# Properties

 Label 1872.4.a.m.1.1 Level $1872$ Weight $4$ Character 1872.1 Self dual yes Analytic conductor $110.452$ 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] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1872.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: $$110.451575531$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1872.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+12.0000 q^{5} -2.00000 q^{7} +O(q^{10})$$ $$q+12.0000 q^{5} -2.00000 q^{7} -36.0000 q^{11} +13.0000 q^{13} +78.0000 q^{17} -74.0000 q^{19} -96.0000 q^{23} +19.0000 q^{25} -18.0000 q^{29} +214.000 q^{31} -24.0000 q^{35} -286.000 q^{37} +384.000 q^{41} -524.000 q^{43} +300.000 q^{47} -339.000 q^{49} -558.000 q^{53} -432.000 q^{55} +576.000 q^{59} +74.0000 q^{61} +156.000 q^{65} -38.0000 q^{67} -456.000 q^{71} -682.000 q^{73} +72.0000 q^{77} -704.000 q^{79} -888.000 q^{83} +936.000 q^{85} +1020.00 q^{89} -26.0000 q^{91} -888.000 q^{95} +110.000 q^{97} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 12.0000 1.07331 0.536656 0.843801i $$-0.319687\pi$$
0.536656 + 0.843801i $$0.319687\pi$$
$$6$$ 0 0
$$7$$ −2.00000 −0.107990 −0.0539949 0.998541i $$-0.517195\pi$$
−0.0539949 + 0.998541i $$0.517195\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −36.0000 −0.986764 −0.493382 0.869813i $$-0.664240\pi$$
−0.493382 + 0.869813i $$0.664240\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 78.0000 1.11281 0.556405 0.830911i $$-0.312180\pi$$
0.556405 + 0.830911i $$0.312180\pi$$
$$18$$ 0 0
$$19$$ −74.0000 −0.893514 −0.446757 0.894655i $$-0.647421\pi$$
−0.446757 + 0.894655i $$0.647421\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −96.0000 −0.870321 −0.435161 0.900353i $$-0.643308\pi$$
−0.435161 + 0.900353i $$0.643308\pi$$
$$24$$ 0 0
$$25$$ 19.0000 0.152000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −18.0000 −0.115259 −0.0576296 0.998338i $$-0.518354\pi$$
−0.0576296 + 0.998338i $$0.518354\pi$$
$$30$$ 0 0
$$31$$ 214.000 1.23986 0.619928 0.784659i $$-0.287162\pi$$
0.619928 + 0.784659i $$0.287162\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −24.0000 −0.115907
$$36$$ 0 0
$$37$$ −286.000 −1.27076 −0.635380 0.772200i $$-0.719156\pi$$
−0.635380 + 0.772200i $$0.719156\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 384.000 1.46270 0.731350 0.682002i $$-0.238890\pi$$
0.731350 + 0.682002i $$0.238890\pi$$
$$42$$ 0 0
$$43$$ −524.000 −1.85835 −0.929177 0.369634i $$-0.879483\pi$$
−0.929177 + 0.369634i $$0.879483\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 300.000 0.931053 0.465527 0.885034i $$-0.345865\pi$$
0.465527 + 0.885034i $$0.345865\pi$$
$$48$$ 0 0
$$49$$ −339.000 −0.988338
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −558.000 −1.44617 −0.723087 0.690757i $$-0.757277\pi$$
−0.723087 + 0.690757i $$0.757277\pi$$
$$54$$ 0 0
$$55$$ −432.000 −1.05911
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 576.000 1.27100 0.635498 0.772102i $$-0.280795\pi$$
0.635498 + 0.772102i $$0.280795\pi$$
$$60$$ 0 0
$$61$$ 74.0000 0.155323 0.0776617 0.996980i $$-0.475255\pi$$
0.0776617 + 0.996980i $$0.475255\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 156.000 0.297683
$$66$$ 0 0
$$67$$ −38.0000 −0.0692901 −0.0346451 0.999400i $$-0.511030\pi$$
−0.0346451 + 0.999400i $$0.511030\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −456.000 −0.762215 −0.381107 0.924531i $$-0.624457\pi$$
−0.381107 + 0.924531i $$0.624457\pi$$
$$72$$ 0 0
$$73$$ −682.000 −1.09345 −0.546726 0.837311i $$-0.684126\pi$$
−0.546726 + 0.837311i $$0.684126\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 72.0000 0.106561
$$78$$ 0 0
$$79$$ −704.000 −1.00261 −0.501305 0.865271i $$-0.667147\pi$$
−0.501305 + 0.865271i $$0.667147\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −888.000 −1.17435 −0.587173 0.809462i $$-0.699759\pi$$
−0.587173 + 0.809462i $$0.699759\pi$$
$$84$$ 0 0
$$85$$ 936.000 1.19439
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1020.00 1.21483 0.607415 0.794385i $$-0.292207\pi$$
