Properties

 Label 528.6.a.i.1.1 Level $528$ Weight $6$ Character 528.1 Self dual yes Analytic conductor $84.683$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("528.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 528.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: $$84.6826568613$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+9.00000 q^{3} +46.0000 q^{5} -148.000 q^{7} +81.0000 q^{9} +O(q^{10})$$ $$q+9.00000 q^{3} +46.0000 q^{5} -148.000 q^{7} +81.0000 q^{9} -121.000 q^{11} +574.000 q^{13} +414.000 q^{15} -722.000 q^{17} -2160.00 q^{19} -1332.00 q^{21} +2536.00 q^{23} -1009.00 q^{25} +729.000 q^{27} +4650.00 q^{29} -5032.00 q^{31} -1089.00 q^{33} -6808.00 q^{35} +8118.00 q^{37} +5166.00 q^{39} -5138.00 q^{41} -8304.00 q^{43} +3726.00 q^{45} -24728.0 q^{47} +5097.00 q^{49} -6498.00 q^{51} -28746.0 q^{53} -5566.00 q^{55} -19440.0 q^{57} +5860.00 q^{59} -53658.0 q^{61} -11988.0 q^{63} +26404.0 q^{65} -30908.0 q^{67} +22824.0 q^{69} +69648.0 q^{71} -18446.0 q^{73} -9081.00 q^{75} +17908.0 q^{77} +25300.0 q^{79} +6561.00 q^{81} +17556.0 q^{83} -33212.0 q^{85} +41850.0 q^{87} +132570. q^{89} -84952.0 q^{91} -45288.0 q^{93} -99360.0 q^{95} +70658.0 q^{97} -9801.00 q^{99} +O(q^{100})$$

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
$$3$$ 9.00000 0.577350
$$4$$ 0 0
$$5$$ 46.0000 0.822873 0.411437 0.911438i $$-0.365027\pi$$
0.411437 + 0.911438i $$0.365027\pi$$
$$6$$ 0 0
$$7$$ −148.000 −1.14161 −0.570803 0.821087i $$-0.693368\pi$$
−0.570803 + 0.821087i $$0.693368\pi$$
$$8$$ 0 0
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ −121.000 −0.301511
$$12$$ 0 0
$$13$$ 574.000 0.942006 0.471003 0.882132i $$-0.343892\pi$$
0.471003 + 0.882132i $$0.343892\pi$$
$$14$$ 0 0
$$15$$ 414.000 0.475086
$$16$$ 0 0
$$17$$ −722.000 −0.605919 −0.302960 0.953003i $$-0.597975\pi$$
−0.302960 + 0.953003i $$0.597975\pi$$
$$18$$ 0 0
$$19$$ −2160.00 −1.37268 −0.686341 0.727280i $$-0.740784\pi$$
−0.686341 + 0.727280i $$0.740784\pi$$
$$20$$ 0 0
$$21$$ −1332.00 −0.659107
$$22$$ 0 0
$$23$$ 2536.00 0.999608 0.499804 0.866139i $$-0.333405\pi$$
0.499804 + 0.866139i $$0.333405\pi$$
$$24$$ 0 0
$$25$$ −1009.00 −0.322880
$$26$$ 0 0
$$27$$ 729.000 0.192450
$$28$$ 0 0
$$29$$ 4650.00 1.02673 0.513367 0.858169i $$-0.328398\pi$$
0.513367 + 0.858169i $$0.328398\pi$$
$$30$$ 0 0
$$31$$ −5032.00 −0.940451 −0.470226 0.882546i $$-0.655828\pi$$
−0.470226 + 0.882546i $$0.655828\pi$$
$$32$$ 0 0
$$33$$ −1089.00 −0.174078
$$34$$ 0 0
$$35$$ −6808.00 −0.939398
$$36$$ 0 0
$$37$$ 8118.00 0.974866 0.487433 0.873161i $$-0.337933\pi$$
0.487433 + 0.873161i $$0.337933\pi$$
$$38$$ 0 0
$$39$$ 5166.00 0.543867
$$40$$ 0 0
$$41$$ −5138.00 −0.477347 −0.238674 0.971100i $$-0.576713\pi$$
−0.238674 + 0.971100i $$0.576713\pi$$
$$42$$ 0 0
$$43$$ −8304.00 −0.684883 −0.342441 0.939539i $$-0.611254\pi$$
−0.342441 + 0.939539i $$0.611254\pi$$
$$44$$ 0 0
$$45$$ 3726.00 0.274291
$$46$$ 0 0
$$47$$ −24728.0 −1.63284 −0.816421 0.577457i $$-0.804045\pi$$
−0.816421 + 0.577457i $$0.804045\pi$$
$$48$$ 0 0
$$49$$ 5097.00 0.303266
$$50$$ 0 0
$$51$$ −6498.00 −0.349828
$$52$$ 0 0
$$53$$ −28746.0 −1.40568 −0.702842 0.711346i $$-0.748086\pi$$
−0.702842 + 0.711346i $$0.748086\pi$$
$$54$$ 0 0
$$55$$ −5566.00 −0.248106
$$56$$ 0 0
$$57$$ −19440.0 −0.792518
$$58$$ 0 0
$$59$$ 5860.00 0.219163 0.109582 0.993978i $$-0.465049\pi$$
0.109582 + 0.993978i $$0.465049\pi$$
$$60$$ 0 0
$$61$$ −53658.0 −1.84633 −0.923166 0.384401i $$-0.874408\pi$$
−0.923166 + 0.384401i $$0.874408\pi$$
$$62$$ 0 0
$$63$$ −11988.0 −0.380536
$$64$$ 0 0
$$65$$ 26404.0 0.775151
$$66$$ 0 0
$$67$$ −30908.0 −0.841170 −0.420585 0.907253i $$-0.638175\pi$$
−0.420585 + 0.907253i $$0.638175\pi$$
$$68$$ 0 0
$$69$$ 22824.0 0.577124
$$70$$ 0 0
$$71$$ 69648.0 1.63969 0.819847 0.572583i $$-0.194058\pi$$
0.819847 + 0.572583i $$0.194058\pi$$
$$72$$ 0 0
$$73$$ −18446.0 −0.405131 −0.202565 0.979269i $$-0.564928\pi$$
−0.202565 + 0.979269i $$0.564928\pi$$
$$74$$ 0 0
$$75$$ −9081.00 −0.186415
$$76$$ 0 0
$$77$$ 17908.0 0.344207
$$78$$ 0 0
$$79$$ 25300.0 0.456092 0.228046 0.973650i $$-0.426766\pi$$
0.228046 + 0.973650i $$0.426766\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 0 0
$$83$$ 17556.0 0.279724 0.139862 0.990171i $$-0.455334\pi$$
