# Properties

 Label 1521.4.a.k.1.1 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ 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] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.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: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} +8.00000 q^{4} +17.0000 q^{5} -20.0000 q^{7} +O(q^{10})$$ $$q+4.00000 q^{2} +8.00000 q^{4} +17.0000 q^{5} -20.0000 q^{7} +68.0000 q^{10} -32.0000 q^{11} -80.0000 q^{14} -64.0000 q^{16} +13.0000 q^{17} -30.0000 q^{19} +136.000 q^{20} -128.000 q^{22} -78.0000 q^{23} +164.000 q^{25} -160.000 q^{28} -197.000 q^{29} +74.0000 q^{31} -256.000 q^{32} +52.0000 q^{34} -340.000 q^{35} +227.000 q^{37} -120.000 q^{38} -165.000 q^{41} -156.000 q^{43} -256.000 q^{44} -312.000 q^{46} -162.000 q^{47} +57.0000 q^{49} +656.000 q^{50} -93.0000 q^{53} -544.000 q^{55} -788.000 q^{58} -864.000 q^{59} +145.000 q^{61} +296.000 q^{62} -512.000 q^{64} -862.000 q^{67} +104.000 q^{68} -1360.00 q^{70} +654.000 q^{71} -215.000 q^{73} +908.000 q^{74} -240.000 q^{76} +640.000 q^{77} -76.0000 q^{79} -1088.00 q^{80} -660.000 q^{82} +628.000 q^{83} +221.000 q^{85} -624.000 q^{86} -266.000 q^{89} -624.000 q^{92} -648.000 q^{94} -510.000 q^{95} -238.000 q^{97} +228.000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ 0 0
$$4$$ 8.00000 1.00000
$$5$$ 17.0000 1.52053 0.760263 0.649615i $$-0.225070\pi$$
0.760263 + 0.649615i $$0.225070\pi$$
$$6$$ 0 0
$$7$$ −20.0000 −1.07990 −0.539949 0.841698i $$-0.681557\pi$$
−0.539949 + 0.841698i $$0.681557\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 68.0000 2.15035
$$11$$ −32.0000 −0.877124 −0.438562 0.898701i $$-0.644512\pi$$
−0.438562 + 0.898701i $$0.644512\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −80.0000 −1.52721
$$15$$ 0 0
$$16$$ −64.0000 −1.00000
$$17$$ 13.0000 0.185468 0.0927342 0.995691i $$-0.470439\pi$$
0.0927342 + 0.995691i $$0.470439\pi$$
$$18$$ 0 0
$$19$$ −30.0000 −0.362235 −0.181118 0.983461i $$-0.557971\pi$$
−0.181118 + 0.983461i $$0.557971\pi$$
$$20$$ 136.000 1.52053
$$21$$ 0 0
$$22$$ −128.000 −1.24044
$$23$$ −78.0000 −0.707136 −0.353568 0.935409i $$-0.615032\pi$$
−0.353568 + 0.935409i $$0.615032\pi$$
$$24$$ 0 0
$$25$$ 164.000 1.31200
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −160.000 −1.07990
$$29$$ −197.000 −1.26145 −0.630724 0.776007i $$-0.717242\pi$$
−0.630724 + 0.776007i $$0.717242\pi$$
$$30$$ 0 0
$$31$$ 74.0000 0.428735 0.214368 0.976753i $$-0.431231\pi$$
0.214368 + 0.976753i $$0.431231\pi$$
$$32$$ −256.000 −1.41421
$$33$$ 0 0
$$34$$ 52.0000 0.262292
$$35$$ −340.000 −1.64201
$$36$$ 0 0
$$37$$ 227.000 1.00861 0.504305 0.863526i $$-0.331749\pi$$
0.504305 + 0.863526i $$0.331749\pi$$
$$38$$ −120.000 −0.512278
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −165.000 −0.628504 −0.314252 0.949340i $$-0.601754\pi$$
−0.314252 + 0.949340i $$0.601754\pi$$
$$42$$ 0 0
$$43$$ −156.000 −0.553251 −0.276625 0.960978i $$-0.589216\pi$$
−0.276625 + 0.960978i $$0.589216\pi$$
$$44$$ −256.000 −0.877124
$$45$$ 0 0
$$46$$ −312.000 −1.00004
$$47$$ −162.000 −0.502769 −0.251384 0.967887i $$-0.580886\pi$$
−0.251384 + 0.967887i $$0.580886\pi$$
$$48$$ 0 0
$$49$$ 57.0000 0.166181
$$50$$ 656.000 1.85545
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −93.0000 −0.241029 −0.120514 0.992712i $$-0.538454\pi$$
−0.120514 + 0.992712i $$0.538454\pi$$
$$54$$ 0 0
$$55$$ −544.000 −1.33369
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −788.000 −1.78396
$$59$$ −864.000 −1.90650 −0.953248 0.302190i $$-0.902282\pi$$
−0.953248 + 0.302190i $$0.902282\pi$$
$$60$$ 0 0
$$61$$ 145.000 0.304350 0.152175 0.988354i $$-0.451372\pi$$
0.152175 + 0.988354i $$0.451372\pi$$
$$62$$ 296.000 0.606323
$$63$$ 0 0
$$64$$ −512.000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −862.000 −1.57179 −0.785896 0.618359i $$-0.787798\pi$$
−0.785896 + 0.618359i $$0.787798\pi$$
$$68$$ 104.000 0.185468
$$69$$ 0 0
$$70$$ −1360.00 −2.32216
$$71$$ 654.000 1.09318 0.546588 0.837402i $$-0.315926\pi$$
0.546588 + 0.837402i $$0.315926\pi$$
$$72$$ 0 0
$$73$$ −215.000 −0.344710 −0.172355 0.985035i $$-0.555138\pi$$
−0.172355 + 0.985035i $$0.555138\pi$$
$$74$$ 908.000 1.42639
$$75$$ 0 0
$$76$$ −240.000 −0.362235
$$77$$ 640.000 0.947205
$$78$$ 0 0
$$79$$ −76.0000 −0.108236 −0.0541182 0.998535i $$-0.517235\pi$$
−0.0541182 + 0.998535i $$0.517235\pi$$
$$80$$ −1088.00 −1.52053
$$81$$ 0 0
$$82$$ −660.000 −0.888839
$$83$$ 628.000 0.830505 0.415253 0.909706i $$-0.363693\pi$$
0.415253 + 0.909706i $$0.363693\pi$$
$$84$$ 0 0
$$85$$ 221.000 0.282010
