# Properties

 Label 720.4.a.r.1.1 Level $720$ Weight $4$ Character 720.1 Self dual yes Analytic conductor $42.481$ 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] = [720,4,Mod(1,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("720.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 720.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: $$42.4813752041$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 720.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{5} -20.0000 q^{7} +O(q^{10})$$ $$q+5.00000 q^{5} -20.0000 q^{7} -24.0000 q^{11} +74.0000 q^{13} -54.0000 q^{17} +124.000 q^{19} -120.000 q^{23} +25.0000 q^{25} +78.0000 q^{29} -200.000 q^{31} -100.000 q^{35} -70.0000 q^{37} -330.000 q^{41} -92.0000 q^{43} -24.0000 q^{47} +57.0000 q^{49} -450.000 q^{53} -120.000 q^{55} +24.0000 q^{59} -322.000 q^{61} +370.000 q^{65} +196.000 q^{67} -288.000 q^{71} -430.000 q^{73} +480.000 q^{77} +520.000 q^{79} +156.000 q^{83} -270.000 q^{85} -1026.00 q^{89} -1480.00 q^{91} +620.000 q^{95} -286.000 q^{97} +O(q^{100})$$

## 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
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −20.0000 −1.07990 −0.539949 0.841698i $$-0.681557\pi$$
−0.539949 + 0.841698i $$0.681557\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −24.0000 −0.657843 −0.328921 0.944357i $$-0.606685\pi$$
−0.328921 + 0.944357i $$0.606685\pi$$
$$12$$ 0 0
$$13$$ 74.0000 1.57876 0.789381 0.613904i $$-0.210402\pi$$
0.789381 + 0.613904i $$0.210402\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −54.0000 −0.770407 −0.385204 0.922832i $$-0.625869\pi$$
−0.385204 + 0.922832i $$0.625869\pi$$
$$18$$ 0 0
$$19$$ 124.000 1.49724 0.748620 0.663000i $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −120.000 −1.08790 −0.543951 0.839117i $$-0.683072\pi$$
−0.543951 + 0.839117i $$0.683072\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 78.0000 0.499456 0.249728 0.968316i $$-0.419659\pi$$
0.249728 + 0.968316i $$0.419659\pi$$
$$30$$ 0 0
$$31$$ −200.000 −1.15874 −0.579372 0.815063i $$-0.696702\pi$$
−0.579372 + 0.815063i $$0.696702\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −100.000 −0.482945
$$36$$ 0 0
$$37$$ −70.0000 −0.311025 −0.155513 0.987834i $$-0.549703\pi$$
−0.155513 + 0.987834i $$0.549703\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −330.000 −1.25701 −0.628504 0.777806i $$-0.716332\pi$$
−0.628504 + 0.777806i $$0.716332\pi$$
$$42$$ 0 0
$$43$$ −92.0000 −0.326276 −0.163138 0.986603i $$-0.552162\pi$$
−0.163138 + 0.986603i $$0.552162\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −24.0000 −0.0744843 −0.0372421 0.999306i $$-0.511857\pi$$
−0.0372421 + 0.999306i $$0.511857\pi$$
$$48$$ 0 0
$$49$$ 57.0000 0.166181
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −450.000 −1.16627 −0.583134 0.812376i $$-0.698174\pi$$
−0.583134 + 0.812376i $$0.698174\pi$$
$$54$$ 0 0
$$55$$ −120.000 −0.294196
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 24.0000 0.0529582 0.0264791 0.999649i $$-0.491570\pi$$
0.0264791 + 0.999649i $$0.491570\pi$$
$$60$$ 0 0
$$61$$ −322.000 −0.675867 −0.337933 0.941170i $$-0.609728\pi$$
−0.337933 + 0.941170i $$0.609728\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 370.000 0.706044
$$66$$ 0 0
$$67$$ 196.000 0.357391 0.178696 0.983904i $$-0.442812\pi$$
0.178696 + 0.983904i $$0.442812\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −288.000 −0.481399 −0.240699 0.970600i $$-0.577377\pi$$
−0.240699 + 0.970600i $$0.577377\pi$$
$$72$$ 0 0
$$73$$ −430.000 −0.689420 −0.344710 0.938709i $$-0.612023\pi$$
−0.344710 + 0.938709i $$0.612023\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 480.000 0.710404
$$78$$ 0 0
$$79$$ 520.000 0.740564 0.370282 0.928919i $$-0.379261\pi$$
0.370282 + 0.928919i $$0.379261\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 156.000 0.206304 0.103152 0.994666i $$-0.467107\pi$$
0.103152 + 0.994666i $$0.467107\pi$$
$$84$$ 0 0
$$85$$ −270.000 −0.344537
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1026.00 −1.22198 −0.610988 0.791640i $$-0.709227\pi$$
