# Properties

 Label 72.4.a.d.1.1 Level $72$ Weight $4$ Character 72.1 Self dual yes Analytic conductor $4.248$ 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] = [72,4,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, 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("72.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 72.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: $$4.24813752041$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 72.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+16.0000 q^{5} -12.0000 q^{7} +O(q^{10})$$ $$q+16.0000 q^{5} -12.0000 q^{7} +64.0000 q^{11} +58.0000 q^{13} +32.0000 q^{17} -136.000 q^{19} -128.000 q^{23} +131.000 q^{25} -144.000 q^{29} +20.0000 q^{31} -192.000 q^{35} -18.0000 q^{37} -288.000 q^{41} -200.000 q^{43} +384.000 q^{47} -199.000 q^{49} +496.000 q^{53} +1024.00 q^{55} -128.000 q^{59} -458.000 q^{61} +928.000 q^{65} -496.000 q^{67} +512.000 q^{71} -602.000 q^{73} -768.000 q^{77} +1108.00 q^{79} +704.000 q^{83} +512.000 q^{85} -960.000 q^{89} -696.000 q^{91} -2176.00 q^{95} +206.000 q^{97} +O(q^{100})$$

## 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
$$5$$ 16.0000 1.43108 0.715542 0.698570i $$-0.246180\pi$$
0.715542 + 0.698570i $$0.246180\pi$$
$$6$$ 0 0
$$7$$ −12.0000 −0.647939 −0.323970 0.946068i $$-0.605018\pi$$
−0.323970 + 0.946068i $$0.605018\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 64.0000 1.75425 0.877124 0.480264i $$-0.159459\pi$$
0.877124 + 0.480264i $$0.159459\pi$$
$$12$$ 0 0
$$13$$ 58.0000 1.23741 0.618704 0.785624i $$-0.287658\pi$$
0.618704 + 0.785624i $$0.287658\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 32.0000 0.456538 0.228269 0.973598i $$-0.426693\pi$$
0.228269 + 0.973598i $$0.426693\pi$$
$$18$$ 0 0
$$19$$ −136.000 −1.64213 −0.821067 0.570832i $$-0.806621\pi$$
−0.821067 + 0.570832i $$0.806621\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −128.000 −1.16043 −0.580214 0.814464i $$-0.697031\pi$$
−0.580214 + 0.814464i $$0.697031\pi$$
$$24$$ 0 0
$$25$$ 131.000 1.04800
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −144.000 −0.922073 −0.461037 0.887381i $$-0.652522\pi$$
−0.461037 + 0.887381i $$0.652522\pi$$
$$30$$ 0 0
$$31$$ 20.0000 0.115874 0.0579372 0.998320i $$-0.481548\pi$$
0.0579372 + 0.998320i $$0.481548\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −192.000 −0.927255
$$36$$ 0 0
$$37$$ −18.0000 −0.0799779 −0.0399889 0.999200i $$-0.512732\pi$$
−0.0399889 + 0.999200i $$0.512732\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −288.000 −1.09703 −0.548513 0.836142i $$-0.684806\pi$$
−0.548513 + 0.836142i $$0.684806\pi$$
$$42$$ 0 0
$$43$$ −200.000 −0.709296 −0.354648 0.935000i $$-0.615399\pi$$
−0.354648 + 0.935000i $$0.615399\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 384.000 1.19175 0.595874 0.803078i $$-0.296806\pi$$
0.595874 + 0.803078i $$0.296806\pi$$
$$48$$ 0 0
$$49$$ −199.000 −0.580175
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 496.000 1.28549 0.642744 0.766081i $$-0.277796\pi$$
0.642744 + 0.766081i $$0.277796\pi$$
$$54$$ 0 0
$$55$$ 1024.00 2.51048
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −128.000 −0.282444 −0.141222 0.989978i $$-0.545103\pi$$
−0.141222 + 0.989978i $$0.545103\pi$$
$$60$$ 0 0
$$61$$ −458.000 −0.961326 −0.480663 0.876905i $$-0.659604\pi$$
−0.480663 + 0.876905i $$0.659604\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 928.000 1.77083
$$66$$ 0 0
$$67$$ −496.000 −0.904419 −0.452209 0.891912i $$-0.649364\pi$$
−0.452209 + 0.891912i $$0.649364\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 512.000 0.855820 0.427910 0.903821i $$-0.359250\pi$$
0.427910 + 0.903821i $$0.359250\pi$$
$$72$$ 0 0
$$73$$ −602.000 −0.965189 −0.482594 0.875844i $$-0.660305\pi$$
−0.482594 + 0.875844i $$0.660305\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −768.000 −1.13665
$$78$$ 0 0
$$79$$ 1108.00 1.57797 0.788986 0.614412i $$-0.210607\pi$$
0.788986 + 0.614412i $$0.210607\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 704.000 0.931013 0.465506 0.885045i $$-0.345872\pi$$
