# Properties

 Label 1083.6.a.c.1.1 Level $1083$ Weight $6$ Character 1083.1 Self dual yes Analytic conductor $173.696$ 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] = [1083,6,Mod(1,1083)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1083, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1083.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1083 = 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1083.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: $$173.695676857$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1083.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} +6.00000 q^{5} -54.0000 q^{6} -40.0000 q^{7} -168.000 q^{8} +81.0000 q^{9} +O(q^{10})$$ $$q+6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} +6.00000 q^{5} -54.0000 q^{6} -40.0000 q^{7} -168.000 q^{8} +81.0000 q^{9} +36.0000 q^{10} -564.000 q^{11} -36.0000 q^{12} -638.000 q^{13} -240.000 q^{14} -54.0000 q^{15} -1136.00 q^{16} +882.000 q^{17} +486.000 q^{18} +24.0000 q^{20} +360.000 q^{21} -3384.00 q^{22} -840.000 q^{23} +1512.00 q^{24} -3089.00 q^{25} -3828.00 q^{26} -729.000 q^{27} -160.000 q^{28} -4638.00 q^{29} -324.000 q^{30} -4400.00 q^{31} -1440.00 q^{32} +5076.00 q^{33} +5292.00 q^{34} -240.000 q^{35} +324.000 q^{36} +2410.00 q^{37} +5742.00 q^{39} -1008.00 q^{40} +6870.00 q^{41} +2160.00 q^{42} +9644.00 q^{43} -2256.00 q^{44} +486.000 q^{45} -5040.00 q^{46} -18672.0 q^{47} +10224.0 q^{48} -15207.0 q^{49} -18534.0 q^{50} -7938.00 q^{51} -2552.00 q^{52} -33750.0 q^{53} -4374.00 q^{54} -3384.00 q^{55} +6720.00 q^{56} -27828.0 q^{58} +18084.0 q^{59} -216.000 q^{60} +39758.0 q^{61} -26400.0 q^{62} -3240.00 q^{63} +27712.0 q^{64} -3828.00 q^{65} +30456.0 q^{66} +23068.0 q^{67} +3528.00 q^{68} +7560.00 q^{69} -1440.00 q^{70} +4248.00 q^{71} -13608.0 q^{72} -41110.0 q^{73} +14460.0 q^{74} +27801.0 q^{75} +22560.0 q^{77} +34452.0 q^{78} -21920.0 q^{79} -6816.00 q^{80} +6561.00 q^{81} +41220.0 q^{82} +82452.0 q^{83} +1440.00 q^{84} +5292.00 q^{85} +57864.0 q^{86} +41742.0 q^{87} +94752.0 q^{88} +94086.0 q^{89} +2916.00 q^{90} +25520.0 q^{91} -3360.00 q^{92} +39600.0 q^{93} -112032. q^{94} +12960.0 q^{96} -49442.0 q^{97} -91242.0 q^{98} -45684.0 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 6.00000 1.06066 0.530330 0.847791i $$-0.322068\pi$$
0.530330 + 0.847791i $$0.322068\pi$$
$$3$$ −9.00000 −0.577350
$$4$$ 4.00000 0.125000
$$5$$ 6.00000 0.107331 0.0536656 0.998559i $$-0.482909\pi$$
0.0536656 + 0.998559i $$0.482909\pi$$
$$6$$ −54.0000 −0.612372
$$7$$ −40.0000 −0.308542 −0.154271 0.988029i $$-0.549303\pi$$
−0.154271 + 0.988029i $$0.549303\pi$$
$$8$$ −168.000 −0.928078
$$9$$ 81.0000 0.333333
$$10$$ 36.0000 0.113842
$$11$$ −564.000 −1.40539 −0.702696 0.711490i $$-0.748021\pi$$
−0.702696 + 0.711490i $$0.748021\pi$$
$$12$$ −36.0000 −0.0721688
$$13$$ −638.000 −1.04704 −0.523519 0.852014i $$-0.675381\pi$$
−0.523519 + 0.852014i $$0.675381\pi$$
$$14$$ −240.000 −0.327259
$$15$$ −54.0000 −0.0619677
$$16$$ −1136.00 −1.10938
$$17$$ 882.000 0.740195 0.370098 0.928993i $$-0.379324\pi$$
0.370098 + 0.928993i $$0.379324\pi$$
$$18$$ 486.000 0.353553
$$19$$ 0 0
$$20$$ 24.0000 0.0134164
$$21$$ 360.000 0.178137
$$22$$ −3384.00 −1.49064
$$23$$ −840.000 −0.331100 −0.165550 0.986201i $$-0.552940\pi$$
−0.165550 + 0.986201i $$0.552940\pi$$
$$24$$ 1512.00 0.535826
$$25$$ −3089.00 −0.988480
$$26$$ −3828.00 −1.11055
$$27$$ −729.000 −0.192450
$$28$$ −160.000 −0.0385678
$$29$$ −4638.00 −1.02408 −0.512042 0.858960i $$-0.671111\pi$$
−0.512042 + 0.858960i $$0.671111\pi$$
$$30$$ −324.000 −0.0657267
$$31$$ −4400.00 −0.822334 −0.411167 0.911560i $$-0.634879\pi$$
−0.411167 + 0.911560i $$0.634879\pi$$
$$32$$ −1440.00 −0.248592
$$33$$ 5076.00 0.811403
$$34$$ 5292.00 0.785096
$$35$$ −240.000 −0.0331162
$$36$$ 324.000 0.0416667
$$37$$ 2410.00 0.289409 0.144705 0.989475i $$-0.453777\pi$$
0.144705 + 0.989475i $$0.453777\pi$$
$$38$$ 0 0
$$39$$ 5742.00 0.604507
$$40$$ −1008.00 −0.0996117
$$41$$ 6870.00 0.638259 0.319130 0.947711i $$-0.396609\pi$$
0.319130 + 0.947711i $$0.396609\pi$$
$$42$$ 2160.00 0.188943
$$43$$ 9644.00 0.795401 0.397700 0.917515i $$-0.369808\pi$$
0.397700 + 0.917515i $$0.369808\pi$$
$$44$$ −2256.00 −0.175674
$$45$$ 486.000 0.0357771
$$46$$ −5040.00 −0.351185
$$47$$ −18672.0 −1.23295 −0.616476 0.787374i $$-0.711440\pi$$
−0.616476 + 0.787374i $$0.711440\pi$$
$$48$$ 10224.0 0.640498
$$49$$ −15207.0 −0.904802
$$50$$ −18534.0 −1.04844
$$51$$ −7938.00 −0.427352
$$52$$ −2552.00 −0.130880
$$53$$ −33750.0 −1.65038 −0.825190 0.564855i $$-0.808932\pi$$
−0.825190 + 0.564855i $$0.808932\pi$$
$$54$$ −4374.00 −0.204124
$$55$$ −3384.00 −0.150842
$$56$$ 6720.00 0.286351
$$57$$ 0 0
$$58$$ −27828.0 −1.08621
$$59$$ 18084.0 0.676339 0.338170 0.941085i $$-0.390192\pi$$
0.338170 + 0.941085i $$0.390192\pi$$
$$60$$ −216.000 −0.00774597
$$61$$ 39758.0 1.36804 0.684022 0.729462i $$-0.260229\pi$$
0.684022 + 0.729462i $$0.260229\pi$$
$$62$$ −26400.0 −0.872217
$$63$$ −3240.00 −0.102847
$$64$$ 27712.0 0.845703
$$65$$ −3828.00 −0.112380
$$66$$ 30456.0 0.860623
$$67$$ 23068.0 0.627802 0.313901 0.949456i $$-0.398364\pi$$
0.313901 + 0.949456i $$0.398364\pi$$
$$68$$ 3528.00 0.0925244
$$69$$ 7560.00 0.191161
$$70$$ −1440.00 −0.0351251
$$71$$ 4248.00 0.100009 0.0500044 0.998749i $$-0.484076\pi$$
0.0500044 + 0.998749i $$0.484076\pi$$
$$72$$ −13608.0 −0.309359
$$73$$ −41110.0 −0.902901 −0.451451 0.892296i $$-0.649093\pi$$
