# Properties

 Label 240.4.a.f.1.1 Level $240$ Weight $4$ Character 240.1 Self dual yes Analytic conductor $14.160$ 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] = [240,4,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, 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("240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 240.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: $$14.1604584014$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 240.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -5.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} +24.0000 q^{11} +74.0000 q^{13} -15.0000 q^{15} +54.0000 q^{17} +124.000 q^{19} -60.0000 q^{21} +120.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -78.0000 q^{29} -200.000 q^{31} +72.0000 q^{33} +100.000 q^{35} -70.0000 q^{37} +222.000 q^{39} +330.000 q^{41} -92.0000 q^{43} -45.0000 q^{45} +24.0000 q^{47} +57.0000 q^{49} +162.000 q^{51} +450.000 q^{53} -120.000 q^{55} +372.000 q^{57} -24.0000 q^{59} -322.000 q^{61} -180.000 q^{63} -370.000 q^{65} +196.000 q^{67} +360.000 q^{69} +288.000 q^{71} -430.000 q^{73} +75.0000 q^{75} -480.000 q^{77} +520.000 q^{79} +81.0000 q^{81} -156.000 q^{83} -270.000 q^{85} -234.000 q^{87} +1026.00 q^{89} -1480.00 q^{91} -600.000 q^{93} -620.000 q^{95} -286.000 q^{97} +216.000 q^{99} +O(q^{100})$$

## 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
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ −20.0000 −1.07990 −0.539949 0.841698i $$-0.681557\pi$$
−0.539949 + 0.841698i $$0.681557\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 24.0000 0.657843 0.328921 0.944357i $$-0.393315\pi$$
0.328921 + 0.944357i $$0.393315\pi$$
$$12$$ 0 0
$$13$$ 74.0000 1.57876 0.789381 0.613904i $$-0.210402\pi$$
0.789381 + 0.613904i $$0.210402\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −0.258199
$$16$$ 0 0
$$17$$ 54.0000 0.770407 0.385204 0.922832i $$-0.374131\pi$$
0.385204 + 0.922832i $$0.374131\pi$$
$$18$$ 0 0
$$19$$ 124.000 1.49724 0.748620 0.663000i $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ 0 0
$$21$$ −60.0000 −0.623480
$$22$$ 0 0
$$23$$ 120.000 1.08790 0.543951 0.839117i $$-0.316928\pi$$
0.543951 + 0.839117i $$0.316928\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −78.0000 −0.499456 −0.249728 0.968316i $$-0.580341\pi$$
−0.249728 + 0.968316i $$0.580341\pi$$
$$30$$ 0 0
$$31$$ −200.000 −1.15874 −0.579372 0.815063i $$-0.696702\pi$$
−0.579372 + 0.815063i $$0.696702\pi$$
$$32$$ 0 0
$$33$$ 72.0000 0.379806
$$34$$ 0 0
$$35$$ 100.000 0.482945
$$36$$ 0 0
$$37$$ −70.0000 −0.311025 −0.155513 0.987834i $$-0.549703\pi$$
−0.155513 + 0.987834i $$0.549703\pi$$
$$38$$ 0 0
$$39$$ 222.000 0.911499
$$40$$ 0 0
$$41$$ 330.000 1.25701 0.628504 0.777806i $$-0.283668\pi$$
0.628504 + 0.777806i $$0.283668\pi$$
$$42$$ 0 0
$$43$$ −92.0000 −0.326276 −0.163138 0.986603i $$-0.552162\pi$$
−0.163138 + 0.986603i $$0.552162\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ 24.0000 0.0744843 0.0372421 0.999306i $$-0.488143\pi$$
0.0372421 + 0.999306i $$0.488143\pi$$
$$48$$ 0 0
$$49$$ 57.0000 0.166181
$$50$$ 0 0
$$51$$ 162.000 0.444795
$$52$$ 0 0
$$53$$ 450.000 1.16627 0.583134 0.812376i $$-0.301826\pi$$
0.583134 + 0.812376i $$0.301826\pi$$
$$54$$ 0 0
$$55$$ −120.000 −0.294196
$$56$$ 0 0
$$57$$ 372.000 0.864432
$$58$$ 0 0
$$59$$ −24.0000 −0.0529582 −0.0264791 0.999649i $$-0.508430\pi$$
−0.0264791 + 0.999649i $$0.508430\pi$$
$$60$$ 0 0
$$61$$ −322.000 −0.675867 −0.337933 0.941170i $$-0.609728\pi$$
−0.337933 + 0.941170i $$0.609728\pi$$
$$62$$ 0 0
$$63$$ −180.000 −0.359966
$$64$$ 0 0
$$65$$ −370.000 −0.706044
$$66$$ 0 0
$$67$$ 196.000 0.357391 0.178696 0.983904i $$-0.442812\pi$$
0.178696 + 0.983904i $$0.442812\pi$$
$$68$$ 0 0
$$69$$ 360.000 0.628100
$$70$$ 0 0
$$71$$ 288.000 0.481399 0.240699 0.970600i $$-0.422623\pi$$
0.240699 + 0.970600i $$0.422623\pi$$
$$72$$ 0 0
$$73$$ −430.000 −0.689420 −0.344710 0.938709i $$-0.612023\pi$$
−0.344710 + 0.938709i $$0.612023\pi$$
$$74$$ 0 0
$$75$$ 75.0000 0.115470
$$76$$ 0 0
$$77$$ −480.000 −0.710404
$$78$$ 0 0
$$79$$ 520.000 0.740564 0.370282 0.928919i $$-0.379261\pi$$
0.370282 + 0.928919i $$0.379261\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −156.000 −0.206304 −0.103152 0.994666i $$-0.532893\pi$$
