# Properties

 Label 2112.4.a.u.1.1 Level $2112$ Weight $4$ Character 2112.1 Self dual yes Analytic conductor $124.612$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,4,Mod(1,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2112.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2112 = 2^{6} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2112.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: $$124.612033932$$ 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 $$\chi$$ $$=$$ 2112.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +4.00000 q^{5} -26.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +4.00000 q^{5} -26.0000 q^{7} +9.00000 q^{9} -11.0000 q^{11} +32.0000 q^{13} +12.0000 q^{15} +74.0000 q^{17} +60.0000 q^{19} -78.0000 q^{21} -182.000 q^{23} -109.000 q^{25} +27.0000 q^{27} +90.0000 q^{29} -8.00000 q^{31} -33.0000 q^{33} -104.000 q^{35} +66.0000 q^{37} +96.0000 q^{39} +422.000 q^{41} -408.000 q^{43} +36.0000 q^{45} -506.000 q^{47} +333.000 q^{49} +222.000 q^{51} -348.000 q^{53} -44.0000 q^{55} +180.000 q^{57} +200.000 q^{59} -132.000 q^{61} -234.000 q^{63} +128.000 q^{65} +1036.00 q^{67} -546.000 q^{69} +762.000 q^{71} -542.000 q^{73} -327.000 q^{75} +286.000 q^{77} -550.000 q^{79} +81.0000 q^{81} +132.000 q^{83} +296.000 q^{85} +270.000 q^{87} +570.000 q^{89} -832.000 q^{91} -24.0000 q^{93} +240.000 q^{95} +14.0000 q^{97} -99.0000 q^{99} +O(q^{100})$$

## 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
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ 4.00000 0.357771 0.178885 0.983870i $$-0.442751\pi$$
0.178885 + 0.983870i $$0.442751\pi$$
$$6$$ 0 0
$$7$$ −26.0000 −1.40387 −0.701934 0.712242i $$-0.747680\pi$$
−0.701934 + 0.712242i $$0.747680\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ 32.0000 0.682708 0.341354 0.939935i $$-0.389115\pi$$
0.341354 + 0.939935i $$0.389115\pi$$
$$14$$ 0 0
$$15$$ 12.0000 0.206559
$$16$$ 0 0
$$17$$ 74.0000 1.05574 0.527872 0.849324i $$-0.322990\pi$$
0.527872 + 0.849324i $$0.322990\pi$$
$$18$$ 0 0
$$19$$ 60.0000 0.724471 0.362235 0.932087i $$-0.382014\pi$$
0.362235 + 0.932087i $$0.382014\pi$$
$$20$$ 0 0
$$21$$ −78.0000 −0.810524
$$22$$ 0 0
$$23$$ −182.000 −1.64998 −0.824992 0.565145i $$-0.808820\pi$$
−0.824992 + 0.565145i $$0.808820\pi$$
$$24$$ 0 0
$$25$$ −109.000 −0.872000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 90.0000 0.576296 0.288148 0.957586i $$-0.406961\pi$$
0.288148 + 0.957586i $$0.406961\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −0.0463498 −0.0231749 0.999731i $$-0.507377\pi$$
−0.0231749 + 0.999731i $$0.507377\pi$$
$$32$$ 0 0
$$33$$ −33.0000 −0.174078
$$34$$ 0 0
$$35$$ −104.000 −0.502263
$$36$$ 0 0
$$37$$ 66.0000 0.293252 0.146626 0.989192i $$-0.453159\pi$$
0.146626 + 0.989192i $$0.453159\pi$$
$$38$$ 0 0
$$39$$ 96.0000 0.394162
$$40$$ 0 0
$$41$$ 422.000 1.60745 0.803724 0.595003i $$-0.202849\pi$$
0.803724 + 0.595003i $$0.202849\pi$$
$$42$$ 0 0
$$43$$ −408.000 −1.44696 −0.723482 0.690344i $$-0.757459\pi$$
−0.723482 + 0.690344i $$0.757459\pi$$
$$44$$ 0 0
$$45$$ 36.0000 0.119257
$$46$$ 0 0
$$47$$ −506.000 −1.57038 −0.785188 0.619257i $$-0.787434\pi$$
−0.785188 + 0.619257i $$0.787434\pi$$
$$48$$ 0 0
$$49$$ 333.000 0.970845
$$50$$ 0 0
$$51$$ 222.000 0.609534
$$52$$ 0 0
$$53$$ −348.000 −0.901915 −0.450957 0.892546i $$-0.648917\pi$$
−0.450957 + 0.892546i $$0.648917\pi$$
$$54$$ 0 0
$$55$$ −44.0000 −0.107872
$$56$$ 0 0
$$57$$ 180.000 0.418273
$$58$$ 0 0
$$59$$ 200.000 0.441318 0.220659 0.975351i $$-0.429179\pi$$
0.220659 + 0.975351i $$0.429179\pi$$
$$60$$ 0 0
$$61$$ −132.000 −0.277063 −0.138532 0.990358i $$-0.544238\pi$$
−0.138532 + 0.990358i $$0.544238\pi$$
$$62$$ 0 0
$$63$$ −234.000 −0.467956
$$64$$ 0 0
$$65$$ 128.000 0.244253
$$66$$ 0 0
$$67$$ 1036.00 1.88907 0.944534 0.328414i $$-0.106514\pi$$
0.944534 + 0.328414i $$0.106514\pi$$
$$68$$ 0 0
$$69$$ −546.000 −0.952618
$$70$$ 0 0
$$71$$ 762.000 1.27370 0.636850 0.770987i $$-0.280237\pi$$
0.636850 + 0.770987i $$0.280237\pi$$
$$72$$ 0 0
$$73$$ −542.000 −0.868990 −0.434495 0.900674i $$-0.643073\pi$$
−0.434495 + 0.900674i $$0.643073\pi$$
$$74$$ 0 0
$$75$$ −327.000 −0.503449
$$76$$ 0 0
$$77$$ 286.000 0.423282
$$78$$ 0 0