−0.451451 + 0.892296i $$0.649093\pi$$
$$74$$ 14460.0 0.306965
$$75$$ 27801.0 0.570699
$$76$$ 0 0
$$77$$ 22560.0 0.433623
$$78$$ 34452.0 0.641177
$$79$$ −21920.0 −0.395160 −0.197580 0.980287i $$-0.563308\pi$$
−0.197580 + 0.980287i $$0.563308\pi$$
$$80$$ −6816.00 −0.119071
$$81$$ 6561.00 0.111111
$$82$$ 41220.0 0.676976
$$83$$ 82452.0 1.31373 0.656865 0.754008i $$-0.271882\pi$$
0.656865 + 0.754008i $$0.271882\pi$$
$$84$$ 1440.00 0.0222671
$$85$$ 5292.00 0.0794461
$$86$$ 57864.0 0.843650
$$87$$ 41742.0 0.591255
$$88$$ 94752.0 1.30431
$$89$$ 94086.0 1.25907 0.629535 0.776972i $$-0.283245\pi$$
0.629535 + 0.776972i $$0.283245\pi$$
$$90$$ 2916.00 0.0379473
$$91$$ 25520.0 0.323056
$$92$$ −3360.00 −0.0413875
$$93$$ 39600.0 0.474775
$$94$$ −112032. −1.30774
$$95$$ 0 0
$$96$$ 12960.0 0.143525
$$97$$ −49442.0 −0.533540 −0.266770 0.963760i $$-0.585956\pi$$
−0.266770 + 0.963760i $$0.585956\pi$$
$$98$$ −91242.0 −0.959687
$$99$$ −45684.0 −0.468464
$$100$$ −12356.0 −0.123560
$$101$$ −143034. −1.39520 −0.697599 0.716488i $$-0.745748\pi$$
−0.697599 + 0.716488i $$0.745748\pi$$
$$102$$ −47628.0 −0.453275
$$103$$ −53144.0 −0.493584 −0.246792 0.969068i $$-0.579376\pi$$
−0.246792 + 0.969068i $$0.579376\pi$$
$$104$$ 107184. 0.971732
$$105$$ 2160.00 0.0191197
$$106$$ −202500. −1.75049
$$107$$ −90828.0 −0.766938 −0.383469 0.923554i $$-0.625271\pi$$
−0.383469 + 0.923554i $$0.625271\pi$$
$$108$$ −2916.00 −0.0240563
$$109$$ 61666.0 0.497141 0.248570 0.968614i $$-0.420039\pi$$
0.248570 + 0.968614i $$0.420039\pi$$
$$110$$ −20304.0 −0.159993
$$111$$ −21690.0 −0.167091
$$112$$ 45440.0 0.342289
$$113$$ −10482.0 −0.0772232 −0.0386116 0.999254i $$-0.512294\pi$$
−0.0386116 + 0.999254i $$0.512294\pi$$
$$114$$ 0 0
$$115$$ −5040.00 −0.0355374
$$116$$ −18552.0 −0.128011
$$117$$ −51678.0 −0.349013
$$118$$ 108504. 0.717366
$$119$$ −35280.0 −0.228382
$$120$$ 9072.00 0.0575109
$$121$$ 157045. 0.975126
$$122$$ 238548. 1.45103
$$123$$ −61830.0 −0.368499
$$124$$ −17600.0 −0.102792
$$125$$ −37284.0 −0.213426
$$126$$ −19440.0 −0.109086
$$127$$ 171088. 0.941261 0.470631 0.882330i $$-0.344026\pi$$
0.470631 + 0.882330i $$0.344026\pi$$
$$128$$ 212352. 1.14560
$$129$$ −86796.0 −0.459225
$$130$$ −22968.0 −0.119197
$$131$$ 258468. 1.31592 0.657959 0.753054i $$-0.271420\pi$$
0.657959 + 0.753054i $$0.271420\pi$$
$$132$$ 20304.0 0.101425
$$133$$ 0 0
$$134$$ 138408. 0.665885
$$135$$ −4374.00 −0.0206559
$$136$$ −148176. −0.686959
$$137$$ 300234. 1.36665 0.683327 0.730113i $$-0.260532\pi$$
0.683327 + 0.730113i $$0.260532\pi$$
$$138$$ 45360.0 0.202757
$$139$$ −350164. −1.53721 −0.768607 0.639721i $$-0.779050\pi$$
−0.768607 + 0.639721i $$0.779050\pi$$
$$140$$ −960.000 −0.00413953
$$141$$ 168048. 0.711845
$$142$$ 25488.0 0.106075
$$143$$ 359832. 1.47150
$$144$$ −92016.0 −0.369792
$$145$$ −27828.0 −0.109916
$$146$$ −246660. −0.957672
$$147$$ 136863. 0.522387
$$148$$ 9640.00 0.0361762
$$149$$ −105258. −0.388409 −0.194205 0.980961i $$-0.562213\pi$$
−0.194205 + 0.980961i $$0.562213\pi$$
$$150$$ 166806. 0.605318
$$151$$ −396392. −1.41476 −0.707380 0.706834i $$-0.750123\pi$$
−0.707380 + 0.706834i $$0.750123\pi$$
$$152$$ 0 0
$$153$$ 71442.0 0.246732
$$154$$ 135360. 0.459927
$$155$$ −26400.0 −0.0882622
$$156$$ 22968.0 0.0755634
$$157$$ −137746. −0.445995 −0.222997 0.974819i $$-0.571584\pi$$
−0.222997 + 0.974819i $$0.571584\pi$$
$$158$$ −131520. −0.419130
$$159$$ 303750. 0.952848
$$160$$ −8640.00 −0.0266817
$$161$$ 33600.0 0.102159
$$162$$ 39366.0 0.117851
$$163$$ 352676. 1.03970 0.519849 0.854258i $$-0.325988\pi$$
0.519849 + 0.854258i $$0.325988\pi$$
$$164$$ 27480.0 0.0797824
$$165$$ 30456.0 0.0870889
$$166$$ 494712. 1.39342
$$167$$ 217560. 0.603654 0.301827 0.953363i $$-0.402404\pi$$
0.301827 + 0.953363i $$0.402404\pi$$
$$168$$ −60480.0 −0.165325
$$169$$ 35751.0 0.0962878
$$170$$ 31752.0 0.0842653
$$171$$ 0 0
$$172$$ 38576.0 0.0994251
$$173$$ 163698. 0.415842 0.207921 0.978146i $$-0.433330\pi$$
0.207921 + 0.978146i $$0.433330\pi$$
$$174$$ 250452. 0.627121
$$175$$ 123560. 0.304988
$$176$$ 640704. 1.55911
$$177$$ −162756. −0.390485
$$178$$ 564516. 1.33545
$$179$$ −358740. −0.836849 −0.418425 0.908252i $$-0.637418\pi$$
−0.418425 + 0.908252i $$0.637418\pi$$
$$180$$ 1944.00 0.00447214
$$181$$ 507130. 1.15060 0.575298 0.817944i $$-0.304886\pi$$
0.575298 + 0.817944i $$0.304886\pi$$
$$182$$ 153120. 0.342652
$$183$$ −357822. −0.789840
$$184$$ 141120. 0.307287
$$185$$ 14460.0 0.0310627
$$186$$ 237600. 0.503575
$$187$$ −497448. −1.04026
$$188$$ −74688.0 −0.154119
$$189$$ 29160.0 0.0593790
$$190$$ 0 0
$$191$$ −648384. −1.28602 −0.643012 0.765856i $$-0.722315\pi$$
−0.643012 + 0.765856i $$0.722315\pi$$
$$192$$ −249408. −0.488267
$$193$$ 27838.0 0.0537954 0.0268977 0.999638i $$-0.491437\pi$$
0.0268977 + 0.999638i $$0.491437\pi$$
$$194$$ −296652. −0.565904
$$195$$ 34452.0 0.0648826
$$196$$ −60828.0 −0.113100
$$197$$ 611046. 1.12178 0.560891 0.827890i $$-0.310459\pi$$
0.560891 + 0.827890i $$0.310459\pi$$
$$198$$ −274104. −0.496881
$$199$$ 879032. 1.57352 0.786760 0.617260i $$-0.211757\pi$$
0.786760 + 0.617260i $$0.211757\pi$$
$$200$$ 518952. 0.917386
$$201$$ −207612. −0.362462
$$202$$ −858204. −1.47983
$$203$$ 185520. 0.315973
$$204$$ −31752.0 −0.0534190
$$205$$ 41220.0 0.0685052
$$206$$ −318864. −0.523525
$$207$$ −68040.0 −0.110367
$$208$$ 724768. 1.16156
$$209$$ 0 0
$$210$$ 12960.0 0.0202795
$$211$$ −48500.0 −0.0749956 −0.0374978 0.999297i $$-0.511939\pi$$
