# Properties

 Label 5445.2.a.bs.1.4 Level $5445$ Weight $2$ Character 5445.1 Self dual yes Analytic conductor $43.479$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.48704.2 Defining polynomial: $$x^{4} - 2 x^{3} - 6 x^{2} + 4 x + 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 495) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.69696$$ of defining polynomial Character $$\chi$$ $$=$$ 5445.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.69696 q^{2} +5.27358 q^{4} -1.00000 q^{5} +4.13176 q^{7} +8.82872 q^{8} +O(q^{10})$$ $$q+2.69696 q^{2} +5.27358 q^{4} -1.00000 q^{5} +4.13176 q^{7} +8.82872 q^{8} -2.69696 q^{10} -2.73785 q^{13} +11.1432 q^{14} +13.2635 q^{16} +2.41541 q^{17} +1.15325 q^{19} -5.27358 q^{20} +8.54717 q^{23} +1.00000 q^{25} -7.38386 q^{26} +21.7892 q^{28} -1.67756 q^{29} -10.2635 q^{31} +18.1137 q^{32} +6.51425 q^{34} -4.13176 q^{35} +5.71636 q^{37} +3.11028 q^{38} -8.82872 q^{40} -9.11028 q^{41} -8.13176 q^{43} +23.0514 q^{46} -1.47569 q^{47} +10.0715 q^{49} +2.69696 q^{50} -14.4383 q^{52} +2.54717 q^{53} +36.4782 q^{56} -4.52431 q^{58} +8.24066 q^{59} -3.47569 q^{61} -27.6803 q^{62} +22.3249 q^{64} +2.73785 q^{65} +15.3350 q^{67} +12.7378 q^{68} -11.1432 q^{70} -2.54717 q^{71} -2.73785 q^{73} +15.4168 q^{74} +6.08178 q^{76} -5.15325 q^{79} -13.2635 q^{80} -24.5700 q^{82} +9.28502 q^{83} -2.41541 q^{85} -21.9310 q^{86} -4.78783 q^{89} -11.3121 q^{91} +45.0742 q^{92} -3.97989 q^{94} -1.15325 q^{95} +4.24066 q^{97} +27.1624 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 8 q^{4} - 4 q^{5} - 4 q^{7} + 6 q^{8} + O(q^{10})$$ $$4 q + 2 q^{2} + 8 q^{4} - 4 q^{5} - 4 q^{7} + 6 q^{8} - 2 q^{10} - 8 q^{13} + 8 q^{14} + 12 q^{16} + 4 q^{17} - 4 q^{19} - 8 q^{20} + 8 q^{23} + 4 q^{25} + 16 q^{26} + 8 q^{28} - 4 q^{29} + 14 q^{32} + 4 q^{34} + 4 q^{35} + 8 q^{37} - 20 q^{38} - 6 q^{40} - 4 q^{41} - 12 q^{43} + 16 q^{46} + 20 q^{49} + 2 q^{50} - 20 q^{52} - 16 q^{53} + 48 q^{56} - 24 q^{58} + 24 q^{59} - 8 q^{61} - 20 q^{62} + 8 q^{65} + 48 q^{68} - 8 q^{70} + 16 q^{71} - 8 q^{73} + 12 q^{74} + 36 q^{76} - 12 q^{79} - 12 q^{80} - 40 q^{82} + 8 q^{83} - 4 q^{85} - 16 q^{86} + 16 q^{89} - 16 q^{91} + 72 q^{92} + 40 q^{94} + 4 q^{95} + 8 q^{97} + 62 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.69696 1.90704 0.953519 0.301334i $$-0.0974317\pi$$
0.953519 + 0.301334i $$0.0974317\pi$$
$$3$$ 0 0
$$4$$ 5.27358 2.63679
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.13176 1.56166 0.780830 0.624744i $$-0.214796\pi$$
0.780830 + 0.624744i $$0.214796\pi$$
$$8$$ 8.82872 3.12142
$$9$$ 0 0
$$10$$ −2.69696 −0.852853
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −2.73785 −0.759342 −0.379671 0.925122i $$-0.623963\pi$$
−0.379671 + 0.925122i $$0.623963\pi$$
$$14$$ 11.1432 2.97814
$$15$$ 0 0
$$16$$ 13.2635 3.31588
$$17$$ 2.41541 0.585822 0.292911 0.956140i $$-0.405376\pi$$
0.292911 + 0.956140i $$0.405376\pi$$
$$18$$ 0 0
$$19$$ 1.15325 0.264574 0.132287 0.991211i $$-0.457768\pi$$
0.132287 + 0.991211i $$0.457768\pi$$
$$20$$ −5.27358 −1.17921
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.54717 1.78221 0.891104 0.453799i $$-0.149932\pi$$
0.891104 + 0.453799i $$0.149932\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −7.38386 −1.44809
$$27$$ 0 0
$$28$$ 21.7892 4.11777
$$29$$ −1.67756 −0.311515 −0.155757 0.987795i $$-0.549782\pi$$
−0.155757 + 0.987795i $$0.549782\pi$$
$$30$$ 0 0
$$31$$ −10.2635 −1.84338 −0.921692 0.387922i $$-0.873193\pi$$
−0.921692 + 0.387922i $$0.873193\pi$$
$$32$$ 18.1137 3.20209
$$33$$ 0 0
$$34$$ 6.51425 1.11718
$$35$$ −4.13176 −0.698396
$$36$$ 0 0
$$37$$ 5.71636 0.939764 0.469882 0.882729i $$-0.344297\pi$$
0.469882 + 0.882729i $$0.344297\pi$$
$$38$$ 3.11028 0.504553
$$39$$ 0 0
$$40$$ −8.82872 −1.39594
$$41$$ −9.11028 −1.42279 −0.711393 0.702794i $$-0.751935\pi$$
−0.711393 + 0.702794i $$0.751935\pi$$
$$42$$ 0 0
$$43$$ −8.13176 −1.24008 −0.620041 0.784569i $$-0.712884\pi$$
−0.620041 + 0.784569i $$0.712884\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 23.0514 3.39874
$$47$$ −1.47569 −0.215252 −0.107626 0.994191i $$-0.534325\pi$$
−0.107626 + 0.994191i $$0.534325\pi$$
$$48$$ 0 0
$$49$$ 10.0715 1.43878
$$50$$ 2.69696 0.381408
$$51$$ 0 0
$$52$$ −14.4383 −2.00223
$$53$$ 2.54717 0.349881 0.174940 0.984579i $$-0.444027\pi$$
0.174940 + 0.984579i $$0.444027\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 36.4782 4.87460
$$57$$ 0 0
$$58$$ −4.52431 −0.594070
$$59$$ 8.24066 1.07284 0.536422 0.843950i $$-0.319776\pi$$
0.536422 + 0.843950i $$0.319776\pi$$
$$60$$ 0 0
$$61$$ −3.47569 −0.445017 −0.222509 0.974931i $$-0.571425\pi$$
