# Properties

 Label 2850.2.d.n.799.1 Level $2850$ Weight $2$ Character 2850.799 Analytic conductor $22.757$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.7573645761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 570) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2850.799 Dual form 2850.2.d.n.799.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{12} -2.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} +2.00000 q^{21} +2.00000i q^{22} +8.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} +2.00000i q^{28} -1.00000i q^{32} -2.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} +1.00000i q^{38} -8.00000 q^{41} -2.00000i q^{42} +6.00000i q^{43} +2.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +2.00000 q^{51} +10.0000i q^{53} -1.00000 q^{54} +2.00000 q^{56} -1.00000i q^{57} +8.00000 q^{59} +2.00000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} +2.00000i q^{68} -8.00000 q^{69} +8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} +4.00000 q^{74} +1.00000 q^{76} +4.00000i q^{77} +8.00000 q^{79} +1.00000 q^{81} +8.00000i q^{82} +16.0000i q^{83} -2.00000 q^{84} +6.00000 q^{86} -2.00000i q^{88} -16.0000 q^{89} -8.00000i q^{92} -8.00000 q^{94} +1.00000 q^{96} +8.00000i q^{97} -3.00000i q^{98} +2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} - 4 q^{34} + 2 q^{36} - 16 q^{41} + 4 q^{44} + 16 q^{46} + 6 q^{49} + 4 q^{51} - 2 q^{54} + 4 q^{56} + 16 q^{59} + 4 q^{61} - 2 q^{64} - 4 q^{66} - 16 q^{69} + 16 q^{71} + 8 q^{74} + 2 q^{76} + 16 q^{79} + 2 q^{81} - 4 q^{84} + 12 q^{86} - 32 q^{89} - 16 q^{94} + 2 q^{96} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^11 - 4 * q^14 + 2 * q^16 - 2 * q^19 + 4 * q^21 - 2 * q^24 - 4 * q^34 + 2 * q^36 - 16 * q^41 + 4 * q^44 + 16 * q^46 + 6 * q^49 + 4 * q^51 - 2 * q^54 + 4 * q^56 + 16 * q^59 + 4 * q^61 - 2 * q^64 - 4 * q^66 - 16 * q^69 + 16 * q^71 + 8 * q^74 + 2 * q^76 + 16 * q^79 + 2 * q^81 - 4 * q^84 + 12 * q^86 - 32 * q^89 - 16 * q^94 + 2 * q^96 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1027$$ $$1351$$ $$1901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 2.00000i 0.426401i
$$23$$ 8.00000i 1.66812i 0.551677 + 0.834058i $$0.313988\pi$$
−0.551677 + 0.834058i $$0.686012\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 2.00000i 0.377964i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 2.00000i − 0.348155i
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 6.00000i 0.914991i 0.889212 + 0.457496i $$0.151253\pi$$
−0.889212 + 0.457496i $$0.848747\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ − 1.00000i − 0.132453i
$$58$$ 0 0
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 4.00000i 0.455842i
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 8.00000i 0.883452i
$$83$$ 16.0000i 1.75623i 0.478451 + 0.878114i $$0.341198\pi$$
−0.478451 + 0.878114i $$0.658802\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 6.00000 0.646997
$$87$$ 0 0
$$88$$ − 2.00000i − 0.213201i
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 8.00000i − 0.834058i
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 8.00000i 0.812277i 0.913812 + 0.406138i $$0.133125\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ − 12.0000i − 1.18240i −0.806527 0.591198i $$-0.798655\pi$$
0.806527 0.591198i $$-0.201345\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ 20.0000i 1.93347i 0.255774 + 0.966736i $$0.417670\pi$$
−0.255774 + 0.966736i $$0.582330\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ − 2.00000i − 0.188982i
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ − 8.00000i − 0.736460i
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 8.00000i − 0.721336i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −6.00000 −0.528271
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ 2.00000i 0.173422i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ − 8.00000i − 0.671345i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 3.00000i 0.247436i
$$148$$ − 4.00000i − 0.328798i