$$86$$ −624.000 −0.782415
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −266.000 −0.316808 −0.158404 0.987374i $$-0.550635\pi$$
−0.158404 + 0.987374i $$0.550635\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −624.000 −0.707136
$$93$$ 0 0
$$94$$ −648.000 −0.711022
$$95$$ −510.000 −0.550788
$$96$$ 0 0
$$97$$ −238.000 −0.249126 −0.124563 0.992212i $$-0.539753\pi$$
−0.124563 + 0.992212i $$0.539753\pi$$
$$98$$ 228.000 0.235015
$$99$$ 0 0
$$100$$ 1312.00 1.31200
$$101$$ 819.000 0.806867 0.403433 0.915009i $$-0.367817\pi$$
0.403433 + 0.915009i $$0.367817\pi$$
$$102$$ 0 0
$$103$$ 1638.00 1.56696 0.783480 0.621417i $$-0.213443\pi$$
0.783480 + 0.621417i $$0.213443\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −372.000 −0.340866
$$107$$ −522.000 −0.471623 −0.235811 0.971799i $$-0.575775\pi$$
−0.235811 + 0.971799i $$0.575775\pi$$
$$108$$ 0 0
$$109$$ 1634.00 1.43586 0.717930 0.696115i $$-0.245090\pi$$
0.717930 + 0.696115i $$0.245090\pi$$
$$110$$ −2176.00 −1.88612
$$111$$ 0 0
$$112$$ 1280.00 1.07990
$$113$$ −327.000 −0.272226 −0.136113 0.990693i $$-0.543461\pi$$
−0.136113 + 0.990693i $$0.543461\pi$$
$$114$$ 0 0
$$115$$ −1326.00 −1.07522
$$116$$ −1576.00 −1.26145
$$117$$ 0 0
$$118$$ −3456.00 −2.69619
$$119$$ −260.000 −0.200287
$$120$$ 0 0
$$121$$ −307.000 −0.230654
$$122$$ 580.000 0.430416
$$123$$ 0 0
$$124$$ 592.000 0.428735
$$125$$ 663.000 0.474404
$$126$$ 0 0
$$127$$ −2158.00 −1.50781 −0.753904 0.656985i $$-0.771831\pi$$
−0.753904 + 0.656985i $$0.771831\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −730.000 −0.486873 −0.243437 0.969917i $$-0.578275\pi$$
−0.243437 + 0.969917i $$0.578275\pi$$
$$132$$ 0 0
$$133$$ 600.000 0.391177
$$134$$ −3448.00 −2.22285
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1671.00 1.04207 0.521033 0.853536i $$-0.325547\pi$$
0.521033 + 0.853536i $$0.325547\pi$$
$$138$$ 0 0
$$139$$ 912.000 0.556510 0.278255 0.960507i $$-0.410244\pi$$
0.278255 + 0.960507i $$0.410244\pi$$
$$140$$ −2720.00 −1.64201
$$141$$ 0 0
$$142$$ 2616.00 1.54598
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −3349.00 −1.91806
$$146$$ −860.000 −0.487494
$$147$$ 0 0
$$148$$ 1816.00 1.00861
$$149$$ −2115.00 −1.16287 −0.581435 0.813593i $$-0.697508\pi$$
−0.581435 + 0.813593i $$0.697508\pi$$
$$150$$ 0 0
$$151$$ −514.000 −0.277011 −0.138506 0.990362i $$-0.544230\pi$$
−0.138506 + 0.990362i $$0.544230\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 2560.00 1.33955
$$155$$ 1258.00 0.651903
$$156$$ 0 0
$$157$$ 2901.00 1.47468 0.737341 0.675521i $$-0.236081\pi$$
0.737341 + 0.675521i $$0.236081\pi$$
$$158$$ −304.000 −0.153069
$$159$$ 0 0
$$160$$ −4352.00 −2.15035
$$161$$ 1560.00 0.763635
$$162$$ 0 0
$$163$$ −2360.00 −1.13405 −0.567023 0.823702i $$-0.691905\pi$$
−0.567023 + 0.823702i $$0.691905\pi$$
$$164$$ −1320.00 −0.628504
$$165$$ 0 0
$$166$$ 2512.00 1.17451
$$167$$ 280.000 0.129743 0.0648714 0.997894i $$-0.479336\pi$$
0.0648714 + 0.997894i $$0.479336\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 884.000 0.398822
$$171$$ 0 0
$$172$$ −1248.00 −0.553251
$$173$$ −1326.00 −0.582739 −0.291370 0.956611i $$-0.594111\pi$$
−0.291370 + 0.956611i $$0.594111\pi$$
$$174$$ 0 0
$$175$$ −3280.00 −1.41683
$$176$$ 2048.00 0.877124
$$177$$ 0 0
$$178$$ −1064.00 −0.448035
$$179$$ −4264.00 −1.78048 −0.890241 0.455490i $$-0.849464\pi$$
−0.890241 + 0.455490i $$0.849464\pi$$
$$180$$ 0 0
$$181$$ −403.000 −0.165496 −0.0827479 0.996571i $$-0.526370\pi$$
−0.0827479 + 0.996571i $$0.526370\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 3859.00 1.53362
$$186$$ 0 0
$$187$$ −416.000 −0.162679
$$188$$ −1296.00 −0.502769
$$189$$ 0 0
$$190$$ −2040.00 −0.778932
$$191$$ 1246.00 0.472028 0.236014 0.971750i $$-0.424159\pi$$
0.236014 + 0.971750i $$0.424159\pi$$
$$192$$ 0 0
$$193$$ −267.000 −0.0995807 −0.0497904 0.998760i $$-0.515855\pi$$
−0.0497904 + 0.998760i $$0.515855\pi$$
$$194$$ −952.000 −0.352318
$$195$$ 0 0
$$196$$ 456.000 0.166181
$$197$$ 1278.00 0.462202 0.231101 0.972930i $$-0.425767\pi$$
0.231101 + 0.972930i $$0.425767\pi$$
$$198$$ 0 0
$$199$$ 4238.00 1.50967 0.754834 0.655916i $$-0.227717\pi$$
0.754834 + 0.655916i $$0.227717\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 3276.00 1.14108
$$203$$ 3940.00 1.36224
$$204$$ 0 0
$$205$$ −2805.00 −0.955657
$$206$$ 6552.00 2.21602
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 960.000 0.317725
$$210$$ 0 0
$$211$$ 3070.00 1.00165 0.500823 0.865549i $$-0.333031\pi$$
0.500823 + 0.865549i $$0.333031\pi$$
$$212$$ −744.000 −0.241029
$$213$$ 0 0
$$214$$ −2088.00 −0.666975