0.139862 + 0.990171i $$0.455334\pi$$
$$84$$ 0 0
$$85$$ −33212.0 −0.498595
$$86$$ 0 0
$$87$$ 41850.0 0.592785
$$88$$ 0 0
$$89$$ 132570. 1.77407 0.887034 0.461704i $$-0.152762\pi$$
0.887034 + 0.461704i $$0.152762\pi$$
$$90$$ 0 0
$$91$$ −84952.0 −1.07540
$$92$$ 0 0
$$93$$ −45288.0 −0.542970
$$94$$ 0 0
$$95$$ −99360.0 −1.12954
$$96$$ 0 0
$$97$$ 70658.0 0.762486 0.381243 0.924475i $$-0.375496\pi$$
0.381243 + 0.924475i $$0.375496\pi$$
$$98$$ 0 0
$$99$$ −9801.00 −0.100504
$$100$$ 0 0
$$101$$ −101998. −0.994920 −0.497460 0.867487i $$-0.665734\pi$$
−0.497460 + 0.867487i $$0.665734\pi$$
$$102$$ 0 0
$$103$$ −130904. −1.21579 −0.607897 0.794016i $$-0.707987\pi$$
−0.607897 + 0.794016i $$0.707987\pi$$
$$104$$ 0 0
$$105$$ −61272.0 −0.542361
$$106$$ 0 0
$$107$$ 141612. 1.19575 0.597875 0.801589i $$-0.296012\pi$$
0.597875 + 0.801589i $$0.296012\pi$$
$$108$$ 0 0
$$109$$ −239810. −1.93331 −0.966654 0.256086i $$-0.917567\pi$$
−0.966654 + 0.256086i $$0.917567\pi$$
$$110$$ 0 0
$$111$$ 73062.0 0.562839
$$112$$ 0 0
$$113$$ −42726.0 −0.314772 −0.157386 0.987537i $$-0.550307\pi$$
−0.157386 + 0.987537i $$0.550307\pi$$
$$114$$ 0 0
$$115$$ 116656. 0.822550
$$116$$ 0 0
$$117$$ 46494.0 0.314002
$$118$$ 0 0
$$119$$ 106856. 0.691722
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ 0 0
$$123$$ −46242.0 −0.275597
$$124$$ 0 0
$$125$$ −190164. −1.08856
$$126$$ 0 0
$$127$$ −51788.0 −0.284918 −0.142459 0.989801i $$-0.545501\pi$$
−0.142459 + 0.989801i $$0.545501\pi$$
$$128$$ 0 0
$$129$$ −74736.0 −0.395417
$$130$$ 0 0
$$131$$ −53652.0 −0.273154 −0.136577 0.990629i $$-0.543610\pi$$
−0.136577 + 0.990629i $$0.543610\pi$$
$$132$$ 0 0
$$133$$ 319680. 1.56706
$$134$$ 0 0
$$135$$ 33534.0 0.158362
$$136$$ 0 0
$$137$$ −228862. −1.04177 −0.520886 0.853627i $$-0.674398\pi$$
−0.520886 + 0.853627i $$0.674398\pi$$
$$138$$ 0 0
$$139$$ −374920. −1.64589 −0.822947 0.568119i $$-0.807671\pi$$
−0.822947 + 0.568119i $$0.807671\pi$$
$$140$$ 0 0
$$141$$ −222552. −0.942722
$$142$$ 0 0
$$143$$ −69454.0 −0.284025
$$144$$ 0 0
$$145$$ 213900. 0.844872
$$146$$ 0 0
$$147$$ 45873.0 0.175091
$$148$$ 0 0
$$149$$ −65830.0 −0.242917 −0.121459 0.992597i $$-0.538757\pi$$
−0.121459 + 0.992597i $$0.538757\pi$$
$$150$$ 0 0
$$151$$ −154052. −0.549826 −0.274913 0.961469i $$-0.588649\pi$$
−0.274913 + 0.961469i $$0.588649\pi$$
$$152$$ 0 0
$$153$$ −58482.0 −0.201973
$$154$$ 0 0
$$155$$ −231472. −0.773872
$$156$$ 0 0
$$157$$ 287678. 0.931446 0.465723 0.884931i $$-0.345794\pi$$
0.465723 + 0.884931i $$0.345794\pi$$
$$158$$ 0 0
$$159$$ −258714. −0.811572
$$160$$ 0 0
$$161$$ −375328. −1.14116
$$162$$ 0 0
$$163$$ −105124. −0.309908 −0.154954 0.987922i $$-0.549523\pi$$
−0.154954 + 0.987922i $$0.549523\pi$$
$$164$$ 0 0
$$165$$ −50094.0 −0.143244
$$166$$ 0 0
$$167$$ −150528. −0.417663 −0.208832 0.977952i $$-0.566966\pi$$
−0.208832 + 0.977952i $$0.566966\pi$$
$$168$$ 0 0
$$169$$ −41817.0 −0.112625
$$170$$ 0 0
$$171$$ −174960. −0.457560
$$172$$ 0 0
$$173$$ −2166.00 −0.00550229 −0.00275114 0.999996i $$-0.500876\pi$$
−0.00275114 + 0.999996i $$0.500876\pi$$
$$174$$ 0 0
$$175$$ 149332. 0.368602
$$176$$ 0 0
$$177$$ 52740.0 0.126534
$$178$$ 0 0
$$179$$ −672780. −1.56942 −0.784712 0.619860i $$-0.787189\pi$$
−0.784712 + 0.619860i $$0.787189\pi$$
$$180$$ 0 0
$$181$$ −526778. −1.19517 −0.597587 0.801804i $$-0.703874\pi$$
−0.597587 + 0.801804i $$0.703874\pi$$
$$182$$ 0 0
$$183$$ −482922. −1.06598
$$184$$ 0 0
$$185$$ 373428. 0.802191
$$186$$ 0 0
$$187$$ 87362.0 0.182692
$$188$$ 0 0
$$189$$ −107892. −0.219702
$$190$$ 0 0
$$191$$ 305608. 0.606152 0.303076 0.952966i $$-0.401986\pi$$
0.303076 + 0.952966i $$0.401986\pi$$
$$192$$ 0 0
$$193$$ 116434. 0.225002 0.112501 0.993652i $$-0.464114\pi$$
0.112501 + 0.993652i $$0.464114\pi$$
$$194$$ 0 0
$$195$$ 237636. 0.447534
$$196$$ 0 0
$$197$$ −247742. −0.454814 −0.227407 0.973800i $$-0.573025\pi$$
−0.227407 + 0.973800i $$0.573025\pi$$
$$198$$ 0 0
$$199$$ 513360. 0.918945 0.459472 0.888192i $$-0.348039\pi$$
0.459472 + 0.888192i $$0.348039\pi$$
$$200$$ 0 0
$$201$$ −278172. −0.485650
$$202$$ 0 0
$$203$$ −688200. −1.17213
$$204$$ 0 0
$$205$$ −236348. −0.392796
$$206$$ 0 0
$$207$$ 205416. 0.333203
$$208$$ 0 0
$$209$$ 261360. 0.413879
$$210$$ 0 0
