# Properties

 Label 5850.2.a.a.1.1 Level $5850$ Weight $2$ Character 5850.1 Self dual yes Analytic conductor $46.712$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.7124851824$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1950) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5850.1

## $q$-expansion

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

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 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
$$13$$ 1.00000 0.277350
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ −4.00000 −0.755929
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.00000 1.47959 0.739795 0.672832i $$-0.234922\pi$$
0.739795 + 0.672832i $$0.234922\pi$$
$$38$$ −7.00000 −1.13555
$$39$$ 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
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ 0 0
$$58$$ 5.00000 0.656532
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7.00000 −0.855186 −0.427593 0.903971i $$-0.640638\pi$$
−0.427593 + 0.903971i $$0.640638\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 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
$$73$$ 12.0000 1.40449 0.702247 0.711934i $$-0.252180\pi$$
0.702247 + 0.711934i $$0.252180\pi$$
$$74$$ −9.00000 −1.04623
$$75$$ 0 0
$$76$$ 7.00000 0.802955
$$77$$ 16.0000 1.82337
$$78$$ 0 0
$$79$$ 7.00000 0.787562 0.393781 0.919204i $$-0.371167\pi$$
0.393781 + 0.919204i $$0.371167\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −5.00000 −0.552158
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 10.0000 1.07833
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ 3.00000 0.309426
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ −9.00000 −0.909137
$$99$$ 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
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ −5.00000 −0.483368 −0.241684 0.970355i $$-0.577700\pi$$
−0.241684 + 0.970355i $$0.577700\pi$$
$$108$$ 0 0
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −4.00000 −0.377964
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −5.00000 −0.464238
$$117$$ 0 0
$$118$$ −6.00000 −0.552345
$$119$$ −16.0000 −1.46672
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −4.00000 −0.362143
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −5.00000 −0.443678 −0.221839 0.975083i $$-0.571206\pi$$
−0.221839 + 0.975083i $$0.571206\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 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
$$133$$ −28.0000 −2.42791
$$134$$ 7.00000 0.604708
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 19.0000 1.62328 0.811640 0.584158i $$-0.198575\pi$$
0.811640 + 0.584158i $$0.198575\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −15.0000 −1.25877
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −12.0000 −0.993127
$$147$$ 0 0
$$148$$ 9.00000 0.739795
$$149$$ 4.00000 0.327693 0.163846 0.986486i $$-0.447610\pi$$
0.163846 + 0.986486i $$0.447610\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ −7.00000 −0.567775
$$153$$ 0 0
$$154$$ −16.0000 −1.28932
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ −7.00000 −0.556890
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ −24.0000 −1.87983 −0.939913 0.341415i $$-0.889094\pi$$
−0.939913 + 0.341415i $$0.889094\pi$$
$$164$$ 5.00000 0.390434
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ −9.00000 −0.696441 −0.348220 0.937413i $$-0.613214\pi$$
−0.348220 + 0.937413i $$0.613214\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −10.0000 −0.762493
$$173$$ −13.0000 −0.988372 −0.494186 0.869356i $$-0.664534\pi$$
−0.494186 + 0.869356i $$0.664534\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 14.0000 1.04934
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −4.00000 −0.297318 −0.148659 0.988889i $$-0.547496\pi$$
−0.148659 + 0.988889i $$0.547496\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ 4.00000 0.294884
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −16.0000 −1.17004
$$188$$ −3.00000 −0.218797
$$189$$ 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
$$193$$ −12.0000 −0.863779 −0.431889 0.901927i $$-0.642153\pi$$
−0.431889 + 0.901927i $$0.642153\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 16.0000 1.13995 0.569976 0.821661i $$-0.306952\pi$$
