Properties

 Label 825.4.a.e.1.1 Level $825$ Weight $4$ Character 825.1 Self dual yes Analytic conductor $48.677$ 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] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$48.6765757547$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -8.00000 q^{4} -2.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -8.00000 q^{4} -2.00000 q^{7} +9.00000 q^{9} -11.0000 q^{11} -24.0000 q^{12} +22.0000 q^{13} +64.0000 q^{16} -72.0000 q^{17} +122.000 q^{19} -6.00000 q^{21} -72.0000 q^{23} +27.0000 q^{27} +16.0000 q^{28} +96.0000 q^{29} -112.000 q^{31} -33.0000 q^{33} -72.0000 q^{36} -266.000 q^{37} +66.0000 q^{39} -96.0000 q^{41} +382.000 q^{43} +88.0000 q^{44} -360.000 q^{47} +192.000 q^{48} -339.000 q^{49} -216.000 q^{51} -176.000 q^{52} -318.000 q^{53} +366.000 q^{57} +660.000 q^{59} -430.000 q^{61} -18.0000 q^{63} -512.000 q^{64} -380.000 q^{67} +576.000 q^{68} -216.000 q^{69} +168.000 q^{71} -218.000 q^{73} -976.000 q^{76} +22.0000 q^{77} -706.000 q^{79} +81.0000 q^{81} -1068.00 q^{83} +48.0000 q^{84} +288.000 q^{87} -6.00000 q^{89} -44.0000 q^{91} +576.000 q^{92} -336.000 q^{93} -686.000 q^{97} -99.0000 q^{99} +O(q^{100})$$

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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −8.00000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.00000 −0.107990 −0.0539949 0.998541i $$-0.517195\pi$$
−0.0539949 + 0.998541i $$0.517195\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ −24.0000 −0.577350
$$13$$ 22.0000 0.469362 0.234681 0.972072i $$-0.424595\pi$$
0.234681 + 0.972072i $$0.424595\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 64.0000 1.00000
$$17$$ −72.0000 −1.02721 −0.513605 0.858027i $$-0.671690\pi$$
−0.513605 + 0.858027i $$0.671690\pi$$
$$18$$ 0 0
$$19$$ 122.000 1.47309 0.736545 0.676388i $$-0.236456\pi$$
0.736545 + 0.676388i $$0.236456\pi$$
$$20$$ 0 0
$$21$$ −6.00000 −0.0623480
$$22$$ 0 0
$$23$$ −72.0000 −0.652741 −0.326370 0.945242i $$-0.605826\pi$$
−0.326370 + 0.945242i $$0.605826\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 16.0000 0.107990
$$29$$ 96.0000 0.614716 0.307358 0.951594i $$-0.400555\pi$$
0.307358 + 0.951594i $$0.400555\pi$$
$$30$$ 0 0
$$31$$ −112.000 −0.648897 −0.324448 0.945903i $$-0.605179\pi$$
−0.324448 + 0.945903i $$0.605179\pi$$
$$32$$ 0 0
$$33$$ −33.0000 −0.174078
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −72.0000 −0.333333
$$37$$ −266.000 −1.18190 −0.590948 0.806710i $$-0.701246\pi$$
−0.590948 + 0.806710i $$0.701246\pi$$
$$38$$ 0 0
$$39$$ 66.0000 0.270986
$$40$$ 0 0
$$41$$ −96.0000 −0.365675 −0.182838 0.983143i $$-0.558528\pi$$
−0.182838 + 0.983143i $$0.558528\pi$$
$$42$$ 0 0
$$43$$ 382.000 1.35475 0.677377 0.735636i $$-0.263116\pi$$
0.677377 + 0.735636i $$0.263116\pi$$
$$44$$ 88.0000 0.301511
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −360.000 −1.11726 −0.558632 0.829416i $$-0.688674\pi$$
−0.558632 + 0.829416i $$0.688674\pi$$
$$48$$ 192.000 0.577350
$$49$$ −339.000 −0.988338
$$50$$ 0 0
$$51$$ −216.000 −0.593060
$$52$$ −176.000 −0.469362
$$53$$ −318.000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 366.000 0.850489
$$58$$ 0 0
$$59$$ 660.000 1.45635 0.728175 0.685391i $$-0.240369\pi$$
0.728175 + 0.685391i $$0.240369\pi$$
$$60$$ 0 0
$$61$$ −430.000 −0.902555 −0.451278 0.892384i $$-0.649032\pi$$
−0.451278 + 0.892384i $$0.649032\pi$$
$$62$$ 0 0
$$63$$ −18.0000 −0.0359966
$$64$$ −512.000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −380.000 −0.692901 −0.346451 0.938068i $$-0.612613\pi$$
−0.346451 + 0.938068i $$0.612613\pi$$
$$68$$ 576.000 1.02721
$$69$$ −216.000 −0.376860
$$70$$ 0 0
$$71$$ 168.000 0.280816 0.140408 0.990094i $$-0.455159\pi$$
0.140408 + 0.990094i $$0.455159\pi$$
$$72$$ 0 0
$$73$$ −218.000 −0.349520 −0.174760 0.984611i $$-0.555915\pi$$
−0.174760 + 0.984611i $$0.555915\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ −976.000 −1.47309
$$77$$ 22.0000 0.0325602
$$78$$ 0 0
$$79$$ −706.000 −1.00546 −0.502729 0.864444i $$-0.667671\pi$$
−0.502729 + 0.864444i $$0.667671\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −1068.00 −1.41239 −0.706194 0.708018i $$-0.749589\pi$$
−0.706194 + 0.708018i $$0.749589\pi$$
$$84$$ 48.0000 0.0623480
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 288.000 0.354906
$$88$$ 0 0
$$89$$ −6.00000 −0.00714605 −0.00357303 0.999994i $$-0.501137\pi$$
