Properties

 Label 69.6.a.a.1.1 Level $69$ Weight $6$ Character 69.1 Self dual yes Analytic conductor $11.066$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 69.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$11.0664835671$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{29})$$ Defining polynomial: $$x^{2} - x - 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$3.19258$$ of defining polynomial Character $$\chi$$ $$=$$ 69.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +41.6148 q^{5} +96.9330 q^{6} +0.236813 q^{7} -560.057 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q-10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +41.6148 q^{5} +96.9330 q^{6} +0.236813 q^{7} -560.057 q^{8} +81.0000 q^{9} -448.205 q^{10} -421.598 q^{11} -756.000 q^{12} +254.502 q^{13} -2.55055 q^{14} -374.534 q^{15} +3344.00 q^{16} -975.007 q^{17} -872.397 q^{18} +2039.79 q^{19} +3495.65 q^{20} -2.13132 q^{21} +4540.75 q^{22} +529.000 q^{23} +5040.51 q^{24} -1393.21 q^{25} -2741.07 q^{26} -729.000 q^{27} +19.8923 q^{28} +2671.55 q^{29} +4033.85 q^{30} +9039.14 q^{31} -18094.2 q^{32} +3794.38 q^{33} +10501.1 q^{34} +9.85493 q^{35} +6804.00 q^{36} -12665.3 q^{37} -21969.2 q^{38} -2290.52 q^{39} -23306.7 q^{40} +10146.4 q^{41} +22.9550 q^{42} +19523.2 q^{43} -35414.2 q^{44} +3370.80 q^{45} -5697.50 q^{46} +27679.7 q^{47} -30096.0 q^{48} -16806.9 q^{49} +15005.3 q^{50} +8775.06 q^{51} +21378.2 q^{52} +10852.8 q^{53} +7851.57 q^{54} -17544.7 q^{55} -132.629 q^{56} -18358.1 q^{57} -28773.4 q^{58} +11907.7 q^{59} -31460.8 q^{60} +39861.9 q^{61} -97354.5 q^{62} +19.1818 q^{63} +87872.0 q^{64} +10591.1 q^{65} -40866.7 q^{66} -28550.5 q^{67} -81900.6 q^{68} -4761.00 q^{69} -106.141 q^{70} +52179.8 q^{71} -45364.6 q^{72} +56918.0 q^{73} +136410. q^{74} +12538.8 q^{75} +171342. q^{76} -99.8398 q^{77} +24669.7 q^{78} +23178.3 q^{79} +139160. q^{80} +6561.00 q^{81} -109281. q^{82} -18344.6 q^{83} -179.031 q^{84} -40574.8 q^{85} -210272. q^{86} -24043.9 q^{87} +236119. q^{88} +47362.4 q^{89} -36304.6 q^{90} +60.2694 q^{91} +44436.0 q^{92} -81352.3 q^{93} -298120. q^{94} +84885.4 q^{95} +162847. q^{96} -140379. q^{97} +181016. q^{98} -34149.4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 18q^{3} + 168q^{4} + 94q^{5} - 118q^{7} + 162q^{9} + O(q^{10})$$ $$2q - 18q^{3} + 168q^{4} + 94q^{5} - 118q^{7} + 162q^{9} + 116q^{10} + 320q^{11} - 1512q^{12} - 288q^{13} - 1276q^{14} - 846q^{15} + 6688q^{16} - 1810q^{17} + 730q^{19} + 7896q^{20} + 1062q^{21} + 12528q^{22} + 1058q^{23} - 1774q^{25} - 8584q^{26} - 1458q^{27} - 9912q^{28} + 8208q^{29} - 1044q^{30} + 1772q^{31} - 2880q^{33} + 1508q^{34} - 6184q^{35} + 13608q^{36} - 23112q^{37} - 36076q^{38} + 2592q^{39} + 6032q^{40} + 5516q^{41} + 11484q^{42} + 10322q^{43} + 26880q^{44} + 7614q^{45} + 42952q^{47} - 60192q^{48} - 19634q^{49} + 10904q^{50} + 16290q^{51} - 24192q^{52} - 25350q^{53} + 21304q^{55} - 66352q^{56} - 6570q^{57} + 30856q^{58} + 18344q^{59} - 71064q^{60} + 37224q^{61} - 175624q^{62} - 9558q^{63} + 175744q^{64} - 17828q^{65} - 112752q^{66} - 7482q^{67} - 152040q^{68} - 9522q^{69} - 66816q^{70} + 126848q^{71} + 137660q^{73} + 23896q^{74} + 15966q^{75} + 61320q^{76} - 87784q^{77} + 77256q^{78} + 62286q^{79} + 314336q^{80} + 13122q^{81} - 159152q^{82} + 83120q^{83} + 89208q^{84} - 84316q^{85} - 309372q^{86} - 73872q^{87} + 651456q^{88} + 69770q^{89} + 9396q^{90} + 64204q^{91} + 88872q^{92} - 15948q^{93} - 133632q^{94} + 16272q^{95} - 170104q^{97} + 150568q^{98} + 25920q^{99} + O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −10.7703 −1.90394 −0.951972 0.306186i $$-0.900947\pi$$
−0.951972 + 0.306186i $$0.900947\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ 84.0000 2.62500
$$5$$ 41.6148 0.744429 0.372214 0.928147i $$-0.378599\pi$$
0.372214 + 0.928147i $$0.378599\pi$$
$$6$$ 96.9330 1.09924
$$7$$ 0.236813 0.00182667 0.000913335 1.00000i $$-0.499709\pi$$
0.000913335 1.00000i $$0.499709\pi$$
$$8$$ −560.057 −3.09391
$$9$$ 81.0000 0.333333
$$10$$ −448.205 −1.41735
$$11$$ −421.598 −1.05055 −0.525275 0.850933i $$-0.676037\pi$$
−0.525275 + 0.850933i $$0.676037\pi$$
$$12$$ −756.000 −1.51554
$$13$$ 254.502 0.417670 0.208835 0.977951i $$-0.433033\pi$$
0.208835 + 0.977951i $$0.433033\pi$$
$$14$$ −2.55055 −0.00347788
$$15$$ −374.534 −0.429796
$$16$$ 3344.00 3.26562
$$17$$ −975.007 −0.818249 −0.409125 0.912479i $$-0.634166\pi$$
−0.409125 + 0.912479i $$0.634166\pi$$
$$18$$ −872.397 −0.634648
$$19$$ 2039.79 1.29629 0.648143 0.761519i $$-0.275546\pi$$
0.648143 + 0.761519i $$0.275546\pi$$
$$20$$ 3495.65 1.95413
$$21$$ −2.13132 −0.00105463
$$22$$ 4540.75 2.00019
$$23$$ 529.000 0.208514
$$24$$ 5040.51 1.78627
$$25$$ −1393.21 −0.445826
$$26$$ −2741.07 −0.795220
$$27$$ −729.000 −0.192450
$$28$$ 19.8923 0.00479501
$$29$$ 2671.55 0.589885 0.294943 0.955515i $$-0.404699\pi$$
0.294943 + 0.955515i $$0.404699\pi$$
$$30$$ 4033.85 0.818308
$$31$$ 9039.14 1.68936 0.844681 0.535270i $$-0.179790\pi$$
0.844681 + 0.535270i $$0.179790\pi$$
$$32$$ −18094.2 −3.12366
$$33$$ 3794.38 0.606535
$$34$$ 10501.1 1.55790
$$35$$ 9.85493 0.00135983
$$36$$ 6804.00 0.875000
$$37$$ −12665.3 −1.52094 −0.760471 0.649372i $$-0.775032\pi$$
−0.760471 + 0.649372i $$0.775032\pi$$
$$38$$ −21969.2 −2.46805
$$39$$ −2290.52 −0.241142
$$40$$ −23306.7 −2.30319
$$41$$ 10146.4 0.942658 0.471329 0.881957i $$-0.343774\pi$$
0.471329 + 0.881957i $$0.343774\pi$$
$$42$$ 22.9550 0.00200795
