# Properties

 Label 1950.2.e.p.1249.1 Level $1950$ Weight $2$ Character 1950.1249 Analytic conductor $15.571$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ Defining polynomial: $$x^{4} + 21 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1249.1 Root $$-3.70156i$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.1249 Dual form 1950.2.e.p.1249.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.70156i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.70156i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.70156 q^{11} -1.00000i q^{12} -1.00000i q^{13} -4.70156 q^{14} +1.00000 q^{16} +0.701562i q^{17} +1.00000i q^{18} +1.70156 q^{19} +4.70156 q^{21} -4.70156i q^{22} -1.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} +4.70156i q^{28} +6.40312 q^{29} -10.1047 q^{31} -1.00000i q^{32} +4.70156i q^{33} +0.701562 q^{34} +1.00000 q^{36} +1.70156i q^{37} -1.70156i q^{38} +1.00000 q^{39} -3.70156 q^{41} -4.70156i q^{42} -11.4031i q^{43} -4.70156 q^{44} -7.00000i q^{47} +1.00000i q^{48} -15.1047 q^{49} -0.701562 q^{51} +1.00000i q^{52} +2.40312i q^{53} -1.00000 q^{54} +4.70156 q^{56} +1.70156i q^{57} -6.40312i q^{58} -2.70156 q^{59} +14.1047 q^{61} +10.1047i q^{62} +4.70156i q^{63} -1.00000 q^{64} +4.70156 q^{66} +6.40312i q^{67} -0.701562i q^{68} -1.70156 q^{71} -1.00000i q^{72} -12.0000i q^{73} +1.70156 q^{74} -1.70156 q^{76} -22.1047i q^{77} -1.00000i q^{78} +5.70156 q^{79} +1.00000 q^{81} +3.70156i q^{82} -10.7016i q^{83} -4.70156 q^{84} -11.4031 q^{86} +6.40312i q^{87} +4.70156i q^{88} -11.4031 q^{89} -4.70156 q^{91} -10.1047i q^{93} -7.00000 q^{94} +1.00000 q^{96} +2.59688i q^{97} +15.1047i q^{98} -4.70156 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 6 q^{11} - 6 q^{14} + 4 q^{16} - 6 q^{19} + 6 q^{21} - 4 q^{24} - 4 q^{26} - 2 q^{31} - 10 q^{34} + 4 q^{36} + 4 q^{39} - 2 q^{41} - 6 q^{44} - 22 q^{49} + 10 q^{51} - 4 q^{54} + 6 q^{56} + 2 q^{59} + 18 q^{61} - 4 q^{64} + 6 q^{66} + 6 q^{71} - 6 q^{74} + 6 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} - 20 q^{86} - 20 q^{89} - 6 q^{91} - 28 q^{94} + 4 q^{96} - 6 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 1.00000 0.408248
$$7$$ − 4.70156i − 1.77702i −0.458854 0.888512i $$-0.651740\pi$$
0.458854 0.888512i $$-0.348260\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.70156 1.41757 0.708787 0.705422i $$-0.249243\pi$$
0.708787 + 0.705422i $$0.249243\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 1.00000i − 0.277350i
$$14$$ −4.70156 −1.25655
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0.701562i 0.170154i 0.996374 + 0.0850769i $$0.0271136\pi$$
−0.996374 + 0.0850769i $$0.972886\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 1.70156 0.390365 0.195183 0.980767i $$-0.437470\pi$$
0.195183 + 0.980767i $$0.437470\pi$$
$$20$$ 0 0
$$21$$ 4.70156 1.02596
$$22$$ − 4.70156i − 1.00238i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ − 1.00000i − 0.192450i
$$28$$ 4.70156i 0.888512i
$$29$$ 6.40312 1.18903 0.594515 0.804084i $$-0.297344\pi$$
0.594515 + 0.804084i $$0.297344\pi$$
$$30$$ 0 0
$$31$$ −10.1047 −1.81486 −0.907428 0.420208i $$-0.861957\pi$$
−0.907428 + 0.420208i $$0.861957\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 4.70156i 0.818437i
$$34$$ 0.701562 0.120317
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.70156i 0.279735i 0.990170 + 0.139868i $$0.0446677\pi$$
−0.990170 + 0.139868i $$0.955332\pi$$
$$38$$ − 1.70156i − 0.276030i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −3.70156 −0.578087 −0.289043 0.957316i $$-0.593337\pi$$
−0.289043 + 0.957316i $$0.593337\pi$$
$$42$$ − 4.70156i − 0.725467i
$$43$$ − 11.4031i − 1.73896i −0.493968 0.869480i $$-0.664454\pi$$
0.493968 0.869480i $$-0.335546\pi$$
$$44$$ −4.70156 −0.708787
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 7.00000i − 1.02105i −0.859861 0.510527i $$-0.829450\pi$$
0.859861 0.510527i $$-0.170550\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −15.1047 −2.15781
$$50$$ 0 0
$$51$$ −0.701562 −0.0982383
$$52$$ 1.00000i 0.138675i
$$53$$ 2.40312i 0.330095i 0.986286 + 0.165047i $$0.0527777\pi$$
−0.986286 + 0.165047i $$0.947222\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 4.70156 0.628273
$$57$$ 1.70156i 0.225377i
$$58$$ − 6.40312i − 0.840771i
$$59$$ −2.70156 −0.351713 −0.175857 0.984416i $$-0.556270\pi$$
−0.175857 + 0.984416i $$0.556270\pi$$
$$60$$ 0 0
$$61$$ 14.1047 1.80592 0.902960 0.429725i $$-0.141389\pi$$
0.902960 + 0.429725i $$0.141389\pi$$
$$62$$ 10.1047i 1.28330i
$$63$$ 4.70156i 0.592341i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.70156 0.578722
$$67$$ 6.40312i 0.782266i 0.920334 + 0.391133i $$0.127917\pi$$
−0.920334 + 0.391133i $$0.872083\pi$$
$$68$$ − 0.701562i − 0.0850769i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −1.70156 −0.201938 −0.100969 0.994890i $$-0.532194\pi$$
