# Properties

 Label 2200.4.a.j.1.1 Level $2200$ Weight $4$ Character 2200.1 Self dual yes Analytic conductor $129.804$ 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] = [2200,4,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$129.804202013$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{3} +13.0000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+5.00000 q^{3} +13.0000 q^{7} -2.00000 q^{9} -11.0000 q^{11} -12.0000 q^{13} -67.0000 q^{17} +151.000 q^{19} +65.0000 q^{21} +12.0000 q^{23} -145.000 q^{27} -143.000 q^{29} -337.000 q^{31} -55.0000 q^{33} -125.000 q^{37} -60.0000 q^{39} -240.000 q^{41} +270.000 q^{43} +448.000 q^{47} -174.000 q^{49} -335.000 q^{51} -45.0000 q^{53} +755.000 q^{57} +704.000 q^{59} -217.000 q^{61} -26.0000 q^{63} +284.000 q^{67} +60.0000 q^{69} -515.000 q^{71} -1162.00 q^{73} -143.000 q^{77} -944.000 q^{79} -671.000 q^{81} -124.000 q^{83} -715.000 q^{87} +361.000 q^{89} -156.000 q^{91} -1685.00 q^{93} -916.000 q^{97} +22.0000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 5.00000 0.962250 0.481125 0.876652i $$-0.340228\pi$$
0.481125 + 0.876652i $$0.340228\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 13.0000 0.701934 0.350967 0.936388i $$-0.385853\pi$$
0.350967 + 0.936388i $$0.385853\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.0740741
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ −12.0000 −0.256015 −0.128008 0.991773i $$-0.540858\pi$$
−0.128008 + 0.991773i $$0.540858\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −67.0000 −0.955876 −0.477938 0.878394i $$-0.658616\pi$$
−0.477938 + 0.878394i $$0.658616\pi$$
$$18$$ 0 0
$$19$$ 151.000 1.82325 0.911626 0.411021i $$-0.134828\pi$$
0.911626 + 0.411021i $$0.134828\pi$$
$$20$$ 0 0
$$21$$ 65.0000 0.675436
$$22$$ 0 0
$$23$$ 12.0000 0.108790 0.0543951 0.998519i $$-0.482677\pi$$
0.0543951 + 0.998519i $$0.482677\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −145.000 −1.03353
$$28$$ 0 0
$$29$$ −143.000 −0.915670 −0.457835 0.889037i $$-0.651375\pi$$
−0.457835 + 0.889037i $$0.651375\pi$$
$$30$$ 0 0
$$31$$ −337.000 −1.95248 −0.976242 0.216684i $$-0.930476\pi$$
−0.976242 + 0.216684i $$0.930476\pi$$
$$32$$ 0 0
$$33$$ −55.0000 −0.290129
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −125.000 −0.555402 −0.277701 0.960668i $$-0.589572\pi$$
−0.277701 + 0.960668i $$0.589572\pi$$
$$38$$ 0 0
$$39$$ −60.0000 −0.246351
$$40$$ 0 0
$$41$$ −240.000 −0.914188 −0.457094 0.889418i $$-0.651110\pi$$
−0.457094 + 0.889418i $$0.651110\pi$$
$$42$$ 0 0
$$43$$ 270.000 0.957549 0.478775 0.877938i $$-0.341081\pi$$
0.478775 + 0.877938i $$0.341081\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 448.000 1.39037 0.695186 0.718830i $$-0.255322\pi$$
0.695186 + 0.718830i $$0.255322\pi$$
$$48$$ 0 0
$$49$$ −174.000 −0.507289
$$50$$ 0 0
$$51$$ −335.000 −0.919792
$$52$$ 0 0
$$53$$ −45.0000 −0.116627 −0.0583134 0.998298i $$-0.518572\pi$$
−0.0583134 + 0.998298i $$0.518572\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 755.000 1.75442
$$58$$ 0 0
$$59$$ 704.000 1.55344 0.776720 0.629846i $$-0.216882\pi$$
0.776720 + 0.629846i $$0.216882\pi$$
$$60$$ 0 0
$$61$$ −217.000 −0.455475 −0.227738 0.973723i $$-0.573133\pi$$
−0.227738 + 0.973723i $$0.573133\pi$$
$$62$$ 0 0
$$63$$ −26.0000 −0.0519951
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 284.000 0.517853 0.258926 0.965897i $$-0.416631\pi$$
0.258926 + 0.965897i $$0.416631\pi$$
$$68$$ 0 0
$$69$$ 60.0000 0.104683
$$70$$ 0 0
$$71$$ −515.000 −0.860835 −0.430417 0.902630i $$-0.641634\pi$$
−0.430417 + 0.902630i $$0.641634\pi$$
$$72$$ 0 0
$$73$$ −1162.00 −1.86304 −0.931519 0.363692i $$-0.881516\pi$$
−0.931519 + 0.363692i $$0.881516\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −143.000 −0.211641
$$78$$ 0 0
$$79$$ −944.000 −1.34441 −0.672204 0.740366i $$-0.734652\pi$$
−0.672204 + 0.740366i $$0.734652\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ 0 0
$$83$$ −124.000 −0.163985 −0.0819926 0.996633i $$-0.526128\pi$$
