# Properties

 Label 2028.4.a.a.1.1 Level $2028$ Weight $4$ Character 2028.1 Self dual yes Analytic conductor $119.656$ 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] = [2028,4,Mod(1,2028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2028, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2028.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2028 = 2^{2} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2028.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: $$119.655873492$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 156) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2028.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +6.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +6.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} -36.0000 q^{11} -18.0000 q^{15} +66.0000 q^{17} -56.0000 q^{19} -12.0000 q^{21} +96.0000 q^{23} -89.0000 q^{25} -27.0000 q^{27} +222.000 q^{29} -260.000 q^{31} +108.000 q^{33} +24.0000 q^{35} +106.000 q^{37} +90.0000 q^{41} +44.0000 q^{43} +54.0000 q^{45} -168.000 q^{47} -327.000 q^{49} -198.000 q^{51} +30.0000 q^{53} -216.000 q^{55} +168.000 q^{57} -348.000 q^{59} -346.000 q^{61} +36.0000 q^{63} +256.000 q^{67} -288.000 q^{69} +168.000 q^{71} +814.000 q^{73} +267.000 q^{75} -144.000 q^{77} +200.000 q^{79} +81.0000 q^{81} -1236.00 q^{83} +396.000 q^{85} -666.000 q^{87} -318.000 q^{89} +780.000 q^{93} -336.000 q^{95} +502.000 q^{97} -324.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ 6.00000 0.536656 0.268328 0.963328i $$-0.413529\pi$$
0.268328 + 0.963328i $$0.413529\pi$$
$$6$$ 0 0
$$7$$ 4.00000 0.215980 0.107990 0.994152i $$-0.465559\pi$$
0.107990 + 0.994152i $$0.465559\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −36.0000 −0.986764 −0.493382 0.869813i $$-0.664240\pi$$
−0.493382 + 0.869813i $$0.664240\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ −18.0000 −0.309839
$$16$$ 0 0
$$17$$ 66.0000 0.941609 0.470804 0.882238i $$-0.343964\pi$$
0.470804 + 0.882238i $$0.343964\pi$$
$$18$$ 0 0
$$19$$ −56.0000 −0.676173 −0.338086 0.941115i $$-0.609780\pi$$
−0.338086 + 0.941115i $$0.609780\pi$$
$$20$$ 0 0
$$21$$ −12.0000 −0.124696
$$22$$ 0 0
$$23$$ 96.0000 0.870321 0.435161 0.900353i $$-0.356692\pi$$
0.435161 + 0.900353i $$0.356692\pi$$
$$24$$ 0 0
$$25$$ −89.0000 −0.712000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 222.000 1.42153 0.710765 0.703430i $$-0.248349\pi$$
0.710765 + 0.703430i $$0.248349\pi$$
$$30$$ 0 0
$$31$$ −260.000 −1.50637 −0.753184 0.657810i $$-0.771483\pi$$
−0.753184 + 0.657810i $$0.771483\pi$$
$$32$$ 0 0
$$33$$ 108.000 0.569709
$$34$$ 0 0
$$35$$ 24.0000 0.115907
$$36$$ 0 0
$$37$$ 106.000 0.470981 0.235490 0.971877i $$-0.424330\pi$$
0.235490 + 0.971877i $$0.424330\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 90.0000 0.342820 0.171410 0.985200i $$-0.445168\pi$$
0.171410 + 0.985200i $$0.445168\pi$$
$$42$$ 0 0
$$43$$ 44.0000 0.156045 0.0780225 0.996952i $$-0.475139\pi$$
0.0780225 + 0.996952i $$0.475139\pi$$
$$44$$ 0 0
$$45$$ 54.0000 0.178885
$$46$$ 0 0
$$47$$ −168.000 −0.521390 −0.260695 0.965421i $$-0.583952\pi$$
−0.260695 + 0.965421i $$0.583952\pi$$
$$48$$ 0 0
$$49$$ −327.000 −0.953353
$$50$$ 0 0
$$51$$ −198.000 −0.543638
$$52$$ 0 0
$$53$$ 30.0000 0.0777513 0.0388756 0.999244i $$-0.487622\pi$$
0.0388756 + 0.999244i $$0.487622\pi$$
$$54$$ 0 0
$$55$$ −216.000 −0.529553
$$56$$ 0 0
$$57$$ 168.000 0.390388
$$58$$ 0 0
$$59$$ −348.000 −0.767894 −0.383947 0.923355i $$-0.625435\pi$$
−0.383947 + 0.923355i $$0.625435\pi$$
$$60$$ 0 0
$$61$$ −346.000 −0.726242 −0.363121 0.931742i $$-0.618289\pi$$
−0.363121 + 0.931742i $$0.618289\pi$$
$$62$$ 0 0
$$63$$ 36.0000 0.0719932
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 256.000 0.466797 0.233398 0.972381i $$-0.425015\pi$$
0.233398 + 0.972381i $$0.425015\pi$$
$$68$$ 0 0
$$69$$ −288.000 −0.502480
$$70$$ 0 0
$$71$$ 168.000 0.280816 0.140408 0.990094i $$-0.455159\pi$$
