# Properties

 Label 950.2.a.k.1.3 Level $950$ Weight $2$ Character 950.1 Self dual yes Analytic conductor $7.586$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.257.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.19869$$ of defining polynomial Character $$\chi$$ $$=$$ 950.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +3.03293 q^{3} +1.00000 q^{4} -3.03293 q^{6} -2.46980 q^{7} -1.00000 q^{8} +6.19869 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +3.03293 q^{3} +1.00000 q^{4} -3.03293 q^{6} -2.46980 q^{7} -1.00000 q^{8} +6.19869 q^{9} +0.728896 q^{11} +3.03293 q^{12} +6.23163 q^{13} +2.46980 q^{14} +1.00000 q^{16} -0.563139 q^{17} -6.19869 q^{18} -1.00000 q^{19} -7.49073 q^{21} -0.728896 q^{22} -4.63555 q^{23} -3.03293 q^{24} -6.23163 q^{26} +9.70142 q^{27} -2.46980 q^{28} +10.2316 q^{29} +6.06587 q^{31} -1.00000 q^{32} +2.21069 q^{33} +0.563139 q^{34} +6.19869 q^{36} +5.72890 q^{37} +1.00000 q^{38} +18.9001 q^{39} +4.79476 q^{41} +7.49073 q^{42} -8.06587 q^{43} +0.728896 q^{44} +4.63555 q^{46} -8.12628 q^{47} +3.03293 q^{48} -0.900112 q^{49} -1.70796 q^{51} +6.23163 q^{52} +1.53020 q^{53} -9.70142 q^{54} +2.46980 q^{56} -3.03293 q^{57} -10.2316 q^{58} -5.76183 q^{59} +10.9396 q^{61} -6.06587 q^{62} -15.3095 q^{63} +1.00000 q^{64} -2.21069 q^{66} -12.9330 q^{67} -0.563139 q^{68} -14.0593 q^{69} -4.39738 q^{71} -6.19869 q^{72} +4.09334 q^{73} -5.72890 q^{74} -1.00000 q^{76} -1.80022 q^{77} -18.9001 q^{78} +15.3370 q^{79} +10.8277 q^{81} -4.79476 q^{82} -7.85517 q^{83} -7.49073 q^{84} +8.06587 q^{86} +31.0318 q^{87} -0.728896 q^{88} -10.0000 q^{89} -15.3908 q^{91} -4.63555 q^{92} +18.3974 q^{93} +8.12628 q^{94} -3.03293 q^{96} -11.0055 q^{97} +0.900112 q^{98} +4.51820 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 2q^{7} - 3q^{8} + 13q^{9} + O(q^{10})$$ $$3q - 3q^{2} - 2q^{3} + 3q^{4} + 2q^{6} - 2q^{7} - 3q^{8} + 13q^{9} + 2q^{11} - 2q^{12} + 2q^{13} + 2q^{14} + 3q^{16} + 4q^{17} - 13q^{18} - 3q^{19} - 11q^{21} - 2q^{22} - 14q^{23} + 2q^{24} - 2q^{26} + 7q^{27} - 2q^{28} + 14q^{29} - 4q^{31} - 3q^{32} - 4q^{33} - 4q^{34} + 13q^{36} + 17q^{37} + 3q^{38} + 29q^{39} - 8q^{41} + 11q^{42} - 2q^{43} + 2q^{44} + 14q^{46} - 13q^{47} - 2q^{48} + 25q^{49} - 11q^{51} + 2q^{52} + 10q^{53} - 7q^{54} + 2q^{56} + 2q^{57} - 14q^{58} - 6q^{59} + 22q^{61} + 4q^{62} - 2q^{63} + 3q^{64} + 4q^{66} + 4q^{68} + 8q^{69} - 2q^{71} - 13q^{72} + 12q^{73} - 17q^{74} - 3q^{76} + 50q^{77} - 29q^{78} + 24q^{79} - q^{81} + 8q^{82} - 12q^{83} - 11q^{84} + 2q^{86} + 21q^{87} - 2q^{88} - 30q^{89} - 7q^{91} - 14q^{92} + 44q^{93} + 13q^{94} + 2q^{96} - 25q^{98} + 24q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 3.03293 1.75107 0.875533 0.483159i $$-0.160511\pi$$
0.875533 + 0.483159i $$0.160511\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −3.03293 −1.23819
$$7$$ −2.46980 −0.933495 −0.466747 0.884391i $$-0.654574\pi$$
−0.466747 + 0.884391i $$0.654574\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 6.19869 2.06623
$$10$$ 0 0
$$11$$ 0.728896 0.219770 0.109885 0.993944i $$-0.464952\pi$$
0.109885 + 0.993944i $$0.464952\pi$$
$$12$$ 3.03293 0.875533
$$13$$ 6.23163 1.72834 0.864171 0.503198i $$-0.167843\pi$$
0.864171 + 0.503198i $$0.167843\pi$$
$$14$$ 2.46980 0.660081
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −0.563139 −0.136581 −0.0682907 0.997665i $$-0.521755\pi$$
−0.0682907 + 0.997665i $$0.521755\pi$$
$$18$$ −6.19869 −1.46105
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −7.49073 −1.63461
$$22$$ −0.728896 −0.155401
$$23$$ −4.63555 −0.966579 −0.483290 0.875460i $$-0.660558\pi$$
−0.483290 + 0.875460i $$0.660558\pi$$
$$24$$ −3.03293 −0.619095
$$25$$ 0 0
$$26$$ −6.23163 −1.22212
$$27$$ 9.70142 1.86704
$$28$$ −2.46980 −0.466747
$$29$$ 10.2316 1.89997 0.949983 0.312303i $$-0.101100\pi$$
0.949983 + 0.312303i $$0.101100\pi$$
$$30$$ 0 0
$$31$$ 6.06587 1.08946 0.544731 0.838611i $$-0.316632\pi$$
0.544731 + 0.838611i $$0.316632\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 2.21069 0.384832
$$34$$ 0.563139 0.0965776
$$35$$ 0 0
$$36$$ 6.19869 1.03312
$$37$$ 5.72890 0.941825 0.470912 0.882180i $$-0.343925\pi$$
0.470912 + 0.882180i $$0.343925\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 18.9001 3.02644
$$40$$ 0 0
$$41$$ 4.79476 0.748816 0.374408 0.927264i $$-0.377846\pi$$
0.374408 + 0.927264i $$0.377846\pi$$
$$42$$ 7.49073 1.15584
$$43$$ −8.06587 −1.23003 −0.615017 0.788514i $$-0.710851\pi$$
−0.615017 + 0.788514i $$0.710851\pi$$
$$44$$ 0.728896 0.109885
$$45$$ 0 0
$$46$$ 4.63555 0.683475
$$47$$ −8.12628 −1.18534 −0.592670 0.805446i $$-0.701926\pi$$
−0.592670 + 0.805446i $$0.701926\pi$$
$$48$$ 3.03293 0.437766
$$49$$ −0.900112 −0.128587
$$50$$ 0 0
$$51$$ −1.70796 −0.239163
$$52$$ 6.23163 0.864171
$$53$$ 1.53020 0.210190 0.105095 0.994462i $$-0.466485\pi$$
0.105095 + 0.994462i $$0.466485\pi$$
