# Properties

 Label 5070.2.b.d.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ 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: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.d.1351.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000i q^{11} +1.00000 q^{12} -1.00000i q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000i q^{18} +4.00000i q^{19} -1.00000i q^{20} +4.00000 q^{22} -8.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} +1.00000 q^{30} -8.00000i q^{31} +1.00000i q^{32} +4.00000i q^{33} +6.00000i q^{34} -1.00000 q^{36} +10.0000i q^{37} -4.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} -4.00000 q^{43} +4.00000i q^{44} +1.00000i q^{45} -8.00000i q^{46} -1.00000 q^{48} +7.00000 q^{49} -1.00000i q^{50} -6.00000 q^{51} -10.0000 q^{53} -1.00000i q^{54} +4.00000 q^{55} -4.00000i q^{57} +6.00000i q^{58} -4.00000i q^{59} +1.00000i q^{60} -2.00000 q^{61} +8.00000 q^{62} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} -6.00000 q^{68} +8.00000 q^{69} +16.0000i q^{71} -1.00000i q^{72} -2.00000i q^{73} -10.0000 q^{74} +1.00000 q^{75} -4.00000i q^{76} -16.0000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000i q^{83} +6.00000i q^{85} -4.00000i q^{86} -6.00000 q^{87} -4.00000 q^{88} -10.0000i q^{89} -1.00000 q^{90} +8.00000 q^{92} +8.00000i q^{93} -4.00000 q^{95} -1.00000i q^{96} -6.00000i q^{97} +7.00000i q^{98} -4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} - 2q^{10} + 2q^{12} + 2q^{16} + 12q^{17} + 8q^{22} - 16q^{23} - 2q^{25} - 2q^{27} + 12q^{29} + 2q^{30} - 2q^{36} - 8q^{38} + 2q^{40} - 8q^{43} - 2q^{48} + 14q^{49} - 12q^{51} - 20q^{53} + 8q^{55} - 4q^{61} + 16q^{62} - 2q^{64} - 8q^{66} - 12q^{68} + 16q^{69} - 20q^{74} + 2q^{75} - 32q^{79} + 2q^{81} + 12q^{82} - 12q^{87} - 8q^{88} - 2q^{90} + 16q^{92} - 8q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000i 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ − 4.00000i − 1.20605i −0.797724 0.603023i $$-0.793963\pi$$
0.797724 0.603023i $$-0.206037\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 0 0
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 4.00000i 0.917663i 0.888523 + 0.458831i $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ − 1.00000i − 0.223607i
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 1.00000 0.182574
$$31$$ − 8.00000i − 1.43684i −0.695608 0.718421i $$-0.744865\pi$$
0.695608 0.718421i $$-0.255135\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ 6.00000i 1.02899i
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ − 6.00000i − 0.937043i −0.883452 0.468521i $$-0.844787\pi$$
0.883452 0.468521i $$-0.155213\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 4.00000i 0.603023i
$$45$$ 1.00000i 0.149071i
$$46$$ − 8.00000i − 1.17954i
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 7.00000 1.00000
$$50$$ − 1.00000i − 0.141421i
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ −10.0000 −1.37361 −0.686803 0.726844i $$-0.740986\pi$$
−0.686803 + 0.726844i $$0.740986\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ 6.00000i 0.787839i
$$59$$ − 4.00000i − 0.520756i −0.965507 0.260378i $$-0.916153\pi$$
0.965507 0.260378i $$-0.0838471\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 12.0000i − 1.46603i −0.680211 0.733017i $$-0.738112\pi$$
0.680211 0.733017i $$-0.261888\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 16.0000i 1.89885i 0.313993 + 0.949425i $$0.398333\pi$$
−0.313993 + 0.949425i $$0.601667\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 1.00000 0.115470
$$76$$ − 4.00000i − 0.458831i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 1.00000i 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 6.00000i 0.650791i
$$86$$ − 4.00000i − 0.431331i
$$87$$ −6.00000 −0.643268
$$88$$ −4.00000 −0.426401
$$89$$ − 10.0000i − 1.06000i −0.847998 0.529999i $$-0.822192\pi$$
