# Properties

 Label 5070.2.a.bz.1.4 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.131472.2 Defining polynomial: $$x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 52$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-3.64466$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.64466 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.64466 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.66808 q^{11} +1.00000 q^{12} -3.64466 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +6.31274 q^{19} +1.00000 q^{20} +3.64466 q^{21} +1.66808 q^{22} +1.24453 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +3.64466 q^{28} +10.0448 q^{29} -1.00000 q^{30} -4.21957 q^{31} -1.00000 q^{32} -1.66808 q^{33} -4.00000 q^{34} +3.64466 q^{35} +1.00000 q^{36} +9.86423 q^{37} -6.31274 q^{38} -1.00000 q^{40} -9.28932 q^{41} -3.64466 q^{42} -7.57286 q^{43} -1.66808 q^{44} +1.00000 q^{45} -1.24453 q^{46} -6.82522 q^{47} +1.00000 q^{48} +6.28354 q^{49} -1.00000 q^{50} +4.00000 q^{51} -0.848634 q^{53} -1.00000 q^{54} -1.66808 q^{55} -3.64466 q^{56} +6.31274 q^{57} -10.0448 q^{58} -6.10876 q^{59} +1.00000 q^{60} +7.46410 q^{61} +4.21957 q^{62} +3.64466 q^{63} +1.00000 q^{64} +1.66808 q^{66} -14.7534 q^{67} +4.00000 q^{68} +1.24453 q^{69} -3.64466 q^{70} -3.51093 q^{71} -1.00000 q^{72} +12.2175 q^{73} -9.86423 q^{74} +1.00000 q^{75} +6.31274 q^{76} -6.07957 q^{77} +9.93398 q^{79} +1.00000 q^{80} +1.00000 q^{81} +9.28932 q^{82} -7.95317 q^{83} +3.64466 q^{84} +4.00000 q^{85} +7.57286 q^{86} +10.0448 q^{87} +1.66808 q^{88} +5.95162 q^{89} -1.00000 q^{90} +1.24453 q^{92} -4.21957 q^{93} +6.82522 q^{94} +6.31274 q^{95} -1.00000 q^{96} -2.75342 q^{97} -6.28354 q^{98} -1.66808 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 2q^{7} - 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 2q^{7} - 4q^{8} + 4q^{9} - 4q^{10} + 2q^{11} + 4q^{12} + 2q^{14} + 4q^{15} + 4q^{16} + 16q^{17} - 4q^{18} + 4q^{20} - 2q^{21} - 2q^{22} + 4q^{23} - 4q^{24} + 4q^{25} + 4q^{27} - 2q^{28} + 8q^{29} - 4q^{30} - 4q^{31} - 4q^{32} + 2q^{33} - 16q^{34} - 2q^{35} + 4q^{36} + 10q^{37} - 4q^{40} - 4q^{41} + 2q^{42} + 14q^{43} + 2q^{44} + 4q^{45} - 4q^{46} - 8q^{47} + 4q^{48} + 14q^{49} - 4q^{50} + 16q^{51} + 8q^{53} - 4q^{54} + 2q^{55} + 2q^{56} - 8q^{58} + 6q^{59} + 4q^{60} + 16q^{61} + 4q^{62} - 2q^{63} + 4q^{64} - 2q^{66} - 12q^{67} + 16q^{68} + 4q^{69} + 2q^{70} - 16q^{71} - 4q^{72} - 12q^{73} - 10q^{74} + 4q^{75} - 8q^{77} - 10q^{79} + 4q^{80} + 4q^{81} + 4q^{82} - 16q^{83} - 2q^{84} + 16q^{85} - 14q^{86} + 8q^{87} - 2q^{88} + 4q^{89} - 4q^{90} + 4q^{92} - 4q^{93} + 8q^{94} - 4q^{96} + 36q^{97} - 14q^{98} + 2q^{99} + O(q^{100})$$

## 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
$$7$$ 3.64466 1.37755 0.688776 0.724974i $$-0.258148\pi$$
0.688776 + 0.724974i $$0.258148\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −1.66808 −0.502944 −0.251472 0.967865i $$-0.580915\pi$$
−0.251472 + 0.967865i $$0.580915\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ −3.64466 −0.974076
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 6.31274 1.44824 0.724120 0.689674i $$-0.242246\pi$$
0.724120 + 0.689674i $$0.242246\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 3.64466 0.795330
$$22$$ 1.66808 0.355635
$$23$$ 1.24453 0.259503 0.129752 0.991547i $$-0.458582\pi$$
0.129752 + 0.991547i $$0.458582\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 3.64466 0.688776
$$29$$ 10.0448 1.86527 0.932635 0.360821i $$-0.117504\pi$$
0.932635 + 0.360821i $$0.117504\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −4.21957 −0.757857 −0.378928 0.925426i $$-0.623707\pi$$
−0.378928 + 0.925426i $$0.623707\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.66808 −0.290375
$$34$$ −4.00000 −0.685994
$$35$$ 3.64466 0.616060
$$36$$ 1.00000 0.166667
$$37$$ 9.86423 1.62167 0.810835 0.585275i $$-0.199014\pi$$
0.810835 + 0.585275i $$0.199014\pi$$
$$38$$ −6.31274 −1.02406
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ −9.28932 −1.45075 −0.725374 0.688355i $$-0.758333\pi$$
−0.725374 + 0.688355i $$0.758333\pi$$
$$42$$ −3.64466 −0.562383
$$43$$ −7.57286 −1.15485 −0.577425 0.816443i $$-0.695943\pi$$
−0.577425 + 0.816443i $$0.695943\pi$$
$$44$$ −1.66808 −0.251472
$$45$$ 1.00000 0.149071
$$46$$ −1.24453 −0.183496
$$47$$ −6.82522 −0.995560 −0.497780 0.867303i $$-0.665851\pi$$
−0.497780 + 0.867303i $$0.665851\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 6.28354 0.897649
$$50$$ −1.00000 −0.141421
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ −0.848634 −0.116569 −0.0582844 0.998300i $$-0.518563\pi$$
−0.0582844 + 0.998300i $$0.518563\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ −1.66808 −0.224923
$$56$$ −3.64466 −0.487038
$$57$$ 6.31274 0.836142
$$58$$ −10.0448 −1.31895
$$59$$ −6.10876 −0.795293 −0.397646 0.917539i $$-0.630173\pi$$
−0.397646 + 0.917539i $$0.630173\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 7.46410 0.955680 0.477840 0.878447i $$-0.341420\pi$$
0.477840 + 0.878447i $$0.341420\pi$$
$$62$$ 4.21957 0.535886
$$63$$ 3.64466 0.459184
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.66808 0.205326
