# Properties

 Label 7098.2.a.bk.1.2 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## 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 1.2 Root $$6.17891$$ of defining polynomial Character $$\chi$$ $$=$$ 7098.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +5.17891 q^{11} -1.00000 q^{12} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -3.17891 q^{19} +2.00000 q^{20} -1.00000 q^{21} -5.17891 q^{22} -8.17891 q^{23} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} -3.17891 q^{29} +2.00000 q^{30} -4.17891 q^{31} -1.00000 q^{32} -5.17891 q^{33} -5.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +3.17891 q^{38} -2.00000 q^{40} +7.17891 q^{41} +1.00000 q^{42} +12.1789 q^{43} +5.17891 q^{44} +2.00000 q^{45} +8.17891 q^{46} +7.17891 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -5.00000 q^{51} -3.00000 q^{53} +1.00000 q^{54} +10.3578 q^{55} -1.00000 q^{56} +3.17891 q^{57} +3.17891 q^{58} +12.1789 q^{59} -2.00000 q^{60} -3.00000 q^{61} +4.17891 q^{62} +1.00000 q^{63} +1.00000 q^{64} +5.17891 q^{66} +3.82109 q^{67} +5.00000 q^{68} +8.17891 q^{69} -2.00000 q^{70} +2.17891 q^{71} -1.00000 q^{72} -4.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -3.17891 q^{76} +5.17891 q^{77} +13.1789 q^{79} +2.00000 q^{80} +1.00000 q^{81} -7.17891 q^{82} -6.17891 q^{83} -1.00000 q^{84} +10.0000 q^{85} -12.1789 q^{86} +3.17891 q^{87} -5.17891 q^{88} -9.00000 q^{89} -2.00000 q^{90} -8.17891 q^{92} +4.17891 q^{93} -7.17891 q^{94} -6.35782 q^{95} +1.00000 q^{96} +10.3578 q^{97} -1.00000 q^{98} +5.17891 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 4q^{10} - q^{11} - 2q^{12} - 2q^{14} - 4q^{15} + 2q^{16} + 10q^{17} - 2q^{18} + 5q^{19} + 4q^{20} - 2q^{21} + q^{22} - 5q^{23} + 2q^{24} - 2q^{25} - 2q^{27} + 2q^{28} + 5q^{29} + 4q^{30} + 3q^{31} - 2q^{32} + q^{33} - 10q^{34} + 4q^{35} + 2q^{36} - 4q^{37} - 5q^{38} - 4q^{40} + 3q^{41} + 2q^{42} + 13q^{43} - q^{44} + 4q^{45} + 5q^{46} + 3q^{47} - 2q^{48} + 2q^{49} + 2q^{50} - 10q^{51} - 6q^{53} + 2q^{54} - 2q^{55} - 2q^{56} - 5q^{57} - 5q^{58} + 13q^{59} - 4q^{60} - 6q^{61} - 3q^{62} + 2q^{63} + 2q^{64} - q^{66} + 19q^{67} + 10q^{68} + 5q^{69} - 4q^{70} - 7q^{71} - 2q^{72} - 8q^{73} + 4q^{74} + 2q^{75} + 5q^{76} - q^{77} + 15q^{79} + 4q^{80} + 2q^{81} - 3q^{82} - q^{83} - 2q^{84} + 20q^{85} - 13q^{86} - 5q^{87} + q^{88} - 18q^{89} - 4q^{90} - 5q^{92} - 3q^{93} - 3q^{94} + 10q^{95} + 2q^{96} - 2q^{97} - 2q^{98} - q^{99} + O(q^{100})$$

## 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
$$11$$ 5.17891 1.56150 0.780750 0.624844i $$-0.214837\pi$$
0.780750 + 0.624844i $$0.214837\pi$$
$$12$$ −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
$$19$$ −3.17891 −0.729292 −0.364646 0.931146i $$-0.618810\pi$$
−0.364646 + 0.931146i $$0.618810\pi$$
$$20$$ 2.00000 0.447214
$$21$$ −1.00000 −0.218218
$$22$$ −5.17891 −1.10415
$$23$$ −8.17891 −1.70542 −0.852710 0.522384i $$-0.825043\pi$$
−0.852710 + 0.522384i $$0.825043\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ −3.17891 −0.590308 −0.295154 0.955450i $$-0.595371\pi$$
−0.295154 + 0.955450i $$0.595371\pi$$
$$30$$ 2.00000 0.365148
$$31$$ −4.17891 −0.750554 −0.375277 0.926913i $$-0.622452\pi$$
−0.375277 + 0.926913i $$0.622452\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −5.17891 −0.901532
$$34$$ −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 $$0.552568\pi$$
$$38$$ 3.17891 0.515687
$$39$$ 0 0
$$40$$ −2.00000 −0.316228
$$41$$ 7.17891 1.12116 0.560579 0.828101i $$-0.310579\pi$$
0.560579 + 0.828101i $$0.310579\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 12.1789 1.85727 0.928633 0.371000i $$-0.120985\pi$$
0.928633 + 0.371000i $$0.120985\pi$$
$$44$$ 5.17891 0.780750
$$45$$ 2.00000 0.298142
$$46$$ 8.17891 1.20591
$$47$$ 7.17891 1.04715 0.523576 0.851979i $$-0.324598\pi$$
0.523576 + 0.851979i $$0.324598\pi$$
$$48$$ −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
$$55$$ 10.3578 1.39665
$$56$$ −1.00000 −0.133631
$$57$$ 3.17891 0.421057
$$58$$ 3.17891 0.417411
$$59$$ 12.1789 1.58556 0.792779 0.609509i $$-0.208633\pi$$
0.792779 + 0.609509i $$0.208633\pi$$
$$60$$ −2.00000 −0.258199
$$61$$ −3.00000 −0.384111 −0.192055 0.981384i $$-0.561515\pi$$
−0.192055 + 0.981384i $$0.561515\pi$$
$$62$$ 4.17891 0.530722
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 5.17891 0.637480
$$67$$ 3.82109 0.466821 0.233410 0.972378i $$-0.425011\pi$$
0.233410 + 0.972378i $$0.425011\pi$$
$$68$$ 5.00000 0.606339
$$69$$ 8.17891 0.984625
$$70$$ −2.00000 −0.239046
$$71$$ 2.17891 0.258589 0.129294 0.991606i $$-0.458729\pi$$
0.129294 + 0.991606i $$0.458729\pi$$
$$72$$ −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
$$76$$ −3.17891 −0.364646
$$77$$ 5.17891 0.590191
$$78$$ 0 0
$$79$$ 13.1789 1.48274 0.741372 0.671095i $$-0.234176\pi$$
