# Properties

 Label 45.4.a.a.1.1 Level $45$ Weight $4$ Character 45.1 Self dual yes Analytic conductor $2.655$ 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] = [45,4,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("45.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 45.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: $$2.65508595026$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 45.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{2} +17.0000 q^{4} +5.00000 q^{5} -30.0000 q^{7} -45.0000 q^{8} +O(q^{10})$$ $$q-5.00000 q^{2} +17.0000 q^{4} +5.00000 q^{5} -30.0000 q^{7} -45.0000 q^{8} -25.0000 q^{10} -50.0000 q^{11} -20.0000 q^{13} +150.000 q^{14} +89.0000 q^{16} +10.0000 q^{17} -44.0000 q^{19} +85.0000 q^{20} +250.000 q^{22} -120.000 q^{23} +25.0000 q^{25} +100.000 q^{26} -510.000 q^{28} +50.0000 q^{29} +108.000 q^{31} -85.0000 q^{32} -50.0000 q^{34} -150.000 q^{35} -40.0000 q^{37} +220.000 q^{38} -225.000 q^{40} -400.000 q^{41} +280.000 q^{43} -850.000 q^{44} +600.000 q^{46} +280.000 q^{47} +557.000 q^{49} -125.000 q^{50} -340.000 q^{52} +610.000 q^{53} -250.000 q^{55} +1350.00 q^{56} -250.000 q^{58} -50.0000 q^{59} -518.000 q^{61} -540.000 q^{62} -287.000 q^{64} -100.000 q^{65} -180.000 q^{67} +170.000 q^{68} +750.000 q^{70} -700.000 q^{71} -410.000 q^{73} +200.000 q^{74} -748.000 q^{76} +1500.00 q^{77} -516.000 q^{79} +445.000 q^{80} +2000.00 q^{82} -660.000 q^{83} +50.0000 q^{85} -1400.00 q^{86} +2250.00 q^{88} +1500.00 q^{89} +600.000 q^{91} -2040.00 q^{92} -1400.00 q^{94} -220.000 q^{95} -1630.00 q^{97} -2785.00 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.00000 −1.76777 −0.883883 0.467707i $$-0.845080\pi$$
−0.883883 + 0.467707i $$0.845080\pi$$
$$3$$ 0 0
$$4$$ 17.0000 2.12500
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ −30.0000 −1.61985 −0.809924 0.586535i $$-0.800492\pi$$
−0.809924 + 0.586535i $$0.800492\pi$$
$$8$$ −45.0000 −1.98874
$$9$$ 0 0
$$10$$ −25.0000 −0.790569
$$11$$ −50.0000 −1.37051 −0.685253 0.728305i $$-0.740308\pi$$
−0.685253 + 0.728305i $$0.740308\pi$$
$$12$$ 0 0
$$13$$ −20.0000 −0.426692 −0.213346 0.976977i $$-0.568436\pi$$
−0.213346 + 0.976977i $$0.568436\pi$$
$$14$$ 150.000 2.86351
$$15$$ 0 0
$$16$$ 89.0000 1.39062
$$17$$ 10.0000 0.142668 0.0713340 0.997452i $$-0.477274\pi$$
0.0713340 + 0.997452i $$0.477274\pi$$
$$18$$ 0 0
$$19$$ −44.0000 −0.531279 −0.265639 0.964072i $$-0.585583\pi$$
−0.265639 + 0.964072i $$0.585583\pi$$
$$20$$ 85.0000 0.950329
$$21$$ 0 0
$$22$$ 250.000 2.42274
$$23$$ −120.000 −1.08790 −0.543951 0.839117i $$-0.683072\pi$$
−0.543951 + 0.839117i $$0.683072\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 100.000 0.754293
$$27$$ 0 0
$$28$$ −510.000 −3.44218
$$29$$ 50.0000 0.320164 0.160082 0.987104i $$-0.448824\pi$$
0.160082 + 0.987104i $$0.448824\pi$$
$$30$$ 0 0
$$31$$ 108.000 0.625722 0.312861 0.949799i $$-0.398713\pi$$
0.312861 + 0.949799i $$0.398713\pi$$
$$32$$ −85.0000 −0.469563
$$33$$ 0 0
$$34$$ −50.0000 −0.252204
$$35$$ −150.000 −0.724418
$$36$$ 0 0
$$37$$ −40.0000 −0.177729 −0.0888643 0.996044i $$-0.528324\pi$$
−0.0888643 + 0.996044i $$0.528324\pi$$
$$38$$ 220.000 0.939177
$$39$$ 0 0
$$40$$ −225.000 −0.889391
$$41$$ −400.000 −1.52365 −0.761823 0.647785i $$-0.775696\pi$$
−0.761823 + 0.647785i $$0.775696\pi$$
$$42$$ 0 0
$$43$$ 280.000 0.993014 0.496507 0.868033i $$-0.334616\pi$$
0.496507 + 0.868033i $$0.334616\pi$$
$$44$$ −850.000 −2.91233
$$45$$ 0 0
$$46$$ 600.000 1.92316
$$47$$ 280.000 0.868983 0.434491 0.900676i $$-0.356928\pi$$
0.434491 + 0.900676i $$0.356928\pi$$
$$48$$ 0 0
$$49$$ 557.000 1.62391
$$50$$ −125.000 −0.353553
$$51$$ 0 0
$$52$$ −340.000 −0.906721
$$53$$ 610.000 1.58094 0.790471 0.612499i $$-0.209836\pi$$
0.790471 + 0.612499i $$0.209836\pi$$
$$54$$ 0 0
$$55$$ −250.000 −0.612909
$$56$$ 1350.00 3.22145
$$57$$ 0 0
$$58$$ −250.000 −0.565976
$$59$$ −50.0000 −0.110330 −0.0551648 0.998477i $$-0.517568\pi$$
−0.0551648 + 0.998477i $$0.517568\pi$$
$$60$$ 0 0
$$61$$ −518.000 −1.08726 −0.543632 0.839324i $$-0.682951\pi$$
−0.543632 + 0.839324i $$0.682951\pi$$
$$62$$ −540.000 −1.10613
$$63$$ 0 0
$$64$$ −287.000 −0.560547
$$65$$ −100.000 −0.190823
$$66$$ 0 0
$$67$$ −180.000 −0.328216 −0.164108 0.986442i $$-0.552475\pi$$
−0.164108 + 0.986442i $$0.552475\pi$$
$$68$$ 170.000 0.303170
$$69$$ 0 0
$$70$$ 750.000 1.28060
$$71$$ −700.000 −1.17007 −0.585033 0.811009i $$-0.698919\pi$$
−0.585033 + 0.811009i $$0.698919\pi$$
$$72$$ 0 0
$$73$$ −410.000 −0.657354 −0.328677 0.944442i $$-0.606603\pi$$
