# Properties

 Label 144.4.a.e.1.1 Level $144$ Weight $4$ Character 144.1 Self dual yes Analytic conductor $8.496$ 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] = [144,4,Mod(1,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, 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("144.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 144.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: $$8.49627504083$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 144.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{5} -24.0000 q^{7} +O(q^{10})$$ $$q+2.00000 q^{5} -24.0000 q^{7} -44.0000 q^{11} +22.0000 q^{13} -50.0000 q^{17} -44.0000 q^{19} -56.0000 q^{23} -121.000 q^{25} -198.000 q^{29} +160.000 q^{31} -48.0000 q^{35} -162.000 q^{37} +198.000 q^{41} -52.0000 q^{43} +528.000 q^{47} +233.000 q^{49} +242.000 q^{53} -88.0000 q^{55} -668.000 q^{59} +550.000 q^{61} +44.0000 q^{65} -188.000 q^{67} +728.000 q^{71} +154.000 q^{73} +1056.00 q^{77} +656.000 q^{79} +236.000 q^{83} -100.000 q^{85} -714.000 q^{89} -528.000 q^{91} -88.0000 q^{95} -478.000 q^{97} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 2.00000 0.178885 0.0894427 0.995992i $$-0.471491\pi$$
0.0894427 + 0.995992i $$0.471491\pi$$
$$6$$ 0 0
$$7$$ −24.0000 −1.29588 −0.647939 0.761692i $$-0.724369\pi$$
−0.647939 + 0.761692i $$0.724369\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −44.0000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 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
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −56.0000 −0.507687 −0.253844 0.967245i $$-0.581695\pi$$
−0.253844 + 0.967245i $$0.581695\pi$$
$$24$$ 0 0
$$25$$ −121.000 −0.968000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 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 $$-0.346594\pi$$
0.463498 + 0.886098i $$0.346594\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −48.0000 −0.231814
$$36$$ 0 0
$$37$$ −162.000 −0.719801 −0.359900 0.932991i $$-0.617189\pi$$
−0.359900 + 0.932991i $$0.617189\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 198.000 0.754205 0.377102 0.926172i $$-0.376920\pi$$
0.377102 + 0.926172i $$0.376920\pi$$
$$42$$ 0 0
$$43$$ −52.0000 −0.184417 −0.0922084 0.995740i $$-0.529393\pi$$
−0.0922084 + 0.995740i $$0.529393\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 528.000 1.63865 0.819327 0.573327i $$-0.194347\pi$$
0.819327 + 0.573327i $$0.194347\pi$$
$$48$$ 0 0
$$49$$ 233.000 0.679300
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 242.000 0.627194 0.313597 0.949556i $$-0.398466\pi$$
0.313597 + 0.949556i $$0.398466\pi$$
$$54$$ 0 0
$$55$$ −88.0000 −0.215744
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −668.000 −1.47400 −0.737002 0.675891i $$-0.763759\pi$$
−0.737002 + 0.675891i $$0.763759\pi$$
$$60$$ 0 0
$$61$$ 550.000 1.15443 0.577215 0.816592i $$-0.304139\pi$$
0.577215 + 0.816592i $$0.304139\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 44.0000 0.0839620
$$66$$ 0 0
$$67$$ −188.000 −0.342804 −0.171402 0.985201i $$-0.554830\pi$$
−0.171402 + 0.985201i $$0.554830\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 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 $$-0.460603\pi$$
0.123454 + 0.992350i $$0.460603\pi$$
$$74$$ 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 $$-0.345290\pi$$
0.467125 + 0.884191i $$0.345290\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 236.000 0.312101 0.156050 0.987749i $$-0.450124\pi$$
0.156050 + 0.987749i $$0.450124\pi$$
$$84$$ 0 0
$$85$$ −100.000 −0.127606
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −714.000 −0.850380 −0.425190 0.905104i $$-0.639793\pi$$
−0.425190 + 0.905104i $$0.639793\pi$$
$$90$$ 0 0
$$91$$ −528.000 −0.608236
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −88.0000 −0.0950380
