Properties

 Label 2205.4.a.c.1.1 Level $2205$ Weight $4$ Character 2205.1 Self dual yes Analytic conductor $130.099$ 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] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2205.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: $$130.099211563$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} +21.0000 q^{8} +O(q^{10})$$ $$q-3.00000 q^{2} +1.00000 q^{4} -5.00000 q^{5} +21.0000 q^{8} +15.0000 q^{10} +24.0000 q^{11} -74.0000 q^{13} -71.0000 q^{16} +54.0000 q^{17} +124.000 q^{19} -5.00000 q^{20} -72.0000 q^{22} +120.000 q^{23} +25.0000 q^{25} +222.000 q^{26} +78.0000 q^{29} -200.000 q^{31} +45.0000 q^{32} -162.000 q^{34} -70.0000 q^{37} -372.000 q^{38} -105.000 q^{40} +330.000 q^{41} +92.0000 q^{43} +24.0000 q^{44} -360.000 q^{46} -24.0000 q^{47} -75.0000 q^{50} -74.0000 q^{52} -450.000 q^{53} -120.000 q^{55} -234.000 q^{58} +24.0000 q^{59} +322.000 q^{61} +600.000 q^{62} +433.000 q^{64} +370.000 q^{65} -196.000 q^{67} +54.0000 q^{68} +288.000 q^{71} +430.000 q^{73} +210.000 q^{74} +124.000 q^{76} -520.000 q^{79} +355.000 q^{80} -990.000 q^{82} +156.000 q^{83} -270.000 q^{85} -276.000 q^{86} +504.000 q^{88} +1026.00 q^{89} +120.000 q^{92} +72.0000 q^{94} -620.000 q^{95} +286.000 q^{97} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.00000 −1.06066 −0.530330 0.847791i $$-0.677932\pi$$
−0.530330 + 0.847791i $$0.677932\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.125000
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 21.0000 0.928078
$$9$$ 0 0
$$10$$ 15.0000 0.474342
$$11$$ 24.0000 0.657843 0.328921 0.944357i $$-0.393315\pi$$
0.328921 + 0.944357i $$0.393315\pi$$
$$12$$ 0 0
$$13$$ −74.0000 −1.57876 −0.789381 0.613904i $$-0.789598\pi$$
−0.789381 + 0.613904i $$0.789598\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ 54.0000 0.770407 0.385204 0.922832i $$-0.374131\pi$$
0.385204 + 0.922832i $$0.374131\pi$$
$$18$$ 0 0
$$19$$ 124.000 1.49724 0.748620 0.663000i $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ −5.00000 −0.0559017
$$21$$ 0 0
$$22$$ −72.0000 −0.697748
$$23$$ 120.000 1.08790 0.543951 0.839117i $$-0.316928\pi$$
0.543951 + 0.839117i $$0.316928\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 222.000 1.67453
$$27$$ 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 $$0.696702\pi$$
$$32$$ 45.0000 0.248592
$$33$$ 0 0
$$34$$ −162.000 −0.817140
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −70.0000 −0.311025 −0.155513 0.987834i $$-0.549703\pi$$
−0.155513 + 0.987834i $$0.549703\pi$$
$$38$$ −372.000 −1.58806
$$39$$ 0 0
$$40$$ −105.000 −0.415049
$$41$$ 330.000 1.25701 0.628504 0.777806i $$-0.283668\pi$$
0.628504 + 0.777806i $$0.283668\pi$$
$$42$$ 0 0
$$43$$ 92.0000 0.326276 0.163138 0.986603i $$-0.447838\pi$$
0.163138 + 0.986603i $$0.447838\pi$$
$$44$$ 24.0000 0.0822304
$$45$$ 0 0
$$46$$ −360.000 −1.15389
$$47$$ −24.0000 −0.0744843 −0.0372421 0.999306i $$-0.511857\pi$$
−0.0372421 + 0.999306i $$0.511857\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −75.0000 −0.212132
$$51$$ 0 0
$$52$$ −74.0000 −0.197345
$$53$$ −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
$$58$$ −234.000 −0.529754
$$59$$ 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 $$-0.390272\pi$$
0.337933 + 0.941170i $$0.390272\pi$$
$$62$$ 600.000 1.22903
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ 370.000 0.706044
$$66$$ 0 0
$$67$$ −196.000 −0.357391 −0.178696 0.983904i $$-0.557188\pi$$
−0.178696 + 0.983904i $$0.557188\pi$$
$$68$$ 54.0000 0.0963009
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 288.000 0.481399 0.240699 0.970600i $$-0.422623\pi$$
0.240699 + 0.970600i $$0.422623\pi$$
$$72$$ 0 0
$$73$$ 430.000 0.689420 0.344710 0.938709i $$-0.387977\pi$$
0.344710 + 0.938709i $$0.387977\pi$$
$$74$$ 210.000 0.329892
$$75$$ 0 0
$$76$$ 124.000 0.187155
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −520.000 −0.740564 −0.370282 0.928919i $$-0.620739\pi$$
−0.370282 + 0.928919i $$0.620739\pi$$
$$80$$ 355.000 0.496128
$$81$$ 0 0
$$82$$ −990.000 −1.33326
