# Properties

 Label 50.6.b.b.49.2 Level $50$ Weight $6$ Character 50.49 Analytic conductor $8.019$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 50.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.01919099065$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 10) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 50.49 Dual form 50.6.b.b.49.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000i q^{2} -6.00000i q^{3} -16.0000 q^{4} +24.0000 q^{6} -118.000i q^{7} -64.0000i q^{8} +207.000 q^{9} +O(q^{10})$$ $$q+4.00000i q^{2} -6.00000i q^{3} -16.0000 q^{4} +24.0000 q^{6} -118.000i q^{7} -64.0000i q^{8} +207.000 q^{9} +192.000 q^{11} +96.0000i q^{12} -1106.00i q^{13} +472.000 q^{14} +256.000 q^{16} +762.000i q^{17} +828.000i q^{18} +2740.00 q^{19} -708.000 q^{21} +768.000i q^{22} -1566.00i q^{23} -384.000 q^{24} +4424.00 q^{26} -2700.00i q^{27} +1888.00i q^{28} -5910.00 q^{29} -6868.00 q^{31} +1024.00i q^{32} -1152.00i q^{33} -3048.00 q^{34} -3312.00 q^{36} -5518.00i q^{37} +10960.0i q^{38} -6636.00 q^{39} -378.000 q^{41} -2832.00i q^{42} +2434.00i q^{43} -3072.00 q^{44} +6264.00 q^{46} +13122.0i q^{47} -1536.00i q^{48} +2883.00 q^{49} +4572.00 q^{51} +17696.0i q^{52} +9174.00i q^{53} +10800.0 q^{54} -7552.00 q^{56} -16440.0i q^{57} -23640.0i q^{58} +34980.0 q^{59} -9838.00 q^{61} -27472.0i q^{62} -24426.0i q^{63} -4096.00 q^{64} +4608.00 q^{66} +33722.0i q^{67} -12192.0i q^{68} -9396.00 q^{69} +70212.0 q^{71} -13248.0i q^{72} -21986.0i q^{73} +22072.0 q^{74} -43840.0 q^{76} -22656.0i q^{77} -26544.0i q^{78} -4520.00 q^{79} +34101.0 q^{81} -1512.00i q^{82} +109074. i q^{83} +11328.0 q^{84} -9736.00 q^{86} +35460.0i q^{87} -12288.0i q^{88} -38490.0 q^{89} -130508. q^{91} +25056.0i q^{92} +41208.0i q^{93} -52488.0 q^{94} +6144.00 q^{96} -1918.00i q^{97} +11532.0i q^{98} +39744.0 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + 48q^{6} + 414q^{9} + O(q^{10})$$ $$2q - 32q^{4} + 48q^{6} + 414q^{9} + 384q^{11} + 944q^{14} + 512q^{16} + 5480q^{19} - 1416q^{21} - 768q^{24} + 8848q^{26} - 11820q^{29} - 13736q^{31} - 6096q^{34} - 6624q^{36} - 13272q^{39} - 756q^{41} - 6144q^{44} + 12528q^{46} + 5766q^{49} + 9144q^{51} + 21600q^{54} - 15104q^{56} + 69960q^{59} - 19676q^{61} - 8192q^{64} + 9216q^{66} - 18792q^{69} + 140424q^{71} + 44144q^{74} - 87680q^{76} - 9040q^{79} + 68202q^{81} + 22656q^{84} - 19472q^{86} - 76980q^{89} - 261016q^{91} - 104976q^{94} + 12288q^{96} + 79488q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$\chi(n)$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000i 0.707107i
$$3$$ − 6.00000i − 0.384900i −0.981307 0.192450i $$-0.938357\pi$$
0.981307 0.192450i $$-0.0616434\pi$$
$$4$$ −16.0000 −0.500000
$$5$$ 0 0
$$6$$ 24.0000 0.272166
$$7$$ − 118.000i − 0.910200i −0.890440 0.455100i $$-0.849603\pi$$
0.890440 0.455100i $$-0.150397\pi$$
$$8$$ − 64.0000i − 0.353553i
$$9$$ 207.000 0.851852
$$10$$ 0 0
$$11$$ 192.000 0.478431 0.239216 0.970966i $$-0.423110\pi$$
0.239216 + 0.970966i $$0.423110\pi$$
$$12$$ 96.0000i 0.192450i
$$13$$ − 1106.00i − 1.81508i −0.419961 0.907542i $$-0.637956\pi$$
0.419961 0.907542i $$-0.362044\pi$$
$$14$$ 472.000 0.643609
$$15$$ 0 0
$$16$$ 256.000 0.250000
$$17$$ 762.000i 0.639488i 0.947504 + 0.319744i $$0.103597\pi$$
−0.947504 + 0.319744i $$0.896403\pi$$
$$18$$ 828.000i 0.602350i
$$19$$ 2740.00 1.74127 0.870636 0.491928i $$-0.163708\pi$$
0.870636 + 0.491928i $$0.163708\pi$$
$$20$$ 0 0
$$21$$ −708.000 −0.350336
$$22$$ 768.000i 0.338302i
$$23$$ − 1566.00i − 0.617266i −0.951181 0.308633i $$-0.900129\pi$$
0.951181 0.308633i $$-0.0998714\pi$$
$$24$$ −384.000 −0.136083
$$25$$ 0 0
$$26$$ 4424.00 1.28346
$$27$$ − 2700.00i − 0.712778i
$$28$$ 1888.00i 0.455100i
$$29$$ −5910.00 −1.30495 −0.652473 0.757812i $$-0.726268\pi$$
−0.652473 + 0.757812i $$0.726268\pi$$
$$30$$ 0 0
$$31$$ −6868.00 −1.28359 −0.641795 0.766877i $$-0.721810\pi$$
−0.641795 + 0.766877i $$0.721810\pi$$
$$32$$ 1024.00i 0.176777i
$$33$$ − 1152.00i − 0.184148i
$$34$$ −3048.00 −0.452187
$$35$$ 0 0
$$36$$ −3312.00 −0.425926
$$37$$ − 5518.00i − 0.662640i −0.943519 0.331320i $$-0.892506\pi$$
0.943519 0.331320i $$-0.107494\pi$$
$$38$$ 10960.0i 1.23127i
$$39$$ −6636.00 −0.698626
$$40$$ 0 0
$$41$$ −378.000 −0.0351182 −0.0175591 0.999846i $$-0.505590\pi$$
−0.0175591 + 0.999846i $$0.505590\pi$$
$$42$$ − 2832.00i − 0.247725i
$$43$$ 2434.00i 0.200747i 0.994950 + 0.100374i $$0.0320038\pi$$
−0.994950 + 0.100374i $$0.967996\pi$$
$$44$$ −3072.00 −0.239216
$$45$$ 0 0
$$46$$ 6264.00 0.436473
$$47$$ 13122.0i 0.866474i 0.901280 + 0.433237i $$0.142629\pi$$
−0.901280 + 0.433237i $$0.857371\pi$$
$$48$$ − 1536.00i − 0.0962250i
$$49$$ 2883.00 0.171536
$$50$$ 0 0
$$51$$ 4572.00 0.246139
$$52$$ 17696.0i 0.907542i
$$53$$ 9174.00i 0.448610i 0.974519 + 0.224305i $$0.0720112\pi$$
−0.974519 + 0.224305i $$0.927989\pi$$
$$54$$ 10800.0 0.504010
$$55$$ 0 0
$$56$$ −7552.00 −0.321804
$$57$$ − 16440.0i − 0.670216i
$$58$$ − 23640.0i − 0.922736i
$$59$$ 34980.0 1.30825 0.654124 0.756388i $$-0.273038\pi$$
0.654124 + 0.756388i $$0.273038\pi$$
$$60$$ 0 0
$$61$$ −9838.00 −0.338518 −0.169259 0.985572i $$-0.554137\pi$$
−0.169259 + 0.985572i $$0.554137\pi$$
$$62$$ − 27472.0i − 0.907635i