0.140408 + 0.990094i $$0.455159\pi$$
$$72$$ 0 0
$$73$$ 814.000 1.30509 0.652544 0.757750i $$-0.273702\pi$$
0.652544 + 0.757750i $$0.273702\pi$$
$$74$$ 0 0
$$75$$ 267.000 0.411073
$$76$$ 0 0
$$77$$ −144.000 −0.213121
$$78$$ 0 0
$$79$$ 200.000 0.284832 0.142416 0.989807i $$-0.454513\pi$$
0.142416 + 0.989807i $$0.454513\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −1236.00 −1.63456 −0.817281 0.576240i $$-0.804520\pi$$
−0.817281 + 0.576240i $$0.804520\pi$$
$$84$$ 0 0
$$85$$ 396.000 0.505320
$$86$$ 0 0
$$87$$ −666.000 −0.820721
$$88$$ 0 0
$$89$$ −318.000 −0.378741 −0.189370 0.981906i $$-0.560645\pi$$
−0.189370 + 0.981906i $$0.560645\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 780.000 0.869701
$$94$$ 0 0
$$95$$ −336.000 −0.362872
$$96$$ 0 0
$$97$$ 502.000 0.525468 0.262734 0.964868i $$-0.415376\pi$$
0.262734 + 0.964868i $$0.415376\pi$$
$$98$$ 0 0
$$99$$ −324.000 −0.328921
$$100$$ 0 0
$$101$$ 1062.00 1.04627 0.523133 0.852251i $$-0.324763\pi$$
0.523133 + 0.852251i $$0.324763\pi$$
$$102$$ 0 0
$$103$$ −64.0000 −0.0612243 −0.0306122 0.999531i $$-0.509746\pi$$
−0.0306122 + 0.999531i $$0.509746\pi$$
$$104$$ 0 0
$$105$$ −72.0000 −0.0669189
$$106$$ 0 0
$$107$$ −444.000 −0.401150 −0.200575 0.979678i $$-0.564281\pi$$
−0.200575 + 0.979678i $$0.564281\pi$$
$$108$$ 0 0
$$109$$ −1382.00 −1.21442 −0.607209 0.794542i $$-0.707711\pi$$
−0.607209 + 0.794542i $$0.707711\pi$$
$$110$$ 0 0
$$111$$ −318.000 −0.271921
$$112$$ 0 0
$$113$$ −870.000 −0.724272 −0.362136 0.932125i $$-0.617952\pi$$
−0.362136 + 0.932125i $$0.617952\pi$$
$$114$$ 0 0
$$115$$ 576.000 0.467063
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 264.000 0.203368
$$120$$ 0 0
$$121$$ −35.0000 −0.0262960
$$122$$ 0 0
$$123$$ −270.000 −0.197927
$$124$$ 0 0
$$125$$ −1284.00 −0.918756
$$126$$ 0 0
$$127$$ 464.000 0.324200 0.162100 0.986774i $$-0.448173\pi$$
0.162100 + 0.986774i $$0.448173\pi$$
$$128$$ 0 0
$$129$$ −132.000 −0.0900927
$$130$$ 0 0
$$131$$ 1548.00 1.03244 0.516219 0.856457i $$-0.327339\pi$$
0.516219 + 0.856457i $$0.327339\pi$$
$$132$$ 0 0
$$133$$ −224.000 −0.146040
$$134$$ 0 0
$$135$$ −162.000 −0.103280
$$136$$ 0 0
$$137$$ −294.000 −0.183344 −0.0916720 0.995789i $$-0.529221\pi$$
−0.0916720 + 0.995789i $$0.529221\pi$$
$$138$$ 0 0
$$139$$ 2564.00 1.56457 0.782286 0.622919i $$-0.214053\pi$$
0.782286 + 0.622919i $$0.214053\pi$$
$$140$$ 0 0
$$141$$ 504.000 0.301025
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1332.00 0.762873
$$146$$ 0 0
$$147$$ 981.000 0.550418
$$148$$ 0 0
$$149$$ −114.000 −0.0626795 −0.0313397 0.999509i $$-0.509977\pi$$
−0.0313397 + 0.999509i $$0.509977\pi$$
$$150$$ 0 0
$$151$$ −2036.00 −1.09727 −0.548634 0.836063i $$-0.684852\pi$$
−0.548634 + 0.836063i $$0.684852\pi$$
$$152$$ 0 0
$$153$$ 594.000 0.313870
$$154$$ 0 0
$$155$$ −1560.00 −0.808401
$$156$$ 0 0
$$157$$ 2870.00 1.45892 0.729462 0.684022i $$-0.239771\pi$$
0.729462 + 0.684022i $$0.239771\pi$$
$$158$$ 0 0
$$159$$ −90.0000 −0.0448897
$$160$$ 0 0
$$161$$ 384.000 0.187972
$$162$$ 0 0
$$163$$ −1472.00 −0.707337 −0.353669 0.935371i $$-0.615066\pi$$
−0.353669 + 0.935371i $$0.615066\pi$$
$$164$$ 0 0
$$165$$ 648.000 0.305738
$$166$$ 0 0
$$167$$ 240.000 0.111208 0.0556041 0.998453i $$-0.482292\pi$$
0.0556041 + 0.998453i $$0.482292\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −504.000 −0.225391
$$172$$ 0 0
$$173$$ −306.000 −0.134478 −0.0672392 0.997737i $$-0.521419\pi$$
−0.0672392 + 0.997737i $$0.521419\pi$$
$$174$$ 0 0
$$175$$ −356.000 −0.153778
$$176$$ 0 0
$$177$$ 1044.00 0.443344
$$178$$ 0 0
$$179$$ −2052.00 −0.856836 −0.428418 0.903581i $$-0.640929\pi$$
−0.428418 + 0.903581i $$0.640929\pi$$
$$180$$ 0 0
$$181$$ −4498.00 −1.84715 −0.923574 0.383421i $$-0.874746\pi$$
−0.923574 + 0.383421i $$0.874746\pi$$
$$182$$ 0 0
$$183$$ 1038.00 0.419296
$$184$$ 0 0
$$185$$ 636.000 0.252755
$$186$$ 0 0
$$187$$ −2376.00 −0.929146
$$188$$ 0 0
$$189$$ −108.000 −0.0415653
$$190$$ 0 0
