# Properties

 Label 960.4.a.u.1.1 Level $960$ Weight $4$ Character 960.1 Self dual yes Analytic conductor $56.642$ 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] = [960,4,Mod(1,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, 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("960.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 960.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: $$56.6418336055$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 960.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -5.00000 q^{5} -16.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -5.00000 q^{5} -16.0000 q^{7} +9.00000 q^{9} +28.0000 q^{11} +26.0000 q^{13} -15.0000 q^{15} -62.0000 q^{17} +68.0000 q^{19} -48.0000 q^{21} -208.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +58.0000 q^{29} +160.000 q^{31} +84.0000 q^{33} +80.0000 q^{35} -270.000 q^{37} +78.0000 q^{39} +282.000 q^{41} -76.0000 q^{43} -45.0000 q^{45} -280.000 q^{47} -87.0000 q^{49} -186.000 q^{51} +210.000 q^{53} -140.000 q^{55} +204.000 q^{57} -196.000 q^{59} -742.000 q^{61} -144.000 q^{63} -130.000 q^{65} -836.000 q^{67} -624.000 q^{69} -504.000 q^{71} -1062.00 q^{73} +75.0000 q^{75} -448.000 q^{77} +768.000 q^{79} +81.0000 q^{81} +1052.00 q^{83} +310.000 q^{85} +174.000 q^{87} -726.000 q^{89} -416.000 q^{91} +480.000 q^{93} -340.000 q^{95} -1406.00 q^{97} +252.000 q^{99} +O(q^{100})$$

## 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
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ −16.0000 −0.863919 −0.431959 0.901893i $$-0.642178\pi$$
−0.431959 + 0.901893i $$0.642178\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 28.0000 0.767483 0.383742 0.923440i $$-0.374635\pi$$
0.383742 + 0.923440i $$0.374635\pi$$
$$12$$ 0 0
$$13$$ 26.0000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −0.258199
$$16$$ 0 0
$$17$$ −62.0000 −0.884542 −0.442271 0.896882i $$-0.645827\pi$$
−0.442271 + 0.896882i $$0.645827\pi$$
$$18$$ 0 0
$$19$$ 68.0000 0.821067 0.410533 0.911846i $$-0.365343\pi$$
0.410533 + 0.911846i $$0.365343\pi$$
$$20$$ 0 0
$$21$$ −48.0000 −0.498784
$$22$$ 0 0
$$23$$ −208.000 −1.88570 −0.942848 0.333224i $$-0.891864\pi$$
−0.942848 + 0.333224i $$0.891864\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 58.0000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 160.000 0.926995 0.463498 0.886098i $$-0.346594\pi$$
0.463498 + 0.886098i $$0.346594\pi$$
$$32$$ 0 0
$$33$$ 84.0000 0.443107
$$34$$ 0 0
$$35$$ 80.0000 0.386356
$$36$$ 0 0
$$37$$ −270.000 −1.19967 −0.599834 0.800124i $$-0.704767\pi$$
−0.599834 + 0.800124i $$0.704767\pi$$
$$38$$ 0 0
$$39$$ 78.0000 0.320256
$$40$$ 0 0
$$41$$ 282.000 1.07417 0.537085 0.843528i $$-0.319525\pi$$
0.537085 + 0.843528i $$0.319525\pi$$
$$42$$ 0 0
$$43$$ −76.0000 −0.269532 −0.134766 0.990877i $$-0.543028\pi$$
−0.134766 + 0.990877i $$0.543028\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ −280.000 −0.868983 −0.434491 0.900676i $$-0.643072\pi$$
−0.434491 + 0.900676i $$0.643072\pi$$
$$48$$ 0 0
$$49$$ −87.0000 −0.253644
$$50$$ 0 0
$$51$$ −186.000 −0.510690
$$52$$ 0 0
$$53$$ 210.000 0.544259 0.272129 0.962261i $$-0.412272\pi$$
0.272129 + 0.962261i $$0.412272\pi$$
$$54$$ 0 0
$$55$$ −140.000 −0.343229
$$56$$ 0 0
$$57$$ 204.000 0.474043
$$58$$ 0 0
$$59$$ −196.000 −0.432492 −0.216246 0.976339i $$-0.569381\pi$$
−0.216246 + 0.976339i $$0.569381\pi$$
$$60$$ 0 0
$$61$$ −742.000 −1.55743 −0.778716 0.627376i $$-0.784129\pi$$
−0.778716 + 0.627376i $$0.784129\pi$$
$$62$$ 0 0
$$63$$ −144.000 −0.287973
$$64$$ 0 0
$$65$$ −130.000 −0.248069
$$66$$ 0 0
$$67$$ −836.000 −1.52438 −0.762191 0.647352i $$-0.775877\pi$$
−0.762191 + 0.647352i $$0.775877\pi$$
$$68$$ 0 0
$$69$$ −624.000 −1.08871
$$70$$ 0 0
$$71$$ −504.000 −0.842448 −0.421224 0.906957i $$-0.638399\pi$$
−0.421224 + 0.906957i $$0.638399\pi$$
$$72$$ 0 0
$$73$$ −1062.00 −1.70271 −0.851354 0.524591i $$-0.824218\pi$$
−0.851354 + 0.524591i $$0.824218\pi$$
$$74$$ 0 0
$$75$$ 75.0000 0.115470
$$76$$ 0 0
$$77$$ −448.000 −0.663043
$$78$$ 0 0
$$79$$ 768.000 1.09376 0.546878 0.837212i $$-0.315816\pi$$
0.546878 + 0.837212i $$0.315816\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1052.00 1.39123 0.695614 0.718415i $$-0.255132\pi$$
