# Properties

 Label 5850.2.e.c.5149.2 Level $5850$ Weight $2$ Character 5850.5149 Analytic conductor $46.712$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5850.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.7124851824$$ 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 1950) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 5149.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5850.5149 Dual form 5850.2.e.c.5149.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{4} +4.00000i q^{7} -1.00000i q^{8} -4.00000 q^{11} +1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -7.00000 q^{19} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{26} -4.00000i q^{28} +5.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000 q^{34} -9.00000i q^{37} -7.00000i q^{38} +5.00000 q^{41} -10.0000i q^{43} +4.00000 q^{44} +4.00000 q^{46} +3.00000i q^{47} -9.00000 q^{49} -1.00000i q^{52} -9.00000i q^{53} +4.00000 q^{56} +5.00000i q^{58} -6.00000 q^{59} +4.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +7.00000i q^{67} +4.00000i q^{68} +15.0000 q^{71} +12.0000i q^{73} +9.00000 q^{74} +7.00000 q^{76} -16.0000i q^{77} -7.00000 q^{79} +5.00000i q^{82} -6.00000i q^{83} +10.0000 q^{86} +4.00000i q^{88} +14.0000 q^{89} -4.00000 q^{91} +4.00000i q^{92} -3.00000 q^{94} +16.0000i q^{97} -9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 8q^{11} - 8q^{14} + 2q^{16} - 14q^{19} - 2q^{26} + 10q^{29} + 8q^{31} + 8q^{34} + 10q^{41} + 8q^{44} + 8q^{46} - 18q^{49} + 8q^{56} - 12q^{59} + 8q^{61} - 2q^{64} + 30q^{71} + 18q^{74} + 14q^{76} - 14q^{79} + 20q^{86} + 28q^{89} - 8q^{91} - 6q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$$.

 $$n$$ $$2251$$ $$3251$$ $$3277$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 4.00000i − 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ 0 0
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ − 4.00000i − 0.755929i
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 9.00000i − 1.47959i −0.672832 0.739795i $$-0.734922\pi$$
0.672832 0.739795i $$-0.265078\pi$$
$$38$$ − 7.00000i − 1.13555i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.00000 0.780869 0.390434 0.920631i $$-0.372325\pi$$
0.390434 + 0.920631i $$0.372325\pi$$
$$42$$ 0 0
$$43$$ − 10.0000i − 1.52499i −0.646997 0.762493i $$-0.723975\pi$$
0.646997 0.762493i $$-0.276025\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 3.00000i 0.437595i 0.975770 + 0.218797i $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 1.00000i − 0.138675i
$$53$$ − 9.00000i − 1.23625i −0.786082 0.618123i $$-0.787894\pi$$
0.786082 0.618123i $$-0.212106\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ 0 0
$$58$$ 5.00000i 0.656532i
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.00000i 0.855186i 0.903971 + 0.427593i $$0.140638\pi$$
−0.903971 + 0.427593i $$0.859362\pi$$
$$68$$ 4.00000i 0.485071i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 15.0000 1.78017 0.890086 0.455792i $$-0.150644\pi$$
0.890086 + 0.455792i $$0.150644\pi$$
$$72$$ 0 0
$$73$$ 12.0000i 1.40449i 0.711934 + 0.702247i $$0.247820\pi$$
−0.711934 + 0.702247i $$0.752180\pi$$
$$74$$ 9.00000 1.04623
$$75$$ 0 0
$$76$$ 7.00000 0.802955
$$77$$ − 16.0000i − 1.82337i
$$78$$ 0 0
$$79$$ −7.00000 −0.787562 −0.393781 0.919204i $$-0.628833\pi$$
−0.393781 + 0.919204i $$0.628833\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 5.00000i 0.552158i
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 10.0000 1.07833
$$87$$ 0 0
$$88$$ 4.00000i 0.426401i
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 4.00000i 0.417029i
$$93$$ 0 0
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 16.0000i 1.62455i 0.583272 + 0.812277i $$0.301772\pi$$
−0.583272 + 0.812277i $$0.698228\pi$$
$$98$$ − 9.00000i − 0.909137i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ 5.00000i 0.483368i 0.970355 + 0.241684i $$0.0776998\pi$$