$$215$$ −2652.00 −0.841232
$$216$$ 0 0
$$217$$ −1480.00 −0.462991
$$218$$ 6536.00 2.03061
$$219$$ 0 0
$$220$$ −4352.00 −1.33369
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 5378.00 1.61497 0.807483 0.589891i $$-0.200829\pi$$
0.807483 + 0.589891i $$0.200829\pi$$
$$224$$ 5120.00 1.52721
$$225$$ 0 0
$$226$$ −1308.00 −0.384986
$$227$$ −3974.00 −1.16195 −0.580977 0.813920i $$-0.697329\pi$$
−0.580977 + 0.813920i $$0.697329\pi$$
$$228$$ 0 0
$$229$$ 6298.00 1.81740 0.908698 0.417455i $$-0.137078\pi$$
0.908698 + 0.417455i $$0.137078\pi$$
$$230$$ −5304.00 −1.52059
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −4030.00 −1.13311 −0.566554 0.824025i $$-0.691724\pi$$
−0.566554 + 0.824025i $$0.691724\pi$$
$$234$$ 0 0
$$235$$ −2754.00 −0.764473
$$236$$ −6912.00 −1.90650
$$237$$ 0 0
$$238$$ −1040.00 −0.283249
$$239$$ −984.000 −0.266317 −0.133158 0.991095i $$-0.542512\pi$$
−0.133158 + 0.991095i $$0.542512\pi$$
$$240$$ 0 0
$$241$$ −943.000 −0.252050 −0.126025 0.992027i $$-0.540222\pi$$
−0.126025 + 0.992027i $$0.540222\pi$$
$$242$$ −1228.00 −0.326194
$$243$$ 0 0
$$244$$ 1160.00 0.304350
$$245$$ 969.000 0.252682
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 2652.00 0.670909
$$251$$ 2730.00 0.686518 0.343259 0.939241i $$-0.388469\pi$$
0.343259 + 0.939241i $$0.388469\pi$$
$$252$$ 0 0
$$253$$ 2496.00 0.620246
$$254$$ −8632.00 −2.13236
$$255$$ 0 0
$$256$$ 4096.00 1.00000
$$257$$ 1885.00 0.457522 0.228761 0.973483i $$-0.426533\pi$$
0.228761 + 0.973483i $$0.426533\pi$$
$$258$$ 0 0
$$259$$ −4540.00 −1.08920
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −2920.00 −0.688543
$$263$$ −4032.00 −0.945338 −0.472669 0.881240i $$-0.656709\pi$$
−0.472669 + 0.881240i $$0.656709\pi$$
$$264$$ 0 0
$$265$$ −1581.00 −0.366491
$$266$$ 2400.00 0.553208
$$267$$ 0 0
$$268$$ −6896.00 −1.57179
$$269$$ −4006.00 −0.907993 −0.453997 0.891003i $$-0.650002\pi$$
−0.453997 + 0.891003i $$0.650002\pi$$
$$270$$ 0 0
$$271$$ 4296.00 0.962965 0.481482 0.876456i $$-0.340099\pi$$
0.481482 + 0.876456i $$0.340099\pi$$
$$272$$ −832.000 −0.185468
$$273$$ 0 0
$$274$$ 6684.00 1.47371
$$275$$ −5248.00 −1.15079
$$276$$ 0 0
$$277$$ −5551.00 −1.20407 −0.602035 0.798470i $$-0.705643\pi$$
−0.602035 + 0.798470i $$0.705643\pi$$
$$278$$ 3648.00 0.787023
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5557.00 −1.17973 −0.589863 0.807504i $$-0.700818\pi$$
−0.589863 + 0.807504i $$0.700818\pi$$
$$282$$ 0 0
$$283$$ 3120.00 0.655352 0.327676 0.944790i $$-0.393734\pi$$
0.327676 + 0.944790i $$0.393734\pi$$
$$284$$ 5232.00 1.09318
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3300.00 0.678721
$$288$$ 0 0
$$289$$ −4744.00 −0.965601
$$290$$ −13396.0 −2.71255
$$291$$ 0 0
$$292$$ −1720.00 −0.344710
$$293$$ 8301.00 1.65512 0.827559 0.561379i $$-0.189729\pi$$
0.827559 + 0.561379i $$0.189729\pi$$
$$294$$ 0 0
$$295$$ −14688.0 −2.89888
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −8460.00 −1.64455
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 3120.00 0.597455
$$302$$ −2056.00 −0.391753
$$303$$ 0 0
$$304$$ 1920.00 0.362235
$$305$$ 2465.00 0.462772
$$306$$ 0 0
$$307$$ −8678.00 −1.61329 −0.806644 0.591037i $$-0.798719\pi$$
−0.806644 + 0.591037i $$0.798719\pi$$
$$308$$ 5120.00 0.947205
$$309$$ 0 0
$$310$$ 5032.00 0.921930
$$311$$ −8658.00 −1.57862 −0.789309 0.613996i $$-0.789561\pi$$
−0.789309 + 0.613996i $$0.789561\pi$$
$$312$$ 0 0
$$313$$ −5250.00 −0.948075 −0.474038 0.880505i $$-0.657204\pi$$
−0.474038 + 0.880505i $$0.657204\pi$$
$$314$$ 11604.0 2.08551
$$315$$ 0 0
$$316$$ −608.000 −0.108236
$$317$$ 6413.00 1.13625 0.568123 0.822944i $$-0.307670\pi$$
0.568123 + 0.822944i $$0.307670\pi$$
$$318$$ 0 0
$$319$$ 6304.00 1.10645
$$320$$ −8704.00 −1.52053
$$321$$ 0 0
$$322$$ 6240.00 1.07994
$$323$$ −390.000 −0.0671832
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −9440.00 −1.60378
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3240.00 0.542939
$$330$$ 0 0
$$331$$ −3488.00 −0.579208 −0.289604 0.957147i $$-0.593524\pi$$
−0.289604 + 0.957147i $$0.593524\pi$$
$$332$$ 5024.00 0.830505
$$333$$ 0 0
$$334$$ 1120.00 0.183484
$$335$$ −14654.0 −2.38995
$$336$$ 0 0
$$337$$ −1833.00 −0.296290 −0.148145 0.988966i $$-0.547330\pi$$
−0.148145 + 0.988966i $$0.547330\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 1768.00 0.282010
$$341$$ −2368.00 −0.376054
$$342$$ 0 0
$$343$$ 5720.00 0.900440
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −5304.00 −0.824118
$$347$$ −7230.00 −1.11852 −0.559260 0.828992i $$-0.688915\pi$$
−0.559260 + 0.828992i $$0.688915\pi$$