0.695614 + 0.718415i $$0.255132\pi$$
$$84$$ 0 0
$$85$$ 310.000 0.395579
$$86$$ 0 0
$$87$$ 174.000 0.214423
$$88$$ 0 0
$$89$$ −726.000 −0.864672 −0.432336 0.901712i $$-0.642311\pi$$
−0.432336 + 0.901712i $$0.642311\pi$$
$$90$$ 0 0
$$91$$ −416.000 −0.479216
$$92$$ 0 0
$$93$$ 480.000 0.535201
$$94$$ 0 0
$$95$$ −340.000 −0.367192
$$96$$ 0 0
$$97$$ −1406.00 −1.47173 −0.735864 0.677129i $$-0.763224\pi$$
−0.735864 + 0.677129i $$0.763224\pi$$
$$98$$ 0 0
$$99$$ 252.000 0.255828
$$100$$ 0 0
$$101$$ −990.000 −0.975333 −0.487667 0.873030i $$-0.662152\pi$$
−0.487667 + 0.873030i $$0.662152\pi$$
$$102$$ 0 0
$$103$$ 736.000 0.704080 0.352040 0.935985i $$-0.385488\pi$$
0.352040 + 0.935985i $$0.385488\pi$$
$$104$$ 0 0
$$105$$ 240.000 0.223063
$$106$$ 0 0
$$107$$ −1212.00 −1.09503 −0.547516 0.836795i $$-0.684427\pi$$
−0.547516 + 0.836795i $$0.684427\pi$$
$$108$$ 0 0
$$109$$ 1834.00 1.61161 0.805804 0.592182i $$-0.201733\pi$$
0.805804 + 0.592182i $$0.201733\pi$$
$$110$$ 0 0
$$111$$ −810.000 −0.692629
$$112$$ 0 0
$$113$$ −2046.00 −1.70329 −0.851644 0.524121i $$-0.824394\pi$$
−0.851644 + 0.524121i $$0.824394\pi$$
$$114$$ 0 0
$$115$$ 1040.00 0.843309
$$116$$ 0 0
$$117$$ 234.000 0.184900
$$118$$ 0 0
$$119$$ 992.000 0.764172
$$120$$ 0 0
$$121$$ −547.000 −0.410969
$$122$$ 0 0
$$123$$ 846.000 0.620173
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 1176.00 0.821678 0.410839 0.911708i $$-0.365236\pi$$
0.410839 + 0.911708i $$0.365236\pi$$
$$128$$ 0 0
$$129$$ −228.000 −0.155615
$$130$$ 0 0
$$131$$ −12.0000 −0.00800340 −0.00400170 0.999992i $$-0.501274\pi$$
−0.00400170 + 0.999992i $$0.501274\pi$$
$$132$$ 0 0
$$133$$ −1088.00 −0.709335
$$134$$ 0 0
$$135$$ −135.000 −0.0860663
$$136$$ 0 0
$$137$$ −790.000 −0.492659 −0.246329 0.969186i $$-0.579225\pi$$
−0.246329 + 0.969186i $$0.579225\pi$$
$$138$$ 0 0
$$139$$ 924.000 0.563832 0.281916 0.959439i $$-0.409030\pi$$
0.281916 + 0.959439i $$0.409030\pi$$
$$140$$ 0 0
$$141$$ −840.000 −0.501708
$$142$$ 0 0
$$143$$ 728.000 0.425723
$$144$$ 0 0
$$145$$ −290.000 −0.166091
$$146$$ 0 0
$$147$$ −261.000 −0.146442
$$148$$ 0 0
$$149$$ −3022.00 −1.66156 −0.830778 0.556604i $$-0.812104\pi$$
−0.830778 + 0.556604i $$0.812104\pi$$
$$150$$ 0 0
$$151$$ 1736.00 0.935587 0.467794 0.883838i $$-0.345049\pi$$
0.467794 + 0.883838i $$0.345049\pi$$
$$152$$ 0 0
$$153$$ −558.000 −0.294847
$$154$$ 0 0
$$155$$ −800.000 −0.414565
$$156$$ 0 0
$$157$$ 1322.00 0.672020 0.336010 0.941858i $$-0.390922\pi$$
0.336010 + 0.941858i $$0.390922\pi$$
$$158$$ 0 0
$$159$$ 630.000 0.314228
$$160$$ 0 0
$$161$$ 3328.00 1.62909
$$162$$ 0 0
$$163$$ 908.000 0.436319 0.218160 0.975913i $$-0.429995\pi$$
0.218160 + 0.975913i $$0.429995\pi$$
$$164$$ 0 0
$$165$$ −420.000 −0.198163
$$166$$ 0 0
$$167$$ 1296.00 0.600524 0.300262 0.953857i $$-0.402926\pi$$
0.300262 + 0.953857i $$0.402926\pi$$
$$168$$ 0 0
$$169$$ −1521.00 −0.692308
$$170$$ 0 0
$$171$$ 612.000 0.273689
$$172$$ 0 0
$$173$$ −2134.00 −0.937832 −0.468916 0.883243i $$-0.655355\pi$$
−0.468916 + 0.883243i $$0.655355\pi$$
$$174$$ 0 0
$$175$$ −400.000 −0.172784
$$176$$ 0 0
$$177$$ −588.000 −0.249699
$$178$$ 0 0
$$179$$ −1612.00 −0.673109 −0.336555 0.941664i $$-0.609262\pi$$
−0.336555 + 0.941664i $$0.609262\pi$$
$$180$$ 0 0
$$181$$ −3086.00 −1.26730 −0.633648 0.773621i $$-0.718443\pi$$
−0.633648 + 0.773621i $$0.718443\pi$$
$$182$$ 0 0
$$183$$ −2226.00 −0.899184
$$184$$ 0 0
$$185$$ 1350.00 0.536508
$$186$$ 0 0
$$187$$ −1736.00 −0.678871
$$188$$ 0 0
$$189$$ −432.000 −0.166261
$$190$$ 0 0
$$191$$ −4208.00 −1.59414 −0.797069 0.603889i $$-0.793617\pi$$
−0.797069 + 0.603889i $$0.793617\pi$$
$$192$$ 0 0
$$193$$ 2818.00 1.05101 0.525503 0.850792i $$-0.323877\pi$$
0.525503 + 0.850792i $$0.323877\pi$$
$$194$$ 0 0
$$195$$ −390.000 −0.143223
$$196$$ 0 0
$$197$$ 418.000 0.151174 0.0755870 0.997139i $$-0.475917\pi$$
0.0755870 + 0.997139i $$0.475917\pi$$
$$198$$ 0 0
$$199$$ −3352.00 −1.19406 −0.597028 0.802221i $$-0.703652\pi$$
−0.597028 + 0.802221i $$0.703652\pi$$
$$200$$ 0 0
$$201$$ −2508.00 −0.880103
$$202$$ 0 0
$$203$$ −928.000 −0.320851
$$204$$ 0 0
$$205$$ −1410.00 −0.480384
$$206$$ 0 0
$$207$$ −1872.00 −0.628565
$$208$$ 0 0
$$209$$ 1904.00 0.630155
$$210$$ 0 0
