# Properties

 Label 80.4.a.e.1.1 Level $80$ Weight $4$ Character 80.1 Self dual yes Analytic conductor $4.720$ 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] = [80,4,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("80.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 80.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: $$4.72015280046$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 80.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+6.00000 q^{3} -5.00000 q^{5} +34.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+6.00000 q^{3} -5.00000 q^{5} +34.0000 q^{7} +9.00000 q^{9} -16.0000 q^{11} +58.0000 q^{13} -30.0000 q^{15} -70.0000 q^{17} -4.00000 q^{19} +204.000 q^{21} +134.000 q^{23} +25.0000 q^{25} -108.000 q^{27} -242.000 q^{29} -100.000 q^{31} -96.0000 q^{33} -170.000 q^{35} -438.000 q^{37} +348.000 q^{39} -138.000 q^{41} -178.000 q^{43} -45.0000 q^{45} -22.0000 q^{47} +813.000 q^{49} -420.000 q^{51} +162.000 q^{53} +80.0000 q^{55} -24.0000 q^{57} +268.000 q^{59} +250.000 q^{61} +306.000 q^{63} -290.000 q^{65} -422.000 q^{67} +804.000 q^{69} +852.000 q^{71} +306.000 q^{73} +150.000 q^{75} -544.000 q^{77} +456.000 q^{79} -891.000 q^{81} -434.000 q^{83} +350.000 q^{85} -1452.00 q^{87} -726.000 q^{89} +1972.00 q^{91} -600.000 q^{93} +20.0000 q^{95} +1378.00 q^{97} -144.000 q^{99} +O(q^{100})$$

## 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
$$3$$ 6.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 0 0
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 34.0000 1.83583 0.917914 0.396780i $$-0.129872\pi$$
0.917914 + 0.396780i $$0.129872\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −16.0000 −0.438562 −0.219281 0.975662i $$-0.570371\pi$$
−0.219281 + 0.975662i $$0.570371\pi$$
$$12$$ 0 0
$$13$$ 58.0000 1.23741 0.618704 0.785624i $$-0.287658\pi$$
0.618704 + 0.785624i $$0.287658\pi$$
$$14$$ 0 0
$$15$$ −30.0000 −0.516398
$$16$$ 0 0
$$17$$ −70.0000 −0.998676 −0.499338 0.866407i $$-0.666423\pi$$
−0.499338 + 0.866407i $$0.666423\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.0482980 −0.0241490 0.999708i $$-0.507688\pi$$
−0.0241490 + 0.999708i $$0.507688\pi$$
$$20$$ 0 0
$$21$$ 204.000 2.11983
$$22$$ 0 0
$$23$$ 134.000 1.21482 0.607412 0.794387i $$-0.292208\pi$$
0.607412 + 0.794387i $$0.292208\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −108.000 −0.769800
$$28$$ 0 0
$$29$$ −242.000 −1.54960 −0.774798 0.632209i $$-0.782148\pi$$
−0.774798 + 0.632209i $$0.782148\pi$$
$$30$$ 0 0
$$31$$ −100.000 −0.579372 −0.289686 0.957122i $$-0.593551\pi$$
−0.289686 + 0.957122i $$0.593551\pi$$
$$32$$ 0 0
$$33$$ −96.0000 −0.506408
$$34$$ 0 0
$$35$$ −170.000 −0.821007
$$36$$ 0 0
$$37$$ −438.000 −1.94613 −0.973064 0.230534i $$-0.925953\pi$$
−0.973064 + 0.230534i $$0.925953\pi$$
$$38$$ 0 0
$$39$$ 348.000 1.42884
$$40$$ 0 0
$$41$$ −138.000 −0.525658 −0.262829 0.964842i $$-0.584656\pi$$
−0.262829 + 0.964842i $$0.584656\pi$$
$$42$$ 0 0
$$43$$ −178.000 −0.631273 −0.315637 0.948880i $$-0.602218\pi$$
−0.315637 + 0.948880i $$0.602218\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ −22.0000 −0.0682772 −0.0341386 0.999417i $$-0.510869\pi$$
−0.0341386 + 0.999417i $$0.510869\pi$$
$$48$$ 0 0
$$49$$ 813.000 2.37026
$$50$$ 0 0
$$51$$ −420.000 −1.15317
$$52$$ 0 0
$$53$$ 162.000 0.419857 0.209928 0.977717i $$-0.432677\pi$$
0.209928 + 0.977717i $$0.432677\pi$$
$$54$$ 0 0
$$55$$ 80.0000 0.196131
$$56$$ 0 0
$$57$$ −24.0000 −0.0557698
$$58$$ 0 0
$$59$$ 268.000 0.591367 0.295683 0.955286i $$-0.404453\pi$$
0.295683 + 0.955286i $$0.404453\pi$$
$$60$$ 0 0
$$61$$ 250.000 0.524741 0.262371 0.964967i $$-0.415496\pi$$
0.262371 + 0.964967i $$0.415496\pi$$
$$62$$ 0 0
$$63$$ 306.000 0.611942
$$64$$ 0 0
$$65$$ −290.000 −0.553386
$$66$$ 0 0
$$67$$ −422.000 −0.769485 −0.384743 0.923024i $$-0.625710\pi$$
−0.384743 + 0.923024i $$0.625710\pi$$
$$68$$ 0 0
$$69$$ 804.000 1.40276
$$70$$ 0 0
$$71$$ 852.000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 306.000 0.490611 0.245305 0.969446i $$-0.421112\pi$$
0.245305 + 0.969446i $$0.421112\pi$$
$$74$$ 0 0
$$75$$ 150.000 0.230940
$$76$$ 0 0
$$77$$ −544.000 −0.805124
$$78$$ 0 0
$$79$$ 456.000 0.649418 0.324709 0.945814i $$-0.394734\pi$$
0.324709 + 0.945814i $$0.394734\pi$$
$$80$$ 0 0
