# Properties

 Label 3850.2.c.ba.1849.1 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 770) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.1 Root $$0.264658 - 1.38923i$$ of defining polynomial Character $$\chi$$ $$=$$ 3850.1849 Dual form 3850.2.c.ba.1849.6

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -2.24914i q^{3} -1.00000 q^{4} -2.24914 q^{6} +1.00000i q^{7} +1.00000i q^{8} -2.05863 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -2.24914i q^{3} -1.00000 q^{4} -2.24914 q^{6} +1.00000i q^{7} +1.00000i q^{8} -2.05863 q^{9} +1.00000 q^{11} +2.24914i q^{12} +0.941367i q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.49828i q^{17} +2.05863i q^{18} +4.36641 q^{19} +2.24914 q^{21} -1.00000i q^{22} +6.24914i q^{23} +2.24914 q^{24} +0.941367 q^{26} -2.11727i q^{27} -1.00000i q^{28} -8.74742 q^{29} -9.55691 q^{31} -1.00000i q^{32} -2.24914i q^{33} -6.49828 q^{34} +2.05863 q^{36} -4.24914i q^{37} -4.36641i q^{38} +2.11727 q^{39} -2.13187 q^{41} -2.24914i q^{42} +7.67418i q^{43} -1.00000 q^{44} +6.24914 q^{46} -11.1138i q^{47} -2.24914i q^{48} -1.00000 q^{49} -14.6155 q^{51} -0.941367i q^{52} -4.74742i q^{53} -2.11727 q^{54} -1.00000 q^{56} -9.82066i q^{57} +8.74742i q^{58} +1.88273 q^{59} +9.11383 q^{61} +9.55691i q^{62} -2.05863i q^{63} -1.00000 q^{64} -2.24914 q^{66} -12.9966i q^{67} +6.49828i q^{68} +14.0552 q^{69} -14.6155 q^{71} -2.05863i q^{72} -10.4983i q^{73} -4.24914 q^{74} -4.36641 q^{76} +1.00000i q^{77} -2.11727i q^{78} -8.36641 q^{79} -10.9379 q^{81} +2.13187i q^{82} -8.49828i q^{83} -2.24914 q^{84} +7.67418 q^{86} +19.6742i q^{87} +1.00000i q^{88} -12.3810 q^{89} -0.941367 q^{91} -6.24914i q^{92} +21.4948i q^{93} -11.1138 q^{94} -2.24914 q^{96} +15.3630i q^{97} +1.00000i q^{98} -2.05863 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} + 4 q^{6} - 14 q^{9} + O(q^{10})$$ $$6 q - 6 q^{4} + 4 q^{6} - 14 q^{9} + 6 q^{11} + 6 q^{14} + 6 q^{16} + 12 q^{19} - 4 q^{21} - 4 q^{24} + 4 q^{26} - 24 q^{31} - 4 q^{34} + 14 q^{36} + 16 q^{39} + 8 q^{41} - 6 q^{44} + 20 q^{46} - 6 q^{49} - 56 q^{51} - 16 q^{54} - 6 q^{56} + 8 q^{59} - 12 q^{61} - 6 q^{64} + 4 q^{66} + 16 q^{69} - 56 q^{71} - 8 q^{74} - 12 q^{76} - 36 q^{79} + 6 q^{81} + 4 q^{84} + 16 q^{86} - 36 q^{89} - 4 q^{91} + 4 q^{96} - 14 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ − 2.24914i − 1.29854i −0.760557 0.649271i $$-0.775074\pi$$
0.760557 0.649271i $$-0.224926\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −2.24914 −0.918208
$$7$$ 1.00000i 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ −2.05863 −0.686211
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 2.24914i 0.649271i
$$13$$ 0.941367i 0.261088i 0.991443 + 0.130544i $$0.0416724\pi$$
−0.991443 + 0.130544i $$0.958328\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 6.49828i − 1.57606i −0.615634 0.788032i $$-0.711100\pi$$
0.615634 0.788032i $$-0.288900\pi$$
$$18$$ 2.05863i 0.485224i
$$19$$ 4.36641 1.00172 0.500861 0.865528i $$-0.333017\pi$$
0.500861 + 0.865528i $$0.333017\pi$$
$$20$$ 0 0
$$21$$ 2.24914 0.490803
$$22$$ − 1.00000i − 0.213201i
$$23$$ 6.24914i 1.30304i 0.758633 + 0.651518i $$0.225867\pi$$
−0.758633 + 0.651518i $$0.774133\pi$$
$$24$$ 2.24914 0.459104
$$25$$ 0 0
$$26$$ 0.941367 0.184617
$$27$$ − 2.11727i − 0.407468i
$$28$$ − 1.00000i − 0.188982i
$$29$$ −8.74742 −1.62436 −0.812178 0.583410i $$-0.801718\pi$$
−0.812178 + 0.583410i $$0.801718\pi$$
$$30$$ 0 0
$$31$$ −9.55691 −1.71647 −0.858236 0.513255i $$-0.828440\pi$$
−0.858236 + 0.513255i $$0.828440\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 2.24914i − 0.391525i
$$34$$ −6.49828 −1.11445
$$35$$ 0 0
$$36$$ 2.05863 0.343106
$$37$$ − 4.24914i − 0.698554i −0.937019 0.349277i $$-0.886427\pi$$
0.937019 0.349277i $$-0.113573\pi$$
$$38$$ − 4.36641i − 0.708325i
$$39$$ 2.11727 0.339034
$$40$$ 0 0
$$41$$ −2.13187 −0.332943 −0.166471 0.986046i $$-0.553237\pi$$
−0.166471 + 0.986046i $$0.553237\pi$$
$$42$$ − 2.24914i − 0.347050i
$$43$$ 7.67418i 1.17030i 0.810925 + 0.585151i $$0.198965\pi$$
−0.810925 + 0.585151i $$0.801035\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ 6.24914 0.921386
$$47$$ − 11.1138i − 1.62112i −0.585657 0.810559i $$-0.699163\pi$$
0.585657 0.810559i $$-0.300837\pi$$
$$48$$ − 2.24914i − 0.324635i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −14.6155 −2.04659
$$52$$ − 0.941367i − 0.130544i
$$53$$ − 4.74742i − 0.652109i −0.945351 0.326054i $$-0.894281\pi$$
0.945351 0.326054i $$-0.105719\pi$$
$$54$$ −2.11727 −0.288123
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ − 9.82066i − 1.30078i
$$58$$ 8.74742i 1.14859i
$$59$$ 1.88273 0.245111 0.122556 0.992462i $$-0.460891\pi$$
0.122556 + 0.992462i $$0.460891\pi$$
$$60$$ 0 0
$$61$$ 9.11383 1.16691 0.583453 0.812147i $$-0.301701\pi$$
0.583453 + 0.812147i $$0.301701\pi$$
$$62$$ 9.55691i 1.21373i
$$63$$ − 2.05863i − 0.259363i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.24914 −0.276850
$$67$$ − 12.9966i − 1.58778i −0.608060 0.793891i $$-0.708052\pi$$