−0.00357303 + 0.999994i $$0.501137\pi$$
$$90$$ 0 0
$$91$$ −44.0000 −0.0506863
$$92$$ 576.000 0.652741
$$93$$ −336.000 −0.374641
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −686.000 −0.718070 −0.359035 0.933324i $$-0.616894\pi$$
−0.359035 + 0.933324i $$0.616894\pi$$
$$98$$ 0 0
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ −960.000 −0.945778 −0.472889 0.881122i $$-0.656789\pi$$
−0.472889 + 0.881122i $$0.656789\pi$$
$$102$$ 0 0
$$103$$ 844.000 0.807396 0.403698 0.914892i $$-0.367725\pi$$
0.403698 + 0.914892i $$0.367725\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −2172.00 −1.96238 −0.981192 0.193033i $$-0.938168\pi$$
−0.981192 + 0.193033i $$0.938168\pi$$
$$108$$ −216.000 −0.192450
$$109$$ 614.000 0.539546 0.269773 0.962924i $$-0.413051\pi$$
0.269773 + 0.962924i $$0.413051\pi$$
$$110$$ 0 0
$$111$$ −798.000 −0.682368
$$112$$ −128.000 −0.107990
$$113$$ −1254.00 −1.04395 −0.521975 0.852961i $$-0.674805\pi$$
−0.521975 + 0.852961i $$0.674805\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −768.000 −0.614716
$$117$$ 198.000 0.156454
$$118$$ 0 0
$$119$$ 144.000 0.110928
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −288.000 −0.211123
$$124$$ 896.000 0.648897
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1394.00 −0.973996 −0.486998 0.873403i $$-0.661908\pi$$
−0.486998 + 0.873403i $$0.661908\pi$$
$$128$$ 0 0
$$129$$ 1146.00 0.782168
$$130$$ 0 0
$$131$$ −252.000 −0.168071 −0.0840357 0.996463i $$-0.526781\pi$$
−0.0840357 + 0.996463i $$0.526781\pi$$
$$132$$ 264.000 0.174078
$$133$$ −244.000 −0.159079
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1050.00 0.654800 0.327400 0.944886i $$-0.393828\pi$$
0.327400 + 0.944886i $$0.393828\pi$$
$$138$$ 0 0
$$139$$ 1874.00 1.14353 0.571765 0.820418i $$-0.306259\pi$$
0.571765 + 0.820418i $$0.306259\pi$$
$$140$$ 0 0
$$141$$ −1080.00 −0.645053
$$142$$ 0 0
$$143$$ −242.000 −0.141518
$$144$$ 576.000 0.333333
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1017.00 −0.570617
$$148$$ 2128.00 1.18190
$$149$$ 1476.00 0.811534 0.405767 0.913976i $$-0.367004\pi$$
0.405767 + 0.913976i $$0.367004\pi$$
$$150$$ 0 0
$$151$$ 1478.00 0.796543 0.398271 0.917268i $$-0.369610\pi$$
0.398271 + 0.917268i $$0.369610\pi$$
$$152$$ 0 0
$$153$$ −648.000 −0.342403
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −528.000 −0.270986
$$157$$ −854.000 −0.434119 −0.217059 0.976158i $$-0.569646\pi$$
−0.217059 + 0.976158i $$0.569646\pi$$
$$158$$ 0 0
$$159$$ −954.000 −0.475831
$$160$$ 0 0
$$161$$ 144.000 0.0704894
$$162$$ 0 0
$$163$$ −1544.00 −0.741935 −0.370968 0.928646i $$-0.620974\pi$$
−0.370968 + 0.928646i $$0.620974\pi$$
$$164$$ 768.000 0.365675
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −240.000 −0.111208 −0.0556041 0.998453i $$-0.517708\pi$$
−0.0556041 + 0.998453i $$0.517708\pi$$
$$168$$ 0 0
$$169$$ −1713.00 −0.779700
$$170$$ 0 0
$$171$$ 1098.00 0.491030
$$172$$ −3056.00 −1.35475
$$173$$ −2532.00 −1.11274 −0.556371 0.830934i $$-0.687807\pi$$
−0.556371 + 0.830934i $$0.687807\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −704.000 −0.301511
$$177$$ 1980.00 0.840824
$$178$$ 0 0
$$179$$ 1092.00 0.455977 0.227989 0.973664i $$-0.426785\pi$$
0.227989 + 0.973664i $$0.426785\pi$$
$$180$$ 0 0
$$181$$ −2290.00 −0.940411 −0.470205 0.882557i $$-0.655820\pi$$
−0.470205 + 0.882557i $$0.655820\pi$$
$$182$$ 0 0
$$183$$ −1290.00 −0.521090
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 792.000 0.309715
$$188$$ 2880.00 1.11726
$$189$$ −54.0000 −0.0207827
$$190$$ 0 0
$$191$$ −4392.00 −1.66384 −0.831921 0.554894i $$-0.812759\pi$$
−0.831921 + 0.554894i $$0.812759\pi$$
$$192$$ −1536.00 −0.577350
$$193$$ 5074.00 1.89241 0.946203 0.323572i $$-0.104884\pi$$
0.946203 + 0.323572i $$0.104884\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 2712.00 0.988338
$$197$$ −1692.00 −0.611929 −0.305964 0.952043i $$-0.598979\pi$$
−0.305964 + 0.952043i $$0.598979\pi$$
$$198$$ 0 0
$$199$$ 4664.00 1.66142 0.830709 0.556707i $$-0.187935\pi$$
0.830709 + 0.556707i $$0.187935\pi$$
$$200$$ 0 0
$$201$$ −1140.00 −0.400047
$$202$$ 0 0
$$203$$ −192.000 −0.0663830
$$204$$ 1728.00 0.593060
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −648.000 −0.217580
$$208$$ 1408.00 0.469362
$$209$$ −1342.00 −0.444153
$$210$$ 0 0
$$211$$ −1870.00 −0.610124 −0.305062 0.952333i $$-0.598677\pi$$
