# Properties

 Label 735.4.a.i.1.1 Level $735$ Weight $4$ Character 735.1 Self dual yes Analytic conductor $43.366$ 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] = [735,4,Mod(1,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("735.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 735.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: $$43.3664038542$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 735.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +9.00000 q^{6} -21.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +5.00000 q^{5} +9.00000 q^{6} -21.0000 q^{8} +9.00000 q^{9} +15.0000 q^{10} -24.0000 q^{11} +3.00000 q^{12} -74.0000 q^{13} +15.0000 q^{15} -71.0000 q^{16} -54.0000 q^{17} +27.0000 q^{18} +124.000 q^{19} +5.00000 q^{20} -72.0000 q^{22} -120.000 q^{23} -63.0000 q^{24} +25.0000 q^{25} -222.000 q^{26} +27.0000 q^{27} -78.0000 q^{29} +45.0000 q^{30} -200.000 q^{31} -45.0000 q^{32} -72.0000 q^{33} -162.000 q^{34} +9.00000 q^{36} -70.0000 q^{37} +372.000 q^{38} -222.000 q^{39} -105.000 q^{40} -330.000 q^{41} +92.0000 q^{43} -24.0000 q^{44} +45.0000 q^{45} -360.000 q^{46} +24.0000 q^{47} -213.000 q^{48} +75.0000 q^{50} -162.000 q^{51} -74.0000 q^{52} +450.000 q^{53} +81.0000 q^{54} -120.000 q^{55} +372.000 q^{57} -234.000 q^{58} -24.0000 q^{59} +15.0000 q^{60} +322.000 q^{61} -600.000 q^{62} +433.000 q^{64} -370.000 q^{65} -216.000 q^{66} -196.000 q^{67} -54.0000 q^{68} -360.000 q^{69} -288.000 q^{71} -189.000 q^{72} +430.000 q^{73} -210.000 q^{74} +75.0000 q^{75} +124.000 q^{76} -666.000 q^{78} -520.000 q^{79} -355.000 q^{80} +81.0000 q^{81} -990.000 q^{82} -156.000 q^{83} -270.000 q^{85} +276.000 q^{86} -234.000 q^{87} +504.000 q^{88} -1026.00 q^{89} +135.000 q^{90} -120.000 q^{92} -600.000 q^{93} +72.0000 q^{94} +620.000 q^{95} -135.000 q^{96} +286.000 q^{97} -216.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.00000 1.06066 0.530330 0.847791i $$-0.322068\pi$$
0.530330 + 0.847791i $$0.322068\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 1.00000 0.125000
$$5$$ 5.00000 0.447214
$$6$$ 9.00000 0.612372
$$7$$ 0 0
$$8$$ −21.0000 −0.928078
$$9$$ 9.00000 0.333333
$$10$$ 15.0000 0.474342
$$11$$ −24.0000 −0.657843 −0.328921 0.944357i $$-0.606685\pi$$
−0.328921 + 0.944357i $$0.606685\pi$$
$$12$$ 3.00000 0.0721688
$$13$$ −74.0000 −1.57876 −0.789381 0.613904i $$-0.789598\pi$$
−0.789381 + 0.613904i $$0.789598\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ −71.0000 −1.10938
$$17$$ −54.0000 −0.770407 −0.385204 0.922832i $$-0.625869\pi$$
−0.385204 + 0.922832i $$0.625869\pi$$
$$18$$ 27.0000 0.353553
$$19$$ 124.000 1.49724 0.748620 0.663000i $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ 5.00000 0.0559017
$$21$$ 0 0
$$22$$ −72.0000 −0.697748
$$23$$ −120.000 −1.08790 −0.543951 0.839117i $$-0.683072\pi$$
−0.543951 + 0.839117i $$0.683072\pi$$
$$24$$ −63.0000 −0.535826
$$25$$ 25.0000 0.200000
$$26$$ −222.000 −1.67453
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −78.0000 −0.499456 −0.249728 0.968316i $$-0.580341\pi$$
−0.249728 + 0.968316i $$0.580341\pi$$
$$30$$ 45.0000 0.273861
$$31$$ −200.000 −1.15874 −0.579372 0.815063i $$-0.696702\pi$$
−0.579372 + 0.815063i $$0.696702\pi$$
$$32$$ −45.0000 −0.248592
$$33$$ −72.0000 −0.379806
$$34$$ −162.000 −0.817140
$$35$$ 0 0
$$36$$ 9.00000 0.0416667
$$37$$ −70.0000 −0.311025 −0.155513 0.987834i $$-0.549703\pi$$
−0.155513 + 0.987834i $$0.549703\pi$$
$$38$$ 372.000 1.58806
$$39$$ −222.000 −0.911499
$$40$$ −105.000 −0.415049
$$41$$ −330.000 −1.25701 −0.628504 0.777806i $$-0.716332\pi$$
−0.628504 + 0.777806i $$0.716332\pi$$
$$42$$ 0 0
$$43$$ 92.0000 0.326276 0.163138 0.986603i $$-0.447838\pi$$
0.163138 + 0.986603i $$0.447838\pi$$
$$44$$ −24.0000 −0.0822304
$$45$$ 45.0000 0.149071
$$46$$ −360.000 −1.15389
$$47$$ 24.0000 0.0744843 0.0372421 0.999306i $$-0.488143\pi$$
0.0372421 + 0.999306i $$0.488143\pi$$
$$48$$ −213.000 −0.640498
$$49$$ 0 0
$$50$$ 75.0000 0.212132
$$51$$ −162.000 −0.444795
$$52$$ −74.0000 −0.197345
$$53$$ 450.000 1.16627 0.583134 0.812376i $$-0.301826\pi$$
0.583134 + 0.812376i $$0.301826\pi$$
$$54$$ 81.0000 0.204124
$$55$$ −120.000 −0.294196
$$56$$ 0 0
$$57$$ 372.000 0.864432
$$58$$ −234.000 −0.529754
$$59$$ −24.0000 −0.0529582 −0.0264791 0.999649i $$-0.508430\pi$$
−0.0264791 + 0.999649i $$0.508430\pi$$
$$60$$ 15.0000 0.0322749
$$61$$ 322.000 0.675867 0.337933 0.941170i $$-0.390272\pi$$
0.337933 + 0.941170i $$0.390272\pi$$
$$62$$ −600.000 −1.22903
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ −370.000 −0.706044
$$66$$ −216.000 −0.402845
$$67$$ −196.000 −0.357391 −0.178696 0.983904i $$-0.557188\pi$$
−0.178696 + 0.983904i $$0.557188\pi$$
$$68$$ −54.0000 −0.0963009
$$69$$ −360.000 −0.628100
$$70$$ 0 0
$$71$$ −288.000 −0.481399 −0.240699 0.970600i $$-0.577377\pi$$
−0.240699 + 0.970600i $$0.577377\pi$$