−0.103152 + 0.994666i $$0.532893\pi$$
$$84$$ 0 0
$$85$$ −270.000 −0.344537
$$86$$ 0 0
$$87$$ −234.000 −0.288361
$$88$$ 0 0
$$89$$ 1026.00 1.22198 0.610988 0.791640i $$-0.290773\pi$$
0.610988 + 0.791640i $$0.290773\pi$$
$$90$$ 0 0
$$91$$ −1480.00 −1.70490
$$92$$ 0 0
$$93$$ −600.000 −0.669001
$$94$$ 0 0
$$95$$ −620.000 −0.669586
$$96$$ 0 0
$$97$$ −286.000 −0.299370 −0.149685 0.988734i $$-0.547826\pi$$
−0.149685 + 0.988734i $$0.547826\pi$$
$$98$$ 0 0
$$99$$ 216.000 0.219281
$$100$$ 0 0
$$101$$ −1734.00 −1.70831 −0.854156 0.520017i $$-0.825925\pi$$
−0.854156 + 0.520017i $$0.825925\pi$$
$$102$$ 0 0
$$103$$ −452.000 −0.432397 −0.216198 0.976349i $$-0.569366\pi$$
−0.216198 + 0.976349i $$0.569366\pi$$
$$104$$ 0 0
$$105$$ 300.000 0.278829
$$106$$ 0 0
$$107$$ 1404.00 1.26850 0.634251 0.773127i $$-0.281308\pi$$
0.634251 + 0.773127i $$0.281308\pi$$
$$108$$ 0 0
$$109$$ −1474.00 −1.29526 −0.647631 0.761954i $$-0.724240\pi$$
−0.647631 + 0.761954i $$0.724240\pi$$
$$110$$ 0 0
$$111$$ −210.000 −0.179570
$$112$$ 0 0
$$113$$ 1086.00 0.904091 0.452046 0.891995i $$-0.350694\pi$$
0.452046 + 0.891995i $$0.350694\pi$$
$$114$$ 0 0
$$115$$ −600.000 −0.486524
$$116$$ 0 0
$$117$$ 666.000 0.526254
$$118$$ 0 0
$$119$$ −1080.00 −0.831962
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 0 0
$$123$$ 990.000 0.725734
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −1244.00 −0.869190 −0.434595 0.900626i $$-0.643109\pi$$
−0.434595 + 0.900626i $$0.643109\pi$$
$$128$$ 0 0
$$129$$ −276.000 −0.188376
$$130$$ 0 0
$$131$$ −2328.00 −1.55266 −0.776329 0.630327i $$-0.782921\pi$$
−0.776329 + 0.630327i $$0.782921\pi$$
$$132$$ 0 0
$$133$$ −2480.00 −1.61687
$$134$$ 0 0
$$135$$ −135.000 −0.0860663
$$136$$ 0 0
$$137$$ 2118.00 1.32082 0.660412 0.750903i $$-0.270382\pi$$
0.660412 + 0.750903i $$0.270382\pi$$
$$138$$ 0 0
$$139$$ −2324.00 −1.41812 −0.709062 0.705147i $$-0.750881\pi$$
−0.709062 + 0.705147i $$0.750881\pi$$
$$140$$ 0 0
$$141$$ 72.0000 0.0430035
$$142$$ 0 0
$$143$$ 1776.00 1.03858
$$144$$ 0 0
$$145$$ 390.000 0.223364
$$146$$ 0 0
$$147$$ 171.000 0.0959445
$$148$$ 0 0
$$149$$ 258.000 0.141854 0.0709268 0.997482i $$-0.477404\pi$$
0.0709268 + 0.997482i $$0.477404\pi$$
$$150$$ 0 0
$$151$$ 808.000 0.435458 0.217729 0.976009i $$-0.430135\pi$$
0.217729 + 0.976009i $$0.430135\pi$$
$$152$$ 0 0
$$153$$ 486.000 0.256802
$$154$$ 0 0
$$155$$ 1000.00 0.518206
$$156$$ 0 0
$$157$$ 2378.00 1.20882 0.604411 0.796673i $$-0.293408\pi$$
0.604411 + 0.796673i $$0.293408\pi$$
$$158$$ 0 0
$$159$$ 1350.00 0.673346
$$160$$ 0 0
$$161$$ −2400.00 −1.17482
$$162$$ 0 0
$$163$$ 52.0000 0.0249874 0.0124937 0.999922i $$-0.496023\pi$$
0.0124937 + 0.999922i $$0.496023\pi$$
$$164$$ 0 0
$$165$$ −360.000 −0.169854
$$166$$ 0 0
$$167$$ 3720.00 1.72373 0.861863 0.507141i $$-0.169298\pi$$
0.861863 + 0.507141i $$0.169298\pi$$
$$168$$ 0 0
$$169$$ 3279.00 1.49249
$$170$$ 0 0
$$171$$ 1116.00 0.499080
$$172$$ 0 0
$$173$$ 426.000 0.187215 0.0936075 0.995609i $$-0.470160\pi$$
0.0936075 + 0.995609i $$0.470160\pi$$
$$174$$ 0 0
$$175$$ −500.000 −0.215980
$$176$$ 0 0
$$177$$ −72.0000 −0.0305754
$$178$$ 0 0
$$179$$ 1440.00 0.601289 0.300644 0.953736i $$-0.402798\pi$$
0.300644 + 0.953736i $$0.402798\pi$$
$$180$$ 0 0
$$181$$ −3130.00 −1.28537 −0.642683 0.766133i $$-0.722179\pi$$
−0.642683 + 0.766133i $$0.722179\pi$$
$$182$$ 0 0
$$183$$ −966.000 −0.390212
$$184$$ 0 0
$$185$$ 350.000 0.139095
$$186$$ 0 0
$$187$$ 1296.00 0.506807
$$188$$ 0 0
$$189$$ −540.000 −0.207827
$$190$$ 0 0
$$191$$ −3576.00 −1.35471 −0.677357 0.735655i $$-0.736875\pi$$
−0.677357 + 0.735655i $$0.736875\pi$$
$$192$$ 0 0
$$193$$ 2666.00 0.994315 0.497158 0.867660i $$-0.334377\pi$$
0.497158 + 0.867660i $$0.334377\pi$$
$$194$$ 0 0
$$195$$ −1110.00 −0.407635
$$196$$ 0 0
$$197$$ −2718.00 −0.982992 −0.491496 0.870880i $$-0.663550\pi$$
−0.491496 + 0.870880i $$0.663550\pi$$
$$198$$ 0 0
$$199$$ 3832.00 1.36504 0.682521 0.730866i $$-0.260884\pi$$
0.682521 + 0.730866i $$0.260884\pi$$
$$200$$ 0 0
$$201$$ 588.000 0.206340
$$202$$ 0 0
$$203$$ 1560.00 0.539362
$$204$$ 0 0
$$205$$ −1650.00 −0.562151
$$206$$ 0 0
$$207$$ 1080.00 0.362634
$$208$$ 0 0
$$209$$ 2976.00 0.984948
$$210$$ 0 0
$$211$$ −1100.00 −0.358896 −0.179448 0.983767i $$-0.557431\pi$$