−0.100969 + 0.994890i $$0.532194\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 12.0000i − 1.40449i −0.711934 0.702247i $$-0.752180\pi$$
0.711934 0.702247i $$-0.247820\pi$$
$$74$$ 1.70156 0.197803
$$75$$ 0 0
$$76$$ −1.70156 −0.195183
$$77$$ − 22.1047i − 2.51906i
$$78$$ − 1.00000i − 0.113228i
$$79$$ 5.70156 0.641476 0.320738 0.947168i $$-0.396069\pi$$
0.320738 + 0.947168i $$0.396069\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 3.70156i 0.408769i
$$83$$ − 10.7016i − 1.17465i −0.809352 0.587325i $$-0.800181\pi$$
0.809352 0.587325i $$-0.199819\pi$$
$$84$$ −4.70156 −0.512982
$$85$$ 0 0
$$86$$ −11.4031 −1.22963
$$87$$ 6.40312i 0.686487i
$$88$$ 4.70156i 0.501188i
$$89$$ −11.4031 −1.20873 −0.604364 0.796708i $$-0.706573\pi$$
−0.604364 + 0.796708i $$0.706573\pi$$
$$90$$ 0 0
$$91$$ −4.70156 −0.492858
$$92$$ 0 0
$$93$$ − 10.1047i − 1.04781i
$$94$$ −7.00000 −0.721995
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 2.59688i 0.263673i 0.991271 + 0.131836i $$0.0420874\pi$$
−0.991271 + 0.131836i $$0.957913\pi$$
$$98$$ 15.1047i 1.52580i
$$99$$ −4.70156 −0.472525
$$100$$ 0 0
$$101$$ 6.70156 0.666830 0.333415 0.942780i $$-0.391799\pi$$
0.333415 + 0.942780i $$0.391799\pi$$
$$102$$ 0.701562i 0.0694650i
$$103$$ 1.40312i 0.138254i 0.997608 + 0.0691270i $$0.0220214\pi$$
−0.997608 + 0.0691270i $$0.977979\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 2.40312 0.233412
$$107$$ − 19.1047i − 1.84692i −0.383695 0.923460i $$-0.625349\pi$$
0.383695 0.923460i $$-0.374651\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ −4.29844 −0.411716 −0.205858 0.978582i $$-0.565998\pi$$
−0.205858 + 0.978582i $$0.565998\pi$$
$$110$$ 0 0
$$111$$ −1.70156 −0.161505
$$112$$ − 4.70156i − 0.444256i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 1.70156 0.159366
$$115$$ 0 0
$$116$$ −6.40312 −0.594515
$$117$$ 1.00000i 0.0924500i
$$118$$ 2.70156i 0.248699i
$$119$$ 3.29844 0.302367
$$120$$ 0 0
$$121$$ 11.1047 1.00952
$$122$$ − 14.1047i − 1.27698i
$$123$$ − 3.70156i − 0.333759i
$$124$$ 10.1047 0.907428
$$125$$ 0 0
$$126$$ 4.70156 0.418848
$$127$$ 6.29844i 0.558896i 0.960161 + 0.279448i $$0.0901515\pi$$
−0.960161 + 0.279448i $$0.909849\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 11.4031 1.00399
$$130$$ 0 0
$$131$$ 22.5078 1.96652 0.983258 0.182217i $$-0.0583275\pi$$
0.983258 + 0.182217i $$0.0583275\pi$$
$$132$$ − 4.70156i − 0.409218i
$$133$$ − 8.00000i − 0.693688i
$$134$$ 6.40312 0.553146
$$135$$ 0 0
$$136$$ −0.701562 −0.0601585
$$137$$ − 13.7016i − 1.17060i −0.810816 0.585302i $$-0.800976\pi$$
0.810816 0.585302i $$-0.199024\pi$$
$$138$$ 0 0
$$139$$ −9.40312 −0.797563 −0.398781 0.917046i $$-0.630567\pi$$
−0.398781 + 0.917046i $$0.630567\pi$$
$$140$$ 0 0
$$141$$ 7.00000 0.589506
$$142$$ 1.70156i 0.142792i
$$143$$ − 4.70156i − 0.393164i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −12.0000 −0.993127
$$147$$ − 15.1047i − 1.24581i
$$148$$ − 1.70156i − 0.139868i
$$149$$ −6.59688 −0.540437 −0.270219 0.962799i $$-0.587096\pi$$
−0.270219 + 0.962799i $$0.587096\pi$$
$$150$$ 0 0
$$151$$ −14.1047 −1.14782 −0.573912 0.818917i $$-0.694575\pi$$
−0.573912 + 0.818917i $$0.694575\pi$$
$$152$$ 1.70156i 0.138015i
$$153$$ − 0.701562i − 0.0567179i
$$154$$ −22.1047 −1.78125
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ 22.7016i 1.81178i 0.423511 + 0.905891i $$0.360797\pi$$
−0.423511 + 0.905891i $$0.639203\pi$$
$$158$$ − 5.70156i − 0.453592i
$$159$$ −2.40312 −0.190580
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 18.8062i − 1.47302i −0.676427 0.736510i $$-0.736473\pi$$
0.676427 0.736510i $$-0.263527\pi$$
$$164$$ 3.70156 0.289043
$$165$$ 0 0
$$166$$ −10.7016 −0.830602
$$167$$ 1.10469i 0.0854832i 0.999086 + 0.0427416i $$0.0136092\pi$$
−0.999086 + 0.0427416i $$0.986391\pi$$
$$168$$ 4.70156i 0.362733i
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ −1.70156 −0.130122
$$172$$ 11.4031i 0.869480i
$$173$$ − 0.193752i − 0.0147307i −0.999973 0.00736533i $$-0.997656\pi$$
0.999973 0.00736533i $$-0.00234448\pi$$
$$174$$ 6.40312 0.485420
$$175$$ 0 0
$$176$$ 4.70156 0.354394
$$177$$ − 2.70156i − 0.203062i
$$178$$ 11.4031i 0.854700i
$$179$$ 24.2094 1.80949 0.904747 0.425950i $$-0.140060\pi$$
0.904747 + 0.425950i $$0.140060\pi$$
$$180$$ 0 0
$$181$$ 11.2984 0.839806 0.419903 0.907569i $$-0.362064\pi$$
0.419903 + 0.907569i $$0.362064\pi$$
$$182$$ 4.70156i 0.348503i
$$183$$ 14.1047i 1.04265i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −10.1047 −0.740912
$$187$$ 3.29844i 0.241206i
$$188$$ 7.00000i 0.510527i
$$189$$ −4.70156 −0.341988
$$190$$ 0 0
$$191$$ −12.8062 −0.926628 −0.463314 0.886194i $$-0.653340\pi$$
−0.463314 + 0.886194i $$0.653340\pi$$
$$192$$ − 1.00000i − 0.0721688i
$$193$$ 16.2094i 1.16678i 0.812194 + 0.583388i $$0.198273\pi$$
−0.812194 + 0.583388i $$0.801727\pi$$
$$194$$ 2.59688 0.186445
$$195$$ 0 0
$$196$$ 15.1047 1.07891
