# Properties

 Label 845.2.c.g.506.6 Level $845$ Weight $2$ Character 845.506 Analytic conductor $6.747$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 506.6 Root $$-1.27597 - 0.609843i$$ of defining polynomial Character $$\chi$$ $$=$$ 845.506 Dual form 845.2.c.g.506.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.21969i q^{2} +2.33225 q^{3} +0.512364 q^{4} -1.00000i q^{5} +2.84461i q^{6} -3.60020i q^{7} +3.06430i q^{8} +2.43937 q^{9} +O(q^{10})$$ $$q+1.21969i q^{2} +2.33225 q^{3} +0.512364 q^{4} -1.00000i q^{5} +2.84461i q^{6} -3.60020i q^{7} +3.06430i q^{8} +2.43937 q^{9} +1.21969 q^{10} -5.37182i q^{11} +1.19496 q^{12} +4.39111 q^{14} -2.33225i q^{15} -2.71276 q^{16} +1.13186 q^{17} +2.97527i q^{18} +2.26795i q^{19} -0.512364i q^{20} -8.39654i q^{21} +6.55193 q^{22} +3.89287 q^{23} +7.14670i q^{24} -1.00000 q^{25} -1.30752 q^{27} -1.84461i q^{28} -0.0247279 q^{29} +2.84461 q^{30} +5.46410i q^{31} +2.81988i q^{32} -12.5284i q^{33} +1.38051i q^{34} -3.60020 q^{35} +1.24985 q^{36} +8.70406i q^{37} -2.76619 q^{38} +3.06430 q^{40} +3.73205i q^{41} +10.2412 q^{42} +1.13186 q^{43} -2.75232i q^{44} -2.43937i q^{45} +4.74809i q^{46} -2.58535i q^{47} -6.32681 q^{48} -5.96141 q^{49} -1.21969i q^{50} +2.63977 q^{51} -4.43937 q^{53} -1.59476i q^{54} -5.37182 q^{55} +11.0321 q^{56} +5.28942i q^{57} -0.0301603i q^{58} +0.171425i q^{59} -1.19496i q^{60} +3.36023 q^{61} -6.66449 q^{62} -8.78222i q^{63} -8.86488 q^{64} +15.2807 q^{66} +6.39980i q^{67} +0.579922 q^{68} +9.07914 q^{69} -4.39111i q^{70} -10.7973i q^{71} +7.47497i q^{72} +4.70308i q^{73} -10.6162 q^{74} -2.33225 q^{75} +1.16202i q^{76} -19.3396 q^{77} -11.9826 q^{79} +2.71276i q^{80} -10.3676 q^{81} -4.55193 q^{82} -12.1286i q^{83} -4.30209i q^{84} -1.13186i q^{85} +1.38051i q^{86} -0.0576715 q^{87} +16.4608 q^{88} +16.1540i q^{89} +2.97527 q^{90} +1.99457 q^{92} +12.7436i q^{93} +3.15332 q^{94} +2.26795 q^{95} +6.57666i q^{96} +12.1682i q^{97} -7.27105i q^{98} -13.1039i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} - 4q^{4} + 8q^{9} + O(q^{10})$$ $$8q - 4q^{3} - 4q^{4} + 8q^{9} + 4q^{10} + 20q^{12} + 4q^{14} + 4q^{16} + 4q^{17} + 24q^{22} + 20q^{23} - 8q^{25} - 4q^{27} + 16q^{29} - 8q^{30} - 20q^{35} - 40q^{36} - 16q^{38} - 12q^{40} - 8q^{42} + 4q^{43} - 56q^{48} - 24q^{49} - 8q^{51} - 24q^{53} - 24q^{56} + 56q^{61} - 8q^{62} - 8q^{64} + 12q^{66} + 28q^{68} + 32q^{69} - 20q^{74} + 4q^{75} - 36q^{77} - 16q^{79} - 16q^{81} - 8q^{82} - 44q^{87} + 36q^{88} + 40q^{90} + 44q^{92} - 64q^{94} + 32q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-1$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.21969i 0.862449i 0.902245 + 0.431224i $$0.141918\pi$$
−0.902245 + 0.431224i $$0.858082\pi$$
$$3$$ 2.33225 1.34652 0.673262 0.739404i $$-0.264893\pi$$
0.673262 + 0.739404i $$0.264893\pi$$
$$4$$ 0.512364 0.256182
$$5$$ − 1.00000i − 0.447214i
$$6$$ 2.84461i 1.16131i
$$7$$ − 3.60020i − 1.36075i −0.732866 0.680373i $$-0.761818\pi$$
0.732866 0.680373i $$-0.238182\pi$$
$$8$$ 3.06430i 1.08339i
$$9$$ 2.43937 0.813125
$$10$$ 1.21969 0.385699
$$11$$ − 5.37182i − 1.61966i −0.586662 0.809832i $$-0.699558\pi$$
0.586662 0.809832i $$-0.300442\pi$$
$$12$$ 1.19496 0.344955
$$13$$ 0 0
$$14$$ 4.39111 1.17357
$$15$$ − 2.33225i − 0.602183i
$$16$$ −2.71276 −0.678189
$$17$$ 1.13186 0.274515 0.137258 0.990535i $$-0.456171\pi$$
0.137258 + 0.990535i $$0.456171\pi$$
$$18$$ 2.97527i 0.701278i
$$19$$ 2.26795i 0.520303i 0.965568 + 0.260152i $$0.0837725\pi$$
−0.965568 + 0.260152i $$0.916227\pi$$
$$20$$ − 0.512364i − 0.114568i
$$21$$ − 8.39654i − 1.83228i
$$22$$ 6.55193 1.39688
$$23$$ 3.89287 0.811720 0.405860 0.913935i $$-0.366972\pi$$
0.405860 + 0.913935i $$0.366972\pi$$
$$24$$ 7.14670i 1.45881i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.30752 −0.251632
$$28$$ − 1.84461i − 0.348599i
$$29$$ −0.0247279 −0.00459185 −0.00229593 0.999997i $$-0.500731\pi$$
−0.00229593 + 0.999997i $$0.500731\pi$$
$$30$$ 2.84461 0.519352
$$31$$ 5.46410i 0.981382i 0.871334 + 0.490691i $$0.163256\pi$$
−0.871334 + 0.490691i $$0.836744\pi$$
$$32$$ 2.81988i 0.498490i
$$33$$ − 12.5284i − 2.18091i
$$34$$ 1.38051i 0.236755i
$$35$$ −3.60020 −0.608544
$$36$$ 1.24985 0.208308
$$37$$ 8.70406i 1.43094i 0.698644 + 0.715470i $$0.253787\pi$$
−0.698644 + 0.715470i $$0.746213\pi$$
$$38$$ −2.76619 −0.448735
$$39$$ 0 0
$$40$$ 3.06430 0.484508
$$41$$ 3.73205i 0.582848i 0.956594 + 0.291424i $$0.0941291\pi$$
−0.956594 + 0.291424i $$0.905871\pi$$
$$42$$ 10.2412 1.58024
$$43$$ 1.13186 0.172606 0.0863031 0.996269i $$-0.472495\pi$$
0.0863031 + 0.996269i $$0.472495\pi$$
$$44$$ − 2.75232i − 0.414929i
$$45$$ − 2.43937i − 0.363640i
$$46$$ 4.74809i 0.700067i
$$47$$ − 2.58535i − 0.377113i −0.982062 0.188556i $$-0.939619\pi$$
0.982062 0.188556i $$-0.0603808\pi$$
$$48$$ −6.32681 −0.913197
$$49$$ −5.96141 −0.851630
$$50$$ − 1.21969i − 0.172490i
$$51$$ 2.63977 0.369641
$$52$$ 0 0
$$53$$ −4.43937 −0.609795 −0.304897 0.952385i $$-0.598622\pi$$
−0.304897 + 0.952385i $$0.598622\pi$$
$$54$$ − 1.59476i − 0.217020i
$$55$$ −5.37182 −0.724336
$$56$$ 11.0321 1.47422
$$57$$ 5.28942i 0.700600i
$$58$$ − 0.0301603i − 0.00396024i
$$59$$ 0.171425i 0.0223176i 0.999938 + 0.0111588i $$0.00355203\pi$$
−0.999938 + 0.0111588i $$0.996448\pi$$
$$60$$ − 1.19496i − 0.154269i
$$61$$ 3.36023 0.430234 0.215117 0.976588i $$-0.430987\pi$$
0.215117 + 0.976588i $$0.430987\pi$$
$$62$$ −6.66449 −0.846391