$$83$$ 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
$$86$$ −276.000 −0.346068
$$87$$ 0 0
$$88$$ 504.000 0.610529
$$89$$ 1026.00 1.22198 0.610988 0.791640i $$-0.290773\pi$$
0.610988 + 0.791640i $$0.290773\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 120.000 0.135988
$$93$$ 0 0
$$94$$ 72.0000 0.0790025
$$95$$ −620.000 −0.669586
$$96$$ 0 0
$$97$$ 286.000 0.299370 0.149685 0.988734i $$-0.452174\pi$$
0.149685 + 0.988734i $$0.452174\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 25.0000 0.0250000
$$101$$ −1734.00 −1.70831 −0.854156 0.520017i $$-0.825925\pi$$
−0.854156 + 0.520017i $$0.825925\pi$$
$$102$$ 0 0
$$103$$ −452.000 −0.432397 −0.216198 0.976349i $$-0.569366\pi$$
−0.216198 + 0.976349i $$0.569366\pi$$
$$104$$ −1554.00 −1.46521
$$105$$ 0 0
$$106$$ 1350.00 1.23702
$$107$$ 1404.00 1.26850 0.634251 0.773127i $$-0.281308\pi$$
0.634251 + 0.773127i $$0.281308\pi$$
$$108$$ 0 0
$$109$$ −1474.00 −1.29526 −0.647631 0.761954i $$-0.724240\pi$$
−0.647631 + 0.761954i $$0.724240\pi$$
$$110$$ 360.000 0.312042
$$111$$ 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
$$116$$ 78.0000 0.0624321
$$117$$ 0 0
$$118$$ −72.0000 −0.0561707
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ −966.000 −0.716865
$$123$$ 0 0
$$124$$ −200.000 −0.144843
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 1244.00 0.869190 0.434595 0.900626i $$-0.356891\pi$$
0.434595 + 0.900626i $$0.356891\pi$$
$$128$$ −1659.00 −1.14560
$$129$$ 0 0
$$130$$ −1110.00 −0.748873
$$131$$ 2328.00 1.55266 0.776329 0.630327i $$-0.217079\pi$$
0.776329 + 0.630327i $$0.217079\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 588.000 0.379071
$$135$$ 0 0
$$136$$ 1134.00 0.714998
$$137$$ −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
$$142$$ −864.000 −0.510600
$$143$$ −1776.00 −1.03858
$$144$$ 0 0
$$145$$ −390.000 −0.223364
$$146$$ −1290.00 −0.731241
$$147$$ 0 0
$$148$$ −70.0000 −0.0388781
$$149$$ −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 $$-0.569865\pi$$
−0.217729 + 0.976009i $$0.569865\pi$$
$$152$$ 2604.00 1.38955
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1000.00 0.518206
$$156$$ 0 0
$$157$$ −2378.00 −1.20882 −0.604411 0.796673i $$-0.706592\pi$$
−0.604411 + 0.796673i $$0.706592\pi$$
$$158$$ 1560.00 0.785487
$$159$$ 0 0
$$160$$ −225.000 −0.111174
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −52.0000 −0.0249874 −0.0124937 0.999922i $$-0.503977\pi$$
−0.0124937 + 0.999922i $$0.503977\pi$$
$$164$$ 330.000 0.157126
$$165$$ 0 0
$$166$$ −468.000 −0.218818
$$167$$ −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
$$170$$ 810.000 0.365436
$$171$$ 0 0
$$172$$ 92.0000 0.0407845
$$173$$ 426.000 0.187215 0.0936075 0.995609i $$-0.470160\pi$$
0.0936075 + 0.995609i $$0.470160\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1704.00 −0.729795
$$177$$ 0 0
$$178$$ −3078.00 −1.29610
$$179$$ 1440.00 0.601289 0.300644 0.953736i $$-0.402798\pi$$
0.300644 + 0.953736i $$0.402798\pi$$
$$180$$ 0 0
$$181$$ 3130.00 1.28537 0.642683 0.766133i $$-0.277821\pi$$
0.642683 + 0.766133i $$0.277821\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 2520.00 1.00966
$$185$$ 350.000 0.139095
$$186$$ 0 0
$$187$$ 1296.00 0.506807
$$188$$ −24.0000 −0.00931053
$$189$$ 0 0
$$190$$ 1860.00 0.710203
$$191$$ −3576.00 −1.35471 −0.677357 0.735655i $$-0.736875\pi$$
−0.677357 + 0.735655i $$0.736875\pi$$
$$192$$ 0 0
$$193$$ 2666.00 0.994315 0.497158 0.867660i $$-0.334377\pi$$
0.497158 + 0.867660i $$0.334377\pi$$
$$194$$ −858.000 −0.317530
$$195$$ 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 $$0.260884\pi$$
$$200$$ 525.000 0.185616
$$201$$ 0 0
$$202$$ 5202.00 1.81194
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −1650.00 −0.562151
$$206$$ 1356.00 0.458626
$$207$$ 0 0
$$208$$ 5254.00 1.75144
$$209$$ 2976.00 0.984948
$$210$$ 0 0
$$211$$ 1100.00 0.358896 0.179448 0.983767i $$-0.442569\pi$$
0.179448 + 0.983767i $$0.442569\pi$$
$$212$$ −450.000 −0.145784
$$213$$ 0 0
$$214$$ −4212.00 −1.34545
$$215$$ −460.000 −0.145915