0.608060 0.793891i $$-0.291948\pi$$
$$68$$ 6.49828i 0.788032i
$$69$$ 14.0552 1.69205
$$70$$ 0 0
$$71$$ −14.6155 −1.73455 −0.867273 0.497833i $$-0.834129\pi$$
−0.867273 + 0.497833i $$0.834129\pi$$
$$72$$ − 2.05863i − 0.242612i
$$73$$ − 10.4983i − 1.22873i −0.789022 0.614365i $$-0.789412\pi$$
0.789022 0.614365i $$-0.210588\pi$$
$$74$$ −4.24914 −0.493953
$$75$$ 0 0
$$76$$ −4.36641 −0.500861
$$77$$ 1.00000i 0.113961i
$$78$$ − 2.11727i − 0.239733i
$$79$$ −8.36641 −0.941294 −0.470647 0.882322i $$-0.655980\pi$$
−0.470647 + 0.882322i $$0.655980\pi$$
$$80$$ 0 0
$$81$$ −10.9379 −1.21533
$$82$$ 2.13187i 0.235426i
$$83$$ − 8.49828i − 0.932808i −0.884572 0.466404i $$-0.845549\pi$$
0.884572 0.466404i $$-0.154451\pi$$
$$84$$ −2.24914 −0.245401
$$85$$ 0 0
$$86$$ 7.67418 0.827528
$$87$$ 19.6742i 2.10929i
$$88$$ 1.00000i 0.106600i
$$89$$ −12.3810 −1.31238 −0.656192 0.754594i $$-0.727834\pi$$
−0.656192 + 0.754594i $$0.727834\pi$$
$$90$$ 0 0
$$91$$ −0.941367 −0.0986821
$$92$$ − 6.24914i − 0.651518i
$$93$$ 21.4948i 2.22891i
$$94$$ −11.1138 −1.14630
$$95$$ 0 0
$$96$$ −2.24914 −0.229552
$$97$$ 15.3630i 1.55987i 0.625859 + 0.779937i $$0.284749\pi$$
−0.625859 + 0.779937i $$0.715251\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ −2.05863 −0.206900
$$100$$ 0 0
$$101$$ −16.8793 −1.67955 −0.839776 0.542932i $$-0.817314\pi$$
−0.839776 + 0.542932i $$0.817314\pi$$
$$102$$ 14.6155i 1.44715i
$$103$$ − 6.61555i − 0.651849i −0.945396 0.325925i $$-0.894324\pi$$
0.945396 0.325925i $$-0.105676\pi$$
$$104$$ −0.941367 −0.0923086
$$105$$ 0 0
$$106$$ −4.74742 −0.461110
$$107$$ 14.5535i 1.40694i 0.710726 + 0.703469i $$0.248367\pi$$
−0.710726 + 0.703469i $$0.751633\pi$$
$$108$$ 2.11727i 0.203734i
$$109$$ 0.249141 0.0238633 0.0119317 0.999929i $$-0.496202\pi$$
0.0119317 + 0.999929i $$0.496202\pi$$
$$110$$ 0 0
$$111$$ −9.55691 −0.907102
$$112$$ 1.00000i 0.0944911i
$$113$$ 10.9966i 1.03447i 0.855844 + 0.517235i $$0.173039\pi$$
−0.855844 + 0.517235i $$0.826961\pi$$
$$114$$ −9.82066 −0.919789
$$115$$ 0 0
$$116$$ 8.74742 0.812178
$$117$$ − 1.93793i − 0.179162i
$$118$$ − 1.88273i − 0.173320i
$$119$$ 6.49828 0.595696
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 9.11383i − 0.825127i
$$123$$ 4.79488i 0.432340i
$$124$$ 9.55691 0.858236
$$125$$ 0 0
$$126$$ −2.05863 −0.183398
$$127$$ 5.88273i 0.522008i 0.965338 + 0.261004i $$0.0840536\pi$$
−0.965338 + 0.261004i $$0.915946\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 17.2603 1.51969
$$130$$ 0 0
$$131$$ −1.25258 −0.109438 −0.0547191 0.998502i $$-0.517426\pi$$
−0.0547191 + 0.998502i $$0.517426\pi$$
$$132$$ 2.24914i 0.195763i
$$133$$ 4.36641i 0.378615i
$$134$$ −12.9966 −1.12273
$$135$$ 0 0
$$136$$ 6.49828 0.557223
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ − 14.0552i − 1.19646i
$$139$$ −10.9820 −0.931477 −0.465739 0.884922i $$-0.654211\pi$$
−0.465739 + 0.884922i $$0.654211\pi$$
$$140$$ 0 0
$$141$$ −24.9966 −2.10509
$$142$$ 14.6155i 1.22651i
$$143$$ 0.941367i 0.0787210i
$$144$$ −2.05863 −0.171553
$$145$$ 0 0
$$146$$ −10.4983 −0.868844
$$147$$ 2.24914i 0.185506i
$$148$$ 4.24914i 0.349277i
$$149$$ −0.0146079 −0.00119673 −0.000598363 1.00000i $$-0.500190\pi$$
−0.000598363 1.00000i $$0.500190\pi$$
$$150$$ 0 0
$$151$$ −15.2457 −1.24068 −0.620339 0.784334i $$-0.713005\pi$$
−0.620339 + 0.784334i $$0.713005\pi$$
$$152$$ 4.36641i 0.354162i
$$153$$ 13.3776i 1.08151i
$$154$$ 1.00000 0.0805823
$$155$$ 0 0
$$156$$ −2.11727 −0.169517
$$157$$ 5.50172i 0.439085i 0.975603 + 0.219542i $$0.0704565\pi$$
−0.975603 + 0.219542i $$0.929544\pi$$
$$158$$ 8.36641i 0.665596i
$$159$$ −10.6776 −0.846790
$$160$$ 0 0
$$161$$ −6.24914 −0.492501
$$162$$ 10.9379i 0.859365i
$$163$$ − 18.2277i − 1.42770i −0.700298 0.713850i $$-0.746950\pi$$
0.700298 0.713850i $$-0.253050\pi$$
$$164$$ 2.13187 0.166471
$$165$$ 0 0
$$166$$ −8.49828 −0.659595
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 2.24914i 0.173525i
$$169$$ 12.1138 0.931833
$$170$$ 0 0
$$171$$ −8.98883 −0.687393
$$172$$ − 7.67418i − 0.585151i
$$173$$ 0.117266i 0.00891559i 0.999990 + 0.00445780i $$0.00141897\pi$$
−0.999990 + 0.00445780i $$0.998581\pi$$
$$174$$ 19.6742 1.49150
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ − 4.23453i − 0.318287i
$$178$$ 12.3810i 0.927996i
$$179$$ 22.5535 1.68573 0.842863 0.538128i $$-0.180868\pi$$
0.842863 + 0.538128i $$0.180868\pi$$
$$180$$ 0 0
$$181$$ 20.8793 1.55195 0.775973 0.630766i $$-0.217259\pi$$
0.775973 + 0.630766i $$0.217259\pi$$
$$182$$ 0.941367i 0.0697788i
$$183$$ − 20.4983i − 1.51528i
$$184$$ −6.24914 −0.460693
$$185$$ 0 0
$$186$$ 21.4948 1.57608
$$187$$ − 6.49828i − 0.475201i
$$188$$ 11.1138i 0.810559i
$$189$$ 2.11727 0.154008
$$190$$ 0 0
$$191$$ 3.11383 0.225309 0.112654 0.993634i $$-0.464065\pi$$
0.112654 + 0.993634i $$0.464065\pi$$
$$192$$ 2.24914i 0.162318i
$$193$$ − 6.17246i − 0.444304i −0.975012 0.222152i $$-0.928692\pi$$
0.975012 0.222152i $$-0.0713080\pi$$
$$194$$ 15.3630 1.10300