0.569976 + 0.821661i $$0.306952\pi$$
$$198$$ 0 0
$$199$$ −3.00000 −0.212664 −0.106332 0.994331i $$-0.533911\pi$$
−0.106332 + 0.994331i $$0.533911\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ 20.0000 1.40372
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −28.0000 −1.93680
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ −9.00000 −0.618123
$$213$$ 0 0
$$214$$ 5.00000 0.341793
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 11.0000 0.745014
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ −6.00000 −0.398234 −0.199117 0.979976i $$-0.563807\pi$$
−0.199117 + 0.979976i $$0.563807\pi$$
$$228$$ 0 0
$$229$$ −17.0000 −1.12339 −0.561696 0.827344i $$-0.689851\pi$$
−0.561696 + 0.827344i $$0.689851\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 5.00000 0.328266
$$233$$ −8.00000 −0.524097 −0.262049 0.965055i $$-0.584398\pi$$
−0.262049 + 0.965055i $$0.584398\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ 0 0
$$238$$ 16.0000 1.03713
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 0 0
$$244$$ 4.00000 0.256074
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 7.00000 0.445399
$$248$$ −4.00000 −0.254000
$$249$$ 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
$$253$$ 16.0000 1.00591
$$254$$ 5.00000 0.313728
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −8.00000 −0.499026 −0.249513 0.968371i $$-0.580271\pi$$
−0.249513 + 0.968371i $$0.580271\pi$$
$$258$$ 0 0
$$259$$ −36.0000 −2.23693
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 17.0000 1.05026
$$263$$ −30.0000 −1.84988 −0.924940 0.380114i $$-0.875885\pi$$
−0.924940 + 0.380114i $$0.875885\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 28.0000 1.71679
$$267$$ 0 0
$$268$$ −7.00000 −0.427593
$$269$$ −5.00000 −0.304855 −0.152428 0.988315i $$-0.548709\pi$$
−0.152428 + 0.988315i $$0.548709\pi$$
$$270$$ 0 0
$$271$$ 32.0000 1.94386 0.971931 0.235267i $$-0.0755965\pi$$
0.971931 + 0.235267i $$0.0755965\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −19.0000 −1.14783
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 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
$$283$$ 8.00000 0.475551 0.237775 0.971320i $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ 15.0000 0.890086
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ −20.0000 −1.18056
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 12.0000 0.702247
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −9.00000 −0.523114
$$297$$ 0 0
$$298$$ −4.00000 −0.231714
$$299$$ −4.00000 −0.231326
$$300$$ 0 0
$$301$$ 40.0000 2.30556
$$302$$ 4.00000 0.230174
$$303$$ 0 0
$$304$$ 7.00000 0.401478
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 21.0000 1.19853 0.599267 0.800549i $$-0.295459\pi$$
0.599267 + 0.800549i $$0.295459\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 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
$$313$$ −23.0000 −1.30004 −0.650018 0.759918i $$-0.725239\pi$$
−0.650018 + 0.759918i $$0.725239\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 7.00000 0.393781
$$317$$ 28.0000 1.57264 0.786318 0.617822i $$-0.211985\pi$$
0.786318 + 0.617822i $$0.211985\pi$$
$$318$$ 0 0
$$319$$ 20.0000 1.11979
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −16.0000 −0.891645
$$323$$ 28.0000 1.55796
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 24.0000 1.32924
$$327$$ 0 0
$$328$$ −5.00000 −0.276079
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ 0 0
$$334$$ 9.00000 0.492458
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 10.0000 0.539164
$$345$$ 0 0
$$346$$ 13.0000 0.698884
$$347$$ −11.0000 −0.590511 −0.295255 0.955418i $$-0.595405\pi$$
−0.295255 + 0.955418i $$0.595405\pi$$
$$348$$ 0 0
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ 21.0000 1.11772 0.558859 0.829263i $$-0.311239\pi$$
0.558859 + 0.829263i $$0.311239\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ −4.00000 −0.211407
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 4.00000 0.210235
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 7.00000 0.365397 0.182699 0.983169i $$-0.441517\pi$$
0.182699 + 0.983169i $$0.441517\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 36.0000 1.86903
