# Properties

 Label 950.2.b.g.799.6 Level $950$ Weight $2$ Character 950.799 Analytic conductor $7.586$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.4227136.2 Defining polynomial: $$x^{6} + 9 x^{4} + 22 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.6 Root $$2.19869i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.799 Dual form 950.2.b.g.799.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +3.03293i q^{3} -1.00000 q^{4} -3.03293 q^{6} +2.46980i q^{7} -1.00000i q^{8} -6.19869 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +3.03293i q^{3} -1.00000 q^{4} -3.03293 q^{6} +2.46980i q^{7} -1.00000i q^{8} -6.19869 q^{9} +0.728896 q^{11} -3.03293i q^{12} +6.23163i q^{13} -2.46980 q^{14} +1.00000 q^{16} +0.563139i q^{17} -6.19869i q^{18} +1.00000 q^{19} -7.49073 q^{21} +0.728896i q^{22} -4.63555i q^{23} +3.03293 q^{24} -6.23163 q^{26} -9.70142i q^{27} -2.46980i q^{28} -10.2316 q^{29} +6.06587 q^{31} +1.00000i q^{32} +2.21069i q^{33} -0.563139 q^{34} +6.19869 q^{36} -5.72890i q^{37} +1.00000i q^{38} -18.9001 q^{39} +4.79476 q^{41} -7.49073i q^{42} -8.06587i q^{43} -0.728896 q^{44} +4.63555 q^{46} +8.12628i q^{47} +3.03293i q^{48} +0.900112 q^{49} -1.70796 q^{51} -6.23163i q^{52} +1.53020i q^{53} +9.70142 q^{54} +2.46980 q^{56} +3.03293i q^{57} -10.2316i q^{58} +5.76183 q^{59} +10.9396 q^{61} +6.06587i q^{62} -15.3095i q^{63} -1.00000 q^{64} -2.21069 q^{66} +12.9330i q^{67} -0.563139i q^{68} +14.0593 q^{69} -4.39738 q^{71} +6.19869i q^{72} +4.09334i q^{73} +5.72890 q^{74} -1.00000 q^{76} +1.80022i q^{77} -18.9001i q^{78} -15.3370 q^{79} +10.8277 q^{81} +4.79476i q^{82} -7.85517i q^{83} +7.49073 q^{84} +8.06587 q^{86} -31.0318i q^{87} -0.728896i q^{88} +10.0000 q^{89} -15.3908 q^{91} +4.63555i q^{92} +18.3974i q^{93} -8.12628 q^{94} -3.03293 q^{96} +11.0055i q^{97} +0.900112i q^{98} -4.51820 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} + 4 q^{6} - 26 q^{9} + O(q^{10})$$ $$6 q - 6 q^{4} + 4 q^{6} - 26 q^{9} + 4 q^{11} - 4 q^{14} + 6 q^{16} + 6 q^{19} - 22 q^{21} - 4 q^{24} - 4 q^{26} - 28 q^{29} - 8 q^{31} + 8 q^{34} + 26 q^{36} - 58 q^{39} - 16 q^{41} - 4 q^{44} + 28 q^{46} - 50 q^{49} - 22 q^{51} + 14 q^{54} + 4 q^{56} + 12 q^{59} + 44 q^{61} - 6 q^{64} + 8 q^{66} - 16 q^{69} - 4 q^{71} + 34 q^{74} - 6 q^{76} - 48 q^{79} - 2 q^{81} + 22 q^{84} + 4 q^{86} + 60 q^{89} - 14 q^{91} - 26 q^{94} + 4 q^{96} - 48 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$3$$ 3.03293i 1.75107i 0.483159 + 0.875533i $$0.339489\pi$$
−0.483159 + 0.875533i $$0.660511\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −3.03293 −1.23819
$$7$$ 2.46980i 0.933495i 0.884391 + 0.466747i $$0.154574\pi$$
−0.884391 + 0.466747i $$0.845426\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −6.19869 −2.06623
$$10$$ 0 0
$$11$$ 0.728896 0.219770 0.109885 0.993944i $$-0.464952\pi$$
0.109885 + 0.993944i $$0.464952\pi$$
$$12$$ − 3.03293i − 0.875533i
$$13$$ 6.23163i 1.72834i 0.503198 + 0.864171i $$0.332157\pi$$
−0.503198 + 0.864171i $$0.667843\pi$$
$$14$$ −2.46980 −0.660081
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0.563139i 0.136581i 0.997665 + 0.0682907i $$0.0217545\pi$$
−0.997665 + 0.0682907i $$0.978245\pi$$
$$18$$ − 6.19869i − 1.46105i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −7.49073 −1.63461
$$22$$ 0.728896i 0.155401i
$$23$$ − 4.63555i − 0.966579i −0.875460 0.483290i $$-0.839442\pi$$
0.875460 0.483290i $$-0.160558\pi$$
$$24$$ 3.03293 0.619095
$$25$$ 0 0
$$26$$ −6.23163 −1.22212
$$27$$ − 9.70142i − 1.86704i
$$28$$ − 2.46980i − 0.466747i
$$29$$ −10.2316 −1.89997 −0.949983 0.312303i $$-0.898900\pi$$
−0.949983 + 0.312303i $$0.898900\pi$$
$$30$$ 0 0
$$31$$ 6.06587 1.08946 0.544731 0.838611i $$-0.316632\pi$$
0.544731 + 0.838611i $$0.316632\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 2.21069i 0.384832i
$$34$$ −0.563139 −0.0965776
$$35$$ 0 0
$$36$$ 6.19869 1.03312
$$37$$ − 5.72890i − 0.941825i −0.882180 0.470912i $$-0.843925\pi$$
0.882180 0.470912i $$-0.156075\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −18.9001 −3.02644
$$40$$ 0 0
$$41$$ 4.79476 0.748816 0.374408 0.927264i $$-0.377846\pi$$
0.374408 + 0.927264i $$0.377846\pi$$
$$42$$ − 7.49073i − 1.15584i
$$43$$ − 8.06587i − 1.23003i −0.788514 0.615017i $$-0.789149\pi$$
0.788514 0.615017i $$-0.210851\pi$$
$$44$$ −0.728896 −0.109885
$$45$$ 0 0
$$46$$ 4.63555 0.683475
$$47$$ 8.12628i 1.18534i 0.805446 + 0.592670i $$0.201926\pi$$
−0.805446 + 0.592670i $$0.798074\pi$$
$$48$$ 3.03293i 0.437766i
$$49$$ 0.900112 0.128587
$$50$$ 0 0
$$51$$ −1.70796 −0.239163
$$52$$ − 6.23163i − 0.864171i
$$53$$ 1.53020i 0.210190i 0.994462 + 0.105095i $$0.0335146\pi$$
−0.994462 + 0.105095i $$0.966485\pi$$
$$54$$ 9.70142 1.32020
$$55$$ 0 0
$$56$$ 2.46980 0.330040
$$57$$ 3.03293i 0.401722i
$$58$$ − 10.2316i − 1.34348i
$$59$$ 5.76183 0.750126 0.375063 0.926999i $$-0.377621\pi$$
0.375063 + 0.926999i $$0.377621\pi$$
$$60$$ 0 0
$$61$$ 10.9396 1.40067 0.700336 0.713814i $$-0.253034\pi$$