$$211$$ 4276.00 1.39513 0.697564 0.716523i $$-0.254267\pi$$
0.697564 + 0.716523i $$0.254267\pi$$
$$212$$ 0 0
$$213$$ −1512.00 −0.486387
$$214$$ 0 0
$$215$$ 380.000 0.120539
$$216$$ 0 0
$$217$$ −2560.00 −0.800848
$$218$$ 0 0
$$219$$ −3186.00 −0.983059
$$220$$ 0 0
$$221$$ −1612.00 −0.490655
$$222$$ 0 0
$$223$$ 4712.00 1.41497 0.707486 0.706727i $$-0.249829\pi$$
0.707486 + 0.706727i $$0.249829\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ 732.000 0.214029 0.107014 0.994257i $$-0.465871\pi$$
0.107014 + 0.994257i $$0.465871\pi$$
$$228$$ 0 0
$$229$$ 5186.00 1.49651 0.748254 0.663412i $$-0.230892\pi$$
0.748254 + 0.663412i $$0.230892\pi$$
$$230$$ 0 0
$$231$$ −1344.00 −0.382808
$$232$$ 0 0
$$233$$ −3798.00 −1.06788 −0.533938 0.845523i $$-0.679289\pi$$
−0.533938 + 0.845523i $$0.679289\pi$$
$$234$$ 0 0
$$235$$ 1400.00 0.388621
$$236$$ 0 0
$$237$$ 2304.00 0.631481
$$238$$ 0 0
$$239$$ −3120.00 −0.844419 −0.422209 0.906498i $$-0.638745\pi$$
−0.422209 + 0.906498i $$0.638745\pi$$
$$240$$ 0 0
$$241$$ 1490.00 0.398255 0.199127 0.979974i $$-0.436189\pi$$
0.199127 + 0.979974i $$0.436189\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 435.000 0.113433
$$246$$ 0 0
$$247$$ 1768.00 0.455446
$$248$$ 0 0
$$249$$ 3156.00 0.803226
$$250$$ 0 0
$$251$$ 5292.00 1.33079 0.665395 0.746492i $$-0.268263\pi$$
0.665395 + 0.746492i $$0.268263\pi$$
$$252$$ 0 0
$$253$$ −5824.00 −1.44724
$$254$$ 0 0
$$255$$ 930.000 0.228388
$$256$$ 0 0
$$257$$ −3918.00 −0.950965 −0.475483 0.879725i $$-0.657727\pi$$
−0.475483 + 0.879725i $$0.657727\pi$$
$$258$$ 0 0
$$259$$ 4320.00 1.03642
$$260$$ 0 0
$$261$$ 522.000 0.123797
$$262$$ 0 0
$$263$$ 6624.00 1.55305 0.776527 0.630084i $$-0.216979\pi$$
0.776527 + 0.630084i $$0.216979\pi$$
$$264$$ 0 0
$$265$$ −1050.00 −0.243400
$$266$$ 0 0
$$267$$ −2178.00 −0.499219
$$268$$ 0 0
$$269$$ 2954.00 0.669549 0.334774 0.942298i $$-0.391340\pi$$
0.334774 + 0.942298i $$0.391340\pi$$
$$270$$ 0 0
$$271$$ −6576.00 −1.47404 −0.737018 0.675874i $$-0.763767\pi$$
−0.737018 + 0.675874i $$0.763767\pi$$
$$272$$ 0 0
$$273$$ −1248.00 −0.276675
$$274$$ 0 0
$$275$$ 700.000 0.153497
$$276$$ 0 0
$$277$$ −4478.00 −0.971325 −0.485662 0.874146i $$-0.661422\pi$$
−0.485662 + 0.874146i $$0.661422\pi$$
$$278$$ 0 0
$$279$$ 1440.00 0.308998
$$280$$ 0 0
$$281$$ −6358.00 −1.34977 −0.674887 0.737921i $$-0.735808\pi$$
−0.674887 + 0.737921i $$0.735808\pi$$
$$282$$ 0 0
$$283$$ −860.000 −0.180642 −0.0903210 0.995913i $$-0.528789\pi$$
−0.0903210 + 0.995913i $$0.528789\pi$$
$$284$$ 0 0
$$285$$ −1020.00 −0.211999
$$286$$ 0 0
$$287$$ −4512.00 −0.927996
$$288$$ 0 0
$$289$$ −1069.00 −0.217586
$$290$$ 0 0
$$291$$ −4218.00 −0.849703
$$292$$ 0 0
$$293$$ 5794.00 1.15525 0.577626 0.816301i $$-0.303979\pi$$
0.577626 + 0.816301i $$0.303979\pi$$
$$294$$ 0 0
$$295$$ 980.000 0.193416
$$296$$ 0 0
$$297$$ 756.000 0.147702
$$298$$ 0 0
$$299$$ −5408.00 −1.04600
$$300$$ 0 0
$$301$$ 1216.00 0.232854
$$302$$ 0 0
$$303$$ −2970.00 −0.563109
$$304$$ 0 0
$$305$$ 3710.00 0.696505
$$306$$ 0 0
$$307$$ 6860.00 1.27531 0.637656 0.770321i $$-0.279904\pi$$
0.637656 + 0.770321i $$0.279904\pi$$
$$308$$ 0 0
$$309$$ 2208.00 0.406501
$$310$$ 0 0
$$311$$ −6248.00 −1.13920 −0.569601 0.821922i $$-0.692902\pi$$
−0.569601 + 0.821922i $$0.692902\pi$$
$$312$$ 0 0
$$313$$ 11018.0 1.98969 0.994847 0.101388i $$-0.0323284\pi$$
0.994847 + 0.101388i $$0.0323284\pi$$
$$314$$ 0 0
$$315$$ 720.000 0.128785
$$316$$ 0 0
$$317$$ 954.000 0.169028 0.0845142 0.996422i $$-0.473066\pi$$
0.0845142 + 0.996422i $$0.473066\pi$$
$$318$$ 0 0
$$319$$ 1624.00 0.285036
$$320$$ 0 0
$$321$$ −3636.00 −0.632217
$$322$$ 0 0
$$323$$ −4216.00 −0.726268
$$324$$ 0 0
$$325$$ 650.000 0.110940
$$326$$ 0 0
$$327$$ 5502.00 0.930463
$$328$$ 0 0
$$329$$ 4480.00 0.750731
$$330$$ 0 0
$$331$$ −9396.00 −1.56027 −0.780137 0.625608i $$-0.784851\pi$$
−0.780137 + 0.625608i $$0.784851\pi$$
$$332$$ 0 0
$$333$$ −2430.00 −0.399889
$$334$$ 0 0
$$335$$ 4180.00 0.681725
$$336$$ 0 0
$$337$$ 5074.00 0.820173 0.410087 0.912047i $$-0.365498\pi$$
0.410087 + 0.912047i $$0.365498\pi$$
$$338$$ 0 0
$$339$$ −6138.00 −0.983394
$$340$$ 0 0
$$341$$ 4480.00 0.711453
$$342$$ 0 0
$$343$$ 6880.00 1.08305
$$344$$ 0 0
$$345$$ 3120.00 0.486885
$$346$$ 0 0
