# Properties

 Label 338.4.a.a.1.1 Level $338$ Weight $4$ Character 338.1 Self dual yes Analytic conductor $19.943$ 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] = [338,4,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$19.9426455819$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 338.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} +6.00000 q^{6} -5.00000 q^{7} -8.00000 q^{8} -18.0000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} +6.00000 q^{6} -5.00000 q^{7} -8.00000 q^{8} -18.0000 q^{9} -4.00000 q^{10} +13.0000 q^{11} -12.0000 q^{12} +10.0000 q^{14} -6.00000 q^{15} +16.0000 q^{16} +27.0000 q^{17} +36.0000 q^{18} +75.0000 q^{19} +8.00000 q^{20} +15.0000 q^{21} -26.0000 q^{22} -187.000 q^{23} +24.0000 q^{24} -121.000 q^{25} +135.000 q^{27} -20.0000 q^{28} -13.0000 q^{29} +12.0000 q^{30} -104.000 q^{31} -32.0000 q^{32} -39.0000 q^{33} -54.0000 q^{34} -10.0000 q^{35} -72.0000 q^{36} +423.000 q^{37} -150.000 q^{38} -16.0000 q^{40} +195.000 q^{41} -30.0000 q^{42} +199.000 q^{43} +52.0000 q^{44} -36.0000 q^{45} +374.000 q^{46} +388.000 q^{47} -48.0000 q^{48} -318.000 q^{49} +242.000 q^{50} -81.0000 q^{51} +618.000 q^{53} -270.000 q^{54} +26.0000 q^{55} +40.0000 q^{56} -225.000 q^{57} +26.0000 q^{58} +491.000 q^{59} -24.0000 q^{60} +175.000 q^{61} +208.000 q^{62} +90.0000 q^{63} +64.0000 q^{64} +78.0000 q^{66} +817.000 q^{67} +108.000 q^{68} +561.000 q^{69} +20.0000 q^{70} +79.0000 q^{71} +144.000 q^{72} +230.000 q^{73} -846.000 q^{74} +363.000 q^{75} +300.000 q^{76} -65.0000 q^{77} +764.000 q^{79} +32.0000 q^{80} +81.0000 q^{81} -390.000 q^{82} -732.000 q^{83} +60.0000 q^{84} +54.0000 q^{85} -398.000 q^{86} +39.0000 q^{87} -104.000 q^{88} -1041.00 q^{89} +72.0000 q^{90} -748.000 q^{92} +312.000 q^{93} -776.000 q^{94} +150.000 q^{95} +96.0000 q^{96} -97.0000 q^{97} +636.000 q^{98} -234.000 q^{99} +O(q^{100})$$

## 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 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 2.00000 0.178885 0.0894427 0.995992i $$-0.471491\pi$$
0.0894427 + 0.995992i $$0.471491\pi$$
$$6$$ 6.00000 0.408248
$$7$$ −5.00000 −0.269975 −0.134987 0.990847i $$-0.543099\pi$$
−0.134987 + 0.990847i $$0.543099\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ −18.0000 −0.666667
$$10$$ −4.00000 −0.126491
$$11$$ 13.0000 0.356332 0.178166 0.984000i $$-0.442984\pi$$
0.178166 + 0.984000i $$0.442984\pi$$
$$12$$ −12.0000 −0.288675
$$13$$ 0 0
$$14$$ 10.0000 0.190901
$$15$$ −6.00000 −0.103280
$$16$$ 16.0000 0.250000
$$17$$ 27.0000 0.385204 0.192602 0.981277i $$-0.438307\pi$$
0.192602 + 0.981277i $$0.438307\pi$$
$$18$$ 36.0000 0.471405
$$19$$ 75.0000 0.905588 0.452794 0.891615i $$-0.350427\pi$$
0.452794 + 0.891615i $$0.350427\pi$$
$$20$$ 8.00000 0.0894427
$$21$$ 15.0000 0.155870
$$22$$ −26.0000 −0.251964
$$23$$ −187.000 −1.69531 −0.847656 0.530546i $$-0.821987\pi$$
−0.847656 + 0.530546i $$0.821987\pi$$
$$24$$ 24.0000 0.204124
$$25$$ −121.000 −0.968000
$$26$$ 0 0
$$27$$ 135.000 0.962250
$$28$$ −20.0000 −0.134987
$$29$$ −13.0000 −0.0832427 −0.0416214 0.999133i $$-0.513252\pi$$
−0.0416214 + 0.999133i $$0.513252\pi$$
$$30$$ 12.0000 0.0730297
$$31$$ −104.000 −0.602547 −0.301273 0.953538i $$-0.597412\pi$$
−0.301273 + 0.953538i $$0.597412\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ −39.0000 −0.205728
$$34$$ −54.0000 −0.272380
$$35$$ −10.0000 −0.0482945
$$36$$ −72.0000 −0.333333
$$37$$ 423.000 1.87948 0.939740 0.341890i $$-0.111067\pi$$
0.939740 + 0.341890i $$0.111067\pi$$
$$38$$ −150.000 −0.640348
$$39$$ 0 0
$$40$$ −16.0000 −0.0632456
$$41$$ 195.000 0.742778 0.371389 0.928477i $$-0.378882\pi$$
0.371389 + 0.928477i $$0.378882\pi$$
$$42$$ −30.0000 −0.110217
$$43$$ 199.000 0.705749 0.352875 0.935671i $$-0.385204\pi$$
0.352875 + 0.935671i $$0.385204\pi$$
$$44$$ 52.0000 0.178166
$$45$$ −36.0000 −0.119257
$$46$$ 374.000 1.19877
$$47$$ 388.000 1.20416 0.602081 0.798435i $$-0.294338\pi$$
0.602081 + 0.798435i $$0.294338\pi$$
$$48$$ −48.0000 −0.144338
$$49$$ −318.000 −0.927114
$$50$$ 242.000 0.684479
$$51$$ −81.0000 −0.222397
$$52$$ 0 0
$$53$$ 618.000 1.60168 0.800838 0.598881i $$-0.204388\pi$$
0.800838 + 0.598881i $$0.204388\pi$$
$$54$$ −270.000 −0.680414
$$55$$ 26.0000 0.0637425
$$56$$ 40.0000 0.0954504
$$57$$ −225.000 −0.522842
$$58$$ 26.0000 0.0588615
$$59$$ 491.000 1.08344 0.541718 0.840560i $$-0.317774\pi$$
0.541718 + 0.840560i $$0.317774\pi$$
$$60$$ −24.0000 −0.0516398
$$61$$ 175.000 0.367319 0.183659 0.982990i $$-0.441206\pi$$
0.183659 + 0.982990i $$0.441206\pi$$
$$62$$ 208.000 0.426065
$$63$$ 90.0000 0.179983
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ 78.0000 0.145472
$$67$$ 817.000 1.48974 0.744869 0.667211i $$-0.232512\pi$$
0.744869 + 0.667211i $$0.232512\pi$$
$$68$$ 108.000 0.192602
$$69$$ 561.000 0.978789
$$70$$ 20.0000 0.0341494
$$71$$ 79.0000 0.132050 0.0660252 0.997818i $$-0.478968\pi$$
0.0660252 + 0.997818i $$0.478968\pi$$
$$72$$ 144.000 0.235702
$$73$$ 230.000 0.368760 0.184380 0.982855i $$-0.440972\pi$$
0.184380 + 0.982855i $$0.440972\pi$$
$$74$$ −846.000 −1.32899
