# Properties

 Label 1872.4.a.r.1.1 Level $1872$ Weight $4$ Character 1872.1 Self dual yes Analytic conductor $110.452$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q+20.0000 q^{5} +32.0000 q^{7} +O(q^{10})$$ $$q+20.0000 q^{5} +32.0000 q^{7} +50.0000 q^{11} -13.0000 q^{13} +30.0000 q^{17} +120.000 q^{19} -20.0000 q^{23} +275.000 q^{25} -82.0000 q^{29} +44.0000 q^{31} +640.000 q^{35} -306.000 q^{37} -108.000 q^{41} +356.000 q^{43} -178.000 q^{47} +681.000 q^{49} -198.000 q^{53} +1000.00 q^{55} +94.0000 q^{59} -62.0000 q^{61} -260.000 q^{65} +140.000 q^{67} -778.000 q^{71} +62.0000 q^{73} +1600.00 q^{77} +1096.00 q^{79} -462.000 q^{83} +600.000 q^{85} -1224.00 q^{89} -416.000 q^{91} +2400.00 q^{95} +614.000 q^{97} +O(q^{100})$$

## 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
$$5$$ 20.0000 1.78885 0.894427 0.447214i $$-0.147584\pi$$
0.894427 + 0.447214i $$0.147584\pi$$
$$6$$ 0 0
$$7$$ 32.0000 1.72784 0.863919 0.503631i $$-0.168003\pi$$
0.863919 + 0.503631i $$0.168003\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 50.0000 1.37051 0.685253 0.728305i $$-0.259692\pi$$
0.685253 + 0.728305i $$0.259692\pi$$
$$12$$ 0 0
$$13$$ −13.0000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 30.0000 0.428004 0.214002 0.976833i $$-0.431350\pi$$
0.214002 + 0.976833i $$0.431350\pi$$
$$18$$ 0 0
$$19$$ 120.000 1.44894 0.724471 0.689306i $$-0.242084\pi$$
0.724471 + 0.689306i $$0.242084\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −20.0000 −0.181317 −0.0906584 0.995882i $$-0.528897\pi$$
−0.0906584 + 0.995882i $$0.528897\pi$$
$$24$$ 0 0
$$25$$ 275.000 2.20000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −82.0000 −0.525070 −0.262535 0.964923i $$-0.584558\pi$$
−0.262535 + 0.964923i $$0.584558\pi$$
$$30$$ 0 0
$$31$$ 44.0000 0.254924 0.127462 0.991843i $$-0.459317\pi$$
0.127462 + 0.991843i $$0.459317\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 640.000 3.09085
$$36$$ 0 0
$$37$$ −306.000 −1.35962 −0.679812 0.733386i $$-0.737939\pi$$
−0.679812 + 0.733386i $$0.737939\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −108.000 −0.411385 −0.205692 0.978617i $$-0.565945\pi$$
−0.205692 + 0.978617i $$0.565945\pi$$
$$42$$ 0 0
$$43$$ 356.000 1.26255 0.631273 0.775561i $$-0.282533\pi$$
0.631273 + 0.775561i $$0.282533\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −178.000 −0.552425 −0.276212 0.961097i $$-0.589079\pi$$
−0.276212 + 0.961097i $$0.589079\pi$$
$$48$$ 0 0
$$49$$ 681.000 1.98542
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −198.000 −0.513158 −0.256579 0.966523i $$-0.582595\pi$$
−0.256579 + 0.966523i $$0.582595\pi$$
$$54$$ 0 0
$$55$$ 1000.00 2.45164
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 94.0000 0.207420 0.103710 0.994608i $$-0.466929\pi$$
0.103710 + 0.994608i $$0.466929\pi$$
$$60$$ 0 0
$$61$$ −62.0000 −0.130136 −0.0650679 0.997881i $$-0.520726\pi$$
−0.0650679 + 0.997881i $$0.520726\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −260.000 −0.496139
$$66$$ 0 0
$$67$$ 140.000 0.255279 0.127640 0.991821i $$-0.459260\pi$$
0.127640 + 0.991821i $$0.459260\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −778.000 −1.30045 −0.650223 0.759744i $$-0.725324\pi$$
−0.650223 + 0.759744i $$0.725324\pi$$
$$72$$ 0 0
$$73$$ 62.0000 0.0994048 0.0497024 0.998764i $$-0.484173\pi$$
0.0497024 + 0.998764i $$0.484173\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1600.00 2.36801
$$78$$ 0 0
$$79$$ 1096.00 1.56088 0.780441 0.625230i $$-0.214995\pi$$
0.780441 + 0.625230i $$0.214995\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −462.000 −0.610977 −0.305488 0.952196i $$-0.598820\pi$$
−0.305488 + 0.952196i $$0.598820\pi$$
$$84$$ 0 0
$$85$$ 600.000 0.765637
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1224.00 −1.45779 −0.728897 0.684623i $$-0.759967\pi$$
−0.728897 + 0.684623i $$0.759967\pi$$
$$90$$ 0 0
$$91$$ −416.000 −0.479216
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2400.00 2.59195