$$63$$ − 24426.0i − 0.775356i
$$64$$ −4096.00 −0.125000
$$65$$ 0 0
$$66$$ 4608.00 0.130212
$$67$$ 33722.0i 0.917754i 0.888500 + 0.458877i $$0.151748\pi$$
−0.888500 + 0.458877i $$0.848252\pi$$
$$68$$ − 12192.0i − 0.319744i
$$69$$ −9396.00 −0.237586
$$70$$ 0 0
$$71$$ 70212.0 1.65297 0.826486 0.562957i $$-0.190336\pi$$
0.826486 + 0.562957i $$0.190336\pi$$
$$72$$ − 13248.0i − 0.301175i
$$73$$ − 21986.0i − 0.482880i −0.970416 0.241440i $$-0.922380\pi$$
0.970416 0.241440i $$-0.0776197\pi$$
$$74$$ 22072.0 0.468557
$$75$$ 0 0
$$76$$ −43840.0 −0.870636
$$77$$ − 22656.0i − 0.435468i
$$78$$ − 26544.0i − 0.494003i
$$79$$ −4520.00 −0.0814837 −0.0407418 0.999170i $$-0.512972\pi$$
−0.0407418 + 0.999170i $$0.512972\pi$$
$$80$$ 0 0
$$81$$ 34101.0 0.577503
$$82$$ − 1512.00i − 0.0248323i
$$83$$ 109074.i 1.73790i 0.494896 + 0.868952i $$0.335206\pi$$
−0.494896 + 0.868952i $$0.664794\pi$$
$$84$$ 11328.0 0.175168
$$85$$ 0 0
$$86$$ −9736.00 −0.141950
$$87$$ 35460.0i 0.502274i
$$88$$ − 12288.0i − 0.169151i
$$89$$ −38490.0 −0.515078 −0.257539 0.966268i $$-0.582912\pi$$
−0.257539 + 0.966268i $$0.582912\pi$$
$$90$$ 0 0
$$91$$ −130508. −1.65209
$$92$$ 25056.0i 0.308633i
$$93$$ 41208.0i 0.494054i
$$94$$ −52488.0 −0.612689
$$95$$ 0 0
$$96$$ 6144.00 0.0680414
$$97$$ − 1918.00i − 0.0206976i −0.999946 0.0103488i $$-0.996706\pi$$
0.999946 0.0103488i $$-0.00329418\pi$$
$$98$$ 11532.0i 0.121294i
$$99$$ 39744.0 0.407553
$$100$$ 0 0
$$101$$ 77622.0 0.757149 0.378575 0.925571i $$-0.376414\pi$$
0.378575 + 0.925571i $$0.376414\pi$$
$$102$$ 18288.0i 0.174047i
$$103$$ 46714.0i 0.433864i 0.976187 + 0.216932i $$0.0696051\pi$$
−0.976187 + 0.216932i $$0.930395\pi$$
$$104$$ −70784.0 −0.641729
$$105$$ 0 0
$$106$$ −36696.0 −0.317215
$$107$$ − 1038.00i − 0.00876472i −0.999990 0.00438236i $$-0.998605\pi$$
0.999990 0.00438236i $$-0.00139495\pi$$
$$108$$ 43200.0i 0.356389i
$$109$$ −206930. −1.66823 −0.834117 0.551587i $$-0.814023\pi$$
−0.834117 + 0.551587i $$0.814023\pi$$
$$110$$ 0 0
$$111$$ −33108.0 −0.255050
$$112$$ − 30208.0i − 0.227550i
$$113$$ − 139386.i − 1.02689i −0.858123 0.513444i $$-0.828369\pi$$
0.858123 0.513444i $$-0.171631\pi$$
$$114$$ 65760.0 0.473914
$$115$$ 0 0
$$116$$ 94560.0 0.652473
$$117$$ − 228942.i − 1.54618i
$$118$$ 139920.i 0.925070i
$$119$$ 89916.0 0.582062
$$120$$ 0 0
$$121$$ −124187. −0.771104
$$122$$ − 39352.0i − 0.239369i
$$123$$ 2268.00i 0.0135170i
$$124$$ 109888. 0.641795
$$125$$ 0 0
$$126$$ 97704.0 0.548259
$$127$$ 299882.i 1.64984i 0.565252 + 0.824919i $$0.308779\pi$$
−0.565252 + 0.824919i $$0.691221\pi$$
$$128$$ − 16384.0i − 0.0883883i
$$129$$ 14604.0 0.0772676
$$130$$ 0 0
$$131$$ 7872.00 0.0400781 0.0200390 0.999799i $$-0.493621\pi$$
0.0200390 + 0.999799i $$0.493621\pi$$
$$132$$ 18432.0i 0.0920741i
$$133$$ − 323320.i − 1.58491i
$$134$$ −134888. −0.648950
$$135$$ 0 0
$$136$$ 48768.0 0.226093
$$137$$ − 164238.i − 0.747605i −0.927508 0.373803i $$-0.878054\pi$$
0.927508 0.373803i $$-0.121946\pi$$
$$138$$ − 37584.0i − 0.167998i
$$139$$ 282100. 1.23841 0.619207 0.785228i $$-0.287454\pi$$
0.619207 + 0.785228i $$0.287454\pi$$
$$140$$ 0 0
$$141$$ 78732.0 0.333506
$$142$$ 280848.i 1.16883i
$$143$$ − 212352.i − 0.868393i
$$144$$ 52992.0 0.212963
$$145$$ 0 0
$$146$$ 87944.0 0.341448
$$147$$ − 17298.0i − 0.0660241i
$$148$$ 88288.0i 0.331320i
$$149$$ 388950. 1.43525 0.717626 0.696429i $$-0.245229\pi$$
0.717626 + 0.696429i $$0.245229\pi$$
$$150$$ 0 0
$$151$$ −97948.0 −0.349585 −0.174793 0.984605i $$-0.555926\pi$$
−0.174793 + 0.984605i $$0.555926\pi$$
$$152$$ − 175360.i − 0.615633i
$$153$$ 157734.i 0.544749i
$$154$$ 90624.0 0.307923
$$155$$ 0 0
$$156$$ 106176. 0.349313
$$157$$ − 3718.00i − 0.0120382i −0.999982 0.00601908i $$-0.998084\pi$$
0.999982 0.00601908i $$-0.00191594\pi$$
$$158$$ − 18080.0i − 0.0576177i
$$159$$ 55044.0 0.172670
$$160$$ 0 0
$$161$$ −184788. −0.561835
$$162$$ 136404.i 0.408357i
$$163$$ 43234.0i 0.127455i 0.997967 + 0.0637274i $$0.0202988\pi$$
−0.997967 + 0.0637274i $$0.979701\pi$$
$$164$$ 6048.00 0.0175591
$$165$$ 0 0
$$166$$ −436296. −1.22888
$$167$$ 186522.i 0.517534i 0.965940 + 0.258767i $$0.0833162\pi$$
−0.965940 + 0.258767i $$0.916684\pi$$
$$168$$ 45312.0i 0.123863i
$$169$$ −851943. −2.29453
$$170$$ 0 0
$$171$$ 567180. 1.48331
$$172$$ − 38944.0i − 0.100374i
$$173$$ 374454.i 0.951225i 0.879655 + 0.475612i $$0.157774\pi$$
−0.879655 + 0.475612i $$0.842226\pi$$
$$174$$ −141840. −0.355161
$$175$$ 0 0
$$176$$ 49152.0 0.119608
$$177$$ − 209880.i − 0.503545i
$$178$$ − 153960.i − 0.364215i
$$179$$ −272100. −0.634740 −0.317370 0.948302i $$-0.602800\pi$$
−0.317370 + 0.948302i $$0.602800\pi$$
$$180$$ 0 0
$$181$$ −75418.0 −0.171111 −0.0855556 0.996333i $$-0.527267\pi$$
−0.0855556 + 0.996333i $$0.527267\pi$$
$$182$$ − 522032.i − 1.16820i
$$183$$ 59028.0i 0.130296i
$$184$$ −100224. −0.218236
$$185$$ 0 0
$$186$$ −164832. −0.349349
$$187$$ 146304.i 0.305951i
$$188$$ − 209952.i − 0.433237i
$$189$$ −318600. −0.648771
$$190$$ 0 0
$$191$$ −356988. −0.708060 −0.354030 0.935234i $$-0.615189\pi$$
−0.354030 + 0.935234i $$0.615189\pi$$
$$192$$ 24576.0i 0.0481125i
$$193$$ 438694.i 0.847751i 0.905720 + 0.423876i $$0.139331\pi$$
−0.905720 + 0.423876i $$0.860669\pi$$
