# Properties

 Label 2200.4.a.d.1.1 Level $2200$ Weight $4$ Character 2200.1 Self dual yes Analytic conductor $129.804$ Analytic rank $1$ 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] = [2200,4,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$129.804202013$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} +8.00000 q^{7} -23.0000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} +8.00000 q^{7} -23.0000 q^{9} -11.0000 q^{11} +25.0000 q^{13} -52.0000 q^{17} +19.0000 q^{19} -16.0000 q^{21} -75.0000 q^{23} +100.000 q^{27} +241.000 q^{29} +189.000 q^{31} +22.0000 q^{33} +68.0000 q^{37} -50.0000 q^{39} +240.000 q^{41} -183.000 q^{43} -56.0000 q^{47} -279.000 q^{49} +104.000 q^{51} -142.000 q^{53} -38.0000 q^{57} -584.000 q^{59} -622.000 q^{61} -184.000 q^{63} -142.000 q^{67} +150.000 q^{69} +435.000 q^{71} +628.000 q^{73} -88.0000 q^{77} -824.000 q^{79} +421.000 q^{81} +837.000 q^{83} -482.000 q^{87} -53.0000 q^{89} +200.000 q^{91} -378.000 q^{93} +1677.00 q^{97} +253.000 q^{99} +O(q^{100})$$

## 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
$$3$$ −2.00000 −0.384900 −0.192450 0.981307i $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 8.00000 0.431959 0.215980 0.976398i $$-0.430705\pi$$
0.215980 + 0.976398i $$0.430705\pi$$
$$8$$ 0 0
$$9$$ −23.0000 −0.851852
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ 25.0000 0.533366 0.266683 0.963784i $$-0.414072\pi$$
0.266683 + 0.963784i $$0.414072\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −52.0000 −0.741874 −0.370937 0.928658i $$-0.620963\pi$$
−0.370937 + 0.928658i $$0.620963\pi$$
$$18$$ 0 0
$$19$$ 19.0000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ −16.0000 −0.166261
$$22$$ 0 0
$$23$$ −75.0000 −0.679938 −0.339969 0.940437i $$-0.610417\pi$$
−0.339969 + 0.940437i $$0.610417\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 100.000 0.712778
$$28$$ 0 0
$$29$$ 241.000 1.54319 0.771596 0.636113i $$-0.219459\pi$$
0.771596 + 0.636113i $$0.219459\pi$$
$$30$$ 0 0
$$31$$ 189.000 1.09501 0.547506 0.836801i $$-0.315577\pi$$
0.547506 + 0.836801i $$0.315577\pi$$
$$32$$ 0 0
$$33$$ 22.0000 0.116052
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 68.0000 0.302139 0.151069 0.988523i $$-0.451728\pi$$
0.151069 + 0.988523i $$0.451728\pi$$
$$38$$ 0 0
$$39$$ −50.0000 −0.205293
$$40$$ 0 0
$$41$$ 240.000 0.914188 0.457094 0.889418i $$-0.348890\pi$$
0.457094 + 0.889418i $$0.348890\pi$$
$$42$$ 0 0
$$43$$ −183.000 −0.649006 −0.324503 0.945885i $$-0.605197\pi$$
−0.324503 + 0.945885i $$0.605197\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −56.0000 −0.173797 −0.0868983 0.996217i $$-0.527696\pi$$
−0.0868983 + 0.996217i $$0.527696\pi$$
$$48$$ 0 0
$$49$$ −279.000 −0.813411
$$50$$ 0 0
$$51$$ 104.000 0.285547
$$52$$ 0 0
$$53$$ −142.000 −0.368023 −0.184011 0.982924i $$-0.558908\pi$$
−0.184011 + 0.982924i $$0.558908\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −38.0000 −0.0883022
$$58$$ 0 0
$$59$$ −584.000 −1.28865 −0.644325 0.764752i $$-0.722861\pi$$
−0.644325 + 0.764752i $$0.722861\pi$$
$$60$$ 0 0
$$61$$ −622.000 −1.30556 −0.652778 0.757549i $$-0.726397\pi$$
−0.652778 + 0.757549i $$0.726397\pi$$
$$62$$ 0 0
$$63$$ −184.000 −0.367965
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −142.000 −0.258926 −0.129463 0.991584i $$-0.541325\pi$$
−0.129463 + 0.991584i $$0.541325\pi$$
$$68$$ 0 0
$$69$$ 150.000 0.261708
$$70$$ 0 0
$$71$$ 435.000 0.727113 0.363556 0.931572i $$-0.381562\pi$$
0.363556 + 0.931572i $$0.381562\pi$$
$$72$$ 0 0
$$73$$ 628.000 1.00687 0.503437 0.864032i $$-0.332069\pi$$
0.503437 + 0.864032i $$0.332069\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −88.0000 −0.130241
$$78$$ 0 0
$$79$$ −824.000 −1.17351 −0.586755 0.809765i $$-0.699595\pi$$
−0.586755 + 0.809765i $$0.699595\pi$$
$$80$$ 0 0
$$81$$ 421.000 0.577503
$$82$$ 0 0
$$83$$ 837.000 1.10690 0.553450 0.832882i $$-0.313311\pi$$
0.553450 + 0.832882i $$0.313311\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −482.000 −0.593975
