# Properties

 Label 1600.4.a.o.1.1 Level $1600$ Weight $4$ Character 1600.1 Self dual yes Analytic conductor $94.403$ 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] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1600.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: $$94.4030560092$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 8) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.00000 q^{3} -24.0000 q^{7} -11.0000 q^{9} +O(q^{10})$$ $$q-4.00000 q^{3} -24.0000 q^{7} -11.0000 q^{9} +44.0000 q^{11} +22.0000 q^{13} -50.0000 q^{17} -44.0000 q^{19} +96.0000 q^{21} +56.0000 q^{23} +152.000 q^{27} -198.000 q^{29} -160.000 q^{31} -176.000 q^{33} -162.000 q^{37} -88.0000 q^{39} -198.000 q^{41} +52.0000 q^{43} -528.000 q^{47} +233.000 q^{49} +200.000 q^{51} -242.000 q^{53} +176.000 q^{57} +668.000 q^{59} -550.000 q^{61} +264.000 q^{63} +188.000 q^{67} -224.000 q^{69} +728.000 q^{71} -154.000 q^{73} -1056.00 q^{77} -656.000 q^{79} -311.000 q^{81} +236.000 q^{83} +792.000 q^{87} +714.000 q^{89} -528.000 q^{91} +640.000 q^{93} +478.000 q^{97} -484.000 q^{99} +O(q^{100})$$

## 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
$$3$$ −4.00000 −0.769800 −0.384900 0.922958i $$-0.625764\pi$$
−0.384900 + 0.922958i $$0.625764\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −24.0000 −1.29588 −0.647939 0.761692i $$-0.724369\pi$$
−0.647939 + 0.761692i $$0.724369\pi$$
$$8$$ 0 0
$$9$$ −11.0000 −0.407407
$$10$$ 0 0
$$11$$ 44.0000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 22.0000 0.469362 0.234681 0.972072i $$-0.424595\pi$$
0.234681 + 0.972072i $$0.424595\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −50.0000 −0.713340 −0.356670 0.934230i $$-0.616088\pi$$
−0.356670 + 0.934230i $$0.616088\pi$$
$$18$$ 0 0
$$19$$ −44.0000 −0.531279 −0.265639 0.964072i $$-0.585583\pi$$
−0.265639 + 0.964072i $$0.585583\pi$$
$$20$$ 0 0
$$21$$ 96.0000 0.997567
$$22$$ 0 0
$$23$$ 56.0000 0.507687 0.253844 0.967245i $$-0.418305\pi$$
0.253844 + 0.967245i $$0.418305\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 152.000 1.08342
$$28$$ 0 0
$$29$$ −198.000 −1.26785 −0.633925 0.773394i $$-0.718557\pi$$
−0.633925 + 0.773394i $$0.718557\pi$$
$$30$$ 0 0
$$31$$ −160.000 −0.926995 −0.463498 0.886098i $$-0.653406\pi$$
−0.463498 + 0.886098i $$0.653406\pi$$
$$32$$ 0 0
$$33$$ −176.000 −0.928414
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −162.000 −0.719801 −0.359900 0.932991i $$-0.617189\pi$$
−0.359900 + 0.932991i $$0.617189\pi$$
$$38$$ 0 0
$$39$$ −88.0000 −0.361315
$$40$$ 0 0
$$41$$ −198.000 −0.754205 −0.377102 0.926172i $$-0.623080\pi$$
−0.377102 + 0.926172i $$0.623080\pi$$
$$42$$ 0 0
$$43$$ 52.0000 0.184417 0.0922084 0.995740i $$-0.470607\pi$$
0.0922084 + 0.995740i $$0.470607\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −528.000 −1.63865 −0.819327 0.573327i $$-0.805653\pi$$
−0.819327 + 0.573327i $$0.805653\pi$$
$$48$$ 0 0
$$49$$ 233.000 0.679300
$$50$$ 0 0
$$51$$ 200.000 0.549129
$$52$$ 0 0
$$53$$ −242.000 −0.627194 −0.313597 0.949556i $$-0.601534\pi$$
−0.313597 + 0.949556i $$0.601534\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 176.000 0.408978
$$58$$ 0 0
$$59$$ 668.000 1.47400 0.737002 0.675891i $$-0.236241\pi$$
0.737002 + 0.675891i $$0.236241\pi$$
$$60$$ 0 0
$$61$$ −550.000 −1.15443 −0.577215 0.816592i $$-0.695861\pi$$
−0.577215 + 0.816592i $$0.695861\pi$$
$$62$$ 0 0
$$63$$ 264.000 0.527950
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 188.000 0.342804 0.171402 0.985201i $$-0.445170\pi$$
0.171402 + 0.985201i $$0.445170\pi$$
$$68$$ 0 0
$$69$$ −224.000 −0.390818
$$70$$ 0 0
$$71$$ 728.000 1.21687 0.608435 0.793604i $$-0.291798\pi$$
0.608435 + 0.793604i $$0.291798\pi$$
$$72$$ 0 0
$$73$$ −154.000 −0.246909 −0.123454 0.992350i $$-0.539397\pi$$
−0.123454 + 0.992350i $$0.539397\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1056.00 −1.56289
$$78$$ 0 0
$$79$$ −656.000 −0.934250 −0.467125 0.884191i $$-0.654710\pi$$
−0.467125 + 0.884191i $$0.654710\pi$$
$$80$$ 0 0
$$81$$ −311.000 −0.426612
$$82$$ 0 0
$$83$$ 236.000 0.312101 0.156050 0.987749i $$-0.450124\pi$$
0.156050 + 0.987749i $$0.450124\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 792.000 0.975992
$$88$$ 0 0
$$89$$ 714.000 0.850380 0.425190 0.905104i $$-0.360207\pi$$