$$348$$ 0 0
$$349$$ 5258.00 0.806459 0.403230 0.915099i $$-0.367888\pi$$
0.403230 + 0.915099i $$0.367888\pi$$
$$350$$ −13120.0 −2.00370
$$351$$ 0 0
$$352$$ 8192.00 1.24044
$$353$$ 3163.00 0.476911 0.238455 0.971153i $$-0.423359\pi$$
0.238455 + 0.971153i $$0.423359\pi$$
$$354$$ 0 0
$$355$$ 11118.0 1.66220
$$356$$ −2128.00 −0.316808
$$357$$ 0 0
$$358$$ −17056.0 −2.51798
$$359$$ −10068.0 −1.48014 −0.740068 0.672532i $$-0.765207\pi$$
−0.740068 + 0.672532i $$0.765207\pi$$
$$360$$ 0 0
$$361$$ −5959.00 −0.868786
$$362$$ −1612.00 −0.234047
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −3655.00 −0.524141
$$366$$ 0 0
$$367$$ 7438.00 1.05793 0.528965 0.848644i $$-0.322580\pi$$
0.528965 + 0.848644i $$0.322580\pi$$
$$368$$ 4992.00 0.707136
$$369$$ 0 0
$$370$$ 15436.0 2.16886
$$371$$ 1860.00 0.260287
$$372$$ 0 0
$$373$$ −9683.00 −1.34415 −0.672073 0.740485i $$-0.734596\pi$$
−0.672073 + 0.740485i $$0.734596\pi$$
$$374$$ −1664.00 −0.230063
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 1062.00 0.143935 0.0719674 0.997407i $$-0.477072\pi$$
0.0719674 + 0.997407i $$0.477072\pi$$
$$380$$ −4080.00 −0.550788
$$381$$ 0 0
$$382$$ 4984.00 0.667549
$$383$$ −3532.00 −0.471219 −0.235609 0.971848i $$-0.575709\pi$$
−0.235609 + 0.971848i $$0.575709\pi$$
$$384$$ 0 0
$$385$$ 10880.0 1.44025
$$386$$ −1068.00 −0.140828
$$387$$ 0 0
$$388$$ −1904.00 −0.249126
$$389$$ 11063.0 1.44194 0.720972 0.692964i $$-0.243696\pi$$
0.720972 + 0.692964i $$0.243696\pi$$
$$390$$ 0 0
$$391$$ −1014.00 −0.131151
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 5112.00 0.653652
$$395$$ −1292.00 −0.164576
$$396$$ 0 0
$$397$$ 5986.00 0.756747 0.378374 0.925653i $$-0.376483\pi$$
0.378374 + 0.925653i $$0.376483\pi$$
$$398$$ 16952.0 2.13499
$$399$$ 0 0
$$400$$ −10496.0 −1.31200
$$401$$ 5935.00 0.739102 0.369551 0.929211i $$-0.379512\pi$$
0.369551 + 0.929211i $$0.379512\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 6552.00 0.806867
$$405$$ 0 0
$$406$$ 15760.0 1.92649
$$407$$ −7264.00 −0.884676
$$408$$ 0 0
$$409$$ 15089.0 1.82421 0.912106 0.409954i $$-0.134455\pi$$
0.912106 + 0.409954i $$0.134455\pi$$
$$410$$ −11220.0 −1.35150
$$411$$ 0 0
$$412$$ 13104.0 1.56696
$$413$$ 17280.0 2.05882
$$414$$ 0 0
$$415$$ 10676.0 1.26281
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 3840.00 0.449331
$$419$$ 10814.0 1.26086 0.630428 0.776248i $$-0.282880\pi$$
0.630428 + 0.776248i $$0.282880\pi$$
$$420$$ 0 0
$$421$$ 6535.00 0.756524 0.378262 0.925699i $$-0.376522\pi$$
0.378262 + 0.925699i $$0.376522\pi$$
$$422$$ 12280.0 1.41654
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 2132.00 0.243335
$$426$$ 0 0
$$427$$ −2900.00 −0.328667
$$428$$ −4176.00 −0.471623
$$429$$ 0 0
$$430$$ −10608.0 −1.18968
$$431$$ 1980.00 0.221284 0.110642 0.993860i $$-0.464709\pi$$
0.110642 + 0.993860i $$0.464709\pi$$
$$432$$ 0 0
$$433$$ −6929.00 −0.769022 −0.384511 0.923120i $$-0.625630\pi$$
−0.384511 + 0.923120i $$0.625630\pi$$
$$434$$ −5920.00 −0.654767
$$435$$ 0 0
$$436$$ 13072.0 1.43586
$$437$$ 2340.00 0.256150
$$438$$ 0 0
$$439$$ −4576.00 −0.497496 −0.248748 0.968568i $$-0.580019\pi$$
−0.248748 + 0.968568i $$0.580019\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 8812.00 0.945081 0.472540 0.881309i $$-0.343337\pi$$
0.472540 + 0.881309i $$0.343337\pi$$
$$444$$ 0 0
$$445$$ −4522.00 −0.481715
$$446$$ 21512.0 2.28391
$$447$$ 0 0
$$448$$ 10240.0 1.07990
$$449$$ 1918.00 0.201595 0.100797 0.994907i $$-0.467861\pi$$
0.100797 + 0.994907i $$0.467861\pi$$
$$450$$ 0 0
$$451$$ 5280.00 0.551276
$$452$$ −2616.00 −0.272226
$$453$$ 0 0
$$454$$ −15896.0 −1.64325
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 11761.0 1.20384 0.601922 0.798555i $$-0.294402\pi$$
0.601922 + 0.798555i $$0.294402\pi$$
$$458$$ 25192.0 2.57019
$$459$$ 0 0
$$460$$ −10608.0 −1.07522
$$461$$ 901.000 0.0910277 0.0455138 0.998964i $$-0.485507\pi$$
0.0455138 + 0.998964i $$0.485507\pi$$
$$462$$ 0 0
$$463$$ −1372.00 −0.137715 −0.0688577 0.997626i $$-0.521935\pi$$
−0.0688577 + 0.997626i $$0.521935\pi$$
$$464$$ 12608.0 1.26145
$$465$$ 0 0
$$466$$ −16120.0 −1.60246
$$467$$ 6396.00 0.633772 0.316886 0.948464i $$-0.397363\pi$$
0.316886 + 0.948464i $$0.397363\pi$$
$$468$$ 0 0
$$469$$ 17240.0 1.69738
$$470$$ −11016.0 −1.08113
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 4992.00 0.485269
$$474$$ 0 0
$$475$$ −4920.00 −0.475253
$$476$$ −2080.00 −0.200287
$$477$$ 0 0
$$478$$ −3936.00 −0.376629
$$479$$ 3270.00 0.311921 0.155960 0.987763i $$-0.450153\pi$$
0.155960 + 0.987763i $$0.450153\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −3772.00 −0.356452