−0.222509 + 0.974931i $$0.571425\pi$$
$$62$$ −27.6803 −3.51540
$$63$$ 0 0
$$64$$ 22.3249 2.79062
$$65$$ 2.73785 0.339588
$$66$$ 0 0
$$67$$ 15.3350 1.87347 0.936734 0.350041i $$-0.113832\pi$$
0.936734 + 0.350041i $$0.113832\pi$$
$$68$$ 12.7378 1.54469
$$69$$ 0 0
$$70$$ −11.1432 −1.33187
$$71$$ −2.54717 −0.302293 −0.151147 0.988511i $$-0.548297\pi$$
−0.151147 + 0.988511i $$0.548297\pi$$
$$72$$ 0 0
$$73$$ −2.73785 −0.320441 −0.160220 0.987081i $$-0.551220\pi$$
−0.160220 + 0.987081i $$0.551220\pi$$
$$74$$ 15.4168 1.79216
$$75$$ 0 0
$$76$$ 6.08178 0.697628
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −5.15325 −0.579786 −0.289893 0.957059i $$-0.593620\pi$$
−0.289893 + 0.957059i $$0.593620\pi$$
$$80$$ −13.2635 −1.48291
$$81$$ 0 0
$$82$$ −24.5700 −2.71331
$$83$$ 9.28502 1.01916 0.509582 0.860422i $$-0.329800\pi$$
0.509582 + 0.860422i $$0.329800\pi$$
$$84$$ 0 0
$$85$$ −2.41541 −0.261988
$$86$$ −21.9310 −2.36488
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −4.78783 −0.507509 −0.253755 0.967269i $$-0.581666\pi$$
−0.253755 + 0.967269i $$0.581666\pi$$
$$90$$ 0 0
$$91$$ −11.3121 −1.18583
$$92$$ 45.0742 4.69931
$$93$$ 0 0
$$94$$ −3.97989 −0.410494
$$95$$ −1.15325 −0.118321
$$96$$ 0 0
$$97$$ 4.24066 0.430574 0.215287 0.976551i $$-0.430931\pi$$
0.215287 + 0.976551i $$0.430931\pi$$
$$98$$ 27.1624 2.74381
$$99$$ 0 0
$$100$$ 5.27358 0.527358
$$101$$ −13.9411 −1.38719 −0.693595 0.720365i $$-0.743974\pi$$
−0.693595 + 0.720365i $$0.743974\pi$$
$$102$$ 0 0
$$103$$ −18.5042 −1.82327 −0.911636 0.410998i $$-0.865180\pi$$
−0.911636 + 0.410998i $$0.865180\pi$$
$$104$$ −24.1717 −2.37023
$$105$$ 0 0
$$106$$ 6.86961 0.667236
$$107$$ 13.7663 1.33084 0.665421 0.746468i $$-0.268252\pi$$
0.665421 + 0.746468i $$0.268252\pi$$
$$108$$ 0 0
$$109$$ −12.7878 −1.22485 −0.612426 0.790528i $$-0.709806\pi$$
−0.612426 + 0.790528i $$0.709806\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 54.8018 5.17828
$$113$$ 14.8107 1.39327 0.696637 0.717424i $$-0.254679\pi$$
0.696637 + 0.717424i $$0.254679\pi$$
$$114$$ 0 0
$$115$$ −8.54717 −0.797028
$$116$$ −8.84675 −0.821400
$$117$$ 0 0
$$118$$ 22.2247 2.04595
$$119$$ 9.97989 0.914855
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −9.37380 −0.848664
$$123$$ 0 0
$$124$$ −54.1256 −4.86062
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −12.9626 −1.15024 −0.575121 0.818068i $$-0.695045\pi$$
−0.575121 + 0.818068i $$0.695045\pi$$
$$128$$ 23.9820 2.11973
$$129$$ 0 0
$$130$$ 7.38386 0.647607
$$131$$ −1.47569 −0.128932 −0.0644660 0.997920i $$-0.520534\pi$$
−0.0644660 + 0.997920i $$0.520534\pi$$
$$132$$ 0 0
$$133$$ 4.76497 0.413175
$$134$$ 41.3579 3.57278
$$135$$ 0 0
$$136$$ 21.3249 1.82860
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 0.889725 0.0754655 0.0377327 0.999288i $$-0.487986\pi$$
0.0377327 + 0.999288i $$0.487986\pi$$
$$140$$ −21.7892 −1.84152
$$141$$ 0 0
$$142$$ −6.86961 −0.576485
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1.67756 0.139314
$$146$$ −7.38386 −0.611093
$$147$$ 0 0
$$148$$ 30.1457 2.47796
$$149$$ 12.7289 1.04279 0.521397 0.853314i $$-0.325411\pi$$
0.521397 + 0.853314i $$0.325411\pi$$
$$150$$ 0 0
$$151$$ 1.15325 0.0938504 0.0469252 0.998898i $$-0.485058\pi$$
0.0469252 + 0.998898i $$0.485058\pi$$
$$152$$ 10.1818 0.825849
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 10.2635 0.824386
$$156$$ 0 0
$$157$$ −3.95702 −0.315805 −0.157902 0.987455i $$-0.550473\pi$$
−0.157902 + 0.987455i $$0.550473\pi$$
$$158$$ −13.8981 −1.10567
$$159$$ 0 0
$$160$$ −18.1137 −1.43202
$$161$$ 35.3149 2.78320
$$162$$ 0 0
$$163$$ 3.07148 0.240577 0.120288 0.992739i $$-0.461618\pi$$
0.120288 + 0.992739i $$0.461618\pi$$
$$164$$ −48.0438 −3.75159
$$165$$ 0 0
$$166$$ 25.0413 1.94358
$$167$$ 2.71498 0.210092 0.105046 0.994467i $$-0.466501\pi$$
0.105046 + 0.994467i $$0.466501\pi$$
$$168$$ 0 0
$$169$$ −5.50419 −0.423399
$$170$$ −6.51425 −0.499620
$$171$$ 0 0
$$172$$ −42.8835 −3.26984
$$173$$ 14.4154 1.09598 0.547992 0.836484i $$-0.315393\pi$$
0.547992 + 0.836484i $$0.315393\pi$$
$$174$$ 0 0
$$175$$ 4.13176 0.312332
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −12.9126 −0.967839
$$179$$ −10.7878 −0.806321 −0.403160 0.915129i $$-0.632088\pi$$
−0.403160 + 0.915129i $$0.632088\pi$$
$$180$$ 0 0
$$181$$ −6.50419 −0.483453 −0.241726 0.970344i $$-0.577714\pi$$
−0.241726 + 0.970344i $$0.577714\pi$$
$$182$$ −30.5084 −2.26143
$$183$$ 0 0
$$184$$ 75.4606 5.56303
$$185$$ −5.71636 −0.420275
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −7.78220 −0.567575
$$189$$ 0 0
$$190$$ −3.11028 −0.225643
$$191$$ −15.8822 −1.14919 −0.574597 0.818437i $$-0.694841\pi$$