0.700336 + 0.713814i $$0.253034\pi$$
$$62$$ 6.06587i 0.770366i
$$63$$ − 15.3095i − 1.92882i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.21069 −0.272118
$$67$$ 12.9330i 1.58002i 0.613092 + 0.790012i $$0.289926\pi$$
−0.613092 + 0.790012i $$0.710074\pi$$
$$68$$ − 0.563139i − 0.0682907i
$$69$$ 14.0593 1.69254
$$70$$ 0 0
$$71$$ −4.39738 −0.521873 −0.260937 0.965356i $$-0.584031\pi$$
−0.260937 + 0.965356i $$0.584031\pi$$
$$72$$ 6.19869i 0.730523i
$$73$$ 4.09334i 0.479090i 0.970885 + 0.239545i $$0.0769982\pi$$
−0.970885 + 0.239545i $$0.923002\pi$$
$$74$$ 5.72890 0.665971
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 1.80022i 0.205155i
$$78$$ − 18.9001i − 2.14002i
$$79$$ −15.3370 −1.72554 −0.862772 0.505593i $$-0.831274\pi$$
−0.862772 + 0.505593i $$0.831274\pi$$
$$80$$ 0 0
$$81$$ 10.8277 1.20308
$$82$$ 4.79476i 0.529493i
$$83$$ − 7.85517i − 0.862217i −0.902300 0.431109i $$-0.858123\pi$$
0.902300 0.431109i $$-0.141877\pi$$
$$84$$ 7.49073 0.817305
$$85$$ 0 0
$$86$$ 8.06587 0.869765
$$87$$ − 31.0318i − 3.32696i
$$88$$ − 0.728896i − 0.0777006i
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −15.3908 −1.61340
$$92$$ 4.63555i 0.483290i
$$93$$ 18.3974i 1.90772i
$$94$$ −8.12628 −0.838162
$$95$$ 0 0
$$96$$ −3.03293 −0.309548
$$97$$ 11.0055i 1.11744i 0.829358 + 0.558718i $$0.188706\pi$$
−0.829358 + 0.558718i $$0.811294\pi$$
$$98$$ 0.900112i 0.0909250i
$$99$$ −4.51820 −0.454096
$$100$$ 0 0
$$101$$ −9.73436 −0.968605 −0.484302 0.874901i $$-0.660926\pi$$
−0.484302 + 0.874901i $$0.660926\pi$$
$$102$$ − 1.70796i − 0.169114i
$$103$$ 8.33151i 0.820928i 0.911877 + 0.410464i $$0.134633\pi$$
−0.911877 + 0.410464i $$0.865367\pi$$
$$104$$ 6.23163 0.611061
$$105$$ 0 0
$$106$$ −1.53020 −0.148627
$$107$$ − 15.0264i − 1.45266i −0.687348 0.726328i $$-0.741225\pi$$
0.687348 0.726328i $$-0.258775\pi$$
$$108$$ 9.70142i 0.933520i
$$109$$ 13.6619 1.30858 0.654288 0.756245i $$-0.272968\pi$$
0.654288 + 0.756245i $$0.272968\pi$$
$$110$$ 0 0
$$111$$ 17.3754 1.64920
$$112$$ 2.46980i 0.233374i
$$113$$ − 1.20524i − 0.113379i −0.998392 0.0566895i $$-0.981945\pi$$
0.998392 0.0566895i $$-0.0180545\pi$$
$$114$$ −3.03293 −0.284060
$$115$$ 0 0
$$116$$ 10.2316 0.949983
$$117$$ − 38.6279i − 3.57115i
$$118$$ 5.76183i 0.530419i
$$119$$ −1.39084 −0.127498
$$120$$ 0 0
$$121$$ −10.4687 −0.951701
$$122$$ 10.9396i 0.990424i
$$123$$ 14.5422i 1.31123i
$$124$$ −6.06587 −0.544731
$$125$$ 0 0
$$126$$ 15.3095 1.36388
$$127$$ 9.52366i 0.845088i 0.906342 + 0.422544i $$0.138863\pi$$
−0.906342 + 0.422544i $$0.861137\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 24.4633 2.15387
$$130$$ 0 0
$$131$$ −17.4028 −1.52049 −0.760247 0.649635i $$-0.774922\pi$$
−0.760247 + 0.649635i $$0.774922\pi$$
$$132$$ − 2.21069i − 0.192416i
$$133$$ 2.46980i 0.214158i
$$134$$ −12.9330 −1.11725
$$135$$ 0 0
$$136$$ 0.563139 0.0482888
$$137$$ 2.25910i 0.193008i 0.995333 + 0.0965040i $$0.0307661\pi$$
−0.995333 + 0.0965040i $$0.969234\pi$$
$$138$$ 14.0593i 1.19681i
$$139$$ −16.8606 −1.43010 −0.715050 0.699073i $$-0.753596\pi$$
−0.715050 + 0.699073i $$0.753596\pi$$
$$140$$ 0 0
$$141$$ −24.6465 −2.07561
$$142$$ − 4.39738i − 0.369020i
$$143$$ 4.54221i 0.379838i
$$144$$ −6.19869 −0.516558
$$145$$ 0 0
$$146$$ −4.09334 −0.338768
$$147$$ 2.72998i 0.225165i
$$148$$ 5.72890i 0.470912i
$$149$$ 13.0055 1.06545 0.532724 0.846289i $$-0.321168\pi$$
0.532724 + 0.846289i $$0.321168\pi$$
$$150$$ 0 0
$$151$$ 17.5895 1.43142 0.715708 0.698400i $$-0.246104\pi$$
0.715708 + 0.698400i $$0.246104\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ − 3.49073i − 0.282209i
$$154$$ −1.80022 −0.145066
$$155$$ 0 0
$$156$$ 18.9001 1.51322
$$157$$ 9.52366i 0.760071i 0.924972 + 0.380035i $$0.124088\pi$$
−0.924972 + 0.380035i $$0.875912\pi$$
$$158$$ − 15.3370i − 1.22014i
$$159$$ −4.64101 −0.368056
$$160$$ 0 0
$$161$$ 11.4489 0.902297
$$162$$ 10.8277i 0.850704i
$$163$$ − 13.1921i − 1.03329i −0.856200 0.516644i $$-0.827181\pi$$
0.856200 0.516644i $$-0.172819\pi$$
$$164$$ −4.79476 −0.374408
$$165$$ 0 0
$$166$$ 7.85517 0.609680
$$167$$ 1.81331i 0.140318i 0.997536 + 0.0701591i $$0.0223507\pi$$
−0.997536 + 0.0701591i $$0.977649\pi$$
$$168$$ 7.49073i 0.577922i
$$169$$ −25.8332 −1.98717
$$170$$ 0 0
$$171$$ −6.19869 −0.474026
$$172$$ 8.06587i 0.615017i
$$173$$ 12.5237i 0.952156i 0.879403 + 0.476078i $$0.157942\pi$$
−0.879403 + 0.476078i $$0.842058\pi$$
$$174$$ 31.0318 2.35252
$$175$$ 0 0
$$176$$ 0.728896 0.0549426
$$177$$ 17.4753i 1.31352i
$$178$$ 10.0000i 0.749532i
$$179$$ −1.39738 −0.104445 −0.0522226 0.998635i $$-0.516631\pi$$
−0.0522226 + 0.998635i $$0.516631\pi$$
$$180$$ 0 0
$$181$$ 5.72890 0.425825 0.212913 0.977071i $$-0.431705\pi$$
0.212913 + 0.977071i $$0.431705\pi$$
$$182$$ − 15.3908i − 1.14084i
$$183$$ 33.1791i 2.45267i
$$184$$ −4.63555 −0.341737
$$185$$ 0 0
$$186$$ −18.3974 −1.34896
$$187$$ 0.410470i 0.0300165i
$$188$$ − 8.12628i − 0.592670i
$$189$$ 23.9605 1.74287
$$190$$ 0 0
$$191$$ −27.0198 −1.95509 −0.977544 0.210733i $$-0.932415\pi$$