$$483$$ 0 0
$$484$$ −2456.00 −0.230654
$$485$$ −4046.00 −0.378803
$$486$$ 0 0
$$487$$ −19920.0 −1.85351 −0.926757 0.375661i $$-0.877416\pi$$
−0.926757 + 0.375661i $$0.877416\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 3876.00 0.357347
$$491$$ −6552.00 −0.602215 −0.301108 0.953590i $$-0.597356\pi$$
−0.301108 + 0.953590i $$0.597356\pi$$
$$492$$ 0 0
$$493$$ −2561.00 −0.233959
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −4736.00 −0.428735
$$497$$ −13080.0 −1.18052
$$498$$ 0 0
$$499$$ −1746.00 −0.156637 −0.0783183 0.996928i $$-0.524955\pi$$
−0.0783183 + 0.996928i $$0.524955\pi$$
$$500$$ 5304.00 0.474404
$$501$$ 0 0
$$502$$ 10920.0 0.970883
$$503$$ −14692.0 −1.30235 −0.651177 0.758926i $$-0.725724\pi$$
−0.651177 + 0.758926i $$0.725724\pi$$
$$504$$ 0 0
$$505$$ 13923.0 1.22686
$$506$$ 9984.00 0.877160
$$507$$ 0 0
$$508$$ −17264.0 −1.50781
$$509$$ 8077.00 0.703353 0.351677 0.936122i $$-0.385612\pi$$
0.351677 + 0.936122i $$0.385612\pi$$
$$510$$ 0 0
$$511$$ 4300.00 0.372252
$$512$$ 16384.0 1.41421
$$513$$ 0 0
$$514$$ 7540.00 0.647033
$$515$$ 27846.0 2.38260
$$516$$ 0 0
$$517$$ 5184.00 0.440990
$$518$$ −18160.0 −1.54036
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −11247.0 −0.945758 −0.472879 0.881127i $$-0.656785\pi$$
−0.472879 + 0.881127i $$0.656785\pi$$
$$522$$ 0 0
$$523$$ 2732.00 0.228417 0.114208 0.993457i $$-0.463567\pi$$
0.114208 + 0.993457i $$0.463567\pi$$
$$524$$ −5840.00 −0.486873
$$525$$ 0 0
$$526$$ −16128.0 −1.33691
$$527$$ 962.000 0.0795168
$$528$$ 0 0
$$529$$ −6083.00 −0.499959
$$530$$ −6324.00 −0.518296
$$531$$ 0 0
$$532$$ 4800.00 0.391177
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −8874.00 −0.717115
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −16024.0 −1.28410
$$539$$ −1824.00 −0.145761
$$540$$ 0 0
$$541$$ 18375.0 1.46026 0.730132 0.683306i $$-0.239458\pi$$
0.730132 + 0.683306i $$0.239458\pi$$
$$542$$ 17184.0 1.36184
$$543$$ 0 0
$$544$$ −3328.00 −0.262292
$$545$$ 27778.0 2.18326
$$546$$ 0 0
$$547$$ −10346.0 −0.808708 −0.404354 0.914603i $$-0.632504\pi$$
−0.404354 + 0.914603i $$0.632504\pi$$
$$548$$ 13368.0 1.04207
$$549$$ 0 0
$$550$$ −20992.0 −1.62746
$$551$$ 5910.00 0.456941
$$552$$ 0 0
$$553$$ 1520.00 0.116884
$$554$$ −22204.0 −1.70281
$$555$$ 0 0
$$556$$ 7296.00 0.556510
$$557$$ 345.000 0.0262444 0.0131222 0.999914i $$-0.495823\pi$$
0.0131222 + 0.999914i $$0.495823\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 21760.0 1.64201
$$561$$ 0 0
$$562$$ −22228.0 −1.66838
$$563$$ 8580.00 0.642280 0.321140 0.947032i $$-0.395934\pi$$
0.321140 + 0.947032i $$0.395934\pi$$
$$564$$ 0 0
$$565$$ −5559.00 −0.413927
$$566$$ 12480.0 0.926808
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 19682.0 1.45011 0.725055 0.688691i $$-0.241814\pi$$
0.725055 + 0.688691i $$0.241814\pi$$
$$570$$ 0 0
$$571$$ 26624.0 1.95128 0.975639 0.219382i $$-0.0704042\pi$$
0.975639 + 0.219382i $$0.0704042\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 13200.0 0.959856
$$575$$ −12792.0 −0.927762
$$576$$ 0 0
$$577$$ 14101.0 1.01739 0.508694 0.860948i $$-0.330129\pi$$
0.508694 + 0.860948i $$0.330129\pi$$
$$578$$ −18976.0 −1.36557
$$579$$ 0 0
$$580$$ −26792.0 −1.91806
$$581$$ −12560.0 −0.896862
$$582$$ 0 0
$$583$$ 2976.00 0.211412
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 33204.0 2.34069
$$587$$ 1408.00 0.0990023 0.0495012 0.998774i $$-0.484237\pi$$
0.0495012 + 0.998774i $$0.484237\pi$$
$$588$$ 0 0
$$589$$ −2220.00 −0.155303
$$590$$ −58752.0 −4.09963
$$591$$ 0 0
$$592$$ −14528.0 −1.00861
$$593$$ −1241.00 −0.0859389 −0.0429694 0.999076i $$-0.513682\pi$$
−0.0429694 + 0.999076i $$0.513682\pi$$
$$594$$ 0 0
$$595$$ −4420.00 −0.304542
$$596$$ −16920.0 −1.16287
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −11078.0 −0.755651 −0.377825 0.925877i $$-0.623328\pi$$
−0.377825 + 0.925877i $$0.623328\pi$$
$$600$$ 0 0
$$601$$ −13817.0 −0.937782 −0.468891 0.883256i $$-0.655346\pi$$
−0.468891 + 0.883256i $$0.655346\pi$$
$$602$$ 12480.0 0.844928
$$603$$ 0 0
$$604$$ −4112.00 −0.277011
$$605$$ −5219.00 −0.350715
$$606$$ 0 0
$$607$$ 8270.00 0.552997 0.276498 0.961014i $$-0.410826\pi$$
0.276498 + 0.961014i $$0.410826\pi$$
$$608$$ 7680.00 0.512278
$$609$$ 0 0
$$610$$ 9860.00 0.654459
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −22273.0 −1.46753 −0.733767 0.679402i $$-0.762239\pi$$
−0.733767 + 0.679402i $$0.762239\pi$$
$$614$$ −34712.0 −2.28153
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18989.0 −1.23901 −0.619504 0.784993i $$-0.712666\pi$$