$$63$$ − 8.78222i − 1.10646i
$$64$$ −8.86488 −1.10811
$$65$$ 0 0
$$66$$ 15.2807 1.88093
$$67$$ 6.39980i 0.781861i 0.920420 + 0.390930i $$0.127847\pi$$
−0.920420 + 0.390930i $$0.872153\pi$$
$$68$$ 0.579922 0.0703258
$$69$$ 9.07914 1.09300
$$70$$ − 4.39111i − 0.524838i
$$71$$ − 10.7973i − 1.28141i −0.767788 0.640703i $$-0.778643\pi$$
0.767788 0.640703i $$-0.221357\pi$$
$$72$$ 7.47497i 0.880933i
$$73$$ 4.70308i 0.550454i 0.961379 + 0.275227i $$0.0887531\pi$$
−0.961379 + 0.275227i $$0.911247\pi$$
$$74$$ −10.6162 −1.23411
$$75$$ −2.33225 −0.269305
$$76$$ 1.16202i 0.133292i
$$77$$ −19.3396 −2.20395
$$78$$ 0 0
$$79$$ −11.9826 −1.34815 −0.674075 0.738663i $$-0.735457\pi$$
−0.674075 + 0.738663i $$0.735457\pi$$
$$80$$ 2.71276i 0.303295i
$$81$$ −10.3676 −1.15195
$$82$$ −4.55193 −0.502677
$$83$$ − 12.1286i − 1.33129i −0.746270 0.665643i $$-0.768157\pi$$
0.746270 0.665643i $$-0.231843\pi$$
$$84$$ − 4.30209i − 0.469396i
$$85$$ − 1.13186i − 0.122767i
$$86$$ 1.38051i 0.148864i
$$87$$ −0.0576715 −0.00618303
$$88$$ 16.4608 1.75473
$$89$$ 16.1540i 1.71232i 0.516707 + 0.856162i $$0.327158\pi$$
−0.516707 + 0.856162i $$0.672842\pi$$
$$90$$ 2.97527 0.313621
$$91$$ 0 0
$$92$$ 1.99457 0.207948
$$93$$ 12.7436i 1.32145i
$$94$$ 3.15332 0.325240
$$95$$ 2.26795 0.232687
$$96$$ 6.57666i 0.671228i
$$97$$ 12.1682i 1.23549i 0.786379 + 0.617745i $$0.211954\pi$$
−0.786379 + 0.617745i $$0.788046\pi$$
$$98$$ − 7.27105i − 0.734487i
$$99$$ − 13.1039i − 1.31699i
$$100$$ −0.512364 −0.0512364
$$101$$ 4.05441 0.403429 0.201714 0.979444i $$-0.435349\pi$$
0.201714 + 0.979444i $$0.435349\pi$$
$$102$$ 3.21969i 0.318797i
$$103$$ 17.9035 1.76408 0.882041 0.471173i $$-0.156169\pi$$
0.882041 + 0.471173i $$0.156169\pi$$
$$104$$ 0 0
$$105$$ −8.39654 −0.819419
$$106$$ − 5.41465i − 0.525917i
$$107$$ −9.13186 −0.882810 −0.441405 0.897308i $$-0.645520\pi$$
−0.441405 + 0.897308i $$0.645520\pi$$
$$108$$ −0.669925 −0.0644636
$$109$$ 7.37605i 0.706498i 0.935529 + 0.353249i $$0.114923\pi$$
−0.935529 + 0.353249i $$0.885077\pi$$
$$110$$ − 6.55193i − 0.624702i
$$111$$ 20.3000i 1.92679i
$$112$$ 9.76645i 0.922843i
$$113$$ 7.07588 0.665643 0.332821 0.942990i $$-0.391999\pi$$
0.332821 + 0.942990i $$0.391999\pi$$
$$114$$ −6.45143 −0.604232
$$115$$ − 3.89287i − 0.363012i
$$116$$ −0.0126697 −0.00117635
$$117$$ 0 0
$$118$$ −0.209084 −0.0192478
$$119$$ − 4.07490i − 0.373545i
$$120$$ 7.14670 0.652401
$$121$$ −17.8564 −1.62331
$$122$$ 4.09843i 0.371055i
$$123$$ 8.70406i 0.784819i
$$124$$ 2.79961i 0.251412i
$$125$$ 1.00000i 0.0894427i
$$126$$ 10.7116 0.954262
$$127$$ 11.4361 1.01479 0.507395 0.861713i $$-0.330608\pi$$
0.507395 + 0.861713i $$0.330608\pi$$
$$128$$ − 5.17262i − 0.457199i
$$129$$ 2.63977 0.232418
$$130$$ 0 0
$$131$$ −10.5680 −0.923328 −0.461664 0.887055i $$-0.652747\pi$$
−0.461664 + 0.887055i $$0.652747\pi$$
$$132$$ − 6.41910i − 0.558711i
$$133$$ 8.16506 0.708001
$$134$$ −7.80576 −0.674315
$$135$$ 1.30752i 0.112533i
$$136$$ 3.46834i 0.297408i
$$137$$ − 3.78672i − 0.323522i −0.986830 0.161761i $$-0.948283\pi$$
0.986830 0.161761i $$-0.0517173\pi$$
$$138$$ 11.0737i 0.942656i
$$139$$ 2.01386 0.170814 0.0854068 0.996346i $$-0.472781\pi$$
0.0854068 + 0.996346i $$0.472781\pi$$
$$140$$ −1.84461 −0.155898
$$141$$ − 6.02968i − 0.507791i
$$142$$ 13.1694 1.10515
$$143$$ 0 0
$$144$$ −6.61742 −0.551452
$$145$$ 0.0247279i 0.00205354i
$$146$$ −5.73629 −0.474739
$$147$$ −13.9035 −1.14674
$$148$$ 4.45965i 0.366581i
$$149$$ − 5.51780i − 0.452035i −0.974123 0.226018i $$-0.927429\pi$$
0.974123 0.226018i $$-0.0725707\pi$$
$$150$$ − 2.84461i − 0.232261i
$$151$$ 4.88961i 0.397911i 0.980009 + 0.198956i $$0.0637549\pi$$
−0.980009 + 0.198956i $$0.936245\pi$$
$$152$$ −6.94967 −0.563693
$$153$$ 2.76102 0.223215
$$154$$ − 23.5882i − 1.90079i
$$155$$ 5.46410 0.438887
$$156$$ 0 0
$$157$$ 10.0405 0.801323 0.400661 0.916226i $$-0.368780\pi$$
0.400661 + 0.916226i $$0.368780\pi$$
$$158$$ − 14.6150i − 1.16271i
$$159$$ −10.3537 −0.821103
$$160$$ 2.81988 0.222931
$$161$$ − 14.0151i − 1.10454i
$$162$$ − 12.6452i − 0.993501i
$$163$$ − 6.78124i − 0.531148i −0.964090 0.265574i $$-0.914439\pi$$
0.964090 0.265574i $$-0.0855615\pi$$
$$164$$ 1.91217i 0.149315i
$$165$$ −12.5284 −0.975335
$$166$$ 14.7931 1.14817
$$167$$ − 10.4898i − 0.811726i −0.913934 0.405863i $$-0.866971\pi$$
0.913934 0.405863i $$-0.133029\pi$$
$$168$$ 25.7295 1.98507
$$169$$ 0 0
$$170$$ 1.38051 0.105880
$$171$$ 5.53238i 0.423071i
$$172$$ 0.579922 0.0442186
$$173$$ 4.45845 0.338970 0.169485 0.985533i $$-0.445790\pi$$
0.169485 + 0.985533i $$0.445790\pi$$
$$174$$ − 0.0703412i − 0.00533255i
$$175$$ 3.60020i 0.272149i
$$176$$ 14.5724i 1.09844i
$$177$$ 0.399804i 0.0300511i
$$178$$ −19.7029 −1.47679
$$179$$ −18.6313 −1.39257 −0.696284 0.717766i $$-0.745165\pi$$
−0.696284 + 0.717766i $$0.745165\pi$$
$$180$$ − 1.24985i − 0.0931581i
$$181$$ −18.0900 −1.34462 −0.672310 0.740270i $$-0.734698\pi$$
−0.672310 + 0.740270i $$0.734698\pi$$
$$182$$ 0 0
$$183$$ 7.83690 0.579320
$$184$$ 11.9289i 0.879412i
$$185$$ 8.70406 0.639935
$$186$$ −15.5432 −1.13969
$$187$$ − 6.08012i − 0.444622i
$$188$$ − 1.32464i − 0.0966095i
$$189$$ 4.70732i 0.342407i
$$190$$ 2.76619i 0.200680i
$$191$$ 27.3363 1.97799 0.988994 0.147958i $$-0.0472701\pi$$
0.988994 + 0.147958i $$0.0472701\pi$$
$$192$$ −20.6751 −1.49210
$$193$$ − 21.7674i − 1.56685i −0.621486 0.783425i $$-0.713471\pi$$
0.621486 0.783425i $$-0.286529\pi$$
$$194$$ −14.8413 −1.06555
