# Properties

 Label 1014.4.a.j.1.1 Level $1014$ Weight $4$ Character 1014.1 Self dual yes Analytic conductor $59.828$ 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] = [1014,4,Mod(1,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, 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("1014.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1014.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: $$59.8279367458$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1014.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -10.0000 q^{5} +6.00000 q^{6} +8.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -10.0000 q^{5} +6.00000 q^{6} +8.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -20.0000 q^{10} -40.0000 q^{11} +12.0000 q^{12} +16.0000 q^{14} -30.0000 q^{15} +16.0000 q^{16} +130.000 q^{17} +18.0000 q^{18} +20.0000 q^{19} -40.0000 q^{20} +24.0000 q^{21} -80.0000 q^{22} +24.0000 q^{24} -25.0000 q^{25} +27.0000 q^{27} +32.0000 q^{28} -18.0000 q^{29} -60.0000 q^{30} +184.000 q^{31} +32.0000 q^{32} -120.000 q^{33} +260.000 q^{34} -80.0000 q^{35} +36.0000 q^{36} +74.0000 q^{37} +40.0000 q^{38} -80.0000 q^{40} +362.000 q^{41} +48.0000 q^{42} +76.0000 q^{43} -160.000 q^{44} -90.0000 q^{45} +452.000 q^{47} +48.0000 q^{48} -279.000 q^{49} -50.0000 q^{50} +390.000 q^{51} +382.000 q^{53} +54.0000 q^{54} +400.000 q^{55} +64.0000 q^{56} +60.0000 q^{57} -36.0000 q^{58} -464.000 q^{59} -120.000 q^{60} +358.000 q^{61} +368.000 q^{62} +72.0000 q^{63} +64.0000 q^{64} -240.000 q^{66} +700.000 q^{67} +520.000 q^{68} -160.000 q^{70} +748.000 q^{71} +72.0000 q^{72} -1058.00 q^{73} +148.000 q^{74} -75.0000 q^{75} +80.0000 q^{76} -320.000 q^{77} -976.000 q^{79} -160.000 q^{80} +81.0000 q^{81} +724.000 q^{82} +1008.00 q^{83} +96.0000 q^{84} -1300.00 q^{85} +152.000 q^{86} -54.0000 q^{87} -320.000 q^{88} +386.000 q^{89} -180.000 q^{90} +552.000 q^{93} +904.000 q^{94} -200.000 q^{95} +96.0000 q^{96} +614.000 q^{97} -558.000 q^{98} -360.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.707107
$$3$$ 3.00000 0.577350
$$4$$ 4.00000 0.500000
$$5$$ −10.0000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 6.00000 0.408248
$$7$$ 8.00000 0.431959 0.215980 0.976398i $$-0.430705\pi$$
0.215980 + 0.976398i $$0.430705\pi$$
$$8$$ 8.00000 0.353553
$$9$$ 9.00000 0.333333
$$10$$ −20.0000 −0.632456
$$11$$ −40.0000 −1.09640 −0.548202 0.836346i $$-0.684688\pi$$
−0.548202 + 0.836346i $$0.684688\pi$$
$$12$$ 12.0000 0.288675
$$13$$ 0 0
$$14$$ 16.0000 0.305441
$$15$$ −30.0000 −0.516398
$$16$$ 16.0000 0.250000
$$17$$ 130.000 1.85468 0.927342 0.374215i $$-0.122088\pi$$
0.927342 + 0.374215i $$0.122088\pi$$
$$18$$ 18.0000 0.235702
$$19$$ 20.0000 0.241490 0.120745 0.992684i $$-0.461472\pi$$
0.120745 + 0.992684i $$0.461472\pi$$
$$20$$ −40.0000 −0.447214
$$21$$ 24.0000 0.249392
$$22$$ −80.0000 −0.775275
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 24.0000 0.204124
$$25$$ −25.0000 −0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 32.0000 0.215980
$$29$$ −18.0000 −0.115259 −0.0576296 0.998338i $$-0.518354\pi$$
−0.0576296 + 0.998338i $$0.518354\pi$$
$$30$$ −60.0000 −0.365148
$$31$$ 184.000 1.06604 0.533022 0.846101i $$-0.321056\pi$$
0.533022 + 0.846101i $$0.321056\pi$$
$$32$$ 32.0000 0.176777
$$33$$ −120.000 −0.633010
$$34$$ 260.000 1.31146
$$35$$ −80.0000 −0.386356
$$36$$ 36.0000 0.166667
$$37$$ 74.0000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 40.0000 0.170759
$$39$$ 0 0
$$40$$ −80.0000 −0.316228
$$41$$ 362.000 1.37890 0.689450 0.724333i $$-0.257852\pi$$
0.689450 + 0.724333i $$0.257852\pi$$
$$42$$ 48.0000 0.176347
$$43$$ 76.0000 0.269532 0.134766 0.990877i $$-0.456972\pi$$
0.134766 + 0.990877i $$0.456972\pi$$
$$44$$ −160.000 −0.548202
$$45$$ −90.0000 −0.298142
$$46$$ 0 0
$$47$$ 452.000 1.40279 0.701393 0.712774i $$-0.252562\pi$$
0.701393 + 0.712774i $$0.252562\pi$$
$$48$$ 48.0000 0.144338
$$49$$ −279.000 −0.813411
$$50$$ −50.0000 −0.141421
$$51$$ 390.000 1.07080
$$52$$ 0 0
$$53$$ 382.000 0.990033 0.495016 0.868884i $$-0.335162\pi$$
0.495016 + 0.868884i $$0.335162\pi$$
$$54$$ 54.0000 0.136083
$$55$$ 400.000 0.980654
$$56$$ 64.0000 0.152721
$$57$$ 60.0000 0.139424
$$58$$ −36.0000 −0.0815005
$$59$$ −464.000 −1.02386 −0.511929 0.859028i $$-0.671069\pi$$
−0.511929 + 0.859028i $$0.671069\pi$$
$$60$$ −120.000 −0.258199
$$61$$ 358.000 0.751430 0.375715 0.926735i $$-0.377397\pi$$
0.375715 + 0.926735i $$0.377397\pi$$
$$62$$ 368.000 0.753807
$$63$$ 72.0000 0.143986
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ −240.000 −0.447605
$$67$$ 700.000 1.27640 0.638199 0.769872i $$-0.279680\pi$$
0.638199 + 0.769872i $$0.279680\pi$$
$$68$$ 520.000 0.927342
$$69$$ 0 0
$$70$$ −160.000 −0.273195
$$71$$ 748.000 1.25030 0.625150 0.780505i $$-0.285038\pi$$
0.625150 + 0.780505i $$0.285038\pi$$
$$72$$ 72.0000 0.117851
$$73$$ −1058.00 −1.69629 −0.848147 0.529760i $$-0.822282\pi$$
−0.848147 + 0.529760i $$0.822282\pi$$
