# Properties

 Label 784.6.a.e.1.1 Level $784$ Weight $6$ Character 784.1 Self dual yes Analytic conductor $125.741$ Analytic rank $0$ 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] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.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: $$125.740914733$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.00000 q^{3} -4.00000 q^{5} -207.000 q^{9} +O(q^{10})$$ $$q-6.00000 q^{3} -4.00000 q^{5} -207.000 q^{9} +240.000 q^{11} +744.000 q^{13} +24.0000 q^{15} +1042.00 q^{17} -986.000 q^{19} -184.000 q^{23} -3109.00 q^{25} +2700.00 q^{27} -734.000 q^{29} +5140.00 q^{31} -1440.00 q^{33} -6054.00 q^{37} -4464.00 q^{39} -7598.00 q^{41} -13016.0 q^{43} +828.000 q^{45} +14668.0 q^{47} -6252.00 q^{51} -14522.0 q^{53} -960.000 q^{55} +5916.00 q^{57} -13362.0 q^{59} -9676.00 q^{61} -2976.00 q^{65} +62124.0 q^{67} +1104.00 q^{69} +2112.00 q^{71} +28910.0 q^{73} +18654.0 q^{75} +101768. q^{79} +34101.0 q^{81} -23922.0 q^{83} -4168.00 q^{85} +4404.00 q^{87} -141674. q^{89} -30840.0 q^{93} +3944.00 q^{95} -99982.0 q^{97} -49680.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −6.00000 −0.384900 −0.192450 0.981307i $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$4$$ 0 0
$$5$$ −4.00000 −0.0715542 −0.0357771 0.999360i $$-0.511391\pi$$
−0.0357771 + 0.999360i $$0.511391\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −207.000 −0.851852
$$10$$ 0 0
$$11$$ 240.000 0.598039 0.299020 0.954247i $$-0.403340\pi$$
0.299020 + 0.954247i $$0.403340\pi$$
$$12$$ 0 0
$$13$$ 744.000 1.22100 0.610498 0.792017i $$-0.290969\pi$$
0.610498 + 0.792017i $$0.290969\pi$$
$$14$$ 0 0
$$15$$ 24.0000 0.0275412
$$16$$ 0 0
$$17$$ 1042.00 0.874471 0.437236 0.899347i $$-0.355958\pi$$
0.437236 + 0.899347i $$0.355958\pi$$
$$18$$ 0 0
$$19$$ −986.000 −0.626604 −0.313302 0.949654i $$-0.601435\pi$$
−0.313302 + 0.949654i $$0.601435\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −184.000 −0.0725268 −0.0362634 0.999342i $$-0.511546\pi$$
−0.0362634 + 0.999342i $$0.511546\pi$$
$$24$$ 0 0
$$25$$ −3109.00 −0.994880
$$26$$ 0 0
$$27$$ 2700.00 0.712778
$$28$$ 0 0
$$29$$ −734.000 −0.162069 −0.0810347 0.996711i $$-0.525822\pi$$
−0.0810347 + 0.996711i $$0.525822\pi$$
$$30$$ 0 0
$$31$$ 5140.00 0.960636 0.480318 0.877094i $$-0.340521\pi$$
0.480318 + 0.877094i $$0.340521\pi$$
$$32$$ 0 0
$$33$$ −1440.00 −0.230185
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6054.00 −0.727006 −0.363503 0.931593i $$-0.618419\pi$$
−0.363503 + 0.931593i $$0.618419\pi$$
$$38$$ 0 0
$$39$$ −4464.00 −0.469962
$$40$$ 0 0
$$41$$ −7598.00 −0.705894 −0.352947 0.935643i $$-0.614820\pi$$
−0.352947 + 0.935643i $$0.614820\pi$$
$$42$$ 0 0
$$43$$ −13016.0 −1.07351 −0.536755 0.843738i $$-0.680350\pi$$
−0.536755 + 0.843738i $$0.680350\pi$$
$$44$$ 0 0
$$45$$ 828.000 0.0609536
$$46$$ 0 0
$$47$$ 14668.0 0.968559 0.484280 0.874913i $$-0.339082\pi$$
0.484280 + 0.874913i $$0.339082\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −6252.00 −0.336584
$$52$$ 0 0
$$53$$ −14522.0 −0.710128 −0.355064 0.934842i $$-0.615541\pi$$
−0.355064 + 0.934842i $$0.615541\pi$$
$$54$$ 0 0
$$55$$ −960.000 −0.0427922
$$56$$ 0 0
$$57$$ 5916.00 0.241180
$$58$$ 0 0
$$59$$ −13362.0 −0.499737 −0.249868 0.968280i $$-0.580387\pi$$
−0.249868 + 0.968280i $$0.580387\pi$$
$$60$$ 0 0
$$61$$ −9676.00 −0.332944 −0.166472 0.986046i $$-0.553238\pi$$
−0.166472 + 0.986046i $$0.553238\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −2976.00 −0.0873674
$$66$$ 0 0
$$67$$ 62124.0 1.69072 0.845361 0.534195i $$-0.179385\pi$$
0.845361 + 0.534195i $$0.179385\pi$$
$$68$$ 0 0
$$69$$ 1104.00 0.0279156
$$70$$ 0 0
$$71$$ 2112.00 0.0497219 0.0248610 0.999691i $$-0.492086\pi$$
0.0248610 + 0.999691i $$0.492086\pi$$
$$72$$ 0 0
$$73$$ 28910.0 0.634952 0.317476 0.948266i $$-0.397165\pi$$
0.317476 + 0.948266i $$0.397165\pi$$
$$74$$ 0 0
$$75$$ 18654.0 0.382929
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 101768. 1.83461 0.917304 0.398186i $$-0.130360\pi$$
0.917304 + 0.398186i $$0.130360\pi$$
$$80$$ 0 0
$$81$$ 34101.0 0.577503
$$82$$ 0 0
$$83$$ −23922.0 −0.381156 −0.190578 0.981672i $$-0.561036\pi$$
−0.190578 + 0.981672i $$0.561036\pi$$