$$194$$ 7672.00 0.0146354
$$195$$ 0 0
$$196$$ −46128.0 −0.0857678
$$197$$ − 156798.i − 0.287856i −0.989588 0.143928i $$-0.954027\pi$$
0.989588 0.143928i $$-0.0459733\pi$$
$$198$$ 158976.i 0.288183i
$$199$$ 162520. 0.290920 0.145460 0.989364i $$-0.453534\pi$$
0.145460 + 0.989364i $$0.453534\pi$$
$$200$$ 0 0
$$201$$ 202332. 0.353244
$$202$$ 310488.i 0.535385i
$$203$$ 697380.i 1.18776i
$$204$$ −73152.0 −0.123070
$$205$$ 0 0
$$206$$ −186856. −0.306788
$$207$$ − 324162.i − 0.525819i
$$208$$ − 283136.i − 0.453771i
$$209$$ 526080. 0.833079
$$210$$ 0 0
$$211$$ −181648. −0.280882 −0.140441 0.990089i $$-0.544852\pi$$
−0.140441 + 0.990089i $$0.544852\pi$$
$$212$$ − 146784.i − 0.224305i
$$213$$ − 421272.i − 0.636229i
$$214$$ 4152.00 0.00619759
$$215$$ 0 0
$$216$$ −172800. −0.252005
$$217$$ 810424.i 1.16832i
$$218$$ − 827720.i − 1.17962i
$$219$$ −131916. −0.185861
$$220$$ 0 0
$$221$$ 842772. 1.16073
$$222$$ − 132432.i − 0.180348i
$$223$$ 288274.i 0.388189i 0.980983 + 0.194095i $$0.0621769\pi$$
−0.980983 + 0.194095i $$0.937823\pi$$
$$224$$ 120832. 0.160902
$$225$$ 0 0
$$226$$ 557544. 0.726119
$$227$$ 1.12552e6i 1.44974i 0.688887 + 0.724869i $$0.258100\pi$$
−0.688887 + 0.724869i $$0.741900\pi$$
$$228$$ 263040.i 0.335108i
$$229$$ 415810. 0.523970 0.261985 0.965072i $$-0.415623\pi$$
0.261985 + 0.965072i $$0.415623\pi$$
$$230$$ 0 0
$$231$$ −135936. −0.167612
$$232$$ 378240.i 0.461368i
$$233$$ − 770586.i − 0.929889i −0.885340 0.464945i $$-0.846074\pi$$
0.885340 0.464945i $$-0.153926\pi$$
$$234$$ 915768. 1.09332
$$235$$ 0 0
$$236$$ −559680. −0.654124
$$237$$ 27120.0i 0.0313631i
$$238$$ 359664.i 0.411580i
$$239$$ 595320. 0.674149 0.337074 0.941478i $$-0.390563\pi$$
0.337074 + 0.941478i $$0.390563\pi$$
$$240$$ 0 0
$$241$$ 273902. 0.303775 0.151888 0.988398i $$-0.451465\pi$$
0.151888 + 0.988398i $$0.451465\pi$$
$$242$$ − 496748.i − 0.545253i
$$243$$ − 860706.i − 0.935059i
$$244$$ 157408. 0.169259
$$245$$ 0 0
$$246$$ −9072.00 −0.00955796
$$247$$ − 3.03044e6i − 3.16055i
$$248$$ 439552.i 0.453817i
$$249$$ 654444. 0.668920
$$250$$ 0 0
$$251$$ 850752. 0.852351 0.426176 0.904640i $$-0.359861\pi$$
0.426176 + 0.904640i $$0.359861\pi$$
$$252$$ 390816.i 0.387678i
$$253$$ − 300672.i − 0.295319i
$$254$$ −1.19953e6 −1.16661
$$255$$ 0 0
$$256$$ 65536.0 0.0625000
$$257$$ 825402.i 0.779530i 0.920914 + 0.389765i $$0.127444\pi$$
−0.920914 + 0.389765i $$0.872556\pi$$
$$258$$ 58416.0i 0.0546365i
$$259$$ −651124. −0.603135
$$260$$ 0 0
$$261$$ −1.22337e6 −1.11162
$$262$$ 31488.0i 0.0283395i
$$263$$ − 1.36465e6i − 1.21655i −0.793726 0.608276i $$-0.791861\pi$$
0.793726 0.608276i $$-0.208139\pi$$
$$264$$ −73728.0 −0.0651062
$$265$$ 0 0
$$266$$ 1.29328e6 1.12070
$$267$$ 230940.i 0.198254i
$$268$$ − 539552.i − 0.458877i
$$269$$ 113310. 0.0954745 0.0477373 0.998860i $$-0.484799\pi$$
0.0477373 + 0.998860i $$0.484799\pi$$
$$270$$ 0 0
$$271$$ −849628. −0.702758 −0.351379 0.936233i $$-0.614287\pi$$
−0.351379 + 0.936233i $$0.614287\pi$$
$$272$$ 195072.i 0.159872i
$$273$$ 783048.i 0.635890i
$$274$$ 656952. 0.528637
$$275$$ 0 0
$$276$$ 150336. 0.118793
$$277$$ 438602.i 0.343456i 0.985144 + 0.171728i $$0.0549350\pi$$
−0.985144 + 0.171728i $$0.945065\pi$$
$$278$$ 1.12840e6i 0.875691i
$$279$$ −1.42168e6 −1.09343
$$280$$ 0 0
$$281$$ −1.45670e6 −1.10053 −0.550267 0.834989i $$-0.685474\pi$$
−0.550267 + 0.834989i $$0.685474\pi$$
$$282$$ 314928.i 0.235824i
$$283$$ 120394.i 0.0893591i 0.999001 + 0.0446795i $$0.0142267\pi$$
−0.999001 + 0.0446795i $$0.985773\pi$$
$$284$$ −1.12339e6 −0.826486
$$285$$ 0 0
$$286$$ 849408. 0.614047
$$287$$ 44604.0i 0.0319646i
$$288$$ 211968.i 0.150588i
$$289$$ 839213. 0.591055
$$290$$ 0 0
$$291$$ −11508.0 −0.00796650
$$292$$ 351776.i 0.241440i
$$293$$ 2.64209e6i 1.79796i 0.437993 + 0.898978i $$0.355689\pi$$
−0.437993 + 0.898978i $$0.644311\pi$$
$$294$$ 69192.0 0.0466861
$$295$$ 0 0
$$296$$ −353152. −0.234278
$$297$$ − 518400.i − 0.341015i
$$298$$ 1.55580e6i 1.01488i
$$299$$ −1.73200e6 −1.12039
$$300$$ 0 0
$$301$$ 287212. 0.182720
$$302$$ − 391792.i − 0.247194i
$$303$$ − 465732.i − 0.291427i
$$304$$ 701440. 0.435318
$$305$$ 0 0
$$306$$ −630936. −0.385196
$$307$$ − 1.44756e6i − 0.876577i −0.898834 0.438288i $$-0.855585\pi$$
0.898834 0.438288i $$-0.144415\pi$$
$$308$$ 362496.i 0.217734i
$$309$$ 280284. 0.166994
$$310$$ 0 0
$$311$$ −928068. −0.544100 −0.272050 0.962283i $$-0.587702\pi$$
−0.272050 + 0.962283i $$0.587702\pi$$
$$312$$ 424704.i 0.247002i
$$313$$ − 2.29563e6i − 1.32446i −0.749299 0.662232i $$-0.769609\pi$$
0.749299 0.662232i $$-0.230391\pi$$
$$314$$ 14872.0 0.00851227
$$315$$ 0 0
$$316$$ 72320.0 0.0407418
$$317$$ 2.73652e6i 1.52950i 0.644324 + 0.764752i $$0.277139\pi$$
−0.644324 + 0.764752i $$0.722861\pi$$
$$318$$ 220176.i 0.122096i
$$319$$ −1.13472e6 −0.624327
$$320$$ 0 0
$$321$$ −6228.00 −0.00337354
$$322$$ − 739152.i − 0.397278i
$$323$$ 2.08788e6i 1.11352i
$$324$$ −545616. −0.288752
$$325$$ 0 0
$$326$$ −172936. −0.0901242
$$327$$ 1.24158e6i 0.642104i
$$328$$ 24192.0i 0.0124162i
$$329$$ 1.54840e6 0.788665
$$330$$ 0 0
$$331$$ 3.81879e6 1.91583 0.957913 0.287059i $$-0.0926776\pi$$
0.957913 + 0.287059i $$0.0926776\pi$$
$$332$$ − 1.74518e6i − 0.868952i