0.465506 + 0.885045i $$0.345872\pi$$
$$84$$ 0 0
$$85$$ 512.000 0.653343
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −960.000 −1.14337 −0.571684 0.820474i $$-0.693710\pi$$
−0.571684 + 0.820474i $$0.693710\pi$$
$$90$$ 0 0
$$91$$ −696.000 −0.801765
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −2176.00 −2.35003
$$96$$ 0 0
$$97$$ 206.000 0.215630 0.107815 0.994171i $$-0.465615\pi$$
0.107815 + 0.994171i $$0.465615\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 432.000 0.425600 0.212800 0.977096i $$-0.431742\pi$$
0.212800 + 0.977096i $$0.431742\pi$$
$$102$$ 0 0
$$103$$ −68.0000 −0.0650509 −0.0325254 0.999471i $$-0.510355\pi$$
−0.0325254 + 0.999471i $$0.510355\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −384.000 −0.346941 −0.173470 0.984839i $$-0.555498\pi$$
−0.173470 + 0.984839i $$0.555498\pi$$
$$108$$ 0 0
$$109$$ −518.000 −0.455187 −0.227594 0.973756i $$-0.573086\pi$$
−0.227594 + 0.973756i $$0.573086\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −960.000 −0.799196 −0.399598 0.916690i $$-0.630850\pi$$
−0.399598 + 0.916690i $$0.630850\pi$$
$$114$$ 0 0
$$115$$ −2048.00 −1.66067
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −384.000 −0.295809
$$120$$ 0 0
$$121$$ 2765.00 2.07739
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 96.0000 0.0686920
$$126$$ 0 0
$$127$$ 796.000 0.556170 0.278085 0.960556i $$-0.410300\pi$$
0.278085 + 0.960556i $$0.410300\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 512.000 0.341478 0.170739 0.985316i $$-0.445384\pi$$
0.170739 + 0.985316i $$0.445384\pi$$
$$132$$ 0 0
$$133$$ 1632.00 1.06400
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1824.00 1.13748 0.568740 0.822517i $$-0.307431\pi$$
0.568740 + 0.822517i $$0.307431\pi$$
$$138$$ 0 0
$$139$$ −2160.00 −1.31805 −0.659024 0.752121i $$-0.729031\pi$$
−0.659024 + 0.752121i $$0.729031\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 3712.00 2.17072
$$144$$ 0 0
$$145$$ −2304.00 −1.31956
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −688.000 −0.378276 −0.189138 0.981950i $$-0.560569\pi$$
−0.189138 + 0.981950i $$0.560569\pi$$
$$150$$ 0 0
$$151$$ −844.000 −0.454859 −0.227430 0.973795i $$-0.573032\pi$$
−0.227430 + 0.973795i $$0.573032\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 320.000 0.165826
$$156$$ 0 0
$$157$$ 118.000 0.0599836 0.0299918 0.999550i $$-0.490452\pi$$
0.0299918 + 0.999550i $$0.490452\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1536.00 0.751887
$$162$$ 0 0
$$163$$ 3576.00 1.71837 0.859184 0.511667i $$-0.170972\pi$$
0.859184 + 0.511667i $$0.170972\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 384.000 0.177933 0.0889665 0.996035i $$-0.471644\pi$$
0.0889665 + 0.996035i $$0.471644\pi$$
$$168$$ 0 0
$$169$$ 1167.00 0.531179
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2448.00 1.07583 0.537913 0.843000i $$-0.319213\pi$$
0.537913 + 0.843000i $$0.319213\pi$$
$$174$$ 0 0
$$175$$ −1572.00 −0.679040
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −4224.00 −1.76378 −0.881890 0.471455i $$-0.843729\pi$$
−0.881890 + 0.471455i $$0.843729\pi$$
$$180$$ 0 0
$$181$$ −510.000 −0.209436 −0.104718 0.994502i $$-0.533394\pi$$
−0.104718 + 0.994502i $$0.533394\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −288.000 −0.114455
$$186$$ 0 0
$$187$$ 2048.00 0.800880
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −384.000 −0.145473 −0.0727363 0.997351i $$-0.523173\pi$$
−0.0727363 + 0.997351i $$0.523173\pi$$
$$192$$ 0 0
$$193$$ −3454.00 −1.28821 −0.644105 0.764937i $$-0.722770\pi$$
−0.644105 + 0.764937i $$0.722770\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3216.00 −1.16310 −0.581550 0.813511i $$-0.697553\pi$$
−0.581550 + 0.813511i $$0.697553\pi$$
$$198$$ 0 0
$$199$$ 1708.00 0.608427 0.304213 0.952604i $$-0.401606\pi$$
0.304213 + 0.952604i $$0.401606\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1728.00 0.597447
$$204$$ 0 0
$$205$$ −4608.00 −1.56994
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −8704.00 −2.88071
$$210$$ 0 0