$$79$$ −550.000 −0.783289 −0.391645 0.920117i $$-0.628094\pi$$
−0.391645 + 0.920117i $$0.628094\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 132.000 0.174565 0.0872824 0.996184i $$-0.472182\pi$$
0.0872824 + 0.996184i $$0.472182\pi$$
$$84$$ 0 0
$$85$$ 296.000 0.377714
$$86$$ 0 0
$$87$$ 270.000 0.332725
$$88$$ 0 0
$$89$$ 570.000 0.678875 0.339438 0.940629i $$-0.389763\pi$$
0.339438 + 0.940629i $$0.389763\pi$$
$$90$$ 0 0
$$91$$ −832.000 −0.958432
$$92$$ 0 0
$$93$$ −24.0000 −0.0267600
$$94$$ 0 0
$$95$$ 240.000 0.259195
$$96$$ 0 0
$$97$$ 14.0000 0.0146545 0.00732724 0.999973i $$-0.497668\pi$$
0.00732724 + 0.999973i $$0.497668\pi$$
$$98$$ 0 0
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ −1702.00 −1.67679 −0.838393 0.545067i $$-0.816504\pi$$
−0.838393 + 0.545067i $$0.816504\pi$$
$$102$$ 0 0
$$103$$ −1132.00 −1.08291 −0.541453 0.840731i $$-0.682126\pi$$
−0.541453 + 0.840731i $$0.682126\pi$$
$$104$$ 0 0
$$105$$ −312.000 −0.289982
$$106$$ 0 0
$$107$$ −564.000 −0.509570 −0.254785 0.966998i $$-0.582005\pi$$
−0.254785 + 0.966998i $$0.582005\pi$$
$$108$$ 0 0
$$109$$ 320.000 0.281197 0.140598 0.990067i $$-0.455097\pi$$
0.140598 + 0.990067i $$0.455097\pi$$
$$110$$ 0 0
$$111$$ 198.000 0.169309
$$112$$ 0 0
$$113$$ −2142.00 −1.78321 −0.891604 0.452817i $$-0.850419\pi$$
−0.891604 + 0.452817i $$0.850419\pi$$
$$114$$ 0 0
$$115$$ −728.000 −0.590316
$$116$$ 0 0
$$117$$ 288.000 0.227569
$$118$$ 0 0
$$119$$ −1924.00 −1.48212
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ 1266.00 0.928060
$$124$$ 0 0
$$125$$ −936.000 −0.669747
$$126$$ 0 0
$$127$$ −1606.00 −1.12212 −0.561061 0.827775i $$-0.689607\pi$$
−0.561061 + 0.827775i $$0.689607\pi$$
$$128$$ 0 0
$$129$$ −1224.00 −0.835405
$$130$$ 0 0
$$131$$ 1908.00 1.27254 0.636270 0.771466i $$-0.280476\pi$$
0.636270 + 0.771466i $$0.280476\pi$$
$$132$$ 0 0
$$133$$ −1560.00 −1.01706
$$134$$ 0 0
$$135$$ 108.000 0.0688530
$$136$$ 0 0
$$137$$ −2186.00 −1.36323 −0.681615 0.731711i $$-0.738722\pi$$
−0.681615 + 0.731711i $$0.738722\pi$$
$$138$$ 0 0
$$139$$ −2740.00 −1.67197 −0.835985 0.548753i $$-0.815103\pi$$
−0.835985 + 0.548753i $$0.815103\pi$$
$$140$$ 0 0
$$141$$ −1518.00 −0.906657
$$142$$ 0 0
$$143$$ −352.000 −0.205844
$$144$$ 0 0
$$145$$ 360.000 0.206182
$$146$$ 0 0
$$147$$ 999.000 0.560518
$$148$$ 0 0
$$149$$ 1310.00 0.720264 0.360132 0.932901i $$-0.382732\pi$$
0.360132 + 0.932901i $$0.382732\pi$$
$$150$$ 0 0
$$151$$ −1198.00 −0.645641 −0.322821 0.946460i $$-0.604631\pi$$
−0.322821 + 0.946460i $$0.604631\pi$$
$$152$$ 0 0
$$153$$ 666.000 0.351914
$$154$$ 0 0
$$155$$ −32.0000 −0.0165826
$$156$$ 0 0
$$157$$ −2114.00 −1.07462 −0.537311 0.843384i $$-0.680560\pi$$
−0.537311 + 0.843384i $$0.680560\pi$$
$$158$$ 0 0
$$159$$ −1044.00 −0.520721
$$160$$ 0 0
$$161$$ 4732.00 2.31636
$$162$$ 0 0
$$163$$ −3868.00 −1.85868 −0.929341 0.369223i $$-0.879624\pi$$
−0.929341 + 0.369223i $$0.879624\pi$$
$$164$$ 0 0
$$165$$ −132.000 −0.0622799
$$166$$ 0 0
$$167$$ 2004.00 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −1173.00 −0.533910
$$170$$ 0 0
$$171$$ 540.000 0.241490
$$172$$ 0 0
$$173$$ −678.000 −0.297962 −0.148981 0.988840i $$-0.547599\pi$$
−0.148981 + 0.988840i $$0.547599\pi$$
$$174$$ 0 0
$$175$$ 2834.00 1.22417
$$176$$ 0 0
$$177$$ 600.000 0.254795
$$178$$ 0 0
$$179$$ 1680.00 0.701503 0.350752 0.936469i $$-0.385926\pi$$
0.350752 + 0.936469i $$0.385926\pi$$
$$180$$ 0 0
$$181$$ 4358.00 1.78966 0.894828 0.446412i $$-0.147298\pi$$
0.894828 + 0.446412i $$0.147298\pi$$
$$182$$ 0 0
$$183$$ −396.000 −0.159963
$$184$$ 0 0
$$185$$ 264.000 0.104917
$$186$$ 0 0
$$187$$ −814.000 −0.318319
$$188$$ 0 0
$$189$$ −702.000 −0.270175
$$190$$ 0 0
$$191$$ −1778.00 −0.673568 −0.336784 0.941582i $$-0.609339\pi$$
−0.336784 + 0.941582i $$0.609339\pi$$
$$192$$ 0 0
$$193$$ −3962.00 −1.47767 −0.738837 0.673884i $$-0.764625\pi$$
−0.738837 + 0.673884i $$0.764625\pi$$
$$194$$ 0 0
$$195$$ 384.000 0.141020
$$196$$ 0 0
$$197$$ −374.000 −0.135261 −0.0676304 0.997710i $$-0.521544\pi$$
−0.0676304 + 0.997710i $$0.521544\pi$$
$$198$$ 0 0
$$199$$ 2100.00 0.748066 0.374033 0.927415i $$-0.377975\pi$$
0.374033 + 0.927415i $$0.377975\pi$$
$$200$$ 0 0
$$201$$ 3108.00 1.09065
$$202$$ 0 0
$$203$$ −2340.00 −0.809043
$$204$$ 0 0