−0.328677 + 0.944442i $$0.606603\pi$$
$$74$$ 200.000 0.314183
$$75$$ 0 0
$$76$$ −748.000 −1.12897
$$77$$ 1500.00 2.22001
$$78$$ 0 0
$$79$$ −516.000 −0.734868 −0.367434 0.930050i $$-0.619764\pi$$
−0.367434 + 0.930050i $$0.619764\pi$$
$$80$$ 445.000 0.621906
$$81$$ 0 0
$$82$$ 2000.00 2.69345
$$83$$ −660.000 −0.872824 −0.436412 0.899747i $$-0.643751\pi$$
−0.436412 + 0.899747i $$0.643751\pi$$
$$84$$ 0 0
$$85$$ 50.0000 0.0638031
$$86$$ −1400.00 −1.75542
$$87$$ 0 0
$$88$$ 2250.00 2.72558
$$89$$ 1500.00 1.78651 0.893257 0.449547i $$-0.148415\pi$$
0.893257 + 0.449547i $$0.148415\pi$$
$$90$$ 0 0
$$91$$ 600.000 0.691177
$$92$$ −2040.00 −2.31179
$$93$$ 0 0
$$94$$ −1400.00 −1.53616
$$95$$ −220.000 −0.237595
$$96$$ 0 0
$$97$$ −1630.00 −1.70620 −0.853100 0.521747i $$-0.825280\pi$$
−0.853100 + 0.521747i $$0.825280\pi$$
$$98$$ −2785.00 −2.87069
$$99$$ 0 0
$$100$$ 425.000 0.425000
$$101$$ 450.000 0.443333 0.221667 0.975122i $$-0.428850\pi$$
0.221667 + 0.975122i $$0.428850\pi$$
$$102$$ 0 0
$$103$$ 770.000 0.736605 0.368303 0.929706i $$-0.379939\pi$$
0.368303 + 0.929706i $$0.379939\pi$$
$$104$$ 900.000 0.848579
$$105$$ 0 0
$$106$$ −3050.00 −2.79474
$$107$$ −660.000 −0.596305 −0.298152 0.954518i $$-0.596370\pi$$
−0.298152 + 0.954518i $$0.596370\pi$$
$$108$$ 0 0
$$109$$ 1754.00 1.54131 0.770655 0.637253i $$-0.219929\pi$$
0.770655 + 0.637253i $$0.219929\pi$$
$$110$$ 1250.00 1.08348
$$111$$ 0 0
$$112$$ −2670.00 −2.25260
$$113$$ 310.000 0.258074 0.129037 0.991640i $$-0.458811\pi$$
0.129037 + 0.991640i $$0.458811\pi$$
$$114$$ 0 0
$$115$$ −600.000 −0.486524
$$116$$ 850.000 0.680349
$$117$$ 0 0
$$118$$ 250.000 0.195037
$$119$$ −300.000 −0.231100
$$120$$ 0 0
$$121$$ 1169.00 0.878287
$$122$$ 2590.00 1.92203
$$123$$ 0 0
$$124$$ 1836.00 1.32966
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −1070.00 −0.747615 −0.373808 0.927506i $$-0.621948\pi$$
−0.373808 + 0.927506i $$0.621948\pi$$
$$128$$ 2115.00 1.46048
$$129$$ 0 0
$$130$$ 500.000 0.337330
$$131$$ −1950.00 −1.30055 −0.650276 0.759698i $$-0.725347\pi$$
−0.650276 + 0.759698i $$0.725347\pi$$
$$132$$ 0 0
$$133$$ 1320.00 0.860590
$$134$$ 900.000 0.580210
$$135$$ 0 0
$$136$$ −450.000 −0.283729
$$137$$ −1050.00 −0.654800 −0.327400 0.944886i $$-0.606172\pi$$
−0.327400 + 0.944886i $$0.606172\pi$$
$$138$$ 0 0
$$139$$ 1676.00 1.02271 0.511354 0.859370i $$-0.329144\pi$$
0.511354 + 0.859370i $$0.329144\pi$$
$$140$$ −2550.00 −1.53939
$$141$$ 0 0
$$142$$ 3500.00 2.06840
$$143$$ 1000.00 0.584785
$$144$$ 0 0
$$145$$ 250.000 0.143182
$$146$$ 2050.00 1.16205
$$147$$ 0 0
$$148$$ −680.000 −0.377673
$$149$$ −2050.00 −1.12713 −0.563566 0.826071i $$-0.690571\pi$$
−0.563566 + 0.826071i $$0.690571\pi$$
$$150$$ 0 0
$$151$$ 448.000 0.241442 0.120721 0.992686i $$-0.461479\pi$$
0.120721 + 0.992686i $$0.461479\pi$$
$$152$$ 1980.00 1.05657
$$153$$ 0 0
$$154$$ −7500.00 −3.92446
$$155$$ 540.000 0.279831
$$156$$ 0 0
$$157$$ −100.000 −0.0508336 −0.0254168 0.999677i $$-0.508091\pi$$
−0.0254168 + 0.999677i $$0.508091\pi$$
$$158$$ 2580.00 1.29907
$$159$$ 0 0
$$160$$ −425.000 −0.209995
$$161$$ 3600.00 1.76223
$$162$$ 0 0
$$163$$ −1900.00 −0.913003 −0.456501 0.889723i $$-0.650898\pi$$
−0.456501 + 0.889723i $$0.650898\pi$$
$$164$$ −6800.00 −3.23775
$$165$$ 0 0
$$166$$ 3300.00 1.54295
$$167$$ −1920.00 −0.889665 −0.444833 0.895614i $$-0.646737\pi$$
−0.444833 + 0.895614i $$0.646737\pi$$
$$168$$ 0 0
$$169$$ −1797.00 −0.817934
$$170$$ −250.000 −0.112789
$$171$$ 0 0
$$172$$ 4760.00 2.11015
$$173$$ −2550.00 −1.12065 −0.560326 0.828272i $$-0.689324\pi$$
−0.560326 + 0.828272i $$0.689324\pi$$
$$174$$ 0 0
$$175$$ −750.000 −0.323970
$$176$$ −4450.00 −1.90586
$$177$$ 0 0
$$178$$ −7500.00 −3.15814
$$179$$ −3650.00 −1.52410 −0.762050 0.647518i $$-0.775807\pi$$
−0.762050 + 0.647518i $$0.775807\pi$$
$$180$$ 0 0
$$181$$ −4342.00 −1.78308 −0.891542 0.452937i $$-0.850376\pi$$
−0.891542 + 0.452937i $$0.850376\pi$$
$$182$$ −3000.00 −1.22184
$$183$$ 0 0
$$184$$ 5400.00 2.16355
$$185$$ −200.000 −0.0794827
$$186$$ 0 0
$$187$$ −500.000 −0.195527
$$188$$ 4760.00 1.84659
$$189$$ 0 0
$$190$$ 1100.00 0.420013
$$191$$ 3500.00 1.32592 0.662961 0.748654i $$-0.269299\pi$$
0.662961 + 0.748654i $$0.269299\pi$$
$$192$$ 0 0
$$193$$ 3350.00 1.24942 0.624711 0.780856i $$-0.285217\pi$$
0.624711 + 0.780856i $$0.285217\pi$$
$$194$$ 8150.00 3.01616
$$195$$ 0 0
$$196$$ 9469.00 3.45080
$$197$$ −90.0000 −0.0325494 −0.0162747 0.999868i $$-0.505181\pi$$
−0.0162747 + 0.999868i $$0.505181\pi$$
$$198$$ 0 0
$$199$$ 3664.00 1.30520 0.652598 0.757704i $$-0.273679\pi$$
0.652598 + 0.757704i $$0.273679\pi$$