−0.970355 + 0.241684i $$0.922300\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.00000i 0.377964i
$$113$$ 14.0000i 1.31701i 0.752577 + 0.658505i $$0.228811\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −5.00000 −0.464238
$$117$$ 0 0
$$118$$ − 6.00000i − 0.552345i
$$119$$ 16.0000 1.46672
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 4.00000i 0.362143i
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 5.00000i 0.443678i 0.975083 + 0.221839i $$0.0712060\pi$$
−0.975083 + 0.221839i $$0.928794\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −17.0000 −1.48530 −0.742648 0.669681i $$-0.766431\pi$$
−0.742648 + 0.669681i $$0.766431\pi$$
$$132$$ 0 0
$$133$$ − 28.0000i − 2.42791i
$$134$$ −7.00000 −0.604708
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ − 19.0000i − 1.62328i −0.584158 0.811640i $$-0.698575\pi$$
0.584158 0.811640i $$-0.301425\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 15.0000i 1.25877i
$$143$$ − 4.00000i − 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −12.0000 −0.993127
$$147$$ 0 0
$$148$$ 9.00000i 0.739795i
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 7.00000i 0.567775i
$$153$$ 0 0
$$154$$ 16.0000 1.28932
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ − 7.00000i − 0.556890i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ − 24.0000i − 1.87983i −0.341415 0.939913i $$-0.610906\pi$$
0.341415 0.939913i $$-0.389094\pi$$
$$164$$ −5.00000 −0.390434
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ 9.00000i 0.696441i 0.937413 + 0.348220i $$0.113214\pi$$
−0.937413 + 0.348220i $$0.886786\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 10.0000i 0.762493i
$$173$$ − 13.0000i − 0.988372i −0.869356 0.494186i $$-0.835466\pi$$
0.869356 0.494186i $$-0.164534\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 14.0000i 1.04934i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −4.00000 −0.297318 −0.148659 0.988889i $$-0.547496\pi$$
−0.148659 + 0.988889i $$0.547496\pi$$
$$182$$ − 4.00000i − 0.296500i
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 16.0000i 1.17004i
$$188$$ − 3.00000i − 0.218797i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.0000 −0.723575 −0.361787 0.932261i $$-0.617833\pi$$
−0.361787 + 0.932261i $$0.617833\pi$$
$$192$$ 0 0
$$193$$ − 12.0000i − 0.863779i −0.901927 0.431889i $$-0.857847\pi$$
0.901927 0.431889i $$-0.142153\pi$$
$$194$$ −16.0000 −1.14873
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ − 16.0000i − 1.13995i −0.821661 0.569976i $$-0.806952\pi$$
0.821661 0.569976i $$-0.193048\pi$$
$$198$$ 0 0
$$199$$ 3.00000 0.212664 0.106332 0.994331i $$-0.466089\pi$$
0.106332 + 0.994331i $$0.466089\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 10.0000i 0.703598i
$$203$$ 20.0000i 1.40372i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 1.00000i 0.0693375i
$$209$$ 28.0000 1.93680
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 9.00000i 0.618123i
$$213$$ 0 0
$$214$$ −5.00000 −0.341793
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 16.0000i 1.08615i
$$218$$ 11.0000i 0.745014i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ − 2.00000i − 0.133930i −0.997755 0.0669650i $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ 6.00000i 0.398234i 0.979976 + 0.199117i $$0.0638074\pi$$
−0.979976 + 0.199117i $$0.936193\pi$$
$$228$$ 0 0
$$229$$ 17.0000 1.12339 0.561696 0.827344i $$-0.310149\pi$$
0.561696 + 0.827344i $$0.310149\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 5.00000i − 0.328266i
$$233$$ − 8.00000i − 0.524097i −0.965055 0.262049i $$-0.915602\pi$$
0.965055 0.262049i $$-0.0843981\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ 0 0
$$238$$ 16.0000i 1.03713i
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ 0 0
$$244$$ −4.00000 −0.256074