$$43$$ 19523.2 1.61020 0.805102 0.593137i $$-0.202111\pi$$
0.805102 + 0.593137i $$0.202111\pi$$
$$44$$ −35414.2 −2.75769
$$45$$ 3370.80 0.248143
$$46$$ −5697.50 −0.397000
$$47$$ 27679.7 1.82775 0.913875 0.405995i $$-0.133075\pi$$
0.913875 + 0.405995i $$0.133075\pi$$
$$48$$ −30096.0 −1.88541
$$49$$ −16806.9 −0.999997
$$50$$ 15005.3 0.848827
$$51$$ 8775.06 0.472416
$$52$$ 21378.2 1.09638
$$53$$ 10852.8 0.530703 0.265351 0.964152i $$-0.414512\pi$$
0.265351 + 0.964152i $$0.414512\pi$$
$$54$$ 7851.57 0.366414
$$55$$ −17544.7 −0.782059
$$56$$ −132.629 −0.00565155
$$57$$ −18358.1 −0.748411
$$58$$ −28773.4 −1.12311
$$59$$ 11907.7 0.445345 0.222672 0.974893i $$-0.428522\pi$$
0.222672 + 0.974893i $$0.428522\pi$$
$$60$$ −31460.8 −1.12821
$$61$$ 39861.9 1.37162 0.685809 0.727782i $$-0.259449\pi$$
0.685809 + 0.727782i $$0.259449\pi$$
$$62$$ −97354.5 −3.21645
$$63$$ 19.1818 0.000608890 0
$$64$$ 87872.0 2.68164
$$65$$ 10591.1 0.310925
$$66$$ −40866.7 −1.15481
$$67$$ −28550.5 −0.777009 −0.388504 0.921447i $$-0.627008\pi$$
−0.388504 + 0.921447i $$0.627008\pi$$
$$68$$ −81900.6 −2.14790
$$69$$ −4761.00 −0.120386
$$70$$ −106.141 −0.00258903
$$71$$ 52179.8 1.22845 0.614223 0.789132i $$-0.289469\pi$$
0.614223 + 0.789132i $$0.289469\pi$$
$$72$$ −45364.6 −1.03130
$$73$$ 56918.0 1.25009 0.625047 0.780587i $$-0.285080\pi$$
0.625047 + 0.780587i $$0.285080\pi$$
$$74$$ 136410. 2.89579
$$75$$ 12538.8 0.257398
$$76$$ 171342. 3.40275
$$77$$ −99.8398 −0.00191901
$$78$$ 24669.7 0.459120
$$79$$ 23178.3 0.417844 0.208922 0.977932i $$-0.433004\pi$$
0.208922 + 0.977932i $$0.433004\pi$$
$$80$$ 139160. 2.43103
$$81$$ 6561.00 0.111111
$$82$$ −109281. −1.79477
$$83$$ −18344.6 −0.292289 −0.146144 0.989263i $$-0.546686\pi$$
−0.146144 + 0.989263i $$0.546686\pi$$
$$84$$ −179.031 −0.00276840
$$85$$ −40574.8 −0.609128
$$86$$ −210272. −3.06574
$$87$$ −24043.9 −0.340571
$$88$$ 236119. 3.25030
$$89$$ 47362.4 0.633810 0.316905 0.948457i $$-0.397356\pi$$
0.316905 + 0.948457i $$0.397356\pi$$
$$90$$ −36304.6 −0.472450
$$91$$ 60.2694 0.000762945 0
$$92$$ 44436.0 0.547350
$$93$$ −81352.3 −0.975354
$$94$$ −298120. −3.47993
$$95$$ 84885.4 0.964992
$$96$$ 162847. 1.80344
$$97$$ −140379. −1.51486 −0.757432 0.652915i $$-0.773546\pi$$
−0.757432 + 0.652915i $$0.773546\pi$$
$$98$$ 181016. 1.90394
$$99$$ −34149.4 −0.350183
$$100$$ −117029. −1.17029
$$101$$ 87080.3 0.849408 0.424704 0.905332i $$-0.360378\pi$$
0.424704 + 0.905332i $$0.360378\pi$$
$$102$$ −94510.3 −0.899454
$$103$$ 17988.9 0.167075 0.0835375 0.996505i $$-0.473378\pi$$
0.0835375 + 0.996505i $$0.473378\pi$$
$$104$$ −142536. −1.29223
$$105$$ −88.6944 −0.000785096 0
$$106$$ −116888. −1.01043
$$107$$ −127485. −1.07646 −0.538231 0.842797i $$-0.680907\pi$$
−0.538231 + 0.842797i $$0.680907\pi$$
$$108$$ −61236.0 −0.505181
$$109$$ −136418. −1.09978 −0.549890 0.835237i $$-0.685330\pi$$
−0.549890 + 0.835237i $$0.685330\pi$$
$$110$$ 188962. 1.48900
$$111$$ 113988. 0.878116
$$112$$ 791.902 0.00596522
$$113$$ −104012. −0.766279 −0.383139 0.923691i $$-0.625157\pi$$
−0.383139 + 0.923691i $$0.625157\pi$$
$$114$$ 197723. 1.42493
$$115$$ 22014.2 0.155224
$$116$$ 224410. 1.54845
$$117$$ 20614.7 0.139223
$$118$$ −128249. −0.847912
$$119$$ −230.894 −0.00149467
$$120$$ 209760. 1.32975
$$121$$ 16693.7 0.103655
$$122$$ −429325. −2.61148
$$123$$ −91318.0 −0.544244
$$124$$ 759288. 4.43458
$$125$$ −188024. −1.07631
$$126$$ −206.595 −0.00115929
$$127$$ 203763. 1.12103 0.560513 0.828145i $$-0.310604\pi$$
0.560513 + 0.828145i $$0.310604\pi$$
$$128$$ −367397. −1.98203
$$129$$ −175709. −0.929651
$$130$$ −114069. −0.591984
$$131$$ 60674.1 0.308905 0.154453 0.988000i $$-0.450639\pi$$
0.154453 + 0.988000i $$0.450639\pi$$
$$132$$ 318728. 1.59215
$$133$$ 483.048 0.00236789
$$134$$ 307498. 1.47938
$$135$$ −30337.2 −0.143265
$$136$$ 546060. 2.53159
$$137$$ 94446.5 0.429917 0.214958 0.976623i $$-0.431038\pi$$
0.214958 + 0.976623i $$0.431038\pi$$
$$138$$ 51277.5 0.229208
$$139$$ 403508. 1.77139 0.885697 0.464263i $$-0.153681\pi$$
0.885697 + 0.464263i $$0.153681\pi$$
$$140$$ 827.814 0.00356954
$$141$$ −249117. −1.05525
$$142$$ −561993. −2.33889
$$143$$ −107298. −0.438783
$$144$$ 270864. 1.08854
$$145$$ 111176. 0.439128
$$146$$ −613026. −2.38011
$$147$$ 151262. 0.577348
$$148$$ −1.06389e6 −3.99247
$$149$$ 384528. 1.41893 0.709467 0.704738i $$-0.248936\pi$$
0.709467 + 0.704738i $$0.248936\pi$$
$$150$$ −135048. −0.490070
$$151$$ −442508. −1.57935 −0.789675 0.613525i $$-0.789751\pi$$
−0.789675 + 0.613525i $$0.789751\pi$$
$$152$$ −1.14240e6 −4.01059
$$153$$ −78975.6 −0.272750
$$154$$ 1075.31 0.00365368
$$155$$ 376162. 1.25761
$$156$$ −192404. −0.632997
$$157$$ 240724. 0.779416 0.389708 0.920938i $$-0.372576\pi$$
0.389708 + 0.920938i $$0.372576\pi$$
$$158$$ −249638. −0.795552
$$159$$ −97675.1 −0.306402
$$160$$ −752985. −2.32534
$$161$$ 125.274 0.000380887 0
$$162$$ −70664.1 −0.211549
$$163$$ −178188. −0.525301 −0.262651 0.964891i $$-0.584597\pi$$
−0.262651 + 0.964891i $$0.584597\pi$$
$$164$$ 852301. 2.47448
$$165$$ 157903. 0.451522
$$166$$ 197577. 0.556502
$$167$$ 25525.2 0.0708236 0.0354118 0.999373i $$-0.488726\pi$$
0.0354118 + 0.999373i $$0.488726\pi$$
$$168$$ 1193.66 0.00326292
$$169$$ −306522. −0.825552
$$170$$ 437004. 1.15975
$$171$$ 165223. 0.432095
$$172$$ 1.63995e6 4.22678
$$173$$ 701865. 1.78295 0.891474 0.453072i $$-0.149672\pi$$
0.891474 + 0.453072i $$0.149672\pi$$
$$174$$ 258961. 0.648427
$$175$$ −329.929 −0.000814377 0
$$176$$ −1.40982e6 −3.43070