$$347$$ −3916.00 −0.605827 −0.302913 0.953018i $$-0.597959\pi$$
−0.302913 + 0.953018i $$0.597959\pi$$
$$348$$ 0 0
$$349$$ 1818.00 0.278840 0.139420 0.990233i $$-0.455476\pi$$
0.139420 + 0.990233i $$0.455476\pi$$
$$350$$ 0 0
$$351$$ 702.000 0.106752
$$352$$ 0 0
$$353$$ −7118.00 −1.07324 −0.536619 0.843825i $$-0.680299\pi$$
−0.536619 + 0.843825i $$0.680299\pi$$
$$354$$ 0 0
$$355$$ 2520.00 0.376754
$$356$$ 0 0
$$357$$ 2976.00 0.441195
$$358$$ 0 0
$$359$$ 5304.00 0.779762 0.389881 0.920865i $$-0.372516\pi$$
0.389881 + 0.920865i $$0.372516\pi$$
$$360$$ 0 0
$$361$$ −2235.00 −0.325849
$$362$$ 0 0
$$363$$ −1641.00 −0.237273
$$364$$ 0 0
$$365$$ 5310.00 0.761474
$$366$$ 0 0
$$367$$ 5672.00 0.806747 0.403373 0.915036i $$-0.367838\pi$$
0.403373 + 0.915036i $$0.367838\pi$$
$$368$$ 0 0
$$369$$ 2538.00 0.358057
$$370$$ 0 0
$$371$$ −3360.00 −0.470195
$$372$$ 0 0
$$373$$ −7774.00 −1.07915 −0.539574 0.841938i $$-0.681415\pi$$
−0.539574 + 0.841938i $$0.681415\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 0 0
$$377$$ 1508.00 0.206010
$$378$$ 0 0
$$379$$ 5516.00 0.747593 0.373797 0.927511i $$-0.378056\pi$$
0.373797 + 0.927511i $$0.378056\pi$$
$$380$$ 0 0
$$381$$ 3528.00 0.474396
$$382$$ 0 0
$$383$$ −7128.00 −0.950976 −0.475488 0.879722i $$-0.657728\pi$$
−0.475488 + 0.879722i $$0.657728\pi$$
$$384$$ 0 0
$$385$$ 2240.00 0.296522
$$386$$ 0 0
$$387$$ −684.000 −0.0898441
$$388$$ 0 0
$$389$$ 10722.0 1.39750 0.698749 0.715367i $$-0.253740\pi$$
0.698749 + 0.715367i $$0.253740\pi$$
$$390$$ 0 0
$$391$$ 12896.0 1.66798
$$392$$ 0 0
$$393$$ −36.0000 −0.00462076
$$394$$ 0 0
$$395$$ −3840.00 −0.489143
$$396$$ 0 0
$$397$$ 12122.0 1.53246 0.766229 0.642568i $$-0.222131\pi$$
0.766229 + 0.642568i $$0.222131\pi$$
$$398$$ 0 0
$$399$$ −3264.00 −0.409535
$$400$$ 0 0
$$401$$ 10482.0 1.30535 0.652676 0.757637i $$-0.273646\pi$$
0.652676 + 0.757637i $$0.273646\pi$$
$$402$$ 0 0
$$403$$ 4160.00 0.514204
$$404$$ 0 0
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ −7560.00 −0.920726
$$408$$ 0 0
$$409$$ 3850.00 0.465453 0.232726 0.972542i $$-0.425235\pi$$
0.232726 + 0.972542i $$0.425235\pi$$
$$410$$ 0 0
$$411$$ −2370.00 −0.284437
$$412$$ 0 0
$$413$$ 3136.00 0.373638
$$414$$ 0 0
$$415$$ −5260.00 −0.622176
$$416$$ 0 0
$$417$$ 2772.00 0.325529
$$418$$ 0 0
$$419$$ 5796.00 0.675783 0.337892 0.941185i $$-0.390286\pi$$
0.337892 + 0.941185i $$0.390286\pi$$
$$420$$ 0 0
$$421$$ −3294.00 −0.381330 −0.190665 0.981655i $$-0.561064\pi$$
−0.190665 + 0.981655i $$0.561064\pi$$
$$422$$ 0 0
$$423$$ −2520.00 −0.289661
$$424$$ 0 0
$$425$$ −1550.00 −0.176908
$$426$$ 0 0
$$427$$ 11872.0 1.34549
$$428$$ 0 0
$$429$$ 2184.00 0.245791
$$430$$ 0 0
$$431$$ 1696.00 0.189544 0.0947720 0.995499i $$-0.469788\pi$$
0.0947720 + 0.995499i $$0.469788\pi$$
$$432$$ 0 0
$$433$$ −12334.0 −1.36890 −0.684451 0.729059i $$-0.739958\pi$$
−0.684451 + 0.729059i $$0.739958\pi$$
$$434$$ 0 0
$$435$$ −870.000 −0.0958927
$$436$$ 0 0
$$437$$ −14144.0 −1.54828
$$438$$ 0 0
$$439$$ 376.000 0.0408781 0.0204391 0.999791i $$-0.493494\pi$$
0.0204391 + 0.999791i $$0.493494\pi$$
$$440$$ 0 0
$$441$$ −783.000 −0.0845481
$$442$$ 0 0
$$443$$ −8028.00 −0.860997 −0.430499 0.902591i $$-0.641662\pi$$
−0.430499 + 0.902591i $$0.641662\pi$$
$$444$$ 0 0
$$445$$ 3630.00 0.386693
$$446$$ 0 0
$$447$$ −9066.00 −0.959300
$$448$$ 0 0
$$449$$ 8898.00 0.935240 0.467620 0.883930i $$-0.345112\pi$$
0.467620 + 0.883930i $$0.345112\pi$$
$$450$$ 0 0
$$451$$ 7896.00 0.824408
$$452$$ 0 0
$$453$$ 5208.00 0.540162
$$454$$ 0 0
$$455$$ 2080.00 0.214312
$$456$$ 0 0
$$457$$ 10330.0 1.05737 0.528684 0.848819i $$-0.322686\pi$$
0.528684 + 0.848819i $$0.322686\pi$$
$$458$$ 0 0
$$459$$ −1674.00 −0.170230
$$460$$ 0 0
$$461$$ −1878.00 −0.189734 −0.0948668 0.995490i $$-0.530243\pi$$
−0.0948668 + 0.995490i $$0.530243\pi$$
$$462$$ 0 0
$$463$$ −13224.0 −1.32737 −0.663684 0.748013i $$-0.731008\pi$$
−0.663684 + 0.748013i $$0.731008\pi$$
$$464$$ 0 0
$$465$$ −2400.00 −0.239349
$$466$$ 0 0
$$467$$ 8012.00 0.793900 0.396950 0.917840i $$-0.370069\pi$$
0.396950 + 0.917840i $$0.370069\pi$$
$$468$$ 0 0
$$469$$ 13376.0 1.31694
$$470$$ 0 0
$$471$$ 3966.00 0.387991
$$472$$ 0 0
$$473$$ −2128.00 −0.206862
$$474$$ 0 0
$$475$$ 1700.00 0.164213