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 6.73281i − 0.479693i −0.970811 0.239847i $$-0.922903\pi$$
0.970811 0.239847i $$-0.0770971\pi$$
$$198$$ 2.05863i 0.146301i
$$199$$ 13.2311 0.937927 0.468964 0.883217i $$-0.344627\pi$$
0.468964 + 0.883217i $$0.344627\pi$$
$$200$$ 0 0
$$201$$ −29.2311 −2.06180
$$202$$ 16.8793i 1.18762i
$$203$$ − 8.74742i − 0.613949i
$$204$$ 14.6155 1.02329
$$205$$ 0 0
$$206$$ −6.61555 −0.460927
$$207$$ − 12.8647i − 0.894158i
$$208$$ 0.941367i 0.0652720i
$$209$$ 4.36641 0.302031
$$210$$ 0 0
$$211$$ 23.1138 1.59122 0.795611 0.605808i $$-0.207150\pi$$
0.795611 + 0.605808i $$0.207150\pi$$
$$212$$ 4.74742i 0.326054i
$$213$$ 32.8724i 2.25238i
$$214$$ 14.5535 0.994855
$$215$$ 0 0
$$216$$ 2.11727 0.144062
$$217$$ − 9.55691i − 0.648766i
$$218$$ − 0.249141i − 0.0168739i
$$219$$ −23.6121 −1.59556
$$220$$ 0 0
$$221$$ 6.11727 0.411492
$$222$$ 9.55691i 0.641418i
$$223$$ 10.3810i 0.695164i 0.937650 + 0.347582i $$0.112997\pi$$
−0.937650 + 0.347582i $$0.887003\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 10.9966 0.731480
$$227$$ − 16.1104i − 1.06928i −0.845079 0.534642i $$-0.820446\pi$$
0.845079 0.534642i $$-0.179554\pi$$
$$228$$ 9.82066i 0.650389i
$$229$$ −24.2897 −1.60511 −0.802555 0.596578i $$-0.796527\pi$$
−0.802555 + 0.596578i $$0.796527\pi$$
$$230$$ 0 0
$$231$$ 2.24914 0.147983
$$232$$ − 8.74742i − 0.574296i
$$233$$ 6.88617i 0.451128i 0.974228 + 0.225564i $$0.0724225\pi$$
−0.974228 + 0.225564i $$0.927578\pi$$
$$234$$ −1.93793 −0.126686
$$235$$ 0 0
$$236$$ −1.88273 −0.122556
$$237$$ 18.8172i 1.22231i
$$238$$ − 6.49828i − 0.421221i
$$239$$ 11.1353 0.720283 0.360142 0.932898i $$-0.382728\pi$$
0.360142 + 0.932898i $$0.382728\pi$$
$$240$$ 0 0
$$241$$ 2.36641 0.152434 0.0762168 0.997091i $$-0.475716\pi$$
0.0762168 + 0.997091i $$0.475716\pi$$
$$242$$ − 1.00000i − 0.0642824i
$$243$$ 18.2491i 1.17068i
$$244$$ −9.11383 −0.583453
$$245$$ 0 0
$$246$$ 4.79488 0.305711
$$247$$ 4.11039i 0.261538i
$$248$$ − 9.55691i − 0.606865i
$$249$$ −19.1138 −1.21129
$$250$$ 0 0
$$251$$ 25.1070 1.58474 0.792368 0.610043i $$-0.208848\pi$$
0.792368 + 0.610043i $$0.208848\pi$$
$$252$$ 2.05863i 0.129682i
$$253$$ 6.24914i 0.392880i
$$254$$ 5.88273 0.369116
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 2.86469i 0.178694i 0.996001 + 0.0893472i $$0.0284781\pi$$
−0.996001 + 0.0893472i $$0.971522\pi$$
$$258$$ − 17.2603i − 1.07458i
$$259$$ 4.24914 0.264029
$$260$$ 0 0
$$261$$ 18.0077 1.11465
$$262$$ 1.25258i 0.0773846i
$$263$$ 3.76547i 0.232189i 0.993238 + 0.116094i $$0.0370375\pi$$
−0.993238 + 0.116094i $$0.962963\pi$$
$$264$$ 2.24914 0.138425
$$265$$ 0 0
$$266$$ 4.36641 0.267722
$$267$$ 27.8466i 1.70419i
$$268$$ 12.9966i 0.793891i
$$269$$ −8.94137 −0.545165 −0.272582 0.962132i $$-0.587878\pi$$
−0.272582 + 0.962132i $$0.587878\pi$$
$$270$$ 0 0
$$271$$ 21.4948 1.30572 0.652859 0.757479i $$-0.273569\pi$$
0.652859 + 0.757479i $$0.273569\pi$$
$$272$$ − 6.49828i − 0.394016i
$$273$$ 2.11727i 0.128143i
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ −14.0552 −0.846023
$$277$$ − 7.64820i − 0.459536i −0.973245 0.229768i $$-0.926203\pi$$
0.973245 0.229768i $$-0.0737967\pi$$
$$278$$ 10.9820i 0.658654i
$$279$$ 19.6742 1.17786
$$280$$ 0 0
$$281$$ −28.6155 −1.70706 −0.853530 0.521043i $$-0.825543\pi$$
−0.853530 + 0.521043i $$0.825543\pi$$
$$282$$ 24.9966i 1.48852i
$$283$$ − 2.87930i − 0.171156i −0.996331 0.0855782i $$-0.972726\pi$$
0.996331 0.0855782i $$-0.0272737\pi$$
$$284$$ 14.6155 0.867273
$$285$$ 0 0
$$286$$ 0.941367 0.0556642
$$287$$ − 2.13187i − 0.125841i
$$288$$ 2.05863i 0.121306i
$$289$$ −25.2277 −1.48398
$$290$$ 0 0
$$291$$ 34.5535 2.02556
$$292$$ 10.4983i 0.614365i
$$293$$ − 12.9414i − 0.756043i −0.925797 0.378021i $$-0.876605\pi$$
0.925797 0.378021i $$-0.123395\pi$$
$$294$$ 2.24914 0.131173
$$295$$ 0 0
$$296$$ 4.24914 0.246976
$$297$$ − 2.11727i − 0.122856i
$$298$$ 0.0146079i 0 0.000846213i
$$299$$ −5.88273 −0.340207
$$300$$ 0 0
$$301$$ −7.67418 −0.442332
$$302$$ 15.2457i 0.877292i
$$303$$ 37.9639i 2.18097i
$$304$$ 4.36641 0.250431
$$305$$ 0 0
$$306$$ 13.3776 0.764745
$$307$$ − 0.498281i − 0.0284384i −0.999899 0.0142192i $$-0.995474\pi$$
0.999899 0.0142192i $$-0.00452626\pi$$
$$308$$ − 1.00000i − 0.0569803i
$$309$$ −14.8793 −0.846454
$$310$$ 0 0
$$311$$ −23.4396 −1.32914 −0.664570 0.747226i $$-0.731385\pi$$
−0.664570 + 0.747226i $$0.731385\pi$$
$$312$$ 2.11727i 0.119867i
$$313$$ 9.36984i 0.529615i 0.964301 + 0.264807i $$0.0853084\pi$$
−0.964301 + 0.264807i $$0.914692\pi$$
$$314$$ 5.50172 0.310480
$$315$$ 0 0
$$316$$ 8.36641 0.470647
$$317$$ 9.97852i 0.560449i 0.959934 + 0.280225i $$0.0904090\pi$$
−0.959934 + 0.280225i $$0.909591\pi$$
$$318$$ 10.6776i 0.598771i
$$319$$ −8.74742 −0.489762
$$320$$ 0 0
$$321$$ 32.7328 1.82697
$$322$$ 6.24914i 0.348251i
$$323$$ − 28.3741i − 1.57878i
$$324$$ 10.9379 0.607663
$$325$$ 0 0
$$326$$ −18.2277 −1.00954
$$327$$ − 0.560352i − 0.0309875i
$$328$$ − 2.13187i − 0.117713i
$$329$$ 11.1138 0.612725
$$330$$ 0 0