−0.610988 + 0.791640i $$0.709227\pi$$
$$90$$ 0 0
$$91$$ −1480.00 −1.70490
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 620.000 0.669586
$$96$$ 0 0
$$97$$ −286.000 −0.299370 −0.149685 0.988734i $$-0.547826\pi$$
−0.149685 + 0.988734i $$0.547826\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1734.00 1.70831 0.854156 0.520017i $$-0.174075\pi$$
0.854156 + 0.520017i $$0.174075\pi$$
$$102$$ 0 0
$$103$$ −452.000 −0.432397 −0.216198 0.976349i $$-0.569366\pi$$
−0.216198 + 0.976349i $$0.569366\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1404.00 −1.26850 −0.634251 0.773127i $$-0.718692\pi$$
−0.634251 + 0.773127i $$0.718692\pi$$
$$108$$ 0 0
$$109$$ −1474.00 −1.29526 −0.647631 0.761954i $$-0.724240\pi$$
−0.647631 + 0.761954i $$0.724240\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1086.00 −0.904091 −0.452046 0.891995i $$-0.649306\pi$$
−0.452046 + 0.891995i $$0.649306\pi$$
$$114$$ 0 0
$$115$$ −600.000 −0.486524
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1080.00 0.831962
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −1244.00 −0.869190 −0.434595 0.900626i $$-0.643109\pi$$
−0.434595 + 0.900626i $$0.643109\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2328.00 1.55266 0.776329 0.630327i $$-0.217079\pi$$
0.776329 + 0.630327i $$0.217079\pi$$
$$132$$ 0 0
$$133$$ −2480.00 −1.61687
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2118.00 −1.32082 −0.660412 0.750903i $$-0.729618\pi$$
−0.660412 + 0.750903i $$0.729618\pi$$
$$138$$ 0 0
$$139$$ −2324.00 −1.41812 −0.709062 0.705147i $$-0.750881\pi$$
−0.709062 + 0.705147i $$0.750881\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −1776.00 −1.03858
$$144$$ 0 0
$$145$$ 390.000 0.223364
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −258.000 −0.141854 −0.0709268 0.997482i $$-0.522596\pi$$
−0.0709268 + 0.997482i $$0.522596\pi$$
$$150$$ 0 0
$$151$$ 808.000 0.435458 0.217729 0.976009i $$-0.430135\pi$$
0.217729 + 0.976009i $$0.430135\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1000.00 −0.518206
$$156$$ 0 0
$$157$$ 2378.00 1.20882 0.604411 0.796673i $$-0.293408\pi$$
0.604411 + 0.796673i $$0.293408\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 2400.00 1.17482
$$162$$ 0 0
$$163$$ 52.0000 0.0249874 0.0124937 0.999922i $$-0.496023\pi$$
0.0124937 + 0.999922i $$0.496023\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −3720.00 −1.72373 −0.861863 0.507141i $$-0.830702\pi$$
−0.861863 + 0.507141i $$0.830702\pi$$
$$168$$ 0 0
$$169$$ 3279.00 1.49249
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −426.000 −0.187215 −0.0936075 0.995609i $$-0.529840\pi$$
−0.0936075 + 0.995609i $$0.529840\pi$$
$$174$$ 0 0
$$175$$ −500.000 −0.215980
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −1440.00 −0.601289 −0.300644 0.953736i $$-0.597202\pi$$
−0.300644 + 0.953736i $$0.597202\pi$$
$$180$$ 0 0
$$181$$ −3130.00 −1.28537 −0.642683 0.766133i $$-0.722179\pi$$
−0.642683 + 0.766133i $$0.722179\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −350.000 −0.139095
$$186$$ 0 0
$$187$$ 1296.00 0.506807
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3576.00 1.35471 0.677357 0.735655i $$-0.263125\pi$$
0.677357 + 0.735655i $$0.263125\pi$$
$$192$$ 0 0
$$193$$ 2666.00 0.994315 0.497158 0.867660i $$-0.334377\pi$$
0.497158 + 0.867660i $$0.334377\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2718.00 0.982992 0.491496 0.870880i $$-0.336450\pi$$
0.491496 + 0.870880i $$0.336450\pi$$
$$198$$ 0 0
$$199$$ 3832.00 1.36504 0.682521 0.730866i $$-0.260884\pi$$
0.682521 + 0.730866i $$0.260884\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −1560.00 −0.539362
$$204$$ 0 0
$$205$$ −1650.00 −0.562151
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2976.00 −0.984948
$$210$$ 0 0
$$211$$ −1100.00 −0.358896 −0.179448 0.983767i $$-0.557431\pi$$
−0.179448 + 0.983767i $$0.557431\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −460.000 −0.145915
$$216$$ 0 0
$$217$$ 4000.00 1.25133
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −3996.00 −1.21629
