# Properties

 Label 1050.2.g.j.799.1 Level $1050$ Weight $2$ Character 1050.799 Analytic conductor $8.384$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1050.799 Dual form 1050.2.g.j.799.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -1.00000i q^{12} +1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} -1.00000 q^{21} -2.00000i q^{22} +7.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} -1.00000i q^{28} -1.00000 q^{29} +3.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} +6.00000i q^{37} +4.00000i q^{38} -1.00000 q^{39} -3.00000 q^{41} +1.00000i q^{42} -1.00000i q^{43} -2.00000 q^{44} +7.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} -1.00000 q^{49} -1.00000 q^{51} -1.00000i q^{52} +11.0000i q^{53} -1.00000 q^{54} -1.00000 q^{56} -4.00000i q^{57} +1.00000i q^{58} +3.00000 q^{59} +5.00000 q^{61} -3.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +12.0000i q^{67} -1.00000i q^{68} -7.00000 q^{69} +4.00000 q^{71} -1.00000i q^{72} -14.0000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +2.00000i q^{77} +1.00000i q^{78} +2.00000 q^{79} +1.00000 q^{81} +3.00000i q^{82} -3.00000i q^{83} +1.00000 q^{84} -1.00000 q^{86} -1.00000i q^{87} +2.00000i q^{88} -10.0000 q^{89} -1.00000 q^{91} -7.00000i q^{92} +3.00000i q^{93} +12.0000 q^{94} +1.00000 q^{96} +10.0000i q^{97} +1.00000i q^{98} -2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} + 2q^{14} + 2q^{16} - 8q^{19} - 2q^{21} - 2q^{24} + 2q^{26} - 2q^{29} + 6q^{31} + 2q^{34} + 2q^{36} - 2q^{39} - 6q^{41} - 4q^{44} + 14q^{46} - 2q^{49} - 2q^{51} - 2q^{54} - 2q^{56} + 6q^{59} + 10q^{61} - 2q^{64} + 4q^{66} - 14q^{69} + 8q^{71} + 12q^{74} + 8q^{76} + 4q^{79} + 2q^{81} + 2q^{84} - 2q^{86} - 20q^{89} - 2q^{91} + 24q^{94} + 2q^{96} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ 1.00000i 0.377964i
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000i 0.242536i 0.992620 + 0.121268i $$0.0386960\pi$$
−0.992620 + 0.121268i $$0.961304\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ − 2.00000i − 0.426401i
$$23$$ 7.00000i 1.45960i 0.683660 + 0.729800i $$0.260387\pi$$
−0.683660 + 0.729800i $$0.739613\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 1.00000i − 0.188982i
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 2.00000i 0.348155i
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −3.00000 −0.468521 −0.234261 0.972174i $$-0.575267\pi$$
−0.234261 + 0.972174i $$0.575267\pi$$
$$42$$ 1.00000i 0.154303i
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 7.00000 1.03209
$$47$$ 12.0000i 1.75038i 0.483779 + 0.875190i $$0.339264\pi$$
−0.483779 + 0.875190i $$0.660736\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ − 1.00000i − 0.138675i
$$53$$ 11.0000i 1.51097i 0.655168 + 0.755483i $$0.272598\pi$$
−0.655168 + 0.755483i $$0.727402\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ − 4.00000i − 0.529813i
$$58$$ 1.00000i 0.131306i
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ − 3.00000i − 0.381000i
$$63$$ − 1.00000i − 0.125988i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ − 1.00000i − 0.121268i
$$69$$ −7.00000 −0.842701
$$70$$ 0 0
$$71$$ 4.00000 0.474713 0.237356 0.971423i $$-0.423719\pi$$
0.237356 + 0.971423i $$0.423719\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 2.00000i 0.227921i
$$78$$ 1.00000i 0.113228i
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 3.00000i 0.331295i
$$83$$ − 3.00000i − 0.329293i −0.986353 0.164646i $$-0.947352\pi$$
0.986353 0.164646i $$-0.0526483\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ −1.00000 −0.107833
$$87$$ − 1.00000i − 0.107211i
$$88$$ 2.00000i 0.213201i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ − 7.00000i − 0.729800i
$$93$$ 3.00000i 0.311086i
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 1.00000i 0.101015i
$$99$$ −2.00000 −0.201008
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ − 17.0000i − 1.67506i −0.546392 0.837530i $$-0.683999\pi$$
0.546392 0.837530i $$-0.316001\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 11.0000 1.06841
$$107$$ 18.0000i 1.74013i 0.492941 + 0.870063i $$0.335922\pi$$
−0.492941 + 0.870063i $$0.664078\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 1.00000i 0.0944911i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 1.00000 0.0928477