0.847998 0.529999i $$-0.177808\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ 8.00000i 0.829561i
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ − 1.00000i − 0.102062i
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ − 4.00000i − 0.402015i
$$100$$ 1.00000 0.100000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ − 10.0000i − 0.971286i
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ − 10.0000i − 0.957826i −0.877862 0.478913i $$-0.841031\pi$$
0.877862 0.478913i $$-0.158969\pi$$
$$110$$ 4.00000i 0.381385i
$$111$$ − 10.0000i − 0.949158i
$$112$$ 0 0
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 4.00000 0.374634
$$115$$ − 8.00000i − 0.746004i
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ −5.00000 −0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ 6.00000i 0.541002i
$$124$$ 8.00000i 0.718421i
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ 0 0
$$134$$ 12.0000 1.03664
$$135$$ − 1.00000i − 0.0860663i
$$136$$ − 6.00000i − 0.514496i
$$137$$ − 10.0000i − 0.854358i −0.904167 0.427179i $$-0.859507\pi$$
0.904167 0.427179i $$-0.140493\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −16.0000 −1.34269
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 6.00000i 0.498273i
$$146$$ 2.00000 0.165521
$$147$$ −7.00000 −0.577350
$$148$$ − 10.0000i − 0.821995i
$$149$$ 6.00000i 0.491539i 0.969328 + 0.245770i $$0.0790407\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ − 16.0000i − 1.27289i
$$159$$ 10.0000 0.793052
$$160$$ −1.00000 −0.0790569
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 6.00000i 0.468521i
$$165$$ −4.00000 −0.311400
$$166$$ 12.0000 0.931381
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −6.00000 −0.460179
$$171$$ 4.00000i 0.305888i
$$172$$ 4.00000 0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ − 6.00000i − 0.454859i
$$175$$ 0 0
$$176$$ − 4.00000i − 0.301511i
$$177$$ 4.00000i 0.300658i
$$178$$ 10.0000 0.749532
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ − 1.00000i − 0.0745356i
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 8.00000i 0.589768i
$$185$$ −10.0000 −0.735215
$$186$$ −8.00000 −0.586588
$$187$$ − 24.0000i − 1.75505i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ − 4.00000i − 0.290191i
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ − 10.0000i − 0.719816i −0.932988 0.359908i $$-0.882808\pi$$
0.932988 0.359908i $$-0.117192\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ 12.0000i 0.846415i
$$202$$ 2.00000i 0.140720i
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 6.00000 0.419058
$$206$$ 0 0
$$207$$ −8.00000 −0.556038
$$208$$ 0 0
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 10.0000 0.686803
$$213$$ − 16.0000i − 1.09630i
$$214$$ − 12.0000i − 0.820303i
$$215$$ − 4.00000i − 0.272798i
$$216$$ 1.00000i 0.0680414i
$$217$$ 0 0
$$218$$ 10.0000 0.677285
$$219$$ 2.00000i 0.135147i
$$220$$ −4.00000 −0.269680
$$221$$ 0 0
$$222$$ 10.0000 0.671156
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 10.0000i 0.665190i
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ − 14.0000i − 0.925146i −0.886581 0.462573i $$-0.846926\pi$$
0.886581 0.462573i $$-0.153074\pi$$
$$230$$ 8.00000 0.527504
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 4.00000i 0.260378i
$$237$$ 16.0000 1.03931
$$238$$ 0 0
$$239$$ − 8.00000i − 0.517477i −0.965947 0.258738i $$-0.916693\pi$$
0.965947 0.258738i $$-0.0833068\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ − 18.0000i − 1.15948i −0.814801 0.579741i $$-0.803154\pi$$
0.814801 0.579741i $$-0.196846\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 7.00000i 0.447214i
$$246$$ −6.00000 −0.382546
$$247$$ 0 0
$$248$$ −8.00000 −0.508001
$$249$$ 12.0000i 0.760469i
$$250$$ 1.00000 0.0632456
$$251$$ 28.0000 1.76734 0.883672 0.468106i $$-0.155064\pi$$
0.883672 + 0.468106i $$0.155064\pi$$
$$252$$ 0 0