$$75$$ 363.000 0.558875
$$76$$ 300.000 0.452794
$$77$$ −65.0000 −0.0962005
$$78$$ 0 0
$$79$$ 764.000 1.08806 0.544030 0.839066i $$-0.316898\pi$$
0.544030 + 0.839066i $$0.316898\pi$$
$$80$$ 32.0000 0.0447214
$$81$$ 81.0000 0.111111
$$82$$ −390.000 −0.525223
$$83$$ −732.000 −0.968041 −0.484021 0.875057i $$-0.660824\pi$$
−0.484021 + 0.875057i $$0.660824\pi$$
$$84$$ 60.0000 0.0779350
$$85$$ 54.0000 0.0689073
$$86$$ −398.000 −0.499040
$$87$$ 39.0000 0.0480602
$$88$$ −104.000 −0.125982
$$89$$ −1041.00 −1.23984 −0.619920 0.784665i $$-0.712835\pi$$
−0.619920 + 0.784665i $$0.712835\pi$$
$$90$$ 72.0000 0.0843274
$$91$$ 0 0
$$92$$ −748.000 −0.847656
$$93$$ 312.000 0.347881
$$94$$ −776.000 −0.851471
$$95$$ 150.000 0.161997
$$96$$ 96.0000 0.102062
$$97$$ −97.0000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ 636.000 0.655568
$$99$$ −234.000 −0.237554
$$100$$ −484.000 −0.484000
$$101$$ −809.000 −0.797015 −0.398507 0.917165i $$-0.630472\pi$$
−0.398507 + 0.917165i $$0.630472\pi$$
$$102$$ 162.000 0.157259
$$103$$ 1288.00 1.23214 0.616070 0.787691i $$-0.288724\pi$$
0.616070 + 0.787691i $$0.288724\pi$$
$$104$$ 0 0
$$105$$ 30.0000 0.0278829
$$106$$ −1236.00 −1.13256
$$107$$ 1277.00 1.15376 0.576880 0.816829i $$-0.304270\pi$$
0.576880 + 0.816829i $$0.304270\pi$$
$$108$$ 540.000 0.481125
$$109$$ 826.000 0.725839 0.362920 0.931820i $$-0.381780\pi$$
0.362920 + 0.931820i $$0.381780\pi$$
$$110$$ −52.0000 −0.0450728
$$111$$ −1269.00 −1.08512
$$112$$ −80.0000 −0.0674937
$$113$$ 947.000 0.788374 0.394187 0.919030i $$-0.371026\pi$$
0.394187 + 0.919030i $$0.371026\pi$$
$$114$$ 450.000 0.369705
$$115$$ −374.000 −0.303267
$$116$$ −52.0000 −0.0416214
$$117$$ 0 0
$$118$$ −982.000 −0.766105
$$119$$ −135.000 −0.103995
$$120$$ 48.0000 0.0365148
$$121$$ −1162.00 −0.873028
$$122$$ −350.000 −0.259734
$$123$$ −585.000 −0.428843
$$124$$ −416.000 −0.301273
$$125$$ −492.000 −0.352047
$$126$$ −180.000 −0.127267
$$127$$ 1177.00 0.822377 0.411188 0.911550i $$-0.365114\pi$$
0.411188 + 0.911550i $$0.365114\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ −597.000 −0.407464
$$130$$ 0 0
$$131$$ −1420.00 −0.947069 −0.473534 0.880775i $$-0.657022\pi$$
−0.473534 + 0.880775i $$0.657022\pi$$
$$132$$ −156.000 −0.102864
$$133$$ −375.000 −0.244486
$$134$$ −1634.00 −1.05340
$$135$$ 270.000 0.172133
$$136$$ −216.000 −0.136190
$$137$$ −2409.00 −1.50230 −0.751149 0.660133i $$-0.770500\pi$$
−0.751149 + 0.660133i $$0.770500\pi$$
$$138$$ −1122.00 −0.692109
$$139$$ 2827.00 1.72506 0.862529 0.506008i $$-0.168879\pi$$
0.862529 + 0.506008i $$0.168879\pi$$
$$140$$ −40.0000 −0.0241473
$$141$$ −1164.00 −0.695223
$$142$$ −158.000 −0.0933737
$$143$$ 0 0
$$144$$ −288.000 −0.166667
$$145$$ −26.0000 −0.0148909
$$146$$ −460.000 −0.260753
$$147$$ 954.000 0.535269
$$148$$ 1692.00 0.939740
$$149$$ 855.000 0.470096 0.235048 0.971984i $$-0.424475\pi$$
0.235048 + 0.971984i $$0.424475\pi$$
$$150$$ −726.000 −0.395184
$$151$$ 2064.00 1.11236 0.556179 0.831063i $$-0.312267\pi$$
0.556179 + 0.831063i $$0.312267\pi$$
$$152$$ −600.000 −0.320174
$$153$$ −486.000 −0.256802
$$154$$ 130.000 0.0680240
$$155$$ −208.000 −0.107787
$$156$$ 0 0
$$157$$ −1894.00 −0.962788 −0.481394 0.876504i $$-0.659869\pi$$
−0.481394 + 0.876504i $$0.659869\pi$$
$$158$$ −1528.00 −0.769374
$$159$$ −1854.00 −0.924728
$$160$$ −64.0000 −0.0316228
$$161$$ 935.000 0.457691
$$162$$ −162.000 −0.0785674
$$163$$ −985.000 −0.473320 −0.236660 0.971593i $$-0.576053\pi$$
−0.236660 + 0.971593i $$0.576053\pi$$
$$164$$ 780.000 0.371389
$$165$$ −78.0000 −0.0368018
$$166$$ 1464.00 0.684509
$$167$$ −2355.00 −1.09123 −0.545615 0.838036i $$-0.683704\pi$$
−0.545615 + 0.838036i $$0.683704\pi$$
$$168$$ −120.000 −0.0551083
$$169$$ 0 0
$$170$$ −108.000 −0.0487248
$$171$$ −1350.00 −0.603726
$$172$$ 796.000 0.352875
$$173$$ −3889.00 −1.70911 −0.854553 0.519365i $$-0.826169\pi$$
−0.854553 + 0.519365i $$0.826169\pi$$
$$174$$ −78.0000 −0.0339837
$$175$$ 605.000 0.261335
$$176$$ 208.000 0.0890829
$$177$$ −1473.00 −0.625522
$$178$$ 2082.00 0.876699
$$179$$ 2229.00 0.930745 0.465372 0.885115i $$-0.345920\pi$$
0.465372 + 0.885115i $$0.345920\pi$$
$$180$$ −144.000 −0.0596285
$$181$$ −1038.00 −0.426265 −0.213132 0.977023i $$-0.568367\pi$$
−0.213132 + 0.977023i $$0.568367\pi$$
$$182$$ 0 0
$$183$$ −525.000 −0.212072
$$184$$ 1496.00 0.599384
$$185$$ 846.000 0.336212
$$186$$ −624.000 −0.245989
$$187$$ 351.000 0.137260
$$188$$ 1552.00 0.602081
$$189$$ −675.000 −0.259783
$$190$$ −300.000 −0.114549
$$191$$ −2141.00 −0.811085 −0.405543 0.914076i $$-0.632917\pi$$
−0.405543 + 0.914076i $$0.632917\pi$$
$$192$$ −192.000 −0.0721688
$$193$$ 2627.00 0.979770 0.489885 0.871787i $$-0.337039\pi$$
0.489885 + 0.871787i $$0.337039\pi$$
$$194$$ 194.000 0.0717958
$$195$$ 0 0
$$196$$ −1272.00 −0.463557
$$197$$ 1203.00 0.435077 0.217539 0.976052i $$-0.430197\pi$$
0.217539 + 0.976052i $$0.430197\pi$$
$$198$$ 468.000 0.167976
$$199$$ 743.000 0.264673 0.132336 0.991205i $$-0.457752\pi$$
0.132336 + 0.991205i $$0.457752\pi$$
$$200$$ 968.000 0.342240
$$201$$ −2451.00 −0.860101