$$191$$ −4056.00 −1.53655 −0.768277 0.640117i $$-0.778886\pi$$
−0.768277 + 0.640117i $$0.778886\pi$$
$$192$$ 0 0
$$193$$ 2062.00 0.769047 0.384523 0.923115i $$-0.374366\pi$$
0.384523 + 0.923115i $$0.374366\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4374.00 1.58190 0.790951 0.611880i $$-0.209586\pi$$
0.790951 + 0.611880i $$0.209586\pi$$
$$198$$ 0 0
$$199$$ −2536.00 −0.903378 −0.451689 0.892175i $$-0.649178\pi$$
−0.451689 + 0.892175i $$0.649178\pi$$
$$200$$ 0 0
$$201$$ −768.000 −0.269505
$$202$$ 0 0
$$203$$ 888.000 0.307022
$$204$$ 0 0
$$205$$ 540.000 0.183977
$$206$$ 0 0
$$207$$ 864.000 0.290107
$$208$$ 0 0
$$209$$ 2016.00 0.667223
$$210$$ 0 0
$$211$$ −4444.00 −1.44994 −0.724971 0.688780i $$-0.758147\pi$$
−0.724971 + 0.688780i $$0.758147\pi$$
$$212$$ 0 0
$$213$$ −504.000 −0.162129
$$214$$ 0 0
$$215$$ 264.000 0.0837426
$$216$$ 0 0
$$217$$ −1040.00 −0.325345
$$218$$ 0 0
$$219$$ −2442.00 −0.753493
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 2716.00 0.815591 0.407796 0.913073i $$-0.366298\pi$$
0.407796 + 0.913073i $$0.366298\pi$$
$$224$$ 0 0
$$225$$ −801.000 −0.237333
$$226$$ 0 0
$$227$$ 4692.00 1.37189 0.685945 0.727653i $$-0.259389\pi$$
0.685945 + 0.727653i $$0.259389\pi$$
$$228$$ 0 0
$$229$$ −6446.00 −1.86010 −0.930052 0.367429i $$-0.880238\pi$$
−0.930052 + 0.367429i $$0.880238\pi$$
$$230$$ 0 0
$$231$$ 432.000 0.123046
$$232$$ 0 0
$$233$$ −3102.00 −0.872184 −0.436092 0.899902i $$-0.643638\pi$$
−0.436092 + 0.899902i $$0.643638\pi$$
$$234$$ 0 0
$$235$$ −1008.00 −0.279807
$$236$$ 0 0
$$237$$ −600.000 −0.164448
$$238$$ 0 0
$$239$$ −816.000 −0.220848 −0.110424 0.993885i $$-0.535221\pi$$
−0.110424 + 0.993885i $$0.535221\pi$$
$$240$$ 0 0
$$241$$ −3818.00 −1.02049 −0.510247 0.860028i $$-0.670446\pi$$
−0.510247 + 0.860028i $$0.670446\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −1962.00 −0.511623
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 3708.00 0.943715
$$250$$ 0 0
$$251$$ −6612.00 −1.66273 −0.831366 0.555725i $$-0.812441\pi$$
−0.831366 + 0.555725i $$0.812441\pi$$
$$252$$ 0 0
$$253$$ −3456.00 −0.858802
$$254$$ 0 0
$$255$$ −1188.00 −0.291747
$$256$$ 0 0
$$257$$ −4806.00 −1.16650 −0.583249 0.812293i $$-0.698219\pi$$
−0.583249 + 0.812293i $$0.698219\pi$$
$$258$$ 0 0
$$259$$ 424.000 0.101722
$$260$$ 0 0
$$261$$ 1998.00 0.473843
$$262$$ 0 0
$$263$$ −4584.00 −1.07476 −0.537379 0.843341i $$-0.680586\pi$$
−0.537379 + 0.843341i $$0.680586\pi$$
$$264$$ 0 0
$$265$$ 180.000 0.0417257
$$266$$ 0 0
$$267$$ 954.000 0.218666
$$268$$ 0 0
$$269$$ 7134.00 1.61698 0.808490 0.588510i $$-0.200285\pi$$
0.808490 + 0.588510i $$0.200285\pi$$
$$270$$ 0 0
$$271$$ −3140.00 −0.703843 −0.351921 0.936030i $$-0.614472\pi$$
−0.351921 + 0.936030i $$0.614472\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 3204.00 0.702576
$$276$$ 0 0
$$277$$ −4786.00 −1.03813 −0.519067 0.854734i $$-0.673720\pi$$
−0.519067 + 0.854734i $$0.673720\pi$$
$$278$$ 0 0
$$279$$ −2340.00 −0.502122
$$280$$ 0 0
$$281$$ −3798.00 −0.806298 −0.403149 0.915134i $$-0.632084\pi$$
−0.403149 + 0.915134i $$0.632084\pi$$
$$282$$ 0 0
$$283$$ 3572.00 0.750295 0.375147 0.926965i $$-0.377592\pi$$
0.375147 + 0.926965i $$0.377592\pi$$
$$284$$ 0 0
$$285$$ 1008.00 0.209504
$$286$$ 0 0
$$287$$ 360.000 0.0740423
$$288$$ 0 0
$$289$$ −557.000 −0.113373
$$290$$ 0 0
$$291$$ −1506.00 −0.303379
$$292$$ 0 0
$$293$$ −7122.00 −1.42004 −0.710020 0.704182i $$-0.751314\pi$$
−0.710020 + 0.704182i $$0.751314\pi$$
$$294$$ 0 0
$$295$$ −2088.00 −0.412095
$$296$$ 0 0
$$297$$ 972.000 0.189903
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 176.000 0.0337026
$$302$$ 0 0
$$303$$ −3186.00 −0.604062
$$304$$ 0 0
$$305$$ −2076.00 −0.389742
$$306$$ 0 0
$$307$$ 6856.00 1.27457 0.637284 0.770629i $$-0.280058\pi$$
0.637284 + 0.770629i $$0.280058\pi$$
$$308$$ 0 0
$$309$$ 192.000 0.0353479
$$310$$ 0 0
$$311$$ −8832.00 −1.61034 −0.805172 0.593042i $$-0.797927\pi$$
−0.805172 + 0.593042i $$0.797927\pi$$
$$312$$ 0 0