$$200$$ −1125.00 −0.397748
$$201$$ 0 0
$$202$$ −2250.00 −0.783710
$$203$$ −1500.00 −0.518618
$$204$$ 0 0
$$205$$ −2000.00 −0.681395
$$206$$ −3850.00 −1.30215
$$207$$ 0 0
$$208$$ −1780.00 −0.593369
$$209$$ 2200.00 0.728120
$$210$$ 0 0
$$211$$ −268.000 −0.0874402 −0.0437201 0.999044i $$-0.513921\pi$$
−0.0437201 + 0.999044i $$0.513921\pi$$
$$212$$ 10370.0 3.35950
$$213$$ 0 0
$$214$$ 3300.00 1.05413
$$215$$ 1400.00 0.444089
$$216$$ 0 0
$$217$$ −3240.00 −1.01357
$$218$$ −8770.00 −2.72468
$$219$$ 0 0
$$220$$ −4250.00 −1.30243
$$221$$ −200.000 −0.0608754
$$222$$ 0 0
$$223$$ −3670.00 −1.10207 −0.551034 0.834482i $$-0.685767\pi$$
−0.551034 + 0.834482i $$0.685767\pi$$
$$224$$ 2550.00 0.760621
$$225$$ 0 0
$$226$$ −1550.00 −0.456214
$$227$$ −3760.00 −1.09938 −0.549692 0.835368i $$-0.685255\pi$$
−0.549692 + 0.835368i $$0.685255\pi$$
$$228$$ 0 0
$$229$$ −1434.00 −0.413805 −0.206903 0.978362i $$-0.566338\pi$$
−0.206903 + 0.978362i $$0.566338\pi$$
$$230$$ 3000.00 0.860061
$$231$$ 0 0
$$232$$ −2250.00 −0.636723
$$233$$ 3450.00 0.970030 0.485015 0.874506i $$-0.338814\pi$$
0.485015 + 0.874506i $$0.338814\pi$$
$$234$$ 0 0
$$235$$ 1400.00 0.388621
$$236$$ −850.000 −0.234450
$$237$$ 0 0
$$238$$ 1500.00 0.408532
$$239$$ 4900.00 1.32617 0.663085 0.748544i $$-0.269247\pi$$
0.663085 + 0.748544i $$0.269247\pi$$
$$240$$ 0 0
$$241$$ 4822.00 1.28885 0.644424 0.764668i $$-0.277097\pi$$
0.644424 + 0.764668i $$0.277097\pi$$
$$242$$ −5845.00 −1.55261
$$243$$ 0 0
$$244$$ −8806.00 −2.31044
$$245$$ 2785.00 0.726233
$$246$$ 0 0
$$247$$ 880.000 0.226693
$$248$$ −4860.00 −1.24440
$$249$$ 0 0
$$250$$ −625.000 −0.158114
$$251$$ 4650.00 1.16934 0.584672 0.811270i $$-0.301223\pi$$
0.584672 + 0.811270i $$0.301223\pi$$
$$252$$ 0 0
$$253$$ 6000.00 1.49098
$$254$$ 5350.00 1.32161
$$255$$ 0 0
$$256$$ −8279.00 −2.02124
$$257$$ −5130.00 −1.24514 −0.622569 0.782565i $$-0.713911\pi$$
−0.622569 + 0.782565i $$0.713911\pi$$
$$258$$ 0 0
$$259$$ 1200.00 0.287893
$$260$$ −1700.00 −0.405498
$$261$$ 0 0
$$262$$ 9750.00 2.29907
$$263$$ 1280.00 0.300107 0.150054 0.988678i $$-0.452055\pi$$
0.150054 + 0.988678i $$0.452055\pi$$
$$264$$ 0 0
$$265$$ 3050.00 0.707019
$$266$$ −6600.00 −1.52132
$$267$$ 0 0
$$268$$ −3060.00 −0.697460
$$269$$ −3350.00 −0.759305 −0.379653 0.925129i $$-0.623956\pi$$
−0.379653 + 0.925129i $$0.623956\pi$$
$$270$$ 0 0
$$271$$ 5512.00 1.23554 0.617768 0.786361i $$-0.288037\pi$$
0.617768 + 0.786361i $$0.288037\pi$$
$$272$$ 890.000 0.198398
$$273$$ 0 0
$$274$$ 5250.00 1.15753
$$275$$ −1250.00 −0.274101
$$276$$ 0 0
$$277$$ 4920.00 1.06720 0.533600 0.845737i $$-0.320839\pi$$
0.533600 + 0.845737i $$0.320839\pi$$
$$278$$ −8380.00 −1.80791
$$279$$ 0 0
$$280$$ 6750.00 1.44068
$$281$$ −4500.00 −0.955329 −0.477665 0.878542i $$-0.658517\pi$$
−0.477665 + 0.878542i $$0.658517\pi$$
$$282$$ 0 0
$$283$$ −6900.00 −1.44934 −0.724669 0.689098i $$-0.758007\pi$$
−0.724669 + 0.689098i $$0.758007\pi$$
$$284$$ −11900.0 −2.48639
$$285$$ 0 0
$$286$$ −5000.00 −1.03376
$$287$$ 12000.0 2.46808
$$288$$ 0 0
$$289$$ −4813.00 −0.979646
$$290$$ −1250.00 −0.253112
$$291$$ 0 0
$$292$$ −6970.00 −1.39688
$$293$$ −1530.00 −0.305063 −0.152532 0.988299i $$-0.548743\pi$$
−0.152532 + 0.988299i $$0.548743\pi$$
$$294$$ 0 0
$$295$$ −250.000 −0.0493409
$$296$$ 1800.00 0.353456
$$297$$ 0 0
$$298$$ 10250.0 1.99251
$$299$$ 2400.00 0.464199
$$300$$ 0 0
$$301$$ −8400.00 −1.60853
$$302$$ −2240.00 −0.426813
$$303$$ 0 0
$$304$$ −3916.00 −0.738809
$$305$$ −2590.00 −0.486239
$$306$$ 0 0
$$307$$ 3040.00 0.565153 0.282576 0.959245i $$-0.408811\pi$$
0.282576 + 0.959245i $$0.408811\pi$$
$$308$$ 25500.0 4.71752
$$309$$ 0 0
$$310$$ −2700.00 −0.494676
$$311$$ 5700.00 1.03928 0.519642 0.854384i $$-0.326065\pi$$
0.519642 + 0.854384i $$0.326065\pi$$
$$312$$ 0 0
$$313$$ 3110.00 0.561622 0.280811 0.959763i $$-0.409397\pi$$
0.280811 + 0.959763i $$0.409397\pi$$
$$314$$ 500.000 0.0898619
$$315$$ 0 0
$$316$$ −8772.00 −1.56159
$$317$$ 950.000 0.168320 0.0841598 0.996452i $$-0.473179\pi$$
0.0841598 + 0.996452i $$0.473179\pi$$
$$318$$ 0 0
$$319$$ −2500.00 −0.438787
$$320$$ −1435.00 −0.250684
$$321$$ 0 0
$$322$$ −18000.0 −3.11522
$$323$$ −440.000 −0.0757965
$$324$$ 0 0
$$325$$ −500.000 −0.0853385
$$326$$ 9500.00 1.61398
$$327$$ 0 0
$$328$$ 18000.0 3.03013
$$329$$ −8400.00 −1.40762
$$330$$ 0 0
$$331$$ 2292.00 0.380603 0.190302 0.981726i $$-0.439053\pi$$
0.190302 + 0.981726i $$0.439053\pi$$
$$332$$ −11220.0 −1.85475
$$333$$ 0 0
$$334$$ 9600.00 1.57272
$$335$$ −900.000 −0.146783
$$336$$ 0 0