$$476$$ 0 0
$$477$$ 1890.00 0.181420
$$478$$ 0 0
$$479$$ −1792.00 −0.170936 −0.0854682 0.996341i $$-0.527239\pi$$
−0.0854682 + 0.996341i $$0.527239\pi$$
$$480$$ 0 0
$$481$$ −7020.00 −0.665456
$$482$$ 0 0
$$483$$ 9984.00 0.940554
$$484$$ 0 0
$$485$$ 7030.00 0.658177
$$486$$ 0 0
$$487$$ 8272.00 0.769692 0.384846 0.922981i $$-0.374255\pi$$
0.384846 + 0.922981i $$0.374255\pi$$
$$488$$ 0 0
$$489$$ 2724.00 0.251909
$$490$$ 0 0
$$491$$ −516.000 −0.0474272 −0.0237136 0.999719i $$-0.507549\pi$$
−0.0237136 + 0.999719i $$0.507549\pi$$
$$492$$ 0 0
$$493$$ −3596.00 −0.328511
$$494$$ 0 0
$$495$$ −1260.00 −0.114410
$$496$$ 0 0
$$497$$ 8064.00 0.727807
$$498$$ 0 0
$$499$$ 14020.0 1.25776 0.628879 0.777503i $$-0.283514\pi$$
0.628879 + 0.777503i $$0.283514\pi$$
$$500$$ 0 0
$$501$$ 3888.00 0.346713
$$502$$ 0 0
$$503$$ 1872.00 0.165941 0.0829705 0.996552i $$-0.473559\pi$$
0.0829705 + 0.996552i $$0.473559\pi$$
$$504$$ 0 0
$$505$$ 4950.00 0.436182
$$506$$ 0 0
$$507$$ −4563.00 −0.399704
$$508$$ 0 0
$$509$$ −8678.00 −0.755689 −0.377844 0.925869i $$-0.623335\pi$$
−0.377844 + 0.925869i $$0.623335\pi$$
$$510$$ 0 0
$$511$$ 16992.0 1.47100
$$512$$ 0 0
$$513$$ 1836.00 0.158014
$$514$$ 0 0
$$515$$ −3680.00 −0.314874
$$516$$ 0 0
$$517$$ −7840.00 −0.666930
$$518$$ 0 0
$$519$$ −6402.00 −0.541458
$$520$$ 0 0
$$521$$ 18074.0 1.51984 0.759920 0.650017i $$-0.225238\pi$$
0.759920 + 0.650017i $$0.225238\pi$$
$$522$$ 0 0
$$523$$ 20852.0 1.74339 0.871696 0.490047i $$-0.163020\pi$$
0.871696 + 0.490047i $$0.163020\pi$$
$$524$$ 0 0
$$525$$ −1200.00 −0.0997567
$$526$$ 0 0
$$527$$ −9920.00 −0.819966
$$528$$ 0 0
$$529$$ 31097.0 2.55585
$$530$$ 0 0
$$531$$ −1764.00 −0.144164
$$532$$ 0 0
$$533$$ 7332.00 0.595843
$$534$$ 0 0
$$535$$ 6060.00 0.489713
$$536$$ 0 0
$$537$$ −4836.00 −0.388620
$$538$$ 0 0
$$539$$ −2436.00 −0.194668
$$540$$ 0 0
$$541$$ 12410.0 0.986225 0.493112 0.869966i $$-0.335859\pi$$
0.493112 + 0.869966i $$0.335859\pi$$
$$542$$ 0 0
$$543$$ −9258.00 −0.731674
$$544$$ 0 0
$$545$$ −9170.00 −0.720733
$$546$$ 0 0
$$547$$ −3620.00 −0.282962 −0.141481 0.989941i $$-0.545186\pi$$
−0.141481 + 0.989941i $$0.545186\pi$$
$$548$$ 0 0
$$549$$ −6678.00 −0.519144
$$550$$ 0 0
$$551$$ 3944.00 0.304937
$$552$$ 0 0
$$553$$ −12288.0 −0.944917
$$554$$ 0 0
$$555$$ 4050.00 0.309753
$$556$$ 0 0
$$557$$ −11734.0 −0.892613 −0.446307 0.894880i $$-0.647261\pi$$
−0.446307 + 0.894880i $$0.647261\pi$$
$$558$$ 0 0
$$559$$ −1976.00 −0.149510
$$560$$ 0 0
$$561$$ −5208.00 −0.391946
$$562$$ 0 0
$$563$$ 1372.00 0.102705 0.0513525 0.998681i $$-0.483647\pi$$
0.0513525 + 0.998681i $$0.483647\pi$$
$$564$$ 0 0
$$565$$ 10230.0 0.761733
$$566$$ 0 0
$$567$$ −1296.00 −0.0959910
$$568$$ 0 0
$$569$$ 18922.0 1.39412 0.697058 0.717015i $$-0.254492\pi$$
0.697058 + 0.717015i $$0.254492\pi$$
$$570$$ 0 0
$$571$$ −14596.0 −1.06974 −0.534872 0.844933i $$-0.679640\pi$$
−0.534872 + 0.844933i $$0.679640\pi$$
$$572$$ 0 0
$$573$$ −12624.0 −0.920376
$$574$$ 0 0
$$575$$ −5200.00 −0.377139
$$576$$ 0 0
$$577$$ −2302.00 −0.166089 −0.0830446 0.996546i $$-0.526464\pi$$
−0.0830446 + 0.996546i $$0.526464\pi$$
$$578$$ 0 0
$$579$$ 8454.00 0.606798
$$580$$ 0 0
$$581$$ −16832.0 −1.20191
$$582$$ 0 0
$$583$$ 5880.00 0.417710
$$584$$ 0 0
$$585$$ −1170.00 −0.0826898
$$586$$ 0 0
$$587$$ −23292.0 −1.63776 −0.818879 0.573966i $$-0.805404\pi$$
−0.818879 + 0.573966i $$0.805404\pi$$
$$588$$ 0 0
$$589$$ 10880.0 0.761125
$$590$$ 0 0
$$591$$ 1254.00 0.0872803
$$592$$ 0 0
$$593$$ −16542.0 −1.14553 −0.572764 0.819720i $$-0.694129\pi$$
−0.572764 + 0.819720i $$0.694129\pi$$
$$594$$ 0 0
$$595$$ −4960.00 −0.341748
$$596$$ 0 0
$$597$$ −10056.0 −0.689388
$$598$$ 0 0
$$599$$ 7464.00 0.509133 0.254567 0.967055i $$-0.418067\pi$$
0.254567 + 0.967055i $$0.418067\pi$$
$$600$$ 0 0
$$601$$ −17270.0 −1.17214 −0.586072 0.810259i $$-0.699326\pi$$
−0.586072 + 0.810259i $$0.699326\pi$$
$$602$$ 0 0
$$603$$ −7524.00 −0.508128
$$604$$ 0 0
$$605$$ 2735.00 0.183791
$$606$$ 0 0
$$607$$ 984.000 0.0657979 0.0328990 0.999459i $$-0.489526\pi$$
0.0328990 + 0.999459i $$0.489526\pi$$
$$608$$ 0 0
$$609$$ −2784.00 −0.185244
$$610$$ 0 0
$$611$$ −7280.00 −0.482025
$$612$$ 0 0
$$613$$ −7278.00 −0.479536 −0.239768 0.970830i $$-0.577071\pi$$