$$74$$ 148.000 0.232495
$$75$$ −75.0000 −0.115470
$$76$$ 80.0000 0.120745
$$77$$ −320.000 −0.473602
$$78$$ 0 0
$$79$$ −976.000 −1.38998 −0.694991 0.719018i $$-0.744592\pi$$
−0.694991 + 0.719018i $$0.744592\pi$$
$$80$$ −160.000 −0.223607
$$81$$ 81.0000 0.111111
$$82$$ 724.000 0.975030
$$83$$ 1008.00 1.33304 0.666520 0.745487i $$-0.267783\pi$$
0.666520 + 0.745487i $$0.267783\pi$$
$$84$$ 96.0000 0.124696
$$85$$ −1300.00 −1.65888
$$86$$ 152.000 0.190588
$$87$$ −54.0000 −0.0665449
$$88$$ −320.000 −0.387638
$$89$$ 386.000 0.459729 0.229865 0.973223i $$-0.426172\pi$$
0.229865 + 0.973223i $$0.426172\pi$$
$$90$$ −180.000 −0.210819
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 552.000 0.615481
$$94$$ 904.000 0.991920
$$95$$ −200.000 −0.215995
$$96$$ 96.0000 0.102062
$$97$$ 614.000 0.642704 0.321352 0.946960i $$-0.395863\pi$$
0.321352 + 0.946960i $$0.395863\pi$$
$$98$$ −558.000 −0.575168
$$99$$ −360.000 −0.365468
$$100$$ −100.000 −0.100000
$$101$$ 518.000 0.510326 0.255163 0.966898i $$-0.417871\pi$$
0.255163 + 0.966898i $$0.417871\pi$$
$$102$$ 780.000 0.757172
$$103$$ 112.000 0.107143 0.0535713 0.998564i $$-0.482940\pi$$
0.0535713 + 0.998564i $$0.482940\pi$$
$$104$$ 0 0
$$105$$ −240.000 −0.223063
$$106$$ 764.000 0.700059
$$107$$ −372.000 −0.336099 −0.168050 0.985779i $$-0.553747\pi$$
−0.168050 + 0.985779i $$0.553747\pi$$
$$108$$ 108.000 0.0962250
$$109$$ −934.000 −0.820743 −0.410371 0.911918i $$-0.634601\pi$$
−0.410371 + 0.911918i $$0.634601\pi$$
$$110$$ 800.000 0.693427
$$111$$ 222.000 0.189832
$$112$$ 128.000 0.107990
$$113$$ 1914.00 1.59340 0.796699 0.604376i $$-0.206578\pi$$
0.796699 + 0.604376i $$0.206578\pi$$
$$114$$ 120.000 0.0985880
$$115$$ 0 0
$$116$$ −72.0000 −0.0576296
$$117$$ 0 0
$$118$$ −928.000 −0.723977
$$119$$ 1040.00 0.801148
$$120$$ −240.000 −0.182574
$$121$$ 269.000 0.202104
$$122$$ 716.000 0.531341
$$123$$ 1086.00 0.796108
$$124$$ 736.000 0.533022
$$125$$ 1500.00 1.07331
$$126$$ 144.000 0.101814
$$127$$ 1296.00 0.905523 0.452761 0.891632i $$-0.350439\pi$$
0.452761 + 0.891632i $$0.350439\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 228.000 0.155615
$$130$$ 0 0
$$131$$ −892.000 −0.594919 −0.297460 0.954734i $$-0.596139\pi$$
−0.297460 + 0.954734i $$0.596139\pi$$
$$132$$ −480.000 −0.316505
$$133$$ 160.000 0.104314
$$134$$ 1400.00 0.902549
$$135$$ −270.000 −0.172133
$$136$$ 1040.00 0.655730
$$137$$ −2326.00 −1.45054 −0.725269 0.688466i $$-0.758284\pi$$
−0.725269 + 0.688466i $$0.758284\pi$$
$$138$$ 0 0
$$139$$ 1932.00 1.17892 0.589461 0.807797i $$-0.299340\pi$$
0.589461 + 0.807797i $$0.299340\pi$$
$$140$$ −320.000 −0.193178
$$141$$ 1356.00 0.809899
$$142$$ 1496.00 0.884095
$$143$$ 0 0
$$144$$ 144.000 0.0833333
$$145$$ 180.000 0.103091
$$146$$ −2116.00 −1.19946
$$147$$ −837.000 −0.469623
$$148$$ 296.000 0.164399
$$149$$ −882.000 −0.484941 −0.242471 0.970159i $$-0.577958\pi$$
−0.242471 + 0.970159i $$0.577958\pi$$
$$150$$ −150.000 −0.0816497
$$151$$ 1776.00 0.957145 0.478572 0.878048i $$-0.341154\pi$$
0.478572 + 0.878048i $$0.341154\pi$$
$$152$$ 160.000 0.0853797
$$153$$ 1170.00 0.618228
$$154$$ −640.000 −0.334887
$$155$$ −1840.00 −0.953499
$$156$$ 0 0
$$157$$ −2410.00 −1.22509 −0.612544 0.790436i $$-0.709854\pi$$
−0.612544 + 0.790436i $$0.709854\pi$$
$$158$$ −1952.00 −0.982866
$$159$$ 1146.00 0.571596
$$160$$ −320.000 −0.158114
$$161$$ 0 0
$$162$$ 162.000 0.0785674
$$163$$ −3212.00 −1.54346 −0.771728 0.635953i $$-0.780607\pi$$
−0.771728 + 0.635953i $$0.780607\pi$$
$$164$$ 1448.00 0.689450
$$165$$ 1200.00 0.566181
$$166$$ 2016.00 0.942602
$$167$$ −1668.00 −0.772896 −0.386448 0.922311i $$-0.626298\pi$$
−0.386448 + 0.922311i $$0.626298\pi$$
$$168$$ 192.000 0.0881733
$$169$$ 0 0
$$170$$ −2600.00 −1.17301
$$171$$ 180.000 0.0804967
$$172$$ 304.000 0.134766
$$173$$ 3598.00 1.58122 0.790609 0.612321i $$-0.209764\pi$$
0.790609 + 0.612321i $$0.209764\pi$$
$$174$$ −108.000 −0.0470544
$$175$$ −200.000 −0.0863919
$$176$$ −640.000 −0.274101
$$177$$ −1392.00 −0.591125
$$178$$ 772.000 0.325078
$$179$$ 1068.00 0.445956 0.222978 0.974824i $$-0.428422\pi$$
0.222978 + 0.974824i $$0.428422\pi$$
$$180$$ −360.000 −0.149071
$$181$$ −4786.00 −1.96542 −0.982709 0.185158i $$-0.940720\pi$$
−0.982709 + 0.185158i $$0.940720\pi$$
$$182$$ 0 0
$$183$$ 1074.00 0.433838
$$184$$ 0 0
$$185$$ −740.000 −0.294086
$$186$$ 1104.00 0.435211
$$187$$ −5200.00 −2.03348
$$188$$ 1808.00 0.701393
$$189$$ 216.000 0.0831306
$$190$$ −400.000 −0.152732
$$191$$ −1312.00 −0.497031 −0.248516 0.968628i $$-0.579943\pi$$
−0.248516 + 0.968628i $$0.579943\pi$$
$$192$$ 192.000 0.0721688
$$193$$ 350.000 0.130537 0.0652683 0.997868i $$-0.479210\pi$$
0.0652683 + 0.997868i $$0.479210\pi$$
$$194$$ 1228.00 0.454460
$$195$$ 0 0
$$196$$ −1116.00 −0.406706
$$197$$ 342.000 0.123688 0.0618439 0.998086i $$-0.480302\pi$$
0.0618439 + 0.998086i $$0.480302\pi$$
$$198$$ −720.000 −0.258425
$$199$$ −3368.00 −1.19975 −0.599877 0.800092i $$-0.704784\pi$$
−0.599877 + 0.800092i $$0.704784\pi$$