$$211$$ 2320.00 0.756945 0.378472 0.925613i $$-0.376449\pi$$
0.378472 + 0.925613i $$0.376449\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −3200.00 −1.01506
$$216$$ 0 0
$$217$$ −240.000 −0.0750795
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1856.00 0.564923
$$222$$ 0 0
$$223$$ −116.000 −0.0348338 −0.0174169 0.999848i $$-0.505544\pi$$
−0.0174169 + 0.999848i $$0.505544\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1344.00 0.392971 0.196485 0.980507i $$-0.437047\pi$$
0.196485 + 0.980507i $$0.437047\pi$$
$$228$$ 0 0
$$229$$ 4594.00 1.32568 0.662839 0.748762i $$-0.269352\pi$$
0.662839 + 0.748762i $$0.269352\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5056.00 1.42159 0.710793 0.703401i $$-0.248336\pi$$
0.710793 + 0.703401i $$0.248336\pi$$
$$234$$ 0 0
$$235$$ 6144.00 1.70549
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −3712.00 −1.00464 −0.502321 0.864681i $$-0.667520\pi$$
−0.502321 + 0.864681i $$0.667520\pi$$
$$240$$ 0 0
$$241$$ −978.000 −0.261405 −0.130702 0.991422i $$-0.541723\pi$$
−0.130702 + 0.991422i $$0.541723\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −3184.00 −0.830279
$$246$$ 0 0
$$247$$ −7888.00 −2.03199
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1856.00 0.466732 0.233366 0.972389i $$-0.425026\pi$$
0.233366 + 0.972389i $$0.425026\pi$$
$$252$$ 0 0
$$253$$ −8192.00 −2.03568
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 7808.00 1.89513 0.947567 0.319556i $$-0.103534\pi$$
0.947567 + 0.319556i $$0.103534\pi$$
$$258$$ 0 0
$$259$$ 216.000 0.0518208
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 1024.00 0.240086 0.120043 0.992769i $$-0.461697\pi$$
0.120043 + 0.992769i $$0.461697\pi$$
$$264$$ 0 0
$$265$$ 7936.00 1.83964
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 1328.00 0.301002 0.150501 0.988610i $$-0.451911\pi$$
0.150501 + 0.988610i $$0.451911\pi$$
$$270$$ 0 0
$$271$$ −5812.00 −1.30278 −0.651391 0.758742i $$-0.725814\pi$$
−0.651391 + 0.758742i $$0.725814\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 8384.00 1.83845
$$276$$ 0 0
$$277$$ 8386.00 1.81901 0.909505 0.415692i $$-0.136461\pi$$
0.909505 + 0.415692i $$0.136461\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −640.000 −0.135869 −0.0679345 0.997690i $$-0.521641\pi$$
−0.0679345 + 0.997690i $$0.521641\pi$$
$$282$$ 0 0
$$283$$ 4832.00 1.01496 0.507478 0.861665i $$-0.330578\pi$$
0.507478 + 0.861665i $$0.330578\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3456.00 0.710806
$$288$$ 0 0
$$289$$ −3889.00 −0.791573
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 6384.00 1.27289 0.636446 0.771321i $$-0.280404\pi$$
0.636446 + 0.771321i $$0.280404\pi$$
$$294$$ 0 0
$$295$$ −2048.00 −0.404201
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −7424.00 −1.43592
$$300$$ 0 0
$$301$$ 2400.00 0.459580
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −7328.00 −1.37574
$$306$$ 0 0
$$307$$ −3312.00 −0.615719 −0.307860 0.951432i $$-0.599613\pi$$
−0.307860 + 0.951432i $$0.599613\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 9984.00 1.82039 0.910194 0.414182i $$-0.135932\pi$$
0.910194 + 0.414182i $$0.135932\pi$$
$$312$$ 0 0
$$313$$ 2586.00 0.466995 0.233497 0.972357i $$-0.424983\pi$$
0.233497 + 0.972357i $$0.424983\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2832.00 −0.501770 −0.250885 0.968017i $$-0.580722\pi$$
−0.250885 + 0.968017i $$0.580722\pi$$
$$318$$ 0 0
$$319$$ −9216.00 −1.61755
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4352.00 −0.749696
$$324$$ 0 0
$$325$$ 7598.00 1.29680
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −4608.00 −0.772180
$$330$$ 0 0
$$331$$ −5920.00 −0.983059 −0.491530 0.870861i $$-0.663562\pi$$
−0.491530 + 0.870861i $$0.663562\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −7936.00 −1.29430
$$336$$ 0 0
$$337$$ −4674.00 −0.755516 −0.377758 0.925904i $$-0.623305\pi$$
−0.377758 + 0.925904i $$0.623305\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1280.00 0.203272
$$342$$ 0 0
$$343$$ 6504.00 1.02386