$$88$$ 0 0
$$89$$ −53.0000 −0.0631235 −0.0315617 0.999502i $$-0.510048\pi$$
−0.0315617 + 0.999502i $$0.510048\pi$$
$$90$$ 0 0
$$91$$ 200.000 0.230392
$$92$$ 0 0
$$93$$ −378.000 −0.421471
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1677.00 1.75540 0.877699 0.479213i $$-0.159078\pi$$
0.877699 + 0.479213i $$0.159078\pi$$
$$98$$ 0 0
$$99$$ 253.000 0.256843
$$100$$ 0 0
$$101$$ −1275.00 −1.25611 −0.628056 0.778168i $$-0.716149\pi$$
−0.628056 + 0.778168i $$0.716149\pi$$
$$102$$ 0 0
$$103$$ 391.000 0.374042 0.187021 0.982356i $$-0.440117\pi$$
0.187021 + 0.982356i $$0.440117\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1099.00 0.992938 0.496469 0.868055i $$-0.334630\pi$$
0.496469 + 0.868055i $$0.334630\pi$$
$$108$$ 0 0
$$109$$ −905.000 −0.795259 −0.397630 0.917546i $$-0.630167\pi$$
−0.397630 + 0.917546i $$0.630167\pi$$
$$110$$ 0 0
$$111$$ −136.000 −0.116293
$$112$$ 0 0
$$113$$ −226.000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −575.000 −0.454348
$$118$$ 0 0
$$119$$ −416.000 −0.320459
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −480.000 −0.351871
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1532.00 1.07042 0.535209 0.844720i $$-0.320233\pi$$
0.535209 + 0.844720i $$0.320233\pi$$
$$128$$ 0 0
$$129$$ 366.000 0.249802
$$130$$ 0 0
$$131$$ −2089.00 −1.39326 −0.696629 0.717432i $$-0.745318\pi$$
−0.696629 + 0.717432i $$0.745318\pi$$
$$132$$ 0 0
$$133$$ 152.000 0.0990983
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1979.00 −1.23414 −0.617071 0.786908i $$-0.711681\pi$$
−0.617071 + 0.786908i $$0.711681\pi$$
$$138$$ 0 0
$$139$$ −47.0000 −0.0286798 −0.0143399 0.999897i $$-0.504565\pi$$
−0.0143399 + 0.999897i $$0.504565\pi$$
$$140$$ 0 0
$$141$$ 112.000 0.0668943
$$142$$ 0 0
$$143$$ −275.000 −0.160816
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 558.000 0.313082
$$148$$ 0 0
$$149$$ −1602.00 −0.880812 −0.440406 0.897799i $$-0.645165\pi$$
−0.440406 + 0.897799i $$0.645165\pi$$
$$150$$ 0 0
$$151$$ −160.000 −0.0862292 −0.0431146 0.999070i $$-0.513728\pi$$
−0.0431146 + 0.999070i $$0.513728\pi$$
$$152$$ 0 0
$$153$$ 1196.00 0.631966
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1934.00 0.983121 0.491561 0.870843i $$-0.336427\pi$$
0.491561 + 0.870843i $$0.336427\pi$$
$$158$$ 0 0
$$159$$ 284.000 0.141652
$$160$$ 0 0
$$161$$ −600.000 −0.293706
$$162$$ 0 0
$$163$$ −2692.00 −1.29358 −0.646791 0.762668i $$-0.723889\pi$$
−0.646791 + 0.762668i $$0.723889\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −504.000 −0.233537 −0.116769 0.993159i $$-0.537254\pi$$
−0.116769 + 0.993159i $$0.537254\pi$$
$$168$$ 0 0
$$169$$ −1572.00 −0.715521
$$170$$ 0 0
$$171$$ −437.000 −0.195428
$$172$$ 0 0
$$173$$ −2759.00 −1.21250 −0.606251 0.795273i $$-0.707327\pi$$
−0.606251 + 0.795273i $$0.707327\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1168.00 0.496001
$$178$$ 0 0
$$179$$ −2400.00 −1.00215 −0.501074 0.865405i $$-0.667061\pi$$
−0.501074 + 0.865405i $$0.667061\pi$$
$$180$$ 0 0
$$181$$ −232.000 −0.0952731 −0.0476365 0.998865i $$-0.515169\pi$$
−0.0476365 + 0.998865i $$0.515169\pi$$
$$182$$ 0 0
$$183$$ 1244.00 0.502509
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 572.000 0.223683
$$188$$ 0 0
$$189$$ 800.000 0.307891
$$190$$ 0 0
$$191$$ 2039.00 0.772444 0.386222 0.922406i $$-0.373780\pi$$
0.386222 + 0.922406i $$0.373780\pi$$
$$192$$ 0 0
$$193$$ −484.000 −0.180513 −0.0902567 0.995919i $$-0.528769\pi$$
−0.0902567 + 0.995919i $$0.528769\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2081.00 −0.752615 −0.376307 0.926495i $$-0.622806\pi$$
−0.376307 + 0.926495i $$0.622806\pi$$
$$198$$ 0 0
$$199$$ −2437.00 −0.868112 −0.434056 0.900886i $$-0.642918\pi$$
−0.434056 + 0.900886i $$0.642918\pi$$
$$200$$ 0 0
$$201$$ 284.000 0.0996608
$$202$$ 0 0
$$203$$ 1928.00 0.666596
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1725.00 0.579207
$$208$$ 0 0
$$209$$ −209.000 −0.0691714
$$210$$ 0 0
$$211$$ 4332.00 1.41340 0.706699 0.707514i $$-0.250183\pi$$