0.607415 + 0.794385i $$0.292207\pi$$
$$90$$ 0 0
$$91$$ −26.0000 −0.0299510
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −888.000 −0.959020
$$96$$ 0 0
$$97$$ 110.000 0.115142 0.0575712 0.998341i $$-0.481664\pi$$
0.0575712 + 0.998341i $$0.481664\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 990.000 0.975333 0.487667 0.873030i $$-0.337848\pi$$
0.487667 + 0.873030i $$0.337848\pi$$
$$102$$ 0 0
$$103$$ −1208.00 −1.15561 −0.577805 0.816175i $$-0.696090\pi$$
−0.577805 + 0.816175i $$0.696090\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 996.000 0.899878 0.449939 0.893059i $$-0.351446\pi$$
0.449939 + 0.893059i $$0.351446\pi$$
$$108$$ 0 0
$$109$$ −1402.00 −1.23199 −0.615997 0.787749i $$-0.711246\pi$$
−0.615997 + 0.787749i $$0.711246\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1926.00 −1.60339 −0.801694 0.597735i $$-0.796068\pi$$
−0.801694 + 0.597735i $$0.796068\pi$$
$$114$$ 0 0
$$115$$ −1152.00 −0.934127
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −156.000 −0.120172
$$120$$ 0 0
$$121$$ −35.0000 −0.0262960
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1272.00 −0.910169
$$126$$ 0 0
$$127$$ 988.000 0.690321 0.345161 0.938544i $$-0.387824\pi$$
0.345161 + 0.938544i $$0.387824\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2100.00 −1.40059 −0.700297 0.713851i $$-0.746949\pi$$
−0.700297 + 0.713851i $$0.746949\pi$$
$$132$$ 0 0
$$133$$ 148.000 0.0964904
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2496.00 1.55655 0.778276 0.627922i $$-0.216094\pi$$
0.778276 + 0.627922i $$0.216094\pi$$
$$138$$ 0 0
$$139$$ 2464.00 1.50355 0.751776 0.659418i $$-0.229197\pi$$
0.751776 + 0.659418i $$0.229197\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −468.000 −0.273679
$$144$$ 0 0
$$145$$ −216.000 −0.123709
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −216.000 −0.118761 −0.0593806 0.998235i $$-0.518913\pi$$
−0.0593806 + 0.998235i $$0.518913\pi$$
$$150$$ 0 0
$$151$$ 898.000 0.483962 0.241981 0.970281i $$-0.422203\pi$$
0.241981 + 0.970281i $$0.422203\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 2568.00 1.33075
$$156$$ 0 0
$$157$$ −1510.00 −0.767587 −0.383793 0.923419i $$-0.625383\pi$$
−0.383793 + 0.923419i $$0.625383\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 192.000 0.0939858
$$162$$ 0 0
$$163$$ 394.000 0.189328 0.0946640 0.995509i $$-0.469822\pi$$
0.0946640 + 0.995509i $$0.469822\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 84.0000 0.0389228 0.0194614 0.999811i $$-0.493805\pi$$
0.0194614 + 0.999811i $$0.493805\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −1194.00 −0.524729 −0.262365 0.964969i $$-0.584502\pi$$
−0.262365 + 0.964969i $$0.584502\pi$$
$$174$$ 0 0
$$175$$ −38.0000 −0.0164145
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 3156.00 1.31782 0.658912 0.752220i $$-0.271017\pi$$
0.658912 + 0.752220i $$0.271017\pi$$
$$180$$ 0 0
$$181$$ −1078.00 −0.442691 −0.221346 0.975195i $$-0.571045\pi$$
−0.221346 + 0.975195i $$0.571045\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −3432.00 −1.36392
$$186$$ 0 0
$$187$$ −2808.00 −1.09808
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3192.00 1.20924 0.604620 0.796514i $$-0.293325\pi$$
0.604620 + 0.796514i $$0.293325\pi$$
$$192$$ 0 0
$$193$$ 722.000 0.269278 0.134639 0.990895i $$-0.457012\pi$$
0.134639 + 0.990895i $$0.457012\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2796.00 −1.01120 −0.505601 0.862767i $$-0.668729\pi$$
−0.505601 + 0.862767i $$0.668729\pi$$
$$198$$ 0 0
$$199$$ 340.000 0.121115 0.0605577 0.998165i $$-0.480712\pi$$
0.0605577 + 0.998165i $$0.480712\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 36.0000 0.0124468
$$204$$ 0 0
$$205$$ 4608.00 1.56994
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 2664.00 0.881688
$$210$$ 0 0
$$211$$ 1924.00 0.627742 0.313871 0.949466i $$-0.398374\pi$$
0.313871 + 0.949466i $$0.398374\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −6288.00 −1.99460