$$200$$ −200.000 −0.0707107
$$201$$ 2100.00 0.736928
$$202$$ 1036.00 0.360855
$$203$$ −144.000 −0.0497873
$$204$$ 1560.00 0.535401
$$205$$ −3620.00 −1.23333
$$206$$ 224.000 0.0757613
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −800.000 −0.264771
$$210$$ −480.000 −0.157729
$$211$$ −2004.00 −0.653844 −0.326922 0.945051i $$-0.606011\pi$$
−0.326922 + 0.945051i $$0.606011\pi$$
$$212$$ 1528.00 0.495016
$$213$$ 2244.00 0.721861
$$214$$ −744.000 −0.237658
$$215$$ −760.000 −0.241077
$$216$$ 216.000 0.0680414
$$217$$ 1472.00 0.460488
$$218$$ −1868.00 −0.580353
$$219$$ −3174.00 −0.979356
$$220$$ 1600.00 0.490327
$$221$$ 0 0
$$222$$ 444.000 0.134231
$$223$$ 5608.00 1.68403 0.842017 0.539451i $$-0.181368\pi$$
0.842017 + 0.539451i $$0.181368\pi$$
$$224$$ 256.000 0.0763604
$$225$$ −225.000 −0.0666667
$$226$$ 3828.00 1.12670
$$227$$ 1928.00 0.563726 0.281863 0.959455i $$-0.409048\pi$$
0.281863 + 0.959455i $$0.409048\pi$$
$$228$$ 240.000 0.0697122
$$229$$ 3938.00 1.13638 0.568189 0.822898i $$-0.307644\pi$$
0.568189 + 0.822898i $$0.307644\pi$$
$$230$$ 0 0
$$231$$ −960.000 −0.273434
$$232$$ −144.000 −0.0407503
$$233$$ 2562.00 0.720353 0.360176 0.932884i $$-0.382717\pi$$
0.360176 + 0.932884i $$0.382717\pi$$
$$234$$ 0 0
$$235$$ −4520.00 −1.25469
$$236$$ −1856.00 −0.511929
$$237$$ −2928.00 −0.802506
$$238$$ 2080.00 0.566497
$$239$$ −7164.00 −1.93891 −0.969457 0.245260i $$-0.921127\pi$$
−0.969457 + 0.245260i $$0.921127\pi$$
$$240$$ −480.000 −0.129099
$$241$$ 6182.00 1.65236 0.826178 0.563410i $$-0.190511\pi$$
0.826178 + 0.563410i $$0.190511\pi$$
$$242$$ 538.000 0.142909
$$243$$ 243.000 0.0641500
$$244$$ 1432.00 0.375715
$$245$$ 2790.00 0.727537
$$246$$ 2172.00 0.562934
$$247$$ 0 0
$$248$$ 1472.00 0.376904
$$249$$ 3024.00 0.769631
$$250$$ 3000.00 0.758947
$$251$$ −1396.00 −0.351055 −0.175527 0.984475i $$-0.556163\pi$$
−0.175527 + 0.984475i $$0.556163\pi$$
$$252$$ 288.000 0.0719932
$$253$$ 0 0
$$254$$ 2592.00 0.640301
$$255$$ −3900.00 −0.957755
$$256$$ 256.000 0.0625000
$$257$$ 6906.00 1.67620 0.838102 0.545514i $$-0.183665\pi$$
0.838102 + 0.545514i $$0.183665\pi$$
$$258$$ 456.000 0.110036
$$259$$ 592.000 0.142027
$$260$$ 0 0
$$261$$ −162.000 −0.0384197
$$262$$ −1784.00 −0.420671
$$263$$ −6848.00 −1.60557 −0.802787 0.596266i $$-0.796650\pi$$
−0.802787 + 0.596266i $$0.796650\pi$$
$$264$$ −960.000 −0.223803
$$265$$ −3820.00 −0.885512
$$266$$ 320.000 0.0737611
$$267$$ 1158.00 0.265425
$$268$$ 2800.00 0.638199
$$269$$ −6034.00 −1.36766 −0.683828 0.729643i $$-0.739686\pi$$
−0.683828 + 0.729643i $$0.739686\pi$$
$$270$$ −540.000 −0.121716
$$271$$ −4832.00 −1.08311 −0.541556 0.840665i $$-0.682164\pi$$
−0.541556 + 0.840665i $$0.682164\pi$$
$$272$$ 2080.00 0.463671
$$273$$ 0 0
$$274$$ −4652.00 −1.02568
$$275$$ 1000.00 0.219281
$$276$$ 0 0
$$277$$ −4082.00 −0.885428 −0.442714 0.896663i $$-0.645984\pi$$
−0.442714 + 0.896663i $$0.645984\pi$$
$$278$$ 3864.00 0.833623
$$279$$ 1656.00 0.355348
$$280$$ −640.000 −0.136598
$$281$$ −3350.00 −0.711189 −0.355595 0.934640i $$-0.615722\pi$$
−0.355595 + 0.934640i $$0.615722\pi$$
$$282$$ 2712.00 0.572685
$$283$$ 7796.00 1.63754 0.818770 0.574121i $$-0.194656\pi$$
0.818770 + 0.574121i $$0.194656\pi$$
$$284$$ 2992.00 0.625150
$$285$$ −600.000 −0.124705
$$286$$ 0 0
$$287$$ 2896.00 0.595629
$$288$$ 288.000 0.0589256
$$289$$ 11987.0 2.43985
$$290$$ 360.000 0.0728963
$$291$$ 1842.00 0.371065
$$292$$ −4232.00 −0.848147
$$293$$ −3922.00 −0.781999 −0.390999 0.920391i $$-0.627871\pi$$
−0.390999 + 0.920391i $$0.627871\pi$$
$$294$$ −1674.00 −0.332074
$$295$$ 4640.00 0.915767
$$296$$ 592.000 0.116248
$$297$$ −1080.00 −0.211003
$$298$$ −1764.00 −0.342905
$$299$$ 0 0
$$300$$ −300.000 −0.0577350
$$301$$ 608.000 0.116427
$$302$$ 3552.00 0.676803
$$303$$ 1554.00 0.294637
$$304$$ 320.000 0.0603726
$$305$$ −3580.00 −0.672099
$$306$$ 2340.00 0.437153
$$307$$ −5956.00 −1.10725 −0.553627 0.832765i $$-0.686757\pi$$
−0.553627 + 0.832765i $$0.686757\pi$$
$$308$$ −1280.00 −0.236801
$$309$$ 336.000 0.0618588
$$310$$ −3680.00 −0.674226
$$311$$ 2352.00 0.428841 0.214421 0.976741i $$-0.431214\pi$$
0.214421 + 0.976741i $$0.431214\pi$$
$$312$$ 0 0
$$313$$ 8442.00 1.52450 0.762252 0.647280i $$-0.224093\pi$$
0.762252 + 0.647280i $$0.224093\pi$$
$$314$$ −4820.00 −0.866269
$$315$$ −720.000 −0.128785
$$316$$ −3904.00 −0.694991
$$317$$ 5550.00 0.983341 0.491670 0.870781i $$-0.336386\pi$$
0.491670 + 0.870781i $$0.336386\pi$$
$$318$$ 2292.00 0.404179
$$319$$ 720.000 0.126371
$$320$$ −640.000 −0.111803
$$321$$ −1116.00 −0.194047
$$322$$ 0 0
$$323$$ 2600.00 0.447888
$$324$$ 324.000 0.0555556
$$325$$ 0 0
$$326$$ −6424.00 −1.09139
$$327$$ −2802.00 −0.473856
$$328$$ 2896.00 0.487515
$$329$$ 3616.00 0.605947
$$330$$ 2400.00 0.400350
$$331$$ −140.000 −0.0232480 −0.0116240 0.999932i $$-0.503700\pi$$
−0.0116240 + 0.999932i $$0.503700\pi$$
$$332$$ 4032.00 0.666520
$$333$$ 666.000 0.109599
$$334$$ −3336.00 −0.546520
$$335$$ −7000.00 −1.14164
