Properties

 Label 3850.2.c.x.1849.2 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 770) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 1849.2 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3850.1849 Dual form 3850.2.c.x.1849.3

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +2.73205i q^{3} -1.00000 q^{4} +2.73205 q^{6} -1.00000i q^{7} +1.00000i q^{8} -4.46410 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +2.73205i q^{3} -1.00000 q^{4} +2.73205 q^{6} -1.00000i q^{7} +1.00000i q^{8} -4.46410 q^{9} +1.00000 q^{11} -2.73205i q^{12} -1.46410i q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.46410i q^{17} +4.46410i q^{18} -6.73205 q^{19} +2.73205 q^{21} -1.00000i q^{22} -8.19615i q^{23} -2.73205 q^{24} -1.46410 q^{26} -4.00000i q^{27} +1.00000i q^{28} +4.73205 q^{29} +2.00000 q^{31} -1.00000i q^{32} +2.73205i q^{33} +3.46410 q^{34} +4.46410 q^{36} -0.732051i q^{37} +6.73205i q^{38} +4.00000 q^{39} -2.19615 q^{41} -2.73205i q^{42} +2.00000i q^{43} -1.00000 q^{44} -8.19615 q^{46} -6.92820i q^{47} +2.73205i q^{48} -1.00000 q^{49} -9.46410 q^{51} +1.46410i q^{52} -7.26795i q^{53} -4.00000 q^{54} +1.00000 q^{56} -18.3923i q^{57} -4.73205i q^{58} -6.92820 q^{59} -4.92820 q^{61} -2.00000i q^{62} +4.46410i q^{63} -1.00000 q^{64} +2.73205 q^{66} +4.00000i q^{67} -3.46410i q^{68} +22.3923 q^{69} +9.46410 q^{71} -4.46410i q^{72} -14.3923i q^{73} -0.732051 q^{74} +6.73205 q^{76} -1.00000i q^{77} -4.00000i q^{78} +12.1962 q^{79} -2.46410 q^{81} +2.19615i q^{82} -16.3923i q^{83} -2.73205 q^{84} +2.00000 q^{86} +12.9282i q^{87} +1.00000i q^{88} -3.46410 q^{89} -1.46410 q^{91} +8.19615i q^{92} +5.46410i q^{93} -6.92820 q^{94} +2.73205 q^{96} -14.5885i q^{97} +1.00000i q^{98} -4.46410 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} - 20 q^{19} + 4 q^{21} - 4 q^{24} + 8 q^{26} + 12 q^{29} + 8 q^{31} + 4 q^{36} + 16 q^{39} + 12 q^{41} - 4 q^{44} - 12 q^{46} - 4 q^{49} - 24 q^{51} - 16 q^{54} + 4 q^{56} + 8 q^{61} - 4 q^{64} + 4 q^{66} + 48 q^{69} + 24 q^{71} + 4 q^{74} + 20 q^{76} + 28 q^{79} + 4 q^{81} - 4 q^{84} + 8 q^{86} + 8 q^{91} + 4 q^{96} - 4 q^{99} + O(q^{100})$$

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
$$3$$ 2.73205i 1.57735i 0.614810 + 0.788675i $$0.289233\pi$$
−0.614810 + 0.788675i $$0.710767\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 2.73205 1.11536
$$7$$ − 1.00000i − 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ −4.46410 −1.48803
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ − 2.73205i − 0.788675i
$$13$$ − 1.46410i − 0.406069i −0.979172 0.203034i $$-0.934920\pi$$
0.979172 0.203034i $$-0.0650803\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.46410i 0.840168i 0.907485 + 0.420084i $$0.137999\pi$$
−0.907485 + 0.420084i $$0.862001\pi$$
$$18$$ 4.46410i 1.05220i
$$19$$ −6.73205 −1.54444 −0.772219 0.635356i $$-0.780853\pi$$
−0.772219 + 0.635356i $$0.780853\pi$$
$$20$$ 0 0
$$21$$ 2.73205 0.596182
$$22$$ − 1.00000i − 0.213201i
$$23$$ − 8.19615i − 1.70902i −0.519438 0.854508i $$-0.673859\pi$$
0.519438 0.854508i $$-0.326141\pi$$
$$24$$ −2.73205 −0.557678
$$25$$ 0 0
$$26$$ −1.46410 −0.287134
$$27$$ − 4.00000i − 0.769800i
$$28$$ 1.00000i 0.188982i
$$29$$ 4.73205 0.878720 0.439360 0.898311i $$-0.355205\pi$$
0.439360 + 0.898311i $$0.355205\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 2.73205i 0.475589i
$$34$$ 3.46410 0.594089
$$35$$ 0 0
$$36$$ 4.46410 0.744017
$$37$$ − 0.732051i − 0.120348i −0.998188 0.0601742i $$-0.980834\pi$$
0.998188 0.0601742i $$-0.0191656\pi$$
$$38$$ 6.73205i 1.09208i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −2.19615 −0.342981 −0.171491 0.985186i $$-0.554858\pi$$
−0.171491 + 0.985186i $$0.554858\pi$$
$$42$$ − 2.73205i − 0.421565i
$$43$$ 2.00000i 0.304997i 0.988304 + 0.152499i $$0.0487319\pi$$
−0.988304 + 0.152499i $$0.951268\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ 0 0
$$46$$ −8.19615 −1.20846
$$47$$ − 6.92820i − 1.01058i −0.862949 0.505291i $$-0.831385\pi$$
0.862949 0.505291i $$-0.168615\pi$$
$$48$$ 2.73205i 0.394338i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −9.46410 −1.32524
$$52$$ 1.46410i 0.203034i
$$53$$ − 7.26795i − 0.998330i −0.866507 0.499165i $$-0.833640\pi$$
0.866507 0.499165i $$-0.166360\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ − 18.3923i − 2.43612i
$$58$$ − 4.73205i − 0.621349i
$$59$$ −6.92820 −0.901975 −0.450988 0.892530i $$-0.648928\pi$$
−0.450988 + 0.892530i $$0.648928\pi$$
$$60$$ 0 0
$$61$$ −4.92820 −0.630992 −0.315496 0.948927i $$-0.602171\pi$$
−0.315496 + 0.948927i $$0.602171\pi$$
$$62$$ − 2.00000i − 0.254000i
$$63$$ 4.46410i 0.562424i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.73205 0.336292
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ − 3.46410i − 0.420084i
$$69$$ 22.3923 2.69572
$$70$$ 0 0
$$71$$ 9.46410 1.12318 0.561591 0.827415i $$-0.310189\pi$$
0.561591 + 0.827415i $$0.310189\pi$$