$$211$$ 620688. 0.959770 0.479885 0.877331i $$-0.340678\pi$$
0.479885 + 0.877331i $$0.340678\pi$$
$$212$$ 0 0
$$213$$ 626832. 0.946678
$$214$$ 0 0
$$215$$ −381984. −0.563571
$$216$$ 0 0
$$217$$ 744736. 1.07363
$$218$$ 0 0
$$219$$ −166014. −0.233902
$$220$$ 0 0
$$221$$ −414428. −0.570780
$$222$$ 0 0
$$223$$ 1.31802e6 1.77484 0.887419 0.460964i $$-0.152496\pi$$
0.887419 + 0.460964i $$0.152496\pi$$
$$224$$ 0 0
$$225$$ −81729.0 −0.107627
$$226$$ 0 0
$$227$$ 887412. 1.14304 0.571519 0.820589i $$-0.306354\pi$$
0.571519 + 0.820589i $$0.306354\pi$$
$$228$$ 0 0
$$229$$ −237450. −0.299215 −0.149608 0.988745i $$-0.547801\pi$$
−0.149608 + 0.988745i $$0.547801\pi$$
$$230$$ 0 0
$$231$$ 161172. 0.198728
$$232$$ 0 0
$$233$$ −914706. −1.10380 −0.551902 0.833909i $$-0.686098\pi$$
−0.551902 + 0.833909i $$0.686098\pi$$
$$234$$ 0 0
$$235$$ −1.13749e6 −1.34362
$$236$$ 0 0
$$237$$ 227700. 0.263325
$$238$$ 0 0
$$239$$ −1.40892e6 −1.59548 −0.797740 0.603001i $$-0.793971\pi$$
−0.797740 + 0.603001i $$0.793971\pi$$
$$240$$ 0 0
$$241$$ −826358. −0.916486 −0.458243 0.888827i $$-0.651521\pi$$
−0.458243 + 0.888827i $$0.651521\pi$$
$$242$$ 0 0
$$243$$ 59049.0 0.0641500
$$244$$ 0 0
$$245$$ 234462. 0.249550
$$246$$ 0 0
$$247$$ −1.23984e6 −1.29307
$$248$$ 0 0
$$249$$ 158004. 0.161499
$$250$$ 0 0
$$251$$ 1.60387e6 1.60688 0.803442 0.595384i $$-0.203000\pi$$
0.803442 + 0.595384i $$0.203000\pi$$
$$252$$ 0 0
$$253$$ −306856. −0.301393
$$254$$ 0 0
$$255$$ −298908. −0.287864
$$256$$ 0 0
$$257$$ 397618. 0.375520 0.187760 0.982215i $$-0.439877\pi$$
0.187760 + 0.982215i $$0.439877\pi$$
$$258$$ 0 0
$$259$$ −1.20146e6 −1.11291
$$260$$ 0 0
$$261$$ 376650. 0.342245
$$262$$ 0 0
$$263$$ −2.13166e6 −1.90033 −0.950166 0.311745i $$-0.899087\pi$$
−0.950166 + 0.311745i $$0.899087\pi$$
$$264$$ 0 0
$$265$$ −1.32232e6 −1.15670
$$266$$ 0 0
$$267$$ 1.19313e6 1.02426
$$268$$ 0 0
$$269$$ −725810. −0.611564 −0.305782 0.952101i $$-0.598918\pi$$
−0.305782 + 0.952101i $$0.598918\pi$$
$$270$$ 0 0
$$271$$ 1.46787e6 1.21413 0.607063 0.794654i $$-0.292348\pi$$
0.607063 + 0.794654i $$0.292348\pi$$
$$272$$ 0 0
$$273$$ −764568. −0.620883
$$274$$ 0 0
$$275$$ 122089. 0.0973520
$$276$$ 0 0
$$277$$ 1.52100e6 1.19105 0.595524 0.803338i $$-0.296944\pi$$
0.595524 + 0.803338i $$0.296944\pi$$
$$278$$ 0 0
$$279$$ −407592. −0.313484
$$280$$ 0 0
$$281$$ 464382. 0.350840 0.175420 0.984494i $$-0.443872\pi$$
0.175420 + 0.984494i $$0.443872\pi$$
$$282$$ 0 0
$$283$$ 415136. 0.308123 0.154062 0.988061i $$-0.450765\pi$$
0.154062 + 0.988061i $$0.450765\pi$$
$$284$$ 0 0
$$285$$ −894240. −0.652142
$$286$$ 0 0
$$287$$ 760424. 0.544943
$$288$$ 0 0
$$289$$ −898573. −0.632862
$$290$$ 0 0
$$291$$ 635922. 0.440222
$$292$$ 0 0
$$293$$ −2.59321e6 −1.76469 −0.882344 0.470605i $$-0.844036\pi$$
−0.882344 + 0.470605i $$0.844036\pi$$
$$294$$ 0 0
$$295$$ 269560. 0.180343
$$296$$ 0 0
$$297$$ −88209.0 −0.0580259
$$298$$ 0 0
$$299$$ 1.45566e6 0.941636
$$300$$ 0 0
$$301$$ 1.22899e6 0.781867
$$302$$ 0 0
$$303$$ −917982. −0.574417
$$304$$ 0 0
$$305$$ −2.46827e6 −1.51930
$$306$$ 0 0
$$307$$ 930832. 0.563671 0.281835 0.959463i $$-0.409057\pi$$
0.281835 + 0.959463i $$0.409057\pi$$
$$308$$ 0 0
$$309$$ −1.17814e6 −0.701939
$$310$$ 0 0
$$311$$ −2.48527e6 −1.45704 −0.728522 0.685022i $$-0.759793\pi$$
−0.728522 + 0.685022i $$0.759793\pi$$
$$312$$ 0 0
$$313$$ 1.31719e6 0.759957 0.379978 0.924995i $$-0.375931\pi$$
0.379978 + 0.924995i $$0.375931\pi$$
$$314$$ 0 0
$$315$$ −551448. −0.313133
$$316$$ 0 0
$$317$$ 2.25540e6 1.26059 0.630297 0.776354i $$-0.282933\pi$$
0.630297 + 0.776354i $$0.282933\pi$$
$$318$$ 0 0
$$319$$ −562650. −0.309572
$$320$$ 0 0
$$321$$ 1.27451e6 0.690367
$$322$$ 0 0
$$323$$ 1.55952e6 0.831734
$$324$$ 0 0
$$325$$ −579166. −0.304155
$$326$$ 0 0
$$327$$ −2.15829e6 −1.11620
$$328$$ 0 0
$$329$$ 3.65974e6 1.86406
$$330$$ 0 0
$$331$$ 3.17071e6 1.59069 0.795346 0.606155i $$-0.207289\pi$$
0.795346 + 0.606155i $$0.207289\pi$$
$$332$$ 0 0
$$333$$ 657558. 0.324955
$$334$$ 0 0
$$335$$ −1.42177e6 −0.692176
$$336$$ 0 0
$$337$$ 1.27630e6 0.612177 0.306089 0.952003i $$-0.400980\pi$$
0.306089 + 0.952003i $$0.400980\pi$$
$$338$$ 0 0
$$339$$ −384534. −0.181734
$$340$$ 0 0
$$341$$ 608872. 0.283557
$$342$$ 0 0
