# Properties

 Label 1568.4.a.c.1.1 Level $1568$ Weight $4$ Character 1568.1 Self dual yes Analytic conductor $92.515$ 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] = [1568,4,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, 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("1568.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1568.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: $$92.5149948890$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1568.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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 $$-0.779644\pi$$
−0.769800 + 0.638285i $$0.779644\pi$$
$$4$$ 0 0
$$5$$ 10.0000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 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 $$-0.320927\pi$$
0.533366 + 0.845885i $$0.320927\pi$$
$$14$$ 0 0
$$15$$ −80.0000 −1.37706
$$16$$ 0 0
$$17$$ 30.0000 0.428004 0.214002 0.976833i $$-0.431350\pi$$
0.214002 + 0.976833i $$0.431350\pi$$
$$18$$ 0 0
$$19$$ −40.0000 −0.482980 −0.241490 0.970403i $$-0.577636\pi$$
−0.241490 + 0.970403i $$0.577636\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 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 $$-0.877617\pi$$
−0.926995 + 0.375073i $$0.877617\pi$$
$$32$$ 0 0
$$33$$ 320.000 1.68803
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 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 $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 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 $$-0.276636\pi$$
0.645530 + 0.763735i $$0.276636\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 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 $$-0.429179\pi$$
0.220659 + 0.975351i $$0.429179\pi$$
$$60$$ 0 0
$$61$$ −30.0000 −0.0629690 −0.0314845 0.999504i $$-0.510023\pi$$
−0.0314845 + 0.999504i $$0.510023\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 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 $$-0.331478\pi$$
0.505041 + 0.863096i $$0.331478\pi$$
$$74$$ 0 0
$$75$$ 200.000 0.307920
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 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 $$-0.618931\pi$$
−0.364999 + 0.931008i $$0.618931\pi$$
$$84$$ 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 $$-0.437810\pi$$
0.194134 + 0.980975i $$0.437810\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 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 $$-0.481664\pi$$
0.0575712 + 0.998341i $$0.481664\pi$$
$$98$$ 0 0
$$99$$ −1480.00 −1.50248
$$100$$ 0 0
$$101$$ 1098.00 1.08173 0.540867 0.841108i $$-0.318096\pi$$
0.540867 + 0.841108i $$0.318096\pi$$
$$102$$ 0 0
$$103$$ 48.0000 0.0459183 0.0229591 0.999736i $$-0.492691\pi$$
0.0229591 + 0.999736i $$0.492691\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 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
$$119$$ 0 0
$$120$$ 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 $$-0.626430\pi$$
−0.386831 + 0.922151i $$0.626430\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 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 $$-0.295961\pi$$
0.598004 + 0.801493i $$0.295961\pi$$
$$140$$ 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
$$147$$ 0 0
$$148$$ 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 $$-0.650347\pi$$
−0.454960 + 0.890512i $$0.650347\pi$$
$$158$$ 0 0
$$159$$ 3280.00 1.63598
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 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 $$-0.734116\pi$$
−0.670956 + 0.741497i $$0.734116\pi$$
$$168$$ 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 $$-0.552698\pi$$
−0.164802 + 0.986327i $$0.552698\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 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 $$-0.471072\pi$$
0.0907558 + 0.995873i $$0.471072\pi$$
$$182$$ 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
$$189$$ 0 0
$$190$$ 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 $$-0.676761\pi$$
−0.527208 + 0.849736i $$0.676761\pi$$
$$200$$ 0 0
$$201$$ −6208.00 −2.17850
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 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
$$217$$ 0 0
$$218$$ 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 $$-0.722653\pi$$