$$337$$ −7730.00 −1.24950 −0.624748 0.780827i $$-0.714798\pi$$
−0.624748 + 0.780827i $$0.714798\pi$$
$$338$$ 8985.00 1.44592
$$339$$ 0 0
$$340$$ 850.000 0.135582
$$341$$ −5400.00 −0.857555
$$342$$ 0 0
$$343$$ −6420.00 −1.01063
$$344$$ −12600.0 −1.97484
$$345$$ 0 0
$$346$$ 12750.0 1.98105
$$347$$ 1120.00 0.173270 0.0866351 0.996240i $$-0.472389\pi$$
0.0866351 + 0.996240i $$0.472389\pi$$
$$348$$ 0 0
$$349$$ 1186.00 0.181906 0.0909529 0.995855i $$-0.471009\pi$$
0.0909529 + 0.995855i $$0.471009\pi$$
$$350$$ 3750.00 0.572703
$$351$$ 0 0
$$352$$ 4250.00 0.643539
$$353$$ −3630.00 −0.547324 −0.273662 0.961826i $$-0.588235\pi$$
−0.273662 + 0.961826i $$0.588235\pi$$
$$354$$ 0 0
$$355$$ −3500.00 −0.523270
$$356$$ 25500.0 3.79634
$$357$$ 0 0
$$358$$ 18250.0 2.69425
$$359$$ 1800.00 0.264625 0.132312 0.991208i $$-0.457760\pi$$
0.132312 + 0.991208i $$0.457760\pi$$
$$360$$ 0 0
$$361$$ −4923.00 −0.717743
$$362$$ 21710.0 3.15208
$$363$$ 0 0
$$364$$ 10200.0 1.46875
$$365$$ −2050.00 −0.293978
$$366$$ 0 0
$$367$$ 8490.00 1.20756 0.603780 0.797151i $$-0.293661\pi$$
0.603780 + 0.797151i $$0.293661\pi$$
$$368$$ −10680.0 −1.51286
$$369$$ 0 0
$$370$$ 1000.00 0.140507
$$371$$ −18300.0 −2.56089
$$372$$ 0 0
$$373$$ 100.000 0.0138815 0.00694076 0.999976i $$-0.497791\pi$$
0.00694076 + 0.999976i $$0.497791\pi$$
$$374$$ 2500.00 0.345647
$$375$$ 0 0
$$376$$ −12600.0 −1.72818
$$377$$ −1000.00 −0.136612
$$378$$ 0 0
$$379$$ −8084.00 −1.09564 −0.547820 0.836597i $$-0.684542\pi$$
−0.547820 + 0.836597i $$0.684542\pi$$
$$380$$ −3740.00 −0.504889
$$381$$ 0 0
$$382$$ −17500.0 −2.34392
$$383$$ 9480.00 1.26477 0.632383 0.774656i $$-0.282077\pi$$
0.632383 + 0.774656i $$0.282077\pi$$
$$384$$ 0 0
$$385$$ 7500.00 0.992819
$$386$$ −16750.0 −2.20869
$$387$$ 0 0
$$388$$ −27710.0 −3.62568
$$389$$ 10950.0 1.42722 0.713608 0.700545i $$-0.247060\pi$$
0.713608 + 0.700545i $$0.247060\pi$$
$$390$$ 0 0
$$391$$ −1200.00 −0.155209
$$392$$ −25065.0 −3.22952
$$393$$ 0 0
$$394$$ 450.000 0.0575398
$$395$$ −2580.00 −0.328643
$$396$$ 0 0
$$397$$ 13840.0 1.74965 0.874823 0.484442i $$-0.160977\pi$$
0.874823 + 0.484442i $$0.160977\pi$$
$$398$$ −18320.0 −2.30728
$$399$$ 0 0
$$400$$ 2225.00 0.278125
$$401$$ −9300.00 −1.15815 −0.579077 0.815273i $$-0.696587\pi$$
−0.579077 + 0.815273i $$0.696587\pi$$
$$402$$ 0 0
$$403$$ −2160.00 −0.266991
$$404$$ 7650.00 0.942083
$$405$$ 0 0
$$406$$ 7500.00 0.916795
$$407$$ 2000.00 0.243578
$$408$$ 0 0
$$409$$ −2854.00 −0.345040 −0.172520 0.985006i $$-0.555191\pi$$
−0.172520 + 0.985006i $$0.555191\pi$$
$$410$$ 10000.0 1.20455
$$411$$ 0 0
$$412$$ 13090.0 1.56529
$$413$$ 1500.00 0.178717
$$414$$ 0 0
$$415$$ −3300.00 −0.390339
$$416$$ 1700.00 0.200359
$$417$$ 0 0
$$418$$ −11000.0 −1.28715
$$419$$ −1150.00 −0.134084 −0.0670420 0.997750i $$-0.521356\pi$$
−0.0670420 + 0.997750i $$0.521356\pi$$
$$420$$ 0 0
$$421$$ −11162.0 −1.29217 −0.646084 0.763266i $$-0.723594\pi$$
−0.646084 + 0.763266i $$0.723594\pi$$
$$422$$ 1340.00 0.154574
$$423$$ 0 0
$$424$$ −27450.0 −3.14408
$$425$$ 250.000 0.0285336
$$426$$ 0 0
$$427$$ 15540.0 1.76120
$$428$$ −11220.0 −1.26715
$$429$$ 0 0
$$430$$ −7000.00 −0.785046
$$431$$ 1200.00 0.134111 0.0670556 0.997749i $$-0.478639\pi$$
0.0670556 + 0.997749i $$0.478639\pi$$
$$432$$ 0 0
$$433$$ 1510.00 0.167589 0.0837944 0.996483i $$-0.473296\pi$$
0.0837944 + 0.996483i $$0.473296\pi$$
$$434$$ 16200.0 1.79176
$$435$$ 0 0
$$436$$ 29818.0 3.27528
$$437$$ 5280.00 0.577979
$$438$$ 0 0
$$439$$ 424.000 0.0460966 0.0230483 0.999734i $$-0.492663\pi$$
0.0230483 + 0.999734i $$0.492663\pi$$
$$440$$ 11250.0 1.21892
$$441$$ 0 0
$$442$$ 1000.00 0.107613
$$443$$ −12360.0 −1.32560 −0.662801 0.748796i $$-0.730632\pi$$
−0.662801 + 0.748796i $$0.730632\pi$$
$$444$$ 0 0
$$445$$ 7500.00 0.798953
$$446$$ 18350.0 1.94820
$$447$$ 0 0
$$448$$ 8610.00 0.908001
$$449$$ 1300.00 0.136639 0.0683194 0.997664i $$-0.478236\pi$$
0.0683194 + 0.997664i $$0.478236\pi$$
$$450$$ 0 0
$$451$$ 20000.0 2.08817
$$452$$ 5270.00 0.548407
$$453$$ 0 0
$$454$$ 18800.0 1.94345
$$455$$ 3000.00 0.309104
$$456$$ 0 0
$$457$$ −7190.00 −0.735961 −0.367980 0.929834i $$-0.619951\pi$$
−0.367980 + 0.929834i $$0.619951\pi$$
$$458$$ 7170.00 0.731511
$$459$$ 0 0
$$460$$ −10200.0 −1.03386
$$461$$ 150.000 0.0151544 0.00757722 0.999971i $$-0.497588\pi$$
0.00757722 + 0.999971i $$0.497588\pi$$
$$462$$ 0 0
$$463$$ 2670.00 0.268003 0.134002 0.990981i $$-0.457217\pi$$
0.134002 + 0.990981i $$0.457217\pi$$
$$464$$ 4450.00 0.445229
$$465$$ 0 0
$$466$$ −17250.0 −1.71479
$$467$$ −1180.00 −0.116925 −0.0584624 0.998290i $$-0.518620\pi$$