$$81$$ −891.000 −1.22222
$$82$$ 0 0
$$83$$ −434.000 −0.573948 −0.286974 0.957938i $$-0.592649\pi$$
−0.286974 + 0.957938i $$0.592649\pi$$
$$84$$ 0 0
$$85$$ 350.000 0.446622
$$86$$ 0 0
$$87$$ −1452.00 −1.78932
$$88$$ 0 0
$$89$$ −726.000 −0.864672 −0.432336 0.901712i $$-0.642311\pi$$
−0.432336 + 0.901712i $$0.642311\pi$$
$$90$$ 0 0
$$91$$ 1972.00 2.27167
$$92$$ 0 0
$$93$$ −600.000 −0.669001
$$94$$ 0 0
$$95$$ 20.0000 0.0215995
$$96$$ 0 0
$$97$$ 1378.00 1.44242 0.721210 0.692717i $$-0.243586\pi$$
0.721210 + 0.692717i $$0.243586\pi$$
$$98$$ 0 0
$$99$$ −144.000 −0.146187
$$100$$ 0 0
$$101$$ 126.000 0.124133 0.0620667 0.998072i $$-0.480231\pi$$
0.0620667 + 0.998072i $$0.480231\pi$$
$$102$$ 0 0
$$103$$ 1262.00 1.20727 0.603634 0.797262i $$-0.293719\pi$$
0.603634 + 0.797262i $$0.293719\pi$$
$$104$$ 0 0
$$105$$ −1020.00 −0.948017
$$106$$ 0 0
$$107$$ −510.000 −0.460781 −0.230390 0.973098i $$-0.574000\pi$$
−0.230390 + 0.973098i $$0.574000\pi$$
$$108$$ 0 0
$$109$$ 26.0000 0.0228472 0.0114236 0.999935i $$-0.496364\pi$$
0.0114236 + 0.999935i $$0.496364\pi$$
$$110$$ 0 0
$$111$$ −2628.00 −2.24720
$$112$$ 0 0
$$113$$ 1242.00 1.03396 0.516980 0.855997i $$-0.327056\pi$$
0.516980 + 0.855997i $$0.327056\pi$$
$$114$$ 0 0
$$115$$ −670.000 −0.543285
$$116$$ 0 0
$$117$$ 522.000 0.412469
$$118$$ 0 0
$$119$$ −2380.00 −1.83340
$$120$$ 0 0
$$121$$ −1075.00 −0.807663
$$122$$ 0 0
$$123$$ −828.000 −0.606978
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 978.000 0.683334 0.341667 0.939821i $$-0.389008\pi$$
0.341667 + 0.939821i $$0.389008\pi$$
$$128$$ 0 0
$$129$$ −1068.00 −0.728931
$$130$$ 0 0
$$131$$ 912.000 0.608258 0.304129 0.952631i $$-0.401635\pi$$
0.304129 + 0.952631i $$0.401635\pi$$
$$132$$ 0 0
$$133$$ −136.000 −0.0886669
$$134$$ 0 0
$$135$$ 540.000 0.344265
$$136$$ 0 0
$$137$$ −926.000 −0.577471 −0.288735 0.957409i $$-0.593235\pi$$
−0.288735 + 0.957409i $$0.593235\pi$$
$$138$$ 0 0
$$139$$ −516.000 −0.314867 −0.157434 0.987530i $$-0.550322\pi$$
−0.157434 + 0.987530i $$0.550322\pi$$
$$140$$ 0 0
$$141$$ −132.000 −0.0788398
$$142$$ 0 0
$$143$$ −928.000 −0.542680
$$144$$ 0 0
$$145$$ 1210.00 0.693000
$$146$$ 0 0
$$147$$ 4878.00 2.73694
$$148$$ 0 0
$$149$$ −958.000 −0.526728 −0.263364 0.964697i $$-0.584832\pi$$
−0.263364 + 0.964697i $$0.584832\pi$$
$$150$$ 0 0
$$151$$ −332.000 −0.178926 −0.0894628 0.995990i $$-0.528515\pi$$
−0.0894628 + 0.995990i $$0.528515\pi$$
$$152$$ 0 0
$$153$$ −630.000 −0.332892
$$154$$ 0 0
$$155$$ 500.000 0.259103
$$156$$ 0 0
$$157$$ −1022.00 −0.519519 −0.259759 0.965673i $$-0.583643\pi$$
−0.259759 + 0.965673i $$0.583643\pi$$
$$158$$ 0 0
$$159$$ 972.000 0.484809
$$160$$ 0 0
$$161$$ 4556.00 2.23021
$$162$$ 0 0
$$163$$ 926.000 0.444969 0.222484 0.974936i $$-0.428583\pi$$
0.222484 + 0.974936i $$0.428583\pi$$
$$164$$ 0 0
$$165$$ 480.000 0.226472
$$166$$ 0 0
$$167$$ −654.000 −0.303042 −0.151521 0.988454i $$-0.548417\pi$$
−0.151521 + 0.988454i $$0.548417\pi$$
$$168$$ 0 0
$$169$$ 1167.00 0.531179
$$170$$ 0 0
$$171$$ −36.0000 −0.0160993
$$172$$ 0 0
$$173$$ −1294.00 −0.568676 −0.284338 0.958724i $$-0.591774\pi$$
−0.284338 + 0.958724i $$0.591774\pi$$
$$174$$ 0 0
$$175$$ 850.000 0.367165
$$176$$ 0 0
$$177$$ 1608.00 0.682851
$$178$$ 0 0
$$179$$ 2836.00 1.18420 0.592102 0.805863i $$-0.298298\pi$$
0.592102 + 0.805863i $$0.298298\pi$$
$$180$$ 0 0
$$181$$ 1742.00 0.715369 0.357685 0.933842i $$-0.383566\pi$$
0.357685 + 0.933842i $$0.383566\pi$$
$$182$$ 0 0
$$183$$ 1500.00 0.605919
$$184$$ 0 0
$$185$$ 2190.00 0.870335
$$186$$ 0 0
$$187$$ 1120.00 0.437981
$$188$$ 0 0
$$189$$ −3672.00 −1.41322
$$190$$ 0 0
$$191$$ −4460.00 −1.68960 −0.844802 0.535079i $$-0.820282\pi$$
−0.844802 + 0.535079i $$0.820282\pi$$
$$192$$ 0 0
$$193$$ −3782.00 −1.41054 −0.705270 0.708939i $$-0.749174\pi$$
−0.705270 + 0.708939i $$0.749174\pi$$
$$194$$ 0 0
$$195$$ −1740.00 −0.638995
$$196$$ 0 0
$$197$$ 4474.00 1.61807 0.809034 0.587762i $$-0.199991\pi$$
0.809034 + 0.587762i $$0.199991\pi$$
$$198$$ 0 0
$$199$$ −3608.00 −1.28525 −0.642624 0.766182i $$-0.722154\pi$$
−0.642624 + 0.766182i $$0.722154\pi$$
$$200$$ 0 0
$$201$$ −2532.00 −0.888525
$$202$$ 0 0
$$203$$ −8228.00 −2.84479
$$204$$ 0 0
$$205$$ 690.000 0.235081
$$206$$ 0 0
$$207$$ 1206.00 0.404941
$$208$$ 0 0
$$209$$ 64.0000 0.0211817
$$210$$ 0 0
