# Properties

 Label 784.6.a.l.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 8) 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)

 $$f(q)$$ $$=$$ $$q+20.0000 q^{3} +74.0000 q^{5} +157.000 q^{9} +O(q^{10})$$ $$q+20.0000 q^{3} +74.0000 q^{5} +157.000 q^{9} -124.000 q^{11} -478.000 q^{13} +1480.00 q^{15} +1198.00 q^{17} +3044.00 q^{19} -184.000 q^{23} +2351.00 q^{25} -1720.00 q^{27} -3282.00 q^{29} -5728.00 q^{31} -2480.00 q^{33} +10326.0 q^{37} -9560.00 q^{39} +8886.00 q^{41} +9188.00 q^{43} +11618.0 q^{45} +23664.0 q^{47} +23960.0 q^{51} +11686.0 q^{53} -9176.00 q^{55} +60880.0 q^{57} +16876.0 q^{59} +18482.0 q^{61} -35372.0 q^{65} +15532.0 q^{67} -3680.00 q^{69} +31960.0 q^{71} +4886.00 q^{73} +47020.0 q^{75} -44560.0 q^{79} -72551.0 q^{81} +67364.0 q^{83} +88652.0 q^{85} -65640.0 q^{87} -71994.0 q^{89} -114560. q^{93} +225256. q^{95} -48866.0 q^{97} -19468.0 q^{99} +O(q^{100})$$

## 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
$$3$$ 20.0000 1.28300 0.641500 0.767123i $$-0.278312\pi$$
0.641500 + 0.767123i $$0.278312\pi$$
$$4$$ 0 0
$$5$$ 74.0000 1.32375 0.661876 0.749613i $$-0.269760\pi$$
0.661876 + 0.749613i $$0.269760\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 157.000 0.646091
$$10$$ 0 0
$$11$$ −124.000 −0.308987 −0.154493 0.987994i $$-0.549375\pi$$
−0.154493 + 0.987994i $$0.549375\pi$$
$$12$$ 0 0
$$13$$ −478.000 −0.784458 −0.392229 0.919868i $$-0.628296\pi$$
−0.392229 + 0.919868i $$0.628296\pi$$
$$14$$ 0 0
$$15$$ 1480.00 1.69837
$$16$$ 0 0
$$17$$ 1198.00 1.00539 0.502695 0.864464i $$-0.332342\pi$$
0.502695 + 0.864464i $$0.332342\pi$$
$$18$$ 0 0
$$19$$ 3044.00 1.93446 0.967232 0.253894i $$-0.0817115\pi$$
0.967232 + 0.253894i $$0.0817115\pi$$
$$20$$ 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
$$25$$ 2351.00 0.752320
$$26$$ 0 0
$$27$$ −1720.00 −0.454066
$$28$$ 0 0
$$29$$ −3282.00 −0.724676 −0.362338 0.932047i $$-0.618021\pi$$
−0.362338 + 0.932047i $$0.618021\pi$$
$$30$$ 0 0
$$31$$ −5728.00 −1.07053 −0.535265 0.844684i $$-0.679788\pi$$
−0.535265 + 0.844684i $$0.679788\pi$$
$$32$$ 0 0
$$33$$ −2480.00 −0.396430
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10326.0 1.24002 0.620009 0.784595i $$-0.287129\pi$$
0.620009 + 0.784595i $$0.287129\pi$$
$$38$$ 0 0
$$39$$ −9560.00 −1.00646
$$40$$ 0 0
$$41$$ 8886.00 0.825556 0.412778 0.910832i $$-0.364558\pi$$
0.412778 + 0.910832i $$0.364558\pi$$
$$42$$ 0 0
$$43$$ 9188.00 0.757792 0.378896 0.925439i $$-0.376304\pi$$
0.378896 + 0.925439i $$0.376304\pi$$
$$44$$ 0 0
$$45$$ 11618.0 0.855264
$$46$$ 0 0
$$47$$ 23664.0 1.56258 0.781292 0.624165i $$-0.214561\pi$$
0.781292 + 0.624165i $$0.214561\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 23960.0 1.28992
$$52$$ 0 0
$$53$$ 11686.0 0.571447 0.285724 0.958312i $$-0.407766\pi$$
0.285724 + 0.958312i $$0.407766\pi$$
$$54$$ 0 0
$$55$$ −9176.00 −0.409022
$$56$$ 0 0
$$57$$ 60880.0 2.48192
$$58$$ 0 0
$$59$$ 16876.0 0.631160 0.315580 0.948899i $$-0.397801\pi$$
0.315580 + 0.948899i $$0.397801\pi$$
$$60$$ 0 0
$$61$$ 18482.0 0.635952 0.317976 0.948099i $$-0.396997\pi$$
0.317976 + 0.948099i $$0.396997\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −35372.0 −1.03843
$$66$$ 0 0
$$67$$ 15532.0 0.422708 0.211354 0.977410i $$-0.432213\pi$$
0.211354 + 0.977410i $$0.432213\pi$$
$$68$$ 0 0
$$69$$ −3680.00 −0.0930519
$$70$$ 0 0
$$71$$ 31960.0 0.752421 0.376210 0.926534i $$-0.377227\pi$$
0.376210 + 0.926534i $$0.377227\pi$$
$$72$$ 0 0
$$73$$ 4886.00 0.107312 0.0536558 0.998559i $$-0.482913\pi$$
0.0536558 + 0.998559i $$0.482913\pi$$
$$74$$ 0 0
$$75$$ 47020.0 0.965227
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −44560.0 −0.803299 −0.401650 0.915793i $$-0.631563\pi$$
−0.401650 + 0.915793i $$0.631563\pi$$
$$80$$ 0 0
$$81$$ −72551.0 −1.22866
$$82$$ 0 0
$$83$$ 67364.0 1.07333 0.536664 0.843796i $$-0.319684\pi$$
0.536664 + 0.843796i $$0.319684\pi$$
$$84$$ 0 0
$$85$$ 88652.0 1.33089
$$86$$ 0 0
$$87$$ −65640.0 −0.929759
$$88$$ 0 0
$$89$$ −71994.0 −0.963432 −0.481716 0.876327i $$-0.659986\pi$$
−0.481716 + 0.876327i $$0.659986\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −114560. −1.37349
$$94$$ 0 0
$$95$$ 225256. 2.56075
$$96$$ 0 0
$$97$$ −48866.0 −0.527324 −0.263662 0.964615i $$-0.584930\pi$$
