# Properties

 Label 32.4.a.c.1.1 Level $32$ Weight $4$ Character 32.1 Self dual yes Analytic conductor $1.888$ 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] = [32,4,Mod(1,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("32.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 32.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: $$1.88806112018$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 32.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+8.00000 q^{3} -10.0000 q^{5} +16.0000 q^{7} +37.0000 q^{9} +O(q^{10})$$ $$q+8.00000 q^{3} -10.0000 q^{5} +16.0000 q^{7} +37.0000 q^{9} -40.0000 q^{11} -50.0000 q^{13} -80.0000 q^{15} -30.0000 q^{17} +40.0000 q^{19} +128.000 q^{21} +48.0000 q^{23} -25.0000 q^{25} +80.0000 q^{27} -34.0000 q^{29} +320.000 q^{31} -320.000 q^{33} -160.000 q^{35} +310.000 q^{37} -400.000 q^{39} +410.000 q^{41} +152.000 q^{43} -370.000 q^{45} -416.000 q^{47} -87.0000 q^{49} -240.000 q^{51} -410.000 q^{53} +400.000 q^{55} +320.000 q^{57} -200.000 q^{59} +30.0000 q^{61} +592.000 q^{63} +500.000 q^{65} +776.000 q^{67} +384.000 q^{69} +400.000 q^{71} -630.000 q^{73} -200.000 q^{75} -640.000 q^{77} -1120.00 q^{79} -359.000 q^{81} +552.000 q^{83} +300.000 q^{85} -272.000 q^{87} -326.000 q^{89} -800.000 q^{91} +2560.00 q^{93} -400.000 q^{95} -110.000 q^{97} -1480.00 q^{99} +O(q^{100})$$

## 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
$$3$$ 8.00000 1.53960 0.769800 0.638285i $$-0.220356\pi$$
0.769800 + 0.638285i $$0.220356\pi$$
$$4$$ 0 0
$$5$$ −10.0000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ 16.0000 0.863919 0.431959 0.901893i $$-0.357822\pi$$
0.431959 + 0.901893i $$0.357822\pi$$
$$8$$ 0 0
$$9$$ 37.0000 1.37037
$$10$$ 0 0
$$11$$ −40.0000 −1.09640 −0.548202 0.836346i $$-0.684688\pi$$
−0.548202 + 0.836346i $$0.684688\pi$$
$$12$$ 0 0
$$13$$ −50.0000 −1.06673 −0.533366 0.845885i $$-0.679073\pi$$
−0.533366 + 0.845885i $$0.679073\pi$$
$$14$$ 0 0
$$15$$ −80.0000 −1.37706
$$16$$ 0 0
$$17$$ −30.0000 −0.428004 −0.214002 0.976833i $$-0.568650\pi$$
−0.214002 + 0.976833i $$0.568650\pi$$
$$18$$ 0 0
$$19$$ 40.0000 0.482980 0.241490 0.970403i $$-0.422364\pi$$
0.241490 + 0.970403i $$0.422364\pi$$
$$20$$ 0 0
$$21$$ 128.000 1.33009
$$22$$ 0 0
$$23$$ 48.0000 0.435161 0.217580 0.976042i $$-0.430184\pi$$
0.217580 + 0.976042i $$0.430184\pi$$
$$24$$ 0 0
$$25$$ −25.0000 −0.200000
$$26$$ 0 0
$$27$$ 80.0000 0.570222
$$28$$ 0 0
$$29$$ −34.0000 −0.217712 −0.108856 0.994058i $$-0.534719\pi$$
−0.108856 + 0.994058i $$0.534719\pi$$
$$30$$ 0 0
$$31$$ 320.000 1.85399 0.926995 0.375073i $$-0.122383\pi$$
0.926995 + 0.375073i $$0.122383\pi$$
$$32$$ 0 0
$$33$$ −320.000 −1.68803
$$34$$ 0 0
$$35$$ −160.000 −0.772712
$$36$$ 0 0
$$37$$ 310.000 1.37740 0.688698 0.725048i $$-0.258182\pi$$
0.688698 + 0.725048i $$0.258182\pi$$
$$38$$ 0 0
$$39$$ −400.000 −1.64234
$$40$$ 0 0
$$41$$ 410.000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 152.000 0.539065 0.269532 0.962991i $$-0.413131\pi$$
0.269532 + 0.962991i $$0.413131\pi$$
$$44$$ 0 0
$$45$$ −370.000 −1.22570
$$46$$ 0 0
$$47$$ −416.000 −1.29106 −0.645530 0.763735i $$-0.723364\pi$$
−0.645530 + 0.763735i $$0.723364\pi$$
$$48$$ 0 0
$$49$$ −87.0000 −0.253644
$$50$$ 0 0
$$51$$ −240.000 −0.658955
$$52$$ 0 0
$$53$$ −410.000 −1.06260 −0.531300 0.847184i $$-0.678296\pi$$
−0.531300 + 0.847184i $$0.678296\pi$$
$$54$$ 0 0
$$55$$ 400.000 0.980654
$$56$$ 0 0
$$57$$ 320.000 0.743597
$$58$$ 0 0
$$59$$ −200.000 −0.441318 −0.220659 0.975351i $$-0.570821\pi$$
−0.220659 + 0.975351i $$0.570821\pi$$
$$60$$ 0 0
$$61$$ 30.0000 0.0629690 0.0314845 0.999504i $$-0.489977\pi$$
0.0314845 + 0.999504i $$0.489977\pi$$
$$62$$ 0 0
$$63$$ 592.000 1.18389
$$64$$ 0 0
$$65$$ 500.000 0.954113
$$66$$ 0 0
$$67$$ 776.000 1.41498 0.707489 0.706725i $$-0.249828\pi$$
0.707489 + 0.706725i $$0.249828\pi$$
$$68$$ 0 0
$$69$$ 384.000 0.669973
$$70$$ 0 0
$$71$$ 400.000 0.668609 0.334305 0.942465i $$-0.391499\pi$$
0.334305 + 0.942465i $$0.391499\pi$$
$$72$$ 0 0
$$73$$ −630.000 −1.01008 −0.505041 0.863096i $$-0.668522\pi$$
−0.505041 + 0.863096i $$0.668522\pi$$
$$74$$ 0 0
$$75$$ −200.000 −0.307920
$$76$$ 0 0
$$77$$ −640.000 −0.947205
$$78$$ 0 0
$$79$$ −1120.00 −1.59506 −0.797531 0.603278i $$-0.793861\pi$$
−0.797531 + 0.603278i $$0.793861\pi$$