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 7.00000i − 0.445399i
$$248$$ − 4.00000i − 0.254000i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 5.00000 0.315597 0.157799 0.987471i $$-0.449560\pi$$
0.157799 + 0.987471i $$0.449560\pi$$
$$252$$ 0 0
$$253$$ 16.0000i 1.00591i
$$254$$ −5.00000 −0.313728
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 8.00000i 0.499026i 0.968371 + 0.249513i $$0.0802706\pi$$
−0.968371 + 0.249513i $$0.919729\pi$$
$$258$$ 0 0
$$259$$ 36.0000 2.23693
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 17.0000i − 1.05026i
$$263$$ − 30.0000i − 1.84988i −0.380114 0.924940i $$-0.624115\pi$$
0.380114 0.924940i $$-0.375885\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 28.0000 1.71679
$$267$$ 0 0
$$268$$ − 7.00000i − 0.427593i
$$269$$ 5.00000 0.304855 0.152428 0.988315i $$-0.451291\pi$$
0.152428 + 0.988315i $$0.451291\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ − 4.00000i − 0.242536i
$$273$$ 0 0
$$274$$ 19.0000 1.14783
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 26.0000i − 1.56219i −0.624413 0.781094i $$-0.714662\pi$$
0.624413 0.781094i $$-0.285338\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −17.0000 −1.01413 −0.507067 0.861906i $$-0.669271\pi$$
−0.507067 + 0.861906i $$0.669271\pi$$
$$282$$ 0 0
$$283$$ 8.00000i 0.475551i 0.971320 + 0.237775i $$0.0764182\pi$$
−0.971320 + 0.237775i $$0.923582\pi$$
$$284$$ −15.0000 −0.890086
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 20.0000i 1.18056i
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 12.0000i − 0.702247i
$$293$$ 12.0000i 0.701047i 0.936554 + 0.350524i $$0.113996\pi$$
−0.936554 + 0.350524i $$0.886004\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −9.00000 −0.523114
$$297$$ 0 0
$$298$$ − 4.00000i − 0.231714i
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 40.0000 2.30556
$$302$$ − 4.00000i − 0.230174i
$$303$$ 0 0
$$304$$ −7.00000 −0.401478
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 21.0000i − 1.19853i −0.800549 0.599267i $$-0.795459\pi$$
0.800549 0.599267i $$-0.204541\pi$$
$$308$$ 16.0000i 0.911685i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ 0 0
$$313$$ − 23.0000i − 1.30004i −0.759918 0.650018i $$-0.774761\pi$$
0.759918 0.650018i $$-0.225239\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 7.00000 0.393781
$$317$$ − 28.0000i − 1.57264i −0.617822 0.786318i $$-0.711985\pi$$
0.617822 0.786318i $$-0.288015\pi$$
$$318$$ 0 0
$$319$$ −20.0000 −1.11979
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 16.0000i 0.891645i
$$323$$ 28.0000i 1.55796i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 24.0000 1.32924
$$327$$ 0 0
$$328$$ − 5.00000i − 0.276079i
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ 0 0
$$334$$ −9.00000 −0.492458
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 34.0000i 1.85210i 0.377403 + 0.926049i $$0.376817\pi$$
−0.377403 + 0.926049i $$0.623183\pi$$
$$338$$ − 1.00000i − 0.0543928i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ − 8.00000i − 0.431959i
$$344$$ −10.0000 −0.539164
$$345$$ 0 0
$$346$$ 13.0000 0.698884
$$347$$ 11.0000i 0.590511i 0.955418 + 0.295255i $$0.0954048\pi$$
−0.955418 + 0.295255i $$0.904595\pi$$
$$348$$ 0 0
$$349$$ 34.0000 1.81998 0.909989 0.414632i $$-0.136090\pi$$
0.909989 + 0.414632i $$0.136090\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 4.00000i − 0.213201i
$$353$$ 21.0000i 1.11772i 0.829263 + 0.558859i $$0.188761\pi$$
−0.829263 + 0.558859i $$0.811239\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ − 4.00000i − 0.211407i
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ − 4.00000i − 0.210235i
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 7.00000i − 0.365397i −0.983169 0.182699i $$-0.941517\pi$$
0.983169 0.182699i $$-0.0584832\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 36.0000 1.86903
$$372$$ 0 0