$$96$$ 0 0
$$97$$ 614.000 0.642704 0.321352 0.946960i $$-0.395863\pi$$
0.321352 + 0.946960i $$0.395863\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1058.00 −1.04233 −0.521163 0.853457i $$-0.674502\pi$$
−0.521163 + 0.853457i $$0.674502\pi$$
$$102$$ 0 0
$$103$$ −1768.00 −1.69132 −0.845661 0.533720i $$-0.820794\pi$$
−0.845661 + 0.533720i $$0.820794\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1808.00 −1.63351 −0.816757 0.576982i $$-0.804230\pi$$
−0.816757 + 0.576982i $$0.804230\pi$$
$$108$$ 0 0
$$109$$ −1886.00 −1.65730 −0.828652 0.559765i $$-0.810891\pi$$
−0.828652 + 0.559765i $$0.810891\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1246.00 −1.03729 −0.518645 0.854990i $$-0.673563\pi$$
−0.518645 + 0.854990i $$0.673563\pi$$
$$114$$ 0 0
$$115$$ −400.000 −0.324349
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 960.000 0.739521
$$120$$ 0 0
$$121$$ 1169.00 0.878287
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 3000.00 2.14663
$$126$$ 0 0
$$127$$ −1624.00 −1.13470 −0.567349 0.823477i $$-0.692031\pi$$
−0.567349 + 0.823477i $$0.692031\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2072.00 −1.38192 −0.690960 0.722893i $$-0.742812\pi$$
−0.690960 + 0.722893i $$0.742812\pi$$
$$132$$ 0 0
$$133$$ 3840.00 2.50354
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 756.000 0.471456 0.235728 0.971819i $$-0.424253\pi$$
0.235728 + 0.971819i $$0.424253\pi$$
$$138$$ 0 0
$$139$$ −172.000 −0.104956 −0.0524779 0.998622i $$-0.516712\pi$$
−0.0524779 + 0.998622i $$0.516712\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −650.000 −0.380110
$$144$$ 0 0
$$145$$ −1640.00 −0.939273
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1272.00 −0.699371 −0.349686 0.936867i $$-0.613712\pi$$
−0.349686 + 0.936867i $$0.613712\pi$$
$$150$$ 0 0
$$151$$ −1404.00 −0.756662 −0.378331 0.925670i $$-0.623502\pi$$
−0.378331 + 0.925670i $$0.623502\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 880.000 0.456021
$$156$$ 0 0
$$157$$ −2170.00 −1.10309 −0.551544 0.834146i $$-0.685961\pi$$
−0.551544 + 0.834146i $$0.685961\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −640.000 −0.313286
$$162$$ 0 0
$$163$$ −248.000 −0.119171 −0.0595855 0.998223i $$-0.518978\pi$$
−0.0595855 + 0.998223i $$0.518978\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 102.000 0.0472635 0.0236317 0.999721i $$-0.492477\pi$$
0.0236317 + 0.999721i $$0.492477\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −682.000 −0.299720 −0.149860 0.988707i $$-0.547882\pi$$
−0.149860 + 0.988707i $$0.547882\pi$$
$$174$$ 0 0
$$175$$ 8800.00 3.80124
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −612.000 −0.255548 −0.127774 0.991803i $$-0.540783\pi$$
−0.127774 + 0.991803i $$0.540783\pi$$
$$180$$ 0 0
$$181$$ −66.0000 −0.0271035 −0.0135518 0.999908i $$-0.504314\pi$$
−0.0135518 + 0.999908i $$0.504314\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6120.00 −2.43217
$$186$$ 0 0
$$187$$ 1500.00 0.586582
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 608.000 0.230332 0.115166 0.993346i $$-0.463260\pi$$
0.115166 + 0.993346i $$0.463260\pi$$
$$192$$ 0 0
$$193$$ 1370.00 0.510957 0.255479 0.966815i $$-0.417767\pi$$
0.255479 + 0.966815i $$0.417767\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4908.00 1.77503 0.887514 0.460781i $$-0.152431\pi$$
0.887514 + 0.460781i $$0.152431\pi$$
$$198$$ 0 0
$$199$$ 328.000 0.116841 0.0584204 0.998292i $$-0.481394\pi$$
0.0584204 + 0.998292i $$0.481394\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −2624.00 −0.907235
$$204$$ 0 0
$$205$$ −2160.00 −0.735907
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 6000.00 1.98578
$$210$$ 0 0
$$211$$ −1316.00 −0.429371 −0.214685 0.976683i $$-0.568873\pi$$
−0.214685 + 0.976683i $$0.568873\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 7120.00 2.25851
$$216$$ 0 0
$$217$$ 1408.00 0.440467
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −390.000 −0.118707