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −9024.00 −1.39606 −0.698031 0.716067i $$-0.745940\pi$$
−0.698031 + 0.716067i $$0.745940\pi$$
$$348$$ 0 0
$$349$$ −4362.00 −0.669033 −0.334516 0.942390i $$-0.608573\pi$$
−0.334516 + 0.942390i $$0.608573\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 8768.00 1.32202 0.661011 0.750376i $$-0.270128\pi$$
0.661011 + 0.750376i $$0.270128\pi$$
$$354$$ 0 0
$$355$$ 8192.00 1.22475
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6144.00 −0.903253 −0.451627 0.892207i $$-0.649156\pi$$
−0.451627 + 0.892207i $$0.649156\pi$$
$$360$$ 0 0
$$361$$ 11637.0 1.69660
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −9632.00 −1.38127
$$366$$ 0 0
$$367$$ 4564.00 0.649152 0.324576 0.945860i $$-0.394778\pi$$
0.324576 + 0.945860i $$0.394778\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −5952.00 −0.832918
$$372$$ 0 0
$$373$$ −8770.00 −1.21741 −0.608704 0.793397i $$-0.708310\pi$$
−0.608704 + 0.793397i $$0.708310\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −8352.00 −1.14098
$$378$$ 0 0
$$379$$ −1096.00 −0.148543 −0.0742714 0.997238i $$-0.523663\pi$$
−0.0742714 + 0.997238i $$0.523663\pi$$
$$380$$ 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
$$385$$ −12288.0 −1.62663
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −3248.00 −0.423342 −0.211671 0.977341i $$-0.567891\pi$$
−0.211671 + 0.977341i $$0.567891\pi$$
$$390$$ 0 0
$$391$$ −4096.00 −0.529779
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 17728.0 2.25821
$$396$$ 0 0
$$397$$ −6106.00 −0.771918 −0.385959 0.922516i $$-0.626129\pi$$
−0.385959 + 0.922516i $$0.626129\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7008.00 0.872725 0.436363 0.899771i $$-0.356266\pi$$
0.436363 + 0.899771i $$0.356266\pi$$
$$402$$ 0 0
$$403$$ 1160.00 0.143384
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1152.00 −0.140301
$$408$$ 0 0
$$409$$ 1590.00 0.192226 0.0961130 0.995370i $$-0.469359\pi$$
0.0961130 + 0.995370i $$0.469359\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1536.00 0.183006
$$414$$ 0 0
$$415$$ 11264.0 1.33236
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −192.000 −0.0223862 −0.0111931 0.999937i $$-0.503563\pi$$
−0.0111931 + 0.999937i $$0.503563\pi$$
$$420$$ 0 0
$$421$$ 9074.00 1.05045 0.525225 0.850963i $$-0.323981\pi$$
0.525225 + 0.850963i $$0.323981\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 4192.00 0.478451
$$426$$ 0 0
$$427$$ 5496.00 0.622881
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5248.00 −0.586513 −0.293257 0.956034i $$-0.594739\pi$$
−0.293257 + 0.956034i $$0.594739\pi$$
$$432$$ 0 0
$$433$$ −8222.00 −0.912527 −0.456263 0.889845i $$-0.650813\pi$$
−0.456263 + 0.889845i $$0.650813\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 17408.0 1.90558
$$438$$ 0 0
$$439$$ 16236.0 1.76515 0.882576 0.470169i $$-0.155807\pi$$
0.882576 + 0.470169i $$0.155807\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 14528.0 1.55812 0.779059 0.626951i $$-0.215697\pi$$
0.779059 + 0.626951i $$0.215697\pi$$
$$444$$ 0 0
$$445$$ −15360.0 −1.63626
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6304.00 0.662593 0.331296 0.943527i $$-0.392514\pi$$
0.331296 + 0.943527i $$0.392514\pi$$
$$450$$ 0 0
$$451$$ −18432.0 −1.92445
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −11136.0 −1.14739
$$456$$ 0 0
$$457$$ 1958.00 0.200419 0.100209 0.994966i $$-0.468049\pi$$
0.100209 + 0.994966i $$0.468049\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 4048.00 0.408968 0.204484 0.978870i $$-0.434448\pi$$
0.204484 + 0.978870i $$0.434448\pi$$
$$462$$ 0 0
$$463$$ 16988.0 1.70518 0.852591 0.522579i $$-0.175030\pi$$
0.852591 + 0.522579i $$0.175030\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6720.00 0.665877 0.332938 0.942949i $$-0.391960\pi$$
0.332938 + 0.942949i $$0.391960\pi$$
$$468$$ 0 0
$$469$$ 5952.00 0.586008
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −12800.0 −1.24428
$$474$$ 0 0
$$475$$ −17816.0 −1.72096
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 9728.00 0.927941 0.463970 0.885851i $$-0.346424\pi$$