$$331$$ −20.4362 −1.12328 −0.561638 0.827383i $$-0.689829\pi$$
−0.561638 + 0.827383i $$0.689829\pi$$
$$332$$ 8.49828i 0.466404i
$$333$$ 8.74742i 0.479356i
$$334$$ 8.00000 0.437741
$$335$$ 0 0
$$336$$ 2.24914 0.122701
$$337$$ − 7.88273i − 0.429400i −0.976680 0.214700i $$-0.931123\pi$$
0.976680 0.214700i $$-0.0688774\pi$$
$$338$$ − 12.1138i − 0.658905i
$$339$$ 24.7328 1.34330
$$340$$ 0 0
$$341$$ −9.55691 −0.517536
$$342$$ 8.98883i 0.486060i
$$343$$ − 1.00000i − 0.0539949i
$$344$$ −7.67418 −0.413764
$$345$$ 0 0
$$346$$ 0.117266 0.00630428
$$347$$ − 13.5569i − 0.727773i −0.931443 0.363887i $$-0.881450\pi$$
0.931443 0.363887i $$-0.118550\pi$$
$$348$$ − 19.6742i − 1.05465i
$$349$$ 10.7328 0.574514 0.287257 0.957853i $$-0.407257\pi$$
0.287257 + 0.957853i $$0.407257\pi$$
$$350$$ 0 0
$$351$$ 1.99312 0.106385
$$352$$ − 1.00000i − 0.0533002i
$$353$$ − 20.8578i − 1.11015i −0.831801 0.555075i $$-0.812690\pi$$
0.831801 0.555075i $$-0.187310\pi$$
$$354$$ −4.23453 −0.225063
$$355$$ 0 0
$$356$$ 12.3810 0.656192
$$357$$ − 14.6155i − 0.773537i
$$358$$ − 22.5535i − 1.19199i
$$359$$ −0.366407 −0.0193382 −0.00966911 0.999953i $$-0.503078\pi$$
−0.00966911 + 0.999953i $$0.503078\pi$$
$$360$$ 0 0
$$361$$ 0.0655089 0.00344783
$$362$$ − 20.8793i − 1.09739i
$$363$$ − 2.24914i − 0.118049i
$$364$$ 0.941367 0.0493410
$$365$$ 0 0
$$366$$ −20.4983 −1.07146
$$367$$ − 20.8432i − 1.08801i −0.839083 0.544003i $$-0.816908\pi$$
0.839083 0.544003i $$-0.183092\pi$$
$$368$$ 6.24914i 0.325759i
$$369$$ 4.38875 0.228469
$$370$$ 0 0
$$371$$ 4.74742 0.246474
$$372$$ − 21.4948i − 1.11446i
$$373$$ − 5.37758i − 0.278440i −0.990261 0.139220i $$-0.955540\pi$$
0.990261 0.139220i $$-0.0444596\pi$$
$$374$$ −6.49828 −0.336018
$$375$$ 0 0
$$376$$ 11.1138 0.573152
$$377$$ − 8.23453i − 0.424100i
$$378$$ − 2.11727i − 0.108900i
$$379$$ −3.53093 −0.181372 −0.0906860 0.995880i $$-0.528906\pi$$
−0.0906860 + 0.995880i $$0.528906\pi$$
$$380$$ 0 0
$$381$$ 13.2311 0.677850
$$382$$ − 3.11383i − 0.159317i
$$383$$ 1.38445i 0.0707422i 0.999374 + 0.0353711i $$0.0112613\pi$$
−0.999374 + 0.0353711i $$0.988739\pi$$
$$384$$ 2.24914 0.114776
$$385$$ 0 0
$$386$$ −6.17246 −0.314170
$$387$$ − 15.7983i − 0.803074i
$$388$$ − 15.3630i − 0.779937i
$$389$$ −15.7294 −0.797511 −0.398756 0.917057i $$-0.630558\pi$$
−0.398756 + 0.917057i $$0.630558\pi$$
$$390$$ 0 0
$$391$$ 40.6087 2.05367
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 2.81722i 0.142110i
$$394$$ −6.73281 −0.339194
$$395$$ 0 0
$$396$$ 2.05863 0.103450
$$397$$ 17.6121i 0.883926i 0.897033 + 0.441963i $$0.145718\pi$$
−0.897033 + 0.441963i $$0.854282\pi$$
$$398$$ − 13.2311i − 0.663215i
$$399$$ 9.82066 0.491648
$$400$$ 0 0
$$401$$ −32.8172 −1.63881 −0.819407 0.573212i $$-0.805697\pi$$
−0.819407 + 0.573212i $$0.805697\pi$$
$$402$$ 29.2311i 1.45791i
$$403$$ − 8.99656i − 0.448151i
$$404$$ 16.8793 0.839776
$$405$$ 0 0
$$406$$ −8.74742 −0.434127
$$407$$ − 4.24914i − 0.210622i
$$408$$ − 14.6155i − 0.723577i
$$409$$ 33.3561 1.64935 0.824676 0.565605i $$-0.191357\pi$$
0.824676 + 0.565605i $$0.191357\pi$$
$$410$$ 0 0
$$411$$ −22.4914 −1.10942
$$412$$ 6.61555i 0.325925i
$$413$$ 1.88273i 0.0926433i
$$414$$ −12.8647 −0.632265
$$415$$ 0 0
$$416$$ 0.941367 0.0461543
$$417$$ 24.7000i 1.20956i
$$418$$ − 4.36641i − 0.213568i
$$419$$ −12.3449 −0.603089 −0.301544 0.953452i $$-0.597502\pi$$
−0.301544 + 0.953452i $$0.597502\pi$$
$$420$$ 0 0
$$421$$ −8.87930 −0.432750 −0.216375 0.976310i $$-0.569423\pi$$
−0.216375 + 0.976310i $$0.569423\pi$$
$$422$$ − 23.1138i − 1.12516i
$$423$$ 22.8793i 1.11243i
$$424$$ 4.74742 0.230555
$$425$$ 0 0
$$426$$ 32.8724 1.59267
$$427$$ 9.11383i 0.441049i
$$428$$ − 14.5535i − 0.703469i
$$429$$ 2.11727 0.102223
$$430$$ 0 0
$$431$$ 18.2784 0.880437 0.440219 0.897891i $$-0.354901\pi$$
0.440219 + 0.897891i $$0.354901\pi$$
$$432$$ − 2.11727i − 0.101867i
$$433$$ 6.86469i 0.329896i 0.986302 + 0.164948i $$0.0527456\pi$$
−0.986302 + 0.164948i $$0.947254\pi$$
$$434$$ −9.55691 −0.458747
$$435$$ 0 0
$$436$$ −0.249141 −0.0119317
$$437$$ 27.2863i 1.30528i
$$438$$ 23.6121i 1.12823i
$$439$$ 11.8466 0.565409 0.282705 0.959207i $$-0.408768\pi$$
0.282705 + 0.959207i $$0.408768\pi$$
$$440$$ 0 0
$$441$$ 2.05863 0.0980302
$$442$$ − 6.11727i − 0.290969i
$$443$$ 17.8827i 0.849634i 0.905279 + 0.424817i $$0.139662\pi$$
−0.905279 + 0.424817i $$0.860338\pi$$
$$444$$ 9.55691 0.453551
$$445$$ 0 0
$$446$$ 10.3810 0.491555
$$447$$ 0.0328552i 0.00155400i
$$448$$ − 1.00000i − 0.0472456i
$$449$$ 16.8172 0.793654 0.396827 0.917893i $$-0.370111\pi$$
0.396827 + 0.917893i $$0.370111\pi$$
$$450$$ 0 0
$$451$$ −2.13187 −0.100386
$$452$$ − 10.9966i − 0.517235i
$$453$$ 34.2897i 1.61107i
$$454$$ −16.1104 −0.756098
$$455$$ 0 0
$$456$$ 9.82066 0.459895
$$457$$ 12.2277i 0.571986i 0.958232 + 0.285993i $$0.0923234\pi$$
−0.958232 + 0.285993i $$0.907677\pi$$
$$458$$ 24.2897i 1.13498i
$$459$$ −13.7586 −0.642196
$$460$$ 0 0
$$461$$ −16.6155 −0.773863 −0.386932 0.922108i $$-0.626465\pi$$
−0.386932 + 0.922108i $$0.626465\pi$$