$$216$$ 0 0
$$217$$ −428.000 −0.133892
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1014.00 0.308638
$$222$$ 0 0
$$223$$ −5042.00 −1.51407 −0.757034 0.653375i $$-0.773352\pi$$
−0.757034 + 0.653375i $$0.773352\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −2676.00 −0.782433 −0.391217 0.920299i $$-0.627946\pi$$
−0.391217 + 0.920299i $$0.627946\pi$$
$$228$$ 0 0
$$229$$ −2410.00 −0.695447 −0.347723 0.937597i $$-0.613045\pi$$
−0.347723 + 0.937597i $$0.613045\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −3726.00 −1.04763 −0.523816 0.851831i $$-0.675492\pi$$
−0.523816 + 0.851831i $$0.675492\pi$$
$$234$$ 0 0
$$235$$ 3600.00 0.999311
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 1248.00 0.337767 0.168884 0.985636i $$-0.445984\pi$$
0.168884 + 0.985636i $$0.445984\pi$$
$$240$$ 0 0
$$241$$ −4210.00 −1.12527 −0.562635 0.826706i $$-0.690212\pi$$
−0.562635 + 0.826706i $$0.690212\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −4068.00 −1.06080
$$246$$ 0 0
$$247$$ −962.000 −0.247816
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −7692.00 −1.93432 −0.967161 0.254165i $$-0.918199\pi$$
−0.967161 + 0.254165i $$0.918199\pi$$
$$252$$ 0 0
$$253$$ 3456.00 0.858802
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1326.00 −0.321843 −0.160921 0.986967i $$-0.551447\pi$$
−0.160921 + 0.986967i $$0.551447\pi$$
$$258$$ 0 0
$$259$$ 572.000 0.137229
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −6048.00 −1.41801 −0.709003 0.705205i $$-0.750855\pi$$
−0.709003 + 0.705205i $$0.750855\pi$$
$$264$$ 0 0
$$265$$ −6696.00 −1.55220
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −6474.00 −1.46739 −0.733693 0.679481i $$-0.762205\pi$$
−0.733693 + 0.679481i $$0.762205\pi$$
$$270$$ 0 0
$$271$$ −5978.00 −1.33999 −0.669996 0.742365i $$-0.733704\pi$$
−0.669996 + 0.742365i $$0.733704\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −684.000 −0.149988
$$276$$ 0 0
$$277$$ 8750.00 1.89797 0.948983 0.315327i $$-0.102114\pi$$
0.948983 + 0.315327i $$0.102114\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −8976.00 −1.90556 −0.952782 0.303656i $$-0.901793\pi$$
−0.952782 + 0.303656i $$0.901793\pi$$
$$282$$ 0 0
$$283$$ 592.000 0.124349 0.0621745 0.998065i $$-0.480196\pi$$
0.0621745 + 0.998065i $$0.480196\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −768.000 −0.157957
$$288$$ 0 0
$$289$$ 1171.00 0.238347
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 4608.00 0.918779 0.459389 0.888235i $$-0.348068\pi$$
0.459389 + 0.888235i $$0.348068\pi$$
$$294$$ 0 0
$$295$$ 6912.00 1.36418
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −1248.00 −0.241384
$$300$$ 0 0
$$301$$ 1048.00 0.200683
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 888.000 0.166711
$$306$$ 0 0
$$307$$ 3166.00 0.588577 0.294289 0.955717i $$-0.404917\pi$$
0.294289 + 0.955717i $$0.404917\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2472.00 0.450721 0.225361 0.974275i $$-0.427644\pi$$
0.225361 + 0.974275i $$0.427644\pi$$
$$312$$ 0 0
$$313$$ −3094.00 −0.558732 −0.279366 0.960185i $$-0.590124\pi$$
−0.279366 + 0.960185i $$0.590124\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2316.00 −0.410345 −0.205173 0.978726i $$-0.565776\pi$$
−0.205173 + 0.978726i $$0.565776\pi$$
$$318$$ 0 0
$$319$$ 648.000 0.113734
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −5772.00 −0.994312
$$324$$ 0 0
$$325$$ 247.000 0.0421572
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −600.000 −0.100544
$$330$$ 0 0
$$331$$ 4426.00 0.734970 0.367485 0.930030i $$-0.380219\pi$$
0.367485 + 0.930030i $$0.380219\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −456.000 −0.0743700
$$336$$ 0 0
$$337$$ 866.000 0.139982 0.0699911 0.997548i $$-0.477703\pi$$
0.0699911 + 0.997548i $$0.477703\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −7704.00 −1.22345
$$342$$ 0 0
$$343$$ 1364.00 0.214720
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −2556.00 −0.395427 −0.197714 0.980260i $$-0.563352\pi$$