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ 2.00000i 0.161690i
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 6.00000i − 0.478852i −0.970915 0.239426i $$-0.923041\pi$$
0.970915 0.239426i $$-0.0769593\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −10.0000 −0.793052
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 10.0000i 0.783260i 0.920123 + 0.391630i $$0.128089\pi$$
−0.920123 + 0.391630i $$0.871911\pi$$
$$164$$ 8.00000 0.624695
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 6.00000i − 0.457496i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 8.00000i 0.601317i
$$178$$ 16.0000i 1.19925i
$$179$$ 8.00000 0.597948 0.298974 0.954261i $$-0.403356\pi$$
0.298974 + 0.954261i $$0.403356\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 2.00000i 0.147844i
$$184$$ −8.00000 −0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 4.00000i 0.292509i
$$188$$ 8.00000i 0.583460i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 4.00000i 0.287926i 0.989583 + 0.143963i $$0.0459847\pi$$
−0.989583 + 0.143963i $$0.954015\pi$$
$$194$$ 8.00000 0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 10.0000i 0.712470i 0.934396 + 0.356235i $$0.115940\pi$$
−0.934396 + 0.356235i $$0.884060\pi$$
$$198$$ − 2.00000i − 0.142134i
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 2.00000i − 0.140720i
$$203$$ 0 0
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ −12.0000 −0.836080
$$207$$ − 8.00000i − 0.556038i
$$208$$ 0 0
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ − 10.0000i − 0.686803i
$$213$$ 8.00000i 0.548151i
$$214$$ 20.0000 1.36717
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ 2.00000i 0.135457i
$$219$$ −2.00000 −0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 4.00000i 0.268462i
$$223$$ − 12.0000i − 0.803579i −0.915732 0.401790i $$-0.868388\pi$$
0.915732 0.401790i $$-0.131612\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 2.00000 0.133038
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 0 0
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −8.00000 −0.520756
$$237$$ 8.00000i 0.519656i
$$238$$ 4.00000i 0.259281i
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 7.00000i 0.449977i
$$243$$ 1.00000i 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −8.00000 −0.510061
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −16.0000 −1.01396
$$250$$ 0 0
$$251$$ 6.00000 0.378717 0.189358 0.981908i $$-0.439359\pi$$
0.189358 + 0.981908i $$0.439359\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ − 16.0000i − 1.00591i
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 30.0000i 1.87135i 0.352865 + 0.935674i $$0.385208\pi$$
−0.352865 + 0.935674i $$0.614792\pi$$
$$258$$ 6.00000i 0.373544i
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 6.00000i 0.370681i
$$263$$ 12.0000i 0.739952i 0.929041 + 0.369976i $$0.120634\pi$$
−0.929041 + 0.369976i $$0.879366\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ − 16.0000i − 0.979184i
$$268$$ 0 0
$$269$$ −8.00000 −0.487769 −0.243884 0.969804i $$-0.578422\pi$$
−0.243884 + 0.969804i $$0.578422\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ − 14.0000i − 0.841178i −0.907251 0.420589i $$-0.861823\pi$$
0.907251 0.420589i $$-0.138177\pi$$
$$278$$ 8.00000i 0.479808i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −20.0000 −1.19310 −0.596550 0.802576i $$-0.703462\pi$$
−0.596550 + 0.802576i $$0.703462\pi$$
$$282$$ − 8.00000i − 0.476393i
$$283$$ 2.00000i 0.118888i 0.998232 + 0.0594438i $$0.0189327\pi$$
−0.998232 + 0.0594438i $$0.981067\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 16.0000i 0.944450i
$$288$$ 1.00000i 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −8.00000 −0.468968
$$292$$ − 2.00000i − 0.117041i
$$293$$ 22.0000i 1.28525i 0.766179 + 0.642627i $$0.222155\pi$$
−0.766179 + 0.642627i $$0.777845\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ −4.00000 −0.232495
$$297$$ 2.00000i 0.116052i
$$298$$ 2.00000i 0.115857i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 8.00000i 0.460348i