$$211$$ 256.000 0.0835250 0.0417625 0.999128i $$-0.486703\pi$$
0.0417625 + 0.999128i $$0.486703\pi$$
$$212$$ 0 0
$$213$$ 5112.00 1.64445
$$214$$ 0 0
$$215$$ 890.000 0.282314
$$216$$ 0 0
$$217$$ −3400.00 −1.06363
$$218$$ 0 0
$$219$$ 1836.00 0.566509
$$220$$ 0 0
$$221$$ −4060.00 −1.23577
$$222$$ 0 0
$$223$$ 5158.00 1.54890 0.774451 0.632634i $$-0.218026\pi$$
0.774451 + 0.632634i $$0.218026\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ 2226.00 0.650858 0.325429 0.945566i $$-0.394491\pi$$
0.325429 + 0.945566i $$0.394491\pi$$
$$228$$ 0 0
$$229$$ 2086.00 0.601951 0.300975 0.953632i $$-0.402688\pi$$
0.300975 + 0.953632i $$0.402688\pi$$
$$230$$ 0 0
$$231$$ −3264.00 −0.929677
$$232$$ 0 0
$$233$$ −5718.00 −1.60772 −0.803860 0.594819i $$-0.797224\pi$$
−0.803860 + 0.594819i $$0.797224\pi$$
$$234$$ 0 0
$$235$$ 110.000 0.0305345
$$236$$ 0 0
$$237$$ 2736.00 0.749883
$$238$$ 0 0
$$239$$ 3624.00 0.980825 0.490412 0.871491i $$-0.336846\pi$$
0.490412 + 0.871491i $$0.336846\pi$$
$$240$$ 0 0
$$241$$ −82.0000 −0.0219174 −0.0109587 0.999940i $$-0.503488\pi$$
−0.0109587 + 0.999940i $$0.503488\pi$$
$$242$$ 0 0
$$243$$ −2430.00 −0.641500
$$244$$ 0 0
$$245$$ −4065.00 −1.06001
$$246$$ 0 0
$$247$$ −232.000 −0.0597644
$$248$$ 0 0
$$249$$ −2604.00 −0.662738
$$250$$ 0 0
$$251$$ 5040.00 1.26742 0.633709 0.773571i $$-0.281532\pi$$
0.633709 + 0.773571i $$0.281532\pi$$
$$252$$ 0 0
$$253$$ −2144.00 −0.532775
$$254$$ 0 0
$$255$$ 2100.00 0.515714
$$256$$ 0 0
$$257$$ −2310.00 −0.560676 −0.280338 0.959901i $$-0.590447\pi$$
−0.280338 + 0.959901i $$0.590447\pi$$
$$258$$ 0 0
$$259$$ −14892.0 −3.57276
$$260$$ 0 0
$$261$$ −2178.00 −0.516532
$$262$$ 0 0
$$263$$ 4110.00 0.963625 0.481813 0.876274i $$-0.339979\pi$$
0.481813 + 0.876274i $$0.339979\pi$$
$$264$$ 0 0
$$265$$ −810.000 −0.187766
$$266$$ 0 0
$$267$$ −4356.00 −0.998438
$$268$$ 0 0
$$269$$ 746.000 0.169087 0.0845435 0.996420i $$-0.473057\pi$$
0.0845435 + 0.996420i $$0.473057\pi$$
$$270$$ 0 0
$$271$$ 4596.00 1.03021 0.515105 0.857127i $$-0.327753\pi$$
0.515105 + 0.857127i $$0.327753\pi$$
$$272$$ 0 0
$$273$$ 11832.0 2.62310
$$274$$ 0 0
$$275$$ −400.000 −0.0877124
$$276$$ 0 0
$$277$$ −2206.00 −0.478504 −0.239252 0.970957i $$-0.576902\pi$$
−0.239252 + 0.970957i $$0.576902\pi$$
$$278$$ 0 0
$$279$$ −900.000 −0.193124
$$280$$ 0 0
$$281$$ 8278.00 1.75738 0.878691 0.477392i $$-0.158418\pi$$
0.878691 + 0.477392i $$0.158418\pi$$
$$282$$ 0 0
$$283$$ −1178.00 −0.247438 −0.123719 0.992317i $$-0.539482\pi$$
−0.123719 + 0.992317i $$0.539482\pi$$
$$284$$ 0 0
$$285$$ 120.000 0.0249410
$$286$$ 0 0
$$287$$ −4692.00 −0.965017
$$288$$ 0 0
$$289$$ −13.0000 −0.00264604
$$290$$ 0 0
$$291$$ 8268.00 1.66556
$$292$$ 0 0
$$293$$ 106.000 0.0211351 0.0105676 0.999944i $$-0.496636\pi$$
0.0105676 + 0.999944i $$0.496636\pi$$
$$294$$ 0 0
$$295$$ −1340.00 −0.264467
$$296$$ 0 0
$$297$$ 1728.00 0.337605
$$298$$ 0 0
$$299$$ 7772.00 1.50323
$$300$$ 0 0
$$301$$ −6052.00 −1.15891
$$302$$ 0 0
$$303$$ 756.000 0.143337
$$304$$ 0 0
$$305$$ −1250.00 −0.234671
$$306$$ 0 0
$$307$$ −8134.00 −1.51216 −0.756078 0.654482i $$-0.772887\pi$$
−0.756078 + 0.654482i $$0.772887\pi$$
$$308$$ 0 0
$$309$$ 7572.00 1.39403
$$310$$ 0 0
$$311$$ 4396.00 0.801525 0.400763 0.916182i $$-0.368745\pi$$
0.400763 + 0.916182i $$0.368745\pi$$
$$312$$ 0 0
$$313$$ 4826.00 0.871507 0.435753 0.900066i $$-0.356482\pi$$
0.435753 + 0.900066i $$0.356482\pi$$
$$314$$ 0 0
$$315$$ −1530.00 −0.273669
$$316$$ 0 0
$$317$$ 7026.00 1.24486 0.622428 0.782677i $$-0.286146\pi$$
0.622428 + 0.782677i $$0.286146\pi$$
$$318$$ 0 0
$$319$$ 3872.00 0.679594
$$320$$ 0 0
$$321$$ −3060.00 −0.532064
$$322$$ 0 0
$$323$$ 280.000 0.0482341
$$324$$ 0 0
$$325$$ 1450.00 0.247482
$$326$$ 0 0
$$327$$ 156.000 0.0263817
$$328$$ 0 0
$$329$$ −748.000 −0.125345
$$330$$ 0 0
$$331$$ −8808.00 −1.46263 −0.731316 0.682038i $$-0.761094\pi$$
−0.731316 + 0.682038i $$0.761094\pi$$
$$332$$ 0 0
$$333$$ −3942.00 −0.648710
$$334$$ 0 0
$$335$$ 2110.00 0.344124
$$336$$ 0 0
$$337$$ 5602.00 0.905520 0.452760 0.891632i $$-0.350439\pi$$
0.452760 + 0.891632i $$0.350439\pi$$
$$338$$ 0 0
$$339$$ 7452.00 1.19391
$$340$$ 0 0
$$341$$ 1600.00 0.254090
$$342$$ 0 0
$$343$$ 15980.0 2.51557
$$344$$ 0 0
$$345$$ −4020.00 −0.627332
$$346$$ 0 0
$$347$$ 6634.00 1.02632 0.513158 0.858294i $$-0.328475\pi$$
