# Properties

 Label 1600.4.a.bn.1.1 Level $1600$ Weight $4$ Character 1600.1 Self dual yes Analytic conductor $94.403$ 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] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1600.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: $$94.4030560092$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000 q^{3} -2.00000 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+5.00000 q^{3} -2.00000 q^{7} -2.00000 q^{9} -39.0000 q^{11} +84.0000 q^{13} +61.0000 q^{17} -151.000 q^{19} -10.0000 q^{21} +58.0000 q^{23} -145.000 q^{27} -192.000 q^{29} -18.0000 q^{31} -195.000 q^{33} -138.000 q^{37} +420.000 q^{39} +229.000 q^{41} -164.000 q^{43} +212.000 q^{47} -339.000 q^{49} +305.000 q^{51} +578.000 q^{53} -755.000 q^{57} +336.000 q^{59} -858.000 q^{61} +4.00000 q^{63} -209.000 q^{67} +290.000 q^{69} -780.000 q^{71} +403.000 q^{73} +78.0000 q^{77} -230.000 q^{79} -671.000 q^{81} -1293.00 q^{83} -960.000 q^{87} -1369.00 q^{89} -168.000 q^{91} -90.0000 q^{93} -382.000 q^{97} +78.0000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 5.00000 0.962250 0.481125 0.876652i $$-0.340228\pi$$
0.481125 + 0.876652i $$0.340228\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.00000 −0.107990 −0.0539949 0.998541i $$-0.517195\pi$$
−0.0539949 + 0.998541i $$0.517195\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.0740741
$$10$$ 0 0
$$11$$ −39.0000 −1.06899 −0.534497 0.845170i $$-0.679499\pi$$
−0.534497 + 0.845170i $$0.679499\pi$$
$$12$$ 0 0
$$13$$ 84.0000 1.79211 0.896054 0.443945i $$-0.146421\pi$$
0.896054 + 0.443945i $$0.146421\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 61.0000 0.870275 0.435137 0.900364i $$-0.356700\pi$$
0.435137 + 0.900364i $$0.356700\pi$$
$$18$$ 0 0
$$19$$ −151.000 −1.82325 −0.911626 0.411021i $$-0.865172\pi$$
−0.911626 + 0.411021i $$0.865172\pi$$
$$20$$ 0 0
$$21$$ −10.0000 −0.103913
$$22$$ 0 0
$$23$$ 58.0000 0.525819 0.262909 0.964821i $$-0.415318\pi$$
0.262909 + 0.964821i $$0.415318\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −145.000 −1.03353
$$28$$ 0 0
$$29$$ −192.000 −1.22943 −0.614716 0.788749i $$-0.710729\pi$$
−0.614716 + 0.788749i $$0.710729\pi$$
$$30$$ 0 0
$$31$$ −18.0000 −0.104287 −0.0521435 0.998640i $$-0.516605\pi$$
−0.0521435 + 0.998640i $$0.516605\pi$$
$$32$$ 0 0
$$33$$ −195.000 −1.02864
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −138.000 −0.613164 −0.306582 0.951844i $$-0.599185\pi$$
−0.306582 + 0.951844i $$0.599185\pi$$
$$38$$ 0 0
$$39$$ 420.000 1.72446
$$40$$ 0 0
$$41$$ 229.000 0.872288 0.436144 0.899877i $$-0.356344\pi$$
0.436144 + 0.899877i $$0.356344\pi$$
$$42$$ 0 0
$$43$$ −164.000 −0.581622 −0.290811 0.956780i $$-0.593925\pi$$
−0.290811 + 0.956780i $$0.593925\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 212.000 0.657944 0.328972 0.944340i $$-0.393298\pi$$
0.328972 + 0.944340i $$0.393298\pi$$
$$48$$ 0 0
$$49$$ −339.000 −0.988338
$$50$$ 0 0
$$51$$ 305.000 0.837422
$$52$$ 0 0
$$53$$ 578.000 1.49801 0.749004 0.662566i $$-0.230532\pi$$
0.749004 + 0.662566i $$0.230532\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −755.000 −1.75442
$$58$$ 0 0
$$59$$ 336.000 0.741415 0.370707 0.928750i $$-0.379115\pi$$
0.370707 + 0.928750i $$0.379115\pi$$
$$60$$ 0 0
$$61$$ −858.000 −1.80091 −0.900456 0.434947i $$-0.856767\pi$$
−0.900456 + 0.434947i $$0.856767\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.00799925
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −209.000 −0.381096 −0.190548 0.981678i $$-0.561026\pi$$
−0.190548 + 0.981678i $$0.561026\pi$$
$$68$$ 0 0
$$69$$ 290.000 0.505970
$$70$$ 0 0
$$71$$ −780.000 −1.30379 −0.651894 0.758310i $$-0.726025\pi$$
−0.651894 + 0.758310i $$0.726025\pi$$
$$72$$ 0 0
$$73$$ 403.000 0.646131 0.323066 0.946377i $$-0.395287\pi$$
0.323066 + 0.946377i $$0.395287\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 78.0000 0.115441
$$78$$ 0 0
$$79$$ −230.000 −0.327557 −0.163779 0.986497i $$-0.552368\pi$$
−0.163779 + 0.986497i $$0.552368\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ 0 0
$$83$$ −1293.00 −1.70994 −0.854971 0.518676i $$-0.826425\pi$$
