# Properties

 Label 1110.2.a.s.1.4 Level $1110$ Weight $2$ Character 1110.1 Self dual yes Analytic conductor $8.863$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.54764.1 Defining polynomial: $$x^{4} - x^{3} - 9 x^{2} + 3 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-2.67673$$ of defining polynomial Character $$\chi$$ $$=$$ 1110.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +5.13277 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +5.13277 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.22069 q^{11} +1.00000 q^{12} -6.60629 q^{13} +5.13277 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.63841 q^{17} +1.00000 q^{18} -7.51836 q^{19} +1.00000 q^{20} +5.13277 q^{21} +2.22069 q^{22} -6.10064 q^{23} +1.00000 q^{24} +1.00000 q^{25} -6.60629 q^{26} +1.00000 q^{27} +5.13277 q^{28} -7.29767 q^{29} +1.00000 q^{30} -0.967873 q^{31} +1.00000 q^{32} +2.22069 q^{33} +5.63841 q^{34} +5.13277 q^{35} +1.00000 q^{36} -1.00000 q^{37} -7.51836 q^{38} -6.60629 q^{39} +1.00000 q^{40} -1.03213 q^{41} +5.13277 q^{42} +5.29767 q^{43} +2.22069 q^{44} +1.00000 q^{45} -6.10064 q^{46} -5.85911 q^{47} +1.00000 q^{48} +19.3453 q^{49} +1.00000 q^{50} +5.63841 q^{51} -6.60629 q^{52} -9.63841 q^{53} +1.00000 q^{54} +2.22069 q^{55} +5.13277 q^{56} -7.51836 q^{57} -7.29767 q^{58} -10.2655 q^{59} +1.00000 q^{60} +6.32134 q^{61} -0.967873 q^{62} +5.13277 q^{63} +1.00000 q^{64} -6.60629 q^{65} +2.22069 q^{66} +1.49436 q^{67} +5.63841 q^{68} -6.10064 q^{69} +5.13277 q^{70} +0.329796 q^{71} +1.00000 q^{72} +12.8718 q^{73} -1.00000 q^{74} +1.00000 q^{75} -7.51836 q^{76} +11.3983 q^{77} -6.60629 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.03213 q^{82} +9.04767 q^{83} +5.13277 q^{84} +5.63841 q^{85} +5.29767 q^{86} -7.29767 q^{87} +2.22069 q^{88} -2.27649 q^{89} +1.00000 q^{90} -33.9086 q^{91} -6.10064 q^{92} -0.967873 q^{93} -5.85911 q^{94} -7.51836 q^{95} +1.00000 q^{96} +17.5982 q^{97} +19.3453 q^{98} +2.22069 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + 4q^{6} + 4q^{7} + 4q^{8} + 4q^{9} + 4q^{10} + 2q^{11} + 4q^{12} - 3q^{13} + 4q^{14} + 4q^{15} + 4q^{16} + 6q^{17} + 4q^{18} + 3q^{19} + 4q^{20} + 4q^{21} + 2q^{22} - q^{23} + 4q^{24} + 4q^{25} - 3q^{26} + 4q^{27} + 4q^{28} - 3q^{29} + 4q^{30} + 3q^{31} + 4q^{32} + 2q^{33} + 6q^{34} + 4q^{35} + 4q^{36} - 4q^{37} + 3q^{38} - 3q^{39} + 4q^{40} - 11q^{41} + 4q^{42} - 5q^{43} + 2q^{44} + 4q^{45} - q^{46} + 4q^{48} + 14q^{49} + 4q^{50} + 6q^{51} - 3q^{52} - 22q^{53} + 4q^{54} + 2q^{55} + 4q^{56} + 3q^{57} - 3q^{58} - 8q^{59} + 4q^{60} - 5q^{61} + 3q^{62} + 4q^{63} + 4q^{64} - 3q^{65} + 2q^{66} + 6q^{67} + 6q^{68} - q^{69} + 4q^{70} - 18q^{71} + 4q^{72} - 5q^{73} - 4q^{74} + 4q^{75} + 3q^{76} - 4q^{77} - 3q^{78} + 16q^{79} + 4q^{80} + 4q^{81} - 11q^{82} - q^{83} + 4q^{84} + 6q^{85} - 5q^{86} - 3q^{87} + 2q^{88} - 5q^{89} + 4q^{90} - 13q^{91} - q^{92} + 3q^{93} + 3q^{95} + 4q^{96} + 7q^{97} + 14q^{98} + 2q^{99} + O(q^{100})$$

## 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
$$5$$ 1.00000 0.447214
$$6$$ 1.00000 0.408248
$$7$$ 5.13277 1.94001 0.970003 0.243094i $$-0.0781625\pi$$
0.970003 + 0.243094i $$0.0781625\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 2.22069 0.669564 0.334782 0.942296i $$-0.391337\pi$$
0.334782 + 0.942296i $$0.391337\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −6.60629 −1.83225 −0.916127 0.400888i $$-0.868702\pi$$
−0.916127 + 0.400888i $$0.868702\pi$$
$$14$$ 5.13277 1.37179
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 5.63841 1.36752 0.683758 0.729709i $$-0.260344\pi$$
0.683758 + 0.729709i $$0.260344\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −7.51836 −1.72483 −0.862415 0.506201i $$-0.831049\pi$$
−0.862415 + 0.506201i $$0.831049\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 5.13277 1.12006
$$22$$ 2.22069 0.473454
$$23$$ −6.10064 −1.27207 −0.636036 0.771659i $$-0.719427\pi$$
−0.636036 + 0.771659i $$0.719427\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ −6.60629 −1.29560
$$27$$ 1.00000 0.192450
$$28$$ 5.13277 0.970003
$$29$$ −7.29767 −1.35514 −0.677572 0.735457i $$-0.736968\pi$$
−0.677572 + 0.735457i $$0.736968\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −0.967873 −0.173835 −0.0869176 0.996216i $$-0.527702\pi$$
−0.0869176 + 0.996216i $$0.527702\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 2.22069 0.386573
$$34$$ 5.63841 0.966980
$$35$$ 5.13277 0.867597
$$36$$ 1.00000 0.166667
$$37$$ −1.00000 −0.164399
$$38$$ −7.51836 −1.21964
$$39$$ −6.60629 −1.05785
$$40$$ 1.00000 0.158114
$$41$$ −1.03213 −0.161191 −0.0805955 0.996747i $$-0.525682\pi$$
−0.0805955 + 0.996747i $$0.525682\pi$$
$$42$$ 5.13277 0.792004
$$43$$ 5.29767 0.807887 0.403944 0.914784i $$-0.367639\pi$$
0.403944 + 0.914784i $$0.367639\pi$$
$$44$$ 2.22069 0.334782
$$45$$ 1.00000 0.149071
$$46$$ −6.10064 −0.899491
$$47$$ −5.85911 −0.854638 −0.427319 0.904101i $$-0.640542\pi$$
−0.427319 + 0.904101i $$0.640542\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 19.3453 2.76362
$$50$$ 1.00000 0.141421
$$51$$ 5.63841 0.789536
$$52$$ −6.60629 −0.916127
$$53$$ −9.63841 −1.32394 −0.661969 0.749531i $$-0.730279\pi$$
−0.661969 + 0.749531i $$0.730279\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 2.22069 0.299438