$$336$$ 384.000 0.0623480
$$337$$ −6174.00 −0.997980 −0.498990 0.866608i $$-0.666296\pi$$
−0.498990 + 0.866608i $$0.666296\pi$$
$$338$$ 0 0
$$339$$ 5742.00 0.919949
$$340$$ −5200.00 −0.829440
$$341$$ −7360.00 −1.16882
$$342$$ 360.000 0.0569198
$$343$$ −4976.00 −0.783320
$$344$$ 608.000 0.0952941
$$345$$ 0 0
$$346$$ 7196.00 1.11809
$$347$$ −2988.00 −0.462260 −0.231130 0.972923i $$-0.574242\pi$$
−0.231130 + 0.972923i $$0.574242\pi$$
$$348$$ −216.000 −0.0332725
$$349$$ 162.000 0.0248472 0.0124236 0.999923i $$-0.496045\pi$$
0.0124236 + 0.999923i $$0.496045\pi$$
$$350$$ −400.000 −0.0610883
$$351$$ 0 0
$$352$$ −1280.00 −0.193819
$$353$$ 10754.0 1.62147 0.810733 0.585416i $$-0.199069\pi$$
0.810733 + 0.585416i $$0.199069\pi$$
$$354$$ −2784.00 −0.417989
$$355$$ −7480.00 −1.11830
$$356$$ 1544.00 0.229865
$$357$$ 3120.00 0.462543
$$358$$ 2136.00 0.315338
$$359$$ −3588.00 −0.527486 −0.263743 0.964593i $$-0.584957\pi$$
−0.263743 + 0.964593i $$0.584957\pi$$
$$360$$ −720.000 −0.105409
$$361$$ −6459.00 −0.941682
$$362$$ −9572.00 −1.38976
$$363$$ 807.000 0.116685
$$364$$ 0 0
$$365$$ 10580.0 1.51721
$$366$$ 2148.00 0.306770
$$367$$ 11272.0 1.60325 0.801626 0.597826i $$-0.203968\pi$$
0.801626 + 0.597826i $$0.203968\pi$$
$$368$$ 0 0
$$369$$ 3258.00 0.459633
$$370$$ −1480.00 −0.207950
$$371$$ 3056.00 0.427654
$$372$$ 2208.00 0.307741
$$373$$ −10914.0 −1.51503 −0.757514 0.652819i $$-0.773586\pi$$
−0.757514 + 0.652819i $$0.773586\pi$$
$$374$$ −10400.0 −1.43789
$$375$$ 4500.00 0.619677
$$376$$ 3616.00 0.495960
$$377$$ 0 0
$$378$$ 432.000 0.0587822
$$379$$ −8100.00 −1.09781 −0.548904 0.835886i $$-0.684955\pi$$
−0.548904 + 0.835886i $$0.684955\pi$$
$$380$$ −800.000 −0.107998
$$381$$ 3888.00 0.522804
$$382$$ −2624.00 −0.351454
$$383$$ −6180.00 −0.824499 −0.412250 0.911071i $$-0.635257\pi$$
−0.412250 + 0.911071i $$0.635257\pi$$
$$384$$ 384.000 0.0510310
$$385$$ 3200.00 0.423603
$$386$$ 700.000 0.0923033
$$387$$ 684.000 0.0898441
$$388$$ 2456.00 0.321352
$$389$$ −7522.00 −0.980413 −0.490206 0.871606i $$-0.663079\pi$$
−0.490206 + 0.871606i $$0.663079\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −2232.00 −0.287584
$$393$$ −2676.00 −0.343477
$$394$$ 684.000 0.0874605
$$395$$ 9760.00 1.24324
$$396$$ −1440.00 −0.182734
$$397$$ −6078.00 −0.768378 −0.384189 0.923254i $$-0.625519\pi$$
−0.384189 + 0.923254i $$0.625519\pi$$
$$398$$ −6736.00 −0.848355
$$399$$ 480.000 0.0602257
$$400$$ −400.000 −0.0500000
$$401$$ −1830.00 −0.227895 −0.113947 0.993487i $$-0.536350\pi$$
−0.113947 + 0.993487i $$0.536350\pi$$
$$402$$ 4200.00 0.521087
$$403$$ 0 0
$$404$$ 2072.00 0.255163
$$405$$ −810.000 −0.0993808
$$406$$ −288.000 −0.0352049
$$407$$ −2960.00 −0.360496
$$408$$ 3120.00 0.378586
$$409$$ −12434.0 −1.50323 −0.751616 0.659601i $$-0.770725\pi$$
−0.751616 + 0.659601i $$0.770725\pi$$
$$410$$ −7240.00 −0.872093
$$411$$ −6978.00 −0.837468
$$412$$ 448.000 0.0535713
$$413$$ −3712.00 −0.442265
$$414$$ 0 0
$$415$$ −10080.0 −1.19231
$$416$$ 0 0
$$417$$ 5796.00 0.680651
$$418$$ −1600.00 −0.187221
$$419$$ −14188.0 −1.65425 −0.827123 0.562021i $$-0.810024\pi$$
−0.827123 + 0.562021i $$0.810024\pi$$
$$420$$ −960.000 −0.111531
$$421$$ −8638.00 −0.999977 −0.499989 0.866032i $$-0.666662\pi$$
−0.499989 + 0.866032i $$0.666662\pi$$
$$422$$ −4008.00 −0.462337
$$423$$ 4068.00 0.467596
$$424$$ 3056.00 0.350029
$$425$$ −3250.00 −0.370937
$$426$$ 4488.00 0.510433
$$427$$ 2864.00 0.324587
$$428$$ −1488.00 −0.168050
$$429$$ 0 0
$$430$$ −1520.00 −0.170467
$$431$$ −4292.00 −0.479671 −0.239836 0.970813i $$-0.577094\pi$$
−0.239836 + 0.970813i $$0.577094\pi$$
$$432$$ 432.000 0.0481125
$$433$$ −5982.00 −0.663918 −0.331959 0.943294i $$-0.607710\pi$$
−0.331959 + 0.943294i $$0.607710\pi$$
$$434$$ 2944.00 0.325614
$$435$$ 540.000 0.0595196
$$436$$ −3736.00 −0.410371
$$437$$ 0 0
$$438$$ −6348.00 −0.692510
$$439$$ 256.000 0.0278319 0.0139160 0.999903i $$-0.495570\pi$$
0.0139160 + 0.999903i $$0.495570\pi$$
$$440$$ 3200.00 0.346714
$$441$$ −2511.00 −0.271137
$$442$$ 0 0
$$443$$ 12556.0 1.34662 0.673311 0.739359i $$-0.264872\pi$$
0.673311 + 0.739359i $$0.264872\pi$$
$$444$$ 888.000 0.0949158
$$445$$ −3860.00 −0.411194
$$446$$ 11216.0 1.19079
$$447$$ −2646.00 −0.279981
$$448$$ 512.000 0.0539949
$$449$$ −5574.00 −0.585865 −0.292932 0.956133i $$-0.594631\pi$$
−0.292932 + 0.956133i $$0.594631\pi$$
$$450$$ −450.000 −0.0471405
$$451$$ −14480.0 −1.51183
$$452$$ 7656.00 0.796699
$$453$$ 5328.00 0.552608
$$454$$ 3856.00 0.398615
$$455$$ 0 0
$$456$$ 480.000 0.0492940
$$457$$ −1266.00 −0.129586 −0.0647932 0.997899i $$-0.520639\pi$$
−0.0647932 + 0.997899i $$0.520639\pi$$
$$458$$ 7876.00 0.803540
$$459$$ 3510.00 0.356934
$$460$$ 0 0
$$461$$ −7554.00 −0.763178 −0.381589 0.924332i $$-0.624623\pi$$
−0.381589 + 0.924332i $$0.624623\pi$$
$$462$$ −1920.00 −0.193347
$$463$$ 6752.00 0.677737 0.338868 0.940834i $$-0.389956\pi$$
0.338868 + 0.940834i $$0.389956\pi$$
$$464$$ −288.000 −0.0288148
$$465$$ −5520.00 −0.550503