$$72$$ − 4.46410i − 0.526099i
$$73$$ − 14.3923i − 1.68449i −0.539093 0.842246i $$-0.681233\pi$$
0.539093 0.842246i $$-0.318767\pi$$
$$74$$ −0.732051 −0.0850992
$$75$$ 0 0
$$76$$ 6.73205 0.772219
$$77$$ − 1.00000i − 0.113961i
$$78$$ − 4.00000i − 0.452911i
$$79$$ 12.1962 1.37217 0.686087 0.727519i $$-0.259327\pi$$
0.686087 + 0.727519i $$0.259327\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ 2.19615i 0.242524i
$$83$$ − 16.3923i − 1.79929i −0.436623 0.899645i $$-0.643826\pi$$
0.436623 0.899645i $$-0.356174\pi$$
$$84$$ −2.73205 −0.298091
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 12.9282i 1.38605i
$$88$$ 1.00000i 0.106600i
$$89$$ −3.46410 −0.367194 −0.183597 0.983002i $$-0.558774\pi$$
−0.183597 + 0.983002i $$0.558774\pi$$
$$90$$ 0 0
$$91$$ −1.46410 −0.153480
$$92$$ 8.19615i 0.854508i
$$93$$ 5.46410i 0.566601i
$$94$$ −6.92820 −0.714590
$$95$$ 0 0
$$96$$ 2.73205 0.278839
$$97$$ − 14.5885i − 1.48123i −0.671928 0.740617i $$-0.734533\pi$$
0.671928 0.740617i $$-0.265467\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ −4.46410 −0.448659
$$100$$ 0 0
$$101$$ −7.85641 −0.781742 −0.390871 0.920446i $$-0.627826\pi$$
−0.390871 + 0.920446i $$0.627826\pi$$
$$102$$ 9.46410i 0.937086i
$$103$$ 12.3923i 1.22105i 0.791997 + 0.610525i $$0.209042\pi$$
−0.791997 + 0.610525i $$0.790958\pi$$
$$104$$ 1.46410 0.143567
$$105$$ 0 0
$$106$$ −7.26795 −0.705926
$$107$$ − 7.85641i − 0.759507i −0.925088 0.379754i $$-0.876009\pi$$
0.925088 0.379754i $$-0.123991\pi$$
$$108$$ 4.00000i 0.384900i
$$109$$ 15.6603 1.49998 0.749990 0.661449i $$-0.230058\pi$$
0.749990 + 0.661449i $$0.230058\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ − 1.00000i − 0.0944911i
$$113$$ 7.85641i 0.739069i 0.929217 + 0.369534i $$0.120483\pi$$
−0.929217 + 0.369534i $$0.879517\pi$$
$$114$$ −18.3923 −1.72260
$$115$$ 0 0
$$116$$ −4.73205 −0.439360
$$117$$ 6.53590i 0.604244i
$$118$$ 6.92820i 0.637793i
$$119$$ 3.46410 0.317554
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 4.92820i 0.446179i
$$123$$ − 6.00000i − 0.541002i
$$124$$ −2.00000 −0.179605
$$125$$ 0 0
$$126$$ 4.46410 0.397694
$$127$$ − 1.07180i − 0.0951066i −0.998869 0.0475533i $$-0.984858\pi$$
0.998869 0.0475533i $$-0.0151424\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −5.46410 −0.481087
$$130$$ 0 0
$$131$$ 5.66025 0.494539 0.247269 0.968947i $$-0.420467\pi$$
0.247269 + 0.968947i $$0.420467\pi$$
$$132$$ − 2.73205i − 0.237795i
$$133$$ 6.73205i 0.583743i
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −3.46410 −0.297044
$$137$$ 0.928203i 0.0793018i 0.999214 + 0.0396509i $$0.0126246\pi$$
−0.999214 + 0.0396509i $$0.987375\pi$$
$$138$$ − 22.3923i − 1.90616i
$$139$$ −13.6603 −1.15865 −0.579324 0.815097i $$-0.696683\pi$$
−0.579324 + 0.815097i $$0.696683\pi$$
$$140$$ 0 0
$$141$$ 18.9282 1.59404
$$142$$ − 9.46410i − 0.794210i
$$143$$ − 1.46410i − 0.122434i
$$144$$ −4.46410 −0.372008
$$145$$ 0 0
$$146$$ −14.3923 −1.19112
$$147$$ − 2.73205i − 0.225336i
$$148$$ 0.732051i 0.0601742i
$$149$$ 7.26795 0.595414 0.297707 0.954657i $$-0.403778\pi$$
0.297707 + 0.954657i $$0.403778\pi$$
$$150$$ 0 0
$$151$$ 11.1244 0.905287 0.452644 0.891692i $$-0.350481\pi$$
0.452644 + 0.891692i $$0.350481\pi$$
$$152$$ − 6.73205i − 0.546041i
$$153$$ − 15.4641i − 1.25020i
$$154$$ −1.00000 −0.0805823
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ − 4.53590i − 0.362004i −0.983483 0.181002i $$-0.942066\pi$$
0.983483 0.181002i $$-0.0579341\pi$$
$$158$$ − 12.1962i − 0.970274i
$$159$$ 19.8564 1.57472
$$160$$ 0 0
$$161$$ −8.19615 −0.645947
$$162$$ 2.46410i 0.193598i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 2.19615 0.171491
$$165$$ 0 0
$$166$$ −16.3923 −1.27229
$$167$$ − 13.8564i − 1.07224i −0.844141 0.536120i $$-0.819889\pi$$
0.844141 0.536120i $$-0.180111\pi$$
$$168$$ 2.73205i 0.210782i
$$169$$ 10.8564 0.835108
$$170$$ 0 0
$$171$$ 30.0526 2.29818
$$172$$ − 2.00000i − 0.152499i
$$173$$ 0.928203i 0.0705700i 0.999377 + 0.0352850i $$0.0112339\pi$$
−0.999377 + 0.0352850i $$0.988766\pi$$
$$174$$ 12.9282 0.980085
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ − 18.9282i − 1.42273i
$$178$$ 3.46410i 0.259645i
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 1.46410i 0.108526i
$$183$$ − 13.4641i − 0.995295i
$$184$$ 8.19615 0.604228
$$185$$ 0 0
$$186$$ 5.46410 0.400647
$$187$$ 3.46410i 0.253320i
$$188$$ 6.92820i 0.505291i
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ − 2.73205i − 0.197169i
$$193$$ 12.3923i 0.892018i 0.895029 + 0.446009i $$0.147155\pi$$
−0.895029 + 0.446009i $$0.852845\pi$$
$$194$$ −14.5885 −1.04739
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 24.2487i 1.72765i 0.503793 + 0.863825i $$0.331938\pi$$
−0.503793 + 0.863825i $$0.668062\pi$$
$$198$$ 4.46410i 0.317250i
$$199$$ −2.92820 −0.207575 −0.103787 0.994600i $$-0.533096\pi$$
−0.103787 + 0.994600i $$0.533096\pi$$
$$200$$ 0 0
$$201$$ −10.9282 −0.770816