−0.305062 + 0.952333i $$0.598677\pi$$
$$212$$ 2544.00 0.824163
$$213$$ 504.000 0.162129
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 224.000 0.0700742
$$218$$ 0 0
$$219$$ −654.000 −0.201796
$$220$$ 0 0
$$221$$ −1584.00 −0.482133
$$222$$ 0 0
$$223$$ −2300.00 −0.690670 −0.345335 0.938479i $$-0.612235\pi$$
−0.345335 + 0.938479i $$0.612235\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 1332.00 0.389462 0.194731 0.980857i $$-0.437617\pi$$
0.194731 + 0.980857i $$0.437617\pi$$
$$228$$ −2928.00 −0.850489
$$229$$ −6022.00 −1.73775 −0.868875 0.495031i $$-0.835157\pi$$
−0.868875 + 0.495031i $$0.835157\pi$$
$$230$$ 0 0
$$231$$ 66.0000 0.0187986
$$232$$ 0 0
$$233$$ 4716.00 1.32599 0.662994 0.748624i $$-0.269285\pi$$
0.662994 + 0.748624i $$0.269285\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −5280.00 −1.45635
$$237$$ −2118.00 −0.580502
$$238$$ 0 0
$$239$$ −6420.00 −1.73755 −0.868777 0.495204i $$-0.835093\pi$$
−0.868777 + 0.495204i $$0.835093\pi$$
$$240$$ 0 0
$$241$$ 3302.00 0.882575 0.441287 0.897366i $$-0.354522\pi$$
0.441287 + 0.897366i $$0.354522\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 3440.00 0.902555
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2684.00 0.691412
$$248$$ 0 0
$$249$$ −3204.00 −0.815443
$$250$$ 0 0
$$251$$ 732.000 0.184077 0.0920387 0.995755i $$-0.470662\pi$$
0.0920387 + 0.995755i $$0.470662\pi$$
$$252$$ 144.000 0.0359966
$$253$$ 792.000 0.196809
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 4096.00 1.00000
$$257$$ 3438.00 0.834461 0.417231 0.908801i $$-0.363001\pi$$
0.417231 + 0.908801i $$0.363001\pi$$
$$258$$ 0 0
$$259$$ 532.000 0.127633
$$260$$ 0 0
$$261$$ 864.000 0.204905
$$262$$ 0 0
$$263$$ 696.000 0.163183 0.0815916 0.996666i $$-0.474000\pi$$
0.0815916 + 0.996666i $$0.474000\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −18.0000 −0.00412578
$$268$$ 3040.00 0.692901
$$269$$ −7338.00 −1.66322 −0.831609 0.555361i $$-0.812580\pi$$
−0.831609 + 0.555361i $$0.812580\pi$$
$$270$$ 0 0
$$271$$ 5114.00 1.14632 0.573161 0.819443i $$-0.305717\pi$$
0.573161 + 0.819443i $$0.305717\pi$$
$$272$$ −4608.00 −1.02721
$$273$$ −132.000 −0.0292637
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 1728.00 0.376860
$$277$$ −986.000 −0.213874 −0.106937 0.994266i $$-0.534104\pi$$
−0.106937 + 0.994266i $$0.534104\pi$$
$$278$$ 0 0
$$279$$ −1008.00 −0.216299
$$280$$ 0 0
$$281$$ 3312.00 0.703122 0.351561 0.936165i $$-0.385651\pi$$
0.351561 + 0.936165i $$0.385651\pi$$
$$282$$ 0 0
$$283$$ −4298.00 −0.902790 −0.451395 0.892324i $$-0.649073\pi$$
−0.451395 + 0.892324i $$0.649073\pi$$
$$284$$ −1344.00 −0.280816
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 192.000 0.0394892
$$288$$ 0 0
$$289$$ 271.000 0.0551598
$$290$$ 0 0
$$291$$ −2058.00 −0.414578
$$292$$ 1744.00 0.349520
$$293$$ −2736.00 −0.545525 −0.272763 0.962081i $$-0.587937\pi$$
−0.272763 + 0.962081i $$0.587937\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −297.000 −0.0580259
$$298$$ 0 0
$$299$$ −1584.00 −0.306372
$$300$$ 0 0
$$301$$ −764.000 −0.146300
$$302$$ 0 0
$$303$$ −2880.00 −0.546045
$$304$$ 7808.00 1.47309
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 250.000 0.0464764 0.0232382 0.999730i $$-0.492602\pi$$
0.0232382 + 0.999730i $$0.492602\pi$$
$$308$$ −176.000 −0.0325602
$$309$$ 2532.00 0.466150
$$310$$ 0 0
$$311$$ 7248.00 1.32153 0.660766 0.750592i $$-0.270232\pi$$
0.660766 + 0.750592i $$0.270232\pi$$
$$312$$ 0 0
$$313$$ 7786.00 1.40604 0.703020 0.711170i $$-0.251834\pi$$
0.703020 + 0.711170i $$0.251834\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 5648.00 1.00546
$$317$$ 4230.00 0.749465 0.374733 0.927133i $$-0.377735\pi$$
0.374733 + 0.927133i $$0.377735\pi$$
$$318$$ 0 0
$$319$$ −1056.00 −0.185344
$$320$$ 0 0
$$321$$ −6516.00 −1.13298
$$322$$ 0 0
$$323$$ −8784.00 −1.51317
$$324$$ −648.000 −0.111111
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 1842.00 0.311507
$$328$$ 0 0
$$329$$ 720.000 0.120653
$$330$$ 0 0
$$331$$ 7736.00 1.28462 0.642310 0.766445i $$-0.277976\pi$$
0.642310 + 0.766445i $$0.277976\pi$$
$$332$$ 8544.00 1.41239
$$333$$ −2394.00 −0.393965
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −384.000 −0.0623480
$$337$$ 2014.00 0.325548 0.162774 0.986663i $$-0.447956\pi$$
0.162774 + 0.986663i $$0.447956\pi$$
$$338$$ 0 0
$$339$$ −3762.00 −0.602725
$$340$$ 0 0
$$341$$ 1232.00 0.195650
$$342$$ 0 0