$$96$$ 0 0
$$97$$ −478.000 −0.500346 −0.250173 0.968201i $$-0.580487\pi$$
−0.250173 + 0.968201i $$0.580487\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 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 $$-0.839866\pi$$
−0.876103 + 0.482123i $$0.839866\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 942.000 0.784212 0.392106 0.919920i $$-0.371747\pi$$
0.392106 + 0.919920i $$0.371747\pi$$
$$114$$ 0 0
$$115$$ −112.000 −0.0908179
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1200.00 0.924402
$$120$$ 0 0
$$121$$ 605.000 0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −492.000 −0.352047
$$126$$ 0 0
$$127$$ −1408.00 −0.983778 −0.491889 0.870658i $$-0.663693\pi$$
−0.491889 + 0.870658i $$0.663693\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2692.00 −1.79543 −0.897714 0.440578i $$-0.854773\pi$$
−0.897714 + 0.440578i $$0.854773\pi$$
$$132$$ 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
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −968.000 −0.566072
$$144$$ 0 0
$$145$$ −396.000 −0.226800
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 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 $$-0.618698\pi$$
−0.364319 + 0.931274i $$0.618698\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 320.000 0.165826
$$156$$ 0 0
$$157$$ 3142.00 1.59719 0.798595 0.601868i $$-0.205577\pi$$
0.798595 + 0.601868i $$0.205577\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1344.00 0.657901
$$162$$ 0 0
$$163$$ −3036.00 −1.45888 −0.729441 0.684043i $$-0.760220\pi$$
−0.729441 + 0.684043i $$0.760220\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −264.000 −0.122329 −0.0611645 0.998128i $$-0.519481\pi$$
−0.0611645 + 0.998128i $$0.519481\pi$$
$$168$$ 0 0
$$169$$ −1713.00 −0.779700
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2826.00 1.24195 0.620973 0.783832i $$-0.286737\pi$$
0.620973 + 0.783832i $$0.286737\pi$$
$$174$$ 0 0
$$175$$ 2904.00 1.25441
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 3084.00 1.28776 0.643880 0.765127i $$-0.277324\pi$$
0.643880 + 0.765127i $$0.277324\pi$$
$$180$$ 0 0
$$181$$ −2418.00 −0.992975 −0.496488 0.868044i $$-0.665377\pi$$
−0.496488 + 0.868044i $$0.665377\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −324.000 −0.128762
$$186$$ 0 0
$$187$$ 2200.00 0.860320
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 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 $$-0.319392\pi$$
0.537438 + 0.843304i $$0.319392\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1086.00 −0.392763 −0.196381 0.980528i $$-0.562919\pi$$
−0.196381 + 0.980528i $$0.562919\pi$$
$$198$$ 0 0
$$199$$ −88.0000 −0.0313475 −0.0156738 0.999877i $$-0.504989\pi$$
−0.0156738 + 0.999877i $$0.504989\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 4752.00 1.64298
$$204$$ 0 0
$$205$$ 396.000 0.134916
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 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
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −104.000 −0.0329895
$$216$$ 0 0
$$217$$ −3840.00 −1.20127
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 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 $$-0.575758\pi$$
−0.235759 + 0.971811i $$0.575758\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 902.000 0.253614 0.126807 0.991927i $$-0.459527\pi$$
0.126807 + 0.991927i $$0.459527\pi$$
$$234$$ 0 0
$$235$$ 1056.00 0.293131
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 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
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 466.000 0.121517
$$246$$ 0 0
$$247$$ −968.000 −0.249362
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −2140.00 −0.538150 −0.269075 0.963119i $$-0.586718\pi$$
−0.269075 + 0.963119i $$0.586718\pi$$