$$54$$ −9.70142 −1.32020
$$55$$ 0 0
$$56$$ 2.46980 0.330040
$$57$$ −3.03293 −0.401722
$$58$$ −10.2316 −1.34348
$$59$$ −5.76183 −0.750126 −0.375063 0.926999i $$-0.622379\pi$$
−0.375063 + 0.926999i $$0.622379\pi$$
$$60$$ 0 0
$$61$$ 10.9396 1.40067 0.700336 0.713814i $$-0.253034\pi$$
0.700336 + 0.713814i $$0.253034\pi$$
$$62$$ −6.06587 −0.770366
$$63$$ −15.3095 −1.92882
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −2.21069 −0.272118
$$67$$ −12.9330 −1.58002 −0.790012 0.613092i $$-0.789926\pi$$
−0.790012 + 0.613092i $$0.789926\pi$$
$$68$$ −0.563139 −0.0682907
$$69$$ −14.0593 −1.69254
$$70$$ 0 0
$$71$$ −4.39738 −0.521873 −0.260937 0.965356i $$-0.584031\pi$$
−0.260937 + 0.965356i $$0.584031\pi$$
$$72$$ −6.19869 −0.730523
$$73$$ 4.09334 0.479090 0.239545 0.970885i $$-0.423002\pi$$
0.239545 + 0.970885i $$0.423002\pi$$
$$74$$ −5.72890 −0.665971
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ −1.80022 −0.205155
$$78$$ −18.9001 −2.14002
$$79$$ 15.3370 1.72554 0.862772 0.505593i $$-0.168726\pi$$
0.862772 + 0.505593i $$0.168726\pi$$
$$80$$ 0 0
$$81$$ 10.8277 1.20308
$$82$$ −4.79476 −0.529493
$$83$$ −7.85517 −0.862217 −0.431109 0.902300i $$-0.641877\pi$$
−0.431109 + 0.902300i $$0.641877\pi$$
$$84$$ −7.49073 −0.817305
$$85$$ 0 0
$$86$$ 8.06587 0.869765
$$87$$ 31.0318 3.32696
$$88$$ −0.728896 −0.0777006
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −15.3908 −1.61340
$$92$$ −4.63555 −0.483290
$$93$$ 18.3974 1.90772
$$94$$ 8.12628 0.838162
$$95$$ 0 0
$$96$$ −3.03293 −0.309548
$$97$$ −11.0055 −1.11744 −0.558718 0.829358i $$-0.688706\pi$$
−0.558718 + 0.829358i $$0.688706\pi$$
$$98$$ 0.900112 0.0909250
$$99$$ 4.51820 0.454096
$$100$$ 0 0
$$101$$ −9.73436 −0.968605 −0.484302 0.874901i $$-0.660926\pi$$
−0.484302 + 0.874901i $$0.660926\pi$$
$$102$$ 1.70796 0.169114
$$103$$ 8.33151 0.820928 0.410464 0.911877i $$-0.365367\pi$$
0.410464 + 0.911877i $$0.365367\pi$$
$$104$$ −6.23163 −0.611061
$$105$$ 0 0
$$106$$ −1.53020 −0.148627
$$107$$ 15.0264 1.45266 0.726328 0.687348i $$-0.241225\pi$$
0.726328 + 0.687348i $$0.241225\pi$$
$$108$$ 9.70142 0.933520
$$109$$ −13.6619 −1.30858 −0.654288 0.756245i $$-0.727032\pi$$
−0.654288 + 0.756245i $$0.727032\pi$$
$$110$$ 0 0
$$111$$ 17.3754 1.64920
$$112$$ −2.46980 −0.233374
$$113$$ −1.20524 −0.113379 −0.0566895 0.998392i $$-0.518055\pi$$
−0.0566895 + 0.998392i $$0.518055\pi$$
$$114$$ 3.03293 0.284060
$$115$$ 0 0
$$116$$ 10.2316 0.949983
$$117$$ 38.6279 3.57115
$$118$$ 5.76183 0.530419
$$119$$ 1.39084 0.127498
$$120$$ 0 0
$$121$$ −10.4687 −0.951701
$$122$$ −10.9396 −0.990424
$$123$$ 14.5422 1.31123
$$124$$ 6.06587 0.544731
$$125$$ 0 0
$$126$$ 15.3095 1.36388
$$127$$ −9.52366 −0.845088 −0.422544 0.906342i $$-0.638863\pi$$
−0.422544 + 0.906342i $$0.638863\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −24.4633 −2.15387
$$130$$ 0 0
$$131$$ −17.4028 −1.52049 −0.760247 0.649635i $$-0.774922\pi$$
−0.760247 + 0.649635i $$0.774922\pi$$
$$132$$ 2.21069 0.192416
$$133$$ 2.46980 0.214158
$$134$$ 12.9330 1.11725
$$135$$ 0 0
$$136$$ 0.563139 0.0482888
$$137$$ −2.25910 −0.193008 −0.0965040 0.995333i $$-0.530766\pi$$
−0.0965040 + 0.995333i $$0.530766\pi$$
$$138$$ 14.0593 1.19681
$$139$$ 16.8606 1.43010 0.715050 0.699073i $$-0.246404\pi$$
0.715050 + 0.699073i $$0.246404\pi$$
$$140$$ 0 0
$$141$$ −24.6465 −2.07561
$$142$$ 4.39738 0.369020
$$143$$ 4.54221 0.379838
$$144$$ 6.19869 0.516558
$$145$$ 0 0
$$146$$ −4.09334 −0.338768
$$147$$ −2.72998 −0.225165
$$148$$ 5.72890 0.470912
$$149$$ −13.0055 −1.06545 −0.532724 0.846289i $$-0.678832\pi$$
−0.532724 + 0.846289i $$0.678832\pi$$
$$150$$ 0 0
$$151$$ 17.5895 1.43142 0.715708 0.698400i $$-0.246104\pi$$
0.715708 + 0.698400i $$0.246104\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ −3.49073 −0.282209
$$154$$ 1.80022 0.145066
$$155$$ 0 0
$$156$$ 18.9001 1.51322
$$157$$ −9.52366 −0.760071 −0.380035 0.924972i $$-0.624088\pi$$
−0.380035 + 0.924972i $$0.624088\pi$$
$$158$$ −15.3370 −1.22014
$$159$$ 4.64101 0.368056
$$160$$ 0 0
$$161$$ 11.4489 0.902297
$$162$$ −10.8277 −0.850704
$$163$$ −13.1921 −1.03329 −0.516644 0.856200i $$-0.672819\pi$$
−0.516644 + 0.856200i $$0.672819\pi$$
$$164$$ 4.79476 0.374408
$$165$$ 0 0
$$166$$ 7.85517 0.609680
$$167$$ −1.81331 −0.140318 −0.0701591 0.997536i $$-0.522351\pi$$
−0.0701591 + 0.997536i $$0.522351\pi$$
$$168$$ 7.49073 0.577922
$$169$$ 25.8332 1.98717
$$170$$ 0 0
$$171$$ −6.19869 −0.474026
$$172$$ −8.06587 −0.615017
$$173$$ 12.5237 0.952156 0.476078 0.879403i $$-0.342058\pi$$
0.476078 + 0.879403i $$0.342058\pi$$
$$174$$ −31.0318 −2.35252
$$175$$ 0 0
$$176$$ 0.728896 0.0549426
$$177$$ −17.4753 −1.31352
$$178$$ 10.0000 0.749532
$$179$$ 1.39738 0.104445 0.0522226 0.998635i $$-0.483369\pi$$
0.0522226 + 0.998635i $$0.483369\pi$$
$$180$$ 0 0
$$181$$ 5.72890 0.425825 0.212913 0.977071i $$-0.431705\pi$$
0.212913 + 0.977071i $$0.431705\pi$$