$$72$$ −189.000 −0.309359
$$73$$ 430.000 0.689420 0.344710 0.938709i $$-0.387977\pi$$
0.344710 + 0.938709i $$0.387977\pi$$
$$74$$ −210.000 −0.329892
$$75$$ 75.0000 0.115470
$$76$$ 124.000 0.187155
$$77$$ 0 0
$$78$$ −666.000 −0.966790
$$79$$ −520.000 −0.740564 −0.370282 0.928919i $$-0.620739\pi$$
−0.370282 + 0.928919i $$0.620739\pi$$
$$80$$ −355.000 −0.496128
$$81$$ 81.0000 0.111111
$$82$$ −990.000 −1.33326
$$83$$ −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
$$86$$ 276.000 0.346068
$$87$$ −234.000 −0.288361
$$88$$ 504.000 0.610529
$$89$$ −1026.00 −1.22198 −0.610988 0.791640i $$-0.709227\pi$$
−0.610988 + 0.791640i $$0.709227\pi$$
$$90$$ 135.000 0.158114
$$91$$ 0 0
$$92$$ −120.000 −0.135988
$$93$$ −600.000 −0.669001
$$94$$ 72.0000 0.0790025
$$95$$ 620.000 0.669586
$$96$$ −135.000 −0.143525
$$97$$ 286.000 0.299370 0.149685 0.988734i $$-0.452174\pi$$
0.149685 + 0.988734i $$0.452174\pi$$
$$98$$ 0 0
$$99$$ −216.000 −0.219281
$$100$$ 25.0000 0.0250000
$$101$$ 1734.00 1.70831 0.854156 0.520017i $$-0.174075\pi$$
0.854156 + 0.520017i $$0.174075\pi$$
$$102$$ −486.000 −0.471776
$$103$$ −452.000 −0.432397 −0.216198 0.976349i $$-0.569366\pi$$
−0.216198 + 0.976349i $$0.569366\pi$$
$$104$$ 1554.00 1.46521
$$105$$ 0 0
$$106$$ 1350.00 1.23702
$$107$$ −1404.00 −1.26850 −0.634251 0.773127i $$-0.718692\pi$$
−0.634251 + 0.773127i $$0.718692\pi$$
$$108$$ 27.0000 0.0240563
$$109$$ −1474.00 −1.29526 −0.647631 0.761954i $$-0.724240\pi$$
−0.647631 + 0.761954i $$0.724240\pi$$
$$110$$ −360.000 −0.312042
$$111$$ −210.000 −0.179570
$$112$$ 0 0
$$113$$ 1086.00 0.904091 0.452046 0.891995i $$-0.350694\pi$$
0.452046 + 0.891995i $$0.350694\pi$$
$$114$$ 1116.00 0.916868
$$115$$ −600.000 −0.486524
$$116$$ −78.0000 −0.0624321
$$117$$ −666.000 −0.526254
$$118$$ −72.0000 −0.0561707
$$119$$ 0 0
$$120$$ −315.000 −0.239629
$$121$$ −755.000 −0.567243
$$122$$ 966.000 0.716865
$$123$$ −990.000 −0.725734
$$124$$ −200.000 −0.144843
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 1244.00 0.869190 0.434595 0.900626i $$-0.356891\pi$$
0.434595 + 0.900626i $$0.356891\pi$$
$$128$$ 1659.00 1.14560
$$129$$ 276.000 0.188376
$$130$$ −1110.00 −0.748873
$$131$$ −2328.00 −1.55266 −0.776329 0.630327i $$-0.782921\pi$$
−0.776329 + 0.630327i $$0.782921\pi$$
$$132$$ −72.0000 −0.0474757
$$133$$ 0 0
$$134$$ −588.000 −0.379071
$$135$$ 135.000 0.0860663
$$136$$ 1134.00 0.714998
$$137$$ 2118.00 1.32082 0.660412 0.750903i $$-0.270382\pi$$
0.660412 + 0.750903i $$0.270382\pi$$
$$138$$ −1080.00 −0.666201
$$139$$ −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
$$142$$ −864.000 −0.510600
$$143$$ 1776.00 1.03858
$$144$$ −639.000 −0.369792
$$145$$ −390.000 −0.223364
$$146$$ 1290.00 0.731241
$$147$$ 0 0
$$148$$ −70.0000 −0.0388781
$$149$$ 258.000 0.141854 0.0709268 0.997482i $$-0.477404\pi$$
0.0709268 + 0.997482i $$0.477404\pi$$
$$150$$ 225.000 0.122474
$$151$$ −808.000 −0.435458 −0.217729 0.976009i $$-0.569865\pi$$
−0.217729 + 0.976009i $$0.569865\pi$$
$$152$$ −2604.00 −1.38955
$$153$$ −486.000 −0.256802
$$154$$ 0 0
$$155$$ −1000.00 −0.518206
$$156$$ −222.000 −0.113937
$$157$$ −2378.00 −1.20882 −0.604411 0.796673i $$-0.706592\pi$$
−0.604411 + 0.796673i $$0.706592\pi$$
$$158$$ −1560.00 −0.785487
$$159$$ 1350.00 0.673346
$$160$$ −225.000 −0.111174
$$161$$ 0 0
$$162$$ 243.000 0.117851
$$163$$ −52.0000 −0.0249874 −0.0124937 0.999922i $$-0.503977\pi$$
−0.0124937 + 0.999922i $$0.503977\pi$$
$$164$$ −330.000 −0.157126
$$165$$ −360.000 −0.169854
$$166$$ −468.000 −0.218818
$$167$$ 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
$$170$$ −810.000 −0.365436
$$171$$ 1116.00 0.499080
$$172$$ 92.0000 0.0407845
$$173$$ −426.000 −0.187215 −0.0936075 0.995609i $$-0.529840\pi$$
−0.0936075 + 0.995609i $$0.529840\pi$$
$$174$$ −702.000 −0.305853
$$175$$ 0 0
$$176$$ 1704.00 0.729795
$$177$$ −72.0000 −0.0305754
$$178$$ −3078.00 −1.29610
$$179$$ −1440.00 −0.601289 −0.300644 0.953736i $$-0.597202\pi$$
−0.300644 + 0.953736i $$0.597202\pi$$
$$180$$ 45.0000 0.0186339
$$181$$ 3130.00 1.28537 0.642683 0.766133i $$-0.277821\pi$$
0.642683 + 0.766133i $$0.277821\pi$$
$$182$$ 0 0
$$183$$ 966.000 0.390212
$$184$$ 2520.00 1.00966
$$185$$ −350.000 −0.139095
$$186$$ −1800.00 −0.709583
$$187$$ 1296.00 0.506807
$$188$$ 24.0000 0.00931053
$$189$$ 0 0
$$190$$ 1860.00 0.710203
$$191$$ 3576.00 1.35471 0.677357 0.735655i $$-0.263125\pi$$
0.677357 + 0.735655i $$0.263125\pi$$
$$192$$ 1299.00 0.488267
$$193$$ 2666.00 0.994315 0.497158 0.867660i $$-0.334377\pi$$
0.497158 + 0.867660i $$0.334377\pi$$
$$194$$ 858.000 0.317530
$$195$$ −1110.00 −0.407635
$$196$$ 0 0
$$197$$ −2718.00 −0.982992 −0.491496 0.870880i $$-0.663550\pi$$
−0.491496 + 0.870880i $$0.663550\pi$$
$$198$$ −648.000 −0.232583
$$199$$ 3832.00 1.36504 0.682521 0.730866i $$-0.260884\pi$$
0.682521 + 0.730866i $$0.260884\pi$$
$$200$$ −525.000 −0.185616