$$80$$ 0 0
$$81$$ −359.000 −0.492455
$$82$$ 0 0
$$83$$ 552.000 0.729998 0.364999 0.931008i $$-0.381069\pi$$
0.364999 + 0.931008i $$0.381069\pi$$
$$84$$ 0 0
$$85$$ 300.000 0.382818
$$86$$ 0 0
$$87$$ −272.000 −0.335189
$$88$$ 0 0
$$89$$ −326.000 −0.388269 −0.194134 0.980975i $$-0.562190\pi$$
−0.194134 + 0.980975i $$0.562190\pi$$
$$90$$ 0 0
$$91$$ −800.000 −0.921569
$$92$$ 0 0
$$93$$ 2560.00 2.85440
$$94$$ 0 0
$$95$$ −400.000 −0.431991
$$96$$ 0 0
$$97$$ −110.000 −0.115142 −0.0575712 0.998341i $$-0.518336\pi$$
−0.0575712 + 0.998341i $$0.518336\pi$$
$$98$$ 0 0
$$99$$ −1480.00 −1.50248
$$100$$ 0 0
$$101$$ −1098.00 −1.08173 −0.540867 0.841108i $$-0.681904\pi$$
−0.540867 + 0.841108i $$0.681904\pi$$
$$102$$ 0 0
$$103$$ −48.0000 −0.0459183 −0.0229591 0.999736i $$-0.507309\pi$$
−0.0229591 + 0.999736i $$0.507309\pi$$
$$104$$ 0 0
$$105$$ −1280.00 −1.18967
$$106$$ 0 0
$$107$$ 664.000 0.599919 0.299959 0.953952i $$-0.403027\pi$$
0.299959 + 0.953952i $$0.403027\pi$$
$$108$$ 0 0
$$109$$ −370.000 −0.325134 −0.162567 0.986698i $$-0.551977\pi$$
−0.162567 + 0.986698i $$0.551977\pi$$
$$110$$ 0 0
$$111$$ 2480.00 2.12064
$$112$$ 0 0
$$113$$ 1490.00 1.24042 0.620210 0.784436i $$-0.287047\pi$$
0.620210 + 0.784436i $$0.287047\pi$$
$$114$$ 0 0
$$115$$ −480.000 −0.389219
$$116$$ 0 0
$$117$$ −1850.00 −1.46182
$$118$$ 0 0
$$119$$ −480.000 −0.369761
$$120$$ 0 0
$$121$$ 269.000 0.202104
$$122$$ 0 0
$$123$$ 3280.00 2.40445
$$124$$ 0 0
$$125$$ 1500.00 1.07331
$$126$$ 0 0
$$127$$ −1024.00 −0.715475 −0.357737 0.933822i $$-0.616452\pi$$
−0.357737 + 0.933822i $$0.616452\pi$$
$$128$$ 0 0
$$129$$ 1216.00 0.829944
$$130$$ 0 0
$$131$$ 1160.00 0.773662 0.386831 0.922151i $$-0.373570\pi$$
0.386831 + 0.922151i $$0.373570\pi$$
$$132$$ 0 0
$$133$$ 640.000 0.417256
$$134$$ 0 0
$$135$$ −800.000 −0.510022
$$136$$ 0 0
$$137$$ 570.000 0.355463 0.177731 0.984079i $$-0.443124\pi$$
0.177731 + 0.984079i $$0.443124\pi$$
$$138$$ 0 0
$$139$$ −1960.00 −1.19601 −0.598004 0.801493i $$-0.704039\pi$$
−0.598004 + 0.801493i $$0.704039\pi$$
$$140$$ 0 0
$$141$$ −3328.00 −1.98772
$$142$$ 0 0
$$143$$ 2000.00 1.16957
$$144$$ 0 0
$$145$$ 340.000 0.194727
$$146$$ 0 0
$$147$$ −696.000 −0.390511
$$148$$ 0 0
$$149$$ −2010.00 −1.10514 −0.552569 0.833467i $$-0.686352\pi$$
−0.552569 + 0.833467i $$0.686352\pi$$
$$150$$ 0 0
$$151$$ −720.000 −0.388032 −0.194016 0.980998i $$-0.562151\pi$$
−0.194016 + 0.980998i $$0.562151\pi$$
$$152$$ 0 0
$$153$$ −1110.00 −0.586524
$$154$$ 0 0
$$155$$ −3200.00 −1.65826
$$156$$ 0 0
$$157$$ 1790.00 0.909921 0.454960 0.890512i $$-0.349653\pi$$
0.454960 + 0.890512i $$0.349653\pi$$
$$158$$ 0 0
$$159$$ −3280.00 −1.63598
$$160$$ 0 0
$$161$$ 768.000 0.375943
$$162$$ 0 0
$$163$$ −1208.00 −0.580478 −0.290239 0.956954i $$-0.593735\pi$$
−0.290239 + 0.956954i $$0.593735\pi$$
$$164$$ 0 0
$$165$$ 3200.00 1.50982
$$166$$ 0 0
$$167$$ 2896.00 1.34191 0.670956 0.741497i $$-0.265884\pi$$
0.670956 + 0.741497i $$0.265884\pi$$
$$168$$ 0 0
$$169$$ 303.000 0.137915
$$170$$ 0 0
$$171$$ 1480.00 0.661862
$$172$$ 0 0
$$173$$ 750.000 0.329604 0.164802 0.986327i $$-0.447302\pi$$
0.164802 + 0.986327i $$0.447302\pi$$
$$174$$ 0 0
$$175$$ −400.000 −0.172784
$$176$$ 0 0
$$177$$ −1600.00 −0.679454
$$178$$ 0 0
$$179$$ 2280.00 0.952040 0.476020 0.879434i $$-0.342079\pi$$
0.476020 + 0.879434i $$0.342079\pi$$
$$180$$ 0 0
$$181$$ −442.000 −0.181512 −0.0907558 0.995873i $$-0.528928\pi$$
−0.0907558 + 0.995873i $$0.528928\pi$$
$$182$$ 0 0
$$183$$ 240.000 0.0969471
$$184$$ 0 0
$$185$$ −3100.00 −1.23198
$$186$$ 0 0
$$187$$ 1200.00 0.469266
$$188$$ 0 0
$$189$$ 1280.00 0.492626
$$190$$ 0 0
$$191$$ −1920.00 −0.727363 −0.363681 0.931523i $$-0.618480\pi$$
−0.363681 + 0.931523i $$0.618480\pi$$
$$192$$ 0 0
$$193$$ −5070.00 −1.89091 −0.945457 0.325746i $$-0.894385\pi$$
−0.945457 + 0.325746i $$0.894385\pi$$
$$194$$ 0 0
$$195$$ 4000.00 1.46895
$$196$$ 0 0
$$197$$ 1910.00 0.690771 0.345385 0.938461i $$-0.387748\pi$$
0.345385 + 0.938461i $$0.387748\pi$$
$$198$$ 0 0
$$199$$ 2960.00 1.05442 0.527208 0.849736i $$-0.323239\pi$$
0.527208 + 0.849736i $$0.323239\pi$$
$$200$$ 0 0
$$201$$ 6208.00 2.17850
$$202$$ 0 0
$$203$$ −544.000 −0.188085
$$204$$ 0 0
$$205$$ −4100.00 −1.39686
$$206$$ 0 0
$$207$$ 1776.00 0.596331
$$208$$ 0 0