−0.197714 + 0.980260i $$0.563352\pi$$
$$348$$ 0 0
$$349$$ −11014.0 −1.68930 −0.844650 0.535318i $$-0.820192\pi$$
−0.844650 + 0.535318i $$0.820192\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 9720.00 1.46556 0.732781 0.680465i $$-0.238222\pi$$
0.732781 + 0.680465i $$0.238222\pi$$
$$354$$ 0 0
$$355$$ −5472.00 −0.818095
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −2988.00 −0.439277 −0.219639 0.975581i $$-0.570488\pi$$
−0.219639 + 0.975581i $$0.570488\pi$$
$$360$$ 0 0
$$361$$ −1383.00 −0.201633
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8184.00 −1.17362
$$366$$ 0 0
$$367$$ 2068.00 0.294138 0.147069 0.989126i $$-0.453016\pi$$
0.147069 + 0.989126i $$0.453016\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1116.00 0.156172
$$372$$ 0 0
$$373$$ 902.000 0.125211 0.0626056 0.998038i $$-0.480059\pi$$
0.0626056 + 0.998038i $$0.480059\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −234.000 −0.0319671
$$378$$ 0 0
$$379$$ −12818.0 −1.73725 −0.868623 0.495473i $$-0.834995\pi$$
−0.868623 + 0.495473i $$0.834995\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −1332.00 −0.177708 −0.0888538 0.996045i $$-0.528320\pi$$
−0.0888538 + 0.996045i $$0.528320\pi$$
$$384$$ 0 0
$$385$$ 864.000 0.114373
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −3054.00 −0.398056 −0.199028 0.979994i $$-0.563779\pi$$
−0.199028 + 0.979994i $$0.563779\pi$$
$$390$$ 0 0
$$391$$ −7488.00 −0.968502
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −8448.00 −1.07611
$$396$$ 0 0
$$397$$ 11162.0 1.41110 0.705548 0.708663i $$-0.250701\pi$$
0.705548 + 0.708663i $$0.250701\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 14820.0 1.84557 0.922787 0.385310i $$-0.125905\pi$$
0.922787 + 0.385310i $$0.125905\pi$$
$$402$$ 0 0
$$403$$ 2782.00 0.343874
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 10296.0 1.25394
$$408$$ 0 0
$$409$$ −9682.00 −1.17052 −0.585262 0.810844i $$-0.699008\pi$$
−0.585262 + 0.810844i $$0.699008\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −1152.00 −0.137255
$$414$$ 0 0
$$415$$ −10656.0 −1.26044
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −348.000 −0.0405750 −0.0202875 0.999794i $$-0.506458\pi$$
−0.0202875 + 0.999794i $$0.506458\pi$$
$$420$$ 0 0
$$421$$ 2486.00 0.287792 0.143896 0.989593i $$-0.454037\pi$$
0.143896 + 0.989593i $$0.454037\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 1482.00 0.169147
$$426$$ 0 0
$$427$$ −148.000 −0.0167734
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −1812.00 −0.202508 −0.101254 0.994861i $$-0.532285\pi$$
−0.101254 + 0.994861i $$0.532285\pi$$
$$432$$ 0 0
$$433$$ −6226.00 −0.690999 −0.345499 0.938419i $$-0.612290\pi$$
−0.345499 + 0.938419i $$0.612290\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7104.00 0.777644
$$438$$ 0 0
$$439$$ 12544.0 1.36376 0.681882 0.731462i $$-0.261162\pi$$
0.681882 + 0.731462i $$0.261162\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −8556.00 −0.917625 −0.458812 0.888533i $$-0.651725\pi$$
−0.458812 + 0.888533i $$0.651725\pi$$
$$444$$ 0 0
$$445$$ 12240.0 1.30389
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −4116.00 −0.432619 −0.216310 0.976325i $$-0.569402\pi$$
−0.216310 + 0.976325i $$0.569402\pi$$
$$450$$ 0 0
$$451$$ −13824.0 −1.44334
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −312.000 −0.0321468
$$456$$ 0 0
$$457$$ −6514.00 −0.666766 −0.333383 0.942791i $$-0.608190\pi$$
−0.333383 + 0.942791i $$0.608190\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −10500.0 −1.06081 −0.530405 0.847744i $$-0.677960\pi$$
−0.530405 + 0.847744i $$0.677960\pi$$
$$462$$ 0 0
$$463$$ 5542.00 0.556282 0.278141 0.960540i $$-0.410282\pi$$
0.278141 + 0.960540i $$0.410282\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −5220.00 −0.517244 −0.258622 0.965979i $$-0.583268\pi$$
−0.258622 + 0.965979i $$0.583268\pi$$
$$468$$ 0 0
$$469$$ 76.0000 0.00748263
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 18864.0 1.83376
$$474$$ 0 0
$$475$$ −1406.00 −0.135814