$$201$$ −588.000 −0.206340
$$202$$ 5202.00 1.81194
$$203$$ 0 0
$$204$$ −162.000 −0.0555994
$$205$$ −1650.00 −0.562151
$$206$$ −1356.00 −0.458626
$$207$$ −1080.00 −0.362634
$$208$$ 5254.00 1.75144
$$209$$ −2976.00 −0.984948
$$210$$ 0 0
$$211$$ 1100.00 0.358896 0.179448 0.983767i $$-0.442569\pi$$
0.179448 + 0.983767i $$0.442569\pi$$
$$212$$ 450.000 0.145784
$$213$$ −864.000 −0.277936
$$214$$ −4212.00 −1.34545
$$215$$ 460.000 0.145915
$$216$$ −567.000 −0.178609
$$217$$ 0 0
$$218$$ −4422.00 −1.37383
$$219$$ 1290.00 0.398037
$$220$$ −120.000 −0.0367745
$$221$$ 3996.00 1.21629
$$222$$ −630.000 −0.190463
$$223$$ −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
$$226$$ 3258.00 0.958933
$$227$$ −660.000 −0.192977 −0.0964884 0.995334i $$-0.530761\pi$$
−0.0964884 + 0.995334i $$0.530761\pi$$
$$228$$ 372.000 0.108054
$$229$$ 1906.00 0.550009 0.275004 0.961443i $$-0.411321\pi$$
0.275004 + 0.961443i $$0.411321\pi$$
$$230$$ −1800.00 −0.516037
$$231$$ 0 0
$$232$$ 1638.00 0.463534
$$233$$ −1458.00 −0.409943 −0.204972 0.978768i $$-0.565710\pi$$
−0.204972 + 0.978768i $$0.565710\pi$$
$$234$$ −1998.00 −0.558177
$$235$$ 120.000 0.0333104
$$236$$ −24.0000 −0.00661978
$$237$$ −1560.00 −0.427565
$$238$$ 0 0
$$239$$ 1176.00 0.318281 0.159140 0.987256i $$-0.449128\pi$$
0.159140 + 0.987256i $$0.449128\pi$$
$$240$$ −1065.00 −0.286439
$$241$$ −866.000 −0.231469 −0.115734 0.993280i $$-0.536922\pi$$
−0.115734 + 0.993280i $$0.536922\pi$$
$$242$$ −2265.00 −0.601652
$$243$$ 243.000 0.0641500
$$244$$ 322.000 0.0844834
$$245$$ 0 0
$$246$$ −2970.00 −0.769757
$$247$$ −9176.00 −2.36379
$$248$$ 4200.00 1.07540
$$249$$ −468.000 −0.119110
$$250$$ 375.000 0.0948683
$$251$$ −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
$$254$$ 3732.00 0.921915
$$255$$ −810.000 −0.198918
$$256$$ 1513.00 0.369385
$$257$$ −2526.00 −0.613103 −0.306552 0.951854i $$-0.599175\pi$$
−0.306552 + 0.951854i $$0.599175\pi$$
$$258$$ 828.000 0.199802
$$259$$ 0 0
$$260$$ −370.000 −0.0882555
$$261$$ −702.000 −0.166485
$$262$$ −6984.00 −1.64684
$$263$$ 5448.00 1.27733 0.638666 0.769484i $$-0.279487\pi$$
0.638666 + 0.769484i $$0.279487\pi$$
$$264$$ 1512.00 0.352489
$$265$$ 2250.00 0.521571
$$266$$ 0 0
$$267$$ −3078.00 −0.705508
$$268$$ −196.000 −0.0446739
$$269$$ 2574.00 0.583418 0.291709 0.956507i $$-0.405776\pi$$
0.291709 + 0.956507i $$0.405776\pi$$
$$270$$ 405.000 0.0912871
$$271$$ 3184.00 0.713706 0.356853 0.934161i $$-0.383850\pi$$
0.356853 + 0.934161i $$0.383850\pi$$
$$272$$ 3834.00 0.854671
$$273$$ 0 0
$$274$$ 6354.00 1.40095
$$275$$ −600.000 −0.131569
$$276$$ −360.000 −0.0785125
$$277$$ 3962.00 0.859399 0.429699 0.902972i $$-0.358620\pi$$
0.429699 + 0.902972i $$0.358620\pi$$
$$278$$ −6972.00 −1.50415
$$279$$ −1800.00 −0.386248
$$280$$ 0 0
$$281$$ −8286.00 −1.75908 −0.879540 0.475825i $$-0.842149\pi$$
−0.879540 + 0.475825i $$0.842149\pi$$
$$282$$ 216.000 0.0456121
$$283$$ 2716.00 0.570493 0.285246 0.958454i $$-0.407925\pi$$
0.285246 + 0.958454i $$0.407925\pi$$
$$284$$ −288.000 −0.0601748
$$285$$ 1860.00 0.386586
$$286$$ 5328.00 1.10158
$$287$$ 0 0
$$288$$ −405.000 −0.0828641
$$289$$ −1997.00 −0.406473
$$290$$ −1170.00 −0.236913
$$291$$ 858.000 0.172841
$$292$$ 430.000 0.0861776
$$293$$ −6018.00 −1.19992 −0.599958 0.800032i $$-0.704816\pi$$
−0.599958 + 0.800032i $$0.704816\pi$$
$$294$$ 0 0
$$295$$ −120.000 −0.0236836
$$296$$ 1470.00 0.288655
$$297$$ −648.000 −0.126602
$$298$$ 774.000 0.150458
$$299$$ 8880.00 1.71754
$$300$$ 75.0000 0.0144338
$$301$$ 0 0
$$302$$ −2424.00 −0.461873
$$303$$ 5202.00 0.986294
$$304$$ −8804.00 −1.66100
$$305$$ 1610.00 0.302257
$$306$$ −1458.00 −0.272380
$$307$$ −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
$$310$$ −3000.00 −0.549640
$$311$$ −1536.00 −0.280060 −0.140030 0.990147i $$-0.544720\pi$$
−0.140030 + 0.990147i $$0.544720\pi$$
$$312$$ 4662.00 0.845942
$$313$$ 7342.00 1.32586 0.662930 0.748681i $$-0.269313\pi$$
0.662930 + 0.748681i $$0.269313\pi$$
$$314$$ −7134.00 −1.28215
$$315$$ 0 0
$$316$$ −520.000 −0.0925705
$$317$$ −3894.00 −0.689933 −0.344967 0.938615i $$-0.612110\pi$$
−0.344967 + 0.938615i $$0.612110\pi$$
$$318$$ 4050.00 0.714191
$$319$$ 1872.00 0.328564
$$320$$ 2165.00 0.378210
$$321$$ −4212.00 −0.732370
$$322$$ 0 0
$$323$$ −6696.00 −1.15348
$$324$$ 81.0000 0.0138889
$$325$$ −1850.00 −0.315752
$$326$$ −156.000 −0.0265032
$$327$$ −4422.00 −0.747820
$$328$$ 6930.00 1.16660
$$329$$ 0 0
$$330$$ −1080.00 −0.180158
$$331$$ 3692.00 0.613084 0.306542 0.951857i $$-0.400828\pi$$
0.306542 + 0.951857i $$0.400828\pi$$
$$332$$ −156.000 −0.0257880
$$333$$ −630.000 −0.103675
$$334$$ 11160.0 1.82829
$$335$$ −980.000 −0.159830
$$336$$ 0 0
$$337$$ −8998.00 −1.45446 −0.727229 0.686395i $$-0.759192\pi$$
−0.727229 + 0.686395i $$0.759192\pi$$
$$338$$ 9837.00 1.58302
$$339$$ 3258.00 0.521977
$$340$$ −270.000 −0.0430671
$$341$$ 4800.00 0.762271