−0.977544 + 0.210733i $$0.932415\pi$$
$$192$$ − 3.03293i − 0.218883i
$$193$$ 23.0713i 1.66071i 0.557234 + 0.830355i $$0.311862\pi$$
−0.557234 + 0.830355i $$0.688138\pi$$
$$194$$ −11.0055 −0.790146
$$195$$ 0 0
$$196$$ −0.900112 −0.0642937
$$197$$ 0.794765i 0.0566247i 0.999599 + 0.0283123i $$0.00901330\pi$$
−0.999599 + 0.0283123i $$0.990987\pi$$
$$198$$ − 4.51820i − 0.321095i
$$199$$ −8.07241 −0.572238 −0.286119 0.958194i $$-0.592365\pi$$
−0.286119 + 0.958194i $$0.592365\pi$$
$$200$$ 0 0
$$201$$ −39.2251 −2.76672
$$202$$ − 9.73436i − 0.684907i
$$203$$ − 25.2700i − 1.77361i
$$204$$ 1.70796 0.119581
$$205$$ 0 0
$$206$$ −8.33151 −0.580484
$$207$$ 28.7344i 1.99718i
$$208$$ 6.23163i 0.432085i
$$209$$ 0.728896 0.0504188
$$210$$ 0 0
$$211$$ −11.6410 −0.801400 −0.400700 0.916209i $$-0.631233\pi$$
−0.400700 + 0.916209i $$0.631233\pi$$
$$212$$ − 1.53020i − 0.105095i
$$213$$ − 13.3370i − 0.913834i
$$214$$ 15.0264 1.02718
$$215$$ 0 0
$$216$$ −9.70142 −0.660098
$$217$$ 14.9815i 1.01701i
$$218$$ 13.6619i 0.925304i
$$219$$ −12.4148 −0.838917
$$220$$ 0 0
$$221$$ −3.50927 −0.236059
$$222$$ 17.3754i 1.16616i
$$223$$ 15.6554i 1.04836i 0.851607 + 0.524182i $$0.175629\pi$$
−0.851607 + 0.524182i $$0.824371\pi$$
$$224$$ −2.46980 −0.165020
$$225$$ 0 0
$$226$$ 1.20524 0.0801710
$$227$$ 4.80131i 0.318674i 0.987224 + 0.159337i $$0.0509357\pi$$
−0.987224 + 0.159337i $$0.949064\pi$$
$$228$$ − 3.03293i − 0.200861i
$$229$$ −2.79476 −0.184683 −0.0923416 0.995727i $$-0.529435\pi$$
−0.0923416 + 0.995727i $$0.529435\pi$$
$$230$$ 0 0
$$231$$ −5.45996 −0.359239
$$232$$ 10.2316i 0.671739i
$$233$$ − 11.5422i − 0.756155i −0.925774 0.378078i $$-0.876585\pi$$
0.925774 0.378078i $$-0.123415\pi$$
$$234$$ 38.6279 2.52519
$$235$$ 0 0
$$236$$ −5.76183 −0.375063
$$237$$ − 46.5160i − 3.02154i
$$238$$ − 1.39084i − 0.0901547i
$$239$$ 26.2251 1.69636 0.848180 0.529708i $$-0.177699\pi$$
0.848180 + 0.529708i $$0.177699\pi$$
$$240$$ 0 0
$$241$$ −12.0659 −0.777231 −0.388615 0.921400i $$-0.627047\pi$$
−0.388615 + 0.921400i $$0.627047\pi$$
$$242$$ − 10.4687i − 0.672954i
$$243$$ 3.73544i 0.239629i
$$244$$ −10.9396 −0.700336
$$245$$ 0 0
$$246$$ −14.5422 −0.927177
$$247$$ 6.23163i 0.396509i
$$248$$ − 6.06587i − 0.385183i
$$249$$ 23.8242 1.50980
$$250$$ 0 0
$$251$$ 13.5237 0.853606 0.426803 0.904345i $$-0.359640\pi$$
0.426803 + 0.904345i $$0.359640\pi$$
$$252$$ 15.3095i 0.964408i
$$253$$ − 3.37884i − 0.212426i
$$254$$ −9.52366 −0.597568
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 22.7398i 1.41847i 0.704972 + 0.709235i $$0.250960\pi$$
−0.704972 + 0.709235i $$0.749040\pi$$
$$258$$ 24.4633i 1.52302i
$$259$$ 14.1492 0.879188
$$260$$ 0 0
$$261$$ 63.4227 3.92577
$$262$$ − 17.4028i − 1.07515i
$$263$$ 6.67395i 0.411533i 0.978601 + 0.205767i $$0.0659688\pi$$
−0.978601 + 0.205767i $$0.934031\pi$$
$$264$$ 2.21069 0.136059
$$265$$ 0 0
$$266$$ −2.46980 −0.151433
$$267$$ 30.3293i 1.85613i
$$268$$ − 12.9330i − 0.790012i
$$269$$ 29.7398 1.81327 0.906634 0.421917i $$-0.138643\pi$$
0.906634 + 0.421917i $$0.138643\pi$$
$$270$$ 0 0
$$271$$ 1.11189 0.0675426 0.0337713 0.999430i $$-0.489248\pi$$
0.0337713 + 0.999430i $$0.489248\pi$$
$$272$$ 0.563139i 0.0341453i
$$273$$ − 46.6794i − 2.82517i
$$274$$ −2.25910 −0.136477
$$275$$ 0 0
$$276$$ −14.0593 −0.846272
$$277$$ 13.1263i 0.788682i 0.918964 + 0.394341i $$0.129027\pi$$
−0.918964 + 0.394341i $$0.870973\pi$$
$$278$$ − 16.8606i − 1.01123i
$$279$$ −37.6004 −2.25108
$$280$$ 0 0
$$281$$ 22.9265 1.36768 0.683840 0.729632i $$-0.260309\pi$$
0.683840 + 0.729632i $$0.260309\pi$$
$$282$$ − 24.6465i − 1.46768i
$$283$$ − 0.860634i − 0.0511594i −0.999673 0.0255797i $$-0.991857\pi$$
0.999673 0.0255797i $$-0.00814315\pi$$
$$284$$ 4.39738 0.260937
$$285$$ 0 0
$$286$$ −4.54221 −0.268586
$$287$$ 11.8421i 0.699016i
$$288$$ − 6.19869i − 0.365261i
$$289$$ 16.6829 0.981346
$$290$$ 0 0
$$291$$ −33.3788 −1.95670
$$292$$ − 4.09334i − 0.239545i
$$293$$ − 19.8212i − 1.15796i −0.815340 0.578982i $$-0.803450\pi$$
0.815340 0.578982i $$-0.196550\pi$$
$$294$$ −2.72998 −0.159216
$$295$$ 0 0
$$296$$ −5.72890 −0.332985
$$297$$ − 7.07133i − 0.410320i
$$298$$ 13.0055i 0.753386i
$$299$$ 28.8870 1.67058
$$300$$ 0 0
$$301$$ 19.9210 1.14823
$$302$$ 17.5895i 1.01216i
$$303$$ − 29.5237i − 1.69609i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 3.49073 0.199552
$$307$$ − 16.8661i − 0.962599i −0.876556 0.481299i $$-0.840165\pi$$
0.876556 0.481299i $$-0.159835\pi$$
$$308$$ − 1.80022i − 0.102577i
$$309$$ −25.2689 −1.43750
$$310$$ 0 0
$$311$$ 10.3250 0.585475 0.292738 0.956193i $$-0.405434\pi$$
0.292738 + 0.956193i $$0.405434\pi$$
$$312$$ 18.9001i 1.07001i
$$313$$ 15.7684i 0.891281i 0.895212 + 0.445641i $$0.147024\pi$$
−0.895212 + 0.445641i $$0.852976\pi$$
$$314$$ −9.52366 −0.537451
$$315$$ 0 0
$$316$$ 15.3370 0.862772
$$317$$ − 2.17230i − 0.122009i −0.998138 0.0610043i $$-0.980570\pi$$
0.998138 0.0610043i $$-0.0194303\pi$$
$$318$$ − 4.64101i − 0.260255i
$$319$$ −7.45779 −0.417556
$$320$$ 0 0
$$321$$ 45.5741 2.54370