$$67$$ −14.7534 −1.80242 −0.901209 0.433386i $$-0.857319\pi$$
−0.901209 + 0.433386i $$0.857319\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 1.24453 0.149824
$$70$$ −3.64466 −0.435620
$$71$$ −3.51093 −0.416671 −0.208336 0.978057i $$-0.566805\pi$$
−0.208336 + 0.978057i $$0.566805\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 12.2175 1.42995 0.714976 0.699149i $$-0.246437\pi$$
0.714976 + 0.699149i $$0.246437\pi$$
$$74$$ −9.86423 −1.14669
$$75$$ 1.00000 0.115470
$$76$$ 6.31274 0.724120
$$77$$ −6.07957 −0.692831
$$78$$ 0 0
$$79$$ 9.93398 1.11766 0.558830 0.829282i $$-0.311250\pi$$
0.558830 + 0.829282i $$0.311250\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 9.28932 1.02583
$$83$$ −7.95317 −0.872974 −0.436487 0.899711i $$-0.643777\pi$$
−0.436487 + 0.899711i $$0.643777\pi$$
$$84$$ 3.64466 0.397665
$$85$$ 4.00000 0.433861
$$86$$ 7.57286 0.816603
$$87$$ 10.0448 1.07691
$$88$$ 1.66808 0.177817
$$89$$ 5.95162 0.630870 0.315435 0.948947i $$-0.397850\pi$$
0.315435 + 0.948947i $$0.397850\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ 1.24453 0.129752
$$93$$ −4.21957 −0.437549
$$94$$ 6.82522 0.703967
$$95$$ 6.31274 0.647673
$$96$$ −1.00000 −0.102062
$$97$$ −2.75342 −0.279568 −0.139784 0.990182i $$-0.544641\pi$$
−0.139784 + 0.990182i $$0.544641\pi$$
$$98$$ −6.28354 −0.634734
$$99$$ −1.66808 −0.167648
$$100$$ 1.00000 0.100000
$$101$$ 5.33615 0.530967 0.265483 0.964115i $$-0.414468\pi$$
0.265483 + 0.964115i $$0.414468\pi$$
$$102$$ −4.00000 −0.396059
$$103$$ 7.51248 0.740227 0.370113 0.928987i $$-0.379319\pi$$
0.370113 + 0.928987i $$0.379319\pi$$
$$104$$ 0 0
$$105$$ 3.64466 0.355682
$$106$$ 0.848634 0.0824266
$$107$$ 16.9282 1.63651 0.818256 0.574855i $$-0.194941\pi$$
0.818256 + 0.574855i $$0.194941\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −0.663848 −0.0635851 −0.0317926 0.999494i $$-0.510122\pi$$
−0.0317926 + 0.999494i $$0.510122\pi$$
$$110$$ 1.66808 0.159045
$$111$$ 9.86423 0.936271
$$112$$ 3.64466 0.344388
$$113$$ −17.8700 −1.68107 −0.840534 0.541758i $$-0.817759\pi$$
−0.840534 + 0.541758i $$0.817759\pi$$
$$114$$ −6.31274 −0.591242
$$115$$ 1.24453 0.116053
$$116$$ 10.0448 0.932635
$$117$$ 0 0
$$118$$ 6.10876 0.562357
$$119$$ 14.5786 1.33642
$$120$$ −1.00000 −0.0912871
$$121$$ −8.21752 −0.747047
$$122$$ −7.46410 −0.675768
$$123$$ −9.28932 −0.837590
$$124$$ −4.21957 −0.378928
$$125$$ 1.00000 0.0894427
$$126$$ −3.64466 −0.324692
$$127$$ 14.4407 1.28140 0.640702 0.767790i $$-0.278643\pi$$
0.640702 + 0.767790i $$0.278643\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −7.57286 −0.666753
$$130$$ 0 0
$$131$$ −10.8892 −0.951393 −0.475697 0.879609i $$-0.657804\pi$$
−0.475697 + 0.879609i $$0.657804\pi$$
$$132$$ −1.66808 −0.145187
$$133$$ 23.0078 1.99503
$$134$$ 14.7534 1.27450
$$135$$ 1.00000 0.0860663
$$136$$ −4.00000 −0.342997
$$137$$ −7.03696 −0.601208 −0.300604 0.953749i $$-0.597188\pi$$
−0.300604 + 0.953749i $$0.597188\pi$$
$$138$$ −1.24453 −0.105942
$$139$$ 11.6447 0.987687 0.493844 0.869551i $$-0.335592\pi$$
0.493844 + 0.869551i $$0.335592\pi$$
$$140$$ 3.64466 0.308030
$$141$$ −6.82522 −0.574787
$$142$$ 3.51093 0.294631
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 10.0448 0.834174
$$146$$ −12.2175 −1.01113
$$147$$ 6.28354 0.518258
$$148$$ 9.86423 0.810835
$$149$$ −0.772609 −0.0632946 −0.0316473 0.999499i $$-0.510075\pi$$
−0.0316473 + 0.999499i $$0.510075\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −9.77838 −0.795754 −0.397877 0.917439i $$-0.630253\pi$$
−0.397877 + 0.917439i $$0.630253\pi$$
$$152$$ −6.31274 −0.512030
$$153$$ 4.00000 0.323381
$$154$$ 6.07957 0.489906
$$155$$ −4.21957 −0.338924
$$156$$ 0 0
$$157$$ −12.0135 −0.958786 −0.479393 0.877600i $$-0.659143\pi$$
−0.479393 + 0.877600i $$0.659143\pi$$
$$158$$ −9.93398 −0.790305
$$159$$ −0.848634 −0.0673011
$$160$$ −1.00000 −0.0790569
$$161$$ 4.53590 0.357479
$$162$$ −1.00000 −0.0785674
$$163$$ 10.0916 0.790437 0.395218 0.918587i $$-0.370669\pi$$
0.395218 + 0.918587i $$0.370669\pi$$
$$164$$ −9.28932 −0.725374
$$165$$ −1.66808 −0.129860
$$166$$ 7.95317 0.617285
$$167$$ −7.89701 −0.611089 −0.305545 0.952178i $$-0.598839\pi$$
−0.305545 + 0.952178i $$0.598839\pi$$
$$168$$ −3.64466 −0.281192
$$169$$ 0 0
$$170$$ −4.00000 −0.306786
$$171$$ 6.31274 0.482747
$$172$$ −7.57286 −0.577425
$$173$$ −0.440685 −0.0335047 −0.0167523 0.999860i $$-0.505333\pi$$
−0.0167523 + 0.999860i $$0.505333\pi$$
$$174$$ −10.0448 −0.761493
$$175$$ 3.64466 0.275510
$$176$$ −1.66808 −0.125736
$$177$$ −6.10876 −0.459163
$$178$$ −5.95162 −0.446093
$$179$$ −19.6368 −1.46773 −0.733863 0.679297i $$-0.762285\pi$$
−0.733863 + 0.679297i $$0.762285\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −6.21752 −0.462145 −0.231072 0.972937i $$-0.574223\pi$$
−0.231072 + 0.972937i $$0.574223\pi$$
$$182$$ 0 0
$$183$$ 7.46410 0.551762
$$184$$ −1.24453 −0.0917482
$$185$$ 9.86423 0.725232
$$186$$ 4.21957 0.309394
$$187$$ −6.67230 −0.487927
$$188$$ −6.82522 −0.497780
$$189$$ 3.64466 0.265110
$$190$$ −6.31274 −0.457974
$$191$$ 15.6816 1.13468 0.567341 0.823483i $$-0.307972\pi$$
0.567341 + 0.823483i $$0.307972\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 8.12795 0.585063 0.292531 0.956256i $$-0.405502\pi$$