$$202$$ 1618.00 0.563575
$$203$$ 65.0000 0.0224734
$$204$$ −324.000 −0.111199
$$205$$ 390.000 0.132872
$$206$$ −2576.00 −0.871254
$$207$$ 3366.00 1.13021
$$208$$ 0 0
$$209$$ 975.000 0.322690
$$210$$ −60.0000 −0.0197162
$$211$$ −355.000 −0.115826 −0.0579128 0.998322i $$-0.518445\pi$$
−0.0579128 + 0.998322i $$0.518445\pi$$
$$212$$ 2472.00 0.800838
$$213$$ −237.000 −0.0762393
$$214$$ −2554.00 −0.815831
$$215$$ 398.000 0.126248
$$216$$ −1080.00 −0.340207
$$217$$ 520.000 0.162672
$$218$$ −1652.00 −0.513246
$$219$$ −690.000 −0.212904
$$220$$ 104.000 0.0318713
$$221$$ 0 0
$$222$$ 2538.00 0.767295
$$223$$ −2283.00 −0.685565 −0.342782 0.939415i $$-0.611369\pi$$
−0.342782 + 0.939415i $$0.611369\pi$$
$$224$$ 160.000 0.0477252
$$225$$ 2178.00 0.645333
$$226$$ −1894.00 −0.557465
$$227$$ 2451.00 0.716646 0.358323 0.933598i $$-0.383349\pi$$
0.358323 + 0.933598i $$0.383349\pi$$
$$228$$ −900.000 −0.261421
$$229$$ −1878.00 −0.541929 −0.270964 0.962589i $$-0.587343\pi$$
−0.270964 + 0.962589i $$0.587343\pi$$
$$230$$ 748.000 0.214442
$$231$$ 195.000 0.0555414
$$232$$ 104.000 0.0294308
$$233$$ 1630.00 0.458304 0.229152 0.973391i $$-0.426405\pi$$
0.229152 + 0.973391i $$0.426405\pi$$
$$234$$ 0 0
$$235$$ 776.000 0.215407
$$236$$ 1964.00 0.541718
$$237$$ −2292.00 −0.628192
$$238$$ 270.000 0.0735357
$$239$$ −5544.00 −1.50047 −0.750233 0.661173i $$-0.770059\pi$$
−0.750233 + 0.661173i $$0.770059\pi$$
$$240$$ −96.0000 −0.0258199
$$241$$ 5523.00 1.47621 0.738107 0.674683i $$-0.235720\pi$$
0.738107 + 0.674683i $$0.235720\pi$$
$$242$$ 2324.00 0.617324
$$243$$ −3888.00 −1.02640
$$244$$ 700.000 0.183659
$$245$$ −636.000 −0.165847
$$246$$ 1170.00 0.303238
$$247$$ 0 0
$$248$$ 832.000 0.213032
$$249$$ 2196.00 0.558899
$$250$$ 984.000 0.248934
$$251$$ 2175.00 0.546951 0.273476 0.961879i $$-0.411827\pi$$
0.273476 + 0.961879i $$0.411827\pi$$
$$252$$ 360.000 0.0899915
$$253$$ −2431.00 −0.604094
$$254$$ −2354.00 −0.581508
$$255$$ −162.000 −0.0397837
$$256$$ 256.000 0.0625000
$$257$$ −5685.00 −1.37985 −0.689923 0.723883i $$-0.742356\pi$$
−0.689923 + 0.723883i $$0.742356\pi$$
$$258$$ 1194.00 0.288121
$$259$$ −2115.00 −0.507412
$$260$$ 0 0
$$261$$ 234.000 0.0554952
$$262$$ 2840.00 0.669679
$$263$$ 6117.00 1.43418 0.717092 0.696979i $$-0.245473\pi$$
0.717092 + 0.696979i $$0.245473\pi$$
$$264$$ 312.000 0.0727359
$$265$$ 1236.00 0.286517
$$266$$ 750.000 0.172878
$$267$$ 3123.00 0.715822
$$268$$ 3268.00 0.744869
$$269$$ −5109.00 −1.15800 −0.578999 0.815329i $$-0.696556\pi$$
−0.578999 + 0.815329i $$0.696556\pi$$
$$270$$ −540.000 −0.121716
$$271$$ 7549.00 1.69214 0.846068 0.533074i $$-0.178963\pi$$
0.846068 + 0.533074i $$0.178963\pi$$
$$272$$ 432.000 0.0963009
$$273$$ 0 0
$$274$$ 4818.00 1.06228
$$275$$ −1573.00 −0.344929
$$276$$ 2244.00 0.489395
$$277$$ −981.000 −0.212789 −0.106395 0.994324i $$-0.533931\pi$$
−0.106395 + 0.994324i $$0.533931\pi$$
$$278$$ −5654.00 −1.21980
$$279$$ 1872.00 0.401698
$$280$$ 80.0000 0.0170747
$$281$$ −2762.00 −0.586360 −0.293180 0.956057i $$-0.594713\pi$$
−0.293180 + 0.956057i $$0.594713\pi$$
$$282$$ 2328.00 0.491597
$$283$$ 3925.00 0.824442 0.412221 0.911084i $$-0.364753\pi$$
0.412221 + 0.911084i $$0.364753\pi$$
$$284$$ 316.000 0.0660252
$$285$$ −450.000 −0.0935288
$$286$$ 0 0
$$287$$ −975.000 −0.200531
$$288$$ 576.000 0.117851
$$289$$ −4184.00 −0.851618
$$290$$ 52.0000 0.0105295
$$291$$ 291.000 0.0586210
$$292$$ 920.000 0.184380
$$293$$ 7711.00 1.53748 0.768740 0.639562i $$-0.220884\pi$$
0.768740 + 0.639562i $$0.220884\pi$$
$$294$$ −1908.00 −0.378493
$$295$$ 982.000 0.193811
$$296$$ −3384.00 −0.664497
$$297$$ 1755.00 0.342880
$$298$$ −1710.00 −0.332408
$$299$$ 0 0
$$300$$ 1452.00 0.279438
$$301$$ −995.000 −0.190534
$$302$$ −4128.00 −0.786555
$$303$$ 2427.00 0.460157
$$304$$ 1200.00 0.226397
$$305$$ 350.000 0.0657080
$$306$$ 972.000 0.181587
$$307$$ 10388.0 1.93119 0.965594 0.260056i $$-0.0837409\pi$$
0.965594 + 0.260056i $$0.0837409\pi$$
$$308$$ −260.000 −0.0481002
$$309$$ −3864.00 −0.711376
$$310$$ 416.000 0.0762168
$$311$$ −7272.00 −1.32591 −0.662954 0.748660i $$-0.730697\pi$$
−0.662954 + 0.748660i $$0.730697\pi$$
$$312$$ 0 0
$$313$$ 7910.00 1.42843 0.714217 0.699925i $$-0.246783\pi$$
0.714217 + 0.699925i $$0.246783\pi$$
$$314$$ 3788.00 0.680794
$$315$$ 180.000 0.0321964
$$316$$ 3056.00 0.544030
$$317$$ 7398.00 1.31077 0.655383 0.755296i $$-0.272507\pi$$
0.655383 + 0.755296i $$0.272507\pi$$
$$318$$ 3708.00 0.653881
$$319$$ −169.000 −0.0296620
$$320$$ 128.000 0.0223607
$$321$$ −3831.00 −0.666123
$$322$$ −1870.00 −0.323637
$$323$$ 2025.00 0.348836
$$324$$ 324.000 0.0555556
$$325$$ 0 0
$$326$$ 1970.00 0.334688
$$327$$ −2478.00 −0.419063
$$328$$ −1560.00 −0.262612
$$329$$ −1940.00 −0.325093
$$330$$ 156.000 0.0260228
$$331$$ −2377.00 −0.394718 −0.197359 0.980331i $$-0.563236\pi$$
−0.197359 + 0.980331i $$0.563236\pi$$
$$332$$ −2928.00 −0.484021
$$333$$ −7614.00 −1.25299
$$334$$ 4710.00 0.771616
$$335$$ 1634.00 0.266492
$$336$$ 240.000 0.0389675
$$337$$ −7618.00 −1.23139 −0.615696 0.787984i $$-0.711125\pi$$
−0.615696 + 0.787984i $$0.711125\pi$$
$$338$$ 0 0