$$177$$ −107169. −0.257120
$$178$$ −510109. −1.20674
$$179$$ −292917. −0.683302 −0.341651 0.939827i $$-0.610986\pi$$
−0.341651 + 0.939827i $$0.610986\pi$$
$$180$$ 283147. 0.651375
$$181$$ −38932.7 −0.0883321 −0.0441660 0.999024i $$-0.514063\pi$$
−0.0441660 + 0.999024i $$0.514063\pi$$
$$182$$ −649.121 −0.00145260
$$183$$ −358757. −0.791904
$$184$$ −296270. −0.645124
$$185$$ −527066. −1.13223
$$186$$ 876191. 1.85702
$$187$$ 411061. 0.859611
$$188$$ 2.32510e6 4.79784
$$189$$ −172.637 −0.000351543 0
$$190$$ −914243. −1.83729
$$191$$ 37244.8 0.0738724 0.0369362 0.999318i $$-0.488240\pi$$
0.0369362 + 0.999318i $$0.488240\pi$$
$$192$$ −790848. −1.54825
$$193$$ −42454.3 −0.0820406 −0.0410203 0.999158i $$-0.513061\pi$$
−0.0410203 + 0.999158i $$0.513061\pi$$
$$194$$ 1.51193e6 2.88421
$$195$$ −95319.6 −0.179513
$$196$$ −1.41178e6 −2.62499
$$197$$ 87335.1 0.160333 0.0801666 0.996781i $$-0.474455\pi$$
0.0801666 + 0.996781i $$0.474455\pi$$
$$198$$ 367801. 0.666729
$$199$$ 1.08779e6 1.94721 0.973604 0.228245i $$-0.0732989\pi$$
0.973604 + 0.228245i $$0.0732989\pi$$
$$200$$ 780275. 1.37934
$$201$$ 256954. 0.448606
$$202$$ −937883. −1.61722
$$203$$ 632.657 0.00107753
$$204$$ 737105. 1.24009
$$205$$ 422243. 0.701742
$$206$$ −193746. −0.318101
$$207$$ 42849.0 0.0695048
$$208$$ 851055. 1.36395
$$209$$ −859969. −1.36181
$$210$$ 955.267 0.00149478
$$211$$ −1.02777e6 −1.58923 −0.794617 0.607111i $$-0.792328\pi$$
−0.794617 + 0.607111i $$0.792328\pi$$
$$212$$ 911634. 1.39310
$$213$$ −469618. −0.709244
$$214$$ 1.37305e6 2.04952
$$215$$ 812456. 1.19868
$$216$$ 408282. 0.595423
$$217$$ 2140.58 0.00308591
$$218$$ 1.46927e6 2.09392
$$219$$ −512262. −0.721742
$$220$$ −1.47376e6 −2.05291
$$221$$ −248141. −0.341758
$$222$$ −1.22769e6 −1.67188
$$223$$ −209627. −0.282284 −0.141142 0.989989i $$-0.545077\pi$$
−0.141142 + 0.989989i $$0.545077\pi$$
$$224$$ −4284.93 −0.00570589
$$225$$ −112850. −0.148609
$$226$$ 1.12024e6 1.45895
$$227$$ −347103. −0.447088 −0.223544 0.974694i $$-0.571763\pi$$
−0.223544 + 0.974694i $$0.571763\pi$$
$$228$$ −1.54208e6 −1.96458
$$229$$ −474094. −0.597414 −0.298707 0.954345i $$-0.596555\pi$$
−0.298707 + 0.954345i $$0.596555\pi$$
$$230$$ −237101. −0.295538
$$231$$ 898.558 0.00110794
$$232$$ −1.49622e6 −1.82505
$$233$$ 67426.2 0.0813652 0.0406826 0.999172i $$-0.487047\pi$$
0.0406826 + 0.999172i $$0.487047\pi$$
$$234$$ −222027. −0.265073
$$235$$ 1.15189e6 1.36063
$$236$$ 1.00024e6 1.16903
$$237$$ −208605. −0.241243
$$238$$ 2486.81 0.00284577
$$239$$ −631686. −0.715331 −0.357665 0.933850i $$-0.616427\pi$$
−0.357665 + 0.933850i $$0.616427\pi$$
$$240$$ −1.25244e6 −1.40355
$$241$$ −582036. −0.645516 −0.322758 0.946482i $$-0.604610\pi$$
−0.322758 + 0.946482i $$0.604610\pi$$
$$242$$ −179797. −0.197353
$$243$$ −59049.0 −0.0641500
$$244$$ 3.34840e6 3.60050
$$245$$ −699418. −0.744426
$$246$$ 983525. 1.03621
$$247$$ 519130. 0.541419
$$248$$ −5.06243e6 −5.22673
$$249$$ 165101. 0.168753
$$250$$ 2.02508e6 2.04924
$$251$$ −1.28835e6 −1.29077 −0.645386 0.763856i $$-0.723304\pi$$
−0.645386 + 0.763856i $$0.723304\pi$$
$$252$$ 1611.27 0.00159834
$$253$$ −223025. −0.219055
$$254$$ −2.19459e6 −2.13437
$$255$$ 365173. 0.351680
$$256$$ 1.14509e6 1.09204
$$257$$ −204908. −0.193520 −0.0967601 0.995308i $$-0.530848\pi$$
−0.0967601 + 0.995308i $$0.530848\pi$$
$$258$$ 1.89245e6 1.77000
$$259$$ −2999.32 −0.00277826
$$260$$ 889650. 0.816179
$$261$$ 216395. 0.196628
$$262$$ −653480. −0.588138
$$263$$ 418862. 0.373407 0.186703 0.982416i $$-0.440220\pi$$
0.186703 + 0.982416i $$0.440220\pi$$
$$264$$ −2.12507e6 −1.87656
$$265$$ 451637. 0.395071
$$266$$ −5202.58 −0.00450832
$$267$$ −426262. −0.365930
$$268$$ −2.39824e6 −2.03965
$$269$$ −1.48662e6 −1.25262 −0.626308 0.779576i $$-0.715435\pi$$
−0.626308 + 0.779576i $$0.715435\pi$$
$$270$$ 326742. 0.272769
$$271$$ −658640. −0.544785 −0.272392 0.962186i $$-0.587815\pi$$
−0.272392 + 0.962186i $$0.587815\pi$$
$$272$$ −3.26042e6 −2.67209
$$273$$ −542.425 −0.000440487 0
$$274$$ −1.01722e6 −0.818537
$$275$$ 587372. 0.468362
$$276$$ −399924. −0.316013
$$277$$ 236526. 0.185216 0.0926082 0.995703i $$-0.470480\pi$$
0.0926082 + 0.995703i $$0.470480\pi$$
$$278$$ −4.34592e6 −3.37263
$$279$$ 732170. 0.563121
$$280$$ −5519.32 −0.00420718
$$281$$ −1.66253e6 −1.25604 −0.628020 0.778197i $$-0.716134\pi$$
−0.628020 + 0.778197i $$0.716134\pi$$
$$282$$ 2.68308e6 2.00914
$$283$$ 966812. 0.717589 0.358795 0.933417i $$-0.383188\pi$$
0.358795 + 0.933417i $$0.383188\pi$$
$$284$$ 4.38310e6 3.22467
$$285$$ −763968. −0.557139
$$286$$ 1.15563e6 0.835418
$$287$$ 2402.81 0.00172193
$$288$$ −1.46563e6 −1.04122
$$289$$ −469218. −0.330469
$$290$$ −1.19740e6 −0.836074
$$291$$ 1.26341e6 0.874607
$$292$$ 4.78111e6 3.28150
$$293$$ −82436.0 −0.0560981 −0.0280490 0.999607i $$-0.508929\pi$$
−0.0280490 + 0.999607i $$0.508929\pi$$
$$294$$ −1.62915e6 −1.09924
$$295$$ 495535. 0.331528
$$296$$ 7.09332e6 4.70565
$$297$$ 307345. 0.202178
$$298$$ −4.14149e6 −2.70157
$$299$$ 134632. 0.0870902
$$300$$ 1.05326e6 0.675669
$$301$$ 4623.35 0.00294131
$$302$$ 4.76595e6 3.00699
$$303$$ −783722. −0.490406
$$304$$ 6.82105e6 4.23318
$$305$$ 1.65884e6 1.02107
$$306$$ 850593. 0.519300
$$307$$ 471104. 0.285280 0.142640 0.989775i $$-0.454441\pi$$
0.142640 + 0.989775i $$0.454441\pi$$
$$308$$ −8386.54 −0.00503740
$$309$$ −161900. −0.0964608
$$310$$ −4.05139e6 −2.39442
$$311$$ 974141. 0.571112 0.285556 0.958362i $$-0.407822\pi$$
0.285556 + 0.958362i $$0.407822\pi$$
