# Properties

 Label 1950.2.a.bc.1.1 Level $1950$ Weight $2$ Character 1950.1 Self dual yes Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.70156$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.70156 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.70156 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.70156 q^{11} -1.00000 q^{12} +1.00000 q^{13} +4.70156 q^{14} +1.00000 q^{16} +0.701562 q^{17} -1.00000 q^{18} -1.70156 q^{19} +4.70156 q^{21} -4.70156 q^{22} +1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} -4.70156 q^{28} -6.40312 q^{29} -10.1047 q^{31} -1.00000 q^{32} -4.70156 q^{33} -0.701562 q^{34} +1.00000 q^{36} +1.70156 q^{37} +1.70156 q^{38} -1.00000 q^{39} -3.70156 q^{41} -4.70156 q^{42} +11.4031 q^{43} +4.70156 q^{44} -7.00000 q^{47} -1.00000 q^{48} +15.1047 q^{49} -0.701562 q^{51} +1.00000 q^{52} -2.40312 q^{53} +1.00000 q^{54} +4.70156 q^{56} +1.70156 q^{57} +6.40312 q^{58} +2.70156 q^{59} +14.1047 q^{61} +10.1047 q^{62} -4.70156 q^{63} +1.00000 q^{64} +4.70156 q^{66} +6.40312 q^{67} +0.701562 q^{68} -1.70156 q^{71} -1.00000 q^{72} +12.0000 q^{73} -1.70156 q^{74} -1.70156 q^{76} -22.1047 q^{77} +1.00000 q^{78} -5.70156 q^{79} +1.00000 q^{81} +3.70156 q^{82} +10.7016 q^{83} +4.70156 q^{84} -11.4031 q^{86} +6.40312 q^{87} -4.70156 q^{88} +11.4031 q^{89} -4.70156 q^{91} +10.1047 q^{93} +7.00000 q^{94} +1.00000 q^{96} +2.59688 q^{97} -15.1047 q^{98} +4.70156 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 3 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{11} - 2 q^{12} + 2 q^{13} + 3 q^{14} + 2 q^{16} - 5 q^{17} - 2 q^{18} + 3 q^{19} + 3 q^{21} - 3 q^{22} + 2 q^{24} - 2 q^{26} - 2 q^{27} - 3 q^{28} - q^{31} - 2 q^{32} - 3 q^{33} + 5 q^{34} + 2 q^{36} - 3 q^{37} - 3 q^{38} - 2 q^{39} - q^{41} - 3 q^{42} + 10 q^{43} + 3 q^{44} - 14 q^{47} - 2 q^{48} + 11 q^{49} + 5 q^{51} + 2 q^{52} + 8 q^{53} + 2 q^{54} + 3 q^{56} - 3 q^{57} - q^{59} + 9 q^{61} + q^{62} - 3 q^{63} + 2 q^{64} + 3 q^{66} - 5 q^{68} + 3 q^{71} - 2 q^{72} + 24 q^{73} + 3 q^{74} + 3 q^{76} - 25 q^{77} + 2 q^{78} - 5 q^{79} + 2 q^{81} + q^{82} + 15 q^{83} + 3 q^{84} - 10 q^{86} - 3 q^{88} + 10 q^{89} - 3 q^{91} + q^{93} + 14 q^{94} + 2 q^{96} + 18 q^{97} - 11 q^{98} + 3 q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ −4.70156 −1.77702 −0.888512 0.458854i $$-0.848260\pi$$
−0.888512 + 0.458854i $$0.848260\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.70156 1.41757 0.708787 0.705422i $$-0.249243\pi$$
0.708787 + 0.705422i $$0.249243\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.00000 0.277350
$$14$$ 4.70156 1.25655
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0.701562 0.170154 0.0850769 0.996374i $$-0.472886\pi$$
0.0850769 + 0.996374i $$0.472886\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −1.70156 −0.390365 −0.195183 0.980767i $$-0.562530\pi$$
−0.195183 + 0.980767i $$0.562530\pi$$
$$20$$ 0 0
$$21$$ 4.70156 1.02596
$$22$$ −4.70156 −1.00238
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ −1.00000 −0.192450
$$28$$ −4.70156 −0.888512
$$29$$ −6.40312 −1.18903 −0.594515 0.804084i $$-0.702656\pi$$
−0.594515 + 0.804084i $$0.702656\pi$$
$$30$$ 0 0
$$31$$ −10.1047 −1.81486 −0.907428 0.420208i $$-0.861957\pi$$
−0.907428 + 0.420208i $$0.861957\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −4.70156 −0.818437
$$34$$ −0.701562 −0.120317
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.70156 0.279735 0.139868 0.990170i $$-0.455332\pi$$
0.139868 + 0.990170i $$0.455332\pi$$
$$38$$ 1.70156 0.276030
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −3.70156 −0.578087 −0.289043 0.957316i $$-0.593337\pi$$
−0.289043 + 0.957316i $$0.593337\pi$$
$$42$$ −4.70156 −0.725467
$$43$$ 11.4031 1.73896 0.869480 0.493968i $$-0.164454\pi$$
0.869480 + 0.493968i $$0.164454\pi$$
$$44$$ 4.70156 0.708787
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −7.00000 −1.02105 −0.510527 0.859861i $$-0.670550\pi$$
−0.510527 + 0.859861i $$0.670550\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 15.1047 2.15781
$$50$$ 0 0
$$51$$ −0.701562 −0.0982383
$$52$$ 1.00000 0.138675
$$53$$ −2.40312 −0.330095 −0.165047 0.986286i $$-0.552778\pi$$
−0.165047 + 0.986286i $$0.552778\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 4.70156 0.628273
$$57$$ 1.70156 0.225377
$$58$$ 6.40312 0.840771
$$59$$ 2.70156 0.351713 0.175857 0.984416i $$-0.443730\pi$$
0.175857 + 0.984416i $$0.443730\pi$$
$$60$$ 0 0
$$61$$ 14.1047 1.80592 0.902960 0.429725i $$-0.141389\pi$$
0.902960 + 0.429725i $$0.141389\pi$$
$$62$$ 10.1047 1.28330
$$63$$ −4.70156 −0.592341
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.70156 0.578722
$$67$$ 6.40312 0.782266 0.391133 0.920334i $$-0.372083\pi$$