0.706699 + 0.707514i $$0.250183\pi$$
$$212$$ 0 0
$$213$$ −870.000 −0.279866
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1512.00 0.473001
$$218$$ 0 0
$$219$$ −1256.00 −0.387546
$$220$$ 0 0
$$221$$ −1300.00 −0.395690
$$222$$ 0 0
$$223$$ −120.000 −0.0360350 −0.0180175 0.999838i $$-0.505735\pi$$
−0.0180175 + 0.999838i $$0.505735\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 5001.00 1.46224 0.731119 0.682250i $$-0.238998\pi$$
0.731119 + 0.682250i $$0.238998\pi$$
$$228$$ 0 0
$$229$$ 278.000 0.0802216 0.0401108 0.999195i $$-0.487229\pi$$
0.0401108 + 0.999195i $$0.487229\pi$$
$$230$$ 0 0
$$231$$ 176.000 0.0501297
$$232$$ 0 0
$$233$$ −5358.00 −1.50650 −0.753249 0.657735i $$-0.771515\pi$$
−0.753249 + 0.657735i $$0.771515\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1648.00 0.451684
$$238$$ 0 0
$$239$$ 126.000 0.0341015 0.0170508 0.999855i $$-0.494572\pi$$
0.0170508 + 0.999855i $$0.494572\pi$$
$$240$$ 0 0
$$241$$ −510.000 −0.136315 −0.0681577 0.997675i $$-0.521712\pi$$
−0.0681577 + 0.997675i $$0.521712\pi$$
$$242$$ 0 0
$$243$$ −3542.00 −0.935059
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 475.000 0.122362
$$248$$ 0 0
$$249$$ −1674.00 −0.426046
$$250$$ 0 0
$$251$$ 2676.00 0.672939 0.336469 0.941694i $$-0.390767\pi$$
0.336469 + 0.941694i $$0.390767\pi$$
$$252$$ 0 0
$$253$$ 825.000 0.205009
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 617.000 0.149756 0.0748782 0.997193i $$-0.476143\pi$$
0.0748782 + 0.997193i $$0.476143\pi$$
$$258$$ 0 0
$$259$$ 544.000 0.130512
$$260$$ 0 0
$$261$$ −5543.00 −1.31457
$$262$$ 0 0
$$263$$ −3732.00 −0.875000 −0.437500 0.899218i $$-0.644136\pi$$
−0.437500 + 0.899218i $$0.644136\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 106.000 0.0242962
$$268$$ 0 0
$$269$$ −8418.00 −1.90801 −0.954005 0.299792i $$-0.903083\pi$$
−0.954005 + 0.299792i $$0.903083\pi$$
$$270$$ 0 0
$$271$$ −7786.00 −1.74526 −0.872631 0.488381i $$-0.837588\pi$$
−0.872631 + 0.488381i $$0.837588\pi$$
$$272$$ 0 0
$$273$$ −400.000 −0.0886780
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1974.00 −0.428181 −0.214091 0.976814i $$-0.568679\pi$$
−0.214091 + 0.976814i $$0.568679\pi$$
$$278$$ 0 0
$$279$$ −4347.00 −0.932789
$$280$$ 0 0
$$281$$ 428.000 0.0908624 0.0454312 0.998967i $$-0.485534\pi$$
0.0454312 + 0.998967i $$0.485534\pi$$
$$282$$ 0 0
$$283$$ 3220.00 0.676357 0.338179 0.941082i $$-0.390189\pi$$
0.338179 + 0.941082i $$0.390189\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1920.00 0.394892
$$288$$ 0 0
$$289$$ −2209.00 −0.449623
$$290$$ 0 0
$$291$$ −3354.00 −0.675653
$$292$$ 0 0
$$293$$ −8258.00 −1.64654 −0.823272 0.567647i $$-0.807854\pi$$
−0.823272 + 0.567647i $$0.807854\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −1100.00 −0.214911
$$298$$ 0 0
$$299$$ −1875.00 −0.362656
$$300$$ 0 0
$$301$$ −1464.00 −0.280344
$$302$$ 0 0
$$303$$ 2550.00 0.483477
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 3632.00 0.675209 0.337605 0.941288i $$-0.390383\pi$$
0.337605 + 0.941288i $$0.390383\pi$$
$$308$$ 0 0
$$309$$ −782.000 −0.143969
$$310$$ 0 0
$$311$$ −4413.00 −0.804625 −0.402312 0.915502i $$-0.631793\pi$$
−0.402312 + 0.915502i $$0.631793\pi$$
$$312$$ 0 0
$$313$$ −2266.00 −0.409207 −0.204604 0.978845i $$-0.565591\pi$$
−0.204604 + 0.978845i $$0.565591\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3632.00 0.643512 0.321756 0.946823i $$-0.395727\pi$$
0.321756 + 0.946823i $$0.395727\pi$$
$$318$$ 0 0
$$319$$ −2651.00 −0.465290
$$320$$ 0 0
$$321$$ −2198.00 −0.382182
$$322$$ 0 0
$$323$$ −988.000 −0.170197
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 1810.00 0.306096
$$328$$ 0 0
$$329$$ −448.000 −0.0750731
$$330$$ 0 0
$$331$$ 8476.00 1.40750 0.703751 0.710447i $$-0.251507\pi$$
0.703751 + 0.710447i $$0.251507\pi$$
$$332$$ 0 0
$$333$$ −1564.00 −0.257377
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −8834.00 −1.42795 −0.713974 0.700172i $$-0.753107\pi$$
−0.713974 + 0.700172i $$0.753107\pi$$
$$338$$ 0 0
$$339$$ 452.000 0.0724167