$$303$$ 2.00000i 0.114897i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ − 24.0000i − 1.36975i −0.728659 0.684876i $$-0.759856\pi$$
0.728659 0.684876i $$-0.240144\pi$$
$$308$$ − 4.00000i − 0.227921i
$$309$$ 12.0000 0.682656
$$310$$ 0 0
$$311$$ −14.0000 −0.793867 −0.396934 0.917847i $$-0.629926\pi$$
−0.396934 + 0.917847i $$0.629926\pi$$
$$312$$ 0 0
$$313$$ − 18.0000i − 1.01742i −0.860938 0.508710i $$-0.830123\pi$$
0.860938 0.508710i $$-0.169877\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ 10.0000i 0.560772i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −20.0000 −1.11629
$$322$$ − 16.0000i − 0.891645i
$$323$$ 2.00000i 0.111283i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 10.0000 0.553849
$$327$$ − 2.00000i − 0.110600i
$$328$$ − 8.00000i − 0.441726i
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ − 16.0000i − 0.878114i
$$333$$ − 4.00000i − 0.219199i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ − 16.0000i − 0.871576i −0.900049 0.435788i $$-0.856470\pi$$
0.900049 0.435788i $$-0.143530\pi$$
$$338$$ − 13.0000i − 0.707107i
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 1.00000i − 0.0540738i
$$343$$ − 20.0000i − 1.07990i
$$344$$ −6.00000 −0.323498
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ − 8.00000i − 0.429463i −0.976673 0.214731i $$-0.931112\pi$$
0.976673 0.214731i $$-0.0688876\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.00000i 0.106600i
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 8.00000 0.425195
$$355$$ 0 0
$$356$$ 16.0000 0.847998
$$357$$ − 4.00000i − 0.211702i
$$358$$ − 8.00000i − 0.422813i
$$359$$ 30.0000 1.58334 0.791670 0.610949i $$-0.209212\pi$$
0.791670 + 0.610949i $$0.209212\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 6.00000i 0.315353i
$$363$$ − 7.00000i − 0.367405i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 26.0000i 1.35719i 0.734513 + 0.678594i $$0.237411\pi$$
−0.734513 + 0.678594i $$0.762589\pi$$
$$368$$ 8.00000i 0.417029i
$$369$$ 8.00000 0.416463
$$370$$ 0 0
$$371$$ 20.0000 1.03835
$$372$$ 0 0
$$373$$ − 32.0000i − 1.65690i −0.560065 0.828449i $$-0.689224\pi$$
0.560065 0.828449i $$-0.310776\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 0 0
$$378$$ 2.00000i 0.102869i
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ − 10.0000i − 0.511645i
$$383$$ − 8.00000i − 0.408781i −0.978889 0.204390i $$-0.934479\pi$$
0.978889 0.204390i $$-0.0655212\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 4.00000 0.203595
$$387$$ − 6.00000i − 0.304997i
$$388$$ − 8.00000i − 0.406138i
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 3.00000i 0.151523i
$$393$$ − 6.00000i − 0.302660i
$$394$$ 10.0000 0.503793
$$395$$ 0 0
$$396$$ −2.00000 −0.100504
$$397$$ − 6.00000i − 0.301131i −0.988600 0.150566i $$-0.951890\pi$$
0.988600 0.150566i $$-0.0481095\pi$$
$$398$$ − 4.00000i − 0.200502i
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 8.00000i − 0.396545i
$$408$$ 2.00000i 0.0990148i
$$409$$ 18.0000 0.890043 0.445021 0.895520i $$-0.353196\pi$$
0.445021 + 0.895520i $$0.353196\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 12.0000i 0.591198i
$$413$$ − 16.0000i − 0.787309i
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 8.00000i − 0.391762i
$$418$$ − 2.00000i − 0.0978232i
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 8.00000i 0.388973i
$$424$$ −10.0000 −0.485643
$$425$$ 0 0
$$426$$ 8.00000 0.387601
$$427$$ − 4.00000i − 0.193574i
$$428$$ − 20.0000i − 0.966736i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 20.0000 0.963366 0.481683 0.876346i $$-0.340026\pi$$
0.481683 + 0.876346i $$0.340026\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 12.0000i 0.576683i 0.957528 + 0.288342i $$0.0931039\pi$$
−0.957528 + 0.288342i $$0.906896\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ − 8.00000i − 0.382692i
$$438$$ 2.00000i 0.0955637i
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ −12.0000 −0.568216
$$447$$ − 2.00000i − 0.0945968i
$$448$$ 2.00000i 0.0944911i