$$372$$ 0 0
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 16.0000 0.827340
$$375$$ 0 0
$$376$$ 3.00000 0.154713
$$377$$ −5.00000 −0.257513
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 10.0000 0.511645
$$383$$ −5.00000 −0.255488 −0.127744 0.991807i $$-0.540774\pi$$
−0.127744 + 0.991807i $$0.540774\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 12.0000 0.610784
$$387$$ 0 0
$$388$$ −16.0000 −0.812277
$$389$$ −5.00000 −0.253510 −0.126755 0.991934i $$-0.540456\pi$$
−0.126755 + 0.991934i $$0.540456\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ −9.00000 −0.454569
$$393$$ 0 0
$$394$$ −16.0000 −0.806068
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −15.0000 −0.752828 −0.376414 0.926451i $$-0.622843\pi$$
−0.376414 + 0.926451i $$0.622843\pi$$
$$398$$ 3.00000 0.150376
$$399$$ 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
$$403$$ 4.00000 0.199254
$$404$$ 10.0000 0.497519
$$405$$ 0 0
$$406$$ −20.0000 −0.992583
$$407$$ −36.0000 −1.78445
$$408$$ 0 0
$$409$$ −24.0000 −1.18672 −0.593362 0.804936i $$-0.702200\pi$$
−0.593362 + 0.804936i $$0.702200\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ −24.0000 −1.18096
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ 28.0000 1.36952
$$419$$ −23.0000 −1.12362 −0.561812 0.827265i $$-0.689895\pi$$
−0.561812 + 0.827265i $$0.689895\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 16.0000 0.778868
$$423$$ 0 0
$$424$$ 9.00000 0.437079
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −16.0000 −0.774294
$$428$$ −5.00000 −0.241684
$$429$$ 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
$$433$$ −19.0000 −0.913082 −0.456541 0.889702i $$-0.650912\pi$$
−0.456541 + 0.889702i $$0.650912\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ −11.0000 −0.526804
$$437$$ −28.0000 −1.33942
$$438$$ 0 0
$$439$$ −7.00000 −0.334092 −0.167046 0.985949i $$-0.553423\pi$$
−0.167046 + 0.985949i $$0.553423\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −4.00000 −0.190261
$$443$$ −25.0000 −1.18779 −0.593893 0.804544i $$-0.702410\pi$$
−0.593893 + 0.804544i $$0.702410\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 2.00000 0.0947027
$$447$$ 0 0
$$448$$ −4.00000 −0.188982
$$449$$ 27.0000 1.27421 0.637104 0.770778i $$-0.280132\pi$$
0.637104 + 0.770778i $$0.280132\pi$$
$$450$$ 0 0
$$451$$ −20.0000 −0.941763
$$452$$ 14.0000 0.658505
$$453$$ 0 0
$$454$$ 6.00000 0.281594
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.00000 0.187112 0.0935561 0.995614i $$-0.470177\pi$$
0.0935561 + 0.995614i $$0.470177\pi$$
$$458$$ 17.0000 0.794358
$$459$$ 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
$$463$$ −26.0000 −1.20832 −0.604161 0.796862i $$-0.706492\pi$$
−0.604161 + 0.796862i $$0.706492\pi$$
$$464$$ −5.00000 −0.232119
$$465$$ 0 0
$$466$$ 8.00000 0.370593
$$467$$ −3.00000 −0.138823 −0.0694117 0.997588i $$-0.522112\pi$$
−0.0694117 + 0.997588i $$0.522112\pi$$
$$468$$ 0 0
$$469$$ 28.0000 1.29292
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −6.00000 −0.276172
$$473$$ 40.0000 1.83920
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −16.0000 −0.733359
$$477$$ 0 0
$$478$$ 16.0000 0.731823
$$479$$ −15.0000 −0.685367 −0.342684 0.939451i $$-0.611336\pi$$
−0.342684 + 0.939451i $$0.611336\pi$$
$$480$$ 0 0
$$481$$ 9.00000 0.410365
$$482$$ −8.00000 −0.364390
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 22.0000 0.996915 0.498458 0.866914i $$-0.333900\pi$$
0.498458 + 0.866914i $$0.333900\pi$$
$$488$$ −4.00000 −0.181071
$$489$$ 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
$$493$$ −20.0000 −0.900755
$$494$$ −7.00000 −0.314945
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ −60.0000 −2.69137
$$498$$ 0 0
$$499$$ −23.0000 −1.02962 −0.514811 0.857304i $$-0.672138\pi$$
−0.514811 + 0.857304i $$0.672138\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −5.00000 −0.223161
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −16.0000 −0.711287
$$507$$ 0 0
$$508$$ −5.00000 −0.221839
$$509$$ 4.00000 0.177297 0.0886484 0.996063i $$-0.471745\pi$$
0.0886484 + 0.996063i $$0.471745\pi$$
$$510$$ 0 0
$$511$$ −48.0000 −2.12339
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 8.00000 0.352865