−0.239768 + 0.970830i $$0.577071\pi$$
$$614$$ 0 0
$$615$$ −4230.00 −0.277350
$$616$$ 0 0
$$617$$ 18090.0 1.18035 0.590175 0.807275i $$-0.299059\pi$$
0.590175 + 0.807275i $$0.299059\pi$$
$$618$$ 0 0
$$619$$ −24740.0 −1.60644 −0.803219 0.595684i $$-0.796881\pi$$
−0.803219 + 0.595684i $$0.796881\pi$$
$$620$$ 0 0
$$621$$ −5616.00 −0.362902
$$622$$ 0 0
$$623$$ 11616.0 0.747007
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 5712.00 0.363820
$$628$$ 0 0
$$629$$ 16740.0 1.06116
$$630$$ 0 0
$$631$$ 19720.0 1.24412 0.622061 0.782969i $$-0.286296\pi$$
0.622061 + 0.782969i $$0.286296\pi$$
$$632$$ 0 0
$$633$$ 12828.0 0.805477
$$634$$ 0 0
$$635$$ −5880.00 −0.367466
$$636$$ 0 0
$$637$$ −2262.00 −0.140697
$$638$$ 0 0
$$639$$ −4536.00 −0.280816
$$640$$ 0 0
$$641$$ −16542.0 −1.01930 −0.509649 0.860383i $$-0.670225\pi$$
−0.509649 + 0.860383i $$0.670225\pi$$
$$642$$ 0 0
$$643$$ 10092.0 0.618957 0.309479 0.950906i $$-0.399845\pi$$
0.309479 + 0.950906i $$0.399845\pi$$
$$644$$ 0 0
$$645$$ 1140.00 0.0695930
$$646$$ 0 0
$$647$$ −14544.0 −0.883746 −0.441873 0.897078i $$-0.645686\pi$$
−0.441873 + 0.897078i $$0.645686\pi$$
$$648$$ 0 0
$$649$$ −5488.00 −0.331930
$$650$$ 0 0
$$651$$ −7680.00 −0.462370
$$652$$ 0 0
$$653$$ −23062.0 −1.38206 −0.691030 0.722826i $$-0.742843\pi$$
−0.691030 + 0.722826i $$0.742843\pi$$
$$654$$ 0 0
$$655$$ 60.0000 0.00357923
$$656$$ 0 0
$$657$$ −9558.00 −0.567569
$$658$$ 0 0
$$659$$ 28020.0 1.65630 0.828152 0.560504i $$-0.189392\pi$$
0.828152 + 0.560504i $$0.189392\pi$$
$$660$$ 0 0
$$661$$ 6738.00 0.396487 0.198243 0.980153i $$-0.436476\pi$$
0.198243 + 0.980153i $$0.436476\pi$$
$$662$$ 0 0
$$663$$ −4836.00 −0.283280
$$664$$ 0 0
$$665$$ 5440.00 0.317224
$$666$$ 0 0
$$667$$ −12064.0 −0.700330
$$668$$ 0 0
$$669$$ 14136.0 0.816935
$$670$$ 0 0
$$671$$ −20776.0 −1.19530
$$672$$ 0 0
$$673$$ −14430.0 −0.826502 −0.413251 0.910617i $$-0.635607\pi$$
−0.413251 + 0.910617i $$0.635607\pi$$
$$674$$ 0 0
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ 17890.0 1.01561 0.507805 0.861472i $$-0.330457\pi$$
0.507805 + 0.861472i $$0.330457\pi$$
$$678$$ 0 0
$$679$$ 22496.0 1.27145
$$680$$ 0 0
$$681$$ 2196.00 0.123570
$$682$$ 0 0
$$683$$ −10860.0 −0.608413 −0.304207 0.952606i $$-0.598391\pi$$
−0.304207 + 0.952606i $$0.598391\pi$$
$$684$$ 0 0
$$685$$ 3950.00 0.220324
$$686$$ 0 0
$$687$$ 15558.0 0.864010
$$688$$ 0 0
$$689$$ 5460.00 0.301900
$$690$$ 0 0
$$691$$ 8692.00 0.478523 0.239261 0.970955i $$-0.423095\pi$$
0.239261 + 0.970955i $$0.423095\pi$$
$$692$$ 0 0
$$693$$ −4032.00 −0.221014
$$694$$ 0 0
$$695$$ −4620.00 −0.252153
$$696$$ 0 0
$$697$$ −17484.0 −0.950149
$$698$$ 0 0
$$699$$ −11394.0 −0.616539
$$700$$ 0 0
$$701$$ 698.000 0.0376078 0.0188039 0.999823i $$-0.494014\pi$$
0.0188039 + 0.999823i $$0.494014\pi$$
$$702$$ 0 0
$$703$$ −18360.0 −0.985008
$$704$$ 0 0
$$705$$ 4200.00 0.224370
$$706$$ 0 0
$$707$$ 15840.0 0.842609
$$708$$ 0 0
$$709$$ −2654.00 −0.140583 −0.0702913 0.997527i $$-0.522393\pi$$
−0.0702913 + 0.997527i $$0.522393\pi$$
$$710$$ 0 0
$$711$$ 6912.00 0.364585
$$712$$ 0 0
$$713$$ −33280.0 −1.74803
$$714$$ 0 0
$$715$$ −3640.00 −0.190389
$$716$$ 0 0
$$717$$ −9360.00 −0.487525
$$718$$ 0 0
$$719$$ −28240.0 −1.46478 −0.732388 0.680887i $$-0.761594\pi$$
−0.732388 + 0.680887i $$0.761594\pi$$
$$720$$ 0 0
$$721$$ −11776.0 −0.608268
$$722$$ 0 0
$$723$$ 4470.00 0.229932
$$724$$ 0 0
$$725$$ 1450.00 0.0742781
$$726$$ 0 0
$$727$$ −8320.00 −0.424445 −0.212223 0.977221i $$-0.568070\pi$$
−0.212223 + 0.977221i $$0.568070\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 4712.00 0.238413
$$732$$ 0 0
$$733$$ 2154.00 0.108540 0.0542700 0.998526i $$-0.482717\pi$$
0.0542700 + 0.998526i $$0.482717\pi$$
$$734$$ 0 0
$$735$$ 1305.00 0.0654907
$$736$$ 0 0
$$737$$ −23408.0 −1.16994
$$738$$ 0 0
$$739$$ −22380.0 −1.11402 −0.557011 0.830505i $$-0.688052\pi$$
−0.557011 + 0.830505i $$0.688052\pi$$
$$740$$ 0 0
$$741$$ 5304.00 0.262952
$$742$$ 0 0
$$743$$ −5760.00 −0.284406 −0.142203 0.989837i $$-0.545419\pi$$
−0.142203 + 0.989837i $$0.545419\pi$$
$$744$$ 0 0
$$745$$ 15110.0 0.743071
$$746$$ 0 0
$$747$$ 9468.00 0.463743
$$748$$ 0 0
$$749$$ 19392.0 0.946019
$$750$$ 0 0
$$751$$ −6192.00 −0.300865 −0.150432 0.988620i $$-0.548067\pi$$