$$312$$ 1.28282e6 0.746071
$$313$$ 203897. 0.117639 0.0588194 0.998269i $$-0.481266\pi$$
0.0588194 + 0.998269i $$0.481266\pi$$
$$314$$ −2.59267e6 −1.48396
$$315$$ 798.249 0.000453275 0
$$316$$ 1.94698e6 1.09684
$$317$$ −1.11600e6 −0.623757 −0.311879 0.950122i $$-0.600958\pi$$
−0.311879 + 0.950122i $$0.600958\pi$$
$$318$$ 1.05199e6 0.583371
$$319$$ −1.12632e6 −0.619704
$$320$$ 3.65678e6 1.99629
$$321$$ 1.14736e6 0.621496
$$322$$ −1349.24 −0.000725187 0
$$323$$ −1.98881e6 −1.06068
$$324$$ 551124. 0.291667
$$325$$ −354574. −0.186208
$$326$$ 1.91914e6 1.00014
$$327$$ 1.22776e6 0.634958
$$328$$ −5.68259e6 −2.91650
$$329$$ 6554.91 0.00333870
$$330$$ −1.70066e6 −0.859673
$$331$$ 2.08998e6 1.04851 0.524254 0.851562i $$-0.324344\pi$$
0.524254 + 0.851562i $$0.324344\pi$$
$$332$$ −1.54094e6 −0.767258
$$333$$ −1.02589e6 −0.506981
$$334$$ −274915. −0.134844
$$335$$ −1.18812e6 −0.578428
$$336$$ −7127.12 −0.00344402
$$337$$ 901699. 0.432501 0.216250 0.976338i $$-0.430617\pi$$
0.216250 + 0.976338i $$0.430617\pi$$
$$338$$ 3.30134e6 1.57180
$$339$$ 936107. 0.442411
$$340$$ −3.40828e6 −1.59896
$$341$$ −3.81088e6 −1.77476
$$342$$ −1.77950e6 −0.822685
$$343$$ −7960.21 −0.00365333
$$344$$ −1.09341e7 −4.98182
$$345$$ −198128. −0.0896187
$$346$$ −7.55932e6 −3.39463
$$347$$ −1.53207e6 −0.683053 −0.341526 0.939872i $$-0.610944\pi$$
−0.341526 + 0.939872i $$0.610944\pi$$
$$348$$ −2.01969e6 −0.893998
$$349$$ 1.28353e6 0.564084 0.282042 0.959402i $$-0.408988\pi$$
0.282042 + 0.959402i $$0.408988\pi$$
$$350$$ 3553.44 0.00155053
$$351$$ −185532. −0.0803806
$$352$$ 7.62846e6 3.28156
$$353$$ −4.45978e6 −1.90492 −0.952460 0.304663i $$-0.901456\pi$$
−0.952460 + 0.304663i $$0.901456\pi$$
$$354$$ 1.15425e6 0.489542
$$355$$ 2.17145e6 0.914491
$$356$$ 3.97844e6 1.66375
$$357$$ 2078.05 0.000862949 0
$$358$$ 3.15482e6 1.30097
$$359$$ 4.67506e6 1.91448 0.957240 0.289296i $$-0.0934213\pi$$
0.957240 + 0.289296i $$0.0934213\pi$$
$$360$$ −1.88784e6 −0.767731
$$361$$ 1.68463e6 0.680356
$$362$$ 419318. 0.168179
$$363$$ −150243. −0.0598451
$$364$$ 5062.63 0.00200273
$$365$$ 2.36863e6 0.930606
$$366$$ 3.86393e6 1.50774
$$367$$ 2.43229e6 0.942650 0.471325 0.881960i $$-0.343776\pi$$
0.471325 + 0.881960i $$0.343776\pi$$
$$368$$ 1.76898e6 0.680930
$$369$$ 821862. 0.314219
$$370$$ 5.67668e6 2.15571
$$371$$ 2570.08 0.000969419 0
$$372$$ −6.83359e6 −2.56030
$$373$$ 1.37458e6 0.511561 0.255780 0.966735i $$-0.417668\pi$$
0.255780 + 0.966735i $$0.417668\pi$$
$$374$$ −4.42726e6 −1.63665
$$375$$ 1.69222e6 0.621410
$$376$$ −1.55022e7 −5.65489
$$377$$ 679914. 0.246377
$$378$$ 1859.35 0.000669318 0
$$379$$ 2.01997e6 0.722350 0.361175 0.932498i $$-0.382376\pi$$
0.361175 + 0.932498i $$0.382376\pi$$
$$380$$ 7.13037e6 2.53310
$$381$$ −1.83387e6 −0.647225
$$382$$ −401139. −0.140649
$$383$$ 2.00834e6 0.699584 0.349792 0.936827i $$-0.386252\pi$$
0.349792 + 0.936827i $$0.386252\pi$$
$$384$$ 3.30658e6 1.14433
$$385$$ −4154.82 −0.00142857
$$386$$ 457247. 0.156201
$$387$$ 1.58138e6 0.536734
$$388$$ −1.17919e7 −3.97652
$$389$$ 3.25055e6 1.08914 0.544569 0.838716i $$-0.316693\pi$$
0.544569 + 0.838716i $$0.316693\pi$$
$$390$$ 1.02662e6 0.341782
$$391$$ −515779. −0.170617
$$392$$ 9.41285e6 3.09390
$$393$$ −546067. −0.178346
$$394$$ −940628. −0.305265
$$395$$ 964563. 0.311055
$$396$$ −2.86855e6 −0.919231
$$397$$ 5.43707e6 1.73137 0.865683 0.500592i $$-0.166884\pi$$
0.865683 + 0.500592i $$0.166884\pi$$
$$398$$ −1.17159e7 −3.70737
$$399$$ −4347.43 −0.00136710
$$400$$ −4.65888e6 −1.45590
$$401$$ 3.27208e6 1.01616 0.508081 0.861309i $$-0.330355\pi$$
0.508081 + 0.861309i $$0.330355\pi$$
$$402$$ −2.76748e6 −0.854121
$$403$$ 2.30048e6 0.705596
$$404$$ 7.31474e6 2.22970
$$405$$ 273035. 0.0827143
$$406$$ −6813.92 −0.00205155
$$407$$ 5.33968e6 1.59783
$$408$$ −4.91454e6 −1.46161
$$409$$ −1.56299e6 −0.462007 −0.231003 0.972953i $$-0.574201\pi$$
−0.231003 + 0.972953i $$0.574201\pi$$
$$410$$ −4.54769e6 −1.33608
$$411$$ −850019. −0.248213
$$412$$ 1.51107e6 0.438572
$$413$$ 2819.89 0.000813498 0
$$414$$ −461498. −0.132333
$$415$$ −763406. −0.217588
$$416$$ −4.60500e6 −1.30466
$$417$$ −3.63157e6 −1.02272
$$418$$ 9.26215e6 2.59281
$$419$$ −4.76111e6 −1.32487 −0.662434 0.749120i $$-0.730477\pi$$
−0.662434 + 0.749120i $$0.730477\pi$$
$$420$$ −7450.33 −0.00206088
$$421$$ −4.56989e6 −1.25661 −0.628305 0.777967i $$-0.716251\pi$$
−0.628305 + 0.777967i $$0.716251\pi$$
$$422$$ 1.10694e7 3.02581
$$423$$ 2.24206e6 0.609250
$$424$$ −6.07818e6 −1.64195
$$425$$ 1.35839e6 0.364796
$$426$$ 5.05794e6 1.35036
$$427$$ 9439.80 0.00250549
$$428$$ −1.07087e7 −2.82571
$$429$$ 965678. 0.253331
$$430$$ −8.75042e6 −2.28222
$$431$$ −3.28891e6 −0.852824 −0.426412 0.904529i $$-0.640223\pi$$
−0.426412 + 0.904529i $$0.640223\pi$$
$$432$$ −2.43778e6 −0.628470
$$433$$ 2.11805e6 0.542895 0.271447 0.962453i $$-0.412498\pi$$
0.271447 + 0.962453i $$0.412498\pi$$
$$434$$ −23054.8 −0.00587540
$$435$$ −1.00058e6 −0.253531
$$436$$ −1.14591e7 −2.88692
$$437$$ 1.07905e6 0.270294
$$438$$ 5.51723e6 1.37416
$$439$$ 758443. 0.187829 0.0939143 0.995580i $$-0.470062\pi$$
0.0939143 + 0.995580i $$0.470062\pi$$
$$440$$ 9.82605e6 2.41962
$$441$$ −1.36136e6 −0.333332
$$442$$ 2.67257e6 0.650688
$$443$$ 650377. 0.157455 0.0787274 0.996896i $$-0.474914\pi$$
0.0787274 + 0.996896i $$0.474914\pi$$
$$444$$ 9.57500e6 2.30506
$$445$$ 1.97098e6 0.471826
$$446$$ 2.25776e6 0.537452
$$447$$ −3.46075e6 −0.819222
$$448$$ 20809.2 0.00489847