0.513158 + 0.858294i $$0.328475\pi$$
$$348$$ 0 0
$$349$$ 3198.00 0.490501 0.245251 0.969460i $$-0.421130\pi$$
0.245251 + 0.969460i $$0.421130\pi$$
$$350$$ 0 0
$$351$$ −6264.00 −0.952557
$$352$$ 0 0
$$353$$ −5230.00 −0.788569 −0.394284 0.918988i $$-0.629008\pi$$
−0.394284 + 0.918988i $$0.629008\pi$$
$$354$$ 0 0
$$355$$ −4260.00 −0.636894
$$356$$ 0 0
$$357$$ −14280.0 −2.11702
$$358$$ 0 0
$$359$$ 312.000 0.0458683 0.0229342 0.999737i $$-0.492699\pi$$
0.0229342 + 0.999737i $$0.492699\pi$$
$$360$$ 0 0
$$361$$ −6843.00 −0.997667
$$362$$ 0 0
$$363$$ −6450.00 −0.932609
$$364$$ 0 0
$$365$$ −1530.00 −0.219408
$$366$$ 0 0
$$367$$ −10790.0 −1.53470 −0.767348 0.641231i $$-0.778424\pi$$
−0.767348 + 0.641231i $$0.778424\pi$$
$$368$$ 0 0
$$369$$ −1242.00 −0.175219
$$370$$ 0 0
$$371$$ 5508.00 0.770785
$$372$$ 0 0
$$373$$ −4190.00 −0.581635 −0.290818 0.956778i $$-0.593927\pi$$
−0.290818 + 0.956778i $$0.593927\pi$$
$$374$$ 0 0
$$375$$ −750.000 −0.103280
$$376$$ 0 0
$$377$$ −14036.0 −1.91748
$$378$$ 0 0
$$379$$ 6980.00 0.946012 0.473006 0.881059i $$-0.343169\pi$$
0.473006 + 0.881059i $$0.343169\pi$$
$$380$$ 0 0
$$381$$ 5868.00 0.789047
$$382$$ 0 0
$$383$$ −13962.0 −1.86273 −0.931364 0.364089i $$-0.881380\pi$$
−0.931364 + 0.364089i $$0.881380\pi$$
$$384$$ 0 0
$$385$$ 2720.00 0.360062
$$386$$ 0 0
$$387$$ −1602.00 −0.210424
$$388$$ 0 0
$$389$$ 3810.00 0.496593 0.248296 0.968684i $$-0.420129\pi$$
0.248296 + 0.968684i $$0.420129\pi$$
$$390$$ 0 0
$$391$$ −9380.00 −1.21321
$$392$$ 0 0
$$393$$ 5472.00 0.702356
$$394$$ 0 0
$$395$$ −2280.00 −0.290428
$$396$$ 0 0
$$397$$ −9158.00 −1.15775 −0.578875 0.815416i $$-0.696508\pi$$
−0.578875 + 0.815416i $$0.696508\pi$$
$$398$$ 0 0
$$399$$ −816.000 −0.102384
$$400$$ 0 0
$$401$$ 4866.00 0.605976 0.302988 0.952994i $$-0.402016\pi$$
0.302988 + 0.952994i $$0.402016\pi$$
$$402$$ 0 0
$$403$$ −5800.00 −0.716920
$$404$$ 0 0
$$405$$ 4455.00 0.546594
$$406$$ 0 0
$$407$$ 7008.00 0.853498
$$408$$ 0 0
$$409$$ 13486.0 1.63042 0.815208 0.579169i $$-0.196623\pi$$
0.815208 + 0.579169i $$0.196623\pi$$
$$410$$ 0 0
$$411$$ −5556.00 −0.666806
$$412$$ 0 0
$$413$$ 9112.00 1.08565
$$414$$ 0 0
$$415$$ 2170.00 0.256677
$$416$$ 0 0
$$417$$ −3096.00 −0.363577
$$418$$ 0 0
$$419$$ −5628.00 −0.656195 −0.328098 0.944644i $$-0.606407\pi$$
−0.328098 + 0.944644i $$0.606407\pi$$
$$420$$ 0 0
$$421$$ 7938.00 0.918942 0.459471 0.888193i $$-0.348039\pi$$
0.459471 + 0.888193i $$0.348039\pi$$
$$422$$ 0 0
$$423$$ −198.000 −0.0227591
$$424$$ 0 0
$$425$$ −1750.00 −0.199735
$$426$$ 0 0
$$427$$ 8500.00 0.963334
$$428$$ 0 0
$$429$$ −5568.00 −0.626633
$$430$$ 0 0
$$431$$ −1916.00 −0.214131 −0.107066 0.994252i $$-0.534145\pi$$
−0.107066 + 0.994252i $$0.534145\pi$$
$$432$$ 0 0
$$433$$ −16510.0 −1.83238 −0.916189 0.400746i $$-0.868751\pi$$
−0.916189 + 0.400746i $$0.868751\pi$$
$$434$$ 0 0
$$435$$ 7260.00 0.800208
$$436$$ 0 0
$$437$$ −536.000 −0.0586736
$$438$$ 0 0
$$439$$ 1256.00 0.136550 0.0682752 0.997667i $$-0.478250\pi$$
0.0682752 + 0.997667i $$0.478250\pi$$
$$440$$ 0 0
$$441$$ 7317.00 0.790087
$$442$$ 0 0
$$443$$ 12222.0 1.31080 0.655400 0.755282i $$-0.272500\pi$$
0.655400 + 0.755282i $$0.272500\pi$$
$$444$$ 0 0
$$445$$ 3630.00 0.386693
$$446$$ 0 0
$$447$$ −5748.00 −0.608213
$$448$$ 0 0
$$449$$ −5946.00 −0.624965 −0.312482 0.949924i $$-0.601160\pi$$
−0.312482 + 0.949924i $$0.601160\pi$$
$$450$$ 0 0
$$451$$ 2208.00 0.230534
$$452$$ 0 0
$$453$$ −1992.00 −0.206606
$$454$$ 0 0
$$455$$ −9860.00 −1.01592
$$456$$ 0 0
$$457$$ 1258.00 0.128768 0.0643838 0.997925i $$-0.479492\pi$$
0.0643838 + 0.997925i $$0.479492\pi$$
$$458$$ 0 0
$$459$$ 7560.00 0.768781
$$460$$ 0 0
$$461$$ 16422.0 1.65911 0.829554 0.558426i $$-0.188595\pi$$
0.829554 + 0.558426i $$0.188595\pi$$
$$462$$ 0 0
$$463$$ −2658.00 −0.266799 −0.133399 0.991062i $$-0.542589\pi$$
−0.133399 + 0.991062i $$0.542589\pi$$
$$464$$ 0 0
$$465$$ 3000.00 0.299186
$$466$$ 0 0
$$467$$ −3686.00 −0.365241 −0.182621 0.983183i $$-0.558458\pi$$
−0.182621 + 0.983183i $$0.558458\pi$$
$$468$$ 0 0
$$469$$ −14348.0 −1.41264
$$470$$ 0 0
$$471$$ −6132.00 −0.599889
$$472$$ 0 0
$$473$$ 2848.00 0.276852
$$474$$ 0 0
$$475$$ −100.000 −0.00965961
$$476$$ 0 0
$$477$$ 1458.00 0.139952
$$478$$ 0 0
$$479$$ −88.0000 −0.00839420 −0.00419710 0.999991i $$-0.501336\pi$$
