# Properties

 Label 847.6.a.a.1.1 Level $847$ Weight $6$ Character 847.1 Self dual yes Analytic conductor $135.845$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,6,Mod(1,847)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("847.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 847.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$135.845095382$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 847.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -6.00000 q^{3} -28.0000 q^{4} -74.0000 q^{5} -12.0000 q^{6} +49.0000 q^{7} -120.000 q^{8} -207.000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -6.00000 q^{3} -28.0000 q^{4} -74.0000 q^{5} -12.0000 q^{6} +49.0000 q^{7} -120.000 q^{8} -207.000 q^{9} -148.000 q^{10} +168.000 q^{12} -364.000 q^{13} +98.0000 q^{14} +444.000 q^{15} +656.000 q^{16} -148.000 q^{17} -414.000 q^{18} +1320.00 q^{19} +2072.00 q^{20} -294.000 q^{21} -436.000 q^{23} +720.000 q^{24} +2351.00 q^{25} -728.000 q^{26} +2700.00 q^{27} -1372.00 q^{28} -2970.00 q^{29} +888.000 q^{30} +8842.00 q^{31} +5152.00 q^{32} -296.000 q^{34} -3626.00 q^{35} +5796.00 q^{36} +138.000 q^{37} +2640.00 q^{38} +2184.00 q^{39} +8880.00 q^{40} -532.000 q^{41} -588.000 q^{42} +20676.0 q^{43} +15318.0 q^{45} -872.000 q^{46} -11722.0 q^{47} -3936.00 q^{48} +2401.00 q^{49} +4702.00 q^{50} +888.000 q^{51} +10192.0 q^{52} +5274.00 q^{53} +5400.00 q^{54} -5880.00 q^{56} -7920.00 q^{57} -5940.00 q^{58} -27670.0 q^{59} -12432.0 q^{60} -19512.0 q^{61} +17684.0 q^{62} -10143.0 q^{63} -10688.0 q^{64} +26936.0 q^{65} +64088.0 q^{67} +4144.00 q^{68} +2616.00 q^{69} -7252.00 q^{70} -3708.00 q^{71} +24840.0 q^{72} +24296.0 q^{73} +276.000 q^{74} -14106.0 q^{75} -36960.0 q^{76} +4368.00 q^{78} +2200.00 q^{79} -48544.0 q^{80} +34101.0 q^{81} -1064.00 q^{82} -74424.0 q^{83} +8232.00 q^{84} +10952.0 q^{85} +41352.0 q^{86} +17820.0 q^{87} +34170.0 q^{89} +30636.0 q^{90} -17836.0 q^{91} +12208.0 q^{92} -53052.0 q^{93} -23444.0 q^{94} -97680.0 q^{95} -30912.0 q^{96} +151718. q^{97} +4802.00 q^{98} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.353553 0.176777 0.984251i $$-0.443433\pi$$
0.176777 + 0.984251i $$0.443433\pi$$
$$3$$ −6.00000 −0.384900 −0.192450 0.981307i $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$4$$ −28.0000 −0.875000
$$5$$ −74.0000 −1.32375 −0.661876 0.749613i $$-0.730240\pi$$
−0.661876 + 0.749613i $$0.730240\pi$$
$$6$$ −12.0000 −0.136083
$$7$$ 49.0000 0.377964
$$8$$ −120.000 −0.662913
$$9$$ −207.000 −0.851852
$$10$$ −148.000 −0.468017
$$11$$ 0 0
$$12$$ 168.000 0.336788
$$13$$ −364.000 −0.597369 −0.298685 0.954352i $$-0.596548\pi$$
−0.298685 + 0.954352i $$0.596548\pi$$
$$14$$ 98.0000 0.133631
$$15$$ 444.000 0.509512
$$16$$ 656.000 0.640625
$$17$$ −148.000 −0.124205 −0.0621025 0.998070i $$-0.519781\pi$$
−0.0621025 + 0.998070i $$0.519781\pi$$
$$18$$ −414.000 −0.301175
$$19$$ 1320.00 0.838861 0.419430 0.907787i $$-0.362230\pi$$
0.419430 + 0.907787i $$0.362230\pi$$
$$20$$ 2072.00 1.15828
$$21$$ −294.000 −0.145479
$$22$$ 0 0
$$23$$ −436.000 −0.171857 −0.0859284 0.996301i $$-0.527386\pi$$
−0.0859284 + 0.996301i $$0.527386\pi$$
$$24$$ 720.000 0.255155
$$25$$ 2351.00 0.752320
$$26$$ −728.000 −0.211202
$$27$$ 2700.00 0.712778
$$28$$ −1372.00 −0.330719
$$29$$ −2970.00 −0.655785 −0.327892 0.944715i $$-0.606338\pi$$
−0.327892 + 0.944715i $$0.606338\pi$$
$$30$$ 888.000 0.180140
$$31$$ 8842.00 1.65252 0.826259 0.563290i $$-0.190465\pi$$
0.826259 + 0.563290i $$0.190465\pi$$
$$32$$ 5152.00 0.889408
$$33$$ 0 0
$$34$$ −296.000 −0.0439131
$$35$$ −3626.00 −0.500331
$$36$$ 5796.00 0.745370
$$37$$ 138.000 0.0165720 0.00828600 0.999966i $$-0.497362\pi$$
0.00828600 + 0.999966i $$0.497362\pi$$
$$38$$ 2640.00 0.296582
$$39$$ 2184.00 0.229928
$$40$$ 8880.00 0.877532
$$41$$ −532.000 −0.0494256 −0.0247128 0.999695i $$-0.507867\pi$$
−0.0247128 + 0.999695i $$0.507867\pi$$
$$42$$ −588.000 −0.0514344
$$43$$ 20676.0 1.70528 0.852639 0.522500i $$-0.175000\pi$$
0.852639 + 0.522500i $$0.175000\pi$$
$$44$$ 0 0
$$45$$ 15318.0 1.12764
$$46$$ −872.000 −0.0607606
$$47$$ −11722.0 −0.774029 −0.387014 0.922074i $$-0.626494\pi$$
−0.387014 + 0.922074i $$0.626494\pi$$
$$48$$ −3936.00 −0.246577
$$49$$ 2401.00 0.142857
$$50$$ 4702.00 0.265985
$$51$$ 888.000 0.0478066
$$52$$ 10192.0 0.522698
$$53$$ 5274.00 0.257899 0.128950 0.991651i $$-0.458839\pi$$
0.128950 + 0.991651i $$0.458839\pi$$
$$54$$ 5400.00 0.252005
$$55$$ 0 0
$$56$$ −5880.00 −0.250557
$$57$$ −7920.00 −0.322878
$$58$$ −5940.00 −0.231855
$$59$$ −27670.0 −1.03485 −0.517427 0.855727i $$-0.673110\pi$$
−0.517427 + 0.855727i $$0.673110\pi$$
$$60$$ −12432.0 −0.445823
$$61$$ −19512.0 −0.671394 −0.335697 0.941970i $$-0.608972\pi$$
−0.335697 + 0.941970i $$0.608972\pi$$
$$62$$ 17684.0 0.584253
$$63$$ −10143.0 −0.321970
$$64$$ −10688.0 −0.326172
$$65$$ 26936.0 0.790769
$$66$$ 0 0
$$67$$ 64088.0 1.74417 0.872087 0.489351i $$-0.162766\pi$$
0.872087 + 0.489351i $$0.162766\pi$$
$$68$$ 4144.00 0.108679
$$69$$ 2616.00 0.0661477
$$70$$ −7252.00 −0.176894
$$71$$ −3708.00 −0.0872959 −0.0436480 0.999047i $$-0.513898\pi$$