−0.179448 + 0.983767i $$0.557431\pi$$
$$212$$ 0 0
$$213$$ 864.000 0.277936
$$214$$ 0 0
$$215$$ 460.000 0.145915
$$216$$ 0 0
$$217$$ 4000.00 1.25133
$$218$$ 0 0
$$219$$ −1290.00 −0.398037
$$220$$ 0 0
$$221$$ 3996.00 1.21629
$$222$$ 0 0
$$223$$ −1964.00 −0.589772 −0.294886 0.955532i $$-0.595282\pi$$
−0.294886 + 0.955532i $$0.595282\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −660.000 −0.192977 −0.0964884 0.995334i $$-0.530761\pi$$
−0.0964884 + 0.995334i $$0.530761\pi$$
$$228$$ 0 0
$$229$$ −1906.00 −0.550009 −0.275004 0.961443i $$-0.588679\pi$$
−0.275004 + 0.961443i $$0.588679\pi$$
$$230$$ 0 0
$$231$$ −1440.00 −0.410152
$$232$$ 0 0
$$233$$ −1458.00 −0.409943 −0.204972 0.978768i $$-0.565710\pi$$
−0.204972 + 0.978768i $$0.565710\pi$$
$$234$$ 0 0
$$235$$ −120.000 −0.0333104
$$236$$ 0 0
$$237$$ 1560.00 0.427565
$$238$$ 0 0
$$239$$ −1176.00 −0.318281 −0.159140 0.987256i $$-0.550872\pi$$
−0.159140 + 0.987256i $$0.550872\pi$$
$$240$$ 0 0
$$241$$ 866.000 0.231469 0.115734 0.993280i $$-0.463078\pi$$
0.115734 + 0.993280i $$0.463078\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ −285.000 −0.0743183
$$246$$ 0 0
$$247$$ 9176.00 2.36379
$$248$$ 0 0
$$249$$ −468.000 −0.119110
$$250$$ 0 0
$$251$$ −432.000 −0.108636 −0.0543179 0.998524i $$-0.517298\pi$$
−0.0543179 + 0.998524i $$0.517298\pi$$
$$252$$ 0 0
$$253$$ 2880.00 0.715668
$$254$$ 0 0
$$255$$ −810.000 −0.198918
$$256$$ 0 0
$$257$$ 2526.00 0.613103 0.306552 0.951854i $$-0.400825\pi$$
0.306552 + 0.951854i $$0.400825\pi$$
$$258$$ 0 0
$$259$$ 1400.00 0.335876
$$260$$ 0 0
$$261$$ −702.000 −0.166485
$$262$$ 0 0
$$263$$ −5448.00 −1.27733 −0.638666 0.769484i $$-0.720513\pi$$
−0.638666 + 0.769484i $$0.720513\pi$$
$$264$$ 0 0
$$265$$ −2250.00 −0.521571
$$266$$ 0 0
$$267$$ 3078.00 0.705508
$$268$$ 0 0
$$269$$ −2574.00 −0.583418 −0.291709 0.956507i $$-0.594224\pi$$
−0.291709 + 0.956507i $$0.594224\pi$$
$$270$$ 0 0
$$271$$ 3184.00 0.713706 0.356853 0.934161i $$-0.383850\pi$$
0.356853 + 0.934161i $$0.383850\pi$$
$$272$$ 0 0
$$273$$ −4440.00 −0.984326
$$274$$ 0 0
$$275$$ 600.000 0.131569
$$276$$ 0 0
$$277$$ 3962.00 0.859399 0.429699 0.902972i $$-0.358620\pi$$
0.429699 + 0.902972i $$0.358620\pi$$
$$278$$ 0 0
$$279$$ −1800.00 −0.386248
$$280$$ 0 0
$$281$$ −8286.00 −1.75908 −0.879540 0.475825i $$-0.842149\pi$$
−0.879540 + 0.475825i $$0.842149\pi$$
$$282$$ 0 0
$$283$$ 2716.00 0.570493 0.285246 0.958454i $$-0.407925\pi$$
0.285246 + 0.958454i $$0.407925\pi$$
$$284$$ 0 0
$$285$$ −1860.00 −0.386586
$$286$$ 0 0
$$287$$ −6600.00 −1.35744
$$288$$ 0 0
$$289$$ −1997.00 −0.406473
$$290$$ 0 0
$$291$$ −858.000 −0.172841
$$292$$ 0 0
$$293$$ 6018.00 1.19992 0.599958 0.800032i $$-0.295184\pi$$
0.599958 + 0.800032i $$0.295184\pi$$
$$294$$ 0 0
$$295$$ 120.000 0.0236836
$$296$$ 0 0
$$297$$ 648.000 0.126602
$$298$$ 0 0
$$299$$ 8880.00 1.71754
$$300$$ 0 0
$$301$$ 1840.00 0.352345
$$302$$ 0 0
$$303$$ −5202.00 −0.986294
$$304$$ 0 0
$$305$$ 1610.00 0.302257
$$306$$ 0 0
$$307$$ −9236.00 −1.71702 −0.858512 0.512793i $$-0.828611\pi$$
−0.858512 + 0.512793i $$0.828611\pi$$
$$308$$ 0 0
$$309$$ −1356.00 −0.249644
$$310$$ 0 0
$$311$$ −1536.00 −0.280060 −0.140030 0.990147i $$-0.544720\pi$$
−0.140030 + 0.990147i $$0.544720\pi$$
$$312$$ 0 0
$$313$$ −7342.00 −1.32586 −0.662930 0.748681i $$-0.730687\pi$$
−0.662930 + 0.748681i $$0.730687\pi$$
$$314$$ 0 0
$$315$$ 900.000 0.160982
$$316$$ 0 0
$$317$$ −3894.00 −0.689933 −0.344967 0.938615i $$-0.612110\pi$$
−0.344967 + 0.938615i $$0.612110\pi$$
$$318$$ 0 0
$$319$$ −1872.00 −0.328564
$$320$$ 0 0
$$321$$ 4212.00 0.732370
$$322$$ 0 0
$$323$$ 6696.00 1.15348
$$324$$ 0 0
$$325$$ 1850.00 0.315752
$$326$$ 0 0
$$327$$ −4422.00 −0.747820
$$328$$ 0 0
$$329$$ −480.000 −0.0804354
$$330$$ 0 0
$$331$$ −3692.00 −0.613084 −0.306542 0.951857i $$-0.599172\pi$$
−0.306542 + 0.951857i $$0.599172\pi$$
$$332$$ 0 0
$$333$$ −630.000 −0.103675
$$334$$ 0 0
$$335$$ −980.000 −0.159830
$$336$$ 0 0
$$337$$ −8998.00 −1.45446 −0.727229 0.686395i $$-0.759192\pi$$
−0.727229 + 0.686395i $$0.759192\pi$$
$$338$$ 0 0
$$339$$ 3258.00 0.521977
$$340$$ 0 0
$$341$$ −4800.00 −0.762271
$$342$$ 0 0
$$343$$ 5720.00 0.900440
$$344$$ 0 0
$$345$$ −1800.00 −0.280895
$$346$$ 0 0
$$347$$ −5244.00 −0.811276 −0.405638 0.914034i $$-0.632951\pi$$
−0.405638 + 0.914034i $$0.632951\pi$$