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 11592.0 1.10575 0.552873 0.833266i $$-0.313532\pi$$
0.552873 + 0.833266i $$0.313532\pi$$
$$480$$ 0 0
$$481$$ −3718.00 −0.352445
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1320.00 0.123584
$$486$$ 0 0
$$487$$ −12170.0 −1.13239 −0.566196 0.824270i $$-0.691586\pi$$
−0.566196 + 0.824270i $$0.691586\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1812.00 0.166547 0.0832733 0.996527i $$-0.473463\pi$$
0.0832733 + 0.996527i $$0.473463\pi$$
$$492$$ 0 0
$$493$$ −1404.00 −0.128262
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 912.000 0.0823115
$$498$$ 0 0
$$499$$ 1330.00 0.119317 0.0596583 0.998219i $$-0.480999\pi$$
0.0596583 + 0.998219i $$0.480999\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −2688.00 −0.238274 −0.119137 0.992878i $$-0.538013\pi$$
−0.119137 + 0.992878i $$0.538013\pi$$
$$504$$ 0 0
$$505$$ 11880.0 1.04684
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −5124.00 −0.446203 −0.223101 0.974795i $$-0.571618\pi$$
−0.223101 + 0.974795i $$0.571618\pi$$
$$510$$ 0 0
$$511$$ 1364.00 0.118082
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −14496.0 −1.24033
$$516$$ 0 0
$$517$$ −10800.0 −0.918730
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 882.000 0.0741672 0.0370836 0.999312i $$-0.488193\pi$$
0.0370836 + 0.999312i $$0.488193\pi$$
$$522$$ 0 0
$$523$$ 2320.00 0.193970 0.0969852 0.995286i $$-0.469080\pi$$
0.0969852 + 0.995286i $$0.469080\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16692.0 1.37972
$$528$$ 0 0
$$529$$ −2951.00 −0.242541
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4992.00 0.405680
$$534$$ 0 0
$$535$$ 11952.0 0.965851
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 12204.0 0.975257
$$540$$ 0 0
$$541$$ 21422.0 1.70241 0.851205 0.524833i $$-0.175872\pi$$
0.851205 + 0.524833i $$0.175872\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −16824.0 −1.32231
$$546$$ 0 0
$$547$$ −7040.00 −0.550290 −0.275145 0.961403i $$-0.588726\pi$$
−0.275145 + 0.961403i $$0.588726\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1332.00 0.102986
$$552$$ 0 0
$$553$$ 1408.00 0.108272
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 8400.00 0.638994 0.319497 0.947587i $$-0.396486\pi$$
0.319497 + 0.947587i $$0.396486\pi$$
$$558$$ 0 0
$$559$$ −6812.00 −0.515415
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 19044.0 1.42559 0.712797 0.701371i $$-0.247428\pi$$
0.712797 + 0.701371i $$0.247428\pi$$
$$564$$ 0 0
$$565$$ −23112.0 −1.72094
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 4698.00 0.346134 0.173067 0.984910i $$-0.444632\pi$$
0.173067 + 0.984910i $$0.444632\pi$$
$$570$$ 0 0
$$571$$ 8728.00 0.639677 0.319838 0.947472i $$-0.396371\pi$$
0.319838 + 0.947472i $$0.396371\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1824.00 −0.132289
$$576$$ 0 0
$$577$$ 2018.00 0.145599 0.0727993 0.997347i $$-0.476807\pi$$
0.0727993 + 0.997347i $$0.476807\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 1776.00 0.126817
$$582$$ 0 0
$$583$$ 20088.0 1.42703
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 11376.0 0.799894 0.399947 0.916538i $$-0.369029\pi$$
0.399947 + 0.916538i $$0.369029\pi$$
$$588$$ 0 0
$$589$$ −15836.0 −1.10783
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 25596.0 1.77252 0.886258 0.463192i $$-0.153296\pi$$
0.886258 + 0.463192i $$0.153296\pi$$
$$594$$ 0 0
$$595$$ −1872.00 −0.128982
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 3480.00 0.237377 0.118689 0.992932i $$-0.462131\pi$$
0.118689 + 0.992932i $$0.462131\pi$$
$$600$$ 0 0
$$601$$ 10010.0 0.679395 0.339698 0.940535i $$-0.389675\pi$$
0.339698 + 0.940535i $$0.389675\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −420.000 −0.0282238
$$606$$ 0 0
$$607$$ −3764.00 −0.251690 −0.125845 0.992050i $$-0.540164\pi$$
−0.125845 + 0.992050i $$0.540164\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3900.00 0.258228
$$612$$ 0 0
$$613$$ 13610.0 0.896742 0.448371 0.893848i $$-0.352004\pi$$