$$343$$ 1364.00 0.214720
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −7692.00 −1.18999 −0.594997 0.803728i $$-0.702847\pi$$
−0.594997 + 0.803728i $$0.702847\pi$$
$$348$$ −2304.00 −0.354906
$$349$$ −1750.00 −0.268411 −0.134205 0.990954i $$-0.542848\pi$$
−0.134205 + 0.990954i $$0.542848\pi$$
$$350$$ 0 0
$$351$$ 594.000 0.0903287
$$352$$ 0 0
$$353$$ −8034.00 −1.21135 −0.605675 0.795712i $$-0.707097\pi$$
−0.605675 + 0.795712i $$0.707097\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 48.0000 0.00714605
$$357$$ 432.000 0.0640444
$$358$$ 0 0
$$359$$ 2304.00 0.338720 0.169360 0.985554i $$-0.445830\pi$$
0.169360 + 0.985554i $$0.445830\pi$$
$$360$$ 0 0
$$361$$ 8025.00 1.17000
$$362$$ 0 0
$$363$$ 363.000 0.0524864
$$364$$ 352.000 0.0506863
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 2356.00 0.335101 0.167551 0.985863i $$-0.446414\pi$$
0.167551 + 0.985863i $$0.446414\pi$$
$$368$$ −4608.00 −0.652741
$$369$$ −864.000 −0.121892
$$370$$ 0 0
$$371$$ 636.000 0.0890013
$$372$$ 2688.00 0.374641
$$373$$ 8602.00 1.19409 0.597044 0.802209i $$-0.296342\pi$$
0.597044 + 0.802209i $$0.296342\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 2112.00 0.288524
$$378$$ 0 0
$$379$$ −12016.0 −1.62855 −0.814275 0.580479i $$-0.802865\pi$$
−0.814275 + 0.580479i $$0.802865\pi$$
$$380$$ 0 0
$$381$$ −4182.00 −0.562337
$$382$$ 0 0
$$383$$ −1728.00 −0.230540 −0.115270 0.993334i $$-0.536773\pi$$
−0.115270 + 0.993334i $$0.536773\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3438.00 0.451585
$$388$$ 5488.00 0.718070
$$389$$ 8010.00 1.04402 0.522009 0.852940i $$-0.325183\pi$$
0.522009 + 0.852940i $$0.325183\pi$$
$$390$$ 0 0
$$391$$ 5184.00 0.670502
$$392$$ 0 0
$$393$$ −756.000 −0.0970360
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 792.000 0.100504
$$397$$ 10150.0 1.28316 0.641579 0.767057i $$-0.278280\pi$$
0.641579 + 0.767057i $$0.278280\pi$$
$$398$$ 0 0
$$399$$ −732.000 −0.0918442
$$400$$ 0 0
$$401$$ 11862.0 1.47721 0.738604 0.674140i $$-0.235486\pi$$
0.738604 + 0.674140i $$0.235486\pi$$
$$402$$ 0 0
$$403$$ −2464.00 −0.304567
$$404$$ 7680.00 0.945778
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2926.00 0.356355
$$408$$ 0 0
$$409$$ −682.000 −0.0824517 −0.0412258 0.999150i $$-0.513126\pi$$
−0.0412258 + 0.999150i $$0.513126\pi$$
$$410$$ 0 0
$$411$$ 3150.00 0.378049
$$412$$ −6752.00 −0.807396
$$413$$ −1320.00 −0.157271
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 5622.00 0.660217
$$418$$ 0 0
$$419$$ 10836.0 1.26342 0.631710 0.775205i $$-0.282353\pi$$
0.631710 + 0.775205i $$0.282353\pi$$
$$420$$ 0 0
$$421$$ 12350.0 1.42970 0.714848 0.699280i $$-0.246496\pi$$
0.714848 + 0.699280i $$0.246496\pi$$
$$422$$ 0 0
$$423$$ −3240.00 −0.372421
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 860.000 0.0974668
$$428$$ 17376.0 1.96238
$$429$$ −726.000 −0.0817054
$$430$$ 0 0
$$431$$ −5940.00 −0.663851 −0.331925 0.943306i $$-0.607698\pi$$
−0.331925 + 0.943306i $$0.607698\pi$$
$$432$$ 1728.00 0.192450
$$433$$ 12898.0 1.43150 0.715749 0.698358i $$-0.246086\pi$$
0.715749 + 0.698358i $$0.246086\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −4912.00 −0.539546
$$437$$ −8784.00 −0.961546
$$438$$ 0 0
$$439$$ 11450.0 1.24483 0.622413 0.782689i $$-0.286152\pi$$
0.622413 + 0.782689i $$0.286152\pi$$
$$440$$ 0 0
$$441$$ −3051.00 −0.329446
$$442$$ 0 0
$$443$$ −2100.00 −0.225224 −0.112612 0.993639i $$-0.535922\pi$$
−0.112612 + 0.993639i $$0.535922\pi$$
$$444$$ 6384.00 0.682368
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 4428.00 0.468540
$$448$$ 1024.00 0.107990
$$449$$ −11934.0 −1.25434 −0.627172 0.778881i $$-0.715788\pi$$
−0.627172 + 0.778881i $$0.715788\pi$$
$$450$$ 0 0
$$451$$ 1056.00 0.110255
$$452$$ 10032.0 1.04395
$$453$$ 4434.00 0.459884
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −578.000 −0.0591635 −0.0295817 0.999562i $$-0.509418\pi$$
−0.0295817 + 0.999562i $$0.509418\pi$$
$$458$$ 0 0
$$459$$ −1944.00 −0.197687
$$460$$ 0 0
$$461$$ −324.000 −0.0327336 −0.0163668 0.999866i $$-0.505210\pi$$
−0.0163668 + 0.999866i $$0.505210\pi$$
$$462$$ 0 0
$$463$$ 11788.0 1.18323 0.591614 0.806221i $$-0.298491\pi$$
0.591614 + 0.806221i $$0.298491\pi$$
$$464$$ 6144.00 0.614716
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 14484.0 1.43520 0.717601 0.696454i $$-0.245240\pi$$
0.717601 + 0.696454i $$0.245240\pi$$
$$468$$ −1584.00 −0.156454