0.425190 + 0.905104i $$0.360207\pi$$
$$90$$ 0 0
$$91$$ −528.000 −0.608236
$$92$$ 0 0
$$93$$ 640.000 0.713601
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 478.000 0.500346 0.250173 0.968201i $$-0.419513\pi$$
0.250173 + 0.968201i $$0.419513\pi$$
$$98$$ 0 0
$$99$$ −484.000 −0.491352
$$100$$ 0 0
$$101$$ −1566.00 −1.54280 −0.771400 0.636350i $$-0.780443\pi$$
−0.771400 + 0.636350i $$0.780443\pi$$
$$102$$ 0 0
$$103$$ 968.000 0.926018 0.463009 0.886354i $$-0.346770\pi$$
0.463009 + 0.886354i $$0.346770\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −780.000 −0.704724 −0.352362 0.935864i $$-0.614621\pi$$
−0.352362 + 0.935864i $$0.614621\pi$$
$$108$$ 0 0
$$109$$ 1994.00 1.75221 0.876103 0.482123i $$-0.160134\pi$$
0.876103 + 0.482123i $$0.160134\pi$$
$$110$$ 0 0
$$111$$ 648.000 0.554103
$$112$$ 0 0
$$113$$ 942.000 0.784212 0.392106 0.919920i $$-0.371747\pi$$
0.392106 + 0.919920i $$0.371747\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −242.000 −0.191221
$$118$$ 0 0
$$119$$ 1200.00 0.924402
$$120$$ 0 0
$$121$$ 605.000 0.454545
$$122$$ 0 0
$$123$$ 792.000 0.580587
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1408.00 −0.983778 −0.491889 0.870658i $$-0.663693\pi$$
−0.491889 + 0.870658i $$0.663693\pi$$
$$128$$ 0 0
$$129$$ −208.000 −0.141964
$$130$$ 0 0
$$131$$ 2692.00 1.79543 0.897714 0.440578i $$-0.145227\pi$$
0.897714 + 0.440578i $$0.145227\pi$$
$$132$$ 0 0
$$133$$ 1056.00 0.688472
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1626.00 −1.01400 −0.507002 0.861945i $$-0.669246\pi$$
−0.507002 + 0.861945i $$0.669246\pi$$
$$138$$ 0 0
$$139$$ 684.000 0.417382 0.208691 0.977982i $$-0.433080\pi$$
0.208691 + 0.977982i $$0.433080\pi$$
$$140$$ 0 0
$$141$$ 2112.00 1.26144
$$142$$ 0 0
$$143$$ 968.000 0.566072
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −932.000 −0.522926
$$148$$ 0 0
$$149$$ −302.000 −0.166046 −0.0830228 0.996548i $$-0.526457\pi$$
−0.0830228 + 0.996548i $$0.526457\pi$$
$$150$$ 0 0
$$151$$ 1352.00 0.728637 0.364319 0.931274i $$-0.381302\pi$$
0.364319 + 0.931274i $$0.381302\pi$$
$$152$$ 0 0
$$153$$ 550.000 0.290620
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 3142.00 1.59719 0.798595 0.601868i $$-0.205577\pi$$
0.798595 + 0.601868i $$0.205577\pi$$
$$158$$ 0 0
$$159$$ 968.000 0.482814
$$160$$ 0 0
$$161$$ −1344.00 −0.657901
$$162$$ 0 0
$$163$$ 3036.00 1.45888 0.729441 0.684043i $$-0.239780\pi$$
0.729441 + 0.684043i $$0.239780\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 264.000 0.122329 0.0611645 0.998128i $$-0.480519\pi$$
0.0611645 + 0.998128i $$0.480519\pi$$
$$168$$ 0 0
$$169$$ −1713.00 −0.779700
$$170$$ 0 0
$$171$$ 484.000 0.216447
$$172$$ 0 0
$$173$$ −2826.00 −1.24195 −0.620973 0.783832i $$-0.713263\pi$$
−0.620973 + 0.783832i $$0.713263\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −2672.00 −1.13469
$$178$$ 0 0
$$179$$ −3084.00 −1.28776 −0.643880 0.765127i $$-0.722676\pi$$
−0.643880 + 0.765127i $$0.722676\pi$$
$$180$$ 0 0
$$181$$ 2418.00 0.992975 0.496488 0.868044i $$-0.334623\pi$$
0.496488 + 0.868044i $$0.334623\pi$$
$$182$$ 0 0
$$183$$ 2200.00 0.888681
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −2200.00 −0.860320
$$188$$ 0 0
$$189$$ −3648.00 −1.40398
$$190$$ 0 0
$$191$$ −960.000 −0.363681 −0.181841 0.983328i $$-0.558206\pi$$
−0.181841 + 0.983328i $$0.558206\pi$$
$$192$$ 0 0
$$193$$ −2882.00 −1.07488 −0.537438 0.843304i $$-0.680608\pi$$
−0.537438 + 0.843304i $$0.680608\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1086.00 0.392763 0.196381 0.980528i $$-0.437081\pi$$
0.196381 + 0.980528i $$0.437081\pi$$
$$198$$ 0 0
$$199$$ 88.0000 0.0313475 0.0156738 0.999877i $$-0.495011\pi$$
0.0156738 + 0.999877i $$0.495011\pi$$
$$200$$ 0 0
$$201$$ −752.000 −0.263890
$$202$$ 0 0
$$203$$ 4752.00 1.64298
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −616.000 −0.206836
$$208$$ 0 0
$$209$$ −1936.00 −0.640746
$$210$$ 0 0
$$211$$ 3476.00 1.13411 0.567056 0.823679i $$-0.308082\pi$$
0.567056 + 0.823679i $$0.308082\pi$$
$$212$$ 0 0
$$213$$ −2912.00 −0.936746
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3840.00 1.20127
$$218$$ 0 0
$$219$$ 616.000 0.190070
$$220$$ 0 0
$$221$$ −1100.00 −0.334815
$$222$$ 0 0
$$223$$ −928.000 −0.278670 −0.139335 0.990245i $$-0.544497\pi$$