$$84$$ 0 0
$$85$$ −4168.00 −0.0625721
$$86$$ 0 0
$$87$$ 4404.00 0.0623805
$$88$$ 0 0
$$89$$ −141674. −1.89590 −0.947949 0.318421i $$-0.896847\pi$$
−0.947949 + 0.318421i $$0.896847\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −30840.0 −0.369749
$$94$$ 0 0
$$95$$ 3944.00 0.0448361
$$96$$ 0 0
$$97$$ −99982.0 −1.07893 −0.539464 0.842009i $$-0.681373\pi$$
−0.539464 + 0.842009i $$0.681373\pi$$
$$98$$ 0 0
$$99$$ −49680.0 −0.509441
$$100$$ 0 0
$$101$$ 108684. 1.06014 0.530069 0.847955i $$-0.322166\pi$$
0.530069 + 0.847955i $$0.322166\pi$$
$$102$$ 0 0
$$103$$ 87396.0 0.811706 0.405853 0.913938i $$-0.366975\pi$$
0.405853 + 0.913938i $$0.366975\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −71892.0 −0.607045 −0.303523 0.952824i $$-0.598163\pi$$
−0.303523 + 0.952824i $$0.598163\pi$$
$$108$$ 0 0
$$109$$ −118166. −0.952634 −0.476317 0.879274i $$-0.658029\pi$$
−0.476317 + 0.879274i $$0.658029\pi$$
$$110$$ 0 0
$$111$$ 36324.0 0.279825
$$112$$ 0 0
$$113$$ 252774. 1.86224 0.931121 0.364709i $$-0.118832\pi$$
0.931121 + 0.364709i $$0.118832\pi$$
$$114$$ 0 0
$$115$$ 736.000 0.00518959
$$116$$ 0 0
$$117$$ −154008. −1.04011
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −103451. −0.642349
$$122$$ 0 0
$$123$$ 45588.0 0.271699
$$124$$ 0 0
$$125$$ 24936.0 0.142742
$$126$$ 0 0
$$127$$ 3592.00 0.0197618 0.00988091 0.999951i $$-0.496855\pi$$
0.00988091 + 0.999951i $$0.496855\pi$$
$$128$$ 0 0
$$129$$ 78096.0 0.413194
$$130$$ 0 0
$$131$$ 364534. 1.85592 0.927961 0.372677i $$-0.121560\pi$$
0.927961 + 0.372677i $$0.121560\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −10800.0 −0.0510022
$$136$$ 0 0
$$137$$ −77246.0 −0.351621 −0.175810 0.984424i $$-0.556255\pi$$
−0.175810 + 0.984424i $$0.556255\pi$$
$$138$$ 0 0
$$139$$ −122742. −0.538835 −0.269418 0.963023i $$-0.586831\pi$$
−0.269418 + 0.963023i $$0.586831\pi$$
$$140$$ 0 0
$$141$$ −88008.0 −0.372799
$$142$$ 0 0
$$143$$ 178560. 0.730204
$$144$$ 0 0
$$145$$ 2936.00 0.0115967
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −469234. −1.73151 −0.865753 0.500472i $$-0.833160\pi$$
−0.865753 + 0.500472i $$0.833160\pi$$
$$150$$ 0 0
$$151$$ 411584. 1.46898 0.734490 0.678619i $$-0.237421\pi$$
0.734490 + 0.678619i $$0.237421\pi$$
$$152$$ 0 0
$$153$$ −215694. −0.744920
$$154$$ 0 0
$$155$$ −20560.0 −0.0687375
$$156$$ 0 0
$$157$$ 574632. 1.86055 0.930274 0.366867i $$-0.119569\pi$$
0.930274 + 0.366867i $$0.119569\pi$$
$$158$$ 0 0
$$159$$ 87132.0 0.273328
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 264704. 0.780354 0.390177 0.920740i $$-0.372414\pi$$
0.390177 + 0.920740i $$0.372414\pi$$
$$164$$ 0 0
$$165$$ 5760.00 0.0164707
$$166$$ 0 0
$$167$$ 343356. 0.952694 0.476347 0.879257i $$-0.341961\pi$$
0.476347 + 0.879257i $$0.341961\pi$$
$$168$$ 0 0
$$169$$ 182243. 0.490833
$$170$$ 0 0
$$171$$ 204102. 0.533773
$$172$$ 0 0
$$173$$ 303296. 0.770462 0.385231 0.922820i $$-0.374122\pi$$
0.385231 + 0.922820i $$0.374122\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 80172.0 0.192349
$$178$$ 0 0
$$179$$ 362756. 0.846218 0.423109 0.906079i $$-0.360939\pi$$
0.423109 + 0.906079i $$0.360939\pi$$
$$180$$ 0 0
$$181$$ 146560. 0.332521 0.166260 0.986082i $$-0.446831\pi$$
0.166260 + 0.986082i $$0.446831\pi$$
$$182$$ 0 0
$$183$$ 58056.0 0.128150
$$184$$ 0 0
$$185$$ 24216.0 0.0520203
$$186$$ 0 0
$$187$$ 250080. 0.522968
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −552536. −1.09592 −0.547958 0.836506i $$-0.684595\pi$$
−0.547958 + 0.836506i $$0.684595\pi$$
$$192$$ 0 0
$$193$$ 305358. 0.590087 0.295043 0.955484i $$-0.404666\pi$$
0.295043 + 0.955484i $$0.404666\pi$$
$$194$$ 0 0
$$195$$ 17856.0 0.0336277
$$196$$ 0 0
$$197$$ 743838. 1.36557 0.682783 0.730621i $$-0.260769\pi$$
0.682783 + 0.730621i $$0.260769\pi$$
$$198$$ 0 0
$$199$$ 286220. 0.512351 0.256175 0.966630i $$-0.417538\pi$$
0.256175 + 0.966630i $$0.417538\pi$$
$$200$$ 0 0
$$201$$ −372744. −0.650760
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 30392.0 0.0505097
$$206$$ 0 0
$$207$$ 38088.0 0.0617820
$$208$$ 0 0
$$209$$ −236640. −0.374733
$$210$$ 0 0
$$211$$ 895372. 1.38451 0.692257 0.721651i $$-0.256616\pi$$
0.692257 + 0.721651i $$0.256616\pi$$
$$212$$ 0 0