$$348$$ 0 0
$$349$$ 6302.00 0.966585 0.483293 0.875459i $$-0.339441\pi$$
0.483293 + 0.875459i $$0.339441\pi$$
$$350$$ 0 0
$$351$$ 1998.00 0.303833
$$352$$ 0 0
$$353$$ 3414.00 0.514756 0.257378 0.966311i $$-0.417141\pi$$
0.257378 + 0.966311i $$0.417141\pi$$
$$354$$ 0 0
$$355$$ −1440.00 −0.215288
$$356$$ 0 0
$$357$$ −3240.00 −0.480333
$$358$$ 0 0
$$359$$ −4824.00 −0.709195 −0.354597 0.935019i $$-0.615382\pi$$
−0.354597 + 0.935019i $$0.615382\pi$$
$$360$$ 0 0
$$361$$ 8517.00 1.24173
$$362$$ 0 0
$$363$$ −2265.00 −0.327498
$$364$$ 0 0
$$365$$ 2150.00 0.308318
$$366$$ 0 0
$$367$$ 3508.00 0.498954 0.249477 0.968381i $$-0.419741\pi$$
0.249477 + 0.968381i $$0.419741\pi$$
$$368$$ 0 0
$$369$$ 2970.00 0.419003
$$370$$ 0 0
$$371$$ −9000.00 −1.25945
$$372$$ 0 0
$$373$$ 10802.0 1.49948 0.749740 0.661732i $$-0.230178\pi$$
0.749740 + 0.661732i $$0.230178\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 0 0
$$377$$ −5772.00 −0.788523
$$378$$ 0 0
$$379$$ −1460.00 −0.197876 −0.0989382 0.995094i $$-0.531545\pi$$
−0.0989382 + 0.995094i $$0.531545\pi$$
$$380$$ 0 0
$$381$$ −3732.00 −0.501827
$$382$$ 0 0
$$383$$ 4872.00 0.649994 0.324997 0.945715i $$-0.394637\pi$$
0.324997 + 0.945715i $$0.394637\pi$$
$$384$$ 0 0
$$385$$ 2400.00 0.317702
$$386$$ 0 0
$$387$$ −828.000 −0.108759
$$388$$ 0 0
$$389$$ −14046.0 −1.83075 −0.915373 0.402606i $$-0.868104\pi$$
−0.915373 + 0.402606i $$0.868104\pi$$
$$390$$ 0 0
$$391$$ 6480.00 0.838127
$$392$$ 0 0
$$393$$ −6984.00 −0.896428
$$394$$ 0 0
$$395$$ −2600.00 −0.331190
$$396$$ 0 0
$$397$$ −2734.00 −0.345631 −0.172816 0.984954i $$-0.555286\pi$$
−0.172816 + 0.984954i $$0.555286\pi$$
$$398$$ 0 0
$$399$$ −7440.00 −0.933498
$$400$$ 0 0
$$401$$ −15942.0 −1.98530 −0.992650 0.121019i $$-0.961384\pi$$
−0.992650 + 0.121019i $$0.961384\pi$$
$$402$$ 0 0
$$403$$ −14800.0 −1.82938
$$404$$ 0 0
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ −1680.00 −0.204606
$$408$$ 0 0
$$409$$ 8714.00 1.05350 0.526748 0.850022i $$-0.323411\pi$$
0.526748 + 0.850022i $$0.323411\pi$$
$$410$$ 0 0
$$411$$ 6354.00 0.762578
$$412$$ 0 0
$$413$$ 480.000 0.0571895
$$414$$ 0 0
$$415$$ 780.000 0.0922619
$$416$$ 0 0
$$417$$ −6972.00 −0.818754
$$418$$ 0 0
$$419$$ −11976.0 −1.39634 −0.698169 0.715933i $$-0.746002\pi$$
−0.698169 + 0.715933i $$0.746002\pi$$
$$420$$ 0 0
$$421$$ 11054.0 1.27967 0.639833 0.768514i $$-0.279004\pi$$
0.639833 + 0.768514i $$0.279004\pi$$
$$422$$ 0 0
$$423$$ 216.000 0.0248281
$$424$$ 0 0
$$425$$ 1350.00 0.154081
$$426$$ 0 0
$$427$$ 6440.00 0.729868
$$428$$ 0 0
$$429$$ 5328.00 0.599623
$$430$$ 0 0
$$431$$ −720.000 −0.0804668 −0.0402334 0.999190i $$-0.512810\pi$$
−0.0402334 + 0.999190i $$0.512810\pi$$
$$432$$ 0 0
$$433$$ −15622.0 −1.73382 −0.866912 0.498462i $$-0.833898\pi$$
−0.866912 + 0.498462i $$0.833898\pi$$
$$434$$ 0 0
$$435$$ 1170.00 0.128959
$$436$$ 0 0
$$437$$ 14880.0 1.62885
$$438$$ 0 0
$$439$$ 9880.00 1.07414 0.537069 0.843538i $$-0.319531\pi$$
0.537069 + 0.843538i $$0.319531\pi$$
$$440$$ 0 0
$$441$$ 513.000 0.0553936
$$442$$ 0 0
$$443$$ 16116.0 1.72843 0.864215 0.503123i $$-0.167816\pi$$
0.864215 + 0.503123i $$0.167816\pi$$
$$444$$ 0 0
$$445$$ −5130.00 −0.546484
$$446$$ 0 0
$$447$$ 774.000 0.0818992
$$448$$ 0 0
$$449$$ 9018.00 0.947852 0.473926 0.880565i $$-0.342836\pi$$
0.473926 + 0.880565i $$0.342836\pi$$
$$450$$ 0 0
$$451$$ 7920.00 0.826914
$$452$$ 0 0
$$453$$ 2424.00 0.251412
$$454$$ 0 0
$$455$$ 7400.00 0.762456
$$456$$ 0 0
$$457$$ −3670.00 −0.375657 −0.187829 0.982202i $$-0.560145\pi$$
−0.187829 + 0.982202i $$0.560145\pi$$
$$458$$ 0 0
$$459$$ 1458.00 0.148265
$$460$$ 0 0
$$461$$ 17562.0 1.77428 0.887141 0.461499i $$-0.152688\pi$$
0.887141 + 0.461499i $$0.152688\pi$$
$$462$$ 0 0
$$463$$ −1172.00 −0.117640 −0.0588202 0.998269i $$-0.518734\pi$$
−0.0588202 + 0.998269i $$0.518734\pi$$
$$464$$ 0 0
$$465$$ 3000.00 0.299186
$$466$$ 0 0
$$467$$ −6876.00 −0.681335 −0.340667 0.940184i $$-0.610653\pi$$
−0.340667 + 0.940184i $$0.610653\pi$$
$$468$$ 0 0
$$469$$ −3920.00 −0.385946
$$470$$ 0 0
$$471$$ 7134.00 0.697914
$$472$$ 0 0
$$473$$ −2208.00 −0.214638
$$474$$ 0 0
$$475$$ 3100.00 0.299448
$$476$$ 0 0
$$477$$ 4050.00 0.388756
$$478$$ 0 0
$$479$$ −2280.00 −0.217486 −0.108743 0.994070i $$-0.534683\pi$$
−0.108743 + 0.994070i $$0.534683\pi$$
