519964
$$906$$ 14.5367 0.482950
$$907$$ −52.6102 −1.74689 −0.873446 0.486921i $$-0.838120\pi$$
−0.873446 + 0.486921i $$0.838120\pi$$
$$908$$ −8.35782 −0.277364
$$909$$ −16.3578 −0.542555
$$910$$ 0 0
$$911$$ −16.7156 −0.553814 −0.276907 0.960897i $$-0.589309\pi$$
−0.276907 + 0.960897i $$0.589309\pi$$
$$912$$ −8.17891 −0.270831
$$913$$ −32.0000 −1.05905
$$914$$ −31.8945 −1.05498
$$915$$ 6.00000 0.198354
$$916$$ −1.00000 −0.0330409
$$917$$ 19.5367 0.645159
$$918$$ 5.00000 0.165025
$$919$$ 21.2524 0.701051 0.350525 0.936553i $$-0.386003\pi$$
0.350525 + 0.936553i $$0.386003\pi$$
$$920$$ −6.35782 −0.209611
$$921$$ 24.8945 0.820303
$$922$$ −24.7156 −0.813966
$$923$$ 0 0
$$924$$ 6.17891 0.203271
$$925$$ 2.00000 0.0657596
$$926$$ −28.8945 −0.949534
$$927$$ 3.17891 0.104409
$$928$$ −8.17891 −0.268486
$$929$$ −39.0000 −1.27955 −0.639774 0.768563i $$-0.720972\pi$$
−0.639774 + 0.768563i $$0.720972\pi$$
$$930$$ 14.3578 0.470811
$$931$$ 8.17891 0.268053
$$932$$ −0.357817 −0.0117207
$$933$$ 8.53673 0.279480
$$934$$ −33.8945 −1.10906
$$935$$ −61.7891 −2.02072
$$936$$ 0 0
$$937$$ −47.0735 −1.53782 −0.768911 0.639355i $$-0.779201\pi$$
−0.768911 + 0.639355i $$0.779201\pi$$
$$938$$ −15.1789 −0.495609
$$939$$ −4.00000 −0.130535
$$940$$ −8.35782 −0.272602
$$941$$ 20.0000 0.651981 0.325991 0.945373i $$-0.394302\pi$$
0.325991 + 0.945373i $$0.394302\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ −13.2844 −0.432598
$$944$$ 0.821092 0.0267243
$$945$$ −2.00000 −0.0650600
$$946$$ 5.07345 0.164952
$$947$$ −39.2524 −1.27553 −0.637765 0.770231i $$-0.720141\pi$$
−0.637765 + 0.770231i $$0.720141\pi$$
$$948$$ −1.82109 −0.0591463
$$949$$ 0 0
$$950$$ 8.17891 0.265359
$$951$$ 7.17891 0.232792
$$952$$ −5.00000 −0.162051
$$953$$ 42.7156 1.38370 0.691848 0.722044i $$-0.256797\pi$$
0.691848 + 0.722044i $$0.256797\pi$$
$$954$$ 3.00000 0.0971286
$$955$$ 55.0735 1.78213
$$956$$ 1.17891 0.0381286
$$957$$ 50.5367 1.63362
$$958$$ −29.8211 −0.963476
$$959$$ 16.3578 0.528221
$$960$$ −2.00000 −0.0645497
$$961$$ 20.5367 0.662475
$$962$$ 0 0
$$963$$ 7.82109 0.252031
$$964$$ 18.3578 0.591265
$$965$$ −41.0735 −1.32220
$$966$$ 3.17891 0.102280
$$967$$ −34.3578 −1.10487 −0.552436 0.833555i $$-0.686302\pi$$
−0.552436 + 0.833555i $$0.686302\pi$$
$$968$$ −27.1789 −0.873563
$$969$$ −40.8945 −1.31372
$$970$$ 24.7156 0.793571
$$971$$ −16.8211 −0.539815 −0.269907 0.962886i $$-0.586993\pi$$
−0.269907 + 0.962886i $$0.586993\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −18.1789 −0.582789
$$974$$ −38.1789 −1.22333
$$975$$ 0 0
$$976$$ −3.00000 −0.0960277
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 8.82109 0.282067
$$979$$ 55.6102 1.77731
$$980$$ 2.00000 0.0638877
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ −55.0735 −1.75657 −0.878285 0.478137i $$-0.841312\pi$$
−0.878285 + 0.478137i $$0.841312\pi$$
$$984$$ −4.17891 −0.133219
$$985$$ −34.0000 −1.08333
$$986$$ −40.8945 −1.30235
$$987$$ 4.17891 0.133016
$$988$$ 0 0
$$989$$ 2.61018 0.0829987
$$990$$ 12.3578 0.392757
$$991$$ −15.4633 −0.491207 −0.245604 0.969370i $$-0.578986\pi$$
−0.245604 + 0.969370i $$0.578986\pi$$
$$992$$ −7.17891 −0.227931
$$993$$ −24.3578 −0.772972
$$994$$ 9.17891 0.291137
$$995$$ 25.6422 0.812912
$$996$$ −5.17891 −0.164100
$$997$$ 57.7156 1.82787 0.913936 0.405858i $$-0.133027\pi$$
0.913936 + 0.405858i $$0.133027\pi$$
$$998$$ −3.17891 −0.100627
$$999$$ 2.00000 0.0632772
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 7098.2.a.bk.1.1 2
13.4 even 6 546.2.l.k.211.1 4
13.10 even 6 546.2.l.k.295.1 yes 4
13.12 even 2 7098.2.a.br.1.2 2
39.17 odd 6 1638.2.r.ba.757.2 4
39.23 odd 6 1638.2.r.ba.1387.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.k.211.1 4 13.4 even 6
546.2.l.k.295.1 yes 4 13.10 even 6
1638.2.r.ba.757.2 4 39.17 odd 6
1638.2.r.ba.1387.2 4 39.23 odd 6
7098.2.a.bk.1.1 2 1.1 even 1 trivial
7098.2.a.br.1.2 2 13.12 even 2