$$252$$ 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
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −7400.00 −1.73499 −0.867497 0.497442i $$-0.834273\pi$$
−0.867497 + 0.497442i $$0.834273\pi$$
$$264$$ 0 0
$$265$$ 484.000 0.112196
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 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 $$-0.917439\pi$$
−0.966551 + 0.256474i $$0.917439\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 5324.00 1.16745
$$276$$ 0 0
$$277$$ −1874.00 −0.406490 −0.203245 0.979128i $$-0.565149\pi$$
−0.203245 + 0.979128i $$0.565149\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3338.00 −0.708642 −0.354321 0.935124i $$-0.615288\pi$$
−0.354321 + 0.935124i $$0.615288\pi$$
$$282$$ 0 0
$$283$$ −7172.00 −1.50647 −0.753235 0.657751i $$-0.771508\pi$$
−0.753235 + 0.657751i $$0.771508\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4752.00 −0.977358
$$288$$ 0 0
$$289$$ −2413.00 −0.491146
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −5214.00 −1.03961 −0.519804 0.854286i $$-0.673995\pi$$
−0.519804 + 0.854286i $$0.673995\pi$$
$$294$$ 0 0
$$295$$ −1336.00 −0.263678
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1232.00 −0.238289
$$300$$ 0 0
$$301$$ 1248.00 0.238982
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1100.00 0.206511
$$306$$ 0 0
$$307$$ −396.000 −0.0736186 −0.0368093 0.999322i $$-0.511719\pi$$
−0.0368093 + 0.999322i $$0.511719\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 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 $$-0.437695\pi$$
0.194491 + 0.980904i $$0.437695\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 7386.00 1.30864 0.654320 0.756217i $$-0.272955\pi$$
0.654320 + 0.756217i $$0.272955\pi$$
$$318$$ 0 0
$$319$$ 8712.00 1.52909
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 2200.00 0.378982
$$324$$ 0 0
$$325$$ −2662.00 −0.454342
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −376.000 −0.0613226
$$336$$ 0 0
$$337$$ −3342.00 −0.540209 −0.270104 0.962831i $$-0.587058\pi$$
−0.270104 + 0.962831i $$0.587058\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 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 $$-0.666727\pi$$
−0.500164 + 0.865931i $$0.666727\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 11230.0 1.69324 0.846618 0.532200i $$-0.178635\pi$$
0.846618 + 0.532200i $$0.178635\pi$$
$$354$$ 0 0
$$355$$ 1456.00 0.217680
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 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
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 308.000 0.0441684
$$366$$ 0 0
$$367$$ −7120.00 −1.01270 −0.506350 0.862328i $$-0.669006\pi$$
−0.506350 + 0.862328i $$0.669006\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 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
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 10368.0 1.38324 0.691619 0.722263i $$-0.256898\pi$$
0.691619 + 0.722263i $$0.256898\pi$$
$$384$$ 0 0
$$385$$ 2112.00 0.279578
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 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
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 1312.00 0.167124
$$396$$ 0 0
$$397$$ 9878.00 1.24877 0.624386 0.781116i $$-0.285349\pi$$
0.624386 + 0.781116i $$0.285349\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 13134.0 1.63561 0.817806 0.575494i $$-0.195190\pi$$
0.817806 + 0.575494i $$0.195190\pi$$
$$402$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 16032.0 1.91013
$$414$$ 0 0
$$415$$ 472.000 0.0558303
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −5412.00 −0.631011 −0.315505 0.948924i $$-0.602174\pi$$
−0.315505 + 0.948924i $$0.602174\pi$$
$$420$$ 0 0
$$421$$ −4642.00 −0.537381 −0.268690 0.963227i $$-0.586591\pi$$