$$56$$ 5.13277 0.685895
$$57$$ −7.51836 −0.995832
$$58$$ −7.29767 −0.958231
$$59$$ −10.2655 −1.33646 −0.668230 0.743955i $$-0.732948\pi$$
−0.668230 + 0.743955i $$0.732948\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 6.32134 0.809365 0.404682 0.914457i $$-0.367382\pi$$
0.404682 + 0.914457i $$0.367382\pi$$
$$62$$ −0.967873 −0.122920
$$63$$ 5.13277 0.646668
$$64$$ 1.00000 0.125000
$$65$$ −6.60629 −0.819409
$$66$$ 2.22069 0.273349
$$67$$ 1.49436 0.182565 0.0912825 0.995825i $$-0.470903\pi$$
0.0912825 + 0.995825i $$0.470903\pi$$
$$68$$ 5.63841 0.683758
$$69$$ −6.10064 −0.734431
$$70$$ 5.13277 0.613484
$$71$$ 0.329796 0.0391396 0.0195698 0.999808i $$-0.493770\pi$$
0.0195698 + 0.999808i $$0.493770\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 12.8718 1.50653 0.753267 0.657715i $$-0.228477\pi$$
0.753267 + 0.657715i $$0.228477\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 1.00000 0.115470
$$76$$ −7.51836 −0.862415
$$77$$ 11.3983 1.29896
$$78$$ −6.60629 −0.748015
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ −1.03213 −0.113979
$$83$$ 9.04767 0.993111 0.496556 0.868005i $$-0.334598\pi$$
0.496556 + 0.868005i $$0.334598\pi$$
$$84$$ 5.13277 0.560031
$$85$$ 5.63841 0.611572
$$86$$ 5.29767 0.571262
$$87$$ −7.29767 −0.782392
$$88$$ 2.22069 0.236727
$$89$$ −2.27649 −0.241307 −0.120654 0.992695i $$-0.538499\pi$$
−0.120654 + 0.992695i $$0.538499\pi$$
$$90$$ 1.00000 0.105409
$$91$$ −33.9086 −3.55458
$$92$$ −6.10064 −0.636036
$$93$$ −0.967873 −0.100364
$$94$$ −5.85911 −0.604321
$$95$$ −7.51836 −0.771368
$$96$$ 1.00000 0.102062
$$97$$ 17.5982 1.78682 0.893411 0.449239i $$-0.148305\pi$$
0.893411 + 0.449239i $$0.148305\pi$$
$$98$$ 19.3453 1.95417
$$99$$ 2.22069 0.223188
$$100$$ 1.00000 0.100000
$$101$$ 7.61901 0.758120 0.379060 0.925372i $$-0.376247\pi$$
0.379060 + 0.925372i $$0.376247\pi$$
$$102$$ 5.63841 0.558286
$$103$$ 5.85911 0.577315 0.288657 0.957432i $$-0.406791\pi$$
0.288657 + 0.957432i $$0.406791\pi$$
$$104$$ −6.60629 −0.647800
$$105$$ 5.13277 0.500907
$$106$$ −9.63841 −0.936165
$$107$$ 5.65926 0.547101 0.273551 0.961858i $$-0.411802\pi$$
0.273551 + 0.961858i $$0.411802\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −3.16772 −0.303413 −0.151706 0.988426i $$-0.548477\pi$$
−0.151706 + 0.988426i $$0.548477\pi$$
$$110$$ 2.22069 0.211735
$$111$$ −1.00000 −0.0949158
$$112$$ 5.13277 0.485001
$$113$$ −0.702331 −0.0660697 −0.0330348 0.999454i $$-0.510517\pi$$
−0.0330348 + 0.999454i $$0.510517\pi$$
$$114$$ −7.51836 −0.704159
$$115$$ −6.10064 −0.568888
$$116$$ −7.29767 −0.677572
$$117$$ −6.60629 −0.610751
$$118$$ −10.2655 −0.945020
$$119$$ 28.9407 2.65299
$$120$$ 1.00000 0.0912871
$$121$$ −6.06852 −0.551683
$$122$$ 6.32134 0.572307
$$123$$ −1.03213 −0.0930637
$$124$$ −0.967873 −0.0869176
$$125$$ 1.00000 0.0894427
$$126$$ 5.13277 0.457264
$$127$$ −19.3132 −1.71377 −0.856885 0.515507i $$-0.827604\pi$$
−0.856885 + 0.515507i $$0.827604\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 5.29767 0.466434
$$130$$ −6.60629 −0.579410
$$131$$ −18.5953 −1.62468 −0.812341 0.583183i $$-0.801807\pi$$
−0.812341 + 0.583183i $$0.801807\pi$$
$$132$$ 2.22069 0.193287
$$133$$ −38.5900 −3.34618
$$134$$ 1.49436 0.129093
$$135$$ 1.00000 0.0860663
$$136$$ 5.63841 0.483490
$$137$$ −19.5074 −1.66663 −0.833316 0.552798i $$-0.813560\pi$$
−0.833316 + 0.552798i $$0.813560\pi$$
$$138$$ −6.10064 −0.519321
$$139$$ 10.9036 0.924833 0.462417 0.886663i $$-0.346982\pi$$
0.462417 + 0.886663i $$0.346982\pi$$
$$140$$ 5.13277 0.433798
$$141$$ −5.85911 −0.493426
$$142$$ 0.329796 0.0276759
$$143$$ −14.6705 −1.22681
$$144$$ 1.00000 0.0833333
$$145$$ −7.29767 −0.606038
$$146$$ 12.8718 1.06528
$$147$$ 19.3453 1.59558
$$148$$ −1.00000 −0.0821995
$$149$$ −12.6720 −1.03813 −0.519065 0.854735i $$-0.673720\pi$$
−0.519065 + 0.854735i $$0.673720\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 10.5420 0.857898 0.428949 0.903329i $$-0.358884\pi$$
0.428949 + 0.903329i $$0.358884\pi$$
$$152$$ −7.51836 −0.609820
$$153$$ 5.63841 0.455839
$$154$$ 11.3983 0.918502
$$155$$ −0.967873 −0.0777415
$$156$$ −6.60629 −0.528926
$$157$$ −15.4990 −1.23695 −0.618476 0.785804i $$-0.712250\pi$$
−0.618476 + 0.785804i $$0.712250\pi$$
$$158$$ 4.00000 0.318223
$$159$$ −9.63841 −0.764376
$$160$$ 1.00000 0.0790569
$$161$$ −31.3132 −2.46783
$$162$$ 1.00000 0.0785674
$$163$$ −4.12149 −0.322820 −0.161410 0.986887i $$-0.551604\pi$$
−0.161410 + 0.986887i $$0.551604\pi$$
$$164$$ −1.03213 −0.0805955
$$165$$ 2.22069 0.172881
$$166$$ 9.04767 0.702236
$$167$$ 21.9474 1.69834 0.849169 0.528121i $$-0.177103\pi$$
0.849169 + 0.528121i $$0.177103\pi$$
$$168$$ 5.13277 0.396002
$$169$$ 30.6430 2.35715
$$170$$ 5.63841 0.432446
$$171$$ −7.51836 −0.574944
$$172$$ 5.29767 0.403944
$$173$$ −0.425841 −0.0323761 −0.0161880 0.999869i $$-0.505153\pi$$
−0.0161880 + 0.999869i $$0.505153\pi$$
$$174$$ −7.29767 −0.553235
$$175$$ 5.13277 0.388001
$$176$$ 2.22069 0.167391
$$177$$ −10.2655 −0.771605
$$178$$ −2.27649 −0.170630
$$179$$ −17.5424 −1.31118 −0.655589 0.755118i $$-0.727580\pi$$
−0.655589 + 0.755118i $$0.727580\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 18.5311 1.37740 0.688702 0.725044i $$-0.258181\pi$$
0.688702 + 0.725044i $$0.258181\pi$$