$$322$$ 11.4489i 0.638020i
$$323$$ 0.563139i 0.0313339i
$$324$$ −10.8277 −0.601539
$$325$$ 0 0
$$326$$ 13.1921 0.730645
$$327$$ 41.4358i 2.29140i
$$328$$ − 4.79476i − 0.264747i
$$329$$ −20.0702 −1.10651
$$330$$ 0 0
$$331$$ 11.9791 0.658429 0.329215 0.944255i $$-0.393216\pi$$
0.329215 + 0.944255i $$0.393216\pi$$
$$332$$ 7.85517i 0.431109i
$$333$$ 35.5117i 1.94603i
$$334$$ −1.81331 −0.0992200
$$335$$ 0 0
$$336$$ −7.49073 −0.408653
$$337$$ − 11.1921i − 0.609675i −0.952404 0.304838i $$-0.901398\pi$$
0.952404 0.304838i $$-0.0986022\pi$$
$$338$$ − 25.8332i − 1.40514i
$$339$$ 3.65540 0.198534
$$340$$ 0 0
$$341$$ 4.42139 0.239432
$$342$$ − 6.19869i − 0.335187i
$$343$$ 19.5117i 1.05353i
$$344$$ −8.06587 −0.434883
$$345$$ 0 0
$$346$$ −12.5237 −0.673276
$$347$$ − 20.3843i − 1.09429i −0.837039 0.547143i $$-0.815715\pi$$
0.837039 0.547143i $$-0.184285\pi$$
$$348$$ 31.0318i 1.66348i
$$349$$ 0.252557 0.0135191 0.00675954 0.999977i $$-0.497848\pi$$
0.00675954 + 0.999977i $$0.497848\pi$$
$$350$$ 0 0
$$351$$ 60.4556 3.22688
$$352$$ 0.728896i 0.0388503i
$$353$$ 28.6434i 1.52453i 0.647263 + 0.762267i $$0.275914\pi$$
−0.647263 + 0.762267i $$0.724086\pi$$
$$354$$ −17.4753 −0.928799
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ − 4.21832i − 0.223257i
$$358$$ − 1.39738i − 0.0738540i
$$359$$ 18.5741 0.980301 0.490151 0.871638i $$-0.336942\pi$$
0.490151 + 0.871638i $$0.336942\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 5.72890i 0.301104i
$$363$$ − 31.7509i − 1.66649i
$$364$$ 15.3908 0.806699
$$365$$ 0 0
$$366$$ −33.1791 −1.73430
$$367$$ − 4.79476i − 0.250285i −0.992139 0.125142i $$-0.960061\pi$$
0.992139 0.125142i $$-0.0399388\pi$$
$$368$$ − 4.63555i − 0.241645i
$$369$$ −29.7213 −1.54723
$$370$$ 0 0
$$371$$ −3.77929 −0.196211
$$372$$ − 18.3974i − 0.953860i
$$373$$ − 15.9485i − 0.825783i −0.910780 0.412892i $$-0.864519\pi$$
0.910780 0.412892i $$-0.135481\pi$$
$$374$$ −0.410470 −0.0212249
$$375$$ 0 0
$$376$$ 8.12628 0.419081
$$377$$ − 63.7597i − 3.28379i
$$378$$ 23.9605i 1.23240i
$$379$$ 16.8013 0.863025 0.431513 0.902107i $$-0.357980\pi$$
0.431513 + 0.902107i $$0.357980\pi$$
$$380$$ 0 0
$$381$$ −28.8846 −1.47980
$$382$$ − 27.0198i − 1.38246i
$$383$$ − 26.7948i − 1.36915i −0.728943 0.684574i $$-0.759988\pi$$
0.728943 0.684574i $$-0.240012\pi$$
$$384$$ 3.03293 0.154774
$$385$$ 0 0
$$386$$ −23.0713 −1.17430
$$387$$ 49.9978i 2.54153i
$$388$$ − 11.0055i − 0.558718i
$$389$$ −18.3424 −0.929998 −0.464999 0.885311i $$-0.653945\pi$$
−0.464999 + 0.885311i $$0.653945\pi$$
$$390$$ 0 0
$$391$$ 2.61046 0.132017
$$392$$ − 0.900112i − 0.0454625i
$$393$$ − 52.7817i − 2.66248i
$$394$$ −0.794765 −0.0400397
$$395$$ 0 0
$$396$$ 4.51820 0.227048
$$397$$ 21.2162i 1.06481i 0.846490 + 0.532404i $$0.178711\pi$$
−0.846490 + 0.532404i $$0.821289\pi$$
$$398$$ − 8.07241i − 0.404633i
$$399$$ −7.49073 −0.375005
$$400$$ 0 0
$$401$$ −27.8661 −1.39157 −0.695783 0.718252i $$-0.744943\pi$$
−0.695783 + 0.718252i $$0.744943\pi$$
$$402$$ − 39.2251i − 1.95637i
$$403$$ 37.8002i 1.88296i
$$404$$ 9.73436 0.484302
$$405$$ 0 0
$$406$$ 25.2700 1.25413
$$407$$ − 4.17577i − 0.206985i
$$408$$ 1.70796i 0.0845568i
$$409$$ 18.0899 0.894487 0.447243 0.894412i $$-0.352406\pi$$
0.447243 + 0.894412i $$0.352406\pi$$
$$410$$ 0 0
$$411$$ −6.85171 −0.337970
$$412$$ − 8.33151i − 0.410464i
$$413$$ 14.2305i 0.700239i
$$414$$ −28.7344 −1.41222
$$415$$ 0 0
$$416$$ −6.23163 −0.305531
$$417$$ − 51.1372i − 2.50420i
$$418$$ 0.728896i 0.0356515i
$$419$$ 15.3370 0.749260 0.374630 0.927174i $$-0.377770\pi$$
0.374630 + 0.927174i $$0.377770\pi$$
$$420$$ 0 0
$$421$$ −8.42486 −0.410602 −0.205301 0.978699i $$-0.565817\pi$$
−0.205301 + 0.978699i $$0.565817\pi$$
$$422$$ − 11.6410i − 0.566676i
$$423$$ − 50.3723i − 2.44918i
$$424$$ 1.53020 0.0743133
$$425$$ 0 0
$$426$$ 13.3370 0.646178
$$427$$ 27.0185i 1.30752i
$$428$$ 15.0264i 0.726328i
$$429$$ −13.7762 −0.665122
$$430$$ 0 0
$$431$$ 16.2766 0.784014 0.392007 0.919962i $$-0.371781\pi$$
0.392007 + 0.919962i $$0.371781\pi$$
$$432$$ − 9.70142i − 0.466760i
$$433$$ 18.1976i 0.874521i 0.899335 + 0.437261i $$0.144051\pi$$
−0.899335 + 0.437261i $$0.855949\pi$$
$$434$$ −14.9815 −0.719133
$$435$$ 0 0
$$436$$ −13.6619 −0.654288
$$437$$ − 4.63555i − 0.221749i
$$438$$ − 12.4148i − 0.593204i
$$439$$ −20.4214 −0.974660 −0.487330 0.873218i $$-0.662029\pi$$
−0.487330 + 0.873218i $$0.662029\pi$$
$$440$$ 0 0
$$441$$ −5.57952 −0.265691
$$442$$ − 3.50927i − 0.166919i
$$443$$ − 21.6685i − 1.02950i −0.857340 0.514750i $$-0.827885\pi$$
0.857340 0.514750i $$-0.172115\pi$$
$$444$$ −17.3754 −0.824598
$$445$$ 0 0
$$446$$ −15.6554 −0.741305
$$447$$ 39.4447i 1.86567i
$$448$$ − 2.46980i − 0.116687i
$$449$$ −23.8133 −1.12382 −0.561910 0.827198i $$-0.689933\pi$$
−0.561910 + 0.827198i $$0.689933\pi$$
$$450$$ 0 0
$$451$$ 3.49489 0.164568
$$452$$ 1.20524i 0.0566895i
$$453$$ 53.3479i 2.50650i
$$454$$ −4.80131 −0.225337
$$455$$ 0 0
$$456$$ 3.03293 0.142030
$$457$$ 9.15813i 0.428399i 0.976790 + 0.214200i $$0.0687143\pi$$