$$222$$ 0 0
$$223$$ 1932.00 0.580163 0.290081 0.957002i $$-0.406318\pi$$
0.290081 + 0.957002i $$0.406318\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 4998.00 1.46136 0.730680 0.682720i $$-0.239203\pi$$
0.730680 + 0.682720i $$0.239203\pi$$
$$228$$ 0 0
$$229$$ −78.0000 −0.0225082 −0.0112541 0.999937i $$-0.503582\pi$$
−0.0112541 + 0.999937i $$0.503582\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1282.00 0.360458 0.180229 0.983625i $$-0.442316\pi$$
0.180229 + 0.983625i $$0.442316\pi$$
$$234$$ 0 0
$$235$$ −3560.00 −0.988208
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 294.000 0.0795702 0.0397851 0.999208i $$-0.487333\pi$$
0.0397851 + 0.999208i $$0.487333\pi$$
$$240$$ 0 0
$$241$$ −4962.00 −1.32627 −0.663134 0.748501i $$-0.730774\pi$$
−0.663134 + 0.748501i $$0.730774\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 13620.0 3.55163
$$246$$ 0 0
$$247$$ −1560.00 −0.401864
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 744.000 0.187095 0.0935475 0.995615i $$-0.470179\pi$$
0.0935475 + 0.995615i $$0.470179\pi$$
$$252$$ 0 0
$$253$$ −1000.00 −0.248496
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1026.00 0.249028 0.124514 0.992218i $$-0.460263\pi$$
0.124514 + 0.992218i $$0.460263\pi$$
$$258$$ 0 0
$$259$$ −9792.00 −2.34921
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −5532.00 −1.29703 −0.648513 0.761204i $$-0.724609\pi$$
−0.648513 + 0.761204i $$0.724609\pi$$
$$264$$ 0 0
$$265$$ −3960.00 −0.917966
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 3534.00 0.801010 0.400505 0.916294i $$-0.368835\pi$$
0.400505 + 0.916294i $$0.368835\pi$$
$$270$$ 0 0
$$271$$ −2392.00 −0.536176 −0.268088 0.963394i $$-0.586392\pi$$
−0.268088 + 0.963394i $$0.586392\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 13750.0 3.01511
$$276$$ 0 0
$$277$$ 6102.00 1.32359 0.661794 0.749686i $$-0.269796\pi$$
0.661794 + 0.749686i $$0.269796\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 7540.00 1.60071 0.800354 0.599528i $$-0.204645\pi$$
0.800354 + 0.599528i $$0.204645\pi$$
$$282$$ 0 0
$$283$$ 2756.00 0.578895 0.289447 0.957194i $$-0.406528\pi$$
0.289447 + 0.957194i $$0.406528\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −3456.00 −0.710806
$$288$$ 0 0
$$289$$ −4013.00 −0.816813
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −968.000 −0.193007 −0.0965037 0.995333i $$-0.530766\pi$$
−0.0965037 + 0.995333i $$0.530766\pi$$
$$294$$ 0 0
$$295$$ 1880.00 0.371043
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 260.000 0.0502883
$$300$$ 0 0
$$301$$ 11392.0 2.18147
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −1240.00 −0.232794
$$306$$ 0 0
$$307$$ 6436.00 1.19649 0.598244 0.801314i $$-0.295865\pi$$
0.598244 + 0.801314i $$0.295865\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 7932.00 1.44625 0.723123 0.690719i $$-0.242706\pi$$
0.723123 + 0.690719i $$0.242706\pi$$
$$312$$ 0 0
$$313$$ 10358.0 1.87051 0.935254 0.353978i $$-0.115171\pi$$
0.935254 + 0.353978i $$0.115171\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2820.00 0.499643 0.249822 0.968292i $$-0.419628\pi$$
0.249822 + 0.968292i $$0.419628\pi$$
$$318$$ 0 0
$$319$$ −4100.00 −0.719611
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3600.00 0.620153
$$324$$ 0 0
$$325$$ −3575.00 −0.610170
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −5696.00 −0.954500
$$330$$ 0 0
$$331$$ 4180.00 0.694120 0.347060 0.937843i $$-0.387180\pi$$
0.347060 + 0.937843i $$0.387180\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 2800.00 0.456658
$$336$$ 0 0
$$337$$ −5026.00 −0.812414 −0.406207 0.913781i $$-0.633149\pi$$
−0.406207 + 0.913781i $$0.633149\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2200.00 0.349374
$$342$$ 0 0
$$343$$ 10816.0 1.70265
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −7332.00 −1.13430 −0.567150 0.823614i $$-0.691954\pi$$
−0.567150 + 0.823614i $$0.691954\pi$$
$$348$$ 0 0