$$343$$ 1.73308e6 0.795396
$$344$$ 0 0
$$345$$ 1.04990e6 0.474900
$$346$$ 0 0
$$347$$ −3.69303e6 −1.64649 −0.823245 0.567687i $$-0.807838\pi$$
−0.823245 + 0.567687i $$0.807838\pi$$
$$348$$ 0 0
$$349$$ 1.70919e6 0.751150 0.375575 0.926792i $$-0.377445\pi$$
0.375575 + 0.926792i $$0.377445\pi$$
$$350$$ 0 0
$$351$$ 418446. 0.181289
$$352$$ 0 0
$$353$$ 4.36859e6 1.86597 0.932986 0.359914i $$-0.117194\pi$$
0.932986 + 0.359914i $$0.117194\pi$$
$$354$$ 0 0
$$355$$ 3.20381e6 1.34926
$$356$$ 0 0
$$357$$ 961704. 0.399366
$$358$$ 0 0
$$359$$ 3.51284e6 1.43854 0.719271 0.694730i $$-0.244476\pi$$
0.719271 + 0.694730i $$0.244476\pi$$
$$360$$ 0 0
$$361$$ 2.18950e6 0.884254
$$362$$ 0 0
$$363$$ 131769. 0.0524864
$$364$$ 0 0
$$365$$ −848516. −0.333371
$$366$$ 0 0
$$367$$ 2.15259e6 0.834251 0.417125 0.908849i $$-0.363038\pi$$
0.417125 + 0.908849i $$0.363038\pi$$
$$368$$ 0 0
$$369$$ −416178. −0.159116
$$370$$ 0 0
$$371$$ 4.25441e6 1.60474
$$372$$ 0 0
$$373$$ −2.24247e6 −0.834553 −0.417276 0.908780i $$-0.637015\pi$$
−0.417276 + 0.908780i $$0.637015\pi$$
$$374$$ 0 0
$$375$$ −1.71148e6 −0.628482
$$376$$ 0 0
$$377$$ 2.66910e6 0.967189
$$378$$ 0 0
$$379$$ 2.40986e6 0.861775 0.430887 0.902406i $$-0.358201\pi$$
0.430887 + 0.902406i $$0.358201\pi$$
$$380$$ 0 0
$$381$$ −466092. −0.164497
$$382$$ 0 0
$$383$$ 1.01066e6 0.352052 0.176026 0.984386i $$-0.443676\pi$$
0.176026 + 0.984386i $$0.443676\pi$$
$$384$$ 0 0
$$385$$ 823768. 0.283239
$$386$$ 0 0
$$387$$ −672624. −0.228294
$$388$$ 0 0
$$389$$ 1.27779e6 0.428140 0.214070 0.976818i $$-0.431328\pi$$
0.214070 + 0.976818i $$0.431328\pi$$
$$390$$ 0 0
$$391$$ −1.83099e6 −0.605682
$$392$$ 0 0
$$393$$ −482868. −0.157706
$$394$$ 0 0
$$395$$ 1.16380e6 0.375306
$$396$$ 0 0
$$397$$ 5.45400e6 1.73676 0.868378 0.495903i $$-0.165163\pi$$
0.868378 + 0.495903i $$0.165163\pi$$
$$398$$ 0 0
$$399$$ 2.87712e6 0.904744
$$400$$ 0 0
$$401$$ −1.48980e6 −0.462665 −0.231332 0.972875i $$-0.574308\pi$$
−0.231332 + 0.972875i $$0.574308\pi$$
$$402$$ 0 0
$$403$$ −2.88837e6 −0.885911
$$404$$ 0 0
$$405$$ 301806. 0.0914303
$$406$$ 0 0
$$407$$ −982278. −0.293933
$$408$$ 0 0
$$409$$ −4.39899e6 −1.30030 −0.650152 0.759804i $$-0.725295\pi$$
−0.650152 + 0.759804i $$0.725295\pi$$
$$410$$ 0 0
$$411$$ −2.05976e6 −0.601467
$$412$$ 0 0
$$413$$ −867280. −0.250198
$$414$$ 0 0
$$415$$ 807576. 0.230178
$$416$$ 0 0
$$417$$ −3.37428e6 −0.950257
$$418$$ 0 0
$$419$$ 280420. 0.0780322 0.0390161 0.999239i $$-0.487578\pi$$
0.0390161 + 0.999239i $$0.487578\pi$$
$$420$$ 0 0
$$421$$ 817462. 0.224782 0.112391 0.993664i $$-0.464149\pi$$
0.112391 + 0.993664i $$0.464149\pi$$
$$422$$ 0 0
$$423$$ −2.00297e6 −0.544281
$$424$$ 0 0
$$425$$ 728498. 0.195639
$$426$$ 0 0
$$427$$ 7.94138e6 2.10779
$$428$$ 0 0
$$429$$ −625086. −0.163982
$$430$$ 0 0
$$431$$ −1.88599e6 −0.489043 −0.244521 0.969644i $$-0.578631\pi$$
−0.244521 + 0.969644i $$0.578631\pi$$
$$432$$ 0 0
$$433$$ 5.84067e6 1.49707 0.748537 0.663093i $$-0.230757\pi$$
0.748537 + 0.663093i $$0.230757\pi$$
$$434$$ 0 0
$$435$$ 1.92510e6 0.487787
$$436$$ 0 0
$$437$$ −5.47776e6 −1.37214
$$438$$ 0 0
$$439$$ 509540. 0.126188 0.0630938 0.998008i $$-0.479903\pi$$
0.0630938 + 0.998008i $$0.479903\pi$$
$$440$$ 0 0
$$441$$ 412857. 0.101089
$$442$$ 0 0
$$443$$ −4.10268e6 −0.993250 −0.496625 0.867965i $$-0.665428\pi$$
−0.496625 + 0.867965i $$0.665428\pi$$
$$444$$ 0 0
$$445$$ 6.09822e6 1.45983
$$446$$ 0 0
$$447$$ −592470. −0.140248
$$448$$ 0 0
$$449$$ 513410. 0.120185 0.0600923 0.998193i $$-0.480861\pi$$
0.0600923 + 0.998193i $$0.480861\pi$$
$$450$$ 0 0
$$451$$ 621698. 0.143926
$$452$$ 0 0
$$453$$ −1.38647e6 −0.317442
$$454$$ 0 0
$$455$$ −3.90779e6 −0.884918
$$456$$ 0 0
$$457$$ 1.22738e6 0.274908 0.137454 0.990508i $$-0.456108\pi$$
0.137454 + 0.990508i $$0.456108\pi$$
$$458$$ 0 0
$$459$$ −526338. −0.116609
$$460$$ 0 0
$$461$$ −6.41000e6 −1.40477 −0.702386 0.711797i $$-0.747882\pi$$
−0.702386 + 0.711797i $$0.747882\pi$$
$$462$$ 0 0
$$463$$ −6.63030e6 −1.43741 −0.718705 0.695315i $$-0.755265\pi$$
−0.718705 + 0.695315i $$0.755265\pi$$
$$464$$ 0 0
$$465$$ −2.08325e6 −0.446795
$$466$$ 0 0
$$467$$ 4.14769e6 0.880064 0.440032 0.897982i $$-0.354967\pi$$
0.440032 + 0.897982i $$0.354967\pi$$
$$468$$ 0 0
$$469$$ 4.57438e6 0.960286