−0.0436480 + 0.999047i $$0.513898\pi$$
$$72$$ 24840.0 0.564703
$$73$$ 24296.0 0.533615 0.266807 0.963750i $$-0.414031\pi$$
0.266807 + 0.963750i $$0.414031\pi$$
$$74$$ 276.000 0.00585908
$$75$$ −14106.0 −0.289568
$$76$$ −36960.0 −0.734003
$$77$$ 0 0
$$78$$ 4368.00 0.0812917
$$79$$ 2200.00 0.0396602 0.0198301 0.999803i $$-0.493687\pi$$
0.0198301 + 0.999803i $$0.493687\pi$$
$$80$$ −48544.0 −0.848029
$$81$$ 34101.0 0.577503
$$82$$ −1064.00 −0.0174746
$$83$$ −74424.0 −1.18582 −0.592909 0.805270i $$-0.702020\pi$$
−0.592909 + 0.805270i $$0.702020\pi$$
$$84$$ 8232.00 0.127294
$$85$$ 10952.0 0.164417
$$86$$ 41352.0 0.602907
$$87$$ 17820.0 0.252412
$$88$$ 0 0
$$89$$ 34170.0 0.457267 0.228634 0.973513i $$-0.426574\pi$$
0.228634 + 0.973513i $$0.426574\pi$$
$$90$$ 30636.0 0.398681
$$91$$ −17836.0 −0.225784
$$92$$ 12208.0 0.150375
$$93$$ −53052.0 −0.636055
$$94$$ −23444.0 −0.273660
$$95$$ −97680.0 −1.11044
$$96$$ −30912.0 −0.342333
$$97$$ 151718. 1.63722 0.818611 0.574348i $$-0.194744\pi$$
0.818611 + 0.574348i $$0.194744\pi$$
$$98$$ 4802.00 0.0505076
$$99$$ 0 0
$$100$$ −65828.0 −0.658280
$$101$$ −116852. −1.13981 −0.569905 0.821710i $$-0.693020\pi$$
−0.569905 + 0.821710i $$0.693020\pi$$
$$102$$ 1776.00 0.0169022
$$103$$ 103694. 0.963076 0.481538 0.876425i $$-0.340078\pi$$
0.481538 + 0.876425i $$0.340078\pi$$
$$104$$ 43680.0 0.396004
$$105$$ 21756.0 0.192578
$$106$$ 10548.0 0.0911812
$$107$$ 97092.0 0.819830 0.409915 0.912124i $$-0.365558\pi$$
0.409915 + 0.912124i $$0.365558\pi$$
$$108$$ −75600.0 −0.623681
$$109$$ −52930.0 −0.426713 −0.213356 0.976974i $$-0.568440\pi$$
−0.213356 + 0.976974i $$0.568440\pi$$
$$110$$ 0 0
$$111$$ −828.000 −0.00637856
$$112$$ 32144.0 0.242133
$$113$$ −80526.0 −0.593253 −0.296627 0.954994i $$-0.595862\pi$$
−0.296627 + 0.954994i $$0.595862\pi$$
$$114$$ −15840.0 −0.114155
$$115$$ 32264.0 0.227496
$$116$$ 83160.0 0.573812
$$117$$ 75348.0 0.508870
$$118$$ −55340.0 −0.365876
$$119$$ −7252.00 −0.0469451
$$120$$ −53280.0 −0.337762
$$121$$ 0 0
$$122$$ −39024.0 −0.237373
$$123$$ 3192.00 0.0190239
$$124$$ −247576. −1.44595
$$125$$ 57276.0 0.327867
$$126$$ −20286.0 −0.113833
$$127$$ −221048. −1.21612 −0.608061 0.793890i $$-0.708052\pi$$
−0.608061 + 0.793890i $$0.708052\pi$$
$$128$$ −186240. −1.00473
$$129$$ −124056. −0.656362
$$130$$ 53872.0 0.279579
$$131$$ −37572.0 −0.191287 −0.0956436 0.995416i $$-0.530491\pi$$
−0.0956436 + 0.995416i $$0.530491\pi$$
$$132$$ 0 0
$$133$$ 64680.0 0.317060
$$134$$ 128176. 0.616658
$$135$$ −199800. −0.943542
$$136$$ 17760.0 0.0823371
$$137$$ −290602. −1.32281 −0.661405 0.750029i $$-0.730039\pi$$
−0.661405 + 0.750029i $$0.730039\pi$$
$$138$$ 5232.00 0.0233868
$$139$$ −367360. −1.61270 −0.806352 0.591435i $$-0.798561\pi$$
−0.806352 + 0.591435i $$0.798561\pi$$
$$140$$ 101528. 0.437790
$$141$$ 70332.0 0.297924
$$142$$ −7416.00 −0.0308638
$$143$$ 0 0
$$144$$ −135792. −0.545718
$$145$$ 219780. 0.868097
$$146$$ 48592.0 0.188661
$$147$$ −14406.0 −0.0549857
$$148$$ −3864.00 −0.0145005
$$149$$ 462730. 1.70751 0.853753 0.520679i $$-0.174321\pi$$
0.853753 + 0.520679i $$0.174321\pi$$
$$150$$ −28212.0 −0.102378
$$151$$ 7648.00 0.0272964 0.0136482 0.999907i $$-0.495656\pi$$
0.0136482 + 0.999907i $$0.495656\pi$$
$$152$$ −158400. −0.556091
$$153$$ 30636.0 0.105804
$$154$$ 0 0
$$155$$ −654308. −2.18752
$$156$$ −61152.0 −0.201187
$$157$$ −161482. −0.522847 −0.261424 0.965224i $$-0.584192\pi$$
−0.261424 + 0.965224i $$0.584192\pi$$
$$158$$ 4400.00 0.0140220
$$159$$ −31644.0 −0.0992656
$$160$$ −381248. −1.17736
$$161$$ −21364.0 −0.0649558
$$162$$ 68202.0 0.204178
$$163$$ 179464. 0.529064 0.264532 0.964377i $$-0.414782\pi$$
0.264532 + 0.964377i $$0.414782\pi$$
$$164$$ 14896.0 0.0432474
$$165$$ 0 0
$$166$$ −148848. −0.419250
$$167$$ −316848. −0.879144 −0.439572 0.898207i $$-0.644870\pi$$
−0.439572 + 0.898207i $$0.644870\pi$$
$$168$$ 35280.0 0.0964396
$$169$$ −238797. −0.643150
$$170$$ 21904.0 0.0581301
$$171$$ −273240. −0.714585
$$172$$ −578928. −1.49212
$$173$$ 175116. 0.444847 0.222423 0.974950i $$-0.428603\pi$$
0.222423 + 0.974950i $$0.428603\pi$$
$$174$$ 35640.0 0.0892410
$$175$$ 115199. 0.284350
$$176$$ 0 0
$$177$$ 166020. 0.398316
$$178$$ 68340.0 0.161668
$$179$$ −69780.0 −0.162779 −0.0813895 0.996682i $$-0.525936\pi$$
−0.0813895 + 0.996682i $$0.525936\pi$$
$$180$$ −428904. −0.986686
$$181$$ −78638.0 −0.178417 −0.0892085 0.996013i $$-0.528434\pi$$
−0.0892085 + 0.996013i $$0.528434\pi$$
$$182$$ −35672.0 −0.0798268
$$183$$ 117072. 0.258420
$$184$$ 52320.0 0.113926
$$185$$ −10212.0 −0.0219372
$$186$$ −106104. −0.224879
$$187$$ 0 0
$$188$$ 328216. 0.677275
$$189$$ 132300. 0.269405
$$190$$ −195360. −0.392601
$$191$$ −927208. −1.83905 −0.919525 0.393030i $$-0.871427\pi$$
−0.919525 + 0.393030i $$0.871427\pi$$
$$192$$ 64128.0 0.125544
$$193$$ −877474. −1.69567 −0.847834 0.530261i $$-0.822094\pi$$
−0.847834 + 0.530261i $$0.822094\pi$$
$$194$$ 303436. 0.578846
$$195$$ −161616. −0.304367
$$196$$ −67228.0 −0.125000
$$197$$ 744602. 1.36697 0.683484 0.729965i $$-0.260464\pi$$
0.683484 + 0.729965i $$0.260464\pi$$
$$198$$ 0 0
$$199$$ 1.07931e6 1.93203 0.966014 0.258489i $$-0.0832246\pi$$