−0.00419710 + 0.999991i $$0.501336\pi$$
$$480$$ 0 0
$$481$$ −25404.0 −2.40816
$$482$$ 0 0
$$483$$ 27336.0 2.57522
$$484$$ 0 0
$$485$$ −6890.00 −0.645070
$$486$$ 0 0
$$487$$ 14714.0 1.36911 0.684553 0.728963i $$-0.259997\pi$$
0.684553 + 0.728963i $$0.259997\pi$$
$$488$$ 0 0
$$489$$ 5556.00 0.513806
$$490$$ 0 0
$$491$$ 7344.00 0.675010 0.337505 0.941324i $$-0.390417\pi$$
0.337505 + 0.941324i $$0.390417\pi$$
$$492$$ 0 0
$$493$$ 16940.0 1.54754
$$494$$ 0 0
$$495$$ 720.000 0.0653770
$$496$$ 0 0
$$497$$ 28968.0 2.61447
$$498$$ 0 0
$$499$$ −1604.00 −0.143898 −0.0719488 0.997408i $$-0.522922\pi$$
−0.0719488 + 0.997408i $$0.522922\pi$$
$$500$$ 0 0
$$501$$ −3924.00 −0.349923
$$502$$ 0 0
$$503$$ −14802.0 −1.31210 −0.656052 0.754715i $$-0.727775\pi$$
−0.656052 + 0.754715i $$0.727775\pi$$
$$504$$ 0 0
$$505$$ −630.000 −0.0555141
$$506$$ 0 0
$$507$$ 7002.00 0.613353
$$508$$ 0 0
$$509$$ −22514.0 −1.96054 −0.980271 0.197660i $$-0.936666\pi$$
−0.980271 + 0.197660i $$0.936666\pi$$
$$510$$ 0 0
$$511$$ 10404.0 0.900677
$$512$$ 0 0
$$513$$ 432.000 0.0371799
$$514$$ 0 0
$$515$$ −6310.00 −0.539906
$$516$$ 0 0
$$517$$ 352.000 0.0299438
$$518$$ 0 0
$$519$$ −7764.00 −0.656651
$$520$$ 0 0
$$521$$ −6710.00 −0.564243 −0.282121 0.959379i $$-0.591038\pi$$
−0.282121 + 0.959379i $$0.591038\pi$$
$$522$$ 0 0
$$523$$ −7930.00 −0.663011 −0.331505 0.943453i $$-0.607557\pi$$
−0.331505 + 0.943453i $$0.607557\pi$$
$$524$$ 0 0
$$525$$ 5100.00 0.423966
$$526$$ 0 0
$$527$$ 7000.00 0.578605
$$528$$ 0 0
$$529$$ 5789.00 0.475795
$$530$$ 0 0
$$531$$ 2412.00 0.197122
$$532$$ 0 0
$$533$$ −8004.00 −0.650454
$$534$$ 0 0
$$535$$ 2550.00 0.206068
$$536$$ 0 0
$$537$$ 17016.0 1.36740
$$538$$ 0 0
$$539$$ −13008.0 −1.03951
$$540$$ 0 0
$$541$$ 4918.00 0.390834 0.195417 0.980720i $$-0.437394\pi$$
0.195417 + 0.980720i $$0.437394\pi$$
$$542$$ 0 0
$$543$$ 10452.0 0.826037
$$544$$ 0 0
$$545$$ −130.000 −0.0102176
$$546$$ 0 0
$$547$$ 3922.00 0.306568 0.153284 0.988182i $$-0.451015\pi$$
0.153284 + 0.988182i $$0.451015\pi$$
$$548$$ 0 0
$$549$$ 2250.00 0.174914
$$550$$ 0 0
$$551$$ 968.000 0.0748424
$$552$$ 0 0
$$553$$ 15504.0 1.19222
$$554$$ 0 0
$$555$$ 13140.0 1.00498
$$556$$ 0 0
$$557$$ 17786.0 1.35299 0.676496 0.736446i $$-0.263497\pi$$
0.676496 + 0.736446i $$0.263497\pi$$
$$558$$ 0 0
$$559$$ −10324.0 −0.781143
$$560$$ 0 0
$$561$$ 6720.00 0.505737
$$562$$ 0 0
$$563$$ −20266.0 −1.51707 −0.758535 0.651633i $$-0.774084\pi$$
−0.758535 + 0.651633i $$0.774084\pi$$
$$564$$ 0 0
$$565$$ −6210.00 −0.462401
$$566$$ 0 0
$$567$$ −30294.0 −2.24379
$$568$$ 0 0
$$569$$ 13358.0 0.984177 0.492088 0.870545i $$-0.336234\pi$$
0.492088 + 0.870545i $$0.336234\pi$$
$$570$$ 0 0
$$571$$ −16360.0 −1.19903 −0.599514 0.800364i $$-0.704639\pi$$
−0.599514 + 0.800364i $$0.704639\pi$$
$$572$$ 0 0
$$573$$ −26760.0 −1.95099
$$574$$ 0 0
$$575$$ 3350.00 0.242965
$$576$$ 0 0
$$577$$ −15574.0 −1.12366 −0.561832 0.827251i $$-0.689903\pi$$
−0.561832 + 0.827251i $$0.689903\pi$$
$$578$$ 0 0
$$579$$ −22692.0 −1.62875
$$580$$ 0 0
$$581$$ −14756.0 −1.05367
$$582$$ 0 0
$$583$$ −2592.00 −0.184133
$$584$$ 0 0
$$585$$ −2610.00 −0.184462
$$586$$ 0 0
$$587$$ −6654.00 −0.467870 −0.233935 0.972252i $$-0.575160\pi$$
−0.233935 + 0.972252i $$0.575160\pi$$
$$588$$ 0 0
$$589$$ 400.000 0.0279825
$$590$$ 0 0
$$591$$ 26844.0 1.86838
$$592$$ 0 0
$$593$$ −17742.0 −1.22863 −0.614314 0.789062i $$-0.710567\pi$$
−0.614314 + 0.789062i $$0.710567\pi$$
$$594$$ 0 0
$$595$$ 11900.0 0.819920
$$596$$ 0 0
$$597$$ −21648.0 −1.48408
$$598$$ 0 0
$$599$$ −15840.0 −1.08048 −0.540238 0.841512i $$-0.681666\pi$$
−0.540238 + 0.841512i $$0.681666\pi$$
$$600$$ 0 0
$$601$$ −3002.00 −0.203751 −0.101875 0.994797i $$-0.532484\pi$$
−0.101875 + 0.994797i $$0.532484\pi$$
$$602$$ 0 0
$$603$$ −3798.00 −0.256495
$$604$$ 0 0
$$605$$ 5375.00 0.361198
$$606$$ 0 0
$$607$$ 23610.0 1.57875 0.789374 0.613912i $$-0.210405\pi$$
0.789374 + 0.613912i $$0.210405\pi$$
$$608$$ 0 0
$$609$$ −49368.0 −3.28488
$$610$$ 0 0
$$611$$ −1276.00 −0.0844868
$$612$$ 0 0
$$613$$ 23850.0 1.57144 0.785720 0.618583i $$-0.212293\pi$$
0.785720 + 0.618583i $$0.212293\pi$$
$$614$$ 0 0
$$615$$ 4140.00 0.271449
$$616$$ 0 0
$$617$$ −5334.00 −0.348037 −0.174018 0.984742i $$-0.555675\pi$$
−0.174018 + 0.984742i $$0.555675\pi$$