$$470$$ 0 0
$$471$$ 2.58910e6 0.537770
$$472$$ 0 0
$$473$$ 1.00478e6 0.206500
$$474$$ 0 0
$$475$$ 2.17944e6 0.443211
$$476$$ 0 0
$$477$$ −2.32843e6 −0.468561
$$478$$ 0 0
$$479$$ 5.05132e6 1.00593 0.502963 0.864308i $$-0.332243\pi$$
0.502963 + 0.864308i $$0.332243\pi$$
$$480$$ 0 0
$$481$$ 4.65973e6 0.918329
$$482$$ 0 0
$$483$$ −3.37795e6 −0.658849
$$484$$ 0 0
$$485$$ 3.25027e6 0.627429
$$486$$ 0 0
$$487$$ −2.66221e6 −0.508651 −0.254325 0.967119i $$-0.581853\pi$$
−0.254325 + 0.967119i $$0.581853\pi$$
$$488$$ 0 0
$$489$$ −946116. −0.178925
$$490$$ 0 0
$$491$$ 5.54659e6 1.03830 0.519149 0.854684i $$-0.326249\pi$$
0.519149 + 0.854684i $$0.326249\pi$$
$$492$$ 0 0
$$493$$ −3.35730e6 −0.622118
$$494$$ 0 0
$$495$$ −450846. −0.0827018
$$496$$ 0 0
$$497$$ −1.03079e7 −1.87189
$$498$$ 0 0
$$499$$ 6820.00 0.00122612 0.000613060 1.00000i $$-0.499805\pi$$
0.000613060 1.00000i $$0.499805\pi$$
$$500$$ 0 0
$$501$$ −1.35475e6 −0.241138
$$502$$ 0 0
$$503$$ 451136. 0.0795037 0.0397519 0.999210i $$-0.487343\pi$$
0.0397519 + 0.999210i $$0.487343\pi$$
$$504$$ 0 0
$$505$$ −4.69191e6 −0.818693
$$506$$ 0 0
$$507$$ −376353. −0.0650243
$$508$$ 0 0
$$509$$ 393390. 0.0673021 0.0336511 0.999434i $$-0.489287\pi$$
0.0336511 + 0.999434i $$0.489287\pi$$
$$510$$ 0 0
$$511$$ 2.73001e6 0.462500
$$512$$ 0 0
$$513$$ −1.57464e6 −0.264173
$$514$$ 0 0
$$515$$ −6.02158e6 −1.00044
$$516$$ 0 0
$$517$$ 2.99209e6 0.492321
$$518$$ 0 0
$$519$$ −19494.0 −0.00317675
$$520$$ 0 0
$$521$$ 3.28432e6 0.530092 0.265046 0.964236i $$-0.414613\pi$$
0.265046 + 0.964236i $$0.414613\pi$$
$$522$$ 0 0
$$523$$ 1.68266e6 0.268993 0.134497 0.990914i $$-0.457058\pi$$
0.134497 + 0.990914i $$0.457058\pi$$
$$524$$ 0 0
$$525$$ 1.34399e6 0.212812
$$526$$ 0 0
$$527$$ 3.63310e6 0.569838
$$528$$ 0 0
$$529$$ −5047.00 −0.000784141 0
$$530$$ 0 0
$$531$$ 474660. 0.0730544
$$532$$ 0 0
$$533$$ −2.94921e6 −0.449664
$$534$$ 0 0
$$535$$ 6.51415e6 0.983951
$$536$$ 0 0
$$537$$ −6.05502e6 −0.906108
$$538$$ 0 0
$$539$$ −616737. −0.0914383
$$540$$ 0 0
$$541$$ 9.48158e6 1.39280 0.696398 0.717656i $$-0.254785\pi$$
0.696398 + 0.717656i $$0.254785\pi$$
$$542$$ 0 0
$$543$$ −4.74100e6 −0.690034
$$544$$ 0 0
$$545$$ −1.10313e7 −1.59087
$$546$$ 0 0
$$547$$ 6.09239e6 0.870602 0.435301 0.900285i $$-0.356642\pi$$
0.435301 + 0.900285i $$0.356642\pi$$
$$548$$ 0 0
$$549$$ −4.34630e6 −0.615444
$$550$$ 0 0
$$551$$ −1.00440e7 −1.40938
$$552$$ 0 0
$$553$$ −3.74440e6 −0.520678
$$554$$ 0 0
$$555$$ 3.36085e6 0.463145
$$556$$ 0 0
$$557$$ 8.49594e6 1.16031 0.580154 0.814507i $$-0.302992\pi$$
0.580154 + 0.814507i $$0.302992\pi$$
$$558$$ 0 0
$$559$$ −4.76650e6 −0.645163
$$560$$ 0 0
$$561$$ 786258. 0.105477
$$562$$ 0 0
$$563$$ 7.02216e6 0.933683 0.466842 0.884341i $$-0.345392\pi$$
0.466842 + 0.884341i $$0.345392\pi$$
$$564$$ 0 0
$$565$$ −1.96540e6 −0.259017
$$566$$ 0 0
$$567$$ −971028. −0.126845
$$568$$ 0 0
$$569$$ 9.41847e6 1.21955 0.609775 0.792574i $$-0.291260\pi$$
0.609775 + 0.792574i $$0.291260\pi$$
$$570$$ 0 0
$$571$$ −7.29699e6 −0.936599 −0.468299 0.883570i $$-0.655133\pi$$
−0.468299 + 0.883570i $$0.655133\pi$$
$$572$$ 0 0
$$573$$ 2.75047e6 0.349962
$$574$$ 0 0
$$575$$ −2.55882e6 −0.322753
$$576$$ 0 0
$$577$$ −3.29590e6 −0.412131 −0.206065 0.978538i $$-0.566066\pi$$
−0.206065 + 0.978538i $$0.566066\pi$$
$$578$$ 0 0
$$579$$ 1.04791e6 0.129905
$$580$$ 0 0
$$581$$ −2.59829e6 −0.319335
$$582$$ 0 0
$$583$$ 3.47827e6 0.423830
$$584$$ 0 0
$$585$$ 2.13872e6 0.258384
$$586$$ 0 0
$$587$$ −4.39827e6 −0.526849 −0.263425 0.964680i $$-0.584852\pi$$
−0.263425 + 0.964680i $$0.584852\pi$$
$$588$$ 0 0
$$589$$ 1.08691e7 1.29094
$$590$$ 0 0
$$591$$ −2.22968e6 −0.262587
$$592$$ 0 0
$$593$$ 9.21781e6 1.07644 0.538222 0.842803i $$-0.319096\pi$$
0.538222 + 0.842803i $$0.319096\pi$$
$$594$$ 0 0
$$595$$ 4.91538e6 0.569199
$$596$$ 0 0
$$597$$ 4.62024e6 0.530553
$$598$$ 0 0
$$599$$ −3.77140e6 −0.429473 −0.214736 0.976672i $$-0.568889\pi$$
−0.214736 + 0.976672i $$0.568889\pi$$
$$600$$ 0 0
$$601$$ 4.19724e6 0.473999 0.237000 0.971510i $$-0.423836\pi$$
0.237000 + 0.971510i $$0.423836\pi$$
$$602$$ 0 0
$$603$$ −2.50355e6 −0.280390
$$604$$ 0 0
$$605$$ 673486. 0.0748066
$$606$$ 0 0
$$607$$ 1.00133e6 0.110308 0.0551539 0.998478i $$-0.482435\pi$$