$$117$$ − 1.00000i − 0.0924500i
$$118$$ − 3.00000i − 0.276172i
$$119$$ −1.00000 −0.0916698
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 5.00000i − 0.452679i
$$123$$ − 3.00000i − 0.270501i
$$124$$ −3.00000 −0.269408
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ − 14.0000i − 1.24230i −0.783692 0.621150i $$-0.786666\pi$$
0.783692 0.621150i $$-0.213334\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ − 2.00000i − 0.174078i
$$133$$ − 4.00000i − 0.346844i
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ − 4.00000i − 0.341743i −0.985293 0.170872i $$-0.945342\pi$$
0.985293 0.170872i $$-0.0546583\pi$$
$$138$$ 7.00000i 0.595880i
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −12.0000 −1.01058
$$142$$ − 4.00000i − 0.335673i
$$143$$ 2.00000i 0.167248i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −14.0000 −1.15865
$$147$$ − 1.00000i − 0.0824786i
$$148$$ − 6.00000i − 0.493197i
$$149$$ −5.00000 −0.409616 −0.204808 0.978802i $$-0.565657\pi$$
−0.204808 + 0.978802i $$0.565657\pi$$
$$150$$ 0 0
$$151$$ 22.0000 1.79033 0.895167 0.445730i $$-0.147056\pi$$
0.895167 + 0.445730i $$0.147056\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ − 1.00000i − 0.0808452i
$$154$$ 2.00000 0.161165
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ − 2.00000i − 0.159111i
$$159$$ −11.0000 −0.872357
$$160$$ 0 0
$$161$$ −7.00000 −0.551677
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 19.0000i − 1.48819i −0.668071 0.744097i $$-0.732880\pi$$
0.668071 0.744097i $$-0.267120\pi$$
$$164$$ 3.00000 0.234261
$$165$$ 0 0
$$166$$ −3.00000 −0.232845
$$167$$ 2.00000i 0.154765i 0.997001 + 0.0773823i $$0.0246562\pi$$
−0.997001 + 0.0773823i $$0.975344\pi$$
$$168$$ − 1.00000i − 0.0771517i
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 1.00000i 0.0762493i
$$173$$ − 12.0000i − 0.912343i −0.889892 0.456172i $$-0.849220\pi$$
0.889892 0.456172i $$-0.150780\pi$$
$$174$$ −1.00000 −0.0758098
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 3.00000i 0.225494i
$$178$$ 10.0000i 0.749532i
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 1.00000i 0.0741249i
$$183$$ 5.00000i 0.369611i
$$184$$ −7.00000 −0.516047
$$185$$ 0 0
$$186$$ 3.00000 0.219971
$$187$$ 2.00000i 0.146254i
$$188$$ − 12.0000i − 0.875190i
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −13.0000 −0.940647 −0.470323 0.882494i $$-0.655863\pi$$
−0.470323 + 0.882494i $$0.655863\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 27.0000i − 1.92367i −0.273629 0.961835i $$-0.588224\pi$$
0.273629 0.961835i $$-0.411776\pi$$
$$198$$ 2.00000i 0.142134i
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 0 0
$$203$$ − 1.00000i − 0.0701862i
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ −17.0000 −1.18445
$$207$$ − 7.00000i − 0.486534i
$$208$$ 1.00000i 0.0693375i
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ 15.0000 1.03264 0.516321 0.856395i $$-0.327301\pi$$
0.516321 + 0.856395i $$0.327301\pi$$
$$212$$ − 11.0000i − 0.755483i
$$213$$ 4.00000i 0.274075i
$$214$$ 18.0000 1.23045
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 3.00000i 0.203653i
$$218$$ − 4.00000i − 0.270914i
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ −1.00000 −0.0672673
$$222$$ 6.00000i 0.402694i
$$223$$ − 13.0000i − 0.870544i −0.900299 0.435272i $$-0.856652\pi$$
0.900299 0.435272i $$-0.143348\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ − 7.00000i − 0.464606i −0.972643 0.232303i $$-0.925374\pi$$
0.972643 0.232303i $$-0.0746261\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ − 1.00000i − 0.0656532i
$$233$$ 20.0000i 1.31024i 0.755523 + 0.655122i $$0.227383\pi$$
−0.755523 + 0.655122i $$0.772617\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ −3.00000 −0.195283
$$237$$ 2.00000i 0.129914i
$$238$$ 1.00000i 0.0648204i
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 7.00000i 0.449977i
$$243$$ 1.00000i 0.0641500i
$$244$$ −5.00000 −0.320092
$$245$$ 0 0
$$246$$ −3.00000 −0.191273
$$247$$ − 4.00000i − 0.254514i
$$248$$ 3.00000i 0.190500i
$$249$$ 3.00000 0.190117
$$250$$ 0 0
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 1.00000i 0.0629941i
$$253$$ 14.0000i 0.880172i
$$254$$ −14.0000 −0.878438