−0.0819926 + 0.996633i $$0.526128\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −715.000 −0.881104
$$88$$ 0 0
$$89$$ 361.000 0.429954 0.214977 0.976619i $$-0.431032\pi$$
0.214977 + 0.976619i $$0.431032\pi$$
$$90$$ 0 0
$$91$$ −156.000 −0.179706
$$92$$ 0 0
$$93$$ −1685.00 −1.87878
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −916.000 −0.958822 −0.479411 0.877591i $$-0.659150\pi$$
−0.479411 + 0.877591i $$0.659150\pi$$
$$98$$ 0 0
$$99$$ 22.0000 0.0223342
$$100$$ 0 0
$$101$$ −1190.00 −1.17237 −0.586185 0.810177i $$-0.699371\pi$$
−0.586185 + 0.810177i $$0.699371\pi$$
$$102$$ 0 0
$$103$$ 1460.00 1.39668 0.698340 0.715766i $$-0.253922\pi$$
0.698340 + 0.715766i $$0.253922\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1124.00 −1.01553 −0.507763 0.861497i $$-0.669527\pi$$
−0.507763 + 0.861497i $$0.669527\pi$$
$$108$$ 0 0
$$109$$ 1582.00 1.39017 0.695083 0.718929i $$-0.255368\pi$$
0.695083 + 0.718929i $$0.255368\pi$$
$$110$$ 0 0
$$111$$ −625.000 −0.534436
$$112$$ 0 0
$$113$$ −914.000 −0.760902 −0.380451 0.924801i $$-0.624231\pi$$
−0.380451 + 0.924801i $$0.624231\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 24.0000 0.0189641
$$118$$ 0 0
$$119$$ −871.000 −0.670962
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −1200.00 −0.879678
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −40.0000 −0.0279482 −0.0139741 0.999902i $$-0.504448\pi$$
−0.0139741 + 0.999902i $$0.504448\pi$$
$$128$$ 0 0
$$129$$ 1350.00 0.921402
$$130$$ 0 0
$$131$$ −1005.00 −0.670284 −0.335142 0.942168i $$-0.608784\pi$$
−0.335142 + 0.942168i $$0.608784\pi$$
$$132$$ 0 0
$$133$$ 1963.00 1.27980
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1692.00 −1.05516 −0.527581 0.849504i $$-0.676901\pi$$
−0.527581 + 0.849504i $$0.676901\pi$$
$$138$$ 0 0
$$139$$ 564.000 0.344157 0.172079 0.985083i $$-0.444952\pi$$
0.172079 + 0.985083i $$0.444952\pi$$
$$140$$ 0 0
$$141$$ 2240.00 1.33789
$$142$$ 0 0
$$143$$ 132.000 0.0771916
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −870.000 −0.488139
$$148$$ 0 0
$$149$$ −2681.00 −1.47407 −0.737034 0.675856i $$-0.763774\pi$$
−0.737034 + 0.675856i $$0.763774\pi$$
$$150$$ 0 0
$$151$$ 970.000 0.522765 0.261382 0.965235i $$-0.415822\pi$$
0.261382 + 0.965235i $$0.415822\pi$$
$$152$$ 0 0
$$153$$ 134.000 0.0708056
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2551.00 1.29676 0.648382 0.761315i $$-0.275446\pi$$
0.648382 + 0.761315i $$0.275446\pi$$
$$158$$ 0 0
$$159$$ −225.000 −0.112224
$$160$$ 0 0
$$161$$ 156.000 0.0763635
$$162$$ 0 0
$$163$$ 2377.00 1.14221 0.571107 0.820875i $$-0.306514\pi$$
0.571107 + 0.820875i $$0.306514\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1113.00 0.515728 0.257864 0.966181i $$-0.416981\pi$$
0.257864 + 0.966181i $$0.416981\pi$$
$$168$$ 0 0
$$169$$ −2053.00 −0.934456
$$170$$ 0 0
$$171$$ −302.000 −0.135056
$$172$$ 0 0
$$173$$ −2222.00 −0.976506 −0.488253 0.872702i $$-0.662366\pi$$
−0.488253 + 0.872702i $$0.662366\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 3520.00 1.49480
$$178$$ 0 0
$$179$$ 2986.00 1.24684 0.623419 0.781888i $$-0.285743\pi$$
0.623419 + 0.781888i $$0.285743\pi$$
$$180$$ 0 0
$$181$$ 302.000 0.124019 0.0620096 0.998076i $$-0.480249\pi$$
0.0620096 + 0.998076i $$0.480249\pi$$
$$182$$ 0 0
$$183$$ −1085.00 −0.438281
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 737.000 0.288207
$$188$$ 0 0
$$189$$ −1885.00 −0.725469
$$190$$ 0 0
$$191$$ −1416.00 −0.536430 −0.268215 0.963359i $$-0.586434\pi$$
−0.268215 + 0.963359i $$0.586434\pi$$
$$192$$ 0 0
$$193$$ −4655.00 −1.73614 −0.868068 0.496445i $$-0.834638\pi$$
−0.868068 + 0.496445i $$0.834638\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −4146.00 −1.49944 −0.749721 0.661753i $$-0.769813\pi$$
−0.749721 + 0.661753i $$0.769813\pi$$
$$198$$ 0 0
$$199$$ −3703.00 −1.31909 −0.659544 0.751666i $$-0.729251\pi$$
−0.659544 + 0.751666i $$0.729251\pi$$
$$200$$ 0 0
$$201$$ 1420.00 0.498304
$$202$$ 0 0
$$203$$ −1859.00 −0.642740
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −24.0000 −0.00805853
$$208$$ 0 0
$$209$$ −1661.00 −0.549731