0.741372 + 0.671095i $$0.234176\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ −7.17891 −0.792778
$$83$$ −6.17891 −0.678223 −0.339112 0.940746i $$-0.610126\pi$$
−0.339112 + 0.940746i $$0.610126\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 10.0000 1.08465
$$86$$ −12.1789 −1.31329
$$87$$ 3.17891 0.340815
$$88$$ −5.17891 −0.552073
$$89$$ −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
$$92$$ −8.17891 −0.852710
$$93$$ 4.17891 0.433333
$$94$$ −7.17891 −0.740448
$$95$$ −6.35782 −0.652298
$$96$$ 1.00000 0.102062
$$97$$ 10.3578 1.05168 0.525838 0.850584i $$-0.323752\pi$$
0.525838 + 0.850584i $$0.323752\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 5.17891 0.520500
$$100$$ −1.00000 −0.100000
$$101$$ 6.35782 0.632626 0.316313 0.948655i $$-0.397555\pi$$
0.316313 + 0.948655i $$0.397555\pi$$
$$102$$ 5.00000 0.495074
$$103$$ −8.17891 −0.805892 −0.402946 0.915224i $$-0.632014\pi$$
−0.402946 + 0.915224i $$0.632014\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 3.00000 0.291386
$$107$$ 19.1789 1.85410 0.927048 0.374944i $$-0.122338\pi$$
0.927048 + 0.374944i $$0.122338\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ −10.3578 −0.987579
$$111$$ 2.00000 0.189832
$$112$$ 1.00000 0.0944911
$$113$$ 18.3578 1.72696 0.863479 0.504385i $$-0.168281\pi$$
0.863479 + 0.504385i $$0.168281\pi$$
$$114$$ −3.17891 −0.297732
$$115$$ −16.3578 −1.52537
$$116$$ −3.17891 −0.295154
$$117$$ 0 0
$$118$$ −12.1789 −1.12116
$$119$$ 5.00000 0.458349
$$120$$ 2.00000 0.182574
$$121$$ 15.8211 1.43828
$$122$$ 3.00000 0.271607
$$123$$ −7.17891 −0.647300
$$124$$ −4.17891 −0.375277
$$125$$ −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
$$129$$ −12.1789 −1.07229
$$130$$ 0 0
$$131$$ −14.5367 −1.27008 −0.635040 0.772479i $$-0.719016\pi$$
−0.635040 + 0.772479i $$0.719016\pi$$
$$132$$ −5.17891 −0.450766
$$133$$ −3.17891 −0.275646
$$134$$ −3.82109 −0.330092
$$135$$ −2.00000 −0.172133
$$136$$ −5.00000 −0.428746
$$137$$ −6.35782 −0.543185 −0.271592 0.962412i $$-0.587550\pi$$
−0.271592 + 0.962412i $$0.587550\pi$$
$$138$$ −8.17891 −0.696235
$$139$$ −6.82109 −0.578557 −0.289279 0.957245i $$-0.593415\pi$$
−0.289279 + 0.957245i $$0.593415\pi$$
$$140$$ 2.00000 0.169031
$$141$$ −7.17891 −0.604573
$$142$$ −2.17891 −0.182850
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −6.35782 −0.527988
$$146$$ 4.00000 0.331042
$$147$$ −1.00000 −0.0824786
$$148$$ −2.00000 −0.164399
$$149$$ −16.1789 −1.32543 −0.662714 0.748873i $$-0.730595\pi$$
−0.662714 + 0.748873i $$0.730595\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −19.5367 −1.58988 −0.794938 0.606691i $$-0.792497\pi$$
−0.794938 + 0.606691i $$0.792497\pi$$
$$152$$ 3.17891 0.257844
$$153$$ 5.00000 0.404226
$$154$$ −5.17891 −0.417328
$$155$$ −8.35782 −0.671316
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ −13.1789 −1.04846
$$159$$ 3.00000 0.237915
$$160$$ −2.00000 −0.158114
$$161$$ −8.17891 −0.644588
$$162$$ −1.00000 −0.0785674
$$163$$ 20.1789 1.58053 0.790267 0.612763i $$-0.209942\pi$$
0.790267 + 0.612763i $$0.209942\pi$$
$$164$$ 7.17891 0.560579
$$165$$ −10.3578 −0.806355
$$166$$ 6.17891 0.479576
$$167$$ 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
$$171$$ −3.17891 −0.243097
$$172$$ 12.1789 0.928633
$$173$$ 4.00000 0.304114 0.152057 0.988372i $$-0.451410\pi$$
0.152057 + 0.988372i $$0.451410\pi$$
$$174$$ −3.17891 −0.240992
$$175$$ −1.00000 −0.0755929
$$176$$ 5.17891 0.390375
$$177$$ −12.1789 −0.915423
$$178$$ 9.00000 0.674579
$$179$$ 8.35782 0.624693 0.312346 0.949968i $$-0.398885\pi$$
0.312346 + 0.949968i $$0.398885\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 19.1789 1.42556 0.712779 0.701389i $$-0.247436\pi$$
0.712779 + 0.701389i $$0.247436\pi$$
$$182$$ 0 0
$$183$$ 3.00000 0.221766
$$184$$ 8.17891 0.602957
$$185$$ −4.00000 −0.294086
$$186$$ −4.17891 −0.306412
$$187$$ 25.8945 1.89360
$$188$$ 7.17891 0.523576
$$189$$ −1.00000 −0.0727393
$$190$$ 6.35782 0.461245
$$191$$ −6.53673 −0.472981 −0.236490 0.971634i $$-0.575997\pi$$
−0.236490 + 0.971634i $$0.575997\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 13.5367 0.974395 0.487197 0.873292i $$-0.338019\pi$$
0.487197 + 0.873292i $$0.338019\pi$$
$$194$$ −10.3578 −0.743648
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −17.0000 −1.21120 −0.605600 0.795769i $$-0.707067\pi$$
−0.605600 + 0.795769i $$0.707067\pi$$
$$198$$ −5.17891 −0.368049
$$199$$ 24.1789 1.71400 0.856999 0.515319i $$-0.172326\pi$$
0.856999 + 0.515319i $$0.172326\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −3.82109 −0.269519
$$202$$ −6.35782 −0.447334
$$203$$ −3.17891 −0.223116
$$204$$ −5.00000 −0.350070
$$205$$ 14.3578 1.00279
$$206$$ 8.17891 0.569852
$$207$$ −8.17891 −0.568473
$$208$$ 0 0
$$209$$ −16.4633 −1.13879
$$210$$ 2.00000 0.138013
$$211$$ −6.35782 −0.437690 −0.218845 0.975760i $$-0.570229\pi$$