$$349$$ −8162.00 −1.25187 −0.625934 0.779876i $$-0.715282\pi$$
−0.625934 + 0.779876i $$0.715282\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −1244.00 −0.187568 −0.0937839 0.995593i $$-0.529896\pi$$
−0.0937839 + 0.995593i $$0.529896\pi$$
$$354$$ 0 0
$$355$$ −15560.0 −2.32631
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 9558.00 1.40516 0.702579 0.711605i $$-0.252032\pi$$
0.702579 + 0.711605i $$0.252032\pi$$
$$360$$ 0 0
$$361$$ 7541.00 1.09943
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1240.00 0.177821
$$366$$ 0 0
$$367$$ 11032.0 1.56912 0.784558 0.620055i $$-0.212890\pi$$
0.784558 + 0.620055i $$0.212890\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −6336.00 −0.886654
$$372$$ 0 0
$$373$$ 5474.00 0.759874 0.379937 0.925012i $$-0.375946\pi$$
0.379937 + 0.925012i $$0.375946\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1066.00 0.145628
$$378$$ 0 0
$$379$$ 7040.00 0.954144 0.477072 0.878864i $$-0.341698\pi$$
0.477072 + 0.878864i $$0.341698\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −1830.00 −0.244148 −0.122074 0.992521i $$-0.538955\pi$$
−0.122074 + 0.992521i $$0.538955\pi$$
$$384$$ 0 0
$$385$$ 32000.0 4.23603
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −10158.0 −1.32399 −0.661994 0.749509i $$-0.730289\pi$$
−0.661994 + 0.749509i $$0.730289\pi$$
$$390$$ 0 0
$$391$$ −600.000 −0.0776044
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 21920.0 2.79219
$$396$$ 0 0
$$397$$ −12658.0 −1.60022 −0.800109 0.599854i $$-0.795225\pi$$
−0.800109 + 0.599854i $$0.795225\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −15720.0 −1.95765 −0.978827 0.204689i $$-0.934382\pi$$
−0.978827 + 0.204689i $$0.934382\pi$$
$$402$$ 0 0
$$403$$ −572.000 −0.0707031
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −15300.0 −1.86337
$$408$$ 0 0
$$409$$ 7654.00 0.925345 0.462672 0.886529i $$-0.346891\pi$$
0.462672 + 0.886529i $$0.346891\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 3008.00 0.358387
$$414$$ 0 0
$$415$$ −9240.00 −1.09295
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −1848.00 −0.215467 −0.107734 0.994180i $$-0.534359\pi$$
−0.107734 + 0.994180i $$0.534359\pi$$
$$420$$ 0 0
$$421$$ −12542.0 −1.45192 −0.725962 0.687735i $$-0.758605\pi$$
−0.725962 + 0.687735i $$0.758605\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 8250.00 0.941609
$$426$$ 0 0
$$427$$ −1984.00 −0.224854
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5238.00 −0.585396 −0.292698 0.956205i $$-0.594553\pi$$
−0.292698 + 0.956205i $$0.594553\pi$$
$$432$$ 0 0
$$433$$ −8258.00 −0.916522 −0.458261 0.888818i $$-0.651528\pi$$
−0.458261 + 0.888818i $$0.651528\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −2400.00 −0.262718
$$438$$ 0 0
$$439$$ 6304.00 0.685361 0.342681 0.939452i $$-0.388665\pi$$
0.342681 + 0.939452i $$0.388665\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12744.0 1.36678 0.683392 0.730051i $$-0.260504\pi$$
0.683392 + 0.730051i $$0.260504\pi$$
$$444$$ 0 0
$$445$$ −24480.0 −2.60778
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 11776.0 1.23774 0.618868 0.785495i $$-0.287591\pi$$
0.618868 + 0.785495i $$0.287591\pi$$
$$450$$ 0 0
$$451$$ −5400.00 −0.563805
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −8320.00 −0.857248
$$456$$ 0 0
$$457$$ 2134.00 0.218434 0.109217 0.994018i $$-0.465166\pi$$
0.109217 + 0.994018i $$0.465166\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −2724.00 −0.275205 −0.137602 0.990488i $$-0.543940\pi$$
−0.137602 + 0.990488i $$0.543940\pi$$
$$462$$ 0 0
$$463$$ 5648.00 0.566922 0.283461 0.958984i $$-0.408517\pi$$
0.283461 + 0.958984i $$0.408517\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −18224.0 −1.80579 −0.902897 0.429856i $$-0.858564\pi$$
−0.902897 + 0.429856i $$0.858564\pi$$
$$468$$ 0 0
$$469$$ 4480.00 0.441081
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 17800.0 1.73033
$$474$$ 0 0
$$475$$ 33000.0 3.18767