0.292531 + 0.956256i $$0.405502\pi$$
$$194$$ 2.75342 0.197684
$$195$$ 0 0
$$196$$ 6.28354 0.448825
$$197$$ −0.891239 −0.0634981 −0.0317491 0.999496i $$-0.510108\pi$$
−0.0317491 + 0.999496i $$0.510108\pi$$
$$198$$ 1.66808 0.118545
$$199$$ 0.361116 0.0255988 0.0127994 0.999918i $$-0.495926\pi$$
0.0127994 + 0.999918i $$0.495926\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −14.7534 −1.04063
$$202$$ −5.33615 −0.375450
$$203$$ 36.6098 2.56951
$$204$$ 4.00000 0.280056
$$205$$ −9.28932 −0.648794
$$206$$ −7.51248 −0.523419
$$207$$ 1.24453 0.0865010
$$208$$ 0 0
$$209$$ −10.5301 −0.728384
$$210$$ −3.64466 −0.251505
$$211$$ 2.22739 0.153340 0.0766700 0.997057i $$-0.475571\pi$$
0.0766700 + 0.997057i $$0.475571\pi$$
$$212$$ −0.848634 −0.0582844
$$213$$ −3.51093 −0.240565
$$214$$ −16.9282 −1.15719
$$215$$ −7.57286 −0.516465
$$216$$ −1.00000 −0.0680414
$$217$$ −15.3789 −1.04399
$$218$$ 0.663848 0.0449615
$$219$$ 12.2175 0.825584
$$220$$ −1.66808 −0.112462
$$221$$ 0 0
$$222$$ −9.86423 −0.662044
$$223$$ −6.08380 −0.407401 −0.203701 0.979033i $$-0.565297\pi$$
−0.203701 + 0.979033i $$0.565297\pi$$
$$224$$ −3.64466 −0.243519
$$225$$ 1.00000 0.0666667
$$226$$ 17.8700 1.18869
$$227$$ −15.3205 −1.01686 −0.508429 0.861104i $$-0.669774\pi$$
−0.508429 + 0.861104i $$0.669774\pi$$
$$228$$ 6.31274 0.418071
$$229$$ −22.2644 −1.47127 −0.735635 0.677378i $$-0.763116\pi$$
−0.735635 + 0.677378i $$0.763116\pi$$
$$230$$ −1.24453 −0.0820621
$$231$$ −6.07957 −0.400006
$$232$$ −10.0448 −0.659473
$$233$$ −10.8366 −0.709928 −0.354964 0.934880i $$-0.615507\pi$$
−0.354964 + 0.934880i $$0.615507\pi$$
$$234$$ 0 0
$$235$$ −6.82522 −0.445228
$$236$$ −6.10876 −0.397646
$$237$$ 9.93398 0.645281
$$238$$ −14.5786 −0.944993
$$239$$ −16.4975 −1.06714 −0.533568 0.845757i $$-0.679149\pi$$
−0.533568 + 0.845757i $$0.679149\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ −4.40435 −0.283709 −0.141855 0.989887i $$-0.545307\pi$$
−0.141855 + 0.989887i $$0.545307\pi$$
$$242$$ 8.21752 0.528242
$$243$$ 1.00000 0.0641500
$$244$$ 7.46410 0.477840
$$245$$ 6.28354 0.401441
$$246$$ 9.28932 0.592265
$$247$$ 0 0
$$248$$ 4.21957 0.267943
$$249$$ −7.95317 −0.504011
$$250$$ −1.00000 −0.0632456
$$251$$ 11.9453 0.753984 0.376992 0.926217i $$-0.376959\pi$$
0.376992 + 0.926217i $$0.376959\pi$$
$$252$$ 3.64466 0.229592
$$253$$ −2.07598 −0.130515
$$254$$ −14.4407 −0.906089
$$255$$ 4.00000 0.250490
$$256$$ 1.00000 0.0625000
$$257$$ 19.4621 1.21401 0.607005 0.794698i $$-0.292371\pi$$
0.607005 + 0.794698i $$0.292371\pi$$
$$258$$ 7.57286 0.471466
$$259$$ 35.9518 2.23393
$$260$$ 0 0
$$261$$ 10.0448 0.621757
$$262$$ 10.8892 0.672737
$$263$$ 2.03478 0.125470 0.0627350 0.998030i $$-0.480018\pi$$
0.0627350 + 0.998030i $$0.480018\pi$$
$$264$$ 1.66808 0.102663
$$265$$ −0.848634 −0.0521312
$$266$$ −23.0078 −1.41070
$$267$$ 5.95162 0.364233
$$268$$ −14.7534 −0.901209
$$269$$ 20.5287 1.25166 0.625829 0.779960i $$-0.284761\pi$$
0.625829 + 0.779960i $$0.284761\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 25.5401 1.55145 0.775725 0.631072i $$-0.217385\pi$$
0.775725 + 0.631072i $$0.217385\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ 7.03696 0.425119
$$275$$ −1.66808 −0.100589
$$276$$ 1.24453 0.0749121
$$277$$ 18.0604 1.08514 0.542572 0.840010i $$-0.317451\pi$$
0.542572 + 0.840010i $$0.317451\pi$$
$$278$$ −11.6447 −0.698400
$$279$$ −4.21957 −0.252619
$$280$$ −3.64466 −0.217810
$$281$$ 20.2175 1.20608 0.603038 0.797712i $$-0.293957\pi$$
0.603038 + 0.797712i $$0.293957\pi$$
$$282$$ 6.82522 0.406436
$$283$$ 8.69149 0.516656 0.258328 0.966057i $$-0.416829\pi$$
0.258328 + 0.966057i $$0.416829\pi$$
$$284$$ −3.51093 −0.208336
$$285$$ 6.31274 0.373934
$$286$$ 0 0
$$287$$ −33.8564 −1.99848
$$288$$ −1.00000 −0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ −10.0448 −0.589850
$$291$$ −2.75342 −0.161408
$$292$$ 12.2175 0.714976
$$293$$ −7.54790 −0.440953 −0.220476 0.975392i $$-0.570761\pi$$
−0.220476 + 0.975392i $$0.570761\pi$$
$$294$$ −6.28354 −0.366464
$$295$$ −6.10876 −0.355666
$$296$$ −9.86423 −0.573347
$$297$$ −1.66808 −0.0967916
$$298$$ 0.772609 0.0447560
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ −27.6005 −1.59087
$$302$$ 9.77838 0.562683
$$303$$ 5.33615 0.306554
$$304$$ 6.31274 0.362060
$$305$$ 7.46410 0.427393
$$306$$ −4.00000 −0.228665
$$307$$ 26.0427 1.48634 0.743169 0.669104i $$-0.233322\pi$$
0.743169 + 0.669104i $$0.233322\pi$$
$$308$$ −6.07957 −0.346416
$$309$$ 7.51248 0.427370
$$310$$ 4.21957 0.239655
$$311$$ −25.3789 −1.43910 −0.719552 0.694438i $$-0.755653\pi$$
−0.719552 + 0.694438i $$0.755653\pi$$
$$312$$ 0 0
$$313$$ −31.4600 −1.77822 −0.889112 0.457689i $$-0.848677\pi$$
−0.889112 + 0.457689i $$0.848677\pi$$
$$314$$ 12.0135 0.677964
$$315$$ 3.64466 0.205353
$$316$$ 9.93398 0.558830
$$317$$ 24.7093 1.38781 0.693905 0.720066i $$-0.255889\pi$$
0.693905 + 0.720066i $$0.255889\pi$$
$$318$$ 0.848634 0.0475890
$$319$$ −16.7555 −0.938126
$$320$$ 1.00000 0.0559017
$$321$$ 16.9282 0.944840
$$322$$ −4.53590 −0.252776
$$323$$ 25.2509 1.40500
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −10.0916 −0.558923
$$327$$ −0.663848 −0.0367109