$$462$$ − 2.24914i − 0.104639i
$$463$$ 18.4837i 0.859009i 0.903065 + 0.429505i $$0.141312\pi$$
−0.903065 + 0.429505i $$0.858688\pi$$
$$464$$ −8.74742 −0.406089
$$465$$ 0 0
$$466$$ 6.88617 0.318996
$$467$$ − 24.3956i − 1.12889i −0.825469 0.564447i $$-0.809089\pi$$
0.825469 0.564447i $$-0.190911\pi$$
$$468$$ 1.93793i 0.0895808i
$$469$$ 12.9966 0.600125
$$470$$ 0 0
$$471$$ 12.3741 0.570170
$$472$$ 1.88273i 0.0866598i
$$473$$ 7.67418i 0.352859i
$$474$$ 18.8172 0.864304
$$475$$ 0 0
$$476$$ −6.49828 −0.297848
$$477$$ 9.77320i 0.447484i
$$478$$ − 11.1353i − 0.509317i
$$479$$ 27.7655 1.26864 0.634318 0.773072i $$-0.281281\pi$$
0.634318 + 0.773072i $$0.281281\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ − 2.36641i − 0.107787i
$$483$$ 14.0552i 0.639534i
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 18.2491 0.827798
$$487$$ − 10.0958i − 0.457484i −0.973487 0.228742i $$-0.926539\pi$$
0.973487 0.228742i $$-0.0734612\pi$$
$$488$$ 9.11383i 0.412564i
$$489$$ −40.9966 −1.85393
$$490$$ 0 0
$$491$$ −32.1104 −1.44912 −0.724561 0.689211i $$-0.757957\pi$$
−0.724561 + 0.689211i $$0.757957\pi$$
$$492$$ − 4.79488i − 0.216170i
$$493$$ 56.8432i 2.56009i
$$494$$ 4.11039 0.184935
$$495$$ 0 0
$$496$$ −9.55691 −0.429118
$$497$$ − 14.6155i − 0.655597i
$$498$$ 19.1138i 0.856511i
$$499$$ 8.79488 0.393713 0.196857 0.980432i $$-0.436927\pi$$
0.196857 + 0.980432i $$0.436927\pi$$
$$500$$ 0 0
$$501$$ 17.9931 0.803874
$$502$$ − 25.1070i − 1.12058i
$$503$$ − 15.0034i − 0.668970i −0.942401 0.334485i $$-0.891438\pi$$
0.942401 0.334485i $$-0.108562\pi$$
$$504$$ 2.05863 0.0916988
$$505$$ 0 0
$$506$$ 6.24914 0.277808
$$507$$ − 27.2457i − 1.21002i
$$508$$ − 5.88273i − 0.261004i
$$509$$ −21.8759 −0.969630 −0.484815 0.874617i $$-0.661113\pi$$
−0.484815 + 0.874617i $$0.661113\pi$$
$$510$$ 0 0
$$511$$ 10.4983 0.464417
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 9.24485i − 0.408170i
$$514$$ 2.86469 0.126356
$$515$$ 0 0
$$516$$ −17.2603 −0.759843
$$517$$ − 11.1138i − 0.488786i
$$518$$ − 4.24914i − 0.186697i
$$519$$ 0.263748 0.0115773
$$520$$ 0 0
$$521$$ 14.0292 0.614631 0.307316 0.951608i $$-0.400569\pi$$
0.307316 + 0.951608i $$0.400569\pi$$
$$522$$ − 18.0077i − 0.788177i
$$523$$ 1.14992i 0.0502825i 0.999684 + 0.0251412i $$0.00800355\pi$$
−0.999684 + 0.0251412i $$0.991996\pi$$
$$524$$ 1.25258 0.0547191
$$525$$ 0 0
$$526$$ 3.76547 0.164182
$$527$$ 62.1035i 2.70527i
$$528$$ − 2.24914i − 0.0978813i
$$529$$ −16.0518 −0.697902
$$530$$ 0 0
$$531$$ −3.87586 −0.168198
$$532$$ − 4.36641i − 0.189308i
$$533$$ − 2.00688i − 0.0869274i
$$534$$ 27.8466 1.20504
$$535$$ 0 0
$$536$$ 12.9966 0.561366
$$537$$ − 50.7259i − 2.18899i
$$538$$ 8.94137i 0.385490i
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −6.47680 −0.278459 −0.139230 0.990260i $$-0.544463\pi$$
−0.139230 + 0.990260i $$0.544463\pi$$
$$542$$ − 21.4948i − 0.923283i
$$543$$ − 46.9605i − 2.01527i
$$544$$ −6.49828 −0.278612
$$545$$ 0 0
$$546$$ 2.11727 0.0906106
$$547$$ − 12.9966i − 0.555693i −0.960626 0.277846i $$-0.910379\pi$$
0.960626 0.277846i $$-0.0896206\pi$$
$$548$$ 10.0000i 0.427179i
$$549$$ −18.7620 −0.800744
$$550$$ 0 0
$$551$$ −38.1948 −1.62715
$$552$$ 14.0552i 0.598229i
$$553$$ − 8.36641i − 0.355776i
$$554$$ −7.64820 −0.324941
$$555$$ 0 0
$$556$$ 10.9820 0.465739
$$557$$ 16.4914i 0.698763i 0.936981 + 0.349382i $$0.113608\pi$$
−0.936981 + 0.349382i $$0.886392\pi$$
$$558$$ − 19.6742i − 0.832874i
$$559$$ −7.22422 −0.305552
$$560$$ 0 0
$$561$$ −14.6155 −0.617069
$$562$$ 28.6155i 1.20707i
$$563$$ − 16.7620i − 0.706435i −0.935541 0.353218i $$-0.885088\pi$$
0.935541 0.353218i $$-0.114912\pi$$
$$564$$ 24.9966 1.05255
$$565$$ 0 0
$$566$$ −2.87930 −0.121026
$$567$$ − 10.9379i − 0.459350i
$$568$$ − 14.6155i − 0.613255i
$$569$$ 22.0000 0.922288 0.461144 0.887325i $$-0.347439\pi$$
0.461144 + 0.887325i $$0.347439\pi$$
$$570$$ 0 0
$$571$$ 12.7328 0.532852 0.266426 0.963855i $$-0.414157\pi$$
0.266426 + 0.963855i $$0.414157\pi$$
$$572$$ − 0.941367i − 0.0393605i
$$573$$ − 7.00344i − 0.292573i
$$574$$ −2.13187 −0.0889827
$$575$$ 0 0
$$576$$ 2.05863 0.0857764
$$577$$ 11.8613i 0.493790i 0.969042 + 0.246895i $$0.0794103\pi$$
−0.969042 + 0.246895i $$0.920590\pi$$
$$578$$ 25.2277i 1.04933i
$$579$$ −13.8827 −0.576947
$$580$$ 0 0
$$581$$ 8.49828 0.352568
$$582$$ − 34.5535i − 1.43229i
$$583$$ − 4.74742i − 0.196618i
$$584$$ 10.4983 0.434422
$$585$$ 0 0
$$586$$ −12.9414 −0.534603
$$587$$ 8.60094i 0.354999i 0.984121 + 0.177499i $$0.0568008\pi$$
−0.984121 + 0.177499i $$0.943199\pi$$
$$588$$ − 2.24914i − 0.0927530i
$$589$$ −41.7294 −1.71943
$$590$$ 0 0
$$591$$ −15.1430 −0.622902
$$592$$ − 4.24914i − 0.174639i
$$593$$ 10.7328i 0.440744i 0.975416 + 0.220372i $$0.0707271\pi$$
−0.975416 + 0.220372i $$0.929273\pi$$
$$594$$ −2.11727 −0.0868725
$$595$$ 0 0
$$596$$ 0.0146079 0.000598363 0
$$597$$ − 29.7586i − 1.21794i
$$598$$ 5.88273i 0.240563i
$$599$$ −16.0812 −0.657059 −0.328529 0.944494i $$-0.606553\pi$$
−0.328529 + 0.944494i $$0.606553\pi$$