−0.268690 + 0.963227i $$0.586591\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6050.00 0.690513
$$426$$ 0 0
$$427$$ −13200.0 −1.49600
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 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 $$-0.323456\pi$$
0.526629 + 0.850096i $$0.323456\pi$$
$$434$$ 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 $$-0.597443\pi$$
−0.301368 + 0.953508i $$0.597443\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 7652.00 0.820672 0.410336 0.911935i $$-0.365412\pi$$
0.410336 + 0.911935i $$0.365412\pi$$
$$444$$ 0 0
$$445$$ −1428.00 −0.152121
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 446.000 0.0468776 0.0234388 0.999725i $$-0.492539\pi$$
0.0234388 + 0.999725i $$0.492539\pi$$
$$450$$ 0 0
$$451$$ −8712.00 −0.909605
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −1056.00 −0.108804
$$456$$ 0 0
$$457$$ 1562.00 0.159885 0.0799423 0.996799i $$-0.474526\pi$$
0.0799423 + 0.996799i $$0.474526\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 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
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 2288.00 0.222415
$$474$$ 0 0
$$475$$ 5324.00 0.514278
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 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
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −956.000 −0.0895046
$$486$$ 0 0
$$487$$ 15176.0 1.41209 0.706047 0.708165i $$-0.250477\pi$$
0.706047 + 0.708165i $$0.250477\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −8844.00 −0.812880 −0.406440 0.913677i $$-0.633230\pi$$
−0.406440 + 0.913677i $$0.633230\pi$$
$$492$$ 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
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 16488.0 1.46156 0.730779 0.682614i $$-0.239157\pi$$
0.730779 + 0.682614i $$0.239157\pi$$
$$504$$ 0 0
$$505$$ −3132.00 −0.275984
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 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
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1936.00 0.165651
$$516$$ 0 0
$$517$$ −23232.0 −1.97629
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10970.0 −0.922465 −0.461233 0.887279i $$-0.652593\pi$$
−0.461233 + 0.887279i $$0.652593\pi$$
$$522$$ 0 0
$$523$$ 16940.0 1.41632 0.708159 0.706053i $$-0.249526\pi$$
0.708159 + 0.706053i $$0.249526\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8000.00 −0.661263
$$528$$ 0 0
$$529$$ −9031.00 −0.742254
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4356.00 0.353995
$$534$$ 0 0
$$535$$ −1560.00 −0.126065
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −10252.0 −0.819267
$$540$$ 0 0
$$541$$ 198.000 0.0157351 0.00786755 0.999969i $$-0.497496\pi$$
0.00786755 + 0.999969i $$0.497496\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −3988.00 −0.313444
$$546$$ 0 0
$$547$$ 15268.0 1.19344 0.596721 0.802449i $$-0.296470\pi$$
0.596721 + 0.802449i $$0.296470\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 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 $$-0.791581\pi$$
−0.793189 + 0.608976i $$0.791581\pi$$
$$558$$ 0 0
$$559$$ −1144.00 −0.0865582
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −19316.0 −1.44595 −0.722977 0.690872i $$-0.757227\pi$$
−0.722977 + 0.690872i $$0.757227\pi$$
$$564$$ 0 0
$$565$$ 1884.00 0.140284
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −7018.00 −0.517065 −0.258532 0.966003i $$-0.583239\pi$$
−0.258532 + 0.966003i $$0.583239\pi$$
$$570$$ 0 0
$$571$$ −24420.0 −1.78975 −0.894873 0.446320i $$-0.852734\pi$$
−0.894873 + 0.446320i $$0.852734\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6776.00 0.491441
$$576$$ 0 0
$$577$$ 23234.0 1.67633 0.838166 0.545415i $$-0.183628\pi$$
0.838166 + 0.545415i $$0.183628\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 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