$$195$$ 0 0
$$196$$ −3.05441 −0.218172
$$197$$ − 1.69672i − 0.120886i −0.998172 0.0604432i $$-0.980749\pi$$
0.998172 0.0604432i $$-0.0192514\pi$$
$$198$$ 15.9826 1.13583
$$199$$ −25.3255 −1.79527 −0.897637 0.440735i $$-0.854718\pi$$
−0.897637 + 0.440735i $$0.854718\pi$$
$$200$$ − 3.06430i − 0.216679i
$$201$$ 14.9259i 1.05279i
$$202$$ 4.94511i 0.347937i
$$203$$ 0.0890252i 0.00624834i
$$204$$ 1.35252 0.0946954
$$205$$ 3.73205 0.260658
$$206$$ 21.8366i 1.52143i
$$207$$ 9.49617 0.660030
$$208$$ 0 0
$$209$$ 12.1830 0.842716
$$210$$ − 10.2412i − 0.706707i
$$211$$ −0.335507 −0.0230973 −0.0115486 0.999933i $$-0.503676\pi$$
−0.0115486 + 0.999933i $$0.503676\pi$$
$$212$$ −2.27458 −0.156218
$$213$$ − 25.1820i − 1.72544i
$$214$$ − 11.1380i − 0.761378i
$$215$$ − 1.13186i − 0.0771919i
$$216$$ − 4.00663i − 0.272616i
$$217$$ 19.6718 1.33541
$$218$$ −8.99648 −0.609318
$$219$$ 10.9688i 0.741200i
$$220$$ −2.75232 −0.185562
$$221$$ 0 0
$$222$$ −24.7597 −1.66176
$$223$$ − 12.2968i − 0.823452i −0.911308 0.411726i $$-0.864926\pi$$
0.911308 0.411726i $$-0.135074\pi$$
$$224$$ 10.1521 0.678318
$$225$$ −2.43937 −0.162625
$$226$$ 8.63036i 0.574083i
$$227$$ − 7.63227i − 0.506571i −0.967392 0.253286i $$-0.918489\pi$$
0.967392 0.253286i $$-0.0815112\pi$$
$$228$$ 2.71011i 0.179481i
$$229$$ 14.4008i 0.951631i 0.879545 + 0.475815i $$0.157847\pi$$
−0.879545 + 0.475815i $$0.842153\pi$$
$$230$$ 4.74809 0.313080
$$231$$ −45.1047 −2.96767
$$232$$ − 0.0757736i − 0.00497478i
$$233$$ −9.49617 −0.622115 −0.311057 0.950391i $$-0.600683\pi$$
−0.311057 + 0.950391i $$0.600683\pi$$
$$234$$ 0 0
$$235$$ −2.58535 −0.168650
$$236$$ 0.0878318i 0.00571736i
$$237$$ −27.9464 −1.81531
$$238$$ 4.97010 0.322164
$$239$$ − 19.9143i − 1.28815i −0.764962 0.644076i $$-0.777242\pi$$
0.764962 0.644076i $$-0.222758\pi$$
$$240$$ 6.32681i 0.408394i
$$241$$ − 23.2664i − 1.49872i −0.662163 0.749360i $$-0.730361\pi$$
0.662163 0.749360i $$-0.269639\pi$$
$$242$$ − 21.7792i − 1.40002i
$$243$$ −20.2572 −1.29950
$$244$$ 1.72166 0.110218
$$245$$ 5.96141i 0.380860i
$$246$$ −10.6162 −0.676866
$$247$$ 0 0
$$248$$ −16.7436 −1.06322
$$249$$ − 28.2869i − 1.79261i
$$250$$ −1.21969 −0.0771398
$$251$$ −11.8402 −0.747344 −0.373672 0.927561i $$-0.621901\pi$$
−0.373672 + 0.927561i $$0.621901\pi$$
$$252$$ − 4.49969i − 0.283454i
$$253$$ − 20.9118i − 1.31471i
$$254$$ 13.9485i 0.875205i
$$255$$ − 2.63977i − 0.165309i
$$256$$ −11.4208 −0.713800
$$257$$ 5.55002 0.346201 0.173100 0.984904i $$-0.444621\pi$$
0.173100 + 0.984904i $$0.444621\pi$$
$$258$$ 3.21969i 0.200449i
$$259$$ 31.3363 1.94714
$$260$$ 0 0
$$261$$ −0.0603205 −0.00373375
$$262$$ − 12.8896i − 0.796323i
$$263$$ 6.85967 0.422985 0.211493 0.977380i $$-0.432168\pi$$
0.211493 + 0.977380i $$0.432168\pi$$
$$264$$ 38.3907 2.36279
$$265$$ 4.43937i 0.272709i
$$266$$ 9.95882i 0.610614i
$$267$$ 37.6752i 2.30568i
$$268$$ 3.27903i 0.200299i
$$269$$ −1.42199 −0.0867001 −0.0433501 0.999060i $$-0.513803\pi$$
−0.0433501 + 0.999060i $$0.513803\pi$$
$$270$$ −1.59476 −0.0970542
$$271$$ 9.96947i 0.605602i 0.953054 + 0.302801i $$0.0979218\pi$$
−0.953054 + 0.302801i $$0.902078\pi$$
$$272$$ −3.07045 −0.186173
$$273$$ 0 0
$$274$$ 4.61862 0.279021
$$275$$ 5.37182i 0.323933i
$$276$$ 4.65182 0.280007
$$277$$ 17.5237 1.05290 0.526449 0.850206i $$-0.323523\pi$$
0.526449 + 0.850206i $$0.323523\pi$$
$$278$$ 2.45628i 0.147318i
$$279$$ 13.3290i 0.797986i
$$280$$ − 11.0321i − 0.659292i
$$281$$ − 10.7352i − 0.640406i −0.947349 0.320203i $$-0.896249\pi$$
0.947349 0.320203i $$-0.103751\pi$$
$$282$$ 7.35433 0.437944
$$283$$ −1.31838 −0.0783698 −0.0391849 0.999232i $$-0.512476\pi$$
−0.0391849 + 0.999232i $$0.512476\pi$$
$$284$$ − 5.53216i − 0.328273i
$$285$$ 5.28942 0.313318
$$286$$ 0 0
$$287$$ 13.4361 0.793109
$$288$$ 6.87875i 0.405334i
$$289$$ −15.7189 −0.924641
$$290$$ −0.0301603 −0.00177107
$$291$$ 28.3792i 1.66362i
$$292$$ 2.40969i 0.141016i
$$293$$ − 18.7427i − 1.09496i −0.836820 0.547479i $$-0.815588\pi$$
0.836820 0.547479i $$-0.184412\pi$$
$$294$$ − 16.9579i − 0.989004i
$$295$$ 0.171425 0.00998072
$$296$$ −26.6718 −1.55027
$$297$$ 7.02375i 0.407559i
$$298$$ 6.72998 0.389857
$$299$$ 0 0
$$300$$ −1.19496 −0.0689910
$$301$$ − 4.07490i − 0.234873i
$$302$$ −5.96380 −0.343178
$$303$$ 9.45589 0.543226
$$304$$ − 6.15239i − 0.352864i
$$305$$ − 3.36023i − 0.192406i
$$306$$ 3.36758i 0.192512i
$$307$$ − 14.3043i − 0.816387i −0.912895 0.408194i $$-0.866159\pi$$
0.912895 0.408194i $$-0.133841\pi$$
$$308$$ −9.90891 −0.564612
$$309$$ 41.7553 2.37538
$$310$$ 6.66449i 0.378518i
$$311$$ −2.76102 −0.156563 −0.0782815 0.996931i $$-0.524943\pi$$
−0.0782815 + 0.996931i $$0.524943\pi$$
$$312$$ 0 0
$$313$$ −16.3858 −0.926179 −0.463090 0.886311i $$-0.653259\pi$$
−0.463090 + 0.886311i $$0.653259\pi$$
$$314$$ 12.2463i 0.691100i
$$315$$ −8.78222 −0.494822
$$316$$ −6.13946 −0.345372
$$317$$ 1.78575i 0.100297i 0.998742 + 0.0501487i $$0.0159695\pi$$
−0.998742 + 0.0501487i $$0.984030\pi$$
$$318$$ − 12.6283i − 0.708159i
$$319$$ 0.132834i 0.00743725i
$$320$$ 8.86488i 0.495562i
$$321$$ −21.2977 −1.18872
$$322$$ 17.0940 0.952613
$$323$$ 2.56699i 0.142831i
$$324$$ −5.31197 −0.295110
$$325$$ 0 0
$$326$$ 8.27099 0.458088
$$327$$ 17.2028i 0.951316i
$$328$$ −11.4361 −0.631454
$$329$$ −9.30778 −0.513155
$$330$$ − 15.2807i − 0.841176i
$$331$$ 7.22440i 0.397089i 0.980092 + 0.198545i $$0.0636215\pi$$
−0.980092 + 0.198545i $$0.936379\pi$$
$$332$$ − 6.21425i − 0.341052i