$$480$$ 0 0
$$481$$ −5180.00 −0.491035
$$482$$ 0 0
$$483$$ −7200.00 −0.678284
$$484$$ 0 0
$$485$$ 1430.00 0.133882
$$486$$ 0 0
$$487$$ 3076.00 0.286215 0.143108 0.989707i $$-0.454290\pi$$
0.143108 + 0.989707i $$0.454290\pi$$
$$488$$ 0 0
$$489$$ 156.000 0.0144265
$$490$$ 0 0
$$491$$ 18912.0 1.73826 0.869131 0.494582i $$-0.164679\pi$$
0.869131 + 0.494582i $$0.164679\pi$$
$$492$$ 0 0
$$493$$ −4212.00 −0.384785
$$494$$ 0 0
$$495$$ −1080.00 −0.0980654
$$496$$ 0 0
$$497$$ −5760.00 −0.519862
$$498$$ 0 0
$$499$$ −9956.00 −0.893170 −0.446585 0.894741i $$-0.647360\pi$$
−0.446585 + 0.894741i $$0.647360\pi$$
$$500$$ 0 0
$$501$$ 11160.0 0.995194
$$502$$ 0 0
$$503$$ 10656.0 0.944588 0.472294 0.881441i $$-0.343426\pi$$
0.472294 + 0.881441i $$0.343426\pi$$
$$504$$ 0 0
$$505$$ 8670.00 0.763980
$$506$$ 0 0
$$507$$ 9837.00 0.861689
$$508$$ 0 0
$$509$$ −2766.00 −0.240866 −0.120433 0.992721i $$-0.538428\pi$$
−0.120433 + 0.992721i $$0.538428\pi$$
$$510$$ 0 0
$$511$$ 8600.00 0.744504
$$512$$ 0 0
$$513$$ 3348.00 0.288144
$$514$$ 0 0
$$515$$ 2260.00 0.193374
$$516$$ 0 0
$$517$$ 576.000 0.0489989
$$518$$ 0 0
$$519$$ 1278.00 0.108089
$$520$$ 0 0
$$521$$ 10530.0 0.885466 0.442733 0.896654i $$-0.354009\pi$$
0.442733 + 0.896654i $$0.354009\pi$$
$$522$$ 0 0
$$523$$ −12692.0 −1.06115 −0.530576 0.847637i $$-0.678024\pi$$
−0.530576 + 0.847637i $$0.678024\pi$$
$$524$$ 0 0
$$525$$ −1500.00 −0.124696
$$526$$ 0 0
$$527$$ −10800.0 −0.892705
$$528$$ 0 0
$$529$$ 2233.00 0.183529
$$530$$ 0 0
$$531$$ −216.000 −0.0176527
$$532$$ 0 0
$$533$$ 24420.0 1.98452
$$534$$ 0 0
$$535$$ −7020.00 −0.567292
$$536$$ 0 0
$$537$$ 4320.00 0.347154
$$538$$ 0 0
$$539$$ 1368.00 0.109321
$$540$$ 0 0
$$541$$ 18110.0 1.43920 0.719602 0.694386i $$-0.244324\pi$$
0.719602 + 0.694386i $$0.244324\pi$$
$$542$$ 0 0
$$543$$ −9390.00 −0.742106
$$544$$ 0 0
$$545$$ 7370.00 0.579259
$$546$$ 0 0
$$547$$ −3620.00 −0.282962 −0.141481 0.989941i $$-0.545186\pi$$
−0.141481 + 0.989941i $$0.545186\pi$$
$$548$$ 0 0
$$549$$ −2898.00 −0.225289
$$550$$ 0 0
$$551$$ −9672.00 −0.747806
$$552$$ 0 0
$$553$$ −10400.0 −0.799734
$$554$$ 0 0
$$555$$ 1050.00 0.0803063
$$556$$ 0 0
$$557$$ −14166.0 −1.07762 −0.538809 0.842428i $$-0.681125\pi$$
−0.538809 + 0.842428i $$0.681125\pi$$
$$558$$ 0 0
$$559$$ −6808.00 −0.515112
$$560$$ 0 0
$$561$$ 3888.00 0.292605
$$562$$ 0 0
$$563$$ 13404.0 1.00339 0.501697 0.865043i $$-0.332709\pi$$
0.501697 + 0.865043i $$0.332709\pi$$
$$564$$ 0 0
$$565$$ −5430.00 −0.404322
$$566$$ 0 0
$$567$$ −1620.00 −0.119989
$$568$$ 0 0
$$569$$ −18654.0 −1.37437 −0.687185 0.726483i $$-0.741154\pi$$
−0.687185 + 0.726483i $$0.741154\pi$$
$$570$$ 0 0
$$571$$ 7684.00 0.563162 0.281581 0.959537i $$-0.409141\pi$$
0.281581 + 0.959537i $$0.409141\pi$$
$$572$$ 0 0
$$573$$ −10728.0 −0.782144
$$574$$ 0 0
$$575$$ 3000.00 0.217580
$$576$$ 0 0
$$577$$ −1726.00 −0.124531 −0.0622654 0.998060i $$-0.519833\pi$$
−0.0622654 + 0.998060i $$0.519833\pi$$
$$578$$ 0 0
$$579$$ 7998.00 0.574068
$$580$$ 0 0
$$581$$ 3120.00 0.222787
$$582$$ 0 0
$$583$$ 10800.0 0.767222
$$584$$ 0 0
$$585$$ −3330.00 −0.235348
$$586$$ 0 0
$$587$$ −10596.0 −0.745049 −0.372524 0.928022i $$-0.621508\pi$$
−0.372524 + 0.928022i $$0.621508\pi$$
$$588$$ 0 0
$$589$$ −24800.0 −1.73492
$$590$$ 0 0
$$591$$ −8154.00 −0.567531
$$592$$ 0 0
$$593$$ 2862.00 0.198193 0.0990963 0.995078i $$-0.468405\pi$$
0.0990963 + 0.995078i $$0.468405\pi$$
$$594$$ 0 0
$$595$$ 5400.00 0.372065
$$596$$ 0 0
$$597$$ 11496.0 0.788107
$$598$$ 0 0
$$599$$ 23592.0 1.60925 0.804627 0.593781i $$-0.202365\pi$$
0.804627 + 0.593781i $$0.202365\pi$$
$$600$$ 0 0
$$601$$ −9574.00 −0.649803 −0.324902 0.945748i $$-0.605331\pi$$
−0.324902 + 0.945748i $$0.605331\pi$$
$$602$$ 0 0
$$603$$ 1764.00 0.119130
$$604$$ 0 0
$$605$$ 3775.00 0.253679
$$606$$ 0 0
$$607$$ −17444.0 −1.16644 −0.583221 0.812314i $$-0.698208\pi$$
−0.583221 + 0.812314i $$0.698208\pi$$
$$608$$ 0 0
$$609$$ 4680.00 0.311401
$$610$$ 0 0
$$611$$ 1776.00 0.117593
$$612$$ 0 0
$$613$$ −2374.00 −0.156419 −0.0782096 0.996937i $$-0.524920\pi$$
−0.0782096 + 0.996937i $$0.524920\pi$$
$$614$$ 0 0
$$615$$ −4950.00 −0.324558
$$616$$ 0 0
$$617$$ −12162.0 −0.793555 −0.396778 0.917915i $$-0.629872\pi$$
−0.396778 + 0.917915i $$0.629872\pi$$
$$618$$ 0 0