−0.263662 + 0.964615i $$0.584930\pi$$
$$98$$ 0 0
$$99$$ −19468.0 −0.199633
$$100$$ 0 0
$$101$$ −51606.0 −0.503381 −0.251690 0.967808i $$-0.580986\pi$$
−0.251690 + 0.967808i $$0.580986\pi$$
$$102$$ 0 0
$$103$$ 180424. 1.67572 0.837860 0.545886i $$-0.183807\pi$$
0.837860 + 0.545886i $$0.183807\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 65700.0 0.554761 0.277381 0.960760i $$-0.410534\pi$$
0.277381 + 0.960760i $$0.410534\pi$$
$$108$$ 0 0
$$109$$ −112706. −0.908617 −0.454308 0.890844i $$-0.650114\pi$$
−0.454308 + 0.890844i $$0.650114\pi$$
$$110$$ 0 0
$$111$$ 206520. 1.59094
$$112$$ 0 0
$$113$$ −23502.0 −0.173145 −0.0865723 0.996246i $$-0.527591\pi$$
−0.0865723 + 0.996246i $$0.527591\pi$$
$$114$$ 0 0
$$115$$ −13616.0 −0.0960075
$$116$$ 0 0
$$117$$ −75046.0 −0.506831
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −145675. −0.904527
$$122$$ 0 0
$$123$$ 177720. 1.05919
$$124$$ 0 0
$$125$$ −57276.0 −0.327867
$$126$$ 0 0
$$127$$ 94592.0 0.520409 0.260205 0.965553i $$-0.416210\pi$$
0.260205 + 0.965553i $$0.416210\pi$$
$$128$$ 0 0
$$129$$ 183760. 0.972247
$$130$$ 0 0
$$131$$ 70292.0 0.357872 0.178936 0.983861i $$-0.442735\pi$$
0.178936 + 0.983861i $$0.442735\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −127280. −0.601071
$$136$$ 0 0
$$137$$ 277290. 1.26221 0.631107 0.775696i $$-0.282601\pi$$
0.631107 + 0.775696i $$0.282601\pi$$
$$138$$ 0 0
$$139$$ −130308. −0.572050 −0.286025 0.958222i $$-0.592334\pi$$
−0.286025 + 0.958222i $$0.592334\pi$$
$$140$$ 0 0
$$141$$ 473280. 2.00480
$$142$$ 0 0
$$143$$ 59272.0 0.242387
$$144$$ 0 0
$$145$$ −242868. −0.959291
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −401530. −1.48167 −0.740836 0.671685i $$-0.765571\pi$$
−0.740836 + 0.671685i $$0.765571\pi$$
$$150$$ 0 0
$$151$$ 75976.0 0.271165 0.135583 0.990766i $$-0.456709\pi$$
0.135583 + 0.990766i $$0.456709\pi$$
$$152$$ 0 0
$$153$$ 188086. 0.649573
$$154$$ 0 0
$$155$$ −423872. −1.41712
$$156$$ 0 0
$$157$$ 394322. 1.27674 0.638369 0.769730i $$-0.279609\pi$$
0.638369 + 0.769730i $$0.279609\pi$$
$$158$$ 0 0
$$159$$ 233720. 0.733167
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 11724.0 0.0345626 0.0172813 0.999851i $$-0.494499\pi$$
0.0172813 + 0.999851i $$0.494499\pi$$
$$164$$ 0 0
$$165$$ −183520. −0.524775
$$166$$ 0 0
$$167$$ −551928. −1.53141 −0.765705 0.643192i $$-0.777610\pi$$
−0.765705 + 0.643192i $$0.777610\pi$$
$$168$$ 0 0
$$169$$ −142809. −0.384626
$$170$$ 0 0
$$171$$ 477908. 1.24984
$$172$$ 0 0
$$173$$ −432894. −1.09968 −0.549840 0.835270i $$-0.685311\pi$$
−0.549840 + 0.835270i $$0.685311\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 337520. 0.809779
$$178$$ 0 0
$$179$$ −559620. −1.30545 −0.652726 0.757594i $$-0.726375\pi$$
−0.652726 + 0.757594i $$0.726375\pi$$
$$180$$ 0 0
$$181$$ −604710. −1.37199 −0.685995 0.727607i $$-0.740633\pi$$
−0.685995 + 0.727607i $$0.740633\pi$$
$$182$$ 0 0
$$183$$ 369640. 0.815927
$$184$$ 0 0
$$185$$ 764124. 1.64148
$$186$$ 0 0
$$187$$ −148552. −0.310652
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 409152. 0.811524 0.405762 0.913979i $$-0.367006\pi$$
0.405762 + 0.913979i $$0.367006\pi$$
$$192$$ 0 0
$$193$$ 540866. 1.04519 0.522596 0.852580i $$-0.324963\pi$$
0.522596 + 0.852580i $$0.324963\pi$$
$$194$$ 0 0
$$195$$ −707440. −1.33230
$$196$$ 0 0
$$197$$ −629898. −1.15639 −0.578195 0.815898i $$-0.696243\pi$$
−0.578195 + 0.815898i $$0.696243\pi$$
$$198$$ 0 0
$$199$$ 283048. 0.506673 0.253336 0.967378i $$-0.418472\pi$$
0.253336 + 0.967378i $$0.418472\pi$$
$$200$$ 0 0
$$201$$ 310640. 0.542335
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 657564. 1.09283
$$206$$ 0 0
$$207$$ −28888.0 −0.0468588
$$208$$ 0 0
$$209$$ −377456. −0.597724
$$210$$ 0 0
$$211$$ −142756. −0.220744 −0.110372 0.993890i $$-0.535204\pi$$
−0.110372 + 0.993890i $$0.535204\pi$$
$$212$$ 0 0
$$213$$ 639200. 0.965357
$$214$$ 0 0
$$215$$ 679912. 1.00313
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 97720.0 0.137681
$$220$$ 0 0
$$221$$ −572644. −0.788686
$$222$$ 0 0
$$223$$ 889696. 1.19806 0.599031 0.800726i $$-0.295553\pi$$
0.599031 + 0.800726i $$0.295553\pi$$
$$224$$ 0 0
$$225$$ 369107. 0.486067
$$226$$ 0 0
$$227$$ 1.14316e6 1.47245 0.736226 0.676736i $$-0.236606\pi$$
0.736226 + 0.676736i $$0.236606\pi$$