−0.139335 + 0.990245i $$0.544497\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 156.000 0.0456127 0.0228064 0.999740i $$-0.492740\pi$$
0.0228064 + 0.999740i $$0.492740\pi$$
$$228$$ 0 0
$$229$$ 1634.00 0.471519 0.235759 0.971811i $$-0.424242\pi$$
0.235759 + 0.971811i $$0.424242\pi$$
$$230$$ 0 0
$$231$$ 4224.00 1.20311
$$232$$ 0 0
$$233$$ 902.000 0.253614 0.126807 0.991927i $$-0.459527\pi$$
0.126807 + 0.991927i $$0.459527\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 2624.00 0.719186
$$238$$ 0 0
$$239$$ 1616.00 0.437365 0.218683 0.975796i $$-0.429824\pi$$
0.218683 + 0.975796i $$0.429824\pi$$
$$240$$ 0 0
$$241$$ 4818.00 1.28778 0.643889 0.765119i $$-0.277320\pi$$
0.643889 + 0.765119i $$0.277320\pi$$
$$242$$ 0 0
$$243$$ −2860.00 −0.755017
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −968.000 −0.249362
$$248$$ 0 0
$$249$$ −944.000 −0.240255
$$250$$ 0 0
$$251$$ 2140.00 0.538150 0.269075 0.963119i $$-0.413282\pi$$
0.269075 + 0.963119i $$0.413282\pi$$
$$252$$ 0 0
$$253$$ 2464.00 0.612294
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −770.000 −0.186892 −0.0934461 0.995624i $$-0.529788\pi$$
−0.0934461 + 0.995624i $$0.529788\pi$$
$$258$$ 0 0
$$259$$ 3888.00 0.932774
$$260$$ 0 0
$$261$$ 2178.00 0.516532
$$262$$ 0 0
$$263$$ 7400.00 1.73499 0.867497 0.497442i $$-0.165727\pi$$
0.867497 + 0.497442i $$0.165727\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −2856.00 −0.654623
$$268$$ 0 0
$$269$$ 2794.00 0.633283 0.316642 0.948545i $$-0.397445\pi$$
0.316642 + 0.948545i $$0.397445\pi$$
$$270$$ 0 0
$$271$$ 8624.00 1.93310 0.966551 0.256474i $$-0.0825608\pi$$
0.966551 + 0.256474i $$0.0825608\pi$$
$$272$$ 0 0
$$273$$ 2112.00 0.468220
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1874.00 −0.406490 −0.203245 0.979128i $$-0.565149\pi$$
−0.203245 + 0.979128i $$0.565149\pi$$
$$278$$ 0 0
$$279$$ 1760.00 0.377665
$$280$$ 0 0
$$281$$ 3338.00 0.708642 0.354321 0.935124i $$-0.384712\pi$$
0.354321 + 0.935124i $$0.384712\pi$$
$$282$$ 0 0
$$283$$ 7172.00 1.50647 0.753235 0.657751i $$-0.228492\pi$$
0.753235 + 0.657751i $$0.228492\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 4752.00 0.977358
$$288$$ 0 0
$$289$$ −2413.00 −0.491146
$$290$$ 0 0
$$291$$ −1912.00 −0.385166
$$292$$ 0 0
$$293$$ 5214.00 1.03961 0.519804 0.854286i $$-0.326005\pi$$
0.519804 + 0.854286i $$0.326005\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 6688.00 1.30666
$$298$$ 0 0
$$299$$ 1232.00 0.238289
$$300$$ 0 0
$$301$$ −1248.00 −0.238982
$$302$$ 0 0
$$303$$ 6264.00 1.18765
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 396.000 0.0736186 0.0368093 0.999322i $$-0.488281\pi$$
0.0368093 + 0.999322i $$0.488281\pi$$
$$308$$ 0 0
$$309$$ −3872.00 −0.712849
$$310$$ 0 0
$$311$$ −4056.00 −0.739533 −0.369766 0.929125i $$-0.620562\pi$$
−0.369766 + 0.929125i $$0.620562\pi$$
$$312$$ 0 0
$$313$$ −2154.00 −0.388982 −0.194491 0.980904i $$-0.562305\pi$$
−0.194491 + 0.980904i $$0.562305\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7386.00 −1.30864 −0.654320 0.756217i $$-0.727045\pi$$
−0.654320 + 0.756217i $$0.727045\pi$$
$$318$$ 0 0
$$319$$ −8712.00 −1.52909
$$320$$ 0 0
$$321$$ 3120.00 0.542497
$$322$$ 0 0
$$323$$ 2200.00 0.378982
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −7976.00 −1.34885
$$328$$ 0 0
$$329$$ 12672.0 2.12350
$$330$$ 0 0
$$331$$ 1132.00 0.187977 0.0939884 0.995573i $$-0.470038\pi$$
0.0939884 + 0.995573i $$0.470038\pi$$
$$332$$ 0 0
$$333$$ 1782.00 0.293252
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 3342.00 0.540209 0.270104 0.962831i $$-0.412942\pi$$
0.270104 + 0.962831i $$0.412942\pi$$
$$338$$ 0 0
$$339$$ −3768.00 −0.603686
$$340$$ 0 0
$$341$$ −7040.00 −1.11800
$$342$$ 0 0
$$343$$ 2640.00 0.415588
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2244.00 0.347159 0.173580 0.984820i $$-0.444467\pi$$
0.173580 + 0.984820i $$0.444467\pi$$
$$348$$ 0 0
$$349$$ 6522.00 1.00033 0.500164 0.865931i $$-0.333273\pi$$
0.500164 + 0.865931i $$0.333273\pi$$
$$350$$ 0 0
$$351$$ 3344.00 0.508517
$$352$$ 0 0
$$353$$ 11230.0 1.69324 0.846618 0.532200i $$-0.178635\pi$$
0.846618 + 0.532200i $$0.178635\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4800.00 −0.711605
$$358$$ 0 0
$$359$$ 1848.00 0.271682 0.135841 0.990731i $$-0.456626\pi$$
0.135841 + 0.990731i $$0.456626\pi$$