$$469$$ 760.000 0.0748263
$$470$$ 0 0
$$471$$ −2562.00 −0.250638
$$472$$ 0 0
$$473$$ −4202.00 −0.408474
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1152.00 −0.110928
$$477$$ −2862.00 −0.274721
$$478$$ 0 0
$$479$$ −3084.00 −0.294179 −0.147089 0.989123i $$-0.546990\pi$$
−0.147089 + 0.989123i $$0.546990\pi$$
$$480$$ 0 0
$$481$$ −5852.00 −0.554736
$$482$$ 0 0
$$483$$ 432.000 0.0406971
$$484$$ −968.000 −0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 5584.00 0.519579 0.259790 0.965665i $$-0.416347\pi$$
0.259790 + 0.965665i $$0.416347\pi$$
$$488$$ 0 0
$$489$$ −4632.00 −0.428356
$$490$$ 0 0
$$491$$ −10752.0 −0.988250 −0.494125 0.869391i $$-0.664512\pi$$
−0.494125 + 0.869391i $$0.664512\pi$$
$$492$$ 2304.00 0.211123
$$493$$ −6912.00 −0.631442
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −7168.00 −0.648897
$$497$$ −336.000 −0.0303253
$$498$$ 0 0
$$499$$ −13372.0 −1.19963 −0.599813 0.800141i $$-0.704758\pi$$
−0.599813 + 0.800141i $$0.704758\pi$$
$$500$$ 0 0
$$501$$ −720.000 −0.0642060
$$502$$ 0 0
$$503$$ −9072.00 −0.804176 −0.402088 0.915601i $$-0.631715\pi$$
−0.402088 + 0.915601i $$0.631715\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −5139.00 −0.450160
$$508$$ 11152.0 0.973996
$$509$$ −14586.0 −1.27016 −0.635082 0.772445i $$-0.719034\pi$$
−0.635082 + 0.772445i $$0.719034\pi$$
$$510$$ 0 0
$$511$$ 436.000 0.0377446
$$512$$ 0 0
$$513$$ 3294.00 0.283496
$$514$$ 0 0
$$515$$ 0 0
$$516$$ −9168.00 −0.782168
$$517$$ 3960.00 0.336868
$$518$$ 0 0
$$519$$ −7596.00 −0.642442
$$520$$ 0 0
$$521$$ 2718.00 0.228556 0.114278 0.993449i $$-0.463545\pi$$
0.114278 + 0.993449i $$0.463545\pi$$
$$522$$ 0 0
$$523$$ 2086.00 0.174406 0.0872031 0.996191i $$-0.472207\pi$$
0.0872031 + 0.996191i $$0.472207\pi$$
$$524$$ 2016.00 0.168071
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8064.00 0.666553
$$528$$ −2112.00 −0.174078
$$529$$ −6983.00 −0.573929
$$530$$ 0 0
$$531$$ 5940.00 0.485450
$$532$$ 1952.00 0.159079
$$533$$ −2112.00 −0.171634
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 3276.00 0.263259
$$538$$ 0 0
$$539$$ 3729.00 0.297995
$$540$$ 0 0
$$541$$ 13838.0 1.09971 0.549854 0.835261i $$-0.314683\pi$$
0.549854 + 0.835261i $$0.314683\pi$$
$$542$$ 0 0
$$543$$ −6870.00 −0.542946
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −23546.0 −1.84050 −0.920251 0.391329i $$-0.872015\pi$$
−0.920251 + 0.391329i $$0.872015\pi$$
$$548$$ −8400.00 −0.654800
$$549$$ −3870.00 −0.300852
$$550$$ 0 0
$$551$$ 11712.0 0.905532
$$552$$ 0 0
$$553$$ 1412.00 0.108579
$$554$$ 0 0
$$555$$ 0 0
$$556$$ −14992.0 −1.14353
$$557$$ −15624.0 −1.18853 −0.594264 0.804270i $$-0.702557\pi$$
−0.594264 + 0.804270i $$0.702557\pi$$
$$558$$ 0 0
$$559$$ 8404.00 0.635870
$$560$$ 0 0
$$561$$ 2376.00 0.178814
$$562$$ 0 0
$$563$$ −2400.00 −0.179659 −0.0898294 0.995957i $$-0.528632\pi$$
−0.0898294 + 0.995957i $$0.528632\pi$$
$$564$$ 8640.00 0.645053
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −162.000 −0.0119989
$$568$$ 0 0
$$569$$ −18300.0 −1.34829 −0.674144 0.738600i $$-0.735487\pi$$
−0.674144 + 0.738600i $$0.735487\pi$$
$$570$$ 0 0
$$571$$ 25454.0 1.86553 0.932764 0.360487i $$-0.117389\pi$$
0.932764 + 0.360487i $$0.117389\pi$$
$$572$$ 1936.00 0.141518
$$573$$ −13176.0 −0.960620
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −4608.00 −0.333333
$$577$$ −19802.0 −1.42871 −0.714357 0.699781i $$-0.753281\pi$$
−0.714357 + 0.699781i $$0.753281\pi$$
$$578$$ 0 0
$$579$$ 15222.0 1.09258
$$580$$ 0 0
$$581$$ 2136.00 0.152524
$$582$$ 0 0
$$583$$ 3498.00 0.248495
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −18396.0 −1.29350 −0.646750 0.762702i $$-0.723872\pi$$
−0.646750 + 0.762702i $$0.723872\pi$$
$$588$$ 8136.00 0.570617
$$589$$ −13664.0 −0.955883
$$590$$ 0 0
$$591$$ −5076.00 −0.353297
$$592$$ −17024.0 −1.18190
$$593$$ −15012.0 −1.03958 −0.519788 0.854295i $$-0.673989\pi$$
−0.519788 + 0.854295i $$0.673989\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −11808.0 −0.811534
$$597$$ 13992.0 0.959220
$$598$$ 0 0
$$599$$ 15408.0 1.05101 0.525504 0.850791i $$-0.323877\pi$$
0.525504 + 0.850791i $$0.323877\pi$$
$$600$$ 0 0
$$601$$ −1558.00 −0.105744 −0.0528720 0.998601i $$-0.516838\pi$$
−0.0528720 + 0.998601i $$0.516838\pi$$
$$602$$ 0 0
$$603$$ −3420.00 −0.230967
$$604$$ −11824.0 −0.796543
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −22970.0 −1.53595 −0.767977 0.640478i $$-0.778736\pi$$