0.391133 + 0.920334i $$0.372083\pi$$
$$68$$ 0.701562 0.0850769
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1.70156 −0.201938 −0.100969 0.994890i $$-0.532194\pi$$
−0.100969 + 0.994890i $$0.532194\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 12.0000 1.40449 0.702247 0.711934i $$-0.252180\pi$$
0.702247 + 0.711934i $$0.252180\pi$$
$$74$$ −1.70156 −0.197803
$$75$$ 0 0
$$76$$ −1.70156 −0.195183
$$77$$ −22.1047 −2.51906
$$78$$ 1.00000 0.113228
$$79$$ −5.70156 −0.641476 −0.320738 0.947168i $$-0.603931\pi$$
−0.320738 + 0.947168i $$0.603931\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 3.70156 0.408769
$$83$$ 10.7016 1.17465 0.587325 0.809352i $$-0.300181\pi$$
0.587325 + 0.809352i $$0.300181\pi$$
$$84$$ 4.70156 0.512982
$$85$$ 0 0
$$86$$ −11.4031 −1.22963
$$87$$ 6.40312 0.686487
$$88$$ −4.70156 −0.501188
$$89$$ 11.4031 1.20873 0.604364 0.796708i $$-0.293427\pi$$
0.604364 + 0.796708i $$0.293427\pi$$
$$90$$ 0 0
$$91$$ −4.70156 −0.492858
$$92$$ 0 0
$$93$$ 10.1047 1.04781
$$94$$ 7.00000 0.721995
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 2.59688 0.263673 0.131836 0.991271i $$-0.457913\pi$$
0.131836 + 0.991271i $$0.457913\pi$$
$$98$$ −15.1047 −1.52580
$$99$$ 4.70156 0.472525
$$100$$ 0 0
$$101$$ 6.70156 0.666830 0.333415 0.942780i $$-0.391799\pi$$
0.333415 + 0.942780i $$0.391799\pi$$
$$102$$ 0.701562 0.0694650
$$103$$ −1.40312 −0.138254 −0.0691270 0.997608i $$-0.522021\pi$$
−0.0691270 + 0.997608i $$0.522021\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 2.40312 0.233412
$$107$$ −19.1047 −1.84692 −0.923460 0.383695i $$-0.874651\pi$$
−0.923460 + 0.383695i $$0.874651\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 4.29844 0.411716 0.205858 0.978582i $$-0.434002\pi$$
0.205858 + 0.978582i $$0.434002\pi$$
$$110$$ 0 0
$$111$$ −1.70156 −0.161505
$$112$$ −4.70156 −0.444256
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ −1.70156 −0.159366
$$115$$ 0 0
$$116$$ −6.40312 −0.594515
$$117$$ 1.00000 0.0924500
$$118$$ −2.70156 −0.248699
$$119$$ −3.29844 −0.302367
$$120$$ 0 0
$$121$$ 11.1047 1.00952
$$122$$ −14.1047 −1.27698
$$123$$ 3.70156 0.333759
$$124$$ −10.1047 −0.907428
$$125$$ 0 0
$$126$$ 4.70156 0.418848
$$127$$ 6.29844 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −11.4031 −1.00399
$$130$$ 0 0
$$131$$ 22.5078 1.96652 0.983258 0.182217i $$-0.0583275\pi$$
0.983258 + 0.182217i $$0.0583275\pi$$
$$132$$ −4.70156 −0.409218
$$133$$ 8.00000 0.693688
$$134$$ −6.40312 −0.553146
$$135$$ 0 0
$$136$$ −0.701562 −0.0601585
$$137$$ −13.7016 −1.17060 −0.585302 0.810816i $$-0.699024\pi$$
−0.585302 + 0.810816i $$0.699024\pi$$
$$138$$ 0 0
$$139$$ 9.40312 0.797563 0.398781 0.917046i $$-0.369433\pi$$
0.398781 + 0.917046i $$0.369433\pi$$
$$140$$ 0 0
$$141$$ 7.00000 0.589506
$$142$$ 1.70156 0.142792
$$143$$ 4.70156 0.393164
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −12.0000 −0.993127
$$147$$ −15.1047 −1.24581
$$148$$ 1.70156 0.139868
$$149$$ 6.59688 0.540437 0.270219 0.962799i $$-0.412904\pi$$
0.270219 + 0.962799i $$0.412904\pi$$
$$150$$ 0 0
$$151$$ −14.1047 −1.14782 −0.573912 0.818917i $$-0.694575\pi$$
−0.573912 + 0.818917i $$0.694575\pi$$
$$152$$ 1.70156 0.138015
$$153$$ 0.701562 0.0567179
$$154$$ 22.1047 1.78125
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ 22.7016 1.81178 0.905891 0.423511i $$-0.139203\pi$$
0.905891 + 0.423511i $$0.139203\pi$$
$$158$$ 5.70156 0.453592
$$159$$ 2.40312 0.190580
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 18.8062 1.47302 0.736510 0.676427i $$-0.236473\pi$$
0.736510 + 0.676427i $$0.236473\pi$$
$$164$$ −3.70156 −0.289043
$$165$$ 0 0
$$166$$ −10.7016 −0.830602
$$167$$ 1.10469 0.0854832 0.0427416 0.999086i $$-0.486391\pi$$
0.0427416 + 0.999086i $$0.486391\pi$$
$$168$$ −4.70156 −0.362733
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −1.70156 −0.130122
$$172$$ 11.4031 0.869480
$$173$$ 0.193752 0.0147307 0.00736533 0.999973i $$-0.497656\pi$$
0.00736533 + 0.999973i $$0.497656\pi$$
$$174$$ −6.40312 −0.485420
$$175$$ 0 0
$$176$$ 4.70156 0.354394
$$177$$ −2.70156 −0.203062
$$178$$ −11.4031 −0.854700
$$179$$ −24.2094 −1.80949 −0.904747 0.425950i $$-0.859940\pi$$
−0.904747 + 0.425950i $$0.859940\pi$$
$$180$$ 0 0
$$181$$ 11.2984 0.839806 0.419903 0.907569i $$-0.362064\pi$$
0.419903 + 0.907569i $$0.362064\pi$$
$$182$$ 4.70156 0.348503
$$183$$ −14.1047 −1.04265
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −10.1047 −0.740912
$$187$$ 3.29844 0.241206
$$188$$ −7.00000 −0.510527
$$189$$ 4.70156 0.341988
$$190$$ 0 0
$$191$$ −12.8062 −0.926628 −0.463314 0.886194i $$-0.653340\pi$$
−0.463314 + 0.886194i $$0.653340\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −16.2094 −1.16678 −0.583388 0.812194i $$-0.698273\pi$$