0.0551539 + 0.998478i $$0.482435\pi$$
$$608$$ 0 0
$$609$$ −6.19380e6 −0.676728
$$610$$ 0 0
$$611$$ −1.41939e7 −1.53815
$$612$$ 0 0
$$613$$ −7.38239e6 −0.793498 −0.396749 0.917927i $$-0.629862\pi$$
−0.396749 + 0.917927i $$0.629862\pi$$
$$614$$ 0 0
$$615$$ −2.12713e6 −0.226781
$$616$$ 0 0
$$617$$ −1.54025e7 −1.62884 −0.814418 0.580279i $$-0.802944\pi$$
−0.814418 + 0.580279i $$0.802944\pi$$
$$618$$ 0 0
$$619$$ 7.12402e6 0.747306 0.373653 0.927569i $$-0.378105\pi$$
0.373653 + 0.927569i $$0.378105\pi$$
$$620$$ 0 0
$$621$$ 1.84874e6 0.192375
$$622$$ 0 0
$$623$$ −1.96204e7 −2.02529
$$624$$ 0 0
$$625$$ −5.59442e6 −0.572869
$$626$$ 0 0
$$627$$ 2.35224e6 0.238953
$$628$$ 0 0
$$629$$ −5.86120e6 −0.590690
$$630$$ 0 0
$$631$$ −1.16696e7 −1.16677 −0.583383 0.812197i $$-0.698271\pi$$
−0.583383 + 0.812197i $$0.698271\pi$$
$$632$$ 0 0
$$633$$ 5.58619e6 0.554124
$$634$$ 0 0
$$635$$ −2.38225e6 −0.234451
$$636$$ 0 0
$$637$$ 2.92568e6 0.285679
$$638$$ 0 0
$$639$$ 5.64149e6 0.546565
$$640$$ 0 0
$$641$$ −1.10271e7 −1.06003 −0.530014 0.847989i $$-0.677813\pi$$
−0.530014 + 0.847989i $$0.677813\pi$$
$$642$$ 0 0
$$643$$ 9.56024e6 0.911887 0.455944 0.890009i $$-0.349302\pi$$
0.455944 + 0.890009i $$0.349302\pi$$
$$644$$ 0 0
$$645$$ −3.43786e6 −0.325378
$$646$$ 0 0
$$647$$ 1.09942e7 1.03253 0.516263 0.856430i $$-0.327323\pi$$
0.516263 + 0.856430i $$0.327323\pi$$
$$648$$ 0 0
$$649$$ −709060. −0.0660802
$$650$$ 0 0
$$651$$ 6.70262e6 0.619858
$$652$$ 0 0
$$653$$ −295346. −0.0271049 −0.0135525 0.999908i $$-0.504314\pi$$
−0.0135525 + 0.999908i $$0.504314\pi$$
$$654$$ 0 0
$$655$$ −2.46799e6 −0.224771
$$656$$ 0 0
$$657$$ −1.49413e6 −0.135044
$$658$$ 0 0
$$659$$ 1.65613e7 1.48553 0.742766 0.669551i $$-0.233514\pi$$
0.742766 + 0.669551i $$0.233514\pi$$
$$660$$ 0 0
$$661$$ 1.97042e6 0.175411 0.0877053 0.996146i $$-0.472047\pi$$
0.0877053 + 0.996146i $$0.472047\pi$$
$$662$$ 0 0
$$663$$ −3.72985e6 −0.329540
$$664$$ 0 0
$$665$$ 1.47053e7 1.28949
$$666$$ 0 0
$$667$$ 1.17924e7 1.02633
$$668$$ 0 0
$$669$$ 1.18621e7 1.02470
$$670$$ 0 0
$$671$$ 6.49262e6 0.556690
$$672$$ 0 0
$$673$$ −1.63733e6 −0.139347 −0.0696735 0.997570i $$-0.522196\pi$$
−0.0696735 + 0.997570i $$0.522196\pi$$
$$674$$ 0 0
$$675$$ −735561. −0.0621383
$$676$$ 0 0
$$677$$ −6.35878e6 −0.533215 −0.266607 0.963805i $$-0.585903\pi$$
−0.266607 + 0.963805i $$0.585903\pi$$
$$678$$ 0 0
$$679$$ −1.04574e7 −0.870460
$$680$$ 0 0
$$681$$ 7.98671e6 0.659933
$$682$$ 0 0
$$683$$ −1.11033e7 −0.910751 −0.455376 0.890299i $$-0.650495\pi$$
−0.455376 + 0.890299i $$0.650495\pi$$
$$684$$ 0 0
$$685$$ −1.05277e7 −0.857245
$$686$$ 0 0
$$687$$ −2.13705e6 −0.172752
$$688$$ 0 0
$$689$$ −1.65002e7 −1.32416
$$690$$ 0 0
$$691$$ −1.70189e7 −1.35592 −0.677962 0.735097i $$-0.737136\pi$$
−0.677962 + 0.735097i $$0.737136\pi$$
$$692$$ 0 0
$$693$$ 1.45055e6 0.114736
$$694$$ 0 0
$$695$$ −1.72463e7 −1.35436
$$696$$ 0 0
$$697$$ 3.70964e6 0.289234
$$698$$ 0 0
$$699$$ −8.23235e6 −0.637281
$$700$$ 0 0
$$701$$ 1.58021e7 1.21456 0.607280 0.794488i $$-0.292260\pi$$
0.607280 + 0.794488i $$0.292260\pi$$
$$702$$ 0 0
$$703$$ −1.75349e7 −1.33818
$$704$$ 0 0
$$705$$ −1.02374e7 −0.775741
$$706$$ 0 0
$$707$$ 1.50957e7 1.13581
$$708$$ 0 0
$$709$$ 1.24834e7 0.932643 0.466322 0.884615i $$-0.345579\pi$$
0.466322 + 0.884615i $$0.345579\pi$$
$$710$$ 0 0
$$711$$ 2.04930e6 0.152031
$$712$$ 0 0
$$713$$ −1.27612e7 −0.940083
$$714$$ 0 0
$$715$$ −3.19488e6 −0.233717
$$716$$ 0 0
$$717$$ −1.26803e7 −0.921151
$$718$$ 0 0
$$719$$ −2.00724e6 −0.144803 −0.0724014 0.997376i $$-0.523066\pi$$
−0.0724014 + 0.997376i $$0.523066\pi$$
$$720$$ 0 0
$$721$$ 1.93738e7 1.38796
$$722$$ 0 0
$$723$$ −7.43722e6 −0.529133
$$724$$ 0 0
$$725$$ −4.69185e6 −0.331512
$$726$$ 0 0
$$727$$ −6.97301e6 −0.489310 −0.244655 0.969610i $$-0.578675\pi$$
−0.244655 + 0.969610i $$0.578675\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ 5.99549e6 0.414984
$$732$$ 0 0
$$733$$ −2.34965e7 −1.61527 −0.807633 0.589685i $$-0.799252\pi$$
−0.807633 + 0.589685i $$0.799252\pi$$
$$734$$ 0 0
$$735$$ 2.11016e6 0.144078
$$736$$ 0 0
$$737$$ 3.73987e6 0.253622
$$738$$ 0 0
$$739$$ 1.39901e7 0.942346 0.471173 0.882041i $$-0.343831\pi$$
0.471173 + 0.882041i $$0.343831\pi$$
$$740$$ 0 0