$$222$$ 0 0
$$223$$ −1964.00 −0.589772 −0.294886 0.955532i $$-0.595282\pi$$
−0.294886 + 0.955532i $$0.595282\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 660.000 0.192977 0.0964884 0.995334i $$-0.469239\pi$$
0.0964884 + 0.995334i $$0.469239\pi$$
$$228$$ 0 0
$$229$$ −1906.00 −0.550009 −0.275004 0.961443i $$-0.588679\pi$$
−0.275004 + 0.961443i $$0.588679\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1458.00 0.409943 0.204972 0.978768i $$-0.434290\pi$$
0.204972 + 0.978768i $$0.434290\pi$$
$$234$$ 0 0
$$235$$ −120.000 −0.0333104
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1176.00 0.318281 0.159140 0.987256i $$-0.449128\pi$$
0.159140 + 0.987256i $$0.449128\pi$$
$$240$$ 0 0
$$241$$ 866.000 0.231469 0.115734 0.993280i $$-0.463078\pi$$
0.115734 + 0.993280i $$0.463078\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 285.000 0.0743183
$$246$$ 0 0
$$247$$ 9176.00 2.36379
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 432.000 0.108636 0.0543179 0.998524i $$-0.482702\pi$$
0.0543179 + 0.998524i $$0.482702\pi$$
$$252$$ 0 0
$$253$$ 2880.00 0.715668
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −2526.00 −0.613103 −0.306552 0.951854i $$-0.599175\pi$$
−0.306552 + 0.951854i $$0.599175\pi$$
$$258$$ 0 0
$$259$$ 1400.00 0.335876
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 5448.00 1.27733 0.638666 0.769484i $$-0.279487\pi$$
0.638666 + 0.769484i $$0.279487\pi$$
$$264$$ 0 0
$$265$$ −2250.00 −0.521571
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 2574.00 0.583418 0.291709 0.956507i $$-0.405776\pi$$
0.291709 + 0.956507i $$0.405776\pi$$
$$270$$ 0 0
$$271$$ 3184.00 0.713706 0.356853 0.934161i $$-0.383850\pi$$
0.356853 + 0.934161i $$0.383850\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −600.000 −0.131569
$$276$$ 0 0
$$277$$ 3962.00 0.859399 0.429699 0.902972i $$-0.358620\pi$$
0.429699 + 0.902972i $$0.358620\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8286.00 1.75908 0.879540 0.475825i $$-0.157851\pi$$
0.879540 + 0.475825i $$0.157851\pi$$
$$282$$ 0 0
$$283$$ 2716.00 0.570493 0.285246 0.958454i $$-0.407925\pi$$
0.285246 + 0.958454i $$0.407925\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6600.00 1.35744
$$288$$ 0 0
$$289$$ −1997.00 −0.406473
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6018.00 −1.19992 −0.599958 0.800032i $$-0.704816\pi$$
−0.599958 + 0.800032i $$0.704816\pi$$
$$294$$ 0 0
$$295$$ 120.000 0.0236836
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −8880.00 −1.71754
$$300$$ 0 0
$$301$$ 1840.00 0.352345
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −1610.00 −0.302257
$$306$$ 0 0
$$307$$ −9236.00 −1.71702 −0.858512 0.512793i $$-0.828611\pi$$
−0.858512 + 0.512793i $$0.828611\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 1536.00 0.280060 0.140030 0.990147i $$-0.455280\pi$$
0.140030 + 0.990147i $$0.455280\pi$$
$$312$$ 0 0
$$313$$ −7342.00 −1.32586 −0.662930 0.748681i $$-0.730687\pi$$
−0.662930 + 0.748681i $$0.730687\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3894.00 0.689933 0.344967 0.938615i $$-0.387890\pi$$
0.344967 + 0.938615i $$0.387890\pi$$
$$318$$ 0 0
$$319$$ −1872.00 −0.328564
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6696.00 −1.15348
$$324$$ 0 0
$$325$$ 1850.00 0.315752
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 480.000 0.0804354
$$330$$ 0 0
$$331$$ −3692.00 −0.613084 −0.306542 0.951857i $$-0.599172\pi$$
−0.306542 + 0.951857i $$0.599172\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 980.000 0.159830
$$336$$ 0 0
$$337$$ −8998.00 −1.45446 −0.727229 0.686395i $$-0.759192\pi$$
−0.727229 + 0.686395i $$0.759192\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 4800.00 0.762271
$$342$$ 0 0
$$343$$ 5720.00 0.900440
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 5244.00 0.811276 0.405638 0.914034i $$-0.367049\pi$$
0.405638 + 0.914034i $$0.367049\pi$$
$$348$$ 0 0
$$349$$ 6302.00 0.966585 0.483293 0.875459i $$-0.339441\pi$$
0.483293 + 0.875459i $$0.339441\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −3414.00 −0.514756 −0.257378 0.966311i $$-0.582859\pi$$