$$213$$ −12672.0 −0.0191380
$$214$$ 0 0
$$215$$ 52064.0 0.0768142
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −173460. −0.244393
$$220$$ 0 0
$$221$$ 775248. 1.06773
$$222$$ 0 0
$$223$$ −1.18812e6 −1.59992 −0.799960 0.600054i $$-0.795146\pi$$
−0.799960 + 0.600054i $$0.795146\pi$$
$$224$$ 0 0
$$225$$ 643563. 0.847490
$$226$$ 0 0
$$227$$ 808822. 1.04181 0.520905 0.853615i $$-0.325595\pi$$
0.520905 + 0.853615i $$0.325595\pi$$
$$228$$ 0 0
$$229$$ 344344. 0.433914 0.216957 0.976181i $$-0.430387\pi$$
0.216957 + 0.976181i $$0.430387\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −334022. −0.403074 −0.201537 0.979481i $$-0.564594\pi$$
−0.201537 + 0.979481i $$0.564594\pi$$
$$234$$ 0 0
$$235$$ −58672.0 −0.0693045
$$236$$ 0 0
$$237$$ −610608. −0.706141
$$238$$ 0 0
$$239$$ 954272. 1.08063 0.540316 0.841463i $$-0.318305\pi$$
0.540316 + 0.841463i $$0.318305\pi$$
$$240$$ 0 0
$$241$$ 272882. 0.302644 0.151322 0.988485i $$-0.451647\pi$$
0.151322 + 0.988485i $$0.451647\pi$$
$$242$$ 0 0
$$243$$ −860706. −0.935059
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −733584. −0.765081
$$248$$ 0 0
$$249$$ 143532. 0.146707
$$250$$ 0 0
$$251$$ 43754.0 0.0438363 0.0219181 0.999760i $$-0.493023\pi$$
0.0219181 + 0.999760i $$0.493023\pi$$
$$252$$ 0 0
$$253$$ −44160.0 −0.0433738
$$254$$ 0 0
$$255$$ 25008.0 0.0240840
$$256$$ 0 0
$$257$$ 1.73201e6 1.63576 0.817878 0.575391i $$-0.195150\pi$$
0.817878 + 0.575391i $$0.195150\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 151938. 0.138059
$$262$$ 0 0
$$263$$ −613248. −0.546697 −0.273349 0.961915i $$-0.588131\pi$$
−0.273349 + 0.961915i $$0.588131\pi$$
$$264$$ 0 0
$$265$$ 58088.0 0.0508126
$$266$$ 0 0
$$267$$ 850044. 0.729732
$$268$$ 0 0
$$269$$ 2.01360e6 1.69665 0.848325 0.529475i $$-0.177611\pi$$
0.848325 + 0.529475i $$0.177611\pi$$
$$270$$ 0 0
$$271$$ −1.22138e6 −1.01024 −0.505122 0.863048i $$-0.668552\pi$$
−0.505122 + 0.863048i $$0.668552\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −746160. −0.594977
$$276$$ 0 0
$$277$$ 2.11821e6 1.65871 0.829355 0.558722i $$-0.188708\pi$$
0.829355 + 0.558722i $$0.188708\pi$$
$$278$$ 0 0
$$279$$ −1.06398e6 −0.818320
$$280$$ 0 0
$$281$$ −1.64516e6 −1.24292 −0.621458 0.783447i $$-0.713459\pi$$
−0.621458 + 0.783447i $$0.713459\pi$$
$$282$$ 0 0
$$283$$ 1.66393e6 1.23501 0.617504 0.786567i $$-0.288144\pi$$
0.617504 + 0.786567i $$0.288144\pi$$
$$284$$ 0 0
$$285$$ −23664.0 −0.0172574
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −334093. −0.235300
$$290$$ 0 0
$$291$$ 599892. 0.415280
$$292$$ 0 0
$$293$$ 1.15732e6 0.787559 0.393779 0.919205i $$-0.371167\pi$$
0.393779 + 0.919205i $$0.371167\pi$$
$$294$$ 0 0
$$295$$ 53448.0 0.0357583
$$296$$ 0 0
$$297$$ 648000. 0.426269
$$298$$ 0 0
$$299$$ −136896. −0.0885549
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −652104. −0.408047
$$304$$ 0 0
$$305$$ 38704.0 0.0238235
$$306$$ 0 0
$$307$$ 344998. 0.208915 0.104458 0.994529i $$-0.466689\pi$$
0.104458 + 0.994529i $$0.466689\pi$$
$$308$$ 0 0
$$309$$ −524376. −0.312426
$$310$$ 0 0
$$311$$ 3.28798e6 1.92765 0.963824 0.266540i $$-0.0858804\pi$$
0.963824 + 0.266540i $$0.0858804\pi$$
$$312$$ 0 0
$$313$$ 2.21063e6 1.27542 0.637712 0.770275i $$-0.279881\pi$$
0.637712 + 0.770275i $$0.279881\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.19631e6 0.668645 0.334322 0.942459i $$-0.391493\pi$$
0.334322 + 0.942459i $$0.391493\pi$$
$$318$$ 0 0
$$319$$ −176160. −0.0969238
$$320$$ 0 0
$$321$$ 431352. 0.233652
$$322$$ 0 0
$$323$$ −1.02741e6 −0.547947
$$324$$ 0 0
$$325$$ −2.31310e6 −1.21475
$$326$$ 0 0
$$327$$ 708996. 0.366669
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2.12828e6 −1.06772 −0.533862 0.845572i $$-0.679260\pi$$
−0.533862 + 0.845572i $$0.679260\pi$$
$$332$$ 0 0
$$333$$ 1.25318e6 0.619302
$$334$$ 0 0
$$335$$ −248496. −0.120978
$$336$$ 0 0
$$337$$ 1.89841e6 0.910576 0.455288 0.890344i $$-0.349536\pi$$
0.455288 + 0.890344i $$0.349536\pi$$
$$338$$ 0 0
$$339$$ −1.51664e6 −0.716778
$$340$$ 0 0
$$341$$ 1.23360e6 0.574498
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −4416.00 −0.00199747
$$346$$ 0 0
$$347$$ −2.17548e6 −0.969910 −0.484955 0.874539i $$-0.661164\pi$$
−0.484955 + 0.874539i $$0.661164\pi$$