$$449$$ −7.58111e6 −1.77467 −0.887333 0.461129i $$-0.847445\pi$$
−0.887333 + 0.461129i $$0.847445\pi$$
$$450$$ 1.21543e6 0.282942
$$451$$ −4.27772e6 −0.990309
$$452$$ −8.73699e6 −2.01148
$$453$$ 3.98257e6 0.911838
$$454$$ 3.73841e6 0.851231
$$455$$ 2508.10 0.000567958 0
$$456$$ 1.02816e7 2.31551
$$457$$ −8.04537e6 −1.80200 −0.901002 0.433815i $$-0.857167\pi$$
−0.901002 + 0.433815i $$0.857167\pi$$
$$458$$ 5.10614e6 1.13744
$$459$$ 710780. 0.157472
$$460$$ 1.84920e6 0.407463
$$461$$ 5.63578e6 1.23510 0.617550 0.786532i $$-0.288125\pi$$
0.617550 + 0.786532i $$0.288125\pi$$
$$462$$ −9677.77 −0.00210945
$$463$$ 1.39156e6 0.301682 0.150841 0.988558i $$-0.451802\pi$$
0.150841 + 0.988558i $$0.451802\pi$$
$$464$$ 8.93365e6 1.92634
$$465$$ −3.38546e6 −0.726082
$$466$$ −726202. −0.154915
$$467$$ 4.05063e6 0.859469 0.429734 0.902955i $$-0.358607\pi$$
0.429734 + 0.902955i $$0.358607\pi$$
$$468$$ 1.73163e6 0.365461
$$469$$ −6761.12 −0.00141934
$$470$$ −1.24062e7 −2.59056
$$471$$ −2.16651e6 −0.449996
$$472$$ −6.66897e6 −1.37786
$$473$$ −8.23095e6 −1.69160
$$474$$ 2.24675e6 0.459312
$$475$$ −2.84184e6 −0.577917
$$476$$ −19395.1 −0.00392351
$$477$$ 879076. 0.176901
$$478$$ 6.80347e6 1.36195
$$479$$ 6.55529e6 1.30543 0.652714 0.757605i $$-0.273630\pi$$
0.652714 + 0.757605i $$0.273630\pi$$
$$480$$ 6.77687e6 1.34254
$$481$$ −3.22336e6 −0.635252
$$482$$ 6.26872e6 1.22903
$$483$$ −1127.47 −0.000219905 0
$$484$$ 1.40227e6 0.272094
$$485$$ −5.84186e6 −1.12771
$$486$$ 635977. 0.122138
$$487$$ −1.98924e6 −0.380071 −0.190035 0.981777i $$-0.560860\pi$$
−0.190035 + 0.981777i $$0.560860\pi$$
$$488$$ −2.23249e7 −4.24366
$$489$$ 1.60369e6 0.303283
$$490$$ 7.53296e6 1.41735
$$491$$ 7.99205e6 1.49608 0.748040 0.663654i $$-0.230995\pi$$
0.748040 + 0.663654i $$0.230995\pi$$
$$492$$ −7.67071e6 −1.42864
$$493$$ −2.60478e6 −0.482673
$$494$$ −5.59120e6 −1.03083
$$495$$ −1.42112e6 −0.260686
$$496$$ 3.02269e7 5.51682
$$497$$ 12356.8 0.00224397
$$498$$ −1.77819e6 −0.321296
$$499$$ −1.00308e7 −1.80338 −0.901688 0.432387i $$-0.857671\pi$$
−0.901688 + 0.432387i $$0.857671\pi$$
$$500$$ −1.57940e7 −2.82533
$$501$$ −229727. −0.0408900
$$502$$ 1.38760e7 2.45756
$$503$$ −4.82601e6 −0.850488 −0.425244 0.905079i $$-0.639812\pi$$
−0.425244 + 0.905079i $$0.639812\pi$$
$$504$$ −10742.9 −0.00188385
$$505$$ 3.62383e6 0.632324
$$506$$ 2.40206e6 0.417068
$$507$$ 2.75869e6 0.476633
$$508$$ 1.71161e7 2.94270
$$509$$ 138289. 0.0236589 0.0118294 0.999930i $$-0.496234\pi$$
0.0118294 + 0.999930i $$0.496234\pi$$
$$510$$ −3.93303e6 −0.669579
$$511$$ 13478.9 0.00228351
$$512$$ −576256. −0.0971494
$$513$$ −1.48700e6 −0.249470
$$514$$ 2.20693e6 0.368451
$$515$$ 748605. 0.124375
$$516$$ −1.47596e7 −2.44033
$$517$$ −1.16697e7 −1.92014
$$518$$ 32303.6 0.00528965
$$519$$ −6.31679e6 −1.02939
$$520$$ −5.93160e6 −0.961975
$$521$$ 4.62831e6 0.747012 0.373506 0.927628i $$-0.378155\pi$$
0.373506 + 0.927628i $$0.378155\pi$$
$$522$$ −2.33065e6 −0.374369
$$523$$ −1.54914e6 −0.247649 −0.123825 0.992304i $$-0.539516\pi$$
−0.123825 + 0.992304i $$0.539516\pi$$
$$524$$ 5.09662e6 0.810876
$$525$$ 2969.36 0.000470181 0
$$526$$ −4.51129e6 −0.710945
$$527$$ −8.81323e6 −1.38232
$$528$$ 1.26884e7 1.98072
$$529$$ 279841. 0.0434783
$$530$$ −4.86428e6 −0.752192
$$531$$ 964521. 0.148448
$$532$$ 40576.0 0.00621570
$$533$$ 2.58229e6 0.393720
$$534$$ 4.59098e6 0.696710
$$535$$ −5.30526e6 −0.801349
$$536$$ 1.59899e7 2.40399
$$537$$ 2.63626e6 0.394505
$$538$$ 1.60113e7 2.38491
$$539$$ 7.08577e6 1.05055
$$540$$ −2.54833e6 −0.376072
$$541$$ −6.52702e6 −0.958787 −0.479393 0.877600i $$-0.659143\pi$$
−0.479393 + 0.877600i $$0.659143\pi$$
$$542$$ 7.09377e6 1.03724
$$543$$ 350395. 0.0509985
$$544$$ 1.76419e7 2.55593
$$545$$ −5.67701e6 −0.818707
$$546$$ 5842.09 0.000838662 0
$$547$$ 1.06849e7 1.52686 0.763432 0.645888i $$-0.223513\pi$$
0.763432 + 0.645888i $$0.223513\pi$$
$$548$$ 7.93351e6 1.12853
$$549$$ 3.22881e6 0.457206
$$550$$ −6.32619e6 −0.891735
$$551$$ 5.44938e6 0.764660
$$552$$ 2.66643e6 0.372463
$$553$$ 5488.93 0.000763264 0
$$554$$ −2.54746e6 −0.352642
$$555$$ 4.74360e6 0.653695
$$556$$ 3.38947e7 4.64991
$$557$$ −3.07804e6 −0.420374 −0.210187 0.977661i $$-0.567407\pi$$
−0.210187 + 0.977661i $$0.567407\pi$$
$$558$$ −7.88572e6 −1.07215
$$559$$ 4.96871e6 0.672533
$$560$$ 32954.9 0.00444068
$$561$$ −3.69955e6 −0.496297
$$562$$ 1.79060e7 2.39143
$$563$$ 1.10366e7 1.46745 0.733724 0.679448i $$-0.237781\pi$$
0.733724 + 0.679448i $$0.237781\pi$$
$$564$$ −2.09259e7 −2.77004
$$565$$ −4.32844e6 −0.570440
$$566$$ −1.04129e7 −1.36625
$$567$$ 1553.73 0.000202963 0
$$568$$ −2.92237e7 −3.80070
$$569$$ −9.26308e6 −1.19943 −0.599715 0.800214i $$-0.704719\pi$$
−0.599715 + 0.800214i $$0.704719\pi$$
$$570$$ 8.22819e6 1.06076
$$571$$ −1.31913e7 −1.69315 −0.846576 0.532267i $$-0.821340\pi$$
−0.846576 + 0.532267i $$0.821340\pi$$
$$572$$ −9.01300e6 −1.15181
$$573$$ −335203. −0.0426503
$$574$$ −25879.0 −0.00327845
$$575$$ −737006. −0.0929611
$$576$$ 7.11763e6 0.893880
$$577$$ −1.29741e6 −0.162232 −0.0811160 0.996705i $$-0.525848\pi$$
−0.0811160 + 0.996705i $$0.525848\pi$$
$$578$$ 5.05363e6 0.629193
$$579$$ 382089. 0.0473661
$$580$$ 9.33878e6 1.15271
$$581$$ −4344.23 −0.000533916 0
$$582$$ −1.36074e7 −1.66520
$$583$$ −4.57551e6 −0.557530
$$584$$ −3.18773e7 −3.86767
$$585$$ 857876. 0.103642
$$586$$ 887863. 0.106808
$$587$$ 4.51203e6 0.540477 0.270238 0.962793i $$-0.412897\pi$$
0.270238 + 0.962793i $$0.412897\pi$$
$$588$$ 1.27060e7 1.51554