$$202$$ 7.85641i 0.552775i
$$203$$ − 4.73205i − 0.332125i
$$204$$ 9.46410 0.662620
$$205$$ 0 0
$$206$$ 12.3923 0.863413
$$207$$ 36.5885i 2.54307i
$$208$$ − 1.46410i − 0.101517i
$$209$$ −6.73205 −0.465666
$$210$$ 0 0
$$211$$ 26.9282 1.85381 0.926907 0.375291i $$-0.122457\pi$$
0.926907 + 0.375291i $$0.122457\pi$$
$$212$$ 7.26795i 0.499165i
$$213$$ 25.8564i 1.77165i
$$214$$ −7.85641 −0.537053
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ − 2.00000i − 0.135769i
$$218$$ − 15.6603i − 1.06065i
$$219$$ 39.3205 2.65703
$$220$$ 0 0
$$221$$ 5.07180 0.341166
$$222$$ − 2.00000i − 0.134231i
$$223$$ − 25.4641i − 1.70520i −0.522562 0.852601i $$-0.675024\pi$$
0.522562 0.852601i $$-0.324976\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ 7.85641 0.522600
$$227$$ − 6.92820i − 0.459841i −0.973209 0.229920i $$-0.926153\pi$$
0.973209 0.229920i $$-0.0738466\pi$$
$$228$$ 18.3923i 1.21806i
$$229$$ −24.3923 −1.61189 −0.805944 0.591991i $$-0.798342\pi$$
−0.805944 + 0.591991i $$0.798342\pi$$
$$230$$ 0 0
$$231$$ 2.73205 0.179756
$$232$$ 4.73205i 0.310674i
$$233$$ − 7.85641i − 0.514690i −0.966320 0.257345i $$-0.917152\pi$$
0.966320 0.257345i $$-0.0828477\pi$$
$$234$$ 6.53590 0.427265
$$235$$ 0 0
$$236$$ 6.92820 0.450988
$$237$$ 33.3205i 2.16440i
$$238$$ − 3.46410i − 0.224544i
$$239$$ −1.26795 −0.0820168 −0.0410084 0.999159i $$-0.513057\pi$$
−0.0410084 + 0.999159i $$0.513057\pi$$
$$240$$ 0 0
$$241$$ 3.26795 0.210507 0.105254 0.994445i $$-0.466435\pi$$
0.105254 + 0.994445i $$0.466435\pi$$
$$242$$ − 1.00000i − 0.0642824i
$$243$$ − 18.7321i − 1.20166i
$$244$$ 4.92820 0.315496
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 9.85641i 0.627148i
$$248$$ 2.00000i 0.127000i
$$249$$ 44.7846 2.83811
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ − 4.46410i − 0.281212i
$$253$$ − 8.19615i − 0.515288i
$$254$$ −1.07180 −0.0672505
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 23.6603i 1.47589i 0.674863 + 0.737943i $$0.264203\pi$$
−0.674863 + 0.737943i $$0.735797\pi$$
$$258$$ 5.46410i 0.340180i
$$259$$ −0.732051 −0.0454874
$$260$$ 0 0
$$261$$ −21.1244 −1.30756
$$262$$ − 5.66025i − 0.349692i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ −2.73205 −0.168146
$$265$$ 0 0
$$266$$ 6.73205 0.412769
$$267$$ − 9.46410i − 0.579194i
$$268$$ − 4.00000i − 0.244339i
$$269$$ −28.3923 −1.73111 −0.865555 0.500814i $$-0.833034\pi$$
−0.865555 + 0.500814i $$0.833034\pi$$
$$270$$ 0 0
$$271$$ 0.392305 0.0238308 0.0119154 0.999929i $$-0.496207\pi$$
0.0119154 + 0.999929i $$0.496207\pi$$
$$272$$ 3.46410i 0.210042i
$$273$$ − 4.00000i − 0.242091i
$$274$$ 0.928203 0.0560748
$$275$$ 0 0
$$276$$ −22.3923 −1.34786
$$277$$ − 7.07180i − 0.424903i −0.977172 0.212452i $$-0.931855\pi$$
0.977172 0.212452i $$-0.0681448\pi$$
$$278$$ 13.6603i 0.819288i
$$279$$ −8.92820 −0.534518
$$280$$ 0 0
$$281$$ 22.3923 1.33581 0.667906 0.744245i $$-0.267191\pi$$
0.667906 + 0.744245i $$0.267191\pi$$
$$282$$ − 18.9282i − 1.12716i
$$283$$ − 31.7128i − 1.88513i −0.334021 0.942566i $$-0.608406\pi$$
0.334021 0.942566i $$-0.391594\pi$$
$$284$$ −9.46410 −0.561591
$$285$$ 0 0
$$286$$ −1.46410 −0.0865741
$$287$$ 2.19615i 0.129635i
$$288$$ 4.46410i 0.263050i
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ 39.8564 2.33642
$$292$$ 14.3923i 0.842246i
$$293$$ 21.4641i 1.25395i 0.779041 + 0.626973i $$0.215706\pi$$
−0.779041 + 0.626973i $$0.784294\pi$$
$$294$$ −2.73205 −0.159336
$$295$$ 0 0
$$296$$ 0.732051 0.0425496
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 7.26795i − 0.421021i
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ 2.00000 0.115278
$$302$$ − 11.1244i − 0.640135i
$$303$$ − 21.4641i − 1.23308i
$$304$$ −6.73205 −0.386110
$$305$$ 0 0
$$306$$ −15.4641 −0.884024
$$307$$ − 24.3923i − 1.39214i −0.717973 0.696071i $$-0.754930\pi$$
0.717973 0.696071i $$-0.245070\pi$$
$$308$$ 1.00000i 0.0569803i
$$309$$ −33.8564 −1.92602
$$310$$ 0 0
$$311$$ 24.9282 1.41355 0.706774 0.707439i $$-0.250150\pi$$
0.706774 + 0.707439i $$0.250150\pi$$
$$312$$ 4.00000i 0.226455i
$$313$$ 22.1962i 1.25460i 0.778777 + 0.627300i $$0.215840\pi$$
−0.778777 + 0.627300i $$0.784160\pi$$
$$314$$ −4.53590 −0.255976
$$315$$ 0 0
$$316$$ −12.1962 −0.686087
$$317$$ − 30.5885i − 1.71802i −0.511960 0.859009i $$-0.671080\pi$$
0.511960 0.859009i $$-0.328920\pi$$
$$318$$ − 19.8564i − 1.11349i
$$319$$ 4.73205 0.264944
$$320$$ 0 0
$$321$$ 21.4641 1.19801
$$322$$ 8.19615i 0.456754i
$$323$$ − 23.3205i − 1.29759i
$$324$$ 2.46410 0.136895
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ 42.7846i 2.36599i
$$328$$ − 2.19615i − 0.121262i
$$329$$ −6.92820 −0.381964
$$330$$ 0 0
$$331$$ −18.7846 −1.03250 −0.516248 0.856439i $$-0.672672\pi$$
−0.516248 + 0.856439i $$0.672672\pi$$
$$332$$ 16.3923i 0.899645i
$$333$$ 3.26795i 0.179083i
$$334$$ −13.8564 −0.758189
$$335$$ 0 0
$$336$$ 2.73205 0.149046
$$337$$ − 22.7846i − 1.24116i −0.784144 0.620578i $$-0.786898\pi$$