$$253$$ 32.0000i 2.01182i
$$254$$ − 8.00000i − 0.501965i
$$255$$ − 6.00000i − 0.375735i
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 12.0000i 0.741362i
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 4.00000 0.246183
$$265$$ − 10.0000i − 0.614295i
$$266$$ 0 0
$$267$$ 10.0000i 0.611990i
$$268$$ 12.0000i 0.733017i
$$269$$ −26.0000 −1.58525 −0.792624 0.609711i $$-0.791286\pi$$
−0.792624 + 0.609711i $$0.791286\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ − 24.0000i − 1.45790i −0.684569 0.728948i $$-0.740010\pi$$
0.684569 0.728948i $$-0.259990\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ 4.00000i 0.241209i
$$276$$ −8.00000 −0.481543
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 12.0000i 0.719712i
$$279$$ − 8.00000i − 0.478947i
$$280$$ 0 0
$$281$$ − 10.0000i − 0.596550i −0.954480 0.298275i $$-0.903589\pi$$
0.954480 0.298275i $$-0.0964112\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ − 16.0000i − 0.949425i
$$285$$ 4.00000 0.236940
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 1.00000i 0.0589256i
$$289$$ 19.0000 1.11765
$$290$$ −6.00000 −0.352332
$$291$$ 6.00000i 0.351726i
$$292$$ 2.00000i 0.117041i
$$293$$ 26.0000i 1.51894i 0.650545 + 0.759468i $$0.274541\pi$$
−0.650545 + 0.759468i $$0.725459\pi$$
$$294$$ − 7.00000i − 0.408248i
$$295$$ 4.00000 0.232889
$$296$$ 10.0000 0.581238
$$297$$ 4.00000i 0.232104i
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −2.00000 −0.114897
$$304$$ 4.00000i 0.229416i
$$305$$ − 2.00000i − 0.114520i
$$306$$ 6.00000i 0.342997i
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 8.00000i 0.454369i
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ − 2.00000i − 0.112867i
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 10.0000i 0.560772i
$$319$$ − 24.0000i − 1.34374i
$$320$$ − 1.00000i − 0.0559017i
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 24.0000i 1.33540i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 10.0000i 0.553001i
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ − 4.00000i − 0.220193i
$$331$$ − 20.0000i − 1.09930i −0.835395 0.549650i $$-0.814761\pi$$
0.835395 0.549650i $$-0.185239\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ 10.0000i 0.547997i
$$334$$ 8.00000 0.437741
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 0 0
$$339$$ −10.0000 −0.543125
$$340$$ − 6.00000i − 0.325396i
$$341$$ −32.0000 −1.73290
$$342$$ −4.00000 −0.216295
$$343$$ 0 0
$$344$$ 4.00000i 0.215666i
$$345$$ 8.00000i 0.430706i
$$346$$ 18.0000i 0.967686i
$$347$$ 20.0000 1.07366 0.536828 0.843692i $$-0.319622\pi$$
0.536828 + 0.843692i $$0.319622\pi$$
$$348$$ 6.00000 0.321634
$$349$$ − 6.00000i − 0.321173i −0.987022 0.160586i $$-0.948662\pi$$
0.987022 0.160586i $$-0.0513385\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ − 30.0000i − 1.59674i −0.602168 0.798369i $$-0.705696\pi$$
0.602168 0.798369i $$-0.294304\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ −16.0000 −0.849192
$$356$$ 10.0000i 0.529999i
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ − 16.0000i − 0.844448i −0.906492 0.422224i $$-0.861250\pi$$
0.906492 0.422224i $$-0.138750\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 3.00000 0.157895
$$362$$ − 6.00000i − 0.315353i
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 2.00000 0.104685
$$366$$ 2.00000i 0.104542i
$$367$$ −24.0000 −1.25279 −0.626395 0.779506i $$-0.715470\pi$$
−0.626395 + 0.779506i $$0.715470\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ − 6.00000i − 0.312348i
$$370$$ − 10.0000i − 0.519875i
$$371$$ 0 0
$$372$$ − 8.00000i − 0.414781i
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 24.0000 1.24101
$$375$$ 1.00000i 0.0516398i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ − 4.00000i − 0.205466i −0.994709 0.102733i $$-0.967241\pi$$
0.994709 0.102733i $$-0.0327588\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 8.00000 0.409852