−0.218845 + 0.975760i $$0.570229\pi$$
$$212$$ −3.00000 −0.206041
$$213$$ −2.17891 −0.149296
$$214$$ −19.1789 −1.31104
$$215$$ 24.3578 1.66119
$$216$$ 1.00000 0.0680414
$$217$$ −4.17891 −0.283683
$$218$$ 4.00000 0.270914
$$219$$ 4.00000 0.270295
$$220$$ 10.3578 0.698324
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ 2.17891 0.145910 0.0729552 0.997335i $$-0.476757\pi$$
0.0729552 + 0.997335i $$0.476757\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −1.00000 −0.0666667
$$226$$ −18.3578 −1.22114
$$227$$ 14.3578 0.952962 0.476481 0.879185i $$-0.341912\pi$$
0.476481 + 0.879185i $$0.341912\pi$$
$$228$$ 3.17891 0.210528
$$229$$ −1.00000 −0.0660819 −0.0330409 0.999454i $$-0.510519\pi$$
−0.0330409 + 0.999454i $$0.510519\pi$$
$$230$$ 16.3578 1.07860
$$231$$ −5.17891 −0.340747
$$232$$ 3.17891 0.208706
$$233$$ 22.3578 1.46471 0.732355 0.680923i $$-0.238421\pi$$
0.732355 + 0.680923i $$0.238421\pi$$
$$234$$ 0 0
$$235$$ 14.3578 0.936601
$$236$$ 12.1789 0.792779
$$237$$ −13.1789 −0.856062
$$238$$ −5.00000 −0.324102
$$239$$ −10.1789 −0.658419 −0.329209 0.944257i $$-0.606782\pi$$
−0.329209 + 0.944257i $$0.606782\pi$$
$$240$$ −2.00000 −0.129099
$$241$$ −4.35782 −0.280712 −0.140356 0.990101i $$-0.544825\pi$$
−0.140356 + 0.990101i $$0.544825\pi$$
$$242$$ −15.8211 −1.01702
$$243$$ −1.00000 −0.0641500
$$244$$ −3.00000 −0.192055
$$245$$ 2.00000 0.127775
$$246$$ 7.17891 0.457710
$$247$$ 0 0
$$248$$ 4.17891 0.265361
$$249$$ 6.17891 0.391572
$$250$$ 12.0000 0.758947
$$251$$ 22.1789 1.39992 0.699960 0.714182i $$-0.253201\pi$$
0.699960 + 0.714182i $$0.253201\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ −42.3578 −2.66301
$$254$$ −12.0000 −0.752947
$$255$$ −10.0000 −0.626224
$$256$$ 1.00000 0.0625000
$$257$$ −19.7156 −1.22983 −0.614914 0.788594i $$-0.710809\pi$$
−0.614914 + 0.788594i $$0.710809\pi$$
$$258$$ 12.1789 0.758226
$$259$$ −2.00000 −0.124274
$$260$$ 0 0
$$261$$ −3.17891 −0.196769
$$262$$ 14.5367 0.898082
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 5.17891 0.318740
$$265$$ −6.00000 −0.368577
$$266$$ 3.17891 0.194911
$$267$$ 9.00000 0.550791
$$268$$ 3.82109 0.233410
$$269$$ −4.35782 −0.265701 −0.132850 0.991136i $$-0.542413\pi$$
−0.132850 + 0.991136i $$0.542413\pi$$
$$270$$ 2.00000 0.121716
$$271$$ −19.8211 −1.20405 −0.602023 0.798479i $$-0.705638\pi$$
−0.602023 + 0.798479i $$0.705638\pi$$
$$272$$ 5.00000 0.303170
$$273$$ 0 0
$$274$$ 6.35782 0.384090
$$275$$ −5.17891 −0.312300
$$276$$ 8.17891 0.492312
$$277$$ 5.64218 0.339006 0.169503 0.985530i $$-0.445784\pi$$
0.169503 + 0.985530i $$0.445784\pi$$
$$278$$ 6.82109 0.409102
$$279$$ −4.17891 −0.250185
$$280$$ −2.00000 −0.119523
$$281$$ −28.3578 −1.69169 −0.845843 0.533432i $$-0.820902\pi$$
−0.845843 + 0.533432i $$0.820902\pi$$
$$282$$ 7.17891 0.427498
$$283$$ 7.64218 0.454281 0.227140 0.973862i $$-0.427062\pi$$
0.227140 + 0.973862i $$0.427062\pi$$
$$284$$ 2.17891 0.129294
$$285$$ 6.35782 0.376605
$$286$$ 0 0
$$287$$ 7.17891 0.423758
$$288$$ −1.00000 −0.0589256
$$289$$ 8.00000 0.470588
$$290$$ 6.35782 0.373344
$$291$$ −10.3578 −0.607186
$$292$$ −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
$$295$$ 24.3578 1.41817
$$296$$ 2.00000 0.116248
$$297$$ −5.17891 −0.300511
$$298$$ 16.1789 0.937219
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 12.1789 0.701981
$$302$$ 19.5367 1.12421
$$303$$ −6.35782 −0.365247
$$304$$ −3.17891 −0.182323
$$305$$ −6.00000 −0.343559
$$306$$ −5.00000 −0.285831
$$307$$ 31.8945 1.82032 0.910159 0.414259i $$-0.135959\pi$$
0.910159 + 0.414259i $$0.135959\pi$$
$$308$$ 5.17891 0.295096
$$309$$ 8.17891 0.465282
$$310$$ 8.35782 0.474692
$$311$$ 25.5367 1.44805 0.724027 0.689771i $$-0.242289\pi$$
0.724027 + 0.689771i $$0.242289\pi$$
$$312$$ 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
$$316$$ 13.1789 0.741372
$$317$$ 4.17891 0.234711 0.117355 0.993090i $$-0.462558\pi$$
0.117355 + 0.993090i $$0.462558\pi$$
$$318$$ −3.00000 −0.168232
$$319$$ −16.4633 −0.921766
$$320$$ 2.00000 0.111803
$$321$$ −19.1789 −1.07046
$$322$$ 8.17891 0.455793
$$323$$ −15.8945 −0.884396
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −20.1789 −1.11761
$$327$$ 4.00000 0.221201
$$328$$ −7.17891 −0.396389
$$329$$ 7.17891 0.395786
$$330$$ 10.3578 0.570179
$$331$$ 1.64218 0.0902626 0.0451313 0.998981i $$-0.485629\pi$$
0.0451313 + 0.998981i $$0.485629\pi$$
$$332$$ −6.17891 −0.339112
$$333$$ −2.00000 −0.109599
$$334$$ −8.00000 −0.437741
$$335$$ 7.64218 0.417537
$$336$$ −1.00000 −0.0545545
$$337$$ 33.5367 1.82686 0.913431 0.406994i $$-0.133423\pi$$
0.913431 + 0.406994i $$0.133423\pi$$
$$338$$ 0 0
$$339$$ −18.3578 −0.997060
$$340$$ 10.0000 0.542326
$$341$$ −21.6422 −1.17199
$$342$$ 3.17891 0.171896
$$343$$ 1.00000 0.0539949
$$344$$ −12.1789 −0.656643
$$345$$ 16.3578 0.880675
$$346$$ −4.00000 −0.215041