0.463970 + 0.885851i $$0.346424\pi$$
$$480$$ 0 0
$$481$$ −1044.00 −0.0989653
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3296.00 0.308585
$$486$$ 0 0
$$487$$ −8444.00 −0.785696 −0.392848 0.919603i $$-0.628510\pi$$
−0.392848 + 0.919603i $$0.628510\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −15360.0 −1.41179 −0.705893 0.708318i $$-0.749454\pi$$
−0.705893 + 0.708318i $$0.749454\pi$$
$$492$$ 0 0
$$493$$ −4608.00 −0.420961
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −6144.00 −0.554519
$$498$$ 0 0
$$499$$ 6624.00 0.594250 0.297125 0.954839i $$-0.403972\pi$$
0.297125 + 0.954839i $$0.403972\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −6912.00 −0.612705 −0.306353 0.951918i $$-0.599109\pi$$
−0.306353 + 0.951918i $$0.599109\pi$$
$$504$$ 0 0
$$505$$ 6912.00 0.609069
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −19920.0 −1.73465 −0.867327 0.497739i $$-0.834164\pi$$
−0.867327 + 0.497739i $$0.834164\pi$$
$$510$$ 0 0
$$511$$ 7224.00 0.625383
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −1088.00 −0.0930932
$$516$$ 0 0
$$517$$ 24576.0 2.09062
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3680.00 0.309451 0.154725 0.987958i $$-0.450551\pi$$
0.154725 + 0.987958i $$0.450551\pi$$
$$522$$ 0 0
$$523$$ −11720.0 −0.979885 −0.489942 0.871755i $$-0.662982\pi$$
−0.489942 + 0.871755i $$0.662982\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 640.000 0.0529010
$$528$$ 0 0
$$529$$ 4217.00 0.346593
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −16704.0 −1.35747
$$534$$ 0 0
$$535$$ −6144.00 −0.496501
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −12736.0 −1.01777
$$540$$ 0 0
$$541$$ 11754.0 0.934092 0.467046 0.884233i $$-0.345318\pi$$
0.467046 + 0.884233i $$0.345318\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −8288.00 −0.651411
$$546$$ 0 0
$$547$$ −18904.0 −1.47765 −0.738827 0.673895i $$-0.764620\pi$$
−0.738827 + 0.673895i $$0.764620\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 19584.0 1.51417
$$552$$ 0 0
$$553$$ −13296.0 −1.02243
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3088.00 0.234906 0.117453 0.993078i $$-0.462527\pi$$
0.117453 + 0.993078i $$0.462527\pi$$
$$558$$ 0 0
$$559$$ −11600.0 −0.877688
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −21440.0 −1.60495 −0.802476 0.596684i $$-0.796485\pi$$
−0.802476 + 0.596684i $$0.796485\pi$$
$$564$$ 0 0
$$565$$ −15360.0 −1.14372
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 22624.0 1.66687 0.833434 0.552620i $$-0.186372\pi$$
0.833434 + 0.552620i $$0.186372\pi$$
$$570$$ 0 0
$$571$$ −6000.00 −0.439741 −0.219871 0.975529i $$-0.570564\pi$$
−0.219871 + 0.975529i $$0.570564\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −16768.0 −1.21613
$$576$$ 0 0
$$577$$ 19922.0 1.43737 0.718686 0.695335i $$-0.244744\pi$$
0.718686 + 0.695335i $$0.244744\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −8448.00 −0.603239
$$582$$ 0 0
$$583$$ 31744.0 2.25506
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −3584.00 −0.252006 −0.126003 0.992030i $$-0.540215\pi$$
−0.126003 + 0.992030i $$0.540215\pi$$
$$588$$ 0 0
$$589$$ −2720.00 −0.190281
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 1984.00 0.137391 0.0686957 0.997638i $$-0.478116\pi$$
0.0686957 + 0.997638i $$0.478116\pi$$
$$594$$ 0 0
$$595$$ −6144.00 −0.423327
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 14976.0 1.02154 0.510770 0.859717i $$-0.329360\pi$$
0.510770 + 0.859717i $$0.329360\pi$$
$$600$$ 0 0
$$601$$ 25738.0 1.74688 0.873440 0.486932i $$-0.161884\pi$$
0.873440 + 0.486932i $$0.161884\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 44240.0 2.97291
$$606$$ 0 0
$$607$$ −8548.00 −0.571586 −0.285793 0.958291i $$-0.592257\pi$$
−0.285793 + 0.958291i $$0.592257\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 22272.0 1.47468
$$612$$ 0 0
$$613$$ 8558.00 0.563873 0.281937 0.959433i $$-0.409023\pi$$
0.281937 + 0.959433i $$0.409023\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −10368.0 −0.676499 −0.338250 0.941056i $$-0.609835\pi$$