−0.150432 + 0.988620i $$0.548067\pi$$
$$752$$ 0 0
$$753$$ 15876.0 0.768331
$$754$$ 0 0
$$755$$ −8680.00 −0.418407
$$756$$ 0 0
$$757$$ 13666.0 0.656142 0.328071 0.944653i $$-0.393602\pi$$
0.328071 + 0.944653i $$0.393602\pi$$
$$758$$ 0 0
$$759$$ −17472.0 −0.835564
$$760$$ 0 0
$$761$$ −32022.0 −1.52536 −0.762678 0.646778i $$-0.776116\pi$$
−0.762678 + 0.646778i $$0.776116\pi$$
$$762$$ 0 0
$$763$$ −29344.0 −1.39230
$$764$$ 0 0
$$765$$ 2790.00 0.131860
$$766$$ 0 0
$$767$$ −5096.00 −0.239903
$$768$$ 0 0
$$769$$ 22786.0 1.06851 0.534255 0.845323i $$-0.320592\pi$$
0.534255 + 0.845323i $$0.320592\pi$$
$$770$$ 0 0
$$771$$ −11754.0 −0.549040
$$772$$ 0 0
$$773$$ −8286.00 −0.385546 −0.192773 0.981243i $$-0.561748\pi$$
−0.192773 + 0.981243i $$0.561748\pi$$
$$774$$ 0 0
$$775$$ 4000.00 0.185399
$$776$$ 0 0
$$777$$ 12960.0 0.598375
$$778$$ 0 0
$$779$$ 19176.0 0.881966
$$780$$ 0 0
$$781$$ −14112.0 −0.646565
$$782$$ 0 0
$$783$$ 1566.00 0.0714742
$$784$$ 0 0
$$785$$ −6610.00 −0.300536
$$786$$ 0 0
$$787$$ 25804.0 1.16876 0.584379 0.811481i $$-0.301338\pi$$
0.584379 + 0.811481i $$0.301338\pi$$
$$788$$ 0 0
$$789$$ 19872.0 0.896656
$$790$$ 0 0
$$791$$ 32736.0 1.47150
$$792$$ 0 0
$$793$$ −19292.0 −0.863908
$$794$$ 0 0
$$795$$ −3150.00 −0.140527
$$796$$ 0 0
$$797$$ −17670.0 −0.785324 −0.392662 0.919683i $$-0.628446\pi$$
−0.392662 + 0.919683i $$0.628446\pi$$
$$798$$ 0 0
$$799$$ 17360.0 0.768652
$$800$$ 0 0
$$801$$ −6534.00 −0.288224
$$802$$ 0 0
$$803$$ −29736.0 −1.30680
$$804$$ 0 0
$$805$$ −16640.0 −0.728550
$$806$$ 0 0
$$807$$ 8862.00 0.386564
$$808$$ 0 0
$$809$$ −7398.00 −0.321508 −0.160754 0.986995i $$-0.551393\pi$$
−0.160754 + 0.986995i $$0.551393\pi$$
$$810$$ 0 0
$$811$$ 28108.0 1.21702 0.608511 0.793545i $$-0.291767\pi$$
0.608511 + 0.793545i $$0.291767\pi$$
$$812$$ 0 0
$$813$$ −19728.0 −0.851035
$$814$$ 0 0
$$815$$ −4540.00 −0.195128
$$816$$ 0 0
$$817$$ −5168.00 −0.221304
$$818$$ 0 0
$$819$$ −3744.00 −0.159739
$$820$$ 0 0
$$821$$ −30830.0 −1.31057 −0.655283 0.755384i $$-0.727451\pi$$
−0.655283 + 0.755384i $$0.727451\pi$$
$$822$$ 0 0
$$823$$ 5872.00 0.248706 0.124353 0.992238i $$-0.460314\pi$$
0.124353 + 0.992238i $$0.460314\pi$$
$$824$$ 0 0
$$825$$ 2100.00 0.0886214
$$826$$ 0 0
$$827$$ 16308.0 0.685713 0.342857 0.939388i $$-0.388606\pi$$
0.342857 + 0.939388i $$0.388606\pi$$
$$828$$ 0 0
$$829$$ −28294.0 −1.18539 −0.592697 0.805426i $$-0.701937\pi$$
−0.592697 + 0.805426i $$0.701937\pi$$
$$830$$ 0 0
$$831$$ −13434.0 −0.560795
$$832$$ 0 0
$$833$$ 5394.00 0.224359
$$834$$ 0 0
$$835$$ −6480.00 −0.268562
$$836$$ 0 0
$$837$$ 4320.00 0.178400
$$838$$ 0 0
$$839$$ 20536.0 0.845032 0.422516 0.906356i $$-0.361147\pi$$
0.422516 + 0.906356i $$0.361147\pi$$
$$840$$ 0 0
$$841$$ −21025.0 −0.862069
$$842$$ 0 0
$$843$$ −19074.0 −0.779292
$$844$$ 0 0
$$845$$ 7605.00 0.309609
$$846$$ 0 0
$$847$$ 8752.00 0.355044
$$848$$ 0 0
$$849$$ −2580.00 −0.104294
$$850$$ 0 0
$$851$$ 56160.0 2.26221
$$852$$ 0 0
$$853$$ −27710.0 −1.11228 −0.556139 0.831090i $$-0.687718\pi$$
−0.556139 + 0.831090i $$0.687718\pi$$
$$854$$ 0 0
$$855$$ −3060.00 −0.122397
$$856$$ 0 0
$$857$$ 12858.0 0.512510 0.256255 0.966609i $$-0.417511\pi$$
0.256255 + 0.966609i $$0.417511\pi$$
$$858$$ 0 0
$$859$$ 3148.00 0.125039 0.0625194 0.998044i $$-0.480086\pi$$
0.0625194 + 0.998044i $$0.480086\pi$$
$$860$$ 0 0
$$861$$ −13536.0 −0.535779
$$862$$ 0 0
$$863$$ 48456.0 1.91131 0.955656 0.294487i $$-0.0951487\pi$$
0.955656 + 0.294487i $$0.0951487\pi$$
$$864$$ 0 0
$$865$$ 10670.0 0.419411
$$866$$ 0 0
$$867$$ −3207.00 −0.125623
$$868$$ 0 0
$$869$$ 21504.0 0.839440
$$870$$ 0 0
$$871$$ −21736.0 −0.845576
$$872$$ 0 0
$$873$$ −12654.0 −0.490576
$$874$$ 0 0
$$875$$ 2000.00 0.0772712
$$876$$ 0 0
$$877$$ −9478.00 −0.364937 −0.182468 0.983212i $$-0.558409\pi$$
−0.182468 + 0.983212i $$0.558409\pi$$
$$878$$ 0 0
$$879$$ 17382.0 0.666986
$$880$$ 0 0
$$881$$ 8178.00 0.312740 0.156370 0.987699i $$-0.450021\pi$$
0.156370 + 0.987699i $$0.450021\pi$$
$$882$$ 0 0
$$883$$ 316.000 0.0120433 0.00602166 0.999982i $$-0.498083\pi$$
0.00602166 + 0.999982i $$0.498083\pi$$
$$884$$ 0 0
$$885$$ 2940.00 0.111669
$$886$$ 0 0
$$887$$ −6304.00 −0.238633 −0.119317 0.992856i $$-0.538070\pi$$
−0.119317 + 0.992856i $$0.538070\pi$$