0.966014 + 0.258489i $$0.0832246\pi$$
$$200$$ −282120. −0.498722
$$201$$ −384528. −0.671333
$$202$$ −233704. −0.402984
$$203$$ −145530. −0.247863
$$204$$ −24864.0 −0.0418307
$$205$$ 39368.0 0.0654273
$$206$$ 207388. 0.340499
$$207$$ 90252.0 0.146397
$$208$$ −238784. −0.382690
$$209$$ 0 0
$$210$$ 43512.0 0.0680865
$$211$$ −728772. −1.12690 −0.563450 0.826150i $$-0.690526\pi$$
−0.563450 + 0.826150i $$0.690526\pi$$
$$212$$ −147672. −0.225662
$$213$$ 22248.0 0.0336002
$$214$$ 194184. 0.289854
$$215$$ −1.53002e6 −2.25737
$$216$$ −324000. −0.472510
$$217$$ 433258. 0.624593
$$218$$ −105860. −0.150866
$$219$$ −145776. −0.205388
$$220$$ 0 0
$$221$$ 53872.0 0.0741963
$$222$$ −1656.00 −0.00225516
$$223$$ 38374.0 0.0516743 0.0258372 0.999666i $$-0.491775\pi$$
0.0258372 + 0.999666i $$0.491775\pi$$
$$224$$ 252448. 0.336165
$$225$$ −486657. −0.640865
$$226$$ −161052. −0.209747
$$227$$ −323268. −0.416388 −0.208194 0.978088i $$-0.566759\pi$$
−0.208194 + 0.978088i $$0.566759\pi$$
$$228$$ 221760. 0.282518
$$229$$ 813690. 1.02535 0.512673 0.858584i $$-0.328655\pi$$
0.512673 + 0.858584i $$0.328655\pi$$
$$230$$ 64528.0 0.0804320
$$231$$ 0 0
$$232$$ 356400. 0.434728
$$233$$ −1.10801e6 −1.33707 −0.668537 0.743679i $$-0.733079\pi$$
−0.668537 + 0.743679i $$0.733079\pi$$
$$234$$ 150696. 0.179913
$$235$$ 867428. 1.02462
$$236$$ 774760. 0.905497
$$237$$ −13200.0 −0.0152652
$$238$$ −14504.0 −0.0165976
$$239$$ 1.31352e6 1.48745 0.743724 0.668487i $$-0.233058\pi$$
0.743724 + 0.668487i $$0.233058\pi$$
$$240$$ 291264. 0.326406
$$241$$ 1.05607e6 1.17125 0.585625 0.810582i $$-0.300849\pi$$
0.585625 + 0.810582i $$0.300849\pi$$
$$242$$ 0 0
$$243$$ −860706. −0.935059
$$244$$ 546336. 0.587469
$$245$$ −177674. −0.189107
$$246$$ 6384.00 0.00672597
$$247$$ −480480. −0.501110
$$248$$ −1.06104e6 −1.09548
$$249$$ 446544. 0.456421
$$250$$ 114552. 0.115918
$$251$$ 1.68914e6 1.69232 0.846159 0.532931i $$-0.178909\pi$$
0.846159 + 0.532931i $$0.178909\pi$$
$$252$$ 284004. 0.281724
$$253$$ 0 0
$$254$$ −442096. −0.429964
$$255$$ −65712.0 −0.0632840
$$256$$ −30464.0 −0.0290527
$$257$$ 641938. 0.606262 0.303131 0.952949i $$-0.401968\pi$$
0.303131 + 0.952949i $$0.401968\pi$$
$$258$$ −248112. −0.232059
$$259$$ 6762.00 0.00626363
$$260$$ −754208. −0.691923
$$261$$ 614790. 0.558632
$$262$$ −75144.0 −0.0676303
$$263$$ 1.10150e6 0.981959 0.490980 0.871171i $$-0.336639\pi$$
0.490980 + 0.871171i $$0.336639\pi$$
$$264$$ 0 0
$$265$$ −390276. −0.341395
$$266$$ 129360. 0.112097
$$267$$ −205020. −0.176002
$$268$$ −1.79446e6 −1.52615
$$269$$ −2.15147e6 −1.81282 −0.906410 0.422399i $$-0.861188\pi$$
−0.906410 + 0.422399i $$0.861188\pi$$
$$270$$ −399600. −0.333592
$$271$$ 1.08327e6 0.896010 0.448005 0.894031i $$-0.352135\pi$$
0.448005 + 0.894031i $$0.352135\pi$$
$$272$$ −97088.0 −0.0795689
$$273$$ 107016. 0.0869045
$$274$$ −581204. −0.467684
$$275$$ 0 0
$$276$$ −73248.0 −0.0578793
$$277$$ 2.22372e6 1.74133 0.870665 0.491877i $$-0.163689\pi$$
0.870665 + 0.491877i $$0.163689\pi$$
$$278$$ −734720. −0.570177
$$279$$ −1.83029e6 −1.40770
$$280$$ 435120. 0.331676
$$281$$ 153018. 0.115605 0.0578025 0.998328i $$-0.481591\pi$$
0.0578025 + 0.998328i $$0.481591\pi$$
$$282$$ 140664. 0.105332
$$283$$ −715324. −0.530929 −0.265465 0.964121i $$-0.585525\pi$$
−0.265465 + 0.964121i $$0.585525\pi$$
$$284$$ 103824. 0.0763839
$$285$$ 586080. 0.427410
$$286$$ 0 0
$$287$$ −26068.0 −0.0186811
$$288$$ −1.06646e6 −0.757644
$$289$$ −1.39795e6 −0.984573
$$290$$ 439560. 0.306919
$$291$$ −910308. −0.630167
$$292$$ −680288. −0.466913
$$293$$ −347424. −0.236424 −0.118212 0.992988i $$-0.537716\pi$$
−0.118212 + 0.992988i $$0.537716\pi$$
$$294$$ −28812.0 −0.0194404
$$295$$ 2.04758e6 1.36989
$$296$$ −16560.0 −0.0109858
$$297$$ 0 0
$$298$$ 925460. 0.603694
$$299$$ 158704. 0.102662
$$300$$ 394968. 0.253372
$$301$$ 1.01312e6 0.644535
$$302$$ 15296.0 0.00965074
$$303$$ 701112. 0.438713
$$304$$ 865920. 0.537395
$$305$$ 1.44389e6 0.888759
$$306$$ 61272.0 0.0374075
$$307$$ −2.64043e6 −1.59893 −0.799463 0.600715i $$-0.794883\pi$$
−0.799463 + 0.600715i $$0.794883\pi$$
$$308$$ 0 0
$$309$$ −622164. −0.370688
$$310$$ −1.30862e6 −0.773407
$$311$$ −947778. −0.555656 −0.277828 0.960631i $$-0.589614\pi$$
−0.277828 + 0.960631i $$0.589614\pi$$
$$312$$ −262080. −0.152422
$$313$$ −248686. −0.143480 −0.0717399 0.997423i $$-0.522855\pi$$
−0.0717399 + 0.997423i $$0.522855\pi$$
$$314$$ −322964. −0.184854
$$315$$ 750582. 0.426208
$$316$$ −61600.0 −0.0347027
$$317$$ 2.60904e6 1.45825 0.729125 0.684380i $$-0.239927\pi$$
0.729125 + 0.684380i $$0.239927\pi$$
$$318$$ −63288.0 −0.0350957
$$319$$ 0 0
$$320$$ 790912. 0.431771
$$321$$ −582552. −0.315553
$$322$$ −42728.0 −0.0229653
$$323$$ −195360. −0.104191
$$324$$ −954828. −0.505316
$$325$$ −855764. −0.449413
$$326$$ 358928. 0.187052
$$327$$ 317580. 0.164242
$$328$$ 63840.0 0.0327649
$$329$$ −574378. −0.292555
$$330$$ 0 0
$$331$$ 152332. 0.0764225 0.0382112 0.999270i $$-0.487834\pi$$
0.0382112 + 0.999270i $$0.487834\pi$$
$$332$$ 2.08387e6 1.03759
$$333$$ −28566.0 −0.0141169
$$334$$ −633696. −0.310824
$$335$$ −4.74251e6 −2.30885