−0.854971 + 0.518676i $$0.826425\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −960.000 −1.18302
$$88$$ 0 0
$$89$$ −1369.00 −1.63049 −0.815246 0.579115i $$-0.803398\pi$$
−0.815246 + 0.579115i $$0.803398\pi$$
$$90$$ 0 0
$$91$$ −168.000 −0.193530
$$92$$ 0 0
$$93$$ −90.0000 −0.100350
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −382.000 −0.399858 −0.199929 0.979810i $$-0.564071\pi$$
−0.199929 + 0.979810i $$0.564071\pi$$
$$98$$ 0 0
$$99$$ 78.0000 0.0791848
$$100$$ 0 0
$$101$$ 794.000 0.782237 0.391119 0.920340i $$-0.372088\pi$$
0.391119 + 0.920340i $$0.372088\pi$$
$$102$$ 0 0
$$103$$ 1348.00 1.28954 0.644769 0.764378i $$-0.276954\pi$$
0.644769 + 0.764378i $$0.276954\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −775.000 −0.700206 −0.350103 0.936711i $$-0.613853\pi$$
−0.350103 + 0.936711i $$0.613853\pi$$
$$108$$ 0 0
$$109$$ −446.000 −0.391918 −0.195959 0.980612i $$-0.562782\pi$$
−0.195959 + 0.980612i $$0.562782\pi$$
$$110$$ 0 0
$$111$$ −690.000 −0.590017
$$112$$ 0 0
$$113$$ 231.000 0.192307 0.0961533 0.995367i $$-0.469346\pi$$
0.0961533 + 0.995367i $$0.469346\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −168.000 −0.132749
$$118$$ 0 0
$$119$$ −122.000 −0.0939809
$$120$$ 0 0
$$121$$ 190.000 0.142750
$$122$$ 0 0
$$123$$ 1145.00 0.839359
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2386.00 −1.66711 −0.833556 0.552435i $$-0.813699\pi$$
−0.833556 + 0.552435i $$0.813699\pi$$
$$128$$ 0 0
$$129$$ −820.000 −0.559666
$$130$$ 0 0
$$131$$ −2452.00 −1.63536 −0.817680 0.575673i $$-0.804740\pi$$
−0.817680 + 0.575673i $$0.804740\pi$$
$$132$$ 0 0
$$133$$ 302.000 0.196893
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1125.00 0.701571 0.350786 0.936456i $$-0.385915\pi$$
0.350786 + 0.936456i $$0.385915\pi$$
$$138$$ 0 0
$$139$$ 1377.00 0.840256 0.420128 0.907465i $$-0.361985\pi$$
0.420128 + 0.907465i $$0.361985\pi$$
$$140$$ 0 0
$$141$$ 1060.00 0.633107
$$142$$ 0 0
$$143$$ −3276.00 −1.91575
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −1695.00 −0.951029
$$148$$ 0 0
$$149$$ −1920.00 −1.05565 −0.527827 0.849352i $$-0.676993\pi$$
−0.527827 + 0.849352i $$0.676993\pi$$
$$150$$ 0 0
$$151$$ 1854.00 0.999181 0.499591 0.866262i $$-0.333484\pi$$
0.499591 + 0.866262i $$0.333484\pi$$
$$152$$ 0 0
$$153$$ −122.000 −0.0644648
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −634.000 −0.322285 −0.161142 0.986931i $$-0.551518\pi$$
−0.161142 + 0.986931i $$0.551518\pi$$
$$158$$ 0 0
$$159$$ 2890.00 1.44146
$$160$$ 0 0
$$161$$ −116.000 −0.0567831
$$162$$ 0 0
$$163$$ 103.000 0.0494944 0.0247472 0.999694i $$-0.492122\pi$$
0.0247472 + 0.999694i $$0.492122\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −44.0000 −0.0203882 −0.0101941 0.999948i $$-0.503245\pi$$
−0.0101941 + 0.999948i $$0.503245\pi$$
$$168$$ 0 0
$$169$$ 4859.00 2.21165
$$170$$ 0 0
$$171$$ 302.000 0.135056
$$172$$ 0 0
$$173$$ −1128.00 −0.495724 −0.247862 0.968795i $$-0.579728\pi$$
−0.247862 + 0.968795i $$0.579728\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1680.00 0.713427
$$178$$ 0 0
$$179$$ 2245.00 0.937426 0.468713 0.883351i $$-0.344718\pi$$
0.468713 + 0.883351i $$0.344718\pi$$
$$180$$ 0 0
$$181$$ −3050.00 −1.25251 −0.626256 0.779617i $$-0.715414\pi$$
−0.626256 + 0.779617i $$0.715414\pi$$
$$182$$ 0 0
$$183$$ −4290.00 −1.73293
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −2379.00 −0.930319
$$188$$ 0 0
$$189$$ 290.000 0.111611
$$190$$ 0 0
$$191$$ −4222.00 −1.59944 −0.799720 0.600373i $$-0.795019\pi$$
−0.799720 + 0.600373i $$0.795019\pi$$
$$192$$ 0 0
$$193$$ 3357.00 1.25203 0.626016 0.779810i $$-0.284684\pi$$
0.626016 + 0.779810i $$0.284684\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −166.000 −0.0600356 −0.0300178 0.999549i $$-0.509556\pi$$
−0.0300178 + 0.999549i $$0.509556\pi$$
$$198$$ 0 0
$$199$$ 3372.00 1.20118 0.600590 0.799557i $$-0.294932\pi$$
0.600590 + 0.799557i $$0.294932\pi$$
$$200$$ 0 0
$$201$$ −1045.00 −0.366710
$$202$$ 0 0
$$203$$ 384.000 0.132766
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −116.000 −0.0389496
$$208$$ 0 0
$$209$$ 5889.00 1.94905
$$210$$ 0 0