$$197$$ 5.40312i 0.384957i 0.981301 + 0.192478i $$0.0616525\pi$$
−0.981301 + 0.192478i $$0.938347\pi$$
$$198$$ 4.70156i 0.334125i
$$199$$ −8.29844 −0.588261 −0.294130 0.955765i $$-0.595030\pi$$
−0.294130 + 0.955765i $$0.595030\pi$$
$$200$$ 0 0
$$201$$ −6.40312 −0.451642
$$202$$ − 6.70156i − 0.471520i
$$203$$ − 30.1047i − 2.11293i
$$204$$ 0.701562 0.0491192
$$205$$ 0 0
$$206$$ 1.40312 0.0977603
$$207$$ 0 0
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ 6.80625 0.468561 0.234281 0.972169i $$-0.424727\pi$$
0.234281 + 0.972169i $$0.424727\pi$$
$$212$$ − 2.40312i − 0.165047i
$$213$$ − 1.70156i − 0.116589i
$$214$$ −19.1047 −1.30597
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 47.5078i 3.22504i
$$218$$ 4.29844i 0.291127i
$$219$$ 12.0000 0.810885
$$220$$ 0 0
$$221$$ 0.701562 0.0471922
$$222$$ 1.70156i 0.114201i
$$223$$ 11.4031i 0.763610i 0.924243 + 0.381805i $$0.124697\pi$$
−0.924243 + 0.381805i $$0.875303\pi$$
$$224$$ −4.70156 −0.314136
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ 16.1047i 1.06891i 0.845198 + 0.534453i $$0.179482\pi$$
−0.845198 + 0.534453i $$0.820518\pi$$
$$228$$ − 1.70156i − 0.112689i
$$229$$ −15.7016 −1.03759 −0.518794 0.854899i $$-0.673619\pi$$
−0.518794 + 0.854899i $$0.673619\pi$$
$$230$$ 0 0
$$231$$ 22.1047 1.45438
$$232$$ 6.40312i 0.420386i
$$233$$ − 20.2094i − 1.32396i −0.749521 0.661980i $$-0.769716\pi$$
0.749521 0.661980i $$-0.230284\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ 2.70156 0.175857
$$237$$ 5.70156i 0.370356i
$$238$$ − 3.29844i − 0.213806i
$$239$$ −6.10469 −0.394879 −0.197440 0.980315i $$-0.563263\pi$$
−0.197440 + 0.980315i $$0.563263\pi$$
$$240$$ 0 0
$$241$$ −2.59688 −0.167279 −0.0836397 0.996496i $$-0.526654\pi$$
−0.0836397 + 0.996496i $$0.526654\pi$$
$$242$$ − 11.1047i − 0.713836i
$$243$$ 1.00000i 0.0641500i
$$244$$ −14.1047 −0.902960
$$245$$ 0 0
$$246$$ −3.70156 −0.236003
$$247$$ − 1.70156i − 0.108268i
$$248$$ − 10.1047i − 0.641648i
$$249$$ 10.7016 0.678184
$$250$$ 0 0
$$251$$ 0.298438 0.0188372 0.00941862 0.999956i $$-0.497002\pi$$
0.00941862 + 0.999956i $$0.497002\pi$$
$$252$$ − 4.70156i − 0.296171i
$$253$$ 0 0
$$254$$ 6.29844 0.395199
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 15.2984i 0.954290i 0.878824 + 0.477145i $$0.158328\pi$$
−0.878824 + 0.477145i $$0.841672\pi$$
$$258$$ − 11.4031i − 0.709928i
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ −6.40312 −0.396343
$$262$$ − 22.5078i − 1.39054i
$$263$$ 2.00000i 0.123325i 0.998097 + 0.0616626i $$0.0196403\pi$$
−0.998097 + 0.0616626i $$0.980360\pi$$
$$264$$ −4.70156 −0.289361
$$265$$ 0 0
$$266$$ −8.00000 −0.490511
$$267$$ − 11.4031i − 0.697860i
$$268$$ − 6.40312i − 0.391133i
$$269$$ −31.2094 −1.90287 −0.951435 0.307851i $$-0.900390\pi$$
−0.951435 + 0.307851i $$0.900390\pi$$
$$270$$ 0 0
$$271$$ 19.5078 1.18502 0.592508 0.805565i $$-0.298138\pi$$
0.592508 + 0.805565i $$0.298138\pi$$
$$272$$ 0.701562i 0.0425385i
$$273$$ − 4.70156i − 0.284551i
$$274$$ −13.7016 −0.827742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ 9.40312i 0.563962i
$$279$$ 10.1047 0.604952
$$280$$ 0 0
$$281$$ −21.9109 −1.30710 −0.653548 0.756885i $$-0.726720\pi$$
−0.653548 + 0.756885i $$0.726720\pi$$
$$282$$ − 7.00000i − 0.416844i
$$283$$ 21.4031i 1.27228i 0.771572 + 0.636142i $$0.219471\pi$$
−0.771572 + 0.636142i $$0.780529\pi$$
$$284$$ 1.70156 0.100969
$$285$$ 0 0
$$286$$ −4.70156 −0.278009
$$287$$ 17.4031i 1.02727i
$$288$$ 1.00000i 0.0589256i
$$289$$ 16.5078 0.971048
$$290$$ 0 0
$$291$$ −2.59688 −0.152232
$$292$$ 12.0000i 0.702247i
$$293$$ 21.4031i 1.25038i 0.780471 + 0.625192i $$0.214979\pi$$
−0.780471 + 0.625192i $$0.785021\pi$$
$$294$$ −15.1047 −0.880923
$$295$$ 0 0
$$296$$ −1.70156 −0.0989013
$$297$$ − 4.70156i − 0.272812i
$$298$$ 6.59688i 0.382147i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −53.6125 −3.09017
$$302$$ 14.1047i 0.811633i
$$303$$ 6.70156i 0.384995i
$$304$$ 1.70156 0.0975913
$$305$$ 0 0
$$306$$ −0.701562 −0.0401056
$$307$$ 5.70156i 0.325405i 0.986675 + 0.162703i $$0.0520211\pi$$
−0.986675 + 0.162703i $$0.947979\pi$$
$$308$$ 22.1047i 1.25953i
$$309$$ −1.40312 −0.0798209
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 1.00000i 0.0566139i
$$313$$ − 21.2094i − 1.19882i −0.800440 0.599412i $$-0.795401\pi$$
0.800440 0.599412i $$-0.204599\pi$$
$$314$$ 22.7016 1.28112
$$315$$ 0 0
$$316$$ −5.70156 −0.320738
$$317$$ − 1.19375i − 0.0670478i −0.999438 0.0335239i $$-0.989327\pi$$
0.999438 0.0335239i $$-0.0106730\pi$$
$$318$$ 2.40312i 0.134761i
$$319$$ 30.1047 1.68554
$$320$$ 0 0
$$321$$ 19.1047 1.06632
$$322$$ 0 0
$$323$$ 1.19375i 0.0664221i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −18.8062 −1.04158
$$327$$ − 4.29844i − 0.237704i
$$328$$ − 3.70156i − 0.204385i
$$329$$ −32.9109 −1.81444
$$330$$ 0 0
$$331$$ −12.0000 −0.659580 −0.329790 0.944054i $$-0.606978\pi$$