$$449$$ −24.0000 −1.13263 −0.566315 0.824189i $$-0.691631\pi$$
−0.566315 + 0.824189i $$0.691631\pi$$
$$450$$ 0 0
$$451$$ 16.0000 0.753411
$$452$$ − 2.00000i − 0.0940721i
$$453$$ − 8.00000i − 0.375873i
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ 10.0000i 0.467269i
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 4.00000i 0.186097i
$$463$$ − 18.0000i − 0.836531i −0.908325 0.418265i $$-0.862638\pi$$
0.908325 0.418265i $$-0.137362\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 20.0000i 0.925490i 0.886492 + 0.462745i $$0.153135\pi$$
−0.886492 + 0.462745i $$0.846865\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ 8.00000i 0.368230i
$$473$$ − 12.0000i − 0.551761i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ − 10.0000i − 0.457869i
$$478$$ 6.00000i 0.274434i
$$479$$ 30.0000 1.37073 0.685367 0.728197i $$-0.259642\pi$$
0.685367 + 0.728197i $$0.259642\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ − 10.0000i − 0.455488i
$$483$$ 16.0000i 0.728025i
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 20.0000i 0.906287i 0.891438 + 0.453143i $$0.149697\pi$$
−0.891438 + 0.453143i $$0.850303\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ −10.0000 −0.452216
$$490$$ 0 0
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ 8.00000i 0.360668i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 16.0000i − 0.717698i
$$498$$ 16.0000i 0.716977i
$$499$$ 16.0000 0.716258 0.358129 0.933672i $$-0.383415\pi$$
0.358129 + 0.933672i $$0.383415\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 6.00000i − 0.267793i
$$503$$ − 28.0000i − 1.24846i −0.781241 0.624229i $$-0.785413\pi$$
0.781241 0.624229i $$-0.214587\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ −16.0000 −0.711287
$$507$$ 13.0000i 0.577350i
$$508$$ − 16.0000i − 0.709885i
$$509$$ −20.0000 −0.886484 −0.443242 0.896402i $$-0.646172\pi$$
−0.443242 + 0.896402i $$0.646172\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 1.00000i 0.0441511i
$$514$$ 30.0000 1.32324
$$515$$ 0 0
$$516$$ 6.00000 0.264135
$$517$$ 16.0000i 0.703679i
$$518$$ − 8.00000i − 0.351500i
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −20.0000 −0.876216 −0.438108 0.898922i $$-0.644351\pi$$
−0.438108 + 0.898922i $$0.644351\pi$$
$$522$$ 0 0
$$523$$ − 28.0000i − 1.22435i −0.790721 0.612177i $$-0.790294\pi$$
0.790721 0.612177i $$-0.209706\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 0 0
$$528$$ − 2.00000i − 0.0870388i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ − 2.00000i − 0.0867110i
$$533$$ 0 0
$$534$$ −16.0000 −0.692388
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 8.00000i 0.345225i
$$538$$ 8.00000i 0.344904i
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 4.00000i 0.171815i
$$543$$ − 6.00000i − 0.257485i
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 40.0000i − 1.71028i −0.518400 0.855138i $$-0.673472\pi$$
0.518400 0.855138i $$-0.326528\pi$$
$$548$$ − 18.0000i − 0.768922i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 8.00000i − 0.340503i
$$553$$ − 16.0000i − 0.680389i
$$554$$ −14.0000 −0.594803
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 20.0000i 0.843649i
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ 2.00000 0.0840663
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 8.00000i 0.335673i
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 0 0
$$573$$ 10.0000i 0.417756i
$$574$$ 16.0000 0.667827
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 14.0000i − 0.582828i −0.956597 0.291414i $$-0.905874\pi$$
0.956597 0.291414i $$-0.0941257\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ 32.0000 1.32758
$$582$$ 8.00000i 0.331611i
$$583$$ − 20.0000i − 0.828315i
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 22.0000 0.908812
$$587$$ − 44.0000i − 1.81607i −0.418890 0.908037i $$-0.637581\pi$$
0.418890 0.908037i $$-0.362419\pi$$
$$588$$ − 3.00000i − 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −10.0000 −0.411345
$$592$$ 4.00000i 0.164399i
