# Properties

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

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -11.0000 q^{5} +6.00000 q^{6} -19.0000 q^{7} +8.00000 q^{8} -18.0000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -11.0000 q^{5} +6.00000 q^{6} -19.0000 q^{7} +8.00000 q^{8} -18.0000 q^{9} -22.0000 q^{10} +38.0000 q^{11} +12.0000 q^{12} -38.0000 q^{14} -33.0000 q^{15} +16.0000 q^{16} -51.0000 q^{17} -36.0000 q^{18} -90.0000 q^{19} -44.0000 q^{20} -57.0000 q^{21} +76.0000 q^{22} -52.0000 q^{23} +24.0000 q^{24} -4.00000 q^{25} -135.000 q^{27} -76.0000 q^{28} -190.000 q^{29} -66.0000 q^{30} -292.000 q^{31} +32.0000 q^{32} +114.000 q^{33} -102.000 q^{34} +209.000 q^{35} -72.0000 q^{36} +441.000 q^{37} -180.000 q^{38} -88.0000 q^{40} -312.000 q^{41} -114.000 q^{42} +373.000 q^{43} +152.000 q^{44} +198.000 q^{45} -104.000 q^{46} +41.0000 q^{47} +48.0000 q^{48} +18.0000 q^{49} -8.00000 q^{50} -153.000 q^{51} +468.000 q^{53} -270.000 q^{54} -418.000 q^{55} -152.000 q^{56} -270.000 q^{57} -380.000 q^{58} -530.000 q^{59} -132.000 q^{60} +592.000 q^{61} -584.000 q^{62} +342.000 q^{63} +64.0000 q^{64} +228.000 q^{66} +206.000 q^{67} -204.000 q^{68} -156.000 q^{69} +418.000 q^{70} +863.000 q^{71} -144.000 q^{72} +322.000 q^{73} +882.000 q^{74} -12.0000 q^{75} -360.000 q^{76} -722.000 q^{77} -460.000 q^{79} -176.000 q^{80} +81.0000 q^{81} -624.000 q^{82} -528.000 q^{83} -228.000 q^{84} +561.000 q^{85} +746.000 q^{86} -570.000 q^{87} +304.000 q^{88} -870.000 q^{89} +396.000 q^{90} -208.000 q^{92} -876.000 q^{93} +82.0000 q^{94} +990.000 q^{95} +96.0000 q^{96} +346.000 q^{97} +36.0000 q^{98} -684.000 q^{99} +O(q^{100})$$

## 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 $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 4.00000 0.500000
$$5$$ −11.0000 −0.983870 −0.491935 0.870632i $$-0.663710\pi$$
−0.491935 + 0.870632i $$0.663710\pi$$
$$6$$ 6.00000 0.408248
$$7$$ −19.0000 −1.02590 −0.512952 0.858417i $$-0.671448\pi$$
−0.512952 + 0.858417i $$0.671448\pi$$
$$8$$ 8.00000 0.353553
$$9$$ −18.0000 −0.666667
$$10$$ −22.0000 −0.695701
$$11$$ 38.0000 1.04158 0.520792 0.853683i $$-0.325637\pi$$
0.520792 + 0.853683i $$0.325637\pi$$
$$12$$ 12.0000 0.288675
$$13$$ 0 0
$$14$$ −38.0000 −0.725423
$$15$$ −33.0000 −0.568038
$$16$$ 16.0000 0.250000
$$17$$ −51.0000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ −36.0000 −0.471405
$$19$$ −90.0000 −1.08671 −0.543353 0.839504i $$-0.682845\pi$$
−0.543353 + 0.839504i $$0.682845\pi$$
$$20$$ −44.0000 −0.491935
$$21$$ −57.0000 −0.592306
$$22$$ 76.0000 0.736512
$$23$$ −52.0000 −0.471424 −0.235712 0.971823i $$-0.575742\pi$$
−0.235712 + 0.971823i $$0.575742\pi$$
$$24$$ 24.0000 0.204124
$$25$$ −4.00000 −0.0320000
$$26$$ 0 0
$$27$$ −135.000 −0.962250
$$28$$ −76.0000 −0.512952
$$29$$ −190.000 −1.21662 −0.608312 0.793698i $$-0.708153\pi$$
−0.608312 + 0.793698i $$0.708153\pi$$
$$30$$ −66.0000 −0.401663
$$31$$ −292.000 −1.69177 −0.845883 0.533368i $$-0.820926\pi$$
−0.845883 + 0.533368i $$0.820926\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 114.000 0.601359
$$34$$ −102.000 −0.514496
$$35$$ 209.000 1.00936
$$36$$ −72.0000 −0.333333
$$37$$ 441.000 1.95946 0.979729 0.200327i $$-0.0642004\pi$$
0.979729 + 0.200327i $$0.0642004\pi$$
$$38$$ −180.000 −0.768417
$$39$$ 0 0
$$40$$ −88.0000 −0.347851
$$41$$ −312.000 −1.18844 −0.594222 0.804301i $$-0.702540\pi$$
−0.594222 + 0.804301i $$0.702540\pi$$
$$42$$ −114.000 −0.418823
$$43$$ 373.000 1.32284 0.661418 0.750017i $$-0.269955\pi$$
0.661418 + 0.750017i $$0.269955\pi$$
$$44$$ 152.000 0.520792
$$45$$ 198.000 0.655913
$$46$$ −104.000 −0.333347
$$47$$ 41.0000 0.127244 0.0636220 0.997974i $$-0.479735\pi$$
0.0636220 + 0.997974i $$0.479735\pi$$
$$48$$ 48.0000 0.144338
$$49$$ 18.0000 0.0524781
$$50$$ −8.00000 −0.0226274
$$51$$ −153.000 −0.420084
$$52$$ 0 0
$$53$$ 468.000 1.21292 0.606460 0.795114i $$-0.292589\pi$$
0.606460 + 0.795114i $$0.292589\pi$$
$$54$$ −270.000 −0.680414
$$55$$ −418.000 −1.02478
$$56$$ −152.000 −0.362712
$$57$$ −270.000 −0.627410
$$58$$ −380.000 −0.860284
$$59$$ −530.000 −1.16949 −0.584747 0.811216i $$-0.698806\pi$$
−0.584747 + 0.811216i $$0.698806\pi$$
$$60$$ −132.000 −0.284019
$$61$$ 592.000 1.24259 0.621294 0.783578i $$-0.286607\pi$$
0.621294 + 0.783578i $$0.286607\pi$$
$$62$$ −584.000 −1.19626
$$63$$ 342.000 0.683936
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ 228.000 0.425225
$$67$$ 206.000 0.375625 0.187813 0.982205i $$-0.439860\pi$$
0.187813 + 0.982205i $$0.439860\pi$$
$$68$$ −204.000 −0.363803
$$69$$ −156.000 −0.272177
$$70$$ 418.000 0.713722
$$71$$ 863.000 1.44252 0.721262 0.692662i $$-0.243562\pi$$
0.721262 + 0.692662i $$0.243562\pi$$
$$72$$ −144.000 −0.235702
$$73$$ 322.000 0.516264 0.258132 0.966110i $$-0.416893\pi$$
0.258132 + 0.966110i $$0.416893\pi$$
$$74$$ 882.000 1.38555
$$75$$ −12.0000 −0.0184752
$$76$$ −360.000 −0.543353
$$77$$ −722.000 −1.06857
$$78$$ 0 0
$$79$$ −460.000 −0.655114 −0.327557 0.944831i $$-0.606225\pi$$
−0.327557 + 0.944831i $$0.606225\pi$$
$$80$$ −176.000 −0.245967