0.784144 0.620578i $$-0.213102\pi$$
$$338$$ − 10.8564i − 0.590511i
$$339$$ −21.4641 −1.16577
$$340$$ 0 0
$$341$$ 2.00000 0.108306
$$342$$ − 30.0526i − 1.62506i
$$343$$ 1.00000i 0.0539949i
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ 0.928203 0.0499005
$$347$$ 23.0718i 1.23856i 0.785171 + 0.619279i $$0.212575\pi$$
−0.785171 + 0.619279i $$0.787425\pi$$
$$348$$ − 12.9282i − 0.693024i
$$349$$ 5.60770 0.300173 0.150087 0.988673i $$-0.452045\pi$$
0.150087 + 0.988673i $$0.452045\pi$$
$$350$$ 0 0
$$351$$ −5.85641 −0.312592
$$352$$ − 1.00000i − 0.0533002i
$$353$$ 14.1962i 0.755585i 0.925890 + 0.377792i $$0.123317\pi$$
−0.925890 + 0.377792i $$0.876683\pi$$
$$354$$ −18.9282 −1.00602
$$355$$ 0 0
$$356$$ 3.46410 0.183597
$$357$$ 9.46410i 0.500893i
$$358$$ − 6.00000i − 0.317110i
$$359$$ 34.0526 1.79723 0.898613 0.438743i $$-0.144576\pi$$
0.898613 + 0.438743i $$0.144576\pi$$
$$360$$ 0 0
$$361$$ 26.3205 1.38529
$$362$$ − 14.0000i − 0.735824i
$$363$$ 2.73205i 0.143395i
$$364$$ 1.46410 0.0767398
$$365$$ 0 0
$$366$$ −13.4641 −0.703780
$$367$$ − 12.3923i − 0.646873i −0.946250 0.323437i $$-0.895162\pi$$
0.946250 0.323437i $$-0.104838\pi$$
$$368$$ − 8.19615i − 0.427254i
$$369$$ 9.80385 0.510368
$$370$$ 0 0
$$371$$ −7.26795 −0.377333
$$372$$ − 5.46410i − 0.283300i
$$373$$ 18.3923i 0.952317i 0.879359 + 0.476159i $$0.157971\pi$$
−0.879359 + 0.476159i $$0.842029\pi$$
$$374$$ 3.46410 0.179124
$$375$$ 0 0
$$376$$ 6.92820 0.357295
$$377$$ − 6.92820i − 0.356821i
$$378$$ 4.00000i 0.205738i
$$379$$ −6.14359 −0.315575 −0.157788 0.987473i $$-0.550436\pi$$
−0.157788 + 0.987473i $$0.550436\pi$$
$$380$$ 0 0
$$381$$ 2.92820 0.150016
$$382$$ 12.0000i 0.613973i
$$383$$ − 33.4641i − 1.70994i −0.518681 0.854968i $$-0.673577\pi$$
0.518681 0.854968i $$-0.326423\pi$$
$$384$$ −2.73205 −0.139419
$$385$$ 0 0
$$386$$ 12.3923 0.630752
$$387$$ − 8.92820i − 0.453846i
$$388$$ 14.5885i 0.740617i
$$389$$ 1.60770 0.0815134 0.0407567 0.999169i $$-0.487023\pi$$
0.0407567 + 0.999169i $$0.487023\pi$$
$$390$$ 0 0
$$391$$ 28.3923 1.43586
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 15.4641i 0.780061i
$$394$$ 24.2487 1.22163
$$395$$ 0 0
$$396$$ 4.46410 0.224330
$$397$$ − 30.3923i − 1.52535i −0.646784 0.762673i $$-0.723887\pi$$
0.646784 0.762673i $$-0.276113\pi$$
$$398$$ 2.92820i 0.146778i
$$399$$ −18.3923 −0.920767
$$400$$ 0 0
$$401$$ 2.53590 0.126637 0.0633184 0.997993i $$-0.479832\pi$$
0.0633184 + 0.997993i $$0.479832\pi$$
$$402$$ 10.9282i 0.545049i
$$403$$ − 2.92820i − 0.145864i
$$404$$ 7.85641 0.390871
$$405$$ 0 0
$$406$$ −4.73205 −0.234848
$$407$$ − 0.732051i − 0.0362864i
$$408$$ − 9.46410i − 0.468543i
$$409$$ −10.8756 −0.537766 −0.268883 0.963173i $$-0.586654\pi$$
−0.268883 + 0.963173i $$0.586654\pi$$
$$410$$ 0 0
$$411$$ −2.53590 −0.125087
$$412$$ − 12.3923i − 0.610525i
$$413$$ 6.92820i 0.340915i
$$414$$ 36.5885 1.79822
$$415$$ 0 0
$$416$$ −1.46410 −0.0717835
$$417$$ − 37.3205i − 1.82759i
$$418$$ 6.73205i 0.329275i
$$419$$ −30.9282 −1.51094 −0.755471 0.655182i $$-0.772592\pi$$
−0.755471 + 0.655182i $$0.772592\pi$$
$$420$$ 0 0
$$421$$ −35.8564 −1.74753 −0.873767 0.486344i $$-0.838330\pi$$
−0.873767 + 0.486344i $$0.838330\pi$$
$$422$$ − 26.9282i − 1.31084i
$$423$$ 30.9282i 1.50378i
$$424$$ 7.26795 0.352963
$$425$$ 0 0
$$426$$ 25.8564 1.25275
$$427$$ 4.92820i 0.238492i
$$428$$ 7.85641i 0.379754i
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ 3.12436 0.150495 0.0752475 0.997165i $$-0.476025\pi$$
0.0752475 + 0.997165i $$0.476025\pi$$
$$432$$ − 4.00000i − 0.192450i
$$433$$ − 18.1962i − 0.874451i −0.899352 0.437226i $$-0.855961\pi$$
0.899352 0.437226i $$-0.144039\pi$$
$$434$$ −2.00000 −0.0960031
$$435$$ 0 0
$$436$$ −15.6603 −0.749990
$$437$$ 55.1769i 2.63947i
$$438$$ − 39.3205i − 1.87881i
$$439$$ −26.2487 −1.25278 −0.626391 0.779509i $$-0.715469\pi$$
−0.626391 + 0.779509i $$0.715469\pi$$
$$440$$ 0 0
$$441$$ 4.46410 0.212576
$$442$$ − 5.07180i − 0.241241i
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ −25.4641 −1.20576
$$447$$ 19.8564i 0.939176i
$$448$$ 1.00000i 0.0472456i
$$449$$ −40.3923 −1.90623 −0.953115 0.302607i $$-0.902143\pi$$
−0.953115 + 0.302607i $$0.902143\pi$$
$$450$$ 0 0
$$451$$ −2.19615 −0.103413
$$452$$ − 7.85641i − 0.369534i
$$453$$ 30.3923i 1.42796i
$$454$$ −6.92820 −0.325157
$$455$$ 0 0
$$456$$ 18.3923 0.861299
$$457$$ − 2.00000i − 0.0935561i −0.998905 0.0467780i $$-0.985105\pi$$
0.998905 0.0467780i $$-0.0148953\pi$$
$$458$$ 24.3923i 1.13978i
$$459$$ 13.8564 0.646762
$$460$$ 0 0
$$461$$ 22.3923 1.04291 0.521457 0.853278i $$-0.325389\pi$$
0.521457 + 0.853278i $$0.325389\pi$$
$$462$$ − 2.73205i − 0.127107i
$$463$$ 27.5167i 1.27881i 0.768871 + 0.639404i $$0.220819\pi$$
−0.768871 + 0.639404i $$0.779181\pi$$
$$464$$ 4.73205 0.219680
$$465$$ 0 0
$$466$$ −7.85641 −0.363941
$$467$$ 34.0526i 1.57576i 0.615826 + 0.787882i $$0.288822\pi$$
−0.615826 + 0.787882i $$0.711178\pi$$