−0.643824 + 0.765173i $$0.722653\pi$$
$$224$$ 0 0
$$225$$ −925.000 −0.274074
$$226$$ 0 0
$$227$$ 6456.00 1.88766 0.943832 0.330425i $$-0.107192\pi$$
0.943832 + 0.330425i $$0.107192\pi$$
$$228$$ 0 0
$$229$$ 1066.00 0.307613 0.153806 0.988101i $$-0.450847\pi$$
0.153806 + 0.988101i $$0.450847\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 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 $$-0.678019\pi$$
−0.530561 + 0.847647i $$0.678019\pi$$
$$242$$ 0 0
$$243$$ 5032.00 1.32841
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 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 $$-0.829558\pi$$
−0.860034 + 0.510237i $$0.829558\pi$$
$$252$$ 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 $$-0.688993\pi$$
−0.559463 + 0.828855i $$0.688993\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 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 $$-0.716526\pi$$
−0.628977 + 0.777424i $$0.716526\pi$$
$$270$$ 0 0
$$271$$ 480.000 0.107594 0.0537969 0.998552i $$-0.482868\pi$$
0.0537969 + 0.998552i $$0.482868\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 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 $$-0.573118\pi$$
−0.227693 + 0.973733i $$0.573118\pi$$
$$284$$ 0 0
$$285$$ 3200.00 0.665093
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 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 $$-0.566164\pi$$
−0.206366 + 0.978475i $$0.566164\pi$$
$$294$$ 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
$$301$$ 0 0
$$302$$ 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 $$-0.556393\pi$$
−0.176238 + 0.984347i $$0.556393\pi$$
$$308$$ 0 0
$$309$$ −384.000 −0.0706958
$$310$$ 0 0
$$311$$ 1680.00 0.306315 0.153158 0.988202i $$-0.451056\pi$$
0.153158 + 0.988202i $$0.451056\pi$$
$$312$$ 0 0
$$313$$ −970.000 −0.175168 −0.0875841 0.996157i $$-0.527915\pi$$
−0.0875841 + 0.996157i $$0.527915\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 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
$$329$$ 0 0
$$330$$ 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
$$343$$ 0 0
$$344$$ 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 $$-0.302364\pi$$
0.581761 + 0.813360i $$0.302364\pi$$
$$350$$ 0 0
$$351$$ −4000.00 −0.608274
$$352$$ 0 0
$$353$$ −2530.00 −0.381468 −0.190734 0.981642i $$-0.561087\pi$$
−0.190734 + 0.981642i $$0.561087\pi$$
$$354$$ 0 0
$$355$$ 4000.00 0.598022
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 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 $$-0.563441\pi$$
−0.197989 + 0.980204i $$0.563441\pi$$
$$368$$ 0 0
$$369$$ −15170.0 −2.14016
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 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 $$-0.762495\pi$$
−0.734311 + 0.678813i $$0.762495\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 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 $$-0.410916\pi$$
0.276227 + 0.961093i $$0.410916\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 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 $$-0.314366\pi$$
0.550685 + 0.834713i $$0.314366\pi$$
$$410$$ 0 0
$$411$$ −4560.00 −0.547271
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 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 $$-0.651930\pi$$
−0.459383 + 0.888238i $$0.651930\pi$$
$$420$$ 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
$$427$$ 0 0
$$428$$ 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 $$-0.588208\pi$$
−0.273580 + 0.961849i $$0.588208\pi$$
$$434$$ 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 $$-0.303684\pi$$
0.578382 + 0.815766i $$0.303684\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 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
$$455$$ 0 0
$$456$$ 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 $$-0.689823\pi$$
−0.561624 + 0.827393i $$0.689823\pi$$
$$462$$ 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 $$-0.535018\pi$$
−0.109790 + 0.993955i $$0.535018\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 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 $$-0.331989\pi$$
0.503652 + 0.863907i $$0.331989\pi$$
$$480$$ 0 0
$$481$$ 15500.0 1.46931