$$81$$ 81.0000 0.111111
$$82$$ −624.000 −0.840357
$$83$$ −528.000 −0.698259 −0.349130 0.937074i $$-0.613523\pi$$
−0.349130 + 0.937074i $$0.613523\pi$$
$$84$$ −228.000 −0.296153
$$85$$ 561.000 0.715871
$$86$$ 746.000 0.935387
$$87$$ −570.000 −0.702419
$$88$$ 304.000 0.368256
$$89$$ −870.000 −1.03618 −0.518089 0.855327i $$-0.673356\pi$$
−0.518089 + 0.855327i $$0.673356\pi$$
$$90$$ 396.000 0.463801
$$91$$ 0 0
$$92$$ −208.000 −0.235712
$$93$$ −876.000 −0.976742
$$94$$ 82.0000 0.0899750
$$95$$ 990.000 1.06918
$$96$$ 96.0000 0.102062
$$97$$ 346.000 0.362175 0.181088 0.983467i $$-0.442038\pi$$
0.181088 + 0.983467i $$0.442038\pi$$
$$98$$ 36.0000 0.0371076
$$99$$ −684.000 −0.694390
$$100$$ −16.0000 −0.0160000
$$101$$ 1492.00 1.46990 0.734948 0.678123i $$-0.237206\pi$$
0.734948 + 0.678123i $$0.237206\pi$$
$$102$$ −306.000 −0.297044
$$103$$ −152.000 −0.145408 −0.0727039 0.997354i $$-0.523163\pi$$
−0.0727039 + 0.997354i $$0.523163\pi$$
$$104$$ 0 0
$$105$$ 627.000 0.582752
$$106$$ 936.000 0.857664
$$107$$ 764.000 0.690268 0.345134 0.938553i $$-0.387833\pi$$
0.345134 + 0.938553i $$0.387833\pi$$
$$108$$ −540.000 −0.481125
$$109$$ −1135.00 −0.997370 −0.498685 0.866783i $$-0.666183\pi$$
−0.498685 + 0.866783i $$0.666183\pi$$
$$110$$ −836.000 −0.724632
$$111$$ 1323.00 1.13129
$$112$$ −304.000 −0.256476
$$113$$ −1822.00 −1.51681 −0.758404 0.651785i $$-0.774021\pi$$
−0.758404 + 0.651785i $$0.774021\pi$$
$$114$$ −540.000 −0.443646
$$115$$ 572.000 0.463820
$$116$$ −760.000 −0.608312
$$117$$ 0 0
$$118$$ −1060.00 −0.826957
$$119$$ 969.000 0.746454
$$120$$ −264.000 −0.200832
$$121$$ 113.000 0.0848986
$$122$$ 1184.00 0.878642
$$123$$ −936.000 −0.686149
$$124$$ −1168.00 −0.845883
$$125$$ 1419.00 1.01535
$$126$$ 684.000 0.483616
$$127$$ −1256.00 −0.877575 −0.438787 0.898591i $$-0.644592\pi$$
−0.438787 + 0.898591i $$0.644592\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 1119.00 0.763740
$$130$$ 0 0
$$131$$ 1097.00 0.731644 0.365822 0.930685i $$-0.380788\pi$$
0.365822 + 0.930685i $$0.380788\pi$$
$$132$$ 456.000 0.300680
$$133$$ 1710.00 1.11486
$$134$$ 412.000 0.265607
$$135$$ 1485.00 0.946729
$$136$$ −408.000 −0.257248
$$137$$ 156.000 0.0972845 0.0486423 0.998816i $$-0.484511\pi$$
0.0486423 + 0.998816i $$0.484511\pi$$
$$138$$ −312.000 −0.192458
$$139$$ −2015.00 −1.22957 −0.614784 0.788695i $$-0.710757\pi$$
−0.614784 + 0.788695i $$0.710757\pi$$
$$140$$ 836.000 0.504678
$$141$$ 123.000 0.0734643
$$142$$ 1726.00 1.02002
$$143$$ 0 0
$$144$$ −288.000 −0.166667
$$145$$ 2090.00 1.19700
$$146$$ 644.000 0.365054
$$147$$ 54.0000 0.0302983
$$148$$ 1764.00 0.979729
$$149$$ 1050.00 0.577311 0.288656 0.957433i $$-0.406792\pi$$
0.288656 + 0.957433i $$0.406792\pi$$
$$150$$ −24.0000 −0.0130639
$$151$$ −1917.00 −1.03313 −0.516567 0.856247i $$-0.672790\pi$$
−0.516567 + 0.856247i $$0.672790\pi$$
$$152$$ −720.000 −0.384209
$$153$$ 918.000 0.485071
$$154$$ −1444.00 −0.755590
$$155$$ 3212.00 1.66448
$$156$$ 0 0
$$157$$ −1546.00 −0.785887 −0.392943 0.919563i $$-0.628543\pi$$
−0.392943 + 0.919563i $$0.628543\pi$$
$$158$$ −920.000 −0.463236
$$159$$ 1404.00 0.700280
$$160$$ −352.000 −0.173925
$$161$$ 988.000 0.483635
$$162$$ 162.000 0.0785674
$$163$$ −668.000 −0.320993 −0.160496 0.987036i $$-0.551309\pi$$
−0.160496 + 0.987036i $$0.551309\pi$$
$$164$$ −1248.00 −0.594222
$$165$$ −1254.00 −0.591659
$$166$$ −1056.00 −0.493744
$$167$$ 936.000 0.433712 0.216856 0.976204i $$-0.430420\pi$$
0.216856 + 0.976204i $$0.430420\pi$$
$$168$$ −456.000 −0.209412
$$169$$ 0 0
$$170$$ 1122.00 0.506197
$$171$$ 1620.00 0.724471
$$172$$ 1492.00 0.661418
$$173$$ 1508.00 0.662723 0.331362 0.943504i $$-0.392492\pi$$
0.331362 + 0.943504i $$0.392492\pi$$
$$174$$ −1140.00 −0.496685
$$175$$ 76.0000 0.0328289
$$176$$ 608.000 0.260396
$$177$$ −1590.00 −0.675207
$$178$$ −1740.00 −0.732688
$$179$$ 1785.00 0.745347 0.372674 0.927962i $$-0.378441\pi$$
0.372674 + 0.927962i $$0.378441\pi$$
$$180$$ 792.000 0.327957
$$181$$ −1008.00 −0.413945 −0.206973 0.978347i $$-0.566361\pi$$
−0.206973 + 0.978347i $$0.566361\pi$$
$$182$$ 0 0
$$183$$ 1776.00 0.717408
$$184$$ −416.000 −0.166674
$$185$$ −4851.00 −1.92785
$$186$$ −1752.00 −0.690661
$$187$$ −1938.00 −0.757864
$$188$$ 164.000 0.0636220
$$189$$ 2565.00 0.987176
$$190$$ 1980.00 0.756023
$$191$$ −2138.00 −0.809949 −0.404974 0.914328i $$-0.632720\pi$$
−0.404974 + 0.914328i $$0.632720\pi$$
$$192$$ 192.000 0.0721688
$$193$$ −4688.00 −1.74844 −0.874222 0.485527i $$-0.838628\pi$$
−0.874222 + 0.485527i $$0.838628\pi$$
$$194$$ 692.000 0.256096
$$195$$ 0 0
$$196$$ 72.0000 0.0262391
$$197$$ 891.000 0.322239 0.161120 0.986935i $$-0.448490\pi$$
0.161120 + 0.986935i $$0.448490\pi$$
$$198$$ −1368.00 −0.491008
$$199$$ −1630.00 −0.580641 −0.290321 0.956929i $$-0.593762\pi$$
−0.290321 + 0.956929i $$0.593762\pi$$
$$200$$ −32.0000 −0.0113137
$$201$$ 618.000 0.216867
$$202$$ 2984.00 1.03937
$$203$$ 3610.00 1.24814
$$204$$ −612.000 −0.210042
$$205$$ 3432.00 1.16927
$$206$$ −304.000 −0.102819
$$207$$ 936.000 0.314283
$$208$$ 0 0
$$209$$ −3420.00 −1.13190
$$210$$ 1254.00 0.412068