$$466$$ 5124.00 0.509366
$$467$$ 7924.00 0.785180 0.392590 0.919714i $$-0.371579\pi$$
0.392590 + 0.919714i $$0.371579\pi$$
$$468$$ 0 0
$$469$$ 5600.00 0.551352
$$470$$ −9040.00 −0.887200
$$471$$ −7230.00 −0.707305
$$472$$ −3712.00 −0.361989
$$473$$ −3040.00 −0.295517
$$474$$ −5856.00 −0.567458
$$475$$ −500.000 −0.0482980
$$476$$ 4160.00 0.400574
$$477$$ 3438.00 0.330011
$$478$$ −14328.0 −1.37102
$$479$$ 11084.0 1.05729 0.528644 0.848844i $$-0.322701\pi$$
0.528644 + 0.848844i $$0.322701\pi$$
$$480$$ −960.000 −0.0912871
$$481$$ 0 0
$$482$$ 12364.0 1.16839
$$483$$ 0 0
$$484$$ 1076.00 0.101052
$$485$$ −6140.00 −0.574852
$$486$$ 486.000 0.0453609
$$487$$ −4432.00 −0.412388 −0.206194 0.978511i $$-0.566108\pi$$
−0.206194 + 0.978511i $$0.566108\pi$$
$$488$$ 2864.00 0.265670
$$489$$ −9636.00 −0.891114
$$490$$ 5580.00 0.514446
$$491$$ −1140.00 −0.104781 −0.0523905 0.998627i $$-0.516684\pi$$
−0.0523905 + 0.998627i $$0.516684\pi$$
$$492$$ 4344.00 0.398054
$$493$$ −2340.00 −0.213769
$$494$$ 0 0
$$495$$ 3600.00 0.326885
$$496$$ 2944.00 0.266511
$$497$$ 5984.00 0.540079
$$498$$ 6048.00 0.544212
$$499$$ −1764.00 −0.158251 −0.0791257 0.996865i $$-0.525213\pi$$
−0.0791257 + 0.996865i $$0.525213\pi$$
$$500$$ 6000.00 0.536656
$$501$$ −5004.00 −0.446232
$$502$$ −2792.00 −0.248233
$$503$$ 16976.0 1.50482 0.752408 0.658697i $$-0.228892\pi$$
0.752408 + 0.658697i $$0.228892\pi$$
$$504$$ 576.000 0.0509069
$$505$$ −5180.00 −0.456449
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 5184.00 0.452761
$$509$$ −9474.00 −0.825005 −0.412503 0.910956i $$-0.635345\pi$$
−0.412503 + 0.910956i $$0.635345\pi$$
$$510$$ −7800.00 −0.677235
$$511$$ −8464.00 −0.732731
$$512$$ 512.000 0.0441942
$$513$$ 540.000 0.0464748
$$514$$ 13812.0 1.18526
$$515$$ −1120.00 −0.0958313
$$516$$ 912.000 0.0778073
$$517$$ −18080.0 −1.53802
$$518$$ 1184.00 0.100429
$$519$$ 10794.0 0.912917
$$520$$ 0 0
$$521$$ 14114.0 1.18684 0.593422 0.804892i $$-0.297777\pi$$
0.593422 + 0.804892i $$0.297777\pi$$
$$522$$ −324.000 −0.0271668
$$523$$ 20284.0 1.69590 0.847952 0.530074i $$-0.177836\pi$$
0.847952 + 0.530074i $$0.177836\pi$$
$$524$$ −3568.00 −0.297460
$$525$$ −600.000 −0.0498784
$$526$$ −13696.0 −1.13531
$$527$$ 23920.0 1.97718
$$528$$ −1920.00 −0.158252
$$529$$ −12167.0 −1.00000
$$530$$ −7640.00 −0.626152
$$531$$ −4176.00 −0.341286
$$532$$ 640.000 0.0521570
$$533$$ 0 0
$$534$$ 2316.00 0.187684
$$535$$ 3720.00 0.300616
$$536$$ 5600.00 0.451275
$$537$$ 3204.00 0.257473
$$538$$ −12068.0 −0.967079
$$539$$ 11160.0 0.891828
$$540$$ −1080.00 −0.0860663
$$541$$ 14362.0 1.14135 0.570675 0.821176i $$-0.306682\pi$$
0.570675 + 0.821176i $$0.306682\pi$$
$$542$$ −9664.00 −0.765875
$$543$$ −14358.0 −1.13473
$$544$$ 4160.00 0.327865
$$545$$ 9340.00 0.734095
$$546$$ 0 0
$$547$$ −20956.0 −1.63805 −0.819025 0.573757i $$-0.805485\pi$$
−0.819025 + 0.573757i $$0.805485\pi$$
$$548$$ −9304.00 −0.725269
$$549$$ 3222.00 0.250477
$$550$$ 2000.00 0.155055
$$551$$ −360.000 −0.0278340
$$552$$ 0 0
$$553$$ −7808.00 −0.600416
$$554$$ −8164.00 −0.626092
$$555$$ −2220.00 −0.169791
$$556$$ 7728.00 0.589461
$$557$$ 4134.00 0.314476 0.157238 0.987561i $$-0.449741\pi$$
0.157238 + 0.987561i $$0.449741\pi$$
$$558$$ 3312.00 0.251269
$$559$$ 0 0
$$560$$ −1280.00 −0.0965891
$$561$$ −15600.0 −1.17403
$$562$$ −6700.00 −0.502887
$$563$$ −16228.0 −1.21479 −0.607397 0.794399i $$-0.707786\pi$$
−0.607397 + 0.794399i $$0.707786\pi$$
$$564$$ 5424.00 0.404950
$$565$$ −19140.0 −1.42518
$$566$$ 15592.0 1.15792
$$567$$ 648.000 0.0479955
$$568$$ 5984.00 0.442048
$$569$$ 2514.00 0.185224 0.0926119 0.995702i $$-0.470478\pi$$
0.0926119 + 0.995702i $$0.470478\pi$$
$$570$$ −1200.00 −0.0881798
$$571$$ −11612.0 −0.851046 −0.425523 0.904948i $$-0.639910\pi$$
−0.425523 + 0.904948i $$0.639910\pi$$
$$572$$ 0 0
$$573$$ −3936.00 −0.286961
$$574$$ 5792.00 0.421173
$$575$$ 0 0
$$576$$ 576.000 0.0416667
$$577$$ −6354.00 −0.458441 −0.229221 0.973375i $$-0.573618\pi$$
−0.229221 + 0.973375i $$0.573618\pi$$
$$578$$ 23974.0 1.72524
$$579$$ 1050.00 0.0753653
$$580$$ 720.000 0.0515455
$$581$$ 8064.00 0.575819
$$582$$ 3684.00 0.262383
$$583$$ −15280.0 −1.08548
$$584$$ −8464.00 −0.599731
$$585$$ 0 0
$$586$$ −7844.00 −0.552957
$$587$$ 13240.0 0.930960 0.465480 0.885059i $$-0.345882\pi$$
0.465480 + 0.885059i $$0.345882\pi$$
$$588$$ −3348.00 −0.234812
$$589$$ 3680.00 0.257439
$$590$$ 9280.00 0.647545
$$591$$ 1026.00 0.0714112
$$592$$ 1184.00 0.0821995
$$593$$ 1146.00 0.0793602 0.0396801 0.999212i $$-0.487366\pi$$
0.0396801 + 0.999212i $$0.487366\pi$$
$$594$$ −2160.00 −0.149202
$$595$$ −10400.0 −0.716569
$$596$$ −3528.00 −0.242471
$$597$$ −10104.0 −0.692679
$$598$$ 0 0
$$599$$ 10464.0 0.713769 0.356884 0.934149i $$-0.383839\pi$$
0.356884 + 0.934149i $$0.383839\pi$$
$$600$$ −600.000 −0.0408248
$$601$$ 6650.00 0.451346 0.225673 0.974203i $$-0.427542\pi$$
0.225673 + 0.974203i $$0.427542\pi$$
$$602$$ 1216.00 0.0823263
$$603$$ 6300.00 0.425466