$$333$$ 21.2325i 1.16353i
$$334$$ 12.7943 0.700072
$$335$$ 6.39980 0.349659
$$336$$ 22.7778i 1.24263i
$$337$$ 4.36219 0.237624 0.118812 0.992917i $$-0.462091\pi$$
0.118812 + 0.992917i $$0.462091\pi$$
$$338$$ 0 0
$$339$$ 16.5027 0.896303
$$340$$ − 0.579922i − 0.0314507i
$$341$$ 29.3521 1.58951
$$342$$ −6.74777 −0.364877
$$343$$ − 3.73913i − 0.201894i
$$344$$ 3.46834i 0.187000i
$$345$$ − 9.07914i − 0.488804i
$$346$$ 5.43792i 0.292344i
$$347$$ −26.7072 −1.43372 −0.716858 0.697219i $$-0.754420\pi$$
−0.716858 + 0.697219i $$0.754420\pi$$
$$348$$ −0.0295488 −0.00158398
$$349$$ − 23.5711i − 1.26173i −0.775892 0.630865i $$-0.782700\pi$$
0.775892 0.630865i $$-0.217300\pi$$
$$350$$ −4.39111 −0.234715
$$351$$ 0 0
$$352$$ 15.1479 0.807385
$$353$$ 5.73727i 0.305364i 0.988275 + 0.152682i $$0.0487910\pi$$
−0.988275 + 0.152682i $$0.951209\pi$$
$$354$$ −0.487636 −0.0259176
$$355$$ −10.7973 −0.573063
$$356$$ 8.27675i 0.438667i
$$357$$ − 9.50367i − 0.502988i
$$358$$ − 22.7243i − 1.20102i
$$359$$ 24.7583i 1.30669i 0.757059 + 0.653347i $$0.226636\pi$$
−0.757059 + 0.653347i $$0.773364\pi$$
$$360$$ 7.47497 0.393965
$$361$$ 13.8564 0.729285
$$362$$ − 22.0641i − 1.15967i
$$363$$ −41.6455 −2.18582
$$364$$ 0 0
$$365$$ 4.70308 0.246171
$$366$$ 9.55856i 0.499634i
$$367$$ 26.0535 1.35998 0.679992 0.733220i $$-0.261983\pi$$
0.679992 + 0.733220i $$0.261983\pi$$
$$368$$ −10.5604 −0.550499
$$369$$ 9.10387i 0.473928i
$$370$$ 10.6162i 0.551912i
$$371$$ 15.9826i 0.829776i
$$372$$ 6.52938i 0.338532i
$$373$$ 13.2045 0.683702 0.341851 0.939754i $$-0.388946\pi$$
0.341851 + 0.939754i $$0.388946\pi$$
$$374$$ 7.41584 0.383464
$$375$$ 2.33225i 0.120437i
$$376$$ 7.92229 0.408561
$$377$$ 0 0
$$378$$ −5.74146 −0.295309
$$379$$ 25.9977i 1.33541i 0.744425 + 0.667707i $$0.232724\pi$$
−0.744425 + 0.667707i $$0.767276\pi$$
$$380$$ 1.16202 0.0596101
$$381$$ 26.6718 1.36644
$$382$$ 33.3418i 1.70591i
$$383$$ − 9.60020i − 0.490547i −0.969454 0.245274i $$-0.921122\pi$$
0.969454 0.245274i $$-0.0788778\pi$$
$$384$$ − 12.0638i − 0.615629i
$$385$$ 19.3396i 0.985637i
$$386$$ 26.5494 1.35133
$$387$$ 2.76102 0.140350
$$388$$ 6.23453i 0.316510i
$$389$$ −5.63129 −0.285518 −0.142759 0.989758i $$-0.545597\pi$$
−0.142759 + 0.989758i $$0.545597\pi$$
$$390$$ 0 0
$$391$$ 4.40617 0.222829
$$392$$ − 18.2675i − 0.922650i
$$393$$ −24.6471 −1.24328
$$394$$ 2.06947 0.104258
$$395$$ 11.9826i 0.602911i
$$396$$ − 6.71395i − 0.337389i
$$397$$ 16.7658i 0.841452i 0.907188 + 0.420726i $$0.138225\pi$$
−0.907188 + 0.420726i $$0.861775\pi$$
$$398$$ − 30.8891i − 1.54833i
$$399$$ 19.0429 0.953339
$$400$$ 2.71276 0.135638
$$401$$ 13.8780i 0.693036i 0.938043 + 0.346518i $$0.112636\pi$$
−0.938043 + 0.346518i $$0.887364\pi$$
$$402$$ −18.2050 −0.907980
$$403$$ 0 0
$$404$$ 2.07733 0.103351
$$405$$ 10.3676i 0.515169i
$$406$$ −0.108583 −0.00538888
$$407$$ 46.7566 2.31764
$$408$$ 8.08903i 0.400466i
$$409$$ 29.4251i 1.45498i 0.686120 + 0.727489i $$0.259313\pi$$
−0.686120 + 0.727489i $$0.740687\pi$$
$$410$$ 4.55193i 0.224804i
$$411$$ − 8.83157i − 0.435629i
$$412$$ 9.17310 0.451926
$$413$$ 0.617162 0.0303686
$$414$$ 11.5824i 0.569242i
$$415$$ −12.1286 −0.595369
$$416$$ 0 0
$$417$$ 4.69683 0.230005
$$418$$ 14.8595i 0.726800i
$$419$$ 6.96793 0.340406 0.170203 0.985409i $$-0.445558\pi$$
0.170203 + 0.985409i $$0.445558\pi$$
$$420$$ −4.30209 −0.209920
$$421$$ 7.12125i 0.347069i 0.984828 + 0.173534i $$0.0555188\pi$$
−0.984828 + 0.173534i $$0.944481\pi$$
$$422$$ − 0.409213i − 0.0199202i
$$423$$ − 6.30664i − 0.306640i
$$424$$ − 13.6036i − 0.660647i
$$425$$ −1.13186 −0.0549030
$$426$$ 30.7142 1.48811
$$427$$ − 12.0975i − 0.585439i
$$428$$ −4.67883 −0.226160
$$429$$ 0 0
$$430$$ 1.38051 0.0665740
$$431$$ 30.2144i 1.45537i 0.685909 + 0.727687i $$0.259405\pi$$
−0.685909 + 0.727687i $$0.740595\pi$$
$$432$$ 3.54698 0.170654
$$433$$ −1.20013 −0.0576745 −0.0288373 0.999584i $$-0.509180\pi$$
−0.0288373 + 0.999584i $$0.509180\pi$$
$$434$$ 23.9935i 1.15172i
$$435$$ 0.0576715i 0.00276514i
$$436$$ 3.77922i 0.180992i
$$437$$ 8.82884i 0.422341i
$$438$$ −13.3784 −0.639247
$$439$$ 16.5541 0.790084 0.395042 0.918663i $$-0.370730\pi$$
0.395042 + 0.918663i $$0.370730\pi$$
$$440$$ − 16.4608i − 0.784740i
$$441$$ −14.5421 −0.692481
$$442$$ 0 0
$$443$$ 4.55949 0.216628 0.108314 0.994117i $$-0.465455\pi$$
0.108314 + 0.994117i $$0.465455\pi$$
$$444$$ 10.4010i 0.493610i
$$445$$ 16.1540 0.765775
$$446$$ 14.9982 0.710185
$$447$$ − 12.8689i − 0.608676i
$$448$$ 31.9153i 1.50786i
$$449$$ − 13.8522i − 0.653724i −0.945072 0.326862i $$-0.894009\pi$$
0.945072 0.326862i $$-0.105991\pi$$
$$450$$ − 2.97527i − 0.140256i
$$451$$ 20.0479 0.944018
$$452$$ 3.62542 0.170526
$$453$$ 11.4038i 0.535796i
$$454$$ 9.30897 0.436892
$$455$$ 0 0
$$456$$ −16.2083 −0.759025
$$457$$ 40.1146i 1.87648i 0.345984 + 0.938240i $$0.387545\pi$$
−0.345984 + 0.938240i $$0.612455\pi$$
$$458$$ −17.5644 −0.820733
$$459$$ −1.47992 −0.0690768
$$460$$ − 1.99457i − 0.0929972i
$$461$$ − 7.53900i − 0.351126i −0.984468 0.175563i $$-0.943825\pi$$
0.984468 0.175563i $$-0.0561746\pi$$
$$462$$ − 55.0136i − 2.55946i
$$463$$ − 23.3031i − 1.08299i −0.840705 0.541494i $$-0.817859\pi$$
0.840705 0.541494i $$-0.182141\pi$$
$$464$$ 0.0670807 0.00311414
$$465$$ 12.7436 0.590972
$$466$$ − 11.5824i − 0.536542i
$$467$$ 22.6297 1.04718 0.523589 0.851971i $$-0.324593\pi$$
0.523589 + 0.851971i $$0.324593\pi$$
$$468$$ 0 0
$$469$$ 23.0405 1.06391