−0.257378 + 0.966311i $$0.582859\pi$$
$$354$$ 0 0
$$355$$ −1440.00 −0.215288
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 4824.00 0.709195 0.354597 0.935019i $$-0.384618\pi$$
0.354597 + 0.935019i $$0.384618\pi$$
$$360$$ 0 0
$$361$$ 8517.00 1.24173
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −2150.00 −0.308318
$$366$$ 0 0
$$367$$ 3508.00 0.498954 0.249477 0.968381i $$-0.419741\pi$$
0.249477 + 0.968381i $$0.419741\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 9000.00 1.25945
$$372$$ 0 0
$$373$$ 10802.0 1.49948 0.749740 0.661732i $$-0.230178\pi$$
0.749740 + 0.661732i $$0.230178\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 5772.00 0.788523
$$378$$ 0 0
$$379$$ −1460.00 −0.197876 −0.0989382 0.995094i $$-0.531545\pi$$
−0.0989382 + 0.995094i $$0.531545\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −4872.00 −0.649994 −0.324997 0.945715i $$-0.605363\pi$$
−0.324997 + 0.945715i $$0.605363\pi$$
$$384$$ 0 0
$$385$$ 2400.00 0.317702
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 14046.0 1.83075 0.915373 0.402606i $$-0.131896\pi$$
0.915373 + 0.402606i $$0.131896\pi$$
$$390$$ 0 0
$$391$$ 6480.00 0.838127
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 2600.00 0.331190
$$396$$ 0 0
$$397$$ −2734.00 −0.345631 −0.172816 0.984954i $$-0.555286\pi$$
−0.172816 + 0.984954i $$0.555286\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 15942.0 1.98530 0.992650 0.121019i $$-0.0386161\pi$$
0.992650 + 0.121019i $$0.0386161\pi$$
$$402$$ 0 0
$$403$$ −14800.0 −1.82938
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1680.00 0.204606
$$408$$ 0 0
$$409$$ 8714.00 1.05350 0.526748 0.850022i $$-0.323411\pi$$
0.526748 + 0.850022i $$0.323411\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −480.000 −0.0571895
$$414$$ 0 0
$$415$$ 780.000 0.0922619
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 11976.0 1.39634 0.698169 0.715933i $$-0.253998\pi$$
0.698169 + 0.715933i $$0.253998\pi$$
$$420$$ 0 0
$$421$$ 11054.0 1.27967 0.639833 0.768514i $$-0.279004\pi$$
0.639833 + 0.768514i $$0.279004\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −1350.00 −0.154081
$$426$$ 0 0
$$427$$ 6440.00 0.729868
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 720.000 0.0804668 0.0402334 0.999190i $$-0.487190\pi$$
0.0402334 + 0.999190i $$0.487190\pi$$
$$432$$ 0 0
$$433$$ −15622.0 −1.73382 −0.866912 0.498462i $$-0.833898\pi$$
−0.866912 + 0.498462i $$0.833898\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −14880.0 −1.62885
$$438$$ 0 0
$$439$$ 9880.00 1.07414 0.537069 0.843538i $$-0.319531\pi$$
0.537069 + 0.843538i $$0.319531\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −16116.0 −1.72843 −0.864215 0.503123i $$-0.832184\pi$$
−0.864215 + 0.503123i $$0.832184\pi$$
$$444$$ 0 0
$$445$$ −5130.00 −0.546484
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −9018.00 −0.947852 −0.473926 0.880565i $$-0.657164\pi$$
−0.473926 + 0.880565i $$0.657164\pi$$
$$450$$ 0 0
$$451$$ 7920.00 0.826914
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −7400.00 −0.762456
$$456$$ 0 0
$$457$$ −3670.00 −0.375657 −0.187829 0.982202i $$-0.560145\pi$$
−0.187829 + 0.982202i $$0.560145\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −17562.0 −1.77428 −0.887141 0.461499i $$-0.847312\pi$$
−0.887141 + 0.461499i $$0.847312\pi$$
$$462$$ 0 0
$$463$$ −1172.00 −0.117640 −0.0588202 0.998269i $$-0.518734\pi$$
−0.0588202 + 0.998269i $$0.518734\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6876.00 0.681335 0.340667 0.940184i $$-0.389347\pi$$
0.340667 + 0.940184i $$0.389347\pi$$
$$468$$ 0 0
$$469$$ −3920.00 −0.385946
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 2208.00 0.214638
$$474$$ 0 0
$$475$$ 3100.00 0.299448
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 2280.00 0.217486 0.108743 0.994070i $$-0.465317\pi$$
0.108743 + 0.994070i $$0.465317\pi$$
$$480$$ 0 0
$$481$$ −5180.00 −0.491035
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −1430.00 −0.133882
$$486$$ 0 0