−0.329790 + 0.944054i $$0.606978\pi$$
$$332$$ 10.7016i 0.587325i
$$333$$ − 1.70156i − 0.0932450i
$$334$$ 1.10469 0.0604457
$$335$$ 0 0
$$336$$ 4.70156 0.256491
$$337$$ − 8.10469i − 0.441490i −0.975332 0.220745i $$-0.929151\pi$$
0.975332 0.220745i $$-0.0708489\pi$$
$$338$$ 1.00000i 0.0543928i
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ −47.5078 −2.57269
$$342$$ 1.70156i 0.0920099i
$$343$$ 38.1047i 2.05746i
$$344$$ 11.4031 0.614815
$$345$$ 0 0
$$346$$ −0.193752 −0.0104161
$$347$$ − 10.5078i − 0.564089i −0.959401 0.282044i $$-0.908987\pi$$
0.959401 0.282044i $$-0.0910126\pi$$
$$348$$ − 6.40312i − 0.343243i
$$349$$ 11.4031 0.610395 0.305198 0.952289i $$-0.401277\pi$$
0.305198 + 0.952289i $$0.401277\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ − 4.70156i − 0.250594i
$$353$$ 14.5078i 0.772173i 0.922463 + 0.386086i $$0.126173\pi$$
−0.922463 + 0.386086i $$0.873827\pi$$
$$354$$ −2.70156 −0.143586
$$355$$ 0 0
$$356$$ 11.4031 0.604364
$$357$$ 3.29844i 0.174572i
$$358$$ − 24.2094i − 1.27951i
$$359$$ 34.6125 1.82678 0.913389 0.407088i $$-0.133456\pi$$
0.913389 + 0.407088i $$0.133456\pi$$
$$360$$ 0 0
$$361$$ −16.1047 −0.847615
$$362$$ − 11.2984i − 0.593833i
$$363$$ 11.1047i 0.582845i
$$364$$ 4.70156 0.246429
$$365$$ 0 0
$$366$$ 14.1047 0.737264
$$367$$ 2.29844i 0.119977i 0.998199 + 0.0599887i $$0.0191065\pi$$
−0.998199 + 0.0599887i $$0.980894\pi$$
$$368$$ 0 0
$$369$$ 3.70156 0.192696
$$370$$ 0 0
$$371$$ 11.2984 0.586586
$$372$$ 10.1047i 0.523904i
$$373$$ 5.29844i 0.274343i 0.990547 + 0.137171i $$0.0438011\pi$$
−0.990547 + 0.137171i $$0.956199\pi$$
$$374$$ 3.29844 0.170558
$$375$$ 0 0
$$376$$ 7.00000 0.360997
$$377$$ − 6.40312i − 0.329778i
$$378$$ 4.70156i 0.241822i
$$379$$ 13.8953 0.713754 0.356877 0.934151i $$-0.383841\pi$$
0.356877 + 0.934151i $$0.383841\pi$$
$$380$$ 0 0
$$381$$ −6.29844 −0.322679
$$382$$ 12.8062i 0.655225i
$$383$$ 32.5078i 1.66107i 0.556965 + 0.830536i $$0.311966\pi$$
−0.556965 + 0.830536i $$0.688034\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 16.2094 0.825035
$$387$$ 11.4031i 0.579653i
$$388$$ − 2.59688i − 0.131836i
$$389$$ 27.1047 1.37426 0.687131 0.726533i $$-0.258870\pi$$
0.687131 + 0.726533i $$0.258870\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 15.1047i − 0.762902i
$$393$$ 22.5078i 1.13537i
$$394$$ 5.40312 0.272205
$$395$$ 0 0
$$396$$ 4.70156 0.236262
$$397$$ 21.9109i 1.09968i 0.835271 + 0.549839i $$0.185311\pi$$
−0.835271 + 0.549839i $$0.814689\pi$$
$$398$$ 8.29844i 0.415963i
$$399$$ 8.00000 0.400501
$$400$$ 0 0
$$401$$ 18.2094 0.909333 0.454666 0.890662i $$-0.349759\pi$$
0.454666 + 0.890662i $$0.349759\pi$$
$$402$$ 6.40312i 0.319359i
$$403$$ 10.1047i 0.503350i
$$404$$ −6.70156 −0.333415
$$405$$ 0 0
$$406$$ −30.1047 −1.49407
$$407$$ 8.00000i 0.396545i
$$408$$ − 0.701562i − 0.0347325i
$$409$$ −29.4031 −1.45389 −0.726945 0.686695i $$-0.759061\pi$$
−0.726945 + 0.686695i $$0.759061\pi$$
$$410$$ 0 0
$$411$$ 13.7016 0.675848
$$412$$ − 1.40312i − 0.0691270i
$$413$$ 12.7016i 0.625003i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ − 9.40312i − 0.460473i
$$418$$ − 8.00000i − 0.391293i
$$419$$ −9.91093 −0.484181 −0.242090 0.970254i $$-0.577833\pi$$
−0.242090 + 0.970254i $$0.577833\pi$$
$$420$$ 0 0
$$421$$ −0.596876 −0.0290899 −0.0145450 0.999894i $$-0.504630\pi$$
−0.0145450 + 0.999894i $$0.504630\pi$$
$$422$$ − 6.80625i − 0.331323i
$$423$$ 7.00000i 0.340352i
$$424$$ −2.40312 −0.116706
$$425$$ 0 0
$$426$$ −1.70156 −0.0824410
$$427$$ − 66.3141i − 3.20916i
$$428$$ 19.1047i 0.923460i
$$429$$ 4.70156 0.226994
$$430$$ 0 0
$$431$$ 35.3141 1.70102 0.850509 0.525960i $$-0.176294\pi$$
0.850509 + 0.525960i $$0.176294\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ − 11.1047i − 0.533657i −0.963744 0.266829i $$-0.914024\pi$$
0.963744 0.266829i $$-0.0859758\pi$$
$$434$$ 47.5078 2.28045
$$435$$ 0 0
$$436$$ 4.29844 0.205858
$$437$$ 0 0
$$438$$ − 12.0000i − 0.573382i
$$439$$ 10.2984 0.491518 0.245759 0.969331i $$-0.420963\pi$$
0.245759 + 0.969331i $$0.420963\pi$$
$$440$$ 0 0
$$441$$ 15.1047 0.719271
$$442$$ − 0.701562i − 0.0333699i
$$443$$ 32.5078i 1.54449i 0.635323 + 0.772246i $$0.280867\pi$$
−0.635323 + 0.772246i $$0.719133\pi$$
$$444$$ 1.70156 0.0807526
$$445$$ 0 0
$$446$$ 11.4031 0.539954
$$447$$ − 6.59688i − 0.312022i
$$448$$ 4.70156i 0.222128i
$$449$$ −2.29844 −0.108470 −0.0542350 0.998528i $$-0.517272\pi$$
−0.0542350 + 0.998528i $$0.517272\pi$$
$$450$$ 0 0
$$451$$ −17.4031 −0.819481
$$452$$ 14.0000i 0.658505i
$$453$$ − 14.1047i − 0.662696i
$$454$$ 16.1047 0.755830
$$455$$ 0 0
$$456$$ −1.70156 −0.0796829
$$457$$ 2.59688i 0.121477i 0.998154 + 0.0607384i $$0.0193455\pi$$
−0.998154 + 0.0607384i $$0.980654\pi$$
$$458$$ 15.7016i 0.733686i
$$459$$ 0.701562 0.0327461
$$460$$ 0 0
$$461$$ 36.2094 1.68644 0.843219 0.537570i $$-0.180658\pi$$
0.843219 + 0.537570i $$0.180658\pi$$