$$342$$ 3348.00 0.529354
$$343$$ 0 0
$$344$$ −1932.00 −0.302809
$$345$$ −1800.00 −0.280895
$$346$$ −1278.00 −0.198571
$$347$$ 5244.00 0.811276 0.405638 0.914034i $$-0.367049\pi$$
0.405638 + 0.914034i $$0.367049\pi$$
$$348$$ −234.000 −0.0360452
$$349$$ −6302.00 −0.966585 −0.483293 0.875459i $$-0.660559\pi$$
−0.483293 + 0.875459i $$0.660559\pi$$
$$350$$ 0 0
$$351$$ −1998.00 −0.303833
$$352$$ 1080.00 0.163535
$$353$$ −3414.00 −0.514756 −0.257378 0.966311i $$-0.582859\pi$$
−0.257378 + 0.966311i $$0.582859\pi$$
$$354$$ −216.000 −0.0324301
$$355$$ −1440.00 −0.215288
$$356$$ −1026.00 −0.152747
$$357$$ 0 0
$$358$$ −4320.00 −0.637763
$$359$$ 4824.00 0.709195 0.354597 0.935019i $$-0.384618\pi$$
0.354597 + 0.935019i $$0.384618\pi$$
$$360$$ −945.000 −0.138350
$$361$$ 8517.00 1.24173
$$362$$ 9390.00 1.36334
$$363$$ −2265.00 −0.327498
$$364$$ 0 0
$$365$$ 2150.00 0.308318
$$366$$ 2898.00 0.413882
$$367$$ 3508.00 0.498954 0.249477 0.968381i $$-0.419741\pi$$
0.249477 + 0.968381i $$0.419741\pi$$
$$368$$ 8520.00 1.20689
$$369$$ −2970.00 −0.419003
$$370$$ −1050.00 −0.147532
$$371$$ 0 0
$$372$$ −600.000 −0.0836251
$$373$$ 10802.0 1.49948 0.749740 0.661732i $$-0.230178\pi$$
0.749740 + 0.661732i $$0.230178\pi$$
$$374$$ 3888.00 0.537550
$$375$$ 375.000 0.0516398
$$376$$ −504.000 −0.0691272
$$377$$ 5772.00 0.788523
$$378$$ 0 0
$$379$$ 1460.00 0.197876 0.0989382 0.995094i $$-0.468455\pi$$
0.0989382 + 0.995094i $$0.468455\pi$$
$$380$$ 620.000 0.0836982
$$381$$ 3732.00 0.501827
$$382$$ 10728.0 1.43689
$$383$$ 4872.00 0.649994 0.324997 0.945715i $$-0.394637\pi$$
0.324997 + 0.945715i $$0.394637\pi$$
$$384$$ 4977.00 0.661410
$$385$$ 0 0
$$386$$ 7998.00 1.05463
$$387$$ 828.000 0.108759
$$388$$ 286.000 0.0374213
$$389$$ −14046.0 −1.83075 −0.915373 0.402606i $$-0.868104\pi$$
−0.915373 + 0.402606i $$0.868104\pi$$
$$390$$ −3330.00 −0.432362
$$391$$ 6480.00 0.838127
$$392$$ 0 0
$$393$$ −6984.00 −0.896428
$$394$$ −8154.00 −1.04262
$$395$$ −2600.00 −0.331190
$$396$$ −216.000 −0.0274101
$$397$$ 2734.00 0.345631 0.172816 0.984954i $$-0.444714\pi$$
0.172816 + 0.984954i $$0.444714\pi$$
$$398$$ 11496.0 1.44785
$$399$$ 0 0
$$400$$ −1775.00 −0.221875
$$401$$ −15942.0 −1.98530 −0.992650 0.121019i $$-0.961384\pi$$
−0.992650 + 0.121019i $$0.961384\pi$$
$$402$$ −1764.00 −0.218857
$$403$$ 14800.0 1.82938
$$404$$ 1734.00 0.213539
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ 1680.00 0.204606
$$408$$ 3402.00 0.412804
$$409$$ −8714.00 −1.05350 −0.526748 0.850022i $$-0.676589\pi$$
−0.526748 + 0.850022i $$0.676589\pi$$
$$410$$ −4950.00 −0.596251
$$411$$ 6354.00 0.762578
$$412$$ −452.000 −0.0540496
$$413$$ 0 0
$$414$$ −3240.00 −0.384631
$$415$$ −780.000 −0.0922619
$$416$$ 3330.00 0.392468
$$417$$ −6972.00 −0.818754
$$418$$ −8928.00 −1.04470
$$419$$ −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 $$0.279004\pi$$
$$422$$ 3300.00 0.380667
$$423$$ 216.000 0.0248281
$$424$$ −9450.00 −1.08239
$$425$$ −1350.00 −0.154081
$$426$$ −2592.00 −0.294795
$$427$$ 0 0
$$428$$ −1404.00 −0.158563
$$429$$ 5328.00 0.599623
$$430$$ 1380.00 0.154766
$$431$$ 720.000 0.0804668 0.0402334 0.999190i $$-0.487190\pi$$
0.0402334 + 0.999190i $$0.487190\pi$$
$$432$$ −1917.00 −0.213499
$$433$$ 15622.0 1.73382 0.866912 0.498462i $$-0.166102\pi$$
0.866912 + 0.498462i $$0.166102\pi$$
$$434$$ 0 0
$$435$$ −1170.00 −0.128959
$$436$$ −1474.00 −0.161908
$$437$$ −14880.0 −1.62885
$$438$$ 3870.00 0.422182
$$439$$ 9880.00 1.07414 0.537069 0.843538i $$-0.319531\pi$$
0.537069 + 0.843538i $$0.319531\pi$$
$$440$$ 2520.00 0.273037
$$441$$ 0 0
$$442$$ 11988.0 1.29007
$$443$$ −16116.0 −1.72843 −0.864215 0.503123i $$-0.832184\pi$$
−0.864215 + 0.503123i $$0.832184\pi$$
$$444$$ −210.000 −0.0224463
$$445$$ −5130.00 −0.546484
$$446$$ −5892.00 −0.625548
$$447$$ 774.000 0.0818992
$$448$$ 0 0
$$449$$ 9018.00 0.947852 0.473926 0.880565i $$-0.342836\pi$$
0.473926 + 0.880565i $$0.342836\pi$$
$$450$$ 675.000 0.0707107
$$451$$ 7920.00 0.826914
$$452$$ 1086.00 0.113011
$$453$$ −2424.00 −0.251412
$$454$$ −1980.00 −0.204683
$$455$$ 0 0
$$456$$ −7812.00 −0.802260
$$457$$ −3670.00 −0.375657 −0.187829 0.982202i $$-0.560145\pi$$
−0.187829 + 0.982202i $$0.560145\pi$$
$$458$$ 5718.00 0.583372
$$459$$ −1458.00 −0.148265
$$460$$ −600.000 −0.0608155
$$461$$ −17562.0 −1.77428 −0.887141 0.461499i $$-0.847312\pi$$
−0.887141 + 0.461499i $$0.847312\pi$$
$$462$$ 0 0
$$463$$ 1172.00 0.117640 0.0588202 0.998269i $$-0.481266\pi$$
0.0588202 + 0.998269i $$0.481266\pi$$
$$464$$ 5538.00 0.554084
$$465$$ −3000.00 −0.299186
$$466$$ −4374.00 −0.434810
$$467$$ −6876.00 −0.681335 −0.340667 0.940184i $$-0.610653\pi$$
−0.340667 + 0.940184i $$0.610653\pi$$
$$468$$ −666.000 −0.0657818
$$469$$ 0 0
$$470$$ 360.000 0.0353310
$$471$$ −7134.00 −0.697914
$$472$$ 504.000 0.0491493
$$473$$ −2208.00 −0.214638
$$474$$ −4680.00 −0.453501
$$475$$ 3100.00 0.299448
$$476$$ 0 0