−0.574597 + 0.818437i $$0.694841\pi$$
$$192$$ 0 0
$$193$$ 3.83219 0.275847 0.137923 0.990443i $$-0.455957\pi$$
0.137923 + 0.990443i $$0.455957\pi$$
$$194$$ 11.4369 0.821121
$$195$$ 0 0
$$196$$ 53.1128 3.79377
$$197$$ −2.67893 −0.190866 −0.0954331 0.995436i $$-0.530424\pi$$
−0.0954331 + 0.995436i $$0.530424\pi$$
$$198$$ 0 0
$$199$$ 6.78783 0.481177 0.240588 0.970627i $$-0.422660\pi$$
0.240588 + 0.970627i $$0.422660\pi$$
$$200$$ 8.82872 0.624285
$$201$$ 0 0
$$202$$ −37.5985 −2.64542
$$203$$ −6.93128 −0.486480
$$204$$ 0 0
$$205$$ 9.11028 0.636289
$$206$$ −49.9050 −3.47705
$$207$$ 0 0
$$208$$ −36.3135 −2.51789
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −17.6803 −1.21716 −0.608581 0.793491i $$-0.708261\pi$$
−0.608581 + 0.793491i $$0.708261\pi$$
$$212$$ 13.4327 0.922563
$$213$$ 0 0
$$214$$ 37.1273 2.53797
$$215$$ 8.13176 0.554582
$$216$$ 0 0
$$217$$ −42.4065 −2.87874
$$218$$ −34.4883 −2.33584
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −6.61301 −0.444839
$$222$$ 0 0
$$223$$ −8.83081 −0.591355 −0.295677 0.955288i $$-0.595545\pi$$
−0.295677 + 0.955288i $$0.595545\pi$$
$$224$$ 74.8417 5.00057
$$225$$ 0 0
$$226$$ 39.9438 2.65702
$$227$$ 10.4972 0.696723 0.348361 0.937360i $$-0.386738\pi$$
0.348361 + 0.937360i $$0.386738\pi$$
$$228$$ 0 0
$$229$$ 4.95139 0.327197 0.163598 0.986527i $$-0.447690\pi$$
0.163598 + 0.986527i $$0.447690\pi$$
$$230$$ −23.0514 −1.51996
$$231$$ 0 0
$$232$$ −14.8107 −0.972370
$$233$$ −3.89110 −0.254914 −0.127457 0.991844i $$-0.540682\pi$$
−0.127457 + 0.991844i $$0.540682\pi$$
$$234$$ 0 0
$$235$$ 1.47569 0.0962637
$$236$$ 43.4578 2.82886
$$237$$ 0 0
$$238$$ 26.9153 1.74466
$$239$$ −15.6186 −1.01029 −0.505143 0.863036i $$-0.668560\pi$$
−0.505143 + 0.863036i $$0.668560\pi$$
$$240$$ 0 0
$$241$$ 10.2635 0.661132 0.330566 0.943783i $$-0.392760\pi$$
0.330566 + 0.943783i $$0.392760\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −18.3294 −1.17342
$$245$$ −10.0715 −0.643443
$$246$$ 0 0
$$247$$ −3.15743 −0.200902
$$248$$ −90.6138 −5.75398
$$249$$ 0 0
$$250$$ −2.69696 −0.170571
$$251$$ 19.6415 1.23976 0.619881 0.784696i $$-0.287181\pi$$
0.619881 + 0.784696i $$0.287181\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −34.9595 −2.19356
$$255$$ 0 0
$$256$$ 20.0285 1.25178
$$257$$ 1.16919 0.0729320 0.0364660 0.999335i $$-0.488390\pi$$
0.0364660 + 0.999335i $$0.488390\pi$$
$$258$$ 0 0
$$259$$ 23.6186 1.46759
$$260$$ 14.4383 0.895424
$$261$$ 0 0
$$262$$ −3.97989 −0.245878
$$263$$ 8.93553 0.550989 0.275494 0.961303i $$-0.411158\pi$$
0.275494 + 0.961303i $$0.411158\pi$$
$$264$$ 0 0
$$265$$ −2.54717 −0.156471
$$266$$ 12.8509 0.787941
$$267$$ 0 0
$$268$$ 80.8704 4.93995
$$269$$ 14.9514 0.911602 0.455801 0.890082i $$-0.349353\pi$$
0.455801 + 0.890082i $$0.349353\pi$$
$$270$$ 0 0
$$271$$ 22.7289 1.38068 0.690342 0.723483i $$-0.257460\pi$$
0.690342 + 0.723483i $$0.257460\pi$$
$$272$$ 32.0368 1.94252
$$273$$ 0 0
$$274$$ 16.1818 0.977575
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.1387 1.33019 0.665093 0.746761i $$-0.268392\pi$$
0.665093 + 0.746761i $$0.268392\pi$$
$$278$$ 2.39955 0.143915
$$279$$ 0 0
$$280$$ −36.4782 −2.17999
$$281$$ −19.9841 −1.19215 −0.596075 0.802929i $$-0.703274\pi$$
−0.596075 + 0.802929i $$0.703274\pi$$
$$282$$ 0 0
$$283$$ −24.2775 −1.44315 −0.721573 0.692339i $$-0.756580\pi$$
−0.721573 + 0.692339i $$0.756580\pi$$
$$284$$ −13.4327 −0.797085
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −37.6415 −2.22191
$$288$$ 0 0
$$289$$ −11.1658 −0.656813
$$290$$ 4.52431 0.265676
$$291$$ 0 0
$$292$$ −14.4383 −0.844936
$$293$$ −16.4182 −0.959159 −0.479579 0.877498i $$-0.659211\pi$$
−0.479579 + 0.877498i $$0.659211\pi$$
$$294$$ 0 0
$$295$$ −8.24066 −0.479790
$$296$$ 50.4681 2.93340
$$297$$ 0 0
$$298$$ 34.3294 1.98865
$$299$$ −23.4008 −1.35331
$$300$$ 0 0
$$301$$ −33.5985 −1.93659
$$302$$ 3.11028 0.178976
$$303$$ 0 0
$$304$$ 15.2962 0.877297
$$305$$ 3.47569 0.199018
$$306$$ 0 0
$$307$$ −5.44390 −0.310700 −0.155350 0.987859i $$-0.549651\pi$$
−0.155350 + 0.987859i $$0.549651\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 27.6803 1.57214
$$311$$ −28.2864 −1.60397 −0.801987 0.597341i $$-0.796224\pi$$
−0.801987 + 0.597341i $$0.796224\pi$$
$$312$$ 0 0
$$313$$ 22.8107 1.28934 0.644668 0.764462i $$-0.276995\pi$$
0.644668 + 0.764462i $$0.276995\pi$$
$$314$$ −10.6719 −0.602252
$$315$$ 0 0
$$316$$ −27.1761 −1.52878
$$317$$ −23.8593 −1.34007 −0.670036 0.742328i $$-0.733721\pi$$
−0.670036 + 0.742328i $$0.733721\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −22.3249 −1.24800
$$321$$ 0 0
$$322$$ 95.2428 5.30767
$$323$$ 2.78557 0.154993
$$324$$ 0 0
$$325$$ −2.73785 −0.151868