$$209$$ −1600.00 −0.529542
$$210$$ 0 0
$$211$$ 40.0000 0.0130508 0.00652539 0.999979i $$-0.497923\pi$$
0.00652539 + 0.999979i $$0.497923\pi$$
$$212$$ 0 0
$$213$$ 3200.00 1.02939
$$214$$ 0 0
$$215$$ −1520.00 −0.482154
$$216$$ 0 0
$$217$$ 5120.00 1.60170
$$218$$ 0 0
$$219$$ −5040.00 −1.55512
$$220$$ 0 0
$$221$$ 1500.00 0.456565
$$222$$ 0 0
$$223$$ 4288.00 1.28765 0.643824 0.765173i $$-0.277347\pi$$
0.643824 + 0.765173i $$0.277347\pi$$
$$224$$ 0 0
$$225$$ −925.000 −0.274074
$$226$$ 0 0
$$227$$ −6456.00 −1.88766 −0.943832 0.330425i $$-0.892808\pi$$
−0.943832 + 0.330425i $$0.892808\pi$$
$$228$$ 0 0
$$229$$ −1066.00 −0.307613 −0.153806 0.988101i $$-0.549153\pi$$
−0.153806 + 0.988101i $$0.549153\pi$$
$$230$$ 0 0
$$231$$ −5120.00 −1.45832
$$232$$ 0 0
$$233$$ −5910.00 −1.66170 −0.830852 0.556494i $$-0.812146\pi$$
−0.830852 + 0.556494i $$0.812146\pi$$
$$234$$ 0 0
$$235$$ 4160.00 1.15476
$$236$$ 0 0
$$237$$ −8960.00 −2.45576
$$238$$ 0 0
$$239$$ −3360.00 −0.909374 −0.454687 0.890651i $$-0.650249\pi$$
−0.454687 + 0.890651i $$0.650249\pi$$
$$240$$ 0 0
$$241$$ 3970.00 1.06112 0.530561 0.847647i $$-0.321981\pi$$
0.530561 + 0.847647i $$0.321981\pi$$
$$242$$ 0 0
$$243$$ −5032.00 −1.32841
$$244$$ 0 0
$$245$$ 870.000 0.226866
$$246$$ 0 0
$$247$$ −2000.00 −0.515210
$$248$$ 0 0
$$249$$ 4416.00 1.12391
$$250$$ 0 0
$$251$$ 6840.00 1.72007 0.860034 0.510237i $$-0.170442\pi$$
0.860034 + 0.510237i $$0.170442\pi$$
$$252$$ 0 0
$$253$$ −1920.00 −0.477112
$$254$$ 0 0
$$255$$ 2400.00 0.589388
$$256$$ 0 0
$$257$$ 4610.00 1.11893 0.559463 0.828855i $$-0.311007\pi$$
0.559463 + 0.828855i $$0.311007\pi$$
$$258$$ 0 0
$$259$$ 4960.00 1.18996
$$260$$ 0 0
$$261$$ −1258.00 −0.298346
$$262$$ 0 0
$$263$$ −4848.00 −1.13666 −0.568328 0.822802i $$-0.692409\pi$$
−0.568328 + 0.822802i $$0.692409\pi$$
$$264$$ 0 0
$$265$$ 4100.00 0.950419
$$266$$ 0 0
$$267$$ −2608.00 −0.597779
$$268$$ 0 0
$$269$$ 5550.00 1.25795 0.628977 0.777424i $$-0.283474\pi$$
0.628977 + 0.777424i $$0.283474\pi$$
$$270$$ 0 0
$$271$$ −480.000 −0.107594 −0.0537969 0.998552i $$-0.517132\pi$$
−0.0537969 + 0.998552i $$0.517132\pi$$
$$272$$ 0 0
$$273$$ −6400.00 −1.41885
$$274$$ 0 0
$$275$$ 1000.00 0.219281
$$276$$ 0 0
$$277$$ 1030.00 0.223418 0.111709 0.993741i $$-0.464368\pi$$
0.111709 + 0.993741i $$0.464368\pi$$
$$278$$ 0 0
$$279$$ 11840.0 2.54065
$$280$$ 0 0
$$281$$ −3270.00 −0.694206 −0.347103 0.937827i $$-0.612835\pi$$
−0.347103 + 0.937827i $$0.612835\pi$$
$$282$$ 0 0
$$283$$ 2168.00 0.455386 0.227693 0.973733i $$-0.426882\pi$$
0.227693 + 0.973733i $$0.426882\pi$$
$$284$$ 0 0
$$285$$ −3200.00 −0.665093
$$286$$ 0 0
$$287$$ 6560.00 1.34921
$$288$$ 0 0
$$289$$ −4013.00 −0.816813
$$290$$ 0 0
$$291$$ −880.000 −0.177273
$$292$$ 0 0
$$293$$ 2070.00 0.412733 0.206366 0.978475i $$-0.433836\pi$$
0.206366 + 0.978475i $$0.433836\pi$$
$$294$$ 0 0
$$295$$ 2000.00 0.394727
$$296$$ 0 0
$$297$$ −3200.00 −0.625195
$$298$$ 0 0
$$299$$ −2400.00 −0.464199
$$300$$ 0 0
$$301$$ 2432.00 0.465708
$$302$$ 0 0
$$303$$ −8784.00 −1.66544
$$304$$ 0 0
$$305$$ −300.000 −0.0563211
$$306$$ 0 0
$$307$$ 1896.00 0.352477 0.176238 0.984347i $$-0.443607\pi$$
0.176238 + 0.984347i $$0.443607\pi$$
$$308$$ 0 0
$$309$$ −384.000 −0.0706958
$$310$$ 0 0
$$311$$ −1680.00 −0.306315 −0.153158 0.988202i $$-0.548944\pi$$
−0.153158 + 0.988202i $$0.548944\pi$$
$$312$$ 0 0
$$313$$ 970.000 0.175168 0.0875841 0.996157i $$-0.472085\pi$$
0.0875841 + 0.996157i $$0.472085\pi$$
$$314$$ 0 0
$$315$$ −5920.00 −1.05890
$$316$$ 0 0
$$317$$ 7230.00 1.28100 0.640500 0.767958i $$-0.278727\pi$$
0.640500 + 0.767958i $$0.278727\pi$$
$$318$$ 0 0
$$319$$ 1360.00 0.238700
$$320$$ 0 0
$$321$$ 5312.00 0.923635
$$322$$ 0 0
$$323$$ −1200.00 −0.206718
$$324$$ 0 0
$$325$$ 1250.00 0.213346
$$326$$ 0 0
$$327$$ −2960.00 −0.500576
$$328$$ 0 0
$$329$$ −6656.00 −1.11537
$$330$$ 0 0
$$331$$ −5800.00 −0.963132 −0.481566 0.876410i $$-0.659932\pi$$
−0.481566 + 0.876410i $$0.659932\pi$$
$$332$$ 0 0
$$333$$ 11470.0 1.88754
$$334$$ 0 0
$$335$$ −7760.00 −1.26559
$$336$$ 0 0
$$337$$ −1870.00 −0.302271 −0.151136 0.988513i $$-0.548293\pi$$
−0.151136 + 0.988513i $$0.548293\pi$$
$$338$$ 0 0
$$339$$ 11920.0 1.90975
$$340$$ 0 0
$$341$$ −12800.0 −2.03272
$$342$$ 0 0
$$343$$ −6880.00 −1.08305
$$344$$ 0 0
$$345$$ −3840.00 −0.599242