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 5.00000i 0.311891i 0.987766 + 0.155946i $$0.0498425\pi$$
−0.987766 + 0.155946i $$0.950158\pi$$
$$258$$ − 1.00000i − 0.0622573i
$$259$$ −6.00000 −0.372822
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ − 8.00000i − 0.494242i
$$263$$ 11.0000i 0.678289i 0.940734 + 0.339145i $$0.110138\pi$$
−0.940734 + 0.339145i $$0.889862\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ − 10.0000i − 0.611990i
$$268$$ − 12.0000i − 0.733017i
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ 1.00000i 0.0606339i
$$273$$ − 1.00000i − 0.0605228i
$$274$$ −4.00000 −0.241649
$$275$$ 0 0
$$276$$ 7.00000 0.421350
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ −3.00000 −0.179605
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 12.0000i 0.714590i
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ − 3.00000i − 0.177084i
$$288$$ 1.00000i 0.0589256i
$$289$$ 16.0000 0.941176
$$290$$ 0 0
$$291$$ −10.0000 −0.586210
$$292$$ 14.0000i 0.819288i
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ − 2.00000i − 0.116052i
$$298$$ 5.00000i 0.289642i
$$299$$ −7.00000 −0.404820
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ − 22.0000i − 1.26596i
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ −1.00000 −0.0571662
$$307$$ 22.0000i 1.25561i 0.778372 + 0.627803i $$0.216046\pi$$
−0.778372 + 0.627803i $$0.783954\pi$$
$$308$$ − 2.00000i − 0.113961i
$$309$$ 17.0000 0.967096
$$310$$ 0 0
$$311$$ 34.0000 1.92796 0.963982 0.265969i $$-0.0856919\pi$$
0.963982 + 0.265969i $$0.0856919\pi$$
$$312$$ − 1.00000i − 0.0566139i
$$313$$ 18.0000i 1.01742i 0.860938 + 0.508710i $$0.169877\pi$$
−0.860938 + 0.508710i $$0.830123\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ 17.0000i 0.954815i 0.878682 + 0.477408i $$0.158423\pi$$
−0.878682 + 0.477408i $$0.841577\pi$$
$$318$$ 11.0000i 0.616849i
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ −18.0000 −1.00466
$$322$$ 7.00000i 0.390095i
$$323$$ − 4.00000i − 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −19.0000 −1.05231
$$327$$ 4.00000i 0.221201i
$$328$$ − 3.00000i − 0.165647i
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −3.00000 −0.164895 −0.0824475 0.996595i $$-0.526274\pi$$
−0.0824475 + 0.996595i $$0.526274\pi$$
$$332$$ 3.00000i 0.164646i
$$333$$ − 6.00000i − 0.328798i
$$334$$ 2.00000 0.109435
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ − 27.0000i − 1.47078i −0.677642 0.735392i $$-0.736998\pi$$
0.677642 0.735392i $$-0.263002\pi$$
$$338$$ − 12.0000i − 0.652714i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ − 4.00000i − 0.216295i
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ − 26.0000i − 1.39575i −0.716218 0.697877i $$-0.754128\pi$$
0.716218 0.697877i $$-0.245872\pi$$
$$348$$ 1.00000i 0.0536056i
$$349$$ 1.00000 0.0535288 0.0267644 0.999642i $$-0.491480\pi$$
0.0267644 + 0.999642i $$0.491480\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ − 2.00000i − 0.106600i
$$353$$ 10.0000i 0.532246i 0.963939 + 0.266123i $$0.0857428\pi$$
−0.963939 + 0.266123i $$0.914257\pi$$
$$354$$ 3.00000 0.159448
$$355$$ 0 0
$$356$$ 10.0000 0.529999
$$357$$ − 1.00000i − 0.0529256i
$$358$$ 10.0000i 0.528516i
$$359$$ 9.00000 0.475002 0.237501 0.971387i $$-0.423672\pi$$
0.237501 + 0.971387i $$0.423672\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 18.0000i 0.946059i
$$363$$ − 7.00000i − 0.367405i
$$364$$ 1.00000 0.0524142
$$365$$ 0 0
$$366$$ 5.00000 0.261354
$$367$$ − 13.0000i − 0.678594i −0.940679 0.339297i $$-0.889811\pi$$
0.940679 0.339297i $$-0.110189\pi$$
$$368$$ 7.00000i 0.364900i
$$369$$ 3.00000 0.156174
$$370$$ 0 0
$$371$$ −11.0000 −0.571092
$$372$$ − 3.00000i − 0.155543i
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 2.00000 0.103418
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ − 1.00000i − 0.0515026i
$$378$$ − 1.00000i − 0.0514344i
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ 0 0
$$381$$ 14.0000 0.717242
$$382$$ 13.0000i 0.665138i
$$383$$ − 20.0000i − 1.02195i −0.859595 0.510976i $$-0.829284\pi$$
0.859595 0.510976i $$-0.170716\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 1.00000i 0.0508329i
$$388$$ − 10.0000i − 0.507673i
$$389$$ −22.0000 −1.11544 −0.557722 0.830028i $$-0.688325\pi$$
−0.557722 + 0.830028i $$0.688325\pi$$
$$390$$ 0 0
$$391$$ −7.00000 −0.354005