$$210$$ 0 0
$$211$$ 5231.00 1.70672 0.853358 0.521326i $$-0.174562\pi$$
0.853358 + 0.521326i $$0.174562\pi$$
$$212$$ 0 0
$$213$$ −2575.00 −0.828338
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4381.00 −1.37051
$$218$$ 0 0
$$219$$ −5810.00 −1.79271
$$220$$ 0 0
$$221$$ 804.000 0.244719
$$222$$ 0 0
$$223$$ −3742.00 −1.12369 −0.561845 0.827243i $$-0.689908\pi$$
−0.561845 + 0.827243i $$0.689908\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1260.00 0.368410 0.184205 0.982888i $$-0.441029\pi$$
0.184205 + 0.982888i $$0.441029\pi$$
$$228$$ 0 0
$$229$$ 1704.00 0.491718 0.245859 0.969306i $$-0.420930\pi$$
0.245859 + 0.969306i $$0.420930\pi$$
$$230$$ 0 0
$$231$$ −715.000 −0.203652
$$232$$ 0 0
$$233$$ −4939.00 −1.38869 −0.694345 0.719643i $$-0.744306\pi$$
−0.694345 + 0.719643i $$0.744306\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −4720.00 −1.29366
$$238$$ 0 0
$$239$$ 5224.00 1.41386 0.706930 0.707284i $$-0.250080\pi$$
0.706930 + 0.707284i $$0.250080\pi$$
$$240$$ 0 0
$$241$$ 1128.00 0.301497 0.150749 0.988572i $$-0.451832\pi$$
0.150749 + 0.988572i $$0.451832\pi$$
$$242$$ 0 0
$$243$$ 560.000 0.147835
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1812.00 −0.466781
$$248$$ 0 0
$$249$$ −620.000 −0.157795
$$250$$ 0 0
$$251$$ −4110.00 −1.03355 −0.516775 0.856121i $$-0.672868\pi$$
−0.516775 + 0.856121i $$0.672868\pi$$
$$252$$ 0 0
$$253$$ −132.000 −0.0328015
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1356.00 −0.329124 −0.164562 0.986367i $$-0.552621\pi$$
−0.164562 + 0.986367i $$0.552621\pi$$
$$258$$ 0 0
$$259$$ −1625.00 −0.389856
$$260$$ 0 0
$$261$$ 286.000 0.0678274
$$262$$ 0 0
$$263$$ 3063.00 0.718147 0.359074 0.933309i $$-0.383093\pi$$
0.359074 + 0.933309i $$0.383093\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 1805.00 0.413724
$$268$$ 0 0
$$269$$ −344.000 −0.0779704 −0.0389852 0.999240i $$-0.512413\pi$$
−0.0389852 + 0.999240i $$0.512413\pi$$
$$270$$ 0 0
$$271$$ 7086.00 1.58835 0.794177 0.607687i $$-0.207902\pi$$
0.794177 + 0.607687i $$0.207902\pi$$
$$272$$ 0 0
$$273$$ −780.000 −0.172922
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1156.00 0.250748 0.125374 0.992110i $$-0.459987\pi$$
0.125374 + 0.992110i $$0.459987\pi$$
$$278$$ 0 0
$$279$$ 674.000 0.144628
$$280$$ 0 0
$$281$$ 7174.00 1.52301 0.761503 0.648161i $$-0.224462\pi$$
0.761503 + 0.648161i $$0.224462\pi$$
$$282$$ 0 0
$$283$$ −2072.00 −0.435221 −0.217611 0.976036i $$-0.569826\pi$$
−0.217611 + 0.976036i $$0.569826\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −3120.00 −0.641700
$$288$$ 0 0
$$289$$ −424.000 −0.0863016
$$290$$ 0 0
$$291$$ −4580.00 −0.922627
$$292$$ 0 0
$$293$$ 4736.00 0.944301 0.472150 0.881518i $$-0.343478\pi$$
0.472150 + 0.881518i $$0.343478\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1595.00 0.311620
$$298$$ 0 0
$$299$$ −144.000 −0.0278520
$$300$$ 0 0
$$301$$ 3510.00 0.672136
$$302$$ 0 0
$$303$$ −5950.00 −1.12811
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4216.00 −0.783778 −0.391889 0.920013i $$-0.628178\pi$$
−0.391889 + 0.920013i $$0.628178\pi$$
$$308$$ 0 0
$$309$$ 7300.00 1.34396
$$310$$ 0 0
$$311$$ −6605.00 −1.20429 −0.602147 0.798386i $$-0.705688\pi$$
−0.602147 + 0.798386i $$0.705688\pi$$
$$312$$ 0 0
$$313$$ 4842.00 0.874396 0.437198 0.899365i $$-0.355971\pi$$
0.437198 + 0.899365i $$0.355971\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7211.00 −1.27763 −0.638817 0.769359i $$-0.720576\pi$$
−0.638817 + 0.769359i $$0.720576\pi$$
$$318$$ 0 0
$$319$$ 1573.00 0.276085
$$320$$ 0 0
$$321$$ −5620.00 −0.977189
$$322$$ 0 0
$$323$$ −10117.0 −1.74280
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 7910.00 1.33769
$$328$$ 0 0
$$329$$ 5824.00 0.975950
$$330$$ 0 0
$$331$$ −4426.00 −0.734970 −0.367485 0.930030i $$-0.619781\pi$$
−0.367485 + 0.930030i $$0.619781\pi$$
$$332$$ 0 0
$$333$$ 250.000 0.0411409
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1751.00 0.283036 0.141518 0.989936i $$-0.454802\pi$$
0.141518 + 0.989936i $$0.454802\pi$$
$$338$$ 0 0
$$339$$ −4570.00 −0.732178