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 12.0000 0.527759
$$518$$ 36.0000 1.58175
$$519$$ 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
$$523$$ −14.0000 −0.612177 −0.306089 0.952003i $$-0.599020\pi$$
−0.306089 + 0.952003i $$0.599020\pi$$
$$524$$ −17.0000 −0.742648
$$525$$ 0 0
$$526$$ 30.0000 1.30806
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −28.0000 −1.21395
$$533$$ 5.00000 0.216574
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 7.00000 0.302354
$$537$$ 0 0
$$538$$ 5.00000 0.215565
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ −32.0000 −1.37452
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 6.00000 0.256541 0.128271 0.991739i $$-0.459057\pi$$
0.128271 + 0.991739i $$0.459057\pi$$
$$548$$ 19.0000 0.811640
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −35.0000 −1.49105
$$552$$ 0 0
$$553$$ −28.0000 −1.19068
$$554$$ −26.0000 −1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 38.0000 1.61011 0.805056 0.593199i $$-0.202135\pi$$
0.805056 + 0.593199i $$0.202135\pi$$
$$558$$ 0 0
$$559$$ −10.0000 −0.422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 17.0000 0.717102
$$563$$ −39.0000 −1.64365 −0.821827 0.569737i $$-0.807045\pi$$
−0.821827 + 0.569737i $$0.807045\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −8.00000 −0.336265
$$567$$ 0 0
$$568$$ −15.0000 −0.629386
$$569$$ −8.00000 −0.335377 −0.167689 0.985840i $$-0.553630\pi$$
−0.167689 + 0.985840i $$0.553630\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 0 0
$$574$$ 20.0000 0.834784
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −8.00000 −0.333044 −0.166522 0.986038i $$-0.553254\pi$$
−0.166522 + 0.986038i $$0.553254\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ −12.0000 −0.496564
$$585$$ 0 0
$$586$$ −12.0000 −0.495715
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ 28.0000 1.15372
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 9.00000 0.369898
$$593$$ −29.0000 −1.19089 −0.595444 0.803397i $$-0.703024\pi$$
−0.595444 + 0.803397i $$0.703024\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4.00000 0.163846
$$597$$ 0 0
$$598$$ 4.00000 0.163572
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ −3.00000 −0.122373 −0.0611863 0.998126i $$-0.519488\pi$$
−0.0611863 + 0.998126i $$0.519488\pi$$
$$602$$ −40.0000 −1.63028
$$603$$ 0 0
$$604$$ −4.00000 −0.162758
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −3.00000 −0.121766 −0.0608831 0.998145i $$-0.519392\pi$$
−0.0608831 + 0.998145i $$0.519392\pi$$
$$608$$ −7.00000 −0.283887
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −3.00000 −0.121367
$$612$$ 0 0
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ −21.0000 −0.847491
$$615$$ 0 0
$$616$$ −16.0000 −0.644658
$$617$$ −7.00000 −0.281809 −0.140905 0.990023i $$-0.545001\pi$$
−0.140905 + 0.990023i $$0.545001\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 2.00000 0.0801927
$$623$$ 56.0000 2.24359
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 23.0000 0.919265
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ −7.00000 −0.278445
$$633$$ 0 0
$$634$$ −28.0000 −1.11202
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 9.00000 0.356593
$$638$$ −20.0000 −0.791808
$$639$$ 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
$$643$$ −5.00000 −0.197181 −0.0985904 0.995128i $$-0.531433\pi$$
−0.0985904 + 0.995128i $$0.531433\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ −28.0000 −1.10165
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −24.0000 −0.939913
$$653$$ 10.0000 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 5.00000 0.195217
$$657$$ 0 0
$$658$$ −12.0000 −0.467809
$$659$$ −33.0000 −1.28550 −0.642749 0.766077i $$-0.722206\pi$$
−0.642749 + 0.766077i $$0.722206\pi$$
$$660$$ 0 0
$$661$$ −7.00000 −0.272268 −0.136134 0.990690i $$-0.543468\pi$$
−0.136134 + 0.990690i $$0.543468\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 20.0000 0.774403
$$668$$ −9.00000 −0.348220
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −16.0000 −0.617673
$$672$$ 0 0
$$673$$ 19.0000 0.732396 0.366198 0.930537i $$-0.380659\pi$$
0.366198 + 0.930537i $$0.380659\pi$$
$$674$$ 34.0000 1.30963