$$228$$ 0 0
$$229$$ 695786. 0.876773 0.438386 0.898787i $$-0.355550\pi$$
0.438386 + 0.898787i $$0.355550\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −347126. −0.418887 −0.209444 0.977821i $$-0.567165\pi$$
−0.209444 + 0.977821i $$0.567165\pi$$
$$234$$ 0 0
$$235$$ 1.75114e6 2.06847
$$236$$ 0 0
$$237$$ −891200. −1.03063
$$238$$ 0 0
$$239$$ 1.64296e6 1.86051 0.930255 0.366912i $$-0.119585\pi$$
0.930255 + 0.366912i $$0.119585\pi$$
$$240$$ 0 0
$$241$$ 1.16744e6 1.29477 0.647383 0.762165i $$-0.275863\pi$$
0.647383 + 0.762165i $$0.275863\pi$$
$$242$$ 0 0
$$243$$ −1.03306e6 −1.12230
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.45503e6 −1.51751
$$248$$ 0 0
$$249$$ 1.34728e6 1.37708
$$250$$ 0 0
$$251$$ −790612. −0.792098 −0.396049 0.918229i $$-0.629619\pi$$
−0.396049 + 0.918229i $$0.629619\pi$$
$$252$$ 0 0
$$253$$ 22816.0 0.0224098
$$254$$ 0 0
$$255$$ 1.77304e6 1.70753
$$256$$ 0 0
$$257$$ 129790. 0.122577 0.0612884 0.998120i $$-0.480479\pi$$
0.0612884 + 0.998120i $$0.480479\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −515274. −0.468206
$$262$$ 0 0
$$263$$ −70888.0 −0.0631951 −0.0315975 0.999501i $$-0.510059\pi$$
−0.0315975 + 0.999501i $$0.510059\pi$$
$$264$$ 0 0
$$265$$ 864764. 0.756455
$$266$$ 0 0
$$267$$ −1.43988e6 −1.23608
$$268$$ 0 0
$$269$$ −1.79017e6 −1.50839 −0.754197 0.656649i $$-0.771973\pi$$
−0.754197 + 0.656649i $$0.771973\pi$$
$$270$$ 0 0
$$271$$ −1.77362e6 −1.46702 −0.733511 0.679678i $$-0.762120\pi$$
−0.733511 + 0.679678i $$0.762120\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −291524. −0.232457
$$276$$ 0 0
$$277$$ −275450. −0.215697 −0.107848 0.994167i $$-0.534396\pi$$
−0.107848 + 0.994167i $$0.534396\pi$$
$$278$$ 0 0
$$279$$ −899296. −0.691659
$$280$$ 0 0
$$281$$ 594170. 0.448895 0.224448 0.974486i $$-0.427942\pi$$
0.224448 + 0.974486i $$0.427942\pi$$
$$282$$ 0 0
$$283$$ 1.09243e6 0.810824 0.405412 0.914134i $$-0.367128\pi$$
0.405412 + 0.914134i $$0.367128\pi$$
$$284$$ 0 0
$$285$$ 4.50512e6 3.28545
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 15347.0 0.0108088
$$290$$ 0 0
$$291$$ −977320. −0.676557
$$292$$ 0 0
$$293$$ −333654. −0.227053 −0.113527 0.993535i $$-0.536215\pi$$
−0.113527 + 0.993535i $$0.536215\pi$$
$$294$$ 0 0
$$295$$ 1.24882e6 0.835500
$$296$$ 0 0
$$297$$ 213280. 0.140300
$$298$$ 0 0
$$299$$ 87952.0 0.0568942
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −1.03212e6 −0.645838
$$304$$ 0 0
$$305$$ 1.36767e6 0.841843
$$306$$ 0 0
$$307$$ 1.05997e6 0.641872 0.320936 0.947101i $$-0.396003\pi$$
0.320936 + 0.947101i $$0.396003\pi$$
$$308$$ 0 0
$$309$$ 3.60848e6 2.14995
$$310$$ 0 0
$$311$$ −1.33649e6 −0.783545 −0.391773 0.920062i $$-0.628138\pi$$
−0.391773 + 0.920062i $$0.628138\pi$$
$$312$$ 0 0
$$313$$ −1.64419e6 −0.948615 −0.474308 0.880359i $$-0.657302\pi$$
−0.474308 + 0.880359i $$0.657302\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −1.72370e6 −0.963414 −0.481707 0.876332i $$-0.659983\pi$$
−0.481707 + 0.876332i $$0.659983\pi$$
$$318$$ 0 0
$$319$$ 406968. 0.223915
$$320$$ 0 0
$$321$$ 1.31400e6 0.711759
$$322$$ 0 0
$$323$$ 3.64671e6 1.94489
$$324$$ 0 0
$$325$$ −1.12378e6 −0.590163
$$326$$ 0 0
$$327$$ −2.25412e6 −1.16576
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −2.74963e6 −1.37944 −0.689722 0.724074i $$-0.742267\pi$$
−0.689722 + 0.724074i $$0.742267\pi$$
$$332$$ 0 0
$$333$$ 1.62118e6 0.801164
$$334$$ 0 0
$$335$$ 1.14937e6 0.559561
$$336$$ 0 0
$$337$$ −3.41489e6 −1.63796 −0.818978 0.573824i $$-0.805459\pi$$
−0.818978 + 0.573824i $$0.805459\pi$$
$$338$$ 0 0
$$339$$ −470040. −0.222145
$$340$$ 0 0
$$341$$ 710272. 0.330780
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −272320. −0.123178
$$346$$ 0 0
$$347$$ −730764. −0.325802 −0.162901 0.986642i $$-0.552085\pi$$
−0.162901 + 0.986642i $$0.552085\pi$$
$$348$$ 0 0
$$349$$ 2.29749e6 1.00969 0.504847 0.863209i $$-0.331549\pi$$
0.504847 + 0.863209i $$0.331549\pi$$
$$350$$ 0 0
$$351$$ 822160. 0.356196
$$352$$ 0 0
$$353$$ 1.17072e6 0.500052 0.250026 0.968239i $$-0.419561\pi$$
0.250026 + 0.968239i $$0.419561\pi$$
$$354$$ 0 0
$$355$$ 2.36504e6 0.996019
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −3.88654e6 −1.59157 −0.795787 0.605577i $$-0.792942\pi$$
−0.795787 + 0.605577i $$0.792942\pi$$