$$340$$ 0 0
$$341$$ 3707.00 0.588696
$$342$$ 0 0
$$343$$ −6721.00 −1.05802
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 1494.00 0.231130 0.115565 0.993300i $$-0.463132\pi$$
0.115565 + 0.993300i $$0.463132\pi$$
$$348$$ 0 0
$$349$$ 1822.00 0.279454 0.139727 0.990190i $$-0.455378\pi$$
0.139727 + 0.990190i $$0.455378\pi$$
$$350$$ 0 0
$$351$$ 1740.00 0.264599
$$352$$ 0 0
$$353$$ −1390.00 −0.209581 −0.104791 0.994494i $$-0.533417\pi$$
−0.104791 + 0.994494i $$0.533417\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −4355.00 −0.645633
$$358$$ 0 0
$$359$$ −224.000 −0.0329311 −0.0164656 0.999864i $$-0.505241\pi$$
−0.0164656 + 0.999864i $$0.505241\pi$$
$$360$$ 0 0
$$361$$ 15942.0 2.32425
$$362$$ 0 0
$$363$$ 605.000 0.0874773
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8256.00 1.17428 0.587139 0.809486i $$-0.300254\pi$$
0.587139 + 0.809486i $$0.300254\pi$$
$$368$$ 0 0
$$369$$ 480.000 0.0677176
$$370$$ 0 0
$$371$$ −585.000 −0.0818644
$$372$$ 0 0
$$373$$ −11348.0 −1.57527 −0.787637 0.616140i $$-0.788696\pi$$
−0.787637 + 0.616140i $$0.788696\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1716.00 0.234426
$$378$$ 0 0
$$379$$ 3710.00 0.502823 0.251411 0.967880i $$-0.419105\pi$$
0.251411 + 0.967880i $$0.419105\pi$$
$$380$$ 0 0
$$381$$ −200.000 −0.0268932
$$382$$ 0 0
$$383$$ −14370.0 −1.91716 −0.958581 0.284822i $$-0.908066\pi$$
−0.958581 + 0.284822i $$0.908066\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −540.000 −0.0709296
$$388$$ 0 0
$$389$$ 6186.00 0.806279 0.403140 0.915138i $$-0.367919\pi$$
0.403140 + 0.915138i $$0.367919\pi$$
$$390$$ 0 0
$$391$$ −804.000 −0.103990
$$392$$ 0 0
$$393$$ −5025.00 −0.644981
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −13634.0 −1.72360 −0.861802 0.507245i $$-0.830664\pi$$
−0.861802 + 0.507245i $$0.830664\pi$$
$$398$$ 0 0
$$399$$ 9815.00 1.23149
$$400$$ 0 0
$$401$$ −7237.00 −0.901243 −0.450622 0.892715i $$-0.648798\pi$$
−0.450622 + 0.892715i $$0.648798\pi$$
$$402$$ 0 0
$$403$$ 4044.00 0.499866
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1375.00 0.167460
$$408$$ 0 0
$$409$$ −2420.00 −0.292570 −0.146285 0.989242i $$-0.546732\pi$$
−0.146285 + 0.989242i $$0.546732\pi$$
$$410$$ 0 0
$$411$$ −8460.00 −1.01533
$$412$$ 0 0
$$413$$ 9152.00 1.09041
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 2820.00 0.331165
$$418$$ 0 0
$$419$$ −9140.00 −1.06568 −0.532838 0.846217i $$-0.678874\pi$$
−0.532838 + 0.846217i $$0.678874\pi$$
$$420$$ 0 0
$$421$$ −10884.0 −1.25999 −0.629993 0.776601i $$-0.716942\pi$$
−0.629993 + 0.776601i $$0.716942\pi$$
$$422$$ 0 0
$$423$$ −896.000 −0.102991
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −2821.00 −0.319714
$$428$$ 0 0
$$429$$ 660.000 0.0742776
$$430$$ 0 0
$$431$$ 1752.00 0.195802 0.0979012 0.995196i $$-0.468787\pi$$
0.0979012 + 0.995196i $$0.468787\pi$$
$$432$$ 0 0
$$433$$ −5302.00 −0.588448 −0.294224 0.955737i $$-0.595061\pi$$
−0.294224 + 0.955737i $$0.595061\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1812.00 0.198352
$$438$$ 0 0
$$439$$ −3502.00 −0.380732 −0.190366 0.981713i $$-0.560967\pi$$
−0.190366 + 0.981713i $$0.560967\pi$$
$$440$$ 0 0
$$441$$ 348.000 0.0375769
$$442$$ 0 0
$$443$$ −7508.00 −0.805228 −0.402614 0.915370i $$-0.631898\pi$$
−0.402614 + 0.915370i $$0.631898\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −13405.0 −1.41842
$$448$$ 0 0
$$449$$ 12062.0 1.26780 0.633899 0.773416i $$-0.281454\pi$$
0.633899 + 0.773416i $$0.281454\pi$$
$$450$$ 0 0
$$451$$ 2640.00 0.275638
$$452$$ 0 0
$$453$$ 4850.00 0.503031
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4691.00 0.480166 0.240083 0.970752i $$-0.422825\pi$$
0.240083 + 0.970752i $$0.422825\pi$$
$$458$$ 0 0
$$459$$ 9715.00 0.987925
$$460$$ 0 0
$$461$$ 951.000 0.0960791 0.0480396 0.998845i $$-0.484703\pi$$
0.0480396 + 0.998845i $$0.484703\pi$$
$$462$$ 0 0
$$463$$ −5784.00 −0.580573 −0.290286 0.956940i $$-0.593751\pi$$
−0.290286 + 0.956940i $$0.593751\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 9233.00 0.914887 0.457444 0.889239i $$-0.348765\pi$$