−0.338250 + 0.941056i $$0.609835\pi$$
$$618$$ 0 0
$$619$$ −13088.0 −0.849840 −0.424920 0.905231i $$-0.639698\pi$$
−0.424920 + 0.905231i $$0.639698\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 11520.0 0.740833
$$624$$ 0 0
$$625$$ −14839.0 −0.949696
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −576.000 −0.0365129
$$630$$ 0 0
$$631$$ −4412.00 −0.278350 −0.139175 0.990268i $$-0.544445\pi$$
−0.139175 + 0.990268i $$0.544445\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 12736.0 0.795926
$$636$$ 0 0
$$637$$ −11542.0 −0.717913
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −30176.0 −1.85941 −0.929704 0.368308i $$-0.879937\pi$$
−0.929704 + 0.368308i $$0.879937\pi$$
$$642$$ 0 0
$$643$$ 21288.0 1.30562 0.652812 0.757520i $$-0.273589\pi$$
0.652812 + 0.757520i $$0.273589\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17024.0 1.03444 0.517220 0.855853i $$-0.326967\pi$$
0.517220 + 0.855853i $$0.326967\pi$$
$$648$$ 0 0
$$649$$ −8192.00 −0.495476
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2256.00 0.135198 0.0675989 0.997713i $$-0.478466\pi$$
0.0675989 + 0.997713i $$0.478466\pi$$
$$654$$ 0 0
$$655$$ 8192.00 0.488684
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 23808.0 1.40733 0.703663 0.710534i $$-0.251546\pi$$
0.703663 + 0.710534i $$0.251546\pi$$
$$660$$ 0 0
$$661$$ −26242.0 −1.54417 −0.772084 0.635520i $$-0.780786\pi$$
−0.772084 + 0.635520i $$0.780786\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 26112.0 1.52268
$$666$$ 0 0
$$667$$ 18432.0 1.07000
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −29312.0 −1.68640
$$672$$ 0 0
$$673$$ −24590.0 −1.40843 −0.704216 0.709986i $$-0.748701\pi$$
−0.704216 + 0.709986i $$0.748701\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −2864.00 −0.162589 −0.0812943 0.996690i $$-0.525905\pi$$
−0.0812943 + 0.996690i $$0.525905\pi$$
$$678$$ 0 0
$$679$$ −2472.00 −0.139715
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −7616.00 −0.426674 −0.213337 0.976979i $$-0.568433\pi$$
−0.213337 + 0.976979i $$0.568433\pi$$
$$684$$ 0 0
$$685$$ 29184.0 1.62783
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 28768.0 1.59067
$$690$$ 0 0
$$691$$ 2168.00 0.119355 0.0596777 0.998218i $$-0.480993\pi$$
0.0596777 + 0.998218i $$0.480993\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −34560.0 −1.88624
$$696$$ 0 0
$$697$$ −9216.00 −0.500833
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −18000.0 −0.969830 −0.484915 0.874561i $$-0.661149\pi$$
−0.484915 + 0.874561i $$0.661149\pi$$
$$702$$ 0 0
$$703$$ 2448.00 0.131334
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −5184.00 −0.275763
$$708$$ 0 0
$$709$$ 3506.00 0.185713 0.0928566 0.995679i $$-0.470400\pi$$
0.0928566 + 0.995679i $$0.470400\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −2560.00 −0.134464
$$714$$ 0 0
$$715$$ 59392.0 3.10648
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 15616.0 0.809984 0.404992 0.914320i $$-0.367274\pi$$
0.404992 + 0.914320i $$0.367274\pi$$
$$720$$ 0 0
$$721$$ 816.000 0.0421490
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −18864.0 −0.966333
$$726$$ 0 0
$$727$$ −15036.0 −0.767062 −0.383531 0.923528i $$-0.625292\pi$$
−0.383531 + 0.923528i $$0.625292\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −6400.00 −0.323820
$$732$$ 0 0
$$733$$ −19126.0 −0.963758 −0.481879 0.876238i $$-0.660046\pi$$
−0.481879 + 0.876238i $$0.660046\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −31744.0 −1.58657
$$738$$ 0 0
$$739$$ −17392.0 −0.865731 −0.432865 0.901459i $$-0.642497\pi$$
−0.432865 + 0.901459i $$0.642497\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −32384.0 −1.59900 −0.799498 0.600669i $$-0.794901\pi$$
−0.799498 + 0.600669i $$0.794901\pi$$
$$744$$ 0 0
$$745$$ −11008.0 −0.541345
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 4608.00 0.224797
$$750$$ 0 0
$$751$$ −27708.0 −1.34631 −0.673155 0.739501i $$-0.735061\pi$$
−0.673155 + 0.739501i $$0.735061\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −13504.0 −0.650942
$$756$$ 0 0