$$477$$ 4050.00 0.388756
$$478$$ 3528.00 0.337588
$$479$$ −2280.00 −0.217486 −0.108743 0.994070i $$-0.534683\pi$$
−0.108743 + 0.994070i $$0.534683\pi$$
$$480$$ −675.000 −0.0641862
$$481$$ 5180.00 0.491035
$$482$$ −2598.00 −0.245510
$$483$$ 0 0
$$484$$ −755.000 −0.0709053
$$485$$ 1430.00 0.133882
$$486$$ 729.000 0.0680414
$$487$$ −3076.00 −0.286215 −0.143108 0.989707i $$-0.545710\pi$$
−0.143108 + 0.989707i $$0.545710\pi$$
$$488$$ −6762.00 −0.627257
$$489$$ −156.000 −0.0144265
$$490$$ 0 0
$$491$$ −18912.0 −1.73826 −0.869131 0.494582i $$-0.835321\pi$$
−0.869131 + 0.494582i $$0.835321\pi$$
$$492$$ −990.000 −0.0907168
$$493$$ 4212.00 0.384785
$$494$$ −27528.0 −2.50717
$$495$$ −1080.00 −0.0980654
$$496$$ 14200.0 1.28548
$$497$$ 0 0
$$498$$ −1404.00 −0.126335
$$499$$ 9956.00 0.893170 0.446585 0.894741i $$-0.352640\pi$$
0.446585 + 0.894741i $$0.352640\pi$$
$$500$$ 125.000 0.0111803
$$501$$ 11160.0 0.995194
$$502$$ −1296.00 −0.115226
$$503$$ 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
$$506$$ 8640.00 0.759081
$$507$$ 9837.00 0.861689
$$508$$ 1244.00 0.108649
$$509$$ 2766.00 0.240866 0.120433 0.992721i $$-0.461572\pi$$
0.120433 + 0.992721i $$0.461572\pi$$
$$510$$ −2430.00 −0.210985
$$511$$ 0 0
$$512$$ −8733.00 −0.753804
$$513$$ 3348.00 0.288144
$$514$$ −7578.00 −0.650294
$$515$$ −2260.00 −0.193374
$$516$$ 276.000 0.0235469
$$517$$ −576.000 −0.0489989
$$518$$ 0 0
$$519$$ −1278.00 −0.108089
$$520$$ 7770.00 0.655264
$$521$$ −10530.0 −0.885466 −0.442733 0.896654i $$-0.645991\pi$$
−0.442733 + 0.896654i $$0.645991\pi$$
$$522$$ −2106.00 −0.176585
$$523$$ −12692.0 −1.06115 −0.530576 0.847637i $$-0.678024\pi$$
−0.530576 + 0.847637i $$0.678024\pi$$
$$524$$ −2328.00 −0.194082
$$525$$ 0 0
$$526$$ 16344.0 1.35481
$$527$$ 10800.0 0.892705
$$528$$ 5112.00 0.421347
$$529$$ 2233.00 0.183529
$$530$$ 6750.00 0.553210
$$531$$ −216.000 −0.0176527
$$532$$ 0 0
$$533$$ 24420.0 1.98452
$$534$$ −9234.00 −0.748304
$$535$$ −7020.00 −0.567292
$$536$$ 4116.00 0.331687
$$537$$ −4320.00 −0.347154
$$538$$ 7722.00 0.618809
$$539$$ 0 0
$$540$$ 135.000 0.0107583
$$541$$ 18110.0 1.43920 0.719602 0.694386i $$-0.244324\pi$$
0.719602 + 0.694386i $$0.244324\pi$$
$$542$$ 9552.00 0.756999
$$543$$ 9390.00 0.742106
$$544$$ 2430.00 0.191517
$$545$$ −7370.00 −0.579259
$$546$$ 0 0
$$547$$ 3620.00 0.282962 0.141481 0.989941i $$-0.454814\pi$$
0.141481 + 0.989941i $$0.454814\pi$$
$$548$$ 2118.00 0.165103
$$549$$ 2898.00 0.225289
$$550$$ −1800.00 −0.139550
$$551$$ −9672.00 −0.747806
$$552$$ 7560.00 0.582926
$$553$$ 0 0
$$554$$ 11886.0 0.911530
$$555$$ −1050.00 −0.0803063
$$556$$ −2324.00 −0.177265
$$557$$ −14166.0 −1.07762 −0.538809 0.842428i $$-0.681125\pi$$
−0.538809 + 0.842428i $$0.681125\pi$$
$$558$$ −5400.00 −0.409678
$$559$$ −6808.00 −0.515112
$$560$$ 0 0
$$561$$ 3888.00 0.292605
$$562$$ −24858.0 −1.86579
$$563$$ 13404.0 1.00339 0.501697 0.865043i $$-0.332709\pi$$
0.501697 + 0.865043i $$0.332709\pi$$
$$564$$ 72.0000 0.00537544
$$565$$ 5430.00 0.404322
$$566$$ 8148.00 0.605099
$$567$$ 0 0
$$568$$ 6048.00 0.446775
$$569$$ −18654.0 −1.37437 −0.687185 0.726483i $$-0.741154\pi$$
−0.687185 + 0.726483i $$0.741154\pi$$
$$570$$ 5580.00 0.410036
$$571$$ −7684.00 −0.563162 −0.281581 0.959537i $$-0.590859\pi$$
−0.281581 + 0.959537i $$0.590859\pi$$
$$572$$ 1776.00 0.129822
$$573$$ 10728.0 0.782144
$$574$$ 0 0
$$575$$ −3000.00 −0.217580
$$576$$ 3897.00 0.281901
$$577$$ 1726.00 0.124531 0.0622654 0.998060i $$-0.480167\pi$$
0.0622654 + 0.998060i $$0.480167\pi$$
$$578$$ −5991.00 −0.431129
$$579$$ 7998.00 0.574068
$$580$$ −390.000 −0.0279205
$$581$$ 0 0
$$582$$ 2574.00 0.183326
$$583$$ −10800.0 −0.767222
$$584$$ −9030.00 −0.639836
$$585$$ −3330.00 −0.235348
$$586$$ −18054.0 −1.27270
$$587$$ −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
$$590$$ −360.000 −0.0251203
$$591$$ −8154.00 −0.567531
$$592$$ 4970.00 0.345043
$$593$$ −2862.00 −0.198193 −0.0990963 0.995078i $$-0.531595\pi$$
−0.0990963 + 0.995078i $$0.531595\pi$$
$$594$$ −1944.00 −0.134282
$$595$$ 0 0
$$596$$ 258.000 0.0177317
$$597$$ 11496.0 0.788107
$$598$$ 26640.0 1.82172
$$599$$ −23592.0 −1.60925 −0.804627 0.593781i $$-0.797635\pi$$
−0.804627 + 0.593781i $$0.797635\pi$$
$$600$$ −1575.00 −0.107165
$$601$$ 9574.00 0.649803 0.324902 0.945748i $$-0.394669\pi$$
0.324902 + 0.945748i $$0.394669\pi$$
$$602$$ 0 0
$$603$$ −1764.00 −0.119130
$$604$$ −808.000 −0.0544322
$$605$$ −3775.00 −0.253679
$$606$$ 15606.0 1.04612
$$607$$ −17444.0 −1.16644 −0.583221 0.812314i $$-0.698208\pi$$
−0.583221 + 0.812314i $$0.698208\pi$$
$$608$$ −5580.00 −0.372202
$$609$$ 0 0
$$610$$ 4830.00 0.320592
$$611$$ −1776.00 −0.117593
$$612$$ −486.000 −0.0321003
$$613$$ −2374.00 −0.156419 −0.0782096 0.996937i $$-0.524920\pi$$
−0.0782096 + 0.996937i $$0.524920\pi$$
$$614$$ −27708.0 −1.82118
$$615$$ −4950.00 −0.324558
$$616$$ 0 0