−0.583388 + 0.812194i $$0.698273\pi$$
$$194$$ −2.59688 −0.186445
$$195$$ 0 0
$$196$$ 15.1047 1.07891
$$197$$ 5.40312 0.384957 0.192478 0.981301i $$-0.438347\pi$$
0.192478 + 0.981301i $$0.438347\pi$$
$$198$$ −4.70156 −0.334125
$$199$$ 8.29844 0.588261 0.294130 0.955765i $$-0.404970\pi$$
0.294130 + 0.955765i $$0.404970\pi$$
$$200$$ 0 0
$$201$$ −6.40312 −0.451642
$$202$$ −6.70156 −0.471520
$$203$$ 30.1047 2.11293
$$204$$ −0.701562 −0.0491192
$$205$$ 0 0
$$206$$ 1.40312 0.0977603
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ 6.80625 0.468561 0.234281 0.972169i $$-0.424727\pi$$
0.234281 + 0.972169i $$0.424727\pi$$
$$212$$ −2.40312 −0.165047
$$213$$ 1.70156 0.116589
$$214$$ 19.1047 1.30597
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 47.5078 3.22504
$$218$$ −4.29844 −0.291127
$$219$$ −12.0000 −0.810885
$$220$$ 0 0
$$221$$ 0.701562 0.0471922
$$222$$ 1.70156 0.114201
$$223$$ −11.4031 −0.763610 −0.381805 0.924243i $$-0.624697\pi$$
−0.381805 + 0.924243i $$0.624697\pi$$
$$224$$ 4.70156 0.314136
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ 16.1047 1.06891 0.534453 0.845198i $$-0.320518\pi$$
0.534453 + 0.845198i $$0.320518\pi$$
$$228$$ 1.70156 0.112689
$$229$$ 15.7016 1.03759 0.518794 0.854899i $$-0.326381\pi$$
0.518794 + 0.854899i $$0.326381\pi$$
$$230$$ 0 0
$$231$$ 22.1047 1.45438
$$232$$ 6.40312 0.420386
$$233$$ 20.2094 1.32396 0.661980 0.749521i $$-0.269716\pi$$
0.661980 + 0.749521i $$0.269716\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 2.70156 0.175857
$$237$$ 5.70156 0.370356
$$238$$ 3.29844 0.213806
$$239$$ 6.10469 0.394879 0.197440 0.980315i $$-0.436737\pi$$
0.197440 + 0.980315i $$0.436737\pi$$
$$240$$ 0 0
$$241$$ −2.59688 −0.167279 −0.0836397 0.996496i $$-0.526654\pi$$
−0.0836397 + 0.996496i $$0.526654\pi$$
$$242$$ −11.1047 −0.713836
$$243$$ −1.00000 −0.0641500
$$244$$ 14.1047 0.902960
$$245$$ 0 0
$$246$$ −3.70156 −0.236003
$$247$$ −1.70156 −0.108268
$$248$$ 10.1047 0.641648
$$249$$ −10.7016 −0.678184
$$250$$ 0 0
$$251$$ 0.298438 0.0188372 0.00941862 0.999956i $$-0.497002\pi$$
0.00941862 + 0.999956i $$0.497002\pi$$
$$252$$ −4.70156 −0.296171
$$253$$ 0 0
$$254$$ −6.29844 −0.395199
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 15.2984 0.954290 0.477145 0.878824i $$-0.341672\pi$$
0.477145 + 0.878824i $$0.341672\pi$$
$$258$$ 11.4031 0.709928
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ −6.40312 −0.396343
$$262$$ −22.5078 −1.39054
$$263$$ −2.00000 −0.123325 −0.0616626 0.998097i $$-0.519640\pi$$
−0.0616626 + 0.998097i $$0.519640\pi$$
$$264$$ 4.70156 0.289361
$$265$$ 0 0
$$266$$ −8.00000 −0.490511
$$267$$ −11.4031 −0.697860
$$268$$ 6.40312 0.391133
$$269$$ 31.2094 1.90287 0.951435 0.307851i $$-0.0996099\pi$$
0.951435 + 0.307851i $$0.0996099\pi$$
$$270$$ 0 0
$$271$$ 19.5078 1.18502 0.592508 0.805565i $$-0.298138\pi$$
0.592508 + 0.805565i $$0.298138\pi$$
$$272$$ 0.701562 0.0425385
$$273$$ 4.70156 0.284551
$$274$$ 13.7016 0.827742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ −9.40312 −0.563962
$$279$$ −10.1047 −0.604952
$$280$$ 0 0
$$281$$ −21.9109 −1.30710 −0.653548 0.756885i $$-0.726720\pi$$
−0.653548 + 0.756885i $$0.726720\pi$$
$$282$$ −7.00000 −0.416844
$$283$$ −21.4031 −1.27228 −0.636142 0.771572i $$-0.719471\pi$$
−0.636142 + 0.771572i $$0.719471\pi$$
$$284$$ −1.70156 −0.100969
$$285$$ 0 0
$$286$$ −4.70156 −0.278009
$$287$$ 17.4031 1.02727
$$288$$ −1.00000 −0.0589256
$$289$$ −16.5078 −0.971048
$$290$$ 0 0
$$291$$ −2.59688 −0.152232
$$292$$ 12.0000 0.702247
$$293$$ −21.4031 −1.25038 −0.625192 0.780471i $$-0.714979\pi$$
−0.625192 + 0.780471i $$0.714979\pi$$
$$294$$ 15.1047 0.880923
$$295$$ 0 0
$$296$$ −1.70156 −0.0989013
$$297$$ −4.70156 −0.272812
$$298$$ −6.59688 −0.382147
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −53.6125 −3.09017
$$302$$ 14.1047 0.811633
$$303$$ −6.70156 −0.384995
$$304$$ −1.70156 −0.0975913
$$305$$ 0 0
$$306$$ −0.701562 −0.0401056
$$307$$ 5.70156 0.325405 0.162703 0.986675i $$-0.447979\pi$$
0.162703 + 0.986675i $$0.447979\pi$$
$$308$$ −22.1047 −1.25953
$$309$$ 1.40312 0.0798209
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 1.00000 0.0566139
$$313$$ 21.2094 1.19882 0.599412 0.800440i $$-0.295401\pi$$
0.599412 + 0.800440i $$0.295401\pi$$
$$314$$ −22.7016 −1.28112
$$315$$ 0 0
$$316$$ −5.70156 −0.320738
$$317$$ −1.19375 −0.0670478 −0.0335239 0.999438i $$-0.510673\pi$$
−0.0335239 + 0.999438i $$0.510673\pi$$
$$318$$ −2.40312 −0.134761
$$319$$ −30.1047 −1.68554
$$320$$ 0 0
$$321$$ 19.1047 1.06632
$$322$$ 0 0
$$323$$ −1.19375 −0.0664221
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −18.8062 −1.04158
$$327$$ −4.29844 −0.237704
$$328$$ 3.70156 0.204385
$$329$$ 32.9109 1.81444