$$336$$ −192864. −0.0931972
$$337$$ −206558. −0.0990757 −0.0495379 0.998772i $$-0.515775\pi$$
−0.0495379 + 0.998772i $$0.515775\pi$$
$$338$$ −477594. −0.227388
$$339$$ 483156. 0.228343
$$340$$ −306656. −0.143865
$$341$$ 0 0
$$342$$ −546480. −0.252644
$$343$$ 117649. 0.0539949
$$344$$ −2.48112e6 −1.13045
$$345$$ −193584. −0.0875632
$$346$$ 350232. 0.157277
$$347$$ −2.01807e6 −0.899730 −0.449865 0.893097i $$-0.648528\pi$$
−0.449865 + 0.893097i $$0.648528\pi$$
$$348$$ −498960. −0.220860
$$349$$ 580440. 0.255090 0.127545 0.991833i $$-0.459290\pi$$
0.127545 + 0.991833i $$0.459290\pi$$
$$350$$ 230398. 0.100533
$$351$$ −982800. −0.425792
$$352$$ 0 0
$$353$$ 572034. 0.244335 0.122167 0.992510i $$-0.461016\pi$$
0.122167 + 0.992510i $$0.461016\pi$$
$$354$$ 332040. 0.140826
$$355$$ 274392. 0.115558
$$356$$ −956760. −0.400109
$$357$$ 43512.0 0.0180692
$$358$$ −139560. −0.0575511
$$359$$ −4.56544e6 −1.86959 −0.934795 0.355187i $$-0.884417\pi$$
−0.934795 + 0.355187i $$0.884417\pi$$
$$360$$ −1.83816e6 −0.747527
$$361$$ −733699. −0.296312
$$362$$ −157276. −0.0630799
$$363$$ 0 0
$$364$$ 499408. 0.197561
$$365$$ −1.79790e6 −0.706373
$$366$$ 234144. 0.0913651
$$367$$ 924358. 0.358241 0.179120 0.983827i $$-0.442675\pi$$
0.179120 + 0.983827i $$0.442675\pi$$
$$368$$ −286016. −0.110096
$$369$$ 110124. 0.0421033
$$370$$ −20424.0 −0.00775598
$$371$$ 258426. 0.0974768
$$372$$ 1.48546e6 0.556548
$$373$$ −4.92021e6 −1.83110 −0.915550 0.402205i $$-0.868244\pi$$
−0.915550 + 0.402205i $$0.868244\pi$$
$$374$$ 0 0
$$375$$ −343656. −0.126196
$$376$$ 1.40664e6 0.513113
$$377$$ 1.08108e6 0.391746
$$378$$ 264600. 0.0952490
$$379$$ 3.97540e6 1.42162 0.710809 0.703385i $$-0.248329\pi$$
0.710809 + 0.703385i $$0.248329\pi$$
$$380$$ 2.73504e6 0.971638
$$381$$ 1.32629e6 0.468086
$$382$$ −1.85442e6 −0.650203
$$383$$ −982846. −0.342364 −0.171182 0.985239i $$-0.554759\pi$$
−0.171182 + 0.985239i $$0.554759\pi$$
$$384$$ 1.11744e6 0.386720
$$385$$ 0 0
$$386$$ −1.75495e6 −0.599509
$$387$$ −4.27993e6 −1.45264
$$388$$ −4.24810e6 −1.43257
$$389$$ −744090. −0.249317 −0.124658 0.992200i $$-0.539783\pi$$
−0.124658 + 0.992200i $$0.539783\pi$$
$$390$$ −323232. −0.107610
$$391$$ 64528.0 0.0213455
$$392$$ −288120. −0.0947018
$$393$$ 225432. 0.0736265
$$394$$ 1.48920e6 0.483296
$$395$$ −162800. −0.0525003
$$396$$ 0 0
$$397$$ 5.73024e6 1.82472 0.912360 0.409388i $$-0.134258\pi$$
0.912360 + 0.409388i $$0.134258\pi$$
$$398$$ 2.15862e6 0.683075
$$399$$ −388080. −0.122036
$$400$$ 1.54226e6 0.481955
$$401$$ 4.26756e6 1.32531 0.662657 0.748923i $$-0.269429\pi$$
0.662657 + 0.748923i $$0.269429\pi$$
$$402$$ −769056. −0.237352
$$403$$ −3.21849e6 −0.987164
$$404$$ 3.27186e6 0.997334
$$405$$ −2.52347e6 −0.764471
$$406$$ −291060. −0.0876330
$$407$$ 0 0
$$408$$ −106560. −0.0316916
$$409$$ 5.81772e6 1.71967 0.859834 0.510574i $$-0.170567\pi$$
0.859834 + 0.510574i $$0.170567\pi$$
$$410$$ 78736.0 0.0231320
$$411$$ 1.74361e6 0.509149
$$412$$ −2.90343e6 −0.842692
$$413$$ −1.35583e6 −0.391138
$$414$$ 180504. 0.0517590
$$415$$ 5.50738e6 1.56973
$$416$$ −1.87533e6 −0.531305
$$417$$ 2.20416e6 0.620730
$$418$$ 0 0
$$419$$ −4.38823e6 −1.22111 −0.610554 0.791974i $$-0.709053\pi$$
−0.610554 + 0.791974i $$0.709053\pi$$
$$420$$ −609168. −0.168505
$$421$$ −1.48456e6 −0.408218 −0.204109 0.978948i $$-0.565430\pi$$
−0.204109 + 0.978948i $$0.565430\pi$$
$$422$$ −1.45754e6 −0.398419
$$423$$ 2.42645e6 0.659358
$$424$$ −632880. −0.170965
$$425$$ −347948. −0.0934420
$$426$$ 44496.0 0.0118795
$$427$$ −956088. −0.253763
$$428$$ −2.71858e6 −0.717352
$$429$$ 0 0
$$430$$ −3.06005e6 −0.798100
$$431$$ 206448. 0.0535325 0.0267662 0.999642i $$-0.491479\pi$$
0.0267662 + 0.999642i $$0.491479\pi$$
$$432$$ 1.77120e6 0.456623
$$433$$ −5.67867e6 −1.45555 −0.727774 0.685817i $$-0.759445\pi$$
−0.727774 + 0.685817i $$0.759445\pi$$
$$434$$ 866516. 0.220827
$$435$$ −1.31868e6 −0.334131
$$436$$ 1.48204e6 0.373374
$$437$$ −575520. −0.144164
$$438$$ −291552. −0.0726157
$$439$$ 4.43666e6 1.09874 0.549370 0.835579i $$-0.314868\pi$$
0.549370 + 0.835579i $$0.314868\pi$$
$$440$$ 0 0
$$441$$ −497007. −0.121693
$$442$$ 107744. 0.0262324
$$443$$ 6.17328e6 1.49454 0.747269 0.664522i $$-0.231365\pi$$
0.747269 + 0.664522i $$0.231365\pi$$
$$444$$ 23184.0 0.00558124
$$445$$ −2.52858e6 −0.605308
$$446$$ 76748.0 0.0182696
$$447$$ −2.77638e6 −0.657219
$$448$$ −523712. −0.123281
$$449$$ −4.93105e6 −1.15431 −0.577156 0.816634i $$-0.695838\pi$$
−0.577156 + 0.816634i $$0.695838\pi$$
$$450$$ −973314. −0.226580
$$451$$ 0 0
$$452$$ 2.25473e6 0.519096
$$453$$ −45888.0 −0.0105064
$$454$$ −646536. −0.147215
$$455$$ 1.31986e6 0.298883
$$456$$ 950400. 0.214040
$$457$$ 4.15030e6 0.929585 0.464793 0.885420i $$-0.346129\pi$$
0.464793 + 0.885420i $$0.346129\pi$$
$$458$$ 1.62738e6 0.362514
$$459$$ −399600. −0.0885307
$$460$$ −903392. −0.199059
$$461$$ 4.40345e6 0.965029 0.482515 0.875888i $$-0.339723\pi$$
0.482515 + 0.875888i $$0.339723\pi$$
$$462$$ 0 0
$$463$$ 6.82728e6 1.48012 0.740058 0.672544i $$-0.234798\pi$$
0.740058 + 0.672544i $$0.234798\pi$$
$$464$$ −1.94832e6 −0.420112
$$465$$ 3.92585e6 0.841979