$$617$$ −12162.0 −0.793555 −0.396778 0.917915i $$-0.629872\pi$$
−0.396778 + 0.917915i $$0.629872\pi$$
$$618$$ −4068.00 −0.264788
$$619$$ −8804.00 −0.571668 −0.285834 0.958279i $$-0.592271\pi$$
−0.285834 + 0.958279i $$0.592271\pi$$
$$620$$ −1000.00 −0.0647758
$$621$$ −3240.00 −0.209367
$$622$$ −4608.00 −0.297048
$$623$$ 0 0
$$624$$ 15762.0 1.01119
$$625$$ 625.000 0.0400000
$$626$$ 22026.0 1.40629
$$627$$ −8928.00 −0.568660
$$628$$ −2378.00 −0.151103
$$629$$ 3780.00 0.239616
$$630$$ 0 0
$$631$$ −12688.0 −0.800478 −0.400239 0.916411i $$-0.631073\pi$$
−0.400239 + 0.916411i $$0.631073\pi$$
$$632$$ 10920.0 0.687301
$$633$$ 3300.00 0.207209
$$634$$ −11682.0 −0.731785
$$635$$ 6220.00 0.388714
$$636$$ 1350.00 0.0841682
$$637$$ 0 0
$$638$$ 5616.00 0.348495
$$639$$ −2592.00 −0.160466
$$640$$ 8295.00 0.512326
$$641$$ −9150.00 −0.563812 −0.281906 0.959442i $$-0.590967\pi$$
−0.281906 + 0.959442i $$0.590967\pi$$
$$642$$ −12636.0 −0.776796
$$643$$ −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
$$646$$ −20088.0 −1.22345
$$647$$ 2736.00 0.166249 0.0831246 0.996539i $$-0.473510\pi$$
0.0831246 + 0.996539i $$0.473510\pi$$
$$648$$ −1701.00 −0.103120
$$649$$ 576.000 0.0348382
$$650$$ −5550.00 −0.334906
$$651$$ 0 0
$$652$$ −52.0000 −0.00312343
$$653$$ 22218.0 1.33148 0.665741 0.746183i $$-0.268116\pi$$
0.665741 + 0.746183i $$0.268116\pi$$
$$654$$ −13266.0 −0.793183
$$655$$ −11640.0 −0.694370
$$656$$ 23430.0 1.39449
$$657$$ 3870.00 0.229807
$$658$$ 0 0
$$659$$ 14520.0 0.858299 0.429149 0.903234i $$-0.358813\pi$$
0.429149 + 0.903234i $$0.358813\pi$$
$$660$$ −360.000 −0.0212318
$$661$$ 10618.0 0.624799 0.312400 0.949951i $$-0.398867\pi$$
0.312400 + 0.949951i $$0.398867\pi$$
$$662$$ 11076.0 0.650273
$$663$$ 11988.0 0.702225
$$664$$ 3276.00 0.191466
$$665$$ 0 0
$$666$$ −1890.00 −0.109964
$$667$$ 9360.00 0.543359
$$668$$ 3720.00 0.215466
$$669$$ −5892.00 −0.340505
$$670$$ −2940.00 −0.169526
$$671$$ −7728.00 −0.444614
$$672$$ 0 0
$$673$$ 1370.00 0.0784690 0.0392345 0.999230i $$-0.487508\pi$$
0.0392345 + 0.999230i $$0.487508\pi$$
$$674$$ −26994.0 −1.54269
$$675$$ 675.000 0.0384900
$$676$$ 3279.00 0.186561
$$677$$ 13758.0 0.781038 0.390519 0.920595i $$-0.372296\pi$$
0.390519 + 0.920595i $$0.372296\pi$$
$$678$$ 9774.00 0.553640
$$679$$ 0 0
$$680$$ 5670.00 0.319757
$$681$$ −1980.00 −0.111415
$$682$$ 14400.0 0.808511
$$683$$ 11988.0 0.671608 0.335804 0.941932i $$-0.390992\pi$$
0.335804 + 0.941932i $$0.390992\pi$$
$$684$$ 1116.00 0.0623850
$$685$$ 10590.0 0.590691
$$686$$ 0 0
$$687$$ 5718.00 0.317548
$$688$$ −6532.00 −0.361962
$$689$$ −33300.0 −1.84126
$$690$$ −5400.00 −0.297934
$$691$$ −32996.0 −1.81654 −0.908268 0.418388i $$-0.862595\pi$$
−0.908268 + 0.418388i $$0.862595\pi$$
$$692$$ −426.000 −0.0234019
$$693$$ 0 0
$$694$$ 15732.0 0.860488
$$695$$ −11620.0 −0.634204
$$696$$ 4914.00 0.267622
$$697$$ 17820.0 0.968408
$$698$$ −18906.0 −1.02522
$$699$$ −4374.00 −0.236681
$$700$$ 0 0
$$701$$ −25902.0 −1.39558 −0.697792 0.716300i $$-0.745834\pi$$
−0.697792 + 0.716300i $$0.745834\pi$$
$$702$$ −5994.00 −0.322263
$$703$$ −8680.00 −0.465679
$$704$$ −10392.0 −0.556340
$$705$$ 360.000 0.0192318
$$706$$ −10242.0 −0.545981
$$707$$ 0 0
$$708$$ −72.0000 −0.00382193
$$709$$ −27394.0 −1.45106 −0.725531 0.688189i $$-0.758406\pi$$
−0.725531 + 0.688189i $$0.758406\pi$$
$$710$$ −4320.00 −0.228347
$$711$$ −4680.00 −0.246855
$$712$$ 21546.0 1.13409
$$713$$ 24000.0 1.26060
$$714$$ 0 0
$$715$$ 8880.00 0.464466
$$716$$ −1440.00 −0.0751611
$$717$$ 3528.00 0.183760
$$718$$ 14472.0 0.752215
$$719$$ −34848.0 −1.80753 −0.903763 0.428033i $$-0.859207\pi$$
−0.903763 + 0.428033i $$0.859207\pi$$
$$720$$ −3195.00 −0.165376
$$721$$ 0 0
$$722$$ 25551.0 1.31705
$$723$$ −2598.00 −0.133639
$$724$$ 3130.00 0.160671
$$725$$ −1950.00 −0.0998913
$$726$$ −6795.00 −0.347364
$$727$$ −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
$$730$$ 6450.00 0.327021
$$731$$ −4968.00 −0.251365
$$732$$ 966.000 0.0487765
$$733$$ −18002.0 −0.907120 −0.453560 0.891226i $$-0.649846\pi$$
−0.453560 + 0.891226i $$0.649846\pi$$
$$734$$ 10524.0 0.529221
$$735$$ 0 0
$$736$$ 5400.00 0.270444
$$737$$ 4704.00 0.235107
$$738$$ −8910.00 −0.444420
$$739$$ 15284.0 0.760800 0.380400 0.924822i $$-0.375786\pi$$
0.380400 + 0.924822i $$0.375786\pi$$
$$740$$ −350.000 −0.0173868
$$741$$ −27528.0 −1.36473
$$742$$ 0 0
$$743$$ −18768.0 −0.926691 −0.463345 0.886178i $$-0.653351\pi$$
−0.463345 + 0.886178i $$0.653351\pi$$
$$744$$ 12600.0 0.620885
$$745$$ 1290.00 0.0634388
$$746$$ 32406.0 1.59044
$$747$$ −1404.00 −0.0687680
$$748$$ 1296.00 0.0633509
$$749$$ 0 0
$$750$$ 1125.00 0.0547723
$$751$$ 8696.00 0.422532 0.211266 0.977429i $$-0.432241\pi$$
0.211266 + 0.977429i $$0.432241\pi$$
$$752$$ −1704.00 −0.0826310
$$753$$ −1296.00 −0.0627209
$$754$$ 17316.0 0.836355
$$755$$ −4040.00 −0.194743
$$756$$ 0 0
$$757$$ −38662.0 −1.85627 −0.928134 0.372247i $$-0.878587\pi$$