0.448371 + 0.893848i $$0.352004\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −6408.00 −0.418114 −0.209057 0.977903i $$-0.567039\pi$$
−0.209057 + 0.977903i $$0.567039\pi$$
$$618$$ 0 0
$$619$$ 6694.00 0.434660 0.217330 0.976098i $$-0.430265\pi$$
0.217330 + 0.976098i $$0.430265\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −2040.00 −0.131189
$$624$$ 0 0
$$625$$ −17639.0 −1.12890
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −22308.0 −1.41411
$$630$$ 0 0
$$631$$ 27250.0 1.71918 0.859592 0.510981i $$-0.170718\pi$$
0.859592 + 0.510981i $$0.170718\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 11856.0 0.740931
$$636$$ 0 0
$$637$$ −4407.00 −0.274116
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 12630.0 0.778245 0.389122 0.921186i $$-0.372778\pi$$
0.389122 + 0.921186i $$0.372778\pi$$
$$642$$ 0 0
$$643$$ −14798.0 −0.907583 −0.453792 0.891108i $$-0.649929\pi$$
−0.453792 + 0.891108i $$0.649929\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 26232.0 1.59395 0.796976 0.604012i $$-0.206432\pi$$
0.796976 + 0.604012i $$0.206432\pi$$
$$648$$ 0 0
$$649$$ −20736.0 −1.25417
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 30390.0 1.82121 0.910607 0.413274i $$-0.135615\pi$$
0.910607 + 0.413274i $$0.135615\pi$$
$$654$$ 0 0
$$655$$ −25200.0 −1.50328
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −28740.0 −1.69886 −0.849432 0.527698i $$-0.823055\pi$$
−0.849432 + 0.527698i $$0.823055\pi$$
$$660$$ 0 0
$$661$$ −9214.00 −0.542183 −0.271092 0.962554i $$-0.587385\pi$$
−0.271092 + 0.962554i $$0.587385\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1776.00 0.103564
$$666$$ 0 0
$$667$$ 1728.00 0.100312
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −2664.00 −0.153268
$$672$$ 0 0
$$673$$ 16598.0 0.950677 0.475339 0.879803i $$-0.342326\pi$$
0.475339 + 0.879803i $$0.342326\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 8610.00 0.488788 0.244394 0.969676i $$-0.421411\pi$$
0.244394 + 0.969676i $$0.421411\pi$$
$$678$$ 0 0
$$679$$ −220.000 −0.0124342
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 804.000 0.0450428 0.0225214 0.999746i $$-0.492831\pi$$
0.0225214 + 0.999746i $$0.492831\pi$$
$$684$$ 0 0
$$685$$ 29952.0 1.67067
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −7254.00 −0.401096
$$690$$ 0 0
$$691$$ −2270.00 −0.124971 −0.0624854 0.998046i $$-0.519903\pi$$
−0.0624854 + 0.998046i $$0.519903\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 29568.0 1.61378
$$696$$ 0 0
$$697$$ 29952.0 1.62771
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1782.00 −0.0960131 −0.0480066 0.998847i $$-0.515287\pi$$
−0.0480066 + 0.998847i $$0.515287\pi$$
$$702$$ 0 0
$$703$$ 21164.0 1.13544
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1980.00 −0.105326
$$708$$ 0 0
$$709$$ −10690.0 −0.566250 −0.283125 0.959083i $$-0.591371\pi$$
−0.283125 + 0.959083i $$0.591371\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −20544.0 −1.07907
$$714$$ 0 0
$$715$$ −5616.00 −0.293743
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 11568.0 0.600019 0.300009 0.953936i $$-0.403010\pi$$
0.300009 + 0.953936i $$0.403010\pi$$
$$720$$ 0 0
$$721$$ 2416.00 0.124794
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −342.000 −0.0175194
$$726$$ 0 0
$$727$$ 11644.0 0.594019 0.297010 0.954874i $$-0.404011\pi$$
0.297010 + 0.954874i $$0.404011\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −40872.0 −2.06800
$$732$$ 0 0
$$733$$ −15010.0 −0.756353 −0.378177 0.925733i $$-0.623449\pi$$
−0.378177 + 0.925733i $$0.623449\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1368.00 0.0683730
$$738$$ 0 0
$$739$$ −33410.0 −1.66307 −0.831534 0.555474i $$-0.812537\pi$$
−0.831534 + 0.555474i $$0.812537\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −6504.00 −0.321142 −0.160571 0.987024i $$-0.551334\pi$$
−0.160571 + 0.987024i $$0.551334\pi$$
$$744$$ 0 0
$$745$$ −2592.00 −0.127468
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −1992.00 −0.0971777