$$462$$ − 22.1047i − 1.02840i
$$463$$ − 21.2984i − 0.989822i −0.868944 0.494911i $$-0.835201\pi$$
0.868944 0.494911i $$-0.164799\pi$$
$$464$$ 6.40312 0.297258
$$465$$ 0 0
$$466$$ −20.2094 −0.936181
$$467$$ − 30.2984i − 1.40204i −0.713139 0.701022i $$-0.752727\pi$$
0.713139 0.701022i $$-0.247273\pi$$
$$468$$ − 1.00000i − 0.0462250i
$$469$$ 30.1047 1.39011
$$470$$ 0 0
$$471$$ −22.7016 −1.04603
$$472$$ − 2.70156i − 0.124349i
$$473$$ − 53.6125i − 2.46511i
$$474$$ 5.70156 0.261881
$$475$$ 0 0
$$476$$ −3.29844 −0.151184
$$477$$ − 2.40312i − 0.110032i
$$478$$ 6.10469i 0.279222i
$$479$$ 40.6125 1.85563 0.927816 0.373038i $$-0.121684\pi$$
0.927816 + 0.373038i $$0.121684\pi$$
$$480$$ 0 0
$$481$$ 1.70156 0.0775846
$$482$$ 2.59688i 0.118284i
$$483$$ 0 0
$$484$$ −11.1047 −0.504758
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 7.89531i 0.357771i 0.983870 + 0.178885i $$0.0572491\pi$$
−0.983870 + 0.178885i $$0.942751\pi$$
$$488$$ 14.1047i 0.638489i
$$489$$ 18.8062 0.850448
$$490$$ 0 0
$$491$$ 33.4031 1.50746 0.753731 0.657183i $$-0.228252\pi$$
0.753731 + 0.657183i $$0.228252\pi$$
$$492$$ 3.70156i 0.166879i
$$493$$ 4.49219i 0.202318i
$$494$$ −1.70156 −0.0765569
$$495$$ 0 0
$$496$$ −10.1047 −0.453714
$$497$$ 8.00000i 0.358849i
$$498$$ − 10.7016i − 0.479548i
$$499$$ −17.2094 −0.770397 −0.385199 0.922834i $$-0.625867\pi$$
−0.385199 + 0.922834i $$0.625867\pi$$
$$500$$ 0 0
$$501$$ −1.10469 −0.0493537
$$502$$ − 0.298438i − 0.0133199i
$$503$$ − 12.2094i − 0.544389i −0.962242 0.272195i $$-0.912251\pi$$
0.962242 0.272195i $$-0.0877494\pi$$
$$504$$ −4.70156 −0.209424
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 1.00000i − 0.0444116i
$$508$$ − 6.29844i − 0.279448i
$$509$$ 5.40312 0.239489 0.119745 0.992805i $$-0.461792\pi$$
0.119745 + 0.992805i $$0.461792\pi$$
$$510$$ 0 0
$$511$$ −56.4187 −2.49582
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 1.70156i − 0.0751258i
$$514$$ 15.2984 0.674785
$$515$$ 0 0
$$516$$ −11.4031 −0.501995
$$517$$ − 32.9109i − 1.44742i
$$518$$ − 8.00000i − 0.351500i
$$519$$ 0.193752 0.00850475
$$520$$ 0 0
$$521$$ −36.2094 −1.58636 −0.793181 0.608986i $$-0.791576\pi$$
−0.793181 + 0.608986i $$0.791576\pi$$
$$522$$ 6.40312i 0.280257i
$$523$$ − 24.8062i − 1.08470i −0.840152 0.542351i $$-0.817534\pi$$
0.840152 0.542351i $$-0.182466\pi$$
$$524$$ −22.5078 −0.983258
$$525$$ 0 0
$$526$$ 2.00000 0.0872041
$$527$$ − 7.08907i − 0.308805i
$$528$$ 4.70156i 0.204609i
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 2.70156 0.117238
$$532$$ 8.00000i 0.346844i
$$533$$ 3.70156i 0.160332i
$$534$$ −11.4031 −0.493461
$$535$$ 0 0
$$536$$ −6.40312 −0.276573
$$537$$ 24.2094i 1.04471i
$$538$$ 31.2094i 1.34553i
$$539$$ −71.0156 −3.05886
$$540$$ 0 0
$$541$$ −1.79063 −0.0769851 −0.0384925 0.999259i $$-0.512256\pi$$
−0.0384925 + 0.999259i $$0.512256\pi$$
$$542$$ − 19.5078i − 0.837932i
$$543$$ 11.2984i 0.484862i
$$544$$ 0.701562 0.0300792
$$545$$ 0 0
$$546$$ −4.70156 −0.201208
$$547$$ 31.6125i 1.35165i 0.737061 + 0.675826i $$0.236213\pi$$
−0.737061 + 0.675826i $$0.763787\pi$$
$$548$$ 13.7016i 0.585302i
$$549$$ −14.1047 −0.601973
$$550$$ 0 0
$$551$$ 10.8953 0.464156
$$552$$ 0 0
$$553$$ − 26.8062i − 1.13992i
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ 9.40312 0.398781
$$557$$ − 16.8062i − 0.712104i −0.934466 0.356052i $$-0.884123\pi$$
0.934466 0.356052i $$-0.115877\pi$$
$$558$$ − 10.1047i − 0.427765i
$$559$$ −11.4031 −0.482301
$$560$$ 0 0
$$561$$ −3.29844 −0.139260
$$562$$ 21.9109i 0.924257i
$$563$$ − 32.5078i − 1.37004i −0.728524 0.685020i $$-0.759793\pi$$
0.728524 0.685020i $$-0.240207\pi$$
$$564$$ −7.00000 −0.294753
$$565$$ 0 0
$$566$$ 21.4031 0.899640
$$567$$ − 4.70156i − 0.197447i
$$568$$ − 1.70156i − 0.0713960i
$$569$$ 11.2984 0.473655 0.236828 0.971552i $$-0.423892\pi$$
0.236828 + 0.971552i $$0.423892\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 4.70156i 0.196582i
$$573$$ − 12.8062i − 0.534989i
$$574$$ 17.4031 0.726392
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 24.2094i − 1.00785i −0.863748 0.503925i $$-0.831889\pi$$
0.863748 0.503925i $$-0.168111\pi$$
$$578$$ − 16.5078i − 0.686634i
$$579$$ −16.2094 −0.673639
$$580$$ 0 0
$$581$$ −50.3141 −2.08738
$$582$$ 2.59688i 0.107644i
$$583$$ 11.2984i 0.467933i
$$584$$ 12.0000 0.496564
$$585$$ 0 0
$$586$$ 21.4031 0.884155
$$587$$ 7.50781i 0.309881i 0.987924 + 0.154940i $$0.0495185\pi$$
−0.987924 + 0.154940i $$0.950481\pi$$
$$588$$ 15.1047i 0.622907i
$$589$$ −17.1938 −0.708456
$$590$$ 0 0
$$591$$ −5.40312 −0.222255
$$592$$ 1.70156i 0.0699338i
$$593$$ 17.9109i 0.735514i 0.929922 + 0.367757i $$0.119874\pi$$
−0.929922 + 0.367757i $$0.880126\pi$$
$$594$$ −4.70156 −0.192907
$$595$$ 0 0
$$596$$ 6.59688 0.270219
$$597$$ − 8.29844i − 0.339632i
$$598$$ 0 0
$$599$$ −8.20937 −0.335426 −0.167713 0.985836i $$-0.553638\pi$$
−0.167713 + 0.985836i $$0.553638\pi$$
$$600$$ 0 0