$$348$$ 0 0
$$349$$ −2.12950e6 −0.935869 −0.467934 0.883763i $$-0.655002\pi$$
−0.467934 + 0.883763i $$0.655002\pi$$
$$350$$ 0 0
$$351$$ 2.00880e6 0.870300
$$352$$ 0 0
$$353$$ 2.54144e6 1.08553 0.542766 0.839884i $$-0.317377\pi$$
0.542766 + 0.839884i $$0.317377\pi$$
$$354$$ 0 0
$$355$$ −8448.00 −0.00355781
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 479280. 0.196270 0.0981348 0.995173i $$-0.468712\pi$$
0.0981348 + 0.995173i $$0.468712\pi$$
$$360$$ 0 0
$$361$$ −1.50390e6 −0.607368
$$362$$ 0 0
$$363$$ 620706. 0.247240
$$364$$ 0 0
$$365$$ −115640. −0.0454335
$$366$$ 0 0
$$367$$ 1.89390e6 0.733991 0.366996 0.930223i $$-0.380386\pi$$
0.366996 + 0.930223i $$0.380386\pi$$
$$368$$ 0 0
$$369$$ 1.57279e6 0.601317
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 1.56683e6 0.583109 0.291555 0.956554i $$-0.405828\pi$$
0.291555 + 0.956554i $$0.405828\pi$$
$$374$$ 0 0
$$375$$ −149616. −0.0549414
$$376$$ 0 0
$$377$$ −546096. −0.197886
$$378$$ 0 0
$$379$$ 57360.0 0.0205121 0.0102561 0.999947i $$-0.496735\pi$$
0.0102561 + 0.999947i $$0.496735\pi$$
$$380$$ 0 0
$$381$$ −21552.0 −0.00760633
$$382$$ 0 0
$$383$$ −4.41239e6 −1.53701 −0.768505 0.639844i $$-0.778999\pi$$
−0.768505 + 0.639844i $$0.778999\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 2.69431e6 0.914472
$$388$$ 0 0
$$389$$ −872470. −0.292332 −0.146166 0.989260i $$-0.546693\pi$$
−0.146166 + 0.989260i $$0.546693\pi$$
$$390$$ 0 0
$$391$$ −191728. −0.0634225
$$392$$ 0 0
$$393$$ −2.18720e6 −0.714345
$$394$$ 0 0
$$395$$ −407072. −0.131274
$$396$$ 0 0
$$397$$ −3.63170e6 −1.15647 −0.578233 0.815871i $$-0.696258\pi$$
−0.578233 + 0.815871i $$0.696258\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3.58423e6 −1.11310 −0.556550 0.830814i $$-0.687875\pi$$
−0.556550 + 0.830814i $$0.687875\pi$$
$$402$$ 0 0
$$403$$ 3.82416e6 1.17293
$$404$$ 0 0
$$405$$ −136404. −0.0413228
$$406$$ 0 0
$$407$$ −1.45296e6 −0.434778
$$408$$ 0 0
$$409$$ −2.18309e6 −0.645304 −0.322652 0.946518i $$-0.604574\pi$$
−0.322652 + 0.946518i $$0.604574\pi$$
$$410$$ 0 0
$$411$$ 463476. 0.135339
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 95688.0 0.0272733
$$416$$ 0 0
$$417$$ 736452. 0.207398
$$418$$ 0 0
$$419$$ −4.91137e6 −1.36668 −0.683342 0.730099i $$-0.739474\pi$$
−0.683342 + 0.730099i $$0.739474\pi$$
$$420$$ 0 0
$$421$$ 693766. 0.190769 0.0953845 0.995441i $$-0.469592\pi$$
0.0953845 + 0.995441i $$0.469592\pi$$
$$422$$ 0 0
$$423$$ −3.03628e6 −0.825069
$$424$$ 0 0
$$425$$ −3.23958e6 −0.869994
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −1.07136e6 −0.281056
$$430$$ 0 0
$$431$$ 1.25035e6 0.324219 0.162110 0.986773i $$-0.448170\pi$$
0.162110 + 0.986773i $$0.448170\pi$$
$$432$$ 0 0
$$433$$ −157750. −0.0404343 −0.0202171 0.999796i $$-0.506436\pi$$
−0.0202171 + 0.999796i $$0.506436\pi$$
$$434$$ 0 0
$$435$$ −17616.0 −0.00446359
$$436$$ 0 0
$$437$$ 181424. 0.0454455
$$438$$ 0 0
$$439$$ 263736. 0.0653143 0.0326571 0.999467i $$-0.489603\pi$$
0.0326571 + 0.999467i $$0.489603\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 1.50410e6 0.364139 0.182070 0.983286i $$-0.441720\pi$$
0.182070 + 0.983286i $$0.441720\pi$$
$$444$$ 0 0
$$445$$ 566696. 0.135659
$$446$$ 0 0
$$447$$ 2.81540e6 0.666457
$$448$$ 0 0
$$449$$ 2.11128e6 0.494231 0.247116 0.968986i $$-0.420517\pi$$
0.247116 + 0.968986i $$0.420517\pi$$
$$450$$ 0 0
$$451$$ −1.82352e6 −0.422152
$$452$$ 0 0
$$453$$ −2.46950e6 −0.565411
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.99938e6 0.895782 0.447891 0.894088i $$-0.352175\pi$$
0.447891 + 0.894088i $$0.352175\pi$$
$$458$$ 0 0
$$459$$ 2.81340e6 0.623304
$$460$$ 0 0
$$461$$ 2.24090e6 0.491101 0.245551 0.969384i $$-0.421031\pi$$
0.245551 + 0.969384i $$0.421031\pi$$
$$462$$ 0 0
$$463$$ 1.47304e6 0.319346 0.159673 0.987170i $$-0.448956\pi$$
0.159673 + 0.987170i $$0.448956\pi$$
$$464$$ 0 0
$$465$$ 123360. 0.0264571
$$466$$ 0 0
$$467$$ 8.50472e6 1.80454 0.902272 0.431166i $$-0.141898\pi$$
0.902272 + 0.431166i $$0.141898\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −3.44779e6 −0.716125
$$472$$ 0 0
$$473$$ −3.12384e6 −0.642001
$$474$$ 0 0
$$475$$ 3.06547e6 0.623395
$$476$$ 0 0
$$477$$ 3.00605e6 0.604924