$$468$$ − 6.53590i − 0.302122i
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 12.3923 0.571007
$$472$$ − 6.92820i − 0.318896i
$$473$$ 2.00000i 0.0919601i
$$474$$ 33.3205 1.53046
$$475$$ 0 0
$$476$$ −3.46410 −0.158777
$$477$$ 32.4449i 1.48555i
$$478$$ 1.26795i 0.0579946i
$$479$$ 32.7846 1.49797 0.748984 0.662589i $$-0.230542\pi$$
0.748984 + 0.662589i $$0.230542\pi$$
$$480$$ 0 0
$$481$$ −1.07180 −0.0488697
$$482$$ − 3.26795i − 0.148851i
$$483$$ − 22.3923i − 1.01889i
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ −18.7321 −0.849703
$$487$$ − 4.19615i − 0.190146i −0.995470 0.0950729i $$-0.969692\pi$$
0.995470 0.0950729i $$-0.0303084\pi$$
$$488$$ − 4.92820i − 0.223089i
$$489$$ 10.9282 0.494190
$$490$$ 0 0
$$491$$ −27.7128 −1.25066 −0.625331 0.780360i $$-0.715036\pi$$
−0.625331 + 0.780360i $$0.715036\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ 16.3923i 0.738272i
$$494$$ 9.85641 0.443461
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ − 9.46410i − 0.424523i
$$498$$ − 44.7846i − 2.00685i
$$499$$ −12.1436 −0.543622 −0.271811 0.962351i $$-0.587623\pi$$
−0.271811 + 0.962351i $$0.587623\pi$$
$$500$$ 0 0
$$501$$ 37.8564 1.69130
$$502$$ − 12.0000i − 0.535586i
$$503$$ 8.78461i 0.391686i 0.980635 + 0.195843i $$0.0627444\pi$$
−0.980635 + 0.195843i $$0.937256\pi$$
$$504$$ −4.46410 −0.198847
$$505$$ 0 0
$$506$$ −8.19615 −0.364363
$$507$$ 29.6603i 1.31726i
$$508$$ 1.07180i 0.0475533i
$$509$$ −11.0718 −0.490749 −0.245374 0.969428i $$-0.578911\pi$$
−0.245374 + 0.969428i $$0.578911\pi$$
$$510$$ 0 0
$$511$$ −14.3923 −0.636678
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 26.9282i 1.18891i
$$514$$ 23.6603 1.04361
$$515$$ 0 0
$$516$$ 5.46410 0.240544
$$517$$ − 6.92820i − 0.304702i
$$518$$ 0.732051i 0.0321645i
$$519$$ −2.53590 −0.111314
$$520$$ 0 0
$$521$$ 10.3923 0.455295 0.227648 0.973744i $$-0.426897\pi$$
0.227648 + 0.973744i $$0.426897\pi$$
$$522$$ 21.1244i 0.924588i
$$523$$ − 29.1769i − 1.27582i −0.770112 0.637909i $$-0.779800\pi$$
0.770112 0.637909i $$-0.220200\pi$$
$$524$$ −5.66025 −0.247269
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 6.92820i 0.301797i
$$528$$ 2.73205i 0.118897i
$$529$$ −44.1769 −1.92074
$$530$$ 0 0
$$531$$ 30.9282 1.34217
$$532$$ − 6.73205i − 0.291871i
$$533$$ 3.21539i 0.139274i
$$534$$ −9.46410 −0.409552
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 16.3923i 0.707380i
$$538$$ 28.3923i 1.22408i
$$539$$ −1.00000 −0.0430730
$$540$$ 0 0
$$541$$ −20.7321 −0.891340 −0.445670 0.895197i $$-0.647035\pi$$
−0.445670 + 0.895197i $$0.647035\pi$$
$$542$$ − 0.392305i − 0.0168509i
$$543$$ 38.2487i 1.64141i
$$544$$ 3.46410 0.148522
$$545$$ 0 0
$$546$$ −4.00000 −0.171184
$$547$$ − 28.7846i − 1.23074i −0.788238 0.615371i $$-0.789006\pi$$
0.788238 0.615371i $$-0.210994\pi$$
$$548$$ − 0.928203i − 0.0396509i
$$549$$ 22.0000 0.938937
$$550$$ 0 0
$$551$$ −31.8564 −1.35713
$$552$$ 22.3923i 0.953080i
$$553$$ − 12.1962i − 0.518633i
$$554$$ −7.07180 −0.300452
$$555$$ 0 0
$$556$$ 13.6603 0.579324
$$557$$ 25.6077i 1.08503i 0.840045 + 0.542516i $$0.182528\pi$$
−0.840045 + 0.542516i $$0.817472\pi$$
$$558$$ 8.92820i 0.377961i
$$559$$ 2.92820 0.123850
$$560$$ 0 0
$$561$$ −9.46410 −0.399575
$$562$$ − 22.3923i − 0.944562i
$$563$$ 5.07180i 0.213751i 0.994272 + 0.106875i $$0.0340846\pi$$
−0.994272 + 0.106875i $$0.965915\pi$$
$$564$$ −18.9282 −0.797021
$$565$$ 0 0
$$566$$ −31.7128 −1.33299
$$567$$ 2.46410i 0.103483i
$$568$$ 9.46410i 0.397105i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 3.60770 0.150977 0.0754887 0.997147i $$-0.475948\pi$$
0.0754887 + 0.997147i $$0.475948\pi$$
$$572$$ 1.46410i 0.0612172i
$$573$$ − 32.7846i − 1.36960i
$$574$$ 2.19615 0.0916656
$$575$$ 0 0
$$576$$ 4.46410 0.186004
$$577$$ 34.5885i 1.43994i 0.694007 + 0.719968i $$0.255844\pi$$
−0.694007 + 0.719968i $$0.744156\pi$$
$$578$$ − 5.00000i − 0.207973i
$$579$$ −33.8564 −1.40702
$$580$$ 0 0
$$581$$ −16.3923 −0.680067
$$582$$ − 39.8564i − 1.65210i
$$583$$ − 7.26795i − 0.301008i
$$584$$ 14.3923 0.595558
$$585$$ 0 0
$$586$$ 21.4641 0.886674
$$587$$ 11.4115i 0.471005i 0.971874 + 0.235502i $$0.0756735\pi$$
−0.971874 + 0.235502i $$0.924326\pi$$
$$588$$ 2.73205i 0.112668i
$$589$$ −13.4641 −0.554779
$$590$$ 0 0
$$591$$ −66.2487 −2.72511
$$592$$ − 0.732051i − 0.0300871i
$$593$$ 0.248711i 0.0102133i 0.999987 + 0.00510667i $$0.00162551\pi$$
−0.999987 + 0.00510667i $$0.998374\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ −7.26795 −0.297707
$$597$$ − 8.00000i − 0.327418i
$$598$$ 12.0000i 0.490716i
$$599$$ −37.1769 −1.51901 −0.759504 0.650503i $$-0.774558\pi$$
−0.759504 + 0.650503i $$0.774558\pi$$
$$600$$ 0 0
$$601$$ −5.51666 −0.225029 −0.112515 0.993650i $$-0.535891\pi$$
−0.112515 + 0.993650i $$0.535891\pi$$
$$602$$ − 2.00000i − 0.0815139i
$$603$$ − 17.8564i − 0.727169i
$$604$$ −11.1244 −0.452644
$$605$$ 0 0
$$606$$ −21.4641 −0.871920