$$373$$ 22.0000i 1.13912i 0.821951 + 0.569558i $$0.192886\pi$$
−0.821951 + 0.569558i $$0.807114\pi$$
$$374$$ −16.0000 −0.827340
$$375$$ 0 0
$$376$$ 3.00000 0.154713
$$377$$ 5.00000i 0.257513i
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 10.0000i − 0.511645i
$$383$$ − 5.00000i − 0.255488i −0.991807 0.127744i $$-0.959226\pi$$
0.991807 0.127744i $$-0.0407736\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 12.0000 0.610784
$$387$$ 0 0
$$388$$ − 16.0000i − 0.812277i
$$389$$ 5.00000 0.253510 0.126755 0.991934i $$-0.459544\pi$$
0.126755 + 0.991934i $$0.459544\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 9.00000i 0.454569i
$$393$$ 0 0
$$394$$ 16.0000 0.806068
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 15.0000i 0.752828i 0.926451 + 0.376414i $$0.122843\pi$$
−0.926451 + 0.376414i $$0.877157\pi$$
$$398$$ 3.00000i 0.150376i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 4.00000i 0.199254i
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ −20.0000 −0.992583
$$407$$ 36.0000i 1.78445i
$$408$$ 0 0
$$409$$ 24.0000 1.18672 0.593362 0.804936i $$-0.297800\pi$$
0.593362 + 0.804936i $$0.297800\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 8.00000i − 0.394132i
$$413$$ − 24.0000i − 1.18096i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ 28.0000i 1.36952i
$$419$$ 23.0000 1.12362 0.561812 0.827265i $$-0.310105\pi$$
0.561812 + 0.827265i $$0.310105\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ − 16.0000i − 0.778868i
$$423$$ 0 0
$$424$$ −9.00000 −0.437079
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 16.0000i 0.774294i
$$428$$ − 5.00000i − 0.241684i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −19.0000 −0.915198 −0.457599 0.889159i $$-0.651290\pi$$
−0.457599 + 0.889159i $$0.651290\pi$$
$$432$$ 0 0
$$433$$ − 19.0000i − 0.913082i −0.889702 0.456541i $$-0.849088\pi$$
0.889702 0.456541i $$-0.150912\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ −11.0000 −0.526804
$$437$$ 28.0000i 1.33942i
$$438$$ 0 0
$$439$$ 7.00000 0.334092 0.167046 0.985949i $$-0.446577\pi$$
0.167046 + 0.985949i $$0.446577\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 4.00000i 0.190261i
$$443$$ − 25.0000i − 1.18779i −0.804544 0.593893i $$-0.797590\pi$$
0.804544 0.593893i $$-0.202410\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 2.00000 0.0947027
$$447$$ 0 0
$$448$$ − 4.00000i − 0.188982i
$$449$$ −27.0000 −1.27421 −0.637104 0.770778i $$-0.719868\pi$$
−0.637104 + 0.770778i $$0.719868\pi$$
$$450$$ 0 0
$$451$$ −20.0000 −0.941763
$$452$$ − 14.0000i − 0.658505i
$$453$$ 0 0
$$454$$ −6.00000 −0.281594
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 4.00000i − 0.187112i −0.995614 0.0935561i $$-0.970177\pi$$
0.995614 0.0935561i $$-0.0298234\pi$$
$$458$$ 17.0000i 0.794358i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 24.0000 1.11779 0.558896 0.829238i $$-0.311225\pi$$
0.558896 + 0.829238i $$0.311225\pi$$
$$462$$ 0 0
$$463$$ − 26.0000i − 1.20832i −0.796862 0.604161i $$-0.793508\pi$$
0.796862 0.604161i $$-0.206492\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ 8.00000 0.370593
$$467$$ 3.00000i 0.138823i 0.997588 + 0.0694117i $$0.0221122\pi$$
−0.997588 + 0.0694117i $$0.977888\pi$$
$$468$$ 0 0
$$469$$ −28.0000 −1.29292
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 6.00000i 0.276172i
$$473$$ 40.0000i 1.83920i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −16.0000 −0.733359
$$477$$ 0 0
$$478$$ 16.0000i 0.731823i
$$479$$ 15.0000 0.685367 0.342684 0.939451i $$-0.388664\pi$$
0.342684 + 0.939451i $$0.388664\pi$$
$$480$$ 0 0
$$481$$ 9.00000 0.410365
$$482$$ 8.00000i 0.364390i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 22.0000i − 0.996915i −0.866914 0.498458i $$-0.833900\pi$$
0.866914 0.498458i $$-0.166100\pi$$
$$488$$ − 4.00000i − 0.181071i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 40.0000 1.80517 0.902587 0.430507i $$-0.141665\pi$$