$$347$$ −19.1789 −1.02958 −0.514789 0.857317i $$-0.672130\pi$$
−0.514789 + 0.857317i $$0.672130\pi$$
$$348$$ 3.17891 0.170407
$$349$$ −8.17891 −0.437807 −0.218903 0.975747i $$-0.570248\pi$$
−0.218903 + 0.975747i $$0.570248\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 0 0
$$352$$ −5.17891 −0.276037
$$353$$ −4.17891 −0.222421 −0.111210 0.993797i $$-0.535473\pi$$
−0.111210 + 0.993797i $$0.535473\pi$$
$$354$$ 12.1789 0.647302
$$355$$ 4.35782 0.231289
$$356$$ −9.00000 −0.476999
$$357$$ −5.00000 −0.264628
$$358$$ −8.35782 −0.441724
$$359$$ 10.3578 0.546665 0.273332 0.961920i $$-0.411874\pi$$
0.273332 + 0.961920i $$0.411874\pi$$
$$360$$ −2.00000 −0.105409
$$361$$ −8.89454 −0.468134
$$362$$ −19.1789 −1.00802
$$363$$ −15.8211 −0.830392
$$364$$ 0 0
$$365$$ −8.00000 −0.418739
$$366$$ −3.00000 −0.156813
$$367$$ −8.17891 −0.426936 −0.213468 0.976950i $$-0.568476\pi$$
−0.213468 + 0.976950i $$0.568476\pi$$
$$368$$ −8.17891 −0.426355
$$369$$ 7.17891 0.373719
$$370$$ 4.00000 0.207950
$$371$$ −3.00000 −0.155752
$$372$$ 4.17891 0.216666
$$373$$ −20.0000 −1.03556 −0.517780 0.855514i $$-0.673242\pi$$
−0.517780 + 0.855514i $$0.673242\pi$$
$$374$$ −25.8945 −1.33897
$$375$$ 12.0000 0.619677
$$376$$ −7.17891 −0.370224
$$377$$ 0 0
$$378$$ 1.00000 0.0514344
$$379$$ 24.7156 1.26956 0.634778 0.772694i $$-0.281091\pi$$
0.634778 + 0.772694i $$0.281091\pi$$
$$380$$ −6.35782 −0.326149
$$381$$ −12.0000 −0.614779
$$382$$ 6.53673 0.334448
$$383$$ −21.1789 −1.08219 −0.541096 0.840961i $$-0.681990\pi$$
−0.541096 + 0.840961i $$0.681990\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 10.3578 0.527883
$$386$$ −13.5367 −0.689001
$$387$$ 12.1789 0.619089
$$388$$ 10.3578 0.525838
$$389$$ −12.1789 −0.617495 −0.308748 0.951144i $$-0.599910\pi$$
−0.308748 + 0.951144i $$0.599910\pi$$
$$390$$ 0 0
$$391$$ −40.8945 −2.06813
$$392$$ −1.00000 −0.0505076
$$393$$ 14.5367 0.733281
$$394$$ 17.0000 0.856448
$$395$$ 26.3578 1.32621
$$396$$ 5.17891 0.260250
$$397$$ −3.00000 −0.150566 −0.0752828 0.997162i $$-0.523986\pi$$
−0.0752828 + 0.997162i $$0.523986\pi$$
$$398$$ −24.1789 −1.21198
$$399$$ 3.17891 0.159144
$$400$$ −1.00000 −0.0500000
$$401$$ −9.64218 −0.481508 −0.240754 0.970586i $$-0.577395\pi$$
−0.240754 + 0.970586i $$0.577395\pi$$
$$402$$ 3.82109 0.190579
$$403$$ 0 0
$$404$$ 6.35782 0.316313
$$405$$ 2.00000 0.0993808
$$406$$ 3.17891 0.157767
$$407$$ −10.3578 −0.513418
$$408$$ 5.00000 0.247537
$$409$$ 36.3578 1.79778 0.898889 0.438176i $$-0.144375\pi$$
0.898889 + 0.438176i $$0.144375\pi$$
$$410$$ −14.3578 −0.709082
$$411$$ 6.35782 0.313608
$$412$$ −8.17891 −0.402946
$$413$$ 12.1789 0.599285
$$414$$ 8.17891 0.401971
$$415$$ −12.3578 −0.606621
$$416$$ 0 0
$$417$$ 6.82109 0.334030
$$418$$ 16.4633 0.805245
$$419$$ −26.8945 −1.31388 −0.656942 0.753941i $$-0.728150\pi$$
−0.656942 + 0.753941i $$0.728150\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ −10.7156 −0.522248 −0.261124 0.965305i $$-0.584093\pi$$
−0.261124 + 0.965305i $$0.584093\pi$$
$$422$$ 6.35782 0.309494
$$423$$ 7.17891 0.349050
$$424$$ 3.00000 0.145693
$$425$$ −5.00000 −0.242536
$$426$$ 2.17891 0.105568
$$427$$ −3.00000 −0.145180
$$428$$ 19.1789 0.927048
$$429$$ 0 0
$$430$$ −24.3578 −1.17464
$$431$$ −10.1789 −0.490301 −0.245150 0.969485i $$-0.578837\pi$$
−0.245150 + 0.969485i $$0.578837\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 34.3578 1.65113 0.825566 0.564306i $$-0.190856\pi$$
0.825566 + 0.564306i $$0.190856\pi$$
$$434$$ 4.17891 0.200594
$$435$$ 6.35782 0.304834
$$436$$ −4.00000 −0.191565
$$437$$ 26.0000 1.24375
$$438$$ −4.00000 −0.191127
$$439$$ 40.7156 1.94325 0.971626 0.236524i $$-0.0760083\pi$$
0.971626 + 0.236524i $$0.0760083\pi$$
$$440$$ −10.3578 −0.493790
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 3.17891 0.151034 0.0755172 0.997144i $$-0.475939\pi$$
0.0755172 + 0.997144i $$0.475939\pi$$
$$444$$ 2.00000 0.0949158
$$445$$ −18.0000 −0.853282
$$446$$ −2.17891 −0.103174
$$447$$ 16.1789 0.765236
$$448$$ 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
$$451$$ 37.1789 1.75069
$$452$$ 18.3578 0.863479
$$453$$ 19.5367 0.917915
$$454$$ −14.3578 −0.673846
$$455$$ 0 0
$$456$$ −3.17891 −0.148866
$$457$$ −24.8945 −1.16452 −0.582259 0.813004i $$-0.697831\pi$$
−0.582259 + 0.813004i $$0.697831\pi$$
$$458$$ 1.00000 0.0467269
$$459$$ −5.00000 −0.233380
$$460$$ −16.3578 −0.762687
$$461$$ −20.7156 −0.964823 −0.482412 0.875945i $$-0.660239\pi$$
−0.482412 + 0.875945i $$0.660239\pi$$
$$462$$ 5.17891 0.240945
$$463$$ −27.8945 −1.29637 −0.648185 0.761483i $$-0.724472\pi$$
−0.648185 + 0.761483i $$0.724472\pi$$
$$464$$ −3.17891 −0.147577
$$465$$ 8.35782 0.387584
$$466$$ −22.3578 −1.03571
$$467$$ −22.8945 −1.05943 −0.529717 0.848175i $$-0.677702\pi$$
−0.529717 + 0.848175i $$0.677702\pi$$
$$468$$ 0 0
$$469$$ 3.82109 0.176442