$$466$$ −2.21603e6 −0.472727
$$467$$ 6.88244e6 1.46033 0.730163 0.683272i $$-0.239444\pi$$
0.730163 + 0.683272i $$0.239444\pi$$
$$468$$ −2.10974e6 −0.445261
$$469$$ 3.14031e6 0.659236
$$470$$ 1.73486e6 0.362259
$$471$$ 968892. 0.201244
$$472$$ 3.32040e6 0.686018
$$473$$ 0 0
$$474$$ −26400.0 −0.00539707
$$475$$ 3.10332e6 0.631092
$$476$$ 203056. 0.0410770
$$477$$ −1.09172e6 −0.219692
$$478$$ 2.62704e6 0.525892
$$479$$ −5.02430e6 −1.00055 −0.500273 0.865868i $$-0.666767\pi$$
−0.500273 + 0.865868i $$0.666767\pi$$
$$480$$ 2.28749e6 0.453164
$$481$$ −50232.0 −0.00989960
$$482$$ 2.11214e6 0.414099
$$483$$ 128184. 0.0250015
$$484$$ 0 0
$$485$$ −1.12271e7 −2.16728
$$486$$ −1.72141e6 −0.330593
$$487$$ −4.51601e6 −0.862845 −0.431422 0.902150i $$-0.641988\pi$$
−0.431422 + 0.902150i $$0.641988\pi$$
$$488$$ 2.34144e6 0.445075
$$489$$ −1.07678e6 −0.203637
$$490$$ −355348. −0.0668596
$$491$$ −5.55737e6 −1.04032 −0.520159 0.854070i $$-0.674127\pi$$
−0.520159 + 0.854070i $$0.674127\pi$$
$$492$$ −89376.0 −0.0166459
$$493$$ 439560. 0.0814518
$$494$$ −960960. −0.177169
$$495$$ 0 0
$$496$$ 5.80035e6 1.05864
$$497$$ −181692. −0.0329947
$$498$$ 893088. 0.161369
$$499$$ −3.49744e6 −0.628780 −0.314390 0.949294i $$-0.601800\pi$$
−0.314390 + 0.949294i $$0.601800\pi$$
$$500$$ −1.60373e6 −0.286884
$$501$$ 1.90109e6 0.338383
$$502$$ 3.37828e6 0.598325
$$503$$ −4.61280e6 −0.812915 −0.406457 0.913670i $$-0.633236\pi$$
−0.406457 + 0.913670i $$0.633236\pi$$
$$504$$ 1.21716e6 0.213438
$$505$$ 8.64705e6 1.50883
$$506$$ 0 0
$$507$$ 1.43278e6 0.247548
$$508$$ 6.18934e6 1.06411
$$509$$ 7.41609e6 1.26876 0.634382 0.773020i $$-0.281255\pi$$
0.634382 + 0.773020i $$0.281255\pi$$
$$510$$ −131424. −0.0223743
$$511$$ 1.19050e6 0.201687
$$512$$ 5.89875e6 0.994455
$$513$$ 3.56400e6 0.597922
$$514$$ 1.28388e6 0.214346
$$515$$ −7.67336e6 −1.27487
$$516$$ 3.47357e6 0.574317
$$517$$ 0 0
$$518$$ 13524.0 0.00221453
$$519$$ −1.05070e6 −0.171222
$$520$$ −3.23232e6 −0.524211
$$521$$ 9.75970e6 1.57522 0.787612 0.616172i $$-0.211317\pi$$
0.787612 + 0.616172i $$0.211317\pi$$
$$522$$ 1.22958e6 0.197506
$$523$$ −1.66084e6 −0.265506 −0.132753 0.991149i $$-0.542382\pi$$
−0.132753 + 0.991149i $$0.542382\pi$$
$$524$$ 1.05202e6 0.167376
$$525$$ −691194. −0.109446
$$526$$ 2.20299e6 0.347175
$$527$$ −1.30862e6 −0.205251
$$528$$ 0 0
$$529$$ −6.24625e6 −0.970465
$$530$$ −780552. −0.120701
$$531$$ 5.72769e6 0.881542
$$532$$ −1.81104e6 −0.277427
$$533$$ 193648. 0.0295253
$$534$$ −410040. −0.0622262
$$535$$ −7.18481e6 −1.08525
$$536$$ −7.69056e6 −1.15623
$$537$$ 418680. 0.0626537
$$538$$ −4.30294e6 −0.640929
$$539$$ 0 0
$$540$$ 5.59440e6 0.825599
$$541$$ −6.97250e6 −1.02423 −0.512113 0.858918i $$-0.671137\pi$$
−0.512113 + 0.858918i $$0.671137\pi$$
$$542$$ 2.16654e6 0.316787
$$543$$ 471828. 0.0686727
$$544$$ −762496. −0.110469
$$545$$ 3.91682e6 0.564862
$$546$$ 214032. 0.0307254
$$547$$ −3.08503e6 −0.440850 −0.220425 0.975404i $$-0.570744\pi$$
−0.220425 + 0.975404i $$0.570744\pi$$
$$548$$ 8.13686e6 1.15746
$$549$$ 4.03898e6 0.571928
$$550$$ 0 0
$$551$$ −3.92040e6 −0.550112
$$552$$ −313920. −0.0438502
$$553$$ 107800. 0.0149901
$$554$$ 4.44744e6 0.615653
$$555$$ 61272.0 0.00844364
$$556$$ 1.02861e7 1.41112
$$557$$ 3.29052e6 0.449394 0.224697 0.974429i $$-0.427861\pi$$
0.224697 + 0.974429i $$0.427861\pi$$
$$558$$ −3.66059e6 −0.497697
$$559$$ −7.52606e6 −1.01868
$$560$$ −2.37866e6 −0.320525
$$561$$ 0 0
$$562$$ 306036. 0.0408725
$$563$$ −5.45754e6 −0.725648 −0.362824 0.931858i $$-0.618187\pi$$
−0.362824 + 0.931858i $$0.618187\pi$$
$$564$$ −1.96930e6 −0.260683
$$565$$ 5.95892e6 0.785320
$$566$$ −1.43065e6 −0.187712
$$567$$ 1.67095e6 0.218276
$$568$$ 444960. 0.0578696
$$569$$ −7.28571e6 −0.943390 −0.471695 0.881762i $$-0.656358\pi$$
−0.471695 + 0.881762i $$0.656358\pi$$
$$570$$ 1.17216e6 0.151112
$$571$$ −1.07129e7 −1.37504 −0.687519 0.726166i $$-0.741300\pi$$
−0.687519 + 0.726166i $$0.741300\pi$$
$$572$$ 0 0
$$573$$ 5.56325e6 0.707851
$$574$$ −52136.0 −0.00660477
$$575$$ −1.02504e6 −0.129291
$$576$$ 2.21242e6 0.277850
$$577$$ 4.22024e6 0.527713 0.263856 0.964562i $$-0.415006\pi$$
0.263856 + 0.964562i $$0.415006\pi$$
$$578$$ −2.79591e6 −0.348099
$$579$$ 5.26484e6 0.652663
$$580$$ −6.15384e6 −0.759585
$$581$$ −3.64678e6 −0.448197
$$582$$ −1.82062e6 −0.222798
$$583$$ 0 0
$$584$$ −2.91552e6 −0.353740
$$585$$ −5.57575e6 −0.673618
$$586$$ −694848. −0.0835884
$$587$$ −6.24180e6 −0.747678 −0.373839 0.927494i $$-0.621959\pi$$
−0.373839 + 0.927494i $$0.621959\pi$$
$$588$$ 403368. 0.0481125
$$589$$ 1.16714e7 1.38623
$$590$$ 4.09516e6 0.484329
$$591$$ −4.46761e6 −0.526147
$$592$$ 90528.0 0.0106164
$$593$$ −493664. −0.0576494 −0.0288247 0.999584i $$-0.509176\pi$$
−0.0288247 + 0.999584i $$0.509176\pi$$
$$594$$ 0 0
$$595$$ 536648. 0.0621437
$$596$$ −1.29564e7 −1.49407
$$597$$ −6.47586e6 −0.743638
$$598$$ 317408. 0.0362965
$$599$$ −8.28890e6 −0.943908 −0.471954 0.881623i $$-0.656451\pi$$
−0.471954 + 0.881623i $$0.656451\pi$$
$$600$$ 1.69272e6 0.191958
$$601$$ −80612.0 −0.00910361 −0.00455180 0.999990i $$-0.501449\pi$$
−0.00455180 + 0.999990i $$0.501449\pi$$