$$313$$ 3626.00 0.654804 0.327402 0.944885i $$-0.393827\pi$$
0.327402 + 0.944885i $$0.393827\pi$$
$$314$$ 0 0
$$315$$ 216.000 0.0386356
$$316$$ 0 0
$$317$$ −10146.0 −1.79765 −0.898827 0.438304i $$-0.855579\pi$$
−0.898827 + 0.438304i $$0.855579\pi$$
$$318$$ 0 0
$$319$$ −7992.00 −1.40272
$$320$$ 0 0
$$321$$ 1332.00 0.231604
$$322$$ 0 0
$$323$$ −3696.00 −0.636690
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 4146.00 0.701145
$$328$$ 0 0
$$329$$ −672.000 −0.112610
$$330$$ 0 0
$$331$$ −6536.00 −1.08535 −0.542675 0.839943i $$-0.682589\pi$$
−0.542675 + 0.839943i $$0.682589\pi$$
$$332$$ 0 0
$$333$$ 954.000 0.156994
$$334$$ 0 0
$$335$$ 1536.00 0.250509
$$336$$ 0 0
$$337$$ −6094.00 −0.985048 −0.492524 0.870299i $$-0.663926\pi$$
−0.492524 + 0.870299i $$0.663926\pi$$
$$338$$ 0 0
$$339$$ 2610.00 0.418159
$$340$$ 0 0
$$341$$ 9360.00 1.48643
$$342$$ 0 0
$$343$$ −2680.00 −0.421885
$$344$$ 0 0
$$345$$ −1728.00 −0.269659
$$346$$ 0 0
$$347$$ −2724.00 −0.421418 −0.210709 0.977549i $$-0.567577\pi$$
−0.210709 + 0.977549i $$0.567577\pi$$
$$348$$ 0 0
$$349$$ 1522.00 0.233441 0.116720 0.993165i $$-0.462762\pi$$
0.116720 + 0.993165i $$0.462762\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1362.00 0.205360 0.102680 0.994714i $$-0.467258\pi$$
0.102680 + 0.994714i $$0.467258\pi$$
$$354$$ 0 0
$$355$$ 1008.00 0.150702
$$356$$ 0 0
$$357$$ −792.000 −0.117415
$$358$$ 0 0
$$359$$ −8880.00 −1.30548 −0.652742 0.757581i $$-0.726381\pi$$
−0.652742 + 0.757581i $$0.726381\pi$$
$$360$$ 0 0
$$361$$ −3723.00 −0.542790
$$362$$ 0 0
$$363$$ 105.000 0.0151820
$$364$$ 0 0
$$365$$ 4884.00 0.700384
$$366$$ 0 0
$$367$$ −3712.00 −0.527970 −0.263985 0.964527i $$-0.585037\pi$$
−0.263985 + 0.964527i $$0.585037\pi$$
$$368$$ 0 0
$$369$$ 810.000 0.114273
$$370$$ 0 0
$$371$$ 120.000 0.0167927
$$372$$ 0 0
$$373$$ 5726.00 0.794855 0.397428 0.917634i $$-0.369903\pi$$
0.397428 + 0.917634i $$0.369903\pi$$
$$374$$ 0 0
$$375$$ 3852.00 0.530444
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 13168.0 1.78468 0.892341 0.451361i $$-0.149061\pi$$
0.892341 + 0.451361i $$0.149061\pi$$
$$380$$ 0 0
$$381$$ −1392.00 −0.187177
$$382$$ 0 0
$$383$$ −4872.00 −0.649994 −0.324997 0.945715i $$-0.605363\pi$$
−0.324997 + 0.945715i $$0.605363\pi$$
$$384$$ 0 0
$$385$$ −864.000 −0.114373
$$386$$ 0 0
$$387$$ 396.000 0.0520150
$$388$$ 0 0
$$389$$ −1266.00 −0.165010 −0.0825048 0.996591i $$-0.526292\pi$$
−0.0825048 + 0.996591i $$0.526292\pi$$
$$390$$ 0 0
$$391$$ 6336.00 0.819502
$$392$$ 0 0
$$393$$ −4644.00 −0.596078
$$394$$ 0 0
$$395$$ 1200.00 0.152857
$$396$$ 0 0
$$397$$ 4882.00 0.617180 0.308590 0.951195i $$-0.400143\pi$$
0.308590 + 0.951195i $$0.400143\pi$$
$$398$$ 0 0
$$399$$ 672.000 0.0843160
$$400$$ 0 0
$$401$$ 90.0000 0.0112079 0.00560397 0.999984i $$-0.498216\pi$$
0.00560397 + 0.999984i $$0.498216\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 486.000 0.0596285
$$406$$ 0 0
$$407$$ −3816.00 −0.464747
$$408$$ 0 0
$$409$$ −2354.00 −0.284591 −0.142296 0.989824i $$-0.545448\pi$$
−0.142296 + 0.989824i $$0.545448\pi$$
$$410$$ 0 0
$$411$$ 882.000 0.105854
$$412$$ 0 0
$$413$$ −1392.00 −0.165849
$$414$$ 0 0
$$415$$ −7416.00 −0.877198
$$416$$ 0 0
$$417$$ −7692.00 −0.903307
$$418$$ 0 0
$$419$$ 7020.00 0.818495 0.409248 0.912423i $$-0.365791\pi$$
0.409248 + 0.912423i $$0.365791\pi$$
$$420$$ 0 0
$$421$$ −302.000 −0.0349610 −0.0174805 0.999847i $$-0.505564\pi$$
−0.0174805 + 0.999847i $$0.505564\pi$$
$$422$$ 0 0
$$423$$ −1512.00 −0.173797
$$424$$ 0 0
$$425$$ −5874.00 −0.670426
$$426$$ 0 0
$$427$$ −1384.00 −0.156854
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −9816.00 −1.09703 −0.548515 0.836141i $$-0.684807\pi$$
−0.548515 + 0.836141i $$0.684807\pi$$
$$432$$ 0 0
$$433$$ −14782.0 −1.64059 −0.820297 0.571937i $$-0.806192\pi$$
−0.820297 + 0.571937i $$0.806192\pi$$
$$434$$ 0 0
$$435$$ −3996.00 −0.440445
$$436$$ 0 0
$$437$$ −5376.00 −0.588487
$$438$$ 0 0
$$439$$ 3584.00 0.389647 0.194823 0.980838i $$-0.437587\pi$$