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 10.7016 0.587325
$$333$$ 1.70156 0.0932450
$$334$$ −1.10469 −0.0604457
$$335$$ 0 0
$$336$$ 4.70156 0.256491
$$337$$ −8.10469 −0.441490 −0.220745 0.975332i $$-0.570849\pi$$
−0.220745 + 0.975332i $$0.570849\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ −47.5078 −2.57269
$$342$$ 1.70156 0.0920099
$$343$$ −38.1047 −2.05746
$$344$$ −11.4031 −0.614815
$$345$$ 0 0
$$346$$ −0.193752 −0.0104161
$$347$$ −10.5078 −0.564089 −0.282044 0.959401i $$-0.591013\pi$$
−0.282044 + 0.959401i $$0.591013\pi$$
$$348$$ 6.40312 0.343243
$$349$$ −11.4031 −0.610395 −0.305198 0.952289i $$-0.598723\pi$$
−0.305198 + 0.952289i $$0.598723\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ −4.70156 −0.250594
$$353$$ −14.5078 −0.772173 −0.386086 0.922463i $$-0.626173\pi$$
−0.386086 + 0.922463i $$0.626173\pi$$
$$354$$ 2.70156 0.143586
$$355$$ 0 0
$$356$$ 11.4031 0.604364
$$357$$ 3.29844 0.174572
$$358$$ 24.2094 1.27951
$$359$$ −34.6125 −1.82678 −0.913389 0.407088i $$-0.866544\pi$$
−0.913389 + 0.407088i $$0.866544\pi$$
$$360$$ 0 0
$$361$$ −16.1047 −0.847615
$$362$$ −11.2984 −0.593833
$$363$$ −11.1047 −0.582845
$$364$$ −4.70156 −0.246429
$$365$$ 0 0
$$366$$ 14.1047 0.737264
$$367$$ 2.29844 0.119977 0.0599887 0.998199i $$-0.480894\pi$$
0.0599887 + 0.998199i $$0.480894\pi$$
$$368$$ 0 0
$$369$$ −3.70156 −0.192696
$$370$$ 0 0
$$371$$ 11.2984 0.586586
$$372$$ 10.1047 0.523904
$$373$$ −5.29844 −0.274343 −0.137171 0.990547i $$-0.543801\pi$$
−0.137171 + 0.990547i $$0.543801\pi$$
$$374$$ −3.29844 −0.170558
$$375$$ 0 0
$$376$$ 7.00000 0.360997
$$377$$ −6.40312 −0.329778
$$378$$ −4.70156 −0.241822
$$379$$ −13.8953 −0.713754 −0.356877 0.934151i $$-0.616159\pi$$
−0.356877 + 0.934151i $$0.616159\pi$$
$$380$$ 0 0
$$381$$ −6.29844 −0.322679
$$382$$ 12.8062 0.655225
$$383$$ −32.5078 −1.66107 −0.830536 0.556965i $$-0.811966\pi$$
−0.830536 + 0.556965i $$0.811966\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 16.2094 0.825035
$$387$$ 11.4031 0.579653
$$388$$ 2.59688 0.131836
$$389$$ −27.1047 −1.37426 −0.687131 0.726533i $$-0.741130\pi$$
−0.687131 + 0.726533i $$0.741130\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −15.1047 −0.762902
$$393$$ −22.5078 −1.13537
$$394$$ −5.40312 −0.272205
$$395$$ 0 0
$$396$$ 4.70156 0.236262
$$397$$ 21.9109 1.09968 0.549839 0.835271i $$-0.314689\pi$$
0.549839 + 0.835271i $$0.314689\pi$$
$$398$$ −8.29844 −0.415963
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ 18.2094 0.909333 0.454666 0.890662i $$-0.349759\pi$$
0.454666 + 0.890662i $$0.349759\pi$$
$$402$$ 6.40312 0.319359
$$403$$ −10.1047 −0.503350
$$404$$ 6.70156 0.333415
$$405$$ 0 0
$$406$$ −30.1047 −1.49407
$$407$$ 8.00000 0.396545
$$408$$ 0.701562 0.0347325
$$409$$ 29.4031 1.45389 0.726945 0.686695i $$-0.240939\pi$$
0.726945 + 0.686695i $$0.240939\pi$$
$$410$$ 0 0
$$411$$ 13.7016 0.675848
$$412$$ −1.40312 −0.0691270
$$413$$ −12.7016 −0.625003
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ −9.40312 −0.460473
$$418$$ 8.00000 0.391293
$$419$$ 9.91093 0.484181 0.242090 0.970254i $$-0.422167\pi$$
0.242090 + 0.970254i $$0.422167\pi$$
$$420$$ 0 0
$$421$$ −0.596876 −0.0290899 −0.0145450 0.999894i $$-0.504630\pi$$
−0.0145450 + 0.999894i $$0.504630\pi$$
$$422$$ −6.80625 −0.331323
$$423$$ −7.00000 −0.340352
$$424$$ 2.40312 0.116706
$$425$$ 0 0
$$426$$ −1.70156 −0.0824410
$$427$$ −66.3141 −3.20916
$$428$$ −19.1047 −0.923460
$$429$$ −4.70156 −0.226994
$$430$$ 0 0
$$431$$ 35.3141 1.70102 0.850509 0.525960i $$-0.176294\pi$$
0.850509 + 0.525960i $$0.176294\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 11.1047 0.533657 0.266829 0.963744i $$-0.414024\pi$$
0.266829 + 0.963744i $$0.414024\pi$$
$$434$$ −47.5078 −2.28045
$$435$$ 0 0
$$436$$ 4.29844 0.205858
$$437$$ 0 0
$$438$$ 12.0000 0.573382
$$439$$ −10.2984 −0.491518 −0.245759 0.969331i $$-0.579037\pi$$
−0.245759 + 0.969331i $$0.579037\pi$$
$$440$$ 0 0
$$441$$ 15.1047 0.719271
$$442$$ −0.701562 −0.0333699
$$443$$ −32.5078 −1.54449 −0.772246 0.635323i $$-0.780867\pi$$
−0.772246 + 0.635323i $$0.780867\pi$$
$$444$$ −1.70156 −0.0807526
$$445$$ 0 0
$$446$$ 11.4031 0.539954
$$447$$ −6.59688 −0.312022
$$448$$ −4.70156 −0.222128
$$449$$ 2.29844 0.108470 0.0542350 0.998528i $$-0.482728\pi$$
0.0542350 + 0.998528i $$0.482728\pi$$
$$450$$ 0 0
$$451$$ −17.4031 −0.819481
$$452$$ 14.0000 0.658505
$$453$$ 14.1047 0.662696
$$454$$ −16.1047 −0.755830
$$455$$ 0 0
$$456$$ −1.70156 −0.0796829
$$457$$ 2.59688 0.121477 0.0607384 0.998154i $$-0.480654\pi$$
0.0607384 + 0.998154i $$0.480654\pi$$
$$458$$ −15.7016 −0.733686
$$459$$ −0.701562 −0.0327461
$$460$$ 0 0
$$461$$ 36.2094 1.68644 0.843219 0.537570i $$-0.180658\pi$$
0.843219 + 0.537570i $$0.180658\pi$$