$$602$$ 2.02625e6 0.227877
$$603$$ −1.32662e7 −1.48578
$$604$$ −214144. −0.0238844
$$605$$ 0 0
$$606$$ 1.40222e6 0.155109
$$607$$ −914168. −0.100706 −0.0503529 0.998731i $$-0.516035\pi$$
−0.0503529 + 0.998731i $$0.516035\pi$$
$$608$$ 6.80064e6 0.746089
$$609$$ 873180. 0.0954027
$$610$$ 2.88778e6 0.314224
$$611$$ 4.26681e6 0.462381
$$612$$ −857808. −0.0925788
$$613$$ −9.97327e6 −1.07198 −0.535990 0.844224i $$-0.680061\pi$$
−0.535990 + 0.844224i $$0.680061\pi$$
$$614$$ −5.28086e6 −0.565306
$$615$$ −236208. −0.0251830
$$616$$ 0 0
$$617$$ −4.60636e6 −0.487130 −0.243565 0.969885i $$-0.578317\pi$$
−0.243565 + 0.969885i $$0.578317\pi$$
$$618$$ −1.24433e6 −0.131058
$$619$$ 2.51711e6 0.264044 0.132022 0.991247i $$-0.457853\pi$$
0.132022 + 0.991247i $$0.457853\pi$$
$$620$$ 1.83206e7 1.91408
$$621$$ −1.17720e6 −0.122496
$$622$$ −1.89556e6 −0.196454
$$623$$ 1.67433e6 0.172831
$$624$$ 1.43270e6 0.147297
$$625$$ −1.15853e7 −1.18633
$$626$$ −497372. −0.0507277
$$627$$ 0 0
$$628$$ 4.52150e6 0.457492
$$629$$ −20424.0 −0.00205833
$$630$$ 1.50116e6 0.150687
$$631$$ −2.02153e6 −0.202119 −0.101059 0.994880i $$-0.532223\pi$$
−0.101059 + 0.994880i $$0.532223\pi$$
$$632$$ −264000. −0.0262912
$$633$$ 4.37263e6 0.433744
$$634$$ 5.21808e6 0.515570
$$635$$ 1.63576e7 1.60984
$$636$$ 886032. 0.0868574
$$637$$ −873964. −0.0853385
$$638$$ 0 0
$$639$$ 767556. 0.0743632
$$640$$ 1.37818e7 1.33001
$$641$$ 1.55870e7 1.49837 0.749183 0.662363i $$-0.230446\pi$$
0.749183 + 0.662363i $$0.230446\pi$$
$$642$$ −1.16510e6 −0.111565
$$643$$ −5.88755e6 −0.561574 −0.280787 0.959770i $$-0.590595\pi$$
−0.280787 + 0.959770i $$0.590595\pi$$
$$644$$ 598192. 0.0568363
$$645$$ 9.18014e6 0.868861
$$646$$ −390720. −0.0368370
$$647$$ 5.58958e6 0.524950 0.262475 0.964939i $$-0.415461\pi$$
0.262475 + 0.964939i $$0.415461\pi$$
$$648$$ −4.09212e6 −0.382834
$$649$$ 0 0
$$650$$ −1.71153e6 −0.158891
$$651$$ −2.59955e6 −0.240406
$$652$$ −5.02499e6 −0.462931
$$653$$ 1.76227e7 1.61730 0.808648 0.588293i $$-0.200200\pi$$
0.808648 + 0.588293i $$0.200200\pi$$
$$654$$ 635160. 0.0580683
$$655$$ 2.78033e6 0.253217
$$656$$ −348992. −0.0316633
$$657$$ −5.02927e6 −0.454561
$$658$$ −1.14876e6 −0.103434
$$659$$ 9.75566e6 0.875071 0.437535 0.899201i $$-0.355851\pi$$
0.437535 + 0.899201i $$0.355851\pi$$
$$660$$ 0 0
$$661$$ −9.79522e6 −0.871988 −0.435994 0.899950i $$-0.643603\pi$$
−0.435994 + 0.899950i $$0.643603\pi$$
$$662$$ 304664. 0.0270194
$$663$$ −323232. −0.0285582
$$664$$ 8.93088e6 0.786093
$$665$$ −4.78632e6 −0.419708
$$666$$ −57132.0 −0.00499107
$$667$$ 1.29492e6 0.112701
$$668$$ 8.87174e6 0.769251
$$669$$ −230244. −0.0198895
$$670$$ −9.48502e6 −0.816303
$$671$$ 0 0
$$672$$ −1.51469e6 −0.129390
$$673$$ 4.72727e6 0.402321 0.201160 0.979558i $$-0.435529\pi$$
0.201160 + 0.979558i $$0.435529\pi$$
$$674$$ −413116. −0.0350286
$$675$$ 6.34770e6 0.536237
$$676$$ 6.68632e6 0.562756
$$677$$ 1.93989e7 1.62669 0.813344 0.581783i $$-0.197645\pi$$
0.813344 + 0.581783i $$0.197645\pi$$
$$678$$ 966312. 0.0807315
$$679$$ 7.43418e6 0.618812
$$680$$ −1.31424e6 −0.108994
$$681$$ 1.93961e6 0.160268
$$682$$ 0 0
$$683$$ −6.22004e6 −0.510201 −0.255100 0.966915i $$-0.582109\pi$$
−0.255100 + 0.966915i $$0.582109\pi$$
$$684$$ 7.65072e6 0.625262
$$685$$ 2.15045e7 1.75107
$$686$$ 235298. 0.0190901
$$687$$ −4.88214e6 −0.394656
$$688$$ 1.35635e7 1.09244
$$689$$ −1.91974e6 −0.154061
$$690$$ −387168. −0.0309583
$$691$$ 6.04140e6 0.481330 0.240665 0.970608i $$-0.422635\pi$$
0.240665 + 0.970608i $$0.422635\pi$$
$$692$$ −4.90325e6 −0.389241
$$693$$ 0 0
$$694$$ −4.03614e6 −0.318103
$$695$$ 2.71846e7 2.13482
$$696$$ −2.13840e6 −0.167327
$$697$$ 78736.0 0.00613891
$$698$$ 1.16088e6 0.0901880
$$699$$ 6.64808e6 0.514640
$$700$$ −3.22557e6 −0.248806
$$701$$ −3.97068e6 −0.305190 −0.152595 0.988289i $$-0.548763\pi$$
−0.152595 + 0.988289i $$0.548763\pi$$
$$702$$ −1.96560e6 −0.150540
$$703$$ 182160. 0.0139016
$$704$$ 0 0
$$705$$ −5.20457e6 −0.394377
$$706$$ 1.14407e6 0.0863853
$$707$$ −5.72575e6 −0.430808
$$708$$ −4.64856e6 −0.348526
$$709$$ 5.15939e6 0.385463 0.192732 0.981252i $$-0.438265\pi$$
0.192732 + 0.981252i $$0.438265\pi$$
$$710$$ 548784. 0.0408560
$$711$$ −455400. −0.0337846
$$712$$ −4.10040e6 −0.303128
$$713$$ −3.85511e6 −0.283997
$$714$$ 87024.0 0.00638842
$$715$$ 0 0
$$716$$ 1.95384e6 0.142432
$$717$$ −7.88112e6 −0.572519
$$718$$ −9.13088e6 −0.661000
$$719$$ −1.26734e7 −0.914259 −0.457129 0.889400i $$-0.651122\pi$$
−0.457129 + 0.889400i $$0.651122\pi$$
$$720$$ 1.00486e7 0.722395
$$721$$ 5.08101e6 0.364009
$$722$$ −1.46740e6 −0.104762
$$723$$ −6.33641e6 −0.450814
$$724$$ 2.20186e6 0.156115
$$725$$ −6.98247e6 −0.493360
$$726$$ 0 0
$$727$$ 1.32783e7 0.931762 0.465881 0.884847i $$-0.345737\pi$$
0.465881 + 0.884847i $$0.345737\pi$$
$$728$$ 2.14032e6 0.149675
$$729$$ −3.12231e6 −0.217599
$$730$$ −3.59581e6 −0.249741
$$731$$ −3.06005e6 −0.211804
$$732$$ −3.27802e6 −0.226117
$$733$$ −3.84050e6 −0.264015 −0.132007 0.991249i $$-0.542142\pi$$
−0.132007 + 0.991249i $$0.542142\pi$$
$$734$$ 1.84872e6 0.126657
$$735$$ 1.06604e6 0.0727875
$$736$$ −2.24627e6 −0.152851