$$360$$ 0 0
$$361$$ −4923.00 −0.717743
$$362$$ 0 0
$$363$$ −2420.00 −0.349909
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −7120.00 −1.01270 −0.506350 0.862328i $$-0.669006\pi$$
−0.506350 + 0.862328i $$0.669006\pi$$
$$368$$ 0 0
$$369$$ 2178.00 0.307269
$$370$$ 0 0
$$371$$ 5808.00 0.812766
$$372$$ 0 0
$$373$$ 6350.00 0.881476 0.440738 0.897636i $$-0.354717\pi$$
0.440738 + 0.897636i $$0.354717\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4356.00 −0.595081
$$378$$ 0 0
$$379$$ 7900.00 1.07070 0.535351 0.844630i $$-0.320179\pi$$
0.535351 + 0.844630i $$0.320179\pi$$
$$380$$ 0 0
$$381$$ 5632.00 0.757313
$$382$$ 0 0
$$383$$ −10368.0 −1.38324 −0.691619 0.722263i $$-0.743102\pi$$
−0.691619 + 0.722263i $$0.743102\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −572.000 −0.0751328
$$388$$ 0 0
$$389$$ −8830.00 −1.15090 −0.575448 0.817838i $$-0.695172\pi$$
−0.575448 + 0.817838i $$0.695172\pi$$
$$390$$ 0 0
$$391$$ −2800.00 −0.362154
$$392$$ 0 0
$$393$$ −10768.0 −1.38212
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 9878.00 1.24877 0.624386 0.781116i $$-0.285349\pi$$
0.624386 + 0.781116i $$0.285349\pi$$
$$398$$ 0 0
$$399$$ −4224.00 −0.529986
$$400$$ 0 0
$$401$$ −13134.0 −1.63561 −0.817806 0.575494i $$-0.804810\pi$$
−0.817806 + 0.575494i $$0.804810\pi$$
$$402$$ 0 0
$$403$$ −3520.00 −0.435096
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −7128.00 −0.868113
$$408$$ 0 0
$$409$$ 906.000 0.109533 0.0547663 0.998499i $$-0.482559\pi$$
0.0547663 + 0.998499i $$0.482559\pi$$
$$410$$ 0 0
$$411$$ 6504.00 0.780581
$$412$$ 0 0
$$413$$ −16032.0 −1.91013
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −2736.00 −0.321301
$$418$$ 0 0
$$419$$ 5412.00 0.631011 0.315505 0.948924i $$-0.397826\pi$$
0.315505 + 0.948924i $$0.397826\pi$$
$$420$$ 0 0
$$421$$ 4642.00 0.537381 0.268690 0.963227i $$-0.413409\pi$$
0.268690 + 0.963227i $$0.413409\pi$$
$$422$$ 0 0
$$423$$ 5808.00 0.667600
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 13200.0 1.49600
$$428$$ 0 0
$$429$$ −3872.00 −0.435762
$$430$$ 0 0
$$431$$ 656.000 0.0733142 0.0366571 0.999328i $$-0.488329\pi$$
0.0366571 + 0.999328i $$0.488329\pi$$
$$432$$ 0 0
$$433$$ −9490.00 −1.05326 −0.526629 0.850096i $$-0.676544\pi$$
−0.526629 + 0.850096i $$0.676544\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −2464.00 −0.269723
$$438$$ 0 0
$$439$$ 5544.00 0.602735 0.301368 0.953508i $$-0.402557\pi$$
0.301368 + 0.953508i $$0.402557\pi$$
$$440$$ 0 0
$$441$$ −2563.00 −0.276752
$$442$$ 0 0
$$443$$ 7652.00 0.820672 0.410336 0.911935i $$-0.365412\pi$$
0.410336 + 0.911935i $$0.365412\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 1208.00 0.127822
$$448$$ 0 0
$$449$$ −446.000 −0.0468776 −0.0234388 0.999725i $$-0.507461\pi$$
−0.0234388 + 0.999725i $$0.507461\pi$$
$$450$$ 0 0
$$451$$ −8712.00 −0.909605
$$452$$ 0 0
$$453$$ −5408.00 −0.560905
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1562.00 −0.159885 −0.0799423 0.996799i $$-0.525474\pi$$
−0.0799423 + 0.996799i $$0.525474\pi$$
$$458$$ 0 0
$$459$$ −7600.00 −0.772849
$$460$$ 0 0
$$461$$ −10582.0 −1.06910 −0.534548 0.845138i $$-0.679518\pi$$
−0.534548 + 0.845138i $$0.679518\pi$$
$$462$$ 0 0
$$463$$ 10768.0 1.08085 0.540423 0.841394i $$-0.318264\pi$$
0.540423 + 0.841394i $$0.318264\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −9876.00 −0.978601 −0.489301 0.872115i $$-0.662748\pi$$
−0.489301 + 0.872115i $$0.662748\pi$$
$$468$$ 0 0
$$469$$ −4512.00 −0.444232
$$470$$ 0 0
$$471$$ −12568.0 −1.22952
$$472$$ 0 0
$$473$$ 2288.00 0.222415
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 2662.00 0.255523
$$478$$ 0 0
$$479$$ −352.000 −0.0335768 −0.0167884 0.999859i $$-0.505344\pi$$
−0.0167884 + 0.999859i $$0.505344\pi$$
$$480$$ 0 0
$$481$$ −3564.00 −0.337847
$$482$$ 0 0
$$483$$ 5376.00 0.506452
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 15176.0 1.41209 0.706047 0.708165i $$-0.250477\pi$$
0.706047 + 0.708165i $$0.250477\pi$$
$$488$$ 0 0
$$489$$ −12144.0 −1.12305
$$490$$ 0 0
$$491$$ 8844.00 0.812880 0.406440 0.913677i $$-0.366770\pi$$
0.406440 + 0.913677i $$0.366770\pi$$
$$492$$ 0 0
$$493$$ 9900.00 0.904409
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −17472.0 −1.57691
$$498$$ 0 0
$$499$$ −19404.0 −1.74077 −0.870383 0.492375i $$-0.836129\pi$$