−0.767977 + 0.640478i $$0.778736\pi$$
$$608$$ 0 0
$$609$$ −576.000 −0.0383263
$$610$$ 0 0
$$611$$ −7920.00 −0.524401
$$612$$ 5184.00 0.342403
$$613$$ 11482.0 0.756531 0.378266 0.925697i $$-0.376521\pi$$
0.378266 + 0.925697i $$0.376521\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −246.000 −0.0160512 −0.00802560 0.999968i $$-0.502555\pi$$
−0.00802560 + 0.999968i $$0.502555\pi$$
$$618$$ 0 0
$$619$$ 11648.0 0.756337 0.378169 0.925737i $$-0.376554\pi$$
0.378169 + 0.925737i $$0.376554\pi$$
$$620$$ 0 0
$$621$$ −1944.00 −0.125620
$$622$$ 0 0
$$623$$ 12.0000 0.000771701 0
$$624$$ 4224.00 0.270986
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −4026.00 −0.256432
$$628$$ 6832.00 0.434119
$$629$$ 19152.0 1.21405
$$630$$ 0 0
$$631$$ −22024.0 −1.38948 −0.694740 0.719261i $$-0.744480\pi$$
−0.694740 + 0.719261i $$0.744480\pi$$
$$632$$ 0 0
$$633$$ −5610.00 −0.352255
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 7632.00 0.475831
$$637$$ −7458.00 −0.463888
$$638$$ 0 0
$$639$$ 1512.00 0.0936053
$$640$$ 0 0
$$641$$ 2322.00 0.143079 0.0715394 0.997438i $$-0.477209\pi$$
0.0715394 + 0.997438i $$0.477209\pi$$
$$642$$ 0 0
$$643$$ −14024.0 −0.860113 −0.430056 0.902802i $$-0.641506\pi$$
−0.430056 + 0.902802i $$0.641506\pi$$
$$644$$ −1152.00 −0.0704894
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 7152.00 0.434581 0.217291 0.976107i $$-0.430278\pi$$
0.217291 + 0.976107i $$0.430278\pi$$
$$648$$ 0 0
$$649$$ −7260.00 −0.439106
$$650$$ 0 0
$$651$$ 672.000 0.0404574
$$652$$ 12352.0 0.741935
$$653$$ 3138.00 0.188054 0.0940271 0.995570i $$-0.470026\pi$$
0.0940271 + 0.995570i $$0.470026\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −6144.00 −0.365675
$$657$$ −1962.00 −0.116507
$$658$$ 0 0
$$659$$ −15876.0 −0.938454 −0.469227 0.883078i $$-0.655467\pi$$
−0.469227 + 0.883078i $$0.655467\pi$$
$$660$$ 0 0
$$661$$ −20554.0 −1.20947 −0.604734 0.796428i $$-0.706720\pi$$
−0.604734 + 0.796428i $$0.706720\pi$$
$$662$$ 0 0
$$663$$ −4752.00 −0.278360
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6912.00 −0.401250
$$668$$ 1920.00 0.111208
$$669$$ −6900.00 −0.398758
$$670$$ 0 0
$$671$$ 4730.00 0.272131
$$672$$ 0 0
$$673$$ −27806.0 −1.59263 −0.796317 0.604880i $$-0.793221\pi$$
−0.796317 + 0.604880i $$0.793221\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 13704.0 0.779700
$$677$$ −20820.0 −1.18195 −0.590973 0.806691i $$-0.701256\pi$$
−0.590973 + 0.806691i $$0.701256\pi$$
$$678$$ 0 0
$$679$$ 1372.00 0.0775442
$$680$$ 0 0
$$681$$ 3996.00 0.224856
$$682$$ 0 0
$$683$$ 7020.00 0.393284 0.196642 0.980475i $$-0.436996\pi$$
0.196642 + 0.980475i $$0.436996\pi$$
$$684$$ −8784.00 −0.491030
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −18066.0 −1.00329
$$688$$ 24448.0 1.35475
$$689$$ −6996.00 −0.386831
$$690$$ 0 0
$$691$$ 12536.0 0.690147 0.345074 0.938576i $$-0.387854\pi$$
0.345074 + 0.938576i $$0.387854\pi$$
$$692$$ 20256.0 1.11274
$$693$$ 198.000 0.0108534
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 6912.00 0.375625
$$698$$ 0 0
$$699$$ 14148.0 0.765560
$$700$$ 0 0
$$701$$ −33276.0 −1.79289 −0.896446 0.443153i $$-0.853860\pi$$
−0.896446 + 0.443153i $$0.853860\pi$$
$$702$$ 0 0
$$703$$ −32452.0 −1.74104
$$704$$ 5632.00 0.301511
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1920.00 0.102134
$$708$$ −15840.0 −0.840824
$$709$$ 9818.00 0.520060 0.260030 0.965601i $$-0.416267\pi$$
0.260030 + 0.965601i $$0.416267\pi$$
$$710$$ 0 0
$$711$$ −6354.00 −0.335153
$$712$$ 0 0
$$713$$ 8064.00 0.423561
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −8736.00 −0.455977
$$717$$ −19260.0 −1.00318
$$718$$ 0 0
$$719$$ 3216.00 0.166810 0.0834051 0.996516i $$-0.473420\pi$$
0.0834051 + 0.996516i $$0.473420\pi$$
$$720$$ 0 0
$$721$$ −1688.00 −0.0871906
$$722$$ 0 0
$$723$$ 9906.00 0.509555
$$724$$ 18320.0 0.940411
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 10960.0 0.559125 0.279563 0.960127i $$-0.409811\pi$$
0.279563 + 0.960127i $$0.409811\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −27504.0 −1.39162
$$732$$ 10320.0 0.521090
$$733$$ −14618.0 −0.736600 −0.368300 0.929707i $$-0.620060\pi$$
−0.368300 + 0.929707i $$0.620060\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4180.00 0.208918
$$738$$ 0 0
$$739$$ 36518.0 1.81778 0.908888 0.417040i $$-0.136933\pi$$
0.908888 + 0.417040i $$0.136933\pi$$
$$740$$ 0 0