$$607$$ − 20.9282i − 0.849450i −0.905323 0.424725i $$-0.860371\pi$$
0.905323 0.424725i $$-0.139629\pi$$
$$608$$ 6.73205i 0.273021i
$$609$$ 12.9282 0.523877
$$610$$ 0 0
$$611$$ −10.1436 −0.410366
$$612$$ 15.4641i 0.625099i
$$613$$ − 35.1769i − 1.42078i −0.703807 0.710391i $$-0.748518\pi$$
0.703807 0.710391i $$-0.251482\pi$$
$$614$$ −24.3923 −0.984393
$$615$$ 0 0
$$616$$ 1.00000 0.0402911
$$617$$ − 17.3205i − 0.697297i −0.937253 0.348649i $$-0.886641\pi$$
0.937253 0.348649i $$-0.113359\pi$$
$$618$$ 33.8564i 1.36190i
$$619$$ −28.7846 −1.15695 −0.578476 0.815700i $$-0.696352\pi$$
−0.578476 + 0.815700i $$0.696352\pi$$
$$620$$ 0 0
$$621$$ −32.7846 −1.31560
$$622$$ − 24.9282i − 0.999530i
$$623$$ 3.46410i 0.138786i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ 22.1962 0.887137
$$627$$ − 18.3923i − 0.734518i
$$628$$ 4.53590i 0.181002i
$$629$$ 2.53590 0.101113
$$630$$ 0 0
$$631$$ −46.9282 −1.86818 −0.934091 0.357035i $$-0.883788\pi$$
−0.934091 + 0.357035i $$0.883788\pi$$
$$632$$ 12.1962i 0.485137i
$$633$$ 73.5692i 2.92411i
$$634$$ −30.5885 −1.21482
$$635$$ 0 0
$$636$$ −19.8564 −0.787358
$$637$$ 1.46410i 0.0580098i
$$638$$ − 4.73205i − 0.187344i
$$639$$ −42.2487 −1.67133
$$640$$ 0 0
$$641$$ −35.5692 −1.40490 −0.702450 0.711733i $$-0.747910\pi$$
−0.702450 + 0.711733i $$0.747910\pi$$
$$642$$ − 21.4641i − 0.847121i
$$643$$ 33.2679i 1.31196i 0.754778 + 0.655980i $$0.227744\pi$$
−0.754778 + 0.655980i $$0.772256\pi$$
$$644$$ 8.19615 0.322974
$$645$$ 0 0
$$646$$ −23.3205 −0.917533
$$647$$ 26.5359i 1.04323i 0.853180 + 0.521617i $$0.174671\pi$$
−0.853180 + 0.521617i $$0.825329\pi$$
$$648$$ − 2.46410i − 0.0967991i
$$649$$ −6.92820 −0.271956
$$650$$ 0 0
$$651$$ 5.46410 0.214155
$$652$$ 4.00000i 0.156652i
$$653$$ 11.6603i 0.456301i 0.973626 + 0.228151i $$0.0732678\pi$$
−0.973626 + 0.228151i $$0.926732\pi$$
$$654$$ 42.7846 1.67301
$$655$$ 0 0
$$656$$ −2.19615 −0.0857453
$$657$$ 64.2487i 2.50658i
$$658$$ 6.92820i 0.270089i
$$659$$ −30.2487 −1.17832 −0.589161 0.808015i $$-0.700542\pi$$
−0.589161 + 0.808015i $$0.700542\pi$$
$$660$$ 0 0
$$661$$ −46.0000 −1.78919 −0.894596 0.446875i $$-0.852537\pi$$
−0.894596 + 0.446875i $$0.852537\pi$$
$$662$$ 18.7846i 0.730085i
$$663$$ 13.8564i 0.538138i
$$664$$ 16.3923 0.636145
$$665$$ 0 0
$$666$$ 3.26795 0.126630
$$667$$ − 38.7846i − 1.50175i
$$668$$ 13.8564i 0.536120i
$$669$$ 69.5692 2.68970
$$670$$ 0 0
$$671$$ −4.92820 −0.190251
$$672$$ − 2.73205i − 0.105391i
$$673$$ 2.00000i 0.0770943i 0.999257 + 0.0385472i $$0.0122730\pi$$
−0.999257 + 0.0385472i $$0.987727\pi$$
$$674$$ −22.7846 −0.877630
$$675$$ 0 0
$$676$$ −10.8564 −0.417554
$$677$$ 4.14359i 0.159251i 0.996825 + 0.0796256i $$0.0253725\pi$$
−0.996825 + 0.0796256i $$0.974628\pi$$
$$678$$ 21.4641i 0.824324i
$$679$$ −14.5885 −0.559854
$$680$$ 0 0
$$681$$ 18.9282 0.725330
$$682$$ − 2.00000i − 0.0765840i
$$683$$ 6.24871i 0.239100i 0.992828 + 0.119550i $$0.0381452\pi$$
−0.992828 + 0.119550i $$0.961855\pi$$
$$684$$ −30.0526 −1.14909
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ − 66.6410i − 2.54251i
$$688$$ 2.00000i 0.0762493i
$$689$$ −10.6410 −0.405390
$$690$$ 0 0
$$691$$ −13.4641 −0.512199 −0.256099 0.966650i $$-0.582437\pi$$
−0.256099 + 0.966650i $$0.582437\pi$$
$$692$$ − 0.928203i − 0.0352850i
$$693$$ 4.46410i 0.169577i
$$694$$ 23.0718 0.875793
$$695$$ 0 0
$$696$$ −12.9282 −0.490042
$$697$$ − 7.60770i − 0.288162i
$$698$$ − 5.60770i − 0.212254i
$$699$$ 21.4641 0.811847
$$700$$ 0 0
$$701$$ 41.9090 1.58288 0.791440 0.611247i $$-0.209332\pi$$
0.791440 + 0.611247i $$0.209332\pi$$
$$702$$ 5.85641i 0.221036i
$$703$$ 4.92820i 0.185871i
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 14.1962 0.534279
$$707$$ 7.85641i 0.295471i
$$708$$ 18.9282i 0.711365i
$$709$$ −25.3205 −0.950932 −0.475466 0.879734i $$-0.657720\pi$$
−0.475466 + 0.879734i $$0.657720\pi$$
$$710$$ 0 0
$$711$$ −54.4449 −2.04184
$$712$$ − 3.46410i − 0.129823i
$$713$$ − 16.3923i − 0.613897i
$$714$$ 9.46410 0.354185
$$715$$ 0 0
$$716$$ −6.00000 −0.224231
$$717$$ − 3.46410i − 0.129369i
$$718$$ − 34.0526i − 1.27083i
$$719$$ 27.7128 1.03351 0.516757 0.856132i $$-0.327139\pi$$
0.516757 + 0.856132i $$0.327139\pi$$
$$720$$ 0 0
$$721$$ 12.3923 0.461514
$$722$$ − 26.3205i − 0.979548i
$$723$$ 8.92820i 0.332043i
$$724$$ −14.0000 −0.520306
$$725$$ 0 0
$$726$$ 2.73205 0.101396
$$727$$ 4.00000i 0.148352i 0.997245 + 0.0741759i $$0.0236326\pi$$
−0.997245 + 0.0741759i $$0.976367\pi$$
$$728$$ − 1.46410i − 0.0542632i
$$729$$ 43.7846 1.62165
$$730$$ 0 0
$$731$$ −6.92820 −0.256249
$$732$$ 13.4641i 0.497648i
$$733$$ 26.0000i 0.960332i 0.877178 + 0.480166i $$0.159424\pi$$
−0.877178 + 0.480166i $$0.840576\pi$$
$$734$$ −12.3923 −0.457408
$$735$$ 0 0
$$736$$ −8.19615 −0.302114
$$737$$ 4.00000i 0.147342i
$$738$$ − 9.80385i − 0.360885i
$$739$$ −11.7128 −0.430863 −0.215431 0.976519i $$-0.569116\pi$$
−0.215431 + 0.976519i $$0.569116\pi$$
$$740$$ 0 0
$$741$$ −26.9282 −0.989232