$$205$$ 1688.00 0.575098
$$206$$ 0 0
$$207$$ −1638.00 −0.549995
$$208$$ 0 0
$$209$$ −660.000 −0.218436
$$210$$ 0 0
$$211$$ −2232.00 −0.728233 −0.364117 0.931353i $$-0.618629\pi$$
−0.364117 + 0.931353i $$0.618629\pi$$
$$212$$ 0 0
$$213$$ 2286.00 0.735372
$$214$$ 0 0
$$215$$ −1632.00 −0.517681
$$216$$ 0 0
$$217$$ 208.000 0.0650689
$$218$$ 0 0
$$219$$ −1626.00 −0.501712
$$220$$ 0 0
$$221$$ 2368.00 0.720764
$$222$$ 0 0
$$223$$ 2128.00 0.639020 0.319510 0.947583i $$-0.396482\pi$$
0.319510 + 0.947583i $$0.396482\pi$$
$$224$$ 0 0
$$225$$ −981.000 −0.290667
$$226$$ 0 0
$$227$$ −2964.00 −0.866641 −0.433321 0.901240i $$-0.642658\pi$$
−0.433321 + 0.901240i $$0.642658\pi$$
$$228$$ 0 0
$$229$$ 2550.00 0.735846 0.367923 0.929856i $$-0.380069\pi$$
0.367923 + 0.929856i $$0.380069\pi$$
$$230$$ 0 0
$$231$$ 858.000 0.244382
$$232$$ 0 0
$$233$$ −3042.00 −0.855314 −0.427657 0.903941i $$-0.640661\pi$$
−0.427657 + 0.903941i $$0.640661\pi$$
$$234$$ 0 0
$$235$$ −2024.00 −0.561835
$$236$$ 0 0
$$237$$ −1650.00 −0.452232
$$238$$ 0 0
$$239$$ 2700.00 0.730747 0.365373 0.930861i $$-0.380941\pi$$
0.365373 + 0.930861i $$0.380941\pi$$
$$240$$ 0 0
$$241$$ −578.000 −0.154491 −0.0772453 0.997012i $$-0.524612\pi$$
−0.0772453 + 0.997012i $$0.524612\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 1332.00 0.347340
$$246$$ 0 0
$$247$$ 1920.00 0.494602
$$248$$ 0 0
$$249$$ 396.000 0.100785
$$250$$ 0 0
$$251$$ −3752.00 −0.943522 −0.471761 0.881726i $$-0.656382\pi$$
−0.471761 + 0.881726i $$0.656382\pi$$
$$252$$ 0 0
$$253$$ 2002.00 0.497489
$$254$$ 0 0
$$255$$ 888.000 0.218073
$$256$$ 0 0
$$257$$ 674.000 0.163591 0.0817957 0.996649i $$-0.473935\pi$$
0.0817957 + 0.996649i $$0.473935\pi$$
$$258$$ 0 0
$$259$$ −1716.00 −0.411687
$$260$$ 0 0
$$261$$ 810.000 0.192099
$$262$$ 0 0
$$263$$ −4352.00 −1.02036 −0.510182 0.860066i $$-0.670422\pi$$
−0.510182 + 0.860066i $$0.670422\pi$$
$$264$$ 0 0
$$265$$ −1392.00 −0.322679
$$266$$ 0 0
$$267$$ 1710.00 0.391949
$$268$$ 0 0
$$269$$ −500.000 −0.113329 −0.0566646 0.998393i $$-0.518047\pi$$
−0.0566646 + 0.998393i $$0.518047\pi$$
$$270$$ 0 0
$$271$$ −6538.00 −1.46552 −0.732759 0.680489i $$-0.761768\pi$$
−0.732759 + 0.680489i $$0.761768\pi$$
$$272$$ 0 0
$$273$$ −2496.00 −0.553351
$$274$$ 0 0
$$275$$ 1199.00 0.262918
$$276$$ 0 0
$$277$$ −124.000 −0.0268969 −0.0134484 0.999910i $$-0.504281\pi$$
−0.0134484 + 0.999910i $$0.504281\pi$$
$$278$$ 0 0
$$279$$ −72.0000 −0.0154499
$$280$$ 0 0
$$281$$ 3642.00 0.773180 0.386590 0.922252i $$-0.373653\pi$$
0.386590 + 0.922252i $$0.373653\pi$$
$$282$$ 0 0
$$283$$ −4648.00 −0.976307 −0.488154 0.872758i $$-0.662329\pi$$
−0.488154 + 0.872758i $$0.662329\pi$$
$$284$$ 0 0
$$285$$ 720.000 0.149646
$$286$$ 0 0
$$287$$ −10972.0 −2.25664
$$288$$ 0 0
$$289$$ 563.000 0.114594
$$290$$ 0 0
$$291$$ 42.0000 0.00846077
$$292$$ 0 0
$$293$$ 3102.00 0.618501 0.309250 0.950981i $$-0.399922\pi$$
0.309250 + 0.950981i $$0.399922\pi$$
$$294$$ 0 0
$$295$$ 800.000 0.157891
$$296$$ 0 0
$$297$$ −297.000 −0.0580259
$$298$$ 0 0
$$299$$ −5824.00 −1.12646
$$300$$ 0 0
$$301$$ 10608.0 2.03135
$$302$$ 0 0
$$303$$ −5106.00 −0.968093
$$304$$ 0 0
$$305$$ −528.000 −0.0991252
$$306$$ 0 0
$$307$$ −1244.00 −0.231267 −0.115633 0.993292i $$-0.536890\pi$$
−0.115633 + 0.993292i $$0.536890\pi$$
$$308$$ 0 0
$$309$$ −3396.00 −0.625216
$$310$$ 0 0
$$311$$ 2082.00 0.379612 0.189806 0.981822i $$-0.439214\pi$$
0.189806 + 0.981822i $$0.439214\pi$$
$$312$$ 0 0
$$313$$ 2378.00 0.429433 0.214716 0.976676i $$-0.431117\pi$$
0.214716 + 0.976676i $$0.431117\pi$$
$$314$$ 0 0
$$315$$ −936.000 −0.167421
$$316$$ 0 0
$$317$$ 496.000 0.0878806 0.0439403 0.999034i $$-0.486009\pi$$
0.0439403 + 0.999034i $$0.486009\pi$$
$$318$$ 0 0
$$319$$ −990.000 −0.173760
$$320$$ 0 0
$$321$$ −1692.00 −0.294200
$$322$$ 0 0
$$323$$ 4440.00 0.764855
$$324$$ 0 0
$$325$$ −3488.00 −0.595321
$$326$$ 0 0
$$327$$ 960.000 0.162349
$$328$$ 0 0
$$329$$ 13156.0 2.20460
$$330$$ 0 0
$$331$$ 2708.00 0.449683 0.224842 0.974395i $$-0.427814\pi$$
0.224842 + 0.974395i $$0.427814\pi$$
$$332$$ 0 0
$$333$$ 594.000 0.0977507
$$334$$ 0 0
$$335$$ 4144.00 0.675853
$$336$$ 0 0
$$337$$ 4034.00 0.652065 0.326033 0.945359i $$-0.394288\pi$$
0.326033 + 0.945359i $$0.394288\pi$$