$$339$$ −2841.00 −0.455168
$$340$$ 216.000 0.0344537
$$341$$ −1352.00 −0.214706
$$342$$ 2700.00 0.426898
$$343$$ 3305.00 0.520272
$$344$$ −1592.00 −0.249520
$$345$$ 1122.00 0.175091
$$346$$ 7778.00 1.20852
$$347$$ 375.000 0.0580146 0.0290073 0.999579i $$-0.490765\pi$$
0.0290073 + 0.999579i $$0.490765\pi$$
$$348$$ 156.000 0.0240301
$$349$$ 9727.00 1.49190 0.745952 0.666000i $$-0.231995\pi$$
0.745952 + 0.666000i $$0.231995\pi$$
$$350$$ −1210.00 −0.184792
$$351$$ 0 0
$$352$$ −416.000 −0.0629911
$$353$$ 2263.00 0.341211 0.170605 0.985339i $$-0.445428\pi$$
0.170605 + 0.985339i $$0.445428\pi$$
$$354$$ 2946.00 0.442311
$$355$$ 158.000 0.0236219
$$356$$ −4164.00 −0.619920
$$357$$ 405.000 0.0600417
$$358$$ −4458.00 −0.658136
$$359$$ −4488.00 −0.659798 −0.329899 0.944016i $$-0.607015\pi$$
−0.329899 + 0.944016i $$0.607015\pi$$
$$360$$ 288.000 0.0421637
$$361$$ −1234.00 −0.179910
$$362$$ 2076.00 0.301415
$$363$$ 3486.00 0.504043
$$364$$ 0 0
$$365$$ 460.000 0.0659658
$$366$$ 1050.00 0.149957
$$367$$ −1627.00 −0.231413 −0.115707 0.993283i $$-0.536913\pi$$
−0.115707 + 0.993283i $$0.536913\pi$$
$$368$$ −2992.00 −0.423828
$$369$$ −3510.00 −0.495185
$$370$$ −1692.00 −0.237738
$$371$$ −3090.00 −0.432412
$$372$$ 1248.00 0.173940
$$373$$ 2987.00 0.414641 0.207320 0.978273i $$-0.433526\pi$$
0.207320 + 0.978273i $$0.433526\pi$$
$$374$$ −702.000 −0.0970576
$$375$$ 1476.00 0.203254
$$376$$ −3104.00 −0.425736
$$377$$ 0 0
$$378$$ 1350.00 0.183694
$$379$$ −8867.00 −1.20176 −0.600880 0.799339i $$-0.705183\pi$$
−0.600880 + 0.799339i $$0.705183\pi$$
$$380$$ 600.000 0.0809983
$$381$$ −3531.00 −0.474800
$$382$$ 4282.00 0.573524
$$383$$ 11403.0 1.52132 0.760661 0.649150i $$-0.224875\pi$$
0.760661 + 0.649150i $$0.224875\pi$$
$$384$$ 384.000 0.0510310
$$385$$ −130.000 −0.0172089
$$386$$ −5254.00 −0.692802
$$387$$ −3582.00 −0.470499
$$388$$ −388.000 −0.0507673
$$389$$ 2622.00 0.341750 0.170875 0.985293i $$-0.445341\pi$$
0.170875 + 0.985293i $$0.445341\pi$$
$$390$$ 0 0
$$391$$ −5049.00 −0.653041
$$392$$ 2544.00 0.327784
$$393$$ 4260.00 0.546790
$$394$$ −2406.00 −0.307646
$$395$$ 1528.00 0.194638
$$396$$ −936.000 −0.118777
$$397$$ 659.000 0.0833105 0.0416552 0.999132i $$-0.486737\pi$$
0.0416552 + 0.999132i $$0.486737\pi$$
$$398$$ −1486.00 −0.187152
$$399$$ 1125.00 0.141154
$$400$$ −1936.00 −0.242000
$$401$$ −14685.0 −1.82876 −0.914381 0.404854i $$-0.867322\pi$$
−0.914381 + 0.404854i $$0.867322\pi$$
$$402$$ 4902.00 0.608183
$$403$$ 0 0
$$404$$ −3236.00 −0.398507
$$405$$ 162.000 0.0198762
$$406$$ −130.000 −0.0158911
$$407$$ 5499.00 0.669718
$$408$$ 648.000 0.0786294
$$409$$ −7829.00 −0.946502 −0.473251 0.880928i $$-0.656920\pi$$
−0.473251 + 0.880928i $$0.656920\pi$$
$$410$$ −780.000 −0.0939548
$$411$$ 7227.00 0.867352
$$412$$ 5152.00 0.616070
$$413$$ −2455.00 −0.292500
$$414$$ −6732.00 −0.799178
$$415$$ −1464.00 −0.173169
$$416$$ 0 0
$$417$$ −8481.00 −0.995962
$$418$$ −1950.00 −0.228176
$$419$$ −2919.00 −0.340340 −0.170170 0.985415i $$-0.554432\pi$$
−0.170170 + 0.985415i $$0.554432\pi$$
$$420$$ 120.000 0.0139414
$$421$$ −3110.00 −0.360029 −0.180014 0.983664i $$-0.557614\pi$$
−0.180014 + 0.983664i $$0.557614\pi$$
$$422$$ 710.000 0.0819011
$$423$$ −6984.00 −0.802775
$$424$$ −4944.00 −0.566278
$$425$$ −3267.00 −0.372877
$$426$$ 474.000 0.0539093
$$427$$ −875.000 −0.0991668
$$428$$ 5108.00 0.576880
$$429$$ 0 0
$$430$$ −796.000 −0.0892710
$$431$$ −9135.00 −1.02092 −0.510461 0.859901i $$-0.670525\pi$$
−0.510461 + 0.859901i $$0.670525\pi$$
$$432$$ 2160.00 0.240563
$$433$$ −11669.0 −1.29510 −0.647548 0.762025i $$-0.724205\pi$$
−0.647548 + 0.762025i $$0.724205\pi$$
$$434$$ −1040.00 −0.115027
$$435$$ 78.0000 0.00859727
$$436$$ 3304.00 0.362920
$$437$$ −14025.0 −1.53526
$$438$$ 1380.00 0.150546
$$439$$ 13529.0 1.47085 0.735426 0.677605i $$-0.236982\pi$$
0.735426 + 0.677605i $$0.236982\pi$$
$$440$$ −208.000 −0.0225364
$$441$$ 5724.00 0.618076
$$442$$ 0 0
$$443$$ −1932.00 −0.207206 −0.103603 0.994619i $$-0.533037\pi$$
−0.103603 + 0.994619i $$0.533037\pi$$
$$444$$ −5076.00 −0.542559
$$445$$ −2082.00 −0.221789
$$446$$ 4566.00 0.484768
$$447$$ −2565.00 −0.271410
$$448$$ −320.000 −0.0337468
$$449$$ −5357.00 −0.563057 −0.281528 0.959553i $$-0.590841\pi$$
−0.281528 + 0.959553i $$0.590841\pi$$
$$450$$ −4356.00 −0.456320
$$451$$ 2535.00 0.264675
$$452$$ 3788.00 0.394187
$$453$$ −6192.00 −0.642220
$$454$$ −4902.00 −0.506745
$$455$$ 0 0
$$456$$ 1800.00 0.184852
$$457$$ 19399.0 1.98566 0.992830 0.119532i $$-0.0381394\pi$$
0.992830 + 0.119532i $$0.0381394\pi$$
$$458$$ 3756.00 0.383202
$$459$$ 3645.00 0.370662
$$460$$ −1496.00 −0.151633
$$461$$ −15549.0 −1.57091 −0.785455 0.618919i $$-0.787571\pi$$
−0.785455 + 0.618919i $$0.787571\pi$$
$$462$$ −390.000 −0.0392737
$$463$$ 4072.00 0.408730 0.204365 0.978895i $$-0.434487\pi$$
0.204365 + 0.978895i $$0.434487\pi$$
$$464$$ −208.000 −0.0208107
$$465$$ 624.000 0.0622308
$$466$$ −3260.00 −0.324070
$$467$$ 15224.0 1.50853 0.754264 0.656571i $$-0.227994\pi$$
0.754264 + 0.656571i $$0.227994\pi$$
$$468$$ 0 0
$$469$$ −4085.00 −0.402191