−0.870383 + 0.492375i $$0.836129\pi$$
$$500$$ 0 0
$$501$$ −1056.00 −0.0941689
$$502$$ 0 0
$$503$$ −16488.0 −1.46156 −0.730779 0.682614i $$-0.760843\pi$$
−0.730779 + 0.682614i $$0.760843\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 6852.00 0.600213
$$508$$ 0 0
$$509$$ 12954.0 1.12805 0.564024 0.825759i $$-0.309253\pi$$
0.564024 + 0.825759i $$0.309253\pi$$
$$510$$ 0 0
$$511$$ 3696.00 0.319964
$$512$$ 0 0
$$513$$ −6688.00 −0.575599
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −23232.0 −1.97629
$$518$$ 0 0
$$519$$ 11304.0 0.956051
$$520$$ 0 0
$$521$$ 10970.0 0.922465 0.461233 0.887279i $$-0.347407\pi$$
0.461233 + 0.887279i $$0.347407\pi$$
$$522$$ 0 0
$$523$$ −16940.0 −1.41632 −0.708159 0.706053i $$-0.750474\pi$$
−0.708159 + 0.706053i $$0.750474\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8000.00 0.661263
$$528$$ 0 0
$$529$$ −9031.00 −0.742254
$$530$$ 0 0
$$531$$ −7348.00 −0.600520
$$532$$ 0 0
$$533$$ −4356.00 −0.353995
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 12336.0 0.991318
$$538$$ 0 0
$$539$$ 10252.0 0.819267
$$540$$ 0 0
$$541$$ −198.000 −0.0157351 −0.00786755 0.999969i $$-0.502504\pi$$
−0.00786755 + 0.999969i $$0.502504\pi$$
$$542$$ 0 0
$$543$$ −9672.00 −0.764393
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −15268.0 −1.19344 −0.596721 0.802449i $$-0.703530\pi$$
−0.596721 + 0.802449i $$0.703530\pi$$
$$548$$ 0 0
$$549$$ 6050.00 0.470324
$$550$$ 0 0
$$551$$ 8712.00 0.673582
$$552$$ 0 0
$$553$$ 15744.0 1.21067
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 20854.0 1.58638 0.793189 0.608976i $$-0.208419\pi$$
0.793189 + 0.608976i $$0.208419\pi$$
$$558$$ 0 0
$$559$$ 1144.00 0.0865582
$$560$$ 0 0
$$561$$ 8800.00 0.662275
$$562$$ 0 0
$$563$$ −19316.0 −1.44595 −0.722977 0.690872i $$-0.757227\pi$$
−0.722977 + 0.690872i $$0.757227\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 7464.00 0.552837
$$568$$ 0 0
$$569$$ 7018.00 0.517065 0.258532 0.966003i $$-0.416761\pi$$
0.258532 + 0.966003i $$0.416761\pi$$
$$570$$ 0 0
$$571$$ −24420.0 −1.78975 −0.894873 0.446320i $$-0.852734\pi$$
−0.894873 + 0.446320i $$0.852734\pi$$
$$572$$ 0 0
$$573$$ 3840.00 0.279962
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −23234.0 −1.67633 −0.838166 0.545415i $$-0.816372\pi$$
−0.838166 + 0.545415i $$0.816372\pi$$
$$578$$ 0 0
$$579$$ 11528.0 0.827439
$$580$$ 0 0
$$581$$ −5664.00 −0.404445
$$582$$ 0 0
$$583$$ −10648.0 −0.756424
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −10604.0 −0.745611 −0.372806 0.927909i $$-0.621604\pi$$
−0.372806 + 0.927909i $$0.621604\pi$$
$$588$$ 0 0
$$589$$ 7040.00 0.492493
$$590$$ 0 0
$$591$$ −4344.00 −0.302349
$$592$$ 0 0
$$593$$ 13838.0 0.958277 0.479139 0.877739i $$-0.340949\pi$$
0.479139 + 0.877739i $$0.340949\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −352.000 −0.0241313
$$598$$ 0 0
$$599$$ −3960.00 −0.270119 −0.135059 0.990837i $$-0.543123\pi$$
−0.135059 + 0.990837i $$0.543123\pi$$
$$600$$ 0 0
$$601$$ −5942.00 −0.403293 −0.201647 0.979458i $$-0.564629\pi$$
−0.201647 + 0.979458i $$0.564629\pi$$
$$602$$ 0 0
$$603$$ −2068.00 −0.139661
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 3040.00 0.203278 0.101639 0.994821i $$-0.467591\pi$$
0.101639 + 0.994821i $$0.467591\pi$$
$$608$$ 0 0
$$609$$ −19008.0 −1.26477
$$610$$ 0 0
$$611$$ −11616.0 −0.769121
$$612$$ 0 0
$$613$$ −2530.00 −0.166698 −0.0833489 0.996520i $$-0.526562\pi$$
−0.0833489 + 0.996520i $$0.526562\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 19206.0 1.25317 0.626584 0.779354i $$-0.284453\pi$$
0.626584 + 0.779354i $$0.284453\pi$$
$$618$$ 0 0
$$619$$ −10996.0 −0.714001 −0.357000 0.934104i $$-0.616201\pi$$
−0.357000 + 0.934104i $$0.616201\pi$$
$$620$$ 0 0
$$621$$ 8512.00 0.550040
$$622$$ 0 0
$$623$$ −17136.0 −1.10199
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 7744.00 0.493247
$$628$$ 0 0
$$629$$ 8100.00 0.513463
$$630$$ 0 0
$$631$$ −6680.00 −0.421437 −0.210718 0.977547i $$-0.567580\pi$$
−0.210718 + 0.977547i $$0.567580\pi$$
$$632$$ 0 0
$$633$$ −13904.0 −0.873040
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5126.00 0.318838
$$638$$ 0 0
$$639$$ −8008.00 −0.495761
$$640$$ 0 0
$$641$$ 6274.00 0.386596 0.193298 0.981140i $$-0.438082\pi$$
0.193298 + 0.981140i $$0.438082\pi$$