$$338$$ 0 0
$$339$$ −6426.00 −1.02954
$$340$$ 0 0
$$341$$ 88.0000 0.0139750
$$342$$ 0 0
$$343$$ 260.000 0.0409291
$$344$$ 0 0
$$345$$ −2184.00 −0.340819
$$346$$ 0 0
$$347$$ −11084.0 −1.71476 −0.857378 0.514687i $$-0.827908\pi$$
−0.857378 + 0.514687i $$0.827908\pi$$
$$348$$ 0 0
$$349$$ 3120.00 0.478538 0.239269 0.970953i $$-0.423092\pi$$
0.239269 + 0.970953i $$0.423092\pi$$
$$350$$ 0 0
$$351$$ 864.000 0.131387
$$352$$ 0 0
$$353$$ −5622.00 −0.847674 −0.423837 0.905739i $$-0.639317\pi$$
−0.423837 + 0.905739i $$0.639317\pi$$
$$354$$ 0 0
$$355$$ 3048.00 0.455693
$$356$$ 0 0
$$357$$ −5772.00 −0.855705
$$358$$ 0 0
$$359$$ −8500.00 −1.24962 −0.624809 0.780778i $$-0.714823\pi$$
−0.624809 + 0.780778i $$0.714823\pi$$
$$360$$ 0 0
$$361$$ −3259.00 −0.475142
$$362$$ 0 0
$$363$$ 363.000 0.0524864
$$364$$ 0 0
$$365$$ −2168.00 −0.310899
$$366$$ 0 0
$$367$$ 7144.00 1.01611 0.508057 0.861324i $$-0.330364\pi$$
0.508057 + 0.861324i $$0.330364\pi$$
$$368$$ 0 0
$$369$$ 3798.00 0.535816
$$370$$ 0 0
$$371$$ 9048.00 1.26617
$$372$$ 0 0
$$373$$ 632.000 0.0877312 0.0438656 0.999037i $$-0.486033\pi$$
0.0438656 + 0.999037i $$0.486033\pi$$
$$374$$ 0 0
$$375$$ −2808.00 −0.386679
$$376$$ 0 0
$$377$$ 2880.00 0.393442
$$378$$ 0 0
$$379$$ 4220.00 0.571944 0.285972 0.958238i $$-0.407684\pi$$
0.285972 + 0.958238i $$0.407684\pi$$
$$380$$ 0 0
$$381$$ −4818.00 −0.647857
$$382$$ 0 0
$$383$$ 8458.00 1.12842 0.564208 0.825632i $$-0.309181\pi$$
0.564208 + 0.825632i $$0.309181\pi$$
$$384$$ 0 0
$$385$$ 1144.00 0.151438
$$386$$ 0 0
$$387$$ −3672.00 −0.482321
$$388$$ 0 0
$$389$$ −1740.00 −0.226790 −0.113395 0.993550i $$-0.536173\pi$$
−0.113395 + 0.993550i $$0.536173\pi$$
$$390$$ 0 0
$$391$$ −13468.0 −1.74196
$$392$$ 0 0
$$393$$ 5724.00 0.734701
$$394$$ 0 0
$$395$$ −2200.00 −0.280238
$$396$$ 0 0
$$397$$ 5126.00 0.648027 0.324013 0.946053i $$-0.394968\pi$$
0.324013 + 0.946053i $$0.394968\pi$$
$$398$$ 0 0
$$399$$ −4680.00 −0.587201
$$400$$ 0 0
$$401$$ −3098.00 −0.385802 −0.192901 0.981218i $$-0.561790\pi$$
−0.192901 + 0.981218i $$0.561790\pi$$
$$402$$ 0 0
$$403$$ −256.000 −0.0316433
$$404$$ 0 0
$$405$$ 324.000 0.0397523
$$406$$ 0 0
$$407$$ −726.000 −0.0884189
$$408$$ 0 0
$$409$$ 6390.00 0.772531 0.386265 0.922388i $$-0.373765\pi$$
0.386265 + 0.922388i $$0.373765\pi$$
$$410$$ 0 0
$$411$$ −6558.00 −0.787062
$$412$$ 0 0
$$413$$ −5200.00 −0.619553
$$414$$ 0 0
$$415$$ 528.000 0.0624542
$$416$$ 0 0
$$417$$ −8220.00 −0.965312
$$418$$ 0 0
$$419$$ −9760.00 −1.13796 −0.568982 0.822350i $$-0.692663\pi$$
−0.568982 + 0.822350i $$0.692663\pi$$
$$420$$ 0 0
$$421$$ 5138.00 0.594800 0.297400 0.954753i $$-0.403881\pi$$
0.297400 + 0.954753i $$0.403881\pi$$
$$422$$ 0 0
$$423$$ −4554.00 −0.523459
$$424$$ 0 0
$$425$$ −8066.00 −0.920608
$$426$$ 0 0
$$427$$ 3432.00 0.388960
$$428$$ 0 0
$$429$$ −1056.00 −0.118844
$$430$$ 0 0
$$431$$ −7008.00 −0.783210 −0.391605 0.920133i $$-0.628080\pi$$
−0.391605 + 0.920133i $$0.628080\pi$$
$$432$$ 0 0
$$433$$ 5578.00 0.619080 0.309540 0.950886i $$-0.399825\pi$$
0.309540 + 0.950886i $$0.399825\pi$$
$$434$$ 0 0
$$435$$ 1080.00 0.119039
$$436$$ 0 0
$$437$$ −10920.0 −1.19536
$$438$$ 0 0
$$439$$ −10430.0 −1.13393 −0.566967 0.823741i $$-0.691883\pi$$
−0.566967 + 0.823741i $$0.691883\pi$$
$$440$$ 0 0
$$441$$ 2997.00 0.323615
$$442$$ 0 0
$$443$$ 4432.00 0.475329 0.237664 0.971347i $$-0.423618\pi$$
0.237664 + 0.971347i $$0.423618\pi$$
$$444$$ 0 0
$$445$$ 2280.00 0.242882
$$446$$ 0 0
$$447$$ 3930.00 0.415845
$$448$$ 0 0
$$449$$ −6290.00 −0.661121 −0.330561 0.943785i $$-0.607238\pi$$
−0.330561 + 0.943785i $$0.607238\pi$$
$$450$$ 0 0
$$451$$ −4642.00 −0.484664
$$452$$ 0 0
$$453$$ −3594.00 −0.372761
$$454$$ 0 0
$$455$$ −3328.00 −0.342899
$$456$$ 0 0
$$457$$ 3054.00 0.312604 0.156302 0.987709i $$-0.450043\pi$$
0.156302 + 0.987709i $$0.450043\pi$$
$$458$$ 0 0
$$459$$ 1998.00 0.203178
$$460$$ 0 0
$$461$$ −12882.0 −1.30146 −0.650732 0.759308i $$-0.725538\pi$$
−0.650732 + 0.759308i $$0.725538\pi$$
$$462$$ 0 0
$$463$$ 6148.00 0.617110 0.308555 0.951207i $$-0.400155\pi$$
0.308555 + 0.951207i $$0.400155\pi$$
$$464$$ 0 0
$$465$$ −96.0000 −0.00957396
$$466$$ 0 0
$$467$$ −5124.00 −0.507731 −0.253866 0.967240i $$-0.581702\pi$$
−0.253866 + 0.967240i $$0.581702\pi$$