$$750$$ 0 0
$$751$$ 13912.0 0.675973 0.337987 0.941151i $$-0.390254\pi$$
0.337987 + 0.941151i $$0.390254\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 10776.0 0.519442
$$756$$ 0 0
$$757$$ −23974.0 −1.15106 −0.575528 0.817782i $$-0.695204\pi$$
−0.575528 + 0.817782i $$0.695204\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 288.000 0.0137188 0.00685939 0.999976i $$-0.497817\pi$$
0.00685939 + 0.999976i $$0.497817\pi$$
$$762$$ 0 0
$$763$$ 2804.00 0.133043
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 7488.00 0.352511
$$768$$ 0 0
$$769$$ 1514.00 0.0709964 0.0354982 0.999370i $$-0.488698\pi$$
0.0354982 + 0.999370i $$0.488698\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −15816.0 −0.735915 −0.367957 0.929843i $$-0.619943\pi$$
−0.367957 + 0.929843i $$0.619943\pi$$
$$774$$ 0 0
$$775$$ 4066.00 0.188458
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −28416.0 −1.30694
$$780$$ 0 0
$$781$$ 16416.0 0.752126
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −18120.0 −0.823861
$$786$$ 0 0
$$787$$ −10154.0 −0.459912 −0.229956 0.973201i $$-0.573858\pi$$
−0.229956 + 0.973201i $$0.573858\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3852.00 0.173150
$$792$$ 0 0
$$793$$ 962.000 0.0430790
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 17442.0 0.775191 0.387596 0.921830i $$-0.373306\pi$$
0.387596 + 0.921830i $$0.373306\pi$$
$$798$$ 0 0
$$799$$ 23400.0 1.03609
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 24552.0 1.07898
$$804$$ 0 0
$$805$$ 2304.00 0.100876
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 8778.00 0.381481 0.190740 0.981641i $$-0.438911\pi$$
0.190740 + 0.981641i $$0.438911\pi$$
$$810$$ 0 0
$$811$$ 430.000 0.0186182 0.00930909 0.999957i $$-0.497037\pi$$
0.00930909 + 0.999957i $$0.497037\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 4728.00 0.203208
$$816$$ 0 0
$$817$$ 38776.0 1.66047
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 32976.0 1.40179 0.700895 0.713264i $$-0.252784\pi$$
0.700895 + 0.713264i $$0.252784\pi$$
$$822$$ 0 0
$$823$$ 1168.00 0.0494701 0.0247351 0.999694i $$-0.492126\pi$$
0.0247351 + 0.999694i $$0.492126\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −17172.0 −0.722042 −0.361021 0.932558i $$-0.617572\pi$$
−0.361021 + 0.932558i $$0.617572\pi$$
$$828$$ 0 0
$$829$$ 27146.0 1.13730 0.568649 0.822580i $$-0.307466\pi$$
0.568649 + 0.822580i $$0.307466\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −26442.0 −1.09983
$$834$$ 0 0
$$835$$ 1008.00 0.0417764
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 30696.0 1.26310 0.631552 0.775334i $$-0.282418\pi$$
0.631552 + 0.775334i $$0.282418\pi$$
$$840$$ 0 0
$$841$$ −24065.0 −0.986715
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 2028.00 0.0825625
$$846$$ 0 0
$$847$$ 70.0000 0.00283970
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 27456.0 1.10597
$$852$$ 0 0
$$853$$ 24842.0 0.997156 0.498578 0.866845i $$-0.333856\pi$$
0.498578 + 0.866845i $$0.333856\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −11406.0 −0.454634 −0.227317 0.973821i $$-0.572995\pi$$
−0.227317 + 0.973821i $$0.572995\pi$$
$$858$$ 0 0
$$859$$ −20540.0 −0.815851 −0.407925 0.913015i $$-0.633748\pi$$
−0.407925 + 0.913015i $$0.633748\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 9108.00 0.359258 0.179629 0.983734i $$-0.442510\pi$$
0.179629 + 0.983734i $$0.442510\pi$$
$$864$$ 0 0
$$865$$ −14328.0 −0.563198
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 25344.0 0.989340
$$870$$ 0 0
$$871$$ −494.000 −0.0192176
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 2544.00 0.0982890
$$876$$ 0 0
$$877$$ −24046.0 −0.925856 −0.462928 0.886396i $$-0.653201\pi$$
−0.462928 + 0.886396i $$0.653201\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −7998.00 −0.305856 −0.152928 0.988237i $$-0.548870\pi$$
−0.152928 + 0.988237i $$0.548870\pi$$
$$882$$ 0 0
$$883$$ −24032.0 −0.915902 −0.457951 0.888978i $$-0.651416\pi$$