$$340$$ 0 0
$$341$$ −2079.00 −0.330159
$$342$$ 0 0
$$343$$ −4976.00 −0.783320
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −6724.00 −1.04024 −0.520120 0.854093i $$-0.674113\pi$$
−0.520120 + 0.854093i $$0.674113\pi$$
$$348$$ 0 0
$$349$$ 1825.00 0.279914 0.139957 0.990158i $$-0.455304\pi$$
0.139957 + 0.990158i $$0.455304\pi$$
$$350$$ 0 0
$$351$$ 2500.00 0.380171
$$352$$ 0 0
$$353$$ −3103.00 −0.467864 −0.233932 0.972253i $$-0.575159\pi$$
−0.233932 + 0.972253i $$0.575159\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 832.000 0.123345
$$358$$ 0 0
$$359$$ −9960.00 −1.46426 −0.732129 0.681166i $$-0.761473\pi$$
−0.732129 + 0.681166i $$0.761473\pi$$
$$360$$ 0 0
$$361$$ −6498.00 −0.947368
$$362$$ 0 0
$$363$$ −242.000 −0.0349909
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 5695.00 0.810018 0.405009 0.914313i $$-0.367268\pi$$
0.405009 + 0.914313i $$0.367268\pi$$
$$368$$ 0 0
$$369$$ −5520.00 −0.778753
$$370$$ 0 0
$$371$$ −1136.00 −0.158971
$$372$$ 0 0
$$373$$ −9582.00 −1.33013 −0.665063 0.746787i $$-0.731595\pi$$
−0.665063 + 0.746787i $$0.731595\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6025.00 0.823086
$$378$$ 0 0
$$379$$ −3402.00 −0.461079 −0.230540 0.973063i $$-0.574049\pi$$
−0.230540 + 0.973063i $$0.574049\pi$$
$$380$$ 0 0
$$381$$ −3064.00 −0.412004
$$382$$ 0 0
$$383$$ 1651.00 0.220267 0.110133 0.993917i $$-0.464872\pi$$
0.110133 + 0.993917i $$0.464872\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4209.00 0.552857
$$388$$ 0 0
$$389$$ 8304.00 1.08234 0.541169 0.840914i $$-0.317982\pi$$
0.541169 + 0.840914i $$0.317982\pi$$
$$390$$ 0 0
$$391$$ 3900.00 0.504428
$$392$$ 0 0
$$393$$ 4178.00 0.536265
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4198.00 0.530709 0.265355 0.964151i $$-0.414511\pi$$
0.265355 + 0.964151i $$0.414511\pi$$
$$398$$ 0 0
$$399$$ −304.000 −0.0381429
$$400$$ 0 0
$$401$$ −12809.0 −1.59514 −0.797570 0.603227i $$-0.793881\pi$$
−0.797570 + 0.603227i $$0.793881\pi$$
$$402$$ 0 0
$$403$$ 4725.00 0.584042
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −748.000 −0.0910982
$$408$$ 0 0
$$409$$ −3608.00 −0.436196 −0.218098 0.975927i $$-0.569985\pi$$
−0.218098 + 0.975927i $$0.569985\pi$$
$$410$$ 0 0
$$411$$ 3958.00 0.475021
$$412$$ 0 0
$$413$$ −4672.00 −0.556644
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 94.0000 0.0110388
$$418$$ 0 0
$$419$$ 3462.00 0.403651 0.201825 0.979421i $$-0.435313\pi$$
0.201825 + 0.979421i $$0.435313\pi$$
$$420$$ 0 0
$$421$$ −7406.00 −0.857355 −0.428677 0.903458i $$-0.641020\pi$$
−0.428677 + 0.903458i $$0.641020\pi$$
$$422$$ 0 0
$$423$$ 1288.00 0.148049
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −4976.00 −0.563947
$$428$$ 0 0
$$429$$ 550.000 0.0618980
$$430$$ 0 0
$$431$$ 3320.00 0.371041 0.185521 0.982640i $$-0.440603\pi$$
0.185521 + 0.982640i $$0.440603\pi$$
$$432$$ 0 0
$$433$$ −12043.0 −1.33660 −0.668302 0.743890i $$-0.732979\pi$$
−0.668302 + 0.743890i $$0.732979\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1425.00 −0.155989
$$438$$ 0 0
$$439$$ −9036.00 −0.982380 −0.491190 0.871053i $$-0.663438\pi$$
−0.491190 + 0.871053i $$0.663438\pi$$
$$440$$ 0 0
$$441$$ 6417.00 0.692906
$$442$$ 0 0
$$443$$ 360.000 0.0386097 0.0193049 0.999814i $$-0.493855\pi$$
0.0193049 + 0.999814i $$0.493855\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 3204.00 0.339025
$$448$$ 0 0
$$449$$ 3477.00 0.365456 0.182728 0.983163i $$-0.441507\pi$$
0.182728 + 0.983163i $$0.441507\pi$$
$$450$$ 0 0
$$451$$ −2640.00 −0.275638
$$452$$ 0 0
$$453$$ 320.000 0.0331897
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9212.00 0.942930 0.471465 0.881885i $$-0.343725\pi$$
0.471465 + 0.881885i $$0.343725\pi$$
$$458$$ 0 0
$$459$$ −5200.00 −0.528791
$$460$$ 0 0
$$461$$ −6106.00 −0.616887 −0.308443 0.951243i $$-0.599808\pi$$
−0.308443 + 0.951243i $$0.599808\pi$$
$$462$$ 0 0
$$463$$ 9133.00 0.916731 0.458366 0.888764i $$-0.348435\pi$$