−0.619504 + 0.784993i $$0.712666\pi$$
$$618$$ 0 0
$$619$$ −72.0000 −0.00467516 −0.00233758 0.999997i $$-0.500744\pi$$
−0.00233758 + 0.999997i $$0.500744\pi$$
$$620$$ 10064.0 0.651903
$$621$$ 0 0
$$622$$ −34632.0 −2.23250
$$623$$ 5320.00 0.342121
$$624$$ 0 0
$$625$$ −9229.00 −0.590656
$$626$$ −21000.0 −1.34078
$$627$$ 0 0
$$628$$ 23208.0 1.47468
$$629$$ 2951.00 0.187065
$$630$$ 0 0
$$631$$ 23380.0 1.47503 0.737514 0.675331i $$-0.235999\pi$$
0.737514 + 0.675331i $$0.235999\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 25652.0 1.60689
$$635$$ −36686.0 −2.29266
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 25216.0 1.56475
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −6383.00 −0.393313 −0.196656 0.980472i $$-0.563008\pi$$
−0.196656 + 0.980472i $$0.563008\pi$$
$$642$$ 0 0
$$643$$ 17104.0 1.04901 0.524507 0.851406i $$-0.324250\pi$$
0.524507 + 0.851406i $$0.324250\pi$$
$$644$$ 12480.0 0.763635
$$645$$ 0 0
$$646$$ −1560.00 −0.0950114
$$647$$ −6994.00 −0.424981 −0.212490 0.977163i $$-0.568157\pi$$
−0.212490 + 0.977163i $$0.568157\pi$$
$$648$$ 0 0
$$649$$ 27648.0 1.67223
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −18880.0 −1.13405
$$653$$ 5250.00 0.314622 0.157311 0.987549i $$-0.449717\pi$$
0.157311 + 0.987549i $$0.449717\pi$$
$$654$$ 0 0
$$655$$ −12410.0 −0.740304
$$656$$ 10560.0 0.628504
$$657$$ 0 0
$$658$$ 12960.0 0.767832
$$659$$ 4340.00 0.256544 0.128272 0.991739i $$-0.459057\pi$$
0.128272 + 0.991739i $$0.459057\pi$$
$$660$$ 0 0
$$661$$ 4179.00 0.245907 0.122953 0.992412i $$-0.460763\pi$$
0.122953 + 0.992412i $$0.460763\pi$$
$$662$$ −13952.0 −0.819124
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 10200.0 0.594796
$$666$$ 0 0
$$667$$ 15366.0 0.892015
$$668$$ 2240.00 0.129743
$$669$$ 0 0
$$670$$ −58616.0 −3.37990
$$671$$ −4640.00 −0.266953
$$672$$ 0 0
$$673$$ 22867.0 1.30974 0.654872 0.755740i $$-0.272722\pi$$
0.654872 + 0.755740i $$0.272722\pi$$
$$674$$ −7332.00 −0.419018
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −5410.00 −0.307124 −0.153562 0.988139i $$-0.549075\pi$$
−0.153562 + 0.988139i $$0.549075\pi$$
$$678$$ 0 0
$$679$$ 4760.00 0.269031
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −9472.00 −0.531821
$$683$$ −13578.0 −0.760685 −0.380342 0.924846i $$-0.624194\pi$$
−0.380342 + 0.924846i $$0.624194\pi$$
$$684$$ 0 0
$$685$$ 28407.0 1.58449
$$686$$ 22880.0 1.27341
$$687$$ 0 0
$$688$$ 9984.00 0.553251
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −12744.0 −0.701599 −0.350799 0.936451i $$-0.614090\pi$$
−0.350799 + 0.936451i $$0.614090\pi$$
$$692$$ −10608.0 −0.582739
$$693$$ 0 0
$$694$$ −28920.0 −1.58183
$$695$$ 15504.0 0.846187
$$696$$ 0 0
$$697$$ −2145.00 −0.116568
$$698$$ 21032.0 1.14051
$$699$$ 0 0
$$700$$ −26240.0 −1.41683
$$701$$ −16406.0 −0.883946 −0.441973 0.897028i $$-0.645721\pi$$
−0.441973 + 0.897028i $$0.645721\pi$$
$$702$$ 0 0
$$703$$ −6810.00 −0.365354
$$704$$ 16384.0 0.877124
$$705$$ 0 0
$$706$$ 12652.0 0.674454
$$707$$ −16380.0 −0.871334
$$708$$ 0 0
$$709$$ −709.000 −0.0375558 −0.0187779 0.999824i $$-0.505978\pi$$
−0.0187779 + 0.999824i $$0.505978\pi$$
$$710$$ 44472.0 2.35071
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −5772.00 −0.303174
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −34112.0 −1.78048
$$717$$ 0 0
$$718$$ −40272.0 −2.09323
$$719$$ 7644.00 0.396486 0.198243 0.980153i $$-0.436477\pi$$
0.198243 + 0.980153i $$0.436477\pi$$
$$720$$ 0 0
$$721$$ −32760.0 −1.69216
$$722$$ −23836.0 −1.22865
$$723$$ 0 0
$$724$$ −3224.00 −0.165496
$$725$$ −32308.0 −1.65502
$$726$$ 0 0
$$727$$ −15808.0 −0.806446 −0.403223 0.915102i $$-0.632110\pi$$
−0.403223 + 0.915102i $$0.632110\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −14620.0 −0.741247
$$731$$ −2028.00 −0.102611
$$732$$ 0 0
$$733$$ 2583.00 0.130157 0.0650786 0.997880i $$-0.479270\pi$$
0.0650786 + 0.997880i $$0.479270\pi$$
$$734$$ 29752.0 1.49614
$$735$$ 0 0
$$736$$ 19968.0 1.00004
$$737$$ 27584.0 1.37866
$$738$$ 0 0
$$739$$ −4076.00 −0.202893 −0.101447 0.994841i $$-0.532347\pi$$
−0.101447 + 0.994841i $$0.532347\pi$$
$$740$$ 30872.0 1.53362
$$741$$ 0 0
$$742$$ 7440.00 0.368101
$$743$$ −34056.0 −1.68155 −0.840776 0.541383i $$-0.817901\pi$$
−0.840776 + 0.541383i $$0.817901\pi$$
$$744$$ 0 0
$$745$$ −35955.0 −1.76817
$$746$$ −38732.0 −1.90091
$$747$$ 0 0
$$748$$ −3328.00 −0.162679
$$749$$ 10440.0 0.509305
$$750$$ 0 0
$$751$$ −364.000 −0.0176865 −0.00884324 0.999961i $$-0.502815\pi$$
−0.00884324 + 0.999961i $$0.502815\pi$$
$$752$$ 10368.0 0.502769
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −8738.00 −0.421203