−0.457951 + 0.888978i $$0.651416\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −15648.0 −0.592343 −0.296172 0.955135i $$-0.595710\pi$$
−0.296172 + 0.955135i $$0.595710\pi$$
$$888$$ 0 0
$$889$$ −1976.00 −0.0745477
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −22200.0 −0.831909
$$894$$ 0 0
$$895$$ 37872.0 1.41444
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −3852.00 −0.142905
$$900$$ 0 0
$$901$$ −43524.0 −1.60932
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −12936.0 −0.475146
$$906$$ 0 0
$$907$$ 808.000 0.0295802 0.0147901 0.999891i $$-0.495292\pi$$
0.0147901 + 0.999891i $$0.495292\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 39144.0 1.42360 0.711799 0.702383i $$-0.247880\pi$$
0.711799 + 0.702383i $$0.247880\pi$$
$$912$$ 0 0
$$913$$ 31968.0 1.15880
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4200.00 0.151250
$$918$$ 0 0
$$919$$ 38248.0 1.37289 0.686445 0.727182i $$-0.259170\pi$$
0.686445 + 0.727182i $$0.259170\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −5928.00 −0.211400
$$924$$ 0 0
$$925$$ −5434.00 −0.193155
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 54264.0 1.91641 0.958205 0.286084i $$-0.0923536\pi$$
0.958205 + 0.286084i $$0.0923536\pi$$
$$930$$ 0 0
$$931$$ 25086.0 0.883094
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −33696.0 −1.17859
$$936$$ 0 0
$$937$$ 12206.0 0.425563 0.212782 0.977100i $$-0.431748\pi$$
0.212782 + 0.977100i $$0.431748\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −17664.0 −0.611934 −0.305967 0.952042i $$-0.598980\pi$$
−0.305967 + 0.952042i $$0.598980\pi$$
$$942$$ 0 0
$$943$$ −36864.0 −1.27302
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −51984.0 −1.78379 −0.891897 0.452238i $$-0.850626\pi$$
−0.891897 + 0.452238i $$0.850626\pi$$
$$948$$ 0 0
$$949$$ −8866.00 −0.303269
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −13782.0 −0.468460 −0.234230 0.972181i $$-0.575257\pi$$
−0.234230 + 0.972181i $$0.575257\pi$$
$$954$$ 0 0
$$955$$ 38304.0 1.29789
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −4992.00 −0.168092
$$960$$ 0 0
$$961$$ 16005.0 0.537243
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 8664.00 0.289020
$$966$$ 0 0
$$967$$ −14618.0 −0.486125 −0.243063 0.970011i $$-0.578152\pi$$
−0.243063 + 0.970011i $$0.578152\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −18708.0 −0.618299 −0.309149 0.951013i $$-0.600044\pi$$
−0.309149 + 0.951013i $$0.600044\pi$$
$$972$$ 0 0
$$973$$ −4928.00 −0.162368
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 48804.0 1.59814 0.799068 0.601241i $$-0.205327\pi$$
0.799068 + 0.601241i $$0.205327\pi$$
$$978$$ 0 0
$$979$$ −36720.0 −1.19875
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −44736.0 −1.45153 −0.725766 0.687941i $$-0.758515\pi$$
−0.725766 + 0.687941i $$0.758515\pi$$
$$984$$ 0 0
$$985$$ −33552.0 −1.08534
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 50304.0 1.61737
$$990$$ 0 0
$$991$$ 21004.0 0.673274 0.336637 0.941635i $$-0.390711\pi$$
0.336637 + 0.941635i $$0.390711\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 4080.00 0.129995
$$996$$ 0 0
$$997$$ 9038.00 0.287098 0.143549 0.989643i $$-0.454149\pi$$
0.143549 + 0.989643i $$0.454149\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1872.4.a.m.1.1 1
3.2 odd 2 624.4.a.g.1.1 1
4.3 odd 2 117.4.a.a.1.1 1
12.11 even 2 39.4.a.a.1.1 1
24.5 odd 2 2496.4.a.f.1.1 1
24.11 even 2 2496.4.a.o.1.1 1
52.51 odd 2 1521.4.a.f.1.1 1
60.59 even 2 975.4.a.e.1.1 1
84.83 odd 2 1911.4.a.f.1.1 1
156.47 odd 4 507.4.b.b.337.1 2
156.83 odd 4 507.4.b.b.337.2 2
156.155 even 2 507.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.a.1.1 1 12.11 even 2
117.4.a.a.1.1 1 4.3 odd 2
507.4.a.c.1.1 1 156.155 even 2
507.4.b.b.337.1 2 156.47 odd 4
507.4.b.b.337.2 2 156.83 odd 4
624.4.a.g.1.1 1 3.2 odd 2
975.4.a.e.1.1 1 60.59 even 2
1521.4.a.f.1.1 1 52.51 odd 2
1872.4.a.m.1.1 1 1.1 even 1 trivial
1911.4.a.f.1.1 1 84.83 odd 2
2496.4.a.f.1.1 1 24.5 odd 2
2496.4.a.o.1.1 1 24.11 even 2