$$182$$ −33.9086 −2.51347
$$183$$ 6.32134 0.467287
$$184$$ −6.10064 −0.449746
$$185$$ −1.00000 −0.0735215
$$186$$ −0.967873 −0.0709679
$$187$$ 12.5212 0.915640
$$188$$ −5.85911 −0.427319
$$189$$ 5.13277 0.373354
$$190$$ −7.51836 −0.545439
$$191$$ 16.1695 1.16998 0.584992 0.811039i $$-0.301098\pi$$
0.584992 + 0.811039i $$0.301098\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −11.8591 −0.853637 −0.426819 0.904337i $$-0.640366\pi$$
−0.426819 + 0.904337i $$0.640366\pi$$
$$194$$ 17.5982 1.26347
$$195$$ −6.60629 −0.473086
$$196$$ 19.3453 1.38181
$$197$$ 0.100645 0.00717066 0.00358533 0.999994i $$-0.498859\pi$$
0.00358533 + 0.999994i $$0.498859\pi$$
$$198$$ 2.22069 0.157818
$$199$$ −2.70693 −0.191889 −0.0959446 0.995387i $$-0.530587\pi$$
−0.0959446 + 0.995387i $$0.530587\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 1.49436 0.105404
$$202$$ 7.61901 0.536072
$$203$$ −37.4573 −2.62898
$$204$$ 5.63841 0.394768
$$205$$ −1.03213 −0.0720868
$$206$$ 5.85911 0.408223
$$207$$ −6.10064 −0.424024
$$208$$ −6.60629 −0.458064
$$209$$ −16.6960 −1.15489
$$210$$ 5.13277 0.354195
$$211$$ 11.8033 0.812573 0.406287 0.913746i $$-0.366823\pi$$
0.406287 + 0.913746i $$0.366823\pi$$
$$212$$ −9.63841 −0.661969
$$213$$ 0.329796 0.0225972
$$214$$ 5.65926 0.386859
$$215$$ 5.29767 0.361298
$$216$$ 1.00000 0.0680414
$$217$$ −4.96787 −0.337241
$$218$$ −3.16772 −0.214545
$$219$$ 12.8718 0.869798
$$220$$ 2.22069 0.149719
$$221$$ −37.2490 −2.50564
$$222$$ −1.00000 −0.0671156
$$223$$ 13.1692 0.881872 0.440936 0.897538i $$-0.354646\pi$$
0.440936 + 0.897538i $$0.354646\pi$$
$$224$$ 5.13277 0.342948
$$225$$ 1.00000 0.0666667
$$226$$ −0.702331 −0.0467183
$$227$$ 5.57946 0.370322 0.185161 0.982708i $$-0.440719\pi$$
0.185161 + 0.982708i $$0.440719\pi$$
$$228$$ −7.51836 −0.497916
$$229$$ −1.75990 −0.116298 −0.0581488 0.998308i $$-0.518520\pi$$
−0.0581488 + 0.998308i $$0.518520\pi$$
$$230$$ −6.10064 −0.402265
$$231$$ 11.3983 0.749954
$$232$$ −7.29767 −0.479115
$$233$$ −6.07664 −0.398094 −0.199047 0.979990i $$-0.563785\pi$$
−0.199047 + 0.979990i $$0.563785\pi$$
$$234$$ −6.60629 −0.431866
$$235$$ −5.85911 −0.382206
$$236$$ −10.2655 −0.668230
$$237$$ 4.00000 0.259828
$$238$$ 28.9407 1.87595
$$239$$ 13.9403 0.901726 0.450863 0.892593i $$-0.351116\pi$$
0.450863 + 0.892593i $$0.351116\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −3.89406 −0.250838 −0.125419 0.992104i $$-0.540028\pi$$
−0.125419 + 0.992104i $$0.540028\pi$$
$$242$$ −6.06852 −0.390099
$$243$$ 1.00000 0.0641500
$$244$$ 6.32134 0.404682
$$245$$ 19.3453 1.23593
$$246$$ −1.03213 −0.0658060
$$247$$ 49.6685 3.16033
$$248$$ −0.967873 −0.0614600
$$249$$ 9.04767 0.573373
$$250$$ 1.00000 0.0632456
$$251$$ 20.9308 1.32114 0.660570 0.750765i $$-0.270315\pi$$
0.660570 + 0.750765i $$0.270315\pi$$
$$252$$ 5.13277 0.323334
$$253$$ −13.5477 −0.851734
$$254$$ −19.3132 −1.21182
$$255$$ 5.63841 0.353091
$$256$$ 1.00000 0.0625000
$$257$$ −4.43044 −0.276363 −0.138182 0.990407i $$-0.544126\pi$$
−0.138182 + 0.990407i $$0.544126\pi$$
$$258$$ 5.29767 0.329818
$$259$$ −5.13277 −0.318935
$$260$$ −6.60629 −0.409704
$$261$$ −7.29767 −0.451714
$$262$$ −18.5953 −1.14882
$$263$$ 4.20975 0.259584 0.129792 0.991541i $$-0.458569\pi$$
0.129792 + 0.991541i $$0.458569\pi$$
$$264$$ 2.22069 0.136674
$$265$$ −9.63841 −0.592083
$$266$$ −38.5900 −2.36611
$$267$$ −2.27649 −0.139319
$$268$$ 1.49436 0.0912825
$$269$$ 12.5071 0.762570 0.381285 0.924457i $$-0.375482\pi$$
0.381285 + 0.924457i $$0.375482\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −6.70693 −0.407417 −0.203709 0.979032i $$-0.565299\pi$$
−0.203709 + 0.979032i $$0.565299\pi$$
$$272$$ 5.63841 0.341879
$$273$$ −33.9086 −2.05224
$$274$$ −19.5074 −1.17849
$$275$$ 2.22069 0.133913
$$276$$ −6.10064 −0.367216
$$277$$ −27.0477 −1.62514 −0.812569 0.582866i $$-0.801931\pi$$
−0.812569 + 0.582866i $$0.801931\pi$$
$$278$$ 10.9036 0.653956
$$279$$ −0.967873 −0.0579451
$$280$$ 5.13277 0.306742
$$281$$ −1.39371 −0.0831420 −0.0415710 0.999136i $$-0.513236\pi$$
−0.0415710 + 0.999136i $$0.513236\pi$$
$$282$$ −5.85911 −0.348905
$$283$$ −15.3132 −0.910276 −0.455138 0.890421i $$-0.650410\pi$$
−0.455138 + 0.890421i $$0.650410\pi$$
$$284$$ 0.329796 0.0195698
$$285$$ −7.51836 −0.445349
$$286$$ −14.6705 −0.867487
$$287$$ −5.29767 −0.312712
$$288$$ 1.00000 0.0589256
$$289$$ 14.7917 0.870100
$$290$$ −7.29767 −0.428534
$$291$$ 17.5982 1.03162
$$292$$ 12.8718 0.753267
$$293$$ −12.8672 −0.751712 −0.375856 0.926678i $$-0.622651\pi$$
−0.375856 + 0.926678i $$0.622651\pi$$
$$294$$ 19.3453 1.12824
$$295$$ −10.2655 −0.597683
$$296$$ −1.00000 −0.0581238
$$297$$ 2.22069 0.128858
$$298$$ −12.6720 −0.734068
$$299$$ 40.3026 2.33076
$$300$$ 1.00000 0.0577350
$$301$$ 27.1917 1.56731
$$302$$ 10.5420 0.606626
$$303$$ 7.61901 0.437701
$$304$$ −7.51836 −0.431208
$$305$$ 6.32134 0.361959
$$306$$ 5.63841 0.322327
$$307$$ −4.08970 −0.233411 −0.116706 0.993167i $$-0.537233\pi$$
−0.116706 + 0.993167i $$0.537233\pi$$
$$308$$ 11.3983 0.649479
$$309$$ 5.85911 0.333313
$$310$$ −0.967873 −0.0549715
$$311$$ −10.3090 −0.584567 −0.292283 0.956332i $$-0.594415\pi$$
−0.292283 + 0.956332i $$0.594415\pi$$
$$312$$ −6.60629 −0.374007
$$313$$ −2.64653 −0.149591 −0.0747955 0.997199i $$-0.523830\pi$$