$$328$$ 9.28932 0.512917
$$329$$ −24.8756 −1.37144
$$330$$ 1.66808 0.0918246
$$331$$ −5.60360 −0.308002 −0.154001 0.988071i $$-0.549216\pi$$
−0.154001 + 0.988071i $$0.549216\pi$$
$$332$$ −7.95317 −0.436487
$$333$$ 9.86423 0.540556
$$334$$ 7.89701 0.432105
$$335$$ −14.7534 −0.806065
$$336$$ 3.64466 0.198832
$$337$$ −21.7868 −1.18680 −0.593402 0.804906i $$-0.702216\pi$$
−0.593402 + 0.804906i $$0.702216\pi$$
$$338$$ 0 0
$$339$$ −17.8700 −0.970565
$$340$$ 4.00000 0.216930
$$341$$ 7.03856 0.381159
$$342$$ −6.31274 −0.341354
$$343$$ −2.61124 −0.140994
$$344$$ 7.57286 0.408301
$$345$$ 1.24453 0.0670034
$$346$$ 0.440685 0.0236914
$$347$$ 9.68162 0.519737 0.259868 0.965644i $$-0.416321\pi$$
0.259868 + 0.965644i $$0.416321\pi$$
$$348$$ 10.0448 0.538457
$$349$$ 19.3205 1.03420 0.517102 0.855924i $$-0.327011\pi$$
0.517102 + 0.855924i $$0.327011\pi$$
$$350$$ −3.64466 −0.194815
$$351$$ 0 0
$$352$$ 1.66808 0.0889087
$$353$$ 22.9750 1.22284 0.611419 0.791307i $$-0.290599\pi$$
0.611419 + 0.791307i $$0.290599\pi$$
$$354$$ 6.10876 0.324677
$$355$$ −3.51093 −0.186341
$$356$$ 5.95162 0.315435
$$357$$ 14.5786 0.771583
$$358$$ 19.6368 1.03784
$$359$$ −2.21752 −0.117036 −0.0585182 0.998286i $$-0.518638\pi$$
−0.0585182 + 0.998286i $$0.518638\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 20.8506 1.09740
$$362$$ 6.21752 0.326786
$$363$$ −8.21752 −0.431308
$$364$$ 0 0
$$365$$ 12.2175 0.639494
$$366$$ −7.46410 −0.390155
$$367$$ −6.08112 −0.317432 −0.158716 0.987324i $$-0.550735\pi$$
−0.158716 + 0.987324i $$0.550735\pi$$
$$368$$ 1.24453 0.0648758
$$369$$ −9.28932 −0.483583
$$370$$ −9.86423 −0.512817
$$371$$ −3.09298 −0.160580
$$372$$ −4.21957 −0.218774
$$373$$ 15.6781 0.811780 0.405890 0.913922i $$-0.366962\pi$$
0.405890 + 0.913922i $$0.366962\pi$$
$$374$$ 6.67230 0.345017
$$375$$ 1.00000 0.0516398
$$376$$ 6.82522 0.351984
$$377$$ 0 0
$$378$$ −3.64466 −0.187461
$$379$$ −30.7166 −1.57781 −0.788903 0.614518i $$-0.789350\pi$$
−0.788903 + 0.614518i $$0.789350\pi$$
$$380$$ 6.31274 0.323837
$$381$$ 14.4407 0.739819
$$382$$ −15.6816 −0.802342
$$383$$ 20.0619 1.02512 0.512558 0.858652i $$-0.328698\pi$$
0.512558 + 0.858652i $$0.328698\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ −6.07957 −0.309844
$$386$$ −8.12795 −0.413702
$$387$$ −7.57286 −0.384950
$$388$$ −2.75342 −0.139784
$$389$$ −27.0314 −1.37055 −0.685273 0.728287i $$-0.740317\pi$$
−0.685273 + 0.728287i $$0.740317\pi$$
$$390$$ 0 0
$$391$$ 4.97813 0.251755
$$392$$ −6.28354 −0.317367
$$393$$ −10.8892 −0.549287
$$394$$ 0.891239 0.0449000
$$395$$ 9.93398 0.499833
$$396$$ −1.66808 −0.0838240
$$397$$ 3.73628 0.187518 0.0937592 0.995595i $$-0.470112\pi$$
0.0937592 + 0.995595i $$0.470112\pi$$
$$398$$ −0.361116 −0.0181011
$$399$$ 23.0078 1.15183
$$400$$ 1.00000 0.0500000
$$401$$ 28.0911 1.40280 0.701402 0.712766i $$-0.252558\pi$$
0.701402 + 0.712766i $$0.252558\pi$$
$$402$$ 14.7534 0.735834
$$403$$ 0 0
$$404$$ 5.33615 0.265483
$$405$$ 1.00000 0.0496904
$$406$$ −36.6098 −1.81692
$$407$$ −16.4543 −0.815609
$$408$$ −4.00000 −0.198030
$$409$$ 27.3804 1.35388 0.676938 0.736040i $$-0.263307\pi$$
0.676938 + 0.736040i $$0.263307\pi$$
$$410$$ 9.28932 0.458767
$$411$$ −7.03696 −0.347108
$$412$$ 7.51248 0.370113
$$413$$ −22.2644 −1.09556
$$414$$ −1.24453 −0.0611654
$$415$$ −7.95317 −0.390406
$$416$$ 0 0
$$417$$ 11.6447 0.570241
$$418$$ 10.5301 0.515045
$$419$$ −13.1614 −0.642975 −0.321487 0.946914i $$-0.604183\pi$$
−0.321487 + 0.946914i $$0.604183\pi$$
$$420$$ 3.64466 0.177841
$$421$$ 1.29341 0.0630370 0.0315185 0.999503i $$-0.489966\pi$$
0.0315185 + 0.999503i $$0.489966\pi$$
$$422$$ −2.22739 −0.108428
$$423$$ −6.82522 −0.331853
$$424$$ 0.848634 0.0412133
$$425$$ 4.00000 0.194029
$$426$$ 3.51093 0.170105
$$427$$ 27.2041 1.31650
$$428$$ 16.9282 0.818256
$$429$$ 0 0
$$430$$ 7.57286 0.365196
$$431$$ −12.1279 −0.584183 −0.292091 0.956390i $$-0.594351\pi$$
−0.292091 + 0.956390i $$0.594351\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −2.07802 −0.0998633 −0.0499317 0.998753i $$-0.515900\pi$$
−0.0499317 + 0.998753i $$0.515900\pi$$
$$434$$ 15.3789 0.738210
$$435$$ 10.0448 0.481611
$$436$$ −0.663848 −0.0317926
$$437$$ 7.85641 0.375823
$$438$$ −12.2175 −0.583776
$$439$$ −2.39640 −0.114374 −0.0571869 0.998363i $$-0.518213\pi$$
−0.0571869 + 0.998363i $$0.518213\pi$$
$$440$$ 1.66808 0.0795224
$$441$$ 6.28354 0.299216
$$442$$ 0 0
$$443$$ 21.9959 1.04506 0.522529 0.852622i $$-0.324989\pi$$
0.522529 + 0.852622i $$0.324989\pi$$
$$444$$ 9.86423 0.468136
$$445$$ 5.95162 0.282134
$$446$$ 6.08380 0.288076
$$447$$ −0.772609 −0.0365432
$$448$$ 3.64466 0.172194
$$449$$ −29.2253 −1.37923 −0.689613 0.724178i $$-0.742220\pi$$
−0.689613 + 0.724178i $$0.742220\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 15.4953 0.729645
$$452$$ −17.8700 −0.840534
$$453$$ −9.77838 −0.459429
$$454$$ 15.3205 0.719027
$$455$$ 0 0
$$456$$ −6.31274 −0.295621
$$457$$ 39.6432 1.85443 0.927216 0.374526i $$-0.122195\pi$$
0.927216 + 0.374526i $$0.122195\pi$$
$$458$$ 22.2644 1.04034
$$459$$ 4.00000 0.186704
$$460$$ 1.24453 0.0580266
$$461$$ 16.6758 0.776672 0.388336 0.921518i $$-0.373050\pi$$
0.388336 + 0.921518i $$0.373050\pi$$
$$462$$ 6.07957 0.282847