$$487$$ 3076.00 0.286215 0.143108 0.989707i $$-0.454290\pi$$
0.143108 + 0.989707i $$0.454290\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −18912.0 −1.73826 −0.869131 0.494582i $$-0.835321\pi$$
−0.869131 + 0.494582i $$0.835321\pi$$
$$492$$ 0 0
$$493$$ −4212.00 −0.384785
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 5760.00 0.519862
$$498$$ 0 0
$$499$$ −9956.00 −0.893170 −0.446585 0.894741i $$-0.647360\pi$$
−0.446585 + 0.894741i $$0.647360\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −10656.0 −0.944588 −0.472294 0.881441i $$-0.656574\pi$$
−0.472294 + 0.881441i $$0.656574\pi$$
$$504$$ 0 0
$$505$$ 8670.00 0.763980
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 2766.00 0.240866 0.120433 0.992721i $$-0.461572\pi$$
0.120433 + 0.992721i $$0.461572\pi$$
$$510$$ 0 0
$$511$$ 8600.00 0.744504
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −2260.00 −0.193374
$$516$$ 0 0
$$517$$ 576.000 0.0489989
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10530.0 −0.885466 −0.442733 0.896654i $$-0.645991\pi$$
−0.442733 + 0.896654i $$0.645991\pi$$
$$522$$ 0 0
$$523$$ −12692.0 −1.06115 −0.530576 0.847637i $$-0.678024\pi$$
−0.530576 + 0.847637i $$0.678024\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 10800.0 0.892705
$$528$$ 0 0
$$529$$ 2233.00 0.183529
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −24420.0 −1.98452
$$534$$ 0 0
$$535$$ −7020.00 −0.567292
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1368.00 −0.109321
$$540$$ 0 0
$$541$$ 18110.0 1.43920 0.719602 0.694386i $$-0.244324\pi$$
0.719602 + 0.694386i $$0.244324\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −7370.00 −0.579259
$$546$$ 0 0
$$547$$ −3620.00 −0.282962 −0.141481 0.989941i $$-0.545186\pi$$
−0.141481 + 0.989941i $$0.545186\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 9672.00 0.747806
$$552$$ 0 0
$$553$$ −10400.0 −0.799734
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 14166.0 1.07762 0.538809 0.842428i $$-0.318875\pi$$
0.538809 + 0.842428i $$0.318875\pi$$
$$558$$ 0 0
$$559$$ −6808.00 −0.515112
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −13404.0 −1.00339 −0.501697 0.865043i $$-0.667291\pi$$
−0.501697 + 0.865043i $$0.667291\pi$$
$$564$$ 0 0
$$565$$ −5430.00 −0.404322
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 18654.0 1.37437 0.687185 0.726483i $$-0.258846\pi$$
0.687185 + 0.726483i $$0.258846\pi$$
$$570$$ 0 0
$$571$$ 7684.00 0.563162 0.281581 0.959537i $$-0.409141\pi$$
0.281581 + 0.959537i $$0.409141\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −3000.00 −0.217580
$$576$$ 0 0
$$577$$ −1726.00 −0.124531 −0.0622654 0.998060i $$-0.519833\pi$$
−0.0622654 + 0.998060i $$0.519833\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3120.00 −0.222787
$$582$$ 0 0
$$583$$ 10800.0 0.767222
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 10596.0 0.745049 0.372524 0.928022i $$-0.378492\pi$$
0.372524 + 0.928022i $$0.378492\pi$$
$$588$$ 0 0
$$589$$ −24800.0 −1.73492
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −2862.00 −0.198193 −0.0990963 0.995078i $$-0.531595\pi$$
−0.0990963 + 0.995078i $$0.531595\pi$$
$$594$$ 0 0
$$595$$ 5400.00 0.372065
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −23592.0 −1.60925 −0.804627 0.593781i $$-0.797635\pi$$
−0.804627 + 0.593781i $$0.797635\pi$$
$$600$$ 0 0
$$601$$ −9574.00 −0.649803 −0.324902 0.945748i $$-0.605331\pi$$
−0.324902 + 0.945748i $$0.605331\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −3775.00 −0.253679
$$606$$ 0 0
$$607$$ −17444.0 −1.16644 −0.583221 0.812314i $$-0.698208\pi$$
−0.583221 + 0.812314i $$0.698208\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1776.00 −0.117593
$$612$$ 0 0
$$613$$ −2374.00 −0.156419 −0.0782096 0.996937i $$-0.524920\pi$$
−0.0782096 + 0.996937i $$0.524920\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12162.0 0.793555 0.396778 0.917915i $$-0.370128\pi$$
0.396778 + 0.917915i $$0.370128\pi$$
$$618$$ 0 0
$$619$$ −8804.00 −0.571668 −0.285834 0.958279i $$-0.592271\pi$$