$$756$$ 0 0
$$757$$ −6914.00 −0.331960 −0.165980 0.986129i $$-0.553079\pi$$
−0.165980 + 0.986129i $$0.553079\pi$$
$$758$$ 4248.00 0.203554
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 13982.0 0.666028 0.333014 0.942922i $$-0.391934\pi$$
0.333014 + 0.942922i $$0.391934\pi$$
$$762$$ 0 0
$$763$$ −32680.0 −1.55058
$$764$$ 9968.00 0.472028
$$765$$ 0 0
$$766$$ −14128.0 −0.666404
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 18066.0 0.847174 0.423587 0.905855i $$-0.360771\pi$$
0.423587 + 0.905855i $$0.360771\pi$$
$$770$$ 43520.0 2.03682
$$771$$ 0 0
$$772$$ −2136.00 −0.0995807
$$773$$ 14434.0 0.671610 0.335805 0.941931i $$-0.390992\pi$$
0.335805 + 0.941931i $$0.390992\pi$$
$$774$$ 0 0
$$775$$ 12136.0 0.562501
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 44252.0 2.03922
$$779$$ 4950.00 0.227666
$$780$$ 0 0
$$781$$ −20928.0 −0.958851
$$782$$ −4056.00 −0.185476
$$783$$ 0 0
$$784$$ −3648.00 −0.166181
$$785$$ 49317.0 2.24229
$$786$$ 0 0
$$787$$ 15398.0 0.697433 0.348716 0.937228i $$-0.386618\pi$$
0.348716 + 0.937228i $$0.386618\pi$$
$$788$$ 10224.0 0.462202
$$789$$ 0 0
$$790$$ −5168.00 −0.232746
$$791$$ 6540.00 0.293977
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 23944.0 1.07020
$$795$$ 0 0
$$796$$ 33904.0 1.50967
$$797$$ 36842.0 1.63740 0.818702 0.574219i $$-0.194694\pi$$
0.818702 + 0.574219i $$0.194694\pi$$
$$798$$ 0 0
$$799$$ −2106.00 −0.0932477
$$800$$ −41984.0 −1.85545
$$801$$ 0 0
$$802$$ 23740.0 1.04525
$$803$$ 6880.00 0.302354
$$804$$ 0 0
$$805$$ 26520.0 1.16113
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −41511.0 −1.80402 −0.902008 0.431719i $$-0.857907\pi$$
−0.902008 + 0.431719i $$0.857907\pi$$
$$810$$ 0 0
$$811$$ −23066.0 −0.998714 −0.499357 0.866396i $$-0.666430\pi$$
−0.499357 + 0.866396i $$0.666430\pi$$
$$812$$ 31520.0 1.36224
$$813$$ 0 0
$$814$$ −29056.0 −1.25112
$$815$$ −40120.0 −1.72435
$$816$$ 0 0
$$817$$ 4680.00 0.200407
$$818$$ 60356.0 2.57983
$$819$$ 0 0
$$820$$ −22440.0 −0.955657
$$821$$ 28838.0 1.22589 0.612943 0.790127i $$-0.289985\pi$$
0.612943 + 0.790127i $$0.289985\pi$$
$$822$$ 0 0
$$823$$ 27456.0 1.16289 0.581443 0.813587i $$-0.302488\pi$$
0.581443 + 0.813587i $$0.302488\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 69120.0 2.91161
$$827$$ −33572.0 −1.41162 −0.705812 0.708399i $$-0.749418\pi$$
−0.705812 + 0.708399i $$0.749418\pi$$
$$828$$ 0 0
$$829$$ −45799.0 −1.91878 −0.959388 0.282090i $$-0.908972\pi$$
−0.959388 + 0.282090i $$0.908972\pi$$
$$830$$ 42704.0 1.78588
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 741.000 0.0308213
$$834$$ 0 0
$$835$$ 4760.00 0.197277
$$836$$ 7680.00 0.317725
$$837$$ 0 0
$$838$$ 43256.0 1.78312
$$839$$ 32286.0 1.32853 0.664265 0.747497i $$-0.268745\pi$$
0.664265 + 0.747497i $$0.268745\pi$$
$$840$$ 0 0
$$841$$ 14420.0 0.591250
$$842$$ 26140.0 1.06989
$$843$$ 0 0
$$844$$ 24560.0 1.00165
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 6140.00 0.249083
$$848$$ 5952.00 0.241029
$$849$$ 0 0
$$850$$ 8528.00 0.344127
$$851$$ −17706.0 −0.713224
$$852$$ 0 0
$$853$$ −20937.0 −0.840409 −0.420205 0.907429i $$-0.638042\pi$$
−0.420205 + 0.907429i $$0.638042\pi$$
$$854$$ −11600.0 −0.464805
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 7189.00 0.286548 0.143274 0.989683i $$-0.454237\pi$$
0.143274 + 0.989683i $$0.454237\pi$$
$$858$$ 0 0
$$859$$ −32498.0 −1.29082 −0.645412 0.763835i $$-0.723314\pi$$
−0.645412 + 0.763835i $$0.723314\pi$$
$$860$$ −21216.0 −0.841232
$$861$$ 0 0
$$862$$ 7920.00 0.312942
$$863$$ 8428.00 0.332436 0.166218 0.986089i $$-0.446844\pi$$
0.166218 + 0.986089i $$0.446844\pi$$
$$864$$ 0 0
$$865$$ −22542.0 −0.886071
$$866$$ −27716.0 −1.08756
$$867$$ 0 0
$$868$$ −11840.0 −0.462991
$$869$$ 2432.00 0.0949367
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 9360.00 0.362250
$$875$$ −13260.0 −0.512308
$$876$$ 0 0
$$877$$ 6847.00 0.263634 0.131817 0.991274i $$-0.457919\pi$$
0.131817 + 0.991274i $$0.457919\pi$$
$$878$$ −18304.0 −0.703565
$$879$$ 0 0
$$880$$ 34816.0 1.33369
$$881$$ −29731.0 −1.13696 −0.568481 0.822697i $$-0.692469\pi$$
−0.568481 + 0.822697i $$0.692469\pi$$
$$882$$ 0 0
$$883$$ −23738.0 −0.904697 −0.452348 0.891841i $$-0.649414\pi$$
−0.452348 + 0.891841i $$0.649414\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 35248.0 1.33655
$$887$$ −27588.0 −1.04432 −0.522161 0.852847i $$-0.674874\pi$$
−0.522161 + 0.852847i $$0.674874\pi$$
$$888$$ 0 0
$$889$$ 43160.0 1.62828
$$890$$ −18088.0 −0.681248
$$891$$ 0 0
$$892$$ 43024.0 1.61497
$$893$$ 4860.00 0.182121
$$894$$ 0 0