$$333$$ − 1.14223e6i − 0.564471i
$$334$$ −746088. −0.365952
$$335$$ 0 0
$$336$$ −181248. −0.0875841
$$337$$ − 2.21088e6i − 1.06045i −0.847857 0.530225i $$-0.822108\pi$$
0.847857 0.530225i $$-0.177892\pi$$
$$338$$ − 3.40777e6i − 1.62248i
$$339$$ −836316. −0.395249
$$340$$ 0 0
$$341$$ −1.31866e6 −0.614109
$$342$$ 2.26872e6i 1.04886i
$$343$$ − 2.32342e6i − 1.06633i
$$344$$ 155776. 0.0709748
$$345$$ 0 0
$$346$$ −1.49782e6 −0.672618
$$347$$ − 2.32724e6i − 1.03757i −0.854905 0.518785i $$-0.826385\pi$$
0.854905 0.518785i $$-0.173615\pi$$
$$348$$ − 567360.i − 0.251137i
$$349$$ 311290. 0.136805 0.0684024 0.997658i $$-0.478210\pi$$
0.0684024 + 0.997658i $$0.478210\pi$$
$$350$$ 0 0
$$351$$ −2.98620e6 −1.29375
$$352$$ 196608.i 0.0845755i
$$353$$ 3.08657e6i 1.31838i 0.751977 + 0.659189i $$0.229100\pi$$
−0.751977 + 0.659189i $$0.770900\pi$$
$$354$$ 839520. 0.356060
$$355$$ 0 0
$$356$$ 615840. 0.257539
$$357$$ − 539496.i − 0.224036i
$$358$$ − 1.08840e6i − 0.448829i
$$359$$ 3.53076e6 1.44588 0.722940 0.690911i $$-0.242790\pi$$
0.722940 + 0.690911i $$0.242790\pi$$
$$360$$ 0 0
$$361$$ 5.03150e6 2.03203
$$362$$ − 301672.i − 0.120994i
$$363$$ 745122.i 0.296798i
$$364$$ 2.08813e6 0.826045
$$365$$ 0 0
$$366$$ −236112. −0.0921330
$$367$$ 35762.0i 0.0138598i 0.999976 + 0.00692989i $$0.00220587\pi$$
−0.999976 + 0.00692989i $$0.997794\pi$$
$$368$$ − 400896.i − 0.154316i
$$369$$ −78246.0 −0.0299155
$$370$$ 0 0
$$371$$ 1.08253e6 0.408325
$$372$$ − 659328.i − 0.247027i
$$373$$ 1.71525e6i 0.638346i 0.947696 + 0.319173i $$0.103405\pi$$
−0.947696 + 0.319173i $$0.896595\pi$$
$$374$$ −585216. −0.216340
$$375$$ 0 0
$$376$$ 839808. 0.306345
$$377$$ 6.53646e6i 2.36859i
$$378$$ − 1.27440e6i − 0.458750i
$$379$$ 3.10174e6 1.10919 0.554597 0.832119i $$-0.312873\pi$$
0.554597 + 0.832119i $$0.312873\pi$$
$$380$$ 0 0
$$381$$ 1.79929e6 0.635023
$$382$$ − 1.42795e6i − 0.500674i
$$383$$ − 5.31949e6i − 1.85299i −0.376309 0.926494i $$-0.622807\pi$$
0.376309 0.926494i $$-0.377193\pi$$
$$384$$ −98304.0 −0.0340207
$$385$$ 0 0
$$386$$ −1.75478e6 −0.599451
$$387$$ 503838.i 0.171007i
$$388$$ 30688.0i 0.0103488i
$$389$$ −1.16145e6 −0.389158 −0.194579 0.980887i $$-0.562334\pi$$
−0.194579 + 0.980887i $$0.562334\pi$$
$$390$$ 0 0
$$391$$ 1.19329e6 0.394734
$$392$$ − 184512.i − 0.0606470i
$$393$$ − 47232.0i − 0.0154261i
$$394$$ 627192. 0.203545
$$395$$ 0 0
$$396$$ −635904. −0.203776
$$397$$ 628562.i 0.200157i 0.994980 + 0.100079i $$0.0319095\pi$$
−0.994980 + 0.100079i $$0.968091\pi$$
$$398$$ 650080.i 0.205712i
$$399$$ −1.93992e6 −0.610031
$$400$$ 0 0
$$401$$ −2.72432e6 −0.846052 −0.423026 0.906118i $$-0.639032\pi$$
−0.423026 + 0.906118i $$0.639032\pi$$
$$402$$ 809328.i 0.249781i
$$403$$ 7.59601e6i 2.32982i
$$404$$ −1.24195e6 −0.378575
$$405$$ 0 0
$$406$$ −2.78952e6 −0.839875
$$407$$ − 1.05946e6i − 0.317027i
$$408$$ − 292608.i − 0.0870233i
$$409$$ −1.78019e6 −0.526209 −0.263104 0.964767i $$-0.584746\pi$$
−0.263104 + 0.964767i $$0.584746\pi$$
$$410$$ 0 0
$$411$$ −985428. −0.287753
$$412$$ − 747424.i − 0.216932i
$$413$$ − 4.12764e6i − 1.19077i
$$414$$ 1.29665e6 0.371810
$$415$$ 0 0
$$416$$ 1.13254e6 0.320865
$$417$$ − 1.69260e6i − 0.476666i
$$418$$ 2.10432e6i 0.589076i
$$419$$ −650580. −0.181036 −0.0905181 0.995895i $$-0.528852\pi$$
−0.0905181 + 0.995895i $$0.528852\pi$$
$$420$$ 0 0
$$421$$ −3.54060e6 −0.973579 −0.486790 0.873519i $$-0.661832\pi$$
−0.486790 + 0.873519i $$0.661832\pi$$
$$422$$ − 726592.i − 0.198614i
$$423$$ 2.71625e6i 0.738107i
$$424$$ 587136. 0.158608
$$425$$ 0 0
$$426$$ 1.68509e6 0.449882
$$427$$ 1.16088e6i 0.308119i
$$428$$ 16608.0i 0.00438236i
$$429$$ −1.27411e6 −0.334245
$$430$$ 0 0
$$431$$ −548748. −0.142292 −0.0711459 0.997466i $$-0.522666\pi$$
−0.0711459 + 0.997466i $$0.522666\pi$$
$$432$$ − 691200.i − 0.178195i
$$433$$ 1.49241e6i 0.382534i 0.981538 + 0.191267i $$0.0612596\pi$$
−0.981538 + 0.191267i $$0.938740\pi$$
$$434$$ −3.24170e6 −0.826129
$$435$$ 0 0
$$436$$ 3.31088e6 0.834117
$$437$$ − 4.29084e6i − 1.07483i
$$438$$ − 527664.i − 0.131423i
$$439$$ −4.86212e6 −1.20411 −0.602053 0.798456i $$-0.705650\pi$$
−0.602053 + 0.798456i $$0.705650\pi$$
$$440$$ 0 0
$$441$$ 596781. 0.146123
$$442$$ 3.37109e6i 0.820757i
$$443$$ 1.86155e6i 0.450678i 0.974280 + 0.225339i $$0.0723490\pi$$
−0.974280 + 0.225339i $$0.927651\pi$$
$$444$$ 529728. 0.127525
$$445$$ 0 0
$$446$$ −1.15310e6 −0.274491
$$447$$ − 2.33370e6i − 0.552429i
$$448$$ 483328.i 0.113775i
$$449$$ −3.73719e6 −0.874841 −0.437421 0.899257i $$-0.644108\pi$$
−0.437421 + 0.899257i $$0.644108\pi$$
$$450$$ 0 0
$$451$$ −72576.0 −0.0168016
$$452$$ 2.23018e6i 0.513444i
$$453$$ 587688.i 0.134555i
$$454$$ −4.50209e6 −1.02512
$$455$$ 0 0
$$456$$ −1.05216e6 −0.236957
$$457$$ − 6.48276e6i − 1.45201i −0.687690 0.726005i $$-0.741375\pi$$
0.687690 0.726005i $$-0.258625\pi$$
$$458$$ 1.66324e6i 0.370503i
$$459$$ 2.05740e6 0.455813
$$460$$ 0 0
$$461$$ 1.50910e6 0.330724 0.165362 0.986233i $$-0.447121\pi$$
0.165362 + 0.986233i $$0.447121\pi$$
$$462$$ − 543744.i − 0.118519i
$$463$$ − 8.68401e6i − 1.88264i −0.337513 0.941321i $$-0.609586\pi$$
0.337513 0.941321i $$-0.390414\pi$$
$$464$$ −1.51296e6 −0.326236
$$465$$ 0 0
$$466$$ 3.08234e6 0.657531