$$346$$ 0 0
$$347$$ 376.000 0.0581693 0.0290846 0.999577i $$-0.490741\pi$$
0.0290846 + 0.999577i $$0.490741\pi$$
$$348$$ 0 0
$$349$$ −7586.00 −1.16352 −0.581761 0.813360i $$-0.697636\pi$$
−0.581761 + 0.813360i $$0.697636\pi$$
$$350$$ 0 0
$$351$$ −4000.00 −0.608274
$$352$$ 0 0
$$353$$ 2530.00 0.381468 0.190734 0.981642i $$-0.438913\pi$$
0.190734 + 0.981642i $$0.438913\pi$$
$$354$$ 0 0
$$355$$ −4000.00 −0.598022
$$356$$ 0 0
$$357$$ −3840.00 −0.569284
$$358$$ 0 0
$$359$$ 9680.00 1.42309 0.711547 0.702638i $$-0.247995\pi$$
0.711547 + 0.702638i $$0.247995\pi$$
$$360$$ 0 0
$$361$$ −5259.00 −0.766730
$$362$$ 0 0
$$363$$ 2152.00 0.311159
$$364$$ 0 0
$$365$$ 6300.00 0.903444
$$366$$ 0 0
$$367$$ 2784.00 0.395977 0.197989 0.980204i $$-0.436559\pi$$
0.197989 + 0.980204i $$0.436559\pi$$
$$368$$ 0 0
$$369$$ 15170.0 2.14016
$$370$$ 0 0
$$371$$ −6560.00 −0.918001
$$372$$ 0 0
$$373$$ 7910.00 1.09803 0.549014 0.835813i $$-0.315003\pi$$
0.549014 + 0.835813i $$0.315003\pi$$
$$374$$ 0 0
$$375$$ 12000.0 1.65247
$$376$$ 0 0
$$377$$ 1700.00 0.232240
$$378$$ 0 0
$$379$$ 1720.00 0.233115 0.116557 0.993184i $$-0.462814\pi$$
0.116557 + 0.993184i $$0.462814\pi$$
$$380$$ 0 0
$$381$$ −8192.00 −1.10155
$$382$$ 0 0
$$383$$ 11008.0 1.46862 0.734311 0.678813i $$-0.237505\pi$$
0.734311 + 0.678813i $$0.237505\pi$$
$$384$$ 0 0
$$385$$ 6400.00 0.847206
$$386$$ 0 0
$$387$$ 5624.00 0.738718
$$388$$ 0 0
$$389$$ −12330.0 −1.60708 −0.803542 0.595248i $$-0.797054\pi$$
−0.803542 + 0.595248i $$0.797054\pi$$
$$390$$ 0 0
$$391$$ −1440.00 −0.186250
$$392$$ 0 0
$$393$$ 9280.00 1.19113
$$394$$ 0 0
$$395$$ 11200.0 1.42667
$$396$$ 0 0
$$397$$ −4370.00 −0.552453 −0.276227 0.961093i $$-0.589084\pi$$
−0.276227 + 0.961093i $$0.589084\pi$$
$$398$$ 0 0
$$399$$ 5120.00 0.642408
$$400$$ 0 0
$$401$$ 3298.00 0.410709 0.205354 0.978688i $$-0.434165\pi$$
0.205354 + 0.978688i $$0.434165\pi$$
$$402$$ 0 0
$$403$$ −16000.0 −1.97771
$$404$$ 0 0
$$405$$ 3590.00 0.440466
$$406$$ 0 0
$$407$$ −12400.0 −1.51018
$$408$$ 0 0
$$409$$ −9110.00 −1.10137 −0.550685 0.834713i $$-0.685634\pi$$
−0.550685 + 0.834713i $$0.685634\pi$$
$$410$$ 0 0
$$411$$ 4560.00 0.547271
$$412$$ 0 0
$$413$$ −3200.00 −0.381263
$$414$$ 0 0
$$415$$ −5520.00 −0.652930
$$416$$ 0 0
$$417$$ −15680.0 −1.84137
$$418$$ 0 0
$$419$$ 7880.00 0.918767 0.459383 0.888238i $$-0.348070\pi$$
0.459383 + 0.888238i $$0.348070\pi$$
$$420$$ 0 0
$$421$$ −5290.00 −0.612396 −0.306198 0.951968i $$-0.599057\pi$$
−0.306198 + 0.951968i $$0.599057\pi$$
$$422$$ 0 0
$$423$$ −15392.0 −1.76923
$$424$$ 0 0
$$425$$ 750.000 0.0856008
$$426$$ 0 0
$$427$$ 480.000 0.0544001
$$428$$ 0 0
$$429$$ 16000.0 1.80067
$$430$$ 0 0
$$431$$ 13920.0 1.55569 0.777845 0.628456i $$-0.216313\pi$$
0.777845 + 0.628456i $$0.216313\pi$$
$$432$$ 0 0
$$433$$ 4930.00 0.547161 0.273580 0.961849i $$-0.411792\pi$$
0.273580 + 0.961849i $$0.411792\pi$$
$$434$$ 0 0
$$435$$ 2720.00 0.299802
$$436$$ 0 0
$$437$$ 1920.00 0.210174
$$438$$ 0 0
$$439$$ −10640.0 −1.15676 −0.578382 0.815766i $$-0.696316\pi$$
−0.578382 + 0.815766i $$0.696316\pi$$
$$440$$ 0 0
$$441$$ −3219.00 −0.347587
$$442$$ 0 0
$$443$$ −9288.00 −0.996131 −0.498066 0.867139i $$-0.665956\pi$$
−0.498066 + 0.867139i $$0.665956\pi$$
$$444$$ 0 0
$$445$$ 3260.00 0.347278
$$446$$ 0 0
$$447$$ −16080.0 −1.70147
$$448$$ 0 0
$$449$$ 12850.0 1.35062 0.675311 0.737533i $$-0.264010\pi$$
0.675311 + 0.737533i $$0.264010\pi$$
$$450$$ 0 0
$$451$$ −16400.0 −1.71230
$$452$$ 0 0
$$453$$ −5760.00 −0.597414
$$454$$ 0 0
$$455$$ 8000.00 0.824276
$$456$$ 0 0
$$457$$ 10490.0 1.07375 0.536873 0.843663i $$-0.319606\pi$$
0.536873 + 0.843663i $$0.319606\pi$$
$$458$$ 0 0
$$459$$ −2400.00 −0.244058
$$460$$ 0 0
$$461$$ 11118.0 1.12325 0.561624 0.827393i $$-0.310177\pi$$
0.561624 + 0.827393i $$0.310177\pi$$
$$462$$ 0 0
$$463$$ 5792.00 0.581376 0.290688 0.956818i $$-0.406116\pi$$
0.290688 + 0.956818i $$0.406116\pi$$
$$464$$ 0 0
$$465$$ −25600.0 −2.55306
$$466$$ 0 0
$$467$$ 2216.00 0.219581 0.109790 0.993955i $$-0.464982\pi$$
0.109790 + 0.993955i $$0.464982\pi$$
$$468$$ 0 0
$$469$$ 12416.0 1.22243
$$470$$ 0 0
$$471$$ 14320.0 1.40091
$$472$$ 0 0
$$473$$ −6080.00 −0.591033
$$474$$ 0 0
$$475$$ −1000.00 −0.0965961
$$476$$ 0 0
$$477$$ −15170.0 −1.45616
$$478$$ 0 0
$$479$$ −10560.0 −1.00730 −0.503652 0.863907i $$-0.668011\pi$$