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4422.00 1.37383
$$219$$ 0 0
$$220$$ −120.000 −0.0367745
$$221$$ −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
$$226$$ 3258.00 0.958933
$$227$$ 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 $$-0.411321\pi$$
0.275004 + 0.961443i $$0.411321\pi$$
$$230$$ 1800.00 0.516037
$$231$$ 0 0
$$232$$ 1638.00 0.463534
$$233$$ 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
$$236$$ 24.0000 0.00661978
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1176.00 −0.318281 −0.159140 0.987256i $$-0.550872\pi$$
−0.159140 + 0.987256i $$0.550872\pi$$
$$240$$ 0 0
$$241$$ −866.000 −0.231469 −0.115734 0.993280i $$-0.536922\pi$$
−0.115734 + 0.993280i $$0.536922\pi$$
$$242$$ 2265.00 0.601652
$$243$$ 0 0
$$244$$ 322.000 0.0844834
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −9176.00 −2.36379
$$248$$ −4200.00 −1.07540
$$249$$ 0 0
$$250$$ 375.000 0.0948683
$$251$$ 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
$$254$$ −3732.00 −0.921915
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ 2526.00 0.613103 0.306552 0.951854i $$-0.400825\pi$$
0.306552 + 0.951854i $$0.400825\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 370.000 0.0882555
$$261$$ 0 0
$$262$$ −6984.00 −1.64684
$$263$$ −5448.00 −1.27733 −0.638666 0.769484i $$-0.720513\pi$$
−0.638666 + 0.769484i $$0.720513\pi$$
$$264$$ 0 0
$$265$$ 2250.00 0.521571
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −196.000 −0.0446739
$$269$$ −2574.00 −0.583418 −0.291709 0.956507i $$-0.594224\pi$$
−0.291709 + 0.956507i $$0.594224\pi$$
$$270$$ 0 0
$$271$$ 3184.00 0.713706 0.356853 0.934161i $$-0.383850\pi$$
0.356853 + 0.934161i $$0.383850\pi$$
$$272$$ −3834.00 −0.854671
$$273$$ 0 0
$$274$$ 6354.00 1.40095
$$275$$ 600.000 0.131569
$$276$$ 0 0
$$277$$ 3962.00 0.859399 0.429699 0.902972i $$-0.358620\pi$$
0.429699 + 0.902972i $$0.358620\pi$$
$$278$$ 6972.00 1.50415
$$279$$ 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 $$0.407925\pi$$
$$284$$ 288.000 0.0601748
$$285$$ 0 0
$$286$$ 5328.00 1.10158
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1997.00 −0.406473
$$290$$ 1170.00 0.236913
$$291$$ 0 0
$$292$$ 430.000 0.0861776
$$293$$ 6018.00 1.19992 0.599958 0.800032i $$-0.295184\pi$$
0.599958 + 0.800032i $$0.295184\pi$$
$$294$$ 0 0
$$295$$ −120.000 −0.0236836
$$296$$ −1470.00 −0.288655
$$297$$ 0 0
$$298$$ 774.000 0.150458
$$299$$ −8880.00 −1.71754
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 2424.00 0.461873
$$303$$ 0 0
$$304$$ −8804.00 −1.66100
$$305$$ −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
$$310$$ −3000.00 −0.549640
$$311$$ 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 $$-0.269313\pi$$
0.662930 + 0.748681i $$0.269313\pi$$
$$314$$ 7134.00 1.28215
$$315$$ 0 0
$$316$$ −520.000 −0.0925705
$$317$$ 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
$$320$$ −2165.00 −0.378210
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 6696.00 1.15348
$$324$$ 0 0
$$325$$ −1850.00 −0.315752
$$326$$ 156.000 0.0265032
$$327$$ 0 0
$$328$$ 6930.00 1.16660
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 3692.00 0.613084 0.306542 0.951857i $$-0.400828\pi$$
0.306542 + 0.951857i $$0.400828\pi$$
$$332$$ 156.000 0.0257880
$$333$$ 0 0
$$334$$ 11160.0 1.82829
$$335$$ 980.000 0.159830
$$336$$ 0 0
$$337$$ −8998.00 −1.45446 −0.727229 0.686395i $$-0.759192\pi$$
−0.727229 + 0.686395i $$0.759192\pi$$
$$338$$ −9837.00 −1.58302
$$339$$ 0 0
$$340$$ −270.000 −0.0430671
$$341$$ −4800.00 −0.762271
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 1932.00 0.302809
$$345$$ 0 0
$$346$$ −1278.00 −0.198571
$$347$$ −5244.00 −0.811276 −0.405638 0.914034i $$-0.632951\pi$$
−0.405638 + 0.914034i $$0.632951\pi$$
$$348$$ 0 0
$$349$$ −6302.00 −0.966585 −0.483293 0.875459i $$-0.660559\pi$$
−0.483293 + 0.875459i $$0.660559\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1080.00 0.163535
$$353$$ 3414.00 0.514756 0.257378 0.966311i $$-0.417141\pi$$
0.257378 + 0.966311i $$0.417141\pi$$
$$354$$ 0 0
$$355$$ −1440.00 −0.215288