$$211$$ −5601.00 −1.82743 −0.913717 0.406350i $$-0.866801\pi$$
−0.913717 + 0.406350i $$0.866801\pi$$
$$212$$ 0 0
$$213$$ −3900.00 −1.25457
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 36.0000 0.0112619
$$218$$ 0 0
$$219$$ 2015.00 0.621740
$$220$$ 0 0
$$221$$ 5124.00 1.55963
$$222$$ 0 0
$$223$$ −828.000 −0.248641 −0.124321 0.992242i $$-0.539675\pi$$
−0.124321 + 0.992242i $$0.539675\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −2388.00 −0.698225 −0.349113 0.937081i $$-0.613517\pi$$
−0.349113 + 0.937081i $$0.613517\pi$$
$$228$$ 0 0
$$229$$ 2844.00 0.820685 0.410342 0.911932i $$-0.365409\pi$$
0.410342 + 0.911932i $$0.365409\pi$$
$$230$$ 0 0
$$231$$ 390.000 0.111083
$$232$$ 0 0
$$233$$ −5962.00 −1.67632 −0.838162 0.545421i $$-0.816370\pi$$
−0.838162 + 0.545421i $$0.816370\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −1150.00 −0.315192
$$238$$ 0 0
$$239$$ −4320.00 −1.16919 −0.584597 0.811324i $$-0.698748\pi$$
−0.584597 + 0.811324i $$0.698748\pi$$
$$240$$ 0 0
$$241$$ 3857.00 1.03092 0.515459 0.856914i $$-0.327621\pi$$
0.515459 + 0.856914i $$0.327621\pi$$
$$242$$ 0 0
$$243$$ 560.000 0.147835
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −12684.0 −3.26746
$$248$$ 0 0
$$249$$ −6465.00 −1.64539
$$250$$ 0 0
$$251$$ 287.000 0.0721724 0.0360862 0.999349i $$-0.488511\pi$$
0.0360862 + 0.999349i $$0.488511\pi$$
$$252$$ 0 0
$$253$$ −2262.00 −0.562098
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −2130.00 −0.516987 −0.258494 0.966013i $$-0.583226\pi$$
−0.258494 + 0.966013i $$0.583226\pi$$
$$258$$ 0 0
$$259$$ 276.000 0.0662155
$$260$$ 0 0
$$261$$ 384.000 0.0910690
$$262$$ 0 0
$$263$$ 3066.00 0.718850 0.359425 0.933174i $$-0.382973\pi$$
0.359425 + 0.933174i $$0.382973\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −6845.00 −1.56894
$$268$$ 0 0
$$269$$ 3744.00 0.848609 0.424304 0.905520i $$-0.360519\pi$$
0.424304 + 0.905520i $$0.360519\pi$$
$$270$$ 0 0
$$271$$ −3346.00 −0.750019 −0.375009 0.927021i $$-0.622360\pi$$
−0.375009 + 0.927021i $$0.622360\pi$$
$$272$$ 0 0
$$273$$ −840.000 −0.186224
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7040.00 1.52705 0.763525 0.645779i $$-0.223467\pi$$
0.763525 + 0.645779i $$0.223467\pi$$
$$278$$ 0 0
$$279$$ 36.0000 0.00772496
$$280$$ 0 0
$$281$$ −3010.00 −0.639009 −0.319505 0.947585i $$-0.603516\pi$$
−0.319505 + 0.947585i $$0.603516\pi$$
$$282$$ 0 0
$$283$$ −6001.00 −1.26050 −0.630252 0.776391i $$-0.717048\pi$$
−0.630252 + 0.776391i $$0.717048\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −458.000 −0.0941982
$$288$$ 0 0
$$289$$ −1192.00 −0.242622
$$290$$ 0 0
$$291$$ −1910.00 −0.384764
$$292$$ 0 0
$$293$$ 4802.00 0.957460 0.478730 0.877962i $$-0.341097\pi$$
0.478730 + 0.877962i $$0.341097\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5655.00 1.10484
$$298$$ 0 0
$$299$$ 4872.00 0.942325
$$300$$ 0 0
$$301$$ 328.000 0.0628093
$$302$$ 0 0
$$303$$ 3970.00 0.752708
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 6149.00 1.14313 0.571567 0.820556i $$-0.306336\pi$$
0.571567 + 0.820556i $$0.306336\pi$$
$$308$$ 0 0
$$309$$ 6740.00 1.24086
$$310$$ 0 0
$$311$$ −878.000 −0.160086 −0.0800431 0.996791i $$-0.525506\pi$$
−0.0800431 + 0.996791i $$0.525506\pi$$
$$312$$ 0 0
$$313$$ 4042.00 0.729928 0.364964 0.931022i $$-0.381081\pi$$
0.364964 + 0.931022i $$0.381081\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3844.00 0.681074 0.340537 0.940231i $$-0.389391\pi$$
0.340537 + 0.940231i $$0.389391\pi$$
$$318$$ 0 0
$$319$$ 7488.00 1.31426
$$320$$ 0 0
$$321$$ −3875.00 −0.673774
$$322$$ 0 0
$$323$$ −9211.00 −1.58673
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −2230.00 −0.377123
$$328$$ 0 0
$$329$$ −424.000 −0.0710513
$$330$$ 0 0
$$331$$ 2717.00 0.451178 0.225589 0.974223i $$-0.427569\pi$$
0.225589 + 0.974223i $$0.427569\pi$$
$$332$$ 0 0
$$333$$ 276.000 0.0454195
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1603.00 0.259113 0.129556 0.991572i $$-0.458645\pi$$
0.129556 + 0.991572i $$0.458645\pi$$
$$338$$ 0 0
$$339$$ 1155.00 0.185047
$$340$$ 0 0
$$341$$ 702.000 0.111482
$$342$$ 0 0
$$343$$ 1364.00 0.214720