$$470$$ −1552.00 −0.152316
$$471$$ 5682.00 0.555866
$$472$$ −3928.00 −0.383053
$$473$$ 2587.00 0.251481
$$474$$ 4584.00 0.444199
$$475$$ −9075.00 −0.876610
$$476$$ −540.000 −0.0519976
$$477$$ −11124.0 −1.06778
$$478$$ 11088.0 1.06099
$$479$$ −10335.0 −0.985842 −0.492921 0.870074i $$-0.664071\pi$$
−0.492921 + 0.870074i $$0.664071\pi$$
$$480$$ 192.000 0.0182574
$$481$$ 0 0
$$482$$ −11046.0 −1.04384
$$483$$ −2805.00 −0.264248
$$484$$ −4648.00 −0.436514
$$485$$ −194.000 −0.0181631
$$486$$ 7776.00 0.725775
$$487$$ 6455.00 0.600624 0.300312 0.953841i $$-0.402909\pi$$
0.300312 + 0.953841i $$0.402909\pi$$
$$488$$ −1400.00 −0.129867
$$489$$ 2955.00 0.273271
$$490$$ 1272.00 0.117272
$$491$$ 7777.00 0.714809 0.357404 0.933950i $$-0.383662\pi$$
0.357404 + 0.933950i $$0.383662\pi$$
$$492$$ −2340.00 −0.214421
$$493$$ −351.000 −0.0320654
$$494$$ 0 0
$$495$$ −468.000 −0.0424950
$$496$$ −1664.00 −0.150637
$$497$$ −395.000 −0.0356502
$$498$$ −4392.00 −0.395201
$$499$$ −3044.00 −0.273082 −0.136541 0.990634i $$-0.543599\pi$$
−0.136541 + 0.990634i $$0.543599\pi$$
$$500$$ −1968.00 −0.176023
$$501$$ 7065.00 0.630022
$$502$$ −4350.00 −0.386753
$$503$$ 11347.0 1.00584 0.502920 0.864333i $$-0.332259\pi$$
0.502920 + 0.864333i $$0.332259\pi$$
$$504$$ −720.000 −0.0636336
$$505$$ −1618.00 −0.142574
$$506$$ 4862.00 0.427159
$$507$$ 0 0
$$508$$ 4708.00 0.411188
$$509$$ 727.000 0.0633079 0.0316539 0.999499i $$-0.489923\pi$$
0.0316539 + 0.999499i $$0.489923\pi$$
$$510$$ 324.000 0.0281313
$$511$$ −1150.00 −0.0995558
$$512$$ −512.000 −0.0441942
$$513$$ 10125.0 0.871403
$$514$$ 11370.0 0.975699
$$515$$ 2576.00 0.220412
$$516$$ −2388.00 −0.203732
$$517$$ 5044.00 0.429081
$$518$$ 4230.00 0.358794
$$519$$ 11667.0 0.986752
$$520$$ 0 0
$$521$$ 9582.00 0.805749 0.402874 0.915255i $$-0.368011\pi$$
0.402874 + 0.915255i $$0.368011\pi$$
$$522$$ −468.000 −0.0392410
$$523$$ −10383.0 −0.868101 −0.434051 0.900889i $$-0.642916\pi$$
−0.434051 + 0.900889i $$0.642916\pi$$
$$524$$ −5680.00 −0.473534
$$525$$ −1815.00 −0.150882
$$526$$ −12234.0 −1.01412
$$527$$ −2808.00 −0.232103
$$528$$ −624.000 −0.0514320
$$529$$ 22802.0 1.87409
$$530$$ −2472.00 −0.202598
$$531$$ −8838.00 −0.722291
$$532$$ −1500.00 −0.122243
$$533$$ 0 0
$$534$$ −6246.00 −0.506163
$$535$$ 2554.00 0.206391
$$536$$ −6536.00 −0.526702
$$537$$ −6687.00 −0.537366
$$538$$ 10218.0 0.818828
$$539$$ −4134.00 −0.330360
$$540$$ 1080.00 0.0860663
$$541$$ 12230.0 0.971920 0.485960 0.873981i $$-0.338470\pi$$
0.485960 + 0.873981i $$0.338470\pi$$
$$542$$ −15098.0 −1.19652
$$543$$ 3114.00 0.246104
$$544$$ −864.000 −0.0680950
$$545$$ 1652.00 0.129842
$$546$$ 0 0
$$547$$ −14636.0 −1.14404 −0.572020 0.820239i $$-0.693840\pi$$
−0.572020 + 0.820239i $$0.693840\pi$$
$$548$$ −9636.00 −0.751149
$$549$$ −3150.00 −0.244879
$$550$$ 3146.00 0.243902
$$551$$ −975.000 −0.0753837
$$552$$ −4488.00 −0.346054
$$553$$ −3820.00 −0.293749
$$554$$ 1962.00 0.150465
$$555$$ −2538.00 −0.194112
$$556$$ 11308.0 0.862529
$$557$$ −765.000 −0.0581941 −0.0290970 0.999577i $$-0.509263\pi$$
−0.0290970 + 0.999577i $$0.509263\pi$$
$$558$$ −3744.00 −0.284043
$$559$$ 0 0
$$560$$ −160.000 −0.0120736
$$561$$ −1053.00 −0.0792472
$$562$$ 5524.00 0.414619
$$563$$ 5915.00 0.442784 0.221392 0.975185i $$-0.428940\pi$$
0.221392 + 0.975185i $$0.428940\pi$$
$$564$$ −4656.00 −0.347612
$$565$$ 1894.00 0.141029
$$566$$ −7850.00 −0.582968
$$567$$ −405.000 −0.0299972
$$568$$ −632.000 −0.0466869
$$569$$ −1217.00 −0.0896648 −0.0448324 0.998995i $$-0.514275\pi$$
−0.0448324 + 0.998995i $$0.514275\pi$$
$$570$$ 900.000 0.0661348
$$571$$ −23436.0 −1.71763 −0.858814 0.512287i $$-0.828798\pi$$
−0.858814 + 0.512287i $$0.828798\pi$$
$$572$$ 0 0
$$573$$ 6423.00 0.468280
$$574$$ 1950.00 0.141797
$$575$$ 22627.0 1.64106
$$576$$ −1152.00 −0.0833333
$$577$$ 7854.00 0.566666 0.283333 0.959022i $$-0.408560\pi$$
0.283333 + 0.959022i $$0.408560\pi$$
$$578$$ 8368.00 0.602185
$$579$$ −7881.00 −0.565670
$$580$$ −104.000 −0.00744546
$$581$$ 3660.00 0.261347
$$582$$ −582.000 −0.0414513
$$583$$ 8034.00 0.570728
$$584$$ −1840.00 −0.130376
$$585$$ 0 0
$$586$$ −15422.0 −1.08716
$$587$$ 17033.0 1.19766 0.598831 0.800876i $$-0.295632\pi$$
0.598831 + 0.800876i $$0.295632\pi$$
$$588$$ 3816.00 0.267635
$$589$$ −7800.00 −0.545659
$$590$$ −1964.00 −0.137045
$$591$$ −3609.00 −0.251192
$$592$$ 6768.00 0.469870
$$593$$ −14506.0 −1.00454 −0.502268 0.864712i $$-0.667501\pi$$
−0.502268 + 0.864712i $$0.667501\pi$$
$$594$$ −3510.00 −0.242453
$$595$$ −270.000 −0.0186032
$$596$$ 3420.00 0.235048
$$597$$ −2229.00 −0.152809
$$598$$ 0 0
$$599$$ 15388.0 1.04964 0.524822 0.851212i $$-0.324132\pi$$
0.524822 + 0.851212i $$0.324132\pi$$
$$600$$ −2904.00 −0.197592
$$601$$ −6077.00 −0.412456 −0.206228 0.978504i $$-0.566119\pi$$
−0.206228 + 0.978504i $$0.566119\pi$$
$$602$$ 1990.00 0.134728
$$603$$ −14706.0 −0.993159
$$604$$ 8256.00 0.556179
$$605$$ −2324.00 −0.156172
$$606$$ −4854.00 −0.325380
$$607$$ 10215.0 0.683054 0.341527 0.939872i $$-0.389056\pi$$
0.341527 + 0.939872i $$0.389056\pi$$
$$608$$ −2400.00 −0.160087
$$609$$ −195.000 −0.0129750