−0.0747955 + 0.997199i $$0.523830\pi$$
$$314$$ −15.4990 −0.874657
$$315$$ 5.13277 0.289199
$$316$$ 4.00000 0.225018
$$317$$ −8.06885 −0.453192 −0.226596 0.973989i $$-0.572760\pi$$
−0.226596 + 0.973989i $$0.572760\pi$$
$$318$$ −9.63841 −0.540495
$$319$$ −16.2059 −0.907356
$$320$$ 1.00000 0.0559017
$$321$$ 5.65926 0.315869
$$322$$ −31.3132 −1.74502
$$323$$ −42.3916 −2.35873
$$324$$ 1.00000 0.0555556
$$325$$ −6.60629 −0.366451
$$326$$ −4.12149 −0.228268
$$327$$ −3.16772 −0.175175
$$328$$ −1.03213 −0.0569897
$$329$$ −30.0735 −1.65800
$$330$$ 2.22069 0.122245
$$331$$ 19.6833 1.08189 0.540945 0.841058i $$-0.318067\pi$$
0.540945 + 0.841058i $$0.318067\pi$$
$$332$$ 9.04767 0.496556
$$333$$ −1.00000 −0.0547997
$$334$$ 21.9474 1.20091
$$335$$ 1.49436 0.0816456
$$336$$ 5.13277 0.280016
$$337$$ −23.4891 −1.27953 −0.639765 0.768570i $$-0.720968\pi$$
−0.639765 + 0.768570i $$0.720968\pi$$
$$338$$ 30.6430 1.66676
$$339$$ −0.702331 −0.0381454
$$340$$ 5.63841 0.305786
$$341$$ −2.14935 −0.116394
$$342$$ −7.51836 −0.406547
$$343$$ 63.3658 3.42143
$$344$$ 5.29767 0.285631
$$345$$ −6.10064 −0.328448
$$346$$ −0.425841 −0.0228933
$$347$$ 18.7069 1.00424 0.502120 0.864798i $$-0.332553\pi$$
0.502120 + 0.864798i $$0.332553\pi$$
$$348$$ −7.29767 −0.391196
$$349$$ 16.1123 0.862470 0.431235 0.902240i $$-0.358078\pi$$
0.431235 + 0.902240i $$0.358078\pi$$
$$350$$ 5.13277 0.274358
$$351$$ −6.60629 −0.352617
$$352$$ 2.22069 0.118363
$$353$$ 2.37320 0.126313 0.0631565 0.998004i $$-0.479883\pi$$
0.0631565 + 0.998004i $$0.479883\pi$$
$$354$$ −10.2655 −0.545607
$$355$$ 0.329796 0.0175038
$$356$$ −2.27649 −0.120654
$$357$$ 28.9407 1.53170
$$358$$ −17.5424 −0.927143
$$359$$ −18.5953 −0.981424 −0.490712 0.871322i $$-0.663263\pi$$
−0.490712 + 0.871322i $$0.663263\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 37.5258 1.97504
$$362$$ 18.5311 0.973972
$$363$$ −6.06852 −0.318515
$$364$$ −33.9086 −1.77729
$$365$$ 12.8718 0.673742
$$366$$ 6.32134 0.330422
$$367$$ −29.9520 −1.56348 −0.781740 0.623605i $$-0.785668\pi$$
−0.781740 + 0.623605i $$0.785668\pi$$
$$368$$ −6.10064 −0.318018
$$369$$ −1.03213 −0.0537304
$$370$$ −1.00000 −0.0519875
$$371$$ −49.4718 −2.56845
$$372$$ −0.967873 −0.0501819
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ 12.5212 0.647455
$$375$$ 1.00000 0.0516398
$$376$$ −5.85911 −0.302160
$$377$$ 48.2105 2.48297
$$378$$ 5.13277 0.264001
$$379$$ 23.3185 1.19779 0.598896 0.800827i $$-0.295606\pi$$
0.598896 + 0.800827i $$0.295606\pi$$
$$380$$ −7.51836 −0.385684
$$381$$ −19.3132 −0.989446
$$382$$ 16.1695 0.827304
$$383$$ −16.6063 −0.848542 −0.424271 0.905535i $$-0.639470\pi$$
−0.424271 + 0.905535i $$0.639470\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 11.3983 0.580912
$$386$$ −11.8591 −0.603613
$$387$$ 5.29767 0.269296
$$388$$ 17.5982 0.893411
$$389$$ 15.5795 0.789910 0.394955 0.918701i $$-0.370760\pi$$
0.394955 + 0.918701i $$0.370760\pi$$
$$390$$ −6.60629 −0.334522
$$391$$ −34.3980 −1.73958
$$392$$ 19.3453 0.977087
$$393$$ −18.5953 −0.938011
$$394$$ 0.100645 0.00507042
$$395$$ 4.00000 0.201262
$$396$$ 2.22069 0.111594
$$397$$ −23.2126 −1.16501 −0.582503 0.812829i $$-0.697926\pi$$
−0.582503 + 0.812829i $$0.697926\pi$$
$$398$$ −2.70693 −0.135686
$$399$$ −38.5900 −1.93192
$$400$$ 1.00000 0.0500000
$$401$$ 5.72351 0.285818 0.142909 0.989736i $$-0.454354\pi$$
0.142909 + 0.989736i $$0.454354\pi$$
$$402$$ 1.49436 0.0745319
$$403$$ 6.39405 0.318510
$$404$$ 7.61901 0.379060
$$405$$ 1.00000 0.0496904
$$406$$ −37.4573 −1.85897
$$407$$ −2.22069 −0.110076
$$408$$ 5.63841 0.279143
$$409$$ 28.0254 1.38577 0.692885 0.721049i $$-0.256340\pi$$
0.692885 + 0.721049i $$0.256340\pi$$
$$410$$ −1.03213 −0.0509731
$$411$$ −19.5074 −0.962230
$$412$$ 5.85911 0.288657
$$413$$ −52.6907 −2.59274
$$414$$ −6.10064 −0.299830
$$415$$ 9.04767 0.444133
$$416$$ −6.60629 −0.323900
$$417$$ 10.9036 0.533953
$$418$$ −16.6960 −0.816627
$$419$$ 16.2415 0.793451 0.396726 0.917937i $$-0.370146\pi$$
0.396726 + 0.917937i $$0.370146\pi$$
$$420$$ 5.13277 0.250454
$$421$$ 6.03495 0.294126 0.147063 0.989127i $$-0.453018\pi$$
0.147063 + 0.989127i $$0.453018\pi$$
$$422$$ 11.8033 0.574576
$$423$$ −5.85911 −0.284879
$$424$$ −9.63841 −0.468083
$$425$$ 5.63841 0.273503
$$426$$ 0.329796 0.0159787
$$427$$ 32.4460 1.57017
$$428$$ 5.65926 0.273551
$$429$$ −14.6705 −0.708300
$$430$$ 5.29767 0.255476
$$431$$ −33.2924 −1.60364 −0.801819 0.597568i $$-0.796134\pi$$
−0.801819 + 0.597568i $$0.796134\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 8.67054 0.416680 0.208340 0.978057i $$-0.433194\pi$$
0.208340 + 0.978057i $$0.433194\pi$$
$$434$$ −4.96787 −0.238466
$$435$$ −7.29767 −0.349896
$$436$$ −3.16772 −0.151706
$$437$$ 45.8669 2.19411
$$438$$ 12.8718 0.615040
$$439$$ 32.8874 1.56963 0.784814 0.619731i $$-0.212758\pi$$
0.784814 + 0.619731i $$0.212758\pi$$
$$440$$ 2.22069 0.105867
$$441$$ 19.3453 0.921207
$$442$$ −37.2490 −1.77175
$$443$$ 14.8185 0.704049 0.352025 0.935991i $$-0.385493\pi$$
0.352025 + 0.935991i $$0.385493\pi$$
$$444$$ −1.00000 −0.0474579
$$445$$ −2.27649 −0.107916
$$446$$ 13.1692 0.623578
$$447$$ −12.6720 −0.599364
$$448$$ 5.13277 0.242501
$$449$$ 9.22882 0.435535 0.217767 0.976001i $$-0.430123\pi$$