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −11607.0 −1.79567 −0.897833 0.440335i $$-0.854860\pi$$
−0.897833 + 0.440335i $$0.854860\pi$$
$$348$$ 0 0
$$349$$ −4030.00 −0.618112 −0.309056 0.951044i $$-0.600013\pi$$
−0.309056 + 0.951044i $$0.600013\pi$$
$$350$$ 0 0
$$351$$ −12180.0 −1.85219
$$352$$ 0 0
$$353$$ −2106.00 −0.317538 −0.158769 0.987316i $$-0.550753\pi$$
−0.158769 + 0.987316i $$0.550753\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −610.000 −0.0904331
$$358$$ 0 0
$$359$$ 7394.00 1.08702 0.543510 0.839402i $$-0.317095\pi$$
0.543510 + 0.839402i $$0.317095\pi$$
$$360$$ 0 0
$$361$$ 15942.0 2.32425
$$362$$ 0 0
$$363$$ 950.000 0.137361
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 6940.00 0.987098 0.493549 0.869718i $$-0.335699\pi$$
0.493549 + 0.869718i $$0.335699\pi$$
$$368$$ 0 0
$$369$$ −458.000 −0.0646139
$$370$$ 0 0
$$371$$ −1156.00 −0.161770
$$372$$ 0 0
$$373$$ 7486.00 1.03917 0.519585 0.854419i $$-0.326087\pi$$
0.519585 + 0.854419i $$0.326087\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −16128.0 −2.20327
$$378$$ 0 0
$$379$$ −1285.00 −0.174158 −0.0870792 0.996201i $$-0.527753\pi$$
−0.0870792 + 0.996201i $$0.527753\pi$$
$$380$$ 0 0
$$381$$ −11930.0 −1.60418
$$382$$ 0 0
$$383$$ −9622.00 −1.28371 −0.641855 0.766826i $$-0.721835\pi$$
−0.641855 + 0.766826i $$0.721835\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 328.000 0.0430831
$$388$$ 0 0
$$389$$ −1974.00 −0.257290 −0.128645 0.991691i $$-0.541063\pi$$
−0.128645 + 0.991691i $$0.541063\pi$$
$$390$$ 0 0
$$391$$ 3538.00 0.457607
$$392$$ 0 0
$$393$$ −12260.0 −1.57363
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −8084.00 −1.02198 −0.510988 0.859588i $$-0.670720\pi$$
−0.510988 + 0.859588i $$0.670720\pi$$
$$398$$ 0 0
$$399$$ 1510.00 0.189460
$$400$$ 0 0
$$401$$ −5667.00 −0.705727 −0.352863 0.935675i $$-0.614792\pi$$
−0.352863 + 0.935675i $$0.614792\pi$$
$$402$$ 0 0
$$403$$ −1512.00 −0.186894
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 5382.00 0.655469
$$408$$ 0 0
$$409$$ −4835.00 −0.584536 −0.292268 0.956336i $$-0.594410\pi$$
−0.292268 + 0.956336i $$0.594410\pi$$
$$410$$ 0 0
$$411$$ 5625.00 0.675087
$$412$$ 0 0
$$413$$ −672.000 −0.0800653
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 6885.00 0.808537
$$418$$ 0 0
$$419$$ −4619.00 −0.538551 −0.269276 0.963063i $$-0.586784\pi$$
−0.269276 + 0.963063i $$0.586784\pi$$
$$420$$ 0 0
$$421$$ −7476.00 −0.865458 −0.432729 0.901524i $$-0.642449\pi$$
−0.432729 + 0.901524i $$0.642449\pi$$
$$422$$ 0 0
$$423$$ −424.000 −0.0487366
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 1716.00 0.194480
$$428$$ 0 0
$$429$$ −16380.0 −1.84344
$$430$$ 0 0
$$431$$ 7810.00 0.872841 0.436420 0.899743i $$-0.356246\pi$$
0.436420 + 0.899743i $$0.356246\pi$$
$$432$$ 0 0
$$433$$ 2029.00 0.225191 0.112595 0.993641i $$-0.464084\pi$$
0.112595 + 0.993641i $$0.464084\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −8758.00 −0.958700
$$438$$ 0 0
$$439$$ 3208.00 0.348769 0.174384 0.984678i $$-0.444206\pi$$
0.174384 + 0.984678i $$0.444206\pi$$
$$440$$ 0 0
$$441$$ 678.000 0.0732102
$$442$$ 0 0
$$443$$ 13227.0 1.41859 0.709293 0.704914i $$-0.249014\pi$$
0.709293 + 0.704914i $$0.249014\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −9600.00 −1.01580
$$448$$ 0 0
$$449$$ 3617.00 0.380171 0.190086 0.981768i $$-0.439123\pi$$
0.190086 + 0.981768i $$0.439123\pi$$
$$450$$ 0 0
$$451$$ −8931.00 −0.932471
$$452$$ 0 0
$$453$$ 9270.00 0.961463
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6215.00 −0.636161 −0.318080 0.948064i $$-0.603038\pi$$
−0.318080 + 0.948064i $$0.603038\pi$$
$$458$$ 0 0
$$459$$ −8845.00 −0.899454
$$460$$ 0 0
$$461$$ 7108.00 0.718118 0.359059 0.933315i $$-0.383098\pi$$
0.359059 + 0.933315i $$0.383098\pi$$
$$462$$ 0 0
$$463$$ 3364.00 0.337664 0.168832 0.985645i $$-0.446000\pi$$
0.168832 + 0.985645i $$0.446000\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −18964.0 −1.87912 −0.939560 0.342384i $$-0.888766\pi$$
−0.939560 + 0.342384i $$0.888766\pi$$
$$468$$ 0 0
$$469$$ 418.000 0.0411545
$$470$$ 0 0
$$471$$ −3170.00 −0.310119
$$472$$ 0 0
$$473$$ 6396.00 0.621751