−0.0374978 + 0.999297i $$0.511939\pi$$
$$212$$ −135000. −0.206298
$$213$$ −38232.0 −0.0577402
$$214$$ −544968. −0.813461
$$215$$ 57864.0 0.0853714
$$216$$ 122472. 0.178609
$$217$$ 176000. 0.253725
$$218$$ 369996. 0.527298
$$219$$ 369990. 0.521290
$$220$$ −13536.0 −0.0188553
$$221$$ −562716. −0.775012
$$222$$ −130140. −0.177226
$$223$$ 999472. 1.34589 0.672943 0.739694i $$-0.265030\pi$$
0.672943 + 0.739694i $$0.265030\pi$$
$$224$$ 57600.0 0.0767012
$$225$$ −250209. −0.329493
$$226$$ −62892.0 −0.0819076
$$227$$ −606180. −0.780795 −0.390397 0.920646i $$-0.627662\pi$$
−0.390397 + 0.920646i $$0.627662\pi$$
$$228$$ 0 0
$$229$$ 1.35993e6 1.71367 0.856834 0.515593i $$-0.172428\pi$$
0.856834 + 0.515593i $$0.172428\pi$$
$$230$$ −30240.0 −0.0376931
$$231$$ −203040. −0.250352
$$232$$ 779184. 0.950430
$$233$$ −392886. −0.474107 −0.237054 0.971497i $$-0.576182\pi$$
−0.237054 + 0.971497i $$0.576182\pi$$
$$234$$ −310068. −0.370184
$$235$$ −112032. −0.132334
$$236$$ 72336.0 0.0845424
$$237$$ 197280. 0.228146
$$238$$ −211680. −0.242235
$$239$$ −1.32514e6 −1.50060 −0.750301 0.661096i $$-0.770092\pi$$
−0.750301 + 0.661096i $$0.770092\pi$$
$$240$$ 61344.0 0.0687455
$$241$$ 990094. 1.09808 0.549040 0.835796i $$-0.314994\pi$$
0.549040 + 0.835796i $$0.314994\pi$$
$$242$$ 942270. 1.03428
$$243$$ −59049.0 −0.0641500
$$244$$ 159032. 0.171005
$$245$$ −91242.0 −0.0971135
$$246$$ −370980. −0.390852
$$247$$ 0 0
$$248$$ 739200. 0.763190
$$249$$ −742068. −0.758482
$$250$$ −223704. −0.226373
$$251$$ 147132. 0.147409 0.0737043 0.997280i $$-0.476518\pi$$
0.0737043 + 0.997280i $$0.476518\pi$$
$$252$$ −12960.0 −0.0128559
$$253$$ 473760. 0.465326
$$254$$ 1.02653e6 0.998358
$$255$$ −47628.0 −0.0458682
$$256$$ 387328. 0.369385
$$257$$ 483582. 0.456707 0.228353 0.973578i $$-0.426666\pi$$
0.228353 + 0.973578i $$0.426666\pi$$
$$258$$ −520776. −0.487082
$$259$$ −96400.0 −0.0892951
$$260$$ −15312.0 −0.0140475
$$261$$ −375678. −0.341361
$$262$$ 1.55081e6 1.39574
$$263$$ 813576. 0.725285 0.362643 0.931928i $$-0.381875\pi$$
0.362643 + 0.931928i $$0.381875\pi$$
$$264$$ −852768. −0.753045
$$265$$ −202500. −0.177137
$$266$$ 0 0
$$267$$ −846774. −0.726925
$$268$$ 92272.0 0.0784753
$$269$$ 461106. 0.388526 0.194263 0.980949i $$-0.437768\pi$$
0.194263 + 0.980949i $$0.437768\pi$$
$$270$$ −26244.0 −0.0219089
$$271$$ 1.67514e6 1.38556 0.692782 0.721147i $$-0.256385\pi$$
0.692782 + 0.721147i $$0.256385\pi$$
$$272$$ −1.00195e6 −0.821154
$$273$$ −229680. −0.186516
$$274$$ 1.80140e6 1.44956
$$275$$ 1.74220e6 1.38920
$$276$$ 30240.0 0.0238951
$$277$$ 401126. 0.314110 0.157055 0.987590i $$-0.449800\pi$$
0.157055 + 0.987590i $$0.449800\pi$$
$$278$$ −2.10098e6 −1.63046
$$279$$ −356400. −0.274111
$$280$$ 40320.0 0.0307344
$$281$$ 2.30977e6 1.74503 0.872514 0.488590i $$-0.162489\pi$$
0.872514 + 0.488590i $$0.162489\pi$$
$$282$$ 1.00829e6 0.755026
$$283$$ −1.12877e6 −0.837800 −0.418900 0.908032i $$-0.637584\pi$$
−0.418900 + 0.908032i $$0.637584\pi$$
$$284$$ 16992.0 0.0125011
$$285$$ 0 0
$$286$$ 2.15899e6 1.56076
$$287$$ −274800. −0.196930
$$288$$ −116640. −0.0828641
$$289$$ −641933. −0.452111
$$290$$ −166968. −0.116584
$$291$$ 444978. 0.308039
$$292$$ −164440. −0.112863
$$293$$ 938874. 0.638908 0.319454 0.947602i $$-0.396501\pi$$
0.319454 + 0.947602i $$0.396501\pi$$
$$294$$ 821178. 0.554076
$$295$$ 108504. 0.0725923
$$296$$ −404880. −0.268594
$$297$$ 411156. 0.270468
$$298$$ −631548. −0.411970
$$299$$ 535920. 0.346675
$$300$$ 111204. 0.0713374
$$301$$ −385760. −0.245415
$$302$$ −2.37835e6 −1.50058
$$303$$ 1.28731e6 0.805518
$$304$$ 0 0
$$305$$ 238548. 0.146834
$$306$$ 428652. 0.261699
$$307$$ −692948. −0.419619 −0.209809 0.977742i $$-0.567284\pi$$
−0.209809 + 0.977742i $$0.567284\pi$$
$$308$$ 90240.0 0.0542029
$$309$$ 478296. 0.284971
$$310$$ −158400. −0.0936162
$$311$$ 2.94310e6 1.72545 0.862727 0.505670i $$-0.168755\pi$$
0.862727 + 0.505670i $$0.168755\pi$$
$$312$$ −964656. −0.561030
$$313$$ 885146. 0.510686 0.255343 0.966851i $$-0.417812\pi$$
0.255343 + 0.966851i $$0.417812\pi$$
$$314$$ −826476. −0.473049
$$315$$ −19440.0 −0.0110387
$$316$$ −87680.0 −0.0493950
$$317$$ −2.50880e6 −1.40222 −0.701112 0.713051i $$-0.747313\pi$$
−0.701112 + 0.713051i $$0.747313\pi$$
$$318$$ 1.82250e6 1.01065
$$319$$ 2.61583e6 1.43924
$$320$$ 166272. 0.0907704
$$321$$ 817452. 0.442792
$$322$$ 201600. 0.108355
$$323$$ 0 0
$$324$$ 26244.0 0.0138889
$$325$$ 1.97078e6 1.03498
$$326$$ 2.11606e6 1.10277
$$327$$ −554994. −0.287024
$$328$$ −1.15416e6 −0.592354
$$329$$ 746880. 0.380418
$$330$$ 182736. 0.0923718
$$331$$ 216148. 0.108438 0.0542190 0.998529i $$-0.482733\pi$$
0.0542190 + 0.998529i $$0.482733\pi$$
$$332$$ 329808. 0.164216
$$333$$ 195210. 0.0964698
$$334$$ 1.30536e6 0.640271
$$335$$ 138408. 0.0673828
$$336$$ −408960. −0.197621
$$337$$ −3.25263e6 −1.56012 −0.780062 0.625702i $$-0.784813\pi$$
−0.780062 + 0.625702i $$0.784813\pi$$
$$338$$ 214506. 0.102129
$$339$$ 94338.0 0.0445849
$$340$$ 21168.0 0.00993076
$$341$$ 2.48160e6 1.15570
$$342$$ 0 0
$$343$$ 1.28056e6 0.587712
$$344$$ −1.62019e6 −0.738194
$$345$$ 45360.0 0.0205175
$$346$$ 982188. 0.441067
$$347$$ −2.93207e6 −1.30723 −0.653613 0.756829i $$-0.726747\pi$$
−0.653613 + 0.756829i $$0.726747\pi$$
$$348$$ 166968. 0.0739069
$$349$$ 905198. 0.397814 0.198907 0.980018i $$-0.436261\pi$$
0.198907 + 0.980018i $$0.436261\pi$$
$$350$$ 741360. 0.323489
$$351$$ 465102. 0.201502