$$741$$ −1.11586e7 −0.746556
$$742$$ 0 0
$$743$$ −2.42745e7 −1.61316 −0.806582 0.591123i $$-0.798685\pi$$
−0.806582 + 0.591123i $$0.798685\pi$$
$$744$$ 0 0
$$745$$ −3.02818e6 −0.199890
$$746$$ 0 0
$$747$$ 1.42204e6 0.0932415
$$748$$ 0 0
$$749$$ −2.09586e7 −1.36508
$$750$$ 0 0
$$751$$ −1.53660e7 −0.994170 −0.497085 0.867702i $$-0.665596\pi$$
−0.497085 + 0.867702i $$0.665596\pi$$
$$752$$ 0 0
$$753$$ 1.44348e7 0.927734
$$754$$ 0 0
$$755$$ −7.08639e6 −0.452437
$$756$$ 0 0
$$757$$ 2.07605e7 1.31674 0.658368 0.752697i $$-0.271247\pi$$
0.658368 + 0.752697i $$0.271247\pi$$
$$758$$ 0 0
$$759$$ −2.76170e6 −0.174009
$$760$$ 0 0
$$761$$ 5.83810e6 0.365435 0.182717 0.983165i $$-0.441511\pi$$
0.182717 + 0.983165i $$0.441511\pi$$
$$762$$ 0 0
$$763$$ 3.54919e7 2.20708
$$764$$ 0 0
$$765$$ −2.69017e6 −0.166198
$$766$$ 0 0
$$767$$ 3.36364e6 0.206453
$$768$$ 0 0
$$769$$ −1.39197e7 −0.848818 −0.424409 0.905471i $$-0.639518\pi$$
−0.424409 + 0.905471i $$0.639518\pi$$
$$770$$ 0 0
$$771$$ 3.57856e6 0.216807
$$772$$ 0 0
$$773$$ −4.17883e6 −0.251539 −0.125770 0.992059i $$-0.540140\pi$$
−0.125770 + 0.992059i $$0.540140\pi$$
$$774$$ 0 0
$$775$$ 5.07729e6 0.303653
$$776$$ 0 0
$$777$$ −1.08132e7 −0.642541
$$778$$ 0 0
$$779$$ 1.10981e7 0.655246
$$780$$ 0 0
$$781$$ −8.42741e6 −0.494386
$$782$$ 0 0
$$783$$ 3.38985e6 0.197595
$$784$$ 0 0
$$785$$ 1.32332e7 0.766461
$$786$$ 0 0
$$787$$ −9.66705e6 −0.556361 −0.278181 0.960529i $$-0.589731\pi$$
−0.278181 + 0.960529i $$0.589731\pi$$
$$788$$ 0 0
$$789$$ −1.91850e7 −1.09716
$$790$$ 0 0
$$791$$ 6.32345e6 0.359346
$$792$$ 0 0
$$793$$ −3.07997e7 −1.73926
$$794$$ 0 0
$$795$$ −1.19008e7 −0.667821
$$796$$ 0 0
$$797$$ −5.79884e6 −0.323367 −0.161683 0.986843i $$-0.551692\pi$$
−0.161683 + 0.986843i $$0.551692\pi$$
$$798$$ 0 0
$$799$$ 1.78536e7 0.989371
$$800$$ 0 0
$$801$$ 1.07382e7 0.591356
$$802$$ 0 0
$$803$$ 2.23197e6 0.122151
$$804$$ 0 0
$$805$$ −1.72651e7 −0.939029
$$806$$ 0 0
$$807$$ −6.53229e6 −0.353087
$$808$$ 0 0
$$809$$ −1.92543e7 −1.03433 −0.517163 0.855887i $$-0.673012\pi$$
−0.517163 + 0.855887i $$0.673012\pi$$
$$810$$ 0 0
$$811$$ 1.31938e7 0.704396 0.352198 0.935926i $$-0.385434\pi$$
0.352198 + 0.935926i $$0.385434\pi$$
$$812$$ 0 0
$$813$$ 1.32108e7 0.700976
$$814$$ 0 0
$$815$$ −4.83570e6 −0.255015
$$816$$ 0 0
$$817$$ 1.79366e7 0.940126
$$818$$ 0 0
$$819$$ −6.88111e6 −0.358467
$$820$$ 0 0
$$821$$ 1.33779e7 0.692677 0.346338 0.938110i $$-0.387425\pi$$
0.346338 + 0.938110i $$0.387425\pi$$
$$822$$ 0 0
$$823$$ 1.88613e7 0.970673 0.485336 0.874327i $$-0.338697\pi$$
0.485336 + 0.874327i $$0.338697\pi$$
$$824$$ 0 0
$$825$$ 1.09880e6 0.0562062
$$826$$ 0 0
$$827$$ −1.62680e7 −0.827123 −0.413561 0.910476i $$-0.635715\pi$$
−0.413561 + 0.910476i $$0.635715\pi$$
$$828$$ 0 0
$$829$$ −2.18098e7 −1.10221 −0.551107 0.834435i $$-0.685794\pi$$
−0.551107 + 0.834435i $$0.685794\pi$$
$$830$$ 0 0
$$831$$ 1.36890e7 0.687652
$$832$$ 0 0
$$833$$ −3.68003e6 −0.183755
$$834$$ 0 0
$$835$$ −6.92429e6 −0.343684
$$836$$ 0 0
$$837$$ −3.66833e6 −0.180990
$$838$$ 0 0
$$839$$ −1.17771e7 −0.577607 −0.288804 0.957388i $$-0.593257\pi$$
−0.288804 + 0.957388i $$0.593257\pi$$
$$840$$ 0 0
$$841$$ 1.11135e6 0.0541828
$$842$$ 0 0
$$843$$ 4.17944e6 0.202558
$$844$$ 0 0
$$845$$ −1.92358e6 −0.0926764
$$846$$ 0 0
$$847$$ −2.16687e6 −0.103782
$$848$$ 0 0
$$849$$ 3.73622e6 0.177895
$$850$$ 0 0
$$851$$ 2.05872e7 0.974483
$$852$$ 0 0
$$853$$ −1.43993e7 −0.677591 −0.338796 0.940860i $$-0.610020\pi$$
−0.338796 + 0.940860i $$0.610020\pi$$
$$854$$ 0 0
$$855$$ −8.04816e6 −0.376514
$$856$$ 0 0
$$857$$ 6.27604e6 0.291900 0.145950 0.989292i $$-0.453376\pi$$
0.145950 + 0.989292i $$0.453376\pi$$
$$858$$ 0 0
$$859$$ 4.71738e6 0.218131 0.109066 0.994035i $$-0.465214\pi$$
0.109066 + 0.994035i $$0.465214\pi$$
$$860$$ 0 0
$$861$$ 6.84382e6 0.314623
$$862$$ 0 0
$$863$$ −7.53926e6 −0.344589 −0.172295 0.985045i $$-0.555118\pi$$
−0.172295 + 0.985045i $$0.555118\pi$$
$$864$$ 0 0
$$865$$ −99636.0 −0.00452768
$$866$$ 0 0
$$867$$ −8.08716e6 −0.365383
$$868$$ 0 0
$$869$$ −3.06130e6 −0.137517
$$870$$ 0 0
$$871$$ −1.77412e7 −0.792387
$$872$$ 0 0
$$873$$ 5.72330e6 0.254162
$$874$$ 0 0
$$875$$ 2.81443e7 1.24271
$$876$$ 0 0
$$877$$ −1.04331e7 −0.458051 −0.229025 0.973420i $$-0.573554\pi$$