−0.503652 + 0.863907i $$0.668011\pi$$
$$480$$ 0 0
$$481$$ −15500.0 −1.46931
$$482$$ 0 0
$$483$$ 6144.00 0.578803
$$484$$ 0 0
$$485$$ 1100.00 0.102986
$$486$$ 0 0
$$487$$ 13264.0 1.23419 0.617094 0.786890i $$-0.288310\pi$$
0.617094 + 0.786890i $$0.288310\pi$$
$$488$$ 0 0
$$489$$ −9664.00 −0.893704
$$490$$ 0 0
$$491$$ −4840.00 −0.444860 −0.222430 0.974949i $$-0.571399\pi$$
−0.222430 + 0.974949i $$0.571399\pi$$
$$492$$ 0 0
$$493$$ 1020.00 0.0931815
$$494$$ 0 0
$$495$$ 14800.0 1.34386
$$496$$ 0 0
$$497$$ 6400.00 0.577624
$$498$$ 0 0
$$499$$ 19560.0 1.75476 0.877381 0.479795i $$-0.159289\pi$$
0.877381 + 0.479795i $$0.159289\pi$$
$$500$$ 0 0
$$501$$ 23168.0 2.06601
$$502$$ 0 0
$$503$$ −528.000 −0.0468039 −0.0234019 0.999726i $$-0.507450\pi$$
−0.0234019 + 0.999726i $$0.507450\pi$$
$$504$$ 0 0
$$505$$ 10980.0 0.967532
$$506$$ 0 0
$$507$$ 2424.00 0.212335
$$508$$ 0 0
$$509$$ −19554.0 −1.70278 −0.851391 0.524532i $$-0.824240\pi$$
−0.851391 + 0.524532i $$0.824240\pi$$
$$510$$ 0 0
$$511$$ −10080.0 −0.872628
$$512$$ 0 0
$$513$$ 3200.00 0.275406
$$514$$ 0 0
$$515$$ 480.000 0.0410705
$$516$$ 0 0
$$517$$ 16640.0 1.41552
$$518$$ 0 0
$$519$$ 6000.00 0.507458
$$520$$ 0 0
$$521$$ 15162.0 1.27497 0.637485 0.770463i $$-0.279975\pi$$
0.637485 + 0.770463i $$0.279975\pi$$
$$522$$ 0 0
$$523$$ 10968.0 0.917012 0.458506 0.888691i $$-0.348385\pi$$
0.458506 + 0.888691i $$0.348385\pi$$
$$524$$ 0 0
$$525$$ −3200.00 −0.266018
$$526$$ 0 0
$$527$$ −9600.00 −0.793515
$$528$$ 0 0
$$529$$ −9863.00 −0.810635
$$530$$ 0 0
$$531$$ −7400.00 −0.604770
$$532$$ 0 0
$$533$$ −20500.0 −1.66595
$$534$$ 0 0
$$535$$ −6640.00 −0.536584
$$536$$ 0 0
$$537$$ 18240.0 1.46576
$$538$$ 0 0
$$539$$ 3480.00 0.278097
$$540$$ 0 0
$$541$$ −6722.00 −0.534198 −0.267099 0.963669i $$-0.586065\pi$$
−0.267099 + 0.963669i $$0.586065\pi$$
$$542$$ 0 0
$$543$$ −3536.00 −0.279455
$$544$$ 0 0
$$545$$ 3700.00 0.290808
$$546$$ 0 0
$$547$$ 20424.0 1.59647 0.798233 0.602348i $$-0.205768\pi$$
0.798233 + 0.602348i $$0.205768\pi$$
$$548$$ 0 0
$$549$$ 1110.00 0.0862908
$$550$$ 0 0
$$551$$ −1360.00 −0.105151
$$552$$ 0 0
$$553$$ −17920.0 −1.37800
$$554$$ 0 0
$$555$$ −24800.0 −1.89676
$$556$$ 0 0
$$557$$ −6610.00 −0.502827 −0.251414 0.967880i $$-0.580895\pi$$
−0.251414 + 0.967880i $$0.580895\pi$$
$$558$$ 0 0
$$559$$ −7600.00 −0.575037
$$560$$ 0 0
$$561$$ 9600.00 0.722482
$$562$$ 0 0
$$563$$ −2712.00 −0.203015 −0.101507 0.994835i $$-0.532367\pi$$
−0.101507 + 0.994835i $$0.532367\pi$$
$$564$$ 0 0
$$565$$ −14900.0 −1.10946
$$566$$ 0 0
$$567$$ −5744.00 −0.425441
$$568$$ 0 0
$$569$$ 3530.00 0.260080 0.130040 0.991509i $$-0.458489\pi$$
0.130040 + 0.991509i $$0.458489\pi$$
$$570$$ 0 0
$$571$$ −13640.0 −0.999678 −0.499839 0.866118i $$-0.666608\pi$$
−0.499839 + 0.866118i $$0.666608\pi$$
$$572$$ 0 0
$$573$$ −15360.0 −1.11985
$$574$$ 0 0
$$575$$ −1200.00 −0.0870321
$$576$$ 0 0
$$577$$ −6270.00 −0.452380 −0.226190 0.974083i $$-0.572627\pi$$
−0.226190 + 0.974083i $$0.572627\pi$$
$$578$$ 0 0
$$579$$ −40560.0 −2.91125
$$580$$ 0 0
$$581$$ 8832.00 0.630659
$$582$$ 0 0
$$583$$ 16400.0 1.16504
$$584$$ 0 0
$$585$$ 18500.0 1.30749
$$586$$ 0 0
$$587$$ −8616.00 −0.605827 −0.302913 0.953018i $$-0.597959\pi$$
−0.302913 + 0.953018i $$0.597959\pi$$
$$588$$ 0 0
$$589$$ 12800.0 0.895441
$$590$$ 0 0
$$591$$ 15280.0 1.06351
$$592$$ 0 0
$$593$$ 5490.00 0.380181 0.190090 0.981767i $$-0.439122\pi$$
0.190090 + 0.981767i $$0.439122\pi$$
$$594$$ 0 0
$$595$$ 4800.00 0.330724
$$596$$ 0 0
$$597$$ 23680.0 1.62338
$$598$$ 0 0
$$599$$ −15440.0 −1.05319 −0.526595 0.850116i $$-0.676532\pi$$
−0.526595 + 0.850116i $$0.676532\pi$$
$$600$$ 0 0
$$601$$ 8890.00 0.603379 0.301689 0.953406i $$-0.402449\pi$$
0.301689 + 0.953406i $$0.402449\pi$$
$$602$$ 0 0
$$603$$ 28712.0 1.93904
$$604$$ 0 0
$$605$$ −2690.00 −0.180767
$$606$$ 0 0
$$607$$ 23744.0 1.58771 0.793854 0.608108i $$-0.208071\pi$$
0.793854 + 0.608108i $$0.208071\pi$$
$$608$$ 0 0
$$609$$ −4352.00 −0.289576
$$610$$ 0 0
$$611$$ 20800.0 1.37721
$$612$$ 0 0
$$613$$ −15210.0 −1.00216 −0.501082 0.865400i $$-0.667064\pi$$
−0.501082 + 0.865400i $$0.667064\pi$$
$$614$$ 0 0
$$615$$ −32800.0 −2.15061
$$616$$ 0 0
$$617$$ −12630.0 −0.824092 −0.412046 0.911163i $$-0.635186\pi$$
−0.412046 + 0.911163i $$0.635186\pi$$