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 13838.0 0.958277 0.479139 0.877739i $$-0.340949\pi$$
0.479139 + 0.877739i $$0.340949\pi$$
$$594$$ 0 0
$$595$$ 2400.00 0.165362
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 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
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 1210.00 0.0813116
$$606$$ 0 0
$$607$$ 3040.00 0.203278 0.101639 0.994821i $$-0.467591\pi$$
0.101639 + 0.994821i $$0.467591\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 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
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 17136.0 1.10199
$$624$$ 0 0
$$625$$ 14141.0 0.905024
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 8100.00 0.513463
$$630$$ 0 0
$$631$$ 6680.00 0.421437 0.210718 0.977547i $$-0.432420\pi$$
0.210718 + 0.977547i $$0.432420\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −2816.00 −0.175984
$$636$$ 0 0
$$637$$ 5126.00 0.318838
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −6274.00 −0.386596 −0.193298 0.981140i $$-0.561918\pi$$
−0.193298 + 0.981140i $$0.561918\pi$$
$$642$$ 0 0
$$643$$ −9084.00 −0.557135 −0.278568 0.960417i $$-0.589860\pi$$
−0.278568 + 0.960417i $$0.589860\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −23656.0 −1.43742 −0.718712 0.695308i $$-0.755268\pi$$
−0.718712 + 0.695308i $$0.755268\pi$$
$$648$$ 0 0
$$649$$ 29392.0 1.77771
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 6762.00 0.405234 0.202617 0.979258i $$-0.435055\pi$$
0.202617 + 0.979258i $$0.435055\pi$$
$$654$$ 0 0
$$655$$ −5384.00 −0.321176
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 15276.0 0.902987 0.451494 0.892274i $$-0.350891\pi$$
0.451494 + 0.892274i $$0.350891\pi$$
$$660$$ 0 0
$$661$$ 11054.0 0.650455 0.325228 0.945636i $$-0.394559\pi$$
0.325228 + 0.945636i $$0.394559\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 2112.00 0.123158
$$666$$ 0 0
$$667$$ 11088.0 0.643672
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −24200.0 −1.39230
$$672$$ 0 0
$$673$$ −21278.0 −1.21873 −0.609366 0.792889i $$-0.708576\pi$$
−0.609366 + 0.792889i $$0.708576\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −8926.00 −0.506727 −0.253363 0.967371i $$-0.581537\pi$$
−0.253363 + 0.967371i $$0.581537\pi$$
$$678$$ 0 0
$$679$$ 11472.0 0.648387
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 8116.00 0.454685 0.227343 0.973815i $$-0.426996\pi$$
0.227343 + 0.973815i $$0.426996\pi$$
$$684$$ 0 0
$$685$$ −3252.00 −0.181391
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1368.00 0.0746636
$$696$$ 0 0
$$697$$ −9900.00 −0.538005
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 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 $$-0.273704\pi$$
0.652538 + 0.757756i $$0.273704\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −8960.00 −0.470624
$$714$$ 0 0
$$715$$ −1936.00 −0.101262
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 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
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 23958.0 1.22728
$$726$$ 0 0
$$727$$ −30216.0 −1.54147 −0.770735 0.637155i $$-0.780111\pi$$
−0.770735 + 0.637155i $$0.780111\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 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
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 28600.0 1.41216 0.706078 0.708134i $$-0.250463\pi$$
0.706078 + 0.708134i $$0.250463\pi$$
$$744$$ 0 0
$$745$$ −604.000 −0.0297032
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 18720.0 0.913236
$$750$$ 0 0
$$751$$ 29616.0 1.43902 0.719509 0.694483i $$-0.244367\pi$$
0.719509 + 0.694483i $$0.244367\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −2704.00 −0.130343
$$756$$ 0 0
$$757$$ 2894.00 0.138949 0.0694744 0.997584i $$-0.477868\pi$$