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 14.0000 0.538064 0.269032 0.963131i $$-0.413296\pi$$
0.269032 + 0.963131i $$0.413296\pi$$
$$678$$ 0 0
$$679$$ 64.0000 2.45609
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 16.0000 0.612672
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ 0 0
$$688$$ −10.0000 −0.381246
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ 3.00000 0.114125 0.0570627 0.998371i $$-0.481827\pi$$
0.0570627 + 0.998371i $$0.481827\pi$$
$$692$$ −13.0000 −0.494186
$$693$$ 0 0
$$694$$ 11.0000 0.417554
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 20.0000 0.757554
$$698$$ 34.0000 1.28692
$$699$$ 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
$$703$$ 63.0000 2.37609
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −21.0000 −0.790345
$$707$$ −40.0000 −1.50435
$$708$$ 0 0
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 14.0000 0.524672
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ −15.0000 −0.559795
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ 0 0
$$721$$ −32.0000 −1.19174
$$722$$ −30.0000 −1.11648
$$723$$ 0 0
$$724$$ −4.00000 −0.148659
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −40.0000 −1.47945
$$732$$ 0 0
$$733$$ −3.00000 −0.110808 −0.0554038 0.998464i $$-0.517645\pi$$
−0.0554038 + 0.998464i $$0.517645\pi$$
$$734$$ −7.00000 −0.258375
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 28.0000 1.03139
$$738$$ 0 0
$$739$$ −37.0000 −1.36107 −0.680534 0.732717i $$-0.738252\pi$$
−0.680534 + 0.732717i $$0.738252\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −36.0000 −1.32160
$$743$$ 21.0000 0.770415 0.385208 0.922830i $$-0.374130\pi$$
0.385208 + 0.922830i $$0.374130\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 0 0
$$748$$ −16.0000 −0.585018
$$749$$ 20.0000 0.730784
$$750$$ 0 0
$$751$$ −23.0000 −0.839282 −0.419641 0.907690i $$-0.637844\pi$$
−0.419641 + 0.907690i $$0.637844\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ 0 0
$$754$$ 5.00000 0.182089
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 8.00000 0.290765 0.145382 0.989376i $$-0.453559\pi$$
0.145382 + 0.989376i $$0.453559\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 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
$$763$$ 44.0000 1.59291
$$764$$ −10.0000 −0.361787
$$765$$ 0 0
$$766$$ 5.00000 0.180657
$$767$$ 6.00000 0.216647
$$768$$ 0 0
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −12.0000 −0.431889
$$773$$ 8.00000 0.287740 0.143870 0.989597i $$-0.454045\pi$$
0.143870 + 0.989597i $$0.454045\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 16.0000 0.574367
$$777$$ 0 0
$$778$$ 5.00000 0.179259
$$779$$ 35.0000 1.25401
$$780$$ 0 0
$$781$$ −60.0000 −2.14697
$$782$$ 16.0000 0.572159
$$783$$ 0 0
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ 16.0000 0.569976
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −56.0000 −1.99113
$$792$$ 0 0
$$793$$ 4.00000 0.142044
$$794$$ 15.0000 0.532330
$$795$$ 0 0
$$796$$ −3.00000 −0.106332
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 0 0
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −30.0000 −1.05934
$$803$$ −48.0000 −1.69388
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ −10.0000 −0.351799
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 20.0000 0.701862
$$813$$ 0 0
$$814$$ 36.0000 1.26180
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −70.0000 −2.44899
$$818$$ 24.0000 0.839140
$$819$$ 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
$$823$$ −19.0000 −0.662298 −0.331149 0.943578i $$-0.607436\pi$$
−0.331149 + 0.943578i $$0.607436\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 4.00000 0.139094 0.0695468 0.997579i $$-0.477845\pi$$
0.0695468 + 0.997579i $$0.477845\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 36.0000 1.24733
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −28.0000 −0.968400
$$837$$ 0 0
$$838$$ 23.0000 0.794522
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ −14.0000 −0.482472
$$843$$ 0 0
$$844$$ −16.0000 −0.550743
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −20.0000 −0.687208
$$848$$ −9.00000 −0.309061
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −36.0000 −1.23406
$$852$$ 0 0