0.458366 + 0.888764i $$0.348435\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 9930.00 0.983952 0.491976 0.870609i $$-0.336275\pi$$
0.491976 + 0.870609i $$0.336275\pi$$
$$468$$ 0 0
$$469$$ −1136.00 −0.111846
$$470$$ 0 0
$$471$$ −3868.00 −0.378403
$$472$$ 0 0
$$473$$ 2013.00 0.195683
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 3266.00 0.313501
$$478$$ 0 0
$$479$$ −15906.0 −1.51725 −0.758626 0.651526i $$-0.774129\pi$$
−0.758626 + 0.651526i $$0.774129\pi$$
$$480$$ 0 0
$$481$$ 1700.00 0.161150
$$482$$ 0 0
$$483$$ 1200.00 0.113047
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −3461.00 −0.322039 −0.161019 0.986951i $$-0.551478\pi$$
−0.161019 + 0.986951i $$0.551478\pi$$
$$488$$ 0 0
$$489$$ 5384.00 0.497900
$$490$$ 0 0
$$491$$ −12717.0 −1.16886 −0.584430 0.811444i $$-0.698682\pi$$
−0.584430 + 0.811444i $$0.698682\pi$$
$$492$$ 0 0
$$493$$ −12532.0 −1.14485
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3480.00 0.314083
$$498$$ 0 0
$$499$$ −8178.00 −0.733662 −0.366831 0.930288i $$-0.619557\pi$$
−0.366831 + 0.930288i $$0.619557\pi$$
$$500$$ 0 0
$$501$$ 1008.00 0.0898885
$$502$$ 0 0
$$503$$ −1278.00 −0.113287 −0.0566433 0.998394i $$-0.518040\pi$$
−0.0566433 + 0.998394i $$0.518040\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 3144.00 0.275404
$$508$$ 0 0
$$509$$ 8022.00 0.698564 0.349282 0.937018i $$-0.386426\pi$$
0.349282 + 0.937018i $$0.386426\pi$$
$$510$$ 0 0
$$511$$ 5024.00 0.434929
$$512$$ 0 0
$$513$$ 1900.00 0.163523
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 616.000 0.0524016
$$518$$ 0 0
$$519$$ 5518.00 0.466692
$$520$$ 0 0
$$521$$ 5503.00 0.462746 0.231373 0.972865i $$-0.425678\pi$$
0.231373 + 0.972865i $$0.425678\pi$$
$$522$$ 0 0
$$523$$ −5521.00 −0.461599 −0.230800 0.973001i $$-0.574134\pi$$
−0.230800 + 0.973001i $$0.574134\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −9828.00 −0.812361
$$528$$ 0 0
$$529$$ −6542.00 −0.537684
$$530$$ 0 0
$$531$$ 13432.0 1.09774
$$532$$ 0 0
$$533$$ 6000.00 0.487596
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 4800.00 0.385727
$$538$$ 0 0
$$539$$ 3069.00 0.245253
$$540$$ 0 0
$$541$$ 13763.0 1.09375 0.546874 0.837215i $$-0.315818\pi$$
0.546874 + 0.837215i $$0.315818\pi$$
$$542$$ 0 0
$$543$$ 464.000 0.0366706
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 17939.0 1.40222 0.701112 0.713051i $$-0.252687\pi$$
0.701112 + 0.713051i $$0.252687\pi$$
$$548$$ 0 0
$$549$$ 14306.0 1.11214
$$550$$ 0 0
$$551$$ 4579.00 0.354033
$$552$$ 0 0
$$553$$ −6592.00 −0.506908
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 119.000 0.00905241 0.00452620 0.999990i $$-0.498559\pi$$
0.00452620 + 0.999990i $$0.498559\pi$$
$$558$$ 0 0
$$559$$ −4575.00 −0.346157
$$560$$ 0 0
$$561$$ −1144.00 −0.0860958
$$562$$ 0 0
$$563$$ 9876.00 0.739296 0.369648 0.929172i $$-0.379478\pi$$
0.369648 + 0.929172i $$0.379478\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 3368.00 0.249458
$$568$$ 0 0
$$569$$ −2504.00 −0.184487 −0.0922435 0.995736i $$-0.529404\pi$$
−0.0922435 + 0.995736i $$0.529404\pi$$
$$570$$ 0 0
$$571$$ 3947.00 0.289276 0.144638 0.989485i $$-0.453798\pi$$
0.144638 + 0.989485i $$0.453798\pi$$
$$572$$ 0 0
$$573$$ −4078.00 −0.297314
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −2938.00 −0.211977 −0.105988 0.994367i $$-0.533801\pi$$
−0.105988 + 0.994367i $$0.533801\pi$$
$$578$$ 0 0
$$579$$ 968.000 0.0694796
$$580$$ 0 0
$$581$$ 6696.00 0.478136
$$582$$ 0 0
$$583$$ 1562.00 0.110963
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1484.00 0.104346 0.0521731 0.998638i $$-0.483385\pi$$
0.0521731 + 0.998638i $$0.483385\pi$$
$$588$$ 0 0
$$589$$ 3591.00 0.251213
$$590$$ 0 0
$$591$$ 4162.00 0.289682
$$592$$ 0 0
$$593$$ 534.000 0.0369793 0.0184897 0.999829i $$-0.494114\pi$$
0.0184897 + 0.999829i $$0.494114\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4874.00 0.334137
$$598$$ 0 0
$$599$$ −5460.00 −0.372437 −0.186218 0.982508i $$-0.559623\pi$$
−0.186218 + 0.982508i $$0.559623\pi$$
$$600$$ 0 0
$$601$$ 4844.00 0.328770 0.164385 0.986396i $$-0.447436\pi$$