$$211$$ 5057.00 1.64994 0.824972 0.565173i $$-0.191191\pi$$
0.824972 + 0.565173i $$0.191191\pi$$
$$212$$ 1872.00 0.606460
$$213$$ 2589.00 0.832842
$$214$$ 1528.00 0.488093
$$215$$ −4103.00 −1.30150
$$216$$ −1080.00 −0.340207
$$217$$ 5548.00 1.73559
$$218$$ −2270.00 −0.705247
$$219$$ 966.000 0.298065
$$220$$ −1672.00 −0.512392
$$221$$ 0 0
$$222$$ 2646.00 0.799945
$$223$$ −2913.00 −0.874748 −0.437374 0.899280i $$-0.644091\pi$$
−0.437374 + 0.899280i $$0.644091\pi$$
$$224$$ −608.000 −0.181356
$$225$$ 72.0000 0.0213333
$$226$$ −3644.00 −1.07255
$$227$$ −3744.00 −1.09470 −0.547352 0.836902i $$-0.684364\pi$$
−0.547352 + 0.836902i $$0.684364\pi$$
$$228$$ −1080.00 −0.313705
$$229$$ 1755.00 0.506435 0.253218 0.967409i $$-0.418511\pi$$
0.253218 + 0.967409i $$0.418511\pi$$
$$230$$ 1144.00 0.327970
$$231$$ −2166.00 −0.616937
$$232$$ −1520.00 −0.430142
$$233$$ −2027.00 −0.569928 −0.284964 0.958538i $$-0.591982\pi$$
−0.284964 + 0.958538i $$0.591982\pi$$
$$234$$ 0 0
$$235$$ −451.000 −0.125191
$$236$$ −2120.00 −0.584747
$$237$$ −1380.00 −0.378231
$$238$$ 1938.00 0.527823
$$239$$ 4605.00 1.24633 0.623165 0.782091i $$-0.285847\pi$$
0.623165 + 0.782091i $$0.285847\pi$$
$$240$$ −528.000 −0.142009
$$241$$ 798.000 0.213293 0.106647 0.994297i $$-0.465989\pi$$
0.106647 + 0.994297i $$0.465989\pi$$
$$242$$ 226.000 0.0600324
$$243$$ 3888.00 1.02640
$$244$$ 2368.00 0.621294
$$245$$ −198.000 −0.0516317
$$246$$ −1872.00 −0.485180
$$247$$ 0 0
$$248$$ −2336.00 −0.598130
$$249$$ −1584.00 −0.403140
$$250$$ 2838.00 0.717964
$$251$$ −2088.00 −0.525073 −0.262537 0.964922i $$-0.584559\pi$$
−0.262537 + 0.964922i $$0.584559\pi$$
$$252$$ 1368.00 0.341968
$$253$$ −1976.00 −0.491028
$$254$$ −2512.00 −0.620539
$$255$$ 1683.00 0.413308
$$256$$ 256.000 0.0625000
$$257$$ −6111.00 −1.48324 −0.741622 0.670818i $$-0.765943\pi$$
−0.741622 + 0.670818i $$0.765943\pi$$
$$258$$ 2238.00 0.540046
$$259$$ −8379.00 −2.01022
$$260$$ 0 0
$$261$$ 3420.00 0.811083
$$262$$ 2194.00 0.517350
$$263$$ −2532.00 −0.593649 −0.296825 0.954932i $$-0.595928\pi$$
−0.296825 + 0.954932i $$0.595928\pi$$
$$264$$ 912.000 0.212613
$$265$$ −5148.00 −1.19336
$$266$$ 3420.00 0.788322
$$267$$ −2610.00 −0.598237
$$268$$ 824.000 0.187813
$$269$$ 2400.00 0.543980 0.271990 0.962300i $$-0.412318\pi$$
0.271990 + 0.962300i $$0.412318\pi$$
$$270$$ 2970.00 0.669439
$$271$$ 4793.00 1.07437 0.537185 0.843465i $$-0.319488\pi$$
0.537185 + 0.843465i $$0.319488\pi$$
$$272$$ −816.000 −0.181902
$$273$$ 0 0
$$274$$ 312.000 0.0687905
$$275$$ −152.000 −0.0333307
$$276$$ −624.000 −0.136088
$$277$$ −5676.00 −1.23118 −0.615592 0.788065i $$-0.711083\pi$$
−0.615592 + 0.788065i $$0.711083\pi$$
$$278$$ −4030.00 −0.869436
$$279$$ 5256.00 1.12784
$$280$$ 1672.00 0.356861
$$281$$ −5542.00 −1.17654 −0.588270 0.808664i $$-0.700191\pi$$
−0.588270 + 0.808664i $$0.700191\pi$$
$$282$$ 246.000 0.0519471
$$283$$ −6332.00 −1.33003 −0.665015 0.746830i $$-0.731575\pi$$
−0.665015 + 0.746830i $$0.731575\pi$$
$$284$$ 3452.00 0.721262
$$285$$ 2970.00 0.617290
$$286$$ 0 0
$$287$$ 5928.00 1.21923
$$288$$ −576.000 −0.117851
$$289$$ −2312.00 −0.470588
$$290$$ 4180.00 0.846407
$$291$$ 1038.00 0.209102
$$292$$ 1288.00 0.258132
$$293$$ 3077.00 0.613516 0.306758 0.951788i $$-0.400756\pi$$
0.306758 + 0.951788i $$0.400756\pi$$
$$294$$ 108.000 0.0214241
$$295$$ 5830.00 1.15063
$$296$$ 3528.00 0.692773
$$297$$ −5130.00 −1.00227
$$298$$ 2100.00 0.408221
$$299$$ 0 0
$$300$$ −48.0000 −0.00923760
$$301$$ −7087.00 −1.35710
$$302$$ −3834.00 −0.730536
$$303$$ 4476.00 0.848645
$$304$$ −1440.00 −0.271677
$$305$$ −6512.00 −1.22254
$$306$$ 1836.00 0.342997
$$307$$ 3286.00 0.610886 0.305443 0.952210i $$-0.401195\pi$$
0.305443 + 0.952210i $$0.401195\pi$$
$$308$$ −2888.00 −0.534283
$$309$$ −456.000 −0.0839512
$$310$$ 6424.00 1.17696
$$311$$ 3462.00 0.631228 0.315614 0.948888i $$-0.397789\pi$$
0.315614 + 0.948888i $$0.397789\pi$$
$$312$$ 0 0
$$313$$ −8737.00 −1.57778 −0.788889 0.614536i $$-0.789343\pi$$
−0.788889 + 0.614536i $$0.789343\pi$$
$$314$$ −3092.00 −0.555706
$$315$$ −3762.00 −0.672904
$$316$$ −1840.00 −0.327557
$$317$$ −6054.00 −1.07264 −0.536319 0.844015i $$-0.680186\pi$$
−0.536319 + 0.844015i $$0.680186\pi$$
$$318$$ 2808.00 0.495172
$$319$$ −7220.00 −1.26722
$$320$$ −704.000 −0.122984
$$321$$ 2292.00 0.398526
$$322$$ 1976.00 0.341982
$$323$$ 4590.00 0.790695
$$324$$ 324.000 0.0555556
$$325$$ 0 0
$$326$$ −1336.00 −0.226976
$$327$$ −3405.00 −0.575832
$$328$$ −2496.00 −0.420178
$$329$$ −779.000 −0.130540
$$330$$ −2508.00 −0.418366
$$331$$ −932.000 −0.154765 −0.0773827 0.997001i $$-0.524656\pi$$
−0.0773827 + 0.997001i $$0.524656\pi$$
$$332$$ −2112.00 −0.349130
$$333$$ −7938.00 −1.30631
$$334$$ 1872.00 0.306680
$$335$$ −2266.00 −0.369567
$$336$$ −912.000 −0.148076
$$337$$ −1921.00 −0.310515 −0.155257 0.987874i $$-0.549621\pi$$
−0.155257 + 0.987874i $$0.549621\pi$$
$$338$$ 0 0
$$339$$ −5466.00 −0.875730
$$340$$ 2244.00 0.357935
$$341$$ −11096.0 −1.76212
$$342$$ 3240.00 0.512278
$$343$$ 6175.00 0.972066
$$344$$ 2984.00 0.467693
$$345$$ 1716.00 0.267786