−0.0584624 + 0.998290i $$0.518620\pi$$
$$468$$ 0 0
$$469$$ 5400.00 0.531661
$$470$$ −7000.00 −0.686991
$$471$$ 0 0
$$472$$ 2250.00 0.219417
$$473$$ −14000.0 −1.36093
$$474$$ 0 0
$$475$$ −1100.00 −0.106256
$$476$$ −5100.00 −0.491088
$$477$$ 0 0
$$478$$ −24500.0 −2.34436
$$479$$ 14100.0 1.34498 0.672490 0.740106i $$-0.265225\pi$$
0.672490 + 0.740106i $$0.265225\pi$$
$$480$$ 0 0
$$481$$ 800.000 0.0758355
$$482$$ −24110.0 −2.27838
$$483$$ 0 0
$$484$$ 19873.0 1.86636
$$485$$ −8150.00 −0.763036
$$486$$ 0 0
$$487$$ −9850.00 −0.916522 −0.458261 0.888818i $$-0.651527\pi$$
−0.458261 + 0.888818i $$0.651527\pi$$
$$488$$ 23310.0 2.16228
$$489$$ 0 0
$$490$$ −13925.0 −1.28381
$$491$$ 2450.00 0.225187 0.112594 0.993641i $$-0.464084\pi$$
0.112594 + 0.993641i $$0.464084\pi$$
$$492$$ 0 0
$$493$$ 500.000 0.0456772
$$494$$ −4400.00 −0.400740
$$495$$ 0 0
$$496$$ 9612.00 0.870144
$$497$$ 21000.0 1.89533
$$498$$ 0 0
$$499$$ −17036.0 −1.52833 −0.764164 0.645021i $$-0.776848\pi$$
−0.764164 + 0.645021i $$0.776848\pi$$
$$500$$ 2125.00 0.190066
$$501$$ 0 0
$$502$$ −23250.0 −2.06713
$$503$$ 20600.0 1.82606 0.913030 0.407891i $$-0.133736\pi$$
0.913030 + 0.407891i $$0.133736\pi$$
$$504$$ 0 0
$$505$$ 2250.00 0.198265
$$506$$ −30000.0 −2.63570
$$507$$ 0 0
$$508$$ −18190.0 −1.58868
$$509$$ −5750.00 −0.500716 −0.250358 0.968153i $$-0.580548\pi$$
−0.250358 + 0.968153i $$0.580548\pi$$
$$510$$ 0 0
$$511$$ 12300.0 1.06481
$$512$$ 24475.0 2.11260
$$513$$ 0 0
$$514$$ 25650.0 2.20111
$$515$$ 3850.00 0.329420
$$516$$ 0 0
$$517$$ −14000.0 −1.19095
$$518$$ −6000.00 −0.508928
$$519$$ 0 0
$$520$$ 4500.00 0.379496
$$521$$ 15500.0 1.30339 0.651696 0.758480i $$-0.274058\pi$$
0.651696 + 0.758480i $$0.274058\pi$$
$$522$$ 0 0
$$523$$ −13940.0 −1.16549 −0.582747 0.812653i $$-0.698022\pi$$
−0.582747 + 0.812653i $$0.698022\pi$$
$$524$$ −33150.0 −2.76367
$$525$$ 0 0
$$526$$ −6400.00 −0.530520
$$527$$ 1080.00 0.0892705
$$528$$ 0 0
$$529$$ 2233.00 0.183529
$$530$$ −15250.0 −1.24984
$$531$$ 0 0
$$532$$ 22440.0 1.82875
$$533$$ 8000.00 0.650128
$$534$$ 0 0
$$535$$ −3300.00 −0.266676
$$536$$ 8100.00 0.652736
$$537$$ 0 0
$$538$$ 16750.0 1.34227
$$539$$ −27850.0 −2.22557
$$540$$ 0 0
$$541$$ −20478.0 −1.62739 −0.813695 0.581292i $$-0.802547\pi$$
−0.813695 + 0.581292i $$0.802547\pi$$
$$542$$ −27560.0 −2.18414
$$543$$ 0 0
$$544$$ −850.000 −0.0669916
$$545$$ 8770.00 0.689295
$$546$$ 0 0
$$547$$ 12040.0 0.941121 0.470561 0.882368i $$-0.344052\pi$$
0.470561 + 0.882368i $$0.344052\pi$$
$$548$$ −17850.0 −1.39145
$$549$$ 0 0
$$550$$ 6250.00 0.484547
$$551$$ −2200.00 −0.170096
$$552$$ 0 0
$$553$$ 15480.0 1.19037
$$554$$ −24600.0 −1.88656
$$555$$ 0 0
$$556$$ 28492.0 2.17326
$$557$$ −23550.0 −1.79146 −0.895732 0.444594i $$-0.853348\pi$$
−0.895732 + 0.444594i $$0.853348\pi$$
$$558$$ 0 0
$$559$$ −5600.00 −0.423712
$$560$$ −13350.0 −1.00739
$$561$$ 0 0
$$562$$ 22500.0 1.68880
$$563$$ 6120.00 0.458130 0.229065 0.973411i $$-0.426433\pi$$
0.229065 + 0.973411i $$0.426433\pi$$
$$564$$ 0 0
$$565$$ 1550.00 0.115414
$$566$$ 34500.0 2.56209
$$567$$ 0 0
$$568$$ 31500.0 2.32696
$$569$$ 11700.0 0.862020 0.431010 0.902347i $$-0.358157\pi$$
0.431010 + 0.902347i $$0.358157\pi$$
$$570$$ 0 0
$$571$$ −8188.00 −0.600100 −0.300050 0.953923i $$-0.597003\pi$$
−0.300050 + 0.953923i $$0.597003\pi$$
$$572$$ 17000.0 1.24267
$$573$$ 0 0
$$574$$ −60000.0 −4.36298
$$575$$ −3000.00 −0.217580
$$576$$ 0 0
$$577$$ 11690.0 0.843433 0.421717 0.906728i $$-0.361428\pi$$
0.421717 + 0.906728i $$0.361428\pi$$
$$578$$ 24065.0 1.73179
$$579$$ 0 0
$$580$$ 4250.00 0.304261
$$581$$ 19800.0 1.41384
$$582$$ 0 0
$$583$$ −30500.0 −2.16669
$$584$$ 18450.0 1.30731
$$585$$ 0 0
$$586$$ 7650.00 0.539281
$$587$$ −21060.0 −1.48082 −0.740408 0.672157i $$-0.765368\pi$$
−0.740408 + 0.672157i $$0.765368\pi$$
$$588$$ 0 0
$$589$$ −4752.00 −0.332433
$$590$$ 1250.00 0.0872232
$$591$$ 0 0
$$592$$ −3560.00 −0.247154
$$593$$ 22910.0 1.58651 0.793255 0.608889i $$-0.208385\pi$$
0.793255 + 0.608889i $$0.208385\pi$$
$$594$$ 0 0
$$595$$ −1500.00 −0.103351
$$596$$ −34850.0 −2.39515
$$597$$ 0 0
$$598$$ −12000.0 −0.820596
$$599$$ 1400.00 0.0954966 0.0477483 0.998859i $$-0.484795\pi$$
0.0477483 + 0.998859i $$0.484795\pi$$
$$600$$ 0 0
$$601$$ −11002.0 −0.746724 −0.373362 0.927686i $$-0.621795\pi$$
−0.373362 + 0.927686i $$0.621795\pi$$
$$602$$ 42000.0 2.84351
$$603$$ 0 0
$$604$$ 7616.00 0.513064
$$605$$ 5845.00 0.392782
$$606$$ 0 0
$$607$$ 4630.00 0.309598 0.154799 0.987946i $$-0.450527\pi$$
0.154799 + 0.987946i $$0.450527\pi$$
$$608$$ 3740.00 0.249469
$$609$$ 0 0