$$352$$ 812160. 0.349369
$$353$$ 1.91786e6 0.819181 0.409590 0.912270i $$-0.365672\pi$$
0.409590 + 0.912270i $$0.365672\pi$$
$$354$$ −976536. −0.414171
$$355$$ 25488.0 0.0107341
$$356$$ 376344. 0.157384
$$357$$ 317520. 0.131856
$$358$$ −2.15244e6 −0.887613
$$359$$ −2.43698e6 −0.997968 −0.498984 0.866611i $$-0.666293\pi$$
−0.498984 + 0.866611i $$0.666293\pi$$
$$360$$ −81648.0 −0.0332039
$$361$$ 0 0
$$362$$ 3.04278e6 1.22039
$$363$$ −1.41340e6 −0.562989
$$364$$ 102080. 0.0403819
$$365$$ −246660. −0.0969095
$$366$$ −2.14693e6 −0.837752
$$367$$ −984064. −0.381380 −0.190690 0.981650i $$-0.561073\pi$$
−0.190690 + 0.981650i $$0.561073\pi$$
$$368$$ 954240. 0.367314
$$369$$ 556470. 0.212753
$$370$$ 86760.0 0.0329470
$$371$$ 1.35000e6 0.509212
$$372$$ 158400. 0.0593469
$$373$$ −1.70365e6 −0.634029 −0.317015 0.948421i $$-0.602680\pi$$
−0.317015 + 0.948421i $$0.602680\pi$$
$$374$$ −2.98469e6 −1.10337
$$375$$ 335556. 0.123222
$$376$$ 3.13690e6 1.14428
$$377$$ 2.95904e6 1.07225
$$378$$ 174960. 0.0629810
$$379$$ −2.75654e6 −0.985749 −0.492874 0.870100i $$-0.664054\pi$$
−0.492874 + 0.870100i $$0.664054\pi$$
$$380$$ 0 0
$$381$$ −1.53979e6 −0.543438
$$382$$ −3.89030e6 −1.36403
$$383$$ −456576. −0.159044 −0.0795218 0.996833i $$-0.525339\pi$$
−0.0795218 + 0.996833i $$0.525339\pi$$
$$384$$ −1.91117e6 −0.661410
$$385$$ 135360. 0.0465413
$$386$$ 167028. 0.0570586
$$387$$ 781164. 0.265134
$$388$$ −197768. −0.0666925
$$389$$ −2.00639e6 −0.672268 −0.336134 0.941814i $$-0.609119\pi$$
−0.336134 + 0.941814i $$0.609119\pi$$
$$390$$ 206712. 0.0688183
$$391$$ −740880. −0.245079
$$392$$ 2.55478e6 0.839726
$$393$$ −2.32621e6 −0.759745
$$394$$ 3.66628e6 1.18983
$$395$$ −131520. −0.0424130
$$396$$ −182736. −0.0585580
$$397$$ −5.77040e6 −1.83751 −0.918755 0.394828i $$-0.870804\pi$$
−0.918755 + 0.394828i $$0.870804\pi$$
$$398$$ 5.27419e6 1.66897
$$399$$ 0 0
$$400$$ 3.50910e6 1.09659
$$401$$ −3.00626e6 −0.933610 −0.466805 0.884360i $$-0.654595\pi$$
−0.466805 + 0.884360i $$0.654595\pi$$
$$402$$ −1.24567e6 −0.384449
$$403$$ 2.80720e6 0.861015
$$404$$ −572136. −0.174400
$$405$$ 39366.0 0.0119257
$$406$$ 1.11312e6 0.335140
$$407$$ −1.35924e6 −0.406734
$$408$$ 1.33358e6 0.396616
$$409$$ −1.53363e6 −0.453327 −0.226663 0.973973i $$-0.572782\pi$$
−0.226663 + 0.973973i $$0.572782\pi$$
$$410$$ 247320. 0.0726607
$$411$$ −2.70211e6 −0.789038
$$412$$ −212576. −0.0616980
$$413$$ −723360. −0.208679
$$414$$ −408240. −0.117062
$$415$$ 494712. 0.141004
$$416$$ 918720. 0.260285
$$417$$ 3.15148e6 0.887511
$$418$$ 0 0
$$419$$ −3.87376e6 −1.07795 −0.538973 0.842323i $$-0.681188\pi$$
−0.538973 + 0.842323i $$0.681188\pi$$
$$420$$ 8640.00 0.00238996
$$421$$ 1.33307e6 0.366561 0.183281 0.983061i $$-0.441328\pi$$
0.183281 + 0.983061i $$0.441328\pi$$
$$422$$ −291000. −0.0795448
$$423$$ −1.51243e6 −0.410984
$$424$$ 5.67000e6 1.53168
$$425$$ −2.72450e6 −0.731668
$$426$$ −229392. −0.0612427
$$427$$ −1.59032e6 −0.422099
$$428$$ −363312. −0.0958673
$$429$$ −3.23849e6 −0.849570
$$430$$ 347184. 0.0905500
$$431$$ −6.45192e6 −1.67300 −0.836500 0.547967i $$-0.815402\pi$$
−0.836500 + 0.547967i $$0.815402\pi$$
$$432$$ 828144. 0.213499
$$433$$ 4.16577e6 1.06777 0.533883 0.845558i $$-0.320732\pi$$
0.533883 + 0.845558i $$0.320732\pi$$
$$434$$ 1.05600e6 0.269116
$$435$$ 250452. 0.0634602
$$436$$ 246664. 0.0621426
$$437$$ 0 0
$$438$$ 2.21994e6 0.552912
$$439$$ −792680. −0.196307 −0.0981537 0.995171i $$-0.531294\pi$$
−0.0981537 + 0.995171i $$0.531294\pi$$
$$440$$ 568512. 0.139994
$$441$$ −1.23177e6 −0.301601
$$442$$ −3.37630e6 −0.822025
$$443$$ −1.39981e6 −0.338891 −0.169446 0.985540i $$-0.554198\pi$$
−0.169446 + 0.985540i $$0.554198\pi$$
$$444$$ −86760.0 −0.0208863
$$445$$ 564516. 0.135138
$$446$$ 5.99683e6 1.42753
$$447$$ 947322. 0.224248
$$448$$ −1.10848e6 −0.260935
$$449$$ −2.99248e6 −0.700512 −0.350256 0.936654i $$-0.613905\pi$$
−0.350256 + 0.936654i $$0.613905\pi$$
$$450$$ −1.50125e6 −0.349480
$$451$$ −3.87468e6 −0.897004
$$452$$ −41928.0 −0.00965291
$$453$$ 3.56753e6 0.816812
$$454$$ −3.63708e6 −0.828158
$$455$$ 153120. 0.0346740
$$456$$ 0 0
$$457$$ 6.29969e6 1.41101 0.705503 0.708707i $$-0.250721\pi$$
0.705503 + 0.708707i $$0.250721\pi$$
$$458$$ 8.15956e6 1.81762
$$459$$ −642978. −0.142451
$$460$$ −20160.0 −0.00444218
$$461$$ 3.40318e6 0.745818 0.372909 0.927868i $$-0.378360\pi$$
0.372909 + 0.927868i $$0.378360\pi$$
$$462$$ −1.21824e6 −0.265539
$$463$$ −2.23034e6 −0.483524 −0.241762 0.970336i $$-0.577725\pi$$
−0.241762 + 0.970336i $$0.577725\pi$$
$$464$$ 5.26877e6 1.13609
$$465$$ 237600. 0.0509582
$$466$$ −2.35732e6 −0.502867
$$467$$ −6.51409e6 −1.38217 −0.691085 0.722773i $$-0.742867\pi$$
−0.691085 + 0.722773i $$0.742867\pi$$
$$468$$ −206712. −0.0436266
$$469$$ −922720. −0.193704
$$470$$ −672192. −0.140362
$$471$$ 1.23971e6 0.257495
$$472$$ −3.03811e6 −0.627695
$$473$$ −5.43922e6 −1.11785
$$474$$ 1.18368e6 0.241985
$$475$$ 0 0
$$476$$ −141120. −0.0285477
$$477$$ −2.73375e6 −0.550127
$$478$$ −7.95082e6 −1.59163
$$479$$ 2.39232e6 0.476410 0.238205 0.971215i $$-0.423441\pi$$
0.238205 + 0.971215i $$0.423441\pi$$
$$480$$ 77760.0 0.0154047
$$481$$ −1.53758e6 −0.303023
$$482$$ 5.94056e6 1.16469
$$483$$ −302400. −0.0589812
$$484$$ 628180. 0.121891
$$485$$ −296652. −0.0572655
$$486$$ −354294. −0.0680414
$$487$$ 6.13089e6 1.17139 0.585694 0.810532i $$-0.300822\pi$$
0.585694 + 0.810532i $$0.300822\pi$$
$$488$$ −6.67934e6 −1.26965