$$741$$ 8052.00 0.399187
$$742$$ 0 0
$$743$$ 37452.0 1.84923 0.924617 0.380899i $$-0.124385\pi$$
0.924617 + 0.380899i $$0.124385\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −9612.00 −0.470796
$$748$$ −6336.00 −0.309715
$$749$$ 4344.00 0.211918
$$750$$ 0 0
$$751$$ −10648.0 −0.517378 −0.258689 0.965961i $$-0.583291\pi$$
−0.258689 + 0.965961i $$0.583291\pi$$
$$752$$ −23040.0 −1.11726
$$753$$ 2196.00 0.106277
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 432.000 0.0207827
$$757$$ 1258.00 0.0604000 0.0302000 0.999544i $$-0.490386\pi$$
0.0302000 + 0.999544i $$0.490386\pi$$
$$758$$ 0 0
$$759$$ 2376.00 0.113628
$$760$$ 0 0
$$761$$ 1740.00 0.0828843 0.0414421 0.999141i $$-0.486805\pi$$
0.0414421 + 0.999141i $$0.486805\pi$$
$$762$$ 0 0
$$763$$ −1228.00 −0.0582655
$$764$$ 35136.0 1.66384
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 14520.0 0.683555
$$768$$ 12288.0 0.577350
$$769$$ −10774.0 −0.505228 −0.252614 0.967567i $$-0.581290\pi$$
−0.252614 + 0.967567i $$0.581290\pi$$
$$770$$ 0 0
$$771$$ 10314.0 0.481776
$$772$$ −40592.0 −1.89241
$$773$$ 19146.0 0.890859 0.445429 0.895317i $$-0.353051\pi$$
0.445429 + 0.895317i $$0.353051\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1596.00 0.0736888
$$778$$ 0 0
$$779$$ −11712.0 −0.538673
$$780$$ 0 0
$$781$$ −1848.00 −0.0846692
$$782$$ 0 0
$$783$$ 2592.00 0.118302
$$784$$ −21696.0 −0.988338
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 30670.0 1.38916 0.694579 0.719416i $$-0.255591\pi$$
0.694579 + 0.719416i $$0.255591\pi$$
$$788$$ 13536.0 0.611929
$$789$$ 2088.00 0.0942139
$$790$$ 0 0
$$791$$ 2508.00 0.112736
$$792$$ 0 0
$$793$$ −9460.00 −0.423625
$$794$$ 0 0
$$795$$ 0 0
$$796$$ −37312.0 −1.66142
$$797$$ −11970.0 −0.531994 −0.265997 0.963974i $$-0.585701\pi$$
−0.265997 + 0.963974i $$0.585701\pi$$
$$798$$ 0 0
$$799$$ 25920.0 1.14766
$$800$$ 0 0
$$801$$ −54.0000 −0.00238202
$$802$$ 0 0
$$803$$ 2398.00 0.105384
$$804$$ 9120.00 0.400047
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −22014.0 −0.960260
$$808$$ 0 0
$$809$$ −19932.0 −0.866220 −0.433110 0.901341i $$-0.642584\pi$$
−0.433110 + 0.901341i $$0.642584\pi$$
$$810$$ 0 0
$$811$$ −31462.0 −1.36224 −0.681122 0.732170i $$-0.738508\pi$$
−0.681122 + 0.732170i $$0.738508\pi$$
$$812$$ 1536.00 0.0663830
$$813$$ 15342.0 0.661830
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −13824.0 −0.593060
$$817$$ 46604.0 1.99568
$$818$$ 0 0
$$819$$ −396.000 −0.0168954
$$820$$ 0 0
$$821$$ −39720.0 −1.68847 −0.844237 0.535970i $$-0.819946\pi$$
−0.844237 + 0.535970i $$0.819946\pi$$
$$822$$ 0 0
$$823$$ 28492.0 1.20677 0.603383 0.797451i $$-0.293819\pi$$
0.603383 + 0.797451i $$0.293819\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 18324.0 0.770481 0.385241 0.922816i $$-0.374118\pi$$
0.385241 + 0.922816i $$0.374118\pi$$
$$828$$ 5184.00 0.217580
$$829$$ 21626.0 0.906034 0.453017 0.891502i $$-0.350348\pi$$
0.453017 + 0.891502i $$0.350348\pi$$
$$830$$ 0 0
$$831$$ −2958.00 −0.123480
$$832$$ −11264.0 −0.469362
$$833$$ 24408.0 1.01523
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 10736.0 0.444153
$$837$$ −3024.00 −0.124880
$$838$$ 0 0
$$839$$ −36960.0 −1.52086 −0.760430 0.649420i $$-0.775012\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$840$$ 0 0
$$841$$ −15173.0 −0.622125
$$842$$ 0 0
$$843$$ 9936.00 0.405948
$$844$$ 14960.0 0.610124
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −242.000 −0.00981726
$$848$$ −20352.0 −0.824163
$$849$$ −12894.0 −0.521226
$$850$$ 0 0
$$851$$ 19152.0 0.771471
$$852$$ −4032.00 −0.162129
$$853$$ −31502.0 −1.26449 −0.632244 0.774769i $$-0.717866\pi$$
−0.632244 + 0.774769i $$0.717866\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −5640.00 −0.224806 −0.112403 0.993663i $$-0.535855\pi$$
−0.112403 + 0.993663i $$0.535855\pi$$
$$858$$ 0 0
$$859$$ −2056.00 −0.0816645 −0.0408323 0.999166i $$-0.513001\pi$$
−0.0408323 + 0.999166i $$0.513001\pi$$
$$860$$ 0 0
$$861$$ 576.000 0.0227991
$$862$$ 0 0
$$863$$ −18336.0 −0.723250 −0.361625 0.932324i $$-0.617778\pi$$
−0.361625 + 0.932324i $$0.617778\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 813.000 0.0318465
$$868$$ −1792.00 −0.0700742
$$869$$ 7766.00 0.303157
$$870$$ 0 0
$$871$$ −8360.00 −0.325221
$$872$$ 0 0
$$873$$ −6174.00 −0.239357
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 5232.00 0.201796
$$877$$ 51346.0 1.97700 0.988501 0.151213i $$-0.0483178\pi$$