$$468$$ 0 0
$$469$$ −26936.0 −2.65200
$$470$$ 0 0
$$471$$ −6342.00 −0.620433
$$472$$ 0 0
$$473$$ 4488.00 0.436276
$$474$$ 0 0
$$475$$ −6540.00 −0.631738
$$476$$ 0 0
$$477$$ −3132.00 −0.300638
$$478$$ 0 0
$$479$$ −16520.0 −1.57582 −0.787910 0.615790i $$-0.788837\pi$$
−0.787910 + 0.615790i $$0.788837\pi$$
$$480$$ 0 0
$$481$$ 2112.00 0.200206
$$482$$ 0 0
$$483$$ 14196.0 1.33735
$$484$$ 0 0
$$485$$ 56.0000 0.00524295
$$486$$ 0 0
$$487$$ 524.000 0.0487571 0.0243785 0.999703i $$-0.492239\pi$$
0.0243785 + 0.999703i $$0.492239\pi$$
$$488$$ 0 0
$$489$$ −11604.0 −1.07311
$$490$$ 0 0
$$491$$ 15028.0 1.38127 0.690636 0.723203i $$-0.257331\pi$$
0.690636 + 0.723203i $$0.257331\pi$$
$$492$$ 0 0
$$493$$ 6660.00 0.608421
$$494$$ 0 0
$$495$$ −396.000 −0.0359573
$$496$$ 0 0
$$497$$ −19812.0 −1.78811
$$498$$ 0 0
$$499$$ −9020.00 −0.809200 −0.404600 0.914494i $$-0.632589\pi$$
−0.404600 + 0.914494i $$0.632589\pi$$
$$500$$ 0 0
$$501$$ 6012.00 0.536120
$$502$$ 0 0
$$503$$ −14812.0 −1.31299 −0.656495 0.754330i $$-0.727962\pi$$
−0.656495 + 0.754330i $$0.727962\pi$$
$$504$$ 0 0
$$505$$ −6808.00 −0.599905
$$506$$ 0 0
$$507$$ −3519.00 −0.308253
$$508$$ 0 0
$$509$$ −12660.0 −1.10245 −0.551223 0.834358i $$-0.685839\pi$$
−0.551223 + 0.834358i $$0.685839\pi$$
$$510$$ 0 0
$$511$$ 14092.0 1.21995
$$512$$ 0 0
$$513$$ 1620.00 0.139424
$$514$$ 0 0
$$515$$ −4528.00 −0.387432
$$516$$ 0 0
$$517$$ 5566.00 0.473486
$$518$$ 0 0
$$519$$ −2034.00 −0.172028
$$520$$ 0 0
$$521$$ −3738.00 −0.314328 −0.157164 0.987573i $$-0.550235\pi$$
−0.157164 + 0.987573i $$0.550235\pi$$
$$522$$ 0 0
$$523$$ 6352.00 0.531078 0.265539 0.964100i $$-0.414450\pi$$
0.265539 + 0.964100i $$0.414450\pi$$
$$524$$ 0 0
$$525$$ 8502.00 0.706777
$$526$$ 0 0
$$527$$ −592.000 −0.0489334
$$528$$ 0 0
$$529$$ 20957.0 1.72245
$$530$$ 0 0
$$531$$ 1800.00 0.147106
$$532$$ 0 0
$$533$$ 13504.0 1.09742
$$534$$ 0 0
$$535$$ −2256.00 −0.182309
$$536$$ 0 0
$$537$$ 5040.00 0.405013
$$538$$ 0 0
$$539$$ −3663.00 −0.292721
$$540$$ 0 0
$$541$$ 24728.0 1.96514 0.982569 0.185898i $$-0.0595193\pi$$
0.982569 + 0.185898i $$0.0595193\pi$$
$$542$$ 0 0
$$543$$ 13074.0 1.03326
$$544$$ 0 0
$$545$$ 1280.00 0.100604
$$546$$ 0 0
$$547$$ 22756.0 1.77875 0.889375 0.457178i $$-0.151140\pi$$
0.889375 + 0.457178i $$0.151140\pi$$
$$548$$ 0 0
$$549$$ −1188.00 −0.0923545
$$550$$ 0 0
$$551$$ 5400.00 0.417509
$$552$$ 0 0
$$553$$ 14300.0 1.09963
$$554$$ 0 0
$$555$$ 792.000 0.0605739
$$556$$ 0 0
$$557$$ 9526.00 0.724649 0.362325 0.932052i $$-0.381983\pi$$
0.362325 + 0.932052i $$0.381983\pi$$
$$558$$ 0 0
$$559$$ −13056.0 −0.987853
$$560$$ 0 0
$$561$$ −2442.00 −0.183781
$$562$$ 0 0
$$563$$ −12068.0 −0.903385 −0.451692 0.892174i $$-0.649180\pi$$
−0.451692 + 0.892174i $$0.649180\pi$$
$$564$$ 0 0
$$565$$ −8568.00 −0.637980
$$566$$ 0 0
$$567$$ −2106.00 −0.155985
$$568$$ 0 0
$$569$$ 15090.0 1.11179 0.555893 0.831254i $$-0.312377\pi$$
0.555893 + 0.831254i $$0.312377\pi$$
$$570$$ 0 0
$$571$$ −4412.00 −0.323356 −0.161678 0.986844i $$-0.551691\pi$$
−0.161678 + 0.986844i $$0.551691\pi$$
$$572$$ 0 0
$$573$$ −5334.00 −0.388885
$$574$$ 0 0
$$575$$ 19838.0 1.43879
$$576$$ 0 0
$$577$$ −3906.00 −0.281818 −0.140909 0.990023i $$-0.545002\pi$$
−0.140909 + 0.990023i $$0.545002\pi$$
$$578$$ 0 0
$$579$$ −11886.0 −0.853135
$$580$$ 0 0
$$581$$ −3432.00 −0.245066
$$582$$ 0 0
$$583$$ 3828.00 0.271937
$$584$$ 0 0
$$585$$ 1152.00 0.0814177
$$586$$ 0 0
$$587$$ 12016.0 0.844895 0.422448 0.906387i $$-0.361171\pi$$
0.422448 + 0.906387i $$0.361171\pi$$
$$588$$ 0 0
$$589$$ −480.000 −0.0335790
$$590$$ 0 0
$$591$$ −1122.00 −0.0780929
$$592$$ 0 0
$$593$$ −11342.0 −0.785430 −0.392715 0.919660i $$-0.628464\pi$$
−0.392715 + 0.919660i $$0.628464\pi$$
$$594$$ 0 0
$$595$$ −7696.00 −0.530261
$$596$$ 0 0
$$597$$ 6300.00 0.431896
$$598$$ 0 0
$$599$$ 20690.0 1.41130 0.705651 0.708559i $$-0.250654\pi$$
0.705651 + 0.708559i $$0.250654\pi$$
$$600$$ 0 0
$$601$$ −598.000 −0.0405872 −0.0202936 0.999794i $$-0.506460\pi$$
−0.0202936 + 0.999794i $$0.506460\pi$$
$$602$$ 0 0
$$603$$ 9324.00 0.629689
$$604$$ 0 0
$$605$$ 484.000 0.0325246
$$606$$ 0 0
$$607$$ −166.000 −0.0111001 −0.00555003 0.999985i $$-0.501767\pi$$
−0.00555003 + 0.999985i $$0.501767\pi$$