−0.285834 + 0.958279i $$0.592271\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 20520.0 1.31961
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 3780.00 0.239616
$$630$$ 0 0
$$631$$ 12688.0 0.800478 0.400239 0.916411i $$-0.368927\pi$$
0.400239 + 0.916411i $$0.368927\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −6220.00 −0.388714
$$636$$ 0 0
$$637$$ 4218.00 0.262360
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 9150.00 0.563812 0.281906 0.959442i $$-0.409033\pi$$
0.281906 + 0.959442i $$0.409033\pi$$
$$642$$ 0 0
$$643$$ −25292.0 −1.55120 −0.775598 0.631227i $$-0.782552\pi$$
−0.775598 + 0.631227i $$0.782552\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −2736.00 −0.166249 −0.0831246 0.996539i $$-0.526490\pi$$
−0.0831246 + 0.996539i $$0.526490\pi$$
$$648$$ 0 0
$$649$$ −576.000 −0.0348382
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −22218.0 −1.33148 −0.665741 0.746183i $$-0.731884\pi$$
−0.665741 + 0.746183i $$0.731884\pi$$
$$654$$ 0 0
$$655$$ 11640.0 0.694370
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 14520.0 0.858299 0.429149 0.903234i $$-0.358813\pi$$
0.429149 + 0.903234i $$0.358813\pi$$
$$660$$ 0 0
$$661$$ −10618.0 −0.624799 −0.312400 0.949951i $$-0.601133\pi$$
−0.312400 + 0.949951i $$0.601133\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −12400.0 −0.723085
$$666$$ 0 0
$$667$$ −9360.00 −0.543359
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 7728.00 0.444614
$$672$$ 0 0
$$673$$ 1370.00 0.0784690 0.0392345 0.999230i $$-0.487508\pi$$
0.0392345 + 0.999230i $$0.487508\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 13758.0 0.781038 0.390519 0.920595i $$-0.372296\pi$$
0.390519 + 0.920595i $$0.372296\pi$$
$$678$$ 0 0
$$679$$ 5720.00 0.323289
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 11988.0 0.671608 0.335804 0.941932i $$-0.390992\pi$$
0.335804 + 0.941932i $$0.390992\pi$$
$$684$$ 0 0
$$685$$ −10590.0 −0.590691
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −33300.0 −1.84126
$$690$$ 0 0
$$691$$ −32996.0 −1.81654 −0.908268 0.418388i $$-0.862595\pi$$
−0.908268 + 0.418388i $$0.862595\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −11620.0 −0.634204
$$696$$ 0 0
$$697$$ 17820.0 0.968408
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 25902.0 1.39558 0.697792 0.716300i $$-0.254166\pi$$
0.697792 + 0.716300i $$0.254166\pi$$
$$702$$ 0 0
$$703$$ −8680.00 −0.465679
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −34680.0 −1.84480
$$708$$ 0 0
$$709$$ −27394.0 −1.45106 −0.725531 0.688189i $$-0.758406\pi$$
−0.725531 + 0.688189i $$0.758406\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 24000.0 1.26060
$$714$$ 0 0
$$715$$ −8880.00 −0.464466
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 34848.0 1.80753 0.903763 0.428033i $$-0.140793\pi$$
0.903763 + 0.428033i $$0.140793\pi$$
$$720$$ 0 0
$$721$$ 9040.00 0.466945
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1950.00 0.0998913
$$726$$ 0 0
$$727$$ −28028.0 −1.42985 −0.714925 0.699201i $$-0.753539\pi$$
−0.714925 + 0.699201i $$0.753539\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 4968.00 0.251365
$$732$$ 0 0
$$733$$ 18002.0 0.907120 0.453560 0.891226i $$-0.350154\pi$$
0.453560 + 0.891226i $$0.350154\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −4704.00 −0.235107
$$738$$ 0 0
$$739$$ −15284.0 −0.760800 −0.380400 0.924822i $$-0.624214\pi$$
−0.380400 + 0.924822i $$0.624214\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −18768.0 −0.926691 −0.463345 0.886178i $$-0.653351\pi$$
−0.463345 + 0.886178i $$0.653351\pi$$
$$744$$ 0 0
$$745$$ −1290.00 −0.0634388
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 28080.0 1.36985
$$750$$ 0 0
$$751$$ −8696.00 −0.422532 −0.211266 0.977429i $$-0.567759\pi$$
−0.211266 + 0.977429i $$0.567759\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 4040.00 0.194743
$$756$$ 0 0
$$757$$ −38662.0 −1.85627 −0.928134 0.372247i $$-0.878587\pi$$
−0.928134 + 0.372247i $$0.878587\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −23874.0 −1.13723 −0.568615 0.822604i $$-0.692521\pi$$