$$462$$ −22.1047 −1.02840
$$463$$ 21.2984 0.989822 0.494911 0.868944i $$-0.335201\pi$$
0.494911 + 0.868944i $$0.335201\pi$$
$$464$$ −6.40312 −0.297258
$$465$$ 0 0
$$466$$ −20.2094 −0.936181
$$467$$ −30.2984 −1.40204 −0.701022 0.713139i $$-0.747273\pi$$
−0.701022 + 0.713139i $$0.747273\pi$$
$$468$$ 1.00000 0.0462250
$$469$$ −30.1047 −1.39011
$$470$$ 0 0
$$471$$ −22.7016 −1.04603
$$472$$ −2.70156 −0.124349
$$473$$ 53.6125 2.46511
$$474$$ −5.70156 −0.261881
$$475$$ 0 0
$$476$$ −3.29844 −0.151184
$$477$$ −2.40312 −0.110032
$$478$$ −6.10469 −0.279222
$$479$$ −40.6125 −1.85563 −0.927816 0.373038i $$-0.878316\pi$$
−0.927816 + 0.373038i $$0.878316\pi$$
$$480$$ 0 0
$$481$$ 1.70156 0.0775846
$$482$$ 2.59688 0.118284
$$483$$ 0 0
$$484$$ 11.1047 0.504758
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 7.89531 0.357771 0.178885 0.983870i $$-0.442751\pi$$
0.178885 + 0.983870i $$0.442751\pi$$
$$488$$ −14.1047 −0.638489
$$489$$ −18.8062 −0.850448
$$490$$ 0 0
$$491$$ 33.4031 1.50746 0.753731 0.657183i $$-0.228252\pi$$
0.753731 + 0.657183i $$0.228252\pi$$
$$492$$ 3.70156 0.166879
$$493$$ −4.49219 −0.202318
$$494$$ 1.70156 0.0765569
$$495$$ 0 0
$$496$$ −10.1047 −0.453714
$$497$$ 8.00000 0.358849
$$498$$ 10.7016 0.479548
$$499$$ 17.2094 0.770397 0.385199 0.922834i $$-0.374133\pi$$
0.385199 + 0.922834i $$0.374133\pi$$
$$500$$ 0 0
$$501$$ −1.10469 −0.0493537
$$502$$ −0.298438 −0.0133199
$$503$$ 12.2094 0.544389 0.272195 0.962242i $$-0.412251\pi$$
0.272195 + 0.962242i $$0.412251\pi$$
$$504$$ 4.70156 0.209424
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 6.29844 0.279448
$$509$$ −5.40312 −0.239489 −0.119745 0.992805i $$-0.538208\pi$$
−0.119745 + 0.992805i $$0.538208\pi$$
$$510$$ 0 0
$$511$$ −56.4187 −2.49582
$$512$$ −1.00000 −0.0441942
$$513$$ 1.70156 0.0751258
$$514$$ −15.2984 −0.674785
$$515$$ 0 0
$$516$$ −11.4031 −0.501995
$$517$$ −32.9109 −1.44742
$$518$$ 8.00000 0.351500
$$519$$ −0.193752 −0.00850475
$$520$$ 0 0
$$521$$ −36.2094 −1.58636 −0.793181 0.608986i $$-0.791576\pi$$
−0.793181 + 0.608986i $$0.791576\pi$$
$$522$$ 6.40312 0.280257
$$523$$ 24.8062 1.08470 0.542351 0.840152i $$-0.317534\pi$$
0.542351 + 0.840152i $$0.317534\pi$$
$$524$$ 22.5078 0.983258
$$525$$ 0 0
$$526$$ 2.00000 0.0872041
$$527$$ −7.08907 −0.308805
$$528$$ −4.70156 −0.204609
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 2.70156 0.117238
$$532$$ 8.00000 0.346844
$$533$$ −3.70156 −0.160332
$$534$$ 11.4031 0.493461
$$535$$ 0 0
$$536$$ −6.40312 −0.276573
$$537$$ 24.2094 1.04471
$$538$$ −31.2094 −1.34553
$$539$$ 71.0156 3.05886
$$540$$ 0 0
$$541$$ −1.79063 −0.0769851 −0.0384925 0.999259i $$-0.512256\pi$$
−0.0384925 + 0.999259i $$0.512256\pi$$
$$542$$ −19.5078 −0.837932
$$543$$ −11.2984 −0.484862
$$544$$ −0.701562 −0.0300792
$$545$$ 0 0
$$546$$ −4.70156 −0.201208
$$547$$ 31.6125 1.35165 0.675826 0.737061i $$-0.263787\pi$$
0.675826 + 0.737061i $$0.263787\pi$$
$$548$$ −13.7016 −0.585302
$$549$$ 14.1047 0.601973
$$550$$ 0 0
$$551$$ 10.8953 0.464156
$$552$$ 0 0
$$553$$ 26.8062 1.13992
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ 9.40312 0.398781
$$557$$ −16.8062 −0.712104 −0.356052 0.934466i $$-0.615877\pi$$
−0.356052 + 0.934466i $$0.615877\pi$$
$$558$$ 10.1047 0.427765
$$559$$ 11.4031 0.482301
$$560$$ 0 0
$$561$$ −3.29844 −0.139260
$$562$$ 21.9109 0.924257
$$563$$ 32.5078 1.37004 0.685020 0.728524i $$-0.259793\pi$$
0.685020 + 0.728524i $$0.259793\pi$$
$$564$$ 7.00000 0.294753
$$565$$ 0 0
$$566$$ 21.4031 0.899640
$$567$$ −4.70156 −0.197447
$$568$$ 1.70156 0.0713960
$$569$$ −11.2984 −0.473655 −0.236828 0.971552i $$-0.576108\pi$$
−0.236828 + 0.971552i $$0.576108\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 4.70156 0.196582
$$573$$ 12.8062 0.534989
$$574$$ −17.4031 −0.726392
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −24.2094 −1.00785 −0.503925 0.863748i $$-0.668111\pi$$
−0.503925 + 0.863748i $$0.668111\pi$$
$$578$$ 16.5078 0.686634
$$579$$ 16.2094 0.673639
$$580$$ 0 0
$$581$$ −50.3141 −2.08738
$$582$$ 2.59688 0.107644
$$583$$ −11.2984 −0.467933
$$584$$ −12.0000 −0.496564
$$585$$ 0 0
$$586$$ 21.4031 0.884155
$$587$$ 7.50781 0.309881 0.154940 0.987924i $$-0.450481\pi$$
0.154940 + 0.987924i $$0.450481\pi$$
$$588$$ −15.1047 −0.622907
$$589$$ 17.1938 0.708456
$$590$$ 0 0
$$591$$ −5.40312 −0.222255
$$592$$ 1.70156 0.0699338
$$593$$ −17.9109 −0.735514 −0.367757 0.929922i $$-0.619874\pi$$
−0.367757 + 0.929922i $$0.619874\pi$$
$$594$$ 4.70156 0.192907
$$595$$ 0 0
$$596$$ 6.59688 0.270219
$$597$$ −8.29844 −0.339632
$$598$$ 0 0
$$599$$ 8.20937 0.335426 0.167713 0.985836i $$-0.446362\pi$$
0.167713 + 0.985836i $$0.446362\pi$$
$$600$$ 0 0
$$601$$ 10.6125 0.432893 0.216446 0.976295i $$-0.430553\pi$$
0.216446 + 0.976295i $$0.430553\pi$$