0.217767 + 0.976001i $$0.430123\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ −2.29204 −0.107928
$$452$$ −0.702331 −0.0330348
$$453$$ 10.5420 0.495308
$$454$$ 5.57946 0.261857
$$455$$ −33.9086 −1.58966
$$456$$ −7.51836 −0.352080
$$457$$ −36.4753 −1.70624 −0.853121 0.521713i $$-0.825293\pi$$
−0.853121 + 0.521713i $$0.825293\pi$$
$$458$$ −1.75990 −0.0822348
$$459$$ 5.63841 0.263179
$$460$$ −6.10064 −0.284444
$$461$$ 26.6642 1.24188 0.620938 0.783860i $$-0.286752\pi$$
0.620938 + 0.783860i $$0.286752\pi$$
$$462$$ 11.3983 0.530298
$$463$$ 6.25249 0.290578 0.145289 0.989389i $$-0.453589\pi$$
0.145289 + 0.989389i $$0.453589\pi$$
$$464$$ −7.29767 −0.338786
$$465$$ −0.967873 −0.0448841
$$466$$ −6.07664 −0.281495
$$467$$ 16.1748 0.748480 0.374240 0.927332i $$-0.377904\pi$$
0.374240 + 0.927332i $$0.377904\pi$$
$$468$$ −6.60629 −0.305376
$$469$$ 7.67020 0.354177
$$470$$ −5.85911 −0.270260
$$471$$ −15.4990 −0.714154
$$472$$ −10.2655 −0.472510
$$473$$ 11.7645 0.540932
$$474$$ 4.00000 0.183726
$$475$$ −7.51836 −0.344966
$$476$$ 28.9407 1.32649
$$477$$ −9.63841 −0.441313
$$478$$ 13.9403 0.637617
$$479$$ 7.72351 0.352896 0.176448 0.984310i $$-0.443539\pi$$
0.176448 + 0.984310i $$0.443539\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 6.60629 0.301221
$$482$$ −3.89406 −0.177369
$$483$$ −31.3132 −1.42480
$$484$$ −6.06852 −0.275842
$$485$$ 17.5982 0.799091
$$486$$ 1.00000 0.0453609
$$487$$ 35.8429 1.62420 0.812098 0.583522i $$-0.198326\pi$$
0.812098 + 0.583522i $$0.198326\pi$$
$$488$$ 6.32134 0.286154
$$489$$ −4.12149 −0.186380
$$490$$ 19.3453 0.873934
$$491$$ 2.74718 0.123978 0.0619892 0.998077i $$-0.480256\pi$$
0.0619892 + 0.998077i $$0.480256\pi$$
$$492$$ −1.03213 −0.0465319
$$493$$ −41.1473 −1.85318
$$494$$ 49.6685 2.23469
$$495$$ 2.22069 0.0998128
$$496$$ −0.967873 −0.0434588
$$497$$ 1.69277 0.0759310
$$498$$ 9.04767 0.405436
$$499$$ −3.07698 −0.137744 −0.0688722 0.997625i $$-0.521940\pi$$
−0.0688722 + 0.997625i $$0.521940\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 21.9474 0.980536
$$502$$ 20.9308 0.934187
$$503$$ 8.88278 0.396063 0.198032 0.980196i $$-0.436545\pi$$
0.198032 + 0.980196i $$0.436545\pi$$
$$504$$ 5.13277 0.228632
$$505$$ 7.61901 0.339041
$$506$$ −13.5477 −0.602267
$$507$$ 30.6430 1.36090
$$508$$ −19.3132 −0.856885
$$509$$ 2.32836 0.103203 0.0516013 0.998668i $$-0.483568\pi$$
0.0516013 + 0.998668i $$0.483568\pi$$
$$510$$ 5.63841 0.249673
$$511$$ 66.0682 2.92268
$$512$$ 1.00000 0.0441942
$$513$$ −7.51836 −0.331944
$$514$$ −4.43044 −0.195418
$$515$$ 5.85911 0.258183
$$516$$ 5.29767 0.233217
$$517$$ −13.0113 −0.572236
$$518$$ −5.13277 −0.225521
$$519$$ −0.425841 −0.0186923
$$520$$ −6.60629 −0.289705
$$521$$ −26.3344 −1.15373 −0.576865 0.816839i $$-0.695724\pi$$
−0.576865 + 0.816839i $$0.695724\pi$$
$$522$$ −7.29767 −0.319410
$$523$$ 10.9887 0.480503 0.240252 0.970711i $$-0.422770\pi$$
0.240252 + 0.970711i $$0.422770\pi$$
$$524$$ −18.5953 −0.812341
$$525$$ 5.13277 0.224013
$$526$$ 4.20975 0.183554
$$527$$ −5.45727 −0.237722
$$528$$ 2.22069 0.0966433
$$529$$ 14.2179 0.618168
$$530$$ −9.63841 −0.418666
$$531$$ −10.2655 −0.445487
$$532$$ −38.5900 −1.67309
$$533$$ 6.81852 0.295343
$$534$$ −2.27649 −0.0985134
$$535$$ 5.65926 0.244671
$$536$$ 1.49436 0.0645465
$$537$$ −17.5424 −0.757009
$$538$$ 12.5071 0.539219
$$539$$ 42.9601 1.85042
$$540$$ 1.00000 0.0430331
$$541$$ −6.61934 −0.284588 −0.142294 0.989824i $$-0.545448\pi$$
−0.142294 + 0.989824i $$0.545448\pi$$
$$542$$ −6.70693 −0.288087
$$543$$ 18.5311 0.795245
$$544$$ 5.63841 0.241745
$$545$$ −3.16772 −0.135690
$$546$$ −33.9086 −1.45115
$$547$$ −23.1801 −0.991110 −0.495555 0.868577i $$-0.665035\pi$$
−0.495555 + 0.868577i $$0.665035\pi$$
$$548$$ −19.5074 −0.833316
$$549$$ 6.32134 0.269788
$$550$$ 2.22069 0.0946907
$$551$$ 54.8665 2.33739
$$552$$ −6.10064 −0.259661
$$553$$ 20.5311 0.873071
$$554$$ −27.0477 −1.14915
$$555$$ −1.00000 −0.0424476
$$556$$ 10.9036 0.462417
$$557$$ 11.3411 0.480537 0.240268 0.970706i $$-0.422765\pi$$
0.240268 + 0.970706i $$0.422765\pi$$
$$558$$ −0.967873 −0.0409734
$$559$$ −34.9979 −1.48025
$$560$$ 5.13277 0.216899
$$561$$ 12.5212 0.528645
$$562$$ −1.39371 −0.0587903
$$563$$ 33.3705 1.40640 0.703198 0.710994i $$-0.251755\pi$$
0.703198 + 0.710994i $$0.251755\pi$$
$$564$$ −5.85911 −0.246713
$$565$$ −0.702331 −0.0295473
$$566$$ −15.3132 −0.643663
$$567$$ 5.13277 0.215556
$$568$$ 0.329796 0.0138379
$$569$$ −0.500343 −0.0209755 −0.0104877 0.999945i $$-0.503338\pi$$
−0.0104877 + 0.999945i $$0.503338\pi$$
$$570$$ −7.51836 −0.314910
$$571$$ 27.5463 1.15278 0.576388 0.817176i $$-0.304462\pi$$
0.576388 + 0.817176i $$0.304462\pi$$
$$572$$ −14.6705 −0.613406
$$573$$ 16.1695 0.675490
$$574$$ −5.29767 −0.221120
$$575$$ −6.10064 −0.254414
$$576$$ 1.00000 0.0416667
$$577$$ 12.3902 0.515810 0.257905 0.966170i $$-0.416968\pi$$
0.257905 + 0.966170i $$0.416968\pi$$
$$578$$ 14.7917 0.615253
$$579$$ −11.8591 −0.492848
$$580$$ −7.29767 −0.303019
$$581$$ 46.4396 1.92664
$$582$$ 17.5982 0.729467
$$583$$ −21.4040 −0.886462
$$584$$ 12.8718 0.532640
$$585$$ −6.60629 −0.273136
$$586$$ −12.8672 −0.531540
$$587$$ 0.808273 0.0333610 0.0166805 0.999861i $$-0.494690\pi$$
0.0166805 + 0.999861i $$0.494690\pi$$
$$588$$ 19.3453 0.797789
$$589$$ 7.27682 0.299836
$$590$$ −10.2655 −0.422626