0.457444 + 0.889239i $$0.348765\pi$$
$$468$$ 0 0
$$469$$ 3692.00 0.363498
$$470$$ 0 0
$$471$$ 12755.0 1.24781
$$472$$ 0 0
$$473$$ −2970.00 −0.288712
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 90.0000 0.00863903
$$478$$ 0 0
$$479$$ −7562.00 −0.721329 −0.360665 0.932696i $$-0.617450\pi$$
−0.360665 + 0.932696i $$0.617450\pi$$
$$480$$ 0 0
$$481$$ 1500.00 0.142192
$$482$$ 0 0
$$483$$ 780.000 0.0734808
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −8346.00 −0.776578 −0.388289 0.921538i $$-0.626934\pi$$
−0.388289 + 0.921538i $$0.626934\pi$$
$$488$$ 0 0
$$489$$ 11885.0 1.09910
$$490$$ 0 0
$$491$$ −6045.00 −0.555615 −0.277808 0.960637i $$-0.589608\pi$$
−0.277808 + 0.960637i $$0.589608\pi$$
$$492$$ 0 0
$$493$$ 9581.00 0.875267
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −6695.00 −0.604249
$$498$$ 0 0
$$499$$ 11056.0 0.991853 0.495926 0.868365i $$-0.334829\pi$$
0.495926 + 0.868365i $$0.334829\pi$$
$$500$$ 0 0
$$501$$ 5565.00 0.496259
$$502$$ 0 0
$$503$$ 936.000 0.0829705 0.0414853 0.999139i $$-0.486791\pi$$
0.0414853 + 0.999139i $$0.486791\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −10265.0 −0.899181
$$508$$ 0 0
$$509$$ 1684.00 0.146644 0.0733222 0.997308i $$-0.476640\pi$$
0.0733222 + 0.997308i $$0.476640\pi$$
$$510$$ 0 0
$$511$$ −15106.0 −1.30773
$$512$$ 0 0
$$513$$ −21895.0 −1.88438
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −4928.00 −0.419213
$$518$$ 0 0
$$519$$ −11110.0 −0.939643
$$520$$ 0 0
$$521$$ 13190.0 1.10914 0.554572 0.832136i $$-0.312882\pi$$
0.554572 + 0.832136i $$0.312882\pi$$
$$522$$ 0 0
$$523$$ 5432.00 0.454158 0.227079 0.973876i $$-0.427082\pi$$
0.227079 + 0.973876i $$0.427082\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 22579.0 1.86633
$$528$$ 0 0
$$529$$ −12023.0 −0.988165
$$530$$ 0 0
$$531$$ −1408.00 −0.115070
$$532$$ 0 0
$$533$$ 2880.00 0.234046
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 14930.0 1.19977
$$538$$ 0 0
$$539$$ 1914.00 0.152953
$$540$$ 0 0
$$541$$ 6237.00 0.495655 0.247828 0.968804i $$-0.420283\pi$$
0.247828 + 0.968804i $$0.420283\pi$$
$$542$$ 0 0
$$543$$ 1510.00 0.119338
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 11400.0 0.891095 0.445547 0.895258i $$-0.353009\pi$$
0.445547 + 0.895258i $$0.353009\pi$$
$$548$$ 0 0
$$549$$ 434.000 0.0337389
$$550$$ 0 0
$$551$$ −21593.0 −1.66950
$$552$$ 0 0
$$553$$ −12272.0 −0.943686
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −2588.00 −0.196871 −0.0984354 0.995143i $$-0.531384\pi$$
−0.0984354 + 0.995143i $$0.531384\pi$$
$$558$$ 0 0
$$559$$ −3240.00 −0.245147
$$560$$ 0 0
$$561$$ 3685.00 0.277328
$$562$$ 0 0
$$563$$ 12708.0 0.951294 0.475647 0.879636i $$-0.342214\pi$$
0.475647 + 0.879636i $$0.342214\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −8723.00 −0.646087
$$568$$ 0 0
$$569$$ 9828.00 0.724097 0.362049 0.932159i $$-0.382077\pi$$
0.362049 + 0.932159i $$0.382077\pi$$
$$570$$ 0 0
$$571$$ 16585.0 1.21552 0.607759 0.794122i $$-0.292069\pi$$
0.607759 + 0.794122i $$0.292069\pi$$
$$572$$ 0 0
$$573$$ −7080.00 −0.516180
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2992.00 0.215873 0.107936 0.994158i $$-0.465576\pi$$
0.107936 + 0.994158i $$0.465576\pi$$
$$578$$ 0 0
$$579$$ −23275.0 −1.67060
$$580$$ 0 0
$$581$$ −1612.00 −0.115107
$$582$$ 0 0
$$583$$ 495.000 0.0351643
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −20325.0 −1.42914 −0.714568 0.699566i $$-0.753377\pi$$
−0.714568 + 0.699566i $$0.753377\pi$$
$$588$$ 0 0
$$589$$ −50887.0 −3.55987
$$590$$ 0 0
$$591$$ −20730.0 −1.44284
$$592$$ 0 0
$$593$$ −5226.00 −0.361899 −0.180949 0.983492i $$-0.557917\pi$$
−0.180949 + 0.983492i $$0.557917\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −18515.0 −1.26929
$$598$$ 0 0
$$599$$ 2171.00 0.148088 0.0740440 0.997255i $$-0.476409\pi$$
0.0740440 + 0.997255i $$0.476409\pi$$
$$600$$ 0 0
$$601$$ 20786.0 1.41078 0.705390 0.708820i $$-0.250772\pi$$
0.705390 + 0.708820i $$0.250772\pi$$
$$602$$ 0 0
$$603$$ −568.000 −0.0383594