$$356$$ 1026.00 0.152747
$$357$$ 0 0
$$358$$ −4320.00 −0.637763
$$359$$ −4824.00 −0.709195 −0.354597 0.935019i $$-0.615382\pi$$
−0.354597 + 0.935019i $$0.615382\pi$$
$$360$$ 0 0
$$361$$ 8517.00 1.24173
$$362$$ −9390.00 −1.36334
$$363$$ 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 $$0.419741\pi$$
$$368$$ −8520.00 −1.20689
$$369$$ 0 0
$$370$$ −1050.00 −0.147532
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10802.0 1.49948 0.749740 0.661732i $$-0.230178\pi$$
0.749740 + 0.661732i $$0.230178\pi$$
$$374$$ −3888.00 −0.537550
$$375$$ 0 0
$$376$$ −504.000 −0.0691272
$$377$$ −5772.00 −0.788523
$$378$$ 0 0
$$379$$ 1460.00 0.197876 0.0989382 0.995094i $$-0.468455\pi$$
0.0989382 + 0.995094i $$0.468455\pi$$
$$380$$ −620.000 −0.0836982
$$381$$ 0 0
$$382$$ 10728.0 1.43689
$$383$$ −4872.00 −0.649994 −0.324997 0.945715i $$-0.605363\pi$$
−0.324997 + 0.945715i $$0.605363\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −7998.00 −1.05463
$$387$$ 0 0
$$388$$ 286.000 0.0374213
$$389$$ 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
$$394$$ −8154.00 −1.04262
$$395$$ 2600.00 0.331190
$$396$$ 0 0
$$397$$ 2734.00 0.345631 0.172816 0.984954i $$-0.444714\pi$$
0.172816 + 0.984954i $$0.444714\pi$$
$$398$$ −11496.0 −1.44785
$$399$$ 0 0
$$400$$ −1775.00 −0.221875
$$401$$ 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
$$404$$ −1734.00 −0.213539
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1680.00 −0.204606
$$408$$ 0 0
$$409$$ −8714.00 −1.05350 −0.526748 0.850022i $$-0.676589\pi$$
−0.526748 + 0.850022i $$0.676589\pi$$
$$410$$ 4950.00 0.596251
$$411$$ 0 0
$$412$$ −452.000 −0.0540496
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −780.000 −0.0922619
$$416$$ −3330.00 −0.392468
$$417$$ 0 0
$$418$$ −8928.00 −1.04470
$$419$$ 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 $$0.279004\pi$$
$$422$$ −3300.00 −0.380667
$$423$$ 0 0
$$424$$ −9450.00 −1.08239
$$425$$ 1350.00 0.154081
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 1404.00 0.158563
$$429$$ 0 0
$$430$$ 1380.00 0.154766
$$431$$ −720.000 −0.0804668 −0.0402334 0.999190i $$-0.512810\pi$$
−0.0402334 + 0.999190i $$0.512810\pi$$
$$432$$ 0 0
$$433$$ 15622.0 1.73382 0.866912 0.498462i $$-0.166102\pi$$
0.866912 + 0.498462i $$0.166102\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −1474.00 −0.161908
$$437$$ 14880.0 1.62885
$$438$$ 0 0
$$439$$ 9880.00 1.07414 0.537069 0.843538i $$-0.319531\pi$$
0.537069 + 0.843538i $$0.319531\pi$$
$$440$$ −2520.00 −0.273037
$$441$$ 0 0
$$442$$ 11988.0 1.29007
$$443$$ 16116.0 1.72843 0.864215 0.503123i $$-0.167816\pi$$
0.864215 + 0.503123i $$0.167816\pi$$
$$444$$ 0 0
$$445$$ −5130.00 −0.546484
$$446$$ 5892.00 0.625548
$$447$$ 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
$$452$$ −1086.00 −0.113011
$$453$$ 0 0
$$454$$ −1980.00 −0.204683
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3670.00 −0.375657 −0.187829 0.982202i $$-0.560145\pi$$
−0.187829 + 0.982202i $$0.560145\pi$$
$$458$$ −5718.00 −0.583372
$$459$$ 0 0
$$460$$ −600.000 −0.0608155
$$461$$ 17562.0 1.77428 0.887141 0.461499i $$-0.152688\pi$$
0.887141 + 0.461499i $$0.152688\pi$$
$$462$$ 0 0
$$463$$ 1172.00 0.117640 0.0588202 0.998269i $$-0.481266\pi$$
0.0588202 + 0.998269i $$0.481266\pi$$
$$464$$ −5538.00 −0.554084
$$465$$ 0 0
$$466$$ −4374.00 −0.434810
$$467$$ 6876.00 0.681335 0.340667 0.940184i $$-0.389347\pi$$
0.340667 + 0.940184i $$0.389347\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −360.000 −0.0353310
$$471$$ 0 0
$$472$$ 504.000 0.0491493
$$473$$ 2208.00 0.214638
$$474$$ 0 0
$$475$$ 3100.00 0.299448
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 3528.00 0.337588
$$479$$ 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
$$482$$ 2598.00 0.245510
$$483$$ 0 0
$$484$$ −755.000 −0.0709053
$$485$$ −1430.00 −0.133882
$$486$$ 0 0
$$487$$ −3076.00 −0.286215 −0.143108 0.989707i $$-0.545710\pi$$
−0.143108 + 0.989707i $$0.545710\pi$$
$$488$$ 6762.00 0.627257
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 18912.0 1.73826 0.869131 0.494582i $$-0.164679\pi$$