$$326$$ 8.28364 0.458788
$$327$$ 0 0
$$328$$ −80.4321 −4.44112
$$329$$ −6.09722 −0.336151
$$330$$ 0 0
$$331$$ −18.9084 −1.03930 −0.519650 0.854379i $$-0.673938\pi$$
−0.519650 + 0.854379i $$0.673938\pi$$
$$332$$ 48.9653 2.68732
$$333$$ 0 0
$$334$$ 7.32220 0.400653
$$335$$ −15.3350 −0.837841
$$336$$ 0 0
$$337$$ −24.6630 −1.34348 −0.671740 0.740787i $$-0.734453\pi$$
−0.671740 + 0.740787i $$0.734453\pi$$
$$338$$ −14.8446 −0.807439
$$339$$ 0 0
$$340$$ −12.7378 −0.690807
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 12.6906 0.685229
$$344$$ −71.7931 −3.87082
$$345$$ 0 0
$$346$$ 38.8778 2.09008
$$347$$ −15.5056 −0.832383 −0.416191 0.909277i $$-0.636635\pi$$
−0.416191 + 0.909277i $$0.636635\pi$$
$$348$$ 0 0
$$349$$ −20.4841 −1.09649 −0.548244 0.836319i $$-0.684703\pi$$
−0.548244 + 0.836319i $$0.684703\pi$$
$$350$$ 11.1432 0.595629
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 21.9799 1.16987 0.584936 0.811080i $$-0.301120\pi$$
0.584936 + 0.811080i $$0.301120\pi$$
$$354$$ 0 0
$$355$$ 2.54717 0.135190
$$356$$ −25.2490 −1.33820
$$357$$ 0 0
$$358$$ −29.0943 −1.53768
$$359$$ −19.6962 −1.03953 −0.519764 0.854310i $$-0.673980\pi$$
−0.519764 + 0.854310i $$0.673980\pi$$
$$360$$ 0 0
$$361$$ −17.6700 −0.930000
$$362$$ −17.5415 −0.921963
$$363$$ 0 0
$$364$$ −59.6555 −3.12680
$$365$$ 2.73785 0.143305
$$366$$ 0 0
$$367$$ −28.2635 −1.47534 −0.737672 0.675159i $$-0.764075\pi$$
−0.737672 + 0.675159i $$0.764075\pi$$
$$368$$ 113.366 5.90959
$$369$$ 0 0
$$370$$ −15.4168 −0.801480
$$371$$ 10.5243 0.546395
$$372$$ 0 0
$$373$$ 9.78921 0.506866 0.253433 0.967353i $$-0.418440\pi$$
0.253433 + 0.967353i $$0.418440\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −13.0285 −0.671893
$$377$$ 4.59290 0.236546
$$378$$ 0 0
$$379$$ 5.66162 0.290818 0.145409 0.989372i $$-0.453550\pi$$
0.145409 + 0.989372i $$0.453550\pi$$
$$380$$ −6.08178 −0.311989
$$381$$ 0 0
$$382$$ −42.8336 −2.19156
$$383$$ 26.5042 1.35430 0.677150 0.735845i $$-0.263215\pi$$
0.677150 + 0.735845i $$0.263215\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.3352 0.526050
$$387$$ 0 0
$$388$$ 22.3635 1.13533
$$389$$ 15.3551 0.778535 0.389268 0.921125i $$-0.372728\pi$$
0.389268 + 0.921125i $$0.372728\pi$$
$$390$$ 0 0
$$391$$ 20.6449 1.04406
$$392$$ 88.9183 4.49105
$$393$$ 0 0
$$394$$ −7.22497 −0.363989
$$395$$ 5.15325 0.259288
$$396$$ 0 0
$$397$$ 16.7677 0.841548 0.420774 0.907166i $$-0.361759\pi$$
0.420774 + 0.907166i $$0.361759\pi$$
$$398$$ 18.3065 0.917622
$$399$$ 0 0
$$400$$ 13.2635 0.663176
$$401$$ 26.0457 1.30066 0.650331 0.759651i $$-0.274630\pi$$
0.650331 + 0.759651i $$0.274630\pi$$
$$402$$ 0 0
$$403$$ 28.1000 1.39976
$$404$$ −73.5195 −3.65773
$$405$$ 0 0
$$406$$ −18.6934 −0.927736
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −22.4495 −1.11005 −0.555027 0.831832i $$-0.687292\pi$$
−0.555027 + 0.831832i $$0.687292\pi$$
$$410$$ 24.5700 1.21343
$$411$$ 0 0
$$412$$ −97.5834 −4.80759
$$413$$ 34.0485 1.67542
$$414$$ 0 0
$$415$$ −9.28502 −0.455784
$$416$$ −49.5927 −2.43148
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −27.8822 −1.36213 −0.681067 0.732221i $$-0.738484\pi$$
−0.681067 + 0.732221i $$0.738484\pi$$
$$420$$ 0 0
$$421$$ 15.5985 0.760226 0.380113 0.924940i $$-0.375885\pi$$
0.380113 + 0.924940i $$0.375885\pi$$
$$422$$ −47.6831 −2.32118
$$423$$ 0 0
$$424$$ 22.4883 1.09213
$$425$$ 2.41541 0.117164
$$426$$ 0 0
$$427$$ −14.3608 −0.694965
$$428$$ 72.5980 3.50916
$$429$$ 0 0
$$430$$ 21.9310 1.05761
$$431$$ 2.95139 0.142163 0.0710817 0.997470i $$-0.477355\pi$$
0.0710817 + 0.997470i $$0.477355\pi$$
$$432$$ 0 0
$$433$$ 24.2864 1.16713 0.583565 0.812067i $$-0.301657\pi$$
0.583565 + 0.812067i $$0.301657\pi$$
$$434$$ −114.369 −5.48986
$$435$$ 0 0
$$436$$ −67.4377 −3.22968
$$437$$ 9.85705 0.471527
$$438$$ 0 0
$$439$$ 19.1103 0.912084 0.456042 0.889958i $$-0.349267\pi$$
0.456042 + 0.889958i $$0.349267\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −17.8350 −0.848325
$$443$$ 1.97714 0.0939365 0.0469683 0.998896i $$-0.485044\pi$$
0.0469683 + 0.998896i $$0.485044\pi$$
$$444$$ 0 0
$$445$$ 4.78783 0.226965
$$446$$ −23.8163 −1.12774
$$447$$ 0 0
$$448$$ 92.2414 4.35800
$$449$$ −15.1719 −0.716008 −0.358004 0.933720i $$-0.616543\pi$$
−0.358004 + 0.933720i $$0.616543\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 78.1055 3.67377
$$453$$ 0 0
$$454$$ 28.3105 1.32868
$$455$$ 11.3121 0.530321
$$456$$ 0 0
$$457$$ 6.17056 0.288647 0.144323 0.989531i $$-0.453899\pi$$
0.144323 + 0.989531i $$0.453899\pi$$
$$458$$ 13.3537 0.623977
$$459$$ 0 0
$$460$$ −45.0742 −2.10160
$$461$$ −23.1990 −1.08048 −0.540242 0.841510i $$-0.681667\pi$$