$$346$$ 3016.00 0.468616
$$347$$ 2289.00 0.354121 0.177060 0.984200i $$-0.443341\pi$$
0.177060 + 0.984200i $$0.443341\pi$$
$$348$$ −2280.00 −0.351209
$$349$$ −1195.00 −0.183286 −0.0916431 0.995792i $$-0.529212\pi$$
−0.0916431 + 0.995792i $$0.529212\pi$$
$$350$$ 152.000 0.0232135
$$351$$ 0 0
$$352$$ 1216.00 0.184128
$$353$$ −7588.00 −1.14410 −0.572052 0.820218i $$-0.693852\pi$$
−0.572052 + 0.820218i $$0.693852\pi$$
$$354$$ −3180.00 −0.477444
$$355$$ −9493.00 −1.41926
$$356$$ −3480.00 −0.518089
$$357$$ 2907.00 0.430966
$$358$$ 3570.00 0.527040
$$359$$ −6240.00 −0.917367 −0.458683 0.888600i $$-0.651679\pi$$
−0.458683 + 0.888600i $$0.651679\pi$$
$$360$$ 1584.00 0.231900
$$361$$ 1241.00 0.180930
$$362$$ −2016.00 −0.292703
$$363$$ 339.000 0.0490162
$$364$$ 0 0
$$365$$ −3542.00 −0.507936
$$366$$ 3552.00 0.507284
$$367$$ 9074.00 1.29062 0.645312 0.763919i $$-0.276727\pi$$
0.645312 + 0.763919i $$0.276727\pi$$
$$368$$ −832.000 −0.117856
$$369$$ 5616.00 0.792296
$$370$$ −9702.00 −1.36320
$$371$$ −8892.00 −1.24434
$$372$$ −3504.00 −0.488371
$$373$$ −7732.00 −1.07332 −0.536659 0.843799i $$-0.680314\pi$$
−0.536659 + 0.843799i $$0.680314\pi$$
$$374$$ −3876.00 −0.535891
$$375$$ 4257.00 0.586215
$$376$$ 328.000 0.0449875
$$377$$ 0 0
$$378$$ 5130.00 0.698039
$$379$$ 9320.00 1.26316 0.631578 0.775312i $$-0.282407\pi$$
0.631578 + 0.775312i $$0.282407\pi$$
$$380$$ 3960.00 0.534589
$$381$$ −3768.00 −0.506668
$$382$$ −4276.00 −0.572720
$$383$$ 927.000 0.123675 0.0618375 0.998086i $$-0.480304\pi$$
0.0618375 + 0.998086i $$0.480304\pi$$
$$384$$ 384.000 0.0510310
$$385$$ 7942.00 1.05133
$$386$$ −9376.00 −1.23634
$$387$$ −6714.00 −0.881891
$$388$$ 1384.00 0.181088
$$389$$ −1710.00 −0.222880 −0.111440 0.993771i $$-0.535546\pi$$
−0.111440 + 0.993771i $$0.535546\pi$$
$$390$$ 0 0
$$391$$ 2652.00 0.343011
$$392$$ 144.000 0.0185538
$$393$$ 3291.00 0.422415
$$394$$ 1782.00 0.227858
$$395$$ 5060.00 0.644547
$$396$$ −2736.00 −0.347195
$$397$$ 4606.00 0.582288 0.291144 0.956679i $$-0.405964\pi$$
0.291144 + 0.956679i $$0.405964\pi$$
$$398$$ −3260.00 −0.410575
$$399$$ 5130.00 0.643662
$$400$$ −64.0000 −0.00800000
$$401$$ 7248.00 0.902613 0.451307 0.892369i $$-0.350958\pi$$
0.451307 + 0.892369i $$0.350958\pi$$
$$402$$ 1236.00 0.153348
$$403$$ 0 0
$$404$$ 5968.00 0.734948
$$405$$ −891.000 −0.109319
$$406$$ 7220.00 0.882568
$$407$$ 16758.0 2.04094
$$408$$ −1224.00 −0.148522
$$409$$ −10540.0 −1.27425 −0.637126 0.770759i $$-0.719877\pi$$
−0.637126 + 0.770759i $$0.719877\pi$$
$$410$$ 6864.00 0.826802
$$411$$ 468.000 0.0561672
$$412$$ −608.000 −0.0727039
$$413$$ 10070.0 1.19979
$$414$$ 1872.00 0.222231
$$415$$ 5808.00 0.686996
$$416$$ 0 0
$$417$$ −6045.00 −0.709892
$$418$$ −6840.00 −0.800372
$$419$$ 10635.0 1.23999 0.619993 0.784608i $$-0.287136\pi$$
0.619993 + 0.784608i $$0.287136\pi$$
$$420$$ 2508.00 0.291376
$$421$$ −12487.0 −1.44556 −0.722778 0.691080i $$-0.757135\pi$$
−0.722778 + 0.691080i $$0.757135\pi$$
$$422$$ 10114.0 1.16669
$$423$$ −738.000 −0.0848293
$$424$$ 3744.00 0.428832
$$425$$ 204.000 0.0232834
$$426$$ 5178.00 0.588908
$$427$$ −11248.0 −1.27477
$$428$$ 3056.00 0.345134
$$429$$ 0 0
$$430$$ −8206.00 −0.920299
$$431$$ 14613.0 1.63314 0.816570 0.577246i $$-0.195873\pi$$
0.816570 + 0.577246i $$0.195873\pi$$
$$432$$ −2160.00 −0.240563
$$433$$ −6977.00 −0.774349 −0.387175 0.922006i $$-0.626549\pi$$
−0.387175 + 0.922006i $$0.626549\pi$$
$$434$$ 11096.0 1.22725
$$435$$ 6270.00 0.691088
$$436$$ −4540.00 −0.498685
$$437$$ 4680.00 0.512299
$$438$$ 1932.00 0.210764
$$439$$ 1430.00 0.155467 0.0777337 0.996974i $$-0.475232\pi$$
0.0777337 + 0.996974i $$0.475232\pi$$
$$440$$ −3344.00 −0.362316
$$441$$ −324.000 −0.0349854
$$442$$ 0 0
$$443$$ 12423.0 1.33236 0.666179 0.745792i $$-0.267929\pi$$
0.666179 + 0.745792i $$0.267929\pi$$
$$444$$ 5292.00 0.565647
$$445$$ 9570.00 1.01946
$$446$$ −5826.00 −0.618541
$$447$$ 3150.00 0.333311
$$448$$ −1216.00 −0.128238
$$449$$ 9890.00 1.03951 0.519753 0.854317i $$-0.326024\pi$$
0.519753 + 0.854317i $$0.326024\pi$$
$$450$$ 144.000 0.0150849
$$451$$ −11856.0 −1.23787
$$452$$ −7288.00 −0.758404
$$453$$ −5751.00 −0.596480
$$454$$ −7488.00 −0.774073
$$455$$ 0 0
$$456$$ −2160.00 −0.221823
$$457$$ 9926.00 1.01601 0.508007 0.861353i $$-0.330382\pi$$
0.508007 + 0.861353i $$0.330382\pi$$
$$458$$ 3510.00 0.358104
$$459$$ 6885.00 0.700140
$$460$$ 2288.00 0.231910
$$461$$ 2793.00 0.282176 0.141088 0.989997i $$-0.454940\pi$$
0.141088 + 0.989997i $$0.454940\pi$$
$$462$$ −4332.00 −0.436240
$$463$$ 6872.00 0.689782 0.344891 0.938643i $$-0.387916\pi$$
0.344891 + 0.938643i $$0.387916\pi$$
$$464$$ −3040.00 −0.304156
$$465$$ 9636.00 0.960987
$$466$$ −4054.00 −0.403000
$$467$$ −3676.00 −0.364251 −0.182125 0.983275i $$-0.558298\pi$$
−0.182125 + 0.983275i $$0.558298\pi$$
$$468$$ 0 0
$$469$$ −3914.00 −0.385355
$$470$$ −902.000 −0.0885237
$$471$$ −4638.00 −0.453732
$$472$$ −4240.00 −0.413478
$$473$$ 14174.0 1.37785
$$474$$ −2760.00 −0.267449
$$475$$ 360.000 0.0347746
$$476$$ 3876.00 0.373227
$$477$$ −8424.00 −0.808613
$$478$$ 9210.00 0.881288