0.869131 + 0.494582i $$0.164679\pi$$
$$492$$ 0 0
$$493$$ 4212.00 0.384785
$$494$$ 27528.0 2.50717
$$495$$ 0 0
$$496$$ 14200.0 1.28548
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 9956.00 0.893170 0.446585 0.894741i $$-0.352640\pi$$
0.446585 + 0.894741i $$0.352640\pi$$
$$500$$ −125.000 −0.0111803
$$501$$ 0 0
$$502$$ −1296.00 −0.115226
$$503$$ −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
$$506$$ −8640.00 −0.759081
$$507$$ 0 0
$$508$$ 1244.00 0.108649
$$509$$ −2766.00 −0.240866 −0.120433 0.992721i $$-0.538428\pi$$
−0.120433 + 0.992721i $$0.538428\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 8733.00 0.753804
$$513$$ 0 0
$$514$$ −7578.00 −0.650294
$$515$$ 2260.00 0.193374
$$516$$ 0 0
$$517$$ −576.000 −0.0489989
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 7770.00 0.655264
$$521$$ 10530.0 0.885466 0.442733 0.896654i $$-0.354009\pi$$
0.442733 + 0.896654i $$0.354009\pi$$
$$522$$ 0 0
$$523$$ −12692.0 −1.06115 −0.530576 0.847637i $$-0.678024\pi$$
−0.530576 + 0.847637i $$0.678024\pi$$
$$524$$ 2328.00 0.194082
$$525$$ 0 0
$$526$$ 16344.0 1.35481
$$527$$ −10800.0 −0.892705
$$528$$ 0 0
$$529$$ 2233.00 0.183529
$$530$$ −6750.00 −0.553210
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −24420.0 −1.98452
$$534$$ 0 0
$$535$$ −7020.00 −0.567292
$$536$$ −4116.00 −0.331687
$$537$$ 0 0
$$538$$ 7722.00 0.618809
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 18110.0 1.43920 0.719602 0.694386i $$-0.244324\pi$$
0.719602 + 0.694386i $$0.244324\pi$$
$$542$$ −9552.00 −0.756999
$$543$$ 0 0
$$544$$ 2430.00 0.191517
$$545$$ 7370.00 0.579259
$$546$$ 0 0
$$547$$ 3620.00 0.282962 0.141481 0.989941i $$-0.454814\pi$$
0.141481 + 0.989941i $$0.454814\pi$$
$$548$$ −2118.00 −0.165103
$$549$$ 0 0
$$550$$ −1800.00 −0.139550
$$551$$ 9672.00 0.747806
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −11886.0 −0.911530
$$555$$ 0 0
$$556$$ −2324.00 −0.177265
$$557$$ 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
$$562$$ −24858.0 −1.86579
$$563$$ −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
$$566$$ −8148.00 −0.605099
$$567$$ 0 0
$$568$$ 6048.00 0.446775
$$569$$ 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 $$-0.590859\pi$$
−0.281581 + 0.959537i $$0.590859\pi$$
$$572$$ −1776.00 −0.129822
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3000.00 0.217580
$$576$$ 0 0
$$577$$ 1726.00 0.124531 0.0622654 0.998060i $$-0.480167\pi$$
0.0622654 + 0.998060i $$0.480167\pi$$
$$578$$ 5991.00 0.431129
$$579$$ 0 0
$$580$$ −390.000 −0.0279205
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −10800.0 −0.767222
$$584$$ 9030.00 0.639836
$$585$$ 0 0
$$586$$ −18054.0 −1.27270
$$587$$ 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
$$590$$ 360.000 0.0251203
$$591$$ 0 0
$$592$$ 4970.00 0.345043
$$593$$ 2862.00 0.198193 0.0990963 0.995078i $$-0.468405\pi$$
0.0990963 + 0.995078i $$0.468405\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −258.000 −0.0177317
$$597$$ 0 0
$$598$$ 26640.0 1.82172
$$599$$ 23592.0 1.60925 0.804627 0.593781i $$-0.202365\pi$$
0.804627 + 0.593781i $$0.202365\pi$$
$$600$$ 0 0
$$601$$ 9574.00 0.649803 0.324902 0.945748i $$-0.394669\pi$$
0.324902 + 0.945748i $$0.394669\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −808.000 −0.0544322
$$605$$ 3775.00 0.253679
$$606$$ 0 0
$$607$$ −17444.0 −1.16644 −0.583221 0.812314i $$-0.698208\pi$$
−0.583221 + 0.812314i $$0.698208\pi$$
$$608$$ 5580.00 0.372202
$$609$$ 0 0
$$610$$ 4830.00 0.320592
$$611$$ 1776.00 0.117593
$$612$$ 0 0
$$613$$ −2374.00 −0.156419 −0.0782096 0.996937i $$-0.524920\pi$$
−0.0782096 + 0.996937i $$0.524920\pi$$
$$614$$ 27708.0 1.82118
$$615$$ 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 $$0.592271\pi$$
$$620$$ 1000.00 0.0647758
$$621$$ 0 0
$$622$$ −4608.00 −0.297048
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −22026.0 −1.40629
$$627$$ 0 0
$$628$$ −2378.00 −0.151103
$$629$$ −3780.00 −0.239616
$$630$$ 0 0
$$631$$ −12688.0 −0.800478 −0.400239 0.916411i $$-0.631073\pi$$
−0.400239 + 0.916411i $$0.631073\pi$$