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −1156.00 −0.110964
$$478$$ 0 0
$$479$$ −10926.0 −1.04222 −0.521108 0.853491i $$-0.674481\pi$$
−0.521108 + 0.853491i $$0.674481\pi$$
$$480$$ 0 0
$$481$$ −11592.0 −1.09886
$$482$$ 0 0
$$483$$ −580.000 −0.0546396
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4350.00 0.404758 0.202379 0.979307i $$-0.435133\pi$$
0.202379 + 0.979307i $$0.435133\pi$$
$$488$$ 0 0
$$489$$ 515.000 0.0476260
$$490$$ 0 0
$$491$$ 1324.00 0.121693 0.0608465 0.998147i $$-0.480620\pi$$
0.0608465 + 0.998147i $$0.480620\pi$$
$$492$$ 0 0
$$493$$ −11712.0 −1.06994
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1560.00 0.140796
$$498$$ 0 0
$$499$$ −9068.00 −0.813506 −0.406753 0.913538i $$-0.633339\pi$$
−0.406753 + 0.913538i $$0.633339\pi$$
$$500$$ 0 0
$$501$$ −220.000 −0.0196185
$$502$$ 0 0
$$503$$ 19836.0 1.75834 0.879169 0.476511i $$-0.158099\pi$$
0.879169 + 0.476511i $$0.158099\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 24295.0 2.12816
$$508$$ 0 0
$$509$$ −2682.00 −0.233551 −0.116776 0.993158i $$-0.537256\pi$$
−0.116776 + 0.993158i $$0.537256\pi$$
$$510$$ 0 0
$$511$$ −806.000 −0.0697756
$$512$$ 0 0
$$513$$ 21895.0 1.88438
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −8268.00 −0.703339
$$518$$ 0 0
$$519$$ −5640.00 −0.477011
$$520$$ 0 0
$$521$$ −3035.00 −0.255213 −0.127606 0.991825i $$-0.540729\pi$$
−0.127606 + 0.991825i $$0.540729\pi$$
$$522$$ 0 0
$$523$$ 7701.00 0.643865 0.321932 0.946763i $$-0.395668\pi$$
0.321932 + 0.946763i $$0.395668\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1098.00 −0.0907583
$$528$$ 0 0
$$529$$ −8803.00 −0.723514
$$530$$ 0 0
$$531$$ −672.000 −0.0549196
$$532$$ 0 0
$$533$$ 19236.0 1.56323
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 11225.0 0.902038
$$538$$ 0 0
$$539$$ 13221.0 1.05653
$$540$$ 0 0
$$541$$ 18112.0 1.43936 0.719682 0.694304i $$-0.244288\pi$$
0.719682 + 0.694304i $$0.244288\pi$$
$$542$$ 0 0
$$543$$ −15250.0 −1.20523
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 19541.0 1.52745 0.763723 0.645544i $$-0.223369\pi$$
0.763723 + 0.645544i $$0.223369\pi$$
$$548$$ 0 0
$$549$$ 1716.00 0.133401
$$550$$ 0 0
$$551$$ 28992.0 2.24156
$$552$$ 0 0
$$553$$ 460.000 0.0353729
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 13508.0 1.02756 0.513781 0.857921i $$-0.328244\pi$$
0.513781 + 0.857921i $$0.328244\pi$$
$$558$$ 0 0
$$559$$ −13776.0 −1.04233
$$560$$ 0 0
$$561$$ −11895.0 −0.895200
$$562$$ 0 0
$$563$$ −8712.00 −0.652162 −0.326081 0.945342i $$-0.605728\pi$$
−0.326081 + 0.945342i $$0.605728\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1342.00 0.0993981
$$568$$ 0 0
$$569$$ −9623.00 −0.708993 −0.354497 0.935057i $$-0.615348\pi$$
−0.354497 + 0.935057i $$0.615348\pi$$
$$570$$ 0 0
$$571$$ −604.000 −0.0442673 −0.0221336 0.999755i $$-0.507046\pi$$
−0.0221336 + 0.999755i $$0.507046\pi$$
$$572$$ 0 0
$$573$$ −21110.0 −1.53906
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −3629.00 −0.261832 −0.130916 0.991393i $$-0.541792\pi$$
−0.130916 + 0.991393i $$0.541792\pi$$
$$578$$ 0 0
$$579$$ 16785.0 1.20477
$$580$$ 0 0
$$581$$ 2586.00 0.184656
$$582$$ 0 0
$$583$$ −22542.0 −1.60136
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 9219.00 0.648226 0.324113 0.946018i $$-0.394934\pi$$
0.324113 + 0.946018i $$0.394934\pi$$
$$588$$ 0 0
$$589$$ 2718.00 0.190141
$$590$$ 0 0
$$591$$ −830.000 −0.0577693
$$592$$ 0 0
$$593$$ 19111.0 1.32343 0.661716 0.749755i $$-0.269829\pi$$
0.661716 + 0.749755i $$0.269829\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16860.0 1.15584
$$598$$ 0 0
$$599$$ −17086.0 −1.16547 −0.582734 0.812663i $$-0.698017\pi$$
−0.582734 + 0.812663i $$0.698017\pi$$
$$600$$ 0 0
$$601$$ 9035.00 0.613220 0.306610 0.951835i $$-0.400805\pi$$
0.306610 + 0.951835i $$0.400805\pi$$
$$602$$ 0 0
$$603$$ 418.000 0.0282293
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14784.0 0.988573 0.494287 0.869299i $$-0.335429\pi$$
0.494287 + 0.869299i $$0.335429\pi$$
$$608$$ 0 0
$$609$$ 1920.00 0.127754
$$610$$ 0 0
$$611$$ 17808.0 1.17911
$$612$$ 0 0
$$613$$ −17846.0 −1.17585 −0.587923 0.808917i $$-0.700054\pi$$