$$604$$ 7104.00 0.478572
$$605$$ −2690.00 −0.180767
$$606$$ 3108.00 0.208340
$$607$$ −6664.00 −0.445607 −0.222803 0.974863i $$-0.571521\pi$$
−0.222803 + 0.974863i $$0.571521\pi$$
$$608$$ 640.000 0.0426898
$$609$$ −432.000 −0.0287447
$$610$$ −7160.00 −0.475246
$$611$$ 0 0
$$612$$ 4680.00 0.309114
$$613$$ −2134.00 −0.140606 −0.0703030 0.997526i $$-0.522397\pi$$
−0.0703030 + 0.997526i $$0.522397\pi$$
$$614$$ −11912.0 −0.782947
$$615$$ −10860.0 −0.712061
$$616$$ −2560.00 −0.167444
$$617$$ 714.000 0.0465876 0.0232938 0.999729i $$-0.492585\pi$$
0.0232938 + 0.999729i $$0.492585\pi$$
$$618$$ 672.000 0.0437408
$$619$$ −29228.0 −1.89786 −0.948928 0.315494i $$-0.897830\pi$$
−0.948928 + 0.315494i $$0.897830\pi$$
$$620$$ −7360.00 −0.476750
$$621$$ 0 0
$$622$$ 4704.00 0.303237
$$623$$ 3088.00 0.198584
$$624$$ 0 0
$$625$$ −11875.0 −0.760000
$$626$$ 16884.0 1.07799
$$627$$ −2400.00 −0.152866
$$628$$ −9640.00 −0.612544
$$629$$ 9620.00 0.609816
$$630$$ −1440.00 −0.0910650
$$631$$ 13536.0 0.853977 0.426989 0.904257i $$-0.359574\pi$$
0.426989 + 0.904257i $$0.359574\pi$$
$$632$$ −7808.00 −0.491433
$$633$$ −6012.00 −0.377497
$$634$$ 11100.0 0.695327
$$635$$ −12960.0 −0.809924
$$636$$ 4584.00 0.285798
$$637$$ 0 0
$$638$$ 1440.00 0.0893576
$$639$$ 6732.00 0.416767
$$640$$ −1280.00 −0.0790569
$$641$$ 17218.0 1.06095 0.530476 0.847700i $$-0.322013\pi$$
0.530476 + 0.847700i $$0.322013\pi$$
$$642$$ −2232.00 −0.137212
$$643$$ −15044.0 −0.922671 −0.461335 0.887226i $$-0.652630\pi$$
−0.461335 + 0.887226i $$0.652630\pi$$
$$644$$ 0 0
$$645$$ −2280.00 −0.139186
$$646$$ 5200.00 0.316705
$$647$$ 25176.0 1.52978 0.764892 0.644158i $$-0.222792\pi$$
0.764892 + 0.644158i $$0.222792\pi$$
$$648$$ 648.000 0.0392837
$$649$$ 18560.0 1.12256
$$650$$ 0 0
$$651$$ 4416.00 0.265863
$$652$$ −12848.0 −0.771728
$$653$$ −16034.0 −0.960887 −0.480443 0.877026i $$-0.659524\pi$$
−0.480443 + 0.877026i $$0.659524\pi$$
$$654$$ −5604.00 −0.335067
$$655$$ 8920.00 0.532112
$$656$$ 5792.00 0.344725
$$657$$ −9522.00 −0.565432
$$658$$ 7232.00 0.428469
$$659$$ 25356.0 1.49883 0.749415 0.662100i $$-0.230335\pi$$
0.749415 + 0.662100i $$0.230335\pi$$
$$660$$ 4800.00 0.283091
$$661$$ −18310.0 −1.07742 −0.538711 0.842490i $$-0.681089\pi$$
−0.538711 + 0.842490i $$0.681089\pi$$
$$662$$ −280.000 −0.0164388
$$663$$ 0 0
$$664$$ 8064.00 0.471301
$$665$$ −1600.00 −0.0933013
$$666$$ 1332.00 0.0774984
$$667$$ 0 0
$$668$$ −6672.00 −0.386448
$$669$$ 16824.0 0.972277
$$670$$ −14000.0 −0.807264
$$671$$ −14320.0 −0.823871
$$672$$ 768.000 0.0440867
$$673$$ 24802.0 1.42057 0.710287 0.703912i $$-0.248565\pi$$
0.710287 + 0.703912i $$0.248565\pi$$
$$674$$ −12348.0 −0.705678
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ −22706.0 −1.28901 −0.644507 0.764598i $$-0.722937\pi$$
−0.644507 + 0.764598i $$0.722937\pi$$
$$678$$ 11484.0 0.650502
$$679$$ 4912.00 0.277622
$$680$$ −10400.0 −0.586503
$$681$$ 5784.00 0.325467
$$682$$ −14720.0 −0.826478
$$683$$ 14792.0 0.828697 0.414349 0.910118i $$-0.364009\pi$$
0.414349 + 0.910118i $$0.364009\pi$$
$$684$$ 720.000 0.0402484
$$685$$ 23260.0 1.29740
$$686$$ −9952.00 −0.553891
$$687$$ 11814.0 0.656088
$$688$$ 1216.00 0.0673831
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1148.00 0.0632011 0.0316006 0.999501i $$-0.489940\pi$$
0.0316006 + 0.999501i $$0.489940\pi$$
$$692$$ 14392.0 0.790609
$$693$$ −2880.00 −0.157867
$$694$$ −5976.00 −0.326867
$$695$$ −19320.0 −1.05446
$$696$$ −432.000 −0.0235272
$$697$$ 47060.0 2.55742
$$698$$ 324.000 0.0175696
$$699$$ 7686.00 0.415896
$$700$$ −800.000 −0.0431959
$$701$$ 14870.0 0.801187 0.400594 0.916256i $$-0.368804\pi$$
0.400594 + 0.916256i $$0.368804\pi$$
$$702$$ 0 0
$$703$$ 1480.00 0.0794015
$$704$$ −2560.00 −0.137051
$$705$$ −13560.0 −0.724396
$$706$$ 21508.0 1.14655
$$707$$ 4144.00 0.220440
$$708$$ −5568.00 −0.295563
$$709$$ 6354.00 0.336572 0.168286 0.985738i $$-0.446177\pi$$
0.168286 + 0.985738i $$0.446177\pi$$
$$710$$ −14960.0 −0.790759
$$711$$ −8784.00 −0.463327
$$712$$ 3088.00 0.162539
$$713$$ 0 0
$$714$$ 6240.00 0.327067
$$715$$ 0 0
$$716$$ 4272.00 0.222978
$$717$$ −21492.0 −1.11943
$$718$$ −7176.00 −0.372989
$$719$$ 9288.00 0.481758 0.240879 0.970555i $$-0.422564\pi$$
0.240879 + 0.970555i $$0.422564\pi$$
$$720$$ −1440.00 −0.0745356
$$721$$ 896.000 0.0462813
$$722$$ −12918.0 −0.665870
$$723$$ 18546.0 0.953988
$$724$$ −19144.0 −0.982709
$$725$$ 450.000 0.0230518
$$726$$ 1614.00 0.0825085
$$727$$ −21544.0 −1.09907 −0.549534 0.835471i $$-0.685195\pi$$
−0.549534 + 0.835471i $$0.685195\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 21160.0 1.07283
$$731$$ 9880.00 0.499897
$$732$$ 4296.00 0.216919
$$733$$ −19990.0 −1.00730 −0.503648 0.863909i $$-0.668009\pi$$
−0.503648 + 0.863909i $$0.668009\pi$$
$$734$$ 22544.0 1.13367
$$735$$ 8370.00 0.420044
$$736$$ 0 0
$$737$$ −28000.0 −1.39945
$$738$$ 6516.00 0.325010
$$739$$ −532.000 −0.0264816 −0.0132408 0.999912i $$-0.504215\pi$$
−0.0132408 + 0.999912i $$0.504215\pi$$