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −11841.0 −0.791781 −0.395891 0.918298i $$-0.629564\pi$$
−0.395891 + 0.918298i $$0.629564\pi$$
$$608$$ 0 0
$$609$$ −9295.00 −0.618477
$$610$$ 0 0
$$611$$ −5376.00 −0.355957
$$612$$ 0 0
$$613$$ 21782.0 1.43518 0.717591 0.696465i $$-0.245245\pi$$
0.717591 + 0.696465i $$0.245245\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 25604.0 1.67063 0.835315 0.549772i $$-0.185285\pi$$
0.835315 + 0.549772i $$0.185285\pi$$
$$618$$ 0 0
$$619$$ −3492.00 −0.226745 −0.113373 0.993553i $$-0.536165\pi$$
−0.113373 + 0.993553i $$0.536165\pi$$
$$620$$ 0 0
$$621$$ −1740.00 −0.112438
$$622$$ 0 0
$$623$$ 4693.00 0.301799
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −8305.00 −0.528979
$$628$$ 0 0
$$629$$ 8375.00 0.530895
$$630$$ 0 0
$$631$$ −6027.00 −0.380239 −0.190120 0.981761i $$-0.560888\pi$$
−0.190120 + 0.981761i $$0.560888\pi$$
$$632$$ 0 0
$$633$$ 26155.0 1.64229
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2088.00 0.129874
$$638$$ 0 0
$$639$$ 1030.00 0.0637655
$$640$$ 0 0
$$641$$ −25937.0 −1.59821 −0.799103 0.601194i $$-0.794692\pi$$
−0.799103 + 0.601194i $$0.794692\pi$$
$$642$$ 0 0
$$643$$ −2827.00 −0.173384 −0.0866921 0.996235i $$-0.527630\pi$$
−0.0866921 + 0.996235i $$0.527630\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 228.000 0.0138541 0.00692705 0.999976i $$-0.497795\pi$$
0.00692705 + 0.999976i $$0.497795\pi$$
$$648$$ 0 0
$$649$$ −7744.00 −0.468380
$$650$$ 0 0
$$651$$ −21905.0 −1.31878
$$652$$ 0 0
$$653$$ 7377.00 0.442089 0.221045 0.975264i $$-0.429053\pi$$
0.221045 + 0.975264i $$0.429053\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2324.00 0.138003
$$658$$ 0 0
$$659$$ −6879.00 −0.406628 −0.203314 0.979114i $$-0.565171\pi$$
−0.203314 + 0.979114i $$0.565171\pi$$
$$660$$ 0 0
$$661$$ 20260.0 1.19217 0.596084 0.802922i $$-0.296723\pi$$
0.596084 + 0.802922i $$0.296723\pi$$
$$662$$ 0 0
$$663$$ 4020.00 0.235481
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1716.00 −0.0996159
$$668$$ 0 0
$$669$$ −18710.0 −1.08127
$$670$$ 0 0
$$671$$ 2387.00 0.137331
$$672$$ 0 0
$$673$$ 1489.00 0.0852849 0.0426424 0.999090i $$-0.486422\pi$$
0.0426424 + 0.999090i $$0.486422\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −5640.00 −0.320181 −0.160091 0.987102i $$-0.551179\pi$$
−0.160091 + 0.987102i $$0.551179\pi$$
$$678$$ 0 0
$$679$$ −11908.0 −0.673030
$$680$$ 0 0
$$681$$ 6300.00 0.354503
$$682$$ 0 0
$$683$$ 8019.00 0.449251 0.224626 0.974445i $$-0.427884\pi$$
0.224626 + 0.974445i $$0.427884\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 8520.00 0.473156
$$688$$ 0 0
$$689$$ 540.000 0.0298583
$$690$$ 0 0
$$691$$ 20440.0 1.12529 0.562644 0.826699i $$-0.309784\pi$$
0.562644 + 0.826699i $$0.309784\pi$$
$$692$$ 0 0
$$693$$ 286.000 0.0156771
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 16080.0 0.873850
$$698$$ 0 0
$$699$$ −24695.0 −1.33627
$$700$$ 0 0
$$701$$ 10911.0 0.587878 0.293939 0.955824i $$-0.405034\pi$$
0.293939 + 0.955824i $$0.405034\pi$$
$$702$$ 0 0
$$703$$ −18875.0 −1.01264
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −15470.0 −0.822927
$$708$$ 0 0
$$709$$ 24666.0 1.30656 0.653280 0.757116i $$-0.273392\pi$$
0.653280 + 0.757116i $$0.273392\pi$$
$$710$$ 0 0
$$711$$ 1888.00 0.0995858
$$712$$ 0 0
$$713$$ −4044.00 −0.212411
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 26120.0 1.36049
$$718$$ 0 0
$$719$$ 17399.0 0.902466 0.451233 0.892406i $$-0.350984\pi$$
0.451233 + 0.892406i $$0.350984\pi$$
$$720$$ 0 0
$$721$$ 18980.0 0.980377
$$722$$ 0 0
$$723$$ 5640.00 0.290116
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −19794.0 −1.00979 −0.504896 0.863180i $$-0.668469\pi$$
−0.504896 + 0.863180i $$0.668469\pi$$
$$728$$ 0 0
$$729$$ 20917.0 1.06269
$$730$$ 0 0
$$731$$ −18090.0 −0.915298
$$732$$ 0 0
$$733$$ −32454.0 −1.63536 −0.817678 0.575676i $$-0.804739\pi$$
−0.817678 + 0.575676i $$0.804739\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −3124.00 −0.156138
$$738$$ 0 0
$$739$$ 8024.00 0.399415 0.199707 0.979856i $$-0.436001\pi$$