−0.540242 + 0.841510i $$0.681667\pi$$
$$462$$ 0 0
$$463$$ −1.39809 −0.0649750 −0.0324875 0.999472i $$-0.510343\pi$$
−0.0324875 + 0.999472i $$0.510343\pi$$
$$464$$ −22.2503 −1.03295
$$465$$ 0 0
$$466$$ −10.4941 −0.486131
$$467$$ 22.6901 1.04997 0.524987 0.851110i $$-0.324070\pi$$
0.524987 + 0.851110i $$0.324070\pi$$
$$468$$ 0 0
$$469$$ 63.3606 2.92572
$$470$$ 3.97989 0.183578
$$471$$ 0 0
$$472$$ 72.7545 3.34880
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.15325 0.0529149
$$476$$ 52.6298 2.41228
$$477$$ 0 0
$$478$$ −42.1228 −1.92665
$$479$$ −24.7906 −1.13271 −0.566355 0.824161i $$-0.691647\pi$$
−0.566355 + 0.824161i $$0.691647\pi$$
$$480$$ 0 0
$$481$$ −15.6505 −0.713602
$$482$$ 27.6803 1.26080
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4.24066 −0.192559
$$486$$ 0 0
$$487$$ −33.7191 −1.52796 −0.763979 0.645241i $$-0.776757\pi$$
−0.763979 + 0.645241i $$0.776757\pi$$
$$488$$ −30.6859 −1.38909
$$489$$ 0 0
$$490$$ −27.1624 −1.22707
$$491$$ 6.90566 0.311648 0.155824 0.987785i $$-0.450197\pi$$
0.155824 + 0.987785i $$0.450197\pi$$
$$492$$ 0 0
$$493$$ −4.05198 −0.182492
$$494$$ −8.51546 −0.383129
$$495$$ 0 0
$$496$$ −136.131 −6.11244
$$497$$ −10.5243 −0.472080
$$498$$ 0 0
$$499$$ −2.26078 −0.101206 −0.0506031 0.998719i $$-0.516114\pi$$
−0.0506031 + 0.998719i $$0.516114\pi$$
$$500$$ −5.27358 −0.235842
$$501$$ 0 0
$$502$$ 52.9723 2.36427
$$503$$ 20.5999 0.918504 0.459252 0.888306i $$-0.348117\pi$$
0.459252 + 0.888306i $$0.348117\pi$$
$$504$$ 0 0
$$505$$ 13.9411 0.620370
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −68.3592 −3.03295
$$509$$ 7.43272 0.329449 0.164725 0.986340i $$-0.447326\pi$$
0.164725 + 0.986340i $$0.447326\pi$$
$$510$$ 0 0
$$511$$ −11.3121 −0.500420
$$512$$ 6.05208 0.267466
$$513$$ 0 0
$$514$$ 3.15325 0.139084
$$515$$ 18.5042 0.815392
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 63.6985 2.79875
$$519$$ 0 0
$$520$$ 24.1717 1.06000
$$521$$ 35.0084 1.53375 0.766873 0.641799i $$-0.221812\pi$$
0.766873 + 0.641799i $$0.221812\pi$$
$$522$$ 0 0
$$523$$ 1.52986 0.0668960 0.0334480 0.999440i $$-0.489351\pi$$
0.0334480 + 0.999440i $$0.489351\pi$$
$$524$$ −7.78220 −0.339967
$$525$$ 0 0
$$526$$ 24.0988 1.05076
$$527$$ −24.7906 −1.07989
$$528$$ 0 0
$$529$$ 50.0541 2.17627
$$530$$ −6.86961 −0.298397
$$531$$ 0 0
$$532$$ 25.1285 1.08946
$$533$$ 24.9425 1.08038
$$534$$ 0 0
$$535$$ −13.7663 −0.595171
$$536$$ 135.388 5.84789
$$537$$ 0 0
$$538$$ 40.3233 1.73846
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16.6700 −0.716700 −0.358350 0.933587i $$-0.616660\pi$$
−0.358350 + 0.933587i $$0.616660\pi$$
$$542$$ 61.2990 2.63302
$$543$$ 0 0
$$544$$ 43.7520 1.87585
$$545$$ 12.7878 0.547771
$$546$$ 0 0
$$547$$ 2.25234 0.0963032 0.0481516 0.998840i $$-0.484667\pi$$
0.0481516 + 0.998840i $$0.484667\pi$$
$$548$$ 31.6415 1.35166
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.93465 −0.0824188
$$552$$ 0 0
$$553$$ −21.2920 −0.905429
$$554$$ 59.7071 2.53671
$$555$$ 0 0
$$556$$ 4.69204 0.198987
$$557$$ −11.9911 −0.508078 −0.254039 0.967194i $$-0.581759\pi$$
−0.254039 + 0.967194i $$0.581759\pi$$
$$558$$ 0 0
$$559$$ 22.2635 0.941647
$$560$$ −54.8018 −2.31580
$$561$$ 0 0
$$562$$ −53.8962 −2.27347
$$563$$ −19.4598 −0.820134 −0.410067 0.912055i $$-0.634495\pi$$
−0.410067 + 0.912055i $$0.634495\pi$$
$$564$$ 0 0
$$565$$ −14.8107 −0.623091
$$566$$ −65.4753 −2.75213
$$567$$ 0 0
$$568$$ −22.4883 −0.943586
$$569$$ 19.7205 0.826728 0.413364 0.910566i $$-0.364354\pi$$
0.413364 + 0.910566i $$0.364354\pi$$
$$570$$ 0 0
$$571$$ −5.41678 −0.226685 −0.113343 0.993556i $$-0.536156\pi$$
−0.113343 + 0.993556i $$0.536156\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −101.518 −4.23726
$$575$$ 8.54717 0.356442
$$576$$ 0 0
$$577$$ 6.42708 0.267563 0.133781 0.991011i $$-0.457288\pi$$
0.133781 + 0.991011i $$0.457288\pi$$
$$578$$ −30.1137 −1.25257
$$579$$ 0 0
$$580$$ 8.84675 0.367341
$$581$$ 38.3635 1.59159
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −24.1717 −1.00023
$$585$$ 0 0
$$586$$ −44.2791 −1.82915
$$587$$ −44.9883 −1.85686 −0.928432 0.371502i $$-0.878843\pi$$
−0.928432 + 0.371502i $$0.878843\pi$$
$$588$$ 0 0
$$589$$ −11.8364 −0.487712
$$590$$ −22.2247 −0.914978
$$591$$ 0 0
$$592$$ 75.8191 3.11615
$$593$$ 21.3892 0.878348 0.439174 0.898402i $$-0.355271\pi$$
0.439174 + 0.898402i $$0.355271\pi$$
$$594$$ 0 0
$$595$$ −9.97989 −0.409135
$$596$$ 67.1270 2.74963
$$597$$ 0 0
$$598$$ −63.1111 −2.58081
$$599$$ 26.9514 1.10120 0.550602 0.834768i $$-0.314398\pi$$
0.550602 + 0.834768i $$0.314398\pi$$
$$600$$ 0 0
$$601$$ −6.42708 −0.262166 −0.131083 0.991371i $$-0.541845\pi$$
−0.131083 + 0.991371i $$0.541845\pi$$