$$632$$ −10920.0 −0.687301
$$633$$ 0 0
$$634$$ −11682.0 −0.731785
$$635$$ −6220.00 −0.388714
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −5616.00 −0.348495
$$639$$ 0 0
$$640$$ 8295.00 0.512326
$$641$$ 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
$$646$$ −20088.0 −1.22345
$$647$$ −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
$$650$$ 5550.00 0.334906
$$651$$ 0 0
$$652$$ −52.0000 −0.00312343
$$653$$ −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
$$656$$ −23430.0 −1.39449
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −14520.0 −0.858299 −0.429149 0.903234i $$-0.641187\pi$$
−0.429149 + 0.903234i $$0.641187\pi$$
$$660$$ 0 0
$$661$$ 10618.0 0.624799 0.312400 0.949951i $$-0.398867\pi$$
0.312400 + 0.949951i $$0.398867\pi$$
$$662$$ −11076.0 −0.650273
$$663$$ 0 0
$$664$$ 3276.00 0.191466
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9360.00 0.543359
$$668$$ −3720.00 −0.215466
$$669$$ 0 0
$$670$$ −2940.00 −0.169526
$$671$$ 7728.00 0.444614
$$672$$ 0 0
$$673$$ 1370.00 0.0784690 0.0392345 0.999230i $$-0.487508\pi$$
0.0392345 + 0.999230i $$0.487508\pi$$
$$674$$ 26994.0 1.54269
$$675$$ 0 0
$$676$$ 3279.00 0.186561
$$677$$ −13758.0 −0.781038 −0.390519 0.920595i $$-0.627704\pi$$
−0.390519 + 0.920595i $$0.627704\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −5670.00 −0.319757
$$681$$ 0 0
$$682$$ 14400.0 0.808511
$$683$$ −11988.0 −0.671608 −0.335804 0.941932i $$-0.609008\pi$$
−0.335804 + 0.941932i $$0.609008\pi$$
$$684$$ 0 0
$$685$$ 10590.0 0.590691
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −6532.00 −0.361962
$$689$$ 33300.0 1.84126
$$690$$ 0 0
$$691$$ −32996.0 −1.81654 −0.908268 0.418388i $$-0.862595\pi$$
−0.908268 + 0.418388i $$0.862595\pi$$
$$692$$ 426.000 0.0234019
$$693$$ 0 0
$$694$$ 15732.0 0.860488
$$695$$ 11620.0 0.634204
$$696$$ 0 0
$$697$$ 17820.0 0.968408
$$698$$ 18906.0 1.02522
$$699$$ 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
$$704$$ 10392.0 0.556340
$$705$$ 0 0
$$706$$ −10242.0 −0.545981
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −27394.0 −1.45106 −0.725531 0.688189i $$-0.758406\pi$$
−0.725531 + 0.688189i $$0.758406\pi$$
$$710$$ 4320.00 0.228347
$$711$$ 0 0
$$712$$ 21546.0 1.13409
$$713$$ −24000.0 −1.26060
$$714$$ 0 0
$$715$$ 8880.00 0.464466
$$716$$ 1440.00 0.0751611
$$717$$ 0 0
$$718$$ 14472.0 0.752215
$$719$$ 34848.0 1.80753 0.903763 0.428033i $$-0.140793\pi$$
0.903763 + 0.428033i $$0.140793\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −25551.0 −1.31705
$$723$$ 0 0
$$724$$ 3130.00 0.160671
$$725$$ 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
$$730$$ 6450.00 0.327021
$$731$$ 4968.00 0.251365
$$732$$ 0 0
$$733$$ −18002.0 −0.907120 −0.453560 0.891226i $$-0.649846\pi$$
−0.453560 + 0.891226i $$0.649846\pi$$
$$734$$ −10524.0 −0.529221
$$735$$ 0 0
$$736$$ 5400.00 0.270444
$$737$$ −4704.00 −0.235107
$$738$$ 0 0
$$739$$ 15284.0 0.760800 0.380400 0.924822i $$-0.375786\pi$$
0.380400 + 0.924822i $$0.375786\pi$$
$$740$$ 350.000 0.0173868
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 18768.0 0.926691 0.463345 0.886178i $$-0.346649\pi$$
0.463345 + 0.886178i $$0.346649\pi$$
$$744$$ 0 0
$$745$$ 1290.00 0.0634388
$$746$$ −32406.0 −1.59044
$$747$$ 0 0
$$748$$ 1296.00 0.0633509
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 8696.00 0.422532 0.211266 0.977429i $$-0.432241\pi$$
0.211266 + 0.977429i $$0.432241\pi$$
$$752$$ 1704.00 0.0826310
$$753$$ 0 0
$$754$$ 17316.0 0.836355
$$755$$ 4040.00 0.194743
$$756$$ 0 0
$$757$$ −38662.0 −1.85627 −0.928134 0.372247i $$-0.878587\pi$$
−0.928134 + 0.372247i $$0.878587\pi$$
$$758$$ −4380.00 −0.209880
$$759$$ 0 0
$$760$$ −13020.0 −0.621428
$$761$$ 23874.0 1.13723 0.568615 0.822604i $$-0.307479\pi$$
0.568615 + 0.822604i $$0.307479\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −3576.00 −0.169339
$$765$$ 0 0
$$766$$ 14616.0 0.689422
$$767$$ −1776.00 −0.0836084
$$768$$ 0 0
$$769$$ −23618.0 −1.10753 −0.553763 0.832675i $$-0.686808\pi$$