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 9066.00 0.864794 0.432397 0.901683i $$-0.357668\pi$$
0.432397 + 0.901683i $$0.357668\pi$$
$$480$$ 0 0
$$481$$ 3978.00 0.377092
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 12280.0 1.14970
$$486$$ 0 0
$$487$$ −8948.00 −0.832593 −0.416296 0.909229i $$-0.636672\pi$$
−0.416296 + 0.909229i $$0.636672\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 8720.00 0.801483 0.400741 0.916191i $$-0.368753\pi$$
0.400741 + 0.916191i $$0.368753\pi$$
$$492$$ 0 0
$$493$$ −2460.00 −0.224732
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −24896.0 −2.24696
$$498$$ 0 0
$$499$$ −6604.00 −0.592456 −0.296228 0.955117i $$-0.595729\pi$$
−0.296228 + 0.955117i $$0.595729\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 3404.00 0.301743 0.150872 0.988553i $$-0.451792\pi$$
0.150872 + 0.988553i $$0.451792\pi$$
$$504$$ 0 0
$$505$$ −21160.0 −1.86457
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 76.0000 0.00661815 0.00330908 0.999995i $$-0.498947\pi$$
0.00330908 + 0.999995i $$0.498947\pi$$
$$510$$ 0 0
$$511$$ 1984.00 0.171755
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −35360.0 −3.02553
$$516$$ 0 0
$$517$$ −8900.00 −0.757102
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −12054.0 −1.01362 −0.506809 0.862058i $$-0.669175\pi$$
−0.506809 + 0.862058i $$0.669175\pi$$
$$522$$ 0 0
$$523$$ −276.000 −0.0230758 −0.0115379 0.999933i $$-0.503673\pi$$
−0.0115379 + 0.999933i $$0.503673\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1320.00 0.109108
$$528$$ 0 0
$$529$$ −11767.0 −0.967124
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1404.00 0.114098
$$534$$ 0 0
$$535$$ −36160.0 −2.92212
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 34050.0 2.72103
$$540$$ 0 0
$$541$$ 13778.0 1.09494 0.547470 0.836825i $$-0.315591\pi$$
0.547470 + 0.836825i $$0.315591\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −37720.0 −2.96467
$$546$$ 0 0
$$547$$ 10844.0 0.847634 0.423817 0.905748i $$-0.360690\pi$$
0.423817 + 0.905748i $$0.360690\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −9840.00 −0.760795
$$552$$ 0 0
$$553$$ 35072.0 2.69695
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −20544.0 −1.56280 −0.781398 0.624033i $$-0.785493\pi$$
−0.781398 + 0.624033i $$0.785493\pi$$
$$558$$ 0 0
$$559$$ −4628.00 −0.350167
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 6988.00 0.523107 0.261553 0.965189i $$-0.415765\pi$$
0.261553 + 0.965189i $$0.415765\pi$$
$$564$$ 0 0
$$565$$ −24920.0 −1.85556
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 706.000 0.0520159 0.0260080 0.999662i $$-0.491720\pi$$
0.0260080 + 0.999662i $$0.491720\pi$$
$$570$$ 0 0
$$571$$ 17532.0 1.28492 0.642462 0.766318i $$-0.277913\pi$$
0.642462 + 0.766318i $$0.277913\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −5500.00 −0.398897
$$576$$ 0 0
$$577$$ −14814.0 −1.06883 −0.534415 0.845222i $$-0.679468\pi$$
−0.534415 + 0.845222i $$0.679468\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −14784.0 −1.05567
$$582$$ 0 0
$$583$$ −9900.00 −0.703287
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14170.0 0.996352 0.498176 0.867076i $$-0.334003\pi$$
0.498176 + 0.867076i $$0.334003\pi$$
$$588$$ 0 0
$$589$$ 5280.00 0.369369
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 11744.0 0.813269 0.406634 0.913591i $$-0.366702\pi$$
0.406634 + 0.913591i $$0.366702\pi$$
$$594$$ 0 0
$$595$$ 19200.0 1.32290
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −15076.0 −1.02836 −0.514181 0.857682i $$-0.671904\pi$$
−0.514181 + 0.857682i $$0.671904\pi$$
$$600$$ 0 0
$$601$$ 20230.0 1.37304 0.686522 0.727109i $$-0.259137\pi$$
0.686522 + 0.727109i $$0.259137\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 23380.0 1.57113
$$606$$ 0 0
$$607$$ 28056.0 1.87604 0.938021 0.346577i $$-0.112656\pi$$
0.938021 + 0.346577i $$0.112656\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2314.00 0.153215
$$612$$ 0 0
$$613$$ 27446.0 1.80837 0.904187 0.427136i $$-0.140478\pi$$