$$888$$ 0 0
$$889$$ −18816.0 −0.709863
$$890$$ 0 0
$$891$$ 2268.00 0.0852759
$$892$$ 0 0
$$893$$ −19040.0 −0.713493
$$894$$ 0 0
$$895$$ 8060.00 0.301024
$$896$$ 0 0
$$897$$ −16224.0 −0.603906
$$898$$ 0 0
$$899$$ 9280.00 0.344277
$$900$$ 0 0
$$901$$ −13020.0 −0.481420
$$902$$ 0 0
$$903$$ 3648.00 0.134438
$$904$$ 0 0
$$905$$ 15430.0 0.566752
$$906$$ 0 0
$$907$$ −1596.00 −0.0584281 −0.0292141 0.999573i $$-0.509300\pi$$
−0.0292141 + 0.999573i $$0.509300\pi$$
$$908$$ 0 0
$$909$$ −8910.00 −0.325111
$$910$$ 0 0
$$911$$ −25792.0 −0.938010 −0.469005 0.883196i $$-0.655387\pi$$
−0.469005 + 0.883196i $$0.655387\pi$$
$$912$$ 0 0
$$913$$ 29456.0 1.06775
$$914$$ 0 0
$$915$$ 11130.0 0.402127
$$916$$ 0 0
$$917$$ 192.000 0.00691428
$$918$$ 0 0
$$919$$ −9736.00 −0.349468 −0.174734 0.984616i $$-0.555907\pi$$
−0.174734 + 0.984616i $$0.555907\pi$$
$$920$$ 0 0
$$921$$ 20580.0 0.736302
$$922$$ 0 0
$$923$$ −13104.0 −0.467306
$$924$$ 0 0
$$925$$ −6750.00 −0.239934
$$926$$ 0 0
$$927$$ 6624.00 0.234693
$$928$$ 0 0
$$929$$ −94.0000 −0.00331974 −0.00165987 0.999999i $$-0.500528\pi$$
−0.00165987 + 0.999999i $$0.500528\pi$$
$$930$$ 0 0
$$931$$ −5916.00 −0.208259
$$932$$ 0 0
$$933$$ −18744.0 −0.657718
$$934$$ 0 0
$$935$$ 8680.00 0.303600
$$936$$ 0 0
$$937$$ −8678.00 −0.302559 −0.151280 0.988491i $$-0.548339\pi$$
−0.151280 + 0.988491i $$0.548339\pi$$
$$938$$ 0 0
$$939$$ 33054.0 1.14875
$$940$$ 0 0
$$941$$ −28406.0 −0.984069 −0.492035 0.870576i $$-0.663747\pi$$
−0.492035 + 0.870576i $$0.663747\pi$$
$$942$$ 0 0
$$943$$ −58656.0 −2.02556
$$944$$ 0 0
$$945$$ 2160.00 0.0743543
$$946$$ 0 0
$$947$$ −31988.0 −1.09765 −0.548823 0.835939i $$-0.684924\pi$$
−0.548823 + 0.835939i $$0.684924\pi$$
$$948$$ 0 0
$$949$$ −27612.0 −0.944493
$$950$$ 0 0
$$951$$ 2862.00 0.0975885
$$952$$ 0 0
$$953$$ 6714.00 0.228214 0.114107 0.993468i $$-0.463599\pi$$
0.114107 + 0.993468i $$0.463599\pi$$
$$954$$ 0 0
$$955$$ 21040.0 0.712920
$$956$$ 0 0
$$957$$ 4872.00 0.164566
$$958$$ 0 0
$$959$$ 12640.0 0.425617
$$960$$ 0 0
$$961$$ −4191.00 −0.140680
$$962$$ 0 0
$$963$$ −10908.0 −0.365011
$$964$$ 0 0
$$965$$ −14090.0 −0.470024
$$966$$ 0 0
$$967$$ 15312.0 0.509204 0.254602 0.967046i $$-0.418055\pi$$
0.254602 + 0.967046i $$0.418055\pi$$
$$968$$ 0 0
$$969$$ −12648.0 −0.419311
$$970$$ 0 0
$$971$$ 8540.00 0.282247 0.141123 0.989992i $$-0.454929\pi$$
0.141123 + 0.989992i $$0.454929\pi$$
$$972$$ 0 0
$$973$$ −14784.0 −0.487105
$$974$$ 0 0
$$975$$ 1950.00 0.0640513
$$976$$ 0 0
$$977$$ −8126.00 −0.266094 −0.133047 0.991110i $$-0.542476\pi$$
−0.133047 + 0.991110i $$0.542476\pi$$
$$978$$ 0 0
$$979$$ −20328.0 −0.663622
$$980$$ 0 0
$$981$$ 16506.0 0.537203
$$982$$ 0 0
$$983$$ 1392.00 0.0451657 0.0225829 0.999745i $$-0.492811\pi$$
0.0225829 + 0.999745i $$0.492811\pi$$
$$984$$ 0 0
$$985$$ −2090.00 −0.0676070
$$986$$ 0 0
$$987$$ 13440.0 0.433435
$$988$$ 0 0
$$989$$ 15808.0 0.508256
$$990$$ 0 0
$$991$$ −48832.0 −1.56529 −0.782644 0.622470i $$-0.786129\pi$$
−0.782644 + 0.622470i $$0.786129\pi$$
$$992$$ 0 0
$$993$$ −28188.0 −0.900825
$$994$$ 0 0
$$995$$ 16760.0 0.533998
$$996$$ 0 0
$$997$$ −46926.0 −1.49063 −0.745317 0.666711i $$-0.767702\pi$$
−0.745317 + 0.666711i $$0.767702\pi$$
$$998$$ 0 0
$$999$$ −7290.00 −0.230876
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 960.4.a.u.1.1 1
4.3 odd 2 960.4.a.h.1.1 1
8.3 odd 2 240.4.a.l.1.1 1
8.5 even 2 120.4.a.c.1.1 1
24.5 odd 2 360.4.a.b.1.1 1
24.11 even 2 720.4.a.l.1.1 1
40.3 even 4 1200.4.f.o.49.2 2
40.13 odd 4 600.4.f.c.49.1 2
40.19 odd 2 1200.4.a.c.1.1 1
40.27 even 4 1200.4.f.o.49.1 2
40.29 even 2 600.4.a.q.1.1 1
40.37 odd 4 600.4.f.c.49.2 2
120.29 odd 2 1800.4.a.bb.1.1 1
120.53 even 4 1800.4.f.r.649.2 2
120.77 even 4 1800.4.f.r.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.c.1.1 1 8.5 even 2
240.4.a.l.1.1 1 8.3 odd 2
360.4.a.b.1.1 1 24.5 odd 2
600.4.a.q.1.1 1 40.29 even 2
600.4.f.c.49.1 2 40.13 odd 4
600.4.f.c.49.2 2 40.37 odd 4
720.4.a.l.1.1 1 24.11 even 2
960.4.a.h.1.1 1 4.3 odd 2
960.4.a.u.1.1 1 1.1 even 1 trivial
1200.4.a.c.1.1 1 40.19 odd 2
1200.4.f.o.49.1 2 40.27 even 4
1200.4.f.o.49.2 2 40.3 even 4
1800.4.a.bb.1.1 1 120.29 odd 2
1800.4.f.r.649.1 2 120.77 even 4
1800.4.f.r.649.2 2 120.53 even 4