$$610$$ −700.000 −0.0464626
$$611$$ 0 0
$$612$$ −1944.00 −0.128401
$$613$$ −3457.00 −0.227776 −0.113888 0.993494i $$-0.536331\pi$$
−0.113888 + 0.993494i $$0.536331\pi$$
$$614$$ −20776.0 −1.36556
$$615$$ −1170.00 −0.0767137
$$616$$ 520.000 0.0340120
$$617$$ −7169.00 −0.467768 −0.233884 0.972264i $$-0.575144\pi$$
−0.233884 + 0.972264i $$0.575144\pi$$
$$618$$ 7728.00 0.503019
$$619$$ 20212.0 1.31242 0.656211 0.754578i $$-0.272158\pi$$
0.656211 + 0.754578i $$0.272158\pi$$
$$620$$ −832.000 −0.0538934
$$621$$ −25245.0 −1.63132
$$622$$ 14544.0 0.937558
$$623$$ 5205.00 0.334725
$$624$$ 0 0
$$625$$ 14141.0 0.905024
$$626$$ −15820.0 −1.01005
$$627$$ −2925.00 −0.186305
$$628$$ −7576.00 −0.481394
$$629$$ 11421.0 0.723983
$$630$$ −360.000 −0.0227663
$$631$$ −8945.00 −0.564334 −0.282167 0.959365i $$-0.591053\pi$$
−0.282167 + 0.959365i $$0.591053\pi$$
$$632$$ −6112.00 −0.384687
$$633$$ 1065.00 0.0668720
$$634$$ −14796.0 −0.926852
$$635$$ 2354.00 0.147111
$$636$$ −7416.00 −0.462364
$$637$$ 0 0
$$638$$ 338.000 0.0209742
$$639$$ −1422.00 −0.0880336
$$640$$ −256.000 −0.0158114
$$641$$ 28243.0 1.74030 0.870149 0.492788i $$-0.164022\pi$$
0.870149 + 0.492788i $$0.164022\pi$$
$$642$$ 7662.00 0.471020
$$643$$ 5231.00 0.320825 0.160413 0.987050i $$-0.448718\pi$$
0.160413 + 0.987050i $$0.448718\pi$$
$$644$$ 3740.00 0.228846
$$645$$ −1194.00 −0.0728895
$$646$$ −4050.00 −0.246664
$$647$$ −4871.00 −0.295980 −0.147990 0.988989i $$-0.547280\pi$$
−0.147990 + 0.988989i $$0.547280\pi$$
$$648$$ −648.000 −0.0392837
$$649$$ 6383.00 0.386063
$$650$$ 0 0
$$651$$ −1560.00 −0.0939189
$$652$$ −3940.00 −0.236660
$$653$$ 12255.0 0.734418 0.367209 0.930138i $$-0.380313\pi$$
0.367209 + 0.930138i $$0.380313\pi$$
$$654$$ 4956.00 0.296323
$$655$$ −2840.00 −0.169417
$$656$$ 3120.00 0.185694
$$657$$ −4140.00 −0.245840
$$658$$ 3880.00 0.229876
$$659$$ −2145.00 −0.126794 −0.0633971 0.997988i $$-0.520193\pi$$
−0.0633971 + 0.997988i $$0.520193\pi$$
$$660$$ −312.000 −0.0184009
$$661$$ 2111.00 0.124218 0.0621092 0.998069i $$-0.480217\pi$$
0.0621092 + 0.998069i $$0.480217\pi$$
$$662$$ 4754.00 0.279108
$$663$$ 0 0
$$664$$ 5856.00 0.342254
$$665$$ −750.000 −0.0437350
$$666$$ 15228.0 0.885996
$$667$$ 2431.00 0.141122
$$668$$ −9420.00 −0.545615
$$669$$ 6849.00 0.395811
$$670$$ −3268.00 −0.188439
$$671$$ 2275.00 0.130887
$$672$$ −480.000 −0.0275542
$$673$$ −23273.0 −1.33300 −0.666499 0.745506i $$-0.732208\pi$$
−0.666499 + 0.745506i $$0.732208\pi$$
$$674$$ 15236.0 0.870725
$$675$$ −16335.0 −0.931458
$$676$$ 0 0
$$677$$ −5910.00 −0.335509 −0.167755 0.985829i $$-0.553652\pi$$
−0.167755 + 0.985829i $$0.553652\pi$$
$$678$$ 5682.00 0.321852
$$679$$ 485.000 0.0274118
$$680$$ −432.000 −0.0243624
$$681$$ −7353.00 −0.413756
$$682$$ 2704.00 0.151820
$$683$$ 16747.0 0.938223 0.469111 0.883139i $$-0.344574\pi$$
0.469111 + 0.883139i $$0.344574\pi$$
$$684$$ −5400.00 −0.301863
$$685$$ −4818.00 −0.268739
$$686$$ −6610.00 −0.367888
$$687$$ 5634.00 0.312883
$$688$$ 3184.00 0.176437
$$689$$ 0 0
$$690$$ −2244.00 −0.123808
$$691$$ 10309.0 0.567544 0.283772 0.958892i $$-0.408414\pi$$
0.283772 + 0.958892i $$0.408414\pi$$
$$692$$ −15556.0 −0.854553
$$693$$ 1170.00 0.0641337
$$694$$ −750.000 −0.0410225
$$695$$ 5654.00 0.308588
$$696$$ −312.000 −0.0169919
$$697$$ 5265.00 0.286121
$$698$$ −19454.0 −1.05494
$$699$$ −4890.00 −0.264602
$$700$$ 2420.00 0.130668
$$701$$ −24294.0 −1.30895 −0.654473 0.756085i $$-0.727110\pi$$
−0.654473 + 0.756085i $$0.727110\pi$$
$$702$$ 0 0
$$703$$ 31725.0 1.70204
$$704$$ 832.000 0.0445414
$$705$$ −2328.00 −0.124365
$$706$$ −4526.00 −0.241272
$$707$$ 4045.00 0.215174
$$708$$ −5892.00 −0.312761
$$709$$ 12659.0 0.670548 0.335274 0.942121i $$-0.391171\pi$$
0.335274 + 0.942121i $$0.391171\pi$$
$$710$$ −316.000 −0.0167032
$$711$$ −13752.0 −0.725373
$$712$$ 8328.00 0.438350
$$713$$ 19448.0 1.02151
$$714$$ −810.000 −0.0424559
$$715$$ 0 0
$$716$$ 8916.00 0.465372
$$717$$ 16632.0 0.866295
$$718$$ 8976.00 0.466548
$$719$$ 13091.0 0.679015 0.339508 0.940603i $$-0.389740\pi$$
0.339508 + 0.940603i $$0.389740\pi$$
$$720$$ −576.000 −0.0298142
$$721$$ −6440.00 −0.332647
$$722$$ 2468.00 0.127215
$$723$$ −16569.0 −0.852293
$$724$$ −4152.00 −0.213132
$$725$$ 1573.00 0.0805790
$$726$$ −6972.00 −0.356412
$$727$$ 10792.0 0.550555 0.275277 0.961365i $$-0.411230\pi$$
0.275277 + 0.961365i $$0.411230\pi$$
$$728$$ 0 0
$$729$$ 9477.00 0.481481
$$730$$ −920.000 −0.0466448
$$731$$ 5373.00 0.271857
$$732$$ −2100.00 −0.106036
$$733$$ −2698.00 −0.135952 −0.0679761 0.997687i $$-0.521654\pi$$
−0.0679761 + 0.997687i $$0.521654\pi$$
$$734$$ 3254.00 0.163634
$$735$$ 1908.00 0.0957519
$$736$$ 5984.00 0.299692
$$737$$ 10621.0 0.530841
$$738$$ 7020.00 0.350149
$$739$$ 2841.00 0.141418 0.0707090 0.997497i $$-0.477474\pi$$
0.0707090 + 0.997497i $$0.477474\pi$$
$$740$$ 3384.00 0.168106
$$741$$ 0 0
$$742$$ 6180.00 0.305761
$$743$$ −9191.00 −0.453816 −0.226908 0.973916i $$-0.572862\pi$$
−0.226908 + 0.973916i $$0.572862\pi$$
$$744$$ −2496.00 −0.122994
$$745$$ 1710.00 0.0840934
$$746$$ −5974.00 −0.293195
$$747$$ 13176.0 0.645361