$$467$$ 6.96412e6i 1.47766i 0.673893 + 0.738829i $$0.264621\pi$$
−0.673893 + 0.738829i $$0.735379\pi$$
$$468$$ 3.66307e6i 0.773091i
$$469$$ 3.97920e6 0.835340
$$470$$ 0 0
$$471$$ −22308.0 −0.00463349
$$472$$ − 2.23872e6i − 0.462535i
$$473$$ 467328.i 0.0960437i
$$474$$ −108480. −0.0221771
$$475$$ 0 0
$$476$$ −1.43866e6 −0.291031
$$477$$ 1.89902e6i 0.382149i
$$478$$ 2.38128e6i 0.476695i
$$479$$ 5.51052e6 1.09737 0.548686 0.836029i $$-0.315128\pi$$
0.548686 + 0.836029i $$0.315128\pi$$
$$480$$ 0 0
$$481$$ −6.10291e6 −1.20275
$$482$$ 1.09561e6i 0.214802i
$$483$$ 1.10873e6i 0.216251i
$$484$$ 1.98699e6 0.385552
$$485$$ 0 0
$$486$$ 3.44282e6 0.661187
$$487$$ 5.51808e6i 1.05430i 0.849771 + 0.527152i $$0.176740\pi$$
−0.849771 + 0.527152i $$0.823260\pi$$
$$488$$ 629632.i 0.119684i
$$489$$ 259404. 0.0490574
$$490$$ 0 0
$$491$$ −1.51277e6 −0.283184 −0.141592 0.989925i $$-0.545222\pi$$
−0.141592 + 0.989925i $$0.545222\pi$$
$$492$$ − 36288.0i − 0.00675850i
$$493$$ − 4.50342e6i − 0.834498i
$$494$$ 1.21218e7 2.23485
$$495$$ 0 0
$$496$$ −1.75821e6 −0.320897
$$497$$ − 8.28502e6i − 1.50454i
$$498$$ 2.61778e6i 0.472998i
$$499$$ 1.93042e6 0.347057 0.173528 0.984829i $$-0.444483\pi$$
0.173528 + 0.984829i $$0.444483\pi$$
$$500$$ 0 0
$$501$$ 1.11913e6 0.199199
$$502$$ 3.40301e6i 0.602703i
$$503$$ − 6.73105e6i − 1.18621i −0.805124 0.593106i $$-0.797901\pi$$
0.805124 0.593106i $$-0.202099\pi$$
$$504$$ −1.56326e6 −0.274130
$$505$$ 0 0
$$506$$ 1.20269e6 0.208822
$$507$$ 5.11166e6i 0.883165i
$$508$$ − 4.79811e6i − 0.824919i
$$509$$ 556650. 0.0952331 0.0476165 0.998866i $$-0.484837\pi$$
0.0476165 + 0.998866i $$0.484837\pi$$
$$510$$ 0 0
$$511$$ −2.59435e6 −0.439517
$$512$$ 262144.i 0.0441942i
$$513$$ − 7.39800e6i − 1.24114i
$$514$$ −3.30161e6 −0.551211
$$515$$ 0 0
$$516$$ −233664. −0.0386338
$$517$$ 2.51942e6i 0.414548i
$$518$$ − 2.60450e6i − 0.426481i
$$519$$ 2.24672e6 0.366127
$$520$$ 0 0
$$521$$ 1.01110e7 1.63192 0.815962 0.578106i $$-0.196208\pi$$
0.815962 + 0.578106i $$0.196208\pi$$
$$522$$ − 4.89348e6i − 0.786034i
$$523$$ 7.03719e6i 1.12498i 0.826804 + 0.562491i $$0.190157\pi$$
−0.826804 + 0.562491i $$0.809843\pi$$
$$524$$ −125952. −0.0200390
$$525$$ 0 0
$$526$$ 5.45858e6 0.860232
$$527$$ − 5.23342e6i − 0.820840i
$$528$$ − 294912.i − 0.0460371i
$$529$$ 3.98399e6 0.618983
$$530$$ 0 0
$$531$$ 7.24086e6 1.11443
$$532$$ 5.17312e6i 0.792453i
$$533$$ 418068.i 0.0637425i
$$534$$ −923760. −0.140186
$$535$$ 0 0
$$536$$ 2.15821e6 0.324475
$$537$$ 1.63260e6i 0.244312i
$$538$$ 453240.i 0.0675107i
$$539$$ 553536. 0.0820680
$$540$$ 0 0
$$541$$ −4.23114e6 −0.621533 −0.310766 0.950486i $$-0.600586\pi$$
−0.310766 + 0.950486i $$0.600586\pi$$
$$542$$ − 3.39851e6i − 0.496925i
$$543$$ 452508.i 0.0658608i
$$544$$ −780288. −0.113047
$$545$$ 0 0
$$546$$ −3.13219e6 −0.449642
$$547$$ 4.44024e6i 0.634510i 0.948340 + 0.317255i $$0.102761\pi$$
−0.948340 + 0.317255i $$0.897239\pi$$
$$548$$ 2.62781e6i 0.373803i
$$549$$ −2.03647e6 −0.288367
$$550$$ 0 0
$$551$$ −1.61934e7 −2.27227
$$552$$ 601344.i 0.0839992i
$$553$$ 533360.i 0.0741665i
$$554$$ −1.75441e6 −0.242860
$$555$$ 0 0
$$556$$ −4.51360e6 −0.619207
$$557$$ − 9.01448e6i − 1.23113i −0.788088 0.615563i $$-0.788929\pi$$
0.788088 0.615563i $$-0.211071\pi$$
$$558$$ − 5.68670e6i − 0.773170i
$$559$$ 2.69200e6 0.364373
$$560$$ 0 0
$$561$$ 877824. 0.117761
$$562$$ − 5.82679e6i − 0.778196i
$$563$$ 9.81287e6i 1.30474i 0.757899 + 0.652372i $$0.226226\pi$$
−0.757899 + 0.652372i $$0.773774\pi$$
$$564$$ −1.25971e6 −0.166753
$$565$$ 0 0
$$566$$ −481576. −0.0631864
$$567$$ − 4.02392e6i − 0.525644i
$$568$$ − 4.49357e6i − 0.584414i
$$569$$ −1.33152e7 −1.72412 −0.862061 0.506804i $$-0.830827\pi$$
−0.862061 + 0.506804i $$0.830827\pi$$
$$570$$ 0 0
$$571$$ 9.95895e6 1.27827 0.639136 0.769094i $$-0.279292\pi$$
0.639136 + 0.769094i $$0.279292\pi$$
$$572$$ 3.39763e6i 0.434196i
$$573$$ 2.14193e6i 0.272533i
$$574$$ −178416. −0.0226024
$$575$$ 0 0
$$576$$ −847872. −0.106481
$$577$$ 4.50372e6i 0.563160i 0.959538 + 0.281580i $$0.0908585\pi$$
−0.959538 + 0.281580i $$0.909141\pi$$
$$578$$ 3.35685e6i 0.417939i
$$579$$ 2.63216e6 0.326300
$$580$$ 0 0
$$581$$ 1.28707e7 1.58184
$$582$$ − 46032.0i − 0.00563316i
$$583$$ 1.76141e6i 0.214629i
$$584$$ −1.40710e6 −0.170724
$$585$$ 0 0
$$586$$ −1.05684e7 −1.27135
$$587$$ 625842.i 0.0749669i 0.999297 + 0.0374834i $$0.0119341\pi$$
−0.999297 + 0.0374834i $$0.988066\pi$$
$$588$$ 276768.i 0.0330121i
$$589$$ −1.88183e7 −2.23508
$$590$$ 0 0
$$591$$ −940788. −0.110796
$$592$$ − 1.41261e6i − 0.165660i
$$593$$ 2.50385e6i 0.292397i 0.989255 + 0.146198i $$0.0467038\pi$$
−0.989255 + 0.146198i $$0.953296\pi$$
$$594$$ 2.07360e6 0.241134
$$595$$ 0 0
$$596$$ −6.22320e6 −0.717626
$$597$$ − 975120.i − 0.111975i
$$598$$ − 6.92798e6i − 0.792235i
$$599$$ 756480. 0.0861451 0.0430725 0.999072i $$-0.486285\pi$$
0.0430725 + 0.999072i $$0.486285\pi$$
$$600$$ 0 0
$$601$$ −1.38565e7 −1.56483 −0.782413 0.622760i $$-0.786011\pi$$
−0.782413 + 0.622760i $$0.786011\pi$$
$$602$$ 1.14885e6i 0.129203i
$$603$$ 6.98045e6i 0.781791i
$$604$$ 1.56717e6 0.174793
$$605$$ 0 0
$$606$$ 1.86293e6 0.206070
$$607$$ 1.13772e7i 1.25333i 0.779291 + 0.626663i $$0.215580\pi$$
−0.779291 + 0.626663i $$0.784420\pi$$