$$737$$ 0 0
$$738$$ 220248. 0.0148858
$$739$$ −1.19394e7 −0.804215 −0.402107 0.915592i $$-0.631722\pi$$
−0.402107 + 0.915592i $$0.631722\pi$$
$$740$$ 285936. 0.0191951
$$741$$ 2.88288e6 0.192877
$$742$$ 516852. 0.0344633
$$743$$ −2.53631e7 −1.68551 −0.842754 0.538298i $$-0.819067\pi$$
−0.842754 + 0.538298i $$0.819067\pi$$
$$744$$ 6.36624e6 0.421649
$$745$$ −3.42420e7 −2.26031
$$746$$ −9.84043e6 −0.647391
$$747$$ 1.54058e7 1.01014
$$748$$ 0 0
$$749$$ 4.75751e6 0.309867
$$750$$ −687312. −0.0446170
$$751$$ 1.43903e7 0.931046 0.465523 0.885036i $$-0.345866\pi$$
0.465523 + 0.885036i $$0.345866\pi$$
$$752$$ −7.68963e6 −0.495862
$$753$$ −1.01349e7 −0.651373
$$754$$ 2.16216e6 0.138503
$$755$$ −565952. −0.0361337
$$756$$ −3.70440e6 −0.235729
$$757$$ −1.85431e7 −1.17609 −0.588047 0.808827i $$-0.700103\pi$$
−0.588047 + 0.808827i $$0.700103\pi$$
$$758$$ 7.95080e6 0.502618
$$759$$ 0 0
$$760$$ 1.17216e7 0.736127
$$761$$ −6.43123e6 −0.402562 −0.201281 0.979534i $$-0.564510\pi$$
−0.201281 + 0.979534i $$0.564510\pi$$
$$762$$ 2.65258e6 0.165493
$$763$$ −2.59357e6 −0.161282
$$764$$ 2.59618e7 1.60917
$$765$$ −2.26706e6 −0.140059
$$766$$ −1.96569e6 −0.121044
$$767$$ 1.00719e7 0.618190
$$768$$ 182784. 0.0111824
$$769$$ −1.48249e7 −0.904014 −0.452007 0.892014i $$-0.649292\pi$$
−0.452007 + 0.892014i $$0.649292\pi$$
$$770$$ 0 0
$$771$$ −3.85163e6 −0.233350
$$772$$ 2.45693e7 1.48371
$$773$$ 1.96325e7 1.18175 0.590876 0.806762i $$-0.298782\pi$$
0.590876 + 0.806762i $$0.298782\pi$$
$$774$$ −8.55986e6 −0.513588
$$775$$ 2.07875e7 1.24322
$$776$$ −1.82062e7 −1.08534
$$777$$ −40572.0 −0.00241087
$$778$$ −1.48818e6 −0.0881468
$$779$$ −702240. −0.0414612
$$780$$ 4.52525e6 0.266321
$$781$$ 0 0
$$782$$ 129056. 0.00754677
$$783$$ −8.01900e6 −0.467429
$$784$$ 1.57506e6 0.0915179
$$785$$ 1.19497e7 0.692120
$$786$$ 450864. 0.0260309
$$787$$ 9.48639e6 0.545964 0.272982 0.962019i $$-0.411990\pi$$
0.272982 + 0.962019i $$0.411990\pi$$
$$788$$ −2.08489e7 −1.19610
$$789$$ −6.60898e6 −0.377956
$$790$$ −325600. −0.0185617
$$791$$ −3.94577e6 −0.224229
$$792$$ 0 0
$$793$$ 7.10237e6 0.401070
$$794$$ 1.14605e7 0.645136
$$795$$ 2.34166e6 0.131403
$$796$$ −3.02207e7 −1.69052
$$797$$ 3.32326e7 1.85318 0.926592 0.376068i $$-0.122724\pi$$
0.926592 + 0.376068i $$0.122724\pi$$
$$798$$ −776160. −0.0431463
$$799$$ 1.73486e6 0.0961383
$$800$$ 1.21124e7 0.669119
$$801$$ −7.07319e6 −0.389524
$$802$$ 8.53512e6 0.468569
$$803$$ 0 0
$$804$$ 1.07668e7 0.587416
$$805$$ 1.58094e6 0.0859854
$$806$$ −6.43698e6 −0.349015
$$807$$ 1.29088e7 0.697755
$$808$$ 1.40222e7 0.755595
$$809$$ 2.75792e7 1.48153 0.740764 0.671766i $$-0.234464\pi$$
0.740764 + 0.671766i $$0.234464\pi$$
$$810$$ −5.04695e6 −0.270281
$$811$$ 7.76107e6 0.414352 0.207176 0.978304i $$-0.433573\pi$$
0.207176 + 0.978304i $$0.433573\pi$$
$$812$$ 4.07484e6 0.216880
$$813$$ −6.49961e6 −0.344874
$$814$$ 0 0
$$815$$ −1.32803e7 −0.700350
$$816$$ 582528. 0.0306261
$$817$$ 2.72923e7 1.43049
$$818$$ 1.16354e7 0.607994
$$819$$ 3.69205e6 0.192335
$$820$$ −1.10230e6 −0.0572488
$$821$$ 1.60578e7 0.831434 0.415717 0.909494i $$-0.363531\pi$$
0.415717 + 0.909494i $$0.363531\pi$$
$$822$$ 3.48722e6 0.180012
$$823$$ 1.04666e6 0.0538651 0.0269326 0.999637i $$-0.491426\pi$$
0.0269326 + 0.999637i $$0.491426\pi$$
$$824$$ −1.24433e7 −0.638435
$$825$$ 0 0
$$826$$ −2.71166e6 −0.138288
$$827$$ −8.91799e6 −0.453423 −0.226711 0.973962i $$-0.572797\pi$$
−0.226711 + 0.973962i $$0.572797\pi$$
$$828$$ −2.52706e6 −0.128097
$$829$$ −2.53821e7 −1.28275 −0.641374 0.767229i $$-0.721635\pi$$
−0.641374 + 0.767229i $$0.721635\pi$$
$$830$$ 1.10148e7 0.554983
$$831$$ −1.33423e7 −0.670238
$$832$$ 3.89043e6 0.194845
$$833$$ −355348. −0.0177436
$$834$$ 4.40832e6 0.219461
$$835$$ 2.34468e7 1.16377
$$836$$ 0 0
$$837$$ 2.38734e7 1.17788
$$838$$ −8.77646e6 −0.431727
$$839$$ −3.10636e7 −1.52351 −0.761757 0.647863i $$-0.775663\pi$$
−0.761757 + 0.647863i $$0.775663\pi$$
$$840$$ −2.61072e6 −0.127662
$$841$$ −1.16902e7 −0.569946
$$842$$ −2.96912e6 −0.144327
$$843$$ −918108. −0.0444964
$$844$$ 2.04056e7 0.986038
$$845$$ 1.76710e7 0.851371
$$846$$ 4.85291e6 0.233118
$$847$$ 0 0
$$848$$ 3.45974e6 0.165217
$$849$$ 4.29194e6 0.204355
$$850$$ −695896. −0.0330367
$$851$$ −60168.0 −0.00284801
$$852$$ −622944. −0.0294002
$$853$$ −2.24337e7 −1.05567 −0.527835 0.849347i $$-0.676996\pi$$
−0.527835 + 0.849347i $$0.676996\pi$$
$$854$$ −1.91218e6 −0.0897187
$$855$$ 2.02198e7 0.945934
$$856$$ −1.16510e7 −0.543476
$$857$$ −4.76449e6 −0.221597 −0.110799 0.993843i $$-0.535341\pi$$
−0.110799 + 0.993843i $$0.535341\pi$$
$$858$$ 0 0
$$859$$ 468030. 0.0216417 0.0108208 0.999941i $$-0.496556\pi$$
0.0108208 + 0.999941i $$0.496556\pi$$
$$860$$ 4.28407e7 1.97520
$$861$$ 156408. 0.00719037
$$862$$ 412896. 0.0189266
$$863$$ −1.20487e7 −0.550697 −0.275349 0.961344i $$-0.588793\pi$$
−0.275349 + 0.961344i $$0.588793\pi$$
$$864$$ 1.39104e7 0.633950
$$865$$ −1.29586e7 −0.588867
$$866$$ −1.13573e7 −0.514614
$$867$$ 8.38772e6 0.378962
$$868$$ −1.21312e7 −0.546519
$$869$$ 0 0
$$870$$ −2.63736e6 −0.118133
$$871$$ −2.33280e7 −1.04192