$$748$$ 1404.00 0.0686301
$$749$$ −6385.00 −0.311486
$$750$$ −2952.00 −0.143722
$$751$$ −1659.00 −0.0806095 −0.0403048 0.999187i $$-0.512833\pi$$
−0.0403048 + 0.999187i $$0.512833\pi$$
$$752$$ 6208.00 0.301041
$$753$$ −6525.00 −0.315782
$$754$$ 0 0
$$755$$ 4128.00 0.198985
$$756$$ −2700.00 −0.129892
$$757$$ −13929.0 −0.668769 −0.334384 0.942437i $$-0.608528\pi$$
−0.334384 + 0.942437i $$0.608528\pi$$
$$758$$ 17734.0 0.849773
$$759$$ 7293.00 0.348774
$$760$$ −1200.00 −0.0572744
$$761$$ 4587.00 0.218500 0.109250 0.994014i $$-0.465155\pi$$
0.109250 + 0.994014i $$0.465155\pi$$
$$762$$ 7062.00 0.335734
$$763$$ −4130.00 −0.195958
$$764$$ −8564.00 −0.405543
$$765$$ −972.000 −0.0459382
$$766$$ −22806.0 −1.07574
$$767$$ 0 0
$$768$$ −768.000 −0.0360844
$$769$$ 14499.0 0.679905 0.339953 0.940443i $$-0.389589\pi$$
0.339953 + 0.940443i $$0.389589\pi$$
$$770$$ 260.000 0.0121685
$$771$$ 17055.0 0.796655
$$772$$ 10508.0 0.489885
$$773$$ 3059.00 0.142335 0.0711673 0.997464i $$-0.477328\pi$$
0.0711673 + 0.997464i $$0.477328\pi$$
$$774$$ 7164.00 0.332693
$$775$$ 12584.0 0.583265
$$776$$ 776.000 0.0358979
$$777$$ 6345.00 0.292954
$$778$$ −5244.00 −0.241654
$$779$$ 14625.0 0.672651
$$780$$ 0 0
$$781$$ 1027.00 0.0470537
$$782$$ 10098.0 0.461769
$$783$$ −1755.00 −0.0801004
$$784$$ −5088.00 −0.231778
$$785$$ −3788.00 −0.172229
$$786$$ −8520.00 −0.386639
$$787$$ 36407.0 1.64901 0.824504 0.565856i $$-0.191454\pi$$
0.824504 + 0.565856i $$0.191454\pi$$
$$788$$ 4812.00 0.217539
$$789$$ −18351.0 −0.828026
$$790$$ −3056.00 −0.137630
$$791$$ −4735.00 −0.212841
$$792$$ 1872.00 0.0839882
$$793$$ 0 0
$$794$$ −1318.00 −0.0589094
$$795$$ −3708.00 −0.165420
$$796$$ 2972.00 0.132336
$$797$$ −13137.0 −0.583860 −0.291930 0.956440i $$-0.594297\pi$$
−0.291930 + 0.956440i $$0.594297\pi$$
$$798$$ −2250.00 −0.0998109
$$799$$ 10476.0 0.463848
$$800$$ 3872.00 0.171120
$$801$$ 18738.0 0.826560
$$802$$ 29370.0 1.29313
$$803$$ 2990.00 0.131401
$$804$$ −9804.00 −0.430050
$$805$$ 1870.00 0.0818743
$$806$$ 0 0
$$807$$ 15327.0 0.668570
$$808$$ 6472.00 0.281787
$$809$$ 15411.0 0.669743 0.334871 0.942264i $$-0.391307\pi$$
0.334871 + 0.942264i $$0.391307\pi$$
$$810$$ −324.000 −0.0140546
$$811$$ −27664.0 −1.19780 −0.598899 0.800824i $$-0.704395\pi$$
−0.598899 + 0.800824i $$0.704395\pi$$
$$812$$ 260.000 0.0112367
$$813$$ −22647.0 −0.976956
$$814$$ −10998.0 −0.473562
$$815$$ −1970.00 −0.0846700
$$816$$ −1296.00 −0.0555994
$$817$$ 14925.0 0.639118
$$818$$ 15658.0 0.669278
$$819$$ 0 0
$$820$$ 1560.00 0.0664361
$$821$$ −21397.0 −0.909574 −0.454787 0.890600i $$-0.650285\pi$$
−0.454787 + 0.890600i $$0.650285\pi$$
$$822$$ −14454.0 −0.613310
$$823$$ −24249.0 −1.02706 −0.513528 0.858073i $$-0.671662\pi$$
−0.513528 + 0.858073i $$0.671662\pi$$
$$824$$ −10304.0 −0.435627
$$825$$ 4719.00 0.199145
$$826$$ 4910.00 0.206829
$$827$$ 14028.0 0.589844 0.294922 0.955521i $$-0.404706\pi$$
0.294922 + 0.955521i $$0.404706\pi$$
$$828$$ 13464.0 0.565104
$$829$$ 30451.0 1.27576 0.637881 0.770135i $$-0.279811\pi$$
0.637881 + 0.770135i $$0.279811\pi$$
$$830$$ 2928.00 0.122449
$$831$$ 2943.00 0.122854
$$832$$ 0 0
$$833$$ −8586.00 −0.357128
$$834$$ 16962.0 0.704252
$$835$$ −4710.00 −0.195205
$$836$$ 3900.00 0.161345
$$837$$ −14040.0 −0.579801
$$838$$ 5838.00 0.240657
$$839$$ 20591.0 0.847295 0.423647 0.905827i $$-0.360750\pi$$
0.423647 + 0.905827i $$0.360750\pi$$
$$840$$ −240.000 −0.00985808
$$841$$ −24220.0 −0.993071
$$842$$ 6220.00 0.254579
$$843$$ 8286.00 0.338535
$$844$$ −1420.00 −0.0579128
$$845$$ 0 0
$$846$$ 13968.0 0.567647
$$847$$ 5810.00 0.235695
$$848$$ 9888.00 0.400419
$$849$$ −11775.0 −0.475992
$$850$$ 6534.00 0.263664
$$851$$ −79101.0 −3.18631
$$852$$ −948.000 −0.0381197
$$853$$ −5798.00 −0.232731 −0.116366 0.993206i $$-0.537124\pi$$
−0.116366 + 0.993206i $$0.537124\pi$$
$$854$$ 1750.00 0.0701215
$$855$$ −2700.00 −0.107998
$$856$$ −10216.0 −0.407916
$$857$$ 5686.00 0.226640 0.113320 0.993559i $$-0.463852\pi$$
0.113320 + 0.993559i $$0.463852\pi$$
$$858$$ 0 0
$$859$$ −46708.0 −1.85525 −0.927623 0.373518i $$-0.878152\pi$$
−0.927623 + 0.373518i $$0.878152\pi$$
$$860$$ 1592.00 0.0631241
$$861$$ 2925.00 0.115777
$$862$$ 18270.0 0.721901
$$863$$ 25168.0 0.992733 0.496367 0.868113i $$-0.334667\pi$$
0.496367 + 0.868113i $$0.334667\pi$$
$$864$$ −4320.00 −0.170103
$$865$$ −7778.00 −0.305734
$$866$$ 23338.0 0.915771
$$867$$ 12552.0 0.491682
$$868$$ 2080.00 0.0813362
$$869$$ 9932.00 0.387710
$$870$$ −156.000 −0.00607919
$$871$$ 0 0
$$872$$ −6608.00 −0.256623
$$873$$ 1746.00 0.0676897
$$874$$ 28050.0 1.08559
$$875$$ 2460.00 0.0950436
$$876$$ −2760.00 −0.106452
$$877$$ 18663.0 0.718591 0.359296 0.933224i $$-0.383017\pi$$
0.359296 + 0.933224i $$0.383017\pi$$
$$878$$ −27058.0 −1.04005
$$879$$ −23133.0 −0.887664
$$880$$ 416.000 0.0159356
$$881$$ 4971.00 0.190099 0.0950495 0.995473i $$-0.469699\pi$$
0.0950495 + 0.995473i $$0.469699\pi$$
$$882$$ −11448.0 −0.437046
$$883$$ 6892.00 0.262666 0.131333 0.991338i $$-0.458074\pi$$
0.131333 + 0.991338i $$0.458074\pi$$
$$884$$ 0 0
$$885$$ −2946.00 −0.111897