$$382$$ 0 0
$$383$$ 32.0000i 1.63512i 0.575841 + 0.817562i $$0.304675\pi$$
−0.575841 + 0.817562i $$0.695325\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ −4.00000 −0.203331
$$388$$ 6.00000i 0.304604i
$$389$$ 34.0000 1.72387 0.861934 0.507020i $$-0.169253\pi$$
0.861934 + 0.507020i $$0.169253\pi$$
$$390$$ 0 0
$$391$$ −48.0000 −2.42746
$$392$$ − 7.00000i − 0.353553i
$$393$$ −12.0000 −0.605320
$$394$$ −6.00000 −0.302276
$$395$$ − 16.0000i − 0.805047i
$$396$$ 4.00000i 0.201008i
$$397$$ − 30.0000i − 1.50566i −0.658217 0.752828i $$-0.728689\pi$$
0.658217 0.752828i $$-0.271311\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ − 34.0000i − 1.69788i −0.528490 0.848939i $$-0.677242\pi$$
0.528490 0.848939i $$-0.322758\pi$$
$$402$$ −12.0000 −0.598506
$$403$$ 0 0
$$404$$ −2.00000 −0.0995037
$$405$$ 1.00000i 0.0496904i
$$406$$ 0 0
$$407$$ 40.0000 1.98273
$$408$$ 6.00000i 0.297044i
$$409$$ − 22.0000i − 1.08783i −0.839140 0.543915i $$-0.816941\pi$$
0.839140 0.543915i $$-0.183059\pi$$
$$410$$ 6.00000i 0.296319i
$$411$$ 10.0000i 0.493264i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ − 8.00000i − 0.393179i
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ −12.0000 −0.587643
$$418$$ 16.0000i 0.782586i
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ − 34.0000i − 1.65706i −0.559946 0.828529i $$-0.689178\pi$$
0.559946 0.828529i $$-0.310822\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 0 0
$$424$$ 10.0000i 0.485643i
$$425$$ −6.00000 −0.291043
$$426$$ 16.0000 0.775203
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ − 24.0000i − 1.15604i −0.816023 0.578020i $$-0.803826\pi$$
0.816023 0.578020i $$-0.196174\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 0 0
$$435$$ − 6.00000i − 0.287678i
$$436$$ 10.0000i 0.478913i
$$437$$ − 32.0000i − 1.53077i
$$438$$ −2.00000 −0.0955637
$$439$$ 40.0000 1.90910 0.954548 0.298057i $$-0.0963387\pi$$
0.954548 + 0.298057i $$0.0963387\pi$$
$$440$$ − 4.00000i − 0.190693i
$$441$$ 7.00000 0.333333
$$442$$ 0 0
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 10.0000i 0.474579i
$$445$$ 10.0000 0.474045
$$446$$ −8.00000 −0.378811
$$447$$ − 6.00000i − 0.283790i
$$448$$ 0 0
$$449$$ − 18.0000i − 0.849473i −0.905317 0.424736i $$-0.860367\pi$$
0.905317 0.424736i $$-0.139633\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ −24.0000 −1.13012
$$452$$ −10.0000 −0.470360
$$453$$ 0 0
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 34.0000i 1.59045i 0.606313 + 0.795226i $$0.292648\pi$$
−0.606313 + 0.795226i $$0.707352\pi$$
$$458$$ 14.0000 0.654177
$$459$$ −6.00000 −0.280056
$$460$$ 8.00000i 0.373002i
$$461$$ − 18.0000i − 0.838344i −0.907907 0.419172i $$-0.862320\pi$$
0.907907 0.419172i $$-0.137680\pi$$
$$462$$ 0 0
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ 6.00000 0.278543
$$465$$ −8.00000 −0.370991
$$466$$ 14.0000i 0.648537i
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ −4.00000 −0.184115
$$473$$ 16.0000i 0.735681i
$$474$$ 16.0000i 0.734904i
$$475$$ − 4.00000i − 0.183533i
$$476$$ 0 0
$$477$$ −10.0000 −0.457869
$$478$$ 8.00000 0.365911
$$479$$ 8.00000i 0.365529i 0.983157 + 0.182765i $$0.0585046\pi$$
−0.983157 + 0.182765i $$0.941495\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ 18.0000 0.819878
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 6.00000 0.272446
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ 4.00000i 0.180886i
$$490$$ −7.00000 −0.316228
$$491$$ −4.00000 −0.180517 −0.0902587 0.995918i $$-0.528769\pi$$
−0.0902587 + 0.995918i $$0.528769\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 36.0000 1.62136
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ − 8.00000i − 0.359211i
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ 20.0000i 0.895323i 0.894203 + 0.447661i $$0.147743\pi$$
−0.894203 + 0.447661i $$0.852257\pi$$