$$182$$ 15.3908 1.14084
$$183$$ 33.1791 2.45267
$$184$$ 4.63555 0.341737
$$185$$ 0 0
$$186$$ −18.3974 −1.34896
$$187$$ −0.410470 −0.0300165
$$188$$ −8.12628 −0.592670
$$189$$ −23.9605 −1.74287
$$190$$ 0 0
$$191$$ −27.0198 −1.95509 −0.977544 0.210733i $$-0.932415\pi$$
−0.977544 + 0.210733i $$0.932415\pi$$
$$192$$ 3.03293 0.218883
$$193$$ 23.0713 1.66071 0.830355 0.557234i $$-0.188138\pi$$
0.830355 + 0.557234i $$0.188138\pi$$
$$194$$ 11.0055 0.790146
$$195$$ 0 0
$$196$$ −0.900112 −0.0642937
$$197$$ −0.794765 −0.0566247 −0.0283123 0.999599i $$-0.509013\pi$$
−0.0283123 + 0.999599i $$0.509013\pi$$
$$198$$ −4.51820 −0.321095
$$199$$ 8.07241 0.572238 0.286119 0.958194i $$-0.407635\pi$$
0.286119 + 0.958194i $$0.407635\pi$$
$$200$$ 0 0
$$201$$ −39.2251 −2.76672
$$202$$ 9.73436 0.684907
$$203$$ −25.2700 −1.77361
$$204$$ −1.70796 −0.119581
$$205$$ 0 0
$$206$$ −8.33151 −0.580484
$$207$$ −28.7344 −1.99718
$$208$$ 6.23163 0.432085
$$209$$ −0.728896 −0.0504188
$$210$$ 0 0
$$211$$ −11.6410 −0.801400 −0.400700 0.916209i $$-0.631233\pi$$
−0.400700 + 0.916209i $$0.631233\pi$$
$$212$$ 1.53020 0.105095
$$213$$ −13.3370 −0.913834
$$214$$ −15.0264 −1.02718
$$215$$ 0 0
$$216$$ −9.70142 −0.660098
$$217$$ −14.9815 −1.01701
$$218$$ 13.6619 0.925304
$$219$$ 12.4148 0.838917
$$220$$ 0 0
$$221$$ −3.50927 −0.236059
$$222$$ −17.3754 −1.16616
$$223$$ 15.6554 1.04836 0.524182 0.851607i $$-0.324371\pi$$
0.524182 + 0.851607i $$0.324371\pi$$
$$224$$ 2.46980 0.165020
$$225$$ 0 0
$$226$$ 1.20524 0.0801710
$$227$$ −4.80131 −0.318674 −0.159337 0.987224i $$-0.550936\pi$$
−0.159337 + 0.987224i $$0.550936\pi$$
$$228$$ −3.03293 −0.200861
$$229$$ 2.79476 0.184683 0.0923416 0.995727i $$-0.470565\pi$$
0.0923416 + 0.995727i $$0.470565\pi$$
$$230$$ 0 0
$$231$$ −5.45996 −0.359239
$$232$$ −10.2316 −0.671739
$$233$$ −11.5422 −0.756155 −0.378078 0.925774i $$-0.623415\pi$$
−0.378078 + 0.925774i $$0.623415\pi$$
$$234$$ −38.6279 −2.52519
$$235$$ 0 0
$$236$$ −5.76183 −0.375063
$$237$$ 46.5160 3.02154
$$238$$ −1.39084 −0.0901547
$$239$$ −26.2251 −1.69636 −0.848180 0.529708i $$-0.822301\pi$$
−0.848180 + 0.529708i $$0.822301\pi$$
$$240$$ 0 0
$$241$$ −12.0659 −0.777231 −0.388615 0.921400i $$-0.627047\pi$$
−0.388615 + 0.921400i $$0.627047\pi$$
$$242$$ 10.4687 0.672954
$$243$$ 3.73544 0.239629
$$244$$ 10.9396 0.700336
$$245$$ 0 0
$$246$$ −14.5422 −0.927177
$$247$$ −6.23163 −0.396509
$$248$$ −6.06587 −0.385183
$$249$$ −23.8242 −1.50980
$$250$$ 0 0
$$251$$ 13.5237 0.853606 0.426803 0.904345i $$-0.359640\pi$$
0.426803 + 0.904345i $$0.359640\pi$$
$$252$$ −15.3095 −0.964408
$$253$$ −3.37884 −0.212426
$$254$$ 9.52366 0.597568
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −22.7398 −1.41847 −0.709235 0.704972i $$-0.750960\pi$$
−0.709235 + 0.704972i $$0.750960\pi$$
$$258$$ 24.4633 1.52302
$$259$$ −14.1492 −0.879188
$$260$$ 0 0
$$261$$ 63.4227 3.92577
$$262$$ 17.4028 1.07515
$$263$$ 6.67395 0.411533 0.205767 0.978601i $$-0.434031\pi$$
0.205767 + 0.978601i $$0.434031\pi$$
$$264$$ −2.21069 −0.136059
$$265$$ 0 0
$$266$$ −2.46980 −0.151433
$$267$$ −30.3293 −1.85613
$$268$$ −12.9330 −0.790012
$$269$$ −29.7398 −1.81327 −0.906634 0.421917i $$-0.861357\pi$$
−0.906634 + 0.421917i $$0.861357\pi$$
$$270$$ 0 0
$$271$$ 1.11189 0.0675426 0.0337713 0.999430i $$-0.489248\pi$$
0.0337713 + 0.999430i $$0.489248\pi$$
$$272$$ −0.563139 −0.0341453
$$273$$ −46.6794 −2.82517
$$274$$ 2.25910 0.136477
$$275$$ 0 0
$$276$$ −14.0593 −0.846272
$$277$$ −13.1263 −0.788682 −0.394341 0.918964i $$-0.629027\pi$$
−0.394341 + 0.918964i $$0.629027\pi$$
$$278$$ −16.8606 −1.01123
$$279$$ 37.6004 2.25108
$$280$$ 0 0
$$281$$ 22.9265 1.36768 0.683840 0.729632i $$-0.260309\pi$$
0.683840 + 0.729632i $$0.260309\pi$$
$$282$$ 24.6465 1.46768
$$283$$ −0.860634 −0.0511594 −0.0255797 0.999673i $$-0.508143\pi$$
−0.0255797 + 0.999673i $$0.508143\pi$$
$$284$$ −4.39738 −0.260937
$$285$$ 0 0
$$286$$ −4.54221 −0.268586
$$287$$ −11.8421 −0.699016
$$288$$ −6.19869 −0.365261
$$289$$ −16.6829 −0.981346
$$290$$ 0 0
$$291$$ −33.3788 −1.95670
$$292$$ 4.09334 0.239545
$$293$$ −19.8212 −1.15796 −0.578982 0.815340i $$-0.696550\pi$$
−0.578982 + 0.815340i $$0.696550\pi$$
$$294$$ 2.72998 0.159216
$$295$$ 0 0
$$296$$ −5.72890 −0.332985
$$297$$ 7.07133 0.410320
$$298$$ 13.0055 0.753386
$$299$$ −28.8870 −1.67058
$$300$$ 0 0
$$301$$ 19.9210 1.14823
$$302$$ −17.5895 −1.01216
$$303$$ −29.5237 −1.69609
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 3.49073 0.199552
$$307$$ 16.8661 0.962599 0.481299 0.876556i $$-0.340165\pi$$
0.481299 + 0.876556i $$0.340165\pi$$
$$308$$ −1.80022 −0.102577
$$309$$ 25.2689 1.43750
$$310$$ 0 0
$$311$$ 10.3250 0.585475 0.292738 0.956193i $$-0.405434\pi$$
0.292738 + 0.956193i $$0.405434\pi$$
$$312$$ −18.9001 −1.07001
$$313$$ 15.7684 0.891281 0.445641 0.895212i $$-0.352976\pi$$
0.445641 + 0.895212i $$0.352976\pi$$
$$314$$ 9.52366 0.537451