$$742$$ 7.26795i 0.266815i
$$743$$ − 32.7846i − 1.20275i −0.798967 0.601375i $$-0.794620\pi$$
0.798967 0.601375i $$-0.205380\pi$$
$$744$$ −5.46410 −0.200324
$$745$$ 0 0
$$746$$ 18.3923 0.673390
$$747$$ 73.1769i 2.67740i
$$748$$ − 3.46410i − 0.126660i
$$749$$ −7.85641 −0.287067
$$750$$ 0 0
$$751$$ −11.6077 −0.423571 −0.211785 0.977316i $$-0.567928\pi$$
−0.211785 + 0.977316i $$0.567928\pi$$
$$752$$ − 6.92820i − 0.252646i
$$753$$ 32.7846i 1.19474i
$$754$$ −6.92820 −0.252310
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ 43.3731i 1.57642i 0.615406 + 0.788210i $$0.288992\pi$$
−0.615406 + 0.788210i $$0.711008\pi$$
$$758$$ 6.14359i 0.223145i
$$759$$ 22.3923 0.812789
$$760$$ 0 0
$$761$$ −12.3397 −0.447315 −0.223658 0.974668i $$-0.571800\pi$$
−0.223658 + 0.974668i $$0.571800\pi$$
$$762$$ − 2.92820i − 0.106078i
$$763$$ − 15.6603i − 0.566939i
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ −33.4641 −1.20911
$$767$$ 10.1436i 0.366264i
$$768$$ 2.73205i 0.0985844i
$$769$$ 18.1962 0.656170 0.328085 0.944648i $$-0.393597\pi$$
0.328085 + 0.944648i $$0.393597\pi$$
$$770$$ 0 0
$$771$$ −64.6410 −2.32799
$$772$$ − 12.3923i − 0.446009i
$$773$$ − 0.928203i − 0.0333851i −0.999861 0.0166926i $$-0.994686\pi$$
0.999861 0.0166926i $$-0.00531366\pi$$
$$774$$ −8.92820 −0.320918
$$775$$ 0 0
$$776$$ 14.5885 0.523695
$$777$$ − 2.00000i − 0.0717496i
$$778$$ − 1.60770i − 0.0576387i
$$779$$ 14.7846 0.529714
$$780$$ 0 0
$$781$$ 9.46410 0.338652
$$782$$ − 28.3923i − 1.01531i
$$783$$ − 18.9282i − 0.676439i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 15.4641 0.551586
$$787$$ − 18.1436i − 0.646749i −0.946271 0.323375i $$-0.895183\pi$$
0.946271 0.323375i $$-0.104817\pi$$
$$788$$ − 24.2487i − 0.863825i
$$789$$ 65.5692 2.33433
$$790$$ 0 0
$$791$$ 7.85641 0.279342
$$792$$ − 4.46410i − 0.158625i
$$793$$ 7.21539i 0.256226i
$$794$$ −30.3923 −1.07858
$$795$$ 0 0
$$796$$ 2.92820 0.103787
$$797$$ 25.6077i 0.907071i 0.891238 + 0.453536i $$0.149837\pi$$
−0.891238 + 0.453536i $$0.850163\pi$$
$$798$$ 18.3923i 0.651081i
$$799$$ 24.0000 0.849059
$$800$$ 0 0
$$801$$ 15.4641 0.546397
$$802$$ − 2.53590i − 0.0895457i
$$803$$ − 14.3923i − 0.507893i
$$804$$ 10.9282 0.385408
$$805$$ 0 0
$$806$$ −2.92820 −0.103142
$$807$$ − 77.5692i − 2.73057i
$$808$$ − 7.85641i − 0.276387i
$$809$$ 31.1769 1.09612 0.548061 0.836438i $$-0.315366\pi$$
0.548061 + 0.836438i $$0.315366\pi$$
$$810$$ 0 0
$$811$$ −43.1244 −1.51430 −0.757150 0.653241i $$-0.773409\pi$$
−0.757150 + 0.653241i $$0.773409\pi$$
$$812$$ 4.73205i 0.166062i
$$813$$ 1.07180i 0.0375896i
$$814$$ −0.732051 −0.0256584
$$815$$ 0 0
$$816$$ −9.46410 −0.331310
$$817$$ − 13.4641i − 0.471049i
$$818$$ 10.8756i 0.380258i
$$819$$ 6.53590 0.228383
$$820$$ 0 0
$$821$$ 17.9090 0.625027 0.312514 0.949913i $$-0.398829\pi$$
0.312514 + 0.949913i $$0.398829\pi$$
$$822$$ 2.53590i 0.0884496i
$$823$$ − 0.875644i − 0.0305230i −0.999884 0.0152615i $$-0.995142\pi$$
0.999884 0.0152615i $$-0.00485808\pi$$
$$824$$ −12.3923 −0.431706
$$825$$ 0 0
$$826$$ 6.92820 0.241063
$$827$$ 25.8564i 0.899115i 0.893251 + 0.449558i $$0.148418\pi$$
−0.893251 + 0.449558i $$0.851582\pi$$
$$828$$ − 36.5885i − 1.27154i
$$829$$ −2.24871 −0.0781010 −0.0390505 0.999237i $$-0.512433\pi$$
−0.0390505 + 0.999237i $$0.512433\pi$$
$$830$$ 0 0
$$831$$ 19.3205 0.670221
$$832$$ 1.46410i 0.0507586i
$$833$$ − 3.46410i − 0.120024i
$$834$$ −37.3205 −1.29230
$$835$$ 0 0
$$836$$ 6.73205 0.232833
$$837$$ − 8.00000i − 0.276520i
$$838$$ 30.9282i 1.06840i
$$839$$ 31.8564 1.09981 0.549903 0.835229i $$-0.314665\pi$$
0.549903 + 0.835229i $$0.314665\pi$$
$$840$$ 0 0
$$841$$ −6.60770 −0.227852
$$842$$ 35.8564i 1.23569i
$$843$$ 61.1769i 2.10704i
$$844$$ −26.9282 −0.926907
$$845$$ 0 0
$$846$$ 30.9282 1.06333
$$847$$ − 1.00000i − 0.0343604i
$$848$$ − 7.26795i − 0.249582i
$$849$$ 86.6410 2.97351
$$850$$ 0 0
$$851$$ −6.00000 −0.205677
$$852$$ − 25.8564i − 0.885826i
$$853$$ − 54.7846i − 1.87579i −0.346920 0.937895i $$-0.612773\pi$$
0.346920 0.937895i $$-0.387227\pi$$
$$854$$ 4.92820 0.168640
$$855$$ 0 0
$$856$$ 7.85641 0.268526
$$857$$ 27.4641i 0.938156i 0.883157 + 0.469078i $$0.155414\pi$$
−0.883157 + 0.469078i $$0.844586\pi$$
$$858$$ − 4.00000i − 0.136558i
$$859$$ 12.7846 0.436205 0.218103 0.975926i $$-0.430013\pi$$
0.218103 + 0.975926i $$0.430013\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ − 3.12436i − 0.106416i
$$863$$ − 7.51666i − 0.255870i −0.991783 0.127935i $$-0.959165\pi$$
0.991783 0.127935i $$-0.0408349\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ −18.1962 −0.618330
$$867$$ 13.6603i 0.463927i
$$868$$ 2.00000i 0.0678844i
$$869$$ 12.1962 0.413726
$$870$$ 0 0
$$871$$ 5.85641 0.198437
$$872$$ 15.6603i 0.530323i
$$873$$ 65.1244i 2.20413i
$$874$$ 55.1769 1.86639
$$875$$ 0 0
$$876$$ −39.3205 −1.32852
$$877$$ 21.3205i 0.719942i 0.932963 + 0.359971i $$0.117213\pi$$
−0.932963 + 0.359971i $$0.882787\pi$$