$$470$$ −14.3578 −0.662277
$$471$$ −2.00000 −0.0921551
$$472$$ −12.1789 −0.560580
$$473$$ 63.0735 2.90012
$$474$$ 13.1789 0.605327
$$475$$ 3.17891 0.145858
$$476$$ 5.00000 0.229175
$$477$$ −3.00000 −0.137361
$$478$$ 10.1789 0.465572
$$479$$ 41.1789 1.88151 0.940756 0.339084i $$-0.110117\pi$$
0.940756 + 0.339084i $$0.110117\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ 0 0
$$482$$ 4.35782 0.198493
$$483$$ 8.17891 0.372153
$$484$$ 15.8211 0.719141
$$485$$ 20.7156 0.940648
$$486$$ 1.00000 0.0453609
$$487$$ 26.8211 1.21538 0.607690 0.794174i $$-0.292096\pi$$
0.607690 + 0.794174i $$0.292096\pi$$
$$488$$ 3.00000 0.135804
$$489$$ −20.1789 −0.912522
$$490$$ −2.00000 −0.0903508
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ −7.17891 −0.323650
$$493$$ −15.8945 −0.715854
$$494$$ 0 0
$$495$$ 10.3578 0.465549
$$496$$ −4.17891 −0.187639
$$497$$ 2.17891 0.0977374
$$498$$ −6.17891 −0.276884
$$499$$ −8.17891 −0.366138 −0.183069 0.983100i $$-0.558603\pi$$
−0.183069 + 0.983100i $$0.558603\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ −8.00000 −0.357414
$$502$$ −22.1789 −0.989893
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 12.7156 0.565838
$$506$$ 42.3578 1.88303
$$507$$ 0 0
$$508$$ 12.0000 0.532414
$$509$$ 16.7156 0.740907 0.370454 0.928851i $$-0.379202\pi$$
0.370454 + 0.928851i $$0.379202\pi$$
$$510$$ 10.0000 0.442807
$$511$$ −4.00000 −0.176950
$$512$$ −1.00000 −0.0441942
$$513$$ 3.17891 0.140352
$$514$$ 19.7156 0.869619
$$515$$ −16.3578 −0.720812
$$516$$ −12.1789 −0.536147
$$517$$ 37.1789 1.63513
$$518$$ 2.00000 0.0878750
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ 7.17891 0.314514 0.157257 0.987558i $$-0.449735\pi$$
0.157257 + 0.987558i $$0.449735\pi$$
$$522$$ 3.17891 0.139137
$$523$$ 11.5367 0.504466 0.252233 0.967667i $$-0.418835\pi$$
0.252233 + 0.967667i $$0.418835\pi$$
$$524$$ −14.5367 −0.635040
$$525$$ 1.00000 0.0436436
$$526$$ 12.0000 0.523225
$$527$$ −20.8945 −0.910181
$$528$$ −5.17891 −0.225383
$$529$$ 43.8945 1.90846
$$530$$ 6.00000 0.260623
$$531$$ 12.1789 0.528520
$$532$$ −3.17891 −0.137823
$$533$$ 0 0
$$534$$ −9.00000 −0.389468
$$535$$ 38.3578 1.65835
$$536$$ −3.82109 −0.165046
$$537$$ −8.35782 −0.360666
$$538$$ 4.35782 0.187879
$$539$$ 5.17891 0.223071
$$540$$ −2.00000 −0.0860663
$$541$$ 20.3578 0.875251 0.437625 0.899157i $$-0.355820\pi$$
0.437625 + 0.899157i $$0.355820\pi$$
$$542$$ 19.8211 0.851389
$$543$$ −19.1789 −0.823046
$$544$$ −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 $$0.220443\pi$$
$$548$$ −6.35782 −0.271592
$$549$$ −3.00000 −0.128037
$$550$$ 5.17891 0.220829
$$551$$ 10.1055 0.430507
$$552$$ −8.17891 −0.348117
$$553$$ 13.1789 0.560424
$$554$$ −5.64218 −0.239713
$$555$$ 4.00000 0.169791
$$556$$ −6.82109 −0.289279
$$557$$ −23.7156 −1.00486 −0.502432 0.864617i $$-0.667561\pi$$
−0.502432 + 0.864617i $$0.667561\pi$$
$$558$$ 4.17891 0.176907
$$559$$ 0 0
$$560$$ 2.00000 0.0845154
$$561$$ −25.8945 −1.09327
$$562$$ 28.3578 1.19620
$$563$$ −18.3578 −0.773690 −0.386845 0.922145i $$-0.626435\pi$$
−0.386845 + 0.922145i $$0.626435\pi$$
$$564$$ −7.17891 −0.302287
$$565$$ 36.7156 1.54464
$$566$$ −7.64218 −0.321225
$$567$$ 1.00000 0.0419961
$$568$$ −2.17891 −0.0914250
$$569$$ −3.64218 −0.152688 −0.0763441 0.997082i $$-0.524325\pi$$
−0.0763441 + 0.997082i $$0.524325\pi$$
$$570$$ −6.35782 −0.266300
$$571$$ −34.1789 −1.43034 −0.715171 0.698949i $$-0.753651\pi$$
−0.715171 + 0.698949i $$0.753651\pi$$
$$572$$ 0 0
$$573$$ 6.53673 0.273076
$$574$$ −7.17891 −0.299642
$$575$$ 8.17891 0.341084
$$576$$ 1.00000 0.0416667
$$577$$ 28.3578 1.18055 0.590276 0.807202i $$-0.299019\pi$$
0.590276 + 0.807202i $$0.299019\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ −13.5367 −0.562567
$$580$$ −6.35782 −0.263994
$$581$$ −6.17891 −0.256344
$$582$$ 10.3578 0.429345
$$583$$ −15.5367 −0.643465
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ 12.0000 0.495715
$$587$$ −0.178908 −0.00738434 −0.00369217 0.999993i $$-0.501175\pi$$
−0.00369217 + 0.999993i $$0.501175\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 13.2844 0.547373
$$590$$ −24.3578 −1.00280
$$591$$ 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 $$0.263031\pi$$
$$594$$ 5.17891 0.212493
$$595$$ 10.0000 0.409960
$$596$$ −16.1789 −0.662714
$$597$$ −24.1789 −0.989577
$$598$$ 0 0
$$599$$ 40.1789 1.64167 0.820833 0.571168i $$-0.193510\pi$$
0.820833 + 0.571168i $$0.193510\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ −12.1789 −0.496375
$$603$$ 3.82109 0.155607
$$604$$ −19.5367 −0.794938
$$605$$ 31.6422 1.28644
$$606$$ 6.35782 0.258269
$$607$$ −0.536725 −0.0217850 −0.0108925 0.999941i $$-0.503467\pi$$
−0.0108925 + 0.999941i $$0.503467\pi$$
$$608$$ 3.17891 0.128922
$$609$$ 3.17891 0.128816
$$610$$ 6.00000 0.242933
$$611$$ 0 0
$$612$$ 5.00000 0.202113