$$872$$ 6.35160e6 0.282873
$$873$$ −3.14056e7 −1.39467
$$874$$ −1.15104e6 −0.0509697
$$875$$ 2.80652e6 0.123922
$$876$$ 4.08173e6 0.179715
$$877$$ −3.61718e6 −0.158807 −0.0794037 0.996843i $$-0.525302\pi$$
−0.0794037 + 0.996843i $$0.525302\pi$$
$$878$$ 8.87332e6 0.388463
$$879$$ 2.08454e6 0.0909995
$$880$$ 0 0
$$881$$ −2.27025e7 −0.985448 −0.492724 0.870186i $$-0.663999\pi$$
−0.492724 + 0.870186i $$0.663999\pi$$
$$882$$ −994014. −0.0430250
$$883$$ 2.41926e7 1.04419 0.522097 0.852886i $$-0.325150\pi$$
0.522097 + 0.852886i $$0.325150\pi$$
$$884$$ −1.50842e6 −0.0649218
$$885$$ −1.22855e7 −0.527271
$$886$$ 1.23466e7 0.528399
$$887$$ 4.06125e7 1.73321 0.866604 0.498997i $$-0.166298\pi$$
0.866604 + 0.498997i $$0.166298\pi$$
$$888$$ 99360.0 0.00422843
$$889$$ −1.08314e7 −0.459651
$$890$$ −5.05716e6 −0.214009
$$891$$ 0 0
$$892$$ −1.07447e6 −0.0452150
$$893$$ −1.54730e7 −0.649302
$$894$$ −5.55276e6 −0.232362
$$895$$ 5.16372e6 0.215479
$$896$$ −9.12576e6 −0.379751
$$897$$ −952224. −0.0395146
$$898$$ −9.86210e6 −0.408111
$$899$$ −2.62607e7 −1.08370
$$900$$ 1.36264e7 0.560757
$$901$$ −780552. −0.0320324
$$902$$ 0 0
$$903$$ −6.07874e6 −0.248082
$$904$$ 9.66312e6 0.393275
$$905$$ 5.81921e6 0.236180
$$906$$ −91776.0 −0.00371457
$$907$$ −1.98235e7 −0.800132 −0.400066 0.916486i $$-0.631013\pi$$
−0.400066 + 0.916486i $$0.631013\pi$$
$$908$$ 9.05150e6 0.364339
$$909$$ 2.41884e7 0.970950
$$910$$ 2.63973e6 0.105671
$$911$$ −2.21209e7 −0.883094 −0.441547 0.897238i $$-0.645570\pi$$
−0.441547 + 0.897238i $$0.645570\pi$$
$$912$$ −5.19552e6 −0.206844
$$913$$ 0 0
$$914$$ 8.30060e6 0.328658
$$915$$ −8.66333e6 −0.342083
$$916$$ −2.27833e7 −0.897177
$$917$$ −1.84103e6 −0.0722998
$$918$$ −799200. −0.0313003
$$919$$ −4.42949e7 −1.73008 −0.865038 0.501706i $$-0.832706\pi$$
−0.865038 + 0.501706i $$0.832706\pi$$
$$920$$ −3.87168e6 −0.150810
$$921$$ 1.58426e7 0.615427
$$922$$ 8.80690e6 0.341189
$$923$$ 1.34971e6 0.0521479
$$924$$ 0 0
$$925$$ 324438. 0.0124674
$$926$$ 1.36546e7 0.523300
$$927$$ −2.14647e7 −0.820398
$$928$$ −1.53014e7 −0.583260
$$929$$ 1.13166e7 0.430207 0.215104 0.976591i $$-0.430991\pi$$
0.215104 + 0.976591i $$0.430991\pi$$
$$930$$ 7.85170e6 0.297684
$$931$$ 3.16932e6 0.119837
$$932$$ 3.10244e7 1.16994
$$933$$ 5.68667e6 0.213872
$$934$$ 1.37649e7 0.516304
$$935$$ 0 0
$$936$$ −9.04176e6 −0.337337
$$937$$ −3.37578e7 −1.25610 −0.628052 0.778172i $$-0.716147\pi$$
−0.628052 + 0.778172i $$0.716147\pi$$
$$938$$ 6.28062e6 0.233075
$$939$$ 1.49212e6 0.0552254
$$940$$ −2.42880e7 −0.896544
$$941$$ 3.30036e7 1.21503 0.607516 0.794307i $$-0.292166\pi$$
0.607516 + 0.794307i $$0.292166\pi$$
$$942$$ 1.93778e6 0.0711505
$$943$$ 231952. 0.00849413
$$944$$ −1.81515e7 −0.662953
$$945$$ −9.79020e6 −0.356625
$$946$$ 0 0
$$947$$ 1.92599e7 0.697876 0.348938 0.937146i $$-0.386542\pi$$
0.348938 + 0.937146i $$0.386542\pi$$
$$948$$ 369600. 0.0133571
$$949$$ −8.84374e6 −0.318765
$$950$$ 6.20664e6 0.223125
$$951$$ −1.56542e7 −0.561281
$$952$$ 870240. 0.0311205
$$953$$ −1.18503e7 −0.422665 −0.211332 0.977414i $$-0.567780\pi$$
−0.211332 + 0.977414i $$0.567780\pi$$
$$954$$ −2.18344e6 −0.0776729
$$955$$ 6.86134e7 2.43445
$$956$$ −3.67786e7 −1.30152
$$957$$ 0 0
$$958$$ −1.00486e7 −0.353746
$$959$$ −1.42395e7 −0.499975
$$960$$ −4.74547e6 −0.166189
$$961$$ 4.95518e7 1.73082
$$962$$ −100464. −0.00350004
$$963$$ −2.00980e7 −0.698374
$$964$$ −2.95699e7 −1.02484
$$965$$ 6.49331e7 2.24465
$$966$$ 256368. 0.00883936
$$967$$ 1.44196e7 0.495892 0.247946 0.968774i $$-0.420245\pi$$
0.247946 + 0.968774i $$0.420245\pi$$
$$968$$ 0 0
$$969$$ 1.17216e6 0.0401031
$$970$$ −2.24543e7 −0.766248
$$971$$ −1.37494e7 −0.467990 −0.233995 0.972238i $$-0.575180\pi$$
−0.233995 + 0.972238i $$0.575180\pi$$
$$972$$ 2.40998e7 0.818177
$$973$$ −1.80006e7 −0.609545
$$974$$ −9.03202e6 −0.305062
$$975$$ 5.13458e6 0.172979
$$976$$ −1.27999e7 −0.430112
$$977$$ −9.18802e6 −0.307954 −0.153977 0.988074i $$-0.549208\pi$$
−0.153977 + 0.988074i $$0.549208\pi$$
$$978$$ −2.15357e6 −0.0719965
$$979$$ 0 0
$$980$$ 4.97487e6 0.165469
$$981$$ 1.09565e7 0.363496
$$982$$ −1.11147e7 −0.367808
$$983$$ −3.43732e7 −1.13458 −0.567292 0.823517i $$-0.692009\pi$$
−0.567292 + 0.823517i $$0.692009\pi$$
$$984$$ −383040. −0.0126112
$$985$$ −5.51005e7 −1.80953
$$986$$ 879120. 0.0287976
$$987$$ 3.44627e6 0.112605
$$988$$ 1.34534e7 0.438471
$$989$$ −9.01474e6 −0.293064
$$990$$ 0 0
$$991$$ −4.32179e7 −1.39791 −0.698956 0.715164i $$-0.746352\pi$$
−0.698956 + 0.715164i $$0.746352\pi$$
$$992$$ 4.55540e7 1.46976
$$993$$ −913992. −0.0294150
$$994$$ −363384. −0.0116654
$$995$$ −7.98689e7 −2.55753
$$996$$ −1.25032e7 −0.399369
$$997$$ −2.54793e7 −0.811801 −0.405901 0.913917i $$-0.633042\pi$$
−0.405901 + 0.913917i $$0.633042\pi$$
$$998$$ −6.99488e6 −0.222307
$$999$$ 372600. 0.0118122
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 847.6.a.a.1.1 1
11.10 odd 2 77.6.a.a.1.1 1
33.32 even 2 693.6.a.a.1.1 1
77.76 even 2 539.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
77.6.a.a.1.1 1 11.10 odd 2
539.6.a.d.1.1 1 77.76 even 2
693.6.a.a.1.1 1 33.32 even 2
847.6.a.a.1.1 1 1.1 even 1 trivial