$$608$$ 2.80576e6i 0.307816i
$$609$$ 4.18428e6 0.457170
$$610$$ 0 0
$$611$$ 1.45129e7 1.57272
$$612$$ − 2.52374e6i − 0.272375i
$$613$$ 7.00161e6i 0.752570i 0.926504 + 0.376285i $$0.122799\pi$$
−0.926504 + 0.376285i $$0.877201\pi$$
$$614$$ 5.79023e6 0.619833
$$615$$ 0 0
$$616$$ −1.44998e6 −0.153961
$$617$$ 7.90300e6i 0.835755i 0.908503 + 0.417878i $$0.137226\pi$$
−0.908503 + 0.417878i $$0.862774\pi$$
$$618$$ 1.12114e6i 0.118083i
$$619$$ −4.02362e6 −0.422076 −0.211038 0.977478i $$-0.567684\pi$$
−0.211038 + 0.977478i $$0.567684\pi$$
$$620$$ 0 0
$$621$$ −4.22820e6 −0.439974
$$622$$ − 3.71227e6i − 0.384737i
$$623$$ 4.54182e6i 0.468824i
$$624$$ −1.69882e6 −0.174657
$$625$$ 0 0
$$626$$ 9.18250e6 0.936538
$$627$$ − 3.15648e6i − 0.320652i
$$628$$ 59488.0i 0.00601908i
$$629$$ 4.20472e6 0.423750
$$630$$ 0 0
$$631$$ −1.00227e7 −1.00210 −0.501049 0.865419i $$-0.667052\pi$$
−0.501049 + 0.865419i $$0.667052\pi$$
$$632$$ 289280.i 0.0288088i
$$633$$ 1.08989e6i 0.108112i
$$634$$ −1.09461e7 −1.08152
$$635$$ 0 0
$$636$$ −880704. −0.0863351
$$637$$ − 3.18860e6i − 0.311352i
$$638$$ − 4.53888e6i − 0.441466i
$$639$$ 1.45339e7 1.40809
$$640$$ 0 0
$$641$$ 6.37390e6 0.612718 0.306359 0.951916i $$-0.400889\pi$$
0.306359 + 0.951916i $$0.400889\pi$$
$$642$$ − 24912.0i − 0.00238545i
$$643$$ − 5.00457e6i − 0.477352i −0.971099 0.238676i $$-0.923287\pi$$
0.971099 0.238676i $$-0.0767134\pi$$
$$644$$ 2.95661e6 0.280918
$$645$$ 0 0
$$646$$ −8.35152e6 −0.787380
$$647$$ − 8.71928e6i − 0.818879i −0.912337 0.409440i $$-0.865724\pi$$
0.912337 0.409440i $$-0.134276\pi$$
$$648$$ − 2.18246e6i − 0.204178i
$$649$$ 6.71616e6 0.625906
$$650$$ 0 0
$$651$$ 4.86254e6 0.449688
$$652$$ − 691744.i − 0.0637274i
$$653$$ 1.58477e6i 0.145440i 0.997352 + 0.0727201i $$0.0231680\pi$$
−0.997352 + 0.0727201i $$0.976832\pi$$
$$654$$ −4.96632e6 −0.454036
$$655$$ 0 0
$$656$$ −96768.0 −0.00877955
$$657$$ − 4.55110e6i − 0.411342i
$$658$$ 6.19358e6i 0.557670i
$$659$$ −1.26410e7 −1.13388 −0.566940 0.823759i $$-0.691873\pi$$
−0.566940 + 0.823759i $$0.691873\pi$$
$$660$$ 0 0
$$661$$ −3.61572e6 −0.321878 −0.160939 0.986964i $$-0.551452\pi$$
−0.160939 + 0.986964i $$0.551452\pi$$
$$662$$ 1.52752e7i 1.35469i
$$663$$ − 5.05663e6i − 0.446763i
$$664$$ 6.98074e6 0.614442
$$665$$ 0 0
$$666$$ 4.56890e6 0.399141
$$667$$ 9.25506e6i 0.805498i
$$668$$ − 2.98435e6i − 0.258767i
$$669$$ 1.72964e6 0.149414
$$670$$ 0 0
$$671$$ −1.88890e6 −0.161958
$$672$$ − 724992.i − 0.0619313i
$$673$$ − 1.11313e7i − 0.947349i −0.880700 0.473675i $$-0.842927\pi$$
0.880700 0.473675i $$-0.157073\pi$$
$$674$$ 8.84351e6 0.749851
$$675$$ 0 0
$$676$$ 1.36311e7 1.14727
$$677$$ − 235518.i − 0.0197493i −0.999951 0.00987467i $$-0.996857\pi$$
0.999951 0.00987467i $$-0.00314326\pi$$
$$678$$ − 3.34526e6i − 0.279483i
$$679$$ −226324. −0.0188389
$$680$$ 0 0
$$681$$ 6.75313e6 0.558004
$$682$$ − 5.27462e6i − 0.434241i
$$683$$ − 2.05830e7i − 1.68833i −0.536084 0.844164i $$-0.680097\pi$$
0.536084 0.844164i $$-0.319903\pi$$
$$684$$ −9.07488e6 −0.741653
$$685$$ 0 0
$$686$$ 9.29368e6 0.754011
$$687$$ − 2.49486e6i − 0.201676i
$$688$$ 623104.i 0.0501868i
$$689$$ 1.01464e7 0.814265
$$690$$ 0 0
$$691$$ −9.54825e6 −0.760727 −0.380363 0.924837i $$-0.624201\pi$$
−0.380363 + 0.924837i $$0.624201\pi$$
$$692$$ − 5.99126e6i − 0.475612i
$$693$$ − 4.68979e6i − 0.370954i
$$694$$ 9.30895e6 0.733672
$$695$$ 0 0
$$696$$ 2.26944e6 0.177581
$$697$$ − 288036.i − 0.0224577i
$$698$$ 1.24516e6i 0.0967357i
$$699$$ −4.62352e6 −0.357915
$$700$$ 0 0
$$701$$ 1.29304e6 0.0993843 0.0496921 0.998765i $$-0.484176\pi$$
0.0496921 + 0.998765i $$0.484176\pi$$
$$702$$ − 1.19448e7i − 0.914821i
$$703$$ − 1.51193e7i − 1.15384i
$$704$$ −786432. −0.0598039
$$705$$ 0 0
$$706$$ −1.23463e7 −0.932234
$$707$$ − 9.15940e6i − 0.689157i
$$708$$ 3.35808e6i 0.251772i
$$709$$ 2.12720e7 1.58926 0.794628 0.607097i $$-0.207666\pi$$
0.794628 + 0.607097i $$0.207666\pi$$
$$710$$ 0 0
$$711$$ −935640. −0.0694120
$$712$$ 2.46336e6i 0.182108i
$$713$$ 1.07553e7i 0.792316i
$$714$$ 2.15798e6 0.158417
$$715$$ 0 0
$$716$$ 4.35360e6 0.317370
$$717$$ − 3.57192e6i − 0.259480i
$$718$$ 1.41230e7i 1.02239i
$$719$$ −8.31732e6 −0.600014 −0.300007 0.953937i $$-0.596989\pi$$
−0.300007 + 0.953937i $$0.596989\pi$$
$$720$$ 0 0
$$721$$ 5.51225e6 0.394903
$$722$$ 2.01260e7i 1.43686i
$$723$$ − 1.64341e6i − 0.116923i
$$724$$ 1.20669e6 0.0855556
$$725$$ 0 0
$$726$$ −2.98049e6 −0.209868
$$727$$ − 4.36740e6i − 0.306469i −0.988190 0.153235i $$-0.951031\pi$$
0.988190 0.153235i $$-0.0489690\pi$$
$$728$$ 8.35251e6i 0.584102i
$$729$$ 3.12231e6 0.217599
$$730$$ 0 0
$$731$$ −1.85471e6 −0.128375
$$732$$ − 944448.i − 0.0651479i
$$733$$ 4.05645e6i 0.278860i 0.990232 + 0.139430i $$0.0445271\pi$$
−0.990232 + 0.139430i $$0.955473\pi$$
$$734$$ −143048. −0.00980035
$$735$$ 0 0
$$736$$ 1.60358e6 0.109118
$$737$$ 6.47462e6i 0.439082i
$$738$$ − 312984.i − 0.0211535i
$$739$$ −768260. −0.0517484 −0.0258742 0.999665i $$-0.508237\pi$$
−0.0258742 + 0.999665i $$0.508237\pi$$
$$740$$ 0 0
$$741$$ −1.81826e7 −1.21650
$$742$$ 4.33013e6i 0.288729i
$$743$$ − 6.18781e6i − 0.411211i −0.978635 0.205605i $$-0.934084\pi$$
0.978635 0.205605i $$-0.0659164\pi$$
$$744$$ 2.63731e6 0.174674