0.0694744 + 0.997584i $$0.477868\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −14762.0 −0.703183 −0.351591 0.936154i $$-0.614359\pi$$
−0.351591 + 0.936154i $$0.614359\pi$$
$$762$$ 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
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −27390.0 −1.27445 −0.637225 0.770678i $$-0.719918\pi$$
−0.637225 + 0.770678i $$0.719918\pi$$
$$774$$ 0 0
$$775$$ −19360.0 −0.897331
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −8712.00 −0.400693
$$780$$ 0 0
$$781$$ −32032.0 −1.46760
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 6284.00 0.285714
$$786$$ 0 0
$$787$$ −19756.0 −0.894823 −0.447411 0.894328i $$-0.647654\pi$$
−0.447411 + 0.894328i $$0.647654\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 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 $$-0.831677\pi$$
−0.863412 + 0.504499i $$0.831677\pi$$
$$798$$ 0 0
$$799$$ −26400.0 −1.16892
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −6776.00 −0.297783
$$804$$ 0 0
$$805$$ 2688.00 0.117689
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 14278.0 0.620504 0.310252 0.950654i $$-0.399587\pi$$
0.310252 + 0.950654i $$0.399587\pi$$
$$810$$ 0 0
$$811$$ 716.000 0.0310014 0.0155007 0.999880i $$-0.495066\pi$$
0.0155007 + 0.999880i $$0.495066\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −6072.00 −0.260973
$$816$$ 0 0
$$817$$ 2288.00 0.0979767
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 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 $$-0.418635\pi$$
0.252840 + 0.967508i $$0.418635\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −11650.0 −0.484572
$$834$$ 0 0
$$835$$ −528.000 −0.0218829
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 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
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −3426.00 −0.139477
$$846$$ 0 0
$$847$$ −14520.0 −0.589036
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 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
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −18656.0 −0.735872 −0.367936 0.929851i $$-0.619935\pi$$
−0.367936 + 0.929851i $$0.619935\pi$$
$$864$$ 0 0
$$865$$ 5652.00 0.222166
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −28864.0 −1.12675
$$870$$ 0 0
$$871$$ −4136.00 −0.160899
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 11808.0 0.456209
$$876$$ 0 0
$$877$$ −13002.0 −0.500623 −0.250311 0.968165i $$-0.580533\pi$$
−0.250311 + 0.968165i $$0.580533\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −49490.0 −1.89258 −0.946289 0.323323i $$-0.895200\pi$$
−0.946289 + 0.323323i $$0.895200\pi$$
$$882$$ 0 0
$$883$$ −1100.00 −0.0419229 −0.0209615 0.999780i $$-0.506673\pi$$
−0.0209615 + 0.999780i $$0.506673\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −14104.0 −0.533896 −0.266948 0.963711i $$-0.586015\pi$$
−0.266948 + 0.963711i $$0.586015\pi$$
$$888$$ 0 0
$$889$$ 33792.0 1.27486
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −23232.0 −0.870581
$$894$$ 0 0
$$895$$ 6168.00 0.230361
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −31680.0 −1.17529
$$900$$ 0 0
$$901$$ −12100.0 −0.447402
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −4836.00 −0.177629
$$906$$ 0 0
$$907$$ 12716.0 0.465521 0.232761 0.972534i $$-0.425224\pi$$
0.232761 + 0.972534i $$0.425224\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 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 $$-0.532643\pi$$
−0.102371 + 0.994746i $$0.532643\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 16016.0 0.571152
$$924$$ 0 0
$$925$$ 19602.0 0.696767
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −8162.00 −0.288252 −0.144126 0.989559i $$-0.546037\pi$$
−0.144126 + 0.989559i $$0.546037\pi$$
$$930$$ 0 0