0.904187 + 0.427136i $$0.140478\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −8804.00 −0.574450 −0.287225 0.957863i $$-0.592733\pi$$
−0.287225 + 0.957863i $$0.592733\pi$$
$$618$$ 0 0
$$619$$ −3508.00 −0.227784 −0.113892 0.993493i $$-0.536332\pi$$
−0.113892 + 0.993493i $$0.536332\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −39168.0 −2.51883
$$624$$ 0 0
$$625$$ 25625.0 1.64000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −9180.00 −0.581925
$$630$$ 0 0
$$631$$ −22084.0 −1.39326 −0.696632 0.717428i $$-0.745319\pi$$
−0.696632 + 0.717428i $$0.745319\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −32480.0 −2.02981
$$636$$ 0 0
$$637$$ −8853.00 −0.550657
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 7342.00 0.452405 0.226202 0.974080i $$-0.427369\pi$$
0.226202 + 0.974080i $$0.427369\pi$$
$$642$$ 0 0
$$643$$ −2996.00 −0.183749 −0.0918746 0.995771i $$-0.529286\pi$$
−0.0918746 + 0.995771i $$0.529286\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9344.00 0.567775 0.283888 0.958858i $$-0.408376\pi$$
0.283888 + 0.958858i $$0.408376\pi$$
$$648$$ 0 0
$$649$$ 4700.00 0.284270
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 16686.0 0.999960 0.499980 0.866037i $$-0.333341\pi$$
0.499980 + 0.866037i $$0.333341\pi$$
$$654$$ 0 0
$$655$$ −41440.0 −2.47205
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 31356.0 1.85350 0.926750 0.375679i $$-0.122590\pi$$
0.926750 + 0.375679i $$0.122590\pi$$
$$660$$ 0 0
$$661$$ 590.000 0.0347176 0.0173588 0.999849i $$-0.494474\pi$$
0.0173588 + 0.999849i $$0.494474\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 76800.0 4.47846
$$666$$ 0 0
$$667$$ 1640.00 0.0952040
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −3100.00 −0.178352
$$672$$ 0 0
$$673$$ 5938.00 0.340109 0.170054 0.985435i $$-0.445606\pi$$
0.170054 + 0.985435i $$0.445606\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −9486.00 −0.538518 −0.269259 0.963068i $$-0.586779\pi$$
−0.269259 + 0.963068i $$0.586779\pi$$
$$678$$ 0 0
$$679$$ 19648.0 1.11049
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 26162.0 1.46568 0.732841 0.680400i $$-0.238194\pi$$
0.732841 + 0.680400i $$0.238194\pi$$
$$684$$ 0 0
$$685$$ 15120.0 0.843366
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 2574.00 0.142325
$$690$$ 0 0
$$691$$ 17348.0 0.955064 0.477532 0.878614i $$-0.341532\pi$$
0.477532 + 0.878614i $$0.341532\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3440.00 −0.187751
$$696$$ 0 0
$$697$$ −3240.00 −0.176074
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −0.00161638 −0.000808191 1.00000i $$-0.500257\pi$$
−0.000808191 1.00000i $$0.500257\pi$$
$$702$$ 0 0
$$703$$ −36720.0 −1.97002
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −33856.0 −1.80097
$$708$$ 0 0
$$709$$ 31466.0 1.66676 0.833378 0.552703i $$-0.186404\pi$$
0.833378 + 0.552703i $$0.186404\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −880.000 −0.0462220
$$714$$ 0 0
$$715$$ −13000.0 −0.679961
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −28892.0 −1.49859 −0.749297 0.662234i $$-0.769609\pi$$
−0.749297 + 0.662234i $$0.769609\pi$$
$$720$$ 0 0
$$721$$ −56576.0 −2.92233
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −22550.0 −1.15515
$$726$$ 0 0
$$727$$ −13384.0 −0.682786 −0.341393 0.939921i $$-0.610899\pi$$
−0.341393 + 0.939921i $$0.610899\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 10680.0 0.540375
$$732$$ 0 0
$$733$$ 7130.00 0.359280 0.179640 0.983732i $$-0.442507\pi$$
0.179640 + 0.983732i $$0.442507\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 7000.00 0.349862
$$738$$ 0 0
$$739$$ 29268.0 1.45689 0.728444 0.685105i $$-0.240244\pi$$
0.728444 + 0.685105i $$0.240244\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −9898.00 −0.488725 −0.244362 0.969684i $$-0.578579\pi$$
−0.244362 + 0.969684i $$0.578579\pi$$
$$744$$ 0 0
$$745$$ −25440.0 −1.25107