$$642$$ 0 0
$$643$$ 9084.00 0.557135 0.278568 0.960417i $$-0.410140\pi$$
0.278568 + 0.960417i $$0.410140\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 23656.0 1.43742 0.718712 0.695308i $$-0.244732\pi$$
0.718712 + 0.695308i $$0.244732\pi$$
$$648$$ 0 0
$$649$$ 29392.0 1.77771
$$650$$ 0 0
$$651$$ −15360.0 −0.924740
$$652$$ 0 0
$$653$$ −6762.00 −0.405234 −0.202617 0.979258i $$-0.564945\pi$$
−0.202617 + 0.979258i $$0.564945\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 1694.00 0.100592
$$658$$ 0 0
$$659$$ −15276.0 −0.902987 −0.451494 0.892274i $$-0.649109\pi$$
−0.451494 + 0.892274i $$0.649109\pi$$
$$660$$ 0 0
$$661$$ −11054.0 −0.650455 −0.325228 0.945636i $$-0.605441\pi$$
−0.325228 + 0.945636i $$0.605441\pi$$
$$662$$ 0 0
$$663$$ 4400.00 0.257740
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −11088.0 −0.643672
$$668$$ 0 0
$$669$$ 3712.00 0.214520
$$670$$ 0 0
$$671$$ −24200.0 −1.39230
$$672$$ 0 0
$$673$$ 21278.0 1.21873 0.609366 0.792889i $$-0.291424\pi$$
0.609366 + 0.792889i $$0.291424\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 8926.00 0.506727 0.253363 0.967371i $$-0.418463\pi$$
0.253363 + 0.967371i $$0.418463\pi$$
$$678$$ 0 0
$$679$$ −11472.0 −0.648387
$$680$$ 0 0
$$681$$ −624.000 −0.0351127
$$682$$ 0 0
$$683$$ 8116.00 0.454685 0.227343 0.973815i $$-0.426996\pi$$
0.227343 + 0.973815i $$0.426996\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −6536.00 −0.362975
$$688$$ 0 0
$$689$$ −5324.00 −0.294381
$$690$$ 0 0
$$691$$ 11764.0 0.647646 0.323823 0.946118i $$-0.395032\pi$$
0.323823 + 0.946118i $$0.395032\pi$$
$$692$$ 0 0
$$693$$ 11616.0 0.636732
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 9900.00 0.538005
$$698$$ 0 0
$$699$$ −3608.00 −0.195232
$$700$$ 0 0
$$701$$ 4698.00 0.253126 0.126563 0.991959i $$-0.459605\pi$$
0.126563 + 0.991959i $$0.459605\pi$$
$$702$$ 0 0
$$703$$ 7128.00 0.382415
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 37584.0 1.99928
$$708$$ 0 0
$$709$$ −24638.0 −1.30508 −0.652538 0.757756i $$-0.726296\pi$$
−0.652538 + 0.757756i $$0.726296\pi$$
$$710$$ 0 0
$$711$$ 7216.00 0.380620
$$712$$ 0 0
$$713$$ −8960.00 −0.470624
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −6464.00 −0.336684
$$718$$ 0 0
$$719$$ 16624.0 0.862268 0.431134 0.902288i $$-0.358114\pi$$
0.431134 + 0.902288i $$0.358114\pi$$
$$720$$ 0 0
$$721$$ −23232.0 −1.20001
$$722$$ 0 0
$$723$$ −19272.0 −0.991332
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −30216.0 −1.54147 −0.770735 0.637155i $$-0.780111\pi$$
−0.770735 + 0.637155i $$0.780111\pi$$
$$728$$ 0 0
$$729$$ 19837.0 1.00782
$$730$$ 0 0
$$731$$ −2600.00 −0.131552
$$732$$ 0 0
$$733$$ −3322.00 −0.167395 −0.0836977 0.996491i $$-0.526673\pi$$
−0.0836977 + 0.996491i $$0.526673\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8272.00 0.413437
$$738$$ 0 0
$$739$$ 14692.0 0.731331 0.365666 0.930746i $$-0.380841\pi$$
0.365666 + 0.930746i $$0.380841\pi$$
$$740$$ 0 0
$$741$$ 3872.00 0.191959
$$742$$ 0 0
$$743$$ −28600.0 −1.41216 −0.706078 0.708134i $$-0.749537\pi$$
−0.706078 + 0.708134i $$0.749537\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −2596.00 −0.127152
$$748$$ 0 0
$$749$$ 18720.0 0.913236
$$750$$ 0 0
$$751$$ −29616.0 −1.43902 −0.719509 0.694483i $$-0.755633\pi$$
−0.719509 + 0.694483i $$0.755633\pi$$
$$752$$ 0 0
$$753$$ −8560.00 −0.414268
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2894.00 0.138949 0.0694744 0.997584i $$-0.477868\pi$$
0.0694744 + 0.997584i $$0.477868\pi$$
$$758$$ 0 0
$$759$$ −9856.00 −0.471344
$$760$$ 0 0
$$761$$ 14762.0 0.703183 0.351591 0.936154i $$-0.385641\pi$$
0.351591 + 0.936154i $$0.385641\pi$$
$$762$$ 0 0
$$763$$ −47856.0 −2.27065
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 14696.0 0.691841
$$768$$ 0 0
$$769$$ −7678.00 −0.360047 −0.180023 0.983662i $$-0.557617\pi$$
−0.180023 + 0.983662i $$0.557617\pi$$
$$770$$ 0 0
$$771$$ 3080.00 0.143870
$$772$$ 0 0
$$773$$ 27390.0 1.27445 0.637225 0.770678i $$-0.280082\pi$$
0.637225 + 0.770678i $$0.280082\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −15552.0 −0.718050
$$778$$ 0 0
$$779$$ 8712.00 0.400693
$$780$$ 0 0
$$781$$ 32032.0 1.46760
$$782$$ 0 0
$$783$$ −30096.0 −1.37362
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 19756.0 0.894823 0.447411 0.894328i $$-0.352346\pi$$