$$392$$ − 1.00000i − 0.0505076i
$$393$$ 8.00000i 0.403547i
$$394$$ −27.0000 −1.36024
$$395$$ 0 0
$$396$$ 2.00000 0.100504
$$397$$ − 3.00000i − 0.150566i −0.997162 0.0752828i $$-0.976014\pi$$
0.997162 0.0752828i $$-0.0239860\pi$$
$$398$$ − 24.0000i − 1.20301i
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ 16.0000 0.799002 0.399501 0.916733i $$-0.369183\pi$$
0.399501 + 0.916733i $$0.369183\pi$$
$$402$$ 12.0000i 0.598506i
$$403$$ 3.00000i 0.149441i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ −1.00000 −0.0496292
$$407$$ 12.0000i 0.594818i
$$408$$ − 1.00000i − 0.0495074i
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 4.00000 0.197305
$$412$$ 17.0000i 0.837530i
$$413$$ 3.00000i 0.147620i
$$414$$ −7.00000 −0.344031
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 4.00000i 0.195881i
$$418$$ 8.00000i 0.391293i
$$419$$ 25.0000 1.22133 0.610665 0.791889i $$-0.290902\pi$$
0.610665 + 0.791889i $$0.290902\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ − 15.0000i − 0.730189i
$$423$$ − 12.0000i − 0.583460i
$$424$$ −11.0000 −0.534207
$$425$$ 0 0
$$426$$ 4.00000 0.193801
$$427$$ 5.00000i 0.241967i
$$428$$ − 18.0000i − 0.870063i
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ 27.0000 1.30054 0.650272 0.759701i $$-0.274655\pi$$
0.650272 + 0.759701i $$0.274655\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 3.00000 0.144005
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ − 28.0000i − 1.33942i
$$438$$ − 14.0000i − 0.668946i
$$439$$ −7.00000 −0.334092 −0.167046 0.985949i $$-0.553423\pi$$
−0.167046 + 0.985949i $$0.553423\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 1.00000i 0.0475651i
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ −13.0000 −0.615568
$$447$$ − 5.00000i − 0.236492i
$$448$$ − 1.00000i − 0.0472456i
$$449$$ 16.0000 0.755087 0.377543 0.925992i $$-0.376769\pi$$
0.377543 + 0.925992i $$0.376769\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ − 6.00000i − 0.282216i
$$453$$ 22.0000i 1.03365i
$$454$$ −7.00000 −0.328526
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 17.0000i 0.795226i 0.917553 + 0.397613i $$0.130161\pi$$
−0.917553 + 0.397613i $$0.869839\pi$$
$$458$$ 14.0000i 0.654177i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ 32.0000 1.49039 0.745194 0.666847i $$-0.232357\pi$$
0.745194 + 0.666847i $$0.232357\pi$$
$$462$$ 2.00000i 0.0930484i
$$463$$ 36.0000i 1.67306i 0.547920 + 0.836531i $$0.315420\pi$$
−0.547920 + 0.836531i $$0.684580\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 0 0
$$466$$ 20.0000 0.926482
$$467$$ 39.0000i 1.80470i 0.430999 + 0.902352i $$0.358161\pi$$
−0.430999 + 0.902352i $$0.641839\pi$$
$$468$$ 1.00000i 0.0462250i
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 3.00000i 0.138086i
$$473$$ − 2.00000i − 0.0919601i
$$474$$ 2.00000 0.0918630
$$475$$ 0 0
$$476$$ 1.00000 0.0458349
$$477$$ − 11.0000i − 0.503655i
$$478$$ − 24.0000i − 1.09773i
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ 26.0000i 1.18427i
$$483$$ − 7.00000i − 0.318511i
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ − 22.0000i − 0.996915i −0.866914 0.498458i $$-0.833900\pi$$
0.866914 0.498458i $$-0.166100\pi$$
$$488$$ 5.00000i 0.226339i
$$489$$ 19.0000 0.859210
$$490$$ 0 0
$$491$$ 30.0000 1.35388 0.676941 0.736038i $$-0.263305\pi$$
0.676941 + 0.736038i $$0.263305\pi$$
$$492$$ 3.00000i 0.135250i
$$493$$ − 1.00000i − 0.0450377i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 3.00000 0.134704
$$497$$ 4.00000i 0.179425i
$$498$$ − 3.00000i − 0.134433i
$$499$$ 27.0000 1.20869 0.604343 0.796724i $$-0.293436\pi$$
0.604343 + 0.796724i $$0.293436\pi$$
$$500$$ 0 0
$$501$$ −2.00000 −0.0893534
$$502$$ 15.0000i 0.669483i
$$503$$ − 2.00000i − 0.0891756i −0.999005 0.0445878i $$-0.985803\pi$$
0.999005 0.0445878i $$-0.0141974\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ 14.0000 0.622376
$$507$$ 12.0000i 0.532939i
$$508$$ 14.0000i 0.621150i
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ 5.00000 0.220541
$$515$$ 0 0
$$516$$ −1.00000 −0.0440225
$$517$$ 24.0000i 1.05552i
$$518$$ 6.00000i 0.263625i
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ 39.0000 1.70862 0.854311 0.519763i $$-0.173980\pi$$
0.854311 + 0.519763i $$0.173980\pi$$
$$522$$ − 1.00000i − 0.0437688i