$$618$$ 0 0
$$619$$ 11160.0 0.724650 0.362325 0.932052i $$-0.381983\pi$$
0.362325 + 0.932052i $$0.381983\pi$$
$$620$$ 0 0
$$621$$ 3840.00 0.248138
$$622$$ 0 0
$$623$$ −5216.00 −0.335433
$$624$$ 0 0
$$625$$ −11875.0 −0.760000
$$626$$ 0 0
$$627$$ −12800.0 −0.815284
$$628$$ 0 0
$$629$$ −9300.00 −0.589531
$$630$$ 0 0
$$631$$ 13040.0 0.822685 0.411342 0.911481i $$-0.365060\pi$$
0.411342 + 0.911481i $$0.365060\pi$$
$$632$$ 0 0
$$633$$ 320.000 0.0200930
$$634$$ 0 0
$$635$$ 10240.0 0.639940
$$636$$ 0 0
$$637$$ 4350.00 0.270570
$$638$$ 0 0
$$639$$ 14800.0 0.916242
$$640$$ 0 0
$$641$$ −16910.0 −1.04197 −0.520987 0.853565i $$-0.674436\pi$$
−0.520987 + 0.853565i $$0.674436\pi$$
$$642$$ 0 0
$$643$$ 4488.00 0.275256 0.137628 0.990484i $$-0.456052\pi$$
0.137628 + 0.990484i $$0.456052\pi$$
$$644$$ 0 0
$$645$$ −12160.0 −0.742325
$$646$$ 0 0
$$647$$ 2064.00 0.125416 0.0627080 0.998032i $$-0.480026\pi$$
0.0627080 + 0.998032i $$0.480026\pi$$
$$648$$ 0 0
$$649$$ 8000.00 0.483864
$$650$$ 0 0
$$651$$ 40960.0 2.46597
$$652$$ 0 0
$$653$$ 4270.00 0.255893 0.127946 0.991781i $$-0.459161\pi$$
0.127946 + 0.991781i $$0.459161\pi$$
$$654$$ 0 0
$$655$$ −11600.0 −0.691984
$$656$$ 0 0
$$657$$ −23310.0 −1.38419
$$658$$ 0 0
$$659$$ −19800.0 −1.17041 −0.585204 0.810886i $$-0.698985\pi$$
−0.585204 + 0.810886i $$0.698985\pi$$
$$660$$ 0 0
$$661$$ 27110.0 1.59524 0.797622 0.603157i $$-0.206091\pi$$
0.797622 + 0.603157i $$0.206091\pi$$
$$662$$ 0 0
$$663$$ 12000.0 0.702928
$$664$$ 0 0
$$665$$ −6400.00 −0.373205
$$666$$ 0 0
$$667$$ −1632.00 −0.0947396
$$668$$ 0 0
$$669$$ 34304.0 1.98247
$$670$$ 0 0
$$671$$ −1200.00 −0.0690395
$$672$$ 0 0
$$673$$ 32210.0 1.84488 0.922440 0.386140i $$-0.126192\pi$$
0.922440 + 0.386140i $$0.126192\pi$$
$$674$$ 0 0
$$675$$ −2000.00 −0.114044
$$676$$ 0 0
$$677$$ 27190.0 1.54357 0.771785 0.635884i $$-0.219364\pi$$
0.771785 + 0.635884i $$0.219364\pi$$
$$678$$ 0 0
$$679$$ −1760.00 −0.0994736
$$680$$ 0 0
$$681$$ −51648.0 −2.90625
$$682$$ 0 0
$$683$$ −20328.0 −1.13884 −0.569421 0.822046i $$-0.692833\pi$$
−0.569421 + 0.822046i $$0.692833\pi$$
$$684$$ 0 0
$$685$$ −5700.00 −0.317935
$$686$$ 0 0
$$687$$ −8528.00 −0.473600
$$688$$ 0 0
$$689$$ 20500.0 1.13351
$$690$$ 0 0
$$691$$ 12520.0 0.689267 0.344633 0.938737i $$-0.388003\pi$$
0.344633 + 0.938737i $$0.388003\pi$$
$$692$$ 0 0
$$693$$ −23680.0 −1.29802
$$694$$ 0 0
$$695$$ 19600.0 1.06974
$$696$$ 0 0
$$697$$ −12300.0 −0.668430
$$698$$ 0 0
$$699$$ −47280.0 −2.55836
$$700$$ 0 0
$$701$$ 11550.0 0.622307 0.311154 0.950360i $$-0.399285\pi$$
0.311154 + 0.950360i $$0.399285\pi$$
$$702$$ 0 0
$$703$$ 12400.0 0.665256
$$704$$ 0 0
$$705$$ 33280.0 1.77787
$$706$$ 0 0
$$707$$ −17568.0 −0.934530
$$708$$ 0 0
$$709$$ −34154.0 −1.80914 −0.904570 0.426325i $$-0.859808\pi$$
−0.904570 + 0.426325i $$0.859808\pi$$
$$710$$ 0 0
$$711$$ −41440.0 −2.18582
$$712$$ 0 0
$$713$$ 15360.0 0.806783
$$714$$ 0 0
$$715$$ −20000.0 −1.04609
$$716$$ 0 0
$$717$$ −26880.0 −1.40007
$$718$$ 0 0
$$719$$ −22880.0 −1.18676 −0.593380 0.804923i $$-0.702207\pi$$
−0.593380 + 0.804923i $$0.702207\pi$$
$$720$$ 0 0
$$721$$ −768.000 −0.0396696
$$722$$ 0 0
$$723$$ 31760.0 1.63370
$$724$$ 0 0
$$725$$ 850.000 0.0435424
$$726$$ 0 0
$$727$$ 10416.0 0.531373 0.265686 0.964060i $$-0.414401\pi$$
0.265686 + 0.964060i $$0.414401\pi$$
$$728$$ 0 0
$$729$$ −30563.0 −1.55276
$$730$$ 0 0
$$731$$ −4560.00 −0.230722
$$732$$ 0 0
$$733$$ 14750.0 0.743252 0.371626 0.928383i $$-0.378800\pi$$
0.371626 + 0.928383i $$0.378800\pi$$
$$734$$ 0 0
$$735$$ 6960.00 0.349284
$$736$$ 0 0
$$737$$ −31040.0 −1.55139
$$738$$ 0 0
$$739$$ −2360.00 −0.117475 −0.0587375 0.998273i $$-0.518707\pi$$
−0.0587375 + 0.998273i $$0.518707\pi$$
$$740$$ 0 0
$$741$$ −16000.0 −0.793218
$$742$$ 0 0
$$743$$ 32208.0 1.59031 0.795153 0.606409i $$-0.207391\pi$$
0.795153 + 0.606409i $$0.207391\pi$$
$$744$$ 0 0
$$745$$ 20100.0 0.988466
$$746$$ 0 0
$$747$$ 20424.0 1.00037
$$748$$ 0 0
$$749$$ 10624.0 0.518281
$$750$$ 0 0
$$751$$ −36640.0 −1.78031 −0.890155 0.455658i $$-0.849404\pi$$
−0.890155 + 0.455658i $$0.849404\pi$$
$$752$$ 0 0
$$753$$ 54720.0 2.64822
$$754$$ 0 0
$$755$$ 7200.00 0.347066
$$756$$ 0 0
$$757$$ −12090.0 −0.580474 −0.290237 0.956955i $$-0.593734\pi$$
−0.290237 + 0.956955i $$0.593734\pi$$
$$758$$ 0 0