$$470$$ − 3.15332i − 0.145452i
$$471$$ 23.4170 1.07900
$$472$$ −0.525296 −0.0241787
$$473$$ − 6.08012i − 0.279564i
$$474$$ − 34.0859i − 1.56562i
$$475$$ − 2.26795i − 0.104061i
$$476$$ − 2.08783i − 0.0956956i
$$477$$ −10.8293 −0.495839
$$478$$ 24.2893 1.11096
$$479$$ − 20.6448i − 0.943286i −0.881790 0.471643i $$-0.843661\pi$$
0.881790 0.471643i $$-0.156339\pi$$
$$480$$ 6.57666 0.300182
$$481$$ 0 0
$$482$$ 28.3777 1.29257
$$483$$ − 32.6867i − 1.48730i
$$484$$ −9.14898 −0.415863
$$485$$ 12.1682 0.552528
$$486$$ − 24.7074i − 1.12075i
$$487$$ 3.03605i 0.137576i 0.997631 + 0.0687882i $$0.0219133\pi$$
−0.997631 + 0.0687882i $$0.978087\pi$$
$$488$$ 10.2968i 0.466112i
$$489$$ − 15.8155i − 0.715203i
$$490$$ −7.27105 −0.328473
$$491$$ −10.6680 −0.481441 −0.240720 0.970595i $$-0.577384\pi$$
−0.240720 + 0.970595i $$0.577384\pi$$
$$492$$ 4.45965i 0.201056i
$$493$$ −0.0279884 −0.00126053
$$494$$ 0 0
$$495$$ −13.1039 −0.588975
$$496$$ − 14.8228i − 0.665562i
$$497$$ −38.8725 −1.74367
$$498$$ 34.5011 1.54603
$$499$$ 33.9143i 1.51821i 0.650966 + 0.759107i $$0.274364\pi$$
−0.650966 + 0.759107i $$0.725636\pi$$
$$500$$ 0.512364i 0.0229136i
$$501$$ − 24.4648i − 1.09301i
$$502$$ − 14.4413i − 0.644546i
$$503$$ −12.6276 −0.563037 −0.281518 0.959556i $$-0.590838\pi$$
−0.281518 + 0.959556i $$0.590838\pi$$
$$504$$ 26.9113 1.19873
$$505$$ − 4.05441i − 0.180419i
$$506$$ 25.5058 1.13387
$$507$$ 0 0
$$508$$ 5.85945 0.259971
$$509$$ − 24.1526i − 1.07055i −0.844679 0.535273i $$-0.820209\pi$$
0.844679 0.535273i $$-0.179791\pi$$
$$510$$ 3.21969 0.142570
$$511$$ 16.9320 0.749029
$$512$$ − 24.2750i − 1.07281i
$$513$$ − 2.96539i − 0.130925i
$$514$$ 6.76929i 0.298581i
$$515$$ − 17.9035i − 0.788921i
$$516$$ 1.35252 0.0595414
$$517$$ −13.8880 −0.610796
$$518$$ 38.2205i 1.67931i
$$519$$ 10.3982 0.456431
$$520$$ 0 0
$$521$$ −24.7521 −1.08441 −0.542205 0.840246i $$-0.682410\pi$$
−0.542205 + 0.840246i $$0.682410\pi$$
$$522$$ − 0.0735722i − 0.00322017i
$$523$$ 37.0326 1.61932 0.809662 0.586897i $$-0.199650\pi$$
0.809662 + 0.586897i $$0.199650\pi$$
$$524$$ −5.41465 −0.236540
$$525$$ 8.39654i 0.366455i
$$526$$ 8.36665i 0.364803i
$$527$$ 6.18457i 0.269404i
$$528$$ 33.9865i 1.47907i
$$529$$ −7.84554 −0.341111
$$530$$ −5.41465 −0.235197
$$531$$ 0.418169i 0.0181470i
$$532$$ 4.18348 0.181377
$$533$$ 0 0
$$534$$ −45.9519 −1.98854
$$535$$ 9.13186i 0.394805i
$$536$$ −19.6109 −0.847062
$$537$$ −43.4528 −1.87512
$$538$$ − 1.73438i − 0.0747744i
$$539$$ 32.0236i 1.37935i
$$540$$ 0.669925i 0.0288290i
$$541$$ 8.38144i 0.360346i 0.983635 + 0.180173i $$0.0576658\pi$$
−0.983635 + 0.180173i $$0.942334\pi$$
$$542$$ −12.1596 −0.522301
$$543$$ −42.1903 −1.81056
$$544$$ 3.19170i 0.136843i
$$545$$ 7.37605 0.315955
$$546$$ 0 0
$$547$$ −22.7842 −0.974181 −0.487091 0.873351i $$-0.661942\pi$$
−0.487091 + 0.873351i $$0.661942\pi$$
$$548$$ − 1.94018i − 0.0828804i
$$549$$ 8.19687 0.349834
$$550$$ −6.55193 −0.279375
$$551$$ − 0.0560816i − 0.00238915i
$$552$$ 27.8212i 1.18415i
$$553$$ 43.1398i 1.83449i
$$554$$ 21.3735i 0.908071i
$$555$$ 20.3000 0.861688
$$556$$ 1.03183 0.0437594
$$557$$ 28.1527i 1.19287i 0.802662 + 0.596435i $$0.203417\pi$$
−0.802662 + 0.596435i $$0.796583\pi$$
$$558$$ −16.2572 −0.688222
$$559$$ 0 0
$$560$$ 9.76645 0.412708
$$561$$ − 14.1803i − 0.598694i
$$562$$ 13.0935 0.552317
$$563$$ 18.1303 0.764101 0.382050 0.924142i $$-0.375218\pi$$
0.382050 + 0.924142i $$0.375218\pi$$
$$564$$ − 3.08939i − 0.130087i
$$565$$ − 7.07588i − 0.297684i
$$566$$ − 1.60801i − 0.0675899i
$$567$$ 37.3253i 1.56752i
$$568$$ 33.0862 1.38827
$$569$$ −40.5985 −1.70198 −0.850988 0.525185i $$-0.823996\pi$$
−0.850988 + 0.525185i $$0.823996\pi$$
$$570$$ 6.45143i 0.270221i
$$571$$ −24.7159 −1.03433 −0.517164 0.855886i $$-0.673012\pi$$
−0.517164 + 0.855886i $$0.673012\pi$$
$$572$$ 0 0
$$573$$ 63.7551 2.66341
$$574$$ 16.3879i 0.684016i
$$575$$ −3.89287 −0.162344
$$576$$ −21.6248 −0.901032
$$577$$ 23.0691i 0.960379i 0.877165 + 0.480189i $$0.159432\pi$$
−0.877165 + 0.480189i $$0.840568\pi$$
$$578$$ − 19.1721i − 0.797456i
$$579$$ − 50.7669i − 2.10980i
$$580$$ 0.0126697i 0 0.000526079i
$$581$$ −43.6653 −1.81154
$$582$$ −34.6137 −1.43478
$$583$$ 23.8475i 0.987662i
$$584$$ −14.4116 −0.596358
$$585$$ 0 0
$$586$$ 22.8602 0.944345
$$587$$ − 20.3523i − 0.840030i −0.907517 0.420015i $$-0.862025\pi$$
0.907517 0.420015i $$-0.137975\pi$$
$$588$$ −7.12364 −0.293774
$$589$$ −12.3923 −0.510616
$$590$$ 0.209084i 0.00860786i
$$591$$ − 3.95717i − 0.162776i
$$592$$ − 23.6120i − 0.970447i
$$593$$ − 10.3834i − 0.426395i −0.977009 0.213198i $$-0.931612\pi$$
0.977009 0.213198i $$-0.0683878\pi$$
$$594$$ −8.56677 −0.351499
$$595$$ −4.07490 −0.167055
$$596$$ − 2.82712i − 0.115803i
$$597$$ −59.0652 −2.41738
$$598$$ 0 0
$$599$$ −31.5965 −1.29100 −0.645499 0.763761i $$-0.723351\pi$$
−0.645499 + 0.763761i $$0.723351\pi$$
$$600$$ − 7.14670i − 0.291763i
$$601$$ −43.8845 −1.79009 −0.895044 0.445979i $$-0.852856\pi$$
−0.895044 + 0.445979i $$0.852856\pi$$
$$602$$ 4.97010 0.202566
$$603$$ 15.6115i 0.635750i
$$604$$ 2.50526i 0.101938i
$$605$$ 17.8564i 0.725966i
$$606$$ 11.5332i 0.468505i
$$607$$ 2.17540 0.0882968 0.0441484 0.999025i $$-0.485943\pi$$
0.0441484 + 0.999025i $$0.485943\pi$$
$$608$$ −6.39535 −0.259366
$$609$$ 0.207629i 0.00841354i
$$610$$ 4.09843 0.165941
$$611$$ 0 0
$$612$$ 1.41465 0.0571837
$$613$$ − 14.7620i − 0.596231i −0.954530 0.298116i $$-0.903642\pi$$