$$315$$ 0 0
$$316$$ 15.3370 0.862772
$$317$$ 2.17230 0.122009 0.0610043 0.998138i $$-0.480570\pi$$
0.0610043 + 0.998138i $$0.480570\pi$$
$$318$$ −4.64101 −0.260255
$$319$$ 7.45779 0.417556
$$320$$ 0 0
$$321$$ 45.5741 2.54370
$$322$$ −11.4489 −0.638020
$$323$$ 0.563139 0.0313339
$$324$$ 10.8277 0.601539
$$325$$ 0 0
$$326$$ 13.1921 0.730645
$$327$$ −41.4358 −2.29140
$$328$$ −4.79476 −0.264747
$$329$$ 20.0702 1.10651
$$330$$ 0 0
$$331$$ 11.9791 0.658429 0.329215 0.944255i $$-0.393216\pi$$
0.329215 + 0.944255i $$0.393216\pi$$
$$332$$ −7.85517 −0.431109
$$333$$ 35.5117 1.94603
$$334$$ 1.81331 0.0992200
$$335$$ 0 0
$$336$$ −7.49073 −0.408653
$$337$$ 11.1921 0.609675 0.304838 0.952404i $$-0.401398\pi$$
0.304838 + 0.952404i $$0.401398\pi$$
$$338$$ −25.8332 −1.40514
$$339$$ −3.65540 −0.198534
$$340$$ 0 0
$$341$$ 4.42139 0.239432
$$342$$ 6.19869 0.335187
$$343$$ 19.5117 1.05353
$$344$$ 8.06587 0.434883
$$345$$ 0 0
$$346$$ −12.5237 −0.673276
$$347$$ 20.3843 1.09429 0.547143 0.837039i $$-0.315715\pi$$
0.547143 + 0.837039i $$0.315715\pi$$
$$348$$ 31.0318 1.66348
$$349$$ −0.252557 −0.0135191 −0.00675954 0.999977i $$-0.502152\pi$$
−0.00675954 + 0.999977i $$0.502152\pi$$
$$350$$ 0 0
$$351$$ 60.4556 3.22688
$$352$$ −0.728896 −0.0388503
$$353$$ 28.6434 1.52453 0.762267 0.647263i $$-0.224086\pi$$
0.762267 + 0.647263i $$0.224086\pi$$
$$354$$ 17.4753 0.928799
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 4.21832 0.223257
$$358$$ −1.39738 −0.0738540
$$359$$ −18.5741 −0.980301 −0.490151 0.871638i $$-0.663058\pi$$
−0.490151 + 0.871638i $$0.663058\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −5.72890 −0.301104
$$363$$ −31.7509 −1.66649
$$364$$ −15.3908 −0.806699
$$365$$ 0 0
$$366$$ −33.1791 −1.73430
$$367$$ 4.79476 0.250285 0.125142 0.992139i $$-0.460061\pi$$
0.125142 + 0.992139i $$0.460061\pi$$
$$368$$ −4.63555 −0.241645
$$369$$ 29.7213 1.54723
$$370$$ 0 0
$$371$$ −3.77929 −0.196211
$$372$$ 18.3974 0.953860
$$373$$ −15.9485 −0.825783 −0.412892 0.910780i $$-0.635481\pi$$
−0.412892 + 0.910780i $$0.635481\pi$$
$$374$$ 0.410470 0.0212249
$$375$$ 0 0
$$376$$ 8.12628 0.419081
$$377$$ 63.7597 3.28379
$$378$$ 23.9605 1.23240
$$379$$ −16.8013 −0.863025 −0.431513 0.902107i $$-0.642020\pi$$
−0.431513 + 0.902107i $$0.642020\pi$$
$$380$$ 0 0
$$381$$ −28.8846 −1.47980
$$382$$ 27.0198 1.38246
$$383$$ −26.7948 −1.36915 −0.684574 0.728943i $$-0.740012\pi$$
−0.684574 + 0.728943i $$0.740012\pi$$
$$384$$ −3.03293 −0.154774
$$385$$ 0 0
$$386$$ −23.0713 −1.17430
$$387$$ −49.9978 −2.54153
$$388$$ −11.0055 −0.558718
$$389$$ 18.3424 0.929998 0.464999 0.885311i $$-0.346055\pi$$
0.464999 + 0.885311i $$0.346055\pi$$
$$390$$ 0 0
$$391$$ 2.61046 0.132017
$$392$$ 0.900112 0.0454625
$$393$$ −52.7817 −2.66248
$$394$$ 0.794765 0.0400397
$$395$$ 0 0
$$396$$ 4.51820 0.227048
$$397$$ −21.2162 −1.06481 −0.532404 0.846490i $$-0.678711\pi$$
−0.532404 + 0.846490i $$0.678711\pi$$
$$398$$ −8.07241 −0.404633
$$399$$ 7.49073 0.375005
$$400$$ 0 0
$$401$$ −27.8661 −1.39157 −0.695783 0.718252i $$-0.744943\pi$$
−0.695783 + 0.718252i $$0.744943\pi$$
$$402$$ 39.2251 1.95637
$$403$$ 37.8002 1.88296
$$404$$ −9.73436 −0.484302
$$405$$ 0 0
$$406$$ 25.2700 1.25413
$$407$$ 4.17577 0.206985
$$408$$ 1.70796 0.0845568
$$409$$ −18.0899 −0.894487 −0.447243 0.894412i $$-0.647594\pi$$
−0.447243 + 0.894412i $$0.647594\pi$$
$$410$$ 0 0
$$411$$ −6.85171 −0.337970
$$412$$ 8.33151 0.410464
$$413$$ 14.2305 0.700239
$$414$$ 28.7344 1.41222
$$415$$ 0 0
$$416$$ −6.23163 −0.305531
$$417$$ 51.1372 2.50420
$$418$$ 0.728896 0.0356515
$$419$$ −15.3370 −0.749260 −0.374630 0.927174i $$-0.622230\pi$$
−0.374630 + 0.927174i $$0.622230\pi$$
$$420$$ 0 0
$$421$$ −8.42486 −0.410602 −0.205301 0.978699i $$-0.565817\pi$$
−0.205301 + 0.978699i $$0.565817\pi$$
$$422$$ 11.6410 0.566676
$$423$$ −50.3723 −2.44918
$$424$$ −1.53020 −0.0743133
$$425$$ 0 0
$$426$$ 13.3370 0.646178
$$427$$ −27.0185 −1.30752
$$428$$ 15.0264 0.726328
$$429$$ 13.7762 0.665122
$$430$$ 0 0
$$431$$ 16.2766 0.784014 0.392007 0.919962i $$-0.371781\pi$$
0.392007 + 0.919962i $$0.371781\pi$$
$$432$$ 9.70142 0.466760
$$433$$ 18.1976 0.874521 0.437261 0.899335i $$-0.355949\pi$$
0.437261 + 0.899335i $$0.355949\pi$$
$$434$$ 14.9815 0.719133
$$435$$ 0 0
$$436$$ −13.6619 −0.654288
$$437$$ 4.63555 0.221749
$$438$$ −12.4148 −0.593204
$$439$$ 20.4214 0.974660 0.487330 0.873218i $$-0.337971\pi$$
0.487330 + 0.873218i $$0.337971\pi$$
$$440$$ 0 0
$$441$$ −5.57952 −0.265691
$$442$$ 3.50927 0.166919
$$443$$ −21.6685 −1.02950 −0.514750 0.857340i $$-0.672115\pi$$
−0.514750 + 0.857340i $$0.672115\pi$$
$$444$$ 17.3754 0.824598
$$445$$ 0 0
$$446$$ −15.6554 −0.741305
$$447$$ −39.4447 −1.86567
$$448$$ −2.46980 −0.116687
$$449$$ 23.8133 1.12382 0.561910 0.827198i $$-0.310067\pi$$
0.561910 + 0.827198i $$0.310067\pi$$
$$450$$ 0 0
$$451$$ 3.49489 0.164568