$$523$$ − 8.00000i − 0.349816i −0.984585 0.174908i $$-0.944037\pi$$
0.984585 0.174908i $$-0.0559627\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 11.0000 0.479623
$$527$$ 3.00000i 0.130682i
$$528$$ 2.00000i 0.0870388i
$$529$$ −26.0000 −1.13043
$$530$$ 0 0
$$531$$ −3.00000 −0.130189
$$532$$ 4.00000i 0.173422i
$$533$$ − 3.00000i − 0.129944i
$$534$$ −10.0000 −0.432742
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ − 10.0000i − 0.431532i
$$538$$ 18.0000i 0.776035i
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ −28.0000 −1.20381 −0.601907 0.798566i $$-0.705592\pi$$
−0.601907 + 0.798566i $$0.705592\pi$$
$$542$$ 20.0000i 0.859074i
$$543$$ − 18.0000i − 0.772454i
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ −1.00000 −0.0427960
$$547$$ 15.0000i 0.641354i 0.947189 + 0.320677i $$0.103910\pi$$
−0.947189 + 0.320677i $$0.896090\pi$$
$$548$$ 4.00000i 0.170872i
$$549$$ −5.00000 −0.213395
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ − 7.00000i − 0.297940i
$$553$$ 2.00000i 0.0850487i
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 3.00000i 0.127000i
$$559$$ 1.00000 0.0422955
$$560$$ 0 0
$$561$$ −2.00000 −0.0844401
$$562$$ 0 0
$$563$$ 41.0000i 1.72794i 0.503540 + 0.863972i $$0.332031\pi$$
−0.503540 + 0.863972i $$0.667969\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ −20.0000 −0.840663
$$567$$ 1.00000i 0.0419961i
$$568$$ 4.00000i 0.167836i
$$569$$ −14.0000 −0.586911 −0.293455 0.955973i $$-0.594805\pi$$
−0.293455 + 0.955973i $$0.594805\pi$$
$$570$$ 0 0
$$571$$ 19.0000 0.795125 0.397563 0.917575i $$-0.369856\pi$$
0.397563 + 0.917575i $$0.369856\pi$$
$$572$$ − 2.00000i − 0.0836242i
$$573$$ − 13.0000i − 0.543083i
$$574$$ −3.00000 −0.125218
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 34.0000i − 1.41544i −0.706494 0.707719i $$-0.749724\pi$$
0.706494 0.707719i $$-0.250276\pi$$
$$578$$ − 16.0000i − 0.665512i
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 3.00000 0.124461
$$582$$ 10.0000i 0.414513i
$$583$$ 22.0000i 0.911147i
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 27.0000i 1.11441i 0.830375 + 0.557205i $$0.188126\pi$$
−0.830375 + 0.557205i $$0.811874\pi$$
$$588$$ 1.00000i 0.0412393i
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ 27.0000 1.11063
$$592$$ 6.00000i 0.246598i
$$593$$ − 18.0000i − 0.739171i −0.929197 0.369586i $$-0.879500\pi$$
0.929197 0.369586i $$-0.120500\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ 0 0
$$596$$ 5.00000 0.204808
$$597$$ 24.0000i 0.982255i
$$598$$ 7.00000i 0.286251i
$$599$$ 45.0000 1.83865 0.919325 0.393499i $$-0.128735\pi$$
0.919325 + 0.393499i $$0.128735\pi$$
$$600$$ 0 0
$$601$$ 28.0000 1.14214 0.571072 0.820900i $$-0.306528\pi$$
0.571072 + 0.820900i $$0.306528\pi$$
$$602$$ − 1.00000i − 0.0407570i
$$603$$ − 12.0000i − 0.488678i
$$604$$ −22.0000 −0.895167
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 28.0000i − 1.13648i −0.822861 0.568242i $$-0.807624\pi$$
0.822861 0.568242i $$-0.192376\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 1.00000 0.0405220
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ 1.00000i 0.0404226i
$$613$$ 18.0000i 0.727013i 0.931592 + 0.363507i $$0.118421\pi$$
−0.931592 + 0.363507i $$0.881579\pi$$
$$614$$ 22.0000 0.887848
$$615$$ 0 0
$$616$$ −2.00000 −0.0805823
$$617$$ − 14.0000i − 0.563619i −0.959470 0.281809i $$-0.909065\pi$$
0.959470 0.281809i $$-0.0909346\pi$$
$$618$$ − 17.0000i − 0.683840i
$$619$$ −38.0000 −1.52735 −0.763674 0.645601i $$-0.776607\pi$$
−0.763674 + 0.645601i $$0.776607\pi$$
$$620$$ 0 0
$$621$$ 7.00000 0.280900
$$622$$ − 34.0000i − 1.36328i
$$623$$ − 10.0000i − 0.400642i
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ 18.0000 0.719425
$$627$$ − 8.00000i − 0.319489i
$$628$$ 18.0000i 0.718278i
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 26.0000 1.03504 0.517522 0.855670i $$-0.326855\pi$$
0.517522 + 0.855670i $$0.326855\pi$$
$$632$$ 2.00000i 0.0795557i
$$633$$ 15.0000i 0.596196i
$$634$$ 17.0000 0.675156
$$635$$ 0 0
$$636$$ 11.0000 0.436178
$$637$$ − 1.00000i − 0.0396214i
$$638$$ 2.00000i 0.0791808i
$$639$$ −4.00000 −0.158238
$$640$$ 0 0
$$641$$ −26.0000 −1.02694 −0.513469 0.858108i $$-0.671640\pi$$
−0.513469 + 0.858108i $$0.671640\pi$$
$$642$$ 18.0000i 0.710403i