$$610$$ 12950.0 0.859558
$$611$$ −5600.00 −0.370788
$$612$$ 0 0
$$613$$ 24040.0 1.58396 0.791979 0.610548i $$-0.209051\pi$$
0.791979 + 0.610548i $$0.209051\pi$$
$$614$$ −15200.0 −0.999059
$$615$$ 0 0
$$616$$ −67500.0 −4.41502
$$617$$ 1890.00 0.123320 0.0616601 0.998097i $$-0.480361\pi$$
0.0616601 + 0.998097i $$0.480361\pi$$
$$618$$ 0 0
$$619$$ 19244.0 1.24957 0.624783 0.780798i $$-0.285187\pi$$
0.624783 + 0.780798i $$0.285187\pi$$
$$620$$ 9180.00 0.594641
$$621$$ 0 0
$$622$$ −28500.0 −1.83721
$$623$$ −45000.0 −2.89388
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −15550.0 −0.992816
$$627$$ 0 0
$$628$$ −1700.00 −0.108021
$$629$$ −400.000 −0.0253562
$$630$$ 0 0
$$631$$ 15892.0 1.00262 0.501308 0.865269i $$-0.332852\pi$$
0.501308 + 0.865269i $$0.332852\pi$$
$$632$$ 23220.0 1.46146
$$633$$ 0 0
$$634$$ −4750.00 −0.297550
$$635$$ −5350.00 −0.334344
$$636$$ 0 0
$$637$$ −11140.0 −0.692909
$$638$$ 12500.0 0.775674
$$639$$ 0 0
$$640$$ 10575.0 0.653146
$$641$$ 12600.0 0.776396 0.388198 0.921576i $$-0.373098\pi$$
0.388198 + 0.921576i $$0.373098\pi$$
$$642$$ 0 0
$$643$$ 7260.00 0.445267 0.222633 0.974902i $$-0.428535\pi$$
0.222633 + 0.974902i $$0.428535\pi$$
$$644$$ 61200.0 3.74475
$$645$$ 0 0
$$646$$ 2200.00 0.133990
$$647$$ −7400.00 −0.449651 −0.224825 0.974399i $$-0.572181\pi$$
−0.224825 + 0.974399i $$0.572181\pi$$
$$648$$ 0 0
$$649$$ 2500.00 0.151207
$$650$$ 2500.00 0.150859
$$651$$ 0 0
$$652$$ −32300.0 −1.94013
$$653$$ 4790.00 0.287055 0.143528 0.989646i $$-0.454155\pi$$
0.143528 + 0.989646i $$0.454155\pi$$
$$654$$ 0 0
$$655$$ −9750.00 −0.581624
$$656$$ −35600.0 −2.11882
$$657$$ 0 0
$$658$$ 42000.0 2.48834
$$659$$ 1450.00 0.0857117 0.0428558 0.999081i $$-0.486354\pi$$
0.0428558 + 0.999081i $$0.486354\pi$$
$$660$$ 0 0
$$661$$ 11818.0 0.695411 0.347706 0.937604i $$-0.386961\pi$$
0.347706 + 0.937604i $$0.386961\pi$$
$$662$$ −11460.0 −0.672818
$$663$$ 0 0
$$664$$ 29700.0 1.73582
$$665$$ 6600.00 0.384868
$$666$$ 0 0
$$667$$ −6000.00 −0.348307
$$668$$ −32640.0 −1.89054
$$669$$ 0 0
$$670$$ 4500.00 0.259478
$$671$$ 25900.0 1.49010
$$672$$ 0 0
$$673$$ 5550.00 0.317885 0.158943 0.987288i $$-0.449192\pi$$
0.158943 + 0.987288i $$0.449192\pi$$
$$674$$ 38650.0 2.20882
$$675$$ 0 0
$$676$$ −30549.0 −1.73811
$$677$$ −12930.0 −0.734033 −0.367016 0.930214i $$-0.619621\pi$$
−0.367016 + 0.930214i $$0.619621\pi$$
$$678$$ 0 0
$$679$$ 48900.0 2.76378
$$680$$ −2250.00 −0.126888
$$681$$ 0 0
$$682$$ 27000.0 1.51596
$$683$$ −32580.0 −1.82524 −0.912620 0.408809i $$-0.865944\pi$$
−0.912620 + 0.408809i $$0.865944\pi$$
$$684$$ 0 0
$$685$$ −5250.00 −0.292835
$$686$$ 32100.0 1.78657
$$687$$ 0 0
$$688$$ 24920.0 1.38091
$$689$$ −12200.0 −0.674576
$$690$$ 0 0
$$691$$ 10228.0 0.563085 0.281542 0.959549i $$-0.409154\pi$$
0.281542 + 0.959549i $$0.409154\pi$$
$$692$$ −43350.0 −2.38139
$$693$$ 0 0
$$694$$ −5600.00 −0.306301
$$695$$ 8380.00 0.457369
$$696$$ 0 0
$$697$$ −4000.00 −0.217376
$$698$$ −5930.00 −0.321567
$$699$$ 0 0
$$700$$ −12750.0 −0.688435
$$701$$ −8350.00 −0.449893 −0.224947 0.974371i $$-0.572221\pi$$
−0.224947 + 0.974371i $$0.572221\pi$$
$$702$$ 0 0
$$703$$ 1760.00 0.0944234
$$704$$ 14350.0 0.768233
$$705$$ 0 0
$$706$$ 18150.0 0.967541
$$707$$ −13500.0 −0.718133
$$708$$ 0 0
$$709$$ −14954.0 −0.792115 −0.396057 0.918226i $$-0.629622\pi$$
−0.396057 + 0.918226i $$0.629622\pi$$
$$710$$ 17500.0 0.925019
$$711$$ 0 0
$$712$$ −67500.0 −3.55291
$$713$$ −12960.0 −0.680723
$$714$$ 0 0
$$715$$ 5000.00 0.261524
$$716$$ −62050.0 −3.23871
$$717$$ 0 0
$$718$$ −9000.00 −0.467795
$$719$$ −29400.0 −1.52494 −0.762472 0.647021i $$-0.776015\pi$$
−0.762472 + 0.647021i $$0.776015\pi$$
$$720$$ 0 0
$$721$$ −23100.0 −1.19319
$$722$$ 24615.0 1.26880
$$723$$ 0 0
$$724$$ −73814.0 −3.78905
$$725$$ 1250.00 0.0640329
$$726$$ 0 0
$$727$$ −16330.0 −0.833076 −0.416538 0.909118i $$-0.636757\pi$$
−0.416538 + 0.909118i $$0.636757\pi$$
$$728$$ −27000.0 −1.37457
$$729$$ 0 0
$$730$$ 10250.0 0.519684
$$731$$ 2800.00 0.141671
$$732$$ 0 0
$$733$$ 30800.0 1.55201 0.776005 0.630726i $$-0.217243\pi$$
0.776005 + 0.630726i $$0.217243\pi$$
$$734$$ −42450.0 −2.13468
$$735$$ 0 0
$$736$$ 10200.0 0.510838
$$737$$ 9000.00 0.449823
$$738$$ 0 0
$$739$$ −9524.00 −0.474081 −0.237041 0.971500i $$-0.576177\pi$$
−0.237041 + 0.971500i $$0.576177\pi$$
$$740$$ −3400.00 −0.168901
$$741$$ 0 0
$$742$$ 91500.0 4.52705
$$743$$ 28600.0 1.41216 0.706078 0.708134i $$-0.250463\pi$$
0.706078 + 0.708134i $$0.250463\pi$$
$$744$$ 0 0
$$745$$ −10250.0 −0.504068
$$746$$ −500.000 −0.0245393
$$747$$ 0 0
$$748$$ −8500.00 −0.415496