$$759$$ −15360.0 −0.734562
$$760$$ 0 0
$$761$$ −3318.00 −0.158052 −0.0790259 0.996873i $$-0.525181\pi$$
−0.0790259 + 0.996873i $$0.525181\pi$$
$$762$$ 0 0
$$763$$ −5920.00 −0.280889
$$764$$ 0 0
$$765$$ 11100.0 0.524603
$$766$$ 0 0
$$767$$ 10000.0 0.470768
$$768$$ 0 0
$$769$$ 11506.0 0.539554 0.269777 0.962923i $$-0.413050\pi$$
0.269777 + 0.962923i $$0.413050\pi$$
$$770$$ 0 0
$$771$$ 36880.0 1.72270
$$772$$ 0 0
$$773$$ 22230.0 1.03436 0.517178 0.855878i $$-0.326982\pi$$
0.517178 + 0.855878i $$0.326982\pi$$
$$774$$ 0 0
$$775$$ −8000.00 −0.370798
$$776$$ 0 0
$$777$$ 39680.0 1.83206
$$778$$ 0 0
$$779$$ 16400.0 0.754289
$$780$$ 0 0
$$781$$ −16000.0 −0.733067
$$782$$ 0 0
$$783$$ −2720.00 −0.124144
$$784$$ 0 0
$$785$$ −17900.0 −0.813858
$$786$$ 0 0
$$787$$ −21336.0 −0.966387 −0.483193 0.875514i $$-0.660523\pi$$
−0.483193 + 0.875514i $$0.660523\pi$$
$$788$$ 0 0
$$789$$ −38784.0 −1.75000
$$790$$ 0 0
$$791$$ 23840.0 1.07162
$$792$$ 0 0
$$793$$ −1500.00 −0.0671709
$$794$$ 0 0
$$795$$ 32800.0 1.46327
$$796$$ 0 0
$$797$$ −7170.00 −0.318663 −0.159332 0.987225i $$-0.550934\pi$$
−0.159332 + 0.987225i $$0.550934\pi$$
$$798$$ 0 0
$$799$$ 12480.0 0.552579
$$800$$ 0 0
$$801$$ −12062.0 −0.532072
$$802$$ 0 0
$$803$$ 25200.0 1.10746
$$804$$ 0 0
$$805$$ −7680.00 −0.336254
$$806$$ 0 0
$$807$$ 44400.0 1.93675
$$808$$ 0 0
$$809$$ −23654.0 −1.02797 −0.513987 0.857798i $$-0.671832\pi$$
−0.513987 + 0.857798i $$0.671832\pi$$
$$810$$ 0 0
$$811$$ −30440.0 −1.31799 −0.658997 0.752146i $$-0.729019\pi$$
−0.658997 + 0.752146i $$0.729019\pi$$
$$812$$ 0 0
$$813$$ −3840.00 −0.165652
$$814$$ 0 0
$$815$$ 12080.0 0.519195
$$816$$ 0 0
$$817$$ 6080.00 0.260358
$$818$$ 0 0
$$819$$ −29600.0 −1.26289
$$820$$ 0 0
$$821$$ −19930.0 −0.847213 −0.423606 0.905846i $$-0.639236\pi$$
−0.423606 + 0.905846i $$0.639236\pi$$
$$822$$ 0 0
$$823$$ −9872.00 −0.418124 −0.209062 0.977902i $$-0.567041\pi$$
−0.209062 + 0.977902i $$0.567041\pi$$
$$824$$ 0 0
$$825$$ 8000.00 0.337605
$$826$$ 0 0
$$827$$ −5704.00 −0.239840 −0.119920 0.992784i $$-0.538264\pi$$
−0.119920 + 0.992784i $$0.538264\pi$$
$$828$$ 0 0
$$829$$ 27230.0 1.14082 0.570408 0.821361i $$-0.306785\pi$$
0.570408 + 0.821361i $$0.306785\pi$$
$$830$$ 0 0
$$831$$ 8240.00 0.343974
$$832$$ 0 0
$$833$$ 2610.00 0.108561
$$834$$ 0 0
$$835$$ −28960.0 −1.20024
$$836$$ 0 0
$$837$$ 25600.0 1.05719
$$838$$ 0 0
$$839$$ −18800.0 −0.773597 −0.386799 0.922164i $$-0.626419\pi$$
−0.386799 + 0.922164i $$0.626419\pi$$
$$840$$ 0 0
$$841$$ −23233.0 −0.952602
$$842$$ 0 0
$$843$$ −26160.0 −1.06880
$$844$$ 0 0
$$845$$ −3030.00 −0.123355
$$846$$ 0 0
$$847$$ 4304.00 0.174601
$$848$$ 0 0
$$849$$ 17344.0 0.701113
$$850$$ 0 0
$$851$$ 14880.0 0.599389
$$852$$ 0 0
$$853$$ −12090.0 −0.485292 −0.242646 0.970115i $$-0.578015\pi$$
−0.242646 + 0.970115i $$0.578015\pi$$
$$854$$ 0 0
$$855$$ −14800.0 −0.591988
$$856$$ 0 0
$$857$$ −470.000 −0.0187338 −0.00936692 0.999956i $$-0.502982\pi$$
−0.00936692 + 0.999956i $$0.502982\pi$$
$$858$$ 0 0
$$859$$ 24440.0 0.970759 0.485380 0.874304i $$-0.338681\pi$$
0.485380 + 0.874304i $$0.338681\pi$$
$$860$$ 0 0
$$861$$ 52480.0 2.07725
$$862$$ 0 0
$$863$$ −22592.0 −0.891125 −0.445562 0.895251i $$-0.646996\pi$$
−0.445562 + 0.895251i $$0.646996\pi$$
$$864$$ 0 0
$$865$$ −7500.00 −0.294807
$$866$$ 0 0
$$867$$ −32104.0 −1.25757
$$868$$ 0 0
$$869$$ 44800.0 1.74883
$$870$$ 0 0
$$871$$ −38800.0 −1.50940
$$872$$ 0 0
$$873$$ −4070.00 −0.157788
$$874$$ 0 0
$$875$$ 24000.0 0.927255
$$876$$ 0 0
$$877$$ −17330.0 −0.667266 −0.333633 0.942703i $$-0.608275\pi$$
−0.333633 + 0.942703i $$0.608275\pi$$
$$878$$ 0 0
$$879$$ 16560.0 0.635444
$$880$$ 0 0
$$881$$ −31470.0 −1.20346 −0.601732 0.798698i $$-0.705522\pi$$
−0.601732 + 0.798698i $$0.705522\pi$$
$$882$$ 0 0
$$883$$ −3352.00 −0.127751 −0.0638753 0.997958i $$-0.520346\pi$$
−0.0638753 + 0.997958i $$0.520346\pi$$
$$884$$ 0 0
$$885$$ 16000.0 0.607722
$$886$$ 0 0
$$887$$ −48144.0 −1.82245 −0.911227 0.411904i $$-0.864864\pi$$
−0.911227 + 0.411904i $$0.864864\pi$$
$$888$$ 0 0
$$889$$ −16384.0 −0.618112
$$890$$ 0 0
$$891$$ 14360.0 0.539931
$$892$$ 0 0
$$893$$ −16640.0 −0.623557
$$894$$ 0 0
$$895$$ −22800.0 −0.851531
$$896$$ 0 0
$$897$$ −19200.0 −0.714682
$$898$$ 0 0
$$899$$ −10880.0 −0.403636
$$900$$ 0 0
$$901$$ 12300.0 0.454797
$$902$$ 0 0