$$478$$ 0 0
$$479$$ −6.56984e6 −1.30833 −0.654163 0.756354i $$-0.726979\pi$$
−0.654163 + 0.756354i $$0.726979\pi$$
$$480$$ 0 0
$$481$$ −4.50418e6 −0.887672
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 399928. 0.0772018
$$486$$ 0 0
$$487$$ −7.71038e6 −1.47317 −0.736585 0.676344i $$-0.763563\pi$$
−0.736585 + 0.676344i $$0.763563\pi$$
$$488$$ 0 0
$$489$$ −1.58822e6 −0.300358
$$490$$ 0 0
$$491$$ −8.72147e6 −1.63262 −0.816311 0.577612i $$-0.803985\pi$$
−0.816311 + 0.577612i $$0.803985\pi$$
$$492$$ 0 0
$$493$$ −764828. −0.141725
$$494$$ 0 0
$$495$$ 198720. 0.0364526
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 7.87430e6 1.41567 0.707833 0.706380i $$-0.249673\pi$$
0.707833 + 0.706380i $$0.249673\pi$$
$$500$$ 0 0
$$501$$ −2.06014e6 −0.366692
$$502$$ 0 0
$$503$$ 8.68726e6 1.53096 0.765479 0.643461i $$-0.222502\pi$$
0.765479 + 0.643461i $$0.222502\pi$$
$$504$$ 0 0
$$505$$ −434736. −0.0758573
$$506$$ 0 0
$$507$$ −1.09346e6 −0.188922
$$508$$ 0 0
$$509$$ 1.34131e6 0.229475 0.114737 0.993396i $$-0.463397\pi$$
0.114737 + 0.993396i $$0.463397\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2.66220e6 −0.446629
$$514$$ 0 0
$$515$$ −349584. −0.0580809
$$516$$ 0 0
$$517$$ 3.52032e6 0.579236
$$518$$ 0 0
$$519$$ −1.81978e6 −0.296551
$$520$$ 0 0
$$521$$ −6.00185e6 −0.968704 −0.484352 0.874873i $$-0.660945\pi$$
−0.484352 + 0.874873i $$0.660945\pi$$
$$522$$ 0 0
$$523$$ 1.19109e7 1.90410 0.952048 0.305950i $$-0.0989740\pi$$
0.952048 + 0.305950i $$0.0989740\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 5.35588e6 0.840048
$$528$$ 0 0
$$529$$ −6.40249e6 −0.994740
$$530$$ 0 0
$$531$$ 2.76593e6 0.425702
$$532$$ 0 0
$$533$$ −5.65291e6 −0.861895
$$534$$ 0 0
$$535$$ 287568. 0.0434366
$$536$$ 0 0
$$537$$ −2.17654e6 −0.325709
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −7.20703e6 −1.05868 −0.529338 0.848411i $$-0.677560\pi$$
−0.529338 + 0.848411i $$0.677560\pi$$
$$542$$ 0 0
$$543$$ −879360. −0.127987
$$544$$ 0 0
$$545$$ 472664. 0.0681650
$$546$$ 0 0
$$547$$ −1.65172e6 −0.236030 −0.118015 0.993012i $$-0.537653\pi$$
−0.118015 + 0.993012i $$0.537653\pi$$
$$548$$ 0 0
$$549$$ 2.00293e6 0.283619
$$550$$ 0 0
$$551$$ 723724. 0.101553
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −145296. −0.0200226
$$556$$ 0 0
$$557$$ −7.58171e6 −1.03545 −0.517725 0.855547i $$-0.673221\pi$$
−0.517725 + 0.855547i $$0.673221\pi$$
$$558$$ 0 0
$$559$$ −9.68390e6 −1.31075
$$560$$ 0 0
$$561$$ −1.50048e6 −0.201290
$$562$$ 0 0
$$563$$ −1.26568e7 −1.68288 −0.841440 0.540351i $$-0.818292\pi$$
−0.841440 + 0.540351i $$0.818292\pi$$
$$564$$ 0 0
$$565$$ −1.01110e6 −0.133251
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 4.57445e6 0.592323 0.296162 0.955138i $$-0.404293\pi$$
0.296162 + 0.955138i $$0.404293\pi$$
$$570$$ 0 0
$$571$$ 5.77802e6 0.741632 0.370816 0.928706i $$-0.379078\pi$$
0.370816 + 0.928706i $$0.379078\pi$$
$$572$$ 0 0
$$573$$ 3.31522e6 0.421818
$$574$$ 0 0
$$575$$ 572056. 0.0721554
$$576$$ 0 0
$$577$$ −5.46520e6 −0.683387 −0.341693 0.939811i $$-0.611000\pi$$
−0.341693 + 0.939811i $$0.611000\pi$$
$$578$$ 0 0
$$579$$ −1.83215e6 −0.227125
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −3.48528e6 −0.424684
$$584$$ 0 0
$$585$$ 616032. 0.0744241
$$586$$ 0 0
$$587$$ 9.89386e6 1.18514 0.592571 0.805518i $$-0.298113\pi$$
0.592571 + 0.805518i $$0.298113\pi$$
$$588$$ 0 0
$$589$$ −5.06804e6 −0.601938
$$590$$ 0 0
$$591$$ −4.46303e6 −0.525607
$$592$$ 0 0
$$593$$ −6.12686e6 −0.715486 −0.357743 0.933820i $$-0.616454\pi$$
−0.357743 + 0.933820i $$0.616454\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.71732e6 −0.197204
$$598$$ 0 0
$$599$$ −299432. −0.0340982 −0.0170491 0.999855i $$-0.505427\pi$$
−0.0170491 + 0.999855i $$0.505427\pi$$
$$600$$ 0 0
$$601$$ 4.98133e6 0.562548 0.281274 0.959628i $$-0.409243\pi$$
0.281274 + 0.959628i $$0.409243\pi$$
$$602$$ 0 0
$$603$$ −1.28597e7 −1.44025
$$604$$ 0 0
$$605$$ 413804. 0.0459628
$$606$$ 0 0
$$607$$ −1.10694e7 −1.21942 −0.609709 0.792625i $$-0.708714\pi$$
−0.609709 + 0.792625i $$0.708714\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.09130e7 1.18261
$$612$$ 0 0
$$613$$ −1.37829e7 −1.48146 −0.740729 0.671804i $$-0.765520\pi$$