$$489$$ −3.17408e6 −0.600269
$$490$$ −547452. −0.103004
$$491$$ −1.23589e6 −0.231354 −0.115677 0.993287i $$-0.536904\pi$$
−0.115677 + 0.993287i $$0.536904\pi$$
$$492$$ −247320. −0.0460624
$$493$$ −4.09072e6 −0.758022
$$494$$ 0 0
$$495$$ −274104. −0.0502808
$$496$$ 4.99840e6 0.912277
$$497$$ −169920. −0.0308570
$$498$$ −4.45241e6 −0.804492
$$499$$ −9.85496e6 −1.77175 −0.885877 0.463921i $$-0.846442\pi$$
−0.885877 + 0.463921i $$0.846442\pi$$
$$500$$ −149136. −0.0266783
$$501$$ −1.95804e6 −0.348520
$$502$$ 882792. 0.156350
$$503$$ 1.16777e6 0.205796 0.102898 0.994692i $$-0.467188\pi$$
0.102898 + 0.994692i $$0.467188\pi$$
$$504$$ 544320. 0.0954504
$$505$$ −858204. −0.149748
$$506$$ 2.84256e6 0.493552
$$507$$ −321759. −0.0555918
$$508$$ 684352. 0.117658
$$509$$ −1.04941e6 −0.179535 −0.0897675 0.995963i $$-0.528612\pi$$
−0.0897675 + 0.995963i $$0.528612\pi$$
$$510$$ −285768. −0.0486506
$$511$$ 1.64440e6 0.278583
$$512$$ −4.47130e6 −0.753804
$$513$$ 0 0
$$514$$ 2.90149e6 0.484411
$$515$$ −318864. −0.0529770
$$516$$ −347184. −0.0574031
$$517$$ 1.05310e7 1.73278
$$518$$ −578400. −0.0947118
$$519$$ −1.47328e6 −0.240086
$$520$$ 643104. 0.104297
$$521$$ 9.61407e6 1.55172 0.775859 0.630906i $$-0.217317\pi$$
0.775859 + 0.630906i $$0.217317\pi$$
$$522$$ −2.25407e6 −0.362069
$$523$$ −6.96148e6 −1.11288 −0.556439 0.830888i $$-0.687833\pi$$
−0.556439 + 0.830888i $$0.687833\pi$$
$$524$$ 1.03387e6 0.164490
$$525$$ −1.11204e6 −0.176085
$$526$$ 4.88146e6 0.769281
$$527$$ −3.88080e6 −0.608688
$$528$$ −5.76634e6 −0.900151
$$529$$ −5.73074e6 −0.890373
$$530$$ −1.21500e6 −0.187883
$$531$$ 1.46480e6 0.225446
$$532$$ 0 0
$$533$$ −4.38306e6 −0.668281
$$534$$ −5.08064e6 −0.771020
$$535$$ −544968. −0.0823164
$$536$$ −3.87542e6 −0.582649
$$537$$ 3.22866e6 0.483155
$$538$$ 2.76664e6 0.412094
$$539$$ 8.57675e6 1.27160
$$540$$ −17496.0 −0.00258199
$$541$$ −712690. −0.104691 −0.0523453 0.998629i $$-0.516670\pi$$
−0.0523453 + 0.998629i $$0.516670\pi$$
$$542$$ 1.00508e7 1.46961
$$543$$ −4.56417e6 −0.664297
$$544$$ −1.27008e6 −0.184007
$$545$$ 369996. 0.0533588
$$546$$ −1.37808e6 −0.197830
$$547$$ 3.62614e6 0.518175 0.259087 0.965854i $$-0.416578\pi$$
0.259087 + 0.965854i $$0.416578\pi$$
$$548$$ 1.20094e6 0.170832
$$549$$ 3.22040e6 0.456015
$$550$$ 1.04532e7 1.47347
$$551$$ 0 0
$$552$$ −1.27008e6 −0.177412
$$553$$ 876800. 0.121924
$$554$$ 2.40676e6 0.333164
$$555$$ −130140. −0.0179340
$$556$$ −1.40066e6 −0.192152
$$557$$ 4.84846e6 0.662165 0.331082 0.943602i $$-0.392586\pi$$
0.331082 + 0.943602i $$0.392586\pi$$
$$558$$ −2.13840e6 −0.290739
$$559$$ −6.15287e6 −0.832815
$$560$$ 272640. 0.0367383
$$561$$ 4.47703e6 0.600597
$$562$$ 1.38586e7 1.85088
$$563$$ −8.50405e6 −1.13072 −0.565360 0.824844i $$-0.691263\pi$$
−0.565360 + 0.824844i $$0.691263\pi$$
$$564$$ 672192. 0.0889807
$$565$$ −62892.0 −0.00828847
$$566$$ −6.77263e6 −0.888621
$$567$$ −262440. −0.0342825
$$568$$ −713664. −0.0928160
$$569$$ −362874. −0.0469867 −0.0234934 0.999724i $$-0.507479\pi$$
−0.0234934 + 0.999724i $$0.507479\pi$$
$$570$$ 0 0
$$571$$ 4.11024e6 0.527566 0.263783 0.964582i $$-0.415030\pi$$
0.263783 + 0.964582i $$0.415030\pi$$
$$572$$ 1.43933e6 0.183937
$$573$$ 5.83546e6 0.742486
$$574$$ −1.64880e6 −0.208876
$$575$$ 2.59476e6 0.327286
$$576$$ 2.24467e6 0.281901
$$577$$ −7.87680e6 −0.984941 −0.492470 0.870329i $$-0.663906\pi$$
−0.492470 + 0.870329i $$0.663906\pi$$
$$578$$ −3.85160e6 −0.479536
$$579$$ −250542. −0.0310588
$$580$$ −111312. −0.0137395
$$581$$ −3.29808e6 −0.405341
$$582$$ 2.66987e6 0.326725
$$583$$ 1.90350e7 2.31943
$$584$$ 6.90648e6 0.837963
$$585$$ −310068. −0.0374600
$$586$$ 5.63324e6 0.677664
$$587$$ 603948. 0.0723443 0.0361721 0.999346i $$-0.488484\pi$$
0.0361721 + 0.999346i $$0.488484\pi$$
$$588$$ 547452. 0.0652984
$$589$$ 0 0
$$590$$ 651024. 0.0769958
$$591$$ −5.49941e6 −0.647661
$$592$$ −2.73776e6 −0.321064
$$593$$ −5.39077e6 −0.629526 −0.314763 0.949170i $$-0.601925\pi$$
−0.314763 + 0.949170i $$0.601925\pi$$
$$594$$ 2.46694e6 0.286874
$$595$$ −211680. −0.0245125
$$596$$ −421032. −0.0485511
$$597$$ −7.91129e6 −0.908472
$$598$$ 3.21552e6 0.367704
$$599$$ −4.27999e6 −0.487389 −0.243695 0.969852i $$-0.578359\pi$$
−0.243695 + 0.969852i $$0.578359\pi$$
$$600$$ −4.67057e6 −0.529653
$$601$$ −1.02483e6 −0.115735 −0.0578674 0.998324i $$-0.518430\pi$$
−0.0578674 + 0.998324i $$0.518430\pi$$
$$602$$ −2.31456e6 −0.260302
$$603$$ 1.86851e6 0.209267
$$604$$ −1.58557e6 −0.176845
$$605$$ 942270. 0.104661
$$606$$ 7.72384e6 0.854381
$$607$$ −1.24342e7 −1.36976 −0.684882 0.728654i $$-0.740146\pi$$
−0.684882 + 0.728654i $$0.740146\pi$$
$$608$$ 0 0
$$609$$ −1.66968e6 −0.182427
$$610$$ 1.43129e6 0.155741
$$611$$ 1.19127e7 1.29095
$$612$$ 285768. 0.0308415
$$613$$ 4.21506e6 0.453057 0.226528 0.974005i $$-0.427262\pi$$
0.226528 + 0.974005i $$0.427262\pi$$
$$614$$ −4.15769e6 −0.445073
$$615$$ −370980. −0.0395515
$$616$$ −3.79008e6 −0.402436
$$617$$ −4.40665e6 −0.466010 −0.233005 0.972476i $$-0.574856\pi$$
−0.233005 + 0.972476i $$0.574856\pi$$
$$618$$ 2.86978e6 0.302257
$$619$$ 4.80168e6 0.503693 0.251847 0.967767i $$-0.418962\pi$$
0.251847 + 0.967767i $$0.418962\pi$$
$$620$$ −105600. −0.0110328
$$621$$ 612360. 0.0637203
$$622$$ 1.76586e7 1.83012
$$623$$ −3.76344e6 −0.388477
$$624$$ −6.52291e6 −0.670625
$$625$$ 9.42942e6 0.965573
$$626$$ 5.31088e6 0.541664
$$627$$ 0 0
$$628$$ −550984. −0.0557494
$$629$$ 2.12562e6 0.214220