0.954530 0.298116i $$-0.0963581\pi$$
$$614$$ 17.4467 0.704092
$$615$$ 8.70406 0.350982
$$616$$ − 59.2622i − 2.38774i
$$617$$ − 20.2972i − 0.817134i −0.912728 0.408567i $$-0.866029\pi$$
0.912728 0.408567i $$-0.133971\pi$$
$$618$$ 50.9284i 2.04864i
$$619$$ − 9.94207i − 0.399605i −0.979836 0.199803i $$-0.935970\pi$$
0.979836 0.199803i $$-0.0640301\pi$$
$$620$$ 2.79961 0.112435
$$621$$ −5.09000 −0.204255
$$622$$ − 3.36758i − 0.135028i
$$623$$ 58.1577 2.33004
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 19.9855i − 0.798782i
$$627$$ 28.4138 1.13474
$$628$$ 5.14441 0.205284
$$629$$ 9.85174i 0.392815i
$$630$$ − 10.7116i − 0.426759i
$$631$$ − 0.973420i − 0.0387512i −0.999812 0.0193756i $$-0.993832\pi$$
0.999812 0.0193756i $$-0.00616784\pi$$
$$632$$ − 36.7183i − 1.46058i
$$633$$ −0.782485 −0.0311010
$$634$$ −2.17805 −0.0865014
$$635$$ − 11.4361i − 0.453828i
$$636$$ −5.30487 −0.210352
$$637$$ 0 0
$$638$$ −0.162015 −0.00641425
$$639$$ − 26.3387i − 1.04194i
$$640$$ −5.17262 −0.204466
$$641$$ 12.6209 0.498497 0.249249 0.968440i $$-0.419816\pi$$
0.249249 + 0.968440i $$0.419816\pi$$
$$642$$ − 25.9766i − 1.02521i
$$643$$ 9.96043i 0.392801i 0.980524 + 0.196401i $$0.0629253\pi$$
−0.980524 + 0.196401i $$0.937075\pi$$
$$644$$ − 7.18083i − 0.282964i
$$645$$ − 2.63977i − 0.103941i
$$646$$ −3.13092 −0.123185
$$647$$ 36.2763 1.42617 0.713084 0.701079i $$-0.247298\pi$$
0.713084 + 0.701079i $$0.247298\pi$$
$$648$$ − 31.7693i − 1.24802i
$$649$$ 0.920861 0.0361470
$$650$$ 0 0
$$651$$ 45.8796 1.79816
$$652$$ − 3.47447i − 0.136071i
$$653$$ 13.7554 0.538289 0.269145 0.963100i $$-0.413259\pi$$
0.269145 + 0.963100i $$0.413259\pi$$
$$654$$ −20.9820 −0.820461
$$655$$ 10.5680i 0.412925i
$$656$$ − 10.1241i − 0.395281i
$$657$$ 11.4726i 0.447588i
$$658$$ − 11.3526i − 0.442570i
$$659$$ −2.58183 −0.100574 −0.0502869 0.998735i $$-0.516014\pi$$
−0.0502869 + 0.998735i $$0.516014\pi$$
$$660$$ −6.41910 −0.249863
$$661$$ − 24.8765i − 0.967582i −0.875183 0.483791i $$-0.839259\pi$$
0.875183 0.483791i $$-0.160741\pi$$
$$662$$ −8.81151 −0.342469
$$663$$ 0 0
$$664$$ 37.1656 1.44231
$$665$$ − 8.16506i − 0.316627i
$$666$$ −25.8970 −1.00349
$$667$$ −0.0962625 −0.00372730
$$668$$ − 5.37460i − 0.207949i
$$669$$ − 28.6791i − 1.10880i
$$670$$ 7.80576i 0.301563i
$$671$$ − 18.0506i − 0.696834i
$$672$$ 23.6773 0.913370
$$673$$ −43.3222 −1.66995 −0.834974 0.550289i $$-0.814517\pi$$
−0.834974 + 0.550289i $$0.814517\pi$$
$$674$$ 5.32051i 0.204938i
$$675$$ 1.30752 0.0503264
$$676$$ 0 0
$$677$$ −41.3625 −1.58969 −0.794845 0.606813i $$-0.792448\pi$$
−0.794845 + 0.606813i $$0.792448\pi$$
$$678$$ 20.1281i 0.773016i
$$679$$ 43.8078 1.68119
$$680$$ 3.46834 0.133005
$$681$$ − 17.8003i − 0.682110i
$$682$$ 35.8004i 1.37087i
$$683$$ − 2.62688i − 0.100515i −0.998736 0.0502574i $$-0.983996\pi$$
0.998736 0.0502574i $$-0.0160042\pi$$
$$684$$ 2.83459i 0.108383i
$$685$$ −3.78672 −0.144683
$$686$$ 4.56057 0.174123
$$687$$ 33.5862i 1.28139i
$$688$$ −3.07045 −0.117060
$$689$$ 0 0
$$690$$ 11.0737 0.421569
$$691$$ − 15.2753i − 0.581099i −0.956860 0.290550i $$-0.906162\pi$$
0.956860 0.290550i $$-0.0938382\pi$$
$$692$$ 2.28435 0.0868380
$$693$$ −47.1765 −1.79209
$$694$$ − 32.5744i − 1.23651i
$$695$$ − 2.01386i − 0.0763902i
$$696$$ − 0.176723i − 0.00669865i
$$697$$ 4.22414i 0.160001i
$$698$$ 28.7493 1.08818
$$699$$ −22.1474 −0.837692
$$700$$ 1.84461i 0.0697197i
$$701$$ 48.1947 1.82029 0.910144 0.414292i $$-0.135971\pi$$
0.910144 + 0.414292i $$0.135971\pi$$
$$702$$ 0 0
$$703$$ −19.7404 −0.744522
$$704$$ 47.6205i 1.79477i
$$705$$ −6.02968 −0.227091
$$706$$ −6.99767 −0.263361
$$707$$ − 14.5967i − 0.548964i
$$708$$ 0.204845i 0.00769856i
$$709$$ 38.8699i 1.45979i 0.683559 + 0.729896i $$0.260431\pi$$
−0.683559 + 0.729896i $$0.739569\pi$$
$$710$$ − 13.1694i − 0.494237i
$$711$$ −29.2301 −1.09621
$$712$$ −49.5008 −1.85512
$$713$$ 21.2711i 0.796607i
$$714$$ 11.5915 0.433801
$$715$$ 0 0
$$716$$ −9.54600 −0.356751
$$717$$ − 46.4452i − 1.73453i
$$718$$ −30.1974 −1.12696
$$719$$ −6.61660 −0.246758 −0.123379 0.992360i $$-0.539373\pi$$
−0.123379 + 0.992360i $$0.539373\pi$$
$$720$$ 6.61742i 0.246617i
$$721$$ − 64.4560i − 2.40047i
$$722$$ 16.9005i 0.628971i
$$723$$ − 54.2629i − 2.01806i
$$724$$ −9.26867 −0.344467
$$725$$ 0.0247279 0.000918370 0
$$726$$ − 50.7945i − 1.88516i
$$727$$ 18.3735 0.681435 0.340717 0.940166i $$-0.389330\pi$$
0.340717 + 0.940166i $$0.389330\pi$$
$$728$$ 0 0
$$729$$ −16.1420 −0.597853
$$730$$ 5.73629i 0.212310i
$$731$$ 1.28110 0.0473830
$$732$$ 4.01534 0.148411
$$733$$ 0.791131i 0.0292211i 0.999893 + 0.0146105i $$0.00465084\pi$$
−0.999893 + 0.0146105i $$0.995349\pi$$
$$734$$ 31.7772i 1.17292i
$$735$$ 13.9035i 0.512837i
$$736$$ 10.9774i 0.404634i
$$737$$ 34.3786 1.26635
$$738$$ −11.1039 −0.408739
$$739$$ − 31.1853i − 1.14717i −0.819146 0.573585i $$-0.805552\pi$$
0.819146 0.573585i $$-0.194448\pi$$
$$740$$ 4.45965 0.163940
$$741$$ 0 0
$$742$$ −19.4938 −0.715639
$$743$$ 5.56304i 0.204088i 0.994780 + 0.102044i $$0.0325383\pi$$
−0.994780 + 0.102044i $$0.967462\pi$$
$$744$$ −39.0503 −1.43165
$$745$$ −5.51780 −0.202156
$$746$$ 16.1053i 0.589658i
$$747$$ − 29.5862i − 1.08250i
$$748$$ − 3.11523i − 0.113904i
$$749$$ 32.8765i 1.20128i
$$750$$ −2.84461 −0.103870
$$751$$ 35.2097 1.28482 0.642410 0.766361i $$-0.277935\pi$$
0.642410 + 0.766361i $$0.277935\pi$$
$$752$$ 7.01343i 0.255754i
$$753$$ −27.6142 −1.00632
$$754$$ 0 0
$$755$$ 4.88961 0.177951