−0.740729 + 0.671804i $$0.765520\pi$$
$$614$$ 0 0
$$615$$ −182352. −0.0194412
$$616$$ 0 0
$$617$$ 1.11450e7 1.17861 0.589303 0.807912i $$-0.299403\pi$$
0.589303 + 0.807912i $$0.299403\pi$$
$$618$$ 0 0
$$619$$ −3.00722e6 −0.315456 −0.157728 0.987483i $$-0.550417\pi$$
−0.157728 + 0.987483i $$0.550417\pi$$
$$620$$ 0 0
$$621$$ −496800. −0.0516955
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 9.61588e6 0.984666
$$626$$ 0 0
$$627$$ 1.41984e6 0.144235
$$628$$ 0 0
$$629$$ −6.30827e6 −0.635746
$$630$$ 0 0
$$631$$ 570304. 0.0570208 0.0285104 0.999593i $$-0.490924\pi$$
0.0285104 + 0.999593i $$0.490924\pi$$
$$632$$ 0 0
$$633$$ −5.37223e6 −0.532900
$$634$$ 0 0
$$635$$ −14368.0 −0.00141404
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −437184. −0.0423557
$$640$$ 0 0
$$641$$ 1.37359e7 1.32042 0.660212 0.751080i $$-0.270467\pi$$
0.660212 + 0.751080i $$0.270467\pi$$
$$642$$ 0 0
$$643$$ 2.58692e6 0.246749 0.123375 0.992360i $$-0.460628\pi$$
0.123375 + 0.992360i $$0.460628\pi$$
$$644$$ 0 0
$$645$$ −312384. −0.0295658
$$646$$ 0 0
$$647$$ 6.52446e6 0.612751 0.306375 0.951911i $$-0.400884\pi$$
0.306375 + 0.951911i $$0.400884\pi$$
$$648$$ 0 0
$$649$$ −3.20688e6 −0.298862
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3.75793e6 0.344878 0.172439 0.985020i $$-0.444835\pi$$
0.172439 + 0.985020i $$0.444835\pi$$
$$654$$ 0 0
$$655$$ −1.45814e6 −0.132799
$$656$$ 0 0
$$657$$ −5.98437e6 −0.540885
$$658$$ 0 0
$$659$$ 6.97436e6 0.625591 0.312796 0.949820i $$-0.398734\pi$$
0.312796 + 0.949820i $$0.398734\pi$$
$$660$$ 0 0
$$661$$ −1.17059e7 −1.04208 −0.521042 0.853531i $$-0.674457\pi$$
−0.521042 + 0.853531i $$0.674457\pi$$
$$662$$ 0 0
$$663$$ −4.65149e6 −0.410968
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 135056. 0.0117544
$$668$$ 0 0
$$669$$ 7.12872e6 0.615809
$$670$$ 0 0
$$671$$ −2.32224e6 −0.199114
$$672$$ 0 0
$$673$$ −1.82825e7 −1.55596 −0.777980 0.628289i $$-0.783755\pi$$
−0.777980 + 0.628289i $$0.783755\pi$$
$$674$$ 0 0
$$675$$ −8.39430e6 −0.709129
$$676$$ 0 0
$$677$$ −2.05661e6 −0.172457 −0.0862283 0.996275i $$-0.527481\pi$$
−0.0862283 + 0.996275i $$0.527481\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −4.85293e6 −0.400993
$$682$$ 0 0
$$683$$ −1.24913e7 −1.02461 −0.512303 0.858805i $$-0.671208\pi$$
−0.512303 + 0.858805i $$0.671208\pi$$
$$684$$ 0 0
$$685$$ 308984. 0.0251599
$$686$$ 0 0
$$687$$ −2.06606e6 −0.167014
$$688$$ 0 0
$$689$$ −1.08044e7 −0.867064
$$690$$ 0 0
$$691$$ 176630. 0.0140724 0.00703622 0.999975i $$-0.497760\pi$$
0.00703622 + 0.999975i $$0.497760\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 490968. 0.0385559
$$696$$ 0 0
$$697$$ −7.91712e6 −0.617284
$$698$$ 0 0
$$699$$ 2.00413e6 0.155143
$$700$$ 0 0
$$701$$ 4.03111e6 0.309835 0.154917 0.987927i $$-0.450489\pi$$
0.154917 + 0.987927i $$0.450489\pi$$
$$702$$ 0 0
$$703$$ 5.96924e6 0.455545
$$704$$ 0 0
$$705$$ 352032. 0.0266753
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.41839e7 1.05969 0.529847 0.848093i $$-0.322249\pi$$
0.529847 + 0.848093i $$0.322249\pi$$
$$710$$ 0 0
$$711$$ −2.10660e7 −1.56282
$$712$$ 0 0
$$713$$ −945760. −0.0696718
$$714$$ 0 0
$$715$$ −714240. −0.0522491
$$716$$ 0 0
$$717$$ −5.72563e6 −0.415935
$$718$$ 0 0
$$719$$ −2.46272e7 −1.77661 −0.888306 0.459253i $$-0.848117\pi$$
−0.888306 + 0.459253i $$0.848117\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −1.63729e6 −0.116488
$$724$$ 0 0
$$725$$ 2.28201e6 0.161240
$$726$$ 0 0
$$727$$ −1.30482e7 −0.915615 −0.457808 0.889051i $$-0.651365\pi$$
−0.457808 + 0.889051i $$0.651365\pi$$
$$728$$ 0 0
$$729$$ −3.12231e6 −0.217599
$$730$$ 0 0
$$731$$ −1.35627e7 −0.938754
$$732$$ 0 0
$$733$$ 2.08870e7 1.43587 0.717936 0.696109i $$-0.245087\pi$$
0.717936 + 0.696109i $$0.245087\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.49098e7 1.01112
$$738$$ 0 0
$$739$$ 1.47615e7 0.994303 0.497151 0.867664i $$-0.334379\pi$$
0.497151 + 0.867664i $$0.334379\pi$$
$$740$$ 0 0
$$741$$ 4.40150e6 0.294480
$$742$$ 0 0
$$743$$ 4.44570e6 0.295439 0.147719 0.989029i $$-0.452807\pi$$
0.147719 + 0.989029i $$0.452807\pi$$
$$744$$ 0 0
$$745$$ 1.87694e6 0.123896
$$746$$ 0 0
$$747$$ 4.95185e6 0.324688