$$740$$ −2960.00 −0.147043
$$741$$ 0 0
$$742$$ 6112.00 0.302397
$$743$$ 25452.0 1.25672 0.628360 0.777922i $$-0.283726\pi$$
0.628360 + 0.777922i $$0.283726\pi$$
$$744$$ 4416.00 0.217605
$$745$$ 8820.00 0.433745
$$746$$ −21828.0 −1.07129
$$747$$ 9072.00 0.444347
$$748$$ −20800.0 −1.01674
$$749$$ −2976.00 −0.145181
$$750$$ 9000.00 0.438178
$$751$$ 6440.00 0.312915 0.156457 0.987685i $$-0.449993\pi$$
0.156457 + 0.987685i $$0.449993\pi$$
$$752$$ 7232.00 0.350697
$$753$$ −4188.00 −0.202682
$$754$$ 0 0
$$755$$ −17760.0 −0.856096
$$756$$ 864.000 0.0415653
$$757$$ −786.000 −0.0377380 −0.0188690 0.999822i $$-0.506007\pi$$
−0.0188690 + 0.999822i $$0.506007\pi$$
$$758$$ −16200.0 −0.776267
$$759$$ 0 0
$$760$$ −1600.00 −0.0763659
$$761$$ 1498.00 0.0713567 0.0356784 0.999363i $$-0.488641\pi$$
0.0356784 + 0.999363i $$0.488641\pi$$
$$762$$ 7776.00 0.369678
$$763$$ −7472.00 −0.354528
$$764$$ −5248.00 −0.248516
$$765$$ −11700.0 −0.552960
$$766$$ −12360.0 −0.583009
$$767$$ 0 0
$$768$$ 768.000 0.0360844
$$769$$ −14738.0 −0.691113 −0.345556 0.938398i $$-0.612310\pi$$
−0.345556 + 0.938398i $$0.612310\pi$$
$$770$$ 6400.00 0.299532
$$771$$ 20718.0 0.967757
$$772$$ 1400.00 0.0652683
$$773$$ 3822.00 0.177837 0.0889184 0.996039i $$-0.471659\pi$$
0.0889184 + 0.996039i $$0.471659\pi$$
$$774$$ 1368.00 0.0635294
$$775$$ −4600.00 −0.213209
$$776$$ 4912.00 0.227230
$$777$$ 1776.00 0.0819995
$$778$$ −15044.0 −0.693256
$$779$$ 7240.00 0.332991
$$780$$ 0 0
$$781$$ −29920.0 −1.37083
$$782$$ 0 0
$$783$$ −486.000 −0.0221816
$$784$$ −4464.00 −0.203353
$$785$$ 24100.0 1.09575
$$786$$ −5352.00 −0.242875
$$787$$ 11900.0 0.538995 0.269498 0.963001i $$-0.413142\pi$$
0.269498 + 0.963001i $$0.413142\pi$$
$$788$$ 1368.00 0.0618439
$$789$$ −20544.0 −0.926978
$$790$$ 19520.0 0.879102
$$791$$ 15312.0 0.688283
$$792$$ −2880.00 −0.129213
$$793$$ 0 0
$$794$$ −12156.0 −0.543325
$$795$$ −11460.0 −0.511251
$$796$$ −13472.0 −0.599877
$$797$$ −21274.0 −0.945500 −0.472750 0.881197i $$-0.656739\pi$$
−0.472750 + 0.881197i $$0.656739\pi$$
$$798$$ 960.000 0.0425860
$$799$$ 58760.0 2.60173
$$800$$ −800.000 −0.0353553
$$801$$ 3474.00 0.153243
$$802$$ −3660.00 −0.161146
$$803$$ 42320.0 1.85983
$$804$$ 8400.00 0.368464
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −18102.0 −0.789617
$$808$$ 4144.00 0.180427
$$809$$ −27566.0 −1.19798 −0.598992 0.800755i $$-0.704432\pi$$
−0.598992 + 0.800755i $$0.704432\pi$$
$$810$$ −1620.00 −0.0702728
$$811$$ 11244.0 0.486844 0.243422 0.969921i $$-0.421730\pi$$
0.243422 + 0.969921i $$0.421730\pi$$
$$812$$ −576.000 −0.0248936
$$813$$ −14496.0 −0.625334
$$814$$ −5920.00 −0.254909
$$815$$ 32120.0 1.38051
$$816$$ 6240.00 0.267701
$$817$$ 1520.00 0.0650894
$$818$$ −24868.0 −1.06295
$$819$$ 0 0
$$820$$ −14480.0 −0.616663
$$821$$ −13554.0 −0.576173 −0.288086 0.957604i $$-0.593019\pi$$
−0.288086 + 0.957604i $$0.593019\pi$$
$$822$$ −13956.0 −0.592179
$$823$$ 14384.0 0.609228 0.304614 0.952476i $$-0.401473\pi$$
0.304614 + 0.952476i $$0.401473\pi$$
$$824$$ 896.000 0.0378806
$$825$$ 3000.00 0.126602
$$826$$ −7424.00 −0.312729
$$827$$ 2488.00 0.104615 0.0523073 0.998631i $$-0.483342\pi$$
0.0523073 + 0.998631i $$0.483342\pi$$
$$828$$ 0 0
$$829$$ −20858.0 −0.873858 −0.436929 0.899496i $$-0.643934\pi$$
−0.436929 + 0.899496i $$0.643934\pi$$
$$830$$ −20160.0 −0.843089
$$831$$ −12246.0 −0.511202
$$832$$ 0 0
$$833$$ −36270.0 −1.50862
$$834$$ 11592.0 0.481293
$$835$$ 16680.0 0.691300
$$836$$ −3200.00 −0.132386
$$837$$ 4968.00 0.205160
$$838$$ −28376.0 −1.16973
$$839$$ −23116.0 −0.951195 −0.475598 0.879663i $$-0.657768\pi$$
−0.475598 + 0.879663i $$0.657768\pi$$
$$840$$ −1920.00 −0.0788646
$$841$$ −24065.0 −0.986715
$$842$$ −17276.0 −0.707091
$$843$$ −10050.0 −0.410605
$$844$$ −8016.00 −0.326922
$$845$$ 0 0
$$846$$ 8136.00 0.330640
$$847$$ 2152.00 0.0873006
$$848$$ 6112.00 0.247508
$$849$$ 23388.0 0.945435
$$850$$ −6500.00 −0.262292
$$851$$ 0 0
$$852$$ 8976.00 0.360930
$$853$$ −934.000 −0.0374907 −0.0187453 0.999824i $$-0.505967\pi$$
−0.0187453 + 0.999824i $$0.505967\pi$$
$$854$$ 5728.00 0.229518
$$855$$ −1800.00 −0.0719985
$$856$$ −2976.00 −0.118829
$$857$$ 12642.0 0.503900 0.251950 0.967740i $$-0.418928\pi$$
0.251950 + 0.967740i $$0.418928\pi$$
$$858$$ 0 0
$$859$$ −22796.0 −0.905459 −0.452730 0.891648i $$-0.649550\pi$$
−0.452730 + 0.891648i $$0.649550\pi$$
$$860$$ −3040.00 −0.120539
$$861$$ 8688.00 0.343886
$$862$$ −8584.00 −0.339179
$$863$$ 76.0000 0.00299776 0.00149888 0.999999i $$-0.499523\pi$$
0.00149888 + 0.999999i $$0.499523\pi$$
$$864$$ 864.000 0.0340207
$$865$$ −35980.0 −1.41429
$$866$$ −11964.0 −0.469461
$$867$$ 35961.0 1.40865
$$868$$ 5888.00 0.230244
$$869$$ 39040.0 1.52398
$$870$$ 1080.00 0.0420867
$$871$$ 0 0
$$872$$ −7472.00 −0.290176
$$873$$ 5526.00 0.214235
$$874$$ 0 0
$$875$$ 12000.0 0.463627
$$876$$ −12696.0 −0.489678
$$877$$ 46130.0 1.77617 0.888084 0.459681i $$-0.152036\pi$$
0.888084 + 0.459681i $$0.152036\pi$$
$$878$$ 512.000 0.0196801