$$643$$ 2.00000i 0.0788723i 0.999222 + 0.0394362i $$0.0125562\pi$$
−0.999222 + 0.0394362i $$0.987444\pi$$
$$644$$ 7.00000 0.275839
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ − 6.00000i − 0.235884i −0.993020 0.117942i $$-0.962370\pi$$
0.993020 0.117942i $$-0.0376297\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 6.00000 0.235521
$$650$$ 0 0
$$651$$ −3.00000 −0.117579
$$652$$ 19.0000i 0.744097i
$$653$$ 14.0000i 0.547862i 0.961749 + 0.273931i $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 4.00000 0.156412
$$655$$ 0 0
$$656$$ −3.00000 −0.117130
$$657$$ 14.0000i 0.546192i
$$658$$ 12.0000i 0.467809i
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 3.00000i 0.116598i
$$663$$ − 1.00000i − 0.0388368i
$$664$$ 3.00000 0.116423
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ − 7.00000i − 0.271041i
$$668$$ − 2.00000i − 0.0773823i
$$669$$ 13.0000 0.502609
$$670$$ 0 0
$$671$$ 10.0000 0.386046
$$672$$ 1.00000i 0.0385758i
$$673$$ 27.0000i 1.04077i 0.853931 + 0.520387i $$0.174212\pi$$
−0.853931 + 0.520387i $$0.825788\pi$$
$$674$$ −27.0000 −1.04000
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ 20.0000i 0.768662i 0.923195 + 0.384331i $$0.125568\pi$$
−0.923195 + 0.384331i $$0.874432\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ −10.0000 −0.383765
$$680$$ 0 0
$$681$$ 7.00000 0.268241
$$682$$ − 6.00000i − 0.229752i
$$683$$ − 6.00000i − 0.229584i −0.993390 0.114792i $$-0.963380\pi$$
0.993390 0.114792i $$-0.0366201\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ − 14.0000i − 0.534133i
$$688$$ − 1.00000i − 0.0381246i
$$689$$ −11.0000 −0.419067
$$690$$ 0 0
$$691$$ −42.0000 −1.59776 −0.798878 0.601494i $$-0.794573\pi$$
−0.798878 + 0.601494i $$0.794573\pi$$
$$692$$ 12.0000i 0.456172i
$$693$$ − 2.00000i − 0.0759737i
$$694$$ −26.0000 −0.986947
$$695$$ 0 0
$$696$$ 1.00000 0.0379049
$$697$$ − 3.00000i − 0.113633i
$$698$$ − 1.00000i − 0.0378506i
$$699$$ −20.0000 −0.756469
$$700$$ 0 0
$$701$$ −39.0000 −1.47301 −0.736505 0.676432i $$-0.763525\pi$$
−0.736505 + 0.676432i $$0.763525\pi$$
$$702$$ − 1.00000i − 0.0377426i
$$703$$ − 24.0000i − 0.905177i
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 10.0000 0.376355
$$707$$ 0 0
$$708$$ − 3.00000i − 0.112747i
$$709$$ −4.00000 −0.150223 −0.0751116 0.997175i $$-0.523931\pi$$
−0.0751116 + 0.997175i $$0.523931\pi$$
$$710$$ 0 0
$$711$$ −2.00000 −0.0750059
$$712$$ − 10.0000i − 0.374766i
$$713$$ 21.0000i 0.786456i
$$714$$ −1.00000 −0.0374241
$$715$$ 0 0
$$716$$ 10.0000 0.373718
$$717$$ 24.0000i 0.896296i
$$718$$ − 9.00000i − 0.335877i
$$719$$ 34.0000 1.26799 0.633993 0.773339i $$-0.281415\pi$$
0.633993 + 0.773339i $$0.281415\pi$$
$$720$$ 0 0
$$721$$ 17.0000 0.633113
$$722$$ 3.00000i 0.111648i
$$723$$ − 26.0000i − 0.966950i
$$724$$ 18.0000 0.668965
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ 1.00000i 0.0370879i 0.999828 + 0.0185440i $$0.00590307\pi$$
−0.999828 + 0.0185440i $$0.994097\pi$$
$$728$$ − 1.00000i − 0.0370625i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 1.00000 0.0369863
$$732$$ − 5.00000i − 0.184805i
$$733$$ 19.0000i 0.701781i 0.936416 + 0.350891i $$0.114121\pi$$
−0.936416 + 0.350891i $$0.885879\pi$$
$$734$$ −13.0000 −0.479839
$$735$$ 0 0
$$736$$ 7.00000 0.258023
$$737$$ 24.0000i 0.884051i
$$738$$ − 3.00000i − 0.110432i
$$739$$ −11.0000 −0.404642 −0.202321 0.979319i $$-0.564848\pi$$
−0.202321 + 0.979319i $$0.564848\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ 11.0000i 0.403823i
$$743$$ − 3.00000i − 0.110059i −0.998485 0.0550297i $$-0.982475\pi$$
0.998485 0.0550297i $$-0.0175253\pi$$
$$744$$ −3.00000 −0.109985
$$745$$ 0 0
$$746$$ −4.00000 −0.146450
$$747$$ 3.00000i 0.109764i
$$748$$ − 2.00000i − 0.0731272i
$$749$$ −18.0000 −0.657706
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ 12.0000i 0.437595i
$$753$$ − 15.0000i − 0.546630i
$$754$$ −1.00000 −0.0364179
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ − 5.00000i − 0.181608i
$$759$$ −14.0000 −0.508168
$$760$$ 0 0
$$761$$ 34.0000 1.23250 0.616250 0.787551i $$-0.288651\pi$$
0.616250 + 0.787551i $$0.288651\pi$$
$$762$$ − 14.0000i − 0.507166i
$$763$$ 4.00000i 0.144810i
$$764$$ 13.0000 0.470323
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ 3.00000i 0.108324i
$$768$$ 1.00000i 0.0360844i