$$591$$ 0.100645 0.00413998
$$592$$ −1.00000 −0.0410997
$$593$$ −24.4121 −1.00248 −0.501242 0.865307i $$-0.667123\pi$$
−0.501242 + 0.865307i $$0.667123\pi$$
$$594$$ 2.22069 0.0911162
$$595$$ 28.9407 1.18645
$$596$$ −12.6720 −0.519065
$$597$$ −2.70693 −0.110787
$$598$$ 40.3026 1.64810
$$599$$ −29.3440 −1.19896 −0.599481 0.800389i $$-0.704626\pi$$
−0.599481 + 0.800389i $$0.704626\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −9.55791 −0.389875 −0.194938 0.980816i $$-0.562450\pi$$
−0.194938 + 0.980816i $$0.562450\pi$$
$$602$$ 27.1917 1.10825
$$603$$ 1.49436 0.0608550
$$604$$ 10.5420 0.428949
$$605$$ −6.06852 −0.246720
$$606$$ 7.61901 0.309501
$$607$$ −31.4912 −1.27819 −0.639094 0.769129i $$-0.720690\pi$$
−0.639094 + 0.769129i $$0.720690\pi$$
$$608$$ −7.51836 −0.304910
$$609$$ −37.4573 −1.51785
$$610$$ 6.32134 0.255944
$$611$$ 38.7069 1.56591
$$612$$ 5.63841 0.227919
$$613$$ 6.41489 0.259095 0.129548 0.991573i $$-0.458648\pi$$
0.129548 + 0.991573i $$0.458648\pi$$
$$614$$ −4.08970 −0.165047
$$615$$ −1.03213 −0.0416194
$$616$$ 11.3983 0.459251
$$617$$ −38.8153 −1.56265 −0.781323 0.624127i $$-0.785455\pi$$
−0.781323 + 0.624127i $$0.785455\pi$$
$$618$$ 5.85911 0.235688
$$619$$ −0.00460032 −0.000184902 0 −9.24512e−5 1.00000i $$-0.500029\pi$$
−9.24512e−5 1.00000i $$0.500029\pi$$
$$620$$ −0.967873 −0.0388707
$$621$$ −6.10064 −0.244810
$$622$$ −10.3090 −0.413351
$$623$$ −11.6847 −0.468138
$$624$$ −6.60629 −0.264463
$$625$$ 1.00000 0.0400000
$$626$$ −2.64653 −0.105777
$$627$$ −16.6960 −0.666773
$$628$$ −15.4990 −0.618476
$$629$$ −5.63841 −0.224818
$$630$$ 5.13277 0.204495
$$631$$ 34.5999 1.37740 0.688701 0.725046i $$-0.258181\pi$$
0.688701 + 0.725046i $$0.258181\pi$$
$$632$$ 4.00000 0.159111
$$633$$ 11.8033 0.469139
$$634$$ −8.06885 −0.320455
$$635$$ −19.3132 −0.766422
$$636$$ −9.63841 −0.382188
$$637$$ −127.801 −5.06365
$$638$$ −16.2059 −0.641597
$$639$$ 0.329796 0.0130465
$$640$$ 1.00000 0.0395285
$$641$$ −40.8711 −1.61431 −0.807156 0.590338i $$-0.798995\pi$$
−0.807156 + 0.590338i $$0.798995\pi$$
$$642$$ 5.65926 0.223353
$$643$$ 11.4788 0.452680 0.226340 0.974048i $$-0.427324\pi$$
0.226340 + 0.974048i $$0.427324\pi$$
$$644$$ −31.3132 −1.23391
$$645$$ 5.29767 0.208596
$$646$$ −42.3916 −1.66788
$$647$$ −17.7490 −0.697783 −0.348892 0.937163i $$-0.613442\pi$$
−0.348892 + 0.937163i $$0.613442\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −22.7966 −0.894846
$$650$$ −6.60629 −0.259120
$$651$$ −4.96787 −0.194706
$$652$$ −4.12149 −0.161410
$$653$$ 21.2768 0.832626 0.416313 0.909221i $$-0.363322\pi$$
0.416313 + 0.909221i $$0.363322\pi$$
$$654$$ −3.16772 −0.123868
$$655$$ −18.5953 −0.726580
$$656$$ −1.03213 −0.0402978
$$657$$ 12.8718 0.502178
$$658$$ −30.0735 −1.17239
$$659$$ 23.5548 0.917563 0.458781 0.888549i $$-0.348286\pi$$
0.458781 + 0.888549i $$0.348286\pi$$
$$660$$ 2.22069 0.0864404
$$661$$ 1.34357 0.0522587 0.0261294 0.999659i $$-0.491682\pi$$
0.0261294 + 0.999659i $$0.491682\pi$$
$$662$$ 19.6833 0.765012
$$663$$ −37.2490 −1.44663
$$664$$ 9.04767 0.351118
$$665$$ −38.5900 −1.49646
$$666$$ −1.00000 −0.0387492
$$667$$ 44.5205 1.72384
$$668$$ 21.9474 0.849169
$$669$$ 13.1692 0.509149
$$670$$ 1.49436 0.0577321
$$671$$ 14.0378 0.541922
$$672$$ 5.13277 0.198001
$$673$$ 17.5562 0.676742 0.338371 0.941013i $$-0.390124\pi$$
0.338371 + 0.941013i $$0.390124\pi$$
$$674$$ −23.4891 −0.904765
$$675$$ 1.00000 0.0384900
$$676$$ 30.6430 1.17858
$$677$$ −19.2073 −0.738196 −0.369098 0.929391i $$-0.620333\pi$$
−0.369098 + 0.929391i $$0.620333\pi$$
$$678$$ −0.702331 −0.0269728
$$679$$ 90.3274 3.46645
$$680$$ 5.63841 0.216223
$$681$$ 5.57946 0.213805
$$682$$ −2.14935 −0.0823029
$$683$$ −34.7532 −1.32980 −0.664898 0.746935i $$-0.731525\pi$$
−0.664898 + 0.746935i $$0.731525\pi$$
$$684$$ −7.51836 −0.287472
$$685$$ −19.5074 −0.745340
$$686$$ 63.3658 2.41932
$$687$$ −1.75990 −0.0671445
$$688$$ 5.29767 0.201972
$$689$$ 63.6741 2.42579
$$690$$ −6.10064 −0.232248
$$691$$ 24.3344 0.925724 0.462862 0.886430i $$-0.346823\pi$$
0.462862 + 0.886430i $$0.346823\pi$$
$$692$$ −0.425841 −0.0161880
$$693$$ 11.3983 0.432986
$$694$$ 18.7069 0.710105
$$695$$ 10.9036 0.413598
$$696$$ −7.29767 −0.276617
$$697$$ −5.81955 −0.220431
$$698$$ 16.1123 0.609858
$$699$$ −6.07664 −0.229840
$$700$$ 5.13277 0.194001
$$701$$ −8.81354 −0.332883 −0.166441 0.986051i $$-0.553228\pi$$
−0.166441 + 0.986051i $$0.553228\pi$$
$$702$$ −6.60629 −0.249338
$$703$$ 7.51836 0.283560
$$704$$ 2.22069 0.0836956
$$705$$ −5.85911 −0.220667
$$706$$ 2.37320 0.0893167
$$707$$ 39.1066 1.47076
$$708$$ −10.2655 −0.385803
$$709$$ −9.07238 −0.340720 −0.170360 0.985382i $$-0.554493\pi$$
−0.170360 + 0.985382i $$0.554493\pi$$
$$710$$ 0.329796 0.0123770
$$711$$ 4.00000 0.150012
$$712$$ −2.27649 −0.0853151
$$713$$ 5.90465 0.221131
$$714$$ 28.9407 1.08308
$$715$$ −14.6705 −0.548647
$$716$$ −17.5424 −0.655589
$$717$$ 13.9403 0.520612
$$718$$ −18.5953 −0.693972
$$719$$ 22.8665 0.852778 0.426389 0.904540i $$-0.359786\pi$$
0.426389 + 0.904540i $$0.359786\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 30.0735 1.11999
$$722$$ 37.5258 1.39657
$$723$$ −3.89406 −0.144822
$$724$$ 18.5311 0.688702
$$725$$ −7.29767 −0.271029
$$726$$ −6.06852 −0.225224
$$727$$ −2.86407 −0.106222 −0.0531112 0.998589i $$-0.516914\pi$$