$$618$$ 0 0
$$619$$ 2164.00 0.140515 0.0702573 0.997529i $$-0.477618\pi$$
0.0702573 + 0.997529i $$0.477618\pi$$
$$620$$ 0 0
$$621$$ −14472.0 −0.935171
$$622$$ 0 0
$$623$$ −24684.0 −1.58739
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 384.000 0.0244585
$$628$$ 0 0
$$629$$ 30660.0 1.94355
$$630$$ 0 0
$$631$$ 25220.0 1.59111 0.795557 0.605879i $$-0.207179\pi$$
0.795557 + 0.605879i $$0.207179\pi$$
$$632$$ 0 0
$$633$$ 1536.00 0.0964463
$$634$$ 0 0
$$635$$ −4890.00 −0.305596
$$636$$ 0 0
$$637$$ 47154.0 2.93298
$$638$$ 0 0
$$639$$ 7668.00 0.474713
$$640$$ 0 0
$$641$$ −12306.0 −0.758280 −0.379140 0.925339i $$-0.623780\pi$$
−0.379140 + 0.925339i $$0.623780\pi$$
$$642$$ 0 0
$$643$$ 27414.0 1.68134 0.840671 0.541547i $$-0.182161\pi$$
0.840671 + 0.541547i $$0.182161\pi$$
$$644$$ 0 0
$$645$$ 5340.00 0.325988
$$646$$ 0 0
$$647$$ 21834.0 1.32671 0.663356 0.748304i $$-0.269131\pi$$
0.663356 + 0.748304i $$0.269131\pi$$
$$648$$ 0 0
$$649$$ −4288.00 −0.259351
$$650$$ 0 0
$$651$$ −20400.0 −1.22817
$$652$$ 0 0
$$653$$ −23998.0 −1.43815 −0.719077 0.694931i $$-0.755435\pi$$
−0.719077 + 0.694931i $$0.755435\pi$$
$$654$$ 0 0
$$655$$ −4560.00 −0.272021
$$656$$ 0 0
$$657$$ 2754.00 0.163537
$$658$$ 0 0
$$659$$ 32004.0 1.89180 0.945902 0.324452i $$-0.105180\pi$$
0.945902 + 0.324452i $$0.105180\pi$$
$$660$$ 0 0
$$661$$ −8526.00 −0.501699 −0.250849 0.968026i $$-0.580710\pi$$
−0.250849 + 0.968026i $$0.580710\pi$$
$$662$$ 0 0
$$663$$ −24360.0 −1.42694
$$664$$ 0 0
$$665$$ 680.000 0.0396530
$$666$$ 0 0
$$667$$ −32428.0 −1.88248
$$668$$ 0 0
$$669$$ 30948.0 1.78852
$$670$$ 0 0
$$671$$ −4000.00 −0.230132
$$672$$ 0 0
$$673$$ 8178.00 0.468408 0.234204 0.972187i $$-0.424752\pi$$
0.234204 + 0.972187i $$0.424752\pi$$
$$674$$ 0 0
$$675$$ −2700.00 −0.153960
$$676$$ 0 0
$$677$$ −16646.0 −0.944989 −0.472495 0.881334i $$-0.656646\pi$$
−0.472495 + 0.881334i $$0.656646\pi$$
$$678$$ 0 0
$$679$$ 46852.0 2.64803
$$680$$ 0 0
$$681$$ 13356.0 0.751546
$$682$$ 0 0
$$683$$ 22446.0 1.25750 0.628750 0.777608i $$-0.283567\pi$$
0.628750 + 0.777608i $$0.283567\pi$$
$$684$$ 0 0
$$685$$ 4630.00 0.258253
$$686$$ 0 0
$$687$$ 12516.0 0.695073
$$688$$ 0 0
$$689$$ 9396.00 0.519534
$$690$$ 0 0
$$691$$ −35336.0 −1.94536 −0.972681 0.232147i $$-0.925425\pi$$
−0.972681 + 0.232147i $$0.925425\pi$$
$$692$$ 0 0
$$693$$ −4896.00 −0.268375
$$694$$ 0 0
$$695$$ 2580.00 0.140813
$$696$$ 0 0
$$697$$ 9660.00 0.524962
$$698$$ 0 0
$$699$$ −34308.0 −1.85643
$$700$$ 0 0
$$701$$ 3482.00 0.187608 0.0938041 0.995591i $$-0.470097\pi$$
0.0938041 + 0.995591i $$0.470097\pi$$
$$702$$ 0 0
$$703$$ 1752.00 0.0939942
$$704$$ 0 0
$$705$$ 660.000 0.0352582
$$706$$ 0 0
$$707$$ 4284.00 0.227887
$$708$$ 0 0
$$709$$ −19402.0 −1.02773 −0.513863 0.857872i $$-0.671786\pi$$
−0.513863 + 0.857872i $$0.671786\pi$$
$$710$$ 0 0
$$711$$ 4104.00 0.216473
$$712$$ 0 0
$$713$$ −13400.0 −0.703834
$$714$$ 0 0
$$715$$ 4640.00 0.242694
$$716$$ 0 0
$$717$$ 21744.0 1.13256
$$718$$ 0 0
$$719$$ 9896.00 0.513294 0.256647 0.966505i $$-0.417382\pi$$
0.256647 + 0.966505i $$0.417382\pi$$
$$720$$ 0 0
$$721$$ 42908.0 2.21633
$$722$$ 0 0
$$723$$ −492.000 −0.0253080
$$724$$ 0 0
$$725$$ −6050.00 −0.309919
$$726$$ 0 0
$$727$$ −494.000 −0.0252014 −0.0126007 0.999921i $$-0.504011\pi$$
−0.0126007 + 0.999921i $$0.504011\pi$$
$$728$$ 0 0
$$729$$ 9477.00 0.481481
$$730$$ 0 0
$$731$$ 12460.0 0.630437
$$732$$ 0 0
$$733$$ 9282.00 0.467720 0.233860 0.972270i $$-0.424864\pi$$
0.233860 + 0.972270i $$0.424864\pi$$
$$734$$ 0 0
$$735$$ −24390.0 −1.22400
$$736$$ 0 0
$$737$$ 6752.00 0.337467
$$738$$ 0 0
$$739$$ 3252.00 0.161877 0.0809383 0.996719i $$-0.474208\pi$$
0.0809383 + 0.996719i $$0.474208\pi$$
$$740$$ 0 0
$$741$$ −1392.00 −0.0690100
$$742$$ 0 0
$$743$$ 4710.00 0.232561 0.116281 0.993216i $$-0.462903\pi$$
0.116281 + 0.993216i $$0.462903\pi$$
$$744$$ 0 0
$$745$$ 4790.00 0.235560
$$746$$ 0 0
$$747$$ −3906.00 −0.191316
$$748$$ 0 0
$$749$$ −17340.0 −0.845914
$$750$$ 0 0
$$751$$ −25764.0 −1.25185 −0.625927 0.779882i $$-0.715279\pi$$
−0.625927 + 0.779882i $$0.715279\pi$$
$$752$$ 0 0
$$753$$ 30240.0 1.46349
$$754$$ 0 0
$$755$$ 1660.00 0.0800180
$$756$$ 0 0
$$757$$ −30094.0 −1.44489 −0.722447 0.691426i $$-0.756983\pi$$
−0.722447 + 0.691426i $$0.756983\pi$$
$$758$$ 0 0