0.447411 + 0.894328i $$0.352346\pi$$
$$788$$ 0 0
$$789$$ −29600.0 −1.33560
$$790$$ 0 0
$$791$$ −22608.0 −1.01624
$$792$$ 0 0
$$793$$ −12100.0 −0.541846
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 38854.0 1.72682 0.863412 0.504499i $$-0.168323\pi$$
0.863412 + 0.504499i $$0.168323\pi$$
$$798$$ 0 0
$$799$$ 26400.0 1.16892
$$800$$ 0 0
$$801$$ −7854.00 −0.346451
$$802$$ 0 0
$$803$$ −6776.00 −0.297783
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −11176.0 −0.487502
$$808$$ 0 0
$$809$$ −14278.0 −0.620504 −0.310252 0.950654i $$-0.600413\pi$$
−0.310252 + 0.950654i $$0.600413\pi$$
$$810$$ 0 0
$$811$$ 716.000 0.0310014 0.0155007 0.999880i $$-0.495066\pi$$
0.0155007 + 0.999880i $$0.495066\pi$$
$$812$$ 0 0
$$813$$ −34496.0 −1.48810
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −2288.00 −0.0979767
$$818$$ 0 0
$$819$$ 5808.00 0.247800
$$820$$ 0 0
$$821$$ 23538.0 1.00059 0.500293 0.865856i $$-0.333225\pi$$
0.500293 + 0.865856i $$0.333225\pi$$
$$822$$ 0 0
$$823$$ 6616.00 0.280218 0.140109 0.990136i $$-0.455255\pi$$
0.140109 + 0.990136i $$0.455255\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 27236.0 1.14521 0.572605 0.819831i $$-0.305933\pi$$
0.572605 + 0.819831i $$0.305933\pi$$
$$828$$ 0 0
$$829$$ −12070.0 −0.505680 −0.252840 0.967508i $$-0.581365\pi$$
−0.252840 + 0.967508i $$0.581365\pi$$
$$830$$ 0 0
$$831$$ 7496.00 0.312916
$$832$$ 0 0
$$833$$ −11650.0 −0.484572
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −24320.0 −1.00433
$$838$$ 0 0
$$839$$ −42024.0 −1.72924 −0.864618 0.502429i $$-0.832440\pi$$
−0.864618 + 0.502429i $$0.832440\pi$$
$$840$$ 0 0
$$841$$ 14815.0 0.607446
$$842$$ 0 0
$$843$$ −13352.0 −0.545513
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −14520.0 −0.589036
$$848$$ 0 0
$$849$$ −28688.0 −1.15968
$$850$$ 0 0
$$851$$ −9072.00 −0.365434
$$852$$ 0 0
$$853$$ 2414.00 0.0968978 0.0484489 0.998826i $$-0.484572\pi$$
0.0484489 + 0.998826i $$0.484572\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 37686.0 1.50213 0.751067 0.660226i $$-0.229539\pi$$
0.751067 + 0.660226i $$0.229539\pi$$
$$858$$ 0 0
$$859$$ −40644.0 −1.61438 −0.807192 0.590289i $$-0.799014\pi$$
−0.807192 + 0.590289i $$0.799014\pi$$
$$860$$ 0 0
$$861$$ −19008.0 −0.752370
$$862$$ 0 0
$$863$$ 18656.0 0.735872 0.367936 0.929851i $$-0.380065\pi$$
0.367936 + 0.929851i $$0.380065\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 9652.00 0.378084
$$868$$ 0 0
$$869$$ −28864.0 −1.12675
$$870$$ 0 0
$$871$$ 4136.00 0.160899
$$872$$ 0 0
$$873$$ −5258.00 −0.203845
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13002.0 −0.500623 −0.250311 0.968165i $$-0.580533\pi$$
−0.250311 + 0.968165i $$0.580533\pi$$
$$878$$ 0 0
$$879$$ −20856.0 −0.800291
$$880$$ 0 0
$$881$$ 49490.0 1.89258 0.946289 0.323323i $$-0.104800\pi$$
0.946289 + 0.323323i $$0.104800\pi$$
$$882$$ 0 0
$$883$$ 1100.00 0.0419229 0.0209615 0.999780i $$-0.493327\pi$$
0.0209615 + 0.999780i $$0.493327\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 14104.0 0.533896 0.266948 0.963711i $$-0.413985\pi$$
0.266948 + 0.963711i $$0.413985\pi$$
$$888$$ 0 0
$$889$$ 33792.0 1.27486
$$890$$ 0 0
$$891$$ −13684.0 −0.514513
$$892$$ 0 0
$$893$$ 23232.0 0.870581
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −4928.00 −0.183435
$$898$$ 0 0
$$899$$ 31680.0 1.17529
$$900$$ 0 0
$$901$$ 12100.0 0.447402
$$902$$ 0 0
$$903$$ 4992.00 0.183968
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −12716.0 −0.465521 −0.232761 0.972534i $$-0.574776\pi$$
−0.232761 + 0.972534i $$0.574776\pi$$
$$908$$ 0 0
$$909$$ 17226.0 0.628548
$$910$$ 0 0
$$911$$ −39632.0 −1.44135 −0.720673 0.693275i $$-0.756167\pi$$
−0.720673 + 0.693275i $$0.756167\pi$$
$$912$$ 0 0
$$913$$ 10384.0 0.376408
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −64608.0 −2.32666
$$918$$ 0 0
$$919$$ 5704.00 0.204742 0.102371 0.994746i $$-0.467357\pi$$
0.102371 + 0.994746i $$0.467357\pi$$
$$920$$ 0 0
$$921$$ −1584.00 −0.0566716
$$922$$ 0 0
$$923$$ 16016.0 0.571152
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −10648.0 −0.377267
$$928$$ 0 0
$$929$$ 8162.00 0.288252 0.144126 0.989559i $$-0.453963\pi$$
0.144126 + 0.989559i $$0.453963\pi$$
$$930$$ 0 0
$$931$$ −10252.0 −0.360898
$$932$$ 0 0
$$933$$ 16224.0 0.569293