$$749$$ 19800.0 0.965923
$$750$$ 0 0
$$751$$ −8252.00 −0.400958 −0.200479 0.979698i $$-0.564250\pi$$
−0.200479 + 0.979698i $$0.564250\pi$$
$$752$$ 24920.0 1.20843
$$753$$ 0 0
$$754$$ 5000.00 0.241498
$$755$$ 2240.00 0.107976
$$756$$ 0 0
$$757$$ −24920.0 −1.19648 −0.598238 0.801318i $$-0.704132\pi$$
−0.598238 + 0.801318i $$0.704132\pi$$
$$758$$ 40420.0 1.93683
$$759$$ 0 0
$$760$$ 9900.00 0.472514
$$761$$ −27900.0 −1.32901 −0.664503 0.747285i $$-0.731357\pi$$
−0.664503 + 0.747285i $$0.731357\pi$$
$$762$$ 0 0
$$763$$ −52620.0 −2.49669
$$764$$ 59500.0 2.81758
$$765$$ 0 0
$$766$$ −47400.0 −2.23581
$$767$$ 1000.00 0.0470768
$$768$$ 0 0
$$769$$ −11506.0 −0.539554 −0.269777 0.962923i $$-0.586950\pi$$
−0.269777 + 0.962923i $$0.586950\pi$$
$$770$$ −37500.0 −1.75507
$$771$$ 0 0
$$772$$ 56950.0 2.65502
$$773$$ 12510.0 0.582087 0.291044 0.956710i $$-0.405998\pi$$
0.291044 + 0.956710i $$0.405998\pi$$
$$774$$ 0 0
$$775$$ 2700.00 0.125144
$$776$$ 73350.0 3.39318
$$777$$ 0 0
$$778$$ −54750.0 −2.52299
$$779$$ 17600.0 0.809481
$$780$$ 0 0
$$781$$ 35000.0 1.60358
$$782$$ 6000.00 0.274373
$$783$$ 0 0
$$784$$ 49573.0 2.25825
$$785$$ −500.000 −0.0227335
$$786$$ 0 0
$$787$$ −1100.00 −0.0498231 −0.0249115 0.999690i $$-0.507930\pi$$
−0.0249115 + 0.999690i $$0.507930\pi$$
$$788$$ −1530.00 −0.0691675
$$789$$ 0 0
$$790$$ 12900.0 0.580964
$$791$$ −9300.00 −0.418040
$$792$$ 0 0
$$793$$ 10360.0 0.463927
$$794$$ −69200.0 −3.09297
$$795$$ 0 0
$$796$$ 62288.0 2.77354
$$797$$ 4490.00 0.199553 0.0997766 0.995010i $$-0.468187\pi$$
0.0997766 + 0.995010i $$0.468187\pi$$
$$798$$ 0 0
$$799$$ 2800.00 0.123976
$$800$$ −2125.00 −0.0939126
$$801$$ 0 0
$$802$$ 46500.0 2.04735
$$803$$ 20500.0 0.900908
$$804$$ 0 0
$$805$$ 18000.0 0.788095
$$806$$ 10800.0 0.471977
$$807$$ 0 0
$$808$$ −20250.0 −0.881674
$$809$$ −28600.0 −1.24292 −0.621460 0.783446i $$-0.713460\pi$$
−0.621460 + 0.783446i $$0.713460\pi$$
$$810$$ 0 0
$$811$$ 10068.0 0.435925 0.217963 0.975957i $$-0.430059\pi$$
0.217963 + 0.975957i $$0.430059\pi$$
$$812$$ −25500.0 −1.10206
$$813$$ 0 0
$$814$$ −10000.0 −0.430589
$$815$$ −9500.00 −0.408307
$$816$$ 0 0
$$817$$ −12320.0 −0.527567
$$818$$ 14270.0 0.609950
$$819$$ 0 0
$$820$$ −34000.0 −1.44797
$$821$$ 14250.0 0.605759 0.302880 0.953029i $$-0.402052\pi$$
0.302880 + 0.953029i $$0.402052\pi$$
$$822$$ 0 0
$$823$$ 6830.00 0.289282 0.144641 0.989484i $$-0.453797\pi$$
0.144641 + 0.989484i $$0.453797\pi$$
$$824$$ −34650.0 −1.46491
$$825$$ 0 0
$$826$$ −7500.00 −0.315930
$$827$$ 8920.00 0.375065 0.187533 0.982258i $$-0.439951\pi$$
0.187533 + 0.982258i $$0.439951\pi$$
$$828$$ 0 0
$$829$$ −3534.00 −0.148059 −0.0740295 0.997256i $$-0.523586\pi$$
−0.0740295 + 0.997256i $$0.523586\pi$$
$$830$$ 16500.0 0.690028
$$831$$ 0 0
$$832$$ 5740.00 0.239181
$$833$$ 5570.00 0.231680
$$834$$ 0 0
$$835$$ −9600.00 −0.397870
$$836$$ 37400.0 1.54726
$$837$$ 0 0
$$838$$ 5750.00 0.237029
$$839$$ 8000.00 0.329190 0.164595 0.986361i $$-0.447368\pi$$
0.164595 + 0.986361i $$0.447368\pi$$
$$840$$ 0 0
$$841$$ −21889.0 −0.897495
$$842$$ 55810.0 2.28425
$$843$$ 0 0
$$844$$ −4556.00 −0.185810
$$845$$ −8985.00 −0.365791
$$846$$ 0 0
$$847$$ −35070.0 −1.42269
$$848$$ 54290.0 2.19850
$$849$$ 0 0
$$850$$ −1250.00 −0.0504408
$$851$$ 4800.00 0.193351
$$852$$ 0 0
$$853$$ 5160.00 0.207122 0.103561 0.994623i $$-0.466976\pi$$
0.103561 + 0.994623i $$0.466976\pi$$
$$854$$ −77700.0 −3.11339
$$855$$ 0 0
$$856$$ 29700.0 1.18589
$$857$$ −7670.00 −0.305720 −0.152860 0.988248i $$-0.548848\pi$$
−0.152860 + 0.988248i $$0.548848\pi$$
$$858$$ 0 0
$$859$$ 25804.0 1.02494 0.512469 0.858706i $$-0.328731\pi$$
0.512469 + 0.858706i $$0.328731\pi$$
$$860$$ 23800.0 0.943690
$$861$$ 0 0
$$862$$ −6000.00 −0.237078
$$863$$ −400.000 −0.0157777 −0.00788885 0.999969i $$-0.502511\pi$$
−0.00788885 + 0.999969i $$0.502511\pi$$
$$864$$ 0 0
$$865$$ −12750.0 −0.501171
$$866$$ −7550.00 −0.296258
$$867$$ 0 0
$$868$$ −55080.0 −2.15384
$$869$$ 25800.0 1.00714
$$870$$ 0 0
$$871$$ 3600.00 0.140047
$$872$$ −78930.0 −3.06526
$$873$$ 0 0
$$874$$ −26400.0 −1.02173
$$875$$ −3750.00 −0.144884
$$876$$ 0 0
$$877$$ −35100.0 −1.35147 −0.675737 0.737143i $$-0.736175\pi$$
−0.675737 + 0.737143i $$0.736175\pi$$
$$878$$ −2120.00 −0.0814881
$$879$$ 0 0
$$880$$ −22250.0 −0.852327
$$881$$ 18700.0 0.715118 0.357559 0.933891i $$-0.383609\pi$$
0.357559 + 0.933891i $$0.383609\pi$$
$$882$$ 0 0
$$883$$ −2980.00 −0.113573 −0.0567865 0.998386i $$-0.518085\pi$$
−0.0567865 + 0.998386i $$0.518085\pi$$
$$884$$ −3400.00 −0.129360
$$885$$ 0 0
$$886$$ 61800.0 2.34335