$$602$$ −90.6138 −3.69314
$$603$$ 0 0
$$604$$ 6.08178 0.247464
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −12.9626 −0.526135 −0.263067 0.964777i $$-0.584734\pi$$
−0.263067 + 0.964777i $$0.584734\pi$$
$$608$$ 20.8897 0.847190
$$609$$ 0 0
$$610$$ 9.37380 0.379534
$$611$$ 4.04023 0.163450
$$612$$ 0 0
$$613$$ 24.4771 0.988620 0.494310 0.869286i $$-0.335421\pi$$
0.494310 + 0.869286i $$0.335421\pi$$
$$614$$ −14.6820 −0.592517
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11.9453 0.480898 0.240449 0.970662i $$-0.422705\pi$$
0.240449 + 0.970662i $$0.422705\pi$$
$$618$$ 0 0
$$619$$ −18.9084 −0.759993 −0.379997 0.924988i $$-0.624075\pi$$
−0.379997 + 0.924988i $$0.624075\pi$$
$$620$$ 54.1256 2.17374
$$621$$ 0 0
$$622$$ −76.2872 −3.05884
$$623$$ −19.7822 −0.792557
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 61.5195 2.45881
$$627$$ 0 0
$$628$$ −20.8677 −0.832712
$$629$$ 13.8073 0.550534
$$630$$ 0 0
$$631$$ −11.8364 −0.471201 −0.235601 0.971850i $$-0.575706\pi$$
−0.235601 + 0.971850i $$0.575706\pi$$
$$632$$ −45.4966 −1.80976
$$633$$ 0 0
$$634$$ −64.3476 −2.55557
$$635$$ 12.9626 0.514404
$$636$$ 0 0
$$637$$ −27.5742 −1.09253
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −23.9820 −0.947971
$$641$$ 12.2206 0.482683 0.241341 0.970440i $$-0.422413\pi$$
0.241341 + 0.970440i $$0.422413\pi$$
$$642$$ 0 0
$$643$$ −8.84257 −0.348717 −0.174358 0.984682i $$-0.555785\pi$$
−0.174358 + 0.984682i $$0.555785\pi$$
$$644$$ 186.236 7.33873
$$645$$ 0 0
$$646$$ 7.51258 0.295578
$$647$$ −9.40985 −0.369939 −0.184970 0.982744i $$-0.559219\pi$$
−0.184970 + 0.982744i $$0.559219\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −7.38386 −0.289619
$$651$$ 0 0
$$652$$ 16.1977 0.634350
$$653$$ −22.7449 −0.890075 −0.445038 0.895512i $$-0.646810\pi$$
−0.445038 + 0.895512i $$0.646810\pi$$
$$654$$ 0 0
$$655$$ 1.47569 0.0576602
$$656$$ −120.834 −4.71779
$$657$$ 0 0
$$658$$ −16.4440 −0.641052
$$659$$ −20.3635 −0.793249 −0.396625 0.917981i $$-0.629819\pi$$
−0.396625 + 0.917981i $$0.629819\pi$$
$$660$$ 0 0
$$661$$ 11.6616 0.453585 0.226792 0.973943i $$-0.427176\pi$$
0.226792 + 0.973943i $$0.427176\pi$$
$$662$$ −50.9952 −1.98198
$$663$$ 0 0
$$664$$ 81.9748 3.18124
$$665$$ −4.76497 −0.184778
$$666$$ 0 0
$$667$$ −14.3384 −0.555184
$$668$$ 14.3177 0.553968
$$669$$ 0 0
$$670$$ −41.3579 −1.59779
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −30.0070 −1.15669 −0.578343 0.815794i $$-0.696300\pi$$
−0.578343 + 0.815794i $$0.696300\pi$$
$$674$$ −66.5151 −2.56207
$$675$$ 0 0
$$676$$ −29.0268 −1.11642
$$677$$ 51.4695 1.97813 0.989067 0.147466i $$-0.0471116\pi$$
0.989067 + 0.147466i $$0.0471116\pi$$
$$678$$ 0 0
$$679$$ 17.5214 0.672411
$$680$$ −21.3249 −0.817774
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 27.6186 1.05680 0.528399 0.848996i $$-0.322793\pi$$
0.528399 + 0.848996i $$0.322793\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ 34.2261 1.30676
$$687$$ 0 0
$$688$$ −107.856 −4.11197
$$689$$ −6.97376 −0.265679
$$690$$ 0 0
$$691$$ −34.2344 −1.30234 −0.651169 0.758933i $$-0.725721\pi$$
−0.651169 + 0.758933i $$0.725721\pi$$
$$692$$ 76.0209 2.88988
$$693$$ 0 0
$$694$$ −41.8179 −1.58738
$$695$$ −0.889725 −0.0337492
$$696$$ 0 0
$$697$$ −22.0050 −0.833499
$$698$$ −55.2447 −2.09104
$$699$$ 0 0
$$700$$ 21.7892 0.823555
$$701$$ 11.8524 0.447658 0.223829 0.974628i $$-0.428144\pi$$
0.223829 + 0.974628i $$0.428144\pi$$
$$702$$ 0 0
$$703$$ 6.59241 0.248637
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 59.2788 2.23099
$$707$$ −57.6013 −2.16632
$$708$$ 0 0
$$709$$ −28.1658 −1.05779 −0.528895 0.848687i $$-0.677393\pi$$
−0.528895 + 0.848687i $$0.677393\pi$$
$$710$$ 6.86961 0.257812
$$711$$ 0 0
$$712$$ −42.2705 −1.58415
$$713$$ −87.7241 −3.28529
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −56.8906 −2.12610
$$717$$ 0 0
$$718$$ −53.1200 −1.98242
$$719$$ −26.2663 −0.979567 −0.489783 0.871844i $$-0.662924\pi$$
−0.489783 + 0.871844i $$0.662924\pi$$
$$720$$ 0 0
$$721$$ −76.4550 −2.84733
$$722$$ −47.6553 −1.77355
$$723$$ 0 0
$$724$$ −34.3004 −1.27476
$$725$$ −1.67756 −0.0623030
$$726$$ 0 0
$$727$$ 21.1401 0.784042 0.392021 0.919956i $$-0.371776\pi$$
0.392021 + 0.919956i $$0.371776\pi$$
$$728$$ −99.8717 −3.70149
$$729$$ 0 0
$$730$$ 7.38386 0.273289
$$731$$ −19.6415 −0.726467
$$732$$ 0 0
$$733$$ −3.86406 −0.142722 −0.0713611 0.997451i $$-0.522734\pi$$
−0.0713611 + 0.997451i $$0.522734\pi$$
$$734$$ −76.2256 −2.81354
$$735$$ 0 0
$$736$$ 154.821 5.70679
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 14.0298 0.516094 0.258047 0.966132i $$-0.416921\pi$$
0.258047 + 0.966132i $$0.416921\pi$$