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 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
$$497$$ 0 0
$$498$$ 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 $$-0.492550\pi$$
0.0234019 + 0.999726i $$0.492550\pi$$
$$504$$ 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 $$-0.175760\pi$$
0.851391 + 0.524532i $$0.175760\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 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 $$-0.720025\pi$$
−0.637485 + 0.770463i $$0.720025\pi$$
$$522$$ 0 0
$$523$$ −10968.0 −0.917012 −0.458506 0.888691i $$-0.651615\pi$$
−0.458506 + 0.888691i $$0.651615\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 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
$$539$$ 0 0
$$540$$ 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
$$553$$ 0 0
$$554$$ 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 $$-0.467633\pi$$
0.101507 + 0.994835i $$0.467633\pi$$
$$564$$ 0 0
$$565$$ 14900.0 1.10946
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 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 $$-0.427373\pi$$
0.226190 + 0.974083i $$0.427373\pi$$
$$578$$ 0 0
$$579$$ 40560.0 2.91125
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 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 $$-0.402041\pi$$
0.302913 + 0.953018i $$0.402041\pi$$
$$588$$ 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 $$-0.560878\pi$$
−0.190090 + 0.981767i $$0.560878\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 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 $$-0.597551\pi$$
−0.301689 + 0.953406i $$0.597551\pi$$
$$602$$ 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 $$-0.791929\pi$$
−0.793854 + 0.608108i $$0.791929\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 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 $$-0.618017\pi$$
−0.362325 + 0.932052i $$0.618017\pi$$
$$620$$ 0 0
$$621$$ −3840.00 −0.248138
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 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
$$637$$ 0 0
$$638$$ 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 $$-0.543948\pi$$
−0.137628 + 0.990484i $$0.543948\pi$$
$$644$$ 0 0
$$645$$ −12160.0 −0.742325
$$646$$ 0 0
$$647$$ −2064.00 −0.125416 −0.0627080 0.998032i $$-0.519974\pi$$
−0.0627080 + 0.998032i $$0.519974\pi$$
$$648$$ 0 0
$$649$$ −8000.00 −0.483864
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 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 $$-0.793909\pi$$
−0.797622 + 0.603157i $$0.793909\pi$$
$$662$$ 0 0
$$663$$ −12000.0 −0.702928
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 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 $$-0.780636\pi$$
−0.771785 + 0.635884i $$0.780636\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 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 $$-0.611997\pi$$
−0.344633 + 0.938737i $$0.611997\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 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
$$707$$ 0 0
$$708$$ 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 $$-0.297793\pi$$
0.593380 + 0.804923i $$0.297793\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 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 $$-0.585599\pi$$
−0.265686 + 0.964060i $$0.585599\pi$$
$$728$$ 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 $$-0.621200\pi$$
−0.371626 + 0.928383i $$0.621200\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 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
$$749$$ 0 0
$$750$$ 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 $$-0.474819\pi$$
0.0790259 + 0.996873i $$0.474819\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 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 $$-0.586950\pi$$
−0.269777 + 0.962923i $$0.586950\pi$$
$$770$$ 0 0
$$771$$ 36880.0 1.72270
$$772$$ 0 0
$$773$$ −22230.0 −1.03436 −0.517178 0.855878i $$-0.673018\pi$$
−0.517178 + 0.855878i $$0.673018\pi$$
$$774$$ 0 0
$$775$$ 8000.00 0.370798
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 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 $$-0.339477\pi$$
0.483193 + 0.875514i $$0.339477\pi$$
$$788$$ 0 0
$$789$$ 38784.0 1.75000