$$479$$ −13575.0 −1.29490 −0.647451 0.762108i $$-0.724165\pi$$
−0.647451 + 0.762108i $$0.724165\pi$$
$$480$$ −1056.00 −0.100416
$$481$$ 0 0
$$482$$ 1596.00 0.150821
$$483$$ 2964.00 0.279227
$$484$$ 452.000 0.0424493
$$485$$ −3806.00 −0.356333
$$486$$ 7776.00 0.725775
$$487$$ −11864.0 −1.10392 −0.551960 0.833871i $$-0.686120\pi$$
−0.551960 + 0.833871i $$0.686120\pi$$
$$488$$ 4736.00 0.439321
$$489$$ −2004.00 −0.185325
$$490$$ −396.000 −0.0365091
$$491$$ 4837.00 0.444584 0.222292 0.974980i $$-0.428646\pi$$
0.222292 + 0.974980i $$0.428646\pi$$
$$492$$ −3744.00 −0.343074
$$493$$ 9690.00 0.885224
$$494$$ 0 0
$$495$$ 7524.00 0.683189
$$496$$ −4672.00 −0.422942
$$497$$ −16397.0 −1.47989
$$498$$ −3168.00 −0.285063
$$499$$ −9160.00 −0.821759 −0.410880 0.911690i $$-0.634778\pi$$
−0.410880 + 0.911690i $$0.634778\pi$$
$$500$$ 5676.00 0.507677
$$501$$ 2808.00 0.250404
$$502$$ −4176.00 −0.371283
$$503$$ −842.000 −0.0746380 −0.0373190 0.999303i $$-0.511882\pi$$
−0.0373190 + 0.999303i $$0.511882\pi$$
$$504$$ 2736.00 0.241808
$$505$$ −16412.0 −1.44619
$$506$$ −3952.00 −0.347209
$$507$$ 0 0
$$508$$ −5024.00 −0.438787
$$509$$ −250.000 −0.0217702 −0.0108851 0.999941i $$-0.503465\pi$$
−0.0108851 + 0.999941i $$0.503465\pi$$
$$510$$ 3366.00 0.292253
$$511$$ −6118.00 −0.529637
$$512$$ 512.000 0.0441942
$$513$$ 12150.0 1.04568
$$514$$ −12222.0 −1.04881
$$515$$ 1672.00 0.143062
$$516$$ 4476.00 0.381870
$$517$$ 1558.00 0.132535
$$518$$ −16758.0 −1.42144
$$519$$ 4524.00 0.382623
$$520$$ 0 0
$$521$$ 18087.0 1.52093 0.760466 0.649377i $$-0.224970\pi$$
0.760466 + 0.649377i $$0.224970\pi$$
$$522$$ 6840.00 0.573522
$$523$$ 20028.0 1.67450 0.837250 0.546821i $$-0.184162\pi$$
0.837250 + 0.546821i $$0.184162\pi$$
$$524$$ 4388.00 0.365822
$$525$$ 228.000 0.0189538
$$526$$ −5064.00 −0.419774
$$527$$ 14892.0 1.23094
$$528$$ 1824.00 0.150340
$$529$$ −9463.00 −0.777760
$$530$$ −10296.0 −0.843830
$$531$$ 9540.00 0.779662
$$532$$ 6840.00 0.557428
$$533$$ 0 0
$$534$$ −5220.00 −0.423018
$$535$$ −8404.00 −0.679134
$$536$$ 1648.00 0.132804
$$537$$ 5355.00 0.430326
$$538$$ 4800.00 0.384652
$$539$$ 684.000 0.0546604
$$540$$ 5940.00 0.473365
$$541$$ 6763.00 0.537457 0.268728 0.963216i $$-0.413397\pi$$
0.268728 + 0.963216i $$0.413397\pi$$
$$542$$ 9586.00 0.759694
$$543$$ −3024.00 −0.238991
$$544$$ −1632.00 −0.128624
$$545$$ 12485.0 0.981282
$$546$$ 0 0
$$547$$ 2539.00 0.198464 0.0992320 0.995064i $$-0.468361\pi$$
0.0992320 + 0.995064i $$0.468361\pi$$
$$548$$ 624.000 0.0486423
$$549$$ −10656.0 −0.828392
$$550$$ −304.000 −0.0235684
$$551$$ 17100.0 1.32211
$$552$$ −1248.00 −0.0962290
$$553$$ 8740.00 0.672084
$$554$$ −11352.0 −0.870578
$$555$$ −14553.0 −1.11305
$$556$$ −8060.00 −0.614784
$$557$$ 7611.00 0.578974 0.289487 0.957182i $$-0.406515\pi$$
0.289487 + 0.957182i $$0.406515\pi$$
$$558$$ 10512.0 0.797506
$$559$$ 0 0
$$560$$ 3344.00 0.252339
$$561$$ −5814.00 −0.437553
$$562$$ −11084.0 −0.831940
$$563$$ 3653.00 0.273456 0.136728 0.990609i $$-0.456341\pi$$
0.136728 + 0.990609i $$0.456341\pi$$
$$564$$ 492.000 0.0367322
$$565$$ 20042.0 1.49234
$$566$$ −12664.0 −0.940473
$$567$$ −1539.00 −0.113989
$$568$$ 6904.00 0.510010
$$569$$ 23095.0 1.70157 0.850785 0.525515i $$-0.176127\pi$$
0.850785 + 0.525515i $$0.176127\pi$$
$$570$$ 5940.00 0.436490
$$571$$ −21273.0 −1.55910 −0.779551 0.626339i $$-0.784553\pi$$
−0.779551 + 0.626339i $$0.784553\pi$$
$$572$$ 0 0
$$573$$ −6414.00 −0.467624
$$574$$ 11856.0 0.862125
$$575$$ 208.000 0.0150856
$$576$$ −1152.00 −0.0833333
$$577$$ −15114.0 −1.09047 −0.545237 0.838282i $$-0.683561\pi$$
−0.545237 + 0.838282i $$0.683561\pi$$
$$578$$ −4624.00 −0.332756
$$579$$ −14064.0 −1.00946
$$580$$ 8360.00 0.598500
$$581$$ 10032.0 0.716347
$$582$$ 2076.00 0.147857
$$583$$ 17784.0 1.26336
$$584$$ 2576.00 0.182527
$$585$$ 0 0
$$586$$ 6154.00 0.433821
$$587$$ 22156.0 1.55788 0.778940 0.627098i $$-0.215757\pi$$
0.778940 + 0.627098i $$0.215757\pi$$
$$588$$ 216.000 0.0151491
$$589$$ 26280.0 1.83845
$$590$$ 11660.0 0.813618
$$591$$ 2673.00 0.186045
$$592$$ 7056.00 0.489865
$$593$$ −12338.0 −0.854403 −0.427201 0.904156i $$-0.640500\pi$$
−0.427201 + 0.904156i $$0.640500\pi$$
$$594$$ −10260.0 −0.708709
$$595$$ −10659.0 −0.734414
$$596$$ 4200.00 0.288656
$$597$$ −4890.00 −0.335233
$$598$$ 0 0
$$599$$ −2750.00 −0.187583 −0.0937913 0.995592i $$-0.529899\pi$$
−0.0937913 + 0.995592i $$0.529899\pi$$
$$600$$ −96.0000 −0.00653197
$$601$$ 23317.0 1.58256 0.791281 0.611452i $$-0.209414\pi$$
0.791281 + 0.611452i $$0.209414\pi$$
$$602$$ −14174.0 −0.959616
$$603$$ −3708.00 −0.250417
$$604$$ −7668.00 −0.516567
$$605$$ −1243.00 −0.0835292
$$606$$ 8952.00 0.600083
$$607$$ −19686.0 −1.31636 −0.658180 0.752861i $$-0.728673\pi$$
−0.658180 + 0.752861i $$0.728673\pi$$
$$608$$ −2880.00 −0.192104
$$609$$ 10830.0 0.720614
$$610$$ −13024.0 −0.864469
$$611$$ 0 0
$$612$$ 3672.00 0.242536
$$613$$ 1822.00 0.120049 0.0600244 0.998197i $$-0.480882\pi$$
0.0600244 + 0.998197i $$0.480882\pi$$
$$614$$ 6572.00 0.431961
$$615$$ 10296.0 0.675081
$$616$$ −5776.00 −0.377795