$$757$$ −37246.0 −1.78828 −0.894141 0.447786i $$-0.852213\pi$$
−0.894141 + 0.447786i $$0.852213\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −4192.00 −0.199684 −0.0998422 0.995003i $$-0.531834\pi$$
−0.0998422 + 0.995003i $$0.531834\pi$$
$$762$$ 0 0
$$763$$ 6216.00 0.294934
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −7424.00 −0.349498
$$768$$ 0 0
$$769$$ 26882.0 1.26058 0.630292 0.776358i $$-0.282935\pi$$
0.630292 + 0.776358i $$0.282935\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −17232.0 −0.801801 −0.400900 0.916122i $$-0.631303\pi$$
−0.400900 + 0.916122i $$0.631303\pi$$
$$774$$ 0 0
$$775$$ 2620.00 0.121436
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 39168.0 1.80146
$$780$$ 0 0
$$781$$ 32768.0 1.50132
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1888.00 0.0858415
$$786$$ 0 0
$$787$$ 31816.0 1.44106 0.720532 0.693421i $$-0.243898\pi$$
0.720532 + 0.693421i $$0.243898\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 11520.0 0.517831
$$792$$ 0 0
$$793$$ −26564.0 −1.18955
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 8272.00 0.367640 0.183820 0.982960i $$-0.441154\pi$$
0.183820 + 0.982960i $$0.441154\pi$$
$$798$$ 0 0
$$799$$ 12288.0 0.544078
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −38528.0 −1.69318
$$804$$ 0 0
$$805$$ 24576.0 1.07601
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −9184.00 −0.399125 −0.199563 0.979885i $$-0.563952\pi$$
−0.199563 + 0.979885i $$0.563952\pi$$
$$810$$ 0 0
$$811$$ −19832.0 −0.858688 −0.429344 0.903141i $$-0.641255\pi$$
−0.429344 + 0.903141i $$0.641255\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 57216.0 2.45913
$$816$$ 0 0
$$817$$ 27200.0 1.16476
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 15216.0 0.646823 0.323412 0.946258i $$-0.395170\pi$$
0.323412 + 0.946258i $$0.395170\pi$$
$$822$$ 0 0
$$823$$ −39772.0 −1.68453 −0.842263 0.539067i $$-0.818777\pi$$
−0.842263 + 0.539067i $$0.818777\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −18304.0 −0.769640 −0.384820 0.922992i $$-0.625737\pi$$
−0.384820 + 0.922992i $$0.625737\pi$$
$$828$$ 0 0
$$829$$ 4906.00 0.205540 0.102770 0.994705i $$-0.467229\pi$$
0.102770 + 0.994705i $$0.467229\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −6368.00 −0.264872
$$834$$ 0 0
$$835$$ 6144.00 0.254637
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 15360.0 0.632045 0.316023 0.948752i $$-0.397652\pi$$
0.316023 + 0.948752i $$0.397652\pi$$
$$840$$ 0 0
$$841$$ −3653.00 −0.149781
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 18672.0 0.760161
$$846$$ 0 0
$$847$$ −33180.0 −1.34602
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 2304.00 0.0928086
$$852$$ 0 0
$$853$$ −24802.0 −0.995550 −0.497775 0.867306i $$-0.665850\pi$$
−0.497775 + 0.867306i $$0.665850\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 15072.0 0.600758 0.300379 0.953820i $$-0.402887\pi$$
0.300379 + 0.953820i $$0.402887\pi$$
$$858$$ 0 0
$$859$$ 1800.00 0.0714962 0.0357481 0.999361i $$-0.488619\pi$$
0.0357481 + 0.999361i $$0.488619\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 7552.00 0.297883 0.148942 0.988846i $$-0.452413\pi$$
0.148942 + 0.988846i $$0.452413\pi$$
$$864$$ 0 0
$$865$$ 39168.0 1.53960
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 70912.0 2.76815
$$870$$ 0 0
$$871$$ −28768.0 −1.11913
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1152.00 −0.0445082
$$876$$ 0 0
$$877$$ 20838.0 0.802337 0.401168 0.916004i $$-0.368604\pi$$
0.401168 + 0.916004i $$0.368604\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 47744.0 1.82581 0.912904 0.408175i $$-0.133835\pi$$
0.912904 + 0.408175i $$0.133835\pi$$
$$882$$ 0 0
$$883$$ 28280.0 1.07780 0.538900 0.842370i $$-0.318840\pi$$
0.538900 + 0.842370i $$0.318840\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 7424.00 0.281030 0.140515 0.990079i $$-0.455124\pi$$
0.140515 + 0.990079i $$0.455124\pi$$
$$888$$ 0 0
$$889$$ −9552.00 −0.360364
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −52224.0 −1.95701
$$894$$ 0 0