$$759$$ −12864.0 −0.615196
$$760$$ 0 0
$$761$$ 22362.0 1.06521 0.532603 0.846365i $$-0.321214\pi$$
0.532603 + 0.846365i $$0.321214\pi$$
$$762$$ 0 0
$$763$$ 884.000 0.0419436
$$764$$ 0 0
$$765$$ 3150.00 0.148874
$$766$$ 0 0
$$767$$ 15544.0 0.731762
$$768$$ 0 0
$$769$$ −30398.0 −1.42546 −0.712731 0.701438i $$-0.752542\pi$$
−0.712731 + 0.701438i $$0.752542\pi$$
$$770$$ 0 0
$$771$$ −13860.0 −0.647413
$$772$$ 0 0
$$773$$ 1290.00 0.0600234 0.0300117 0.999550i $$-0.490446\pi$$
0.0300117 + 0.999550i $$0.490446\pi$$
$$774$$ 0 0
$$775$$ −2500.00 −0.115874
$$776$$ 0 0
$$777$$ −89352.0 −4.12546
$$778$$ 0 0
$$779$$ 552.000 0.0253883
$$780$$ 0 0
$$781$$ −13632.0 −0.624573
$$782$$ 0 0
$$783$$ 26136.0 1.19288
$$784$$ 0 0
$$785$$ 5110.00 0.232336
$$786$$ 0 0
$$787$$ −14.0000 −0.000634112 0 −0.000317056 1.00000i $$-0.500101\pi$$
−0.000317056 1.00000i $$0.500101\pi$$
$$788$$ 0 0
$$789$$ 24660.0 1.11270
$$790$$ 0 0
$$791$$ 42228.0 1.89817
$$792$$ 0 0
$$793$$ 14500.0 0.649319
$$794$$ 0 0
$$795$$ −4860.00 −0.216813
$$796$$ 0 0
$$797$$ −38814.0 −1.72505 −0.862523 0.506017i $$-0.831117\pi$$
−0.862523 + 0.506017i $$0.831117\pi$$
$$798$$ 0 0
$$799$$ 1540.00 0.0681868
$$800$$ 0 0
$$801$$ −6534.00 −0.288224
$$802$$ 0 0
$$803$$ −4896.00 −0.215163
$$804$$ 0 0
$$805$$ −22780.0 −0.997378
$$806$$ 0 0
$$807$$ 4476.00 0.195245
$$808$$ 0 0
$$809$$ 27402.0 1.19086 0.595428 0.803408i $$-0.296982\pi$$
0.595428 + 0.803408i $$0.296982\pi$$
$$810$$ 0 0
$$811$$ 28576.0 1.23729 0.618643 0.785672i $$-0.287683\pi$$
0.618643 + 0.785672i $$0.287683\pi$$
$$812$$ 0 0
$$813$$ 27576.0 1.18958
$$814$$ 0 0
$$815$$ −4630.00 −0.198996
$$816$$ 0 0
$$817$$ 712.000 0.0304893
$$818$$ 0 0
$$819$$ 17748.0 0.757223
$$820$$ 0 0
$$821$$ 31762.0 1.35018 0.675092 0.737733i $$-0.264104\pi$$
0.675092 + 0.737733i $$0.264104\pi$$
$$822$$ 0 0
$$823$$ −20506.0 −0.868523 −0.434261 0.900787i $$-0.642991\pi$$
−0.434261 + 0.900787i $$0.642991\pi$$
$$824$$ 0 0
$$825$$ −2400.00 −0.101282
$$826$$ 0 0
$$827$$ −13014.0 −0.547208 −0.273604 0.961842i $$-0.588216\pi$$
−0.273604 + 0.961842i $$0.588216\pi$$
$$828$$ 0 0
$$829$$ −22790.0 −0.954800 −0.477400 0.878686i $$-0.658421\pi$$
−0.477400 + 0.878686i $$0.658421\pi$$
$$830$$ 0 0
$$831$$ −13236.0 −0.552529
$$832$$ 0 0
$$833$$ −56910.0 −2.36712
$$834$$ 0 0
$$835$$ 3270.00 0.135525
$$836$$ 0 0
$$837$$ 10800.0 0.446001
$$838$$ 0 0
$$839$$ −23696.0 −0.975062 −0.487531 0.873106i $$-0.662102\pi$$
−0.487531 + 0.873106i $$0.662102\pi$$
$$840$$ 0 0
$$841$$ 34175.0 1.40125
$$842$$ 0 0
$$843$$ 49668.0 2.02925
$$844$$ 0 0
$$845$$ −5835.00 −0.237550
$$846$$ 0 0
$$847$$ −36550.0 −1.48273
$$848$$ 0 0
$$849$$ −7068.00 −0.285716
$$850$$ 0 0
$$851$$ −58692.0 −2.36420
$$852$$ 0 0
$$853$$ 5306.00 0.212982 0.106491 0.994314i $$-0.466038\pi$$
0.106491 + 0.994314i $$0.466038\pi$$
$$854$$ 0 0
$$855$$ 180.000 0.00719985
$$856$$ 0 0
$$857$$ −21054.0 −0.839196 −0.419598 0.907710i $$-0.637829\pi$$
−0.419598 + 0.907710i $$0.637829\pi$$
$$858$$ 0 0
$$859$$ −7364.00 −0.292499 −0.146249 0.989248i $$-0.546720\pi$$
−0.146249 + 0.989248i $$0.546720\pi$$
$$860$$ 0 0
$$861$$ −28152.0 −1.11431
$$862$$ 0 0
$$863$$ −17226.0 −0.679467 −0.339733 0.940522i $$-0.610337\pi$$
−0.339733 + 0.940522i $$0.610337\pi$$
$$864$$ 0 0
$$865$$ 6470.00 0.254320
$$866$$ 0 0
$$867$$ −78.0000 −0.00305539
$$868$$ 0 0
$$869$$ −7296.00 −0.284810
$$870$$ 0 0
$$871$$ −24476.0 −0.952167
$$872$$ 0 0
$$873$$ 12402.0 0.480807
$$874$$ 0 0
$$875$$ −4250.00 −0.164201
$$876$$ 0 0
$$877$$ 21202.0 0.816352 0.408176 0.912903i $$-0.366165\pi$$
0.408176 + 0.912903i $$0.366165\pi$$
$$878$$ 0 0
$$879$$ 636.000 0.0244047
$$880$$ 0 0
$$881$$ −29490.0 −1.12774 −0.563872 0.825862i $$-0.690689\pi$$
−0.563872 + 0.825862i $$0.690689\pi$$
$$882$$ 0 0
$$883$$ −2570.00 −0.0979472 −0.0489736 0.998800i $$-0.515595\pi$$
−0.0489736 + 0.998800i $$0.515595\pi$$
$$884$$ 0 0
$$885$$ −8040.00 −0.305380
$$886$$ 0 0
$$887$$ −36334.0 −1.37540 −0.687698 0.725997i $$-0.741379\pi$$
−0.687698 + 0.725997i $$0.741379\pi$$
$$888$$ 0 0
$$889$$ 33252.0 1.25448
$$890$$ 0 0
$$891$$ 14256.0 0.536020
$$892$$ 0 0
$$893$$ 88.0000 0.00329766
$$894$$ 0 0
$$895$$ −14180.0 −0.529592
$$896$$ 0 0
$$897$$ 46632.0 1.73578
$$898$$ 0 0
$$899$$ 24200.0 0.897792
$$900$$ 0 0
$$901$$ −11340.0 −0.419301
$$902$$ 0 0