$$602$$ 53.6125 2.18508
$$603$$ 6.40312 0.260755
$$604$$ −14.1047 −0.573912
$$605$$ 0 0
$$606$$ 6.70156 0.272232
$$607$$ 35.1047 1.42486 0.712428 0.701746i $$-0.247596\pi$$
0.712428 + 0.701746i $$0.247596\pi$$
$$608$$ 1.70156 0.0690075
$$609$$ −30.1047 −1.21990
$$610$$ 0 0
$$611$$ −7.00000 −0.283190
$$612$$ 0.701562 0.0283590
$$613$$ 12.8062 0.517240 0.258620 0.965979i $$-0.416732\pi$$
0.258620 + 0.965979i $$0.416732\pi$$
$$614$$ −5.70156 −0.230096
$$615$$ 0 0
$$616$$ 22.1047 0.890623
$$617$$ 27.1047 1.09119 0.545597 0.838048i $$-0.316303\pi$$
0.545597 + 0.838048i $$0.316303\pi$$
$$618$$ −1.40312 −0.0564419
$$619$$ −13.1938 −0.530302 −0.265151 0.964207i $$-0.585422\pi$$
−0.265151 + 0.964207i $$0.585422\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −30.0000 −1.20289
$$623$$ −53.6125 −2.14794
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ −21.2094 −0.847697
$$627$$ 8.00000 0.319489
$$628$$ 22.7016 0.905891
$$629$$ 1.19375 0.0475980
$$630$$ 0 0
$$631$$ 33.6125 1.33809 0.669046 0.743221i $$-0.266703\pi$$
0.669046 + 0.743221i $$0.266703\pi$$
$$632$$ 5.70156 0.226796
$$633$$ −6.80625 −0.270524
$$634$$ 1.19375 0.0474099
$$635$$ 0 0
$$636$$ 2.40312 0.0952901
$$637$$ 15.1047 0.598469
$$638$$ 30.1047 1.19186
$$639$$ −1.70156 −0.0673128
$$640$$ 0 0
$$641$$ 5.50781 0.217545 0.108773 0.994067i $$-0.465308\pi$$
0.108773 + 0.994067i $$0.465308\pi$$
$$642$$ −19.1047 −0.754002
$$643$$ 10.2984 0.406131 0.203065 0.979165i $$-0.434910\pi$$
0.203065 + 0.979165i $$0.434910\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 1.19375 0.0469675
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 12.7016 0.498580
$$650$$ 0 0
$$651$$ −47.5078 −1.86198
$$652$$ 18.8062 0.736510
$$653$$ 21.2984 0.833472 0.416736 0.909027i $$-0.363174\pi$$
0.416736 + 0.909027i $$0.363174\pi$$
$$654$$ 4.29844 0.168082
$$655$$ 0 0
$$656$$ −3.70156 −0.144522
$$657$$ 12.0000 0.468165
$$658$$ −32.9109 −1.28300
$$659$$ −19.3141 −0.752369 −0.376184 0.926545i $$-0.622764\pi$$
−0.376184 + 0.926545i $$0.622764\pi$$
$$660$$ 0 0
$$661$$ −40.1203 −1.56050 −0.780250 0.625468i $$-0.784908\pi$$
−0.780250 + 0.625468i $$0.784908\pi$$
$$662$$ 12.0000 0.466393
$$663$$ −0.701562 −0.0272464
$$664$$ −10.7016 −0.415301
$$665$$ 0 0
$$666$$ −1.70156 −0.0659342
$$667$$ 0 0
$$668$$ 1.10469 0.0427416
$$669$$ 11.4031 0.440870
$$670$$ 0 0
$$671$$ 66.3141 2.56003
$$672$$ −4.70156 −0.181367
$$673$$ −36.0156 −1.38830 −0.694150 0.719830i $$-0.744220\pi$$
−0.694150 + 0.719830i $$0.744220\pi$$
$$674$$ 8.10469 0.312181
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −31.6125 −1.21497 −0.607483 0.794332i $$-0.707821\pi$$
−0.607483 + 0.794332i $$0.707821\pi$$
$$678$$ 14.0000 0.537667
$$679$$ −12.2094 −0.468553
$$680$$ 0 0
$$681$$ −16.1047 −0.617133
$$682$$ 47.5078 1.81917
$$683$$ −44.7016 −1.71046 −0.855229 0.518251i $$-0.826583\pi$$
−0.855229 + 0.518251i $$0.826583\pi$$
$$684$$ −1.70156 −0.0650609
$$685$$ 0 0
$$686$$ 38.1047 1.45484
$$687$$ −15.7016 −0.599052
$$688$$ 11.4031 0.434740
$$689$$ −2.40312 −0.0915517
$$690$$ 0 0
$$691$$ −22.1938 −0.844290 −0.422145 0.906528i $$-0.638723\pi$$
−0.422145 + 0.906528i $$0.638723\pi$$
$$692$$ 0.193752 0.00736533
$$693$$ −22.1047 −0.839688
$$694$$ 10.5078 0.398871
$$695$$ 0 0
$$696$$ −6.40312 −0.242710
$$697$$ −2.59688 −0.0983637
$$698$$ 11.4031 0.431615
$$699$$ −20.2094 −0.764389
$$700$$ 0 0
$$701$$ 30.9109 1.16749 0.583745 0.811937i $$-0.301587\pi$$
0.583745 + 0.811937i $$0.301587\pi$$
$$702$$ 1.00000 0.0377426
$$703$$ −2.89531 −0.109199
$$704$$ 4.70156 0.177197
$$705$$ 0 0
$$706$$ 14.5078 0.546009
$$707$$ −31.5078 −1.18497
$$708$$ −2.70156 −0.101531
$$709$$ 15.6125 0.586340 0.293170 0.956060i $$-0.405290\pi$$
0.293170 + 0.956060i $$0.405290\pi$$
$$710$$ 0 0
$$711$$ −5.70156 −0.213825
$$712$$ −11.4031 −0.427350
$$713$$ 0 0
$$714$$ −3.29844 −0.123441
$$715$$ 0 0
$$716$$ −24.2094 −0.904747
$$717$$ −6.10469 −0.227984
$$718$$ 34.6125 1.29173
$$719$$ −11.0156 −0.410813 −0.205407 0.978677i $$-0.565852\pi$$
−0.205407 + 0.978677i $$0.565852\pi$$
$$720$$ 0 0
$$721$$ 6.59688 0.245680
$$722$$ 16.1047 0.599354
$$723$$ 2.59688 0.0965788
$$724$$ 11.2984 0.419903
$$725$$ 0 0
$$726$$ 11.1047 0.412134
$$727$$ −7.79063 −0.288938 −0.144469 0.989509i $$-0.546147\pi$$
−0.144469 + 0.989509i $$0.546147\pi$$
$$728$$ 4.70156 0.174251
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ −14.1047 −0.521324
$$733$$ −47.9109 −1.76963 −0.884815 0.465942i $$-0.845716\pi$$
−0.884815 + 0.465942i $$0.845716\pi$$
$$734$$ −2.29844 −0.0848369
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 30.1047 1.10892
$$738$$ 3.70156 0.136256
$$739$$ 4.61250 0.169673 0.0848367 0.996395i $$-0.472963\pi$$
0.0848367 + 0.996395i $$0.472963\pi$$