$$360$$ 0 0
$$361$$ 6.78984e6 2.74215
$$362$$ 0 0
$$363$$ −2.91350e6 −1.16051
$$364$$ 0 0
$$365$$ 361564. 0.142054
$$366$$ 0 0
$$367$$ 933040. 0.361606 0.180803 0.983519i $$-0.442130\pi$$
0.180803 + 0.983519i $$0.442130\pi$$
$$368$$ 0 0
$$369$$ 1.39510e6 0.533384
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −392218. −0.145967 −0.0729836 0.997333i $$-0.523252\pi$$
−0.0729836 + 0.997333i $$0.523252\pi$$
$$374$$ 0 0
$$375$$ −1.14552e6 −0.420653
$$376$$ 0 0
$$377$$ 1.56880e6 0.568477
$$378$$ 0 0
$$379$$ 4.72930e6 1.69122 0.845608 0.533805i $$-0.179238\pi$$
0.845608 + 0.533805i $$0.179238\pi$$
$$380$$ 0 0
$$381$$ 1.89184e6 0.667686
$$382$$ 0 0
$$383$$ 1.89734e6 0.660920 0.330460 0.943820i $$-0.392796\pi$$
0.330460 + 0.943820i $$0.392796\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.44252e6 0.489602
$$388$$ 0 0
$$389$$ −3.72295e6 −1.24742 −0.623711 0.781655i $$-0.714376\pi$$
−0.623711 + 0.781655i $$0.714376\pi$$
$$390$$ 0 0
$$391$$ −220432. −0.0729177
$$392$$ 0 0
$$393$$ 1.40584e6 0.459150
$$394$$ 0 0
$$395$$ −3.29744e6 −1.06337
$$396$$ 0 0
$$397$$ −3.33808e6 −1.06297 −0.531484 0.847068i $$-0.678365\pi$$
−0.531484 + 0.847068i $$0.678365\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 4.27490e6 1.32759 0.663796 0.747913i $$-0.268944\pi$$
0.663796 + 0.747913i $$0.268944\pi$$
$$402$$ 0 0
$$403$$ 2.73798e6 0.839785
$$404$$ 0 0
$$405$$ −5.36877e6 −1.62644
$$406$$ 0 0
$$407$$ −1.28042e6 −0.383149
$$408$$ 0 0
$$409$$ 2.57319e6 0.760613 0.380306 0.924861i $$-0.375819\pi$$
0.380306 + 0.924861i $$0.375819\pi$$
$$410$$ 0 0
$$411$$ 5.54580e6 1.61942
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 4.98494e6 1.42082
$$416$$ 0 0
$$417$$ −2.60616e6 −0.733941
$$418$$ 0 0
$$419$$ 5.26828e6 1.46600 0.732999 0.680230i $$-0.238120\pi$$
0.732999 + 0.680230i $$0.238120\pi$$
$$420$$ 0 0
$$421$$ −973354. −0.267649 −0.133824 0.991005i $$-0.542726\pi$$
−0.133824 + 0.991005i $$0.542726\pi$$
$$422$$ 0 0
$$423$$ 3.71525e6 1.00957
$$424$$ 0 0
$$425$$ 2.81650e6 0.756375
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 1.18544e6 0.310983
$$430$$ 0 0
$$431$$ −3.55736e6 −0.922433 −0.461216 0.887288i $$-0.652587\pi$$
−0.461216 + 0.887288i $$0.652587\pi$$
$$432$$ 0 0
$$433$$ 1.95496e6 0.501092 0.250546 0.968105i $$-0.419390\pi$$
0.250546 + 0.968105i $$0.419390\pi$$
$$434$$ 0 0
$$435$$ −4.85736e6 −1.23077
$$436$$ 0 0
$$437$$ −560096. −0.140300
$$438$$ 0 0
$$439$$ −3.29681e6 −0.816455 −0.408228 0.912880i $$-0.633853\pi$$
−0.408228 + 0.912880i $$0.633853\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 5.05820e6 1.22458 0.612289 0.790634i $$-0.290249\pi$$
0.612289 + 0.790634i $$0.290249\pi$$
$$444$$ 0 0
$$445$$ −5.32756e6 −1.27535
$$446$$ 0 0
$$447$$ −8.03060e6 −1.90099
$$448$$ 0 0
$$449$$ 2.12730e6 0.497981 0.248990 0.968506i $$-0.419901\pi$$
0.248990 + 0.968506i $$0.419901\pi$$
$$450$$ 0 0
$$451$$ −1.10186e6 −0.255086
$$452$$ 0 0
$$453$$ 1.51952e6 0.347905
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 289130. 0.0647594 0.0323797 0.999476i $$-0.489691\pi$$
0.0323797 + 0.999476i $$0.489691\pi$$
$$458$$ 0 0
$$459$$ −2.06056e6 −0.456513
$$460$$ 0 0
$$461$$ −2.66870e6 −0.584854 −0.292427 0.956288i $$-0.594463\pi$$
−0.292427 + 0.956288i $$0.594463\pi$$
$$462$$ 0 0
$$463$$ −7.58619e6 −1.64464 −0.822321 0.569024i $$-0.807321\pi$$
−0.822321 + 0.569024i $$0.807321\pi$$
$$464$$ 0 0
$$465$$ −8.47744e6 −1.81816
$$466$$ 0 0
$$467$$ −1.41961e6 −0.301216 −0.150608 0.988594i $$-0.548123\pi$$
−0.150608 + 0.988594i $$0.548123\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 7.88644e6 1.63806
$$472$$ 0 0
$$473$$ −1.13931e6 −0.234148
$$474$$ 0 0
$$475$$ 7.15644e6 1.45534
$$476$$ 0 0
$$477$$ 1.83470e6 0.369207
$$478$$ 0 0
$$479$$ −1.88406e6 −0.375195 −0.187597 0.982246i $$-0.560070\pi$$
−0.187597 + 0.982246i $$0.560070\pi$$
$$480$$ 0 0
$$481$$ −4.93583e6 −0.972741
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −3.61608e6 −0.698046
$$486$$ 0 0
$$487$$ 6.01388e6 1.14903 0.574516 0.818493i $$-0.305190\pi$$
0.574516 + 0.818493i $$0.305190\pi$$
$$488$$ 0 0
$$489$$ 234480. 0.0443439
$$490$$ 0 0
$$491$$ −4.29232e6 −0.803504 −0.401752 0.915749i $$-0.631599\pi$$
−0.401752 + 0.915749i $$0.631599\pi$$
$$492$$ 0 0
$$493$$ −3.93184e6 −0.728581