$$608$$ 0 0
$$609$$ −7020.00 −0.467101
$$610$$ 0 0
$$611$$ −16192.0 −1.07211
$$612$$ 0 0
$$613$$ −20108.0 −1.32488 −0.662442 0.749113i $$-0.730480\pi$$
−0.662442 + 0.749113i $$0.730480\pi$$
$$614$$ 0 0
$$615$$ 5064.00 0.332033
$$616$$ 0 0
$$617$$ −2286.00 −0.149159 −0.0745793 0.997215i $$-0.523761\pi$$
−0.0745793 + 0.997215i $$0.523761\pi$$
$$618$$ 0 0
$$619$$ 25660.0 1.66618 0.833088 0.553141i $$-0.186571\pi$$
0.833088 + 0.553141i $$0.186571\pi$$
$$620$$ 0 0
$$621$$ −4914.00 −0.317539
$$622$$ 0 0
$$623$$ −14820.0 −0.953051
$$624$$ 0 0
$$625$$ 9881.00 0.632384
$$626$$ 0 0
$$627$$ −1980.00 −0.126114
$$628$$ 0 0
$$629$$ 4884.00 0.309599
$$630$$ 0 0
$$631$$ −11408.0 −0.719723 −0.359862 0.933006i $$-0.617176\pi$$
−0.359862 + 0.933006i $$0.617176\pi$$
$$632$$ 0 0
$$633$$ −6696.00 −0.420446
$$634$$ 0 0
$$635$$ −6424.00 −0.401462
$$636$$ 0 0
$$637$$ 10656.0 0.662804
$$638$$ 0 0
$$639$$ 6858.00 0.424567
$$640$$ 0 0
$$641$$ −3378.00 −0.208148 −0.104074 0.994570i $$-0.533188\pi$$
−0.104074 + 0.994570i $$0.533188\pi$$
$$642$$ 0 0
$$643$$ 11212.0 0.687649 0.343824 0.939034i $$-0.388278\pi$$
0.343824 + 0.939034i $$0.388278\pi$$
$$644$$ 0 0
$$645$$ −4896.00 −0.298883
$$646$$ 0 0
$$647$$ −86.0000 −0.00522567 −0.00261284 0.999997i $$-0.500832\pi$$
−0.00261284 + 0.999997i $$0.500832\pi$$
$$648$$ 0 0
$$649$$ −2200.00 −0.133062
$$650$$ 0 0
$$651$$ 624.000 0.0375676
$$652$$ 0 0
$$653$$ 4432.00 0.265601 0.132801 0.991143i $$-0.457603\pi$$
0.132801 + 0.991143i $$0.457603\pi$$
$$654$$ 0 0
$$655$$ 7632.00 0.455278
$$656$$ 0 0
$$657$$ −4878.00 −0.289663
$$658$$ 0 0
$$659$$ −4580.00 −0.270731 −0.135365 0.990796i $$-0.543221\pi$$
−0.135365 + 0.990796i $$0.543221\pi$$
$$660$$ 0 0
$$661$$ −4282.00 −0.251967 −0.125984 0.992032i $$-0.540209\pi$$
−0.125984 + 0.992032i $$0.540209\pi$$
$$662$$ 0 0
$$663$$ 7104.00 0.416133
$$664$$ 0 0
$$665$$ −6240.00 −0.363875
$$666$$ 0 0
$$667$$ −16380.0 −0.950879
$$668$$ 0 0
$$669$$ 6384.00 0.368938
$$670$$ 0 0
$$671$$ 1452.00 0.0835378
$$672$$ 0 0
$$673$$ 8438.00 0.483300 0.241650 0.970363i $$-0.422311\pi$$
0.241650 + 0.970363i $$0.422311\pi$$
$$674$$ 0 0
$$675$$ −2943.00 −0.167816
$$676$$ 0 0
$$677$$ −34494.0 −1.95822 −0.979108 0.203341i $$-0.934820\pi$$
−0.979108 + 0.203341i $$0.934820\pi$$
$$678$$ 0 0
$$679$$ −364.000 −0.0205730
$$680$$ 0 0
$$681$$ −8892.00 −0.500356
$$682$$ 0 0
$$683$$ 13712.0 0.768192 0.384096 0.923293i $$-0.374513\pi$$
0.384096 + 0.923293i $$0.374513\pi$$
$$684$$ 0 0
$$685$$ −8744.00 −0.487724
$$686$$ 0 0
$$687$$ 7650.00 0.424841
$$688$$ 0 0
$$689$$ −11136.0 −0.615744
$$690$$ 0 0
$$691$$ −11372.0 −0.626066 −0.313033 0.949742i $$-0.601345\pi$$
−0.313033 + 0.949742i $$0.601345\pi$$
$$692$$ 0 0
$$693$$ 2574.00 0.141094
$$694$$ 0 0
$$695$$ −10960.0 −0.598182
$$696$$ 0 0
$$697$$ 31228.0 1.69705
$$698$$ 0 0
$$699$$ −9126.00 −0.493815
$$700$$ 0 0
$$701$$ 6398.00 0.344721 0.172360 0.985034i $$-0.444861\pi$$
0.172360 + 0.985034i $$0.444861\pi$$
$$702$$ 0 0
$$703$$ 3960.00 0.212453
$$704$$ 0 0
$$705$$ −6072.00 −0.324376
$$706$$ 0 0
$$707$$ 44252.0 2.35399
$$708$$ 0 0
$$709$$ 5830.00 0.308816 0.154408 0.988007i $$-0.450653\pi$$
0.154408 + 0.988007i $$0.450653\pi$$
$$710$$ 0 0
$$711$$ −4950.00 −0.261096
$$712$$ 0 0
$$713$$ 1456.00 0.0764763
$$714$$ 0 0
$$715$$ −1408.00 −0.0736451
$$716$$ 0 0
$$717$$ 8100.00 0.421897
$$718$$ 0 0
$$719$$ 34530.0 1.79103 0.895516 0.445030i $$-0.146807\pi$$
0.895516 + 0.445030i $$0.146807\pi$$
$$720$$ 0 0
$$721$$ 29432.0 1.52026
$$722$$ 0 0
$$723$$ −1734.00 −0.0891952
$$724$$ 0 0
$$725$$ −9810.00 −0.502530
$$726$$ 0 0
$$727$$ −17316.0 −0.883377 −0.441688 0.897169i $$-0.645620\pi$$
−0.441688 + 0.897169i $$0.645620\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −30192.0 −1.52762
$$732$$ 0 0
$$733$$ 27072.0 1.36416 0.682079 0.731279i $$-0.261076\pi$$
0.682079 + 0.731279i $$0.261076\pi$$
$$734$$ 0 0
$$735$$ 3996.00 0.200537
$$736$$ 0 0
$$737$$ −11396.0 −0.569575
$$738$$ 0 0
$$739$$ 17320.0 0.862147 0.431073 0.902317i $$-0.358135\pi$$
0.431073 + 0.902317i $$0.358135\pi$$
$$740$$ 0 0
$$741$$ 5760.00 0.285559
$$742$$ 0 0
$$743$$ 14588.0 0.720299 0.360149 0.932895i $$-0.382726\pi$$
0.360149 + 0.932895i $$0.382726\pi$$
$$744$$ 0 0
$$745$$ 5240.00 0.257690