0.199707 + 0.979856i $$0.436001\pi$$
$$740$$ 0 0
$$741$$ −9060.00 −0.449160
$$742$$ 0 0
$$743$$ −24159.0 −1.19288 −0.596439 0.802659i $$-0.703418\pi$$
−0.596439 + 0.802659i $$0.703418\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 248.000 0.0121470
$$748$$ 0 0
$$749$$ −14612.0 −0.712832
$$750$$ 0 0
$$751$$ −38477.0 −1.86957 −0.934784 0.355216i $$-0.884407\pi$$
−0.934784 + 0.355216i $$0.884407\pi$$
$$752$$ 0 0
$$753$$ −20550.0 −0.994533
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 5014.00 0.240736 0.120368 0.992729i $$-0.461593\pi$$
0.120368 + 0.992729i $$0.461593\pi$$
$$758$$ 0 0
$$759$$ −660.000 −0.0315632
$$760$$ 0 0
$$761$$ 15764.0 0.750913 0.375456 0.926840i $$-0.377486\pi$$
0.375456 + 0.926840i $$0.377486\pi$$
$$762$$ 0 0
$$763$$ 20566.0 0.975805
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −8448.00 −0.397705
$$768$$ 0 0
$$769$$ 614.000 0.0287925 0.0143962 0.999896i $$-0.495417\pi$$
0.0143962 + 0.999896i $$0.495417\pi$$
$$770$$ 0 0
$$771$$ −6780.00 −0.316700
$$772$$ 0 0
$$773$$ 5943.00 0.276526 0.138263 0.990396i $$-0.455848\pi$$
0.138263 + 0.990396i $$0.455848\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −8125.00 −0.375139
$$778$$ 0 0
$$779$$ −36240.0 −1.66679
$$780$$ 0 0
$$781$$ 5665.00 0.259551
$$782$$ 0 0
$$783$$ 20735.0 0.946371
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 26154.0 1.18461 0.592306 0.805713i $$-0.298218\pi$$
0.592306 + 0.805713i $$0.298218\pi$$
$$788$$ 0 0
$$789$$ 15315.0 0.691037
$$790$$ 0 0
$$791$$ −11882.0 −0.534103
$$792$$ 0 0
$$793$$ 2604.00 0.116609
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 3270.00 0.145332 0.0726658 0.997356i $$-0.476849\pi$$
0.0726658 + 0.997356i $$0.476849\pi$$
$$798$$ 0 0
$$799$$ −30016.0 −1.32902
$$800$$ 0 0
$$801$$ −722.000 −0.0318485
$$802$$ 0 0
$$803$$ 12782.0 0.561727
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −1720.00 −0.0750271
$$808$$ 0 0
$$809$$ 14306.0 0.621721 0.310860 0.950456i $$-0.399383\pi$$
0.310860 + 0.950456i $$0.399383\pi$$
$$810$$ 0 0
$$811$$ −959.000 −0.0415229 −0.0207614 0.999784i $$-0.506609\pi$$
−0.0207614 + 0.999784i $$0.506609\pi$$
$$812$$ 0 0
$$813$$ 35430.0 1.52839
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 40770.0 1.74585
$$818$$ 0 0
$$819$$ 312.000 0.0133116
$$820$$ 0 0
$$821$$ 14190.0 0.603209 0.301604 0.953433i $$-0.402478\pi$$
0.301604 + 0.953433i $$0.402478\pi$$
$$822$$ 0 0
$$823$$ 34010.0 1.44048 0.720239 0.693726i $$-0.244032\pi$$
0.720239 + 0.693726i $$0.244032\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −14218.0 −0.597833 −0.298917 0.954279i $$-0.596625\pi$$
−0.298917 + 0.954279i $$0.596625\pi$$
$$828$$ 0 0
$$829$$ 33172.0 1.38976 0.694880 0.719126i $$-0.255457\pi$$
0.694880 + 0.719126i $$0.255457\pi$$
$$830$$ 0 0
$$831$$ 5780.00 0.241283
$$832$$ 0 0
$$833$$ 11658.0 0.484905
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 48865.0 2.01795
$$838$$ 0 0
$$839$$ 28656.0 1.17916 0.589580 0.807710i $$-0.299293\pi$$
0.589580 + 0.807710i $$0.299293\pi$$
$$840$$ 0 0
$$841$$ −3940.00 −0.161548
$$842$$ 0 0
$$843$$ 35870.0 1.46551
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1573.00 0.0638122
$$848$$ 0 0
$$849$$ −10360.0 −0.418792
$$850$$ 0 0
$$851$$ −1500.00 −0.0604223
$$852$$ 0 0
$$853$$ 10244.0 0.411193 0.205597 0.978637i $$-0.434087\pi$$
0.205597 + 0.978637i $$0.434087\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 11401.0 0.454435 0.227217 0.973844i $$-0.427037\pi$$
0.227217 + 0.973844i $$0.427037\pi$$
$$858$$ 0 0
$$859$$ 26846.0 1.06633 0.533163 0.846013i $$-0.321003\pi$$
0.533163 + 0.846013i $$0.321003\pi$$
$$860$$ 0 0
$$861$$ −15600.0 −0.617476
$$862$$ 0 0
$$863$$ −40972.0 −1.61611 −0.808055 0.589107i $$-0.799480\pi$$
−0.808055 + 0.589107i $$0.799480\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −2120.00 −0.0830438
$$868$$ 0 0
$$869$$ 10384.0 0.405355
$$870$$ 0 0
$$871$$ −3408.00 −0.132578
$$872$$ 0 0
$$873$$ 1832.00 0.0710238
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −15614.0 −0.601194 −0.300597 0.953751i $$-0.597186\pi$$