$$452$$ −1.20524 −0.0566895
$$453$$ 53.3479 2.50650
$$454$$ 4.80131 0.225337
$$455$$ 0 0
$$456$$ 3.03293 0.142030
$$457$$ −9.15813 −0.428399 −0.214200 0.976790i $$-0.568714\pi$$
−0.214200 + 0.976790i $$0.568714\pi$$
$$458$$ −2.79476 −0.130591
$$459$$ −5.46325 −0.255003
$$460$$ 0 0
$$461$$ −7.78931 −0.362784 −0.181392 0.983411i $$-0.558060\pi$$
−0.181392 + 0.983411i $$0.558060\pi$$
$$462$$ 5.45996 0.254020
$$463$$ −0.405011 −0.0188225 −0.00941123 0.999956i $$-0.502996\pi$$
−0.00941123 + 0.999956i $$0.502996\pi$$
$$464$$ 10.2316 0.474991
$$465$$ 0 0
$$466$$ 11.5422 0.534682
$$467$$ 1.95814 0.0906118 0.0453059 0.998973i $$-0.485574\pi$$
0.0453059 + 0.998973i $$0.485574\pi$$
$$468$$ 38.6279 1.78558
$$469$$ 31.9420 1.47494
$$470$$ 0 0
$$471$$ −28.8846 −1.33093
$$472$$ 5.76183 0.265210
$$473$$ −5.87918 −0.270325
$$474$$ −46.5160 −2.13655
$$475$$ 0 0
$$476$$ 1.39084 0.0637490
$$477$$ 9.48527 0.434301
$$478$$ 26.2251 1.19951
$$479$$ −22.9210 −1.04729 −0.523645 0.851937i $$-0.675428\pi$$
−0.523645 + 0.851937i $$0.675428\pi$$
$$480$$ 0 0
$$481$$ 35.7003 1.62780
$$482$$ 12.0659 0.549585
$$483$$ 34.7237 1.57998
$$484$$ −10.4687 −0.475850
$$485$$ 0 0
$$486$$ −3.73544 −0.169443
$$487$$ 23.9450 1.08505 0.542527 0.840038i $$-0.317468\pi$$
0.542527 + 0.840038i $$0.317468\pi$$
$$488$$ −10.9396 −0.495212
$$489$$ −40.0109 −1.80936
$$490$$ 0 0
$$491$$ 3.86826 0.174572 0.0872861 0.996183i $$-0.472181\pi$$
0.0872861 + 0.996183i $$0.472181\pi$$
$$492$$ 14.5422 0.655613
$$493$$ −5.76183 −0.259500
$$494$$ 6.23163 0.280374
$$495$$ 0 0
$$496$$ 6.06587 0.272366
$$497$$ 10.8606 0.487166
$$498$$ 23.8242 1.06759
$$499$$ −7.92104 −0.354595 −0.177297 0.984157i $$-0.556735\pi$$
−0.177297 + 0.984157i $$0.556735\pi$$
$$500$$ 0 0
$$501$$ −5.49966 −0.245706
$$502$$ −13.5237 −0.603591
$$503$$ 16.6949 0.744388 0.372194 0.928155i $$-0.378606\pi$$
0.372194 + 0.928155i $$0.378606\pi$$
$$504$$ 15.3095 0.681939
$$505$$ 0 0
$$506$$ 3.37884 0.150208
$$507$$ 78.3503 3.47966
$$508$$ −9.52366 −0.422544
$$509$$ 24.4028 1.08164 0.540818 0.841139i $$-0.318115\pi$$
0.540818 + 0.841139i $$0.318115\pi$$
$$510$$ 0 0
$$511$$ −10.1097 −0.447228
$$512$$ −1.00000 −0.0441942
$$513$$ −9.70142 −0.428328
$$514$$ 22.7398 1.00301
$$515$$ 0 0
$$516$$ −24.4633 −1.07693
$$517$$ −5.92321 −0.260503
$$518$$ 14.1492 0.621680
$$519$$ 37.9834 1.66729
$$520$$ 0 0
$$521$$ 26.0659 1.14197 0.570983 0.820962i $$-0.306562\pi$$
0.570983 + 0.820962i $$0.306562\pi$$
$$522$$ −63.4227 −2.77594
$$523$$ −17.8726 −0.781516 −0.390758 0.920493i $$-0.627787\pi$$
−0.390758 + 0.920493i $$0.627787\pi$$
$$524$$ −17.4028 −0.760247
$$525$$ 0 0
$$526$$ −6.67395 −0.290998
$$527$$ −3.41593 −0.148800
$$528$$ 2.21069 0.0962081
$$529$$ −1.51166 −0.0657243
$$530$$ 0 0
$$531$$ −35.7158 −1.54993
$$532$$ 2.46980 0.107079
$$533$$ 29.8792 1.29421
$$534$$ 30.3293 1.31248
$$535$$ 0 0
$$536$$ 12.9330 0.558623
$$537$$ 4.23817 0.182891
$$538$$ 29.7398 1.28217
$$539$$ −0.656088 −0.0282597
$$540$$ 0 0
$$541$$ 23.0604 0.991444 0.495722 0.868481i $$-0.334903\pi$$
0.495722 + 0.868481i $$0.334903\pi$$
$$542$$ −1.11189 −0.0477598
$$543$$ 17.3754 0.745648
$$544$$ 0.563139 0.0241444
$$545$$ 0 0
$$546$$ 46.6794 1.99769
$$547$$ 27.7584 1.18686 0.593431 0.804885i $$-0.297773\pi$$
0.593431 + 0.804885i $$0.297773\pi$$
$$548$$ −2.25910 −0.0965040
$$549$$ 67.8111 2.89411
$$550$$ 0 0
$$551$$ −10.2316 −0.435882
$$552$$ 14.0593 0.598405
$$553$$ −37.8792 −1.61079
$$554$$ 13.1263 0.557682
$$555$$ 0 0
$$556$$ 16.8606 0.715050
$$557$$ 8.60808 0.364736 0.182368 0.983230i $$-0.441624\pi$$
0.182368 + 0.983230i $$0.441624\pi$$
$$558$$ −37.6004 −1.59175
$$559$$ −50.2635 −2.12592
$$560$$ 0 0
$$561$$ −1.24493 −0.0525609
$$562$$ −22.9265 −0.967096
$$563$$ −7.93959 −0.334614 −0.167307 0.985905i $$-0.553507\pi$$
−0.167307 + 0.985905i $$0.553507\pi$$
$$564$$ −24.6465 −1.03780
$$565$$ 0 0
$$566$$ 0.860634 0.0361751
$$567$$ −26.7422 −1.12307
$$568$$ 4.39738 0.184510
$$569$$ −1.65757 −0.0694889 −0.0347444 0.999396i $$-0.511062\pi$$
−0.0347444 + 0.999396i $$0.511062\pi$$
$$570$$ 0 0
$$571$$ −14.0528 −0.588091 −0.294045 0.955791i $$-0.595002\pi$$
−0.294045 + 0.955791i $$0.595002\pi$$
$$572$$ 4.54221 0.189919
$$573$$ −81.9494 −3.42349
$$574$$ 11.8421 0.494279
$$575$$ 0 0
$$576$$ 6.19869 0.258279
$$577$$ 1.36445 0.0568027 0.0284014 0.999597i $$-0.490958\pi$$
0.0284014 + 0.999597i $$0.490958\pi$$
$$578$$ 16.6829 0.693916
$$579$$ 69.9738 2.90801
$$580$$ 0 0
$$581$$ 19.4007 0.804876
$$582$$ 33.3788 1.38360
$$583$$ 1.11536 0.0461935
$$584$$ −4.09334 −0.169384
$$585$$ 0 0
$$586$$ 19.8212 0.818804
$$587$$ 24.6609 1.01786 0.508931 0.860807i $$-0.330041\pi$$
0.508931 + 0.860807i $$0.330041\pi$$
$$588$$ −2.72998 −0.112583
$$589$$ −6.06587 −0.249940
$$590$$ 0 0
$$591$$ −2.41047 −0.0991535
$$592$$ 5.72890 0.235456
$$593$$ 0.747443 0.0306938 0.0153469 0.999882i $$-0.495115\pi$$
0.0153469 + 0.999882i $$0.495115\pi$$