$$931$$ −10252.0 −0.360898
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 4400.00 0.153899
$$936$$ 0 0
$$937$$ −55110.0 −1.92141 −0.960707 0.277564i $$-0.910473\pi$$
−0.960707 + 0.277564i $$0.910473\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 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
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 20502.0 0.696878 0.348439 0.937331i $$-0.386712\pi$$
0.348439 + 0.937331i $$0.386712\pi$$
$$954$$ 0 0
$$955$$ −1920.00 −0.0650573
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 39024.0 1.31403
$$960$$ 0 0
$$961$$ −4191.00 −0.140680
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 5764.00 0.192280
$$966$$ 0 0
$$967$$ 36520.0 1.21448 0.607241 0.794518i $$-0.292276\pi$$
0.607241 + 0.794518i $$0.292276\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 20244.0 0.669064 0.334532 0.942384i $$-0.391422\pi$$
0.334532 + 0.942384i $$0.391422\pi$$
$$972$$ 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
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 37128.0 1.20468 0.602339 0.798240i $$-0.294235\pi$$
0.602339 + 0.798240i $$0.294235\pi$$
$$984$$ 0 0
$$985$$ −2172.00 −0.0702596
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 2912.00 0.0936261
$$990$$ 0 0
$$991$$ −27808.0 −0.891373 −0.445686 0.895189i $$-0.647040\pi$$
−0.445686 + 0.895189i $$0.647040\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −176.000 −0.00560761
$$996$$ 0 0
$$997$$ −28514.0 −0.905765 −0.452882 0.891570i $$-0.649604\pi$$
−0.452882 + 0.891570i $$0.649604\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 144.4.a.e.1.1 1
3.2 odd 2 16.4.a.a.1.1 1
4.3 odd 2 72.4.a.c.1.1 1
8.3 odd 2 576.4.a.k.1.1 1
8.5 even 2 576.4.a.j.1.1 1
12.11 even 2 8.4.a.a.1.1 1
15.2 even 4 400.4.c.i.49.1 2
15.8 even 4 400.4.c.i.49.2 2
15.14 odd 2 400.4.a.g.1.1 1
20.3 even 4 1800.4.f.u.649.1 2
20.7 even 4 1800.4.f.u.649.2 2
20.19 odd 2 1800.4.a.d.1.1 1
21.20 even 2 784.4.a.e.1.1 1
24.5 odd 2 64.4.a.b.1.1 1
24.11 even 2 64.4.a.d.1.1 1
33.32 even 2 1936.4.a.l.1.1 1
36.7 odd 6 648.4.i.e.433.1 2
36.11 even 6 648.4.i.h.433.1 2
36.23 even 6 648.4.i.h.217.1 2
36.31 odd 6 648.4.i.e.217.1 2
48.5 odd 4 256.4.b.g.129.1 2
48.11 even 4 256.4.b.a.129.2 2
48.29 odd 4 256.4.b.g.129.2 2
48.35 even 4 256.4.b.a.129.1 2
60.23 odd 4 200.4.c.e.49.1 2
60.47 odd 4 200.4.c.e.49.2 2
60.59 even 2 200.4.a.g.1.1 1
84.11 even 6 392.4.i.g.177.1 2
84.23 even 6 392.4.i.g.361.1 2
84.47 odd 6 392.4.i.b.361.1 2
84.59 odd 6 392.4.i.b.177.1 2
84.83 odd 2 392.4.a.e.1.1 1
120.29 odd 2 1600.4.a.bm.1.1 1
120.59 even 2 1600.4.a.o.1.1 1
132.131 odd 2 968.4.a.a.1.1 1
156.155 even 2 1352.4.a.a.1.1 1
204.203 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 12.11 even 2
16.4.a.a.1.1 1 3.2 odd 2
64.4.a.b.1.1 1 24.5 odd 2
64.4.a.d.1.1 1 24.11 even 2
72.4.a.c.1.1 1 4.3 odd 2
144.4.a.e.1.1 1 1.1 even 1 trivial
200.4.a.g.1.1 1 60.59 even 2
200.4.c.e.49.1 2 60.23 odd 4
200.4.c.e.49.2 2 60.47 odd 4
256.4.b.a.129.1 2 48.35 even 4
256.4.b.a.129.2 2 48.11 even 4
256.4.b.g.129.1 2 48.5 odd 4
256.4.b.g.129.2 2 48.29 odd 4
392.4.a.e.1.1 1 84.83 odd 2
392.4.i.b.177.1 2 84.59 odd 6
392.4.i.b.361.1 2 84.47 odd 6
392.4.i.g.177.1 2 84.11 even 6
392.4.i.g.361.1 2 84.23 even 6
400.4.a.g.1.1 1 15.14 odd 2
400.4.c.i.49.1 2 15.2 even 4
400.4.c.i.49.2 2 15.8 even 4
576.4.a.j.1.1 1 8.5 even 2
576.4.a.k.1.1 1 8.3 odd 2
648.4.i.e.217.1 2 36.31 odd 6
648.4.i.e.433.1 2 36.7 odd 6
648.4.i.h.217.1 2 36.23 even 6
648.4.i.h.433.1 2 36.11 even 6
784.4.a.e.1.1 1 21.20 even 2
968.4.a.a.1.1 1 132.131 odd 2
1352.4.a.a.1.1 1 156.155 even 2
1600.4.a.o.1.1 1 120.59 even 2
1600.4.a.bm.1.1 1 120.29 odd 2
1800.4.a.d.1.1 1 20.19 odd 2
1800.4.f.u.649.1 2 20.3 even 4
1800.4.f.u.649.2 2 20.7 even 4
1936.4.a.l.1.1 1 33.32 even 2
2312.4.a.a.1.1 1 204.203 even 2