$$601$$ 10.6125 0.432893 0.216446 0.976295i $$-0.430553\pi$$
0.216446 + 0.976295i $$0.430553\pi$$
$$602$$ 53.6125i 2.18508i
$$603$$ − 6.40312i − 0.260755i
$$604$$ 14.1047 0.573912
$$605$$ 0 0
$$606$$ 6.70156 0.272232
$$607$$ 35.1047i 1.42486i 0.701746 + 0.712428i $$0.252404\pi$$
−0.701746 + 0.712428i $$0.747596\pi$$
$$608$$ − 1.70156i − 0.0690075i
$$609$$ 30.1047 1.21990
$$610$$ 0 0
$$611$$ −7.00000 −0.283190
$$612$$ 0.701562i 0.0283590i
$$613$$ − 12.8062i − 0.517240i −0.965979 0.258620i $$-0.916732\pi$$
0.965979 0.258620i $$-0.0832677\pi$$
$$614$$ 5.70156 0.230096
$$615$$ 0 0
$$616$$ 22.1047 0.890623
$$617$$ 27.1047i 1.09119i 0.838048 + 0.545597i $$0.183697\pi$$
−0.838048 + 0.545597i $$0.816303\pi$$
$$618$$ 1.40312i 0.0564419i
$$619$$ 13.1938 0.530302 0.265151 0.964207i $$-0.414578\pi$$
0.265151 + 0.964207i $$0.414578\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 30.0000i − 1.20289i
$$623$$ 53.6125i 2.14794i
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ −21.2094 −0.847697
$$627$$ 8.00000i 0.319489i
$$628$$ − 22.7016i − 0.905891i
$$629$$ −1.19375 −0.0475980
$$630$$ 0 0
$$631$$ 33.6125 1.33809 0.669046 0.743221i $$-0.266703\pi$$
0.669046 + 0.743221i $$0.266703\pi$$
$$632$$ 5.70156i 0.226796i
$$633$$ 6.80625i 0.270524i
$$634$$ −1.19375 −0.0474099
$$635$$ 0 0
$$636$$ 2.40312 0.0952901
$$637$$ 15.1047i 0.598469i
$$638$$ − 30.1047i − 1.19186i
$$639$$ 1.70156 0.0673128
$$640$$ 0 0
$$641$$ 5.50781 0.217545 0.108773 0.994067i $$-0.465308\pi$$
0.108773 + 0.994067i $$0.465308\pi$$
$$642$$ − 19.1047i − 0.754002i
$$643$$ − 10.2984i − 0.406131i −0.979165 0.203065i $$-0.934910\pi$$
0.979165 0.203065i $$-0.0650904\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 1.19375 0.0469675
$$647$$ 24.0000i 0.943537i 0.881722 + 0.471769i $$0.156384\pi$$
−0.881722 + 0.471769i $$0.843616\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −12.7016 −0.498580
$$650$$ 0 0
$$651$$ −47.5078 −1.86198
$$652$$ 18.8062i 0.736510i
$$653$$ − 21.2984i − 0.833472i −0.909027 0.416736i $$-0.863174\pi$$
0.909027 0.416736i $$-0.136826\pi$$
$$654$$ −4.29844 −0.168082
$$655$$ 0 0
$$656$$ −3.70156 −0.144522
$$657$$ 12.0000i 0.468165i
$$658$$ 32.9109i 1.28300i
$$659$$ 19.3141 0.752369 0.376184 0.926545i $$-0.377236\pi$$
0.376184 + 0.926545i $$0.377236\pi$$
$$660$$ 0 0
$$661$$ −40.1203 −1.56050 −0.780250 0.625468i $$-0.784908\pi$$
−0.780250 + 0.625468i $$0.784908\pi$$
$$662$$ 12.0000i 0.466393i
$$663$$ 0.701562i 0.0272464i
$$664$$ 10.7016 0.415301
$$665$$ 0 0
$$666$$ −1.70156 −0.0659342
$$667$$ 0 0
$$668$$ − 1.10469i − 0.0427416i
$$669$$ −11.4031 −0.440870
$$670$$ 0 0
$$671$$ 66.3141 2.56003
$$672$$ − 4.70156i − 0.181367i
$$673$$ 36.0156i 1.38830i 0.719830 + 0.694150i $$0.244220\pi$$
−0.719830 + 0.694150i $$0.755780\pi$$
$$674$$ −8.10469 −0.312181
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ − 31.6125i − 1.21497i −0.794332 0.607483i $$-0.792179\pi$$
0.794332 0.607483i $$-0.207821\pi$$
$$678$$ − 14.0000i − 0.537667i
$$679$$ 12.2094 0.468553
$$680$$ 0 0
$$681$$ −16.1047 −0.617133
$$682$$ 47.5078i 1.81917i
$$683$$ 44.7016i 1.71046i 0.518251 + 0.855229i $$0.326583\pi$$
−0.518251 + 0.855229i $$0.673417\pi$$
$$684$$ 1.70156 0.0650609
$$685$$ 0 0
$$686$$ 38.1047 1.45484
$$687$$ − 15.7016i − 0.599052i
$$688$$ − 11.4031i − 0.434740i
$$689$$ 2.40312 0.0915517
$$690$$ 0 0
$$691$$ −22.1938 −0.844290 −0.422145 0.906528i $$-0.638723\pi$$
−0.422145 + 0.906528i $$0.638723\pi$$
$$692$$ 0.193752i 0.00736533i
$$693$$ 22.1047i 0.839688i
$$694$$ −10.5078 −0.398871
$$695$$ 0 0
$$696$$ −6.40312 −0.242710
$$697$$ − 2.59688i − 0.0983637i
$$698$$ − 11.4031i − 0.431615i
$$699$$ 20.2094 0.764389
$$700$$ 0 0
$$701$$ 30.9109 1.16749 0.583745 0.811937i $$-0.301587\pi$$
0.583745 + 0.811937i $$0.301587\pi$$
$$702$$ 1.00000i 0.0377426i
$$703$$ 2.89531i 0.109199i
$$704$$ −4.70156 −0.177197
$$705$$ 0 0
$$706$$ 14.5078 0.546009
$$707$$ − 31.5078i − 1.18497i
$$708$$ 2.70156i 0.101531i
$$709$$ −15.6125 −0.586340 −0.293170 0.956060i $$-0.594710\pi$$
−0.293170 + 0.956060i $$0.594710\pi$$
$$710$$ 0 0
$$711$$ −5.70156 −0.213825
$$712$$ − 11.4031i − 0.427350i
$$713$$ 0 0
$$714$$ 3.29844 0.123441
$$715$$ 0 0
$$716$$ −24.2094 −0.904747
$$717$$ − 6.10469i − 0.227984i
$$718$$ − 34.6125i − 1.29173i
$$719$$ 11.0156 0.410813 0.205407 0.978677i $$-0.434148\pi$$
0.205407 + 0.978677i $$0.434148\pi$$
$$720$$ 0 0
$$721$$ 6.59688 0.245680
$$722$$ 16.1047i 0.599354i
$$723$$ − 2.59688i − 0.0965788i
$$724$$ −11.2984 −0.419903
$$725$$ 0 0
$$726$$ 11.1047 0.412134
$$727$$ − 7.79063i − 0.288938i −0.989509 0.144469i $$-0.953853\pi$$
0.989509 0.144469i $$-0.0461475\pi$$
$$728$$ − 4.70156i − 0.174251i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ − 14.1047i − 0.521324i
$$733$$ 47.9109i 1.76963i 0.465942 + 0.884815i $$0.345716\pi$$
−0.465942 + 0.884815i $$0.654284\pi$$
$$734$$ 2.29844 0.0848369