$$600$$ 0 0
$$601$$ −34.0889 −1.39052 −0.695258 0.718760i $$-0.744710\pi$$
−0.695258 + 0.718760i $$0.744710\pi$$
$$602$$ 7.67418i 0.312776i
$$603$$ 26.7552i 1.08955i
$$604$$ 15.2457 0.620339
$$605$$ 0 0
$$606$$ 37.9639 1.54218
$$607$$ − 11.6742i − 0.473840i −0.971529 0.236920i $$-0.923862\pi$$
0.971529 0.236920i $$-0.0761380\pi$$
$$608$$ − 4.36641i − 0.177081i
$$609$$ −19.6742 −0.797238
$$610$$ 0 0
$$611$$ 10.4622 0.423255
$$612$$ − 13.3776i − 0.540756i
$$613$$ − 34.2637i − 1.38390i −0.721946 0.691950i $$-0.756752\pi$$
0.721946 0.691950i $$-0.243248\pi$$
$$614$$ −0.498281 −0.0201090
$$615$$ 0 0
$$616$$ −1.00000 −0.0402911
$$617$$ − 39.3707i − 1.58500i −0.609869 0.792502i $$-0.708778\pi$$
0.609869 0.792502i $$-0.291222\pi$$
$$618$$ 14.8793i 0.598533i
$$619$$ −16.1104 −0.647531 −0.323766 0.946137i $$-0.604949\pi$$
−0.323766 + 0.946137i $$0.604949\pi$$
$$620$$ 0 0
$$621$$ 13.2311 0.530946
$$622$$ 23.4396i 0.939844i
$$623$$ − 12.3810i − 0.496035i
$$624$$ 2.11727 0.0847585
$$625$$ 0 0
$$626$$ 9.36984 0.374494
$$627$$ − 9.82066i − 0.392199i
$$628$$ − 5.50172i − 0.219542i
$$629$$ −27.6121 −1.10097
$$630$$ 0 0
$$631$$ 13.8827 0.552663 0.276331 0.961062i $$-0.410881\pi$$
0.276331 + 0.961062i $$0.410881\pi$$
$$632$$ − 8.36641i − 0.332798i
$$633$$ − 51.9862i − 2.06627i
$$634$$ 9.97852 0.396298
$$635$$ 0 0
$$636$$ 10.6776 0.423395
$$637$$ − 0.941367i − 0.0372983i
$$638$$ 8.74742i 0.346314i
$$639$$ 30.0881 1.19026
$$640$$ 0 0
$$641$$ −39.3415 −1.55390 −0.776948 0.629565i $$-0.783233\pi$$
−0.776948 + 0.629565i $$0.783233\pi$$
$$642$$ − 32.7328i − 1.29186i
$$643$$ 38.3595i 1.51275i 0.654137 + 0.756376i $$0.273032\pi$$
−0.654137 + 0.756376i $$0.726968\pi$$
$$644$$ 6.24914 0.246251
$$645$$ 0 0
$$646$$ −28.3741 −1.11637
$$647$$ − 25.7294i − 1.01153i −0.862672 0.505763i $$-0.831211\pi$$
0.862672 0.505763i $$-0.168789\pi$$
$$648$$ − 10.9379i − 0.429682i
$$649$$ 1.88273 0.0739038
$$650$$ 0 0
$$651$$ −21.4948 −0.842449
$$652$$ 18.2277i 0.713850i
$$653$$ 18.4768i 0.723053i 0.932362 + 0.361526i $$0.117744\pi$$
−0.932362 + 0.361526i $$0.882256\pi$$
$$654$$ −0.560352 −0.0219115
$$655$$ 0 0
$$656$$ −2.13187 −0.0832357
$$657$$ 21.6121i 0.843169i
$$658$$ − 11.1138i − 0.433262i
$$659$$ 39.3776 1.53393 0.766966 0.641687i $$-0.221765\pi$$
0.766966 + 0.641687i $$0.221765\pi$$
$$660$$ 0 0
$$661$$ −34.2208 −1.33103 −0.665517 0.746383i $$-0.731789\pi$$
−0.665517 + 0.746383i $$0.731789\pi$$
$$662$$ 20.4362i 0.794276i
$$663$$ − 13.7586i − 0.534339i
$$664$$ 8.49828 0.329797
$$665$$ 0 0
$$666$$ 8.74742 0.338956
$$667$$ − 54.6639i − 2.11659i
$$668$$ − 8.00000i − 0.309529i
$$669$$ 23.3484 0.902700
$$670$$ 0 0
$$671$$ 9.11383 0.351835
$$672$$ − 2.24914i − 0.0867625i
$$673$$ − 24.4622i − 0.942948i −0.881880 0.471474i $$-0.843722\pi$$
0.881880 0.471474i $$-0.156278\pi$$
$$674$$ −7.88273 −0.303632
$$675$$ 0 0
$$676$$ −12.1138 −0.465916
$$677$$ 39.1070i 1.50300i 0.659732 + 0.751501i $$0.270670\pi$$
−0.659732 + 0.751501i $$0.729330\pi$$
$$678$$ − 24.7328i − 0.949858i
$$679$$ −15.3630 −0.589577
$$680$$ 0 0
$$681$$ −36.2345 −1.38851
$$682$$ 9.55691i 0.365953i
$$683$$ 23.3776i 0.894518i 0.894404 + 0.447259i $$0.147600\pi$$
−0.894404 + 0.447259i $$0.852400\pi$$
$$684$$ 8.98883 0.343697
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ 54.6310i 2.08430i
$$688$$ 7.67418i 0.292575i
$$689$$ 4.46907 0.170258
$$690$$ 0 0
$$691$$ 8.49828 0.323290 0.161645 0.986849i $$-0.448320\pi$$
0.161645 + 0.986849i $$0.448320\pi$$
$$692$$ − 0.117266i − 0.00445780i
$$693$$ − 2.05863i − 0.0782010i
$$694$$ −13.5569 −0.514613
$$695$$ 0 0
$$696$$ −19.6742 −0.745748
$$697$$ 13.8535i 0.524739i
$$698$$ − 10.7328i − 0.406243i
$$699$$ 15.4880 0.585809
$$700$$ 0 0
$$701$$ 14.9751 0.565601 0.282800 0.959179i $$-0.408737\pi$$
0.282800 + 0.959179i $$0.408737\pi$$
$$702$$ − 1.99312i − 0.0752256i
$$703$$ − 18.5535i − 0.699758i
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ −20.8578 −0.784994
$$707$$ − 16.8793i − 0.634811i
$$708$$ 4.23453i 0.159143i
$$709$$ −40.7259 −1.52949 −0.764747 0.644330i $$-0.777136\pi$$
−0.764747 + 0.644330i $$0.777136\pi$$
$$710$$ 0 0
$$711$$ 17.2234 0.645927
$$712$$ − 12.3810i − 0.463998i
$$713$$ − 59.7225i − 2.23663i
$$714$$ −14.6155 −0.546973
$$715$$ 0 0
$$716$$ −22.5535 −0.842863
$$717$$ − 25.0449i − 0.935318i
$$718$$ 0.366407i 0.0136742i
$$719$$ 8.00000 0.298350 0.149175 0.988811i $$-0.452338\pi$$
0.149175 + 0.988811i $$0.452338\pi$$
$$720$$ 0 0
$$721$$ 6.61555 0.246376
$$722$$ − 0.0655089i − 0.00243799i
$$723$$ − 5.32238i − 0.197942i
$$724$$ −20.8793 −0.775973
$$725$$ 0 0
$$726$$ −2.24914 −0.0834734
$$727$$ 51.6413i 1.91527i 0.287984 + 0.957635i $$0.407015\pi$$
−0.287984 + 0.957635i $$0.592985\pi$$
$$728$$ − 0.941367i − 0.0348894i
$$729$$ 8.23109 0.304855
$$730$$ 0 0
$$731$$ 49.8690 1.84447
$$732$$ 20.4983i 0.757638i
$$733$$ − 42.0000i − 1.55131i −0.631160 0.775653i $$-0.717421\pi$$
0.631160 0.775653i $$-0.282579\pi$$
$$734$$ −20.8432 −0.769337
$$735$$ 0 0
$$736$$ 6.24914 0.230346