$$494$$ 0 0
$$495$$ −1.44063e6 −0.264265
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −1.34509e6 −0.241825 −0.120912 0.992663i $$-0.538582\pi$$
−0.120912 + 0.992663i $$0.538582\pi$$
$$500$$ 0 0
$$501$$ −1.10386e7 −1.96480
$$502$$ 0 0
$$503$$ 202008. 0.0355999 0.0177999 0.999842i $$-0.494334\pi$$
0.0177999 + 0.999842i $$0.494334\pi$$
$$504$$ 0 0
$$505$$ −3.81884e6 −0.666352
$$506$$ 0 0
$$507$$ −2.85618e6 −0.493476
$$508$$ 0 0
$$509$$ −9.78344e6 −1.67377 −0.836887 0.547375i $$-0.815627\pi$$
−0.836887 + 0.547375i $$0.815627\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −5.23568e6 −0.878374
$$514$$ 0 0
$$515$$ 1.33514e7 2.21824
$$516$$ 0 0
$$517$$ −2.93434e6 −0.482818
$$518$$ 0 0
$$519$$ −8.65788e6 −1.41089
$$520$$ 0 0
$$521$$ 1.04830e7 1.69197 0.845985 0.533207i $$-0.179013\pi$$
0.845985 + 0.533207i $$0.179013\pi$$
$$522$$ 0 0
$$523$$ 6.21017e6 0.992772 0.496386 0.868102i $$-0.334660\pi$$
0.496386 + 0.868102i $$0.334660\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −6.86214e6 −1.07630
$$528$$ 0 0
$$529$$ −6.40249e6 −0.994740
$$530$$ 0 0
$$531$$ 2.64953e6 0.407787
$$532$$ 0 0
$$533$$ −4.24751e6 −0.647614
$$534$$ 0 0
$$535$$ 4.86180e6 0.734366
$$536$$ 0 0
$$537$$ −1.11924e7 −1.67489
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 5.08088e6 0.746355 0.373178 0.927760i $$-0.378268\pi$$
0.373178 + 0.927760i $$0.378268\pi$$
$$542$$ 0 0
$$543$$ −1.20942e7 −1.76026
$$544$$ 0 0
$$545$$ −8.34024e6 −1.20278
$$546$$ 0 0
$$547$$ −3.34687e6 −0.478267 −0.239133 0.970987i $$-0.576863\pi$$
−0.239133 + 0.970987i $$0.576863\pi$$
$$548$$ 0 0
$$549$$ 2.90167e6 0.410883
$$550$$ 0 0
$$551$$ −9.99041e6 −1.40186
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 1.52825e7 2.10601
$$556$$ 0 0
$$557$$ 7.00377e6 0.956520 0.478260 0.878218i $$-0.341268\pi$$
0.478260 + 0.878218i $$0.341268\pi$$
$$558$$ 0 0
$$559$$ −4.39186e6 −0.594456
$$560$$ 0 0
$$561$$ −2.97104e6 −0.398567
$$562$$ 0 0
$$563$$ −1.29819e7 −1.72610 −0.863052 0.505116i $$-0.831450\pi$$
−0.863052 + 0.505116i $$0.831450\pi$$
$$564$$ 0 0
$$565$$ −1.73915e6 −0.229200
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1.89942e6 0.245946 0.122973 0.992410i $$-0.460757\pi$$
0.122973 + 0.992410i $$0.460757\pi$$
$$570$$ 0 0
$$571$$ 1.66300e6 0.213452 0.106726 0.994288i $$-0.465963\pi$$
0.106726 + 0.994288i $$0.465963\pi$$
$$572$$ 0 0
$$573$$ 8.18304e6 1.04119
$$574$$ 0 0
$$575$$ −432584. −0.0545633
$$576$$ 0 0
$$577$$ −8.77344e6 −1.09706 −0.548530 0.836131i $$-0.684812\pi$$
−0.548530 + 0.836131i $$0.684812\pi$$
$$578$$ 0 0
$$579$$ 1.08173e7 1.34098
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −1.44906e6 −0.176570
$$584$$ 0 0
$$585$$ −5.55340e6 −0.670918
$$586$$ 0 0
$$587$$ 5.18393e6 0.620961 0.310480 0.950580i $$-0.399510\pi$$
0.310480 + 0.950580i $$0.399510\pi$$
$$588$$ 0 0
$$589$$ −1.74360e7 −2.07090
$$590$$ 0 0
$$591$$ −1.25980e7 −1.48365
$$592$$ 0 0
$$593$$ −8.49858e6 −0.992452 −0.496226 0.868193i $$-0.665281\pi$$
−0.496226 + 0.868193i $$0.665281\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 5.66096e6 0.650061
$$598$$ 0 0
$$599$$ −1.12471e7 −1.28078 −0.640388 0.768051i $$-0.721227\pi$$
−0.640388 + 0.768051i $$0.721227\pi$$
$$600$$ 0 0
$$601$$ 3.46439e6 0.391238 0.195619 0.980680i $$-0.437328\pi$$
0.195619 + 0.980680i $$0.437328\pi$$
$$602$$ 0 0
$$603$$ 2.43852e6 0.273108
$$604$$ 0 0
$$605$$ −1.07799e7 −1.19737
$$606$$ 0 0
$$607$$ −999712. −0.110129 −0.0550647 0.998483i $$-0.517537\pi$$
−0.0550647 + 0.998483i $$0.517537\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.13114e7 −1.22578
$$612$$ 0 0
$$613$$ 9.81340e6 1.05480 0.527398 0.849619i $$-0.323168\pi$$
0.527398 + 0.849619i $$0.323168\pi$$
$$614$$ 0 0
$$615$$ 1.31513e7 1.40210
$$616$$ 0 0
$$617$$ −5.34745e6 −0.565501 −0.282751 0.959193i $$-0.591247\pi$$
−0.282751 + 0.959193i $$0.591247\pi$$
$$618$$ 0 0
$$619$$ −6.82768e6 −0.716221 −0.358110 0.933679i $$-0.616579\pi$$
−0.358110 + 0.933679i $$0.616579\pi$$
$$620$$ 0 0
$$621$$ 316480. 0.0329319
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.15853e7 −1.18633
$$626$$ 0 0
$$627$$ −7.54912e6 −0.766880
$$628$$ 0 0
$$629$$ 1.23705e7 1.24670
$$630$$ 0 0
$$631$$ 3.60970e6 0.360909 0.180455 0.983583i $$-0.442243\pi$$
0.180455 + 0.983583i $$0.442243\pi$$