$$617$$ −6304.00 −0.411328 −0.205664 0.978623i $$-0.565935\pi$$
−0.205664 + 0.978623i $$0.565935\pi$$
$$618$$ −912.000 −0.0593625
$$619$$ −18340.0 −1.19087 −0.595434 0.803404i $$-0.703020\pi$$
−0.595434 + 0.803404i $$0.703020\pi$$
$$620$$ 12848.0 0.832239
$$621$$ 7020.00 0.453628
$$622$$ 6924.00 0.446346
$$623$$ 16530.0 1.06302
$$624$$ 0 0
$$625$$ −15109.0 −0.966976
$$626$$ −17474.0 −1.11566
$$627$$ −10260.0 −0.653501
$$628$$ −6184.00 −0.392943
$$629$$ −22491.0 −1.42572
$$630$$ −7524.00 −0.475815
$$631$$ −10057.0 −0.634489 −0.317245 0.948344i $$-0.602758\pi$$
−0.317245 + 0.948344i $$0.602758\pi$$
$$632$$ −3680.00 −0.231618
$$633$$ 15171.0 0.952596
$$634$$ −12108.0 −0.758470
$$635$$ 13816.0 0.863419
$$636$$ 5616.00 0.350140
$$637$$ 0 0
$$638$$ −14440.0 −0.896058
$$639$$ −15534.0 −0.961683
$$640$$ −1408.00 −0.0869626
$$641$$ −25058.0 −1.54404 −0.772021 0.635596i $$-0.780754\pi$$
−0.772021 + 0.635596i $$0.780754\pi$$
$$642$$ 4584.00 0.281801
$$643$$ −3698.00 −0.226804 −0.113402 0.993549i $$-0.536175\pi$$
−0.113402 + 0.993549i $$0.536175\pi$$
$$644$$ 3952.00 0.241818
$$645$$ −12309.0 −0.751421
$$646$$ 9180.00 0.559106
$$647$$ −11786.0 −0.716160 −0.358080 0.933691i $$-0.616568\pi$$
−0.358080 + 0.933691i $$0.616568\pi$$
$$648$$ 648.000 0.0392837
$$649$$ −20140.0 −1.21813
$$650$$ 0 0
$$651$$ 16644.0 1.00204
$$652$$ −2672.00 −0.160496
$$653$$ −9672.00 −0.579624 −0.289812 0.957084i $$-0.593593\pi$$
−0.289812 + 0.957084i $$0.593593\pi$$
$$654$$ −6810.00 −0.407174
$$655$$ −12067.0 −0.719842
$$656$$ −4992.00 −0.297111
$$657$$ −5796.00 −0.344176
$$658$$ −1558.00 −0.0923057
$$659$$ 17460.0 1.03209 0.516043 0.856563i $$-0.327404\pi$$
0.516043 + 0.856563i $$0.327404\pi$$
$$660$$ −5016.00 −0.295830
$$661$$ −8702.00 −0.512055 −0.256028 0.966669i $$-0.582414\pi$$
−0.256028 + 0.966669i $$0.582414\pi$$
$$662$$ −1864.00 −0.109436
$$663$$ 0 0
$$664$$ −4224.00 −0.246872
$$665$$ −18810.0 −1.09687
$$666$$ −15876.0 −0.923697
$$667$$ 9880.00 0.573546
$$668$$ 3744.00 0.216856
$$669$$ −8739.00 −0.505036
$$670$$ −4532.00 −0.261323
$$671$$ 22496.0 1.29426
$$672$$ −1824.00 −0.104706
$$673$$ −22667.0 −1.29829 −0.649145 0.760665i $$-0.724873\pi$$
−0.649145 + 0.760665i $$0.724873\pi$$
$$674$$ −3842.00 −0.219567
$$675$$ 540.000 0.0307920
$$676$$ 0 0
$$677$$ −18516.0 −1.05115 −0.525574 0.850748i $$-0.676150\pi$$
−0.525574 + 0.850748i $$0.676150\pi$$
$$678$$ −10932.0 −0.619234
$$679$$ −6574.00 −0.371557
$$680$$ 4488.00 0.253098
$$681$$ −11232.0 −0.632028
$$682$$ −22192.0 −1.24601
$$683$$ 4772.00 0.267343 0.133672 0.991026i $$-0.457323\pi$$
0.133672 + 0.991026i $$0.457323\pi$$
$$684$$ 6480.00 0.362235
$$685$$ −1716.00 −0.0957153
$$686$$ 12350.0 0.687355
$$687$$ 5265.00 0.292391
$$688$$ 5968.00 0.330709
$$689$$ 0 0
$$690$$ 3432.00 0.189354
$$691$$ −19672.0 −1.08301 −0.541504 0.840698i $$-0.682145\pi$$
−0.541504 + 0.840698i $$0.682145\pi$$
$$692$$ 6032.00 0.331362
$$693$$ 12996.0 0.712377
$$694$$ 4578.00 0.250401
$$695$$ 22165.0 1.20974
$$696$$ −4560.00 −0.248342
$$697$$ 15912.0 0.864720
$$698$$ −2390.00 −0.129603
$$699$$ −6081.00 −0.329048
$$700$$ 304.000 0.0164145
$$701$$ −9828.00 −0.529527 −0.264764 0.964313i $$-0.585294\pi$$
−0.264764 + 0.964313i $$0.585294\pi$$
$$702$$ 0 0
$$703$$ −39690.0 −2.12936
$$704$$ 2432.00 0.130198
$$705$$ −1353.00 −0.0722793
$$706$$ −15176.0 −0.809003
$$707$$ −28348.0 −1.50797
$$708$$ −6360.00 −0.337604
$$709$$ 15730.0 0.833219 0.416610 0.909085i $$-0.363218\pi$$
0.416610 + 0.909085i $$0.363218\pi$$
$$710$$ −18986.0 −1.00357
$$711$$ 8280.00 0.436743
$$712$$ −6960.00 −0.366344
$$713$$ 15184.0 0.797539
$$714$$ 5814.00 0.304739
$$715$$ 0 0
$$716$$ 7140.00 0.372674
$$717$$ 13815.0 0.719569
$$718$$ −12480.0 −0.648676
$$719$$ −17890.0 −0.927934 −0.463967 0.885853i $$-0.653574\pi$$
−0.463967 + 0.885853i $$0.653574\pi$$
$$720$$ 3168.00 0.163978
$$721$$ 2888.00 0.149174
$$722$$ 2482.00 0.127937
$$723$$ 2394.00 0.123145
$$724$$ −4032.00 −0.206973
$$725$$ 760.000 0.0389320
$$726$$ 678.000 0.0346597
$$727$$ −6386.00 −0.325782 −0.162891 0.986644i $$-0.552082\pi$$
−0.162891 + 0.986644i $$0.552082\pi$$
$$728$$ 0 0
$$729$$ 9477.00 0.481481
$$730$$ −7084.00 −0.359165
$$731$$ −19023.0 −0.962505
$$732$$ 7104.00 0.358704
$$733$$ −2273.00 −0.114536 −0.0572682 0.998359i $$-0.518239\pi$$
−0.0572682 + 0.998359i $$0.518239\pi$$
$$734$$ 18148.0 0.912609
$$735$$ −594.000 −0.0298096
$$736$$ −1664.00 −0.0833368
$$737$$ 7828.00 0.391246
$$738$$ 11232.0 0.560238
$$739$$ 4980.00 0.247892 0.123946 0.992289i $$-0.460445\pi$$
0.123946 + 0.992289i $$0.460445\pi$$
$$740$$ −19404.0 −0.963926
$$741$$ 0 0
$$742$$ −17784.0 −0.879880
$$743$$ −7483.00 −0.369481 −0.184741 0.982787i $$-0.559145\pi$$
−0.184741 + 0.982787i $$0.559145\pi$$
$$744$$ −7008.00 −0.345330
$$745$$ −11550.0 −0.567999
$$746$$ −15464.0 −0.758951
$$747$$ 9504.00 0.465506
$$748$$ −7752.00 −0.378932
$$749$$ −14516.0 −0.708148
$$750$$ 8514.00 0.414516
$$751$$ 31632.0 1.53697 0.768487 0.639865i $$-0.221010\pi$$
0.768487 + 0.639865i $$0.221010\pi$$
$$752$$ 656.000 0.0318110
$$753$$ −6264.00 −0.303151
$$754$$ 0 0
$$755$$ 21087.0 1.01647