−0.553763 + 0.832675i $$0.686808\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2666.00 0.124289
$$773$$ 11538.0 0.536860 0.268430 0.963299i $$-0.413495\pi$$
0.268430 + 0.963299i $$0.413495\pi$$
$$774$$ 0 0
$$775$$ −5000.00 −0.231749
$$776$$ 6006.00 0.277839
$$777$$ 0 0
$$778$$ −42138.0 −1.94180
$$779$$ 40920.0 1.88204
$$780$$ 0 0
$$781$$ 6912.00 0.316685
$$782$$ −19440.0 −0.888968
$$783$$ 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 $$0.390562\pi$$
$$788$$ 2718.00 0.122874
$$789$$ 0 0
$$790$$ −7800.00 −0.351280
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −23828.0 −1.06703
$$794$$ −8202.00 −0.366597
$$795$$ 0 0
$$796$$ 3832.00 0.170630
$$797$$ −11334.0 −0.503728 −0.251864 0.967763i $$-0.581043\pi$$
−0.251864 + 0.967763i $$0.581043\pi$$
$$798$$ 0 0
$$799$$ −1296.00 −0.0573832
$$800$$ 1125.00 0.0497184
$$801$$ 0 0
$$802$$ −47826.0 −2.10573
$$803$$ 10320.0 0.453530
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −44400.0 −1.94035
$$807$$ 0 0
$$808$$ −36414.0 −1.58545
$$809$$ −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
$$814$$ 5040.00 0.217017
$$815$$ 260.000 0.0111747
$$816$$ 0 0
$$817$$ 11408.0 0.488513
$$818$$ 26142.0 1.11740
$$819$$ 0 0
$$820$$ −1650.00 −0.0702689
$$821$$ 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 $$-0.424913\pi$$
0.233713 + 0.972306i $$0.424913\pi$$
$$824$$ −9492.00 −0.401298
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −25884.0 −1.08836 −0.544181 0.838968i $$-0.683159\pi$$
−0.544181 + 0.838968i $$0.683159\pi$$
$$828$$ 0 0
$$829$$ −15950.0 −0.668234 −0.334117 0.942532i $$-0.608438\pi$$
−0.334117 + 0.942532i $$0.608438\pi$$
$$830$$ 2340.00 0.0978585
$$831$$ 0 0
$$832$$ −32042.0 −1.33516
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 18600.0 0.770874
$$836$$ 2976.00 0.123119
$$837$$ 0 0
$$838$$ −35928.0 −1.48104
$$839$$ 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
$$842$$ −33162.0 −1.35729
$$843$$ 0 0
$$844$$ 1100.00 0.0448620
$$845$$ −16395.0 −0.667462
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 31950.0 1.29383
$$849$$ 0 0
$$850$$ −4050.00 −0.163428
$$851$$ −8400.00 −0.338365
$$852$$ 0 0
$$853$$ 27862.0 1.11838 0.559189 0.829040i $$-0.311113\pi$$
0.559189 + 0.829040i $$0.311113\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 29484.0 1.17727
$$857$$ −7314.00 −0.291530 −0.145765 0.989319i $$-0.546564\pi$$
−0.145765 + 0.989319i $$0.546564\pi$$
$$858$$ 0 0
$$859$$ 28780.0 1.14314 0.571572 0.820552i $$-0.306334\pi$$
0.571572 + 0.820552i $$0.306334\pi$$
$$860$$ −460.000 −0.0182394
$$861$$ 0 0
$$862$$ 2160.00 0.0853479
$$863$$ 32688.0 1.28935 0.644677 0.764455i $$-0.276992\pi$$
0.644677 + 0.764455i $$0.276992\pi$$
$$864$$ 0 0
$$865$$ −2130.00 −0.0837251
$$866$$ −46866.0 −1.83900
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −12480.0 −0.487175
$$870$$ 0 0
$$871$$ 14504.0 0.564236
$$872$$ −30954.0 −1.20210
$$873$$ 0 0
$$874$$ −44640.0 −1.72766
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 36650.0 1.41115 0.705577 0.708633i $$-0.250688\pi$$
0.705577 + 0.708633i $$0.250688\pi$$
$$878$$ −29640.0 −1.13930
$$879$$ 0 0
$$880$$ 8520.00 0.326374
$$881$$ −2646.00 −0.101187 −0.0505936 0.998719i $$-0.516111\pi$$
−0.0505936 + 0.998719i $$0.516111\pi$$
$$882$$ 0 0
$$883$$ 10892.0 0.415113 0.207557 0.978223i $$-0.433449\pi$$
0.207557 + 0.978223i $$0.433449\pi$$
$$884$$ −3996.00 −0.152036
$$885$$ 0 0
$$886$$ −48348.0 −1.83328
$$887$$ −43464.0 −1.64530 −0.822648 0.568550i $$-0.807504\pi$$
−0.822648 + 0.568550i $$0.807504\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 15390.0 0.579634
$$891$$ 0 0
$$892$$ −1964.00 −0.0737215
$$893$$ −2976.00 −0.111521
$$894$$ 0 0
$$895$$ −7200.00 −0.268904
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 27054.0 1.00535
$$899$$ −15600.0 −0.578742
$$900$$ 0 0
$$901$$ −24300.0 −0.898502
$$902$$ −23760.0 −0.877075
$$903$$ 0 0
$$904$$ −22806.0 −0.839067
$$905$$ −15650.0 −0.574833
$$906$$ 0 0
$$907$$ −14884.0 −0.544890 −0.272445 0.962171i $$-0.587832\pi$$
−0.272445 + 0.962171i $$0.587832\pi$$
$$908$$ 660.000 0.0241221