0.194823 + 0.980838i $$0.437587\pi$$
$$440$$ 0 0
$$441$$ −2943.00 −0.317784
$$442$$ 0 0
$$443$$ 180.000 0.0193049 0.00965244 0.999953i $$-0.496927\pi$$
0.00965244 + 0.999953i $$0.496927\pi$$
$$444$$ 0 0
$$445$$ −1908.00 −0.203254
$$446$$ 0 0
$$447$$ 342.000 0.0361880
$$448$$ 0 0
$$449$$ 3450.00 0.362618 0.181309 0.983426i $$-0.441967\pi$$
0.181309 + 0.983426i $$0.441967\pi$$
$$450$$ 0 0
$$451$$ −3240.00 −0.338283
$$452$$ 0 0
$$453$$ 6108.00 0.633507
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16654.0 1.70469 0.852343 0.522984i $$-0.175181\pi$$
0.852343 + 0.522984i $$0.175181\pi$$
$$458$$ 0 0
$$459$$ −1782.00 −0.181213
$$460$$ 0 0
$$461$$ 14046.0 1.41906 0.709531 0.704674i $$-0.248907\pi$$
0.709531 + 0.704674i $$0.248907\pi$$
$$462$$ 0 0
$$463$$ 4588.00 0.460524 0.230262 0.973129i $$-0.426042\pi$$
0.230262 + 0.973129i $$0.426042\pi$$
$$464$$ 0 0
$$465$$ 4680.00 0.466731
$$466$$ 0 0
$$467$$ −15372.0 −1.52319 −0.761597 0.648051i $$-0.775584\pi$$
−0.761597 + 0.648051i $$0.775584\pi$$
$$468$$ 0 0
$$469$$ 1024.00 0.100819
$$470$$ 0 0
$$471$$ −8610.00 −0.842310
$$472$$ 0 0
$$473$$ −1584.00 −0.153980
$$474$$ 0 0
$$475$$ 4984.00 0.481435
$$476$$ 0 0
$$477$$ 270.000 0.0259171
$$478$$ 0 0
$$479$$ −12864.0 −1.22708 −0.613540 0.789664i $$-0.710255\pi$$
−0.613540 + 0.789664i $$0.710255\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −1152.00 −0.108525
$$484$$ 0 0
$$485$$ 3012.00 0.281996
$$486$$ 0 0
$$487$$ 10276.0 0.956160 0.478080 0.878316i $$-0.341333\pi$$
0.478080 + 0.878316i $$0.341333\pi$$
$$488$$ 0 0
$$489$$ 4416.00 0.408381
$$490$$ 0 0
$$491$$ −11220.0 −1.03127 −0.515633 0.856810i $$-0.672443\pi$$
−0.515633 + 0.856810i $$0.672443\pi$$
$$492$$ 0 0
$$493$$ 14652.0 1.33853
$$494$$ 0 0
$$495$$ −1944.00 −0.176518
$$496$$ 0 0
$$497$$ 672.000 0.0606505
$$498$$ 0 0
$$499$$ −17264.0 −1.54878 −0.774392 0.632707i $$-0.781944\pi$$
−0.774392 + 0.632707i $$0.781944\pi$$
$$500$$ 0 0
$$501$$ −720.000 −0.0642060
$$502$$ 0 0
$$503$$ −1896.00 −0.168069 −0.0840343 0.996463i $$-0.526781\pi$$
−0.0840343 + 0.996463i $$0.526781\pi$$
$$504$$ 0 0
$$505$$ 6372.00 0.561486
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −5010.00 −0.436276 −0.218138 0.975918i $$-0.569998\pi$$
−0.218138 + 0.975918i $$0.569998\pi$$
$$510$$ 0 0
$$511$$ 3256.00 0.281873
$$512$$ 0 0
$$513$$ 1512.00 0.130129
$$514$$ 0 0
$$515$$ −384.000 −0.0328564
$$516$$ 0 0
$$517$$ 6048.00 0.514489
$$518$$ 0 0
$$519$$ 918.000 0.0776411
$$520$$ 0 0
$$521$$ 8610.00 0.724013 0.362007 0.932176i $$-0.382092\pi$$
0.362007 + 0.932176i $$0.382092\pi$$
$$522$$ 0 0
$$523$$ −5308.00 −0.443791 −0.221895 0.975070i $$-0.571224\pi$$
−0.221895 + 0.975070i $$0.571224\pi$$
$$524$$ 0 0
$$525$$ 1068.00 0.0887835
$$526$$ 0 0
$$527$$ −17160.0 −1.41841
$$528$$ 0 0
$$529$$ −2951.00 −0.242541
$$530$$ 0 0
$$531$$ −3132.00 −0.255965
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −2664.00 −0.215280
$$536$$ 0 0
$$537$$ 6156.00 0.494695
$$538$$ 0 0
$$539$$ 11772.0 0.940735
$$540$$ 0 0
$$541$$ −6182.00 −0.491285 −0.245642 0.969361i $$-0.578999\pi$$
−0.245642 + 0.969361i $$0.578999\pi$$
$$542$$ 0 0
$$543$$ 13494.0 1.06645
$$544$$ 0 0
$$545$$ −8292.00 −0.651725
$$546$$ 0 0
$$547$$ 1292.00 0.100991 0.0504954 0.998724i $$-0.483920\pi$$
0.0504954 + 0.998724i $$0.483920\pi$$
$$548$$ 0 0
$$549$$ −3114.00 −0.242081
$$550$$ 0 0
$$551$$ −12432.0 −0.961200
$$552$$ 0 0
$$553$$ 800.000 0.0615180
$$554$$ 0 0
$$555$$ −1908.00 −0.145928
$$556$$ 0 0
$$557$$ 12774.0 0.971727 0.485863 0.874035i $$-0.338505\pi$$
0.485863 + 0.874035i $$0.338505\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 7128.00 0.536443
$$562$$ 0 0
$$563$$ 16908.0 1.26570 0.632848 0.774276i $$-0.281886\pi$$
0.632848 + 0.774276i $$0.281886\pi$$
$$564$$ 0 0
$$565$$ −5220.00 −0.388685
$$566$$ 0 0
$$567$$ 324.000 0.0239977
$$568$$ 0 0
$$569$$ −11214.0 −0.826213 −0.413107 0.910683i $$-0.635556\pi$$
−0.413107 + 0.910683i $$0.635556\pi$$