$$745$$ 0 0
$$746$$ −6.86102e6 −0.451379
$$747$$ 2.25783e7i 1.48044i
$$748$$ − 2.34086e6i − 0.152976i
$$749$$ −122484. −0.00797765
$$750$$ 0 0
$$751$$ 1.81698e7 1.17557 0.587787 0.809016i $$-0.299999\pi$$
0.587787 + 0.809016i $$0.299999\pi$$
$$752$$ 3.35923e6i 0.216618i
$$753$$ − 5.10451e6i − 0.328070i
$$754$$ −2.61458e7 −1.67484
$$755$$ 0 0
$$756$$ 5.09760e6 0.324385
$$757$$ 1.93494e7i 1.22724i 0.789603 + 0.613618i $$0.210286\pi$$
−0.789603 + 0.613618i $$0.789714\pi$$
$$758$$ 1.24070e7i 0.784318i
$$759$$ −1.80403e6 −0.113668
$$760$$ 0 0
$$761$$ −3.01992e7 −1.89031 −0.945155 0.326621i $$-0.894090\pi$$
−0.945155 + 0.326621i $$0.894090\pi$$
$$762$$ 7.19717e6i 0.449029i
$$763$$ 2.44177e7i 1.51843i
$$764$$ 5.71181e6 0.354030
$$765$$ 0 0
$$766$$ 2.12779e7 1.31026
$$767$$ − 3.86879e7i − 2.37458i
$$768$$ − 393216.i − 0.0240563i
$$769$$ −2.15854e7 −1.31627 −0.658134 0.752901i $$-0.728654\pi$$
−0.658134 + 0.752901i $$0.728654\pi$$
$$770$$ 0 0
$$771$$ 4.95241e6 0.300041
$$772$$ − 7.01910e6i − 0.423876i
$$773$$ − 3.90895e6i − 0.235294i −0.993055 0.117647i $$-0.962465\pi$$
0.993055 0.117647i $$-0.0375351\pi$$
$$774$$ −2.01535e6 −0.120920
$$775$$ 0 0
$$776$$ −122752. −0.00731769
$$777$$ 3.90674e6i 0.232147i
$$778$$ − 4.64580e6i − 0.275177i
$$779$$ −1.03572e6 −0.0611503
$$780$$ 0 0
$$781$$ 1.34807e7 0.790833
$$782$$ 4.77317e6i 0.279119i
$$783$$ 1.59570e7i 0.930137i
$$784$$ 738048. 0.0428839
$$785$$ 0 0
$$786$$ 188928. 0.0109079
$$787$$ − 2.65082e7i − 1.52561i −0.646628 0.762806i $$-0.723821\pi$$
0.646628 0.762806i $$-0.276179\pi$$
$$788$$ 2.50877e6i 0.143928i
$$789$$ −8.18788e6 −0.468251
$$790$$ 0 0
$$791$$ −1.64475e7 −0.934674
$$792$$ − 2.54362e6i − 0.144092i
$$793$$ 1.08808e7i 0.614439i
$$794$$ −2.51425e6 −0.141533
$$795$$ 0 0
$$796$$ −2.60032e6 −0.145460
$$797$$ 1.07940e7i 0.601919i 0.953637 + 0.300960i $$0.0973070\pi$$
−0.953637 + 0.300960i $$0.902693\pi$$
$$798$$ − 7.75968e6i − 0.431357i
$$799$$ −9.99896e6 −0.554100
$$800$$ 0 0
$$801$$ −7.96743e6 −0.438770
$$802$$ − 1.08973e7i − 0.598249i
$$803$$ − 4.22131e6i − 0.231025i
$$804$$ −3.23731e6 −0.176622
$$805$$ 0 0
$$806$$ −3.03840e7 −1.64743
$$807$$ − 679860.i − 0.0367482i
$$808$$ − 4.96781e6i − 0.267693i
$$809$$ 1.11446e7 0.598675 0.299338 0.954147i $$-0.403234\pi$$
0.299338 + 0.954147i $$0.403234\pi$$
$$810$$ 0 0
$$811$$ −1.14866e7 −0.613253 −0.306626 0.951830i $$-0.599200\pi$$
−0.306626 + 0.951830i $$0.599200\pi$$
$$812$$ − 1.11581e7i − 0.593881i
$$813$$ 5.09777e6i 0.270492i
$$814$$ 4.23782e6 0.224172
$$815$$ 0 0
$$816$$ 1.17043e6 0.0615348
$$817$$ 6.66916e6i 0.349555i
$$818$$ − 7.12076e6i − 0.372086i
$$819$$ −2.70152e7 −1.40734
$$820$$ 0 0
$$821$$ 3.04347e7 1.57584 0.787918 0.615781i $$-0.211159\pi$$
0.787918 + 0.615781i $$0.211159\pi$$
$$822$$ − 3.94171e6i − 0.203472i
$$823$$ − 4.09773e6i − 0.210884i −0.994425 0.105442i $$-0.966374\pi$$
0.994425 0.105442i $$-0.0336257\pi$$
$$824$$ 2.98970e6 0.153394
$$825$$ 0 0
$$826$$ 1.65106e7 0.841999
$$827$$ − 1.70652e7i − 0.867654i −0.900996 0.433827i $$-0.857163\pi$$
0.900996 0.433827i $$-0.142837\pi$$
$$828$$ 5.18659e6i 0.262909i
$$829$$ 2.47617e7 1.25139 0.625697 0.780066i $$-0.284815\pi$$
0.625697 + 0.780066i $$0.284815\pi$$
$$830$$ 0 0
$$831$$ 2.63161e6 0.132196
$$832$$ 4.53018e6i 0.226886i
$$833$$ 2.19685e6i 0.109695i
$$834$$ 6.77040e6 0.337054
$$835$$ 0 0
$$836$$ −8.41728e6 −0.416539
$$837$$ 1.85436e7i 0.914914i
$$838$$ − 2.60232e6i − 0.128012i
$$839$$ −3.16529e7 −1.55242 −0.776208 0.630476i $$-0.782860\pi$$
−0.776208 + 0.630476i $$0.782860\pi$$
$$840$$ 0 0
$$841$$ 1.44170e7 0.702884
$$842$$ − 1.41624e7i − 0.688425i
$$843$$ 8.74019e6i 0.423596i
$$844$$ 2.90637e6 0.140441
$$845$$ 0 0
$$846$$ −1.08650e7 −0.521921
$$847$$ 1.46541e7i 0.701859i
$$848$$ 2.34854e6i 0.112153i
$$849$$ 722364. 0.0343943
$$850$$ 0 0
$$851$$ −8.64119e6 −0.409025
$$852$$ 6.74035e6i 0.318115i
$$853$$ − 2.82671e7i − 1.33017i −0.746765 0.665087i $$-0.768394\pi$$
0.746765 0.665087i $$-0.231606\pi$$
$$854$$ −4.64354e6 −0.217873
$$855$$ 0 0
$$856$$ −66432.0 −0.00309880
$$857$$ 2.60870e7i 1.21331i 0.794966 + 0.606655i $$0.207489\pi$$
−0.794966 + 0.606655i $$0.792511\pi$$
$$858$$ − 5.09645e6i − 0.236347i
$$859$$ 3.38111e7 1.56342 0.781710 0.623642i $$-0.214348\pi$$
0.781710 + 0.623642i $$0.214348\pi$$
$$860$$ 0 0
$$861$$ 267624. 0.0123032
$$862$$ − 2.19499e6i − 0.100615i
$$863$$ − 2.22817e7i − 1.01841i −0.860646 0.509204i $$-0.829940\pi$$
0.860646 0.509204i $$-0.170060\pi$$
$$864$$ 2.76480e6 0.126003
$$865$$ 0 0
$$866$$ −5.96966e6 −0.270492
$$867$$ − 5.03528e6i − 0.227497i
$$868$$ − 1.29668e7i − 0.584162i
$$869$$ −867840. −0.0389843
$$870$$ 0 0
$$871$$ 3.72965e7 1.66580
$$872$$ 1.32435e7i 0.589810i
$$873$$ − 397026.i − 0.0176313i
$$874$$ 1.71634e7 0.760018
$$875$$ 0 0
$$876$$ 2.11066e6 0.0929303
$$877$$ − 3.46748e7i − 1.52235i −0.648545 0.761177i $$-0.724622\pi$$
0.648545 0.761177i $$-0.275378\pi$$
$$878$$ − 1.94485e7i − 0.851431i
$$879$$ 1.58526e7 0.692034
$$880$$ 0 0
$$881$$ 1.42603e7 0.618998 0.309499 0.950900i $$-0.399839\pi$$
0.309499 + 0.950900i $$0.399839\pi$$
$$882$$ 2.38712e6i 0.103325i
$$883$$ 3.75177e7i 1.61933i 0.586895 + 0.809663i $$0.300350\pi$$
−0.586895 + 0.809663i $$0.699650\pi$$