$$619$$ −8804.00 −0.571668 −0.285834 0.958279i $$-0.592271\pi$$
−0.285834 + 0.958279i $$0.592271\pi$$
$$620$$ 0 0
$$621$$ 3240.00 0.209367
$$622$$ 0 0
$$623$$ −20520.0 −1.31961
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 8928.00 0.568660
$$628$$ 0 0
$$629$$ −3780.00 −0.239616
$$630$$ 0 0
$$631$$ 12688.0 0.800478 0.400239 0.916411i $$-0.368927\pi$$
0.400239 + 0.916411i $$0.368927\pi$$
$$632$$ 0 0
$$633$$ −3300.00 −0.207209
$$634$$ 0 0
$$635$$ 6220.00 0.388714
$$636$$ 0 0
$$637$$ 4218.00 0.262360
$$638$$ 0 0
$$639$$ 2592.00 0.160466
$$640$$ 0 0
$$641$$ −9150.00 −0.563812 −0.281906 0.959442i $$-0.590967\pi$$
−0.281906 + 0.959442i $$0.590967\pi$$
$$642$$ 0 0
$$643$$ −25292.0 −1.55120 −0.775598 0.631227i $$-0.782552\pi$$
−0.775598 + 0.631227i $$0.782552\pi$$
$$644$$ 0 0
$$645$$ 1380.00 0.0842441
$$646$$ 0 0
$$647$$ 2736.00 0.166249 0.0831246 0.996539i $$-0.473510\pi$$
0.0831246 + 0.996539i $$0.473510\pi$$
$$648$$ 0 0
$$649$$ −576.000 −0.0348382
$$650$$ 0 0
$$651$$ 12000.0 0.722453
$$652$$ 0 0
$$653$$ 22218.0 1.33148 0.665741 0.746183i $$-0.268116\pi$$
0.665741 + 0.746183i $$0.268116\pi$$
$$654$$ 0 0
$$655$$ 11640.0 0.694370
$$656$$ 0 0
$$657$$ −3870.00 −0.229807
$$658$$ 0 0
$$659$$ −14520.0 −0.858299 −0.429149 0.903234i $$-0.641187\pi$$
−0.429149 + 0.903234i $$0.641187\pi$$
$$660$$ 0 0
$$661$$ −10618.0 −0.624799 −0.312400 0.949951i $$-0.601133\pi$$
−0.312400 + 0.949951i $$0.601133\pi$$
$$662$$ 0 0
$$663$$ 11988.0 0.702225
$$664$$ 0 0
$$665$$ 12400.0 0.723085
$$666$$ 0 0
$$667$$ −9360.00 −0.543359
$$668$$ 0 0
$$669$$ −5892.00 −0.340505
$$670$$ 0 0
$$671$$ −7728.00 −0.444614
$$672$$ 0 0
$$673$$ 1370.00 0.0784690 0.0392345 0.999230i $$-0.487508\pi$$
0.0392345 + 0.999230i $$0.487508\pi$$
$$674$$ 0 0
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ −13758.0 −0.781038 −0.390519 0.920595i $$-0.627704\pi$$
−0.390519 + 0.920595i $$0.627704\pi$$
$$678$$ 0 0
$$679$$ 5720.00 0.323289
$$680$$ 0 0
$$681$$ −1980.00 −0.111415
$$682$$ 0 0
$$683$$ −11988.0 −0.671608 −0.335804 0.941932i $$-0.609008\pi$$
−0.335804 + 0.941932i $$0.609008\pi$$
$$684$$ 0 0
$$685$$ −10590.0 −0.590691
$$686$$ 0 0
$$687$$ −5718.00 −0.317548
$$688$$ 0 0
$$689$$ 33300.0 1.84126
$$690$$ 0 0
$$691$$ −32996.0 −1.81654 −0.908268 0.418388i $$-0.862595\pi$$
−0.908268 + 0.418388i $$0.862595\pi$$
$$692$$ 0 0
$$693$$ −4320.00 −0.236801
$$694$$ 0 0
$$695$$ 11620.0 0.634204
$$696$$ 0 0
$$697$$ 17820.0 0.968408
$$698$$ 0 0
$$699$$ −4374.00 −0.236681
$$700$$ 0 0
$$701$$ −25902.0 −1.39558 −0.697792 0.716300i $$-0.745834\pi$$
−0.697792 + 0.716300i $$0.745834\pi$$
$$702$$ 0 0
$$703$$ −8680.00 −0.465679
$$704$$ 0 0
$$705$$ −360.000 −0.0192318
$$706$$ 0 0
$$707$$ 34680.0 1.84480
$$708$$ 0 0
$$709$$ −27394.0 −1.45106 −0.725531 0.688189i $$-0.758406\pi$$
−0.725531 + 0.688189i $$0.758406\pi$$
$$710$$ 0 0
$$711$$ 4680.00 0.246855
$$712$$ 0 0
$$713$$ −24000.0 −1.26060
$$714$$ 0 0
$$715$$ −8880.00 −0.464466
$$716$$ 0 0
$$717$$ −3528.00 −0.183760
$$718$$ 0 0
$$719$$ −34848.0 −1.80753 −0.903763 0.428033i $$-0.859207\pi$$
−0.903763 + 0.428033i $$0.859207\pi$$
$$720$$ 0 0
$$721$$ 9040.00 0.466945
$$722$$ 0 0
$$723$$ 2598.00 0.133639
$$724$$ 0 0
$$725$$ −1950.00 −0.0998913
$$726$$ 0 0
$$727$$ −28028.0 −1.42985 −0.714925 0.699201i $$-0.753539\pi$$
−0.714925 + 0.699201i $$0.753539\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −4968.00 −0.251365
$$732$$ 0 0
$$733$$ 18002.0 0.907120 0.453560 0.891226i $$-0.350154\pi$$
0.453560 + 0.891226i $$0.350154\pi$$
$$734$$ 0 0
$$735$$ −855.000 −0.0429077
$$736$$ 0 0
$$737$$ 4704.00 0.235107
$$738$$ 0 0
$$739$$ −15284.0 −0.760800 −0.380400 0.924822i $$-0.624214\pi$$
−0.380400 + 0.924822i $$0.624214\pi$$
$$740$$ 0 0
$$741$$ 27528.0 1.36473
$$742$$ 0 0
$$743$$ 18768.0 0.926691 0.463345 0.886178i $$-0.346649\pi$$
0.463345 + 0.886178i $$0.346649\pi$$
$$744$$ 0 0
$$745$$ −1290.00 −0.0634388
$$746$$ 0 0
$$747$$ −1404.00 −0.0687680
$$748$$ 0 0
$$749$$ −28080.0 −1.36985
$$750$$ 0 0
$$751$$ −8696.00 −0.422532 −0.211266 0.977429i $$-0.567759\pi$$
−0.211266 + 0.977429i $$0.567759\pi$$
$$752$$ 0 0
$$753$$ −1296.00 −0.0627209
$$754$$ 0 0
$$755$$ −4040.00 −0.194743
$$756$$ 0 0
$$757$$ −38662.0 −1.85627 −0.928134 0.372247i $$-0.878587\pi$$
−0.928134 + 0.372247i $$0.878587\pi$$