−0.976790 + 0.214200i $$0.931286\pi$$
$$458$$ − 2.79476i − 0.130591i
$$459$$ 5.46325 0.255003
$$460$$ 0 0
$$461$$ −7.78931 −0.362784 −0.181392 0.983411i $$-0.558060\pi$$
−0.181392 + 0.983411i $$0.558060\pi$$
$$462$$ − 5.45996i − 0.254020i
$$463$$ − 0.405011i − 0.0188225i −0.999956 0.00941123i $$-0.997004\pi$$
0.999956 0.00941123i $$-0.00299573\pi$$
$$464$$ −10.2316 −0.474991
$$465$$ 0 0
$$466$$ 11.5422 0.534682
$$467$$ − 1.95814i − 0.0906118i −0.998973 0.0453059i $$-0.985574\pi$$
0.998973 0.0453059i $$-0.0144262\pi$$
$$468$$ 38.6279i 1.78558i
$$469$$ −31.9420 −1.47494
$$470$$ 0 0
$$471$$ −28.8846 −1.33093
$$472$$ − 5.76183i − 0.265210i
$$473$$ − 5.87918i − 0.270325i
$$474$$ 46.5160 2.13655
$$475$$ 0 0
$$476$$ 1.39084 0.0637490
$$477$$ − 9.48527i − 0.434301i
$$478$$ 26.2251i 1.19951i
$$479$$ 22.9210 1.04729 0.523645 0.851937i $$-0.324572\pi$$
0.523645 + 0.851937i $$0.324572\pi$$
$$480$$ 0 0
$$481$$ 35.7003 1.62780
$$482$$ − 12.0659i − 0.549585i
$$483$$ 34.7237i 1.57998i
$$484$$ 10.4687 0.475850
$$485$$ 0 0
$$486$$ −3.73544 −0.169443
$$487$$ − 23.9450i − 1.08505i −0.840038 0.542527i $$-0.817468\pi$$
0.840038 0.542527i $$-0.182532\pi$$
$$488$$ − 10.9396i − 0.495212i
$$489$$ 40.0109 1.80936
$$490$$ 0 0
$$491$$ 3.86826 0.174572 0.0872861 0.996183i $$-0.472181\pi$$
0.0872861 + 0.996183i $$0.472181\pi$$
$$492$$ − 14.5422i − 0.655613i
$$493$$ − 5.76183i − 0.259500i
$$494$$ −6.23163 −0.280374
$$495$$ 0 0
$$496$$ 6.06587 0.272366
$$497$$ − 10.8606i − 0.487166i
$$498$$ 23.8242i 1.06759i
$$499$$ 7.92104 0.354595 0.177297 0.984157i $$-0.443265\pi$$
0.177297 + 0.984157i $$0.443265\pi$$
$$500$$ 0 0
$$501$$ −5.49966 −0.245706
$$502$$ 13.5237i 0.603591i
$$503$$ 16.6949i 0.744388i 0.928155 + 0.372194i $$0.121394\pi$$
−0.928155 + 0.372194i $$0.878606\pi$$
$$504$$ −15.3095 −0.681939
$$505$$ 0 0
$$506$$ 3.37884 0.150208
$$507$$ − 78.3503i − 3.47966i
$$508$$ − 9.52366i − 0.422544i
$$509$$ −24.4028 −1.08164 −0.540818 0.841139i $$-0.681885\pi$$
−0.540818 + 0.841139i $$0.681885\pi$$
$$510$$ 0 0
$$511$$ −10.1097 −0.447228
$$512$$ 1.00000i 0.0441942i
$$513$$ − 9.70142i − 0.428328i
$$514$$ −22.7398 −1.00301
$$515$$ 0 0
$$516$$ −24.4633 −1.07693
$$517$$ 5.92321i 0.260503i
$$518$$ 14.1492i 0.621680i
$$519$$ −37.9834 −1.66729
$$520$$ 0 0
$$521$$ 26.0659 1.14197 0.570983 0.820962i $$-0.306562\pi$$
0.570983 + 0.820962i $$0.306562\pi$$
$$522$$ 63.4227i 2.77594i
$$523$$ − 17.8726i − 0.781516i −0.920493 0.390758i $$-0.872213\pi$$
0.920493 0.390758i $$-0.127787\pi$$
$$524$$ 17.4028 0.760247
$$525$$ 0 0
$$526$$ −6.67395 −0.290998
$$527$$ 3.41593i 0.148800i
$$528$$ 2.21069i 0.0962081i
$$529$$ 1.51166 0.0657243
$$530$$ 0 0
$$531$$ −35.7158 −1.54993
$$532$$ − 2.46980i − 0.107079i
$$533$$ 29.8792i 1.29421i
$$534$$ −30.3293 −1.31248
$$535$$ 0 0
$$536$$ 12.9330 0.558623
$$537$$ − 4.23817i − 0.182891i
$$538$$ 29.7398i 1.28217i
$$539$$ 0.656088 0.0282597
$$540$$ 0 0
$$541$$ 23.0604 0.991444 0.495722 0.868481i $$-0.334903\pi$$
0.495722 + 0.868481i $$0.334903\pi$$
$$542$$ 1.11189i 0.0477598i
$$543$$ 17.3754i 0.745648i
$$544$$ −0.563139 −0.0241444
$$545$$ 0 0
$$546$$ 46.6794 1.99769
$$547$$ − 27.7584i − 1.18686i −0.804885 0.593431i $$-0.797773\pi$$
0.804885 0.593431i $$-0.202227\pi$$
$$548$$ − 2.25910i − 0.0965040i
$$549$$ −67.8111 −2.89411
$$550$$ 0 0
$$551$$ −10.2316 −0.435882
$$552$$ − 14.0593i − 0.598405i
$$553$$ − 37.8792i − 1.61079i
$$554$$ −13.1263 −0.557682
$$555$$ 0 0
$$556$$ 16.8606 0.715050
$$557$$ − 8.60808i − 0.364736i −0.983230 0.182368i $$-0.941624\pi$$
0.983230 0.182368i $$-0.0583762\pi$$
$$558$$ − 37.6004i − 1.59175i
$$559$$ 50.2635 2.12592
$$560$$ 0 0
$$561$$ −1.24493 −0.0525609
$$562$$ 22.9265i 0.967096i
$$563$$ − 7.93959i − 0.334614i −0.985905 0.167307i $$-0.946493\pi$$
0.985905 0.167307i $$-0.0535071\pi$$
$$564$$ 24.6465 1.03780
$$565$$ 0 0
$$566$$ 0.860634 0.0361751
$$567$$ 26.7422i 1.12307i
$$568$$ 4.39738i 0.184510i
$$569$$ 1.65757 0.0694889 0.0347444 0.999396i $$-0.488938\pi$$
0.0347444 + 0.999396i $$0.488938\pi$$
$$570$$ 0 0
$$571$$ −14.0528 −0.588091 −0.294045 0.955791i $$-0.595002\pi$$
−0.294045 + 0.955791i $$0.595002\pi$$
$$572$$ − 4.54221i − 0.189919i
$$573$$ − 81.9494i − 3.42349i
$$574$$ −11.8421 −0.494279
$$575$$ 0 0
$$576$$ 6.19869 0.258279
$$577$$ − 1.36445i − 0.0568027i −0.999597 0.0284014i $$-0.990958\pi$$
0.999597 0.0284014i $$-0.00904165\pi$$
$$578$$ 16.6829i 0.693916i
$$579$$ −69.9738 −2.90801
$$580$$ 0 0
$$581$$ 19.4007 0.804876
$$582$$ − 33.3788i − 1.38360i
$$583$$ 1.11536i 0.0461935i
$$584$$ 4.09334 0.169384
$$585$$ 0 0
$$586$$ 19.8212 0.818804
$$587$$ − 24.6609i − 1.01786i −0.860807 0.508931i $$-0.830041\pi$$
0.860807 0.508931i $$-0.169959\pi$$
$$588$$ − 2.72998i − 0.112583i
$$589$$ 6.06587 0.249940
$$590$$ 0 0
$$591$$ −2.41047 −0.0991535
$$592$$ − 5.72890i − 0.235456i
$$593$$ 0.747443i 0.0306938i 0.999882 + 0.0153469i $$0.00488526\pi$$
−0.999882 + 0.0153469i $$0.995115\pi$$
$$594$$ 7.07133 0.290140
$$595$$ 0 0
$$596$$ −13.0055 −0.532724