$$630$$ −116640. −0.0117084
$$631$$ 8.30727e6 0.830587 0.415293 0.909688i $$-0.363679\pi$$
0.415293 + 0.909688i $$0.363679\pi$$
$$632$$ 3.68256e6 0.366739
$$633$$ 436500. 0.0432987
$$634$$ −1.50528e7 −1.48728
$$635$$ 1.02653e6 0.101027
$$636$$ 1.21500e6 0.119106
$$637$$ 9.70207e6 0.947361
$$638$$ 1.56950e7 1.52654
$$639$$ 344088. 0.0333363
$$640$$ 1.27411e6 0.122958
$$641$$ −1.76956e7 −1.70107 −0.850534 0.525921i $$-0.823721\pi$$
−0.850534 + 0.525921i $$0.823721\pi$$
$$642$$ 4.90471e6 0.469652
$$643$$ −1.28394e7 −1.22466 −0.612330 0.790602i $$-0.709768\pi$$
−0.612330 + 0.790602i $$0.709768\pi$$
$$644$$ 134400. 0.0127698
$$645$$ −520776. −0.0492892
$$646$$ 0 0
$$647$$ −2.08468e7 −1.95785 −0.978924 0.204226i $$-0.934532\pi$$
−0.978924 + 0.204226i $$0.934532\pi$$
$$648$$ −1.10225e6 −0.103120
$$649$$ −1.01994e7 −0.950521
$$650$$ 1.18247e7 1.09776
$$651$$ −1.58400e6 −0.146488
$$652$$ 1.41070e6 0.129962
$$653$$ 1.29632e7 1.18968 0.594841 0.803843i $$-0.297215\pi$$
0.594841 + 0.803843i $$0.297215\pi$$
$$654$$ −3.32996e6 −0.304435
$$655$$ 1.55081e6 0.141239
$$656$$ −7.80432e6 −0.708069
$$657$$ −3.32991e6 −0.300967
$$658$$ 4.48128e6 0.403494
$$659$$ 5.66862e6 0.508468 0.254234 0.967143i $$-0.418177\pi$$
0.254234 + 0.967143i $$0.418177\pi$$
$$660$$ 121824. 0.0108861
$$661$$ 3.11430e6 0.277240 0.138620 0.990346i $$-0.455733\pi$$
0.138620 + 0.990346i $$0.455733\pi$$
$$662$$ 1.29689e6 0.115016
$$663$$ 5.06444e6 0.447454
$$664$$ −1.38519e7 −1.21924
$$665$$ 0 0
$$666$$ 1.17126e6 0.102322
$$667$$ 3.89592e6 0.339075
$$668$$ 870240. 0.0754567
$$669$$ −8.99525e6 −0.777048
$$670$$ 830448. 0.0714703
$$671$$ −2.24235e7 −1.92264
$$672$$ −518400. −0.0442835
$$673$$ −105890. −0.00901192 −0.00450596 0.999990i $$-0.501434\pi$$
−0.00450596 + 0.999990i $$0.501434\pi$$
$$674$$ −1.95158e7 −1.65476
$$675$$ 2.25188e6 0.190233
$$676$$ 143004. 0.0120360
$$677$$ 1.60910e7 1.34931 0.674656 0.738132i $$-0.264292\pi$$
0.674656 + 0.738132i $$0.264292\pi$$
$$678$$ 566028. 0.0472894
$$679$$ 1.97768e6 0.164620
$$680$$ −889056. −0.0737321
$$681$$ 5.45562e6 0.450792
$$682$$ 1.48896e7 1.22581
$$683$$ −1.60780e7 −1.31880 −0.659402 0.751791i $$-0.729190\pi$$
−0.659402 + 0.751791i $$0.729190\pi$$
$$684$$ 0 0
$$685$$ 1.80140e6 0.146685
$$686$$ 7.68336e6 0.623363
$$687$$ −1.22393e7 −0.989386
$$688$$ −1.09556e7 −0.882398
$$689$$ 2.15325e7 1.72801
$$690$$ 272160. 0.0217621
$$691$$ −165964. −0.0132227 −0.00661133 0.999978i $$-0.502104\pi$$
−0.00661133 + 0.999978i $$0.502104\pi$$
$$692$$ 654792. 0.0519802
$$693$$ 1.82736e6 0.144541
$$694$$ −1.75924e7 −1.38652
$$695$$ −2.10098e6 −0.164991
$$696$$ −7.01266e6 −0.548731
$$697$$ 6.05934e6 0.472436
$$698$$ 5.43119e6 0.421945
$$699$$ 3.53597e6 0.273726
$$700$$ 494240. 0.0381235
$$701$$ 1.77248e7 1.36234 0.681171 0.732124i $$-0.261471\pi$$
0.681171 + 0.732124i $$0.261471\pi$$
$$702$$ 2.79061e6 0.213726
$$703$$ 0 0
$$704$$ −1.56296e7 −1.18854
$$705$$ 1.00829e6 0.0764032
$$706$$ 1.15071e7 0.868872
$$707$$ 5.72136e6 0.430478
$$708$$ −651024. −0.0488106
$$709$$ −1.06023e7 −0.792112 −0.396056 0.918226i $$-0.629621\pi$$
−0.396056 + 0.918226i $$0.629621\pi$$
$$710$$ 152928. 0.0113852
$$711$$ −1.77552e6 −0.131720
$$712$$ −1.58064e7 −1.16852
$$713$$ 3.69600e6 0.272275
$$714$$ 1.90512e6 0.139855
$$715$$ 2.15899e6 0.157938
$$716$$ −1.43496e6 −0.104606
$$717$$ 1.19262e7 0.866373
$$718$$ −1.46219e7 −1.05850
$$719$$ 9.03211e6 0.651579 0.325790 0.945442i $$-0.394370\pi$$
0.325790 + 0.945442i $$0.394370\pi$$
$$720$$ −552096. −0.0396902
$$721$$ 2.12576e6 0.152292
$$722$$ 0 0
$$723$$ −8.91085e6 −0.633977
$$724$$ 2.02852e6 0.143825
$$725$$ 1.43268e7 1.01229
$$726$$ −8.48043e6 −0.597140
$$727$$ 1.87575e7 1.31625 0.658127 0.752907i $$-0.271349\pi$$
0.658127 + 0.752907i $$0.271349\pi$$
$$728$$ −4.28736e6 −0.299821
$$729$$ 531441. 0.0370370
$$730$$ −1.47996e6 −0.102788
$$731$$ 8.50601e6 0.588752
$$732$$ −1.43129e6 −0.0987300
$$733$$ −1.17773e7 −0.809626 −0.404813 0.914399i $$-0.632663\pi$$
−0.404813 + 0.914399i $$0.632663\pi$$
$$734$$ −5.90438e6 −0.404515
$$735$$ 821178. 0.0560685
$$736$$ 1.20960e6 0.0823090
$$737$$ −1.30104e7 −0.882308
$$738$$ 3.33882e6 0.225659
$$739$$ 5.88948e6 0.396703 0.198352 0.980131i $$-0.436441\pi$$
0.198352 + 0.980131i $$0.436441\pi$$
$$740$$ 57840.0 0.00388284
$$741$$ 0 0
$$742$$ 8.10000e6 0.540101
$$743$$ 1.00476e7 0.667712 0.333856 0.942624i $$-0.391650\pi$$
0.333856 + 0.942624i $$0.391650\pi$$
$$744$$ −6.65280e6 −0.440628
$$745$$ −631548. −0.0416884
$$746$$ −1.02219e7 −0.672490
$$747$$ 6.67861e6 0.437910
$$748$$ −1.98979e6 −0.130033
$$749$$ 3.63312e6 0.236633
$$750$$ 2.01334e6 0.130696
$$751$$ −4.81530e6 −0.311547 −0.155773 0.987793i $$-0.549787\pi$$
−0.155773 + 0.987793i $$0.549787\pi$$
$$752$$ 2.12114e7 1.36781
$$753$$ −1.32419e6 −0.0851064
$$754$$ 1.77543e7 1.13730
$$755$$ −2.37835e6 −0.151848
$$756$$ 116640. 0.00742238
$$757$$ 3.12973e6 0.198503 0.0992516 0.995062i $$-0.468355\pi$$
0.0992516 + 0.995062i $$0.468355\pi$$
$$758$$ −1.65392e7 −1.04554
$$759$$ −4.26384e6 −0.268656
$$760$$ 0 0
$$761$$ −1.17773e7 −0.737197 −0.368599 0.929589i $$-0.620162\pi$$
−0.368599 + 0.929589i $$0.620162\pi$$
$$762$$ −9.23875e6 −0.576403
$$763$$ −2.46664e6 −0.153389
$$764$$ −2.59354e6 −0.160753
$$765$$ 428652. 0.0264820
$$766$$ −2.73946e6 −0.168691
$$767$$ −1.15376e7 −0.708152
$$768$$ −3.48595e6 −0.213264
$$769$$ −1.49376e6 −0.0910887 −0.0455443 0.998962i $$-0.514502\pi$$