0.902587 + 0.430507i $$0.141665\pi$$
$$492$$ 0 0
$$493$$ − 20.0000i − 0.900755i
$$494$$ 7.00000 0.314945
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 60.0000i 2.69137i
$$498$$ 0 0
$$499$$ 23.0000 1.02962 0.514811 0.857304i $$-0.327862\pi$$
0.514811 + 0.857304i $$0.327862\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 5.00000i 0.223161i
$$503$$ − 36.0000i − 1.60516i −0.596544 0.802580i $$-0.703460\pi$$
0.596544 0.802580i $$-0.296540\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −16.0000 −0.711287
$$507$$ 0 0
$$508$$ − 5.00000i − 0.221839i
$$509$$ −4.00000 −0.177297 −0.0886484 0.996063i $$-0.528255\pi$$
−0.0886484 + 0.996063i $$0.528255\pi$$
$$510$$ 0 0
$$511$$ −48.0000 −2.12339
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ −8.00000 −0.352865
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 12.0000i − 0.527759i
$$518$$ 36.0000i 1.58175i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −36.0000 −1.57719 −0.788594 0.614914i $$-0.789191\pi$$
−0.788594 + 0.614914i $$0.789191\pi$$
$$522$$ 0 0
$$523$$ − 14.0000i − 0.612177i −0.952003 0.306089i $$-0.900980\pi$$
0.952003 0.306089i $$-0.0990204\pi$$
$$524$$ 17.0000 0.742648
$$525$$ 0 0
$$526$$ 30.0000 1.30806
$$527$$ − 16.0000i − 0.696971i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 28.0000i 1.21395i
$$533$$ 5.00000i 0.216574i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 7.00000 0.302354
$$537$$ 0 0
$$538$$ 5.00000i 0.215565i
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ 32.0000i 1.37452i
$$543$$ 0 0
$$544$$ 4.00000 0.171499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 6.00000i − 0.256541i −0.991739 0.128271i $$-0.959057\pi$$
0.991739 0.128271i $$-0.0409426\pi$$
$$548$$ 19.0000i 0.811640i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −35.0000 −1.49105
$$552$$ 0 0
$$553$$ − 28.0000i − 1.19068i
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 38.0000i − 1.61011i −0.593199 0.805056i $$-0.702135\pi$$
0.593199 0.805056i $$-0.297865\pi$$
$$558$$ 0 0
$$559$$ 10.0000 0.422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 17.0000i − 0.717102i
$$563$$ − 39.0000i − 1.64365i −0.569737 0.821827i $$-0.692955\pi$$
0.569737 0.821827i $$-0.307045\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −8.00000 −0.336265
$$567$$ 0 0
$$568$$ − 15.0000i − 0.629386i
$$569$$ 8.00000 0.335377 0.167689 0.985840i $$-0.446370\pi$$
0.167689 + 0.985840i $$0.446370\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 0 0
$$574$$ −20.0000 −0.834784
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8.00000i 0.333044i 0.986038 + 0.166522i $$0.0532537\pi$$
−0.986038 + 0.166522i $$0.946746\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ 36.0000i 1.49097i
$$584$$ 12.0000 0.496564
$$585$$ 0 0
$$586$$ −12.0000 −0.495715
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ −28.0000 −1.15372
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 9.00000i − 0.369898i
$$593$$ − 29.0000i − 1.19089i −0.803397 0.595444i $$-0.796976\pi$$
0.803397 0.595444i $$-0.203024\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4.00000 0.163846
$$597$$ 0 0
$$598$$ 4.00000i 0.163572i
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ −3.00000 −0.122373 −0.0611863 0.998126i $$-0.519488\pi$$
−0.0611863 + 0.998126i $$0.519488\pi$$
$$602$$ 40.0000i 1.63028i
$$603$$ 0 0
$$604$$ 4.00000 0.162758
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 3.00000i 0.121766i 0.998145 + 0.0608831i $$0.0193917\pi$$
−0.998145 + 0.0608831i $$0.980608\pi$$
$$608$$ − 7.00000i − 0.283887i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −3.00000 −0.121367
$$612$$ 0 0
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 21.0000 0.847491
$$615$$ 0 0
$$616$$ −16.0000 −0.644658
$$617$$ 7.00000i 0.281809i 0.990023 + 0.140905i $$0.0450011\pi$$