$$740$$ 0 0
$$741$$ 1.70156 0.0625084
$$742$$ −11.2984 −0.414779
$$743$$ 30.6125 1.12306 0.561532 0.827455i $$-0.310212\pi$$
0.561532 + 0.827455i $$0.310212\pi$$
$$744$$ −10.1047 −0.370456
$$745$$ 0 0
$$746$$ 5.29844 0.193990
$$747$$ 10.7016 0.391550
$$748$$ 3.29844 0.120603
$$749$$ 89.8219 3.28202
$$750$$ 0 0
$$751$$ 8.50781 0.310454 0.155227 0.987879i $$-0.450389\pi$$
0.155227 + 0.987879i $$0.450389\pi$$
$$752$$ −7.00000 −0.255264
$$753$$ −0.298438 −0.0108757
$$754$$ 6.40312 0.233188
$$755$$ 0 0
$$756$$ 4.70156 0.170994
$$757$$ 17.8953 0.650416 0.325208 0.945642i $$-0.394566\pi$$
0.325208 + 0.945642i $$0.394566\pi$$
$$758$$ 13.8953 0.504701
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −26.7172 −0.968497 −0.484249 0.874930i $$-0.660907\pi$$
−0.484249 + 0.874930i $$0.660907\pi$$
$$762$$ 6.29844 0.228168
$$763$$ −20.2094 −0.731628
$$764$$ −12.8062 −0.463314
$$765$$ 0 0
$$766$$ 32.5078 1.17455
$$767$$ 2.70156 0.0975478
$$768$$ −1.00000 −0.0360844
$$769$$ −8.00000 −0.288487 −0.144244 0.989542i $$-0.546075\pi$$
−0.144244 + 0.989542i $$0.546075\pi$$
$$770$$ 0 0
$$771$$ −15.2984 −0.550960
$$772$$ −16.2094 −0.583388
$$773$$ −26.5969 −0.956623 −0.478312 0.878190i $$-0.658751\pi$$
−0.478312 + 0.878190i $$0.658751\pi$$
$$774$$ −11.4031 −0.409877
$$775$$ 0 0
$$776$$ −2.59688 −0.0932224
$$777$$ 8.00000 0.286998
$$778$$ 27.1047 0.971750
$$779$$ 6.29844 0.225665
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 0 0
$$783$$ 6.40312 0.228829
$$784$$ 15.1047 0.539453
$$785$$ 0 0
$$786$$ 22.5078 0.802827
$$787$$ −33.8953 −1.20824 −0.604119 0.796894i $$-0.706475\pi$$
−0.604119 + 0.796894i $$0.706475\pi$$
$$788$$ 5.40312 0.192478
$$789$$ 2.00000 0.0712019
$$790$$ 0 0
$$791$$ −65.8219 −2.34036
$$792$$ −4.70156 −0.167063
$$793$$ 14.1047 0.500872
$$794$$ −21.9109 −0.777590
$$795$$ 0 0
$$796$$ 8.29844 0.294130
$$797$$ −2.91093 −0.103111 −0.0515553 0.998670i $$-0.516418\pi$$
−0.0515553 + 0.998670i $$0.516418\pi$$
$$798$$ 8.00000 0.283197
$$799$$ −4.91093 −0.173736
$$800$$ 0 0
$$801$$ 11.4031 0.402910
$$802$$ −18.2094 −0.642995
$$803$$ 56.4187 1.99097
$$804$$ −6.40312 −0.225821
$$805$$ 0 0
$$806$$ 10.1047 0.355922
$$807$$ −31.2094 −1.09862
$$808$$ −6.70156 −0.235760
$$809$$ 7.19375 0.252919 0.126459 0.991972i $$-0.459639\pi$$
0.126459 + 0.991972i $$0.459639\pi$$
$$810$$ 0 0
$$811$$ 48.7016 1.71014 0.855072 0.518510i $$-0.173513\pi$$
0.855072 + 0.518510i $$0.173513\pi$$
$$812$$ 30.1047 1.05647
$$813$$ −19.5078 −0.684169
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ −0.701562 −0.0245596
$$817$$ −19.4031 −0.678829
$$818$$ −29.4031 −1.02806
$$819$$ −4.70156 −0.164286
$$820$$ 0 0
$$821$$ −20.5969 −0.718836 −0.359418 0.933177i $$-0.617025\pi$$
−0.359418 + 0.933177i $$0.617025\pi$$
$$822$$ −13.7016 −0.477897
$$823$$ −17.1047 −0.596232 −0.298116 0.954530i $$-0.596358\pi$$
−0.298116 + 0.954530i $$0.596358\pi$$
$$824$$ 1.40312 0.0488801
$$825$$ 0 0
$$826$$ 12.7016 0.441944
$$827$$ −20.7016 −0.719864 −0.359932 0.932979i $$-0.617200\pi$$
−0.359932 + 0.932979i $$0.617200\pi$$
$$828$$ 0 0
$$829$$ −5.50781 −0.191294 −0.0956471 0.995415i $$-0.530492\pi$$
−0.0956471 + 0.995415i $$0.530492\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 1.00000 0.0346688
$$833$$ 10.5969 0.367160
$$834$$ 9.40312 0.325604
$$835$$ 0 0
$$836$$ −8.00000 −0.276686
$$837$$ 10.1047 0.349269
$$838$$ −9.91093 −0.342368
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 12.0000 0.413793
$$842$$ 0.596876 0.0205697
$$843$$ 21.9109 0.754653
$$844$$ 6.80625 0.234281
$$845$$ 0 0
$$846$$ 7.00000 0.240665
$$847$$ −52.2094 −1.79394
$$848$$ −2.40312 −0.0825236
$$849$$ 21.4031 0.734553
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 1.70156 0.0582946
$$853$$ −0.507811 −0.0173871 −0.00869355 0.999962i $$-0.502767\pi$$
−0.00869355 + 0.999962i $$0.502767\pi$$
$$854$$ 66.3141 2.26922
$$855$$ 0 0
$$856$$ 19.1047 0.652985
$$857$$ 7.61250 0.260038 0.130019 0.991512i $$-0.458496\pi$$
0.130019 + 0.991512i $$0.458496\pi$$
$$858$$ 4.70156 0.160509
$$859$$ −6.20937 −0.211861 −0.105931 0.994374i $$-0.533782\pi$$
−0.105931 + 0.994374i $$0.533782\pi$$
$$860$$ 0 0
$$861$$ −17.4031 −0.593097
$$862$$ −35.3141 −1.20280
$$863$$ −54.2250 −1.84584 −0.922920 0.384991i $$-0.874204\pi$$
−0.922920 + 0.384991i $$0.874204\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −11.1047 −0.377353
$$867$$ 16.5078 0.560635
$$868$$ 47.5078 1.61252
$$869$$ −26.8062 −0.909340
$$870$$ 0 0
$$871$$ 6.40312 0.216962
$$872$$ −4.29844 −0.145563
$$873$$ 2.59688 0.0878909
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −12.0000 −0.405442
$$877$$ 42.7172 1.44246 0.721228 0.692697i $$-0.243578\pi$$
0.721228 + 0.692697i $$0.243578\pi$$
$$878$$ 10.2984 0.347555