$$903$$ 19456.0 0.717005
$$904$$ 0 0
$$905$$ 4420.00 0.162349
$$906$$ 0 0
$$907$$ 16216.0 0.593653 0.296827 0.954931i $$-0.404072\pi$$
0.296827 + 0.954931i $$0.404072\pi$$
$$908$$ 0 0
$$909$$ −40626.0 −1.48238
$$910$$ 0 0
$$911$$ 49440.0 1.79805 0.899023 0.437901i $$-0.144278\pi$$
0.899023 + 0.437901i $$0.144278\pi$$
$$912$$ 0 0
$$913$$ −22080.0 −0.800374
$$914$$ 0 0
$$915$$ −2400.00 −0.0867121
$$916$$ 0 0
$$917$$ 18560.0 0.668381
$$918$$ 0 0
$$919$$ −16080.0 −0.577182 −0.288591 0.957452i $$-0.593187\pi$$
−0.288591 + 0.957452i $$0.593187\pi$$
$$920$$ 0 0
$$921$$ 15168.0 0.542674
$$922$$ 0 0
$$923$$ −20000.0 −0.713226
$$924$$ 0 0
$$925$$ −7750.00 −0.275479
$$926$$ 0 0
$$927$$ −1776.00 −0.0629250
$$928$$ 0 0
$$929$$ −11310.0 −0.399428 −0.199714 0.979854i $$-0.564001\pi$$
−0.199714 + 0.979854i $$0.564001\pi$$
$$930$$ 0 0
$$931$$ −3480.00 −0.122505
$$932$$ 0 0
$$933$$ −13440.0 −0.471603
$$934$$ 0 0
$$935$$ −12000.0 −0.419724
$$936$$ 0 0
$$937$$ 25130.0 0.876159 0.438080 0.898936i $$-0.355659\pi$$
0.438080 + 0.898936i $$0.355659\pi$$
$$938$$ 0 0
$$939$$ 7760.00 0.269689
$$940$$ 0 0
$$941$$ −22322.0 −0.773301 −0.386651 0.922226i $$-0.626368\pi$$
−0.386651 + 0.922226i $$0.626368\pi$$
$$942$$ 0 0
$$943$$ 19680.0 0.679607
$$944$$ 0 0
$$945$$ −12800.0 −0.440618
$$946$$ 0 0
$$947$$ 36456.0 1.25096 0.625481 0.780239i $$-0.284903\pi$$
0.625481 + 0.780239i $$0.284903\pi$$
$$948$$ 0 0
$$949$$ 31500.0 1.07749
$$950$$ 0 0
$$951$$ 57840.0 1.97223
$$952$$ 0 0
$$953$$ 40650.0 1.38172 0.690862 0.722987i $$-0.257231\pi$$
0.690862 + 0.722987i $$0.257231\pi$$
$$954$$ 0 0
$$955$$ 19200.0 0.650573
$$956$$ 0 0
$$957$$ 10880.0 0.367503
$$958$$ 0 0
$$959$$ 9120.00 0.307091
$$960$$ 0 0
$$961$$ 72609.0 2.43728
$$962$$ 0 0
$$963$$ 24568.0 0.822111
$$964$$ 0 0
$$965$$ 50700.0 1.69129
$$966$$ 0 0
$$967$$ 34704.0 1.15409 0.577045 0.816712i $$-0.304206\pi$$
0.577045 + 0.816712i $$0.304206\pi$$
$$968$$ 0 0
$$969$$ −9600.00 −0.318263
$$970$$ 0 0
$$971$$ −30760.0 −1.01662 −0.508309 0.861175i $$-0.669729\pi$$
−0.508309 + 0.861175i $$0.669729\pi$$
$$972$$ 0 0
$$973$$ −31360.0 −1.03325
$$974$$ 0 0
$$975$$ 10000.0 0.328468
$$976$$ 0 0
$$977$$ −38110.0 −1.24795 −0.623975 0.781444i $$-0.714483\pi$$
−0.623975 + 0.781444i $$0.714483\pi$$
$$978$$ 0 0
$$979$$ 13040.0 0.425700
$$980$$ 0 0
$$981$$ −13690.0 −0.445554
$$982$$ 0 0
$$983$$ 19632.0 0.636992 0.318496 0.947924i $$-0.396822\pi$$
0.318496 + 0.947924i $$0.396822\pi$$
$$984$$ 0 0
$$985$$ −19100.0 −0.617844
$$986$$ 0 0
$$987$$ −53248.0 −1.71723
$$988$$ 0 0
$$989$$ 7296.00 0.234580
$$990$$ 0 0
$$991$$ −47680.0 −1.52836 −0.764180 0.645003i $$-0.776856\pi$$
−0.764180 + 0.645003i $$0.776856\pi$$
$$992$$ 0 0
$$993$$ −46400.0 −1.48284
$$994$$ 0 0
$$995$$ −29600.0 −0.943099
$$996$$ 0 0
$$997$$ −39690.0 −1.26078 −0.630389 0.776280i $$-0.717104\pi$$
−0.630389 + 0.776280i $$0.717104\pi$$
$$998$$ 0 0
$$999$$ 24800.0 0.785423
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 32.4.a.c.1.1 yes 1
3.2 odd 2 288.4.a.i.1.1 1
4.3 odd 2 32.4.a.a.1.1 1
5.2 odd 4 800.4.c.a.449.1 2
5.3 odd 4 800.4.c.a.449.2 2
5.4 even 2 800.4.a.a.1.1 1
7.6 odd 2 1568.4.a.c.1.1 1
8.3 odd 2 64.4.a.e.1.1 1
8.5 even 2 64.4.a.a.1.1 1
12.11 even 2 288.4.a.h.1.1 1
16.3 odd 4 256.4.b.e.129.1 2
16.5 even 4 256.4.b.c.129.1 2
16.11 odd 4 256.4.b.e.129.2 2
16.13 even 4 256.4.b.c.129.2 2
20.3 even 4 800.4.c.b.449.1 2
20.7 even 4 800.4.c.b.449.2 2
20.19 odd 2 800.4.a.k.1.1 1
24.5 odd 2 576.4.a.h.1.1 1
24.11 even 2 576.4.a.g.1.1 1
28.27 even 2 1568.4.a.o.1.1 1
40.19 odd 2 1600.4.a.e.1.1 1
40.29 even 2 1600.4.a.bw.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.a.a.1.1 1 4.3 odd 2
32.4.a.c.1.1 yes 1 1.1 even 1 trivial
64.4.a.a.1.1 1 8.5 even 2
64.4.a.e.1.1 1 8.3 odd 2
256.4.b.c.129.1 2 16.5 even 4
256.4.b.c.129.2 2 16.13 even 4
256.4.b.e.129.1 2 16.3 odd 4
256.4.b.e.129.2 2 16.11 odd 4
288.4.a.h.1.1 1 12.11 even 2
288.4.a.i.1.1 1 3.2 odd 2
576.4.a.g.1.1 1 24.11 even 2
576.4.a.h.1.1 1 24.5 odd 2
800.4.a.a.1.1 1 5.4 even 2
800.4.a.k.1.1 1 20.19 odd 2
800.4.c.a.449.1 2 5.2 odd 4
800.4.c.a.449.2 2 5.3 odd 4
800.4.c.b.449.1 2 20.3 even 4
800.4.c.b.449.2 2 20.7 even 4
1568.4.a.c.1.1 1 7.6 odd 2
1568.4.a.o.1.1 1 28.27 even 2
1600.4.a.e.1.1 1 40.19 odd 2
1600.4.a.bw.1.1 1 40.29 even 2