$$884$$ −1.34844e7 −0.580363
$$885$$ 0 0
$$886$$ −7.44622e6 −0.318677
$$887$$ 4.07657e7i 1.73975i 0.493275 + 0.869873i $$0.335800\pi$$
−0.493275 + 0.869873i $$0.664200\pi$$
$$888$$ 2.11891e6i 0.0901738i
$$889$$ 3.53861e7 1.50168
$$890$$ 0 0
$$891$$ 6.54739e6 0.276296
$$892$$ − 4.61238e6i − 0.194095i
$$893$$ 3.59543e7i 1.50877i
$$894$$ 9.33480e6 0.390626
$$895$$ 0 0
$$896$$ −1.93331e6 −0.0804511
$$897$$ 1.03920e7i 0.431238i
$$898$$ − 1.49488e7i − 0.618606i
$$899$$ 4.05899e7 1.67501
$$900$$ 0 0
$$901$$ −6.99059e6 −0.286881
$$902$$ − 290304.i − 0.0118806i
$$903$$ − 1.72327e6i − 0.0703290i
$$904$$ −8.92070e6 −0.363060
$$905$$ 0 0
$$906$$ −2.35075e6 −0.0951451
$$907$$ − 3.57116e7i − 1.44142i −0.693235 0.720712i $$-0.743815\pi$$
0.693235 0.720712i $$-0.256185\pi$$
$$908$$ − 1.80084e7i − 0.724869i
$$909$$ 1.60678e7 0.644979
$$910$$ 0 0
$$911$$ −2.11389e7 −0.843893 −0.421947 0.906621i $$-0.638653\pi$$
−0.421947 + 0.906621i $$0.638653\pi$$
$$912$$ − 4.20864e6i − 0.167554i
$$913$$ 2.09422e7i 0.831468i
$$914$$ 2.59310e7 1.02673
$$915$$ 0 0
$$916$$ −6.65296e6 −0.261985
$$917$$ − 928896.i − 0.0364791i
$$918$$ 8.22960e6i 0.322309i
$$919$$ −1.85996e7 −0.726465 −0.363233 0.931698i $$-0.618327\pi$$
−0.363233 + 0.931698i $$0.618327\pi$$
$$920$$ 0 0
$$921$$ −8.68535e6 −0.337395
$$922$$ 6.03641e6i 0.233857i
$$923$$ − 7.76545e7i − 3.00028i
$$924$$ 2.17498e6 0.0838059
$$925$$ 0 0
$$926$$ 3.47360e7 1.33123
$$927$$ 9.66980e6i 0.369588i
$$928$$ − 6.05184e6i − 0.230684i
$$929$$ −4.45110e7 −1.69211 −0.846055 0.533096i $$-0.821028\pi$$
−0.846055 + 0.533096i $$0.821028\pi$$
$$930$$ 0 0
$$931$$ 7.89942e6 0.298690
$$932$$ 1.23294e7i 0.464945i
$$933$$ 5.56841e6i 0.209424i
$$934$$ −2.78565e7 −1.04486
$$935$$ 0 0
$$936$$ −1.46523e7 −0.546658
$$937$$ − 2.19419e7i − 0.816441i −0.912883 0.408221i $$-0.866149\pi$$
0.912883 0.408221i $$-0.133851\pi$$
$$938$$ 1.59168e7i 0.590675i
$$939$$ −1.37738e7 −0.509787
$$940$$ 0 0
$$941$$ −7.77722e6 −0.286319 −0.143160 0.989700i $$-0.545726\pi$$
−0.143160 + 0.989700i $$0.545726\pi$$
$$942$$ − 89232.0i − 0.00327637i
$$943$$ 591948.i 0.0216773i
$$944$$ 8.95488e6 0.327062
$$945$$ 0 0
$$946$$ −1.86931e6 −0.0679132
$$947$$ 3.17199e7i 1.14936i 0.818378 + 0.574681i $$0.194874\pi$$
−0.818378 + 0.574681i $$0.805126\pi$$
$$948$$ − 433920.i − 0.0156815i
$$949$$ −2.43165e7 −0.876468
$$950$$ 0 0
$$951$$ 1.64191e7 0.588707
$$952$$ − 5.75462e6i − 0.205790i
$$953$$ 5.60285e6i 0.199838i 0.994996 + 0.0999188i $$0.0318583\pi$$
−0.994996 + 0.0999188i $$0.968142\pi$$
$$954$$ −7.59607e6 −0.270220
$$955$$ 0 0
$$956$$ −9.52512e6 −0.337074
$$957$$ 6.80832e6i 0.240304i
$$958$$ 2.20421e7i 0.775959i
$$959$$ −1.93801e7 −0.680470
$$960$$ 0 0
$$961$$ 1.85403e7 0.647601
$$962$$ − 2.44116e7i − 0.850470i
$$963$$ − 214866.i − 0.00746624i
$$964$$ −4.38243e6 −0.151888
$$965$$ 0 0
$$966$$ −4.43491e6 −0.152912
$$967$$ − 2.03532e7i − 0.699949i −0.936759 0.349975i $$-0.886190\pi$$
0.936759 0.349975i $$-0.113810\pi$$
$$968$$ 7.94797e6i 0.272626i
$$969$$ 1.25273e7 0.428595
$$970$$ 0 0
$$971$$ −2.34306e7 −0.797510 −0.398755 0.917057i $$-0.630558\pi$$
−0.398755 + 0.917057i $$0.630558\pi$$
$$972$$ 1.37713e7i 0.467530i
$$973$$ − 3.32878e7i − 1.12721i
$$974$$ −2.20723e7 −0.745505
$$975$$ 0 0
$$976$$ −2.51853e6 −0.0846296
$$977$$ − 4.30412e7i − 1.44261i −0.692619 0.721303i $$-0.743543\pi$$
0.692619 0.721303i $$-0.256457\pi$$
$$978$$ 1.03762e6i 0.0346888i
$$979$$ −7.39008e6 −0.246429
$$980$$ 0 0
$$981$$ −4.28345e7 −1.42109
$$982$$ − 6.05107e6i − 0.200241i
$$983$$ 4.75003e7i 1.56788i 0.620837 + 0.783940i $$0.286793\pi$$
−0.620837 + 0.783940i $$0.713207\pi$$
$$984$$ 145152. 0.00477898
$$985$$ 0 0
$$986$$ 1.80137e7 0.590079
$$987$$ − 9.29038e6i − 0.303557i
$$988$$ 4.84870e7i 1.58028i
$$989$$ 3.81164e6 0.123914
$$990$$ 0 0
$$991$$ 2.09231e7 0.676770 0.338385 0.941008i $$-0.390119\pi$$
0.338385 + 0.941008i $$0.390119\pi$$
$$992$$ − 7.03283e6i − 0.226909i
$$993$$ − 2.29128e7i − 0.737402i
$$994$$ 3.31401e7 1.06387
$$995$$ 0 0
$$996$$ −1.04711e7 −0.334460
$$997$$ 2.96332e7i 0.944148i 0.881559 + 0.472074i $$0.156495\pi$$
−0.881559 + 0.472074i $$0.843505\pi$$
$$998$$ 7.72168e6i 0.245406i
$$999$$ −1.48986e7 −0.472315
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 50.6.b.b.49.2 2
3.2 odd 2 450.6.c.f.199.1 2
4.3 odd 2 400.6.c.i.49.2 2
5.2 odd 4 50.6.a.b.1.1 1
5.3 odd 4 10.6.a.c.1.1 1
5.4 even 2 inner 50.6.b.b.49.1 2
15.2 even 4 450.6.a.u.1.1 1
15.8 even 4 90.6.a.b.1.1 1
15.14 odd 2 450.6.c.f.199.2 2
20.3 even 4 80.6.a.c.1.1 1
20.7 even 4 400.6.a.i.1.1 1
20.19 odd 2 400.6.c.i.49.1 2
35.13 even 4 490.6.a.k.1.1 1
40.3 even 4 320.6.a.k.1.1 1
40.13 odd 4 320.6.a.f.1.1 1
60.23 odd 4 720.6.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.c.1.1 1 5.3 odd 4
50.6.a.b.1.1 1 5.2 odd 4
50.6.b.b.49.1 2 5.4 even 2 inner
50.6.b.b.49.2 2 1.1 even 1 trivial
80.6.a.c.1.1 1 20.3 even 4
90.6.a.b.1.1 1 15.8 even 4
320.6.a.f.1.1 1 40.13 odd 4
320.6.a.k.1.1 1 40.3 even 4
400.6.a.i.1.1 1 20.7 even 4
400.6.c.i.49.1 2 20.19 odd 2
400.6.c.i.49.2 2 4.3 odd 2
450.6.a.u.1.1 1 15.2 even 4
450.6.c.f.199.1 2 3.2 odd 2
450.6.c.f.199.2 2 15.14 odd 2
490.6.a.k.1.1 1 35.13 even 4
720.6.a.v.1.1 1 60.23 odd 4