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 1248.00 0.0453876 0.0226938 0.999742i $$-0.492776\pi$$
0.0226938 + 0.999742i $$0.492776\pi$$
$$912$$ 0 0
$$913$$ 3744.00 0.135716
$$914$$ 11010.0 0.398445
$$915$$ 0 0
$$916$$ 1906.00 0.0687511
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −6640.00 −0.238339 −0.119169 0.992874i $$-0.538023\pi$$
−0.119169 + 0.992874i $$0.538023\pi$$
$$920$$ −12600.0 −0.451532
$$921$$ 0 0
$$922$$ −52686.0 −1.88191
$$923$$ −21312.0 −0.760014
$$924$$ 0 0
$$925$$ −1750.00 −0.0622050
$$926$$ −3516.00 −0.124776
$$927$$ 0 0
$$928$$ 3510.00 0.124161
$$929$$ 29946.0 1.05758 0.528792 0.848751i $$-0.322645\pi$$
0.528792 + 0.848751i $$0.322645\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 1458.00 0.0512429
$$933$$ 0 0
$$934$$ −20628.0 −0.722665
$$935$$ −6480.00 −0.226651
$$936$$ 0 0
$$937$$ −45002.0 −1.56900 −0.784499 0.620130i $$-0.787080\pi$$
−0.784499 + 0.620130i $$0.787080\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 120.000 0.00416380
$$941$$ 6090.00 0.210976 0.105488 0.994421i $$-0.466360\pi$$
0.105488 + 0.994421i $$0.466360\pi$$
$$942$$ 0 0
$$943$$ 39600.0 1.36750
$$944$$ −1704.00 −0.0587505
$$945$$ 0 0
$$946$$ −6624.00 −0.227658
$$947$$ −56388.0 −1.93491 −0.967457 0.253035i $$-0.918571\pi$$
−0.967457 + 0.253035i $$0.918571\pi$$
$$948$$ 0 0
$$949$$ −31820.0 −1.08843
$$950$$ −9300.00 −0.317612
$$951$$ 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
$$956$$ −1176.00 −0.0397851
$$957$$ 0 0
$$958$$ −6840.00 −0.230679
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 10209.0 0.342687
$$962$$ −15540.0 −0.520821
$$963$$ 0 0
$$964$$ −866.000 −0.0289336
$$965$$ −13330.0 −0.444671
$$966$$ 0 0
$$967$$ −42316.0 −1.40723 −0.703615 0.710582i $$-0.748432\pi$$
−0.703615 + 0.710582i $$0.748432\pi$$
$$968$$ −15855.0 −0.526445
$$969$$ 0 0
$$970$$ 4290.00 0.142004
$$971$$ 24480.0 0.809063 0.404532 0.914524i $$-0.367435\pi$$
0.404532 + 0.914524i $$0.367435\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 9228.00 0.303577
$$975$$ 0 0
$$976$$ −22862.0 −0.749790
$$977$$ 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
$$982$$ −56736.0 −1.84371
$$983$$ 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
$$986$$ −12636.0 −0.408126
$$987$$ 0 0
$$988$$ −9176.00 −0.295473
$$989$$ 11040.0 0.354956
$$990$$ 0 0
$$991$$ 47792.0 1.53195 0.765975 0.642870i $$-0.222256\pi$$
0.765975 + 0.642870i $$0.222256\pi$$
$$992$$ −9000.00 −0.288055
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −19160.0 −0.610465
$$996$$ 0 0
$$997$$ −9938.00 −0.315687 −0.157843 0.987464i $$-0.550454\pi$$
−0.157843 + 0.987464i $$0.550454\pi$$
$$998$$ −29868.0 −0.947350
$$999$$ 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 2205.4.a.c.1.1 1
3.2 odd 2 735.4.a.i.1.1 1
7.6 odd 2 45.4.a.b.1.1 1
21.20 even 2 15.4.a.b.1.1 1
28.27 even 2 720.4.a.r.1.1 1
35.13 even 4 225.4.b.d.199.2 2
35.27 even 4 225.4.b.d.199.1 2
35.34 odd 2 225.4.a.g.1.1 1
63.13 odd 6 405.4.e.k.136.1 2
63.20 even 6 405.4.e.d.271.1 2
63.34 odd 6 405.4.e.k.271.1 2
63.41 even 6 405.4.e.d.136.1 2
84.83 odd 2 240.4.a.f.1.1 1
105.62 odd 4 75.4.b.a.49.2 2
105.83 odd 4 75.4.b.a.49.1 2
105.104 even 2 75.4.a.a.1.1 1
168.83 odd 2 960.4.a.l.1.1 1
168.125 even 2 960.4.a.bi.1.1 1
231.230 odd 2 1815.4.a.a.1.1 1
420.83 even 4 1200.4.f.m.49.2 2
420.167 even 4 1200.4.f.m.49.1 2
420.419 odd 2 1200.4.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 21.20 even 2
45.4.a.b.1.1 1 7.6 odd 2
75.4.a.a.1.1 1 105.104 even 2
75.4.b.a.49.1 2 105.83 odd 4
75.4.b.a.49.2 2 105.62 odd 4
225.4.a.g.1.1 1 35.34 odd 2
225.4.b.d.199.1 2 35.27 even 4
225.4.b.d.199.2 2 35.13 even 4
240.4.a.f.1.1 1 84.83 odd 2
405.4.e.d.136.1 2 63.41 even 6
405.4.e.d.271.1 2 63.20 even 6
405.4.e.k.136.1 2 63.13 odd 6
405.4.e.k.271.1 2 63.34 odd 6
720.4.a.r.1.1 1 28.27 even 2
735.4.a.i.1.1 1 3.2 odd 2
960.4.a.l.1.1 1 168.83 odd 2
960.4.a.bi.1.1 1 168.125 even 2
1200.4.a.o.1.1 1 420.419 odd 2
1200.4.f.m.49.1 2 420.167 even 4
1200.4.f.m.49.2 2 420.83 even 4
1815.4.a.a.1.1 1 231.230 odd 2
2205.4.a.c.1.1 1 1.1 even 1 trivial