$$756$$ 10260.0 0.493588
$$757$$ −16116.0 −0.773773 −0.386886 0.922127i $$-0.626449\pi$$
−0.386886 + 0.922127i $$0.626449\pi$$
$$758$$ 18640.0 0.893186
$$759$$ −5928.00 −0.283495
$$760$$ 7920.00 0.378011
$$761$$ −14622.0 −0.696514 −0.348257 0.937399i $$-0.613226\pi$$
−0.348257 + 0.937399i $$0.613226\pi$$
$$762$$ −7536.00 −0.358268
$$763$$ 21565.0 1.02321
$$764$$ −8552.00 −0.404974
$$765$$ −10098.0 −0.477247
$$766$$ 1854.00 0.0874514
$$767$$ 0 0
$$768$$ 768.000 0.0360844
$$769$$ −9840.00 −0.461430 −0.230715 0.973021i $$-0.574106\pi$$
−0.230715 + 0.973021i $$0.574106\pi$$
$$770$$ 15884.0 0.743402
$$771$$ −18333.0 −0.856351
$$772$$ −18752.0 −0.874222
$$773$$ −4823.00 −0.224413 −0.112207 0.993685i $$-0.535792\pi$$
−0.112207 + 0.993685i $$0.535792\pi$$
$$774$$ −13428.0 −0.623591
$$775$$ 1168.00 0.0541365
$$776$$ 2768.00 0.128048
$$777$$ −25137.0 −1.16060
$$778$$ −3420.00 −0.157600
$$779$$ 28080.0 1.29149
$$780$$ 0 0
$$781$$ 32794.0 1.50251
$$782$$ 5304.00 0.242546
$$783$$ 25650.0 1.17070
$$784$$ 288.000 0.0131195
$$785$$ 17006.0 0.773210
$$786$$ 6582.00 0.298692
$$787$$ −28424.0 −1.28743 −0.643714 0.765266i $$-0.722607\pi$$
−0.643714 + 0.765266i $$0.722607\pi$$
$$788$$ 3564.00 0.161120
$$789$$ −7596.00 −0.342744
$$790$$ 10120.0 0.455764
$$791$$ 34618.0 1.55610
$$792$$ −5472.00 −0.245504
$$793$$ 0 0
$$794$$ 9212.00 0.411740
$$795$$ −15444.0 −0.688984
$$796$$ −6520.00 −0.290321
$$797$$ 33294.0 1.47972 0.739858 0.672763i $$-0.234893\pi$$
0.739858 + 0.672763i $$0.234893\pi$$
$$798$$ 10260.0 0.455138
$$799$$ −2091.00 −0.0925836
$$800$$ −128.000 −0.00565685
$$801$$ 15660.0 0.690785
$$802$$ 14496.0 0.638244
$$803$$ 12236.0 0.537732
$$804$$ 2472.00 0.108434
$$805$$ −10868.0 −0.475834
$$806$$ 0 0
$$807$$ 7200.00 0.314067
$$808$$ 11936.0 0.519687
$$809$$ −18585.0 −0.807681 −0.403840 0.914829i $$-0.632325\pi$$
−0.403840 + 0.914829i $$0.632325\pi$$
$$810$$ −1782.00 −0.0773001
$$811$$ 19348.0 0.837731 0.418866 0.908048i $$-0.362428\pi$$
0.418866 + 0.908048i $$0.362428\pi$$
$$812$$ 14440.0 0.624070
$$813$$ 14379.0 0.620287
$$814$$ 33516.0 1.44316
$$815$$ 7348.00 0.315815
$$816$$ −2448.00 −0.105021
$$817$$ −33570.0 −1.43753
$$818$$ −21080.0 −0.901033
$$819$$ 0 0
$$820$$ 13728.0 0.584637
$$821$$ 15853.0 0.673902 0.336951 0.941522i $$-0.390604\pi$$
0.336951 + 0.941522i $$0.390604\pi$$
$$822$$ 936.000 0.0397162
$$823$$ 23898.0 1.01219 0.506095 0.862478i $$-0.331089\pi$$
0.506095 + 0.862478i $$0.331089\pi$$
$$824$$ −1216.00 −0.0514094
$$825$$ −456.000 −0.0192435
$$826$$ 20140.0 0.848378
$$827$$ −35634.0 −1.49833 −0.749163 0.662386i $$-0.769544\pi$$
−0.749163 + 0.662386i $$0.769544\pi$$
$$828$$ 3744.00 0.157141
$$829$$ −29390.0 −1.23131 −0.615656 0.788015i $$-0.711109\pi$$
−0.615656 + 0.788015i $$0.711109\pi$$
$$830$$ 11616.0 0.485780
$$831$$ −17028.0 −0.710824
$$832$$ 0 0
$$833$$ −918.000 −0.0381835
$$834$$ −12090.0 −0.501969
$$835$$ −10296.0 −0.426716
$$836$$ −13680.0 −0.565948
$$837$$ 39420.0 1.62790
$$838$$ 21270.0 0.876802
$$839$$ 27040.0 1.11266 0.556332 0.830960i $$-0.312208\pi$$
0.556332 + 0.830960i $$0.312208\pi$$
$$840$$ 5016.00 0.206034
$$841$$ 11711.0 0.480175
$$842$$ −24974.0 −1.02216
$$843$$ −16626.0 −0.679276
$$844$$ 20228.0 0.824972
$$845$$ 0 0
$$846$$ −1476.00 −0.0599834
$$847$$ −2147.00 −0.0870977
$$848$$ 7488.00 0.303230
$$849$$ −18996.0 −0.767893
$$850$$ 408.000 0.0164639
$$851$$ −22932.0 −0.923735
$$852$$ 10356.0 0.416421
$$853$$ 15467.0 0.620844 0.310422 0.950599i $$-0.399530\pi$$
0.310422 + 0.950599i $$0.399530\pi$$
$$854$$ −22496.0 −0.901402
$$855$$ −17820.0 −0.712785
$$856$$ 6112.00 0.244047
$$857$$ 15694.0 0.625551 0.312775 0.949827i $$-0.398741\pi$$
0.312775 + 0.949827i $$0.398741\pi$$
$$858$$ 0 0
$$859$$ 2180.00 0.0865898 0.0432949 0.999062i $$-0.486214\pi$$
0.0432949 + 0.999062i $$0.486214\pi$$
$$860$$ −16412.0 −0.650749
$$861$$ 17784.0 0.703922
$$862$$ 29226.0 1.15480
$$863$$ 26537.0 1.04673 0.523366 0.852108i $$-0.324676\pi$$
0.523366 + 0.852108i $$0.324676\pi$$
$$864$$ −4320.00 −0.170103
$$865$$ −16588.0 −0.652033
$$866$$ −13954.0 −0.547548
$$867$$ −6936.00 −0.271694
$$868$$ 22192.0 0.867794
$$869$$ −17480.0 −0.682357
$$870$$ 12540.0 0.488673
$$871$$ 0 0
$$872$$ −9080.00 −0.352623
$$873$$ −6228.00 −0.241450
$$874$$ 9360.00 0.362250
$$875$$ −26961.0 −1.04166
$$876$$ 3864.00 0.149032
$$877$$ −23829.0 −0.917501 −0.458750 0.888565i $$-0.651703\pi$$
−0.458750 + 0.888565i $$0.651703\pi$$
$$878$$ 2860.00 0.109932
$$879$$ 9231.00 0.354214
$$880$$ −6688.00 −0.256196
$$881$$ 24117.0 0.922273 0.461136 0.887329i $$-0.347442\pi$$
0.461136 + 0.887329i $$0.347442\pi$$
$$882$$ −648.000 −0.0247384
$$883$$ −14537.0 −0.554031 −0.277015 0.960866i $$-0.589345\pi$$
−0.277015 + 0.960866i $$0.589345\pi$$
$$884$$ 0 0
$$885$$ 17490.0 0.664316
$$886$$ 24846.0 0.942119
$$887$$ 12064.0 0.456674 0.228337 0.973582i $$-0.426671\pi$$
0.228337 + 0.973582i $$0.426671\pi$$
$$888$$ 10584.0 0.399973
$$889$$ 23864.0 0.900307
$$890$$ 19140.0 0.720870
$$891$$ 3078.00 0.115732
$$892$$ −11652.0 −0.437374
$$893$$ −3690.00 −0.138277