−0.0455443 + 0.998962i $$0.514502\pi$$
$$770$$ 812160. 0.0493645
$$771$$ −4.35224e6 −0.263680
$$772$$ 111352. 0.00672442
$$773$$ 2.25125e7 1.35511 0.677555 0.735472i $$-0.263040\pi$$
0.677555 + 0.735472i $$0.263040\pi$$
$$774$$ 4.68698e6 0.281217
$$775$$ 1.35916e7 0.812861
$$776$$ 8.30626e6 0.495166
$$777$$ 867600. 0.0515545
$$778$$ −1.20384e7 −0.713048
$$779$$ 0 0
$$780$$ 137808. 0.00811032
$$781$$ −2.39587e6 −0.140552
$$782$$ −4.44528e6 −0.259945
$$783$$ 3.38110e6 0.197085
$$784$$ 1.72752e7 1.00376
$$785$$ −826476. −0.0478692
$$786$$ −1.39573e7 −0.805831
$$787$$ −1.19547e7 −0.688022 −0.344011 0.938966i $$-0.611786\pi$$
−0.344011 + 0.938966i $$0.611786\pi$$
$$788$$ 2.44418e6 0.140223
$$789$$ −7.32218e6 −0.418744
$$790$$ −789120. −0.0449858
$$791$$ 419280. 0.0238266
$$792$$ 7.67491e6 0.434771
$$793$$ −2.53656e7 −1.43239
$$794$$ −3.46224e7 −1.94897
$$795$$ 1.82250e6 0.102270
$$796$$ 3.51613e6 0.196690
$$797$$ −540798. −0.0301571 −0.0150785 0.999886i $$-0.504800\pi$$
−0.0150785 + 0.999886i $$0.504800\pi$$
$$798$$ 0 0
$$799$$ −1.64687e7 −0.912625
$$800$$ 4.44816e6 0.245728
$$801$$ 7.62097e6 0.419690
$$802$$ −1.80375e7 −0.990243
$$803$$ 2.31860e7 1.26893
$$804$$ −830448. −0.0453077
$$805$$ 201600. 0.0109648
$$806$$ 1.68432e7 0.913244
$$807$$ −4.14995e6 −0.224316
$$808$$ 2.40297e7 1.29485
$$809$$ −6.14223e6 −0.329955 −0.164978 0.986297i $$-0.552755\pi$$
−0.164978 + 0.986297i $$0.552755\pi$$
$$810$$ 236196. 0.0126491
$$811$$ 3.16734e7 1.69100 0.845499 0.533977i $$-0.179303\pi$$
0.845499 + 0.533977i $$0.179303\pi$$
$$812$$ 742080. 0.0394967
$$813$$ −1.50762e7 −0.799956
$$814$$ −8.15544e6 −0.431406
$$815$$ 2.11606e6 0.111592
$$816$$ 9.01757e6 0.474094
$$817$$ 0 0
$$818$$ −9.20176e6 −0.480825
$$819$$ 2.06712e6 0.107685
$$820$$ 164880. 0.00856315
$$821$$ 2.66175e7 1.37819 0.689095 0.724671i $$-0.258008\pi$$
0.689095 + 0.724671i $$0.258008\pi$$
$$822$$ −1.62126e7 −0.836901
$$823$$ 3.62817e7 1.86719 0.933593 0.358335i $$-0.116655\pi$$
0.933593 + 0.358335i $$0.116655\pi$$
$$824$$ 8.92819e6 0.458084
$$825$$ −1.56798e7 −0.802056
$$826$$ −4.34016e6 −0.221338
$$827$$ −1.09033e6 −0.0554364 −0.0277182 0.999616i $$-0.508824\pi$$
−0.0277182 + 0.999616i $$0.508824\pi$$
$$828$$ −272160. −0.0137958
$$829$$ 1.03016e7 0.520620 0.260310 0.965525i $$-0.416175\pi$$
0.260310 + 0.965525i $$0.416175\pi$$
$$830$$ 2.96827e6 0.149558
$$831$$ −3.61013e6 −0.181351
$$832$$ −1.76803e7 −0.885483
$$833$$ −1.34126e7 −0.669730
$$834$$ 1.89089e7 0.941348
$$835$$ 1.30536e6 0.0647909
$$836$$ 0 0
$$837$$ 3.20760e6 0.158258
$$838$$ −2.32425e7 −1.14333
$$839$$ 1.96134e7 0.961940 0.480970 0.876737i $$-0.340285\pi$$
0.480970 + 0.876737i $$0.340285\pi$$
$$840$$ −362880. −0.0177445
$$841$$ 999895. 0.0487489
$$842$$ 7.99840e6 0.388797
$$843$$ −2.07879e7 −1.00749
$$844$$ −194000. −0.00937445
$$845$$ 214506. 0.0103347
$$846$$ −9.07459e6 −0.435914
$$847$$ −6.28180e6 −0.300868
$$848$$ 3.83400e7 1.83089
$$849$$ 1.01589e7 0.483704
$$850$$ −1.63470e7 −0.776051
$$851$$ −2.02440e6 −0.0958236
$$852$$ −152928. −0.00721752
$$853$$ 3.27565e7 1.54143 0.770717 0.637178i $$-0.219898\pi$$
0.770717 + 0.637178i $$0.219898\pi$$
$$854$$ −9.54192e6 −0.447704
$$855$$ 0 0
$$856$$ 1.52591e7 0.711778
$$857$$ 2.57953e7 1.19974 0.599872 0.800096i $$-0.295218\pi$$
0.599872 + 0.800096i $$0.295218\pi$$
$$858$$ −1.94309e7 −0.901105
$$859$$ −1.98548e7 −0.918085 −0.459043 0.888414i $$-0.651807\pi$$
−0.459043 + 0.888414i $$0.651807\pi$$
$$860$$ 231456. 0.0106714
$$861$$ 2.47320e6 0.113698
$$862$$ −3.87115e7 −1.77448
$$863$$ 673056. 0.0307627 0.0153813 0.999882i $$-0.495104\pi$$
0.0153813 + 0.999882i $$0.495104\pi$$
$$864$$ 1.04976e6 0.0478416
$$865$$ 982188. 0.0446328
$$866$$ 2.49946e7 1.13254
$$867$$ 5.77740e6 0.261026
$$868$$ 704000. 0.0317156
$$869$$ 1.23629e7 0.555354
$$870$$ 1.50271e6 0.0673097
$$871$$ −1.47174e7 −0.657333
$$872$$ −1.03599e7 −0.461385
$$873$$ −4.00480e6 −0.177847
$$874$$ 0 0
$$875$$ 1.49136e6 0.0658510
$$876$$ 1.47996e6 0.0651613
$$877$$ −5.32115e6 −0.233618 −0.116809 0.993154i $$-0.537267\pi$$
−0.116809 + 0.993154i $$0.537267\pi$$
$$878$$ −4.75608e6 −0.208215
$$879$$ −8.44987e6 −0.368874
$$880$$ 3.84422e6 0.167341
$$881$$ 2.78891e7 1.21058 0.605291 0.796004i $$-0.293057\pi$$
0.605291 + 0.796004i $$0.293057\pi$$
$$882$$ −7.39060e6 −0.319896
$$883$$ −2.83786e7 −1.22487 −0.612435 0.790521i $$-0.709810\pi$$
−0.612435 + 0.790521i $$0.709810\pi$$
$$884$$ −2.25086e6 −0.0968765
$$885$$ −976536. −0.0419112
$$886$$ −8.39887e6 −0.359448
$$887$$ −4.22678e7 −1.80385 −0.901925 0.431893i $$-0.857846\pi$$
−0.901925 + 0.431893i $$0.857846\pi$$
$$888$$ 3.64392e6 0.155073
$$889$$ −6.84352e6 −0.290419
$$890$$ 3.38710e6 0.143335
$$891$$ −3.70040e6 −0.156155
$$892$$ 3.99789e6 0.168236
$$893$$ 0 0
$$894$$ 5.68393e6 0.237851
$$895$$ −2.15244e6 −0.0898201
$$896$$ −8.49408e6 −0.353465
$$897$$ −4.82328e6 −0.200153
$$898$$ −1.79549e7 −0.743005
$$899$$ 2.04072e7 0.842140
$$900$$ −1.00084e6 −0.0411867
$$901$$ −2.97675e7 −1.22160
$$902$$ −2.32481e7 −0.951417
$$903$$ 3.47184e6 0.141690
$$904$$ 1.76098e6 0.0716692
$$905$$ 3.04278e6 0.123495
$$906$$ 2.14052e7 0.866360
$$907$$ −3.19526e7 −1.28970 −0.644849 0.764310i $$-0.723080\pi$$
−0.644849 + 0.764310i $$0.723080\pi$$
$$908$$ −2.42472e6 −0.0975994
$$909$$ −1.15858e7 −0.465066
$$910$$ 918720. 0.0367773
$$911$$ 1.16429e7 0.464800 0.232400 0.972620i $$-0.425342\pi$$
0.232400 + 0.972620i $$0.425342\pi$$
$$912$$ 0 0