$$740$$ −30.1457 −1.10818
$$741$$ 0 0
$$742$$ 28.3836 1.04200
$$743$$ −26.2934 −0.964611 −0.482306 0.876003i $$-0.660201\pi$$
−0.482306 + 0.876003i $$0.660201\pi$$
$$744$$ 0 0
$$745$$ −12.7289 −0.466352
$$746$$ 26.4011 0.966613
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 56.8793 2.07832
$$750$$ 0 0
$$751$$ 38.2206 1.39469 0.697344 0.716737i $$-0.254365\pi$$
0.697344 + 0.716737i $$0.254365\pi$$
$$752$$ −19.5729 −0.713751
$$753$$ 0 0
$$754$$ 12.3869 0.451103
$$755$$ −1.15325 −0.0419712
$$756$$ 0 0
$$757$$ −36.6158 −1.33082 −0.665411 0.746477i $$-0.731744\pi$$
−0.665411 + 0.746477i $$0.731744\pi$$
$$758$$ 15.2692 0.554601
$$759$$ 0 0
$$760$$ −10.1818 −0.369331
$$761$$ −51.6803 −1.87341 −0.936705 0.350120i $$-0.886141\pi$$
−0.936705 + 0.350120i $$0.886141\pi$$
$$762$$ 0 0
$$763$$ −52.8363 −1.91280
$$764$$ −83.7560 −3.03019
$$765$$ 0 0
$$766$$ 71.4807 2.58270
$$767$$ −22.5617 −0.814655
$$768$$ 0 0
$$769$$ −44.4841 −1.60414 −0.802068 0.597232i $$-0.796267\pi$$
−0.802068 + 0.597232i $$0.796267\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 20.2094 0.727351
$$773$$ 0.973762 0.0350238 0.0175119 0.999847i $$-0.494426\pi$$
0.0175119 + 0.999847i $$0.494426\pi$$
$$774$$ 0 0
$$775$$ −10.2635 −0.368677
$$776$$ 37.4396 1.34400
$$777$$ 0 0
$$778$$ 41.4121 1.48470
$$779$$ −10.5065 −0.376433
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 55.6784 1.99106
$$783$$ 0 0
$$784$$ 133.583 4.77083
$$785$$ 3.95702 0.141232
$$786$$ 0 0
$$787$$ 30.5382 1.08857 0.544285 0.838900i $$-0.316801\pi$$
0.544285 + 0.838900i $$0.316801\pi$$
$$788$$ −14.1276 −0.503274
$$789$$ 0 0
$$790$$ 13.8981 0.494473
$$791$$ 61.1943 2.17582
$$792$$ 0 0
$$793$$ 9.51592 0.337920
$$794$$ 45.2218 1.60486
$$795$$ 0 0
$$796$$ 35.7962 1.26876
$$797$$ 26.7247 0.946639 0.473319 0.880891i $$-0.343056\pi$$
0.473319 + 0.880891i $$0.343056\pi$$
$$798$$ 0 0
$$799$$ −3.56440 −0.126099
$$800$$ 18.1137 0.640417
$$801$$ 0 0
$$802$$ 70.2442 2.48041
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −35.3149 −1.24469
$$806$$ 75.7845 2.66939
$$807$$ 0 0
$$808$$ −123.082 −4.33001
$$809$$ 17.0327 0.598837 0.299418 0.954122i $$-0.403207\pi$$
0.299418 + 0.954122i $$0.403207\pi$$
$$810$$ 0 0
$$811$$ −23.1103 −0.811512 −0.405756 0.913982i $$-0.632992\pi$$
−0.405756 + 0.913982i $$0.632992\pi$$
$$812$$ −36.5527 −1.28275
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −3.07148 −0.107589
$$816$$ 0 0
$$817$$ −9.37798 −0.328094
$$818$$ −60.5453 −2.11692
$$819$$ 0 0
$$820$$ 48.0438 1.67776
$$821$$ 28.3476 0.989337 0.494668 0.869082i $$-0.335290\pi$$
0.494668 + 0.869082i $$0.335290\pi$$
$$822$$ 0 0
$$823$$ −10.4585 −0.364559 −0.182280 0.983247i $$-0.558348\pi$$
−0.182280 + 0.983247i $$0.558348\pi$$
$$824$$ −163.368 −5.69121
$$825$$ 0 0
$$826$$ 91.8273 3.19508
$$827$$ 45.2672 1.57409 0.787047 0.616893i $$-0.211609\pi$$
0.787047 + 0.616893i $$0.211609\pi$$
$$828$$ 0 0
$$829$$ −22.9855 −0.798320 −0.399160 0.916881i $$-0.630698\pi$$
−0.399160 + 0.916881i $$0.630698\pi$$
$$830$$ −25.0413 −0.869196
$$831$$ 0 0
$$832$$ −61.1223 −2.11903
$$833$$ 24.3267 0.842870
$$834$$ 0 0
$$835$$ −2.71498 −0.0939559
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −75.1971 −2.59764
$$839$$ −20.1547 −0.695818 −0.347909 0.937528i $$-0.613108\pi$$
−0.347909 + 0.937528i $$0.613108\pi$$
$$840$$ 0 0
$$841$$ −26.1858 −0.902959
$$842$$ 42.0686 1.44978
$$843$$ 0 0
$$844$$ −93.2386 −3.20941
$$845$$ 5.50419 0.189350
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 33.7845 1.16016
$$849$$ 0 0
$$850$$ 6.51425 0.223437
$$851$$ 48.8587 1.67485
$$852$$ 0 0
$$853$$ 38.2705 1.31036 0.655179 0.755474i $$-0.272593\pi$$
0.655179 + 0.755474i $$0.272593\pi$$
$$854$$ −38.7303 −1.32533
$$855$$ 0 0
$$856$$ 121.539 4.15413
$$857$$ 9.51250 0.324941 0.162470 0.986713i $$-0.448054\pi$$
0.162470 + 0.986713i $$0.448054\pi$$
$$858$$ 0 0
$$859$$ −1.61865 −0.0552275 −0.0276137 0.999619i $$-0.508791\pi$$
−0.0276137 + 0.999619i $$0.508791\pi$$
$$860$$ 42.8835 1.46232
$$861$$ 0 0
$$862$$ 7.95977 0.271111
$$863$$ 46.0229 1.56664 0.783318 0.621621i $$-0.213526\pi$$
0.783318 + 0.621621i $$0.213526\pi$$
$$864$$ 0 0
$$865$$ −14.4154 −0.490138
$$866$$ 65.4994 2.22576
$$867$$ 0 0
$$868$$ −223.634 −7.59064
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −41.9849 −1.42260
$$872$$ −112.900 −3.82328
$$873$$ 0 0
$$874$$ 26.5841 0.899219
$$875$$ −4.13176 −0.139679
$$876$$ 0 0
$$877$$ 28.7630 0.971257 0.485628 0.874165i $$-0.338591\pi$$
0.485628 + 0.874165i $$0.338591\pi$$
$$878$$ 51.5396 1.73938
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −23.3149 −0.785499 −0.392749 0.919646i $$-0.628476\pi$$