−0.300597 + 0.953751i $$0.597186\pi$$
$$878$$ 0 0
$$879$$ 23680.0 0.908654
$$880$$ 0 0
$$881$$ −39938.0 −1.52729 −0.763647 0.645634i $$-0.776593\pi$$
−0.763647 + 0.645634i $$0.776593\pi$$
$$882$$ 0 0
$$883$$ 48467.0 1.84716 0.923581 0.383403i $$-0.125248\pi$$
0.923581 + 0.383403i $$0.125248\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −25776.0 −0.975731 −0.487865 0.872919i $$-0.662224\pi$$
−0.487865 + 0.872919i $$0.662224\pi$$
$$888$$ 0 0
$$889$$ −520.000 −0.0196178
$$890$$ 0 0
$$891$$ 7381.00 0.277523
$$892$$ 0 0
$$893$$ 67648.0 2.53500
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −720.000 −0.0268006
$$898$$ 0 0
$$899$$ 48191.0 1.78783
$$900$$ 0 0
$$901$$ 3015.00 0.111481
$$902$$ 0 0
$$903$$ 17550.0 0.646763
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 23701.0 0.867672 0.433836 0.900992i $$-0.357160\pi$$
0.433836 + 0.900992i $$0.357160\pi$$
$$908$$ 0 0
$$909$$ 2380.00 0.0868423
$$910$$ 0 0
$$911$$ 24189.0 0.879712 0.439856 0.898068i $$-0.355030\pi$$
0.439856 + 0.898068i $$0.355030\pi$$
$$912$$ 0 0
$$913$$ 1364.00 0.0494434
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −13065.0 −0.470495
$$918$$ 0 0
$$919$$ −24590.0 −0.882643 −0.441322 0.897349i $$-0.645490\pi$$
−0.441322 + 0.897349i $$0.645490\pi$$
$$920$$ 0 0
$$921$$ −21080.0 −0.754191
$$922$$ 0 0
$$923$$ 6180.00 0.220387
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −2920.00 −0.103458
$$928$$ 0 0
$$929$$ 16575.0 0.585369 0.292685 0.956209i $$-0.405451\pi$$
0.292685 + 0.956209i $$0.405451\pi$$
$$930$$ 0 0
$$931$$ −26274.0 −0.924915
$$932$$ 0 0
$$933$$ −33025.0 −1.15883
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −15542.0 −0.541873 −0.270937 0.962597i $$-0.587333\pi$$
−0.270937 + 0.962597i $$0.587333\pi$$
$$938$$ 0 0
$$939$$ 24210.0 0.841388
$$940$$ 0 0
$$941$$ 10893.0 0.377366 0.188683 0.982038i $$-0.439578\pi$$
0.188683 + 0.982038i $$0.439578\pi$$
$$942$$ 0 0
$$943$$ −2880.00 −0.0994546
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 23831.0 0.817744 0.408872 0.912592i $$-0.365922\pi$$
0.408872 + 0.912592i $$0.365922\pi$$
$$948$$ 0 0
$$949$$ 13944.0 0.476967
$$950$$ 0 0
$$951$$ −36055.0 −1.22940
$$952$$ 0 0
$$953$$ 3771.00 0.128179 0.0640895 0.997944i $$-0.479586\pi$$
0.0640895 + 0.997944i $$0.479586\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 7865.00 0.265663
$$958$$ 0 0
$$959$$ −21996.0 −0.740655
$$960$$ 0 0
$$961$$ 83778.0 2.81219
$$962$$ 0 0
$$963$$ 2248.00 0.0752241
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −50413.0 −1.67650 −0.838249 0.545288i $$-0.816420\pi$$
−0.838249 + 0.545288i $$0.816420\pi$$
$$968$$ 0 0
$$969$$ −50585.0 −1.67701
$$970$$ 0 0
$$971$$ 492.000 0.0162606 0.00813029 0.999967i $$-0.497412\pi$$
0.00813029 + 0.999967i $$0.497412\pi$$
$$972$$ 0 0
$$973$$ 7332.00 0.241576
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −56748.0 −1.85827 −0.929135 0.369741i $$-0.879446\pi$$
−0.929135 + 0.369741i $$0.879446\pi$$
$$978$$ 0 0
$$979$$ −3971.00 −0.129636
$$980$$ 0 0
$$981$$ −3164.00 −0.102975
$$982$$ 0 0
$$983$$ −49302.0 −1.59968 −0.799842 0.600210i $$-0.795083\pi$$
−0.799842 + 0.600210i $$0.795083\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 29120.0 0.939108
$$988$$ 0 0
$$989$$ 3240.00 0.104172
$$990$$ 0 0
$$991$$ 5064.00 0.162324 0.0811621 0.996701i $$-0.474137\pi$$
0.0811621 + 0.996701i $$0.474137\pi$$
$$992$$ 0 0
$$993$$ −22130.0 −0.707225
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 61018.0 1.93827 0.969137 0.246522i $$-0.0792878\pi$$
0.969137 + 0.246522i $$0.0792878\pi$$
$$998$$ 0 0
$$999$$ 18125.0 0.574024
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 2200.4.a.j.1.1 1
5.2 odd 4 440.4.b.a.89.1 2
5.3 odd 4 440.4.b.a.89.2 yes 2
5.4 even 2 2200.4.a.c.1.1 1
20.3 even 4 880.4.b.c.529.1 2
20.7 even 4 880.4.b.c.529.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
440.4.b.a.89.1 2 5.2 odd 4
440.4.b.a.89.2 yes 2 5.3 odd 4
880.4.b.c.529.1 2 20.3 even 4
880.4.b.c.529.2 2 20.7 even 4
2200.4.a.c.1.1 1 5.4 even 2
2200.4.a.j.1.1 1 1.1 even 1 trivial