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 30.1047i 1.10892i
$$738$$ − 3.70156i − 0.136256i
$$739$$ −4.61250 −0.169673 −0.0848367 0.996395i $$-0.527037\pi$$
−0.0848367 + 0.996395i $$0.527037\pi$$
$$740$$ 0 0
$$741$$ 1.70156 0.0625084
$$742$$ − 11.2984i − 0.414779i
$$743$$ − 30.6125i − 1.12306i −0.827455 0.561532i $$-0.810212\pi$$
0.827455 0.561532i $$-0.189788\pi$$
$$744$$ 10.1047 0.370456
$$745$$ 0 0
$$746$$ 5.29844 0.193990
$$747$$ 10.7016i 0.391550i
$$748$$ − 3.29844i − 0.120603i
$$749$$ −89.8219 −3.28202
$$750$$ 0 0
$$751$$ 8.50781 0.310454 0.155227 0.987879i $$-0.450389\pi$$
0.155227 + 0.987879i $$0.450389\pi$$
$$752$$ − 7.00000i − 0.255264i
$$753$$ 0.298438i 0.0108757i
$$754$$ −6.40312 −0.233188
$$755$$ 0 0
$$756$$ 4.70156 0.170994
$$757$$ 17.8953i 0.650416i 0.945642 + 0.325208i $$0.105434\pi$$
−0.945642 + 0.325208i $$0.894566\pi$$
$$758$$ − 13.8953i − 0.504701i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −26.7172 −0.968497 −0.484249 0.874930i $$-0.660907\pi$$
−0.484249 + 0.874930i $$0.660907\pi$$
$$762$$ 6.29844i 0.228168i
$$763$$ 20.2094i 0.731628i
$$764$$ 12.8062 0.463314
$$765$$ 0 0
$$766$$ 32.5078 1.17455
$$767$$ 2.70156i 0.0975478i
$$768$$ 1.00000i 0.0360844i
$$769$$ 8.00000 0.288487 0.144244 0.989542i $$-0.453925\pi$$
0.144244 + 0.989542i $$0.453925\pi$$
$$770$$ 0 0
$$771$$ −15.2984 −0.550960
$$772$$ − 16.2094i − 0.583388i
$$773$$ 26.5969i 0.956623i 0.878190 + 0.478312i $$0.158751\pi$$
−0.878190 + 0.478312i $$0.841249\pi$$
$$774$$ 11.4031 0.409877
$$775$$ 0 0
$$776$$ −2.59688 −0.0932224
$$777$$ 8.00000i 0.286998i
$$778$$ − 27.1047i − 0.971750i
$$779$$ −6.29844 −0.225665
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 0 0
$$783$$ − 6.40312i − 0.228829i
$$784$$ −15.1047 −0.539453
$$785$$ 0 0
$$786$$ 22.5078 0.802827
$$787$$ − 33.8953i − 1.20824i −0.796894 0.604119i $$-0.793525\pi$$
0.796894 0.604119i $$-0.206475\pi$$
$$788$$ − 5.40312i − 0.192478i
$$789$$ −2.00000 −0.0712019
$$790$$ 0 0
$$791$$ −65.8219 −2.34036
$$792$$ − 4.70156i − 0.167063i
$$793$$ − 14.1047i − 0.500872i
$$794$$ 21.9109 0.777590
$$795$$ 0 0
$$796$$ 8.29844 0.294130
$$797$$ − 2.91093i − 0.103111i −0.998670 0.0515553i $$-0.983582\pi$$
0.998670 0.0515553i $$-0.0164178\pi$$
$$798$$ − 8.00000i − 0.283197i
$$799$$ 4.91093 0.173736
$$800$$ 0 0
$$801$$ 11.4031 0.402910
$$802$$ − 18.2094i − 0.642995i
$$803$$ − 56.4187i − 1.99097i
$$804$$ 6.40312 0.225821
$$805$$ 0 0
$$806$$ 10.1047 0.355922
$$807$$ − 31.2094i − 1.09862i
$$808$$ 6.70156i 0.235760i
$$809$$ −7.19375 −0.252919 −0.126459 0.991972i $$-0.540361\pi$$
−0.126459 + 0.991972i $$0.540361\pi$$
$$810$$ 0 0
$$811$$ 48.7016 1.71014 0.855072 0.518510i $$-0.173513\pi$$
0.855072 + 0.518510i $$0.173513\pi$$
$$812$$ 30.1047i 1.05647i
$$813$$ 19.5078i 0.684169i
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ −0.701562 −0.0245596
$$817$$ − 19.4031i − 0.678829i
$$818$$ 29.4031i 1.02806i
$$819$$ 4.70156 0.164286
$$820$$ 0 0
$$821$$ −20.5969 −0.718836 −0.359418 0.933177i $$-0.617025\pi$$
−0.359418 + 0.933177i $$0.617025\pi$$
$$822$$ − 13.7016i − 0.477897i
$$823$$ 17.1047i 0.596232i 0.954530 + 0.298116i $$0.0963582\pi$$
−0.954530 + 0.298116i $$0.903642\pi$$
$$824$$ −1.40312 −0.0488801
$$825$$ 0 0
$$826$$ 12.7016 0.441944
$$827$$ − 20.7016i − 0.719864i −0.932979 0.359932i $$-0.882800\pi$$
0.932979 0.359932i $$-0.117200\pi$$
$$828$$ 0 0
$$829$$ 5.50781 0.191294 0.0956471 0.995415i $$-0.469508\pi$$
0.0956471 + 0.995415i $$0.469508\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 1.00000i 0.0346688i
$$833$$ − 10.5969i − 0.367160i
$$834$$ −9.40312 −0.325604
$$835$$ 0 0
$$836$$ −8.00000 −0.276686
$$837$$ 10.1047i 0.349269i
$$838$$ 9.91093i 0.342368i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 12.0000 0.413793
$$842$$ 0.596876i 0.0205697i
$$843$$ − 21.9109i − 0.754653i
$$844$$ −6.80625 −0.234281
$$845$$ 0 0
$$846$$ 7.00000 0.240665
$$847$$ − 52.2094i − 1.79394i
$$848$$ 2.40312i 0.0825236i
$$849$$ −21.4031 −0.734553
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 1.70156i 0.0582946i
$$853$$ 0.507811i 0.0173871i 0.999962 + 0.00869355i $$0.00276728\pi$$
−0.999962 + 0.00869355i $$0.997233\pi$$
$$854$$ −66.3141 −2.26922
$$855$$ 0 0
$$856$$ 19.1047 0.652985
$$857$$ 7.61250i 0.260038i 0.991512 + 0.130019i $$0.0415038\pi$$
−0.991512 + 0.130019i $$0.958496\pi$$
$$858$$ − 4.70156i − 0.160509i
$$859$$ 6.20937 0.211861 0.105931 0.994374i $$-0.466218\pi$$
0.105931 + 0.994374i $$0.466218\pi$$
$$860$$ 0 0
$$861$$ −17.4031 −0.593097
$$862$$ − 35.3141i − 1.20280i
$$863$$ 54.2250i 1.84584i 0.384991 + 0.922920i $$0.374204\pi$$
−0.384991 + 0.922920i $$0.625796\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −11.1047 −0.377353
$$867$$ 16.5078i 0.560635i
$$868$$ − 47.5078i − 1.61252i
$$869$$ 26.8062 0.909340
$$870$$ 0 0
$$871$$ 6.40312 0.216962
$$872$$ − 4.29844i − 0.145563i