−0.990023 + 0.140905i $$0.954999\pi$$
$$618$$ 0 0
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 2.00000i − 0.0801927i
$$623$$ 56.0000i 2.24359i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 23.0000 0.919265
$$627$$ 0 0
$$628$$ − 14.0000i − 0.558661i
$$629$$ −36.0000 −1.43541
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 7.00000i 0.278445i
$$633$$ 0 0
$$634$$ 28.0000 1.11202
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 9.00000i − 0.356593i
$$638$$ − 20.0000i − 0.791808i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ − 5.00000i − 0.197181i −0.995128 0.0985904i $$-0.968567\pi$$
0.995128 0.0985904i $$-0.0314334\pi$$
$$644$$ −16.0000 −0.630488
$$645$$ 0 0
$$646$$ −28.0000 −1.10165
$$647$$ − 36.0000i − 1.41531i −0.706560 0.707653i $$-0.749754\pi$$
0.706560 0.707653i $$-0.250246\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 24.0000i 0.939913i
$$653$$ 10.0000i 0.391330i 0.980671 + 0.195665i $$0.0626866\pi$$
−0.980671 + 0.195665i $$0.937313\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 5.00000 0.195217
$$657$$ 0 0
$$658$$ − 12.0000i − 0.467809i
$$659$$ 33.0000 1.28550 0.642749 0.766077i $$-0.277794\pi$$
0.642749 + 0.766077i $$0.277794\pi$$
$$660$$ 0 0
$$661$$ −7.00000 −0.272268 −0.136134 0.990690i $$-0.543468\pi$$
−0.136134 + 0.990690i $$0.543468\pi$$
$$662$$ − 28.0000i − 1.08825i
$$663$$ 0 0
$$664$$ −6.00000 −0.232845
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 20.0000i − 0.774403i
$$668$$ − 9.00000i − 0.348220i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −16.0000 −0.617673
$$672$$ 0 0
$$673$$ 19.0000i 0.732396i 0.930537 + 0.366198i $$0.119341\pi$$
−0.930537 + 0.366198i $$0.880659\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ − 14.0000i − 0.538064i −0.963131 0.269032i $$-0.913296\pi$$
0.963131 0.269032i $$-0.0867037\pi$$
$$678$$ 0 0
$$679$$ −64.0000 −2.45609
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 16.0000i − 0.612672i
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ 0 0
$$688$$ − 10.0000i − 0.381246i
$$689$$ 9.00000 0.342873
$$690$$ 0 0
$$691$$ 3.00000 0.114125 0.0570627 0.998371i $$-0.481827\pi$$
0.0570627 + 0.998371i $$0.481827\pi$$
$$692$$ 13.0000i 0.494186i
$$693$$ 0 0
$$694$$ −11.0000 −0.417554
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 20.0000i − 0.757554i
$$698$$ 34.0000i 1.28692i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ 0 0
$$703$$ 63.0000i 2.37609i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −21.0000 −0.790345
$$707$$ 40.0000i 1.50435i
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 14.0000i − 0.524672i
$$713$$ − 16.0000i − 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ − 15.0000i − 0.559795i
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 0 0
$$721$$ −32.0000 −1.19174
$$722$$ 30.0000i 1.11648i
$$723$$ 0 0
$$724$$ 4.00000 0.148659
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 28.0000i − 1.03846i −0.854634 0.519231i $$-0.826218\pi$$
0.854634 0.519231i $$-0.173782\pi$$
$$728$$ 4.00000i 0.148250i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −40.0000 −1.47945
$$732$$ 0 0
$$733$$ − 3.00000i − 0.110808i −0.998464 0.0554038i $$-0.982355\pi$$
0.998464 0.0554038i $$-0.0176446\pi$$
$$734$$ 7.00000 0.258375
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ − 28.0000i − 1.03139i
$$738$$ 0 0
$$739$$ 37.0000 1.36107 0.680534 0.732717i $$-0.261748\pi$$
0.680534 + 0.732717i $$0.261748\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 36.0000i 1.32160i
$$743$$ 21.0000i 0.770415i 0.922830 + 0.385208i $$0.125870\pi$$
−0.922830 + 0.385208i $$0.874130\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 0 0
$$748$$ − 16.0000i − 0.585018i
$$749$$ −20.0000 −0.730784
$$750$$ 0 0
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ 3.00000i 0.109399i
$$753$$ 0 0
$$754$$ −5.00000 −0.182089