−0.392749 + 0.919646i $$0.628476\pi$$
$$882$$ 0 0
$$883$$ 29.6616 0.998193 0.499097 0.866546i $$-0.333665\pi$$
0.499097 + 0.866546i $$0.333665\pi$$
$$884$$ −34.8743 −1.17295
$$885$$ 0 0
$$886$$ 5.33225 0.179141
$$887$$ −20.3364 −0.682829 −0.341414 0.939913i $$-0.610906\pi$$
−0.341414 + 0.939913i $$0.610906\pi$$
$$888$$ 0 0
$$889$$ −53.5583 −1.79629
$$890$$ 12.9126 0.432831
$$891$$ 0 0
$$892$$ −46.5700 −1.55928
$$893$$ −1.70185 −0.0569502
$$894$$ 0 0
$$895$$ 10.7878 0.360598
$$896$$ 99.0879 3.31029
$$897$$ 0 0
$$898$$ −40.9181 −1.36545
$$899$$ 17.2177 0.574241
$$900$$ 0 0
$$901$$ 6.15245 0.204968
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 130.760 4.34900
$$905$$ 6.50419 0.216207
$$906$$ 0 0
$$907$$ 9.87942 0.328041 0.164020 0.986457i $$-0.447554\pi$$
0.164020 + 0.986457i $$0.447554\pi$$
$$908$$ 55.3578 1.83711
$$909$$ 0 0
$$910$$ 30.5084 1.01134
$$911$$ 7.22343 0.239323 0.119662 0.992815i $$-0.461819\pi$$
0.119662 + 0.992815i $$0.461819\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 16.6418 0.550460
$$915$$ 0 0
$$916$$ 26.1116 0.862751
$$917$$ −6.09722 −0.201348
$$918$$ 0 0
$$919$$ −35.7233 −1.17840 −0.589201 0.807986i $$-0.700557\pi$$
−0.589201 + 0.807986i $$0.700557\pi$$
$$920$$ −75.4606 −2.48786
$$921$$ 0 0
$$922$$ −62.5667 −2.06052
$$923$$ 6.97376 0.229544
$$924$$ 0 0
$$925$$ 5.71636 0.187953
$$926$$ −3.77060 −0.123910
$$927$$ 0 0
$$928$$ −30.3869 −0.997497
$$929$$ −1.12621 −0.0369498 −0.0184749 0.999829i $$-0.505881\pi$$
−0.0184749 + 0.999829i $$0.505881\pi$$
$$930$$ 0 0
$$931$$ 11.6150 0.380665
$$932$$ −20.5200 −0.672157
$$933$$ 0 0
$$934$$ 61.1943 2.00234
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0.127587 0.00416807 0.00208404 0.999998i $$-0.499337\pi$$
0.00208404 + 0.999998i $$0.499337\pi$$
$$938$$ 170.881 5.57946
$$939$$ 0 0
$$940$$ 7.78220 0.253827
$$941$$ −1.41403 −0.0460961 −0.0230480 0.999734i $$-0.507337\pi$$
−0.0230480 + 0.999734i $$0.507337\pi$$
$$942$$ 0 0
$$943$$ −77.8671 −2.53570
$$944$$ 109.300 3.55742
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −37.1943 −1.20865 −0.604326 0.796737i $$-0.706558\pi$$
−0.604326 + 0.796737i $$0.706558\pi$$
$$948$$ 0 0
$$949$$ 7.49581 0.243324
$$950$$ 3.11028 0.100911
$$951$$ 0 0
$$952$$ 88.1097 2.85565
$$953$$ 16.5041 0.534621 0.267310 0.963610i $$-0.413865\pi$$
0.267310 + 0.963610i $$0.413865\pi$$
$$954$$ 0 0
$$955$$ 15.8822 0.513935
$$956$$ −82.3663 −2.66391
$$957$$ 0 0
$$958$$ −66.8592 −2.16012
$$959$$ 24.7906 0.800530
$$960$$ 0 0
$$961$$ 74.3400 2.39807
$$962$$ −42.2088 −1.36087
$$963$$ 0 0
$$964$$ 54.1256 1.74327
$$965$$ −3.83219 −0.123362
$$966$$ 0 0
$$967$$ 3.60471 0.115920 0.0579598 0.998319i $$-0.481540\pi$$
0.0579598 + 0.998319i $$0.481540\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −11.4369 −0.367217
$$971$$ 29.6214 0.950596 0.475298 0.879825i $$-0.342340\pi$$
0.475298 + 0.879825i $$0.342340\pi$$
$$972$$ 0 0
$$973$$ 3.67613 0.117851
$$974$$ −90.9390 −2.91387
$$975$$ 0 0
$$976$$ −46.1000 −1.47562
$$977$$ −52.7157 −1.68653 −0.843263 0.537501i $$-0.819368\pi$$
−0.843263 + 0.537501i $$0.819368\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −53.1128 −1.69663
$$981$$ 0 0
$$982$$ 18.6243 0.594325
$$983$$ 29.2602 0.933254 0.466627 0.884454i $$-0.345469\pi$$
0.466627 + 0.884454i $$0.345469\pi$$
$$984$$ 0 0
$$985$$ 2.67893 0.0853579
$$986$$ −10.9280 −0.348019
$$987$$ 0 0
$$988$$ −16.6510 −0.529738
$$989$$ −69.5036 −2.21008
$$990$$ 0 0
$$991$$ 38.1776 1.21275 0.606375 0.795179i $$-0.292623\pi$$
0.606375 + 0.795179i $$0.292623\pi$$
$$992$$ −185.911 −5.90268
$$993$$ 0 0
$$994$$ −28.3836 −0.900274
$$995$$ −6.78783 −0.215189
$$996$$ 0 0
$$997$$ 40.1500 1.27156 0.635781 0.771870i $$-0.280678\pi$$
0.635781 + 0.771870i $$0.280678\pi$$
$$998$$ −6.09722 −0.193004
$$999$$ 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 5445.2.a.bs.1.4 4
3.2 odd 2 5445.2.a.bh.1.1 4
11.10 odd 2 495.2.a.f.1.1 4
33.32 even 2 495.2.a.g.1.4 yes 4
44.43 even 2 7920.2.a.cm.1.4 4
55.32 even 4 2475.2.c.t.199.1 8
55.43 even 4 2475.2.c.t.199.8 8
55.54 odd 2 2475.2.a.bj.1.4 4
132.131 odd 2 7920.2.a.cn.1.4 4
165.32 odd 4 2475.2.c.s.199.8 8
165.98 odd 4 2475.2.c.s.199.1 8
165.164 even 2 2475.2.a.bf.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.a.f.1.1 4 11.10 odd 2
495.2.a.g.1.4 yes 4 33.32 even 2
2475.2.a.bf.1.1 4 165.164 even 2
2475.2.a.bj.1.4 4 55.54 odd 2
2475.2.c.s.199.1 8 165.98 odd 4
2475.2.c.s.199.8 8 165.32 odd 4
2475.2.c.t.199.1 8 55.32 even 4
2475.2.c.t.199.8 8 55.43 even 4
5445.2.a.bh.1.1 4 3.2 odd 2
5445.2.a.bs.1.4 4 1.1 even 1 trivial
7920.2.a.cm.1.4 4 44.43 even 2
7920.2.a.cn.1.4 4 132.131 odd 2