$$873$$ − 2.59688i − 0.0878909i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −12.0000 −0.405442
$$877$$ 42.7172i 1.44246i 0.692697 + 0.721228i $$0.256422\pi$$
−0.692697 + 0.721228i $$0.743578\pi$$
$$878$$ − 10.2984i − 0.347555i
$$879$$ −21.4031 −0.721909
$$880$$ 0 0
$$881$$ −39.7172 −1.33811 −0.669053 0.743215i $$-0.733300\pi$$
−0.669053 + 0.743215i $$0.733300\pi$$
$$882$$ − 15.1047i − 0.508601i
$$883$$ 45.6125i 1.53498i 0.641059 + 0.767491i $$0.278495\pi$$
−0.641059 + 0.767491i $$0.721505\pi$$
$$884$$ −0.701562 −0.0235961
$$885$$ 0 0
$$886$$ 32.5078 1.09212
$$887$$ 20.0000i 0.671534i 0.941945 + 0.335767i $$0.108996\pi$$
−0.941945 + 0.335767i $$0.891004\pi$$
$$888$$ − 1.70156i − 0.0571007i
$$889$$ 29.6125 0.993171
$$890$$ 0 0
$$891$$ 4.70156 0.157508
$$892$$ − 11.4031i − 0.381805i
$$893$$ − 11.9109i − 0.398584i
$$894$$ −6.59688 −0.220633
$$895$$ 0 0
$$896$$ 4.70156 0.157068
$$897$$ 0 0
$$898$$ 2.29844i 0.0766999i
$$899$$ −64.7016 −2.15792
$$900$$ 0 0
$$901$$ −1.68594 −0.0561668
$$902$$ 17.4031i 0.579461i
$$903$$ − 53.6125i − 1.78411i
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ −14.1047 −0.468597
$$907$$ 25.0156i 0.830630i 0.909678 + 0.415315i $$0.136329\pi$$
−0.909678 + 0.415315i $$0.863671\pi$$
$$908$$ − 16.1047i − 0.534453i
$$909$$ −6.70156 −0.222277
$$910$$ 0 0
$$911$$ 42.2094 1.39846 0.699229 0.714897i $$-0.253527\pi$$
0.699229 + 0.714897i $$0.253527\pi$$
$$912$$ 1.70156i 0.0563444i
$$913$$ − 50.3141i − 1.66515i
$$914$$ 2.59688 0.0858970
$$915$$ 0 0
$$916$$ 15.7016 0.518794
$$917$$ − 105.822i − 3.49455i
$$918$$ − 0.701562i − 0.0231550i
$$919$$ 4.89531 0.161481 0.0807407 0.996735i $$-0.474271\pi$$
0.0807407 + 0.996735i $$0.474271\pi$$
$$920$$ 0 0
$$921$$ −5.70156 −0.187873
$$922$$ − 36.2094i − 1.19249i
$$923$$ 1.70156i 0.0560076i
$$924$$ −22.1047 −0.727191
$$925$$ 0 0
$$926$$ −21.2984 −0.699910
$$927$$ − 1.40312i − 0.0460846i
$$928$$ − 6.40312i − 0.210193i
$$929$$ −52.2984 −1.71586 −0.857928 0.513770i $$-0.828249\pi$$
−0.857928 + 0.513770i $$0.828249\pi$$
$$930$$ 0 0
$$931$$ −25.7016 −0.842335
$$932$$ 20.2094i 0.661980i
$$933$$ 30.0000i 0.982156i
$$934$$ −30.2984 −0.991395
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ − 14.9109i − 0.487119i −0.969886 0.243560i $$-0.921685\pi$$
0.969886 0.243560i $$-0.0783151\pi$$
$$938$$ − 30.1047i − 0.982953i
$$939$$ 21.2094 0.692142
$$940$$ 0 0
$$941$$ 28.4187 0.926425 0.463212 0.886247i $$-0.346697\pi$$
0.463212 + 0.886247i $$0.346697\pi$$
$$942$$ 22.7016i 0.739657i
$$943$$ 0 0
$$944$$ −2.70156 −0.0879284
$$945$$ 0 0
$$946$$ −53.6125 −1.74309
$$947$$ − 47.5078i − 1.54380i −0.635746 0.771898i $$-0.719307\pi$$
0.635746 0.771898i $$-0.280693\pi$$
$$948$$ − 5.70156i − 0.185178i
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 1.19375 0.0387100
$$952$$ 3.29844i 0.106903i
$$953$$ − 31.2984i − 1.01386i −0.861988 0.506928i $$-0.830781\pi$$
0.861988 0.506928i $$-0.169219\pi$$
$$954$$ −2.40312 −0.0778040
$$955$$ 0 0
$$956$$ 6.10469 0.197440
$$957$$ 30.1047i 0.973146i
$$958$$ − 40.6125i − 1.31213i
$$959$$ −64.4187 −2.08019
$$960$$ 0 0
$$961$$ 71.1047 2.29370
$$962$$ − 1.70156i − 0.0548606i
$$963$$ 19.1047i 0.615640i
$$964$$ 2.59688 0.0836397
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 29.8953i − 0.961368i −0.876894 0.480684i $$-0.840388\pi$$
0.876894 0.480684i $$-0.159612\pi$$
$$968$$ 11.1047i 0.356918i
$$969$$ −1.19375 −0.0383488
$$970$$ 0 0
$$971$$ −36.8953 −1.18403 −0.592013 0.805928i $$-0.701667\pi$$
−0.592013 + 0.805928i $$0.701667\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 44.2094i 1.41729i
$$974$$ 7.89531 0.252982
$$975$$ 0 0
$$976$$ 14.1047 0.451480
$$977$$ 23.4031i 0.748732i 0.927281 + 0.374366i $$0.122140\pi$$
−0.927281 + 0.374366i $$0.877860\pi$$
$$978$$ − 18.8062i − 0.601358i
$$979$$ −53.6125 −1.71346
$$980$$ 0 0
$$981$$ 4.29844 0.137239
$$982$$ − 33.4031i − 1.06594i
$$983$$ − 7.50781i − 0.239462i −0.992806 0.119731i $$-0.961797\pi$$
0.992806 0.119731i $$-0.0382032\pi$$
$$984$$ 3.70156 0.118001
$$985$$ 0 0
$$986$$ 4.49219 0.143060
$$987$$ − 32.9109i − 1.04757i
$$988$$ 1.70156i 0.0541339i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −9.10469 −0.289220 −0.144610 0.989489i $$-0.546193\pi$$
−0.144610 + 0.989489i $$0.546193\pi$$
$$992$$ 10.1047i 0.320824i
$$993$$ − 12.0000i − 0.380808i
$$994$$ 8.00000 0.253745
$$995$$ 0 0
$$996$$ −10.7016 −0.339092
$$997$$ 10.4922i 0.332291i 0.986101 + 0.166145i $$0.0531321\pi$$
−0.986101 + 0.166145i $$0.946868\pi$$
$$998$$ 17.2094i 0.544753i
$$999$$ 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.e.p.1249.1 4
3.2 odd 2 5850.2.e.bi.5149.3 4
5.2 odd 4 1950.2.a.bg.1.2 yes 2
5.3 odd 4 1950.2.a.bc.1.1 2
5.4 even 2 inner 1950.2.e.p.1249.4 4
15.2 even 4 5850.2.a.cg.1.2 2
15.8 even 4 5850.2.a.cj.1.1 2
15.14 odd 2 5850.2.e.bi.5149.2 4

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