$$886$$ 3864.00 0.146516
$$887$$ −24047.0 −0.910281 −0.455140 0.890420i $$-0.650411\pi$$
−0.455140 + 0.890420i $$0.650411\pi$$
$$888$$ 10152.0 0.383647
$$889$$ −5885.00 −0.222021
$$890$$ 4164.00 0.156829
$$891$$ 1053.00 0.0395924
$$892$$ −9132.00 −0.342782
$$893$$ 29100.0 1.09048
$$894$$ 5130.00 0.191916
$$895$$ 4458.00 0.166497
$$896$$ 640.000 0.0238626
$$897$$ 0 0
$$898$$ 10714.0 0.398141
$$899$$ 1352.00 0.0501576
$$900$$ 8712.00 0.322667
$$901$$ 16686.0 0.616971
$$902$$ −5070.00 −0.187154
$$903$$ 2985.00 0.110005
$$904$$ −7576.00 −0.278732
$$905$$ −2076.00 −0.0762526
$$906$$ 12384.0 0.454118
$$907$$ −12843.0 −0.470171 −0.235085 0.971975i $$-0.575537\pi$$
−0.235085 + 0.971975i $$0.575537\pi$$
$$908$$ 9804.00 0.358323
$$909$$ 14562.0 0.531343
$$910$$ 0 0
$$911$$ −144.000 −0.00523703 −0.00261851 0.999997i $$-0.500833\pi$$
−0.00261851 + 0.999997i $$0.500833\pi$$
$$912$$ −3600.00 −0.130710
$$913$$ −9516.00 −0.344944
$$914$$ −38798.0 −1.40407
$$915$$ −1050.00 −0.0379365
$$916$$ −7512.00 −0.270964
$$917$$ 7100.00 0.255684
$$918$$ −7290.00 −0.262098
$$919$$ −11061.0 −0.397028 −0.198514 0.980098i $$-0.563612\pi$$
−0.198514 + 0.980098i $$0.563612\pi$$
$$920$$ 2992.00 0.107221
$$921$$ −31164.0 −1.11497
$$922$$ 31098.0 1.11080
$$923$$ 0 0
$$924$$ 780.000 0.0277707
$$925$$ −51183.0 −1.81934
$$926$$ −8144.00 −0.289016
$$927$$ −23184.0 −0.821427
$$928$$ 416.000 0.0147154
$$929$$ 26307.0 0.929069 0.464534 0.885555i $$-0.346222\pi$$
0.464534 + 0.885555i $$0.346222\pi$$
$$930$$ −1248.00 −0.0440038
$$931$$ −23850.0 −0.839583
$$932$$ 6520.00 0.229152
$$933$$ 21816.0 0.765513
$$934$$ −30448.0 −1.06669
$$935$$ 702.000 0.0245539
$$936$$ 0 0
$$937$$ −46074.0 −1.60637 −0.803187 0.595727i $$-0.796864\pi$$
−0.803187 + 0.595727i $$0.796864\pi$$
$$938$$ 8170.00 0.284392
$$939$$ −23730.0 −0.824706
$$940$$ 3104.00 0.107704
$$941$$ 36118.0 1.25124 0.625618 0.780130i $$-0.284847\pi$$
0.625618 + 0.780130i $$0.284847\pi$$
$$942$$ −11364.0 −0.393056
$$943$$ −36465.0 −1.25924
$$944$$ 7856.00 0.270859
$$945$$ −1350.00 −0.0464714
$$946$$ −5174.00 −0.177824
$$947$$ −55515.0 −1.90496 −0.952479 0.304604i $$-0.901476\pi$$
−0.952479 + 0.304604i $$0.901476\pi$$
$$948$$ −9168.00 −0.314096
$$949$$ 0 0
$$950$$ 18150.0 0.619857
$$951$$ −22194.0 −0.756772
$$952$$ 1080.00 0.0367679
$$953$$ −5353.00 −0.181952 −0.0909762 0.995853i $$-0.528999\pi$$
−0.0909762 + 0.995853i $$0.528999\pi$$
$$954$$ 22248.0 0.755037
$$955$$ −4282.00 −0.145091
$$956$$ −22176.0 −0.750233
$$957$$ 507.000 0.0171254
$$958$$ 20670.0 0.697095
$$959$$ 12045.0 0.405582
$$960$$ −384.000 −0.0129099
$$961$$ −18975.0 −0.636937
$$962$$ 0 0
$$963$$ −22986.0 −0.769173
$$964$$ 22092.0 0.738107
$$965$$ 5254.00 0.175267
$$966$$ 5610.00 0.186852
$$967$$ −5488.00 −0.182505 −0.0912524 0.995828i $$-0.529087\pi$$
−0.0912524 + 0.995828i $$0.529087\pi$$
$$968$$ 9296.00 0.308662
$$969$$ −6075.00 −0.201401
$$970$$ 388.000 0.0128432
$$971$$ −37353.0 −1.23452 −0.617258 0.786761i $$-0.711756\pi$$
−0.617258 + 0.786761i $$0.711756\pi$$
$$972$$ −15552.0 −0.513200
$$973$$ −14135.0 −0.465722
$$974$$ −12910.0 −0.424705
$$975$$ 0 0
$$976$$ 2800.00 0.0918297
$$977$$ −12729.0 −0.416824 −0.208412 0.978041i $$-0.566829\pi$$
−0.208412 + 0.978041i $$0.566829\pi$$
$$978$$ −5910.00 −0.193232
$$979$$ −13533.0 −0.441794
$$980$$ −2544.00 −0.0829236
$$981$$ −14868.0 −0.483893
$$982$$ −15554.0 −0.505446
$$983$$ −56128.0 −1.82116 −0.910582 0.413327i $$-0.864367\pi$$
−0.910582 + 0.413327i $$0.864367\pi$$
$$984$$ 4680.00 0.151619
$$985$$ 2406.00 0.0778290
$$986$$ 702.000 0.0226737
$$987$$ 5820.00 0.187693
$$988$$ 0 0
$$989$$ −37213.0 −1.19647
$$990$$ 936.000 0.0300485
$$991$$ 47001.0 1.50660 0.753298 0.657680i $$-0.228462\pi$$
0.753298 + 0.657680i $$0.228462\pi$$
$$992$$ 3328.00 0.106516
$$993$$ 7131.00 0.227891
$$994$$ 790.000 0.0252085
$$995$$ 1486.00 0.0473461
$$996$$ 8784.00 0.279449
$$997$$ −24433.0 −0.776129 −0.388065 0.921632i $$-0.626856\pi$$
−0.388065 + 0.921632i $$0.626856\pi$$
$$998$$ 6088.00 0.193098
$$999$$ 57105.0 1.80853
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 338.4.a.a.1.1 1
13.2 odd 12 338.4.e.d.147.1 4
13.3 even 3 26.4.c.a.9.1 yes 2
13.4 even 6 338.4.c.d.315.1 2
13.5 odd 4 338.4.b.a.337.2 2
13.6 odd 12 338.4.e.d.23.2 4
13.7 odd 12 338.4.e.d.23.1 4
13.8 odd 4 338.4.b.a.337.1 2
13.9 even 3 26.4.c.a.3.1 2
13.10 even 6 338.4.c.d.191.1 2
13.11 odd 12 338.4.e.d.147.2 4
13.12 even 2 338.4.a.d.1.1 1
39.29 odd 6 234.4.h.b.217.1 2
39.35 odd 6 234.4.h.b.55.1 2
52.3 odd 6 208.4.i.a.113.1 2
52.35 odd 6 208.4.i.a.81.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.c.a.3.1 2 13.9 even 3
26.4.c.a.9.1 yes 2 13.3 even 3
208.4.i.a.81.1 2 52.35 odd 6
208.4.i.a.113.1 2 52.3 odd 6
234.4.h.b.55.1 2 39.35 odd 6
234.4.h.b.217.1 2 39.29 odd 6
338.4.a.a.1.1 1 1.1 even 1 trivial
338.4.a.d.1.1 1 13.12 even 2
338.4.b.a.337.1 2 13.8 odd 4
338.4.b.a.337.2 2 13.5 odd 4
338.4.c.d.191.1 2 13.10 even 6
338.4.c.d.315.1 2 13.4 even 6
338.4.e.d.23.1 4 13.7 odd 12
338.4.e.d.23.2 4 13.6 odd 12
338.4.e.d.147.1 4 13.2 odd 12
338.4.e.d.147.2 4 13.11 odd 12