$$613$$ −15.6422 −0.631782 −0.315891 0.948796i $$-0.602303\pi$$
−0.315891 + 0.948796i $$0.602303\pi$$
$$614$$ −31.8945 −1.28716
$$615$$ −14.3578 −0.578963
$$616$$ −5.17891 −0.208664
$$617$$ −32.0000 −1.28827 −0.644136 0.764911i $$-0.722783\pi$$
−0.644136 + 0.764911i $$0.722783\pi$$
$$618$$ −8.17891 −0.329004
$$619$$ −25.5367 −1.02641 −0.513204 0.858267i $$-0.671542\pi$$
−0.513204 + 0.858267i $$0.671542\pi$$
$$620$$ −8.35782 −0.335658
$$621$$ 8.17891 0.328208
$$622$$ −25.5367 −1.02393
$$623$$ −9.00000 −0.360577
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ −4.00000 −0.159872
$$627$$ 16.4633 0.657480
$$628$$ 2.00000 0.0798087
$$629$$ −10.0000 −0.398726
$$630$$ −2.00000 −0.0796819
$$631$$ −25.8945 −1.03085 −0.515423 0.856936i $$-0.672365\pi$$
−0.515423 + 0.856936i $$0.672365\pi$$
$$632$$ −13.1789 −0.524229
$$633$$ 6.35782 0.252701
$$634$$ −4.17891 −0.165966
$$635$$ 24.0000 0.952411
$$636$$ 3.00000 0.118958
$$637$$ 0 0
$$638$$ 16.4633 0.651787
$$639$$ 2.17891 0.0861963
$$640$$ −2.00000 −0.0790569
$$641$$ 27.0735 1.06934 0.534668 0.845062i $$-0.320436\pi$$
0.534668 + 0.845062i $$0.320436\pi$$
$$642$$ 19.1789 0.756931
$$643$$ 8.82109 0.347870 0.173935 0.984757i $$-0.444352\pi$$
0.173935 + 0.984757i $$0.444352\pi$$
$$644$$ −8.17891 −0.322294
$$645$$ −24.3578 −0.959088
$$646$$ 15.8945 0.625362
$$647$$ 47.8945 1.88293 0.941464 0.337113i $$-0.109450\pi$$
0.941464 + 0.337113i $$0.109450\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 63.0735 2.47585
$$650$$ 0 0
$$651$$ 4.17891 0.163784
$$652$$ 20.1789 0.790267
$$653$$ 21.0000 0.821794 0.410897 0.911682i $$-0.365216\pi$$
0.410897 + 0.911682i $$0.365216\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ −29.0735 −1.13599
$$656$$ 7.17891 0.280289
$$657$$ −4.00000 −0.156055
$$658$$ −7.17891 −0.279863
$$659$$ 17.5367 0.683134 0.341567 0.939857i $$-0.389042\pi$$
0.341567 + 0.939857i $$0.389042\pi$$
$$660$$ −10.3578 −0.403177
$$661$$ −16.5367 −0.643204 −0.321602 0.946875i $$-0.604221\pi$$
−0.321602 + 0.946875i $$0.604221\pi$$
$$662$$ −1.64218 −0.0638253
$$663$$ 0 0
$$664$$ 6.17891 0.239788
$$665$$ −6.35782 −0.246546
$$666$$ 2.00000 0.0774984
$$667$$ 26.0000 1.00672
$$668$$ 8.00000 0.309529
$$669$$ −2.17891 −0.0842415
$$670$$ −7.64218 −0.295243
$$671$$ −15.5367 −0.599789
$$672$$ 1.00000 0.0385758
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ −33.5367 −1.29179
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 18.3578 0.705028
$$679$$ 10.3578 0.397497
$$680$$ −10.0000 −0.383482
$$681$$ −14.3578 −0.550193
$$682$$ 21.6422 0.828722
$$683$$ −8.35782 −0.319803 −0.159901 0.987133i $$-0.551118\pi$$
−0.159901 + 0.987133i $$0.551118\pi$$
$$684$$ −3.17891 −0.121549
$$685$$ −12.7156 −0.485839
$$686$$ −1.00000 −0.0381802
$$687$$ 1.00000 0.0381524
$$688$$ 12.1789 0.464317
$$689$$ 0 0
$$690$$ −16.3578 −0.622731
$$691$$ 40.0000 1.52167 0.760836 0.648944i $$-0.224789\pi$$
0.760836 + 0.648944i $$0.224789\pi$$
$$692$$ 4.00000 0.152057
$$693$$ 5.17891 0.196730
$$694$$ 19.1789 0.728021
$$695$$ −13.6422 −0.517478
$$696$$ −3.17891 −0.120496
$$697$$ 35.8945 1.35960
$$698$$ 8.17891 0.309576
$$699$$ −22.3578 −0.845650
$$700$$ −1.00000 −0.0377964
$$701$$ −21.7156 −0.820188 −0.410094 0.912043i $$-0.634504\pi$$
−0.410094 + 0.912043i $$0.634504\pi$$
$$702$$ 0 0
$$703$$ 6.35782 0.239790
$$704$$ 5.17891 0.195187
$$705$$ −14.3578 −0.540747
$$706$$ 4.17891 0.157275
$$707$$ 6.35782 0.239110
$$708$$ −12.1789 −0.457711
$$709$$ 16.0000 0.600893 0.300446 0.953799i $$-0.402864\pi$$
0.300446 + 0.953799i $$0.402864\pi$$
$$710$$ −4.35782 −0.163546
$$711$$ 13.1789 0.494248
$$712$$ 9.00000 0.337289
$$713$$ 34.1789 1.28001
$$714$$ 5.00000 0.187120
$$715$$ 0 0
$$716$$ 8.35782 0.312346
$$717$$ 10.1789 0.380138
$$718$$ −10.3578 −0.386550
$$719$$ −37.1789 −1.38654 −0.693270 0.720678i $$-0.743831\pi$$
−0.693270 + 0.720678i $$0.743831\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ −8.17891 −0.304598
$$722$$ 8.89454 0.331021
$$723$$ 4.35782 0.162069
$$724$$ 19.1789 0.712779
$$725$$ 3.17891 0.118062
$$726$$ 15.8211 0.587176
$$727$$ 34.5367 1.28090 0.640448 0.768001i $$-0.278749\pi$$
0.640448 + 0.768001i $$0.278749\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 8.00000 0.296093
$$731$$ 60.8945 2.25227
$$732$$ 3.00000 0.110883
$$733$$ −19.7156 −0.728214 −0.364107 0.931357i $$-0.618626\pi$$
−0.364107 + 0.931357i $$0.618626\pi$$
$$734$$ 8.17891 0.301889
$$735$$ −2.00000 −0.0737711
$$736$$ 8.17891 0.301479
$$737$$ 19.7891 0.728940
$$738$$ −7.17891 −0.264259
$$739$$ 16.1789 0.595151 0.297575 0.954698i $$-0.403822\pi$$
0.297575 + 0.954698i $$0.403822\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 0 0
$$742$$ 3.00000 0.110133
$$743$$ −31.8211 −1.16740 −0.583701 0.811968i $$-0.698396\pi$$
−0.583701 + 0.811968i $$0.698396\pi$$
$$744$$ −4.17891 −0.153206