−0.0531112 + 0.998589i $$0.516914\pi$$
$$728$$ −33.9086 −1.25673
$$729$$ 1.00000 0.0370370
$$730$$ 12.8718 0.476408
$$731$$ 29.8704 1.10480
$$732$$ 6.32134 0.233643
$$733$$ −21.6360 −0.799144 −0.399572 0.916702i $$-0.630841\pi$$
−0.399572 + 0.916702i $$0.630841\pi$$
$$734$$ −29.9520 −1.10555
$$735$$ 19.3453 0.713564
$$736$$ −6.10064 −0.224873
$$737$$ 3.31851 0.122239
$$738$$ −1.03213 −0.0379931
$$739$$ 16.7278 0.615341 0.307671 0.951493i $$-0.400451\pi$$
0.307671 + 0.951493i $$0.400451\pi$$
$$740$$ −1.00000 −0.0367607
$$741$$ 49.6685 1.82462
$$742$$ −49.4718 −1.81617
$$743$$ −21.6624 −0.794717 −0.397358 0.917663i $$-0.630073\pi$$
−0.397358 + 0.917663i $$0.630073\pi$$
$$744$$ −0.967873 −0.0354840
$$745$$ −12.6720 −0.464265
$$746$$ −6.00000 −0.219676
$$747$$ 9.04767 0.331037
$$748$$ 12.5212 0.457820
$$749$$ 29.0477 1.06138
$$750$$ 1.00000 0.0365148
$$751$$ 24.8129 0.905435 0.452717 0.891654i $$-0.350455\pi$$
0.452717 + 0.891654i $$0.350455\pi$$
$$752$$ −5.85911 −0.213660
$$753$$ 20.9308 0.762760
$$754$$ 48.2105 1.75572
$$755$$ 10.5420 0.383664
$$756$$ 5.13277 0.186677
$$757$$ 1.97281 0.0717030 0.0358515 0.999357i $$-0.488586\pi$$
0.0358515 + 0.999357i $$0.488586\pi$$
$$758$$ 23.3185 0.846967
$$759$$ −13.5477 −0.491749
$$760$$ −7.51836 −0.272720
$$761$$ 25.4735 0.923414 0.461707 0.887032i $$-0.347237\pi$$
0.461707 + 0.887032i $$0.347237\pi$$
$$762$$ −19.3132 −0.699644
$$763$$ −16.2592 −0.588622
$$764$$ 16.1695 0.584992
$$765$$ 5.63841 0.203857
$$766$$ −16.6063 −0.600009
$$767$$ 67.8171 2.44873
$$768$$ 1.00000 0.0360844
$$769$$ −41.9675 −1.51339 −0.756694 0.653770i $$-0.773187\pi$$
−0.756694 + 0.653770i $$0.773187\pi$$
$$770$$ 11.3983 0.410767
$$771$$ −4.43044 −0.159558
$$772$$ −11.8591 −0.426819
$$773$$ 46.0410 1.65598 0.827990 0.560743i $$-0.189485\pi$$
0.827990 + 0.560743i $$0.189485\pi$$
$$774$$ 5.29767 0.190421
$$775$$ −0.967873 −0.0347670
$$776$$ 17.5982 0.631737
$$777$$ −5.13277 −0.184137
$$778$$ 15.5795 0.558551
$$779$$ 7.75990 0.278027
$$780$$ −6.60629 −0.236543
$$781$$ 0.732376 0.0262065
$$782$$ −34.3980 −1.23007
$$783$$ −7.29767 −0.260797
$$784$$ 19.3453 0.690905
$$785$$ −15.4990 −0.553182
$$786$$ −18.5953 −0.663274
$$787$$ −8.04801 −0.286881 −0.143440 0.989659i $$-0.545816\pi$$
−0.143440 + 0.989659i $$0.545816\pi$$
$$788$$ 0.100645 0.00358533
$$789$$ 4.20975 0.149871
$$790$$ 4.00000 0.142314
$$791$$ −3.60490 −0.128176
$$792$$ 2.22069 0.0789089
$$793$$ −41.7606 −1.48296
$$794$$ −23.2126 −0.823783
$$795$$ −9.63841 −0.341839
$$796$$ −2.70693 −0.0959446
$$797$$ −26.2662 −0.930397 −0.465198 0.885206i $$-0.654017\pi$$
−0.465198 + 0.885206i $$0.654017\pi$$
$$798$$ −38.5900 −1.36607
$$799$$ −33.0361 −1.16873
$$800$$ 1.00000 0.0353553
$$801$$ −2.27649 −0.0804358
$$802$$ 5.72351 0.202104
$$803$$ 28.5844 1.00872
$$804$$ 1.49436 0.0527020
$$805$$ −31.3132 −1.10365
$$806$$ 6.39405 0.225221
$$807$$ 12.5071 0.440270
$$808$$ 7.61901 0.268036
$$809$$ 13.3301 0.468662 0.234331 0.972157i $$-0.424710\pi$$
0.234331 + 0.972157i $$0.424710\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −30.4251 −1.06837 −0.534186 0.845367i $$-0.679382\pi$$
−0.534186 + 0.845367i $$0.679382\pi$$
$$812$$ −37.4573 −1.31449
$$813$$ −6.70693 −0.235222
$$814$$ −2.22069 −0.0778353
$$815$$ −4.12149 −0.144369
$$816$$ 5.63841 0.197384
$$817$$ −39.8298 −1.39347
$$818$$ 28.0254 0.979887
$$819$$ −33.9086 −1.18486
$$820$$ −1.03213 −0.0360434
$$821$$ −26.2895 −0.917512 −0.458756 0.888562i $$-0.651705\pi$$
−0.458756 + 0.888562i $$0.651705\pi$$
$$822$$ −19.5074 −0.680399
$$823$$ 24.4050 0.850705 0.425352 0.905028i $$-0.360150\pi$$
0.425352 + 0.905028i $$0.360150\pi$$
$$824$$ 5.85911 0.204112
$$825$$ 2.22069 0.0773146
$$826$$ −52.6907 −1.83334
$$827$$ −37.5887 −1.30709 −0.653543 0.756890i $$-0.726718\pi$$
−0.653543 + 0.756890i $$0.726718\pi$$
$$828$$ −6.10064 −0.212012
$$829$$ 36.4481 1.26589 0.632947 0.774195i $$-0.281845\pi$$
0.632947 + 0.774195i $$0.281845\pi$$
$$830$$ 9.04767 0.314049
$$831$$ −27.0477 −0.938273
$$832$$ −6.60629 −0.229032
$$833$$ 109.077 3.77929
$$834$$ 10.9036 0.377561
$$835$$ 21.9474 0.759520
$$836$$ −16.6960 −0.577443
$$837$$ −0.967873 −0.0334546
$$838$$ 16.2415 0.561055
$$839$$ −43.3277 −1.49584 −0.747919 0.663790i $$-0.768947\pi$$
−0.747919 + 0.663790i $$0.768947\pi$$
$$840$$ 5.13277 0.177097
$$841$$ 24.2560 0.836413
$$842$$ 6.03495 0.207978
$$843$$ −1.39371 −0.0480021
$$844$$ 11.8033 0.406287
$$845$$ 30.6430 1.05415
$$846$$ −5.85911 −0.201440
$$847$$ −31.1483 −1.07027
$$848$$ −9.63841 −0.330984
$$849$$ −15.3132 −0.525548
$$850$$ 5.63841 0.193396
$$851$$ 6.10064 0.209127
$$852$$ 0.329796 0.0112986
$$853$$ 17.6430 0.604085 0.302043 0.953294i $$-0.402332\pi$$
0.302043 + 0.953294i $$0.402332\pi$$
$$854$$ 32.4460 1.11028
$$855$$ −7.51836 −0.257123
$$856$$ 5.65926 0.193429
$$857$$ 35.6222 1.21683 0.608415 0.793619i $$-0.291806\pi$$
0.608415 + 0.793619i $$0.291806\pi$$
$$858$$ −14.6705 −0.500844
$$859$$ −5.49292 −0.187416 −0.0937080 0.995600i $$-0.529872\pi$$
−0.0937080 + 0.995600i $$0.529872\pi$$
$$860$$ 5.29767 0.180649
$$861$$ −5.29767 −0.180544
$$862$$ −33.2924 −1.13394
$$863$$ −1.12501 −0.0382958 −0.0191479 0.999817i $$-0.506095\pi$$
−0.0191479 + 0.999817i $$0.506095\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −0.425841 −0.0144790