$$746$$ 0 0
$$747$$ 1188.00 0.0581883
$$748$$ 0 0
$$749$$ 14664.0 0.715368
$$750$$ 0 0
$$751$$ 26152.0 1.27071 0.635353 0.772222i $$-0.280855\pi$$
0.635353 + 0.772222i $$0.280855\pi$$
$$752$$ 0 0
$$753$$ −11256.0 −0.544743
$$754$$ 0 0
$$755$$ −4792.00 −0.230992
$$756$$ 0 0
$$757$$ 1066.00 0.0511815 0.0255908 0.999673i $$-0.491853\pi$$
0.0255908 + 0.999673i $$0.491853\pi$$
$$758$$ 0 0
$$759$$ 6006.00 0.287225
$$760$$ 0 0
$$761$$ −37518.0 −1.78716 −0.893578 0.448907i $$-0.851813\pi$$
−0.893578 + 0.448907i $$0.851813\pi$$
$$762$$ 0 0
$$763$$ −8320.00 −0.394763
$$764$$ 0 0
$$765$$ 2664.00 0.125905
$$766$$ 0 0
$$767$$ 6400.00 0.301292
$$768$$ 0 0
$$769$$ −17290.0 −0.810785 −0.405392 0.914143i $$-0.632865\pi$$
−0.405392 + 0.914143i $$0.632865\pi$$
$$770$$ 0 0
$$771$$ 2022.00 0.0944495
$$772$$ 0 0
$$773$$ 17172.0 0.799009 0.399504 0.916731i $$-0.369182\pi$$
0.399504 + 0.916731i $$0.369182\pi$$
$$774$$ 0 0
$$775$$ 872.000 0.0404170
$$776$$ 0 0
$$777$$ −5148.00 −0.237688
$$778$$ 0 0
$$779$$ 25320.0 1.16455
$$780$$ 0 0
$$781$$ −8382.00 −0.384035
$$782$$ 0 0
$$783$$ 2430.00 0.110908
$$784$$ 0 0
$$785$$ −8456.00 −0.384468
$$786$$ 0 0
$$787$$ 9536.00 0.431921 0.215960 0.976402i $$-0.430712\pi$$
0.215960 + 0.976402i $$0.430712\pi$$
$$788$$ 0 0
$$789$$ −13056.0 −0.589108
$$790$$ 0 0
$$791$$ 55692.0 2.50339
$$792$$ 0 0
$$793$$ −4224.00 −0.189153
$$794$$ 0 0
$$795$$ −4176.00 −0.186299
$$796$$ 0 0
$$797$$ 20516.0 0.911812 0.455906 0.890028i $$-0.349315\pi$$
0.455906 + 0.890028i $$0.349315\pi$$
$$798$$ 0 0
$$799$$ −37444.0 −1.65791
$$800$$ 0 0
$$801$$ 5130.00 0.226292
$$802$$ 0 0
$$803$$ 5962.00 0.262010
$$804$$ 0 0
$$805$$ 18928.0 0.828726
$$806$$ 0 0
$$807$$ −1500.00 −0.0654306
$$808$$ 0 0
$$809$$ 22470.0 0.976518 0.488259 0.872699i $$-0.337632\pi$$
0.488259 + 0.872699i $$0.337632\pi$$
$$810$$ 0 0
$$811$$ 3368.00 0.145828 0.0729140 0.997338i $$-0.476770\pi$$
0.0729140 + 0.997338i $$0.476770\pi$$
$$812$$ 0 0
$$813$$ −19614.0 −0.846117
$$814$$ 0 0
$$815$$ −15472.0 −0.664982
$$816$$ 0 0
$$817$$ −24480.0 −1.04828
$$818$$ 0 0
$$819$$ −7488.00 −0.319477
$$820$$ 0 0
$$821$$ 10738.0 0.456466 0.228233 0.973607i $$-0.426705\pi$$
0.228233 + 0.973607i $$0.426705\pi$$
$$822$$ 0 0
$$823$$ −15912.0 −0.673946 −0.336973 0.941514i $$-0.609403\pi$$
−0.336973 + 0.941514i $$0.609403\pi$$
$$824$$ 0 0
$$825$$ 3597.00 0.151796
$$826$$ 0 0
$$827$$ −22924.0 −0.963900 −0.481950 0.876199i $$-0.660071\pi$$
−0.481950 + 0.876199i $$0.660071\pi$$
$$828$$ 0 0
$$829$$ 41690.0 1.74663 0.873313 0.487159i $$-0.161967\pi$$
0.873313 + 0.487159i $$0.161967\pi$$
$$830$$ 0 0
$$831$$ −372.000 −0.0155289
$$832$$ 0 0
$$833$$ 24642.0 1.02496
$$834$$ 0 0
$$835$$ 8016.00 0.332222
$$836$$ 0 0
$$837$$ −216.000 −0.00892001
$$838$$ 0 0
$$839$$ −16450.0 −0.676898 −0.338449 0.940985i $$-0.609902\pi$$
−0.338449 + 0.940985i $$0.609902\pi$$
$$840$$ 0 0
$$841$$ −16289.0 −0.667883
$$842$$ 0 0
$$843$$ 10926.0 0.446396
$$844$$ 0 0
$$845$$ −4692.00 −0.191017
$$846$$ 0 0
$$847$$ −3146.00 −0.127624
$$848$$ 0 0
$$849$$ −13944.0 −0.563671
$$850$$ 0 0
$$851$$ −12012.0 −0.483861
$$852$$ 0 0
$$853$$ 30892.0 1.24000 0.620001 0.784601i $$-0.287132\pi$$
0.620001 + 0.784601i $$0.287132\pi$$
$$854$$ 0 0
$$855$$ 2160.00 0.0863982
$$856$$ 0 0
$$857$$ −38906.0 −1.55076 −0.775381 0.631493i $$-0.782442\pi$$
−0.775381 + 0.631493i $$0.782442\pi$$
$$858$$ 0 0
$$859$$ 1020.00 0.0405145 0.0202572 0.999795i $$-0.493551\pi$$
0.0202572 + 0.999795i $$0.493551\pi$$
$$860$$ 0 0
$$861$$ −32916.0 −1.30287
$$862$$ 0 0
$$863$$ 15078.0 0.594741 0.297370 0.954762i $$-0.403890\pi$$
0.297370 + 0.954762i $$0.403890\pi$$
$$864$$ 0 0
$$865$$ −2712.00 −0.106602
$$866$$ 0 0
$$867$$ 1689.00 0.0661608
$$868$$ 0 0
$$869$$ 6050.00 0.236171
$$870$$ 0 0
$$871$$ 33152.0 1.28968
$$872$$ 0 0
$$873$$ 126.000 0.00488483
$$874$$ 0 0
$$875$$ 24336.0 0.940237
$$876$$ 0 0
$$877$$ −22704.0 −0.874184 −0.437092 0.899417i $$-0.643992\pi$$
−0.437092 + 0.899417i $$0.643992\pi$$
$$878$$ 0 0
$$879$$ 9306.00 0.357092
$$880$$ 0 0
$$881$$ −19358.0 −0.740281 −0.370141 0.928976i $$-0.620690\pi$$
−0.370141 + 0.928976i $$0.620690\pi$$
$$882$$ 0 0
$$883$$ 11252.0 0.428833 0.214417 0.976742i $$-0.431215\pi$$
0.214417 + 0.976742i $$0.431215\pi$$