$$632$$ 0 0
$$633$$ −2.85512e6 −0.283214
$$634$$ 0 0
$$635$$ 6.99981e6 0.688893
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 5.01772e6 0.486132
$$640$$ 0 0
$$641$$ −1.33853e7 −1.28672 −0.643361 0.765563i $$-0.722460\pi$$
−0.643361 + 0.765563i $$0.722460\pi$$
$$642$$ 0 0
$$643$$ −9.91115e6 −0.945358 −0.472679 0.881235i $$-0.656713\pi$$
−0.472679 + 0.881235i $$0.656713\pi$$
$$644$$ 0 0
$$645$$ 1.35982e7 1.28701
$$646$$ 0 0
$$647$$ −1.78359e7 −1.67508 −0.837539 0.546378i $$-0.816006\pi$$
−0.837539 + 0.546378i $$0.816006\pi$$
$$648$$ 0 0
$$649$$ −2.09262e6 −0.195020
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −4.32323e6 −0.396758 −0.198379 0.980125i $$-0.563568\pi$$
−0.198379 + 0.980125i $$0.563568\pi$$
$$654$$ 0 0
$$655$$ 5.20161e6 0.473734
$$656$$ 0 0
$$657$$ 767102. 0.0693330
$$658$$ 0 0
$$659$$ −1.97858e7 −1.77476 −0.887382 0.461035i $$-0.847478\pi$$
−0.887382 + 0.461035i $$0.847478\pi$$
$$660$$ 0 0
$$661$$ −1.57772e7 −1.40451 −0.702255 0.711925i $$-0.747824\pi$$
−0.702255 + 0.711925i $$0.747824\pi$$
$$662$$ 0 0
$$663$$ −1.14529e7 −1.01188
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 603888. 0.0525584
$$668$$ 0 0
$$669$$ 1.77939e7 1.53711
$$670$$ 0 0
$$671$$ −2.29177e6 −0.196501
$$672$$ 0 0
$$673$$ 6.78762e6 0.577670 0.288835 0.957379i $$-0.406732\pi$$
0.288835 + 0.957379i $$0.406732\pi$$
$$674$$ 0 0
$$675$$ −4.04372e6 −0.341603
$$676$$ 0 0
$$677$$ 1.49942e7 1.25734 0.628669 0.777673i $$-0.283600\pi$$
0.628669 + 0.777673i $$0.283600\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 2.28631e7 1.88916
$$682$$ 0 0
$$683$$ −1.15580e7 −0.948053 −0.474026 0.880511i $$-0.657200\pi$$
−0.474026 + 0.880511i $$0.657200\pi$$
$$684$$ 0 0
$$685$$ 2.05195e7 1.67086
$$686$$ 0 0
$$687$$ 1.39157e7 1.12490
$$688$$ 0 0
$$689$$ −5.58591e6 −0.448276
$$690$$ 0 0
$$691$$ −220156. −0.0175402 −0.00877012 0.999962i $$-0.502792\pi$$
−0.00877012 + 0.999962i $$0.502792\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −9.64279e6 −0.757253
$$696$$ 0 0
$$697$$ 1.06454e7 0.830006
$$698$$ 0 0
$$699$$ −6.94252e6 −0.537433
$$700$$ 0 0
$$701$$ 4.78933e6 0.368111 0.184056 0.982916i $$-0.441077\pi$$
0.184056 + 0.982916i $$0.441077\pi$$
$$702$$ 0 0
$$703$$ 3.14323e7 2.39877
$$704$$ 0 0
$$705$$ 3.50227e7 2.65385
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 4.26892e6 0.318935 0.159468 0.987203i $$-0.449022\pi$$
0.159468 + 0.987203i $$0.449022\pi$$
$$710$$ 0 0
$$711$$ −6.99592e6 −0.519004
$$712$$ 0 0
$$713$$ 1.05395e6 0.0776421
$$714$$ 0 0
$$715$$ 4.38613e6 0.320860
$$716$$ 0 0
$$717$$ 3.28592e7 2.38704
$$718$$ 0 0
$$719$$ −1.61960e7 −1.16838 −0.584190 0.811617i $$-0.698588\pi$$
−0.584190 + 0.811617i $$0.698588\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 2.33488e7 1.66119
$$724$$ 0 0
$$725$$ −7.71598e6 −0.545188
$$726$$ 0 0
$$727$$ 6.53426e6 0.458522 0.229261 0.973365i $$-0.426369\pi$$
0.229261 + 0.973365i $$0.426369\pi$$
$$728$$ 0 0
$$729$$ −3.03131e6 −0.211257
$$730$$ 0 0
$$731$$ 1.10072e7 0.761876
$$732$$ 0 0
$$733$$ −1.31617e7 −0.904800 −0.452400 0.891815i $$-0.649432\pi$$
−0.452400 + 0.891815i $$0.649432\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1.92597e6 −0.130611
$$738$$ 0 0
$$739$$ 1.42348e7 0.958825 0.479412 0.877590i $$-0.340850\pi$$
0.479412 + 0.877590i $$0.340850\pi$$
$$740$$ 0 0
$$741$$ −2.91006e7 −1.94696
$$742$$ 0 0
$$743$$ 2.15835e7 1.43434 0.717168 0.696901i $$-0.245438\pi$$
0.717168 + 0.696901i $$0.245438\pi$$
$$744$$ 0 0
$$745$$ −2.97132e7 −1.96137
$$746$$ 0 0
$$747$$ 1.05761e7 0.693467
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −1.86594e7 −1.20725 −0.603625 0.797268i $$-0.706278\pi$$
−0.603625 + 0.797268i $$0.706278\pi$$
$$752$$ 0 0
$$753$$ −1.58122e7 −1.01626
$$754$$ 0 0
$$755$$ 5.62222e6 0.358956
$$756$$ 0 0
$$757$$ −2.56681e6 −0.162800 −0.0813999 0.996682i $$-0.525939\pi$$
−0.0813999 + 0.996682i $$0.525939\pi$$
$$758$$ 0 0
$$759$$ 456320. 0.0287518
$$760$$ 0 0
$$761$$ 2.59586e7 1.62487 0.812436 0.583051i $$-0.198141\pi$$
0.812436 + 0.583051i $$0.198141\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 1.39184e7 0.859874
$$766$$ 0 0
$$767$$ −8.06673e6 −0.495118
$$768$$ 0 0
$$769$$ −5.53267e6 −0.337380 −0.168690 0.985669i $$-0.553954\pi$$