$$913$$ −4.65029e7 −1.84630
$$914$$ 3.77981e7 1.49660
$$915$$ −2.14693e6 −0.0847746
$$916$$ 5.43970e6 0.214208
$$917$$ −1.03387e7 −0.406016
$$918$$ −3.85787e6 −0.151092
$$919$$ 1.39844e6 0.0546204 0.0273102 0.999627i $$-0.491306\pi$$
0.0273102 + 0.999627i $$0.491306\pi$$
$$920$$ 846720. 0.0329815
$$921$$ 6.23653e6 0.242267
$$922$$ 2.04191e7 0.791059
$$923$$ −2.71022e6 −0.104713
$$924$$ −812160. −0.0312940
$$925$$ −7.44449e6 −0.286075
$$926$$ −1.33820e7 −0.512854
$$927$$ −4.30466e6 −0.164528
$$928$$ 6.67872e6 0.254579
$$929$$ −1.66792e7 −0.634067 −0.317033 0.948414i $$-0.602687\pi$$
−0.317033 + 0.948414i $$0.602687\pi$$
$$930$$ 1.42560e6 0.0540493
$$931$$ 0 0
$$932$$ −1.57154e6 −0.0592634
$$933$$ −2.64879e7 −0.996191
$$934$$ −3.90846e7 −1.46601
$$935$$ −2.98469e6 −0.111653
$$936$$ 8.68190e6 0.323911
$$937$$ −2.47956e7 −0.922625 −0.461312 0.887238i $$-0.652621\pi$$
−0.461312 + 0.887238i $$0.652621\pi$$
$$938$$ −5.53632e6 −0.205454
$$939$$ −7.96631e6 −0.294845
$$940$$ −448128. −0.0165418
$$941$$ −2.79574e7 −1.02925 −0.514627 0.857414i $$-0.672070\pi$$
−0.514627 + 0.857414i $$0.672070\pi$$
$$942$$ 7.43828e6 0.273115
$$943$$ −5.77080e6 −0.211328
$$944$$ −2.05434e7 −0.750314
$$945$$ 174960. 0.00637322
$$946$$ −3.26353e7 −1.18566
$$947$$ 7.64936e6 0.277173 0.138586 0.990350i $$-0.455744\pi$$
0.138586 + 0.990350i $$0.455744\pi$$
$$948$$ 789120. 0.0285182
$$949$$ 2.62282e7 0.945372
$$950$$ 0 0
$$951$$ 2.25792e7 0.809575
$$952$$ 5.92704e6 0.211956
$$953$$ 4.62179e7 1.64846 0.824228 0.566257i $$-0.191609\pi$$
0.824228 + 0.566257i $$0.191609\pi$$
$$954$$ −1.64025e7 −0.583498
$$955$$ −3.89030e6 −0.138031
$$956$$ −5.30054e6 −0.187575
$$957$$ −2.35425e7 −0.830945
$$958$$ 1.43539e7 0.505309
$$959$$ −1.20094e7 −0.421671
$$960$$ −1.49645e6 −0.0524063
$$961$$ −9.26915e6 −0.323766
$$962$$ −9.22548e6 −0.321404
$$963$$ −7.35707e6 −0.255646
$$964$$ 3.96038e6 0.137260
$$965$$ 167028. 0.00577392
$$966$$ −1.81440e6 −0.0625591
$$967$$ 2.08557e7 0.717229 0.358615 0.933486i $$-0.383249\pi$$
0.358615 + 0.933486i $$0.383249\pi$$
$$968$$ −2.63836e7 −0.904993
$$969$$ 0 0
$$970$$ −1.77991e6 −0.0607392
$$971$$ 4.58152e7 1.55941 0.779707 0.626144i $$-0.215368\pi$$
0.779707 + 0.626144i $$0.215368\pi$$
$$972$$ −236196. −0.00801875
$$973$$ 1.40066e7 0.474296
$$974$$ 3.67853e7 1.24245
$$975$$ −1.77370e7 −0.597544
$$976$$ −4.51651e7 −1.51767
$$977$$ 1.09544e6 0.0367157 0.0183578 0.999831i $$-0.494156\pi$$
0.0183578 + 0.999831i $$0.494156\pi$$
$$978$$ −1.90445e7 −0.636682
$$979$$ −5.30645e7 −1.76949
$$980$$ −364968. −0.0121392
$$981$$ 4.99495e6 0.165714
$$982$$ −7.41535e6 −0.245388
$$983$$ −5.25817e7 −1.73561 −0.867803 0.496909i $$-0.834468\pi$$
−0.867803 + 0.496909i $$0.834468\pi$$
$$984$$ 1.03874e7 0.341996
$$985$$ 3.66628e6 0.120402
$$986$$ −2.45443e7 −0.804004
$$987$$ −6.72192e6 −0.219634
$$988$$ 0 0
$$989$$ −8.10096e6 −0.263358
$$990$$ −1.64462e6 −0.0533309
$$991$$ 4.90389e7 1.58620 0.793098 0.609094i $$-0.208467\pi$$
0.793098 + 0.609094i $$0.208467\pi$$
$$992$$ 6.33600e6 0.204426
$$993$$ −1.94533e6 −0.0626067
$$994$$ −1.01952e6 −0.0327288
$$995$$ 5.27419e6 0.168888
$$996$$ −2.96827e6 −0.0948103
$$997$$ 3.05461e6 0.0973237 0.0486618 0.998815i $$-0.484504\pi$$
0.0486618 + 0.998815i $$0.484504\pi$$
$$998$$ −5.91297e7 −1.87923
$$999$$ −1.75689e6 −0.0556969
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 1083.6.a.c.1.1 1
19.18 odd 2 3.6.a.a.1.1 1
57.56 even 2 9.6.a.a.1.1 1
76.75 even 2 48.6.a.a.1.1 1
95.18 even 4 75.6.b.b.49.2 2
95.37 even 4 75.6.b.b.49.1 2
95.94 odd 2 75.6.a.e.1.1 1
133.18 odd 6 147.6.e.h.79.1 2
133.37 odd 6 147.6.e.h.67.1 2
133.75 even 6 147.6.e.k.67.1 2
133.94 even 6 147.6.e.k.79.1 2
133.132 even 2 147.6.a.a.1.1 1
152.37 odd 2 192.6.a.d.1.1 1
152.75 even 2 192.6.a.l.1.1 1
171.56 even 6 81.6.c.a.28.1 2
171.94 odd 6 81.6.c.c.55.1 2
171.113 even 6 81.6.c.a.55.1 2
171.151 odd 6 81.6.c.c.28.1 2
209.208 even 2 363.6.a.d.1.1 1
228.227 odd 2 144.6.a.f.1.1 1
247.246 odd 2 507.6.a.b.1.1 1
285.113 odd 4 225.6.b.b.199.1 2
285.227 odd 4 225.6.b.b.199.2 2
285.284 even 2 225.6.a.a.1.1 1
304.37 odd 4 768.6.d.k.385.1 2
304.75 even 4 768.6.d.h.385.2 2
304.189 odd 4 768.6.d.k.385.2 2
304.227 even 4 768.6.d.h.385.1 2
323.322 odd 2 867.6.a.a.1.1 1
399.398 odd 2 441.6.a.i.1.1 1
456.227 odd 2 576.6.a.t.1.1 1
456.341 even 2 576.6.a.s.1.1 1
627.626 odd 2 1089.6.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3.6.a.a.1.1 1 19.18 odd 2
9.6.a.a.1.1 1 57.56 even 2
48.6.a.a.1.1 1 76.75 even 2
75.6.a.e.1.1 1 95.94 odd 2
75.6.b.b.49.1 2 95.37 even 4
75.6.b.b.49.2 2 95.18 even 4
81.6.c.a.28.1 2 171.56 even 6
81.6.c.a.55.1 2 171.113 even 6
81.6.c.c.28.1 2 171.151 odd 6
81.6.c.c.55.1 2 171.94 odd 6
144.6.a.f.1.1 1 228.227 odd 2
147.6.a.a.1.1 1 133.132 even 2
147.6.e.h.67.1 2 133.37 odd 6
147.6.e.h.79.1 2 133.18 odd 6
147.6.e.k.67.1 2 133.75 even 6
147.6.e.k.79.1 2 133.94 even 6
192.6.a.d.1.1 1 152.37 odd 2
192.6.a.l.1.1 1 152.75 even 2
225.6.a.a.1.1 1 285.284 even 2
225.6.b.b.199.1 2 285.113 odd 4
225.6.b.b.199.2 2 285.227 odd 4
363.6.a.d.1.1 1 209.208 even 2
441.6.a.i.1.1 1 399.398 odd 2
507.6.a.b.1.1 1 247.246 odd 2
576.6.a.s.1.1 1 456.341 even 2
576.6.a.t.1.1 1 456.227 odd 2
768.6.d.h.385.1 2 304.227 even 4
768.6.d.h.385.2 2 304.75 even 4
768.6.d.k.385.1 2 304.37 odd 4
768.6.d.k.385.2 2 304.189 odd 4
867.6.a.a.1.1 1 323.322 odd 2
1083.6.a.c.1.1 1 1.1 even 1 trivial
1089.6.a.b.1.1 1 627.626 odd 2