$$745$$ −32.3578 −1.18550
$$746$$ 20.0000 0.732252
$$747$$ −6.17891 −0.226074
$$748$$ 25.8945 0.946798
$$749$$ 19.1789 0.700782
$$750$$ −12.0000 −0.438178
$$751$$ −35.8945 −1.30981 −0.654905 0.755711i $$-0.727291\pi$$
−0.654905 + 0.755711i $$0.727291\pi$$
$$752$$ 7.17891 0.261788
$$753$$ −22.1789 −0.808244
$$754$$ 0 0
$$755$$ −39.0735 −1.42203
$$756$$ −1.00000 −0.0363696
$$757$$ −20.3578 −0.739917 −0.369959 0.929048i $$-0.620628\pi$$
−0.369959 + 0.929048i $$0.620628\pi$$
$$758$$ −24.7156 −0.897712
$$759$$ 42.3578 1.53749
$$760$$ 6.35782 0.230622
$$761$$ 43.4313 1.57438 0.787191 0.616709i $$-0.211535\pi$$
0.787191 + 0.616709i $$0.211535\pi$$
$$762$$ 12.0000 0.434714
$$763$$ −4.00000 −0.144810
$$764$$ −6.53673 −0.236490
$$765$$ 10.0000 0.361551
$$766$$ 21.1789 0.765225
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −18.3578 −0.662000 −0.331000 0.943631i $$-0.607386\pi$$
−0.331000 + 0.943631i $$0.607386\pi$$
$$770$$ −10.3578 −0.373270
$$771$$ 19.7156 0.710041
$$772$$ 13.5367 0.487197
$$773$$ −14.7156 −0.529285 −0.264642 0.964347i $$-0.585254\pi$$
−0.264642 + 0.964347i $$0.585254\pi$$
$$774$$ −12.1789 −0.437762
$$775$$ 4.17891 0.150111
$$776$$ −10.3578 −0.371824
$$777$$ 2.00000 0.0717496
$$778$$ 12.1789 0.436635
$$779$$ −22.8211 −0.817650
$$780$$ 0 0
$$781$$ 11.2844 0.403786
$$782$$ 40.8945 1.46239
$$783$$ 3.17891 0.113605
$$784$$ 1.00000 0.0357143
$$785$$ 4.00000 0.142766
$$786$$ −14.5367 −0.518508
$$787$$ 10.8211 0.385730 0.192865 0.981225i $$-0.438222\pi$$
0.192865 + 0.981225i $$0.438222\pi$$
$$788$$ −17.0000 −0.605600
$$789$$ 12.0000 0.427211
$$790$$ −26.3578 −0.937769
$$791$$ 18.3578 0.652729
$$792$$ −5.17891 −0.184024
$$793$$ 0 0
$$794$$ 3.00000 0.106466
$$795$$ 6.00000 0.212798
$$796$$ 24.1789 0.856999
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ −3.17891 −0.112532
$$799$$ 35.8945 1.26986
$$800$$ 1.00000 0.0353553
$$801$$ −9.00000 −0.317999
$$802$$ 9.64218 0.340477
$$803$$ −20.7156 −0.731039
$$804$$ −3.82109 −0.134760
$$805$$ −16.3578 −0.576537
$$806$$ 0 0
$$807$$ 4.35782 0.153402
$$808$$ −6.35782 −0.223667
$$809$$ −20.0000 −0.703163 −0.351581 0.936157i $$-0.614356\pi$$
−0.351581 + 0.936157i $$0.614356\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ 12.7156 0.446506 0.223253 0.974761i $$-0.428332\pi$$
0.223253 + 0.974761i $$0.428332\pi$$
$$812$$ −3.17891 −0.111558
$$813$$ 19.8211 0.695156
$$814$$ 10.3578 0.363041
$$815$$ 40.3578 1.41367
$$816$$ −5.00000 −0.175035
$$817$$ −38.7156 −1.35449
$$818$$ −36.3578 −1.27122
$$819$$ 0 0
$$820$$ 14.3578 0.501397
$$821$$ −0.284367 −0.00992446 −0.00496223 0.999988i $$-0.501580\pi$$
−0.00496223 + 0.999988i $$0.501580\pi$$
$$822$$ −6.35782 −0.221754
$$823$$ 32.7156 1.14040 0.570198 0.821508i $$-0.306867\pi$$
0.570198 + 0.821508i $$0.306867\pi$$
$$824$$ 8.17891 0.284926
$$825$$ 5.17891 0.180306
$$826$$ −12.1789 −0.423758
$$827$$ 40.7156 1.41582 0.707911 0.706302i $$-0.249638\pi$$
0.707911 + 0.706302i $$0.249638\pi$$
$$828$$ −8.17891 −0.284237
$$829$$ −3.17891 −0.110408 −0.0552040 0.998475i $$-0.517581\pi$$
−0.0552040 + 0.998475i $$0.517581\pi$$
$$830$$ 12.3578 0.428946
$$831$$ −5.64218 −0.195725
$$832$$ 0 0
$$833$$ 5.00000 0.173240
$$834$$ −6.82109 −0.236195
$$835$$ 16.0000 0.553703
$$836$$ −16.4633 −0.569394
$$837$$ 4.17891 0.144444
$$838$$ 26.8945 0.929057
$$839$$ −11.6422 −0.401933 −0.200966 0.979598i $$-0.564408\pi$$
−0.200966 + 0.979598i $$0.564408\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ −18.8945 −0.651536
$$842$$ 10.7156 0.369285
$$843$$ 28.3578 0.976695
$$844$$ −6.35782 −0.218845
$$845$$ 0 0
$$846$$ −7.17891 −0.246816
$$847$$ 15.8211 0.543619
$$848$$ −3.00000 −0.103020
$$849$$ −7.64218 −0.262279
$$850$$ 5.00000 0.171499
$$851$$ 16.3578 0.560739
$$852$$ −2.17891 −0.0746482
$$853$$ −27.7156 −0.948965 −0.474483 0.880265i $$-0.657365\pi$$
−0.474483 + 0.880265i $$0.657365\pi$$
$$854$$ 3.00000 0.102658
$$855$$ −6.35782 −0.217433
$$856$$ −19.1789 −0.655522
$$857$$ −10.7156 −0.366039 −0.183020 0.983109i $$-0.558587\pi$$
−0.183020 + 0.983109i $$0.558587\pi$$
$$858$$ 0 0
$$859$$ 19.8945 0.678793 0.339397 0.940643i $$-0.389777\pi$$
0.339397 + 0.940643i $$0.389777\pi$$
$$860$$ 24.3578 0.830595
$$861$$ −7.17891 −0.244657
$$862$$ 10.1789 0.346695
$$863$$ −13.6422 −0.464385 −0.232193 0.972670i $$-0.574590\pi$$
−0.232193 + 0.972670i $$0.574590\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 8.00000 0.272008
$$866$$ −34.3578 −1.16753
$$867$$ −8.00000 −0.271694
$$868$$ −4.17891 −0.141841
$$869$$ 68.2524 2.31530
$$870$$ −6.35782 −0.215550
$$871$$ 0 0
$$872$$ 4.00000 0.135457
$$873$$ 10.3578 0.350559
$$874$$ −26.0000 −0.879463
$$875$$ −12.0000 −0.405674
$$876$$ 4.00000 0.135147
$$877$$ 40.3578 1.36279 0.681393 0.731917i $$-0.261374\pi$$
0.681393 + 0.731917i $$0.261374\pi$$
$$878$$ −40.7156 −1.37409