−0.229025 + 0.973420i $$0.573554\pi$$
$$878$$ 0 0
$$879$$ −2.33389e7 −1.01884
$$880$$ 0 0
$$881$$ 3.91076e7 1.69755 0.848774 0.528756i $$-0.177342\pi$$
0.848774 + 0.528756i $$0.177342\pi$$
$$882$$ 0 0
$$883$$ −1.29282e7 −0.558003 −0.279001 0.960291i $$-0.590003\pi$$
−0.279001 + 0.960291i $$0.590003\pi$$
$$884$$ 0 0
$$885$$ 2.42604e6 0.104121
$$886$$ 0 0
$$887$$ −3.36466e7 −1.43592 −0.717962 0.696082i $$-0.754925\pi$$
−0.717962 + 0.696082i $$0.754925\pi$$
$$888$$ 0 0
$$889$$ 7.66462e6 0.325264
$$890$$ 0 0
$$891$$ −793881. −0.0335013
$$892$$ 0 0
$$893$$ 5.34125e7 2.24137
$$894$$ 0 0
$$895$$ −3.09479e7 −1.29144
$$896$$ 0 0
$$897$$ 1.31010e7 0.543654
$$898$$ 0 0
$$899$$ −2.33988e7 −0.965594
$$900$$ 0 0
$$901$$ 2.07546e7 0.851731
$$902$$ 0 0
$$903$$ 1.10609e7 0.451411
$$904$$ 0 0
$$905$$ −2.42318e7 −0.983477
$$906$$ 0 0
$$907$$ 4.19629e7 1.69374 0.846872 0.531797i $$-0.178483\pi$$
0.846872 + 0.531797i $$0.178483\pi$$
$$908$$ 0 0
$$909$$ −8.26184e6 −0.331640
$$910$$ 0 0
$$911$$ 1.92521e6 0.0768567 0.0384283 0.999261i $$-0.487765\pi$$
0.0384283 + 0.999261i $$0.487765\pi$$
$$912$$ 0 0
$$913$$ −2.12428e6 −0.0843401
$$914$$ 0 0
$$915$$ −2.22144e7 −0.877167
$$916$$ 0 0
$$917$$ 7.94050e6 0.311835
$$918$$ 0 0
$$919$$ 1.72481e7 0.673678 0.336839 0.941562i $$-0.390642\pi$$
0.336839 + 0.941562i $$0.390642\pi$$
$$920$$ 0 0
$$921$$ 8.37749e6 0.325435
$$922$$ 0 0
$$923$$ 3.99780e7 1.54460
$$924$$ 0 0
$$925$$ −8.19106e6 −0.314765
$$926$$ 0 0
$$927$$ −1.06032e7 −0.405265
$$928$$ 0 0
$$929$$ 2.51145e6 0.0954740 0.0477370 0.998860i $$-0.484799\pi$$
0.0477370 + 0.998860i $$0.484799\pi$$
$$930$$ 0 0
$$931$$ −1.10095e7 −0.416288
$$932$$ 0 0
$$933$$ −2.23674e7 −0.841225
$$934$$ 0 0
$$935$$ 4.01865e6 0.150332
$$936$$ 0 0
$$937$$ 1.79853e7 0.669221 0.334611 0.942357i $$-0.391395\pi$$
0.334611 + 0.942357i $$0.391395\pi$$
$$938$$ 0 0
$$939$$ 1.18547e7 0.438761
$$940$$ 0 0
$$941$$ −3.22586e7 −1.18760 −0.593802 0.804611i $$-0.702374\pi$$
−0.593802 + 0.804611i $$0.702374\pi$$
$$942$$ 0 0
$$943$$ −1.30300e7 −0.477160
$$944$$ 0 0
$$945$$ −4.96303e6 −0.180787
$$946$$ 0 0
$$947$$ −4.41659e7 −1.60034 −0.800169 0.599774i $$-0.795257\pi$$
−0.800169 + 0.599774i $$0.795257\pi$$
$$948$$ 0 0
$$949$$ −1.05880e7 −0.381635
$$950$$ 0 0
$$951$$ 2.02986e7 0.727804
$$952$$ 0 0
$$953$$ 1.87488e7 0.668714 0.334357 0.942446i $$-0.391481\pi$$
0.334357 + 0.942446i $$0.391481\pi$$
$$954$$ 0 0
$$955$$ 1.40580e7 0.498786
$$956$$ 0 0
$$957$$ −5.06385e6 −0.178731
$$958$$ 0 0
$$959$$ 3.38716e7 1.18929
$$960$$ 0 0
$$961$$ −3.30813e6 −0.115551
$$962$$ 0 0
$$963$$ 1.14706e7 0.398584
$$964$$ 0 0
$$965$$ 5.35596e6 0.185148
$$966$$ 0 0
$$967$$ −1.08673e7 −0.373730 −0.186865 0.982386i $$-0.559833\pi$$
−0.186865 + 0.982386i $$0.559833\pi$$
$$968$$ 0 0
$$969$$ 1.40357e7 0.480202
$$970$$ 0 0
$$971$$ 4.79123e7 1.63079 0.815397 0.578902i $$-0.196519\pi$$
0.815397 + 0.578902i $$0.196519\pi$$
$$972$$ 0 0
$$973$$ 5.54882e7 1.87896
$$974$$ 0 0
$$975$$ −5.21249e6 −0.175604
$$976$$ 0 0
$$977$$ 4.01385e7 1.34532 0.672658 0.739954i $$-0.265153\pi$$
0.672658 + 0.739954i $$0.265153\pi$$
$$978$$ 0 0
$$979$$ −1.60410e7 −0.534902
$$980$$ 0 0
$$981$$ −1.94246e7 −0.644436
$$982$$ 0 0
$$983$$ −3.22682e6 −0.106510 −0.0532551 0.998581i $$-0.516960\pi$$
−0.0532551 + 0.998581i $$0.516960\pi$$
$$984$$ 0 0
$$985$$ −1.13961e7 −0.374254
$$986$$ 0 0
$$987$$ 3.29377e7 1.07622
$$988$$ 0 0
$$989$$ −2.10589e7 −0.684614
$$990$$ 0 0
$$991$$ 5.95345e6 0.192568 0.0962841 0.995354i $$-0.469304\pi$$
0.0962841 + 0.995354i $$0.469304\pi$$
$$992$$ 0 0
$$993$$ 2.85364e7 0.918387
$$994$$ 0 0
$$995$$ 2.36146e7 0.756175
$$996$$ 0 0
$$997$$ 3.20783e7 1.02205 0.511027 0.859565i $$-0.329265\pi$$
0.511027 + 0.859565i $$0.329265\pi$$
$$998$$ 0 0
$$999$$ 5.91802e6 0.187613
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 528.6.a.i.1.1 1
4.3 odd 2 33.6.a.a.1.1 1
12.11 even 2 99.6.a.b.1.1 1
20.19 odd 2 825.6.a.b.1.1 1
44.43 even 2 363.6.a.c.1.1 1
132.131 odd 2 1089.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.a.1.1 1 4.3 odd 2
99.6.a.b.1.1 1 12.11 even 2
363.6.a.c.1.1 1 44.43 even 2
528.6.a.i.1.1 1 1.1 even 1 trivial
825.6.a.b.1.1 1 20.19 odd 2
1089.6.a.d.1.1 1 132.131 odd 2