$$500$$ 1.00000i 0.0447214i
$$501$$ 8.00000i 0.357414i
$$502$$ 28.0000i 1.24970i
$$503$$ 8.00000 0.356702 0.178351 0.983967i $$-0.442924\pi$$
0.178351 + 0.983967i $$0.442924\pi$$
$$504$$ 0 0
$$505$$ 2.00000i 0.0889988i
$$506$$ −32.0000 −1.42257
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ 30.0000i 1.32973i 0.746965 + 0.664863i $$0.231510\pi$$
−0.746965 + 0.664863i $$0.768490\pi$$
$$510$$ 6.00000 0.265684
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ − 4.00000i − 0.176604i
$$514$$ 6.00000i 0.264649i
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 6.00000i 0.262613i
$$523$$ 36.0000 1.57417 0.787085 0.616844i $$-0.211589\pi$$
0.787085 + 0.616844i $$0.211589\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 24.0000i 1.04645i
$$527$$ − 48.0000i − 2.09091i
$$528$$ 4.00000i 0.174078i
$$529$$ 41.0000 1.78261
$$530$$ 10.0000 0.434372
$$531$$ − 4.00000i − 0.173585i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −10.0000 −0.432742
$$535$$ − 12.0000i − 0.518805i
$$536$$ −12.0000 −0.518321
$$537$$ 12.0000 0.517838
$$538$$ − 26.0000i − 1.12094i
$$539$$ − 28.0000i − 1.20605i
$$540$$ 1.00000i 0.0430331i
$$541$$ − 22.0000i − 0.945854i −0.881102 0.472927i $$-0.843197\pi$$
0.881102 0.472927i $$-0.156803\pi$$
$$542$$ 24.0000 1.03089
$$543$$ 6.00000 0.257485
$$544$$ 6.00000i 0.257248i
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 10.0000i 0.427179i
$$549$$ −2.00000 −0.0853579
$$550$$ −4.00000 −0.170561
$$551$$ 24.0000i 1.02243i
$$552$$ − 8.00000i − 0.340503i
$$553$$ 0 0
$$554$$ 10.0000i 0.424859i
$$555$$ 10.0000 0.424476
$$556$$ −12.0000 −0.508913
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 8.00000 0.338667
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 24.0000i 1.01328i
$$562$$ 10.0000 0.421825
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 10.0000i 0.420703i
$$566$$ − 20.0000i − 0.840663i
$$567$$ 0 0
$$568$$ 16.0000 0.671345
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 4.00000i 0.167542i
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 8.00000 0.333623
$$576$$ −1.00000 −0.0416667
$$577$$ 42.0000i 1.74848i 0.485491 + 0.874241i $$0.338641\pi$$
−0.485491 + 0.874241i $$0.661359\pi$$
$$578$$ 19.0000i 0.790296i
$$579$$ 10.0000i 0.415586i
$$580$$ − 6.00000i − 0.249136i
$$581$$ 0 0
$$582$$ −6.00000 −0.248708
$$583$$ 40.0000i 1.65663i
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −26.0000 −1.07405
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 7.00000 0.288675
$$589$$ 32.0000 1.31854
$$590$$ 4.00000i 0.164677i
$$591$$ − 6.00000i − 0.246807i
$$592$$ 10.0000i 0.410997i
$$593$$ − 34.0000i − 1.39621i −0.715994 0.698106i $$-0.754026\pi$$
0.715994 0.698106i $$-0.245974\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ − 6.00000i − 0.245770i
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 0 0
$$603$$ − 12.0000i − 0.488678i
$$604$$ 0 0
$$605$$ − 5.00000i − 0.203279i
$$606$$ − 2.00000i − 0.0812444i
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ 2.00000 0.0809776
$$611$$ 0 0
$$612$$ −6.00000 −0.242536
$$613$$ − 10.0000i − 0.403896i −0.979396 0.201948i $$-0.935273\pi$$
0.979396 0.201948i $$-0.0647272\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ 42.0000i 1.69086i 0.534089 + 0.845428i $$0.320655\pi$$
−0.534089 + 0.845428i $$0.679345\pi$$
$$618$$ 0 0
$$619$$ 4.00000i 0.160774i 0.996764 + 0.0803868i $$0.0256155\pi$$
−0.996764 + 0.0803868i $$0.974384\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 8.00000 0.321029
$$622$$ 24.0000i 0.962312i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 22.0000i − 0.879297i
$$627$$ −16.0000 −0.638978
$$628$$ 2.00000 0.0798087
$$629$$ 60.0000i 2.39236i
$$630$$ 0 0
$$631$$ 16.0000i 0.636950i 0.947931 + 0.318475i $$0.103171\pi$$
−0.947931 + 0.318475i $$0.896829\pi$$
$$632$$ 16.0000i 0.636446i
$$633$$ 12.0000 0.476957
$$634$$ 2.00000 0.0794301