−0.928134 + 0.372247i $$0.878587\pi$$
$$758$$ 4380.00 0.209880
$$759$$ 8640.00 0.413191
$$760$$ −13020.0 −0.621428
$$761$$ −23874.0 −1.13723 −0.568615 0.822604i $$-0.692521\pi$$
−0.568615 + 0.822604i $$0.692521\pi$$
$$762$$ 11196.0 0.532268
$$763$$ 0 0
$$764$$ 3576.00 0.169339
$$765$$ −2430.00 −0.114846
$$766$$ 14616.0 0.689422
$$767$$ 1776.00 0.0836084
$$768$$ 4539.00 0.213264
$$769$$ −23618.0 −1.10753 −0.553763 0.832675i $$-0.686808\pi$$
−0.553763 + 0.832675i $$0.686808\pi$$
$$770$$ 0 0
$$771$$ −7578.00 −0.353975
$$772$$ 2666.00 0.124289
$$773$$ −11538.0 −0.536860 −0.268430 0.963299i $$-0.586505\pi$$
−0.268430 + 0.963299i $$0.586505\pi$$
$$774$$ 2484.00 0.115356
$$775$$ −5000.00 −0.231749
$$776$$ −6006.00 −0.277839
$$777$$ 0 0
$$778$$ −42138.0 −1.94180
$$779$$ −40920.0 −1.88204
$$780$$ −1110.00 −0.0509543
$$781$$ 6912.00 0.316685
$$782$$ 19440.0 0.888968
$$783$$ −2106.00 −0.0961204
$$784$$ 0 0
$$785$$ −11890.0 −0.540602
$$786$$ −20952.0 −0.950805
$$787$$ 14884.0 0.674152 0.337076 0.941478i $$-0.390562\pi$$
0.337076 + 0.941478i $$0.390562\pi$$
$$788$$ −2718.00 −0.122874
$$789$$ 16344.0 0.737467
$$790$$ −7800.00 −0.351280
$$791$$ 0 0
$$792$$ 4536.00 0.203510
$$793$$ −23828.0 −1.06703
$$794$$ 8202.00 0.366597
$$795$$ 6750.00 0.301129
$$796$$ 3832.00 0.170630
$$797$$ 11334.0 0.503728 0.251864 0.967763i $$-0.418957\pi$$
0.251864 + 0.967763i $$0.418957\pi$$
$$798$$ 0 0
$$799$$ −1296.00 −0.0573832
$$800$$ −1125.00 −0.0497184
$$801$$ −9234.00 −0.407325
$$802$$ −47826.0 −2.10573
$$803$$ −10320.0 −0.453530
$$804$$ −588.000 −0.0257925
$$805$$ 0 0
$$806$$ 44400.0 1.94035
$$807$$ 7722.00 0.336837
$$808$$ −36414.0 −1.58545
$$809$$ 44730.0 1.94391 0.971955 0.235167i $$-0.0755638\pi$$
0.971955 + 0.235167i $$0.0755638\pi$$
$$810$$ 1215.00 0.0527046
$$811$$ 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
$$814$$ 5040.00 0.217017
$$815$$ −260.000 −0.0111747
$$816$$ 11502.0 0.493444
$$817$$ 11408.0 0.488513
$$818$$ −26142.0 −1.11740
$$819$$ 0 0
$$820$$ −1650.00 −0.0702689
$$821$$ −31686.0 −1.34695 −0.673477 0.739208i $$-0.735200\pi$$
−0.673477 + 0.739208i $$0.735200\pi$$
$$822$$ 19062.0 0.808836
$$823$$ 11036.0 0.467425 0.233713 0.972306i $$-0.424913\pi$$
0.233713 + 0.972306i $$0.424913\pi$$
$$824$$ 9492.00 0.401298
$$825$$ −1800.00 −0.0759612
$$826$$ 0 0
$$827$$ 25884.0 1.08836 0.544181 0.838968i $$-0.316841\pi$$
0.544181 + 0.838968i $$0.316841\pi$$
$$828$$ −1080.00 −0.0453292
$$829$$ −15950.0 −0.668234 −0.334117 0.942532i $$-0.608438\pi$$
−0.334117 + 0.942532i $$0.608438\pi$$
$$830$$ −2340.00 −0.0978585
$$831$$ 11886.0 0.496174
$$832$$ −32042.0 −1.33516
$$833$$ 0 0
$$834$$ −20916.0 −0.868419
$$835$$ 18600.0 0.770874
$$836$$ −2976.00 −0.123119
$$837$$ −5400.00 −0.223000
$$838$$ −35928.0 −1.48104
$$839$$ −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
$$842$$ 33162.0 1.35729
$$843$$ −24858.0 −1.01560
$$844$$ 1100.00 0.0448620
$$845$$ 16395.0 0.667462
$$846$$ 648.000 0.0263342
$$847$$ 0 0
$$848$$ −31950.0 −1.29383
$$849$$ 8148.00 0.329374
$$850$$ −4050.00 −0.163428
$$851$$ 8400.00 0.338365
$$852$$ −864.000 −0.0347420
$$853$$ 27862.0 1.11838 0.559189 0.829040i $$-0.311113\pi$$
0.559189 + 0.829040i $$0.311113\pi$$
$$854$$ 0 0
$$855$$ 5580.00 0.223195
$$856$$ 29484.0 1.17727
$$857$$ 7314.00 0.291530 0.145765 0.989319i $$-0.453436\pi$$
0.145765 + 0.989319i $$0.453436\pi$$
$$858$$ 15984.0 0.635996
$$859$$ 28780.0 1.14314 0.571572 0.820552i $$-0.306334\pi$$
0.571572 + 0.820552i $$0.306334\pi$$
$$860$$ 460.000 0.0182394
$$861$$ 0 0
$$862$$ 2160.00 0.0853479
$$863$$ −32688.0 −1.28935 −0.644677 0.764455i $$-0.723008\pi$$
−0.644677 + 0.764455i $$0.723008\pi$$
$$864$$ −1215.00 −0.0478416
$$865$$ −2130.00 −0.0837251
$$866$$ 46866.0 1.83900
$$867$$ −5991.00 −0.234677
$$868$$ 0 0
$$869$$ 12480.0 0.487175
$$870$$ −3510.00 −0.136782
$$871$$ 14504.0 0.564236
$$872$$ 30954.0 1.20210
$$873$$ 2574.00 0.0997900
$$874$$ −44640.0 −1.72766
$$875$$ 0 0
$$876$$ 1290.00 0.0497546
$$877$$ 36650.0 1.41115 0.705577 0.708633i $$-0.250688\pi$$
0.705577 + 0.708633i $$0.250688\pi$$
$$878$$ 29640.0 1.13930
$$879$$ −18054.0 −0.692772
$$880$$ 8520.00 0.326374
$$881$$ 2646.00 0.101187 0.0505936 0.998719i $$-0.483889\pi$$
0.0505936 + 0.998719i $$0.483889\pi$$
$$882$$ 0 0
$$883$$ 10892.0 0.415113 0.207557 0.978223i $$-0.433449\pi$$
0.207557 + 0.978223i $$0.433449\pi$$
$$884$$ 3996.00 0.152036
$$885$$ −360.000 −0.0136737
$$886$$ −48348.0 −1.83328
$$887$$ 43464.0 1.64530 0.822648 0.568550i $$-0.192496\pi$$
0.822648 + 0.568550i $$0.192496\pi$$
$$888$$ 4410.00 0.166655
$$889$$ 0 0
$$890$$ −15390.0 −0.579634
$$891$$ −1944.00 −0.0730937
$$892$$ −1964.00 −0.0737215
$$893$$ 2976.00 0.111521
$$894$$ 2322.00 0.0868672
$$895$$ −7200.00 −0.268904
$$896$$ 0 0
$$897$$ 26640.0 0.991621
$$898$$ 27054.0 1.00535
$$899$$ 15600.0 0.578742
$$900$$ 225.000 0.00833333