$$756$$ 2.41186i 0.0877186i
$$757$$ 50.0446 1.81890 0.909451 0.415810i $$-0.136502\pi$$
0.909451 + 0.415810i $$0.136502\pi$$
$$758$$ −31.7091 −1.15173
$$759$$ − 48.7715i − 1.77029i
$$760$$ 6.94967i 0.252091i
$$761$$ − 44.8209i − 1.62476i −0.583130 0.812379i $$-0.698172\pi$$
0.583130 0.812379i $$-0.301828\pi$$
$$762$$ 32.5313i 1.17848i
$$763$$ 26.5552 0.961364
$$764$$ 14.0061 0.506725
$$765$$ − 2.76102i − 0.0998248i
$$766$$ 11.7092 0.423072
$$767$$ 0 0
$$768$$ −26.6361 −0.961148
$$769$$ − 39.3633i − 1.41948i −0.704465 0.709739i $$-0.748813\pi$$
0.704465 0.709739i $$-0.251187\pi$$
$$770$$ −23.5882 −0.850061
$$771$$ 12.9440 0.466168
$$772$$ − 11.1528i − 0.401399i
$$773$$ 48.7805i 1.75451i 0.480022 + 0.877256i $$0.340629\pi$$
−0.480022 + 0.877256i $$0.659371\pi$$
$$774$$ 3.36758i 0.121045i
$$775$$ − 5.46410i − 0.196276i
$$776$$ −37.2869 −1.33852
$$777$$ 73.0840 2.62188
$$778$$ − 6.86841i − 0.246244i
$$779$$ −8.46410 −0.303258
$$780$$ 0 0
$$781$$ −58.0013 −2.07545
$$782$$ 5.37415i 0.192179i
$$783$$ 0.0323322 0.00115546
$$784$$ 16.1718 0.577566
$$785$$ − 10.0405i − 0.358363i
$$786$$ − 30.0618i − 1.07227i
$$787$$ − 39.8608i − 1.42088i −0.703756 0.710442i $$-0.748495\pi$$
0.703756 0.710442i $$-0.251505\pi$$
$$788$$ − 0.869338i − 0.0309689i
$$789$$ 15.9984 0.569559
$$790$$ −14.6150 −0.519980
$$791$$ − 25.4745i − 0.905771i
$$792$$ 40.1541 1.42682
$$793$$ 0 0
$$794$$ −20.4490 −0.725709
$$795$$ 10.3537i 0.367208i
$$796$$ −12.9759 −0.459917
$$797$$ −26.2118 −0.928470 −0.464235 0.885712i $$-0.653671\pi$$
−0.464235 + 0.885712i $$0.653671\pi$$
$$798$$ 23.2264i 0.822206i
$$799$$ − 2.92625i − 0.103523i
$$800$$ − 2.81988i − 0.0996979i
$$801$$ 39.4057i 1.39233i
$$802$$ −16.9269 −0.597708
$$803$$ 25.2641 0.891551
$$804$$ 7.64750i 0.269707i
$$805$$ −14.0151 −0.493968
$$806$$ 0 0
$$807$$ −3.31643 −0.116744
$$808$$ 12.4239i 0.437072i
$$809$$ 22.2136 0.780990 0.390495 0.920605i $$-0.372304\pi$$
0.390495 + 0.920605i $$0.372304\pi$$
$$810$$ −12.6452 −0.444307
$$811$$ 19.0950i 0.670515i 0.942127 + 0.335257i $$0.108823\pi$$
−0.942127 + 0.335257i $$0.891177\pi$$
$$812$$ 0.0456133i 0.00160071i
$$813$$ 23.2513i 0.815457i
$$814$$ 57.0284i 1.99885i
$$815$$ −6.78124 −0.237537
$$816$$ −7.16104 −0.250686
$$817$$ 2.56699i 0.0898076i
$$818$$ −35.8894 −1.25484
$$819$$ 0 0
$$820$$ 1.91217 0.0667758
$$821$$ − 34.1584i − 1.19214i −0.802934 0.596068i $$-0.796729\pi$$
0.802934 0.596068i $$-0.203271\pi$$
$$822$$ 10.7718 0.375708
$$823$$ −4.07940 −0.142199 −0.0710995 0.997469i $$-0.522651\pi$$
−0.0710995 + 0.997469i $$0.522651\pi$$
$$824$$ 54.8616i 1.91119i
$$825$$ 12.5284i 0.436183i
$$826$$ 0.752744i 0.0261913i
$$827$$ 54.8780i 1.90830i 0.299337 + 0.954148i $$0.403235\pi$$
−0.299337 + 0.954148i $$0.596765\pi$$
$$828$$ 4.86550 0.169088
$$829$$ −8.14950 −0.283044 −0.141522 0.989935i $$-0.545200\pi$$
−0.141522 + 0.989935i $$0.545200\pi$$
$$830$$ − 14.7931i − 0.513476i
$$831$$ 40.8697 1.41775
$$832$$ 0 0
$$833$$ −6.74745 −0.233785
$$834$$ 5.72866i 0.198367i
$$835$$ −10.4898 −0.363015
$$836$$ 6.24213 0.215889
$$837$$ − 7.14441i − 0.246947i
$$838$$ 8.49869i 0.293582i
$$839$$ 21.8865i 0.755606i 0.925886 + 0.377803i $$0.123320\pi$$
−0.925886 + 0.377803i $$0.876680\pi$$
$$840$$ − 25.7295i − 0.887752i
$$841$$ −28.9994 −0.999979
$$842$$ −8.68570 −0.299329
$$843$$ − 25.0370i − 0.862321i
$$844$$ −0.171902 −0.00591710
$$845$$ 0 0
$$846$$ 7.69213 0.264461
$$847$$ 64.2866i 2.20891i
$$848$$ 12.0429 0.413556
$$849$$ −3.07480 −0.105527
$$850$$ − 1.38051i − 0.0473511i
$$851$$ 33.8838i 1.16152i
$$852$$ − 12.9024i − 0.442028i
$$853$$ 19.2240i 0.658217i 0.944292 + 0.329108i $$0.106748\pi$$
−0.944292 + 0.329108i $$0.893252\pi$$
$$854$$ 14.7552 0.504911
$$855$$ 5.53238 0.189203
$$856$$ − 27.9827i − 0.956430i
$$857$$ 27.8197 0.950302 0.475151 0.879904i $$-0.342393\pi$$
0.475151 + 0.879904i $$0.342393\pi$$
$$858$$ 0 0
$$859$$ 45.7355 1.56048 0.780238 0.625482i $$-0.215098\pi$$
0.780238 + 0.625482i $$0.215098\pi$$
$$860$$ − 0.579922i − 0.0197752i
$$861$$ 31.3363 1.06794
$$862$$ −36.8521 −1.25519
$$863$$ 54.8186i 1.86605i 0.359814 + 0.933024i $$0.382840\pi$$
−0.359814 + 0.933024i $$0.617160\pi$$
$$864$$ − 3.68705i − 0.125436i
$$865$$ − 4.45845i − 0.151592i
$$866$$ − 1.46378i − 0.0497413i
$$867$$ −36.6604 −1.24505
$$868$$ 10.0791 0.342108
$$869$$ 64.3684i 2.18355i
$$870$$ −0.0703412 −0.00238479
$$871$$ 0 0
$$872$$ −22.6024 −0.765415
$$873$$ 29.6827i 1.00461i
$$874$$ −10.7684 −0.364247
$$875$$ 3.60020 0.121709
$$876$$ 5.61999i 0.189882i
$$877$$ − 27.3794i − 0.924537i −0.886740 0.462269i $$-0.847036\pi$$
0.886740 0.462269i $$-0.152964\pi$$
$$878$$ 20.1908i 0.681407i
$$879$$ − 43.7125i − 1.47439i
$$880$$ 14.5724 0.491236
$$881$$ 34.4426 1.16040 0.580200 0.814474i $$-0.302974\pi$$
0.580200 + 0.814474i $$0.302974\pi$$
$$882$$ − 17.7368i − 0.597230i
$$883$$ 17.3592 0.584183 0.292092 0.956390i $$-0.405649\pi$$
0.292092 + 0.956390i $$0.405649\pi$$
$$884$$ 0 0
$$885$$ 0.399804 0.0134393
$$886$$ 5.56115i 0.186831i
$$887$$ −31.1427 −1.04567 −0.522835 0.852434i $$-0.675126\pi$$
−0.522835 + 0.852434i $$0.675126\pi$$
$$888$$ −62.2053 −2.08747
$$889$$ − 41.1722i − 1.38087i
$$890$$ 19.7029i 0.660442i
$$891$$ 55.6927i 1.86578i
$$892$$ − 6.30042i − 0.210954i
$$893$$ 5.86345 0.196213
$$894$$ 15.6960 0.524952
$$895$$ 18.6313i 0.622775i
$$896$$ −18.6224 −0.622132
$$897$$ 0 0
$$898$$ 16.8953 0.563804
$$899$$ − 0.135116i − 0.00450636i
$$900$$ −1.24985 −0.0416616