$$570$$ 0 0
$$571$$ 25220.0 1.84838 0.924189 0.381935i $$-0.124742\pi$$
0.924189 + 0.381935i $$0.124742\pi$$
$$572$$ 0 0
$$573$$ 12168.0 0.887130
$$574$$ 0 0
$$575$$ −8544.00 −0.619669
$$576$$ 0 0
$$577$$ 17710.0 1.27778 0.638888 0.769300i $$-0.279395\pi$$
0.638888 + 0.769300i $$0.279395\pi$$
$$578$$ 0 0
$$579$$ −6186.00 −0.444009
$$580$$ 0 0
$$581$$ −4944.00 −0.353032
$$582$$ 0 0
$$583$$ −1080.00 −0.0767222
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −20028.0 −1.40825 −0.704126 0.710075i $$-0.748661\pi$$
−0.704126 + 0.710075i $$0.748661\pi$$
$$588$$ 0 0
$$589$$ 14560.0 1.01856
$$590$$ 0 0
$$591$$ −13122.0 −0.913311
$$592$$ 0 0
$$593$$ −19926.0 −1.37987 −0.689935 0.723871i $$-0.742361\pi$$
−0.689935 + 0.723871i $$0.742361\pi$$
$$594$$ 0 0
$$595$$ 1584.00 0.109139
$$596$$ 0 0
$$597$$ 7608.00 0.521566
$$598$$ 0 0
$$599$$ −1704.00 −0.116233 −0.0581165 0.998310i $$-0.518509\pi$$
−0.0581165 + 0.998310i $$0.518509\pi$$
$$600$$ 0 0
$$601$$ 11018.0 0.747810 0.373905 0.927467i $$-0.378019\pi$$
0.373905 + 0.927467i $$0.378019\pi$$
$$602$$ 0 0
$$603$$ 2304.00 0.155599
$$604$$ 0 0
$$605$$ −210.000 −0.0141119
$$606$$ 0 0
$$607$$ −448.000 −0.0299568 −0.0149784 0.999888i $$-0.504768\pi$$
−0.0149784 + 0.999888i $$0.504768\pi$$
$$608$$ 0 0
$$609$$ −2664.00 −0.177259
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 12586.0 0.829272 0.414636 0.909987i $$-0.363909\pi$$
0.414636 + 0.909987i $$0.363909\pi$$
$$614$$ 0 0
$$615$$ −1620.00 −0.106219
$$616$$ 0 0
$$617$$ 29610.0 1.93202 0.966008 0.258513i $$-0.0832324\pi$$
0.966008 + 0.258513i $$0.0832324\pi$$
$$618$$ 0 0
$$619$$ 7120.00 0.462321 0.231161 0.972916i $$-0.425748\pi$$
0.231161 + 0.972916i $$0.425748\pi$$
$$620$$ 0 0
$$621$$ −2592.00 −0.167493
$$622$$ 0 0
$$623$$ −1272.00 −0.0818003
$$624$$ 0 0
$$625$$ 3421.00 0.218944
$$626$$ 0 0
$$627$$ −6048.00 −0.385221
$$628$$ 0 0
$$629$$ 6996.00 0.443480
$$630$$ 0 0
$$631$$ 15580.0 0.982932 0.491466 0.870897i $$-0.336461\pi$$
0.491466 + 0.870897i $$0.336461\pi$$
$$632$$ 0 0
$$633$$ 13332.0 0.837124
$$634$$ 0 0
$$635$$ 2784.00 0.173984
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 1512.00 0.0936053
$$640$$ 0 0
$$641$$ −19806.0 −1.22042 −0.610211 0.792239i $$-0.708915\pi$$
−0.610211 + 0.792239i $$0.708915\pi$$
$$642$$ 0 0
$$643$$ −24032.0 −1.47392 −0.736959 0.675937i $$-0.763739\pi$$
−0.736959 + 0.675937i $$0.763739\pi$$
$$644$$ 0 0
$$645$$ −792.000 −0.0483488
$$646$$ 0 0
$$647$$ −2808.00 −0.170624 −0.0853121 0.996354i $$-0.527189\pi$$
−0.0853121 + 0.996354i $$0.527189\pi$$
$$648$$ 0 0
$$649$$ 12528.0 0.757730
$$650$$ 0 0
$$651$$ 3120.00 0.187838
$$652$$ 0 0
$$653$$ 23886.0 1.43144 0.715721 0.698386i $$-0.246098\pi$$
0.715721 + 0.698386i $$0.246098\pi$$
$$654$$ 0 0
$$655$$ 9288.00 0.554064
$$656$$ 0 0
$$657$$ 7326.00 0.435030
$$658$$ 0 0
$$659$$ −3948.00 −0.233372 −0.116686 0.993169i $$-0.537227\pi$$
−0.116686 + 0.993169i $$0.537227\pi$$
$$660$$ 0 0
$$661$$ −5750.00 −0.338350 −0.169175 0.985586i $$-0.554110\pi$$
−0.169175 + 0.985586i $$0.554110\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −1344.00 −0.0783731
$$666$$ 0 0
$$667$$ 21312.0 1.23719
$$668$$ 0 0
$$669$$ −8148.00 −0.470882
$$670$$ 0 0
$$671$$ 12456.0 0.716630
$$672$$ 0 0
$$673$$ 28082.0 1.60844 0.804221 0.594330i $$-0.202583\pi$$
0.804221 + 0.594330i $$0.202583\pi$$
$$674$$ 0 0
$$675$$ 2403.00 0.137024
$$676$$ 0 0
$$677$$ −27954.0 −1.58694 −0.793471 0.608608i $$-0.791728\pi$$
−0.793471 + 0.608608i $$0.791728\pi$$
$$678$$ 0 0
$$679$$ 2008.00 0.113490
$$680$$ 0 0
$$681$$ −14076.0 −0.792061
$$682$$ 0 0
$$683$$ 28428.0 1.59263 0.796316 0.604881i $$-0.206779\pi$$
0.796316 + 0.604881i $$0.206779\pi$$
$$684$$ 0 0
$$685$$ −1764.00 −0.0983927
$$686$$ 0 0
$$687$$ 19338.0 1.07393
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −21680.0 −1.19355 −0.596777 0.802407i $$-0.703552\pi$$
−0.596777 + 0.802407i $$0.703552\pi$$
$$692$$ 0 0
$$693$$ −1296.00 −0.0710404
$$694$$ 0 0