0.164385 + 0.986396i $$0.447436\pi$$
$$602$$ 0 0
$$603$$ 3266.00 0.220567
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 4994.00 0.333938 0.166969 0.985962i $$-0.446602\pi$$
0.166969 + 0.985962i $$0.446602\pi$$
$$608$$ 0 0
$$609$$ −3856.00 −0.256573
$$610$$ 0 0
$$611$$ −1400.00 −0.0926971
$$612$$ 0 0
$$613$$ −9502.00 −0.626072 −0.313036 0.949741i $$-0.601346\pi$$
−0.313036 + 0.949741i $$0.601346\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −25029.0 −1.63311 −0.816556 0.577267i $$-0.804119\pi$$
−0.816556 + 0.577267i $$0.804119\pi$$
$$618$$ 0 0
$$619$$ −15180.0 −0.985680 −0.492840 0.870120i $$-0.664041\pi$$
−0.492840 + 0.870120i $$0.664041\pi$$
$$620$$ 0 0
$$621$$ −7500.00 −0.484645
$$622$$ 0 0
$$623$$ −424.000 −0.0272668
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 418.000 0.0266241
$$628$$ 0 0
$$629$$ −3536.00 −0.224149
$$630$$ 0 0
$$631$$ −26624.0 −1.67969 −0.839845 0.542826i $$-0.817354\pi$$
−0.839845 + 0.542826i $$0.817354\pi$$
$$632$$ 0 0
$$633$$ −8664.00 −0.544018
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6975.00 −0.433845
$$638$$ 0 0
$$639$$ −10005.0 −0.619392
$$640$$ 0 0
$$641$$ −2691.00 −0.165816 −0.0829080 0.996557i $$-0.526421\pi$$
−0.0829080 + 0.996557i $$0.526421\pi$$
$$642$$ 0 0
$$643$$ 28748.0 1.76316 0.881579 0.472037i $$-0.156481\pi$$
0.881579 + 0.472037i $$0.156481\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 26864.0 1.63235 0.816177 0.577802i $$-0.196089\pi$$
0.816177 + 0.577802i $$0.196089\pi$$
$$648$$ 0 0
$$649$$ 6424.00 0.388542
$$650$$ 0 0
$$651$$ −3024.00 −0.182058
$$652$$ 0 0
$$653$$ −4286.00 −0.256852 −0.128426 0.991719i $$-0.540992\pi$$
−0.128426 + 0.991719i $$0.540992\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −14444.0 −0.857708
$$658$$ 0 0
$$659$$ −23167.0 −1.36944 −0.684718 0.728808i $$-0.740075\pi$$
−0.684718 + 0.728808i $$0.740075\pi$$
$$660$$ 0 0
$$661$$ −12472.0 −0.733895 −0.366947 0.930242i $$-0.619597\pi$$
−0.366947 + 0.930242i $$0.619597\pi$$
$$662$$ 0 0
$$663$$ 2600.00 0.152301
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −18075.0 −1.04928
$$668$$ 0 0
$$669$$ 240.000 0.0138699
$$670$$ 0 0
$$671$$ 6842.00 0.393640
$$672$$ 0 0
$$673$$ −6940.00 −0.397500 −0.198750 0.980050i $$-0.563688\pi$$
−0.198750 + 0.980050i $$0.563688\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −27187.0 −1.54340 −0.771700 0.635987i $$-0.780593\pi$$
−0.771700 + 0.635987i $$0.780593\pi$$
$$678$$ 0 0
$$679$$ 13416.0 0.758260
$$680$$ 0 0
$$681$$ −10002.0 −0.562816
$$682$$ 0 0
$$683$$ 19880.0 1.11374 0.556872 0.830598i $$-0.312001\pi$$
0.556872 + 0.830598i $$0.312001\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −556.000 −0.0308773
$$688$$ 0 0
$$689$$ −3550.00 −0.196291
$$690$$ 0 0
$$691$$ 13040.0 0.717894 0.358947 0.933358i $$-0.383136\pi$$
0.358947 + 0.933358i $$0.383136\pi$$
$$692$$ 0 0
$$693$$ 2024.00 0.110946
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −12480.0 −0.678212
$$698$$ 0 0
$$699$$ 10716.0 0.579852
$$700$$ 0 0
$$701$$ 5463.00 0.294343 0.147172 0.989111i $$-0.452983\pi$$
0.147172 + 0.989111i $$0.452983\pi$$
$$702$$ 0 0
$$703$$ 1292.00 0.0693154
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −10200.0 −0.542589
$$708$$ 0 0
$$709$$ 14732.0 0.780355 0.390178 0.920740i $$-0.372414\pi$$
0.390178 + 0.920740i $$0.372414\pi$$
$$710$$ 0 0
$$711$$ 18952.0 0.999656
$$712$$ 0 0
$$713$$ −14175.0 −0.744541
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −252.000 −0.0131257
$$718$$ 0 0
$$719$$ 23616.0 1.22493 0.612467 0.790496i $$-0.290177\pi$$
0.612467 + 0.790496i $$0.290177\pi$$
$$720$$ 0 0
$$721$$ 3128.00 0.161571
$$722$$ 0 0
$$723$$ 1020.00 0.0524678
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −3869.00 −0.197377 −0.0986886 0.995118i $$-0.531465\pi$$
−0.0986886 + 0.995118i $$0.531465\pi$$
$$728$$ 0 0
$$729$$ −4283.00 −0.217599
$$730$$ 0 0
$$731$$ 9516.00 0.481480
$$732$$ 0 0
$$733$$ 13993.0 0.705107 0.352553 0.935792i $$-0.385313\pi$$