$$593$$ − 34.0000i − 1.39621i −0.715994 0.698106i $$-0.754026\pi$$
0.715994 0.698106i $$-0.245974\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ 4.00000i 0.163709i
$$598$$ 0 0
$$599$$ 4.00000 0.163436 0.0817178 0.996656i $$-0.473959\pi$$
0.0817178 + 0.996656i $$0.473959\pi$$
$$600$$ 0 0
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ − 12.0000i − 0.489083i
$$603$$ 0 0
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ 44.0000i 1.78590i 0.450151 + 0.892952i $$0.351370\pi$$
−0.450151 + 0.892952i $$0.648630\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 2.00000i − 0.0808452i
$$613$$ 46.0000i 1.85792i 0.370177 + 0.928961i $$0.379297\pi$$
−0.370177 + 0.928961i $$0.620703\pi$$
$$614$$ −24.0000 −0.968561
$$615$$ 0 0
$$616$$ −4.00000 −0.161165
$$617$$ − 42.0000i − 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ − 12.0000i − 0.482711i
$$619$$ 32.0000 1.28619 0.643094 0.765787i $$-0.277650\pi$$
0.643094 + 0.765787i $$0.277650\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 14.0000i 0.561349i
$$623$$ 32.0000i 1.28205i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −18.0000 −0.719425
$$627$$ 2.00000i 0.0798723i
$$628$$ 6.00000i 0.239426i
$$629$$ 8.00000 0.318981
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 4.00000i 0.158986i
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −8.00000 −0.316475
$$640$$ 0 0
$$641$$ 16.0000 0.631962 0.315981 0.948766i $$-0.397666\pi$$
0.315981 + 0.948766i $$0.397666\pi$$
$$642$$ 20.0000i 0.789337i
$$643$$ − 46.0000i − 1.81406i −0.421063 0.907031i $$-0.638343\pi$$
0.421063 0.907031i $$-0.361657\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ 36.0000i 1.41531i 0.706560 + 0.707653i $$0.250246\pi$$
−0.706560 + 0.707653i $$0.749754\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 10.0000i − 0.391630i
$$653$$ 30.0000i 1.17399i 0.809590 + 0.586995i $$0.199689\pi$$
−0.809590 + 0.586995i $$0.800311\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −8.00000 −0.312348
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 16.0000i 0.623745i
$$659$$ 44.0000 1.71400 0.856998 0.515319i $$-0.172327\pi$$
0.856998 + 0.515319i $$0.172327\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ 12.0000i 0.466393i
$$663$$ 0 0
$$664$$ −16.0000 −0.620920
$$665$$ 0 0
$$666$$ −4.00000 −0.154997
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 12.0000 0.463947
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ − 2.00000i − 0.0771517i
$$673$$ 16.0000i 0.616755i 0.951264 + 0.308377i $$0.0997859\pi$$
−0.951264 + 0.308377i $$0.900214\pi$$
$$674$$ −16.0000 −0.616297
$$675$$ 0 0
$$676$$ −13.0000 −0.500000
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 2.00000i 0.0768095i
$$679$$ 16.0000 0.614024
$$680$$ 0 0
$$681$$ 4.00000 0.153280
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ − 10.0000i − 0.381524i
$$688$$ 6.00000i 0.228748i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 40.0000 1.52167 0.760836 0.648944i $$-0.224789\pi$$
0.760836 + 0.648944i $$0.224789\pi$$
$$692$$ − 6.00000i − 0.228086i
$$693$$ − 4.00000i − 0.151947i
$$694$$ −8.00000 −0.303676
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16.0000i 0.606043i
$$698$$ − 26.0000i − 0.984115i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −38.0000 −1.43524 −0.717620 0.696435i $$-0.754769\pi$$
−0.717620 + 0.696435i $$0.754769\pi$$
$$702$$ 0 0
$$703$$ − 4.00000i − 0.150863i
$$704$$ 2.00000 0.0753778
$$705$$ 0 0
$$706$$ −18.0000 −0.677439
$$707$$ − 4.00000i − 0.150435i
$$708$$ − 8.00000i − 0.300658i
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ − 16.0000i − 0.599625i
$$713$$ 0 0
$$714$$ −4.00000 −0.149696
$$715$$ 0 0
$$716$$ −8.00000 −0.298974
$$717$$ − 6.00000i − 0.224074i
$$718$$ − 30.0000i − 1.11959i
$$719$$ −22.0000 −0.820462 −0.410231 0.911982i $$-0.634552\pi$$
−0.410231 + 0.911982i $$0.634552\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ − 1.00000i − 0.0372161i
$$723$$ 10.0000i 0.371904i
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ −7.00000 −0.259794