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −57856.0 −2.82245
$$750$$ 0 0
$$751$$ 15120.0 0.734669 0.367335 0.930089i $$-0.380270\pi$$
0.367335 + 0.930089i $$0.380270\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −28080.0 −1.35356
$$756$$ 0 0
$$757$$ −5454.00 −0.261861 −0.130931 0.991392i $$-0.541797\pi$$
−0.130931 + 0.991392i $$0.541797\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 11988.0 0.571044 0.285522 0.958372i $$-0.407833\pi$$
0.285522 + 0.958372i $$0.407833\pi$$
$$762$$ 0 0
$$763$$ −60352.0 −2.86355
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −1222.00 −0.0575279
$$768$$ 0 0
$$769$$ 1338.00 0.0627432 0.0313716 0.999508i $$-0.490012\pi$$
0.0313716 + 0.999508i $$0.490012\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 14408.0 0.670401 0.335200 0.942147i $$-0.391196\pi$$
0.335200 + 0.942147i $$0.391196\pi$$
$$774$$ 0 0
$$775$$ 12100.0 0.560832
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −12960.0 −0.596072
$$780$$ 0 0
$$781$$ −38900.0 −1.78227
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −43400.0 −1.97326
$$786$$ 0 0
$$787$$ 10660.0 0.482831 0.241415 0.970422i $$-0.422388\pi$$
0.241415 + 0.970422i $$0.422388\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −39872.0 −1.79227
$$792$$ 0 0
$$793$$ 806.000 0.0360932
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −1974.00 −0.0877323 −0.0438662 0.999037i $$-0.513968\pi$$
−0.0438662 + 0.999037i $$0.513968\pi$$
$$798$$ 0 0
$$799$$ −5340.00 −0.236440
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 3100.00 0.136235
$$804$$ 0 0
$$805$$ −12800.0 −0.560423
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −31734.0 −1.37912 −0.689560 0.724229i $$-0.742196\pi$$
−0.689560 + 0.724229i $$0.742196\pi$$
$$810$$ 0 0
$$811$$ 38824.0 1.68100 0.840502 0.541808i $$-0.182260\pi$$
0.840502 + 0.541808i $$0.182260\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −4960.00 −0.213179
$$816$$ 0 0
$$817$$ 42720.0 1.82936
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 16736.0 0.711438 0.355719 0.934593i $$-0.384236\pi$$
0.355719 + 0.934593i $$0.384236\pi$$
$$822$$ 0 0
$$823$$ −42096.0 −1.78296 −0.891479 0.453062i $$-0.850332\pi$$
−0.891479 + 0.453062i $$0.850332\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 24858.0 1.04522 0.522610 0.852572i $$-0.324958\pi$$
0.522610 + 0.852572i $$0.324958\pi$$
$$828$$ 0 0
$$829$$ 922.000 0.0386277 0.0193139 0.999813i $$-0.493852\pi$$
0.0193139 + 0.999813i $$0.493852\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 20430.0 0.849769
$$834$$ 0 0
$$835$$ 2040.00 0.0845474
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −14294.0 −0.588181 −0.294090 0.955778i $$-0.595017\pi$$
−0.294090 + 0.955778i $$0.595017\pi$$
$$840$$ 0 0
$$841$$ −17665.0 −0.724302
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 3380.00 0.137604
$$846$$ 0 0
$$847$$ 37408.0 1.51754
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 6120.00 0.246523
$$852$$ 0 0
$$853$$ 37966.0 1.52395 0.761976 0.647605i $$-0.224229\pi$$
0.761976 + 0.647605i $$0.224229\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −39038.0 −1.55602 −0.778012 0.628249i $$-0.783772\pi$$
−0.778012 + 0.628249i $$0.783772\pi$$
$$858$$ 0 0
$$859$$ −20564.0 −0.816804 −0.408402 0.912802i $$-0.633914\pi$$
−0.408402 + 0.912802i $$0.633914\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 39866.0 1.57248 0.786242 0.617918i $$-0.212024\pi$$
0.786242 + 0.617918i $$0.212024\pi$$
$$864$$ 0 0
$$865$$ −13640.0 −0.536155
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 54800.0 2.13920
$$870$$ 0 0
$$871$$ −1820.00 −0.0708018
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 96000.0 3.70902
$$876$$ 0 0
$$877$$ 30990.0 1.19322 0.596612 0.802530i $$-0.296513\pi$$
0.596612 + 0.802530i $$0.296513\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 4458.00 0.170481 0.0852405 0.996360i $$-0.472834\pi$$
0.0852405 + 0.996360i $$0.472834\pi$$
$$882$$ 0 0