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 55110.0 1.92141 0.960707 0.277564i $$-0.0895270\pi$$
0.960707 + 0.277564i $$0.0895270\pi$$
$$938$$ 0 0
$$939$$ 8616.00 0.299438
$$940$$ 0 0
$$941$$ −16374.0 −0.567245 −0.283622 0.958936i $$-0.591536\pi$$
−0.283622 + 0.958936i $$0.591536\pi$$
$$942$$ 0 0
$$943$$ −11088.0 −0.382900
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8460.00 0.290299 0.145149 0.989410i $$-0.453634\pi$$
0.145149 + 0.989410i $$0.453634\pi$$
$$948$$ 0 0
$$949$$ −3388.00 −0.115889
$$950$$ 0 0
$$951$$ 29544.0 1.00739
$$952$$ 0 0
$$953$$ 20502.0 0.696878 0.348439 0.937331i $$-0.386712\pi$$
0.348439 + 0.937331i $$0.386712\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 34848.0 1.17709
$$958$$ 0 0
$$959$$ 39024.0 1.31403
$$960$$ 0 0
$$961$$ −4191.00 −0.140680
$$962$$ 0 0
$$963$$ 8580.00 0.287110
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 36520.0 1.21448 0.607241 0.794518i $$-0.292276\pi$$
0.607241 + 0.794518i $$0.292276\pi$$
$$968$$ 0 0
$$969$$ −8800.00 −0.291741
$$970$$ 0 0
$$971$$ −20244.0 −0.669064 −0.334532 0.942384i $$-0.608578\pi$$
−0.334532 + 0.942384i $$0.608578\pi$$
$$972$$ 0 0
$$973$$ −16416.0 −0.540876
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −50034.0 −1.63841 −0.819206 0.573499i $$-0.805586\pi$$
−0.819206 + 0.573499i $$0.805586\pi$$
$$978$$ 0 0
$$979$$ 31416.0 1.02560
$$980$$ 0 0
$$981$$ −21934.0 −0.713862
$$982$$ 0 0
$$983$$ −37128.0 −1.20468 −0.602339 0.798240i $$-0.705765\pi$$
−0.602339 + 0.798240i $$0.705765\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −50688.0 −1.63467
$$988$$ 0 0
$$989$$ 2912.00 0.0936261
$$990$$ 0 0
$$991$$ 27808.0 0.891373 0.445686 0.895189i $$-0.352960\pi$$
0.445686 + 0.895189i $$0.352960\pi$$
$$992$$ 0 0
$$993$$ −4528.00 −0.144705
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −28514.0 −0.905765 −0.452882 0.891570i $$-0.649604\pi$$
−0.452882 + 0.891570i $$0.649604\pi$$
$$998$$ 0 0
$$999$$ −24624.0 −0.779849
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 1600.4.a.o.1.1 1
4.3 odd 2 1600.4.a.bm.1.1 1
5.4 even 2 64.4.a.d.1.1 1
8.3 odd 2 400.4.a.g.1.1 1
8.5 even 2 200.4.a.g.1.1 1
15.14 odd 2 576.4.a.k.1.1 1
20.19 odd 2 64.4.a.b.1.1 1
24.5 odd 2 1800.4.a.d.1.1 1
40.3 even 4 400.4.c.i.49.1 2
40.13 odd 4 200.4.c.e.49.2 2
40.19 odd 2 16.4.a.a.1.1 1
40.27 even 4 400.4.c.i.49.2 2
40.29 even 2 8.4.a.a.1.1 1
40.37 odd 4 200.4.c.e.49.1 2
60.59 even 2 576.4.a.j.1.1 1
80.19 odd 4 256.4.b.g.129.1 2
80.29 even 4 256.4.b.a.129.2 2
80.59 odd 4 256.4.b.g.129.2 2
80.69 even 4 256.4.b.a.129.1 2
120.29 odd 2 72.4.a.c.1.1 1
120.53 even 4 1800.4.f.u.649.2 2
120.59 even 2 144.4.a.e.1.1 1
120.77 even 4 1800.4.f.u.649.1 2
280.69 odd 2 392.4.a.e.1.1 1
280.109 even 6 392.4.i.g.177.1 2
280.139 even 2 784.4.a.e.1.1 1
280.149 even 6 392.4.i.g.361.1 2
280.229 odd 6 392.4.i.b.361.1 2
280.269 odd 6 392.4.i.b.177.1 2
360.29 odd 6 648.4.i.e.433.1 2
360.149 odd 6 648.4.i.e.217.1 2
360.229 even 6 648.4.i.h.217.1 2
360.349 even 6 648.4.i.h.433.1 2
440.109 odd 2 968.4.a.a.1.1 1
440.219 even 2 1936.4.a.l.1.1 1
520.389 even 2 1352.4.a.a.1.1 1
680.509 even 2 2312.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
8.4.a.a.1.1 1 40.29 even 2
16.4.a.a.1.1 1 40.19 odd 2
64.4.a.b.1.1 1 20.19 odd 2
64.4.a.d.1.1 1 5.4 even 2
72.4.a.c.1.1 1 120.29 odd 2
144.4.a.e.1.1 1 120.59 even 2
200.4.a.g.1.1 1 8.5 even 2
200.4.c.e.49.1 2 40.37 odd 4
200.4.c.e.49.2 2 40.13 odd 4
256.4.b.a.129.1 2 80.69 even 4
256.4.b.a.129.2 2 80.29 even 4
256.4.b.g.129.1 2 80.19 odd 4
256.4.b.g.129.2 2 80.59 odd 4
392.4.a.e.1.1 1 280.69 odd 2
392.4.i.b.177.1 2 280.269 odd 6
392.4.i.b.361.1 2 280.229 odd 6
392.4.i.g.177.1 2 280.109 even 6
392.4.i.g.361.1 2 280.149 even 6
400.4.a.g.1.1 1 8.3 odd 2
400.4.c.i.49.1 2 40.3 even 4
400.4.c.i.49.2 2 40.27 even 4
576.4.a.j.1.1 1 60.59 even 2
576.4.a.k.1.1 1 15.14 odd 2
648.4.i.e.217.1 2 360.149 odd 6
648.4.i.e.433.1 2 360.29 odd 6
648.4.i.h.217.1 2 360.229 even 6
648.4.i.h.433.1 2 360.349 even 6
784.4.a.e.1.1 1 280.139 even 2
968.4.a.a.1.1 1 440.109 odd 2
1352.4.a.a.1.1 1 520.389 even 2
1600.4.a.o.1.1 1 1.1 even 1 trivial
1600.4.a.bm.1.1 1 4.3 odd 2
1800.4.a.d.1.1 1 24.5 odd 2
1800.4.f.u.649.1 2 120.77 even 4
1800.4.f.u.649.2 2 120.53 even 4
1936.4.a.l.1.1 1 440.219 even 2
2312.4.a.a.1.1 1 680.509 even 2