−0.587923 + 0.808917i $$0.700054\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −11618.0 −0.758060 −0.379030 0.925384i $$-0.623742\pi$$
−0.379030 + 0.925384i $$0.623742\pi$$
$$618$$ 0 0
$$619$$ 9556.00 0.620498 0.310249 0.950655i $$-0.399588\pi$$
0.310249 + 0.950655i $$0.399588\pi$$
$$620$$ 0 0
$$621$$ −8410.00 −0.543449
$$622$$ 0 0
$$623$$ 2738.00 0.176076
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 29445.0 1.87547
$$628$$ 0 0
$$629$$ −8418.00 −0.533621
$$630$$ 0 0
$$631$$ −19394.0 −1.22355 −0.611777 0.791030i $$-0.709545\pi$$
−0.611777 + 0.791030i $$0.709545\pi$$
$$632$$ 0 0
$$633$$ −28005.0 −1.75845
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −28476.0 −1.77121
$$638$$ 0 0
$$639$$ 1560.00 0.0965769
$$640$$ 0 0
$$641$$ 12138.0 0.747929 0.373964 0.927443i $$-0.377998\pi$$
0.373964 + 0.927443i $$0.377998\pi$$
$$642$$ 0 0
$$643$$ 27036.0 1.65816 0.829079 0.559131i $$-0.188865\pi$$
0.829079 + 0.559131i $$0.188865\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17556.0 1.06677 0.533383 0.845874i $$-0.320920\pi$$
0.533383 + 0.845874i $$0.320920\pi$$
$$648$$ 0 0
$$649$$ −13104.0 −0.792569
$$650$$ 0 0
$$651$$ 180.000 0.0108368
$$652$$ 0 0
$$653$$ −17262.0 −1.03448 −0.517239 0.855841i $$-0.673040\pi$$
−0.517239 + 0.855841i $$0.673040\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −806.000 −0.0478616
$$658$$ 0 0
$$659$$ −10517.0 −0.621675 −0.310838 0.950463i $$-0.600610\pi$$
−0.310838 + 0.950463i $$0.600610\pi$$
$$660$$ 0 0
$$661$$ −1408.00 −0.0828515 −0.0414258 0.999142i $$-0.513190\pi$$
−0.0414258 + 0.999142i $$0.513190\pi$$
$$662$$ 0 0
$$663$$ 25620.0 1.50075
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −11136.0 −0.646458
$$668$$ 0 0
$$669$$ −4140.00 −0.239255
$$670$$ 0 0
$$671$$ 33462.0 1.92517
$$672$$ 0 0
$$673$$ 9626.00 0.551345 0.275672 0.961252i $$-0.411100\pi$$
0.275672 + 0.961252i $$0.411100\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 28464.0 1.61589 0.807947 0.589255i $$-0.200579\pi$$
0.807947 + 0.589255i $$0.200579\pi$$
$$678$$ 0 0
$$679$$ 764.000 0.0431806
$$680$$ 0 0
$$681$$ −11940.0 −0.671868
$$682$$ 0 0
$$683$$ −3963.00 −0.222020 −0.111010 0.993819i $$-0.535409\pi$$
−0.111010 + 0.993819i $$0.535409\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 14220.0 0.789704
$$688$$ 0 0
$$689$$ 48552.0 2.68459
$$690$$ 0 0
$$691$$ 31781.0 1.74965 0.874824 0.484442i $$-0.160977\pi$$
0.874824 + 0.484442i $$0.160977\pi$$
$$692$$ 0 0
$$693$$ −156.000 −0.00855115
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 13969.0 0.759130
$$698$$ 0 0
$$699$$ −29810.0 −1.61304
$$700$$ 0 0
$$701$$ 28004.0 1.50884 0.754420 0.656392i $$-0.227918\pi$$
0.754420 + 0.656392i $$0.227918\pi$$
$$702$$ 0 0
$$703$$ 20838.0 1.11795
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1588.00 −0.0844737
$$708$$ 0 0
$$709$$ 35228.0 1.86603 0.933015 0.359837i $$-0.117168\pi$$
0.933015 + 0.359837i $$0.117168\pi$$
$$710$$ 0 0
$$711$$ 460.000 0.0242635
$$712$$ 0 0
$$713$$ −1044.00 −0.0548361
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −21600.0 −1.12506
$$718$$ 0 0
$$719$$ −8658.00 −0.449081 −0.224540 0.974465i $$-0.572088\pi$$
−0.224540 + 0.974465i $$0.572088\pi$$
$$720$$ 0 0
$$721$$ −2696.00 −0.139257
$$722$$ 0 0
$$723$$ 19285.0 0.992001
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −5728.00 −0.292214 −0.146107 0.989269i $$-0.546674\pi$$
−0.146107 + 0.989269i $$0.546674\pi$$
$$728$$ 0 0
$$729$$ 20917.0 1.06269
$$730$$ 0 0
$$731$$ −10004.0 −0.506171
$$732$$ 0 0
$$733$$ 21460.0 1.08137 0.540684 0.841226i $$-0.318165\pi$$
0.540684 + 0.841226i $$0.318165\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8151.00 0.407389
$$738$$ 0 0
$$739$$ −29164.0 −1.45171 −0.725856 0.687847i $$-0.758556\pi$$
−0.725856 + 0.687847i $$0.758556\pi$$
$$740$$ 0 0
$$741$$ −63420.0 −3.14412
$$742$$ 0 0
$$743$$ −29478.0 −1.45551 −0.727754 0.685838i $$-0.759436\pi$$
−0.727754 + 0.685838i $$0.759436\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 2586.00 0.126662
$$748$$ 0 0
$$749$$ 1550.00 0.0756152
$$750$$ 0 0
$$751$$ 576.000 0.0279874 0.0139937 0.999902i $$-0.495546\pi$$
0.0139937 + 0.999902i $$0.495546\pi$$
$$752$$ 0 0