$$758$$ 0 0
$$759$$ 8640.00 0.413191
$$760$$ 0 0
$$761$$ 23874.0 1.13723 0.568615 0.822604i $$-0.307479\pi$$
0.568615 + 0.822604i $$0.307479\pi$$
$$762$$ 0 0
$$763$$ 29480.0 1.39875
$$764$$ 0 0
$$765$$ −2430.00 −0.114846
$$766$$ 0 0
$$767$$ −1776.00 −0.0836084
$$768$$ 0 0
$$769$$ 23618.0 1.10753 0.553763 0.832675i $$-0.313192\pi$$
0.553763 + 0.832675i $$0.313192\pi$$
$$770$$ 0 0
$$771$$ 7578.00 0.353975
$$772$$ 0 0
$$773$$ 11538.0 0.536860 0.268430 0.963299i $$-0.413495\pi$$
0.268430 + 0.963299i $$0.413495\pi$$
$$774$$ 0 0
$$775$$ −5000.00 −0.231749
$$776$$ 0 0
$$777$$ 4200.00 0.193918
$$778$$ 0 0
$$779$$ 40920.0 1.88204
$$780$$ 0 0
$$781$$ 6912.00 0.316685
$$782$$ 0 0
$$783$$ −2106.00 −0.0961204
$$784$$ 0 0
$$785$$ −11890.0 −0.540602
$$786$$ 0 0
$$787$$ 14884.0 0.674152 0.337076 0.941478i $$-0.390562\pi$$
0.337076 + 0.941478i $$0.390562\pi$$
$$788$$ 0 0
$$789$$ −16344.0 −0.737467
$$790$$ 0 0
$$791$$ −21720.0 −0.976327
$$792$$ 0 0
$$793$$ −23828.0 −1.06703
$$794$$ 0 0
$$795$$ −6750.00 −0.301129
$$796$$ 0 0
$$797$$ −11334.0 −0.503728 −0.251864 0.967763i $$-0.581043\pi$$
−0.251864 + 0.967763i $$0.581043\pi$$
$$798$$ 0 0
$$799$$ 1296.00 0.0573832
$$800$$ 0 0
$$801$$ 9234.00 0.407325
$$802$$ 0 0
$$803$$ −10320.0 −0.453530
$$804$$ 0 0
$$805$$ 12000.0 0.525397
$$806$$ 0 0
$$807$$ −7722.00 −0.336837
$$808$$ 0 0
$$809$$ 44730.0 1.94391 0.971955 0.235167i $$-0.0755638\pi$$
0.971955 + 0.235167i $$0.0755638\pi$$
$$810$$ 0 0
$$811$$ 42748.0 1.85091 0.925453 0.378862i $$-0.123684\pi$$
0.925453 + 0.378862i $$0.123684\pi$$
$$812$$ 0 0
$$813$$ 9552.00 0.412058
$$814$$ 0 0
$$815$$ −260.000 −0.0111747
$$816$$ 0 0
$$817$$ −11408.0 −0.488513
$$818$$ 0 0
$$819$$ −13320.0 −0.568301
$$820$$ 0 0
$$821$$ −31686.0 −1.34695 −0.673477 0.739208i $$-0.735200\pi$$
−0.673477 + 0.739208i $$0.735200\pi$$
$$822$$ 0 0
$$823$$ −11036.0 −0.467425 −0.233713 0.972306i $$-0.575087\pi$$
−0.233713 + 0.972306i $$0.575087\pi$$
$$824$$ 0 0
$$825$$ 1800.00 0.0759612
$$826$$ 0 0
$$827$$ −25884.0 −1.08836 −0.544181 0.838968i $$-0.683159\pi$$
−0.544181 + 0.838968i $$0.683159\pi$$
$$828$$ 0 0
$$829$$ 15950.0 0.668234 0.334117 0.942532i $$-0.391562\pi$$
0.334117 + 0.942532i $$0.391562\pi$$
$$830$$ 0 0
$$831$$ 11886.0 0.496174
$$832$$ 0 0
$$833$$ 3078.00 0.128027
$$834$$ 0 0
$$835$$ −18600.0 −0.770874
$$836$$ 0 0
$$837$$ −5400.00 −0.223000
$$838$$ 0 0
$$839$$ −13800.0 −0.567853 −0.283927 0.958846i $$-0.591637\pi$$
−0.283927 + 0.958846i $$0.591637\pi$$
$$840$$ 0 0
$$841$$ −18305.0 −0.750543
$$842$$ 0 0
$$843$$ −24858.0 −1.01560
$$844$$ 0 0
$$845$$ −16395.0 −0.667462
$$846$$ 0 0
$$847$$ 15100.0 0.612565
$$848$$ 0 0
$$849$$ 8148.00 0.329374
$$850$$ 0 0
$$851$$ −8400.00 −0.338365
$$852$$ 0 0
$$853$$ −27862.0 −1.11838 −0.559189 0.829040i $$-0.688887\pi$$
−0.559189 + 0.829040i $$0.688887\pi$$
$$854$$ 0 0
$$855$$ −5580.00 −0.223195
$$856$$ 0 0
$$857$$ −7314.00 −0.291530 −0.145765 0.989319i $$-0.546564\pi$$
−0.145765 + 0.989319i $$0.546564\pi$$
$$858$$ 0 0
$$859$$ 28780.0 1.14314 0.571572 0.820552i $$-0.306334\pi$$
0.571572 + 0.820552i $$0.306334\pi$$
$$860$$ 0 0
$$861$$ −19800.0 −0.783719
$$862$$ 0 0
$$863$$ 32688.0 1.28935 0.644677 0.764455i $$-0.276992\pi$$
0.644677 + 0.764455i $$0.276992\pi$$
$$864$$ 0 0
$$865$$ −2130.00 −0.0837251
$$866$$ 0 0
$$867$$ −5991.00 −0.234677
$$868$$ 0 0
$$869$$ 12480.0 0.487175
$$870$$ 0 0
$$871$$ 14504.0 0.564236
$$872$$ 0 0
$$873$$ −2574.00 −0.0997900
$$874$$ 0 0
$$875$$ 2500.00 0.0965891
$$876$$ 0 0
$$877$$ 36650.0 1.41115 0.705577 0.708633i $$-0.250688\pi$$
0.705577 + 0.708633i $$0.250688\pi$$
$$878$$ 0 0
$$879$$ 18054.0 0.692772
$$880$$ 0 0
$$881$$ −2646.00 −0.101187 −0.0505936 0.998719i $$-0.516111\pi$$
−0.0505936 + 0.998719i $$0.516111\pi$$
$$882$$ 0 0
$$883$$ −10892.0 −0.415113 −0.207557 0.978223i $$-0.566551\pi$$
−0.207557 + 0.978223i $$0.566551\pi$$
$$884$$ 0 0
$$885$$ 360.000 0.0136737
$$886$$ 0 0
$$887$$ 43464.0 1.64530 0.822648 0.568550i $$-0.192496\pi$$
0.822648 + 0.568550i $$0.192496\pi$$
$$888$$ 0 0
$$889$$ 24880.0 0.938637
$$890$$ 0 0
$$891$$ 1944.00 0.0730937
$$892$$ 0 0
$$893$$ 2976.00 0.111521
$$894$$ 0 0
$$895$$ −7200.00 −0.268904
$$896$$ 0 0
$$897$$ 26640.0 0.991621
$$898$$ 0 0
$$899$$ 15600.0 0.578742
$$900$$ 0 0
$$901$$ 24300.0 0.898502