$$895$$ −72488.0 −2.70727
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 7672.00 0.285098
$$899$$ −14578.0 −0.540827
$$900$$ 0 0
$$901$$ −1209.00 −0.0447033
$$902$$ 21120.0 0.779622
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −6851.00 −0.251641
$$906$$ 0 0
$$907$$ −37128.0 −1.35922 −0.679611 0.733572i $$-0.737852\pi$$
−0.679611 + 0.733572i $$0.737852\pi$$
$$908$$ −31792.0 −1.16195
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −20516.0 −0.746131 −0.373066 0.927805i $$-0.621693\pi$$
−0.373066 + 0.927805i $$0.621693\pi$$
$$912$$ 0 0
$$913$$ −20096.0 −0.728456
$$914$$ 47044.0 1.70249
$$915$$ 0 0
$$916$$ 50384.0 1.81740
$$917$$ 14600.0 0.525774
$$918$$ 0 0
$$919$$ −21006.0 −0.753998 −0.376999 0.926214i $$-0.623044\pi$$
−0.376999 + 0.926214i $$0.623044\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 3604.00 0.128733
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 37228.0 1.32330
$$926$$ −5488.00 −0.194759
$$927$$ 0 0
$$928$$ 50432.0 1.78396
$$929$$ 20427.0 0.721408 0.360704 0.932680i $$-0.382536\pi$$
0.360704 + 0.932680i $$0.382536\pi$$
$$930$$ 0 0
$$931$$ −1710.00 −0.0601965
$$932$$ −32240.0 −1.13311
$$933$$ 0 0
$$934$$ 25584.0 0.896289
$$935$$ −7072.00 −0.247357
$$936$$ 0 0
$$937$$ 33191.0 1.15721 0.578603 0.815609i $$-0.303598\pi$$
0.578603 + 0.815609i $$0.303598\pi$$
$$938$$ 68960.0 2.40045
$$939$$ 0 0
$$940$$ −22032.0 −0.764473
$$941$$ −36422.0 −1.26177 −0.630884 0.775877i $$-0.717308\pi$$
−0.630884 + 0.775877i $$0.717308\pi$$
$$942$$ 0 0
$$943$$ 12870.0 0.444438
$$944$$ 55296.0 1.90650
$$945$$ 0 0
$$946$$ 19968.0 0.686275
$$947$$ −39630.0 −1.35988 −0.679938 0.733270i $$-0.737993\pi$$
−0.679938 + 0.733270i $$0.737993\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −19680.0 −0.672109
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −57642.0 −1.95929 −0.979647 0.200727i $$-0.935670\pi$$
−0.979647 + 0.200727i $$0.935670\pi$$
$$954$$ 0 0
$$955$$ 21182.0 0.717731
$$956$$ −7872.00 −0.266317
$$957$$ 0 0
$$958$$ 13080.0 0.441123
$$959$$ −33420.0 −1.12533
$$960$$ 0 0
$$961$$ −24315.0 −0.816186
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −7544.00 −0.252050
$$965$$ −4539.00 −0.151415
$$966$$ 0 0
$$967$$ −2162.00 −0.0718979 −0.0359489 0.999354i $$-0.511445\pi$$
−0.0359489 + 0.999354i $$0.511445\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −16184.0 −0.535708
$$971$$ 19758.0 0.653001 0.326501 0.945197i $$-0.394130\pi$$
0.326501 + 0.945197i $$0.394130\pi$$
$$972$$ 0 0
$$973$$ −18240.0 −0.600974
$$974$$ −79680.0 −2.62126
$$975$$ 0 0
$$976$$ −9280.00 −0.304350
$$977$$ −12489.0 −0.408965 −0.204482 0.978870i $$-0.565551\pi$$
−0.204482 + 0.978870i $$0.565551\pi$$
$$978$$ 0 0
$$979$$ 8512.00 0.277880
$$980$$ 7752.00 0.252682
$$981$$ 0 0
$$982$$ −26208.0 −0.851661
$$983$$ −28658.0 −0.929856 −0.464928 0.885349i $$-0.653920\pi$$
−0.464928 + 0.885349i $$0.653920\pi$$
$$984$$ 0 0
$$985$$ 21726.0 0.702790
$$986$$ −10244.0 −0.330868
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 12168.0 0.391223
$$990$$ 0 0
$$991$$ −42794.0 −1.37174 −0.685871 0.727723i $$-0.740579\pi$$
−0.685871 + 0.727723i $$0.740579\pi$$
$$992$$ −18944.0 −0.606323
$$993$$ 0 0
$$994$$ −52320.0 −1.66951
$$995$$ 72046.0 2.29549
$$996$$ 0 0
$$997$$ −52583.0 −1.67033 −0.835166 0.549998i $$-0.814628\pi$$
−0.835166 + 0.549998i $$0.814628\pi$$
$$998$$ −6984.00 −0.221518
$$999$$ 0 0
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 1521.4.a.k.1.1 1
3.2 odd 2 169.4.a.a.1.1 1
13.4 even 6 117.4.g.c.55.1 2
13.10 even 6 117.4.g.c.100.1 2
13.12 even 2 1521.4.a.b.1.1 1
39.2 even 12 169.4.e.c.147.1 4
39.5 even 4 169.4.b.c.168.2 2
39.8 even 4 169.4.b.c.168.1 2
39.11 even 12 169.4.e.c.147.2 4
39.17 odd 6 13.4.c.a.3.1 2
39.20 even 12 169.4.e.c.23.1 4
39.23 odd 6 13.4.c.a.9.1 yes 2
39.29 odd 6 169.4.c.d.22.1 2
39.32 even 12 169.4.e.c.23.2 4
39.35 odd 6 169.4.c.d.146.1 2
39.38 odd 2 169.4.a.d.1.1 1
156.23 even 6 208.4.i.b.113.1 2
156.95 even 6 208.4.i.b.81.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.a.3.1 2 39.17 odd 6
13.4.c.a.9.1 yes 2 39.23 odd 6
117.4.g.c.55.1 2 13.4 even 6
117.4.g.c.100.1 2 13.10 even 6
169.4.a.a.1.1 1 3.2 odd 2
169.4.a.d.1.1 1 39.38 odd 2
169.4.b.c.168.1 2 39.8 even 4
169.4.b.c.168.2 2 39.5 even 4
169.4.c.d.22.1 2 39.29 odd 6
169.4.c.d.146.1 2 39.35 odd 6
169.4.e.c.23.1 4 39.20 even 12
169.4.e.c.23.2 4 39.32 even 12
169.4.e.c.147.1 4 39.2 even 12
169.4.e.c.147.2 4 39.11 even 12
208.4.i.b.81.1 2 156.95 even 6
208.4.i.b.113.1 2 156.23 even 6
1521.4.a.b.1.1 1 13.12 even 2
1521.4.a.k.1.1 1 1.1 even 1 trivial