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 8.00000i − 0.290765i −0.989376 0.145382i $$-0.953559\pi$$
0.989376 0.145382i $$-0.0464413\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −35.0000 −1.26875 −0.634375 0.773026i $$-0.718742\pi$$
−0.634375 + 0.773026i $$0.718742\pi$$
$$762$$ 0 0
$$763$$ 44.0000i 1.59291i
$$764$$ 10.0000 0.361787
$$765$$ 0 0
$$766$$ 5.00000 0.180657
$$767$$ − 6.00000i − 0.216647i
$$768$$ 0 0
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 12.0000i 0.431889i
$$773$$ 8.00000i 0.287740i 0.989597 + 0.143870i $$0.0459547\pi$$
−0.989597 + 0.143870i $$0.954045\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 16.0000 0.574367
$$777$$ 0 0
$$778$$ 5.00000i 0.179259i
$$779$$ −35.0000 −1.25401
$$780$$ 0 0
$$781$$ −60.0000 −2.14697
$$782$$ − 16.0000i − 0.572159i
$$783$$ 0 0
$$784$$ −9.00000 −0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ 16.0000i 0.569976i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −56.0000 −1.99113
$$792$$ 0 0
$$793$$ 4.00000i 0.142044i
$$794$$ −15.0000 −0.532330
$$795$$ 0 0
$$796$$ −3.00000 −0.106332
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 30.0000i 1.05934i
$$803$$ − 48.0000i − 1.69388i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ − 10.0000i − 0.351799i
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ − 20.0000i − 0.701862i
$$813$$ 0 0
$$814$$ −36.0000 −1.26180
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 70.0000i 2.44899i
$$818$$ 24.0000i 0.839140i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 0 0
$$823$$ − 19.0000i − 0.662298i −0.943578 0.331149i $$-0.892564\pi$$
0.943578 0.331149i $$-0.107436\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ − 4.00000i − 0.139094i −0.997579 0.0695468i $$-0.977845\pi$$
0.997579 0.0695468i $$-0.0221553\pi$$
$$828$$ 0 0
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 1.00000i − 0.0346688i
$$833$$ 36.0000i 1.24733i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −28.0000 −0.968400
$$837$$ 0 0
$$838$$ 23.0000i 0.794522i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 14.0000i 0.482472i
$$843$$ 0 0
$$844$$ 16.0000 0.550743
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 20.0000i 0.687208i
$$848$$ − 9.00000i − 0.309061i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −36.0000 −1.23406
$$852$$ 0 0
$$853$$ 19.0000i 0.650548i 0.945620 + 0.325274i $$0.105456\pi$$
−0.945620 + 0.325274i $$0.894544\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ 5.00000 0.170896
$$857$$ − 50.0000i − 1.70797i −0.520300 0.853984i $$-0.674180\pi$$
0.520300 0.853984i $$-0.325820\pi$$
$$858$$ 0 0
$$859$$ 2.00000 0.0682391 0.0341196 0.999418i $$-0.489137\pi$$
0.0341196 + 0.999418i $$0.489137\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 19.0000i − 0.647143i
$$863$$ − 39.0000i − 1.32758i −0.747921 0.663788i $$-0.768948\pi$$
0.747921 0.663788i $$-0.231052\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 19.0000 0.645646
$$867$$ 0 0
$$868$$ − 16.0000i − 0.543075i
$$869$$ 28.0000 0.949835
$$870$$ 0 0
$$871$$ −7.00000 −0.237186
$$872$$ − 11.0000i − 0.372507i
$$873$$ 0 0
$$874$$ −28.0000 −0.947114
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 43.0000i − 1.45201i −0.687691 0.726003i $$-0.741376\pi$$
0.687691 0.726003i $$-0.258624\pi$$
$$878$$ 7.00000i 0.236239i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 40.0000i 1.34611i 0.739594 + 0.673054i $$0.235018\pi$$
−0.739594 + 0.673054i $$0.764982\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 25.0000 0.839891
$$887$$ − 36.0000i − 1.20876i −0.796696 0.604381i $$-0.793421\pi$$
0.796696 0.604381i $$-0.206579\pi$$
$$888$$ 0 0
$$889$$ −20.0000 −0.670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.00000i 0.0669650i
$$893$$ − 21.0000i − 0.702738i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ − 27.0000i − 0.901002i