$$884$$ 0 0
$$885$$ 2400.00 0.0911583
$$886$$ 0 0
$$887$$ 43684.0 1.65362 0.826812 0.562478i $$-0.190152\pi$$
0.826812 + 0.562478i $$0.190152\pi$$
$$888$$ 0 0
$$889$$ 41756.0 1.57531
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 0 0
$$893$$ −30360.0 −1.13769
$$894$$ 0 0
$$895$$ 6720.00 0.250977
$$896$$ 0 0
$$897$$ −17472.0 −0.650360
$$898$$ 0 0
$$899$$ −720.000 −0.0267112
$$900$$ 0 0
$$901$$ −25752.0 −0.952190
$$902$$ 0 0
$$903$$ 31824.0 1.17280
$$904$$ 0 0
$$905$$ 17432.0 0.640287
$$906$$ 0 0
$$907$$ −45804.0 −1.67684 −0.838422 0.545022i $$-0.816521\pi$$
−0.838422 + 0.545022i $$0.816521\pi$$
$$908$$ 0 0
$$909$$ −15318.0 −0.558928
$$910$$ 0 0
$$911$$ −15318.0 −0.557089 −0.278544 0.960423i $$-0.589852\pi$$
−0.278544 + 0.960423i $$0.589852\pi$$
$$912$$ 0 0
$$913$$ −1452.00 −0.0526333
$$914$$ 0 0
$$915$$ −1584.00 −0.0572300
$$916$$ 0 0
$$917$$ −49608.0 −1.78648
$$918$$ 0 0
$$919$$ 11350.0 0.407401 0.203701 0.979033i $$-0.434703\pi$$
0.203701 + 0.979033i $$0.434703\pi$$
$$920$$ 0 0
$$921$$ −3732.00 −0.133522
$$922$$ 0 0
$$923$$ 24384.0 0.869566
$$924$$ 0 0
$$925$$ −7194.00 −0.255716
$$926$$ 0 0
$$927$$ −10188.0 −0.360969
$$928$$ 0 0
$$929$$ 33030.0 1.16650 0.583250 0.812292i $$-0.301781\pi$$
0.583250 + 0.812292i $$0.301781\pi$$
$$930$$ 0 0
$$931$$ 19980.0 0.703349
$$932$$ 0 0
$$933$$ 6246.00 0.219169
$$934$$ 0 0
$$935$$ −3256.00 −0.113885
$$936$$ 0 0
$$937$$ −10006.0 −0.348860 −0.174430 0.984670i $$-0.555808\pi$$
−0.174430 + 0.984670i $$0.555808\pi$$
$$938$$ 0 0
$$939$$ 7134.00 0.247933
$$940$$ 0 0
$$941$$ −2622.00 −0.0908340 −0.0454170 0.998968i $$-0.514462\pi$$
−0.0454170 + 0.998968i $$0.514462\pi$$
$$942$$ 0 0
$$943$$ −76804.0 −2.65226
$$944$$ 0 0
$$945$$ −2808.00 −0.0966606
$$946$$ 0 0
$$947$$ 39876.0 1.36832 0.684158 0.729334i $$-0.260170\pi$$
0.684158 + 0.729334i $$0.260170\pi$$
$$948$$ 0 0
$$949$$ −17344.0 −0.593267
$$950$$ 0 0
$$951$$ 1488.00 0.0507379
$$952$$ 0 0
$$953$$ 38918.0 1.32285 0.661426 0.750011i $$-0.269952\pi$$
0.661426 + 0.750011i $$0.269952\pi$$
$$954$$ 0 0
$$955$$ −7112.00 −0.240983
$$956$$ 0 0
$$957$$ −2970.00 −0.100320
$$958$$ 0 0
$$959$$ 56836.0 1.91380
$$960$$ 0 0
$$961$$ −29727.0 −0.997852
$$962$$ 0 0
$$963$$ −5076.00 −0.169857
$$964$$ 0 0
$$965$$ −15848.0 −0.528669
$$966$$ 0 0
$$967$$ 1114.00 0.0370464 0.0185232 0.999828i $$-0.494104\pi$$
0.0185232 + 0.999828i $$0.494104\pi$$
$$968$$ 0 0
$$969$$ 13320.0 0.441589
$$970$$ 0 0
$$971$$ 1688.00 0.0557884 0.0278942 0.999611i $$-0.491120\pi$$
0.0278942 + 0.999611i $$0.491120\pi$$
$$972$$ 0 0
$$973$$ 71240.0 2.34722
$$974$$ 0 0
$$975$$ −10464.0 −0.343709
$$976$$ 0 0
$$977$$ −41826.0 −1.36963 −0.684817 0.728715i $$-0.740118\pi$$
−0.684817 + 0.728715i $$0.740118\pi$$
$$978$$ 0 0
$$979$$ −6270.00 −0.204689
$$980$$ 0 0
$$981$$ 2880.00 0.0937322
$$982$$ 0 0
$$983$$ 978.000 0.0317328 0.0158664 0.999874i $$-0.494949\pi$$
0.0158664 + 0.999874i $$0.494949\pi$$
$$984$$ 0 0
$$985$$ −1496.00 −0.0483924
$$986$$ 0 0
$$987$$ 39468.0 1.27283
$$988$$ 0 0
$$989$$ 74256.0 2.38747
$$990$$ 0 0
$$991$$ 47272.0 1.51528 0.757641 0.652671i $$-0.226352\pi$$
0.757641 + 0.652671i $$0.226352\pi$$
$$992$$ 0 0
$$993$$ 8124.00 0.259625
$$994$$ 0 0
$$995$$ 8400.00 0.267636
$$996$$ 0 0
$$997$$ −51104.0 −1.62335 −0.811675 0.584109i $$-0.801444\pi$$
−0.811675 + 0.584109i $$0.801444\pi$$
$$998$$ 0 0
$$999$$ 1782.00 0.0564364
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 2112.4.a.u.1.1 1
4.3 odd 2 2112.4.a.h.1.1 1
8.3 odd 2 528.4.a.h.1.1 1
8.5 even 2 33.4.a.b.1.1 1
24.5 odd 2 99.4.a.a.1.1 1
24.11 even 2 1584.4.a.l.1.1 1
40.13 odd 4 825.4.c.f.199.2 2
40.29 even 2 825.4.a.f.1.1 1
40.37 odd 4 825.4.c.f.199.1 2
56.13 odd 2 1617.4.a.d.1.1 1
88.21 odd 2 363.4.a.d.1.1 1
120.29 odd 2 2475.4.a.e.1.1 1
264.197 even 2 1089.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.b.1.1 1 8.5 even 2
99.4.a.a.1.1 1 24.5 odd 2
363.4.a.d.1.1 1 88.21 odd 2
528.4.a.h.1.1 1 8.3 odd 2
825.4.a.f.1.1 1 40.29 even 2
825.4.c.f.199.1 2 40.37 odd 4
825.4.c.f.199.2 2 40.13 odd 4
1089.4.a.e.1.1 1 264.197 even 2
1584.4.a.l.1.1 1 24.11 even 2
1617.4.a.d.1.1 1 56.13 odd 2
2112.4.a.h.1.1 1 4.3 odd 2
2112.4.a.u.1.1 1 1.1 even 1 trivial
2475.4.a.e.1.1 1 120.29 odd 2