$$737$$ − 12.9966i − 0.478735i
$$738$$ − 4.38875i − 0.161552i
$$739$$ 49.1070 1.80643 0.903214 0.429190i $$-0.141201\pi$$
0.903214 + 0.429190i $$0.141201\pi$$
$$740$$ 0 0
$$741$$ 9.24485 0.339618
$$742$$ − 4.74742i − 0.174283i
$$743$$ − 53.2311i − 1.95286i −0.215836 0.976430i $$-0.569248\pi$$
0.215836 0.976430i $$-0.430752\pi$$
$$744$$ −21.4948 −0.788039
$$745$$ 0 0
$$746$$ −5.37758 −0.196887
$$747$$ 17.4948i 0.640103i
$$748$$ 6.49828i 0.237601i
$$749$$ −14.5535 −0.531772
$$750$$ 0 0
$$751$$ −31.6121 −1.15354 −0.576771 0.816906i $$-0.695688\pi$$
−0.576771 + 0.816906i $$0.695688\pi$$
$$752$$ − 11.1138i − 0.405280i
$$753$$ − 56.4691i − 2.05785i
$$754$$ −8.23453 −0.299884
$$755$$ 0 0
$$756$$ −2.11727 −0.0770042
$$757$$ 1.24570i 0.0452758i 0.999744 + 0.0226379i $$0.00720649\pi$$
−0.999744 + 0.0226379i $$0.992794\pi$$
$$758$$ 3.53093i 0.128249i
$$759$$ 14.0552 0.510171
$$760$$ 0 0
$$761$$ −10.6009 −0.384284 −0.192142 0.981367i $$-0.561543\pi$$
−0.192142 + 0.981367i $$0.561543\pi$$
$$762$$ − 13.2311i − 0.479312i
$$763$$ 0.249141i 0.00901949i
$$764$$ −3.11383 −0.112654
$$765$$ 0 0
$$766$$ 1.38445 0.0500223
$$767$$ 1.77234i 0.0639956i
$$768$$ − 2.24914i − 0.0811589i
$$769$$ 19.1284 0.689789 0.344895 0.938641i $$-0.387915\pi$$
0.344895 + 0.938641i $$0.387915\pi$$
$$770$$ 0 0
$$771$$ 6.44309 0.232042
$$772$$ 6.17246i 0.222152i
$$773$$ 39.9931i 1.43845i 0.694776 + 0.719226i $$0.255504\pi$$
−0.694776 + 0.719226i $$0.744496\pi$$
$$774$$ −15.7983 −0.567859
$$775$$ 0 0
$$776$$ −15.3630 −0.551498
$$777$$ − 9.55691i − 0.342852i
$$778$$ 15.7294i 0.563925i
$$779$$ −9.30863 −0.333516
$$780$$ 0 0
$$781$$ −14.6155 −0.522985
$$782$$ − 40.6087i − 1.45216i
$$783$$ 18.5206i 0.661873i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 2.81722 0.100487
$$787$$ − 37.9931i − 1.35431i −0.735841 0.677154i $$-0.763213\pi$$
0.735841 0.677154i $$-0.236787\pi$$
$$788$$ 6.73281i 0.239847i
$$789$$ 8.46907 0.301507
$$790$$ 0 0
$$791$$ −10.9966 −0.390993
$$792$$ − 2.05863i − 0.0731503i
$$793$$ 8.57946i 0.304665i
$$794$$ 17.6121 0.625030
$$795$$ 0 0
$$796$$ −13.2311 −0.468964
$$797$$ 20.3810i 0.721933i 0.932579 + 0.360966i $$0.117553\pi$$
−0.932579 + 0.360966i $$0.882447\pi$$
$$798$$ − 9.82066i − 0.347648i
$$799$$ −72.2208 −2.55499
$$800$$ 0 0
$$801$$ 25.4880 0.900573
$$802$$ 32.8172i 1.15882i
$$803$$ − 10.4983i − 0.370476i
$$804$$ 29.2311 1.03090
$$805$$ 0 0
$$806$$ −8.99656 −0.316890
$$807$$ 20.1104i 0.707919i
$$808$$ − 16.8793i − 0.593812i
$$809$$ 31.7294 1.11555 0.557773 0.829994i $$-0.311656\pi$$
0.557773 + 0.829994i $$0.311656\pi$$
$$810$$ 0 0
$$811$$ 43.5095 1.52782 0.763912 0.645321i $$-0.223276\pi$$
0.763912 + 0.645321i $$0.223276\pi$$
$$812$$ 8.74742i 0.306974i
$$813$$ − 48.3449i − 1.69553i
$$814$$ −4.24914 −0.148932
$$815$$ 0 0
$$816$$ −14.6155 −0.511646
$$817$$ 33.5086i 1.17232i
$$818$$ − 33.3561i − 1.16627i
$$819$$ 1.93793 0.0677167
$$820$$ 0 0
$$821$$ −24.7766 −0.864711 −0.432355 0.901703i $$-0.642317\pi$$
−0.432355 + 0.901703i $$0.642317\pi$$
$$822$$ 22.4914i 0.784478i
$$823$$ 28.8647i 1.00616i 0.864240 + 0.503080i $$0.167800\pi$$
−0.864240 + 0.503080i $$0.832200\pi$$
$$824$$ 6.61555 0.230464
$$825$$ 0 0
$$826$$ 1.88273 0.0655087
$$827$$ − 34.2277i − 1.19021i −0.803647 0.595106i $$-0.797110\pi$$
0.803647 0.595106i $$-0.202890\pi$$
$$828$$ 12.8647i 0.447079i
$$829$$ −23.6381 −0.820985 −0.410492 0.911864i $$-0.634643\pi$$
−0.410492 + 0.911864i $$0.634643\pi$$
$$830$$ 0 0
$$831$$ −17.2019 −0.596727
$$832$$ − 0.941367i − 0.0326360i
$$833$$ 6.49828i 0.225152i
$$834$$ 24.7000 0.855290
$$835$$ 0 0
$$836$$ −4.36641 −0.151015
$$837$$ 20.2345i 0.699408i
$$838$$ 12.3449i 0.426448i
$$839$$ 24.9053 0.859826 0.429913 0.902870i $$-0.358544\pi$$
0.429913 + 0.902870i $$0.358544\pi$$
$$840$$ 0 0
$$841$$ 47.5174 1.63853
$$842$$ 8.87930i 0.306001i
$$843$$ 64.3604i 2.21669i
$$844$$ −23.1138 −0.795611
$$845$$ 0 0
$$846$$ 22.8793 0.786606
$$847$$ 1.00000i 0.0343604i
$$848$$ − 4.74742i − 0.163027i
$$849$$ −6.47594 −0.222254
$$850$$ 0 0
$$851$$ 26.5535 0.910241
$$852$$ − 32.8724i − 1.12619i
$$853$$ 22.9966i 0.787387i 0.919242 + 0.393694i $$0.128803\pi$$
−0.919242 + 0.393694i $$0.871197\pi$$
$$854$$ 9.11383 0.311869
$$855$$ 0 0
$$856$$ −14.5535 −0.497428
$$857$$ 31.2603i 1.06783i 0.845538 + 0.533916i $$0.179280\pi$$
−0.845538 + 0.533916i $$0.820720\pi$$
$$858$$ − 2.11727i − 0.0722823i
$$859$$ −20.3449 −0.694160 −0.347080 0.937836i $$-0.612827\pi$$
−0.347080 + 0.937836i $$0.612827\pi$$
$$860$$ 0 0
$$861$$ −4.79488 −0.163409
$$862$$ − 18.2784i − 0.622563i
$$863$$ − 58.3595i − 1.98658i −0.115644 0.993291i $$-0.536893\pi$$
0.115644 0.993291i $$-0.463107\pi$$
$$864$$ −2.11727 −0.0720309
$$865$$ 0 0
$$866$$ 6.86469 0.233272
$$867$$ 56.7405i 1.92701i
$$868$$ 9.55691i 0.324383i
$$869$$ −8.36641 −0.283811
$$870$$ 0 0
$$871$$ 12.2345 0.414551
$$872$$ 0.249141i 0.00843696i
$$873$$ − 31.6267i − 1.07040i
$$874$$ 27.2863 0.922973
$$875$$ 0 0
$$876$$ 23.6121 0.797779
$$877$$ − 11.9639i − 0.403992i −0.979386 0.201996i $$-0.935257\pi$$