$$695$$ 15384.0 0.839638
$$696$$ 0 0
$$697$$ 5940.00 0.322803
$$698$$ 0 0
$$699$$ 9306.00 0.503555
$$700$$ 0 0
$$701$$ −7482.00 −0.403126 −0.201563 0.979476i $$-0.564602\pi$$
−0.201563 + 0.979476i $$0.564602\pi$$
$$702$$ 0 0
$$703$$ −5936.00 −0.318464
$$704$$ 0 0
$$705$$ 3024.00 0.161547
$$706$$ 0 0
$$707$$ 4248.00 0.225972
$$708$$ 0 0
$$709$$ −2270.00 −0.120242 −0.0601210 0.998191i $$-0.519149\pi$$
−0.0601210 + 0.998191i $$0.519149\pi$$
$$710$$ 0 0
$$711$$ 1800.00 0.0949441
$$712$$ 0 0
$$713$$ −24960.0 −1.31102
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 2448.00 0.127507
$$718$$ 0 0
$$719$$ 36024.0 1.86852 0.934262 0.356588i $$-0.116060\pi$$
0.934262 + 0.356588i $$0.116060\pi$$
$$720$$ 0 0
$$721$$ −256.000 −0.0132232
$$722$$ 0 0
$$723$$ 11454.0 0.589182
$$724$$ 0 0
$$725$$ −19758.0 −1.01213
$$726$$ 0 0
$$727$$ −21544.0 −1.09907 −0.549534 0.835471i $$-0.685195\pi$$
−0.549534 + 0.835471i $$0.685195\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 2904.00 0.146933
$$732$$ 0 0
$$733$$ 1018.00 0.0512970 0.0256485 0.999671i $$-0.491835\pi$$
0.0256485 + 0.999671i $$0.491835\pi$$
$$734$$ 0 0
$$735$$ 5886.00 0.295386
$$736$$ 0 0
$$737$$ −9216.00 −0.460618
$$738$$ 0 0
$$739$$ 24568.0 1.22293 0.611467 0.791270i $$-0.290580\pi$$
0.611467 + 0.791270i $$0.290580\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −16968.0 −0.837814 −0.418907 0.908029i $$-0.637587\pi$$
−0.418907 + 0.908029i $$0.637587\pi$$
$$744$$ 0 0
$$745$$ −684.000 −0.0336373
$$746$$ 0 0
$$747$$ −11124.0 −0.544854
$$748$$ 0 0
$$749$$ −1776.00 −0.0866404
$$750$$ 0 0
$$751$$ 3224.00 0.156652 0.0783259 0.996928i $$-0.475043\pi$$
0.0783259 + 0.996928i $$0.475043\pi$$
$$752$$ 0 0
$$753$$ 19836.0 0.959979
$$754$$ 0 0
$$755$$ −12216.0 −0.588855
$$756$$ 0 0
$$757$$ −31570.0 −1.51576 −0.757881 0.652393i $$-0.773765\pi$$
−0.757881 + 0.652393i $$0.773765\pi$$
$$758$$ 0 0
$$759$$ 10368.0 0.495829
$$760$$ 0 0
$$761$$ 34890.0 1.66197 0.830987 0.556293i $$-0.187777\pi$$
0.830987 + 0.556293i $$0.187777\pi$$
$$762$$ 0 0
$$763$$ −5528.00 −0.262290
$$764$$ 0 0
$$765$$ 3564.00 0.168440
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −11522.0 −0.540304 −0.270152 0.962818i $$-0.587074\pi$$
−0.270152 + 0.962818i $$0.587074\pi$$
$$770$$ 0 0
$$771$$ 14418.0 0.673478
$$772$$ 0 0
$$773$$ 28158.0 1.31018 0.655092 0.755549i $$-0.272630\pi$$
0.655092 + 0.755549i $$0.272630\pi$$
$$774$$ 0 0
$$775$$ 23140.0 1.07253
$$776$$ 0 0
$$777$$ −1272.00 −0.0587294
$$778$$ 0 0
$$779$$ −5040.00 −0.231806
$$780$$ 0 0
$$781$$ −6048.00 −0.277099
$$782$$ 0 0
$$783$$ −5994.00 −0.273574
$$784$$ 0 0
$$785$$ 17220.0 0.782940
$$786$$ 0 0
$$787$$ −14504.0 −0.656940 −0.328470 0.944514i $$-0.606533\pi$$
−0.328470 + 0.944514i $$0.606533\pi$$
$$788$$ 0 0
$$789$$ 13752.0 0.620512
$$790$$ 0 0
$$791$$ −3480.00 −0.156428
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ −540.000 −0.0240903
$$796$$ 0 0
$$797$$ −18090.0 −0.803991 −0.401995 0.915642i $$-0.631683\pi$$
−0.401995 + 0.915642i $$0.631683\pi$$
$$798$$ 0 0
$$799$$ −11088.0 −0.490945
$$800$$ 0 0
$$801$$ −2862.00 −0.126247
$$802$$ 0 0
$$803$$ −29304.0 −1.28782
$$804$$ 0 0
$$805$$ 2304.00 0.100876
$$806$$ 0 0
$$807$$ −21402.0 −0.933564
$$808$$ 0 0
$$809$$ 36402.0 1.58199 0.790993 0.611826i $$-0.209565\pi$$
0.790993 + 0.611826i $$0.209565\pi$$
$$810$$ 0 0
$$811$$ 32368.0 1.40147 0.700736 0.713420i $$-0.252855\pi$$
0.700736 + 0.713420i $$0.252855\pi$$
$$812$$ 0 0
$$813$$ 9420.00 0.406364
$$814$$ 0 0
$$815$$ −8832.00 −0.379597
$$816$$ 0 0
$$817$$ −2464.00 −0.105513
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −35778.0 −1.52090 −0.760451 0.649395i $$-0.775022\pi$$
−0.760451 + 0.649395i $$0.775022\pi$$
$$822$$ 0 0
$$823$$ −10240.0 −0.433711 −0.216855 0.976204i $$-0.569580\pi$$
−0.216855 + 0.976204i $$0.569580\pi$$
$$824$$ 0 0
$$825$$ −9612.00 −0.405633
$$826$$ 0 0
$$827$$ 16284.0 0.684704 0.342352 0.939572i $$-0.388776\pi$$
0.342352 + 0.939572i $$0.388776\pi$$