0.352553 + 0.935792i $$0.385313\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1562.00 0.0780692
$$738$$ 0 0
$$739$$ 8736.00 0.434857 0.217428 0.976076i $$-0.430233\pi$$
0.217428 + 0.976076i $$0.430233\pi$$
$$740$$ 0 0
$$741$$ −950.000 −0.0470973
$$742$$ 0 0
$$743$$ 32842.0 1.62161 0.810805 0.585316i $$-0.199030\pi$$
0.810805 + 0.585316i $$0.199030\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −19251.0 −0.942915
$$748$$ 0 0
$$749$$ 8792.00 0.428909
$$750$$ 0 0
$$751$$ 7431.00 0.361067 0.180533 0.983569i $$-0.442218\pi$$
0.180533 + 0.983569i $$0.442218\pi$$
$$752$$ 0 0
$$753$$ −5352.00 −0.259014
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 22460.0 1.07837 0.539183 0.842189i $$-0.318733\pi$$
0.539183 + 0.842189i $$0.318733\pi$$
$$758$$ 0 0
$$759$$ −1650.00 −0.0789080
$$760$$ 0 0
$$761$$ −21940.0 −1.04510 −0.522552 0.852607i $$-0.675020\pi$$
−0.522552 + 0.852607i $$0.675020\pi$$
$$762$$ 0 0
$$763$$ −7240.00 −0.343520
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −14600.0 −0.687321
$$768$$ 0 0
$$769$$ 26002.0 1.21932 0.609659 0.792664i $$-0.291306\pi$$
0.609659 + 0.792664i $$0.291306\pi$$
$$770$$ 0 0
$$771$$ −1234.00 −0.0576413
$$772$$ 0 0
$$773$$ 5060.00 0.235441 0.117720 0.993047i $$-0.462441\pi$$
0.117720 + 0.993047i $$0.462441\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −1088.00 −0.0502340
$$778$$ 0 0
$$779$$ 4560.00 0.209729
$$780$$ 0 0
$$781$$ −4785.00 −0.219233
$$782$$ 0 0
$$783$$ 24100.0 1.09995
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 5460.00 0.247304 0.123652 0.992326i $$-0.460539\pi$$
0.123652 + 0.992326i $$0.460539\pi$$
$$788$$ 0 0
$$789$$ 7464.00 0.336788
$$790$$ 0 0
$$791$$ −1808.00 −0.0812706
$$792$$ 0 0
$$793$$ −15550.0 −0.696339
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 10108.0 0.449239 0.224620 0.974447i $$-0.427886\pi$$
0.224620 + 0.974447i $$0.427886\pi$$
$$798$$ 0 0
$$799$$ 2912.00 0.128935
$$800$$ 0 0
$$801$$ 1219.00 0.0537718
$$802$$ 0 0
$$803$$ −6908.00 −0.303584
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 16836.0 0.734393
$$808$$ 0 0
$$809$$ 17466.0 0.759051 0.379525 0.925181i $$-0.376087\pi$$
0.379525 + 0.925181i $$0.376087\pi$$
$$810$$ 0 0
$$811$$ 25388.0 1.09925 0.549626 0.835411i $$-0.314770\pi$$
0.549626 + 0.835411i $$0.314770\pi$$
$$812$$ 0 0
$$813$$ 15572.0 0.671751
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −3477.00 −0.148892
$$818$$ 0 0
$$819$$ −4600.00 −0.196260
$$820$$ 0 0
$$821$$ −4523.00 −0.192270 −0.0961351 0.995368i $$-0.530648\pi$$
−0.0961351 + 0.995368i $$0.530648\pi$$
$$822$$ 0 0
$$823$$ 31824.0 1.34789 0.673946 0.738781i $$-0.264598\pi$$
0.673946 + 0.738781i $$0.264598\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −3207.00 −0.134847 −0.0674234 0.997724i $$-0.521478\pi$$
−0.0674234 + 0.997724i $$0.521478\pi$$
$$828$$ 0 0
$$829$$ −28174.0 −1.18037 −0.590183 0.807269i $$-0.700944\pi$$
−0.590183 + 0.807269i $$0.700944\pi$$
$$830$$ 0 0
$$831$$ 3948.00 0.164807
$$832$$ 0 0
$$833$$ 14508.0 0.603448
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 18900.0 0.780501
$$838$$ 0 0
$$839$$ −20904.0 −0.860174 −0.430087 0.902787i $$-0.641517\pi$$
−0.430087 + 0.902787i $$0.641517\pi$$
$$840$$ 0 0
$$841$$ 33692.0 1.38144
$$842$$ 0 0
$$843$$ −856.000 −0.0349730
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 968.000 0.0392690
$$848$$ 0 0
$$849$$ −6440.00 −0.260330
$$850$$ 0 0
$$851$$ −5100.00 −0.205436
$$852$$ 0 0
$$853$$ 13398.0 0.537795 0.268897 0.963169i $$-0.413341\pi$$
0.268897 + 0.963169i $$0.413341\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1284.00 0.0511792 0.0255896 0.999673i $$-0.491854\pi$$
0.0255896 + 0.999673i $$0.491854\pi$$
$$858$$ 0 0
$$859$$ 27178.0 1.07951 0.539756 0.841821i $$-0.318516\pi$$
0.539756 + 0.841821i $$0.318516\pi$$
$$860$$ 0 0
$$861$$ −3840.00 −0.151994
$$862$$ 0 0
$$863$$ −15321.0 −0.604325 −0.302163 0.953256i $$-0.597709\pi$$
−0.302163 + 0.953256i $$0.597709\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 4418.00 0.173060
$$868$$ 0 0
$$869$$ 9064.00 0.353826