$$463$$ 32.2175 1.49728 0.748638 0.662979i $$-0.230708\pi$$
0.748638 + 0.662979i $$0.230708\pi$$
$$464$$ 10.0448 0.466318
$$465$$ −4.21957 −0.195678
$$466$$ 10.8366 0.501995
$$467$$ 6.88137 0.318432 0.159216 0.987244i $$-0.449103\pi$$
0.159216 + 0.987244i $$0.449103\pi$$
$$468$$ 0 0
$$469$$ −53.7712 −2.48292
$$470$$ 6.82522 0.314824
$$471$$ −12.0135 −0.553555
$$472$$ 6.10876 0.281179
$$473$$ 12.6321 0.580825
$$474$$ −9.93398 −0.456283
$$475$$ 6.31274 0.289648
$$476$$ 14.5786 0.668211
$$477$$ −0.848634 −0.0388563
$$478$$ 16.4975 0.754579
$$479$$ −18.9709 −0.866805 −0.433402 0.901201i $$-0.642687\pi$$
−0.433402 + 0.901201i $$0.642687\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 4.40435 0.200613
$$483$$ 4.53590 0.206391
$$484$$ −8.21752 −0.373524
$$485$$ −2.75342 −0.125026
$$486$$ −1.00000 −0.0453609
$$487$$ 1.91620 0.0868314 0.0434157 0.999057i $$-0.486176\pi$$
0.0434157 + 0.999057i $$0.486176\pi$$
$$488$$ −7.46410 −0.337884
$$489$$ 10.0916 0.456359
$$490$$ −6.28354 −0.283862
$$491$$ −33.6375 −1.51804 −0.759019 0.651069i $$-0.774321\pi$$
−0.759019 + 0.651069i $$0.774321\pi$$
$$492$$ −9.28932 −0.418795
$$493$$ 40.1791 1.80958
$$494$$ 0 0
$$495$$ −1.66808 −0.0749744
$$496$$ −4.21957 −0.189464
$$497$$ −12.7962 −0.573986
$$498$$ 7.95317 0.356390
$$499$$ 1.82522 0.0817080 0.0408540 0.999165i $$-0.486992\pi$$
0.0408540 + 0.999165i $$0.486992\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −7.89701 −0.352813
$$502$$ −11.9453 −0.533147
$$503$$ −19.9985 −0.891687 −0.445843 0.895111i $$-0.647096\pi$$
−0.445843 + 0.895111i $$0.647096\pi$$
$$504$$ −3.64466 −0.162346
$$505$$ 5.33615 0.237456
$$506$$ 2.07598 0.0922884
$$507$$ 0 0
$$508$$ 14.4407 0.640702
$$509$$ −5.89124 −0.261125 −0.130562 0.991440i $$-0.541678\pi$$
−0.130562 + 0.991440i $$0.541678\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ 44.5287 1.96983
$$512$$ −1.00000 −0.0441942
$$513$$ 6.31274 0.278714
$$514$$ −19.4621 −0.858434
$$515$$ 7.51248 0.331040
$$516$$ −7.57286 −0.333377
$$517$$ 11.3850 0.500711
$$518$$ −35.9518 −1.57963
$$519$$ −0.440685 −0.0193439
$$520$$ 0 0
$$521$$ 32.0370 1.40356 0.701782 0.712391i $$-0.252388\pi$$
0.701782 + 0.712391i $$0.252388\pi$$
$$522$$ −10.0448 −0.439648
$$523$$ 38.7186 1.69305 0.846523 0.532352i $$-0.178692\pi$$
0.846523 + 0.532352i $$0.178692\pi$$
$$524$$ −10.8892 −0.475697
$$525$$ 3.64466 0.159066
$$526$$ −2.03478 −0.0887207
$$527$$ −16.8783 −0.735229
$$528$$ −1.66808 −0.0725937
$$529$$ −21.4511 −0.932658
$$530$$ 0.848634 0.0368623
$$531$$ −6.10876 −0.265098
$$532$$ 23.0078 0.997513
$$533$$ 0 0
$$534$$ −5.95162 −0.257552
$$535$$ 16.9282 0.731870
$$536$$ 14.7534 0.637251
$$537$$ −19.6368 −0.847392
$$538$$ −20.5287 −0.885056
$$539$$ −10.4814 −0.451467
$$540$$ 1.00000 0.0430331
$$541$$ −25.9616 −1.11618 −0.558089 0.829781i $$-0.688465\pi$$
−0.558089 + 0.829781i $$0.688465\pi$$
$$542$$ −25.5401 −1.09704
$$543$$ −6.21752 −0.266819
$$544$$ −4.00000 −0.171499
$$545$$ −0.663848 −0.0284361
$$546$$ 0 0
$$547$$ −17.7596 −0.759348 −0.379674 0.925120i $$-0.623964\pi$$
−0.379674 + 0.925120i $$0.623964\pi$$
$$548$$ −7.03696 −0.300604
$$549$$ 7.46410 0.318560
$$550$$ 1.66808 0.0711270
$$551$$ 63.4101 2.70136
$$552$$ −1.24453 −0.0529708
$$553$$ 36.2060 1.53963
$$554$$ −18.0604 −0.767312
$$555$$ 9.86423 0.418713
$$556$$ 11.6447 0.493844
$$557$$ −26.2202 −1.11099 −0.555493 0.831521i $$-0.687470\pi$$
−0.555493 + 0.831521i $$0.687470\pi$$
$$558$$ 4.21957 0.178629
$$559$$ 0 0
$$560$$ 3.64466 0.154015
$$561$$ −6.67230 −0.281705
$$562$$ −20.2175 −0.852825
$$563$$ −25.9928 −1.09547 −0.547733 0.836653i $$-0.684509\pi$$
−0.547733 + 0.836653i $$0.684509\pi$$
$$564$$ −6.82522 −0.287394
$$565$$ −17.8700 −0.751797
$$566$$ −8.69149 −0.365331
$$567$$ 3.64466 0.153061
$$568$$ 3.51093 0.147316
$$569$$ −25.4699 −1.06775 −0.533876 0.845563i $$-0.679265\pi$$
−0.533876 + 0.845563i $$0.679265\pi$$
$$570$$ −6.31274 −0.264411
$$571$$ 8.44491 0.353409 0.176704 0.984264i $$-0.443456\pi$$
0.176704 + 0.984264i $$0.443456\pi$$
$$572$$ 0 0
$$573$$ 15.6816 0.655109
$$574$$ 33.8564 1.41314
$$575$$ 1.24453 0.0519006
$$576$$ 1.00000 0.0416667
$$577$$ −35.4216 −1.47462 −0.737311 0.675554i $$-0.763905\pi$$
−0.737311 + 0.675554i $$0.763905\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 8.12795 0.337786
$$580$$ 10.0448 0.417087
$$581$$ −28.9866 −1.20257
$$582$$ 2.75342 0.114133
$$583$$ 1.41559 0.0586276
$$584$$ −12.2175 −0.505565
$$585$$ 0 0
$$586$$ 7.54790 0.311801
$$587$$ −27.3789 −1.13005 −0.565024 0.825075i $$-0.691133\pi$$
−0.565024 + 0.825075i $$0.691133\pi$$
$$588$$ 6.28354 0.259129
$$589$$ −26.6370 −1.09756
$$590$$ 6.10876 0.251494
$$591$$ −0.891239 −0.0366607
$$592$$ 9.86423 0.405417
$$593$$ −12.0619 −0.495324 −0.247662 0.968846i $$-0.579662\pi$$
−0.247662 + 0.968846i $$0.579662\pi$$
$$594$$ 1.66808 0.0684420
$$595$$ 14.5786 0.597666
$$596$$ −0.772609 −0.0316473
$$597$$ 0.361116 0.0147795
$$598$$ 0 0
$$599$$ −28.6129 −1.16909 −0.584546 0.811360i $$-0.698727\pi$$
−0.584546 + 0.811360i $$0.698727\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −19.1676 −0.781862 −0.390931 0.920420i $$-0.627847\pi$$
−0.390931 + 0.920420i $$0.627847\pi$$
$$602$$ 27.6005 1.12491