$$753$$ 1435.00 0.0694480
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2880.00 0.138277 0.0691383 0.997607i $$-0.477975\pi$$
0.0691383 + 0.997607i $$0.477975\pi$$
$$758$$ 0 0
$$759$$ −11310.0 −0.540879
$$760$$ 0 0
$$761$$ 20789.0 0.990277 0.495138 0.868814i $$-0.335117\pi$$
0.495138 + 0.868814i $$0.335117\pi$$
$$762$$ 0 0
$$763$$ 892.000 0.0423232
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 28224.0 1.32870
$$768$$ 0 0
$$769$$ 26421.0 1.23897 0.619484 0.785010i $$-0.287342\pi$$
0.619484 + 0.785010i $$0.287342\pi$$
$$770$$ 0 0
$$771$$ −10650.0 −0.497471
$$772$$ 0 0
$$773$$ 32504.0 1.51240 0.756202 0.654339i $$-0.227053\pi$$
0.756202 + 0.654339i $$0.227053\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 1380.00 0.0637159
$$778$$ 0 0
$$779$$ −34579.0 −1.59040
$$780$$ 0 0
$$781$$ 30420.0 1.39374
$$782$$ 0 0
$$783$$ 27840.0 1.27065
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 996.000 0.0451125 0.0225563 0.999746i $$-0.492820\pi$$
0.0225563 + 0.999746i $$0.492820\pi$$
$$788$$ 0 0
$$789$$ 15330.0 0.691714
$$790$$ 0 0
$$791$$ −462.000 −0.0207672
$$792$$ 0 0
$$793$$ −72072.0 −3.22743
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 15134.0 0.672615 0.336307 0.941752i $$-0.390822\pi$$
0.336307 + 0.941752i $$0.390822\pi$$
$$798$$ 0 0
$$799$$ 12932.0 0.572592
$$800$$ 0 0
$$801$$ 2738.00 0.120777
$$802$$ 0 0
$$803$$ −15717.0 −0.690711
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18720.0 0.816574
$$808$$ 0 0
$$809$$ −36942.0 −1.60545 −0.802727 0.596347i $$-0.796618\pi$$
−0.802727 + 0.596347i $$0.796618\pi$$
$$810$$ 0 0
$$811$$ 11748.0 0.508666 0.254333 0.967117i $$-0.418144\pi$$
0.254333 + 0.967117i $$0.418144\pi$$
$$812$$ 0 0
$$813$$ −16730.0 −0.721706
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 24764.0 1.06044
$$818$$ 0 0
$$819$$ 336.000 0.0143355
$$820$$ 0 0
$$821$$ 1198.00 0.0509263 0.0254631 0.999676i $$-0.491894\pi$$
0.0254631 + 0.999676i $$0.491894\pi$$
$$822$$ 0 0
$$823$$ 6788.00 0.287503 0.143751 0.989614i $$-0.454083\pi$$
0.143751 + 0.989614i $$0.454083\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 33011.0 1.38803 0.694017 0.719958i $$-0.255839\pi$$
0.694017 + 0.719958i $$0.255839\pi$$
$$828$$ 0 0
$$829$$ −17732.0 −0.742892 −0.371446 0.928454i $$-0.621138\pi$$
−0.371446 + 0.928454i $$0.621138\pi$$
$$830$$ 0 0
$$831$$ 35200.0 1.46940
$$832$$ 0 0
$$833$$ −20679.0 −0.860126
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 2610.00 0.107784
$$838$$ 0 0
$$839$$ 8480.00 0.348942 0.174471 0.984662i $$-0.444179\pi$$
0.174471 + 0.984662i $$0.444179\pi$$
$$840$$ 0 0
$$841$$ 12475.0 0.511501
$$842$$ 0 0
$$843$$ −15050.0 −0.614887
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −380.000 −0.0154155
$$848$$ 0 0
$$849$$ −30005.0 −1.21292
$$850$$ 0 0
$$851$$ −8004.00 −0.322413
$$852$$ 0 0
$$853$$ 30014.0 1.20476 0.602380 0.798210i $$-0.294219\pi$$
0.602380 + 0.798210i $$0.294219\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −21643.0 −0.862673 −0.431337 0.902191i $$-0.641958\pi$$
−0.431337 + 0.902191i $$0.641958\pi$$
$$858$$ 0 0
$$859$$ −2799.00 −0.111177 −0.0555883 0.998454i $$-0.517703\pi$$
−0.0555883 + 0.998454i $$0.517703\pi$$
$$860$$ 0 0
$$861$$ −2290.00 −0.0906423
$$862$$ 0 0
$$863$$ 19384.0 0.764588 0.382294 0.924041i $$-0.375134\pi$$
0.382294 + 0.924041i $$0.375134\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −5960.00 −0.233463
$$868$$ 0 0
$$869$$ 8970.00 0.350157
$$870$$ 0 0
$$871$$ −17556.0 −0.682965
$$872$$ 0 0
$$873$$ 764.000 0.0296191
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 5132.00 0.197600 0.0988001 0.995107i $$-0.468500\pi$$
0.0988001 + 0.995107i $$0.468500\pi$$
$$878$$ 0 0
$$879$$ 24010.0 0.921316
$$880$$ 0 0
$$881$$ 4430.00 0.169410 0.0847052 0.996406i $$-0.473005\pi$$
0.0847052 + 0.996406i $$0.473005\pi$$
$$882$$ 0 0
$$883$$ 24317.0 0.926764 0.463382 0.886159i $$-0.346636\pi$$
0.463382 + 0.886159i $$0.346636\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 26100.0 0.987996 0.493998 0.869463i $$-0.335535\pi$$
0.493998 + 0.869463i $$0.335535\pi$$
$$888$$ 0 0
$$889$$ 4772.00 0.180031
$$890$$ 0 0