$$589$$ 1.84379e7 2.18990
$$590$$ −5.33708e6 −0.631210
$$591$$ −786016. −0.0925684
$$592$$ −4.23529e7 −4.96683
$$593$$ 9.27288e6 1.08287 0.541437 0.840741i $$-0.317880\pi$$
0.541437 + 0.840741i $$0.317880\pi$$
$$594$$ −3.31020e6 −0.384936
$$595$$ −9608.63 −0.00111268
$$596$$ 3.23004e7 3.72470
$$597$$ −9.79011e6 −1.12422
$$598$$ −1.45003e6 −0.165815
$$599$$ −6.32225e6 −0.719954 −0.359977 0.932961i $$-0.617215\pi$$
−0.359977 + 0.932961i $$0.617215\pi$$
$$600$$ −7.02247e6 −0.796365
$$601$$ 3.18785e6 0.360008 0.180004 0.983666i $$-0.442389\pi$$
0.180004 + 0.983666i $$0.442389\pi$$
$$602$$ −49795.0 −0.00560009
$$603$$ −2.31259e6 −0.259003
$$604$$ −3.71706e7 −4.14579
$$605$$ 694706. 0.0771636
$$606$$ 8.44095e6 0.933705
$$607$$ −5.70410e6 −0.628369 −0.314185 0.949362i $$-0.601731\pi$$
−0.314185 + 0.949362i $$0.601731\pi$$
$$608$$ −3.69082e7 −4.04915
$$609$$ −5693.91 −0.000622110 0
$$610$$ −1.78663e7 −1.94406
$$611$$ 7.04455e6 0.763396
$$612$$ −6.63395e6 −0.715968
$$613$$ 1.17835e7 1.26655 0.633276 0.773926i $$-0.281710\pi$$
0.633276 + 0.773926i $$0.281710\pi$$
$$614$$ −5.07394e6 −0.543156
$$615$$ −3.80018e6 −0.405151
$$616$$ 55916.0 0.00593723
$$617$$ 1.41628e6 0.149774 0.0748868 0.997192i $$-0.476140\pi$$
0.0748868 + 0.997192i $$0.476140\pi$$
$$618$$ 1.74372e6 0.183656
$$619$$ −4.37752e6 −0.459200 −0.229600 0.973285i $$-0.573742\pi$$
−0.229600 + 0.973285i $$0.573742\pi$$
$$620$$ 3.15976e7 3.30123
$$621$$ −385641. −0.0401286
$$622$$ −1.04918e7 −1.08736
$$623$$ 11216.0 0.00115776
$$624$$ −7.65950e6 −0.787479
$$625$$ −3.47084e6 −0.355414
$$626$$ −2.19604e6 −0.223978
$$627$$ 7.73972e6 0.786243
$$628$$ 2.02208e7 2.04597
$$629$$ 1.23488e7 1.24451
$$630$$ −8597.41 −0.000863011 0
$$631$$ −1.61975e7 −1.61948 −0.809739 0.586790i $$-0.800391\pi$$
−0.809739 + 0.586790i $$0.800391\pi$$
$$632$$ −1.29812e7 −1.29277
$$633$$ 9.24989e6 0.917545
$$634$$ 1.20197e7 1.18760
$$635$$ 8.47956e6 0.834525
$$636$$ −8.20471e6 −0.804304
$$637$$ −4.27740e6 −0.417668
$$638$$ 1.21308e7 1.17988
$$639$$ 4.22656e6 0.409482
$$640$$ −1.52892e7 −1.47548
$$641$$ −1.56439e7 −1.50383 −0.751915 0.659260i $$-0.770870\pi$$
−0.751915 + 0.659260i $$0.770870\pi$$
$$642$$ −1.23575e7 −1.18329
$$643$$ −1.57570e7 −1.50295 −0.751476 0.659760i $$-0.770658\pi$$
−0.751476 + 0.659760i $$0.770658\pi$$
$$644$$ 10523.0 0.000999829 0
$$645$$ −7.31211e6 −0.692059
$$646$$ 2.14201e7 2.01948
$$647$$ −660291. −0.0620119 −0.0310059 0.999519i $$-0.509871\pi$$
−0.0310059 + 0.999519i $$0.509871\pi$$
$$648$$ −3.67453e6 −0.343768
$$649$$ −5.02024e6 −0.467857
$$650$$ 3.81888e6 0.354529
$$651$$ −19265.3 −0.00178165
$$652$$ −1.49678e7 −1.37892
$$653$$ 1.24310e7 1.14084 0.570419 0.821354i $$-0.306781\pi$$
0.570419 + 0.821354i $$0.306781\pi$$
$$654$$ −1.32234e7 −1.20892
$$655$$ 2.52494e6 0.229958
$$656$$ 3.39297e7 3.07837
$$657$$ 4.61036e6 0.416698
$$658$$ −70598.6 −0.00635669
$$659$$ −13723.9 −0.00123101 −0.000615507 1.00000i $$-0.500196\pi$$
−0.000615507 1.00000i $$0.500196\pi$$
$$660$$ 1.32638e7 1.18525
$$661$$ 6.54194e6 0.582375 0.291187 0.956666i $$-0.405950\pi$$
0.291187 + 0.956666i $$0.405950\pi$$
$$662$$ −2.25098e7 −1.99630
$$663$$ 2.23327e6 0.197314
$$664$$ 1.02740e7 0.904315
$$665$$ 20101.9 0.00176272
$$666$$ 1.10492e7 0.965262
$$667$$ 1.41325e6 0.123000
$$668$$ 2.14412e6 0.185912
$$669$$ 1.88665e6 0.162977
$$670$$ 1.27965e7 1.10129
$$671$$ −1.68057e7 −1.44095
$$672$$ 38564.4 0.00329430
$$673$$ 1.54266e7 1.31290 0.656451 0.754369i $$-0.272057\pi$$
0.656451 + 0.754369i $$0.272057\pi$$
$$674$$ −9.71159e6 −0.823456
$$675$$ 1.01565e6 0.0857992
$$676$$ −2.57478e7 −2.16707
$$677$$ 5.30462e6 0.444818 0.222409 0.974953i $$-0.428608\pi$$
0.222409 + 0.974953i $$0.428608\pi$$
$$678$$ −1.00822e7 −0.842326
$$679$$ −33243.6 −0.00276716
$$680$$ 2.27242e7 1.88459
$$681$$ 3.12392e6 0.258127
$$682$$ 4.10444e7 3.37904
$$683$$ 1.52230e7 1.24867 0.624337 0.781155i $$-0.285369\pi$$
0.624337 + 0.781155i $$0.285369\pi$$
$$684$$ 1.38787e7 1.13425
$$685$$ 3.93038e6 0.320043
$$686$$ 85734.1 0.00695574
$$687$$ 4.26684e6 0.344917
$$688$$ 6.52857e7 5.25832
$$689$$ 2.76206e6 0.221659
$$690$$ 2.13391e6 0.170629
$$691$$ 1.04711e7 0.834248 0.417124 0.908850i $$-0.363038\pi$$
0.417124 + 0.908850i $$0.363038\pi$$
$$692$$ 5.89567e7 4.68024
$$693$$ −8087.02 −0.000639669 0
$$694$$ 1.65009e7 1.30049
$$695$$ 1.67919e7 1.31868
$$696$$ 1.34660e7 1.05369
$$697$$ −9.89286e6 −0.771329
$$698$$ −1.38241e7 −1.07398
$$699$$ −606836. −0.0469762
$$700$$ −27714.0 −0.00213774
$$701$$ −2.28874e7 −1.75914 −0.879572 0.475765i $$-0.842171\pi$$
−0.879572 + 0.475765i $$0.842171\pi$$
$$702$$ 1.99824e6 0.153040
$$703$$ −2.58346e7 −1.97158
$$704$$ −3.70466e7 −2.81720
$$705$$ −1.03670e7 −0.785560
$$706$$ 4.80333e7 3.62686
$$707$$ 20621.7 0.00155159
$$708$$ −9.00219e6 −0.674940
$$709$$ 1.85127e7 1.38310 0.691550 0.722329i $$-0.256928\pi$$
0.691550 + 0.722329i $$0.256928\pi$$
$$710$$ −2.33873e7 −1.74114
$$711$$ 1.87745e6 0.139281
$$712$$ −2.65257e7 −1.96095
$$713$$ 4.78170e6 0.352256
$$714$$ −22381.3 −0.00164301
$$715$$ −4.46517e6 −0.326643
$$716$$ −2.46051e7 −1.79367
$$717$$ 5.68518e6 0.412996
$$718$$ −5.03519e7 −3.64506
$$719$$ 1.25679e7 0.906652 0.453326 0.891345i $$-0.350237\pi$$
0.453326 + 0.891345i $$0.350237\pi$$
$$720$$ 1.12720e7 0.810342
$$721$$ 4260.00 0.000305191 0
$$722$$ −1.81440e7 −1.29536
$$723$$ 5.23832e6 0.372689
$$724$$ −3.27035e6 −0.231872
$$725$$ −3.72201e6 −0.262986
$$726$$ 1.61817e6 0.113942