$$635$$ − 8.00000i − 0.317470i
$$636$$ −10.0000 −0.396526
$$637$$ 0 0
$$638$$ 24.0000 0.950169
$$639$$ 16.0000i 0.632950i
$$640$$ 1.00000 0.0395285
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 0 0
$$645$$ 4.00000i 0.157500i
$$646$$ −24.0000 −0.944267
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ −34.0000 −1.33052 −0.665261 0.746611i $$-0.731680\pi$$
−0.665261 + 0.746611i $$0.731680\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 12.0000i 0.468879i
$$656$$ − 6.00000i − 0.234261i
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 4.00000 0.155700
$$661$$ − 14.0000i − 0.544537i −0.962221 0.272268i $$-0.912226\pi$$
0.962221 0.272268i $$-0.0877739\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ −48.0000 −1.85857
$$668$$ 8.00000i 0.309529i
$$669$$ − 8.00000i − 0.309298i
$$670$$ 12.0000i 0.463600i
$$671$$ 8.00000i 0.308837i
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ − 2.00000i − 0.0770371i
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 22.0000 0.845529 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ − 10.0000i − 0.384048i
$$679$$ 0 0
$$680$$ 6.00000 0.230089
$$681$$ − 4.00000i − 0.153280i
$$682$$ − 32.0000i − 1.22534i
$$683$$ − 28.0000i − 1.07139i −0.844411 0.535695i $$-0.820050\pi$$
0.844411 0.535695i $$-0.179950\pi$$
$$684$$ − 4.00000i − 0.152944i
$$685$$ 10.0000 0.382080
$$686$$ 0 0
$$687$$ 14.0000i 0.534133i
$$688$$ −4.00000 −0.152499
$$689$$ 0 0
$$690$$ −8.00000 −0.304555
$$691$$ − 12.0000i − 0.456502i −0.973602 0.228251i $$-0.926699\pi$$
0.973602 0.228251i $$-0.0733006\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ 20.0000i 0.759190i
$$695$$ 12.0000i 0.455186i
$$696$$ 6.00000i 0.227429i
$$697$$ − 36.0000i − 1.36360i
$$698$$ 6.00000 0.227103
$$699$$ −14.0000 −0.529529
$$700$$ 0 0
$$701$$ −22.0000 −0.830929 −0.415464 0.909610i $$-0.636381\pi$$
−0.415464 + 0.909610i $$0.636381\pi$$
$$702$$ 0 0
$$703$$ −40.0000 −1.50863
$$704$$ 4.00000i 0.150756i
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 0 0
$$708$$ − 4.00000i − 0.150329i
$$709$$ 50.0000i 1.87779i 0.344204 + 0.938895i $$0.388149\pi$$
−0.344204 + 0.938895i $$0.611851\pi$$
$$710$$ − 16.0000i − 0.600469i
$$711$$ −16.0000 −0.600047
$$712$$ −10.0000 −0.374766
$$713$$ 64.0000i 2.39682i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 8.00000i 0.298765i
$$718$$ 16.0000 0.597115
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 1.00000i 0.0372678i
$$721$$ 0 0
$$722$$ 3.00000i 0.111648i
$$723$$ 18.0000i 0.669427i
$$724$$ 6.00000 0.222988
$$725$$ −6.00000 −0.222834
$$726$$ 5.00000i 0.185567i
$$727$$ 16.0000 0.593407 0.296704 0.954970i $$-0.404113\pi$$
0.296704 + 0.954970i $$0.404113\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000i 0.0740233i
$$731$$ −24.0000 −0.887672
$$732$$ −2.00000 −0.0739221
$$733$$ − 50.0000i − 1.84679i −0.383849 0.923396i $$-0.625402\pi$$
0.383849 0.923396i $$-0.374598\pi$$
$$734$$ − 24.0000i − 0.885856i
$$735$$ − 7.00000i − 0.258199i
$$736$$ − 8.00000i − 0.294884i
$$737$$ −48.0000 −1.76810
$$738$$ 6.00000 0.220863
$$739$$ − 20.0000i − 0.735712i −0.929883 0.367856i $$-0.880092\pi$$
0.929883 0.367856i $$-0.119908\pi$$
$$740$$ 10.0000 0.367607
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 8.00000 0.293294
$$745$$ −6.00000 −0.219823
$$746$$ 22.0000i 0.805477i
$$747$$ − 12.0000i − 0.439057i
$$748$$ 24.0000i 0.877527i
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ −28.0000 −1.02038
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −42.0000 −1.52652 −0.763258 0.646094i $$-0.776401\pi$$
−0.763258 + 0.646094i $$0.776401\pi$$
$$758$$ 4.00000 0.145287
$$759$$ − 32.0000i − 1.16153i
$$760$$ 4.00000i 0.145095i
$$761$$ − 42.0000i − 1.52250i −0.648459 0.761249i $$-0.724586\pi$$
0.648459 0.761249i $$-0.275414\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 6.00000i 0.216930i
$$766$$ −32.0000 −1.15621