$$879$$ −11766.0 −0.451487
$$880$$ 6400.00 0.245164
$$881$$ 6682.00 0.255530 0.127765 0.991804i $$-0.459220\pi$$
0.127765 + 0.991804i $$0.459220\pi$$
$$882$$ −5022.00 −0.191723
$$883$$ 47404.0 1.80665 0.903325 0.428957i $$-0.141119\pi$$
0.903325 + 0.428957i $$0.141119\pi$$
$$884$$ 0 0
$$885$$ 13920.0 0.528718
$$886$$ 25112.0 0.952206
$$887$$ 33672.0 1.27463 0.637314 0.770604i $$-0.280045\pi$$
0.637314 + 0.770604i $$0.280045\pi$$
$$888$$ 1776.00 0.0671156
$$889$$ 10368.0 0.391149
$$890$$ −7720.00 −0.290758
$$891$$ −3240.00 −0.121823
$$892$$ 22432.0 0.842017
$$893$$ 9040.00 0.338759
$$894$$ −5292.00 −0.197976
$$895$$ −10680.0 −0.398875
$$896$$ 1024.00 0.0381802
$$897$$ 0 0
$$898$$ −11148.0 −0.414269
$$899$$ −3312.00 −0.122871
$$900$$ −900.000 −0.0333333
$$901$$ 49660.0 1.83620
$$902$$ −28960.0 −1.06903
$$903$$ 1824.00 0.0672192
$$904$$ 15312.0 0.563351
$$905$$ 47860.0 1.75792
$$906$$ 10656.0 0.390753
$$907$$ −14540.0 −0.532296 −0.266148 0.963932i $$-0.585751\pi$$
−0.266148 + 0.963932i $$0.585751\pi$$
$$908$$ 7712.00 0.281863
$$909$$ 4662.00 0.170109
$$910$$ 0 0
$$911$$ −7840.00 −0.285127 −0.142564 0.989786i $$-0.545535\pi$$
−0.142564 + 0.989786i $$0.545535\pi$$
$$912$$ 960.000 0.0348561
$$913$$ −40320.0 −1.46155
$$914$$ −2532.00 −0.0916314
$$915$$ −10740.0 −0.388037
$$916$$ 15752.0 0.568189
$$917$$ −7136.00 −0.256981
$$918$$ 7020.00 0.252391
$$919$$ 47720.0 1.71288 0.856440 0.516246i $$-0.172671\pi$$
0.856440 + 0.516246i $$0.172671\pi$$
$$920$$ 0 0
$$921$$ −17868.0 −0.639273
$$922$$ −15108.0 −0.539648
$$923$$ 0 0
$$924$$ −3840.00 −0.136717
$$925$$ −1850.00 −0.0657596
$$926$$ 13504.0 0.479232
$$927$$ 1008.00 0.0357142
$$928$$ −576.000 −0.0203751
$$929$$ −7502.00 −0.264944 −0.132472 0.991187i $$-0.542291\pi$$
−0.132472 + 0.991187i $$0.542291\pi$$
$$930$$ −11040.0 −0.389264
$$931$$ −5580.00 −0.196431
$$932$$ 10248.0 0.360176
$$933$$ 7056.00 0.247592
$$934$$ 15848.0 0.555206
$$935$$ 52000.0 1.81880
$$936$$ 0 0
$$937$$ 22058.0 0.769054 0.384527 0.923114i $$-0.374365\pi$$
0.384527 + 0.923114i $$0.374365\pi$$
$$938$$ 11200.0 0.389865
$$939$$ 25326.0 0.880173
$$940$$ −18080.0 −0.627345
$$941$$ −23338.0 −0.808498 −0.404249 0.914649i $$-0.632467\pi$$
−0.404249 + 0.914649i $$0.632467\pi$$
$$942$$ −14460.0 −0.500140
$$943$$ 0 0
$$944$$ −7424.00 −0.255965
$$945$$ −2160.00 −0.0743543
$$946$$ −6080.00 −0.208962
$$947$$ 30488.0 1.04617 0.523087 0.852279i $$-0.324780\pi$$
0.523087 + 0.852279i $$0.324780\pi$$
$$948$$ −11712.0 −0.401253
$$949$$ 0 0
$$950$$ −1000.00 −0.0341519
$$951$$ 16650.0 0.567732
$$952$$ 8320.00 0.283249
$$953$$ 9522.00 0.323660 0.161830 0.986819i $$-0.448260\pi$$
0.161830 + 0.986819i $$0.448260\pi$$
$$954$$ 6876.00 0.233353
$$955$$ 13120.0 0.444558
$$956$$ −28656.0 −0.969457
$$957$$ 2160.00 0.0729602
$$958$$ 22168.0 0.747615
$$959$$ −18608.0 −0.626573
$$960$$ −1920.00 −0.0645497
$$961$$ 4065.00 0.136451
$$962$$ 0 0
$$963$$ −3348.00 −0.112033
$$964$$ 24728.0 0.826178
$$965$$ −3500.00 −0.116755
$$966$$ 0 0
$$967$$ 7616.00 0.253272 0.126636 0.991949i $$-0.459582\pi$$
0.126636 + 0.991949i $$0.459582\pi$$
$$968$$ 2152.00 0.0714544
$$969$$ 7800.00 0.258588
$$970$$ −12280.0 −0.406481
$$971$$ 51316.0 1.69599 0.847996 0.530002i $$-0.177809\pi$$
0.847996 + 0.530002i $$0.177809\pi$$
$$972$$ 972.000 0.0320750
$$973$$ 15456.0 0.509246
$$974$$ −8864.00 −0.291603
$$975$$ 0 0
$$976$$ 5728.00 0.187857
$$977$$ 48666.0 1.59362 0.796808 0.604232i $$-0.206520\pi$$
0.796808 + 0.604232i $$0.206520\pi$$
$$978$$ −19272.0 −0.630113
$$979$$ −15440.0 −0.504050
$$980$$ 11160.0 0.363768
$$981$$ −8406.00 −0.273581
$$982$$ −2280.00 −0.0740914
$$983$$ −17388.0 −0.564182 −0.282091 0.959388i $$-0.591028\pi$$
−0.282091 + 0.959388i $$0.591028\pi$$
$$984$$ 8688.00 0.281467
$$985$$ −3420.00 −0.110630
$$986$$ −4680.00 −0.151158
$$987$$ 10848.0 0.349844
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 7200.00 0.231142
$$991$$ 11496.0 0.368499 0.184249 0.982880i $$-0.441015\pi$$
0.184249 + 0.982880i $$0.441015\pi$$
$$992$$ 5888.00 0.188452
$$993$$ −420.000 −0.0134223
$$994$$ 11968.0 0.381893
$$995$$ 33680.0 1.07309
$$996$$ 12096.0 0.384816
$$997$$ 48862.0 1.55213 0.776066 0.630652i $$-0.217212\pi$$
0.776066 + 0.630652i $$0.217212\pi$$
$$998$$ −3528.00 −0.111901
$$999$$ 1998.00 0.0632772
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 1014.4.a.j.1.1 1
13.5 odd 4 1014.4.b.h.337.1 2
13.8 odd 4 1014.4.b.h.337.2 2
13.12 even 2 78.4.a.c.1.1 1
39.38 odd 2 234.4.a.h.1.1 1
52.51 odd 2 624.4.a.d.1.1 1
65.64 even 2 1950.4.a.l.1.1 1
104.51 odd 2 2496.4.a.j.1.1 1
104.77 even 2 2496.4.a.a.1.1 1
156.155 even 2 1872.4.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.c.1.1 1 13.12 even 2
234.4.a.h.1.1 1 39.38 odd 2
624.4.a.d.1.1 1 52.51 odd 2
1014.4.a.j.1.1 1 1.1 even 1 trivial
1014.4.b.h.337.1 2 13.5 odd 4
1014.4.b.h.337.2 2 13.8 odd 4
1872.4.a.d.1.1 1 156.155 even 2
1950.4.a.l.1.1 1 65.64 even 2
2496.4.a.a.1.1 1 104.77 even 2
2496.4.a.j.1.1 1 104.51 odd 2