$$879$$ 21.4031 0.721909
$$880$$ 0 0
$$881$$ −39.7172 −1.33811 −0.669053 0.743215i $$-0.733300\pi$$
−0.669053 + 0.743215i $$0.733300\pi$$
$$882$$ −15.1047 −0.508601
$$883$$ −45.6125 −1.53498 −0.767491 0.641059i $$-0.778495\pi$$
−0.767491 + 0.641059i $$0.778495\pi$$
$$884$$ 0.701562 0.0235961
$$885$$ 0 0
$$886$$ 32.5078 1.09212
$$887$$ 20.0000 0.671534 0.335767 0.941945i $$-0.391004\pi$$
0.335767 + 0.941945i $$0.391004\pi$$
$$888$$ 1.70156 0.0571007
$$889$$ −29.6125 −0.993171
$$890$$ 0 0
$$891$$ 4.70156 0.157508
$$892$$ −11.4031 −0.381805
$$893$$ 11.9109 0.398584
$$894$$ 6.59688 0.220633
$$895$$ 0 0
$$896$$ 4.70156 0.157068
$$897$$ 0 0
$$898$$ −2.29844 −0.0766999
$$899$$ 64.7016 2.15792
$$900$$ 0 0
$$901$$ −1.68594 −0.0561668
$$902$$ 17.4031 0.579461
$$903$$ 53.6125 1.78411
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ −14.1047 −0.468597
$$907$$ 25.0156 0.830630 0.415315 0.909678i $$-0.363671\pi$$
0.415315 + 0.909678i $$0.363671\pi$$
$$908$$ 16.1047 0.534453
$$909$$ 6.70156 0.222277
$$910$$ 0 0
$$911$$ 42.2094 1.39846 0.699229 0.714897i $$-0.253527\pi$$
0.699229 + 0.714897i $$0.253527\pi$$
$$912$$ 1.70156 0.0563444
$$913$$ 50.3141 1.66515
$$914$$ −2.59688 −0.0858970
$$915$$ 0 0
$$916$$ 15.7016 0.518794
$$917$$ −105.822 −3.49455
$$918$$ 0.701562 0.0231550
$$919$$ −4.89531 −0.161481 −0.0807407 0.996735i $$-0.525729\pi$$
−0.0807407 + 0.996735i $$0.525729\pi$$
$$920$$ 0 0
$$921$$ −5.70156 −0.187873
$$922$$ −36.2094 −1.19249
$$923$$ −1.70156 −0.0560076
$$924$$ 22.1047 0.727191
$$925$$ 0 0
$$926$$ −21.2984 −0.699910
$$927$$ −1.40312 −0.0460846
$$928$$ 6.40312 0.210193
$$929$$ 52.2984 1.71586 0.857928 0.513770i $$-0.171751\pi$$
0.857928 + 0.513770i $$0.171751\pi$$
$$930$$ 0 0
$$931$$ −25.7016 −0.842335
$$932$$ 20.2094 0.661980
$$933$$ −30.0000 −0.982156
$$934$$ 30.2984 0.991395
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ −14.9109 −0.487119 −0.243560 0.969886i $$-0.578315\pi$$
−0.243560 + 0.969886i $$0.578315\pi$$
$$938$$ 30.1047 0.982953
$$939$$ −21.2094 −0.692142
$$940$$ 0 0
$$941$$ 28.4187 0.926425 0.463212 0.886247i $$-0.346697\pi$$
0.463212 + 0.886247i $$0.346697\pi$$
$$942$$ 22.7016 0.739657
$$943$$ 0 0
$$944$$ 2.70156 0.0879284
$$945$$ 0 0
$$946$$ −53.6125 −1.74309
$$947$$ −47.5078 −1.54380 −0.771898 0.635746i $$-0.780693\pi$$
−0.771898 + 0.635746i $$0.780693\pi$$
$$948$$ 5.70156 0.185178
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 1.19375 0.0387100
$$952$$ 3.29844 0.106903
$$953$$ 31.2984 1.01386 0.506928 0.861988i $$-0.330781\pi$$
0.506928 + 0.861988i $$0.330781\pi$$
$$954$$ 2.40312 0.0778040
$$955$$ 0 0
$$956$$ 6.10469 0.197440
$$957$$ 30.1047 0.973146
$$958$$ 40.6125 1.31213
$$959$$ 64.4187 2.08019
$$960$$ 0 0
$$961$$ 71.1047 2.29370
$$962$$ −1.70156 −0.0548606
$$963$$ −19.1047 −0.615640
$$964$$ −2.59688 −0.0836397
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −29.8953 −0.961368 −0.480684 0.876894i $$-0.659612\pi$$
−0.480684 + 0.876894i $$0.659612\pi$$
$$968$$ −11.1047 −0.356918
$$969$$ 1.19375 0.0383488
$$970$$ 0 0
$$971$$ −36.8953 −1.18403 −0.592013 0.805928i $$-0.701667\pi$$
−0.592013 + 0.805928i $$0.701667\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −44.2094 −1.41729
$$974$$ −7.89531 −0.252982
$$975$$ 0 0
$$976$$ 14.1047 0.451480
$$977$$ 23.4031 0.748732 0.374366 0.927281i $$-0.377860\pi$$
0.374366 + 0.927281i $$0.377860\pi$$
$$978$$ 18.8062 0.601358
$$979$$ 53.6125 1.71346
$$980$$ 0 0
$$981$$ 4.29844 0.137239
$$982$$ −33.4031 −1.06594
$$983$$ 7.50781 0.239462 0.119731 0.992806i $$-0.461797\pi$$
0.119731 + 0.992806i $$0.461797\pi$$
$$984$$ −3.70156 −0.118001
$$985$$ 0 0
$$986$$ 4.49219 0.143060
$$987$$ −32.9109 −1.04757
$$988$$ −1.70156 −0.0541339
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −9.10469 −0.289220 −0.144610 0.989489i $$-0.546193\pi$$
−0.144610 + 0.989489i $$0.546193\pi$$
$$992$$ 10.1047 0.320824
$$993$$ 12.0000 0.380808
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ −10.7016 −0.339092
$$997$$ 10.4922 0.332291 0.166145 0.986101i $$-0.446868\pi$$
0.166145 + 0.986101i $$0.446868\pi$$
$$998$$ −17.2094 −0.544753
$$999$$ −1.70156 −0.0538350
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 1950.2.a.bc.1.1 2
3.2 odd 2 5850.2.a.cj.1.1 2
5.2 odd 4 1950.2.e.p.1249.1 4
5.3 odd 4 1950.2.e.p.1249.4 4
5.4 even 2 1950.2.a.bg.1.2 yes 2
15.2 even 4 5850.2.e.bi.5149.3 4
15.8 even 4 5850.2.e.bi.5149.2 4
15.14 odd 2 5850.2.a.cg.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.bc.1.1 2 1.1 even 1 trivial
1950.2.a.bg.1.2 yes 2 5.4 even 2
1950.2.e.p.1249.1 4 5.2 odd 4
1950.2.e.p.1249.4 4 5.3 odd 4
5850.2.a.cg.1.2 2 15.14 odd 2
5850.2.a.cj.1.1 2 3.2 odd 2
5850.2.e.bi.5149.2 4 15.8 even 4
5850.2.e.bi.5149.3 4 15.2 even 4