$$594$$ −7.07133 −0.290140
$$595$$ 0 0
$$596$$ −13.0055 −0.532724
$$597$$ 24.4831 1.00203
$$598$$ 28.8870 1.18128
$$599$$ 1.40501 0.0574072 0.0287036 0.999588i $$-0.490862\pi$$
0.0287036 + 0.999588i $$0.490862\pi$$
$$600$$ 0 0
$$601$$ 29.7453 1.21334 0.606668 0.794956i $$-0.292506\pi$$
0.606668 + 0.794956i $$0.292506\pi$$
$$602$$ −19.9210 −0.811921
$$603$$ −80.1680 −3.26469
$$604$$ 17.5895 0.715708
$$605$$ 0 0
$$606$$ 29.5237 1.19932
$$607$$ −5.66849 −0.230077 −0.115038 0.993361i $$-0.536699\pi$$
−0.115038 + 0.993361i $$0.536699\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ −76.6423 −3.10570
$$610$$ 0 0
$$611$$ −50.6399 −2.04867
$$612$$ −3.49073 −0.141104
$$613$$ 2.99454 0.120948 0.0604742 0.998170i $$-0.480739\pi$$
0.0604742 + 0.998170i $$0.480739\pi$$
$$614$$ −16.8661 −0.680660
$$615$$ 0 0
$$616$$ 1.80022 0.0725331
$$617$$ 32.3370 1.30184 0.650919 0.759147i $$-0.274384\pi$$
0.650919 + 0.759147i $$0.274384\pi$$
$$618$$ −25.2689 −1.01647
$$619$$ −20.9265 −0.841107 −0.420554 0.907268i $$-0.638164\pi$$
−0.420554 + 0.907268i $$0.638164\pi$$
$$620$$ 0 0
$$621$$ −44.9714 −1.80464
$$622$$ −10.3250 −0.413994
$$623$$ 24.6980 0.989503
$$624$$ 18.9001 0.756610
$$625$$ 0 0
$$626$$ −15.7684 −0.630231
$$627$$ −2.21069 −0.0882866
$$628$$ −9.52366 −0.380035
$$629$$ −3.22617 −0.128636
$$630$$ 0 0
$$631$$ 22.8002 0.907663 0.453831 0.891088i $$-0.350057\pi$$
0.453831 + 0.891088i $$0.350057\pi$$
$$632$$ −15.3370 −0.610072
$$633$$ −35.3064 −1.40330
$$634$$ −2.17230 −0.0862731
$$635$$ 0 0
$$636$$ 4.64101 0.184028
$$637$$ −5.60916 −0.222243
$$638$$ −7.45779 −0.295257
$$639$$ −27.2580 −1.07831
$$640$$ 0 0
$$641$$ 36.3184 1.43449 0.717246 0.696820i $$-0.245402\pi$$
0.717246 + 0.696820i $$0.245402\pi$$
$$642$$ −45.5741 −1.79866
$$643$$ −13.6135 −0.536865 −0.268433 0.963298i $$-0.586506\pi$$
−0.268433 + 0.963298i $$0.586506\pi$$
$$644$$ 11.4489 0.451148
$$645$$ 0 0
$$646$$ −0.563139 −0.0221564
$$647$$ 0.0724126 0.00284683 0.00142342 0.999999i $$-0.499547\pi$$
0.00142342 + 0.999999i $$0.499547\pi$$
$$648$$ −10.8277 −0.425352
$$649$$ −4.19978 −0.164856
$$650$$ 0 0
$$651$$ −45.4378 −1.78085
$$652$$ −13.1921 −0.516644
$$653$$ 43.5346 1.70364 0.851820 0.523835i $$-0.175499\pi$$
0.851820 + 0.523835i $$0.175499\pi$$
$$654$$ 41.4358 1.62027
$$655$$ 0 0
$$656$$ 4.79476 0.187204
$$657$$ 25.3734 0.989910
$$658$$ −20.0702 −0.782420
$$659$$ −33.4512 −1.30308 −0.651538 0.758616i $$-0.725876\pi$$
−0.651538 + 0.758616i $$0.725876\pi$$
$$660$$ 0 0
$$661$$ −2.89465 −0.112589 −0.0562945 0.998414i $$-0.517929\pi$$
−0.0562945 + 0.998414i $$0.517929\pi$$
$$662$$ −11.9791 −0.465580
$$663$$ −10.6434 −0.413355
$$664$$ 7.85517 0.304840
$$665$$ 0 0
$$666$$ −35.5117 −1.37605
$$667$$ −47.4292 −1.83647
$$668$$ −1.81331 −0.0701591
$$669$$ 47.4818 1.83575
$$670$$ 0 0
$$671$$ 7.97382 0.307826
$$672$$ 7.49073 0.288961
$$673$$ −42.8475 −1.65165 −0.825826 0.563925i $$-0.809291\pi$$
−0.825826 + 0.563925i $$0.809291\pi$$
$$674$$ −11.1921 −0.431105
$$675$$ 0 0
$$676$$ 25.8332 0.993583
$$677$$ −34.4567 −1.32428 −0.662139 0.749381i $$-0.730351\pi$$
−0.662139 + 0.749381i $$0.730351\pi$$
$$678$$ 3.65540 0.140385
$$679$$ 27.1812 1.04312
$$680$$ 0 0
$$681$$ −14.5621 −0.558019
$$682$$ −4.42139 −0.169304
$$683$$ 2.73436 0.104627 0.0523136 0.998631i $$-0.483340\pi$$
0.0523136 + 0.998631i $$0.483340\pi$$
$$684$$ −6.19869 −0.237013
$$685$$ 0 0
$$686$$ −19.5117 −0.744959
$$687$$ 8.47634 0.323393
$$688$$ −8.06587 −0.307508
$$689$$ 9.53566 0.363280
$$690$$ 0 0
$$691$$ −4.74198 −0.180394 −0.0901968 0.995924i $$-0.528750\pi$$
−0.0901968 + 0.995924i $$0.528750\pi$$
$$692$$ 12.5237 0.476078
$$693$$ −11.1590 −0.423897
$$694$$ −20.3843 −0.773777
$$695$$ 0 0
$$696$$ −31.0318 −1.17626
$$697$$ −2.70012 −0.102274
$$698$$ 0.252557 0.00955943
$$699$$ −35.0068 −1.32408
$$700$$ 0 0
$$701$$ 33.1372 1.25157 0.625787 0.779994i $$-0.284778\pi$$
0.625787 + 0.779994i $$0.284778\pi$$
$$702$$ −60.4556 −2.28175
$$703$$ −5.72890 −0.216069
$$704$$ 0.728896 0.0274713
$$705$$ 0 0
$$706$$ −28.6434 −1.07801
$$707$$ 24.0419 0.904187
$$708$$ −17.4753 −0.656760
$$709$$ 5.37884 0.202006 0.101003 0.994886i $$-0.467795\pi$$
0.101003 + 0.994886i $$0.467795\pi$$
$$710$$ 0 0
$$711$$ 95.0692 3.56537
$$712$$ 10.0000 0.374766
$$713$$ −28.1187 −1.05305
$$714$$ −4.21832 −0.157867
$$715$$ 0 0
$$716$$ 1.39738 0.0522226
$$717$$ −79.5390 −2.97044
$$718$$ 18.5741 0.693178
$$719$$ 33.0857 1.23389 0.616944 0.787007i $$-0.288370\pi$$
0.616944 + 0.787007i $$0.288370\pi$$
$$720$$ 0 0
$$721$$ −20.5771 −0.766332
$$722$$ −1.00000 −0.0372161
$$723$$ −36.5950 −1.36098
$$724$$ 5.72890 0.212913
$$725$$ 0 0
$$726$$ 31.7509 1.17839
$$727$$ −2.62463 −0.0973423 −0.0486711 0.998815i $$-0.515499\pi$$
−0.0486711 + 0.998815i $$0.515499\pi$$
$$728$$ 15.3908 0.570422
$$729$$ −21.1538 −0.783472
$$730$$ 0 0
$$731$$ 4.54221 0.168000
$$732$$ 33.1791 1.22633