$$597$$ − 24.4831i − 1.00203i
$$598$$ 28.8870i 1.18128i
$$599$$ −1.40501 −0.0574072 −0.0287036 0.999588i $$-0.509138\pi$$
−0.0287036 + 0.999588i $$0.509138\pi$$
$$600$$ 0 0
$$601$$ 29.7453 1.21334 0.606668 0.794956i $$-0.292506\pi$$
0.606668 + 0.794956i $$0.292506\pi$$
$$602$$ 19.9210i 0.811921i
$$603$$ − 80.1680i − 3.26469i
$$604$$ −17.5895 −0.715708
$$605$$ 0 0
$$606$$ 29.5237 1.19932
$$607$$ 5.66849i 0.230077i 0.993361 + 0.115038i $$0.0366991\pi$$
−0.993361 + 0.115038i $$0.963301\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ 76.6423 3.10570
$$610$$ 0 0
$$611$$ −50.6399 −2.04867
$$612$$ 3.49073i 0.141104i
$$613$$ 2.99454i 0.120948i 0.998170 + 0.0604742i $$0.0192613\pi$$
−0.998170 + 0.0604742i $$0.980739\pi$$
$$614$$ 16.8661 0.680660
$$615$$ 0 0
$$616$$ 1.80022 0.0725331
$$617$$ − 32.3370i − 1.30184i −0.759147 0.650919i $$-0.774384\pi$$
0.759147 0.650919i $$-0.225616\pi$$
$$618$$ − 25.2689i − 1.01647i
$$619$$ 20.9265 0.841107 0.420554 0.907268i $$-0.361836\pi$$
0.420554 + 0.907268i $$0.361836\pi$$
$$620$$ 0 0
$$621$$ −44.9714 −1.80464
$$622$$ 10.3250i 0.413994i
$$623$$ 24.6980i 0.989503i
$$624$$ −18.9001 −0.756610
$$625$$ 0 0
$$626$$ −15.7684 −0.630231
$$627$$ 2.21069i 0.0882866i
$$628$$ − 9.52366i − 0.380035i
$$629$$ 3.22617 0.128636
$$630$$ 0 0
$$631$$ 22.8002 0.907663 0.453831 0.891088i $$-0.350057\pi$$
0.453831 + 0.891088i $$0.350057\pi$$
$$632$$ 15.3370i 0.610072i
$$633$$ − 35.3064i − 1.40330i
$$634$$ 2.17230 0.0862731
$$635$$ 0 0
$$636$$ 4.64101 0.184028
$$637$$ 5.60916i 0.222243i
$$638$$ − 7.45779i − 0.295257i
$$639$$ 27.2580 1.07831
$$640$$ 0 0
$$641$$ 36.3184 1.43449 0.717246 0.696820i $$-0.245402\pi$$
0.717246 + 0.696820i $$0.245402\pi$$
$$642$$ 45.5741i 1.79866i
$$643$$ − 13.6135i − 0.536865i −0.963298 0.268433i $$-0.913494\pi$$
0.963298 0.268433i $$-0.0865057\pi$$
$$644$$ −11.4489 −0.451148
$$645$$ 0 0
$$646$$ −0.563139 −0.0221564
$$647$$ − 0.0724126i − 0.00284683i −0.999999 0.00142342i $$-0.999547\pi$$
0.999999 0.00142342i $$-0.000453088\pi$$
$$648$$ − 10.8277i − 0.425352i
$$649$$ 4.19978 0.164856
$$650$$ 0 0
$$651$$ −45.4378 −1.78085
$$652$$ 13.1921i 0.516644i
$$653$$ 43.5346i 1.70364i 0.523835 + 0.851820i $$0.324501\pi$$
−0.523835 + 0.851820i $$0.675499\pi$$
$$654$$ −41.4358 −1.62027
$$655$$ 0 0
$$656$$ 4.79476 0.187204
$$657$$ − 25.3734i − 0.989910i
$$658$$ − 20.0702i − 0.782420i
$$659$$ 33.4512 1.30308 0.651538 0.758616i $$-0.274124\pi$$
0.651538 + 0.758616i $$0.274124\pi$$
$$660$$ 0 0
$$661$$ −2.89465 −0.112589 −0.0562945 0.998414i $$-0.517929\pi$$
−0.0562945 + 0.998414i $$0.517929\pi$$
$$662$$ 11.9791i 0.465580i
$$663$$ − 10.6434i − 0.413355i
$$664$$ −7.85517 −0.304840
$$665$$ 0 0
$$666$$ −35.5117 −1.37605
$$667$$ 47.4292i 1.83647i
$$668$$ − 1.81331i − 0.0701591i
$$669$$ −47.4818 −1.83575
$$670$$ 0 0
$$671$$ 7.97382 0.307826
$$672$$ − 7.49073i − 0.288961i
$$673$$ − 42.8475i − 1.65165i −0.563925 0.825826i $$-0.690709\pi$$
0.563925 0.825826i $$-0.309291\pi$$
$$674$$ 11.1921 0.431105
$$675$$ 0 0
$$676$$ 25.8332 0.993583
$$677$$ 34.4567i 1.32428i 0.749381 + 0.662139i $$0.230351\pi$$
−0.749381 + 0.662139i $$0.769649\pi$$
$$678$$ 3.65540i 0.140385i
$$679$$ −27.1812 −1.04312
$$680$$ 0 0
$$681$$ −14.5621 −0.558019
$$682$$ 4.42139i 0.169304i
$$683$$ 2.73436i 0.104627i 0.998631 + 0.0523136i $$0.0166595\pi$$
−0.998631 + 0.0523136i $$0.983340\pi$$
$$684$$ 6.19869 0.237013
$$685$$ 0 0
$$686$$ −19.5117 −0.744959
$$687$$ − 8.47634i − 0.323393i
$$688$$ − 8.06587i − 0.307508i
$$689$$ −9.53566 −0.363280
$$690$$ 0 0
$$691$$ −4.74198 −0.180394 −0.0901968 0.995924i $$-0.528750\pi$$
−0.0901968 + 0.995924i $$0.528750\pi$$
$$692$$ − 12.5237i − 0.476078i
$$693$$ − 11.1590i − 0.423897i
$$694$$ 20.3843 0.773777
$$695$$ 0 0
$$696$$ −31.0318 −1.17626
$$697$$ 2.70012i 0.102274i
$$698$$ 0.252557i 0.00955943i
$$699$$ 35.0068 1.32408
$$700$$ 0 0
$$701$$ 33.1372 1.25157 0.625787 0.779994i $$-0.284778\pi$$
0.625787 + 0.779994i $$0.284778\pi$$
$$702$$ 60.4556i 2.28175i
$$703$$ − 5.72890i − 0.216069i
$$704$$ −0.728896 −0.0274713
$$705$$ 0 0
$$706$$ −28.6434 −1.07801
$$707$$ − 24.0419i − 0.904187i
$$708$$ − 17.4753i − 0.656760i
$$709$$ −5.37884 −0.202006 −0.101003 0.994886i $$-0.532205\pi$$
−0.101003 + 0.994886i $$0.532205\pi$$
$$710$$ 0 0
$$711$$ 95.0692 3.56537
$$712$$ − 10.0000i − 0.374766i
$$713$$ − 28.1187i − 1.05305i
$$714$$ 4.21832 0.157867
$$715$$ 0 0
$$716$$ 1.39738 0.0522226
$$717$$ 79.5390i 2.97044i
$$718$$ 18.5741i 0.693178i
$$719$$ −33.0857 −1.23389 −0.616944 0.787007i $$-0.711630\pi$$
−0.616944 + 0.787007i $$0.711630\pi$$
$$720$$ 0 0
$$721$$ −20.5771 −0.766332
$$722$$ 1.00000i 0.0372161i
$$723$$ − 36.5950i − 1.36098i
$$724$$ −5.72890 −0.212913
$$725$$ 0 0
$$726$$ 31.7509 1.17839
$$727$$ 2.62463i 0.0973423i 0.998815 + 0.0486711i $$0.0154986\pi$$
−0.998815 + 0.0486711i $$0.984501\pi$$
$$728$$ 15.3908i 0.570422i
$$729$$ 21.1538 0.783472
$$730$$ 0 0
$$731$$ 4.54221 0.168000
$$732$$ − 33.1791i − 1.22633i
$$733$$ − 0.608077i − 0.0224598i −0.999937 0.0112299i $$-0.996425\pi$$