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 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 $$-0.449066\pi$$
0.159332 + 0.987225i $$0.449066\pi$$
$$798$$ 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
$$805$$ 0 0
$$806$$ 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 $$-0.270981\pi$$
0.658997 + 0.752146i $$0.270981\pi$$
$$812$$ 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
$$819$$ 0 0
$$820$$ 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 $$-0.693215\pi$$
−0.570408 + 0.821361i $$0.693215\pi$$
$$830$$ 0 0
$$831$$ −8240.00 −0.343974
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 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 $$-0.373581\pi$$
0.386799 + 0.922164i $$0.373581\pi$$
$$840$$ 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
$$847$$ 0 0
$$848$$ 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 $$-0.421985\pi$$
0.242646 + 0.970115i $$0.421985\pi$$
$$854$$ 0 0
$$855$$ −14800.0 −0.591988
$$856$$ 0 0
$$857$$ 470.000 0.0187338 0.00936692 0.999956i $$-0.497018\pi$$
0.00936692 + 0.999956i $$0.497018\pi$$
$$858$$ 0 0
$$859$$ −24440.0 −0.970759 −0.485380 0.874304i $$-0.661319\pi$$
−0.485380 + 0.874304i $$0.661319\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 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
$$875$$ 0 0
$$876$$ 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 $$-0.294478\pi$$
0.601732 + 0.798698i $$0.294478\pi$$
$$882$$ 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 $$-0.135136\pi$$
0.911227 + 0.411904i $$0.135136\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 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
$$903$$ 0 0
$$904$$ 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
$$917$$ 0 0
$$918$$ 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 $$-0.435999\pi$$
0.199714 + 0.979854i $$0.435999\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 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 $$-0.644341\pi$$
−0.438080 + 0.898936i $$0.644341\pi$$
$$938$$ 0 0
$$939$$ 7760.00 0.269689
$$940$$ 0 0
$$941$$ 22322.0 0.773301 0.386651 0.922226i $$-0.373632\pi$$
0.386651 + 0.922226i $$0.373632\pi$$
$$942$$ 0 0
$$943$$ −19680.0 −0.679607
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 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
$$959$$ 0 0
$$960$$ 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 $$-0.330271\pi$$
0.508309 + 0.861175i $$0.330271\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 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 $$-0.603178\pi$$
−0.318496 + 0.947924i $$0.603178\pi$$
$$984$$ 0 0
$$985$$ 19100.0 0.617844
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 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 $$-0.282896\pi$$
0.630389 + 0.776280i $$0.282896\pi$$
$$998$$ 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 1568.4.a.c.1.1 1
4.3 odd 2 1568.4.a.o.1.1 1
7.6 odd 2 32.4.a.c.1.1 yes 1
21.20 even 2 288.4.a.i.1.1 1
28.27 even 2 32.4.a.a.1.1 1
35.13 even 4 800.4.c.a.449.2 2
35.27 even 4 800.4.c.a.449.1 2
35.34 odd 2 800.4.a.a.1.1 1
56.13 odd 2 64.4.a.a.1.1 1
56.27 even 2 64.4.a.e.1.1 1
84.83 odd 2 288.4.a.h.1.1 1
112.13 odd 4 256.4.b.c.129.2 2
112.27 even 4 256.4.b.e.129.2 2
112.69 odd 4 256.4.b.c.129.1 2
112.83 even 4 256.4.b.e.129.1 2
140.27 odd 4 800.4.c.b.449.2 2
140.83 odd 4 800.4.c.b.449.1 2
140.139 even 2 800.4.a.k.1.1 1
168.83 odd 2 576.4.a.g.1.1 1
168.125 even 2 576.4.a.h.1.1 1
280.69 odd 2 1600.4.a.bw.1.1 1
280.139 even 2 1600.4.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.a.a.1.1 1 28.27 even 2
32.4.a.c.1.1 yes 1 7.6 odd 2
64.4.a.a.1.1 1 56.13 odd 2
64.4.a.e.1.1 1 56.27 even 2
256.4.b.c.129.1 2 112.69 odd 4
256.4.b.c.129.2 2 112.13 odd 4
256.4.b.e.129.1 2 112.83 even 4
256.4.b.e.129.2 2 112.27 even 4
288.4.a.h.1.1 1 84.83 odd 2
288.4.a.i.1.1 1 21.20 even 2
576.4.a.g.1.1 1 168.83 odd 2
576.4.a.h.1.1 1 168.125 even 2
800.4.a.a.1.1 1 35.34 odd 2
800.4.a.k.1.1 1 140.139 even 2
800.4.c.a.449.1 2 35.27 even 4
800.4.c.a.449.2 2 35.13 even 4
800.4.c.b.449.1 2 140.83 odd 4
800.4.c.b.449.2 2 140.27 odd 4
1568.4.a.c.1.1 1 1.1 even 1 trivial
1568.4.a.o.1.1 1 4.3 odd 2
1600.4.a.e.1.1 1 280.139 even 2
1600.4.a.bw.1.1 1 280.69 odd 2