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ −5.00000 −0.180071
$$772$$ − 2.00000i − 0.0719816i
$$773$$ − 20.0000i − 0.719350i −0.933078 0.359675i $$-0.882888\pi$$
0.933078 0.359675i $$-0.117112\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ − 6.00000i − 0.215249i
$$778$$ 22.0000i 0.788738i
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 7.00000i 0.250319i
$$783$$ 1.00000i 0.0357371i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 8.00000 0.285351
$$787$$ − 38.0000i − 1.35455i −0.735728 0.677277i $$-0.763160\pi$$
0.735728 0.677277i $$-0.236840\pi$$
$$788$$ 27.0000i 0.961835i
$$789$$ −11.0000 −0.391610
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ − 2.00000i − 0.0710669i
$$793$$ 5.00000i 0.177555i
$$794$$ −3.00000 −0.106466
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ − 42.0000i − 1.48772i −0.668338 0.743858i $$-0.732994\pi$$
0.668338 0.743858i $$-0.267006\pi$$
$$798$$ − 4.00000i − 0.141598i
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ − 16.0000i − 0.564980i
$$803$$ − 28.0000i − 0.988099i
$$804$$ 12.0000 0.423207
$$805$$ 0 0
$$806$$ 3.00000 0.105670
$$807$$ − 18.0000i − 0.633630i
$$808$$ 0 0
$$809$$ −20.0000 −0.703163 −0.351581 0.936157i $$-0.614356\pi$$
−0.351581 + 0.936157i $$0.614356\pi$$
$$810$$ 0 0
$$811$$ −14.0000 −0.491606 −0.245803 0.969320i $$-0.579052\pi$$
−0.245803 + 0.969320i $$0.579052\pi$$
$$812$$ 1.00000i 0.0350931i
$$813$$ − 20.0000i − 0.701431i
$$814$$ 12.0000 0.420600
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ 4.00000i 0.139942i
$$818$$ − 14.0000i − 0.489499i
$$819$$ 1.00000 0.0349428
$$820$$ 0 0
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ − 4.00000i − 0.139516i
$$823$$ − 18.0000i − 0.627441i −0.949515 0.313720i $$-0.898425\pi$$
0.949515 0.313720i $$-0.101575\pi$$
$$824$$ 17.0000 0.592223
$$825$$ 0 0
$$826$$ 3.00000 0.104383
$$827$$ 50.0000i 1.73867i 0.494223 + 0.869335i $$0.335453\pi$$
−0.494223 + 0.869335i $$0.664547\pi$$
$$828$$ 7.00000i 0.243267i
$$829$$ −3.00000 −0.104194 −0.0520972 0.998642i $$-0.516591\pi$$
−0.0520972 + 0.998642i $$0.516591\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ − 1.00000i − 0.0346688i
$$833$$ − 1.00000i − 0.0346479i
$$834$$ 4.00000 0.138509
$$835$$ 0 0
$$836$$ 8.00000 0.276686
$$837$$ − 3.00000i − 0.103695i
$$838$$ − 25.0000i − 0.863611i
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ − 34.0000i − 1.17172i
$$843$$ 0 0
$$844$$ −15.0000 −0.516321
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ − 7.00000i − 0.240523i
$$848$$ 11.0000i 0.377742i
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ −42.0000 −1.43974
$$852$$ − 4.00000i − 0.137038i
$$853$$ 43.0000i 1.47229i 0.676823 + 0.736146i $$0.263356\pi$$
−0.676823 + 0.736146i $$0.736644\pi$$
$$854$$ 5.00000 0.171096
$$855$$ 0 0
$$856$$ −18.0000 −0.615227
$$857$$ 46.0000i 1.57133i 0.618652 + 0.785665i $$0.287679\pi$$
−0.618652 + 0.785665i $$0.712321\pi$$
$$858$$ 2.00000i 0.0682789i
$$859$$ −40.0000 −1.36478 −0.682391 0.730987i $$-0.739060\pi$$
−0.682391 + 0.730987i $$0.739060\pi$$
$$860$$ 0 0
$$861$$ 3.00000 0.102240
$$862$$ − 27.0000i − 0.919624i
$$863$$ 32.0000i 1.08929i 0.838666 + 0.544646i $$0.183336\pi$$
−0.838666 + 0.544646i $$0.816664\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 16.0000i 0.543388i
$$868$$ − 3.00000i − 0.101827i
$$869$$ 4.00000 0.135691
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ 4.00000i 0.135457i
$$873$$ − 10.0000i − 0.338449i
$$874$$ −28.0000 −0.947114
$$875$$ 0 0
$$876$$ −14.0000 −0.473016
$$877$$ − 32.0000i − 1.08056i −0.841484 0.540282i $$-0.818318\pi$$
0.841484 0.540282i $$-0.181682\pi$$
$$878$$ 7.00000i 0.236239i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 5.00000 0.168454 0.0842271 0.996447i $$-0.473158\pi$$
0.0842271 + 0.996447i $$0.473158\pi$$
$$882$$ − 1.00000i − 0.0336718i
$$883$$ 29.0000i 0.975928i 0.872864 + 0.487964i $$0.162260\pi$$
−0.872864 + 0.487964i $$0.837740\pi$$
$$884$$ 1.00000 0.0336336
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ − 50.0000i − 1.67884i −0.543487 0.839418i $$-0.682896\pi$$
0.543487 0.839418i $$-0.317104\pi$$
$$888$$ − 6.00000i − 0.201347i
$$889$$ 14.0000 0.469545
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 13.0000i 0.435272i
$$893$$ − 48.0000i − 1.60626i
$$894$$ −5.00000 −0.167225