$$902$$ 0 0
$$903$$ 5520.00 0.203426
$$904$$ 0 0
$$905$$ 15650.0 0.574833
$$906$$ 0 0
$$907$$ 14884.0 0.544890 0.272445 0.962171i $$-0.412168\pi$$
0.272445 + 0.962171i $$0.412168\pi$$
$$908$$ 0 0
$$909$$ −15606.0 −0.569437
$$910$$ 0 0
$$911$$ 1248.00 0.0453876 0.0226938 0.999742i $$-0.492776\pi$$
0.0226938 + 0.999742i $$0.492776\pi$$
$$912$$ 0 0
$$913$$ −3744.00 −0.135716
$$914$$ 0 0
$$915$$ 4830.00 0.174508
$$916$$ 0 0
$$917$$ 46560.0 1.67671
$$918$$ 0 0
$$919$$ 6640.00 0.238339 0.119169 0.992874i $$-0.461977\pi$$
0.119169 + 0.992874i $$0.461977\pi$$
$$920$$ 0 0
$$921$$ −27708.0 −0.991324
$$922$$ 0 0
$$923$$ 21312.0 0.760014
$$924$$ 0 0
$$925$$ −1750.00 −0.0622050
$$926$$ 0 0
$$927$$ −4068.00 −0.144132
$$928$$ 0 0
$$929$$ 29946.0 1.05758 0.528792 0.848751i $$-0.322645\pi$$
0.528792 + 0.848751i $$0.322645\pi$$
$$930$$ 0 0
$$931$$ 7068.00 0.248812
$$932$$ 0 0
$$933$$ −4608.00 −0.161693
$$934$$ 0 0
$$935$$ −6480.00 −0.226651
$$936$$ 0 0
$$937$$ 45002.0 1.56900 0.784499 0.620130i $$-0.212920\pi$$
0.784499 + 0.620130i $$0.212920\pi$$
$$938$$ 0 0
$$939$$ −22026.0 −0.765486
$$940$$ 0 0
$$941$$ 6090.00 0.210976 0.105488 0.994421i $$-0.466360\pi$$
0.105488 + 0.994421i $$0.466360\pi$$
$$942$$ 0 0
$$943$$ 39600.0 1.36750
$$944$$ 0 0
$$945$$ 2700.00 0.0929429
$$946$$ 0 0
$$947$$ −56388.0 −1.93491 −0.967457 0.253035i $$-0.918571\pi$$
−0.967457 + 0.253035i $$0.918571\pi$$
$$948$$ 0 0
$$949$$ −31820.0 −1.08843
$$950$$ 0 0
$$951$$ −11682.0 −0.398333
$$952$$ 0 0
$$953$$ 10854.0 0.368936 0.184468 0.982839i $$-0.440944\pi$$
0.184468 + 0.982839i $$0.440944\pi$$
$$954$$ 0 0
$$955$$ 17880.0 0.605846
$$956$$ 0 0
$$957$$ −5616.00 −0.189696
$$958$$ 0 0
$$959$$ −42360.0 −1.42636
$$960$$ 0 0
$$961$$ 10209.0 0.342687
$$962$$ 0 0
$$963$$ 12636.0 0.422834
$$964$$ 0 0
$$965$$ −13330.0 −0.444671
$$966$$ 0 0
$$967$$ 42316.0 1.40723 0.703615 0.710582i $$-0.251568\pi$$
0.703615 + 0.710582i $$0.251568\pi$$
$$968$$ 0 0
$$969$$ 20088.0 0.665964
$$970$$ 0 0
$$971$$ −24480.0 −0.809063 −0.404532 0.914524i $$-0.632565\pi$$
−0.404532 + 0.914524i $$0.632565\pi$$
$$972$$ 0 0
$$973$$ 46480.0 1.53143
$$974$$ 0 0
$$975$$ 5550.00 0.182300
$$976$$ 0 0
$$977$$ −6906.00 −0.226144 −0.113072 0.993587i $$-0.536069\pi$$
−0.113072 + 0.993587i $$0.536069\pi$$
$$978$$ 0 0
$$979$$ 24624.0 0.803868
$$980$$ 0 0
$$981$$ −13266.0 −0.431754
$$982$$ 0 0
$$983$$ −6960.00 −0.225829 −0.112914 0.993605i $$-0.536019\pi$$
−0.112914 + 0.993605i $$0.536019\pi$$
$$984$$ 0 0
$$985$$ 13590.0 0.439608
$$986$$ 0 0
$$987$$ −1440.00 −0.0464394
$$988$$ 0 0
$$989$$ −11040.0 −0.354956
$$990$$ 0 0
$$991$$ −47792.0 −1.53195 −0.765975 0.642870i $$-0.777744\pi$$
−0.765975 + 0.642870i $$0.777744\pi$$
$$992$$ 0 0
$$993$$ −11076.0 −0.353964
$$994$$ 0 0
$$995$$ −19160.0 −0.610465
$$996$$ 0 0
$$997$$ 9938.00 0.315687 0.157843 0.987464i $$-0.449546\pi$$
0.157843 + 0.987464i $$0.449546\pi$$
$$998$$ 0 0
$$999$$ −1890.00 −0.0598568
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 240.4.a.f.1.1 1
3.2 odd 2 720.4.a.r.1.1 1
4.3 odd 2 15.4.a.b.1.1 1
5.2 odd 4 1200.4.f.m.49.1 2
5.3 odd 4 1200.4.f.m.49.2 2
5.4 even 2 1200.4.a.o.1.1 1
8.3 odd 2 960.4.a.bi.1.1 1
8.5 even 2 960.4.a.l.1.1 1
12.11 even 2 45.4.a.b.1.1 1
20.3 even 4 75.4.b.a.49.1 2
20.7 even 4 75.4.b.a.49.2 2
20.19 odd 2 75.4.a.a.1.1 1
28.27 even 2 735.4.a.i.1.1 1
36.7 odd 6 405.4.e.d.271.1 2
36.11 even 6 405.4.e.k.271.1 2
36.23 even 6 405.4.e.k.136.1 2
36.31 odd 6 405.4.e.d.136.1 2
44.43 even 2 1815.4.a.a.1.1 1
60.23 odd 4 225.4.b.d.199.2 2
60.47 odd 4 225.4.b.d.199.1 2
60.59 even 2 225.4.a.g.1.1 1
84.83 odd 2 2205.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 4.3 odd 2
45.4.a.b.1.1 1 12.11 even 2
75.4.a.a.1.1 1 20.19 odd 2
75.4.b.a.49.1 2 20.3 even 4
75.4.b.a.49.2 2 20.7 even 4
225.4.a.g.1.1 1 60.59 even 2
225.4.b.d.199.1 2 60.47 odd 4
225.4.b.d.199.2 2 60.23 odd 4
240.4.a.f.1.1 1 1.1 even 1 trivial
405.4.e.d.136.1 2 36.31 odd 6
405.4.e.d.271.1 2 36.7 odd 6
405.4.e.k.136.1 2 36.23 even 6
405.4.e.k.271.1 2 36.11 even 6
720.4.a.r.1.1 1 3.2 odd 2
735.4.a.i.1.1 1 28.27 even 2
960.4.a.l.1.1 1 8.5 even 2
960.4.a.bi.1.1 1 8.3 odd 2
1200.4.a.o.1.1 1 5.4 even 2
1200.4.f.m.49.1 2 5.2 odd 4
1200.4.f.m.49.2 2 5.3 odd 4
1815.4.a.a.1.1 1 44.43 even 2
2205.4.a.c.1.1 1 84.83 odd 2