$$727$$ 3.64050e6 0.255462 0.127731 0.991809i $$-0.459231\pi$$
0.127731 + 0.991809i $$0.459231\pi$$
$$728$$ −33754.3 −0.00236048
$$729$$ 531441. 0.0370370
$$730$$ −2.55110e7 −1.77182
$$731$$ −1.90353e7 −1.31755
$$732$$ −3.01356e7 −2.07875
$$733$$ −4.38735e6 −0.301608 −0.150804 0.988564i $$-0.548186\pi$$
−0.150804 + 0.988564i $$0.548186\pi$$
$$734$$ −2.61966e7 −1.79475
$$735$$ 6.29476e6 0.429795
$$736$$ −9.57181e6 −0.651327
$$737$$ 1.20368e7 0.816287
$$738$$ −8.85173e6 −0.598256
$$739$$ 1.95857e7 1.31925 0.659625 0.751595i $$-0.270715\pi$$
0.659625 + 0.751595i $$0.270715\pi$$
$$740$$ −4.42736e7 −2.97211
$$741$$ −4.67217e6 −0.312589
$$742$$ −27680.6 −0.00184572
$$743$$ −2.94833e7 −1.95931 −0.979657 0.200678i $$-0.935685\pi$$
−0.979657 + 0.200678i $$0.935685\pi$$
$$744$$ 4.55619e7 3.01766
$$745$$ 1.60021e7 1.05630
$$746$$ −1.48047e7 −0.973983
$$747$$ −1.48591e6 −0.0974296
$$748$$ 3.45291e7 2.25648
$$749$$ −30190.0 −0.00196634
$$750$$ −1.82258e7 −1.18313
$$751$$ 1.27147e7 0.822634 0.411317 0.911492i $$-0.365069\pi$$
0.411317 + 0.911492i $$0.365069\pi$$
$$752$$ 9.25609e7 5.96875
$$753$$ 1.15952e7 0.745228
$$754$$ −7.32290e6 −0.469089
$$755$$ −1.84149e7 −1.17571
$$756$$ −14501.5 −0.000922800 0
$$757$$ 1.63980e7 1.04004 0.520022 0.854153i $$-0.325924\pi$$
0.520022 + 0.854153i $$0.325924\pi$$
$$758$$ −2.17558e7 −1.37531
$$759$$ 2.00723e6 0.126471
$$760$$ −4.75407e7 −2.98560
$$761$$ 8.50863e6 0.532596 0.266298 0.963891i $$-0.414199\pi$$
0.266298 + 0.963891i $$0.414199\pi$$
$$762$$ 1.97513e7 1.23228
$$763$$ −32305.5 −0.00200893
$$764$$ 3.12856e6 0.193915
$$765$$ −3.28656e6 −0.203043
$$766$$ −2.16305e7 −1.33197
$$767$$ 3.03053e6 0.186007
$$768$$ −1.03058e7 −0.630490
$$769$$ 1.70141e7 1.03751 0.518755 0.854923i $$-0.326396\pi$$
0.518755 + 0.854923i $$0.326396\pi$$
$$770$$ 44748.7 0.00271991
$$771$$ 1.84417e6 0.111729
$$772$$ −3.56616e6 −0.215356
$$773$$ 3.63916e6 0.219055 0.109527 0.993984i $$-0.465066\pi$$
0.109527 + 0.993984i $$0.465066\pi$$
$$774$$ −1.70320e7 −1.02191
$$775$$ −1.25934e7 −0.753161
$$776$$ 7.86204e7 4.68685
$$777$$ 26993.8 0.00160403
$$778$$ −3.50095e7 −2.07366
$$779$$ 2.06966e7 1.22195
$$780$$ −8.00685e6 −0.471221
$$781$$ −2.19989e7 −1.29054
$$782$$ 5.55511e6 0.324845
$$783$$ −1.94756e6 −0.113524
$$784$$ −5.62024e7 −3.26561
$$785$$ 1.00177e7 0.580220
$$786$$ 5.88132e6 0.339561
$$787$$ −5.30127e6 −0.305101 −0.152550 0.988296i $$-0.548749\pi$$
−0.152550 + 0.988296i $$0.548749\pi$$
$$788$$ 7.33615e6 0.420875
$$789$$ −3.76976e6 −0.215586
$$790$$ −1.03887e7 −0.592232
$$791$$ −24631.3 −0.00139974
$$792$$ 1.91256e7 1.08343
$$793$$ 1.01449e7 0.572883
$$794$$ −5.85591e7 −3.29642
$$795$$ −4.06473e6 −0.228094
$$796$$ 9.13743e7 5.11142
$$797$$ 1.41955e7 0.791598 0.395799 0.918337i $$-0.370468\pi$$
0.395799 + 0.918337i $$0.370468\pi$$
$$798$$ 46823.2 0.00260288
$$799$$ −2.69879e7 −1.49555
$$800$$ 2.52089e7 1.39261
$$801$$ 3.83636e6 0.211270
$$802$$ −3.52414e7 −1.93472
$$803$$ −2.39965e7 −1.31329
$$804$$ 2.15841e7 1.17759
$$805$$ 5213.26 0.000283543 0
$$806$$ −2.47769e7 −1.34341
$$807$$ 1.33795e7 0.723198
$$808$$ −4.87699e7 −2.62799
$$809$$ −1.58542e7 −0.851675 −0.425838 0.904800i $$-0.640021\pi$$
−0.425838 + 0.904800i $$0.640021\pi$$
$$810$$ −2.94068e6 −0.157483
$$811$$ −1.40129e7 −0.748127 −0.374063 0.927403i $$-0.622036\pi$$
−0.374063 + 0.927403i $$0.622036\pi$$
$$812$$ 53143.1 0.00282851
$$813$$ 5.92776e6 0.314532
$$814$$ −5.75101e7 −3.04217
$$815$$ −7.41524e6 −0.391049
$$816$$ 2.93438e7 1.54273
$$817$$ 3.98232e7 2.08728
$$818$$ 1.68339e7 0.879634
$$819$$ 4881.82 0.000254315 0
$$820$$ 3.54684e7 1.84207
$$821$$ 1.16279e7 0.602065 0.301033 0.953614i $$-0.402669\pi$$
0.301033 + 0.953614i $$0.402669\pi$$
$$822$$ 9.15498e6 0.472583
$$823$$ −2.76237e7 −1.42161 −0.710807 0.703387i $$-0.751670\pi$$
−0.710807 + 0.703387i $$0.751670\pi$$
$$824$$ −1.00748e7 −0.516914
$$825$$ −5.28635e6 −0.270409
$$826$$ −30371.1 −0.00154885
$$827$$ −2.78146e7 −1.41420 −0.707098 0.707116i $$-0.749996\pi$$
−0.707098 + 0.707116i $$0.749996\pi$$
$$828$$ 3.59932e6 0.182450
$$829$$ −1.23659e7 −0.624943 −0.312471 0.949927i $$-0.601157\pi$$
−0.312471 + 0.949927i $$0.601157\pi$$
$$830$$ 8.22214e6 0.414276
$$831$$ −2.12873e6 −0.106935
$$832$$ 2.23636e7 1.12004
$$833$$ 1.63869e7 0.818246
$$834$$ 3.91132e7 1.94719
$$835$$ 1.06223e6 0.0527231
$$836$$ −7.22374e7 −3.57476
$$837$$ −6.58953e6 −0.325118
$$838$$ 5.12787e7 2.52247
$$839$$ −3.49502e7 −1.71413 −0.857067 0.515205i $$-0.827716\pi$$
−0.857067 + 0.515205i $$0.827716\pi$$
$$840$$ 49673.9 0.00242901
$$841$$ −1.33740e7 −0.652035
$$842$$ 4.92192e7 2.39251
$$843$$ 1.49628e7 0.725175
$$844$$ −8.63323e7 −4.17174
$$845$$ −1.27558e7 −0.614565
$$846$$ −2.41477e7 −1.15998
$$847$$ 3953.28 0.000189343 0
$$848$$ 3.62917e7 1.73308
$$849$$ −8.70131e6 −0.414300
$$850$$ −1.46303e7 −0.694552
$$851$$ −6.69997e6 −0.317138
$$852$$ −3.94479e7 −1.86177
$$853$$ −9.99950e6 −0.470550 −0.235275 0.971929i $$-0.575599\pi$$
−0.235275 + 0.971929i $$0.575599\pi$$
$$854$$ −101670. −0.00477032
$$855$$ 6.87571e6 0.321664
$$856$$ 7.13987e7 3.33047
$$857$$ 9.71240e6 0.451725 0.225863 0.974159i $$-0.427480\pi$$
0.225863 + 0.974159i $$0.427480\pi$$
$$858$$ −1.04007e7 −0.482329
$$859$$ 3.69393e7 1.70807 0.854035 0.520216i $$-0.174149\pi$$
0.854035 + 0.520216i $$0.174149\pi$$
$$860$$ 6.82463e7 3.14654
$$861$$ −21625.3 −0.000994155 0
$$862$$ 3.54227e7 1.62373
$$863$$ −3.31284e7 −1.51416 −0.757082 0.653319i $$-0.773376\pi$$
−0.757082 + 0.653319i $$0.773376\pi$$