−0.168690 + 0.985669i $$0.553954\pi$$
$$770$$ 0 0
$$771$$ 2.59580e6 0.157266
$$772$$ 0 0
$$773$$ −8.32940e6 −0.501378 −0.250689 0.968068i $$-0.580657\pi$$
−0.250689 + 0.968068i $$0.580657\pi$$
$$774$$ 0 0
$$775$$ −1.34665e7 −0.805381
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.70490e7 1.59701
$$780$$ 0 0
$$781$$ −3.96304e6 −0.232488
$$782$$ 0 0
$$783$$ 5.64504e6 0.329051
$$784$$ 0 0
$$785$$ 2.91798e7 1.69009
$$786$$ 0 0
$$787$$ −1.36523e7 −0.785719 −0.392860 0.919598i $$-0.628514\pi$$
−0.392860 + 0.919598i $$0.628514\pi$$
$$788$$ 0 0
$$789$$ −1.41776e6 −0.0810793
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −8.83440e6 −0.498877
$$794$$ 0 0
$$795$$ 1.72953e7 0.970532
$$796$$ 0 0
$$797$$ 8.54626e6 0.476574 0.238287 0.971195i $$-0.423414\pi$$
0.238287 + 0.971195i $$0.423414\pi$$
$$798$$ 0 0
$$799$$ 2.83495e7 1.57101
$$800$$ 0 0
$$801$$ −1.13031e7 −0.622465
$$802$$ 0 0
$$803$$ −605864. −0.0331578
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −3.58035e7 −1.93527
$$808$$ 0 0
$$809$$ 7.58484e6 0.407451 0.203725 0.979028i $$-0.434695\pi$$
0.203725 + 0.979028i $$0.434695\pi$$
$$810$$ 0 0
$$811$$ 6.18473e6 0.330194 0.165097 0.986277i $$-0.447206\pi$$
0.165097 + 0.986277i $$0.447206\pi$$
$$812$$ 0 0
$$813$$ −3.54723e7 −1.88219
$$814$$ 0 0
$$815$$ 867576. 0.0457524
$$816$$ 0 0
$$817$$ 2.79683e7 1.46592
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.78102e6 −0.143995 −0.0719973 0.997405i $$-0.522937\pi$$
−0.0719973 + 0.997405i $$0.522937\pi$$
$$822$$ 0 0
$$823$$ −1.63895e7 −0.843461 −0.421731 0.906721i $$-0.638577\pi$$
−0.421731 + 0.906721i $$0.638577\pi$$
$$824$$ 0 0
$$825$$ −5.83048e6 −0.298242
$$826$$ 0 0
$$827$$ −2.29511e7 −1.16692 −0.583459 0.812142i $$-0.698301\pi$$
−0.583459 + 0.812142i $$0.698301\pi$$
$$828$$ 0 0
$$829$$ 3.50136e6 0.176950 0.0884750 0.996078i $$-0.471801\pi$$
0.0884750 + 0.996078i $$0.471801\pi$$
$$830$$ 0 0
$$831$$ −5.50900e6 −0.276739
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −4.08427e7 −2.02721
$$836$$ 0 0
$$837$$ 9.85216e6 0.486091
$$838$$ 0 0
$$839$$ 5.29668e6 0.259776 0.129888 0.991529i $$-0.458538\pi$$
0.129888 + 0.991529i $$0.458538\pi$$
$$840$$ 0 0
$$841$$ −9.73962e6 −0.474845
$$842$$ 0 0
$$843$$ 1.18834e7 0.575933
$$844$$ 0 0
$$845$$ −1.05679e7 −0.509150
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 2.18486e7 1.04029
$$850$$ 0 0
$$851$$ −1.89998e6 −0.0899344
$$852$$ 0 0
$$853$$ 2.02948e7 0.955021 0.477511 0.878626i $$-0.341539\pi$$
0.477511 + 0.878626i $$0.341539\pi$$
$$854$$ 0 0
$$855$$ 3.53652e7 1.65448
$$856$$ 0 0
$$857$$ 4.82785e6 0.224544 0.112272 0.993678i $$-0.464187\pi$$
0.112272 + 0.993678i $$0.464187\pi$$
$$858$$ 0 0
$$859$$ −1.30210e7 −0.602092 −0.301046 0.953610i $$-0.597336\pi$$
−0.301046 + 0.953610i $$0.597336\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 3.92387e7 1.79344 0.896721 0.442596i $$-0.145942\pi$$
0.896721 + 0.442596i $$0.145942\pi$$
$$864$$ 0 0
$$865$$ −3.20342e7 −1.45570
$$866$$ 0 0
$$867$$ 306940. 0.0138677
$$868$$ 0 0
$$869$$ 5.52544e6 0.248209
$$870$$ 0 0
$$871$$ −7.42430e6 −0.331596
$$872$$ 0 0
$$873$$ −7.67196e6 −0.340699
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.34622e7 0.591041 0.295520 0.955336i $$-0.404507\pi$$
0.295520 + 0.955336i $$0.404507\pi$$
$$878$$ 0 0
$$879$$ −6.67308e6 −0.291309
$$880$$ 0 0
$$881$$ 917710. 0.0398351 0.0199175 0.999802i $$-0.493660\pi$$
0.0199175 + 0.999802i $$0.493660\pi$$
$$882$$ 0 0
$$883$$ −2.45488e7 −1.05957 −0.529784 0.848133i $$-0.677727\pi$$
−0.529784 + 0.848133i $$0.677727\pi$$
$$884$$ 0 0
$$885$$ 2.49765e7 1.07195
$$886$$ 0 0
$$887$$ −1.61463e7 −0.689070 −0.344535 0.938773i $$-0.611963\pi$$
−0.344535 + 0.938773i $$0.611963\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 8.99632e6 0.379639
$$892$$ 0 0
$$893$$ 7.20332e7 3.02276
$$894$$ 0 0
$$895$$ −4.14119e7 −1.72809
$$896$$ 0 0
$$897$$ 1.75904e6 0.0729953
$$898$$ 0 0
$$899$$ 1.87993e7 0.775787
$$900$$ 0 0
$$901$$ 1.39998e7 0.574527
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4.47485e7 −1.81617
$$906$$ 0 0
$$907$$ 2.03361e7 0.820824 0.410412 0.911900i $$-0.365385\pi$$
0.410412 + 0.911900i $$0.365385\pi$$
$$908$$ 0 0
$$909$$ −8.10214e6 −0.325230
$$910$$ 0 0
$$911$$ −1.07726e7 −0.430054 −0.215027 0.976608i $$-0.568984\pi$$