$$603$$ −14.7534 −0.600806
$$604$$ −9.77838 −0.397877
$$605$$ −8.21752 −0.334090
$$606$$ −5.33615 −0.216766
$$607$$ 18.9382 0.768678 0.384339 0.923192i $$-0.374429\pi$$
0.384339 + 0.923192i $$0.374429\pi$$
$$608$$ −6.31274 −0.256015
$$609$$ 36.6098 1.48351
$$610$$ −7.46410 −0.302213
$$611$$ 0 0
$$612$$ 4.00000 0.161690
$$613$$ 17.9694 0.725779 0.362890 0.931832i $$-0.381790\pi$$
0.362890 + 0.931832i $$0.381790\pi$$
$$614$$ −26.0427 −1.05100
$$615$$ −9.28932 −0.374582
$$616$$ 6.07957 0.244953
$$617$$ −18.0151 −0.725260 −0.362630 0.931933i $$-0.618121\pi$$
−0.362630 + 0.931933i $$0.618121\pi$$
$$618$$ −7.51248 −0.302196
$$619$$ −25.0505 −1.00687 −0.503433 0.864035i $$-0.667930\pi$$
−0.503433 + 0.864035i $$0.667930\pi$$
$$620$$ −4.21957 −0.169462
$$621$$ 1.24453 0.0499414
$$622$$ 25.3789 1.01760
$$623$$ 21.6916 0.869057
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 31.4600 1.25739
$$627$$ −10.5301 −0.420533
$$628$$ −12.0135 −0.479393
$$629$$ 39.4569 1.57325
$$630$$ −3.64466 −0.145207
$$631$$ 1.41727 0.0564206 0.0282103 0.999602i $$-0.491019\pi$$
0.0282103 + 0.999602i $$0.491019\pi$$
$$632$$ −9.93398 −0.395152
$$633$$ 2.22739 0.0885308
$$634$$ −24.7093 −0.981330
$$635$$ 14.4407 0.573061
$$636$$ −0.848634 −0.0336505
$$637$$ 0 0
$$638$$ 16.7555 0.663355
$$639$$ −3.51093 −0.138890
$$640$$ −1.00000 −0.0395285
$$641$$ 4.61970 0.182467 0.0912335 0.995830i $$-0.470919\pi$$
0.0912335 + 0.995830i $$0.470919\pi$$
$$642$$ −16.9282 −0.668103
$$643$$ 48.0896 1.89647 0.948234 0.317573i $$-0.102868\pi$$
0.948234 + 0.317573i $$0.102868\pi$$
$$644$$ 4.53590 0.178739
$$645$$ −7.57286 −0.298181
$$646$$ −25.2509 −0.993485
$$647$$ 3.74565 0.147257 0.0736283 0.997286i $$-0.476542\pi$$
0.0736283 + 0.997286i $$0.476542\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 10.1899 0.399988
$$650$$ 0 0
$$651$$ −15.3789 −0.602746
$$652$$ 10.0916 0.395218
$$653$$ 42.2899 1.65493 0.827466 0.561516i $$-0.189782\pi$$
0.827466 + 0.561516i $$0.189782\pi$$
$$654$$ 0.663848 0.0259585
$$655$$ −10.8892 −0.425476
$$656$$ −9.28932 −0.362687
$$657$$ 12.2175 0.476651
$$658$$ 24.8756 0.969752
$$659$$ −29.1750 −1.13650 −0.568248 0.822858i $$-0.692378\pi$$
−0.568248 + 0.822858i $$0.692378\pi$$
$$660$$ −1.66808 −0.0649298
$$661$$ −44.4662 −1.72954 −0.864768 0.502172i $$-0.832535\pi$$
−0.864768 + 0.502172i $$0.832535\pi$$
$$662$$ 5.60360 0.217790
$$663$$ 0 0
$$664$$ 7.95317 0.308643
$$665$$ 23.0078 0.892203
$$666$$ −9.86423 −0.382231
$$667$$ 12.5011 0.484043
$$668$$ −7.89701 −0.305545
$$669$$ −6.08380 −0.235213
$$670$$ 14.7534 0.569974
$$671$$ −12.4507 −0.480654
$$672$$ −3.64466 −0.140596
$$673$$ 7.90633 0.304767 0.152383 0.988321i $$-0.451305\pi$$
0.152383 + 0.988321i $$0.451305\pi$$
$$674$$ 21.7868 0.839198
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −7.05615 −0.271190 −0.135595 0.990764i $$-0.543295\pi$$
−0.135595 + 0.990764i $$0.543295\pi$$
$$678$$ 17.8700 0.686293
$$679$$ −10.0353 −0.385119
$$680$$ −4.00000 −0.153393
$$681$$ −15.3205 −0.587083
$$682$$ −7.03856 −0.269520
$$683$$ 5.95317 0.227792 0.113896 0.993493i $$-0.463667\pi$$
0.113896 + 0.993493i $$0.463667\pi$$
$$684$$ 6.31274 0.241373
$$685$$ −7.03696 −0.268869
$$686$$ 2.61124 0.0996976
$$687$$ −22.2644 −0.849438
$$688$$ −7.57286 −0.288713
$$689$$ 0 0
$$690$$ −1.24453 −0.0473786
$$691$$ −18.7591 −0.713628 −0.356814 0.934175i $$-0.616137\pi$$
−0.356814 + 0.934175i $$0.616137\pi$$
$$692$$ −0.440685 −0.0167523
$$693$$ −6.07957 −0.230944
$$694$$ −9.68162 −0.367509
$$695$$ 11.6447 0.441707
$$696$$ −10.0448 −0.380747
$$697$$ −37.1573 −1.40743
$$698$$ −19.3205 −0.731292
$$699$$ −10.8366 −0.409877
$$700$$ 3.64466 0.137755
$$701$$ 28.5298 1.07755 0.538777 0.842448i $$-0.318887\pi$$
0.538777 + 0.842448i $$0.318887\pi$$
$$702$$ 0 0
$$703$$ 62.2703 2.34857
$$704$$ −1.66808 −0.0628680
$$705$$ −6.82522 −0.257053
$$706$$ −22.9750 −0.864677
$$707$$ 19.4485 0.731435
$$708$$ −6.10876 −0.229581
$$709$$ 23.1926 0.871015 0.435507 0.900185i $$-0.356569\pi$$
0.435507 + 0.900185i $$0.356569\pi$$
$$710$$ 3.51093 0.131763
$$711$$ 9.93398 0.372553
$$712$$ −5.95162 −0.223046
$$713$$ −5.25139 −0.196666
$$714$$ −14.5786 −0.545592
$$715$$ 0 0
$$716$$ −19.6368 −0.733863
$$717$$ −16.4975 −0.616111
$$718$$ 2.21752 0.0827572
$$719$$ 11.7128 0.436814 0.218407 0.975858i $$-0.429914\pi$$
0.218407 + 0.975858i $$0.429914\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 27.3804 1.01970
$$722$$ −20.8506 −0.775980
$$723$$ −4.40435 −0.163800
$$724$$ −6.21752 −0.231072
$$725$$ 10.0448 0.373054
$$726$$ 8.21752 0.304981
$$727$$ −3.82677 −0.141927 −0.0709634 0.997479i $$-0.522607\pi$$
−0.0709634 + 0.997479i $$0.522607\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −12.2175 −0.452191
$$731$$ −30.2915 −1.12037
$$732$$ 7.46410 0.275881
$$733$$ 12.9340 0.477727 0.238864 0.971053i $$-0.423225\pi$$
0.238864 + 0.971053i $$0.423225\pi$$
$$734$$ 6.08112 0.224458
$$735$$ 6.28354 0.231772
$$736$$ −1.24453 −0.0458741
$$737$$ 24.6098 0.906515
$$738$$ 9.28932 0.341945
$$739$$ −35.3462 −1.30023 −0.650115 0.759836i $$-0.725279\pi$$
−0.650115 + 0.759836i $$0.725279\pi$$
$$740$$ 9.86423 0.362616
$$741$$ 0 0
$$742$$ 3.09298 0.113547