$$727$$ − 22.0000i − 0.815935i −0.912996 0.407967i $$-0.866238\pi$$
0.912996 0.407967i $$-0.133762\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ − 2.00000i − 0.0739221i
$$733$$ 26.0000i 0.960332i 0.877178 + 0.480166i $$0.159424\pi$$
−0.877178 + 0.480166i $$0.840576\pi$$
$$734$$ 26.0000 0.959678
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ 0 0
$$738$$ − 8.00000i − 0.294484i
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 20.0000i − 0.734223i
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −32.0000 −1.17160
$$747$$ − 16.0000i − 0.585409i
$$748$$ − 4.00000i − 0.146254i
$$749$$ 40.0000 1.46157
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ 6.00000i 0.218652i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 38.0000i 1.38113i 0.723269 + 0.690567i $$0.242639\pi$$
−0.723269 + 0.690567i $$0.757361\pi$$
$$758$$ − 28.0000i − 1.01701i
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ 4.00000i 0.144810i
$$764$$ −10.0000 −0.361787
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 0 0
$$768$$ 1.00000i 0.0360844i
$$769$$ −18.0000 −0.649097 −0.324548 0.945869i $$-0.605212\pi$$
−0.324548 + 0.945869i $$0.605212\pi$$
$$770$$ 0 0
$$771$$ −30.0000 −1.08042
$$772$$ − 4.00000i − 0.143963i
$$773$$ 42.0000i 1.51064i 0.655359 + 0.755318i $$0.272517\pi$$
−0.655359 + 0.755318i $$0.727483\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ 0 0
$$776$$ −8.00000 −0.287183
$$777$$ 8.00000i 0.286998i
$$778$$ 14.0000i 0.501924i
$$779$$ 8.00000 0.286630
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ − 16.0000i − 0.572159i
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −6.00000 −0.214013
$$787$$ 52.0000i 1.85360i 0.375555 + 0.926800i $$0.377452\pi$$
−0.375555 + 0.926800i $$0.622548\pi$$
$$788$$ − 10.0000i − 0.356235i
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ 4.00000 0.142224
$$792$$ 2.00000i 0.0710669i
$$793$$ 0 0
$$794$$ −6.00000 −0.212932
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ − 22.0000i − 0.779280i −0.920967 0.389640i $$-0.872599\pi$$
0.920967 0.389640i $$-0.127401\pi$$
$$798$$ 2.00000i 0.0707992i
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ 16.0000 0.565332
$$802$$ 12.0000i 0.423735i
$$803$$ − 4.00000i − 0.141157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 8.00000i − 0.281613i
$$808$$ 2.00000i 0.0703598i
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ 28.0000 0.983213 0.491606 0.870817i $$-0.336410\pi$$
0.491606 + 0.870817i $$0.336410\pi$$
$$812$$ 0 0
$$813$$ − 4.00000i − 0.140286i
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ − 6.00000i − 0.209913i
$$818$$ − 18.0000i − 0.629355i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 18.0000i 0.627822i
$$823$$ − 34.0000i − 1.18517i −0.805510 0.592583i $$-0.798108\pi$$
0.805510 0.592583i $$-0.201892\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ 8.00000i 0.278019i
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 0 0
$$831$$ 14.0000 0.485655
$$832$$ 0 0
$$833$$ − 6.00000i − 0.207888i
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ 0 0
$$838$$ 30.0000i 1.03633i
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 26.0000i 0.896019i
$$843$$ − 20.0000i − 0.688837i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 14.0000i 0.481046i
$$848$$ 10.0000i 0.343401i
$$849$$ −2.00000 −0.0686398
$$850$$ 0 0
$$851$$ −32.0000 −1.09695
$$852$$ − 8.00000i − 0.274075i
$$853$$ − 6.00000i − 0.205436i −0.994711 0.102718i $$-0.967246\pi$$
0.994711 0.102718i $$-0.0327539\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −20.0000 −0.683586
$$857$$ − 30.0000i − 1.02478i −0.858753 0.512390i $$-0.828760\pi$$
0.858753 0.512390i $$-0.171240\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ −16.0000 −0.545279
$$862$$ − 20.0000i − 0.681203i
$$863$$ 16.0000i 0.544646i 0.962206 + 0.272323i $$0.0877920\pi$$
−0.962206 + 0.272323i $$0.912208\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 12.0000 0.407777
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 0 0