$$901$$ −5.02473 −0.167398
$$902$$ 24.4521i 0.814167i
$$903$$ − 9.50367i − 0.316262i
$$904$$ 21.6826i 0.721152i
$$905$$ 18.0900i 0.601332i
$$906$$ −13.9090 −0.462097
$$907$$ −17.6057 −0.584587 −0.292294 0.956329i $$-0.594418\pi$$
−0.292294 + 0.956329i $$0.594418\pi$$
$$908$$ − 3.91050i − 0.129774i
$$909$$ 9.89022 0.328038
$$910$$ 0 0
$$911$$ 50.0232 1.65734 0.828671 0.559737i $$-0.189098\pi$$
0.828671 + 0.559737i $$0.189098\pi$$
$$912$$ − 14.3489i − 0.475139i
$$913$$ −65.1526 −2.15624
$$914$$ −48.9272 −1.61837
$$915$$ − 7.83690i − 0.259080i
$$916$$ 7.37844i 0.243791i
$$917$$ 38.0468i 1.25641i
$$918$$ − 1.80504i − 0.0595752i
$$919$$ −7.61556 −0.251214 −0.125607 0.992080i $$-0.540088\pi$$
−0.125607 + 0.992080i $$0.540088\pi$$
$$920$$ 11.9289 0.393285
$$921$$ − 33.3611i − 1.09928i
$$922$$ 9.19522 0.302828
$$923$$ 0 0
$$924$$ −23.1100 −0.760264
$$925$$ − 8.70406i − 0.286188i
$$926$$ 28.4225 0.934022
$$927$$ 43.6733 1.43442
$$928$$ − 0.0697297i − 0.00228899i
$$929$$ − 14.1239i − 0.463391i −0.972788 0.231695i $$-0.925573\pi$$
0.972788 0.231695i $$-0.0744273\pi$$
$$930$$ 15.5432i 0.509683i
$$931$$ − 13.5202i − 0.443106i
$$932$$ −4.86550 −0.159375
$$933$$ −6.43937 −0.210816
$$934$$ 27.6012i 0.903138i
$$935$$ −6.08012 −0.198841
$$936$$ 0 0
$$937$$ −23.9317 −0.781815 −0.390908 0.920430i $$-0.627839\pi$$
−0.390908 + 0.920430i $$0.627839\pi$$
$$938$$ 28.1023i 0.917571i
$$939$$ −38.2157 −1.24712
$$940$$ −1.32464 −0.0432051
$$941$$ − 25.3591i − 0.826683i −0.910576 0.413342i $$-0.864362\pi$$
0.910576 0.413342i $$-0.135638\pi$$
$$942$$ 28.5614i 0.930582i
$$943$$ 14.5284i 0.473110i
$$944$$ − 0.465033i − 0.0151355i
$$945$$ 4.70732 0.153129
$$946$$ 7.41584 0.241110
$$947$$ − 41.4223i − 1.34604i −0.739623 0.673021i $$-0.764996\pi$$
0.739623 0.673021i $$-0.235004\pi$$
$$948$$ −14.3187 −0.465051
$$949$$ 0 0
$$950$$ 2.76619 0.0897470
$$951$$ 4.16480i 0.135053i
$$952$$ 12.4867 0.404696
$$953$$ −24.3026 −0.787237 −0.393619 0.919274i $$-0.628777\pi$$
−0.393619 + 0.919274i $$0.628777\pi$$
$$954$$ − 13.2083i − 0.427636i
$$955$$ − 27.3363i − 0.884583i
$$956$$ − 10.2034i − 0.330001i
$$957$$ 0.309801i 0.0100144i
$$958$$ 25.1802 0.813536
$$959$$ −13.6329 −0.440231
$$960$$ 20.6751i 0.667286i
$$961$$ 1.14359 0.0368901
$$962$$ 0 0
$$963$$ −22.2760 −0.717834
$$964$$ − 11.9209i − 0.383945i
$$965$$ −21.7674 −0.700717
$$966$$ 39.8675 1.28272
$$967$$ 23.6784i 0.761445i 0.924689 + 0.380722i $$0.124325\pi$$
−0.924689 + 0.380722i $$0.875675\pi$$
$$968$$ − 54.7173i − 1.75868i
$$969$$ 5.98685i 0.192325i
$$970$$ 14.8413i 0.476527i
$$971$$ −16.9722 −0.544663 −0.272332 0.962203i $$-0.587795\pi$$
−0.272332 + 0.962203i $$0.587795\pi$$
$$972$$ −10.3791 −0.332908
$$973$$ − 7.25030i − 0.232434i
$$974$$ −3.70303 −0.118653
$$975$$ 0 0
$$976$$ −9.11550 −0.291780
$$977$$ 24.9994i 0.799802i 0.916558 + 0.399901i $$0.130956\pi$$
−0.916558 + 0.399901i $$0.869044\pi$$
$$978$$ 19.2900 0.616826
$$979$$ 86.7765 2.77339
$$980$$ 3.05441i 0.0975696i
$$981$$ 17.9930i 0.574471i
$$982$$ − 13.0116i − 0.415218i
$$983$$ 27.3418i 0.872068i 0.899930 + 0.436034i $$0.143617\pi$$
−0.899930 + 0.436034i $$0.856383\pi$$
$$984$$ −26.6718 −0.850267
$$985$$ −1.69672 −0.0540620
$$986$$ − 0.0341370i − 0.00108715i
$$987$$ −21.7080 −0.690975
$$988$$ 0 0
$$989$$ 4.40617 0.140108
$$990$$ − 15.9826i − 0.507961i
$$991$$ 16.0760 0.510672 0.255336 0.966852i $$-0.417814\pi$$
0.255336 + 0.966852i $$0.417814\pi$$
$$992$$ −15.4081 −0.489208
$$993$$ 16.8491i 0.534690i
$$994$$ − 47.4123i − 1.50383i
$$995$$ 25.3255i 0.802871i
$$996$$ − 14.4932i − 0.459234i
$$997$$ −34.5612 −1.09456 −0.547282 0.836948i $$-0.684337\pi$$
−0.547282 + 0.836948i $$0.684337\pi$$
$$998$$ −41.3649 −1.30938
$$999$$ − 11.3807i − 0.360070i
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 845.2.c.g.506.6 8
13.2 odd 12 845.2.e.m.191.2 8
13.3 even 3 845.2.m.g.316.2 8
13.4 even 6 845.2.m.g.361.2 8
13.5 odd 4 845.2.a.m.1.3 4
13.6 odd 12 845.2.e.m.146.2 8
13.7 odd 12 845.2.e.n.146.3 8
13.8 odd 4 845.2.a.l.1.2 4
13.9 even 3 65.2.m.a.36.3 8
13.10 even 6 65.2.m.a.56.3 yes 8
13.11 odd 12 845.2.e.n.191.3 8
13.12 even 2 inner 845.2.c.g.506.3 8
39.5 even 4 7605.2.a.cf.1.2 4
39.8 even 4 7605.2.a.cj.1.3 4
39.23 odd 6 585.2.bu.c.316.2 8
39.35 odd 6 585.2.bu.c.361.2 8
52.23 odd 6 1040.2.da.b.641.4 8
52.35 odd 6 1040.2.da.b.881.4 8
65.9 even 6 325.2.n.d.101.2 8
65.22 odd 12 325.2.m.c.49.3 8
65.23 odd 12 325.2.m.c.199.3 8
65.34 odd 4 4225.2.a.bl.1.3 4
65.44 odd 4 4225.2.a.bi.1.2 4
65.48 odd 12 325.2.m.b.49.2 8
65.49 even 6 325.2.n.d.251.2 8
65.62 odd 12 325.2.m.b.199.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.3 8 13.9 even 3
65.2.m.a.56.3 yes 8 13.10 even 6
325.2.m.b.49.2 8 65.48 odd 12
325.2.m.b.199.2 8 65.62 odd 12
325.2.m.c.49.3 8 65.22 odd 12
325.2.m.c.199.3 8 65.23 odd 12
325.2.n.d.101.2 8 65.9 even 6
325.2.n.d.251.2 8 65.49 even 6
585.2.bu.c.316.2 8 39.23 odd 6
585.2.bu.c.361.2 8 39.35 odd 6
845.2.a.l.1.2 4 13.8 odd 4
845.2.a.m.1.3 4 13.5 odd 4
845.2.c.g.506.3 8 13.12 even 2 inner
845.2.c.g.506.6 8 1.1 even 1 trivial
845.2.e.m.146.2 8 13.6 odd 12
845.2.e.m.191.2 8 13.2 odd 12
845.2.e.n.146.3 8 13.7 odd 12
845.2.e.n.191.3 8 13.11 odd 12
845.2.m.g.316.2 8 13.3 even 3
845.2.m.g.361.2 8 13.4 even 6
1040.2.da.b.641.4 8 52.23 odd 6
1040.2.da.b.881.4 8 52.35 odd 6
4225.2.a.bi.1.2 4 65.44 odd 4
4225.2.a.bl.1.3 4 65.34 odd 4
7605.2.a.cf.1.2 4 39.5 even 4
7605.2.a.cj.1.3 4 39.8 even 4