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −1.19094e7 −0.770528 −0.385264 0.922806i $$-0.625890\pi$$
−0.385264 + 0.922806i $$0.625890\pi$$
$$752$$ 0 0
$$753$$ −262524. −0.0168726
$$754$$ 0 0
$$755$$ −1.64634e6 −0.105112
$$756$$ 0 0
$$757$$ −2.55035e7 −1.61756 −0.808781 0.588110i $$-0.799872\pi$$
−0.808781 + 0.588110i $$0.799872\pi$$
$$758$$ 0 0
$$759$$ 264960. 0.0166946
$$760$$ 0 0
$$761$$ 1.46925e7 0.919675 0.459837 0.888003i $$-0.347908\pi$$
0.459837 + 0.888003i $$0.347908\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 862776. 0.0533021
$$766$$ 0 0
$$767$$ −9.94133e6 −0.610177
$$768$$ 0 0
$$769$$ 1.92779e7 1.17556 0.587780 0.809021i $$-0.300002\pi$$
0.587780 + 0.809021i $$0.300002\pi$$
$$770$$ 0 0
$$771$$ −1.03921e7 −0.629603
$$772$$ 0 0
$$773$$ 8.56584e6 0.515610 0.257805 0.966197i $$-0.417001\pi$$
0.257805 + 0.966197i $$0.417001\pi$$
$$774$$ 0 0
$$775$$ −1.59803e7 −0.955718
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 7.49163e6 0.442316
$$780$$ 0 0
$$781$$ 506880. 0.0297357
$$782$$ 0 0
$$783$$ −1.98180e6 −0.115520
$$784$$ 0 0
$$785$$ −2.29853e6 −0.133130
$$786$$ 0 0
$$787$$ 1.89027e7 1.08789 0.543947 0.839119i $$-0.316929\pi$$
0.543947 + 0.839119i $$0.316929\pi$$
$$788$$ 0 0
$$789$$ 3.67949e6 0.210424
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −7.19894e6 −0.406524
$$794$$ 0 0
$$795$$ −348528. −0.0195578
$$796$$ 0 0
$$797$$ 1.71259e7 0.955010 0.477505 0.878629i $$-0.341541\pi$$
0.477505 + 0.878629i $$0.341541\pi$$
$$798$$ 0 0
$$799$$ 1.52841e7 0.846977
$$800$$ 0 0
$$801$$ 2.93265e7 1.61502
$$802$$ 0 0
$$803$$ 6.93840e6 0.379726
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −1.20816e7 −0.653041
$$808$$ 0 0
$$809$$ 2.84511e7 1.52837 0.764185 0.644997i $$-0.223142\pi$$
0.764185 + 0.644997i $$0.223142\pi$$
$$810$$ 0 0
$$811$$ 6.55604e6 0.350017 0.175009 0.984567i $$-0.444005\pi$$
0.175009 + 0.984567i $$0.444005\pi$$
$$812$$ 0 0
$$813$$ 7.32826e6 0.388843
$$814$$ 0 0
$$815$$ −1.05882e6 −0.0558376
$$816$$ 0 0
$$817$$ 1.28338e7 0.672666
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −3.14356e6 −0.162766 −0.0813831 0.996683i $$-0.525934\pi$$
−0.0813831 + 0.996683i $$0.525934\pi$$
$$822$$ 0 0
$$823$$ −1.62191e7 −0.834694 −0.417347 0.908747i $$-0.637040\pi$$
−0.417347 + 0.908747i $$0.637040\pi$$
$$824$$ 0 0
$$825$$ 4.47696e6 0.229007
$$826$$ 0 0
$$827$$ 4.74707e6 0.241358 0.120679 0.992692i $$-0.461493\pi$$
0.120679 + 0.992692i $$0.461493\pi$$
$$828$$ 0 0
$$829$$ −3.47333e7 −1.75533 −0.877666 0.479272i $$-0.840901\pi$$
−0.877666 + 0.479272i $$0.840901\pi$$
$$830$$ 0 0
$$831$$ −1.27093e7 −0.638438
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −1.37342e6 −0.0681692
$$836$$ 0 0
$$837$$ 1.38780e7 0.684720
$$838$$ 0 0
$$839$$ −6.10552e6 −0.299445 −0.149723 0.988728i $$-0.547838\pi$$
−0.149723 + 0.988728i $$0.547838\pi$$
$$840$$ 0 0
$$841$$ −1.99724e7 −0.973734
$$842$$ 0 0
$$843$$ 9.87095e6 0.478399
$$844$$ 0 0
$$845$$ −728972. −0.0351212
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −9.98360e6 −0.475355
$$850$$ 0 0
$$851$$ 1.11394e6 0.0527274
$$852$$ 0 0
$$853$$ −2.75613e7 −1.29696 −0.648481 0.761231i $$-0.724595\pi$$
−0.648481 + 0.761231i $$0.724595\pi$$
$$854$$ 0 0
$$855$$ −816408. −0.0381937
$$856$$ 0 0
$$857$$ 1.82100e7 0.846950 0.423475 0.905908i $$-0.360810\pi$$
0.423475 + 0.905908i $$0.360810\pi$$
$$858$$ 0 0
$$859$$ 3.35920e7 1.55329 0.776647 0.629936i $$-0.216919\pi$$
0.776647 + 0.629936i $$0.216919\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −3.26084e7 −1.49040 −0.745199 0.666843i $$-0.767645\pi$$
−0.745199 + 0.666843i $$0.767645\pi$$
$$864$$ 0 0
$$865$$ −1.21318e6 −0.0551298
$$866$$ 0 0
$$867$$ 2.00456e6 0.0905672
$$868$$ 0 0
$$869$$ 2.44243e7 1.09717
$$870$$ 0 0
$$871$$ 4.62203e7 2.06437
$$872$$ 0 0
$$873$$ 2.06963e7 0.919087
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.33352e7 0.585463 0.292732 0.956195i $$-0.405436\pi$$
0.292732 + 0.956195i $$0.405436\pi$$
$$878$$ 0 0
$$879$$ −6.94390e6 −0.303131
$$880$$ 0 0
$$881$$ −1.43194e7 −0.621564 −0.310782 0.950481i $$-0.600591\pi$$
−0.310782 + 0.950481i $$0.600591\pi$$
$$882$$ 0 0
$$883$$ −4.01556e6 −0.173318 −0.0866592 0.996238i $$-0.527619\pi$$
−0.0866592 + 0.996238i $$0.527619\pi$$