$$878$$ 26.2487i 0.885851i
$$879$$ −58.6410 −1.97791
$$880$$ 0 0
$$881$$ 11.0718 0.373018 0.186509 0.982453i $$-0.440283\pi$$
0.186509 + 0.982453i $$0.440283\pi$$
$$882$$ − 4.46410i − 0.150314i
$$883$$ − 35.6077i − 1.19829i −0.800639 0.599147i $$-0.795506\pi$$
0.800639 0.599147i $$-0.204494\pi$$
$$884$$ −5.07180 −0.170583
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 13.8564i − 0.465253i −0.972566 0.232626i $$-0.925268\pi$$
0.972566 0.232626i $$-0.0747319\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ −1.07180 −0.0359469
$$890$$ 0 0
$$891$$ −2.46410 −0.0825505
$$892$$ 25.4641i 0.852601i
$$893$$ 46.6410i 1.56078i
$$894$$ 19.8564 0.664098
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ − 32.7846i − 1.09465i
$$898$$ 40.3923i 1.34791i
$$899$$ 9.46410 0.315645
$$900$$ 0 0
$$901$$ 25.1769 0.838765
$$902$$ 2.19615i 0.0731239i
$$903$$ 5.46410i 0.181834i
$$904$$ −7.85641 −0.261300
$$905$$ 0 0
$$906$$ 30.3923 1.00972
$$907$$ 31.0333i 1.03044i 0.857057 + 0.515222i $$0.172291\pi$$
−0.857057 + 0.515222i $$0.827709\pi$$
$$908$$ 6.92820i 0.229920i
$$909$$ 35.0718 1.16326
$$910$$ 0 0
$$911$$ 9.46410 0.313560 0.156780 0.987634i $$-0.449889\pi$$
0.156780 + 0.987634i $$0.449889\pi$$
$$912$$ − 18.3923i − 0.609030i
$$913$$ − 16.3923i − 0.542506i
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ 24.3923 0.805944
$$917$$ − 5.66025i − 0.186918i
$$918$$ − 13.8564i − 0.457330i
$$919$$ 25.3731 0.836980 0.418490 0.908221i $$-0.362559\pi$$
0.418490 + 0.908221i $$0.362559\pi$$
$$920$$ 0 0
$$921$$ 66.6410 2.19590
$$922$$ − 22.3923i − 0.737451i
$$923$$ − 13.8564i − 0.456089i
$$924$$ −2.73205 −0.0898779
$$925$$ 0 0
$$926$$ 27.5167 0.904254
$$927$$ − 55.3205i − 1.81696i
$$928$$ − 4.73205i − 0.155337i
$$929$$ 25.6077 0.840161 0.420081 0.907487i $$-0.362002\pi$$
0.420081 + 0.907487i $$0.362002\pi$$
$$930$$ 0 0
$$931$$ 6.73205 0.220634
$$932$$ 7.85641i 0.257345i
$$933$$ 68.1051i 2.22966i
$$934$$ 34.0526 1.11423
$$935$$ 0 0
$$936$$ −6.53590 −0.213633
$$937$$ 1.21539i 0.0397051i 0.999803 + 0.0198525i $$0.00631967\pi$$
−0.999803 + 0.0198525i $$0.993680\pi$$
$$938$$ − 4.00000i − 0.130605i
$$939$$ −60.6410 −1.97894
$$940$$ 0 0
$$941$$ −14.7846 −0.481965 −0.240982 0.970530i $$-0.577470\pi$$
−0.240982 + 0.970530i $$0.577470\pi$$
$$942$$ − 12.3923i − 0.403763i
$$943$$ 18.0000i 0.586161i
$$944$$ −6.92820 −0.225494
$$945$$ 0 0
$$946$$ 2.00000 0.0650256
$$947$$ − 47.3205i − 1.53771i −0.639423 0.768855i $$-0.720827\pi$$
0.639423 0.768855i $$-0.279173\pi$$
$$948$$ − 33.3205i − 1.08220i
$$949$$ −21.0718 −0.684019
$$950$$ 0 0
$$951$$ 83.5692 2.70992
$$952$$ 3.46410i 0.112272i
$$953$$ 26.5359i 0.859582i 0.902928 + 0.429791i $$0.141413\pi$$
−0.902928 + 0.429791i $$0.858587\pi$$
$$954$$ 32.4449 1.05044
$$955$$ 0 0
$$956$$ 1.26795 0.0410084
$$957$$ 12.9282i 0.417909i
$$958$$ − 32.7846i − 1.05922i
$$959$$ 0.928203 0.0299732
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 1.07180i 0.0345561i
$$963$$ 35.0718i 1.13017i
$$964$$ −3.26795 −0.105254
$$965$$ 0 0
$$966$$ −22.3923 −0.720461
$$967$$ − 26.9282i − 0.865953i −0.901405 0.432976i $$-0.857463\pi$$
0.901405 0.432976i $$-0.142537\pi$$
$$968$$ 1.00000i 0.0321412i
$$969$$ 63.7128 2.04675
$$970$$ 0 0
$$971$$ 42.9282 1.37763 0.688816 0.724936i $$-0.258131\pi$$
0.688816 + 0.724936i $$0.258131\pi$$
$$972$$ 18.7321i 0.600831i
$$973$$ 13.6603i 0.437928i
$$974$$ −4.19615 −0.134453
$$975$$ 0 0
$$976$$ −4.92820 −0.157748
$$977$$ 33.7128i 1.07857i 0.842124 + 0.539284i $$0.181305\pi$$
−0.842124 + 0.539284i $$0.818695\pi$$
$$978$$ − 10.9282i − 0.349445i
$$979$$ −3.46410 −0.110713
$$980$$ 0 0
$$981$$ −69.9090 −2.23202
$$982$$ 27.7128i 0.884351i
$$983$$ − 37.1769i − 1.18576i −0.805291 0.592880i $$-0.797991\pi$$
0.805291 0.592880i $$-0.202009\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 16.3923 0.522037
$$987$$ − 18.9282i − 0.602491i
$$988$$ − 9.85641i − 0.313574i
$$989$$ 16.3923 0.521245
$$990$$ 0 0
$$991$$ −4.00000 −0.127064 −0.0635321 0.997980i $$-0.520237\pi$$
−0.0635321 + 0.997980i $$0.520237\pi$$
$$992$$ − 2.00000i − 0.0635001i
$$993$$ − 51.3205i − 1.62861i
$$994$$ −9.46410 −0.300183
$$995$$ 0 0
$$996$$ −44.7846 −1.41905
$$997$$ − 17.7128i − 0.560970i −0.959858 0.280485i $$-0.909505\pi$$
0.959858 0.280485i $$-0.0904954\pi$$
$$998$$ 12.1436i 0.384399i
$$999$$ −2.92820 −0.0926443
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.x.1849.2 4
5.2 odd 4 770.2.a.j.1.2 2
5.3 odd 4 3850.2.a.bd.1.1 2
5.4 even 2 inner 3850.2.c.x.1849.3 4
15.2 even 4 6930.2.a.bv.1.1 2
20.7 even 4 6160.2.a.t.1.1 2
35.27 even 4 5390.2.a.bs.1.1 2
55.32 even 4 8470.2.a.br.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.2 2 5.2 odd 4
3850.2.a.bd.1.1 2 5.3 odd 4
3850.2.c.x.1849.2 4 1.1 even 1 trivial
3850.2.c.x.1849.3 4 5.4 even 2 inner
5390.2.a.bs.1.1 2 35.27 even 4
6160.2.a.t.1.1 2 20.7 even 4
6930.2.a.bv.1.1 2 15.2 even 4
8470.2.a.br.1.2 2 55.32 even 4