0.999937 0.0112299i $$-0.00357467\pi$$
$$734$$ 4.79476 0.176978
$$735$$ 0 0
$$736$$ 4.63555 0.170869
$$737$$ 9.42685i 0.347242i
$$738$$ − 29.7213i − 1.09405i
$$739$$ −36.3974 −1.33890 −0.669450 0.742857i $$-0.733470\pi$$
−0.669450 + 0.742857i $$0.733470\pi$$
$$740$$ 0 0
$$741$$ −18.9001 −0.694313
$$742$$ − 3.77929i − 0.138742i
$$743$$ 23.0713i 0.846405i 0.906035 + 0.423202i $$0.139094\pi$$
−0.906035 + 0.423202i $$0.860906\pi$$
$$744$$ 18.3974 0.674481
$$745$$ 0 0
$$746$$ 15.9485 0.583917
$$747$$ 48.6918i 1.78154i
$$748$$ − 0.410470i − 0.0150083i
$$749$$ 37.1121 1.35605
$$750$$ 0 0
$$751$$ 4.75290 0.173436 0.0867179 0.996233i $$-0.472362\pi$$
0.0867179 + 0.996233i $$0.472362\pi$$
$$752$$ 8.12628i 0.296335i
$$753$$ 41.0164i 1.49472i
$$754$$ 63.7597 2.32199
$$755$$ 0 0
$$756$$ −23.9605 −0.871436
$$757$$ 26.8057i 0.974269i 0.873327 + 0.487135i $$0.161958\pi$$
−0.873327 + 0.487135i $$0.838042\pi$$
$$758$$ 16.8013i 0.610251i
$$759$$ 10.2478 0.371971
$$760$$ 0 0
$$761$$ 25.4478 0.922481 0.461241 0.887275i $$-0.347404\pi$$
0.461241 + 0.887275i $$0.347404\pi$$
$$762$$ − 28.8846i − 1.04638i
$$763$$ 33.7422i 1.22155i
$$764$$ 27.0198 0.977544
$$765$$ 0 0
$$766$$ 26.7948 0.968134
$$767$$ 35.9056i 1.29648i
$$768$$ 3.03293i 0.109442i
$$769$$ 14.6421 0.528007 0.264004 0.964522i $$-0.414957\pi$$
0.264004 + 0.964522i $$0.414957\pi$$
$$770$$ 0 0
$$771$$ −68.9684 −2.48384
$$772$$ − 23.0713i − 0.830355i
$$773$$ 16.8462i 0.605917i 0.953004 + 0.302959i $$0.0979744\pi$$
−0.953004 + 0.302959i $$0.902026\pi$$
$$774$$ −49.9978 −1.79713
$$775$$ 0 0
$$776$$ 11.0055 0.395073
$$777$$ 42.9136i 1.53952i
$$778$$ − 18.3424i − 0.657608i
$$779$$ 4.79476 0.171790
$$780$$ 0 0
$$781$$ −3.20524 −0.114692
$$782$$ 2.61046i 0.0933499i
$$783$$ 99.2613i 3.54731i
$$784$$ 0.900112 0.0321469
$$785$$ 0 0
$$786$$ 52.7817 1.88266
$$787$$ 18.7367i 0.667893i 0.942592 + 0.333946i $$0.108380\pi$$
−0.942592 + 0.333946i $$0.891620\pi$$
$$788$$ − 0.794765i − 0.0283123i
$$789$$ −20.2416 −0.720621
$$790$$ 0 0
$$791$$ 2.97668 0.105839
$$792$$ 4.51820i 0.160547i
$$793$$ 68.1714i 2.42084i
$$794$$ −21.2162 −0.752933
$$795$$ 0 0
$$796$$ 8.07241 0.286119
$$797$$ − 37.9900i − 1.34567i −0.739791 0.672837i $$-0.765075\pi$$
0.739791 0.672837i $$-0.234925\pi$$
$$798$$ − 7.49073i − 0.265169i
$$799$$ −4.57623 −0.161895
$$800$$ 0 0
$$801$$ −61.9869 −2.19020
$$802$$ − 27.8661i − 0.983986i
$$803$$ 2.98362i 0.105290i
$$804$$ 39.2251 1.38336
$$805$$ 0 0
$$806$$ −37.8002 −1.33146
$$807$$ 90.1989i 3.17515i
$$808$$ 9.73436i 0.342453i
$$809$$ −21.8857 −0.769461 −0.384731 0.923029i $$-0.625706\pi$$
−0.384731 + 0.923029i $$0.625706\pi$$
$$810$$ 0 0
$$811$$ 37.8595 1.32943 0.664714 0.747098i $$-0.268553\pi$$
0.664714 + 0.747098i $$0.268553\pi$$
$$812$$ 25.2700i 0.886804i
$$813$$ 3.37229i 0.118271i
$$814$$ 4.17577 0.146361
$$815$$ 0 0
$$816$$ −1.70796 −0.0597907
$$817$$ − 8.06587i − 0.282189i
$$818$$ 18.0899i 0.632498i
$$819$$ 95.4031 3.33365
$$820$$ 0 0
$$821$$ −20.1976 −0.704901 −0.352451 0.935830i $$-0.614652\pi$$
−0.352451 + 0.935830i $$0.614652\pi$$
$$822$$ − 6.85171i − 0.238981i
$$823$$ 4.34590i 0.151489i 0.997127 + 0.0757443i $$0.0241333\pi$$
−0.997127 + 0.0757443i $$0.975867\pi$$
$$824$$ 8.33151 0.290242
$$825$$ 0 0
$$826$$ −14.2305 −0.495144
$$827$$ − 56.8375i − 1.97643i −0.153057 0.988217i $$-0.548912\pi$$
0.153057 0.988217i $$-0.451088\pi$$
$$828$$ − 28.7344i − 0.998588i
$$829$$ 17.4543 0.606214 0.303107 0.952957i $$-0.401976\pi$$
0.303107 + 0.952957i $$0.401976\pi$$
$$830$$ 0 0
$$831$$ −39.8111 −1.38103
$$832$$ − 6.23163i − 0.216043i
$$833$$ 0.506888i 0.0175626i
$$834$$ 51.1372 1.77074
$$835$$ 0 0
$$836$$ −0.728896 −0.0252094
$$837$$ − 58.8475i − 2.03407i
$$838$$ 15.3370i 0.529807i
$$839$$ −1.77622 −0.0613219 −0.0306609 0.999530i $$-0.509761\pi$$
−0.0306609 + 0.999530i $$0.509761\pi$$
$$840$$ 0 0
$$841$$ 75.6862 2.60987
$$842$$ − 8.42486i − 0.290340i
$$843$$ 69.5346i 2.39490i
$$844$$ 11.6410 0.400700
$$845$$ 0 0
$$846$$ 50.3723 1.73184
$$847$$ − 25.8556i − 0.888408i
$$848$$ 1.53020i 0.0525475i
$$849$$ 2.61025 0.0895834
$$850$$ 0 0
$$851$$ −26.5566 −0.910348
$$852$$ 13.3370i 0.456917i
$$853$$ 16.1187i 0.551892i 0.961173 + 0.275946i $$0.0889911\pi$$
−0.961173 + 0.275946i $$0.911009\pi$$
$$854$$ −27.0185 −0.924556
$$855$$ 0 0
$$856$$ −15.0264 −0.513591
$$857$$ − 47.2031i − 1.61243i −0.591625 0.806213i $$-0.701514\pi$$
0.591625 0.806213i $$-0.298486\pi$$
$$858$$ − 13.7762i − 0.470312i
$$859$$ −37.3239 −1.27347 −0.636737 0.771081i $$-0.719716\pi$$
−0.636737 + 0.771081i $$0.719716\pi$$
$$860$$ 0 0
$$861$$ −35.9163 −1.22402
$$862$$ 16.2766i 0.554382i
$$863$$ 1.33697i 0.0455111i 0.999741 + 0.0227555i $$0.00724394\pi$$
−0.999741 + 0.0227555i $$0.992756\pi$$
$$864$$ 9.70142 0.330049
$$865$$ 0 0
$$866$$ −18.1976 −0.618380
$$867$$ 50.5981i 1.71840i
$$868$$ − 14.9815i − 0.508504i
$$869$$ −11.1791 −0.379224
$$870$$ 0 0
$$871$$ −80.5939 −2.73082
$$872$$ − 13.6619i − 0.462652i
$$873$$ − 68.2194i − 2.30888i