$$884$$ 0 0
$$885$$ −320688. −0.0137634
$$886$$ 0 0
$$887$$ −4.29049e7 −1.83104 −0.915520 0.402272i $$-0.868221\pi$$
−0.915520 + 0.402272i $$0.868221\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 8.18424e6 0.345370
$$892$$ 0 0
$$893$$ −1.44626e7 −0.606903
$$894$$ 0 0
$$895$$ −1.45102e6 −0.0605504
$$896$$ 0 0
$$897$$ 821376. 0.0340848
$$898$$ 0 0
$$899$$ −3.77276e6 −0.155690
$$900$$ 0 0
$$901$$ −1.51319e7 −0.620987
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −586240. −0.0237933
$$906$$ 0 0
$$907$$ 3.04706e7 1.22988 0.614940 0.788574i $$-0.289180\pi$$
0.614940 + 0.788574i $$0.289180\pi$$
$$908$$ 0 0
$$909$$ −2.24976e7 −0.903080
$$910$$ 0 0
$$911$$ 2.75748e7 1.10082 0.550411 0.834894i $$-0.314471\pi$$
0.550411 + 0.834894i $$0.314471\pi$$
$$912$$ 0 0
$$913$$ −5.74128e6 −0.227946
$$914$$ 0 0
$$915$$ −232224. −0.00916968
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −3.33346e7 −1.30199 −0.650993 0.759084i $$-0.725647\pi$$
−0.650993 + 0.759084i $$0.725647\pi$$
$$920$$ 0 0
$$921$$ −2.06999e6 −0.0804116
$$922$$ 0 0
$$923$$ 1.57133e6 0.0607103
$$924$$ 0 0
$$925$$ 1.88219e7 0.723284
$$926$$ 0 0
$$927$$ −1.80910e7 −0.691453
$$928$$ 0 0
$$929$$ −2.08624e7 −0.793096 −0.396548 0.918014i $$-0.629792\pi$$
−0.396548 + 0.918014i $$0.629792\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −1.97279e7 −0.741952
$$934$$ 0 0
$$935$$ −1.00032e6 −0.0374205
$$936$$ 0 0
$$937$$ 1.66618e7 0.619975 0.309987 0.950741i $$-0.399675\pi$$
0.309987 + 0.950741i $$0.399675\pi$$
$$938$$ 0 0
$$939$$ −1.32638e7 −0.490911
$$940$$ 0 0
$$941$$ 3.36202e7 1.23773 0.618865 0.785497i $$-0.287593\pi$$
0.618865 + 0.785497i $$0.287593\pi$$
$$942$$ 0 0
$$943$$ 1.39803e6 0.0511962
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 4.25335e7 1.54119 0.770595 0.637325i $$-0.219959\pi$$
0.770595 + 0.637325i $$0.219959\pi$$
$$948$$ 0 0
$$949$$ 2.15090e7 0.775274
$$950$$ 0 0
$$951$$ −7.17786e6 −0.257362
$$952$$ 0 0
$$953$$ 3.20613e7 1.14353 0.571767 0.820416i $$-0.306258\pi$$
0.571767 + 0.820416i $$0.306258\pi$$
$$954$$ 0 0
$$955$$ 2.21014e6 0.0784173
$$956$$ 0 0
$$957$$ 1.05696e6 0.0373060
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.20955e6 −0.0771784
$$962$$ 0 0
$$963$$ 1.48816e7 0.517113
$$964$$ 0 0
$$965$$ −1.22143e6 −0.0422232
$$966$$ 0 0
$$967$$ 5.65115e7 1.94344 0.971719 0.236139i $$-0.0758819\pi$$
0.971719 + 0.236139i $$0.0758819\pi$$
$$968$$ 0 0
$$969$$ 6.16447e6 0.210905
$$970$$ 0 0
$$971$$ 4.41580e7 1.50301 0.751504 0.659729i $$-0.229329\pi$$
0.751504 + 0.659729i $$0.229329\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 1.38786e7 0.467556
$$976$$ 0 0
$$977$$ −2.69053e7 −0.901782 −0.450891 0.892579i $$-0.648894\pi$$
−0.450891 + 0.892579i $$0.648894\pi$$
$$978$$ 0 0
$$979$$ −3.40018e7 −1.13382
$$980$$ 0 0
$$981$$ 2.44604e7 0.811503
$$982$$ 0 0
$$983$$ 8.68688e6 0.286735 0.143367 0.989670i $$-0.454207\pi$$
0.143367 + 0.989670i $$0.454207\pi$$
$$984$$ 0 0
$$985$$ −2.97535e6 −0.0977120
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 2.39494e6 0.0778582
$$990$$ 0 0
$$991$$ 1.54909e7 0.501063 0.250532 0.968108i $$-0.419395\pi$$
0.250532 + 0.968108i $$0.419395\pi$$
$$992$$ 0 0
$$993$$ 1.27697e7 0.410967
$$994$$ 0 0
$$995$$ −1.14488e6 −0.0366608
$$996$$ 0 0
$$997$$ 4.47588e6 0.142607 0.0713034 0.997455i $$-0.477284\pi$$
0.0713034 + 0.997455i $$0.477284\pi$$
$$998$$ 0 0
$$999$$ −1.63458e7 −0.518194
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 784.6.a.e.1.1 1
4.3 odd 2 392.6.a.c.1.1 1
7.6 odd 2 112.6.a.f.1.1 1
21.20 even 2 1008.6.a.p.1.1 1
28.3 even 6 392.6.i.d.177.1 2
28.11 odd 6 392.6.i.c.177.1 2
28.19 even 6 392.6.i.d.361.1 2
28.23 odd 6 392.6.i.c.361.1 2
28.27 even 2 56.6.a.a.1.1 1
56.13 odd 2 448.6.a.g.1.1 1
56.27 even 2 448.6.a.j.1.1 1
84.83 odd 2 504.6.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.a.1.1 1 28.27 even 2
112.6.a.f.1.1 1 7.6 odd 2
392.6.a.c.1.1 1 4.3 odd 2
392.6.i.c.177.1 2 28.11 odd 6
392.6.i.c.361.1 2 28.23 odd 6
392.6.i.d.177.1 2 28.3 even 6
392.6.i.d.361.1 2 28.19 even 6
448.6.a.g.1.1 1 56.13 odd 2
448.6.a.j.1.1 1 56.27 even 2
504.6.a.e.1.1 1 84.83 odd 2
784.6.a.e.1.1 1 1.1 even 1 trivial
1008.6.a.p.1.1 1 21.20 even 2