0.979386 0.201996i $$-0.0647429\pi$$
$$878$$ − 11.8466i − 0.399805i
$$879$$ −29.1070 −0.981753
$$880$$ 0 0
$$881$$ 5.88961 0.198426 0.0992130 0.995066i $$-0.468367\pi$$
0.0992130 + 0.995066i $$0.468367\pi$$
$$882$$ − 2.05863i − 0.0693178i
$$883$$ − 19.4880i − 0.655822i −0.944709 0.327911i $$-0.893655\pi$$
0.944709 0.327911i $$-0.106345\pi$$
$$884$$ −6.11727 −0.205746
$$885$$ 0 0
$$886$$ 17.8827 0.600782
$$887$$ − 50.4622i − 1.69435i −0.531310 0.847177i $$-0.678300\pi$$
0.531310 0.847177i $$-0.321700\pi$$
$$888$$ − 9.55691i − 0.320709i
$$889$$ −5.88273 −0.197301
$$890$$ 0 0
$$891$$ −10.9379 −0.366434
$$892$$ − 10.3810i − 0.347582i
$$893$$ − 48.5275i − 1.62391i
$$894$$ 0.0328552 0.00109884
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 13.2311i 0.441773i
$$898$$ − 16.8172i − 0.561198i
$$899$$ 83.5984 2.78816
$$900$$ 0 0
$$901$$ −30.8501 −1.02777
$$902$$ 2.13187i 0.0709836i
$$903$$ 17.2603i 0.574387i
$$904$$ −10.9966 −0.365740
$$905$$ 0 0
$$906$$ 34.2897 1.13920
$$907$$ 16.8432i 0.559269i 0.960106 + 0.279635i $$0.0902134\pi$$
−0.960106 + 0.279635i $$0.909787\pi$$
$$908$$ 16.1104i 0.534642i
$$909$$ 34.7483 1.15253
$$910$$ 0 0
$$911$$ −38.6155 −1.27939 −0.639695 0.768629i $$-0.720939\pi$$
−0.639695 + 0.768629i $$0.720939\pi$$
$$912$$ − 9.82066i − 0.325195i
$$913$$ − 8.49828i − 0.281252i
$$914$$ 12.2277 0.404455
$$915$$ 0 0
$$916$$ 24.2897 0.802555
$$917$$ − 1.25258i − 0.0413638i
$$918$$ 13.7586i 0.454101i
$$919$$ −17.2818 −0.570074 −0.285037 0.958517i $$-0.592006\pi$$
−0.285037 + 0.958517i $$0.592006\pi$$
$$920$$ 0 0
$$921$$ −1.12070 −0.0369285
$$922$$ 16.6155i 0.547204i
$$923$$ − 13.7586i − 0.452870i
$$924$$ −2.24914 −0.0739913
$$925$$ 0 0
$$926$$ 18.4837 0.607411
$$927$$ 13.6190i 0.447306i
$$928$$ 8.74742i 0.287148i
$$929$$ −2.91539 −0.0956508 −0.0478254 0.998856i $$-0.515229\pi$$
−0.0478254 + 0.998856i $$0.515229\pi$$
$$930$$ 0 0
$$931$$ −4.36641 −0.143103
$$932$$ − 6.88617i − 0.225564i
$$933$$ 52.7191i 1.72594i
$$934$$ −24.3956 −0.798249
$$935$$ 0 0
$$936$$ 1.93793 0.0633432
$$937$$ − 22.5795i − 0.737639i −0.929501 0.368819i $$-0.879762\pi$$
0.929501 0.368819i $$-0.120238\pi$$
$$938$$ − 12.9966i − 0.424353i
$$939$$ 21.0741 0.687727
$$940$$ 0 0
$$941$$ −21.7655 −0.709534 −0.354767 0.934955i $$-0.615440\pi$$
−0.354767 + 0.934955i $$0.615440\pi$$
$$942$$ − 12.3741i − 0.403171i
$$943$$ − 13.3224i − 0.433836i
$$944$$ 1.88273 0.0612778
$$945$$ 0 0
$$946$$ 7.67418 0.249509
$$947$$ 1.49484i 0.0485759i 0.999705 + 0.0242879i $$0.00773185\pi$$
−0.999705 + 0.0242879i $$0.992268\pi$$
$$948$$ − 18.8172i − 0.611155i
$$949$$ 9.88273 0.320807
$$950$$ 0 0
$$951$$ 22.4431 0.727767
$$952$$ 6.49828i 0.210610i
$$953$$ − 4.83098i − 0.156491i −0.996934 0.0782453i $$-0.975068\pi$$
0.996934 0.0782453i $$-0.0249317\pi$$
$$954$$ 9.77320 0.316419
$$955$$ 0 0
$$956$$ −11.1353 −0.360142
$$957$$ 19.6742i 0.635976i
$$958$$ − 27.7655i − 0.897062i
$$959$$ 10.0000 0.322917
$$960$$ 0 0
$$961$$ 60.3346 1.94628
$$962$$ − 4.00000i − 0.128965i
$$963$$ − 29.9603i − 0.965456i
$$964$$ −2.36641 −0.0762168
$$965$$ 0 0
$$966$$ 14.0552 0.452218
$$967$$ 51.9278i 1.66989i 0.550336 + 0.834943i $$0.314499\pi$$
−0.550336 + 0.834943i $$0.685501\pi$$
$$968$$ 1.00000i 0.0321412i
$$969$$ −63.8174 −2.05011
$$970$$ 0 0
$$971$$ 59.6933 1.91565 0.957824 0.287354i $$-0.0927757\pi$$
0.957824 + 0.287354i $$0.0927757\pi$$
$$972$$ − 18.2491i − 0.585341i
$$973$$ − 10.9820i − 0.352065i
$$974$$ −10.0958 −0.323490
$$975$$ 0 0
$$976$$ 9.11383 0.291727
$$977$$ − 31.2311i − 0.999171i −0.866265 0.499586i $$-0.833486\pi$$
0.866265 0.499586i $$-0.166514\pi$$
$$978$$ 40.9966i 1.31093i
$$979$$ −12.3810 −0.395699
$$980$$ 0 0
$$981$$ −0.512889 −0.0163753
$$982$$ 32.1104i 1.02468i
$$983$$ − 22.6155i − 0.721324i −0.932697 0.360662i $$-0.882551\pi$$
0.932697 0.360662i $$-0.117449\pi$$
$$984$$ −4.79488 −0.152855
$$985$$ 0 0
$$986$$ 56.8432 1.81026
$$987$$ − 24.9966i − 0.795649i
$$988$$ − 4.11039i − 0.130769i
$$989$$ −47.9570 −1.52494
$$990$$ 0 0
$$991$$ 22.4102 0.711884 0.355942 0.934508i $$-0.384160\pi$$
0.355942 + 0.934508i $$0.384160\pi$$
$$992$$ 9.55691i 0.303432i
$$993$$ 45.9639i 1.45862i
$$994$$ −14.6155 −0.463577
$$995$$ 0 0
$$996$$ 19.1138 0.605645
$$997$$ 27.3415i 0.865914i 0.901415 + 0.432957i $$0.142530\pi$$
−0.901415 + 0.432957i $$0.857470\pi$$
$$998$$ − 8.79488i − 0.278397i
$$999$$ −8.99656 −0.284639
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 3850.2.c.ba.1849.1 6
5.2 odd 4 770.2.a.m.1.1 3
5.3 odd 4 3850.2.a.bt.1.3 3
5.4 even 2 inner 3850.2.c.ba.1849.6 6
15.2 even 4 6930.2.a.ce.1.2 3
20.7 even 4 6160.2.a.bf.1.3 3
35.27 even 4 5390.2.a.ca.1.3 3
55.32 even 4 8470.2.a.ci.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.1 3 5.2 odd 4
3850.2.a.bt.1.3 3 5.3 odd 4
3850.2.c.ba.1849.1 6 1.1 even 1 trivial
3850.2.c.ba.1849.6 6 5.4 even 2 inner
5390.2.a.ca.1.3 3 35.27 even 4
6160.2.a.bf.1.3 3 20.7 even 4
6930.2.a.ce.1.2 3 15.2 even 4
8470.2.a.ci.1.1 3 55.32 even 4