$$743$$ 37.0953 1.36090 0.680448 0.732796i $$-0.261785\pi$$
0.680448 + 0.732796i $$0.261785\pi$$
$$744$$ 4.21957 0.154697
$$745$$ −0.772609 −0.0283062
$$746$$ −15.6781 −0.574015
$$747$$ −7.95317 −0.290991
$$748$$ −6.67230 −0.243964
$$749$$ 61.6975 2.25438
$$750$$ −1.00000 −0.0365148
$$751$$ −9.65807 −0.352428 −0.176214 0.984352i $$-0.556385\pi$$
−0.176214 + 0.984352i $$0.556385\pi$$
$$752$$ −6.82522 −0.248890
$$753$$ 11.9453 0.435313
$$754$$ 0 0
$$755$$ −9.77838 −0.355872
$$756$$ 3.64466 0.132555
$$757$$ −43.6885 −1.58789 −0.793943 0.607992i $$-0.791975\pi$$
−0.793943 + 0.607992i $$0.791975\pi$$
$$758$$ 30.7166 1.11568
$$759$$ −2.07598 −0.0753531
$$760$$ −6.31274 −0.228987
$$761$$ −39.9007 −1.44640 −0.723200 0.690639i $$-0.757329\pi$$
−0.723200 + 0.690639i $$0.757329\pi$$
$$762$$ −14.4407 −0.523131
$$763$$ −2.41950 −0.0875918
$$764$$ 15.6816 0.567341
$$765$$ 4.00000 0.144620
$$766$$ −20.0619 −0.724867
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 17.5588 0.633187 0.316594 0.948561i $$-0.397461\pi$$
0.316594 + 0.948561i $$0.397461\pi$$
$$770$$ 6.07957 0.219092
$$771$$ 19.4621 0.700909
$$772$$ 8.12795 0.292531
$$773$$ 12.8372 0.461723 0.230861 0.972987i $$-0.425846\pi$$
0.230861 + 0.972987i $$0.425846\pi$$
$$774$$ 7.57286 0.272201
$$775$$ −4.21957 −0.151571
$$776$$ 2.75342 0.0988420
$$777$$ 35.9518 1.28976
$$778$$ 27.0314 0.969122
$$779$$ −58.6410 −2.10103
$$780$$ 0 0
$$781$$ 5.85651 0.209562
$$782$$ −4.97813 −0.178018
$$783$$ 10.0448 0.358971
$$784$$ 6.28354 0.224412
$$785$$ −12.0135 −0.428782
$$786$$ 10.8892 0.388405
$$787$$ 6.95735 0.248003 0.124001 0.992282i $$-0.460427\pi$$
0.124001 + 0.992282i $$0.460427\pi$$
$$788$$ −0.891239 −0.0317491
$$789$$ 2.03478 0.0724402
$$790$$ −9.93398 −0.353435
$$791$$ −65.1301 −2.31576
$$792$$ 1.66808 0.0592725
$$793$$ 0 0
$$794$$ −3.73628 −0.132596
$$795$$ −0.848634 −0.0300979
$$796$$ 0.361116 0.0127994
$$797$$ 34.1354 1.20914 0.604569 0.796553i $$-0.293345\pi$$
0.604569 + 0.796553i $$0.293345\pi$$
$$798$$ −23.0078 −0.814466
$$799$$ −27.3009 −0.965835
$$800$$ −1.00000 −0.0353553
$$801$$ 5.95162 0.210290
$$802$$ −28.0911 −0.991932
$$803$$ −20.3798 −0.719186
$$804$$ −14.7534 −0.520313
$$805$$ 4.53590 0.159869
$$806$$ 0 0
$$807$$ 20.5287 0.722645
$$808$$ −5.33615 −0.187725
$$809$$ −3.32770 −0.116996 −0.0584978 0.998288i $$-0.518631\pi$$
−0.0584978 + 0.998288i $$0.518631\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ −20.6083 −0.723655 −0.361827 0.932245i $$-0.617847\pi$$
−0.361827 + 0.932245i $$0.617847\pi$$
$$812$$ 36.6098 1.28475
$$813$$ 25.5401 0.895730
$$814$$ 16.4543 0.576722
$$815$$ 10.0916 0.353494
$$816$$ 4.00000 0.140028
$$817$$ −47.8055 −1.67250
$$818$$ −27.3804 −0.957335
$$819$$ 0 0
$$820$$ −9.28932 −0.324397
$$821$$ −1.58005 −0.0551442 −0.0275721 0.999620i $$-0.508778\pi$$
−0.0275721 + 0.999620i $$0.508778\pi$$
$$822$$ 7.03696 0.245442
$$823$$ 37.5168 1.30776 0.653878 0.756600i $$-0.273141\pi$$
0.653878 + 0.756600i $$0.273141\pi$$
$$824$$ −7.51248 −0.261710
$$825$$ −1.66808 −0.0580750
$$826$$ 22.2644 0.774676
$$827$$ 49.4912 1.72098 0.860489 0.509469i $$-0.170158\pi$$
0.860489 + 0.509469i $$0.170158\pi$$
$$828$$ 1.24453 0.0432505
$$829$$ 49.1385 1.70665 0.853326 0.521378i $$-0.174582\pi$$
0.853326 + 0.521378i $$0.174582\pi$$
$$830$$ 7.95317 0.276058
$$831$$ 18.0604 0.626508
$$832$$ 0 0
$$833$$ 25.1342 0.870848
$$834$$ −11.6447 −0.403222
$$835$$ −7.89701 −0.273287
$$836$$ −10.5301 −0.364192
$$837$$ −4.21957 −0.145850
$$838$$ 13.1614 0.454652
$$839$$ 21.5421 0.743717 0.371858 0.928289i $$-0.378721\pi$$
0.371858 + 0.928289i $$0.378721\pi$$
$$840$$ −3.64466 −0.125753
$$841$$ 71.8977 2.47923
$$842$$ −1.29341 −0.0445739
$$843$$ 20.2175 0.696328
$$844$$ 2.22739 0.0766700
$$845$$ 0 0
$$846$$ 6.82522 0.234656
$$847$$ −29.9501 −1.02910
$$848$$ −0.848634 −0.0291422
$$849$$ 8.69149 0.298291
$$850$$ −4.00000 −0.137199
$$851$$ 12.2764 0.420828
$$852$$ −3.51093 −0.120283
$$853$$ −2.79821 −0.0958088 −0.0479044 0.998852i $$-0.515254\pi$$
−0.0479044 + 0.998852i $$0.515254\pi$$
$$854$$ −27.2041 −0.930905
$$855$$ 6.31274 0.215891
$$856$$ −16.9282 −0.578594
$$857$$ 48.9658 1.67264 0.836320 0.548242i $$-0.184703\pi$$
0.836320 + 0.548242i $$0.184703\pi$$
$$858$$ 0 0
$$859$$ 4.22739 0.144237 0.0721184 0.997396i $$-0.477024\pi$$
0.0721184 + 0.997396i $$0.477024\pi$$
$$860$$ −7.57286 −0.258232
$$861$$ −33.8564 −1.15382
$$862$$ 12.1279 0.413080
$$863$$ −19.0285 −0.647738 −0.323869 0.946102i $$-0.604984\pi$$
−0.323869 + 0.946102i $$0.604984\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −0.440685 −0.0149837
$$866$$ 2.07802 0.0706141
$$867$$ −1.00000 −0.0339618
$$868$$ −15.3789 −0.521994
$$869$$ −16.5706 −0.562120
$$870$$ −10.0448 −0.340550
$$871$$ 0 0
$$872$$ 0.663848 0.0224807
$$873$$ −2.75342 −0.0931892
$$874$$ −7.85641 −0.265747
$$875$$ 3.64466 0.123212
$$876$$ 12.2175 0.412792
$$877$$ 13.5473 0.457459 0.228729 0.973490i $$-0.426543\pi$$
0.228729 + 0.973490i $$0.426543\pi$$
$$878$$ 2.39640 0.0808745
$$879$$ −7.54790 −0.254584
$$880$$ −1.66808 −0.0562308
$$881$$ 4.98081 0.167808 0.0839039 0.996474i $$-0.473261\pi$$
0.0839039 + 0.996474i $$0.473261\pi$$
$$882$$ −6.28354 −0.211578