0.988501 + 0.151213i $$0.0483178\pi$$
$$878$$ 0 0
$$879$$ −8208.00 −0.314959
$$880$$ 0 0
$$881$$ 32910.0 1.25853 0.629266 0.777190i $$-0.283356\pi$$
0.629266 + 0.777190i $$0.283356\pi$$
$$882$$ 0 0
$$883$$ −15356.0 −0.585244 −0.292622 0.956228i $$-0.594528\pi$$
−0.292622 + 0.956228i $$0.594528\pi$$
$$884$$ 12672.0 0.482133
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −18372.0 −0.695458 −0.347729 0.937595i $$-0.613047\pi$$
−0.347729 + 0.937595i $$0.613047\pi$$
$$888$$ 0 0
$$889$$ 2788.00 0.105182
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 18400.0 0.690670
$$893$$ −43920.0 −1.64583
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −4752.00 −0.176884
$$898$$ 0 0
$$899$$ −10752.0 −0.398887
$$900$$ 0 0
$$901$$ 22896.0 0.846589
$$902$$ 0 0
$$903$$ −2292.00 −0.0844662
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −7640.00 −0.279694 −0.139847 0.990173i $$-0.544661\pi$$
−0.139847 + 0.990173i $$0.544661\pi$$
$$908$$ −10656.0 −0.389462
$$909$$ −8640.00 −0.315259
$$910$$ 0 0
$$911$$ −53040.0 −1.92897 −0.964486 0.264134i $$-0.914914\pi$$
−0.964486 + 0.264134i $$0.914914\pi$$
$$912$$ 23424.0 0.850489
$$913$$ 11748.0 0.425851
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 48176.0 1.73775
$$917$$ 504.000 0.0181500
$$918$$ 0 0
$$919$$ −11302.0 −0.405679 −0.202839 0.979212i $$-0.565017\pi$$
−0.202839 + 0.979212i $$0.565017\pi$$
$$920$$ 0 0
$$921$$ 750.000 0.0268332
$$922$$ 0 0
$$923$$ 3696.00 0.131804
$$924$$ −528.000 −0.0187986
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 7596.00 0.269132
$$928$$ 0 0
$$929$$ 19254.0 0.679982 0.339991 0.940429i $$-0.389576\pi$$
0.339991 + 0.940429i $$0.389576\pi$$
$$930$$ 0 0
$$931$$ −41358.0 −1.45591
$$932$$ −37728.0 −1.32599
$$933$$ 21744.0 0.762987
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −22214.0 −0.774493 −0.387246 0.921976i $$-0.626574\pi$$
−0.387246 + 0.921976i $$0.626574\pi$$
$$938$$ 0 0
$$939$$ 23358.0 0.811778
$$940$$ 0 0
$$941$$ 41736.0 1.44586 0.722930 0.690921i $$-0.242795\pi$$
0.722930 + 0.690921i $$0.242795\pi$$
$$942$$ 0 0
$$943$$ 6912.00 0.238691
$$944$$ 42240.0 1.45635
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −42732.0 −1.46632 −0.733159 0.680057i $$-0.761955\pi$$
−0.733159 + 0.680057i $$0.761955\pi$$
$$948$$ 16944.0 0.580502
$$949$$ −4796.00 −0.164051
$$950$$ 0 0
$$951$$ 12690.0 0.432704
$$952$$ 0 0
$$953$$ 25056.0 0.851672 0.425836 0.904800i $$-0.359980\pi$$
0.425836 + 0.904800i $$0.359980\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 51360.0 1.73755
$$957$$ −3168.00 −0.107008
$$958$$ 0 0
$$959$$ −2100.00 −0.0707117
$$960$$ 0 0
$$961$$ −17247.0 −0.578933
$$962$$ 0 0
$$963$$ −19548.0 −0.654128
$$964$$ −26416.0 −0.882575
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 14326.0 0.476415 0.238207 0.971214i $$-0.423440\pi$$
0.238207 + 0.971214i $$0.423440\pi$$
$$968$$ 0 0
$$969$$ −26352.0 −0.873631
$$970$$ 0 0
$$971$$ 45924.0 1.51779 0.758894 0.651215i $$-0.225740\pi$$
0.758894 + 0.651215i $$0.225740\pi$$
$$972$$ −1944.00 −0.0641500
$$973$$ −3748.00 −0.123490
$$974$$ 0 0
$$975$$ 0 0
$$976$$ −27520.0 −0.902555
$$977$$ 38946.0 1.27533 0.637663 0.770316i $$-0.279901\pi$$
0.637663 + 0.770316i $$0.279901\pi$$
$$978$$ 0 0
$$979$$ 66.0000 0.00215462
$$980$$ 0 0
$$981$$ 5526.00 0.179849
$$982$$ 0 0
$$983$$ −21000.0 −0.681379 −0.340690 0.940176i $$-0.610661\pi$$
−0.340690 + 0.940176i $$0.610661\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 2160.00 0.0696591
$$988$$ −21472.0 −0.691412
$$989$$ −27504.0 −0.884304
$$990$$ 0 0
$$991$$ 7760.00 0.248743 0.124372 0.992236i $$-0.460309\pi$$
0.124372 + 0.992236i $$0.460309\pi$$
$$992$$ 0 0
$$993$$ 23208.0 0.741675
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 25632.0 0.815443
$$997$$ −21350.0 −0.678196 −0.339098 0.940751i $$-0.610122\pi$$
−0.339098 + 0.940751i $$0.610122\pi$$
$$998$$ 0 0
$$999$$ −7182.00 −0.227456
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 825.4.a.e.1.1 1
3.2 odd 2 2475.4.a.f.1.1 1
5.2 odd 4 825.4.c.g.199.1 2
5.3 odd 4 825.4.c.g.199.2 2
5.4 even 2 165.4.a.a.1.1 1
15.14 odd 2 495.4.a.c.1.1 1
55.54 odd 2 1815.4.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.a.1.1 1 5.4 even 2
495.4.a.c.1.1 1 15.14 odd 2
825.4.a.e.1.1 1 1.1 even 1 trivial
825.4.c.g.199.1 2 5.2 odd 4
825.4.c.g.199.2 2 5.3 odd 4
1815.4.a.f.1.1 1 55.54 odd 2
2475.4.a.f.1.1 1 3.2 odd 2