−0.568615 + 0.822604i $$0.692521\pi$$
$$762$$ 0 0
$$763$$ 29480.0 1.39875
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1776.00 0.0836084
$$768$$ 0 0
$$769$$ 23618.0 1.10753 0.553763 0.832675i $$-0.313192\pi$$
0.553763 + 0.832675i $$0.313192\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −11538.0 −0.536860 −0.268430 0.963299i $$-0.586505\pi$$
−0.268430 + 0.963299i $$0.586505\pi$$
$$774$$ 0 0
$$775$$ −5000.00 −0.231749
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −40920.0 −1.88204
$$780$$ 0 0
$$781$$ 6912.00 0.316685
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 11890.0 0.540602
$$786$$ 0 0
$$787$$ 14884.0 0.674152 0.337076 0.941478i $$-0.390562\pi$$
0.337076 + 0.941478i $$0.390562\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 21720.0 0.976327
$$792$$ 0 0
$$793$$ −23828.0 −1.06703
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 11334.0 0.503728 0.251864 0.967763i $$-0.418957\pi$$
0.251864 + 0.967763i $$0.418957\pi$$
$$798$$ 0 0
$$799$$ 1296.00 0.0573832
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 10320.0 0.453530
$$804$$ 0 0
$$805$$ 12000.0 0.525397
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −44730.0 −1.94391 −0.971955 0.235167i $$-0.924436\pi$$
−0.971955 + 0.235167i $$0.924436\pi$$
$$810$$ 0 0
$$811$$ 42748.0 1.85091 0.925453 0.378862i $$-0.123684\pi$$
0.925453 + 0.378862i $$0.123684\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 260.000 0.0111747
$$816$$ 0 0
$$817$$ −11408.0 −0.488513
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 31686.0 1.34695 0.673477 0.739208i $$-0.264800\pi$$
0.673477 + 0.739208i $$0.264800\pi$$
$$822$$ 0 0
$$823$$ −11036.0 −0.467425 −0.233713 0.972306i $$-0.575087\pi$$
−0.233713 + 0.972306i $$0.575087\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 25884.0 1.08836 0.544181 0.838968i $$-0.316841\pi$$
0.544181 + 0.838968i $$0.316841\pi$$
$$828$$ 0 0
$$829$$ 15950.0 0.668234 0.334117 0.942532i $$-0.391562\pi$$
0.334117 + 0.942532i $$0.391562\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −3078.00 −0.128027
$$834$$ 0 0
$$835$$ −18600.0 −0.770874
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 13800.0 0.567853 0.283927 0.958846i $$-0.408363\pi$$
0.283927 + 0.958846i $$0.408363\pi$$
$$840$$ 0 0
$$841$$ −18305.0 −0.750543
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 16395.0 0.667462
$$846$$ 0 0
$$847$$ 15100.0 0.612565
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8400.00 0.338365
$$852$$ 0 0
$$853$$ −27862.0 −1.11838 −0.559189 0.829040i $$-0.688887\pi$$
−0.559189 + 0.829040i $$0.688887\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 7314.00 0.291530 0.145765 0.989319i $$-0.453436\pi$$
0.145765 + 0.989319i $$0.453436\pi$$
$$858$$ 0 0
$$859$$ 28780.0 1.14314 0.571572 0.820552i $$-0.306334\pi$$
0.571572 + 0.820552i $$0.306334\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −32688.0 −1.28935 −0.644677 0.764455i $$-0.723008\pi$$
−0.644677 + 0.764455i $$0.723008\pi$$
$$864$$ 0 0
$$865$$ −2130.00 −0.0837251
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −12480.0 −0.487175
$$870$$ 0 0
$$871$$ 14504.0 0.564236
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2500.00 −0.0965891
$$876$$ 0 0
$$877$$ 36650.0 1.41115 0.705577 0.708633i $$-0.250688\pi$$
0.705577 + 0.708633i $$0.250688\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 2646.00 0.101187 0.0505936 0.998719i $$-0.483889\pi$$
0.0505936 + 0.998719i $$0.483889\pi$$
$$882$$ 0 0
$$883$$ −10892.0 −0.415113 −0.207557 0.978223i $$-0.566551\pi$$
−0.207557 + 0.978223i $$0.566551\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −43464.0 −1.64530 −0.822648 0.568550i $$-0.807504\pi$$
−0.822648 + 0.568550i $$0.807504\pi$$
$$888$$ 0 0
$$889$$ 24880.0 0.938637
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −2976.00 −0.111521
$$894$$ 0 0
$$895$$ −7200.00 −0.268904
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −15600.0 −0.578742
$$900$$ 0 0
$$901$$ 24300.0 0.898502
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −15650.0 −0.574833
$$906$$ 0 0