$$733$$ −0.608077 −0.0224598 −0.0112299 0.999937i $$-0.503575\pi$$
−0.0112299 + 0.999937i $$0.503575\pi$$
$$734$$ −4.79476 −0.176978
$$735$$ 0 0
$$736$$ 4.63555 0.170869
$$737$$ −9.42685 −0.347242
$$738$$ −29.7213 −1.09405
$$739$$ 36.3974 1.33890 0.669450 0.742857i $$-0.266530\pi$$
0.669450 + 0.742857i $$0.266530\pi$$
$$740$$ 0 0
$$741$$ −18.9001 −0.694313
$$742$$ 3.77929 0.138742
$$743$$ 23.0713 0.846405 0.423202 0.906035i $$-0.360906\pi$$
0.423202 + 0.906035i $$0.360906\pi$$
$$744$$ −18.3974 −0.674481
$$745$$ 0 0
$$746$$ 15.9485 0.583917
$$747$$ −48.6918 −1.78154
$$748$$ −0.410470 −0.0150083
$$749$$ −37.1121 −1.35605
$$750$$ 0 0
$$751$$ 4.75290 0.173436 0.0867179 0.996233i $$-0.472362\pi$$
0.0867179 + 0.996233i $$0.472362\pi$$
$$752$$ −8.12628 −0.296335
$$753$$ 41.0164 1.49472
$$754$$ −63.7597 −2.32199
$$755$$ 0 0
$$756$$ −23.9605 −0.871436
$$757$$ −26.8057 −0.974269 −0.487135 0.873327i $$-0.661958\pi$$
−0.487135 + 0.873327i $$0.661958\pi$$
$$758$$ 16.8013 0.610251
$$759$$ −10.2478 −0.371971
$$760$$ 0 0
$$761$$ 25.4478 0.922481 0.461241 0.887275i $$-0.347404\pi$$
0.461241 + 0.887275i $$0.347404\pi$$
$$762$$ 28.8846 1.04638
$$763$$ 33.7422 1.22155
$$764$$ −27.0198 −0.977544
$$765$$ 0 0
$$766$$ 26.7948 0.968134
$$767$$ −35.9056 −1.29648
$$768$$ 3.03293 0.109442
$$769$$ −14.6421 −0.528007 −0.264004 0.964522i $$-0.585043\pi$$
−0.264004 + 0.964522i $$0.585043\pi$$
$$770$$ 0 0
$$771$$ −68.9684 −2.48384
$$772$$ 23.0713 0.830355
$$773$$ 16.8462 0.605917 0.302959 0.953004i $$-0.402026\pi$$
0.302959 + 0.953004i $$0.402026\pi$$
$$774$$ 49.9978 1.79713
$$775$$ 0 0
$$776$$ 11.0055 0.395073
$$777$$ −42.9136 −1.53952
$$778$$ −18.3424 −0.657608
$$779$$ −4.79476 −0.171790
$$780$$ 0 0
$$781$$ −3.20524 −0.114692
$$782$$ −2.61046 −0.0933499
$$783$$ 99.2613 3.54731
$$784$$ −0.900112 −0.0321469
$$785$$ 0 0
$$786$$ 52.7817 1.88266
$$787$$ −18.7367 −0.667893 −0.333946 0.942592i $$-0.608380\pi$$
−0.333946 + 0.942592i $$0.608380\pi$$
$$788$$ −0.794765 −0.0283123
$$789$$ 20.2416 0.720621
$$790$$ 0 0
$$791$$ 2.97668 0.105839
$$792$$ −4.51820 −0.160547
$$793$$ 68.1714 2.42084
$$794$$ 21.2162 0.752933
$$795$$ 0 0
$$796$$ 8.07241 0.286119
$$797$$ 37.9900 1.34567 0.672837 0.739791i $$-0.265075\pi$$
0.672837 + 0.739791i $$0.265075\pi$$
$$798$$ −7.49073 −0.265169
$$799$$ 4.57623 0.161895
$$800$$ 0 0
$$801$$ −61.9869 −2.19020
$$802$$ 27.8661 0.983986
$$803$$ 2.98362 0.105290
$$804$$ −39.2251 −1.38336
$$805$$ 0 0
$$806$$ −37.8002 −1.33146
$$807$$ −90.1989 −3.17515
$$808$$ 9.73436 0.342453
$$809$$ 21.8857 0.769461 0.384731 0.923029i $$-0.374294\pi$$
0.384731 + 0.923029i $$0.374294\pi$$
$$810$$ 0 0
$$811$$ 37.8595 1.32943 0.664714 0.747098i $$-0.268553\pi$$
0.664714 + 0.747098i $$0.268553\pi$$
$$812$$ −25.2700 −0.886804
$$813$$ 3.37229 0.118271
$$814$$ −4.17577 −0.146361
$$815$$ 0 0
$$816$$ −1.70796 −0.0597907
$$817$$ 8.06587 0.282189
$$818$$ 18.0899 0.632498
$$819$$ −95.4031 −3.33365
$$820$$ 0 0
$$821$$ −20.1976 −0.704901 −0.352451 0.935830i $$-0.614652\pi$$
−0.352451 + 0.935830i $$0.614652\pi$$
$$822$$ 6.85171 0.238981
$$823$$ 4.34590 0.151489 0.0757443 0.997127i $$-0.475867\pi$$
0.0757443 + 0.997127i $$0.475867\pi$$
$$824$$ −8.33151 −0.290242
$$825$$ 0 0
$$826$$ −14.2305 −0.495144
$$827$$ 56.8375 1.97643 0.988217 0.153057i $$-0.0489119\pi$$
0.988217 + 0.153057i $$0.0489119\pi$$
$$828$$ −28.7344 −0.998588
$$829$$ −17.4543 −0.606214 −0.303107 0.952957i $$-0.598024\pi$$
−0.303107 + 0.952957i $$0.598024\pi$$
$$830$$ 0 0
$$831$$ −39.8111 −1.38103
$$832$$ 6.23163 0.216043
$$833$$ 0.506888 0.0175626
$$834$$ −51.1372 −1.77074
$$835$$ 0 0
$$836$$ −0.728896 −0.0252094
$$837$$ 58.8475 2.03407
$$838$$ 15.3370 0.529807
$$839$$ 1.77622 0.0613219 0.0306609 0.999530i $$-0.490239\pi$$
0.0306609 + 0.999530i $$0.490239\pi$$
$$840$$ 0 0
$$841$$ 75.6862 2.60987
$$842$$ 8.42486 0.290340
$$843$$ 69.5346 2.39490
$$844$$ −11.6410 −0.400700
$$845$$ 0 0
$$846$$ 50.3723 1.73184
$$847$$ 25.8556 0.888408
$$848$$ 1.53020 0.0525475
$$849$$ −2.61025 −0.0895834
$$850$$ 0 0
$$851$$ −26.5566 −0.910348
$$852$$ −13.3370 −0.456917
$$853$$ 16.1187 0.551892 0.275946 0.961173i $$-0.411009\pi$$
0.275946 + 0.961173i $$0.411009\pi$$
$$854$$ 27.0185 0.924556
$$855$$ 0 0
$$856$$ −15.0264 −0.513591
$$857$$ 47.2031 1.61243 0.806213 0.591625i $$-0.201514\pi$$
0.806213 + 0.591625i $$0.201514\pi$$
$$858$$ −13.7762 −0.470312
$$859$$ 37.3239 1.27347 0.636737 0.771081i $$-0.280284\pi$$
0.636737 + 0.771081i $$0.280284\pi$$
$$860$$ 0 0
$$861$$ −35.9163 −1.22402
$$862$$ −16.2766 −0.554382
$$863$$ 1.33697 0.0455111 0.0227555 0.999741i $$-0.492756\pi$$
0.0227555 + 0.999741i $$0.492756\pi$$
$$864$$ −9.70142 −0.330049
$$865$$ 0 0
$$866$$ −18.1976 −0.618380
$$867$$ −50.5981 −1.71840
$$868$$ −14.9815 −0.508504
$$869$$ 11.1791 0.379224
$$870$$ 0 0
$$871$$ −80.5939 −2.73082
$$872$$ 13.6619 0.462652
$$873$$ −68.2194 −2.30888