$$903$$ −36312.0 −1.33819
$$904$$ 0 0
$$905$$ −8710.00 −0.319923
$$906$$ 0 0
$$907$$ 12474.0 0.456662 0.228331 0.973584i $$-0.426673\pi$$
0.228331 + 0.973584i $$0.426673\pi$$
$$908$$ 0 0
$$909$$ 1134.00 0.0413778
$$910$$ 0 0
$$911$$ 41132.0 1.49590 0.747949 0.663756i $$-0.231039\pi$$
0.747949 + 0.663756i $$0.231039\pi$$
$$912$$ 0 0
$$913$$ 6944.00 0.251712
$$914$$ 0 0
$$915$$ −7500.00 −0.270975
$$916$$ 0 0
$$917$$ 31008.0 1.11666
$$918$$ 0 0
$$919$$ 38416.0 1.37892 0.689460 0.724324i $$-0.257848\pi$$
0.689460 + 0.724324i $$0.257848\pi$$
$$920$$ 0 0
$$921$$ −48804.0 −1.74609
$$922$$ 0 0
$$923$$ 49416.0 1.76224
$$924$$ 0 0
$$925$$ −10950.0 −0.389226
$$926$$ 0 0
$$927$$ 11358.0 0.402423
$$928$$ 0 0
$$929$$ 41302.0 1.45864 0.729319 0.684174i $$-0.239837\pi$$
0.729319 + 0.684174i $$0.239837\pi$$
$$930$$ 0 0
$$931$$ −3252.00 −0.114479
$$932$$ 0 0
$$933$$ 26376.0 0.925521
$$934$$ 0 0
$$935$$ −5600.00 −0.195871
$$936$$ 0 0
$$937$$ −26150.0 −0.911722 −0.455861 0.890051i $$-0.650669\pi$$
−0.455861 + 0.890051i $$0.650669\pi$$
$$938$$ 0 0
$$939$$ 28956.0 1.00633
$$940$$ 0 0
$$941$$ 35254.0 1.22130 0.610652 0.791899i $$-0.290907\pi$$
0.610652 + 0.791899i $$0.290907\pi$$
$$942$$ 0 0
$$943$$ −18492.0 −0.638582
$$944$$ 0 0
$$945$$ 18360.0 0.632011
$$946$$ 0 0
$$947$$ −18550.0 −0.636530 −0.318265 0.948002i $$-0.603100\pi$$
−0.318265 + 0.948002i $$0.603100\pi$$
$$948$$ 0 0
$$949$$ 17748.0 0.607086
$$950$$ 0 0
$$951$$ 42156.0 1.43744
$$952$$ 0 0
$$953$$ 17322.0 0.588788 0.294394 0.955684i $$-0.404882\pi$$
0.294394 + 0.955684i $$0.404882\pi$$
$$954$$ 0 0
$$955$$ 22300.0 0.755614
$$956$$ 0 0
$$957$$ 23232.0 0.784727
$$958$$ 0 0
$$959$$ −31484.0 −1.06014
$$960$$ 0 0
$$961$$ −19791.0 −0.664328
$$962$$ 0 0
$$963$$ −4590.00 −0.153594
$$964$$ 0 0
$$965$$ 18910.0 0.630813
$$966$$ 0 0
$$967$$ −35190.0 −1.17025 −0.585126 0.810942i $$-0.698955\pi$$
−0.585126 + 0.810942i $$0.698955\pi$$
$$968$$ 0 0
$$969$$ 1680.00 0.0556960
$$970$$ 0 0
$$971$$ 40696.0 1.34500 0.672501 0.740096i $$-0.265220\pi$$
0.672501 + 0.740096i $$0.265220\pi$$
$$972$$ 0 0
$$973$$ −17544.0 −0.578042
$$974$$ 0 0
$$975$$ 8700.00 0.285767
$$976$$ 0 0
$$977$$ 44306.0 1.45084 0.725422 0.688304i $$-0.241645\pi$$
0.725422 + 0.688304i $$0.241645\pi$$
$$978$$ 0 0
$$979$$ 11616.0 0.379212
$$980$$ 0 0
$$981$$ 234.000 0.00761574
$$982$$ 0 0
$$983$$ 18798.0 0.609932 0.304966 0.952363i $$-0.401355\pi$$
0.304966 + 0.952363i $$0.401355\pi$$
$$984$$ 0 0
$$985$$ −22370.0 −0.723622
$$986$$ 0 0
$$987$$ −4488.00 −0.144736
$$988$$ 0 0
$$989$$ −23852.0 −0.766885
$$990$$ 0 0
$$991$$ −2468.00 −0.0791106 −0.0395553 0.999217i $$-0.512594\pi$$
−0.0395553 + 0.999217i $$0.512594\pi$$
$$992$$ 0 0
$$993$$ −52848.0 −1.68890
$$994$$ 0 0
$$995$$ 18040.0 0.574780
$$996$$ 0 0
$$997$$ −61086.0 −1.94043 −0.970217 0.242237i $$-0.922119\pi$$
−0.970217 + 0.242237i $$0.922119\pi$$
$$998$$ 0 0
$$999$$ 47304.0 1.49813
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 80.4.a.e.1.1 1
3.2 odd 2 720.4.a.bd.1.1 1
4.3 odd 2 40.4.a.a.1.1 1
5.2 odd 4 400.4.c.f.49.1 2
5.3 odd 4 400.4.c.f.49.2 2
5.4 even 2 400.4.a.e.1.1 1
8.3 odd 2 320.4.a.l.1.1 1
8.5 even 2 320.4.a.c.1.1 1
12.11 even 2 360.4.a.h.1.1 1
16.3 odd 4 1280.4.d.p.641.1 2
16.5 even 4 1280.4.d.a.641.1 2
16.11 odd 4 1280.4.d.p.641.2 2
16.13 even 4 1280.4.d.a.641.2 2
20.3 even 4 200.4.c.c.49.1 2
20.7 even 4 200.4.c.c.49.2 2
20.19 odd 2 200.4.a.i.1.1 1
28.27 even 2 1960.4.a.h.1.1 1
40.19 odd 2 1600.4.a.j.1.1 1
40.29 even 2 1600.4.a.br.1.1 1
60.23 odd 4 1800.4.f.j.649.2 2
60.47 odd 4 1800.4.f.j.649.1 2
60.59 even 2 1800.4.a.bi.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.a.1.1 1 4.3 odd 2
80.4.a.e.1.1 1 1.1 even 1 trivial
200.4.a.i.1.1 1 20.19 odd 2
200.4.c.c.49.1 2 20.3 even 4
200.4.c.c.49.2 2 20.7 even 4
320.4.a.c.1.1 1 8.5 even 2
320.4.a.l.1.1 1 8.3 odd 2
360.4.a.h.1.1 1 12.11 even 2
400.4.a.e.1.1 1 5.4 even 2
400.4.c.f.49.1 2 5.2 odd 4
400.4.c.f.49.2 2 5.3 odd 4
720.4.a.bd.1.1 1 3.2 odd 2
1280.4.d.a.641.1 2 16.5 even 4
1280.4.d.a.641.2 2 16.13 even 4
1280.4.d.p.641.1 2 16.3 odd 4
1280.4.d.p.641.2 2 16.11 odd 4
1600.4.a.j.1.1 1 40.19 odd 2
1600.4.a.br.1.1 1 40.29 even 2
1800.4.a.bi.1.1 1 60.59 even 2
1800.4.f.j.649.1 2 60.47 odd 4
1800.4.f.j.649.2 2 60.23 odd 4
1960.4.a.h.1.1 1 28.27 even 2