$$853$$ 19.0000 0.650548 0.325274 0.945620i $$-0.394544\pi$$
0.325274 + 0.945620i $$0.394544\pi$$
$$854$$ 16.0000 0.547509
$$855$$ 0 0
$$856$$ 5.00000 0.170896
$$857$$ 50.0000 1.70797 0.853984 0.520300i $$-0.174180\pi$$
0.853984 + 0.520300i $$0.174180\pi$$
$$858$$ 0 0
$$859$$ −2.00000 −0.0682391 −0.0341196 0.999418i $$-0.510863\pi$$
−0.0341196 + 0.999418i $$0.510863\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 19.0000 0.647143
$$863$$ −39.0000 −1.32758 −0.663788 0.747921i $$-0.731052\pi$$
−0.663788 + 0.747921i $$0.731052\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 19.0000 0.645646
$$867$$ 0 0
$$868$$ −16.0000 −0.543075
$$869$$ −28.0000 −0.949835
$$870$$ 0 0
$$871$$ −7.00000 −0.237186
$$872$$ 11.0000 0.372507
$$873$$ 0 0
$$874$$ 28.0000 0.947114
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 43.0000 1.45201 0.726003 0.687691i $$-0.241376\pi$$
0.726003 + 0.687691i $$0.241376\pi$$
$$878$$ 7.00000 0.236239
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 40.0000 1.34611 0.673054 0.739594i $$-0.264982\pi$$
0.673054 + 0.739594i $$0.264982\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 25.0000 0.839891
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −2.00000 −0.0669650
$$893$$ −21.0000 −0.702738
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ −27.0000 −0.901002
$$899$$ −20.0000 −0.667037
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 20.0000 0.665927
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 50.0000 1.66022 0.830111 0.557598i $$-0.188277\pi$$
0.830111 + 0.557598i $$0.188277\pi$$
$$908$$ −6.00000 −0.199117
$$909$$ 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
$$913$$ 24.0000 0.794284
$$914$$ −4.00000 −0.132308
$$915$$ 0 0
$$916$$ −17.0000 −0.561696
$$917$$ 68.0000 2.24556
$$918$$ 0 0
$$919$$ 29.0000 0.956622 0.478311 0.878191i $$-0.341249\pi$$
0.478311 + 0.878191i $$0.341249\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −24.0000 −0.790398
$$923$$ 15.0000 0.493731
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 26.0000 0.854413
$$927$$ 0 0
$$928$$ 5.00000 0.164133
$$929$$ −11.0000 −0.360898 −0.180449 0.983584i $$-0.557755\pi$$
−0.180449 + 0.983584i $$0.557755\pi$$
$$930$$ 0 0
$$931$$ 63.0000 2.06474
$$932$$ −8.00000 −0.262049
$$933$$ 0 0
$$934$$ 3.00000 0.0981630
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ −28.0000 −0.914232
$$939$$ 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
$$943$$ −20.0000 −0.651290
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ −40.0000 −1.30051
$$947$$ 8.00000 0.259965 0.129983 0.991516i $$-0.458508\pi$$
0.129983 + 0.991516i $$0.458508\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 16.0000 0.518563
$$953$$ −36.0000 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ 0 0
$$958$$ 15.0000 0.484628
$$959$$ −76.0000 −2.45417
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −9.00000 −0.290172
$$963$$ 0 0
$$964$$ 8.00000 0.257663
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −44.0000 −1.41494 −0.707472 0.706741i $$-0.750165\pi$$
−0.707472 + 0.706741i $$0.750165\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 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
$$973$$ 16.0000 0.512936
$$974$$ −22.0000 −0.704925
$$975$$ 0 0
$$976$$ 4.00000 0.128037
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 56.0000 1.78977
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −40.0000 −1.27645
$$983$$ 48.0000 1.53096 0.765481 0.643458i $$-0.222501\pi$$
0.765481 + 0.643458i $$0.222501\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 20.0000 0.636930
$$987$$ 0 0
$$988$$ 7.00000 0.222700
$$989$$ 40.0000 1.27193
$$990$$ 0 0
$$991$$ 13.0000 0.412959 0.206479 0.978451i $$-0.433799\pi$$
0.206479 + 0.978451i $$0.433799\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ 60.0000 1.90308
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 23.0000 0.728052
$$999$$ 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.a.a.1.1 1
3.2 odd 2 1950.2.a.x.1.1 yes 1
5.2 odd 4 5850.2.e.c.5149.1 2
5.3 odd 4 5850.2.e.c.5149.2 2
5.4 even 2 5850.2.a.by.1.1 1
15.2 even 4 1950.2.e.n.1249.2 2
15.8 even 4 1950.2.e.n.1249.1 2
15.14 odd 2 1950.2.a.e.1.1 1

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