$$874$$ −4.63555 −0.156800
$$875$$ 0 0
$$876$$ 12.4148 0.419459
$$877$$ −13.1647 −0.444539 −0.222270 0.974985i $$-0.571347\pi$$
−0.222270 + 0.974985i $$0.571347\pi$$
$$878$$ −20.4214 −0.689188
$$879$$ −60.1163 −2.02767
$$880$$ 0 0
$$881$$ −10.2052 −0.343823 −0.171912 0.985112i $$-0.554994\pi$$
−0.171912 + 0.985112i $$0.554994\pi$$
$$882$$ 5.57952 0.187872
$$883$$ −1.49966 −0.0504674 −0.0252337 0.999682i $$-0.508033\pi$$
−0.0252337 + 0.999682i $$0.508033\pi$$
$$884$$ −3.50927 −0.118030
$$885$$ 0 0
$$886$$ 21.6685 0.727967
$$887$$ 46.9505 1.57644 0.788222 0.615391i $$-0.211002\pi$$
0.788222 + 0.615391i $$0.211002\pi$$
$$888$$ −17.3754 −0.583079
$$889$$ 23.5215 0.788886
$$890$$ 0 0
$$891$$ 7.89227 0.264401
$$892$$ 15.6554 0.524182
$$893$$ 8.12628 0.271936
$$894$$ 39.4447 1.31923
$$895$$ 0 0
$$896$$ 2.46980 0.0825101
$$897$$ −87.6125 −2.92529
$$898$$ −23.8133 −0.794661
$$899$$ 62.0637 2.06994
$$900$$ 0 0
$$901$$ −0.861719 −0.0287080
$$902$$ −3.49489 −0.116367
$$903$$ 60.4192 2.01063
$$904$$ 1.20524 0.0400855
$$905$$ 0 0
$$906$$ −53.3479 −1.77236
$$907$$ −16.3668 −0.543452 −0.271726 0.962375i $$-0.587594\pi$$
−0.271726 + 0.962375i $$0.587594\pi$$
$$908$$ −4.80131 −0.159337
$$909$$ −60.3403 −2.00136
$$910$$ 0 0
$$911$$ 32.3293 1.07112 0.535559 0.844498i $$-0.320101\pi$$
0.535559 + 0.844498i $$0.320101\pi$$
$$912$$ −3.03293 −0.100430
$$913$$ −5.72561 −0.189490
$$914$$ 9.15813 0.302924
$$915$$ 0 0
$$916$$ 2.79476 0.0923416
$$917$$ 42.9815 1.41937
$$918$$ 5.46325 0.180314
$$919$$ −22.8157 −0.752620 −0.376310 0.926494i $$-0.622807\pi$$
−0.376310 + 0.926494i $$0.622807\pi$$
$$920$$ 0 0
$$921$$ 51.1538 1.68557
$$922$$ 7.78931 0.256527
$$923$$ −27.4028 −0.901976
$$924$$ −5.45996 −0.179620
$$925$$ 0 0
$$926$$ 0.405011 0.0133095
$$927$$ 51.6445 1.69623
$$928$$ −10.2316 −0.335870
$$929$$ −27.2436 −0.893834 −0.446917 0.894575i $$-0.647478\pi$$
−0.446917 + 0.894575i $$0.647478\pi$$
$$930$$ 0 0
$$931$$ 0.900112 0.0295000
$$932$$ −11.5422 −0.378078
$$933$$ 31.3150 1.02521
$$934$$ −1.95814 −0.0640722
$$935$$ 0 0
$$936$$ −38.6279 −1.26259
$$937$$ 51.5006 1.68245 0.841225 0.540685i $$-0.181835\pi$$
0.841225 + 0.540685i $$0.181835\pi$$
$$938$$ −31.9420 −1.04294
$$939$$ 47.8244 1.56069
$$940$$ 0 0
$$941$$ −40.8541 −1.33181 −0.665903 0.746039i $$-0.731953\pi$$
−0.665903 + 0.746039i $$0.731953\pi$$
$$942$$ 28.8846 0.941112
$$943$$ −22.2264 −0.723791
$$944$$ −5.76183 −0.187532
$$945$$ 0 0
$$946$$ 5.87918 0.191149
$$947$$ −38.2526 −1.24304 −0.621521 0.783398i $$-0.713485\pi$$
−0.621521 + 0.783398i $$0.713485\pi$$
$$948$$ 46.5160 1.51077
$$949$$ 25.5082 0.828031
$$950$$ 0 0
$$951$$ 6.58845 0.213645
$$952$$ −1.39084 −0.0450773
$$953$$ −54.1187 −1.75308 −0.876538 0.481334i $$-0.840153\pi$$
−0.876538 + 0.481334i $$0.840153\pi$$
$$954$$ −9.48527 −0.307097
$$955$$ 0 0
$$956$$ −26.2251 −0.848180
$$957$$ 22.6190 0.731168
$$958$$ 22.9210 0.740545
$$959$$ 5.57952 0.180172
$$960$$ 0 0
$$961$$ 5.79476 0.186928
$$962$$ −35.7003 −1.15103
$$963$$ 93.1440 3.00152
$$964$$ −12.0659 −0.388615
$$965$$ 0 0
$$966$$ −34.7237 −1.11722
$$967$$ −28.4214 −0.913970 −0.456985 0.889474i $$-0.651071\pi$$
−0.456985 + 0.889474i $$0.651071\pi$$
$$968$$ 10.4687 0.336477
$$969$$ 1.70796 0.0548677
$$970$$ 0 0
$$971$$ 30.8057 0.988601 0.494301 0.869291i $$-0.335424\pi$$
0.494301 + 0.869291i $$0.335424\pi$$
$$972$$ 3.73544 0.119814
$$973$$ −41.6423 −1.33499
$$974$$ −23.9450 −0.767249
$$975$$ 0 0
$$976$$ 10.9396 0.350168
$$977$$ −37.6554 −1.20470 −0.602351 0.798231i $$-0.705769\pi$$
−0.602351 + 0.798231i $$0.705769\pi$$
$$978$$ 40.0109 1.27941
$$979$$ −7.28896 −0.232956
$$980$$ 0 0
$$981$$ −84.6862 −2.70382
$$982$$ −3.86826 −0.123441
$$983$$ 38.7948 1.23736 0.618680 0.785643i $$-0.287668\pi$$
0.618680 + 0.785643i $$0.287668\pi$$
$$984$$ −14.5422 −0.463589
$$985$$ 0 0
$$986$$ 5.76183 0.183494
$$987$$ 60.8717 1.93757
$$988$$ −6.23163 −0.198254
$$989$$ 37.3898 1.18893
$$990$$ 0 0
$$991$$ −15.0295 −0.477427 −0.238713 0.971090i $$-0.576726\pi$$
−0.238713 + 0.971090i $$0.576726\pi$$
$$992$$ −6.06587 −0.192592
$$993$$ 36.3317 1.15295
$$994$$ −10.8606 −0.344478
$$995$$ 0 0
$$996$$ −23.8242 −0.754900
$$997$$ −1.18123 −0.0374099 −0.0187050 0.999825i $$-0.505954\pi$$
−0.0187050 + 0.999825i $$0.505954\pi$$
$$998$$ 7.92104 0.250736
$$999$$ 55.5784 1.75842
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 950.2.a.k.1.3 3
3.2 odd 2 8550.2.a.co.1.2 3
4.3 odd 2 7600.2.a.cb.1.1 3
5.2 odd 4 950.2.b.g.799.1 6
5.3 odd 4 950.2.b.g.799.6 6
5.4 even 2 950.2.a.m.1.1 yes 3
15.14 odd 2 8550.2.a.cj.1.2 3
20.19 odd 2 7600.2.a.bm.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.3 3 1.1 even 1 trivial
950.2.a.m.1.1 yes 3 5.4 even 2
950.2.b.g.799.1 6 5.2 odd 4
950.2.b.g.799.6 6 5.3 odd 4
7600.2.a.bm.1.3 3 20.19 odd 2
7600.2.a.cb.1.1 3 4.3 odd 2
8550.2.a.cj.1.2 3 15.14 odd 2
8550.2.a.co.1.2 3 3.2 odd 2