$$874$$ 4.63555 0.156800
$$875$$ 0 0
$$876$$ 12.4148 0.419459
$$877$$ 13.1647i 0.444539i 0.974985 + 0.222270i $$0.0713465\pi$$
−0.974985 + 0.222270i $$0.928653\pi$$
$$878$$ − 20.4214i − 0.689188i
$$879$$ 60.1163 2.02767
$$880$$ 0 0
$$881$$ −10.2052 −0.343823 −0.171912 0.985112i $$-0.554994\pi$$
−0.171912 + 0.985112i $$0.554994\pi$$
$$882$$ − 5.57952i − 0.187872i
$$883$$ − 1.49966i − 0.0504674i −0.999682 0.0252337i $$-0.991967\pi$$
0.999682 0.0252337i $$-0.00803299\pi$$
$$884$$ 3.50927 0.118030
$$885$$ 0 0
$$886$$ 21.6685 0.727967
$$887$$ − 46.9505i − 1.57644i −0.615391 0.788222i $$-0.711002\pi$$
0.615391 0.788222i $$-0.288998\pi$$
$$888$$ − 17.3754i − 0.583079i
$$889$$ −23.5215 −0.788886
$$890$$ 0 0
$$891$$ 7.89227 0.264401
$$892$$ − 15.6554i − 0.524182i
$$893$$ 8.12628i 0.271936i
$$894$$ −39.4447 −1.31923
$$895$$ 0 0
$$896$$ 2.46980 0.0825101
$$897$$ 87.6125i 2.92529i
$$898$$ − 23.8133i − 0.794661i
$$899$$ −62.0637 −2.06994
$$900$$ 0 0
$$901$$ −0.861719 −0.0287080
$$902$$ 3.49489i 0.116367i
$$903$$ 60.4192i 2.01063i
$$904$$ −1.20524 −0.0400855
$$905$$ 0 0
$$906$$ −53.3479 −1.77236
$$907$$ 16.3668i 0.543452i 0.962375 + 0.271726i $$0.0875944\pi$$
−0.962375 + 0.271726i $$0.912406\pi$$
$$908$$ − 4.80131i − 0.159337i
$$909$$ 60.3403 2.00136
$$910$$ 0 0
$$911$$ 32.3293 1.07112 0.535559 0.844498i $$-0.320101\pi$$
0.535559 + 0.844498i $$0.320101\pi$$
$$912$$ 3.03293i 0.100430i
$$913$$ − 5.72561i − 0.189490i
$$914$$ −9.15813 −0.302924
$$915$$ 0 0
$$916$$ 2.79476 0.0923416
$$917$$ − 42.9815i − 1.41937i
$$918$$ 5.46325i 0.180314i
$$919$$ 22.8157 0.752620 0.376310 0.926494i $$-0.377193\pi$$
0.376310 + 0.926494i $$0.377193\pi$$
$$920$$ 0 0
$$921$$ 51.1538 1.68557
$$922$$ − 7.78931i − 0.256527i
$$923$$ − 27.4028i − 0.901976i
$$924$$ 5.45996 0.179620
$$925$$ 0 0
$$926$$ 0.405011 0.0133095
$$927$$ − 51.6445i − 1.69623i
$$928$$ − 10.2316i − 0.335870i
$$929$$ 27.2436 0.893834 0.446917 0.894575i $$-0.352522\pi$$
0.446917 + 0.894575i $$0.352522\pi$$
$$930$$ 0 0
$$931$$ 0.900112 0.0295000
$$932$$ 11.5422i 0.378078i
$$933$$ 31.3150i 1.02521i
$$934$$ 1.95814 0.0640722
$$935$$ 0 0
$$936$$ −38.6279 −1.26259
$$937$$ − 51.5006i − 1.68245i −0.540685 0.841225i $$-0.681835\pi$$
0.540685 0.841225i $$-0.318165\pi$$
$$938$$ − 31.9420i − 1.04294i
$$939$$ −47.8244 −1.56069
$$940$$ 0 0
$$941$$ −40.8541 −1.33181 −0.665903 0.746039i $$-0.731953\pi$$
−0.665903 + 0.746039i $$0.731953\pi$$
$$942$$ − 28.8846i − 0.941112i
$$943$$ − 22.2264i − 0.723791i
$$944$$ 5.76183 0.187532
$$945$$ 0 0
$$946$$ 5.87918 0.191149
$$947$$ 38.2526i 1.24304i 0.783398 + 0.621521i $$0.213485\pi$$
−0.783398 + 0.621521i $$0.786515\pi$$
$$948$$ 46.5160i 1.51077i
$$949$$ −25.5082 −0.828031
$$950$$ 0 0
$$951$$ 6.58845 0.213645
$$952$$ 1.39084i 0.0450773i
$$953$$ − 54.1187i − 1.75308i −0.481334 0.876538i $$-0.659847\pi$$
0.481334 0.876538i $$-0.340153\pi$$
$$954$$ 9.48527 0.307097
$$955$$ 0 0
$$956$$ −26.2251 −0.848180
$$957$$ − 22.6190i − 0.731168i
$$958$$ 22.9210i 0.740545i
$$959$$ −5.57952 −0.180172
$$960$$ 0 0
$$961$$ 5.79476 0.186928
$$962$$ 35.7003i 1.15103i
$$963$$ 93.1440i 3.00152i
$$964$$ 12.0659 0.388615
$$965$$ 0 0
$$966$$ −34.7237 −1.11722
$$967$$ 28.4214i 0.913970i 0.889474 + 0.456985i $$0.151071\pi$$
−0.889474 + 0.456985i $$0.848929\pi$$
$$968$$ 10.4687i 0.336477i
$$969$$ −1.70796 −0.0548677
$$970$$ 0 0
$$971$$ 30.8057 0.988601 0.494301 0.869291i $$-0.335424\pi$$
0.494301 + 0.869291i $$0.335424\pi$$
$$972$$ − 3.73544i − 0.119814i
$$973$$ − 41.6423i − 1.33499i
$$974$$ 23.9450 0.767249
$$975$$ 0 0
$$976$$ 10.9396 0.350168
$$977$$ 37.6554i 1.20470i 0.798231 + 0.602351i $$0.205769\pi$$
−0.798231 + 0.602351i $$0.794231\pi$$
$$978$$ 40.0109i 1.27941i
$$979$$ 7.28896 0.232956
$$980$$ 0 0
$$981$$ −84.6862 −2.70382
$$982$$ 3.86826i 0.123441i
$$983$$ 38.7948i 1.23736i 0.785643 + 0.618680i $$0.212332\pi$$
−0.785643 + 0.618680i $$0.787668\pi$$
$$984$$ 14.5422 0.463589
$$985$$ 0 0
$$986$$ 5.76183 0.183494
$$987$$ − 60.8717i − 1.93757i
$$988$$ − 6.23163i − 0.198254i
$$989$$ −37.3898 −1.18893
$$990$$ 0 0
$$991$$ −15.0295 −0.477427 −0.238713 0.971090i $$-0.576726\pi$$
−0.238713 + 0.971090i $$0.576726\pi$$
$$992$$ 6.06587i 0.192592i
$$993$$ 36.3317i 1.15295i
$$994$$ 10.8606 0.344478
$$995$$ 0 0
$$996$$ −23.8242 −0.754900
$$997$$ 1.18123i 0.0374099i 0.999825 + 0.0187050i $$0.00595432\pi$$
−0.999825 + 0.0187050i $$0.994046\pi$$
$$998$$ 7.92104i 0.250736i
$$999$$ −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.b.g.799.6 6
5.2 odd 4 950.2.a.k.1.3 3
5.3 odd 4 950.2.a.m.1.1 yes 3
5.4 even 2 inner 950.2.b.g.799.1 6
15.2 even 4 8550.2.a.co.1.2 3
15.8 even 4 8550.2.a.cj.1.2 3
20.3 even 4 7600.2.a.bm.1.3 3
20.7 even 4 7600.2.a.cb.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.3 3 5.2 odd 4
950.2.a.m.1.1 yes 3 5.3 odd 4
950.2.b.g.799.1 6 5.4 even 2 inner
950.2.b.g.799.6 6 1.1 even 1 trivial
7600.2.a.bm.1.3 3 20.3 even 4
7600.2.a.cb.1.1 3 20.7 even 4
8550.2.a.cj.1.2 3 15.8 even 4
8550.2.a.co.1.2 3 15.2 even 4