$$895$$ −67584.0 −2.52412
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −2880.00 −0.106845
$$900$$ 0 0
$$901$$ 15872.0 0.586873
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −8160.00 −0.299721
$$906$$ 0 0
$$907$$ −5912.00 −0.216433 −0.108217 0.994127i $$-0.534514\pi$$
−0.108217 + 0.994127i $$0.534514\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −26240.0 −0.954303 −0.477151 0.878821i $$-0.658331\pi$$
−0.477151 + 0.878821i $$0.658331\pi$$
$$912$$ 0 0
$$913$$ 45056.0 1.63323
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −6144.00 −0.221257
$$918$$ 0 0
$$919$$ 35620.0 1.27856 0.639279 0.768975i $$-0.279233\pi$$
0.639279 + 0.768975i $$0.279233\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 29696.0 1.05900
$$924$$ 0 0
$$925$$ −2358.00 −0.0838168
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −3232.00 −0.114143 −0.0570713 0.998370i $$-0.518176\pi$$
−0.0570713 + 0.998370i $$0.518176\pi$$
$$930$$ 0 0
$$931$$ 27064.0 0.952725
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 32768.0 1.14613
$$936$$ 0 0
$$937$$ −11478.0 −0.400181 −0.200091 0.979777i $$-0.564124\pi$$
−0.200091 + 0.979777i $$0.564124\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −39984.0 −1.38517 −0.692583 0.721338i $$-0.743527\pi$$
−0.692583 + 0.721338i $$0.743527\pi$$
$$942$$ 0 0
$$943$$ 36864.0 1.27302
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 24192.0 0.830131 0.415066 0.909791i $$-0.363759\pi$$
0.415066 + 0.909791i $$0.363759\pi$$
$$948$$ 0 0
$$949$$ −34916.0 −1.19433
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −39456.0 −1.34114 −0.670569 0.741847i $$-0.733950\pi$$
−0.670569 + 0.741847i $$0.733950\pi$$
$$954$$ 0 0
$$955$$ −6144.00 −0.208183
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −21888.0 −0.737018
$$960$$ 0 0
$$961$$ −29391.0 −0.986573
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −55264.0 −1.84353
$$966$$ 0 0
$$967$$ −41668.0 −1.38568 −0.692840 0.721091i $$-0.743641\pi$$
−0.692840 + 0.721091i $$0.743641\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −51648.0 −1.70697 −0.853483 0.521121i $$-0.825514\pi$$
−0.853483 + 0.521121i $$0.825514\pi$$
$$972$$ 0 0
$$973$$ 25920.0 0.854015
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 55776.0 1.82644 0.913220 0.407466i $$-0.133588\pi$$
0.913220 + 0.407466i $$0.133588\pi$$
$$978$$ 0 0
$$979$$ −61440.0 −2.00575
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −36096.0 −1.17119 −0.585597 0.810602i $$-0.699140\pi$$
−0.585597 + 0.810602i $$0.699140\pi$$
$$984$$ 0 0
$$985$$ −51456.0 −1.66449
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 25600.0 0.823087
$$990$$ 0 0
$$991$$ 42532.0 1.36334 0.681672 0.731658i $$-0.261253\pi$$
0.681672 + 0.731658i $$0.261253\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 27328.0 0.870709
$$996$$ 0 0
$$997$$ 29806.0 0.946806 0.473403 0.880846i $$-0.343025\pi$$
0.473403 + 0.880846i $$0.343025\pi$$
$$998$$ 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 72.4.a.d.1.1 yes 1
3.2 odd 2 72.4.a.a.1.1 1
4.3 odd 2 144.4.a.f.1.1 1
5.2 odd 4 1800.4.f.x.649.1 2
5.3 odd 4 1800.4.f.x.649.2 2
5.4 even 2 1800.4.a.ba.1.1 1
8.3 odd 2 576.4.a.d.1.1 1
8.5 even 2 576.4.a.c.1.1 1
9.2 odd 6 648.4.i.l.433.1 2
9.4 even 3 648.4.i.a.217.1 2
9.5 odd 6 648.4.i.l.217.1 2
9.7 even 3 648.4.i.a.433.1 2
12.11 even 2 144.4.a.a.1.1 1
15.2 even 4 1800.4.f.b.649.1 2
15.8 even 4 1800.4.f.b.649.2 2
15.14 odd 2 1800.4.a.z.1.1 1
24.5 odd 2 576.4.a.w.1.1 1
24.11 even 2 576.4.a.x.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.a.a.1.1 1 3.2 odd 2
72.4.a.d.1.1 yes 1 1.1 even 1 trivial
144.4.a.a.1.1 1 12.11 even 2
144.4.a.f.1.1 1 4.3 odd 2
576.4.a.c.1.1 1 8.5 even 2
576.4.a.d.1.1 1 8.3 odd 2
576.4.a.w.1.1 1 24.5 odd 2
576.4.a.x.1.1 1 24.11 even 2
648.4.i.a.217.1 2 9.4 even 3
648.4.i.a.433.1 2 9.7 even 3
648.4.i.l.217.1 2 9.5 odd 6
648.4.i.l.433.1 2 9.2 odd 6
1800.4.a.z.1.1 1 15.14 odd 2
1800.4.a.ba.1.1 1 5.4 even 2
1800.4.f.b.649.1 2 15.2 even 4
1800.4.f.b.649.2 2 15.8 even 4
1800.4.f.x.649.1 2 5.2 odd 4
1800.4.f.x.649.2 2 5.3 odd 4