$$866$$ 8.67054 0.294637
$$867$$ 14.7917 0.502352
$$868$$ −4.96787 −0.168621
$$869$$ 8.88278 0.301328
$$870$$ −7.29767 −0.247414
$$871$$ −9.87216 −0.334506
$$872$$ −3.16772 −0.107273
$$873$$ 17.5982 0.595608
$$874$$ 45.8669 1.55147
$$875$$ 5.13277 0.173519
$$876$$ 12.8718 0.434899
$$877$$ 31.6367 1.06829 0.534147 0.845392i $$-0.320633\pi$$
0.534147 + 0.845392i $$0.320633\pi$$
$$878$$ 32.8874 1.10990
$$879$$ −12.8672 −0.434001
$$880$$ 2.22069 0.0748596
$$881$$ −24.8012 −0.835575 −0.417787 0.908545i $$-0.637194\pi$$
−0.417787 + 0.908545i $$0.637194\pi$$
$$882$$ 19.3453 0.651392
$$883$$ −27.7980 −0.935478 −0.467739 0.883867i $$-0.654931\pi$$
−0.467739 + 0.883867i $$0.654931\pi$$
$$884$$ −37.2490 −1.25282
$$885$$ −10.2655 −0.345072
$$886$$ 14.8185 0.497838
$$887$$ −28.5226 −0.957696 −0.478848 0.877898i $$-0.658946\pi$$
−0.478848 + 0.877898i $$0.658946\pi$$
$$888$$ −1.00000 −0.0335578
$$889$$ −99.1303 −3.32472
$$890$$ −2.27649 −0.0763081
$$891$$ 2.22069 0.0743960
$$892$$ 13.1692 0.440936
$$893$$ 44.0509 1.47411
$$894$$ −12.6720 −0.423814
$$895$$ −17.5424 −0.586377
$$896$$ 5.13277 0.171474
$$897$$ 40.3026 1.34566
$$898$$ 9.22882 0.307970
$$899$$ 7.06322 0.235572
$$900$$ 1.00000 0.0333333
$$901$$ −54.3453 −1.81051
$$902$$ −2.29204 −0.0763165
$$903$$ 27.1917 0.904884
$$904$$ −0.702331 −0.0233592
$$905$$ 18.5311 0.615994
$$906$$ 10.5420 0.350236
$$907$$ 41.1854 1.36754 0.683769 0.729698i $$-0.260340\pi$$
0.683769 + 0.729698i $$0.260340\pi$$
$$908$$ 5.57946 0.185161
$$909$$ 7.61901 0.252707
$$910$$ −33.9086 −1.12406
$$911$$ 21.3440 0.707157 0.353578 0.935405i $$-0.384965\pi$$
0.353578 + 0.935405i $$0.384965\pi$$
$$912$$ −7.51836 −0.248958
$$913$$ 20.0921 0.664952
$$914$$ −36.4753 −1.20650
$$915$$ 6.32134 0.208977
$$916$$ −1.75990 −0.0581488
$$917$$ −95.4456 −3.15189
$$918$$ 5.63841 0.186095
$$919$$ 36.9089 1.21751 0.608756 0.793358i $$-0.291669\pi$$
0.608756 + 0.793358i $$0.291669\pi$$
$$920$$ −6.10064 −0.201132
$$921$$ −4.08970 −0.134760
$$922$$ 26.6642 0.878138
$$923$$ −2.17873 −0.0717137
$$924$$ 11.3983 0.374977
$$925$$ −1.00000 −0.0328798
$$926$$ 6.25249 0.205469
$$927$$ 5.85911 0.192438
$$928$$ −7.29767 −0.239558
$$929$$ 21.0350 0.690136 0.345068 0.938578i $$-0.387856\pi$$
0.345068 + 0.938578i $$0.387856\pi$$
$$930$$ −0.967873 −0.0317378
$$931$$ −145.445 −4.76678
$$932$$ −6.07664 −0.199047
$$933$$ −10.3090 −0.337500
$$934$$ 16.1748 0.529255
$$935$$ 12.5212 0.409487
$$936$$ −6.60629 −0.215933
$$937$$ 9.67652 0.316118 0.158059 0.987430i $$-0.449476\pi$$
0.158059 + 0.987430i $$0.449476\pi$$
$$938$$ 7.67020 0.250441
$$939$$ −2.64653 −0.0863664
$$940$$ −5.85911 −0.191103
$$941$$ −48.1332 −1.56910 −0.784548 0.620068i $$-0.787105\pi$$
−0.784548 + 0.620068i $$0.787105\pi$$
$$942$$ −15.4990 −0.504983
$$943$$ 6.29664 0.205047
$$944$$ −10.2655 −0.334115
$$945$$ 5.13277 0.166969
$$946$$ 11.7645 0.382497
$$947$$ −17.7871 −0.578002 −0.289001 0.957329i $$-0.593323\pi$$
−0.289001 + 0.957329i $$0.593323\pi$$
$$948$$ 4.00000 0.129914
$$949$$ −85.0350 −2.76035
$$950$$ −7.51836 −0.243928
$$951$$ −8.06885 −0.261650
$$952$$ 28.9407 0.937973
$$953$$ 30.0442 0.973226 0.486613 0.873618i $$-0.338232\pi$$
0.486613 + 0.873618i $$0.338232\pi$$
$$954$$ −9.63841 −0.312055
$$955$$ 16.1695 0.523233
$$956$$ 13.9403 0.450863
$$957$$ −16.2059 −0.523862
$$958$$ 7.72351 0.249535
$$959$$ −100.127 −3.23327
$$960$$ 1.00000 0.0322749
$$961$$ −30.0632 −0.969781
$$962$$ 6.60629 0.212995
$$963$$ 5.65926 0.182367
$$964$$ −3.89406 −0.125419
$$965$$ −11.8591 −0.381758
$$966$$ −31.3132 −1.00749
$$967$$ 15.7052 0.505044 0.252522 0.967591i $$-0.418740\pi$$
0.252522 + 0.967591i $$0.418740\pi$$
$$968$$ −6.06852 −0.195050
$$969$$ −42.3916 −1.36182
$$970$$ 17.5982 0.565043
$$971$$ 36.8693 1.18319 0.591597 0.806234i $$-0.298498\pi$$
0.591597 + 0.806234i $$0.298498\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 55.9658 1.79418
$$974$$ 35.8429 1.14848
$$975$$ −6.60629 −0.211570
$$976$$ 6.32134 0.202341
$$977$$ 31.8404 1.01866 0.509332 0.860570i $$-0.329893\pi$$
0.509332 + 0.860570i $$0.329893\pi$$
$$978$$ −4.12149 −0.131791
$$979$$ −5.05539 −0.161571
$$980$$ 19.3453 0.617964
$$981$$ −3.16772 −0.101138
$$982$$ 2.74718 0.0876660
$$983$$ 11.8708 0.378618 0.189309 0.981918i $$-0.439375\pi$$
0.189309 + 0.981918i $$0.439375\pi$$
$$984$$ −1.03213 −0.0329030
$$985$$ 0.100645 0.00320682
$$986$$ −41.1473 −1.31040
$$987$$ −30.0735 −0.957249
$$988$$ 49.6685 1.58016
$$989$$ −32.3192 −1.02769
$$990$$ 2.22069 0.0705783
$$991$$ −13.9990 −0.444691 −0.222346 0.974968i $$-0.571371\pi$$
−0.222346 + 0.974968i $$0.571371\pi$$
$$992$$ −0.967873 −0.0307300
$$993$$ 19.6833 0.624629
$$994$$ 1.69277 0.0536913
$$995$$ −2.70693 −0.0858155
$$996$$ 9.04767 0.286687
$$997$$ −26.4050 −0.836255 −0.418127 0.908388i $$-0.637313\pi$$
−0.418127 + 0.908388i $$0.637313\pi$$
$$998$$ −3.07698 −0.0974000
$$999$$ −1.00000 −0.0316386
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 1110.2.a.s.1.4 4
3.2 odd 2 3330.2.a.bj.1.4 4
4.3 odd 2 8880.2.a.cg.1.1 4
5.4 even 2 5550.2.a.cj.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.s.1.4 4 1.1 even 1 trivial
3330.2.a.bj.1.4 4 3.2 odd 2
5550.2.a.cj.1.1 4 5.4 even 2
8880.2.a.cg.1.1 4 4.3 odd 2