$$899$$ 20.0000 0.667037
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ − 20.0000i − 0.665927i
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 50.0000i − 1.66022i −0.557598 0.830111i $$-0.688277\pi$$
0.557598 0.830111i $$-0.311723\pi$$
$$908$$ − 6.00000i − 0.199117i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 18.0000 0.596367 0.298183 0.954509i $$-0.403619\pi$$
0.298183 + 0.954509i $$0.403619\pi$$
$$912$$ 0 0
$$913$$ 24.0000i 0.794284i
$$914$$ 4.00000 0.132308
$$915$$ 0 0
$$916$$ −17.0000 −0.561696
$$917$$ − 68.0000i − 2.24556i
$$918$$ 0 0
$$919$$ −29.0000 −0.956622 −0.478311 0.878191i $$-0.658751\pi$$
−0.478311 + 0.878191i $$0.658751\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 24.0000i 0.790398i
$$923$$ 15.0000i 0.493731i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 26.0000 0.854413
$$927$$ 0 0
$$928$$ 5.00000i 0.164133i
$$929$$ 11.0000 0.360898 0.180449 0.983584i $$-0.442245\pi$$
0.180449 + 0.983584i $$0.442245\pi$$
$$930$$ 0 0
$$931$$ 63.0000 2.06474
$$932$$ 8.00000i 0.262049i
$$933$$ 0 0
$$934$$ −3.00000 −0.0981630
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000i 0.718709i 0.933201 + 0.359354i $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ − 28.0000i − 0.914232i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −52.0000 −1.69515 −0.847576 0.530674i $$-0.821939\pi$$
−0.847576 + 0.530674i $$0.821939\pi$$
$$942$$ 0 0
$$943$$ − 20.0000i − 0.651290i
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ −40.0000 −1.30051
$$947$$ − 8.00000i − 0.259965i −0.991516 0.129983i $$-0.958508\pi$$
0.991516 0.129983i $$-0.0414921\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 16.0000i − 0.518563i
$$953$$ − 36.0000i − 1.16615i −0.812417 0.583077i $$-0.801849\pi$$
0.812417 0.583077i $$-0.198151\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 15.0000i 0.484628i
$$959$$ 76.0000 2.45417
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 9.00000i 0.290172i
$$963$$ 0 0
$$964$$ −8.00000 −0.257663
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 44.0000i 1.41494i 0.706741 + 0.707472i $$0.250165\pi$$
−0.706741 + 0.707472i $$0.749835\pi$$
$$968$$ − 5.00000i − 0.160706i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 21.0000 0.673922 0.336961 0.941519i $$-0.390601\pi$$
0.336961 + 0.941519i $$0.390601\pi$$
$$972$$ 0 0
$$973$$ 16.0000i 0.512936i
$$974$$ 22.0000 0.704925
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ 18.0000i 0.575871i 0.957650 + 0.287936i $$0.0929689\pi$$
−0.957650 + 0.287936i $$0.907031\pi$$
$$978$$ 0 0
$$979$$ −56.0000 −1.78977
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 40.0000i 1.27645i
$$983$$ 48.0000i 1.53096i 0.643458 + 0.765481i $$0.277499\pi$$
−0.643458 + 0.765481i $$0.722501\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 20.0000 0.636930
$$987$$ 0 0
$$988$$ 7.00000i 0.222700i
$$989$$ −40.0000 −1.27193
$$990$$ 0 0
$$991$$ 13.0000 0.412959 0.206479 0.978451i $$-0.433799\pi$$
0.206479 + 0.978451i $$0.433799\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ 0 0
$$994$$ −60.0000 −1.90308
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 10.0000i 0.316703i 0.987383 + 0.158352i $$0.0506179\pi$$
−0.987383 + 0.158352i $$0.949382\pi$$
$$998$$ 23.0000i 0.728052i
$$999$$ 0 0
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 5850.2.e.c.5149.2 2
3.2 odd 2 1950.2.e.n.1249.1 2
5.2 odd 4 5850.2.a.a.1.1 1
5.3 odd 4 5850.2.a.by.1.1 1
5.4 even 2 inner 5850.2.e.c.5149.1 2
15.2 even 4 1950.2.a.x.1.1 yes 1
15.8 even 4 1950.2.a.e.1.1 1
15.14 odd 2 1950.2.e.n.1249.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.e.1.1 1 15.8 even 4
1950.2.a.x.1.1 yes 1 15.2 even 4
1950.2.e.n.1249.1 2 3.2 odd 2
1950.2.e.n.1249.2 2 15.14 odd 2
5850.2.a.a.1.1 1 5.2 odd 4
5850.2.a.by.1.1 1 5.3 odd 4
5850.2.e.c.5149.1 2 5.4 even 2 inner
5850.2.e.c.5149.2 2 1.1 even 1 trivial