$$891$$ 26169.0 0.983944
$$892$$ 0 0
$$893$$ −32012.0 −1.19960
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 24360.0 0.906752
$$898$$ 0 0
$$899$$ 3456.00 0.128214
$$900$$ 0 0
$$901$$ 35258.0 1.30368
$$902$$ 0 0
$$903$$ 1640.00 0.0604383
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 24356.0 0.891651 0.445826 0.895120i $$-0.352910\pi$$
0.445826 + 0.895120i $$0.352910\pi$$
$$908$$ 0 0
$$909$$ −1588.00 −0.0579435
$$910$$ 0 0
$$911$$ −29900.0 −1.08741 −0.543705 0.839276i $$-0.682979\pi$$
−0.543705 + 0.839276i $$0.682979\pi$$
$$912$$ 0 0
$$913$$ 50427.0 1.82792
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4904.00 0.176602
$$918$$ 0 0
$$919$$ −34838.0 −1.25049 −0.625245 0.780429i $$-0.715001\pi$$
−0.625245 + 0.780429i $$0.715001\pi$$
$$920$$ 0 0
$$921$$ 30745.0 1.09998
$$922$$ 0 0
$$923$$ −65520.0 −2.33653
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −2696.00 −0.0955213
$$928$$ 0 0
$$929$$ 26334.0 0.930022 0.465011 0.885305i $$-0.346050\pi$$
0.465011 + 0.885305i $$0.346050\pi$$
$$930$$ 0 0
$$931$$ 51189.0 1.80199
$$932$$ 0 0
$$933$$ −4390.00 −0.154043
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −30949.0 −1.07904 −0.539520 0.841973i $$-0.681394\pi$$
−0.539520 + 0.841973i $$0.681394\pi$$
$$938$$ 0 0
$$939$$ 20210.0 0.702373
$$940$$ 0 0
$$941$$ −25276.0 −0.875637 −0.437818 0.899063i $$-0.644249\pi$$
−0.437818 + 0.899063i $$0.644249\pi$$
$$942$$ 0 0
$$943$$ 13282.0 0.458665
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1216.00 −0.0417262 −0.0208631 0.999782i $$-0.506641\pi$$
−0.0208631 + 0.999782i $$0.506641\pi$$
$$948$$ 0 0
$$949$$ 33852.0 1.15794
$$950$$ 0 0
$$951$$ 19220.0 0.655364
$$952$$ 0 0
$$953$$ 6033.00 0.205066 0.102533 0.994730i $$-0.467305\pi$$
0.102533 + 0.994730i $$0.467305\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 37440.0 1.26464
$$958$$ 0 0
$$959$$ −2250.00 −0.0757626
$$960$$ 0 0
$$961$$ −29467.0 −0.989124
$$962$$ 0 0
$$963$$ 1550.00 0.0518671
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 41792.0 1.38980 0.694902 0.719105i $$-0.255448\pi$$
0.694902 + 0.719105i $$0.255448\pi$$
$$968$$ 0 0
$$969$$ −46055.0 −1.52683
$$970$$ 0 0
$$971$$ 2105.00 0.0695702 0.0347851 0.999395i $$-0.488925\pi$$
0.0347851 + 0.999395i $$0.488925\pi$$
$$972$$ 0 0
$$973$$ −2754.00 −0.0907391
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −30119.0 −0.986277 −0.493138 0.869951i $$-0.664150\pi$$
−0.493138 + 0.869951i $$0.664150\pi$$
$$978$$ 0 0
$$979$$ 53391.0 1.74299
$$980$$ 0 0
$$981$$ 892.000 0.0290310
$$982$$ 0 0
$$983$$ −18438.0 −0.598251 −0.299126 0.954214i $$-0.596695\pi$$
−0.299126 + 0.954214i $$0.596695\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −2120.00 −0.0683691
$$988$$ 0 0
$$989$$ −9512.00 −0.305828
$$990$$ 0 0
$$991$$ −2230.00 −0.0714816 −0.0357408 0.999361i $$-0.511379\pi$$
−0.0357408 + 0.999361i $$0.511379\pi$$
$$992$$ 0 0
$$993$$ 13585.0 0.434146
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −6804.00 −0.216133 −0.108067 0.994144i $$-0.534466\pi$$
−0.108067 + 0.994144i $$0.534466\pi$$
$$998$$ 0 0
$$999$$ 20010.0 0.633722
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 1600.4.a.bn.1.1 1
4.3 odd 2 1600.4.a.n.1.1 1
5.4 even 2 1600.4.a.m.1.1 1
8.3 odd 2 400.4.a.q.1.1 1
8.5 even 2 200.4.a.c.1.1 1
20.19 odd 2 1600.4.a.bo.1.1 1
24.5 odd 2 1800.4.a.p.1.1 1
40.3 even 4 400.4.c.g.49.2 2
40.13 odd 4 200.4.c.d.49.1 2
40.19 odd 2 400.4.a.f.1.1 1
40.27 even 4 400.4.c.g.49.1 2
40.29 even 2 200.4.a.h.1.1 yes 1
40.37 odd 4 200.4.c.d.49.2 2
120.29 odd 2 1800.4.a.t.1.1 1
120.53 even 4 1800.4.f.c.649.2 2
120.77 even 4 1800.4.f.c.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.c.1.1 1 8.5 even 2
200.4.a.h.1.1 yes 1 40.29 even 2
200.4.c.d.49.1 2 40.13 odd 4
200.4.c.d.49.2 2 40.37 odd 4
400.4.a.f.1.1 1 40.19 odd 2
400.4.a.q.1.1 1 8.3 odd 2
400.4.c.g.49.1 2 40.27 even 4
400.4.c.g.49.2 2 40.3 even 4
1600.4.a.m.1.1 1 5.4 even 2
1600.4.a.n.1.1 1 4.3 odd 2
1600.4.a.bn.1.1 1 1.1 even 1 trivial
1600.4.a.bo.1.1 1 20.19 odd 2
1800.4.a.p.1.1 1 24.5 odd 2
1800.4.a.t.1.1 1 120.29 odd 2
1800.4.f.c.649.1 2 120.77 even 4
1800.4.f.c.649.2 2 120.53 even 4