$$883$$ −30.9829 −1.04266 −0.521330 0.853355i $$-0.674564\pi$$
−0.521330 + 0.853355i $$0.674564\pi$$
$$884$$ 0 0
$$885$$ −6.10876 −0.205344
$$886$$ −21.9959 −0.738967
$$887$$ 17.1765 0.576731 0.288365 0.957520i $$-0.406888\pi$$
0.288365 + 0.957520i $$0.406888\pi$$
$$888$$ −9.86423 −0.331022
$$889$$ 52.6314 1.76520
$$890$$ −5.95162 −0.199499
$$891$$ −1.66808 −0.0558826
$$892$$ −6.08380 −0.203701
$$893$$ −43.0858 −1.44181
$$894$$ 0.772609 0.0258399
$$895$$ −19.6368 −0.656387
$$896$$ −3.64466 −0.121760
$$897$$ 0 0
$$898$$ 29.2253 0.975261
$$899$$ −42.3847 −1.41361
$$900$$ 1.00000 0.0333333
$$901$$ −3.39454 −0.113088
$$902$$ −15.4953 −0.515937
$$903$$ −27.6005 −0.918487
$$904$$ 17.8700 0.594347
$$905$$ −6.21752 −0.206677
$$906$$ 9.77838 0.324865
$$907$$ −25.0797 −0.832758 −0.416379 0.909191i $$-0.636701\pi$$
−0.416379 + 0.909191i $$0.636701\pi$$
$$908$$ −15.3205 −0.508429
$$909$$ 5.33615 0.176989
$$910$$ 0 0
$$911$$ 18.2332 0.604092 0.302046 0.953293i $$-0.402330\pi$$
0.302046 + 0.953293i $$0.402330\pi$$
$$912$$ 6.31274 0.209036
$$913$$ 13.2665 0.439057
$$914$$ −39.6432 −1.31128
$$915$$ 7.46410 0.246756
$$916$$ −22.2644 −0.735635
$$917$$ −39.6874 −1.31059
$$918$$ −4.00000 −0.132020
$$919$$ −34.3882 −1.13436 −0.567181 0.823593i $$-0.691966\pi$$
−0.567181 + 0.823593i $$0.691966\pi$$
$$920$$ −1.24453 −0.0410310
$$921$$ 26.0427 0.858137
$$922$$ −16.6758 −0.549190
$$923$$ 0 0
$$924$$ −6.07957 −0.200003
$$925$$ 9.86423 0.324334
$$926$$ −32.2175 −1.05873
$$927$$ 7.51248 0.246742
$$928$$ −10.0448 −0.329736
$$929$$ 31.0904 1.02004 0.510022 0.860161i $$-0.329637\pi$$
0.510022 + 0.860161i $$0.329637\pi$$
$$930$$ 4.21957 0.138365
$$931$$ 39.6663 1.30001
$$932$$ −10.8366 −0.354964
$$933$$ −25.3789 −0.830868
$$934$$ −6.88137 −0.225165
$$935$$ −6.67230 −0.218208
$$936$$ 0 0
$$937$$ −18.8783 −0.616726 −0.308363 0.951269i $$-0.599781\pi$$
−0.308363 + 0.951269i $$0.599781\pi$$
$$938$$ 53.7712 1.75569
$$939$$ −31.4600 −1.02666
$$940$$ −6.82522 −0.222614
$$941$$ −28.9398 −0.943409 −0.471705 0.881757i $$-0.656361\pi$$
−0.471705 + 0.881757i $$0.656361\pi$$
$$942$$ 12.0135 0.391423
$$943$$ −11.5609 −0.376473
$$944$$ −6.10876 −0.198823
$$945$$ 3.64466 0.118561
$$946$$ −12.6321 −0.410705
$$947$$ 7.86896 0.255707 0.127853 0.991793i $$-0.459191\pi$$
0.127853 + 0.991793i $$0.459191\pi$$
$$948$$ 9.93398 0.322641
$$949$$ 0 0
$$950$$ −6.31274 −0.204812
$$951$$ 24.7093 0.801253
$$952$$ −14.5786 −0.472496
$$953$$ −55.0968 −1.78476 −0.892381 0.451283i $$-0.850967\pi$$
−0.892381 + 0.451283i $$0.850967\pi$$
$$954$$ 0.848634 0.0274755
$$955$$ 15.6816 0.507445
$$956$$ −16.4975 −0.533568
$$957$$ −16.7555 −0.541627
$$958$$ 18.9709 0.612923
$$959$$ −25.6473 −0.828196
$$960$$ 1.00000 0.0322749
$$961$$ −13.1952 −0.425653
$$962$$ 0 0
$$963$$ 16.9282 0.545504
$$964$$ −4.40435 −0.141855
$$965$$ 8.12795 0.261648
$$966$$ −4.53590 −0.145940
$$967$$ 31.5523 1.01465 0.507326 0.861754i $$-0.330634\pi$$
0.507326 + 0.861754i $$0.330634\pi$$
$$968$$ 8.21752 0.264121
$$969$$ 25.2509 0.811177
$$970$$ 2.75342 0.0884070
$$971$$ 3.67585 0.117964 0.0589818 0.998259i $$-0.481215\pi$$
0.0589818 + 0.998259i $$0.481215\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 42.4408 1.36059
$$974$$ −1.91620 −0.0613991
$$975$$ 0 0
$$976$$ 7.46410 0.238920
$$977$$ −42.0205 −1.34435 −0.672177 0.740391i $$-0.734641\pi$$
−0.672177 + 0.740391i $$0.734641\pi$$
$$978$$ −10.0916 −0.322694
$$979$$ −9.92775 −0.317292
$$980$$ 6.28354 0.200720
$$981$$ −0.663848 −0.0211950
$$982$$ 33.6375 1.07341
$$983$$ 13.4307 0.428372 0.214186 0.976793i $$-0.431290\pi$$
0.214186 + 0.976793i $$0.431290\pi$$
$$984$$ 9.28932 0.296133
$$985$$ −0.891239 −0.0283972
$$986$$ −40.1791 −1.27956
$$987$$ −24.8756 −0.791799
$$988$$ 0 0
$$989$$ −9.42468 −0.299687
$$990$$ 1.66808 0.0530149
$$991$$ 43.7988 1.39132 0.695658 0.718373i $$-0.255113\pi$$
0.695658 + 0.718373i $$0.255113\pi$$
$$992$$ 4.21957 0.133971
$$993$$ −5.60360 −0.177825
$$994$$ 12.7962 0.405870
$$995$$ 0.361116 0.0114481
$$996$$ −7.95317 −0.252006
$$997$$ −23.5125 −0.744648 −0.372324 0.928103i $$-0.621439\pi$$
−0.372324 + 0.928103i $$0.621439\pi$$
$$998$$ −1.82522 −0.0577763
$$999$$ 9.86423 0.312090
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 5070.2.a.bz.1.4 4
13.5 odd 4 5070.2.b.ba.1351.8 8
13.6 odd 12 390.2.bb.c.361.3 yes 8
13.8 odd 4 5070.2.b.ba.1351.1 8
13.11 odd 12 390.2.bb.c.121.3 8
13.12 even 2 5070.2.a.ca.1.1 4
39.11 even 12 1170.2.bs.f.901.1 8
39.32 even 12 1170.2.bs.f.361.1 8
65.19 odd 12 1950.2.bc.g.751.2 8
65.24 odd 12 1950.2.bc.g.901.2 8
65.32 even 12 1950.2.y.k.49.2 8
65.37 even 12 1950.2.y.j.199.3 8
65.58 even 12 1950.2.y.j.49.3 8
65.63 even 12 1950.2.y.k.199.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.c.121.3 8 13.11 odd 12
390.2.bb.c.361.3 yes 8 13.6 odd 12
1170.2.bs.f.361.1 8 39.32 even 12
1170.2.bs.f.901.1 8 39.11 even 12
1950.2.y.j.49.3 8 65.58 even 12
1950.2.y.j.199.3 8 65.37 even 12
1950.2.y.k.49.2 8 65.32 even 12
1950.2.y.k.199.2 8 65.63 even 12
1950.2.bc.g.751.2 8 65.19 odd 12
1950.2.bc.g.901.2 8 65.24 odd 12
5070.2.a.bz.1.4 4 1.1 even 1 trivial
5070.2.a.ca.1.1 4 13.12 even 2
5070.2.b.ba.1351.1 8 13.8 odd 4
5070.2.b.ba.1351.8 8 13.5 odd 4