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 34.0000i 1.22607i 0.790055 + 0.613036i $$0.210052\pi$$
−0.790055 + 0.613036i $$0.789948\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 10.0000i 0.359908i
$$773$$ 38.0000i 1.36677i 0.730061 + 0.683383i $$0.239492\pi$$
−0.730061 + 0.683383i $$0.760508\pi$$
$$774$$ − 4.00000i − 0.143777i
$$775$$ 8.00000i 0.287368i
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ 34.0000i 1.21896i
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ 64.0000 2.29010
$$782$$ − 48.0000i − 1.71648i
$$783$$ −6.00000 −0.214423
$$784$$ 7.00000 0.250000
$$785$$ − 2.00000i − 0.0713831i
$$786$$ − 12.0000i − 0.428026i
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ −24.0000 −0.854423
$$790$$ 16.0000 0.569254
$$791$$ 0 0
$$792$$ −4.00000 −0.142134
$$793$$ 0 0
$$794$$ 30.0000 1.06466
$$795$$ 10.0000i 0.354663i
$$796$$ −8.00000 −0.283552
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ − 1.00000i − 0.0353553i
$$801$$ − 10.0000i − 0.353333i
$$802$$ 34.0000 1.20058
$$803$$ −8.00000 −0.282314
$$804$$ − 12.0000i − 0.423207i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 26.0000 0.915243
$$808$$ − 2.00000i − 0.0703598i
$$809$$ 42.0000 1.47664 0.738321 0.674450i $$-0.235619\pi$$
0.738321 + 0.674450i $$0.235619\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 12.0000i 0.421377i 0.977553 + 0.210688i $$0.0675706\pi$$
−0.977553 + 0.210688i $$0.932429\pi$$
$$812$$ 0 0
$$813$$ 24.0000i 0.841717i
$$814$$ 40.0000i 1.40200i
$$815$$ 4.00000 0.140114
$$816$$ −6.00000 −0.210042
$$817$$ − 16.0000i − 0.559769i
$$818$$ 22.0000 0.769212
$$819$$ 0 0
$$820$$ −6.00000 −0.209529
$$821$$ 6.00000i 0.209401i 0.994504 + 0.104701i $$0.0333885\pi$$
−0.994504 + 0.104701i $$0.966612\pi$$
$$822$$ −10.0000 −0.348790
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ 0 0
$$825$$ − 4.00000i − 0.139262i
$$826$$ 0 0
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 8.00000 0.278019
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 12.0000i 0.416526i
$$831$$ −10.0000 −0.346896
$$832$$ 0 0
$$833$$ 42.0000 1.45521
$$834$$ − 12.0000i − 0.415526i
$$835$$ 8.00000 0.276851
$$836$$ −16.0000 −0.553372
$$837$$ 8.00000i 0.276520i
$$838$$ − 4.00000i − 0.138178i
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 34.0000 1.17172
$$843$$ 10.0000i 0.344418i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −10.0000 −0.343401
$$849$$ 20.0000 0.686398
$$850$$ − 6.00000i − 0.205798i
$$851$$ − 80.0000i − 2.74236i
$$852$$ 16.0000i 0.548151i
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ 12.0000i 0.410152i
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 4.00000i 0.136399i
$$861$$ 0 0
$$862$$ 24.0000 0.817443
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 18.0000i 0.612018i
$$866$$ − 34.0000i − 1.15537i
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ 64.0000i 2.17105i
$$870$$ 6.00000 0.203419
$$871$$ 0 0
$$872$$ −10.0000 −0.338643
$$873$$ − 6.00000i − 0.203069i
$$874$$ 32.0000 1.08242
$$875$$ 0 0
$$876$$ − 2.00000i − 0.0675737i
$$877$$ 14.0000i 0.472746i 0.971662 + 0.236373i $$0.0759588\pi$$
−0.971662 + 0.236373i $$0.924041\pi$$
$$878$$ 40.0000i 1.34993i
$$879$$ − 26.0000i − 0.876958i
$$880$$ 4.00000 0.134840
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 7.00000i 0.235702i
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 0 0
$$885$$ −4.00000 −0.134459
$$886$$ 36.0000i 1.20944i
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ −10.0000 −0.335578
$$889$$ 0 0
$$890$$ 10.0000i 0.335201i
$$891$$ − 4.00000i − 0.134005i
$$892$$ − 8.00000i − 0.267860i
$$893$$ 0 0
$$894$$ 6.00000 0.200670
$$895$$ − 12.0000i − 0.401116i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ − 48.0000i − 1.60089i
$$900$$ 1.00000 0.0333333
$$901$$ −60.0000 −1.99889
$$902$$ − 24.0000i − 0.799113i
$$903$$ 0 0
$$904$$ − 10.0000i − 0.332595i