$$907$$ 14884.0 0.544890 0.272445 0.962171i $$-0.412168\pi$$
0.272445 + 0.962171i $$0.412168\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −1248.00 −0.0453876 −0.0226938 0.999742i $$-0.507224\pi$$
−0.0226938 + 0.999742i $$0.507224\pi$$
$$912$$ 0 0
$$913$$ −3744.00 −0.135716
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −46560.0 −1.67671
$$918$$ 0 0
$$919$$ 6640.00 0.238339 0.119169 0.992874i $$-0.461977\pi$$
0.119169 + 0.992874i $$0.461977\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −21312.0 −0.760014
$$924$$ 0 0
$$925$$ −1750.00 −0.0622050
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −29946.0 −1.05758 −0.528792 0.848751i $$-0.677355\pi$$
−0.528792 + 0.848751i $$0.677355\pi$$
$$930$$ 0 0
$$931$$ 7068.00 0.248812
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 6480.00 0.226651
$$936$$ 0 0
$$937$$ 45002.0 1.56900 0.784499 0.620130i $$-0.212920\pi$$
0.784499 + 0.620130i $$0.212920\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −6090.00 −0.210976 −0.105488 0.994421i $$-0.533640\pi$$
−0.105488 + 0.994421i $$0.533640\pi$$
$$942$$ 0 0
$$943$$ 39600.0 1.36750
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 56388.0 1.93491 0.967457 0.253035i $$-0.0814288\pi$$
0.967457 + 0.253035i $$0.0814288\pi$$
$$948$$ 0 0
$$949$$ −31820.0 −1.08843
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −10854.0 −0.368936 −0.184468 0.982839i $$-0.559056\pi$$
−0.184468 + 0.982839i $$0.559056\pi$$
$$954$$ 0 0
$$955$$ 17880.0 0.605846
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 42360.0 1.42636
$$960$$ 0 0
$$961$$ 10209.0 0.342687
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 13330.0 0.444671
$$966$$ 0 0
$$967$$ 42316.0 1.40723 0.703615 0.710582i $$-0.251568\pi$$
0.703615 + 0.710582i $$0.251568\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24480.0 0.809063 0.404532 0.914524i $$-0.367435\pi$$
0.404532 + 0.914524i $$0.367435\pi$$
$$972$$ 0 0
$$973$$ 46480.0 1.53143
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 6906.00 0.226144 0.113072 0.993587i $$-0.463931\pi$$
0.113072 + 0.993587i $$0.463931\pi$$
$$978$$ 0 0
$$979$$ 24624.0 0.803868
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 6960.00 0.225829 0.112914 0.993605i $$-0.463981\pi$$
0.112914 + 0.993605i $$0.463981\pi$$
$$984$$ 0 0
$$985$$ 13590.0 0.439608
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 11040.0 0.354956
$$990$$ 0 0
$$991$$ −47792.0 −1.53195 −0.765975 0.642870i $$-0.777744\pi$$
−0.765975 + 0.642870i $$0.777744\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 19160.0 0.610465
$$996$$ 0 0
$$997$$ 9938.00 0.315687 0.157843 0.987464i $$-0.449546\pi$$
0.157843 + 0.987464i $$0.449546\pi$$
$$998$$ 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 720.4.a.r.1.1 1
3.2 odd 2 240.4.a.f.1.1 1
4.3 odd 2 45.4.a.b.1.1 1
12.11 even 2 15.4.a.b.1.1 1
15.2 even 4 1200.4.f.m.49.1 2
15.8 even 4 1200.4.f.m.49.2 2
15.14 odd 2 1200.4.a.o.1.1 1
20.3 even 4 225.4.b.d.199.2 2
20.7 even 4 225.4.b.d.199.1 2
20.19 odd 2 225.4.a.g.1.1 1
24.5 odd 2 960.4.a.l.1.1 1
24.11 even 2 960.4.a.bi.1.1 1
28.27 even 2 2205.4.a.c.1.1 1
36.7 odd 6 405.4.e.k.271.1 2
36.11 even 6 405.4.e.d.271.1 2
36.23 even 6 405.4.e.d.136.1 2
36.31 odd 6 405.4.e.k.136.1 2
60.23 odd 4 75.4.b.a.49.1 2
60.47 odd 4 75.4.b.a.49.2 2
60.59 even 2 75.4.a.a.1.1 1
84.83 odd 2 735.4.a.i.1.1 1
132.131 odd 2 1815.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 12.11 even 2
45.4.a.b.1.1 1 4.3 odd 2
75.4.a.a.1.1 1 60.59 even 2
75.4.b.a.49.1 2 60.23 odd 4
75.4.b.a.49.2 2 60.47 odd 4
225.4.a.g.1.1 1 20.19 odd 2
225.4.b.d.199.1 2 20.7 even 4
225.4.b.d.199.2 2 20.3 even 4
240.4.a.f.1.1 1 3.2 odd 2
405.4.e.d.136.1 2 36.23 even 6
405.4.e.d.271.1 2 36.11 even 6
405.4.e.k.136.1 2 36.31 odd 6
405.4.e.k.271.1 2 36.7 odd 6
720.4.a.r.1.1 1 1.1 even 1 trivial
735.4.a.i.1.1 1 84.83 odd 2
960.4.a.l.1.1 1 24.5 odd 2
960.4.a.bi.1.1 1 24.11 even 2
1200.4.a.o.1.1 1 15.14 odd 2
1200.4.f.m.49.1 2 15.2 even 4
1200.4.f.m.49.2 2 15.8 even 4
1815.4.a.a.1.1 1 132.131 odd 2
2205.4.a.c.1.1 1 28.27 even 2