$$901$$ −24300.0 −0.898502
$$902$$ 23760.0 0.877075
$$903$$ 0 0
$$904$$ −22806.0 −0.839067
$$905$$ 15650.0 0.574833
$$906$$ −7272.00 −0.266662
$$907$$ −14884.0 −0.544890 −0.272445 0.962171i $$-0.587832\pi$$
−0.272445 + 0.962171i $$0.587832\pi$$
$$908$$ −660.000 −0.0241221
$$909$$ 15606.0 0.569437
$$910$$ 0 0
$$911$$ −1248.00 −0.0453876 −0.0226938 0.999742i $$-0.507224\pi$$
−0.0226938 + 0.999742i $$0.507224\pi$$
$$912$$ −26412.0 −0.958979
$$913$$ 3744.00 0.135716
$$914$$ −11010.0 −0.398445
$$915$$ 4830.00 0.174508
$$916$$ 1906.00 0.0687511
$$917$$ 0 0
$$918$$ −4374.00 −0.157259
$$919$$ −6640.00 −0.238339 −0.119169 0.992874i $$-0.538023\pi$$
−0.119169 + 0.992874i $$0.538023\pi$$
$$920$$ 12600.0 0.451532
$$921$$ −27708.0 −0.991324
$$922$$ −52686.0 −1.88191
$$923$$ 21312.0 0.760014
$$924$$ 0 0
$$925$$ −1750.00 −0.0622050
$$926$$ 3516.00 0.124776
$$927$$ −4068.00 −0.144132
$$928$$ 3510.00 0.124161
$$929$$ −29946.0 −1.05758 −0.528792 0.848751i $$-0.677355\pi$$
−0.528792 + 0.848751i $$0.677355\pi$$
$$930$$ −9000.00 −0.317335
$$931$$ 0 0
$$932$$ −1458.00 −0.0512429
$$933$$ −4608.00 −0.161693
$$934$$ −20628.0 −0.722665
$$935$$ 6480.00 0.226651
$$936$$ 13986.0 0.488405
$$937$$ −45002.0 −1.56900 −0.784499 0.620130i $$-0.787080\pi$$
−0.784499 + 0.620130i $$0.787080\pi$$
$$938$$ 0 0
$$939$$ 22026.0 0.765486
$$940$$ 120.000 0.00416380
$$941$$ −6090.00 −0.210976 −0.105488 0.994421i $$-0.533640\pi$$
−0.105488 + 0.994421i $$0.533640\pi$$
$$942$$ −21402.0 −0.740249
$$943$$ 39600.0 1.36750
$$944$$ 1704.00 0.0587505
$$945$$ 0 0
$$946$$ −6624.00 −0.227658
$$947$$ 56388.0 1.93491 0.967457 0.253035i $$-0.0814288\pi$$
0.967457 + 0.253035i $$0.0814288\pi$$
$$948$$ −1560.00 −0.0534456
$$949$$ −31820.0 −1.08843
$$950$$ 9300.00 0.317612
$$951$$ −11682.0 −0.398333
$$952$$ 0 0
$$953$$ 10854.0 0.368936 0.184468 0.982839i $$-0.440944\pi$$
0.184468 + 0.982839i $$0.440944\pi$$
$$954$$ 12150.0 0.412338
$$955$$ 17880.0 0.605846
$$956$$ 1176.00 0.0397851
$$957$$ 5616.00 0.189696
$$958$$ −6840.00 −0.230679
$$959$$ 0 0
$$960$$ 6495.00 0.218360
$$961$$ 10209.0 0.342687
$$962$$ 15540.0 0.520821
$$963$$ −12636.0 −0.422834
$$964$$ −866.000 −0.0289336
$$965$$ 13330.0 0.444671
$$966$$ 0 0
$$967$$ −42316.0 −1.40723 −0.703615 0.710582i $$-0.748432\pi$$
−0.703615 + 0.710582i $$0.748432\pi$$
$$968$$ 15855.0 0.526445
$$969$$ −20088.0 −0.665964
$$970$$ 4290.00 0.142004
$$971$$ −24480.0 −0.809063 −0.404532 0.914524i $$-0.632565\pi$$
−0.404532 + 0.914524i $$0.632565\pi$$
$$972$$ 243.000 0.00801875
$$973$$ 0 0
$$974$$ −9228.00 −0.303577
$$975$$ −5550.00 −0.182300
$$976$$ −22862.0 −0.749790
$$977$$ −6906.00 −0.226144 −0.113072 0.993587i $$-0.536069\pi$$
−0.113072 + 0.993587i $$0.536069\pi$$
$$978$$ −468.000 −0.0153016
$$979$$ 24624.0 0.803868
$$980$$ 0 0
$$981$$ −13266.0 −0.431754
$$982$$ −56736.0 −1.84371
$$983$$ −6960.00 −0.225829 −0.112914 0.993605i $$-0.536019\pi$$
−0.112914 + 0.993605i $$0.536019\pi$$
$$984$$ 20790.0 0.673538
$$985$$ −13590.0 −0.439608
$$986$$ 12636.0 0.408126
$$987$$ 0 0
$$988$$ −9176.00 −0.295473
$$989$$ −11040.0 −0.354956
$$990$$ −3240.00 −0.104014
$$991$$ 47792.0 1.53195 0.765975 0.642870i $$-0.222256\pi$$
0.765975 + 0.642870i $$0.222256\pi$$
$$992$$ 9000.00 0.288055
$$993$$ 11076.0 0.353964
$$994$$ 0 0
$$995$$ 19160.0 0.610465
$$996$$ −468.000 −0.0148887
$$997$$ −9938.00 −0.315687 −0.157843 0.987464i $$-0.550454\pi$$
−0.157843 + 0.987464i $$0.550454\pi$$
$$998$$ 29868.0 0.947350
$$999$$ −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 735.4.a.i.1.1 1
3.2 odd 2 2205.4.a.c.1.1 1
7.6 odd 2 15.4.a.b.1.1 1
21.20 even 2 45.4.a.b.1.1 1
28.27 even 2 240.4.a.f.1.1 1
35.13 even 4 75.4.b.a.49.1 2
35.27 even 4 75.4.b.a.49.2 2
35.34 odd 2 75.4.a.a.1.1 1
56.13 odd 2 960.4.a.bi.1.1 1
56.27 even 2 960.4.a.l.1.1 1
63.13 odd 6 405.4.e.d.136.1 2
63.20 even 6 405.4.e.k.271.1 2
63.34 odd 6 405.4.e.d.271.1 2
63.41 even 6 405.4.e.k.136.1 2
77.76 even 2 1815.4.a.a.1.1 1
84.83 odd 2 720.4.a.r.1.1 1
105.62 odd 4 225.4.b.d.199.1 2
105.83 odd 4 225.4.b.d.199.2 2
105.104 even 2 225.4.a.g.1.1 1
140.27 odd 4 1200.4.f.m.49.1 2
140.83 odd 4 1200.4.f.m.49.2 2
140.139 even 2 1200.4.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 7.6 odd 2
45.4.a.b.1.1 1 21.20 even 2
75.4.a.a.1.1 1 35.34 odd 2
75.4.b.a.49.1 2 35.13 even 4
75.4.b.a.49.2 2 35.27 even 4
225.4.a.g.1.1 1 105.104 even 2
225.4.b.d.199.1 2 105.62 odd 4
225.4.b.d.199.2 2 105.83 odd 4
240.4.a.f.1.1 1 28.27 even 2
405.4.e.d.136.1 2 63.13 odd 6
405.4.e.d.271.1 2 63.34 odd 6
405.4.e.k.136.1 2 63.41 even 6
405.4.e.k.271.1 2 63.20 even 6
720.4.a.r.1.1 1 84.83 odd 2
735.4.a.i.1.1 1 1.1 even 1 trivial
960.4.a.l.1.1 1 56.27 even 2
960.4.a.bi.1.1 1 56.13 odd 2
1200.4.a.o.1.1 1 140.139 even 2
1200.4.f.m.49.1 2 140.27 odd 4
1200.4.f.m.49.2 2 140.83 odd 4
1815.4.a.a.1.1 1 77.76 even 2
2205.4.a.c.1.1 1 3.2 odd 2