−0.215027 + 0.976608i $$0.568984\pi$$
$$912$$ 0 0
$$913$$ −8.35314e6 −0.331644
$$914$$ 0 0
$$915$$ 2.73534e7 1.08009
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −4.18566e7 −1.63484 −0.817419 0.576043i $$-0.804596\pi$$
−0.817419 + 0.576043i $$0.804596\pi$$
$$920$$ 0 0
$$921$$ 2.11994e7 0.823522
$$922$$ 0 0
$$923$$ −1.52769e7 −0.590242
$$924$$ 0 0
$$925$$ 2.42764e7 0.932890
$$926$$ 0 0
$$927$$ 2.83266e7 1.08267
$$928$$ 0 0
$$929$$ −2.99845e7 −1.13988 −0.569939 0.821687i $$-0.693033\pi$$
−0.569939 + 0.821687i $$0.693033\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −2.67298e7 −1.00529
$$934$$ 0 0
$$935$$ −1.09928e7 −0.411227
$$936$$ 0 0
$$937$$ −1.42402e7 −0.529867 −0.264934 0.964267i $$-0.585350\pi$$
−0.264934 + 0.964267i $$0.585350\pi$$
$$938$$ 0 0
$$939$$ −3.28837e7 −1.21707
$$940$$ 0 0
$$941$$ 4.14546e7 1.52615 0.763077 0.646307i $$-0.223687\pi$$
0.763077 + 0.646307i $$0.223687\pi$$
$$942$$ 0 0
$$943$$ −1.63502e6 −0.0598749
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.54079e7 0.558300 0.279150 0.960248i $$-0.409947\pi$$
0.279150 + 0.960248i $$0.409947\pi$$
$$948$$ 0 0
$$949$$ −2.33551e6 −0.0841813
$$950$$ 0 0
$$951$$ −3.44740e7 −1.23606
$$952$$ 0 0
$$953$$ −2.06328e7 −0.735912 −0.367956 0.929843i $$-0.619942\pi$$
−0.367956 + 0.929843i $$0.619942\pi$$
$$954$$ 0 0
$$955$$ 3.02772e7 1.07426
$$956$$ 0 0
$$957$$ 8.13936e6 0.287283
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 4.18083e6 0.146034
$$962$$ 0 0
$$963$$ 1.03149e7 0.358426
$$964$$ 0 0
$$965$$ 4.00241e7 1.38358
$$966$$ 0 0
$$967$$ −1.18724e7 −0.408294 −0.204147 0.978940i $$-0.565442\pi$$
−0.204147 + 0.978940i $$0.565442\pi$$
$$968$$ 0 0
$$969$$ 7.29342e7 2.49530
$$970$$ 0 0
$$971$$ 1.53222e6 0.0521523 0.0260761 0.999660i $$-0.491699\pi$$
0.0260761 + 0.999660i $$0.491699\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −2.24756e7 −0.757180
$$976$$ 0 0
$$977$$ 1.74321e7 0.584269 0.292135 0.956377i $$-0.405635\pi$$
0.292135 + 0.956377i $$0.405635\pi$$
$$978$$ 0 0
$$979$$ 8.92726e6 0.297688
$$980$$ 0 0
$$981$$ −1.76948e7 −0.587049
$$982$$ 0 0
$$983$$ 2.23270e6 0.0736963 0.0368482 0.999321i $$-0.488268\pi$$
0.0368482 + 0.999321i $$0.488268\pi$$
$$984$$ 0 0
$$985$$ −4.66125e7 −1.53078
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −1.69059e6 −0.0549602
$$990$$ 0 0
$$991$$ −2.22501e7 −0.719693 −0.359847 0.933011i $$-0.617171\pi$$
−0.359847 + 0.933011i $$0.617171\pi$$
$$992$$ 0 0
$$993$$ −5.49926e7 −1.76983
$$994$$ 0 0
$$995$$ 2.09456e7 0.670709
$$996$$ 0 0
$$997$$ −5.32662e7 −1.69712 −0.848562 0.529095i $$-0.822531\pi$$
−0.848562 + 0.529095i $$0.822531\pi$$
$$998$$ 0 0
$$999$$ −1.77607e7 −0.563050
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.l.1.1 1
4.3 odd 2 392.6.a.b.1.1 1
7.6 odd 2 16.6.a.a.1.1 1
21.20 even 2 144.6.a.k.1.1 1
28.3 even 6 392.6.i.b.177.1 2
28.11 odd 6 392.6.i.e.177.1 2
28.19 even 6 392.6.i.b.361.1 2
28.23 odd 6 392.6.i.e.361.1 2
28.27 even 2 8.6.a.a.1.1 1
35.13 even 4 400.6.c.d.49.1 2
35.27 even 4 400.6.c.d.49.2 2
35.34 odd 2 400.6.a.l.1.1 1
56.13 odd 2 64.6.a.g.1.1 1
56.27 even 2 64.6.a.a.1.1 1
84.83 odd 2 72.6.a.f.1.1 1
112.13 odd 4 256.6.b.d.129.1 2
112.27 even 4 256.6.b.f.129.1 2
112.69 odd 4 256.6.b.d.129.2 2
112.83 even 4 256.6.b.f.129.2 2
140.27 odd 4 200.6.c.a.49.1 2
140.83 odd 4 200.6.c.a.49.2 2
140.139 even 2 200.6.a.a.1.1 1
168.83 odd 2 576.6.a.g.1.1 1
168.125 even 2 576.6.a.h.1.1 1
308.307 odd 2 968.6.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8.6.a.a.1.1 1 28.27 even 2
16.6.a.a.1.1 1 7.6 odd 2
64.6.a.a.1.1 1 56.27 even 2
64.6.a.g.1.1 1 56.13 odd 2
72.6.a.f.1.1 1 84.83 odd 2
144.6.a.k.1.1 1 21.20 even 2
200.6.a.a.1.1 1 140.139 even 2
200.6.c.a.49.1 2 140.27 odd 4
200.6.c.a.49.2 2 140.83 odd 4
256.6.b.d.129.1 2 112.13 odd 4
256.6.b.d.129.2 2 112.69 odd 4
256.6.b.f.129.1 2 112.27 even 4
256.6.b.f.129.2 2 112.83 even 4
392.6.a.b.1.1 1 4.3 odd 2
392.6.i.b.177.1 2 28.3 even 6
392.6.i.b.361.1 2 28.19 even 6
392.6.i.e.177.1 2 28.11 odd 6
392.6.i.e.361.1 2 28.23 odd 6
400.6.a.l.1.1 1 35.34 odd 2
400.6.c.d.49.1 2 35.13 even 4
400.6.c.d.49.2 2 35.27 even 4
576.6.a.g.1.1 1 168.83 odd 2
576.6.a.h.1.1 1 168.125 even 2
784.6.a.l.1.1 1 1.1 even 1 trivial
968.6.a.a.1.1 1 308.307 odd 2