$$872$$ − 2.00000i − 0.0677285i
$$873$$ − 8.00000i − 0.270759i
$$874$$ −8.00000 −0.270604
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ 8.00000i 0.270141i 0.990836 + 0.135070i $$0.0431261\pi$$
−0.990836 + 0.135070i $$0.956874\pi$$
$$878$$ 16.0000i 0.539974i
$$879$$ −22.0000 −0.742042
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ 18.0000i 0.605748i 0.953031 + 0.302874i $$0.0979462\pi$$
−0.953031 + 0.302874i $$0.902054\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 24.0000i 0.805841i 0.915235 + 0.402921i $$0.132005\pi$$
−0.915235 + 0.402921i $$0.867995\pi$$
$$888$$ − 4.00000i − 0.134231i
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ 12.0000i 0.401790i
$$893$$ 8.00000i 0.267710i
$$894$$ −2.00000 −0.0668900
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ 24.0000i 0.800890i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 20.0000 0.666297
$$902$$ − 16.0000i − 0.532742i
$$903$$ 12.0000i 0.399335i
$$904$$ −2.00000 −0.0665190
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ − 44.0000i − 1.46100i −0.682915 0.730498i $$-0.739288\pi$$
0.682915 0.730498i $$-0.260712\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ − 32.0000i − 1.05905i
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 12.0000i 0.396275i
$$918$$ 2.00000i 0.0660098i
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ 24.0000 0.790827
$$922$$ 6.00000i 0.197599i
$$923$$ 0 0
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ −18.0000 −0.591517
$$927$$ 12.0000i 0.394132i
$$928$$ 0 0
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ − 6.00000i − 0.196537i
$$933$$ − 14.0000i − 0.458339i
$$934$$ 20.0000 0.654420
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 58.0000i − 1.89478i −0.320085 0.947389i $$-0.603712\pi$$
0.320085 0.947389i $$-0.396288\pi$$
$$938$$ 0 0
$$939$$ 18.0000 0.587408
$$940$$ 0 0
$$941$$ 16.0000 0.521585 0.260793 0.965395i $$-0.416016\pi$$
0.260793 + 0.965395i $$0.416016\pi$$
$$942$$ − 6.00000i − 0.195491i
$$943$$ − 64.0000i − 2.08413i
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ − 32.0000i − 1.03986i −0.854209 0.519930i $$-0.825958\pi$$
0.854209 0.519930i $$-0.174042\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ − 4.00000i − 0.129641i
$$953$$ 18.0000i 0.583077i 0.956559 + 0.291539i $$0.0941672\pi$$
−0.956559 + 0.291539i $$0.905833\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ − 30.0000i − 0.969256i
$$959$$ 36.0000 1.16250
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ − 20.0000i − 0.644491i
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 16.0000 0.514792
$$967$$ − 6.00000i − 0.192947i −0.995336 0.0964735i $$-0.969244\pi$$
0.995336 0.0964735i $$-0.0307563\pi$$
$$968$$ − 7.00000i − 0.224989i
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 16.0000i 0.512936i
$$974$$ 20.0000 0.640841
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ 10.0000i 0.319765i
$$979$$ 32.0000 1.02272
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 30.0000i 0.957338i
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 8.00000 0.255031
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 16.0000i − 0.509286i
$$988$$ 0 0
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 0 0
$$993$$ − 12.0000i − 0.380808i
$$994$$ −16.0000 −0.507489
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ − 22.0000i − 0.696747i −0.937356 0.348373i $$-0.886734\pi$$
0.937356 0.348373i $$-0.113266\pi$$
$$998$$ − 16.0000i − 0.506471i
$$999$$ 4.00000 0.126554
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 2850.2.d.n.799.1 2
5.2 odd 4 2850.2.a.ba.1.1 1
5.3 odd 4 570.2.a.c.1.1 1
5.4 even 2 inner 2850.2.d.n.799.2 2
15.2 even 4 8550.2.a.o.1.1 1
15.8 even 4 1710.2.a.n.1.1 1
20.3 even 4 4560.2.a.bd.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.c.1.1 1 5.3 odd 4
1710.2.a.n.1.1 1 15.8 even 4
2850.2.a.ba.1.1 1 5.2 odd 4
2850.2.d.n.799.1 2 1.1 even 1 trivial
2850.2.d.n.799.2 2 5.4 even 2 inner
4560.2.a.bd.1.1 1 20.3 even 4
8550.2.a.o.1.1 1 15.2 even 4