$$864$$ 1.31906e7 0.601148
$$865$$ 2.92080e7 1.32728
$$866$$ −2.28121e7 −1.03364
$$867$$ 4.22296e6 0.190796
$$868$$ 179809. 0.00810051
$$869$$ −9.77194e6 −0.438966
$$870$$ 1.07766e7 0.482708
$$871$$ −7.26615e6 −0.324533
$$872$$ 7.64019e7 3.40262
$$873$$ −1.13707e7 −0.504954
$$874$$ −1.16217e7 −0.514625
$$875$$ −44526.6 −0.00196607
$$876$$ −4.30300e7 −1.89457
$$877$$ 1.38007e7 0.605902 0.302951 0.953006i $$-0.402028\pi$$
0.302951 + 0.953006i $$0.402028\pi$$
$$878$$ −8.16869e6 −0.357615
$$879$$ 741924. 0.0323882
$$880$$ −5.86696e7 −2.55391
$$881$$ −7.74868e6 −0.336347 −0.168174 0.985757i $$-0.553787\pi$$
−0.168174 + 0.985757i $$0.553787\pi$$
$$882$$ 1.46623e7 0.634646
$$883$$ 3.79838e7 1.63944 0.819722 0.572761i $$-0.194128\pi$$
0.819722 + 0.572761i $$0.194128\pi$$
$$884$$ −2.08439e7 −0.897115
$$885$$ −4.45982e6 −0.191408
$$886$$ −7.00477e6 −0.299785
$$887$$ −1.51721e7 −0.647495 −0.323747 0.946144i $$-0.604943\pi$$
−0.323747 + 0.946144i $$0.604943\pi$$
$$888$$ −6.38398e7 −2.71681
$$889$$ 48253.7 0.00204775
$$890$$ −2.12281e7 −0.898330
$$891$$ −2.76610e6 −0.116728
$$892$$ −1.76087e7 −0.740995
$$893$$ 5.64607e7 2.36929
$$894$$ 3.72734e7 1.55975
$$895$$ −1.21897e7 −0.508670
$$896$$ −87004.5 −0.00362052
$$897$$ −1.21168e6 −0.0502815
$$898$$ 8.16510e7 3.37886
$$899$$ 2.41485e7 0.996530
$$900$$ −9.47937e6 −0.390098
$$901$$ −1.05815e7 −0.434247
$$902$$ 4.60724e7 1.88549
$$903$$ −41610.2 −0.00169817
$$904$$ 5.82526e7 2.37080
$$905$$ −1.62018e6 −0.0657569
$$906$$ −4.28936e7 −1.73609
$$907$$ 8.48315e6 0.342404 0.171202 0.985236i $$-0.445235\pi$$
0.171202 + 0.985236i $$0.445235\pi$$
$$908$$ −2.91566e7 −1.17361
$$909$$ 7.05350e6 0.283136
$$910$$ −27013.1 −0.00108136
$$911$$ 2.96302e7 1.18288 0.591438 0.806351i $$-0.298561\pi$$
0.591438 + 0.806351i $$0.298561\pi$$
$$912$$ −6.13894e7 −2.44403
$$913$$ 7.73403e6 0.307064
$$914$$ 8.66513e7 3.43091
$$915$$ −1.49296e7 −0.589516
$$916$$ −3.98239e7 −1.56821
$$917$$ 14368.4 0.000564268 0
$$918$$ −7.65534e6 −0.299818
$$919$$ −1.27080e7 −0.496352 −0.248176 0.968715i $$-0.579831\pi$$
−0.248176 + 0.968715i $$0.579831\pi$$
$$920$$ −1.23292e7 −0.480249
$$921$$ −4.23993e6 −0.164706
$$922$$ −6.06992e7 −2.35156
$$923$$ 1.32799e7 0.513085
$$924$$ 75478.9 0.00290834
$$925$$ 1.76454e7 0.678075
$$926$$ −1.49875e7 −0.574385
$$927$$ 1.45710e6 0.0556916
$$928$$ −4.83394e7 −1.84260
$$929$$ −4.50533e7 −1.71272 −0.856362 0.516376i $$-0.827281\pi$$
−0.856362 + 0.516376i $$0.827281\pi$$
$$930$$ 3.64625e7 1.38242
$$931$$ −3.42826e7 −1.29628
$$932$$ 5.66380e6 0.213584
$$933$$ −8.76727e6 −0.329731
$$934$$ −4.36266e7 −1.63638
$$935$$ 1.71062e7 0.639919
$$936$$ −1.15454e7 −0.430744
$$937$$ −2.49632e7 −0.928864 −0.464432 0.885609i $$-0.653741\pi$$
−0.464432 + 0.885609i $$0.653741\pi$$
$$938$$ 72819.4 0.00270234
$$939$$ −1.83508e6 −0.0679188
$$940$$ 9.67585e7 3.57165
$$941$$ −3.36141e7 −1.23751 −0.618754 0.785585i $$-0.712362\pi$$
−0.618754 + 0.785585i $$0.712362\pi$$
$$942$$ 2.33340e7 0.856767
$$943$$ 5.36747e6 0.196558
$$944$$ 3.98192e7 1.45433
$$945$$ −7184.24 −0.000261699 0
$$946$$ 8.86501e7 3.22071
$$947$$ −5.33926e6 −0.193467 −0.0967333 0.995310i $$-0.530839\pi$$
−0.0967333 + 0.995310i $$0.530839\pi$$
$$948$$ −1.75228e7 −0.633262
$$949$$ 1.44858e7 0.522127
$$950$$ 3.06076e7 1.10032
$$951$$ 1.00440e7 0.360126
$$952$$ 129314. 0.00462438
$$953$$ −3.69391e7 −1.31751 −0.658755 0.752358i $$-0.728916\pi$$
−0.658755 + 0.752358i $$0.728916\pi$$
$$954$$ −9.46793e6 −0.336809
$$955$$ 1.54994e6 0.0549928
$$956$$ −5.30617e7 −1.87774
$$957$$ 1.01369e7 0.357786
$$958$$ −7.06026e7 −2.48546
$$959$$ 22366.2 0.000785317 0
$$960$$ −3.29110e7 −1.15256
$$961$$ 5.30769e7 1.85395
$$962$$ 3.47166e7 1.20948
$$963$$ −1.03263e7 −0.358821
$$964$$ −4.88910e7 −1.69448
$$965$$ −1.76673e6 −0.0610734
$$966$$ 12143.2 0.000418687 0
$$967$$ −4.47688e7 −1.53960 −0.769802 0.638283i $$-0.779645\pi$$
−0.769802 + 0.638283i $$0.779645\pi$$
$$968$$ −9.34943e6 −0.320698
$$969$$ 1.78993e7 0.612386
$$970$$ 6.29187e7 2.14709
$$971$$ −4.07416e7 −1.38672 −0.693361 0.720590i $$-0.743871\pi$$
−0.693361 + 0.720590i $$0.743871\pi$$
$$972$$ −4.96012e6 −0.168394
$$973$$ 95555.9 0.00323575
$$974$$ 2.14247e7 0.723633
$$975$$ 3.19116e6 0.107507
$$976$$ 1.33298e8 4.47919
$$977$$ 1.07576e7 0.360561 0.180281 0.983615i $$-0.442299\pi$$
0.180281 + 0.983615i $$0.442299\pi$$
$$978$$ −1.72722e7 −0.577433
$$979$$ −1.99679e7 −0.665849
$$980$$ −5.87511e7 −1.95412
$$981$$ −1.10499e7 −0.366593
$$982$$ −8.60770e7 −2.84845
$$983$$ −2.44766e7 −0.807917 −0.403958 0.914777i $$-0.632366\pi$$
−0.403958 + 0.914777i $$0.632366\pi$$
$$984$$ 5.11433e7 1.68384
$$985$$ 3.63444e6 0.119357
$$986$$ 2.80543e7 0.918982
$$987$$ −58994.2 −0.00192760
$$988$$ 4.36069e7 1.42123
$$989$$ 1.03278e7 0.335751
$$990$$ 1.53060e7 0.496332
$$991$$ −2.09086e6 −0.0676301 −0.0338151 0.999428i $$-0.510766\pi$$
−0.0338151 + 0.999428i $$0.510766\pi$$
$$992$$ −1.63556e8 −5.27699
$$993$$ −1.88098e7 −0.605357
$$994$$ −133087. −0.00427239
$$995$$ 4.52682e7 1.44956
$$996$$ 1.38685e7 0.442977
$$997$$ 1.49624e7 0.476721 0.238360 0.971177i $$-0.423390\pi$$
0.238360 + 0.971177i $$0.423390\pi$$
$$998$$ 1.08036e8 3.43353
$$999$$ 9.23304e6 0.292705
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 69.6.a.a.1.1 2
3.2 odd 2 207.6.a.a.1.2 2
4.3 odd 2 1104.6.a.h.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.a.1.1 2 1.1 even 1 trivial
207.6.a.a.1.2 2 3.2 odd 2
1104.6.a.h.1.1 2 4.3 odd 2