$$879$$ 12.0000 0.404750
$$880$$ 10.3578 0.349162
$$881$$ 11.4633 0.386208 0.193104 0.981178i $$-0.438145\pi$$
0.193104 + 0.981178i $$0.438145\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ −3.46327 −0.116548 −0.0582742 0.998301i $$-0.518560\pi$$
−0.0582742 + 0.998301i $$0.518560\pi$$
$$884$$ 0 0
$$885$$ −24.3578 −0.818779
$$886$$ −3.17891 −0.106798
$$887$$ −45.5367 −1.52897 −0.764487 0.644639i $$-0.777008\pi$$
−0.764487 + 0.644639i $$0.777008\pi$$
$$888$$ −2.00000 −0.0671156
$$889$$ 12.0000 0.402467
$$890$$ 18.0000 0.603361
$$891$$ 5.17891 0.173500
$$892$$ 2.17891 0.0729552
$$893$$ −22.8211 −0.763679
$$894$$ −16.1789 −0.541104
$$895$$ 16.7156 0.558742
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −14.0000 −0.467186
$$899$$ 13.2844 0.443058
$$900$$ −1.00000 −0.0333333
$$901$$ −15.0000 −0.499722
$$902$$ −37.1789 −1.23792
$$903$$ −12.1789 −0.405289
$$904$$ −18.3578 −0.610572
$$905$$ 38.3578 1.27506
$$906$$ −19.5367 −0.649064
$$907$$ 49.6102 1.64728 0.823639 0.567114i $$-0.191940\pi$$
0.823639 + 0.567114i $$0.191940\pi$$
$$908$$ 14.3578 0.476481
$$909$$ 6.35782 0.210875
$$910$$ 0 0
$$911$$ 28.7156 0.951391 0.475696 0.879610i $$-0.342196\pi$$
0.475696 + 0.879610i $$0.342196\pi$$
$$912$$ 3.17891 0.105264
$$913$$ −32.0000 −1.05905
$$914$$ 24.8945 0.823438
$$915$$ 6.00000 0.198354
$$916$$ −1.00000 −0.0330409
$$917$$ −14.5367 −0.480045
$$918$$ 5.00000 0.165025
$$919$$ −58.2524 −1.92157 −0.960784 0.277298i $$-0.910561\pi$$
−0.960784 + 0.277298i $$0.910561\pi$$
$$920$$ 16.3578 0.539301
$$921$$ −31.8945 −1.05096
$$922$$ 20.7156 0.682233
$$923$$ 0 0
$$924$$ −5.17891 −0.170374
$$925$$ 2.00000 0.0657596
$$926$$ 27.8945 0.916672
$$927$$ −8.17891 −0.268631
$$928$$ 3.17891 0.104353
$$929$$ −39.0000 −1.27955 −0.639774 0.768563i $$-0.720972\pi$$
−0.639774 + 0.768563i $$0.720972\pi$$
$$930$$ −8.35782 −0.274064
$$931$$ −3.17891 −0.104185
$$932$$ 22.3578 0.732355
$$933$$ −25.5367 −0.836035
$$934$$ 22.8945 0.749132
$$935$$ 51.7891 1.69368
$$936$$ 0 0
$$937$$ 21.0735 0.688440 0.344220 0.938889i $$-0.388143\pi$$
0.344220 + 0.938889i $$0.388143\pi$$
$$938$$ −3.82109 −0.124763
$$939$$ −4.00000 −0.130535
$$940$$ 14.3578 0.468300
$$941$$ 20.0000 0.651981 0.325991 0.945373i $$-0.394302\pi$$
0.325991 + 0.945373i $$0.394302\pi$$
$$942$$ 2.00000 0.0651635
$$943$$ −58.7156 −1.91204
$$944$$ 12.1789 0.396390
$$945$$ −2.00000 −0.0650600
$$946$$ −63.0735 −2.05069
$$947$$ 40.2524 1.30803 0.654013 0.756483i $$-0.273084\pi$$
0.654013 + 0.756483i $$0.273084\pi$$
$$948$$ −13.1789 −0.428031
$$949$$ 0 0
$$950$$ −3.17891 −0.103137
$$951$$ −4.17891 −0.135510
$$952$$ −5.00000 −0.162051
$$953$$ −2.71563 −0.0879680 −0.0439840 0.999032i $$-0.514005\pi$$
−0.0439840 + 0.999032i $$0.514005\pi$$
$$954$$ 3.00000 0.0971286
$$955$$ −13.0735 −0.423047
$$956$$ −10.1789 −0.329209
$$957$$ 16.4633 0.532182
$$958$$ −41.1789 −1.33043
$$959$$ −6.35782 −0.205305
$$960$$ −2.00000 −0.0645497
$$961$$ −13.5367 −0.436669
$$962$$ 0 0
$$963$$ 19.1789 0.618032
$$964$$ −4.35782 −0.140356
$$965$$ 27.0735 0.871525
$$966$$ −8.17891 −0.263152
$$967$$ −11.6422 −0.374387 −0.187194 0.982323i $$-0.559939\pi$$
−0.187194 + 0.982323i $$0.559939\pi$$
$$968$$ −15.8211 −0.508509
$$969$$ 15.8945 0.510606
$$970$$ −20.7156 −0.665139
$$971$$ −28.1789 −0.904304 −0.452152 0.891941i $$-0.649344\pi$$
−0.452152 + 0.891941i $$0.649344\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −6.82109 −0.218674
$$974$$ −26.8211 −0.859403
$$975$$ 0 0
$$976$$ −3.00000 −0.0960277
$$977$$ −12.0000 −0.383914 −0.191957 0.981403i $$-0.561483\pi$$
−0.191957 + 0.981403i $$0.561483\pi$$
$$978$$ 20.1789 0.645250
$$979$$ −46.6102 −1.48967
$$980$$ 2.00000 0.0638877
$$981$$ −4.00000 −0.127710
$$982$$ 0 0
$$983$$ 13.0735 0.416978 0.208489 0.978025i $$-0.433145\pi$$
0.208489 + 0.978025i $$0.433145\pi$$
$$984$$ 7.17891 0.228855
$$985$$ −34.0000 −1.08333
$$986$$ 15.8945 0.506185
$$987$$ −7.17891 −0.228507
$$988$$ 0 0
$$989$$ −99.6102 −3.16742
$$990$$ −10.3578 −0.329193
$$991$$ −49.5367 −1.57359 −0.786793 0.617217i $$-0.788260\pi$$
−0.786793 + 0.617217i $$0.788260\pi$$
$$992$$ 4.17891 0.132680
$$993$$ −1.64218 −0.0521131
$$994$$ −2.17891 −0.0691108
$$995$$ 48.3578 1.53305
$$996$$ 6.17891 0.195786
$$997$$ 12.2844 0.389050 0.194525 0.980898i $$-0.437683\pi$$
0.194525 + 0.980898i $$0.437683\pi$$
$$998$$ 8.17891 0.258899
$$999$$ 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.1.2 2
13.4 even 6 546.2.l.k.211.2 4
13.10 even 6 546.2.l.k.295.2 yes 4
13.12 even 2 7098.2.a.br.1.1 2
39.17 odd 6 1638.2.r.ba.757.1 4
39.23 odd 6 1638.2.r.ba.1387.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.k.211.2 4 13.4 even 6
546.2.l.k.295.2 yes 4 13.10 even 6
1638.2.r.ba.757.1 4 39.17 odd 6
1638.2.r.ba.1387.1 4 39.23 odd 6
7098.2.a.bk.1.2 2 1.1 even 1 trivial
7098.2.a.br.1.1 2 13.12 even 2