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ − 7.00000i − 0.233723i
$$898$$ − 16.0000i − 0.533927i
$$899$$ −3.00000 −0.100056
$$900$$ 0 0
$$901$$ −11.0000 −0.366463
$$902$$ 6.00000i 0.199778i
$$903$$ 1.00000i 0.0332779i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 22.0000 0.730901
$$907$$ 11.0000i 0.365249i 0.983183 + 0.182625i $$0.0584593\pi$$
−0.983183 + 0.182625i $$0.941541\pi$$
$$908$$ 7.00000i 0.232303i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −19.0000 −0.629498 −0.314749 0.949175i $$-0.601920\pi$$
−0.314749 + 0.949175i $$0.601920\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 6.00000i − 0.198571i
$$914$$ 17.0000 0.562310
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 8.00000i 0.264183i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ −22.0000 −0.724925
$$922$$ − 32.0000i − 1.05386i
$$923$$ 4.00000i 0.131662i
$$924$$ 2.00000 0.0657952
$$925$$ 0 0
$$926$$ 36.0000 1.18303
$$927$$ 17.0000i 0.558353i
$$928$$ 1.00000i 0.0328266i
$$929$$ −25.0000 −0.820223 −0.410112 0.912035i $$-0.634510\pi$$
−0.410112 + 0.912035i $$0.634510\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ − 20.0000i − 0.655122i
$$933$$ 34.0000i 1.11311i
$$934$$ 39.0000 1.27612
$$935$$ 0 0
$$936$$ 1.00000 0.0326860
$$937$$ 14.0000i 0.457360i 0.973502 + 0.228680i $$0.0734410\pi$$
−0.973502 + 0.228680i $$0.926559\pi$$
$$938$$ 12.0000i 0.391814i
$$939$$ −18.0000 −0.587408
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ − 18.0000i − 0.586472i
$$943$$ − 21.0000i − 0.683854i
$$944$$ 3.00000 0.0976417
$$945$$ 0 0
$$946$$ −2.00000 −0.0650256
$$947$$ 32.0000i 1.03986i 0.854209 + 0.519930i $$0.174042\pi$$
−0.854209 + 0.519930i $$0.825958\pi$$
$$948$$ − 2.00000i − 0.0649570i
$$949$$ 14.0000 0.454459
$$950$$ 0 0
$$951$$ −17.0000 −0.551263
$$952$$ − 1.00000i − 0.0324102i
$$953$$ − 40.0000i − 1.29573i −0.761756 0.647864i $$-0.775663\pi$$
0.761756 0.647864i $$-0.224337\pi$$
$$954$$ −11.0000 −0.356138
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ − 2.00000i − 0.0646508i
$$958$$ 8.00000i 0.258468i
$$959$$ 4.00000 0.129167
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 6.00000i 0.193448i
$$963$$ − 18.0000i − 0.580042i
$$964$$ 26.0000 0.837404
$$965$$ 0 0
$$966$$ −7.00000 −0.225221
$$967$$ 44.0000i 1.41494i 0.706741 + 0.707472i $$0.250165\pi$$
−0.706741 + 0.707472i $$0.749835\pi$$
$$968$$ − 7.00000i − 0.224989i
$$969$$ 4.00000 0.128499
$$970$$ 0 0
$$971$$ −44.0000 −1.41203 −0.706014 0.708198i $$-0.749508\pi$$
−0.706014 + 0.708198i $$0.749508\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 4.00000i 0.128234i
$$974$$ −22.0000 −0.704925
$$975$$ 0 0
$$976$$ 5.00000 0.160046
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ − 19.0000i − 0.607553i
$$979$$ −20.0000 −0.639203
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ − 30.0000i − 0.957338i
$$983$$ 24.0000i 0.765481i 0.923856 + 0.382741i $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ 3.00000 0.0956365
$$985$$ 0 0
$$986$$ −1.00000 −0.0318465
$$987$$ − 12.0000i − 0.381964i
$$988$$ 4.00000i 0.127257i
$$989$$ 7.00000 0.222587
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ − 3.00000i − 0.0952501i
$$993$$ − 3.00000i − 0.0952021i
$$994$$ 4.00000 0.126872
$$995$$ 0 0
$$996$$ −3.00000 −0.0950586
$$997$$ 2.00000i 0.0633406i 0.999498 + 0.0316703i $$0.0100827\pi$$
−0.999498 + 0.0316703i $$0.989917\pi$$
$$998$$ − 27.0000i − 0.854670i
$$999$$ 6.00000 0.189832
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 1050.2.g.j.799.1 2
3.2 odd 2 3150.2.g.g.2899.2 2
5.2 odd 4 1050.2.a.o.1.1 yes 1
5.3 odd 4 1050.2.a.e.1.1 1
5.4 even 2 inner 1050.2.g.j.799.2 2
15.2 even 4 3150.2.a.c.1.1 1
15.8 even 4 3150.2.a.bl.1.1 1
15.14 odd 2 3150.2.g.g.2899.1 2
20.3 even 4 8400.2.a.bt.1.1 1
20.7 even 4 8400.2.a.t.1.1 1
35.13 even 4 7350.2.a.bj.1.1 1
35.27 even 4 7350.2.a.ca.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.e.1.1 1 5.3 odd 4
1050.2.a.o.1.1 yes 1 5.2 odd 4
1050.2.g.j.799.1 2 1.1 even 1 trivial
1050.2.g.j.799.2 2 5.4 even 2 inner
3150.2.a.c.1.1 1 15.2 even 4
3150.2.a.bl.1.1 1 15.8 even 4
3150.2.g.g.2899.1 2 15.14 odd 2
3150.2.g.g.2899.2 2 3.2 odd 2
7350.2.a.bj.1.1 1 35.13 even 4
7350.2.a.ca.1.1 1 35.27 even 4
8400.2.a.t.1.1 1 20.7 even 4
8400.2.a.bt.1.1 1 20.3 even 4