$$883$$ 3164.00 0.120586 0.0602928 0.998181i $$-0.480797\pi$$
0.0602928 + 0.998181i $$0.480797\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −32512.0 −1.23072 −0.615359 0.788247i $$-0.710989\pi$$
−0.615359 + 0.788247i $$0.710989\pi$$
$$888$$ 0 0
$$889$$ −51968.0 −1.96057
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −21360.0 −0.800431
$$894$$ 0 0
$$895$$ −12240.0 −0.457138
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −3608.00 −0.133853
$$900$$ 0 0
$$901$$ −5940.00 −0.219634
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −1320.00 −0.0484843
$$906$$ 0 0
$$907$$ −10500.0 −0.384396 −0.192198 0.981356i $$-0.561562\pi$$
−0.192198 + 0.981356i $$0.561562\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 9840.00 0.357864 0.178932 0.983861i $$-0.442736\pi$$
0.178932 + 0.983861i $$0.442736\pi$$
$$912$$ 0 0
$$913$$ −23100.0 −0.837348
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −66304.0 −2.38773
$$918$$ 0 0
$$919$$ 35040.0 1.25774 0.628870 0.777511i $$-0.283518\pi$$
0.628870 + 0.777511i $$0.283518\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 10114.0 0.360679
$$924$$ 0 0
$$925$$ −84150.0 −2.99117
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −44172.0 −1.56000 −0.779998 0.625782i $$-0.784780\pi$$
−0.779998 + 0.625782i $$0.784780\pi$$
$$930$$ 0 0
$$931$$ 81720.0 2.87676
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 30000.0 1.04931
$$936$$ 0 0
$$937$$ −54018.0 −1.88334 −0.941671 0.336535i $$-0.890745\pi$$
−0.941671 + 0.336535i $$0.890745\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 1672.00 0.0579231 0.0289616 0.999581i $$-0.490780\pi$$
0.0289616 + 0.999581i $$0.490780\pi$$
$$942$$ 0 0
$$943$$ 2160.00 0.0745910
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 5238.00 0.179738 0.0898691 0.995954i $$-0.471355\pi$$
0.0898691 + 0.995954i $$0.471355\pi$$
$$948$$ 0 0
$$949$$ −806.000 −0.0275699
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 50042.0 1.70096 0.850482 0.526004i $$-0.176310\pi$$
0.850482 + 0.526004i $$0.176310\pi$$
$$954$$ 0 0
$$955$$ 12160.0 0.412030
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 24192.0 0.814599
$$960$$ 0 0
$$961$$ −27855.0 −0.935014
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 27400.0 0.914028
$$966$$ 0 0
$$967$$ −37676.0 −1.25293 −0.626463 0.779452i $$-0.715498\pi$$
−0.626463 + 0.779452i $$0.715498\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −17364.0 −0.573880 −0.286940 0.957949i $$-0.592638\pi$$
−0.286940 + 0.957949i $$0.592638\pi$$
$$972$$ 0 0
$$973$$ −5504.00 −0.181346
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 14904.0 0.488046 0.244023 0.969769i $$-0.421533\pi$$
0.244023 + 0.969769i $$0.421533\pi$$
$$978$$ 0 0
$$979$$ −61200.0 −1.99792
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −18038.0 −0.585272 −0.292636 0.956224i $$-0.594533\pi$$
−0.292636 + 0.956224i $$0.594533\pi$$
$$984$$ 0 0
$$985$$ 98160.0 3.17527
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −7120.00 −0.228921
$$990$$ 0 0
$$991$$ −46176.0 −1.48015 −0.740075 0.672524i $$-0.765210\pi$$
−0.740075 + 0.672524i $$0.765210\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 6560.00 0.209011
$$996$$ 0 0
$$997$$ 55838.0 1.77373 0.886864 0.462030i $$-0.152879\pi$$
0.886864 + 0.462030i $$0.152879\pi$$
$$998$$ 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.r.1.1 1
3.2 odd 2 624.4.a.e.1.1 1
4.3 odd 2 234.4.a.f.1.1 1
12.11 even 2 78.4.a.d.1.1 1
24.5 odd 2 2496.4.a.i.1.1 1
24.11 even 2 2496.4.a.r.1.1 1
60.59 even 2 1950.4.a.h.1.1 1
156.47 odd 4 1014.4.b.e.337.2 2
156.83 odd 4 1014.4.b.e.337.1 2
156.155 even 2 1014.4.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.d.1.1 1 12.11 even 2
234.4.a.f.1.1 1 4.3 odd 2
624.4.a.e.1.1 1 3.2 odd 2
1014.4.a.d.1.1 1 156.155 even 2
1014.4.b.e.337.1 2 156.83 odd 4
1014.4.b.e.337.2 2 156.47 odd 4
1872.4.a.r.1.1 1 1.1 even 1 trivial
1950.4.a.h.1.1 1 60.59 even 2
2496.4.a.i.1.1 1 24.5 odd 2
2496.4.a.r.1.1 1 24.11 even 2