$$894$$ 6300.00 0.235686
$$895$$ −19635.0 −0.733325
$$896$$ −2432.00 −0.0906779
$$897$$ 0 0
$$898$$ 19780.0 0.735041
$$899$$ 55480.0 2.05824
$$900$$ 288.000 0.0106667
$$901$$ −23868.0 −0.882529
$$902$$ −23712.0 −0.875303
$$903$$ −21261.0 −0.783524
$$904$$ −14576.0 −0.536273
$$905$$ 11088.0 0.407268
$$906$$ −11502.0 −0.421775
$$907$$ 38409.0 1.40612 0.703059 0.711131i $$-0.251817\pi$$
0.703059 + 0.711131i $$0.251817\pi$$
$$908$$ −14976.0 −0.547352
$$909$$ −26856.0 −0.979931
$$910$$ 0 0
$$911$$ −49578.0 −1.80307 −0.901533 0.432711i $$-0.857557\pi$$
−0.901533 + 0.432711i $$0.857557\pi$$
$$912$$ −4320.00 −0.156853
$$913$$ −20064.0 −0.727296
$$914$$ 19852.0 0.718431
$$915$$ −19536.0 −0.705836
$$916$$ 7020.00 0.253218
$$917$$ −20843.0 −0.750596
$$918$$ 13770.0 0.495074
$$919$$ 8280.00 0.297206 0.148603 0.988897i $$-0.452522\pi$$
0.148603 + 0.988897i $$0.452522\pi$$
$$920$$ 4576.00 0.163985
$$921$$ 9858.00 0.352695
$$922$$ 5586.00 0.199528
$$923$$ 0 0
$$924$$ −8664.00 −0.308468
$$925$$ −1764.00 −0.0627027
$$926$$ 13744.0 0.487749
$$927$$ 2736.00 0.0969385
$$928$$ −6080.00 −0.215071
$$929$$ −3180.00 −0.112306 −0.0561531 0.998422i $$-0.517883\pi$$
−0.0561531 + 0.998422i $$0.517883\pi$$
$$930$$ 19272.0 0.679520
$$931$$ −1620.00 −0.0570283
$$932$$ −8108.00 −0.284964
$$933$$ 10386.0 0.364440
$$934$$ −7352.00 −0.257564
$$935$$ 21318.0 0.745640
$$936$$ 0 0
$$937$$ 14214.0 0.495572 0.247786 0.968815i $$-0.420297\pi$$
0.247786 + 0.968815i $$0.420297\pi$$
$$938$$ −7828.00 −0.272487
$$939$$ −26211.0 −0.910930
$$940$$ −1804.00 −0.0625957
$$941$$ −8917.00 −0.308912 −0.154456 0.988000i $$-0.549362\pi$$
−0.154456 + 0.988000i $$0.549362\pi$$
$$942$$ −9276.00 −0.320837
$$943$$ 16224.0 0.560261
$$944$$ −8480.00 −0.292373
$$945$$ −28215.0 −0.971253
$$946$$ 28348.0 0.974284
$$947$$ −34074.0 −1.16923 −0.584613 0.811313i $$-0.698753\pi$$
−0.584613 + 0.811313i $$0.698753\pi$$
$$948$$ −5520.00 −0.189115
$$949$$ 0 0
$$950$$ 720.000 0.0245894
$$951$$ −18162.0 −0.619288
$$952$$ 7752.00 0.263912
$$953$$ 24383.0 0.828796 0.414398 0.910096i $$-0.363992\pi$$
0.414398 + 0.910096i $$0.363992\pi$$
$$954$$ −16848.0 −0.571776
$$955$$ 23518.0 0.796884
$$956$$ 18420.0 0.623165
$$957$$ −21660.0 −0.731628
$$958$$ −27150.0 −0.915633
$$959$$ −2964.00 −0.0998045
$$960$$ −2112.00 −0.0710047
$$961$$ 55473.0 1.86207
$$962$$ 0 0
$$963$$ −13752.0 −0.460179
$$964$$ 3192.00 0.106647
$$965$$ 51568.0 1.72024
$$966$$ 5928.00 0.197443
$$967$$ 16171.0 0.537771 0.268885 0.963172i $$-0.413345\pi$$
0.268885 + 0.963172i $$0.413345\pi$$
$$968$$ 904.000 0.0300162
$$969$$ 13770.0 0.456508
$$970$$ −7612.00 −0.251966
$$971$$ −18513.0 −0.611854 −0.305927 0.952055i $$-0.598966\pi$$
−0.305927 + 0.952055i $$0.598966\pi$$
$$972$$ 15552.0 0.513200
$$973$$ 38285.0 1.26142
$$974$$ −23728.0 −0.780589
$$975$$ 0 0
$$976$$ 9472.00 0.310647
$$977$$ 39966.0 1.30873 0.654363 0.756180i $$-0.272937\pi$$
0.654363 + 0.756180i $$0.272937\pi$$
$$978$$ −4008.00 −0.131045
$$979$$ −33060.0 −1.07927
$$980$$ −792.000 −0.0258158
$$981$$ 20430.0 0.664913
$$982$$ 9674.00 0.314368
$$983$$ 44317.0 1.43794 0.718969 0.695042i $$-0.244614\pi$$
0.718969 + 0.695042i $$0.244614\pi$$
$$984$$ −7488.00 −0.242590
$$985$$ −9801.00 −0.317041
$$986$$ 19380.0 0.625948
$$987$$ −2337.00 −0.0753673
$$988$$ 0 0
$$989$$ −19396.0 −0.623617
$$990$$ 15048.0 0.483088
$$991$$ 52422.0 1.68036 0.840181 0.542305i $$-0.182448\pi$$
0.840181 + 0.542305i $$0.182448\pi$$
$$992$$ −9344.00 −0.299065
$$993$$ −2796.00 −0.0893539
$$994$$ −32794.0 −1.04644
$$995$$ 17930.0 0.571276
$$996$$ −6336.00 −0.201570
$$997$$ −2026.00 −0.0643571 −0.0321786 0.999482i $$-0.510245\pi$$
−0.0321786 + 0.999482i $$0.510245\pi$$
$$998$$ −18320.0 −0.581072
$$999$$ −59535.0 −1.88549
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.e.1.1 1
13.2 odd 12 338.4.e.b.147.2 4
13.3 even 3 338.4.c.b.191.1 2
13.4 even 6 338.4.c.f.315.1 2
13.5 odd 4 338.4.b.c.337.1 2
13.6 odd 12 338.4.e.b.23.1 4
13.7 odd 12 338.4.e.b.23.2 4
13.8 odd 4 338.4.b.c.337.2 2
13.9 even 3 338.4.c.b.315.1 2
13.10 even 6 338.4.c.f.191.1 2
13.11 odd 12 338.4.e.b.147.1 4
13.12 even 2 26.4.a.a.1.1 1
39.38 odd 2 234.4.a.g.1.1 1
52.51 odd 2 208.4.a.c.1.1 1
65.12 odd 4 650.4.b.b.599.1 2
65.38 odd 4 650.4.b.b.599.2 2
65.64 even 2 650.4.a.f.1.1 1
91.90 odd 2 1274.4.a.b.1.1 1
104.51 odd 2 832.4.a.m.1.1 1
104.77 even 2 832.4.a.e.1.1 1
156.155 even 2 1872.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.a.a.1.1 1 13.12 even 2
208.4.a.c.1.1 1 52.51 odd 2
234.4.a.g.1.1 1 39.38 odd 2
338.4.a.e.1.1 1 1.1 even 1 trivial
338.4.b.c.337.1 2 13.5 odd 4
338.4.b.c.337.2 2 13.8 odd 4
338.4.c.b.191.1 2 13.3 even 3
338.4.c.b.315.1 2 13.9 even 3
338.4.c.f.191.1 2 13.10 even 6
338.4.c.f.315.1 2 13.4 even 6
338.4.e.b.23.1 4 13.6 odd 12
338.4.e.b.23.2 4 13.7 odd 12
338.4.e.b.147.1 4 13.11 odd 12
338.4.e.b.147.2 4 13.2 odd 12
650.4.a.f.1.1 1 65.64 even 2
650.4.b.b.599.1 2 65.12 odd 4
650.4.b.b.599.2 2 65.38 odd 4
832.4.a.e.1.1 1 104.77 even 2
832.4.a.m.1.1 1 104.51 odd 2
1274.4.a.b.1.1 1 91.90 odd 2
1872.4.a.c.1.1 1 156.155 even 2