$$870$$ 0 0
$$871$$ −3550.00 −0.138102
$$872$$ 0 0
$$873$$ −38571.0 −1.49534
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −27389.0 −1.05457 −0.527287 0.849687i $$-0.676791\pi$$
−0.527287 + 0.849687i $$0.676791\pi$$
$$878$$ 0 0
$$879$$ 16516.0 0.633755
$$880$$ 0 0
$$881$$ 2725.00 0.104208 0.0521042 0.998642i $$-0.483407\pi$$
0.0521042 + 0.998642i $$0.483407\pi$$
$$882$$ 0 0
$$883$$ 45272.0 1.72540 0.862698 0.505720i $$-0.168773\pi$$
0.862698 + 0.505720i $$0.168773\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 23894.0 0.904489 0.452245 0.891894i $$-0.350623\pi$$
0.452245 + 0.891894i $$0.350623\pi$$
$$888$$ 0 0
$$889$$ 12256.0 0.462377
$$890$$ 0 0
$$891$$ −4631.00 −0.174124
$$892$$ 0 0
$$893$$ −1064.00 −0.0398717
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 3750.00 0.139586
$$898$$ 0 0
$$899$$ 45549.0 1.68982
$$900$$ 0 0
$$901$$ 7384.00 0.273026
$$902$$ 0 0
$$903$$ 2928.00 0.107904
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −8134.00 −0.297778 −0.148889 0.988854i $$-0.547570\pi$$
−0.148889 + 0.988854i $$0.547570\pi$$
$$908$$ 0 0
$$909$$ 29325.0 1.07002
$$910$$ 0 0
$$911$$ 11648.0 0.423617 0.211809 0.977311i $$-0.432065\pi$$
0.211809 + 0.977311i $$0.432065\pi$$
$$912$$ 0 0
$$913$$ −9207.00 −0.333743
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −16712.0 −0.601831
$$918$$ 0 0
$$919$$ −11424.0 −0.410058 −0.205029 0.978756i $$-0.565729\pi$$
−0.205029 + 0.978756i $$0.565729\pi$$
$$920$$ 0 0
$$921$$ −7264.00 −0.259888
$$922$$ 0 0
$$923$$ 10875.0 0.387817
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −8993.00 −0.318629
$$928$$ 0 0
$$929$$ 32935.0 1.16315 0.581573 0.813494i $$-0.302438\pi$$
0.581573 + 0.813494i $$0.302438\pi$$
$$930$$ 0 0
$$931$$ −5301.00 −0.186609
$$932$$ 0 0
$$933$$ 8826.00 0.309700
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −42670.0 −1.48769 −0.743846 0.668351i $$-0.767001\pi$$
−0.743846 + 0.668351i $$0.767001\pi$$
$$938$$ 0 0
$$939$$ 4532.00 0.157504
$$940$$ 0 0
$$941$$ 2.00000 6.92860e−5 0 3.46430e−5 1.00000i $$-0.499989\pi$$
3.46430e−5 1.00000i $$0.499989\pi$$
$$942$$ 0 0
$$943$$ −18000.0 −0.621591
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −31390.0 −1.07713 −0.538563 0.842585i $$-0.681033\pi$$
−0.538563 + 0.842585i $$0.681033\pi$$
$$948$$ 0 0
$$949$$ 15700.0 0.537032
$$950$$ 0 0
$$951$$ −7264.00 −0.247688
$$952$$ 0 0
$$953$$ −20902.0 −0.710474 −0.355237 0.934776i $$-0.615600\pi$$
−0.355237 + 0.934776i $$0.615600\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 5302.00 0.179090
$$958$$ 0 0
$$959$$ −15832.0 −0.533099
$$960$$ 0 0
$$961$$ 5930.00 0.199053
$$962$$ 0 0
$$963$$ −25277.0 −0.845836
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −3630.00 −0.120717 −0.0603583 0.998177i $$-0.519224\pi$$
−0.0603583 + 0.998177i $$0.519224\pi$$
$$968$$ 0 0
$$969$$ 1976.00 0.0655090
$$970$$ 0 0
$$971$$ 23282.0 0.769470 0.384735 0.923027i $$-0.374293\pi$$
0.384735 + 0.923027i $$0.374293\pi$$
$$972$$ 0 0
$$973$$ −376.000 −0.0123885
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −42910.0 −1.40513 −0.702565 0.711619i $$-0.747962\pi$$
−0.702565 + 0.711619i $$0.747962\pi$$
$$978$$ 0 0
$$979$$ 583.000 0.0190324
$$980$$ 0 0
$$981$$ 20815.0 0.677443
$$982$$ 0 0
$$983$$ 26849.0 0.871160 0.435580 0.900150i $$-0.356543\pi$$
0.435580 + 0.900150i $$0.356543\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 896.000 0.0288956
$$988$$ 0 0
$$989$$ 13725.0 0.441284
$$990$$ 0 0
$$991$$ 30024.0 0.962405 0.481203 0.876609i $$-0.340200\pi$$
0.481203 + 0.876609i $$0.340200\pi$$
$$992$$ 0 0
$$993$$ −16952.0 −0.541748
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 22810.0 0.724574 0.362287 0.932067i $$-0.381996\pi$$
0.362287 + 0.932067i $$0.381996\pi$$
$$998$$ 0 0
$$999$$ 6800.00 0.215358
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 2200.4.a.d.1.1 1
5.4 even 2 2200.4.a.g.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2200.4.a.d.1.1 1 1.1 even 1 trivial
2200.4.a.g.1.1 yes 1 5.4 even 2