$$887$$ 35880.0 1.35821 0.679105 0.734041i $$-0.262368\pi$$
0.679105 + 0.734041i $$0.262368\pi$$
$$888$$ 0 0
$$889$$ 32100.0 1.21102
$$890$$ −37500.0 −1.41236
$$891$$ 0 0
$$892$$ −62390.0 −2.34190
$$893$$ −12320.0 −0.461672
$$894$$ 0 0
$$895$$ −18250.0 −0.681598
$$896$$ −63450.0 −2.36575
$$897$$ 0 0
$$898$$ −6500.00 −0.241545
$$899$$ 5400.00 0.200334
$$900$$ 0 0
$$901$$ 6100.00 0.225550
$$902$$ −100000. −3.69139
$$903$$ 0 0
$$904$$ −13950.0 −0.513241
$$905$$ −21710.0 −0.797420
$$906$$ 0 0
$$907$$ 45240.0 1.65620 0.828098 0.560584i $$-0.189423\pi$$
0.828098 + 0.560584i $$0.189423\pi$$
$$908$$ −63920.0 −2.33619
$$909$$ 0 0
$$910$$ −15000.0 −0.546423
$$911$$ −33200.0 −1.20743 −0.603713 0.797202i $$-0.706313\pi$$
−0.603713 + 0.797202i $$0.706313\pi$$
$$912$$ 0 0
$$913$$ 33000.0 1.19621
$$914$$ 35950.0 1.30101
$$915$$ 0 0
$$916$$ −24378.0 −0.879336
$$917$$ 58500.0 2.10670
$$918$$ 0 0
$$919$$ 35356.0 1.26908 0.634541 0.772889i $$-0.281189\pi$$
0.634541 + 0.772889i $$0.281189\pi$$
$$920$$ 27000.0 0.967569
$$921$$ 0 0
$$922$$ −750.000 −0.0267895
$$923$$ 14000.0 0.499259
$$924$$ 0 0
$$925$$ −1000.00 −0.0355457
$$926$$ −13350.0 −0.473767
$$927$$ 0 0
$$928$$ −4250.00 −0.150337
$$929$$ 25700.0 0.907631 0.453816 0.891096i $$-0.350062\pi$$
0.453816 + 0.891096i $$0.350062\pi$$
$$930$$ 0 0
$$931$$ −24508.0 −0.862747
$$932$$ 58650.0 2.06131
$$933$$ 0 0
$$934$$ 5900.00 0.206696
$$935$$ −2500.00 −0.0874425
$$936$$ 0 0
$$937$$ 52890.0 1.84401 0.922007 0.387173i $$-0.126549\pi$$
0.922007 + 0.387173i $$0.126549\pi$$
$$938$$ −27000.0 −0.939852
$$939$$ 0 0
$$940$$ 23800.0 0.825820
$$941$$ −38050.0 −1.31817 −0.659083 0.752070i $$-0.729055\pi$$
−0.659083 + 0.752070i $$0.729055\pi$$
$$942$$ 0 0
$$943$$ 48000.0 1.65758
$$944$$ −4450.00 −0.153427
$$945$$ 0 0
$$946$$ 70000.0 2.40581
$$947$$ 29640.0 1.01708 0.508538 0.861040i $$-0.330186\pi$$
0.508538 + 0.861040i $$0.330186\pi$$
$$948$$ 0 0
$$949$$ 8200.00 0.280488
$$950$$ 5500.00 0.187835
$$951$$ 0 0
$$952$$ 13500.0 0.459598
$$953$$ −15170.0 −0.515640 −0.257820 0.966193i $$-0.583004\pi$$
−0.257820 + 0.966193i $$0.583004\pi$$
$$954$$ 0 0
$$955$$ 17500.0 0.592970
$$956$$ 83300.0 2.81811
$$957$$ 0 0
$$958$$ −70500.0 −2.37761
$$959$$ 31500.0 1.06068
$$960$$ 0 0
$$961$$ −18127.0 −0.608472
$$962$$ −4000.00 −0.134059
$$963$$ 0 0
$$964$$ 81974.0 2.73880
$$965$$ 16750.0 0.558758
$$966$$ 0 0
$$967$$ −5470.00 −0.181906 −0.0909531 0.995855i $$-0.528991\pi$$
−0.0909531 + 0.995855i $$0.528991\pi$$
$$968$$ −52605.0 −1.74668
$$969$$ 0 0
$$970$$ 40750.0 1.34887
$$971$$ −15150.0 −0.500707 −0.250354 0.968154i $$-0.580547\pi$$
−0.250354 + 0.968154i $$0.580547\pi$$
$$972$$ 0 0
$$973$$ −50280.0 −1.65663
$$974$$ 49250.0 1.62020
$$975$$ 0 0
$$976$$ −46102.0 −1.51198
$$977$$ −31190.0 −1.02135 −0.510674 0.859775i $$-0.670604\pi$$
−0.510674 + 0.859775i $$0.670604\pi$$
$$978$$ 0 0
$$979$$ −75000.0 −2.44843
$$980$$ 47345.0 1.54325
$$981$$ 0 0
$$982$$ −12250.0 −0.398079
$$983$$ 7560.00 0.245297 0.122648 0.992450i $$-0.460861\pi$$
0.122648 + 0.992450i $$0.460861\pi$$
$$984$$ 0 0
$$985$$ −450.000 −0.0145565
$$986$$ −2500.00 −0.0807467
$$987$$ 0 0
$$988$$ 14960.0 0.481722
$$989$$ −33600.0 −1.08030
$$990$$ 0 0
$$991$$ 32672.0 1.04729 0.523643 0.851938i $$-0.324573\pi$$
0.523643 + 0.851938i $$0.324573\pi$$
$$992$$ −9180.00 −0.293816
$$993$$ 0 0
$$994$$ −105000. −3.35050
$$995$$ 18320.0 0.583702
$$996$$ 0 0
$$997$$ −4740.00 −0.150569 −0.0752845 0.997162i $$-0.523986\pi$$
−0.0752845 + 0.997162i $$0.523986\pi$$
$$998$$ 85180.0 2.70173
$$999$$ 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 45.4.a.a.1.1 1
3.2 odd 2 45.4.a.e.1.1 yes 1
4.3 odd 2 720.4.a.bc.1.1 1
5.2 odd 4 225.4.b.a.199.1 2
5.3 odd 4 225.4.b.a.199.2 2
5.4 even 2 225.4.a.h.1.1 1
7.6 odd 2 2205.4.a.a.1.1 1
9.2 odd 6 405.4.e.b.271.1 2
9.4 even 3 405.4.e.n.136.1 2
9.5 odd 6 405.4.e.b.136.1 2
9.7 even 3 405.4.e.n.271.1 2
12.11 even 2 720.4.a.o.1.1 1
15.2 even 4 225.4.b.b.199.2 2
15.8 even 4 225.4.b.b.199.1 2
15.14 odd 2 225.4.a.a.1.1 1
21.20 even 2 2205.4.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.a.a.1.1 1 1.1 even 1 trivial
45.4.a.e.1.1 yes 1 3.2 odd 2
225.4.a.a.1.1 1 15.14 odd 2
225.4.a.h.1.1 1 5.4 even 2
225.4.b.a.199.1 2 5.2 odd 4
225.4.b.a.199.2 2 5.3 odd 4
225.4.b.b.199.1 2 15.8 even 4
225.4.b.b.199.2 2 15.2 even 4
405.4.e.b.136.1 2 9.5 odd 6
405.4.e.b.271.1 2 9.2 odd 6
405.4.e.n.136.1 2 9.4 even 3
405.4.e.n.271.1 2 9.7 even 3
720.4.a.o.1.1 1 12.11 even 2
720.4.a.bc.1.1 1 4.3 odd 2
2205.4.a.a.1.1 1 7.6 odd 2
2205.4.a.t.1.1 1 21.20 even 2