$$905$$ − 6.00000i − 0.199447i
$$906$$ 0 0
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ − 4.00000i − 0.132745i
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ −48.0000 −1.58857
$$914$$ −34.0000 −1.12462
$$915$$ 2.00000i 0.0661180i
$$916$$ 14.0000i 0.462573i
$$917$$ 0 0
$$918$$ − 6.00000i − 0.198030i
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ −8.00000 −0.263752
$$921$$ − 12.0000i − 0.395413i
$$922$$ 18.0000 0.592798
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 10.0000i − 0.328798i
$$926$$ 8.00000 0.262896
$$927$$ 0 0
$$928$$ 6.00000i 0.196960i
$$929$$ − 14.0000i − 0.459325i −0.973270 0.229663i $$-0.926238\pi$$
0.973270 0.229663i $$-0.0737623\pi$$
$$930$$ − 8.00000i − 0.262330i
$$931$$ 28.0000i 0.917663i
$$932$$ −14.0000 −0.458585
$$933$$ −24.0000 −0.785725
$$934$$ − 12.0000i − 0.392652i
$$935$$ 24.0000 0.784884
$$936$$ 0 0
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ 22.0000 0.717943
$$940$$ 0 0
$$941$$ 30.0000i 0.977972i 0.872292 + 0.488986i $$0.162633\pi$$
−0.872292 + 0.488986i $$0.837367\pi$$
$$942$$ 2.00000i 0.0651635i
$$943$$ 48.0000i 1.56310i
$$944$$ − 4.00000i − 0.130189i
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ − 52.0000i − 1.68977i −0.534946 0.844886i $$-0.679668\pi$$
0.534946 0.844886i $$-0.320332\pi$$
$$948$$ −16.0000 −0.519656
$$949$$ 0 0
$$950$$ 4.00000 0.129777
$$951$$ 2.00000i 0.0648544i
$$952$$ 0 0
$$953$$ −34.0000 −1.10137 −0.550684 0.834714i $$-0.685633\pi$$
−0.550684 + 0.834714i $$0.685633\pi$$
$$954$$ − 10.0000i − 0.323762i
$$955$$ 0 0
$$956$$ 8.00000i 0.258738i
$$957$$ 24.0000i 0.775810i
$$958$$ −8.00000 −0.258468
$$959$$ 0 0
$$960$$ 1.00000i 0.0322749i
$$961$$ −33.0000 −1.06452
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ 18.0000i 0.579741i
$$965$$ 10.0000 0.321911
$$966$$ 0 0
$$967$$ 16.0000i 0.514525i 0.966342 + 0.257263i $$0.0828206\pi$$
−0.966342 + 0.257263i $$0.917179\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ − 24.0000i − 0.770991i
$$970$$ 6.00000i 0.192648i
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 30.0000i − 0.959785i −0.877327 0.479893i $$-0.840676\pi$$
0.877327 0.479893i $$-0.159324\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ −40.0000 −1.27841
$$980$$ − 7.00000i − 0.223607i
$$981$$ − 10.0000i − 0.319275i
$$982$$ − 4.00000i − 0.127645i
$$983$$ − 8.00000i − 0.255160i −0.991828 0.127580i $$-0.959279\pi$$
0.991828 0.127580i $$-0.0407210\pi$$
$$984$$ 6.00000 0.191273
$$985$$ −6.00000 −0.191176
$$986$$ 36.0000i 1.14647i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 32.0000 1.01754
$$990$$ 4.00000i 0.127128i
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 8.00000 0.254000
$$993$$ 20.0000i 0.634681i
$$994$$ 0 0
$$995$$ 8.00000i 0.253617i
$$996$$ − 12.0000i − 0.380235i
$$997$$ −58.0000 −1.83688 −0.918439 0.395562i $$-0.870550\pi$$
−0.918439 + 0.395562i $$0.870550\pi$$
$$998$$ −20.0000 −0.633089
$$999$$ − 10.0000i − 0.316386i
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 5070.2.b.d.1351.2 2
13.5 odd 4 390.2.a.f.1.1 1
13.8 odd 4 5070.2.a.a.1.1 1
13.12 even 2 inner 5070.2.b.d.1351.1 2
39.5 even 4 1170.2.a.a.1.1 1
52.31 even 4 3120.2.a.w.1.1 1
65.18 even 4 1950.2.e.g.1249.1 2
65.44 odd 4 1950.2.a.k.1.1 1
65.57 even 4 1950.2.e.g.1249.2 2
156.83 odd 4 9360.2.a.p.1.1 1
195.44 even 4 5850.2.a.bo.1.1 1
195.83 odd 4 5850.2.e.e.5149.2 2
195.122 odd 4 5850.2.e.e.5149.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.f.1.1 1 13.5 odd 4
1170.2.a.a.1.1 1 39.5 even 4
1950.2.a.k.1.1 1 65.44 odd 4
1950.2.e.g.1249.1 2 65.18 even 4
1950.2.e.g.1249.2 2 65.57 even 4
3120.2.a.w.1.1 1 52.31 even 4
5070.2.a.a.1.1 1 13.8 odd 4
5070.2.b.d.1351.1 2 13.12 even 2 inner
5070.2.b.d.1351.2 2 1.1 even 1 trivial
5850.2.a.bo.1.1 1 195.44 even 4
5850.2.e.e.5149.1 2 195.122 odd 4
5850.2.e.e.5149.2 2 195.83 odd 4
9360.2.a.p.1.1 1 156.83 odd 4