$$828$$ 0 0
$$829$$ 14150.0 0.592822 0.296411 0.955060i $$-0.404210\pi$$
0.296411 + 0.955060i $$0.404210\pi$$
$$830$$ 0 0
$$831$$ 14358.0 0.599366
$$832$$ 0 0
$$833$$ −21582.0 −0.897685
$$834$$ 0 0
$$835$$ 1440.00 0.0596805
$$836$$ 0 0
$$837$$ 7020.00 0.289900
$$838$$ 0 0
$$839$$ −39576.0 −1.62850 −0.814252 0.580511i $$-0.802853\pi$$
−0.814252 + 0.580511i $$0.802853\pi$$
$$840$$ 0 0
$$841$$ 24895.0 1.02075
$$842$$ 0 0
$$843$$ 11394.0 0.465516
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −140.000 −0.00567941
$$848$$ 0 0
$$849$$ −10716.0 −0.433183
$$850$$ 0 0
$$851$$ 10176.0 0.409905
$$852$$ 0 0
$$853$$ 6922.00 0.277848 0.138924 0.990303i $$-0.455636\pi$$
0.138924 + 0.990303i $$0.455636\pi$$
$$854$$ 0 0
$$855$$ −3024.00 −0.120957
$$856$$ 0 0
$$857$$ 48162.0 1.91970 0.959850 0.280514i $$-0.0905050\pi$$
0.959850 + 0.280514i $$0.0905050\pi$$
$$858$$ 0 0
$$859$$ −27652.0 −1.09834 −0.549170 0.835711i $$-0.685056\pi$$
−0.549170 + 0.835711i $$0.685056\pi$$
$$860$$ 0 0
$$861$$ −1080.00 −0.0427483
$$862$$ 0 0
$$863$$ −648.000 −0.0255599 −0.0127799 0.999918i $$-0.504068\pi$$
−0.0127799 + 0.999918i $$0.504068\pi$$
$$864$$ 0 0
$$865$$ −1836.00 −0.0721686
$$866$$ 0 0
$$867$$ 1671.00 0.0654558
$$868$$ 0 0
$$869$$ −7200.00 −0.281062
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 4518.00 0.175156
$$874$$ 0 0
$$875$$ −5136.00 −0.198433
$$876$$ 0 0
$$877$$ −7166.00 −0.275916 −0.137958 0.990438i $$-0.544054\pi$$
−0.137958 + 0.990438i $$0.544054\pi$$
$$878$$ 0 0
$$879$$ 21366.0 0.819860
$$880$$ 0 0
$$881$$ −37062.0 −1.41731 −0.708655 0.705555i $$-0.750698\pi$$
−0.708655 + 0.705555i $$0.750698\pi$$
$$882$$ 0 0
$$883$$ 24716.0 0.941970 0.470985 0.882141i $$-0.343899\pi$$
0.470985 + 0.882141i $$0.343899\pi$$
$$884$$ 0 0
$$885$$ 6264.00 0.237923
$$886$$ 0 0
$$887$$ 48672.0 1.84244 0.921221 0.389040i $$-0.127193\pi$$
0.921221 + 0.389040i $$0.127193\pi$$
$$888$$ 0 0
$$889$$ 1856.00 0.0700205
$$890$$ 0 0
$$891$$ −2916.00 −0.109640
$$892$$ 0 0
$$893$$ 9408.00 0.352550
$$894$$ 0 0
$$895$$ −12312.0 −0.459827
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −57720.0 −2.14135
$$900$$ 0 0
$$901$$ 1980.00 0.0732113
$$902$$ 0 0
$$903$$ −528.000 −0.0194582
$$904$$ 0 0
$$905$$ −26988.0 −0.991283
$$906$$ 0 0
$$907$$ −9484.00 −0.347201 −0.173600 0.984816i $$-0.555540\pi$$
−0.173600 + 0.984816i $$0.555540\pi$$
$$908$$ 0 0
$$909$$ 9558.00 0.348756
$$910$$ 0 0
$$911$$ 12792.0 0.465223 0.232611 0.972570i $$-0.425273\pi$$
0.232611 + 0.972570i $$0.425273\pi$$
$$912$$ 0 0
$$913$$ 44496.0 1.61293
$$914$$ 0 0
$$915$$ 6228.00 0.225018
$$916$$ 0 0
$$917$$ 6192.00 0.222986
$$918$$ 0 0
$$919$$ −18592.0 −0.667349 −0.333674 0.942688i $$-0.608289\pi$$
−0.333674 + 0.942688i $$0.608289\pi$$
$$920$$ 0 0
$$921$$ −20568.0 −0.735873
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −9434.00 −0.335338
$$926$$ 0 0
$$927$$ −576.000 −0.0204081
$$928$$ 0 0
$$929$$ 15378.0 0.543096 0.271548 0.962425i $$-0.412464\pi$$
0.271548 + 0.962425i $$0.412464\pi$$
$$930$$ 0 0
$$931$$ 18312.0 0.644631
$$932$$ 0 0
$$933$$ 26496.0 0.929732
$$934$$ 0 0
$$935$$ −14256.0 −0.498632
$$936$$ 0 0
$$937$$ −37078.0 −1.29273 −0.646364 0.763030i $$-0.723711\pi$$
−0.646364 + 0.763030i $$0.723711\pi$$
$$938$$ 0 0
$$939$$ −10878.0 −0.378051
$$940$$ 0 0
$$941$$ −10842.0 −0.375599 −0.187800 0.982207i $$-0.560136\pi$$
−0.187800 + 0.982207i $$0.560136\pi$$
$$942$$ 0 0
$$943$$ 8640.00 0.298364
$$944$$ 0 0
$$945$$ −648.000 −0.0223063
$$946$$ 0 0
$$947$$ −41508.0 −1.42432 −0.712159 0.702018i $$-0.752282\pi$$
−0.712159 + 0.702018i $$0.752282\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 30438.0 1.03788
$$952$$ 0 0
$$953$$ 38706.0 1.31565 0.657823 0.753173i $$-0.271478\pi$$
0.657823 + 0.753173i $$0.271478\pi$$
$$954$$ 0 0
$$955$$ −24336.0 −0.824602
$$956$$ 0 0
$$957$$ 23976.0 0.809858
$$958$$ 0 0
$$959$$ −1176.00 −0.0395986
$$960$$ 0 0
$$961$$ 37809.0 1.26914
$$962$$ 0 0
$$963$$ −3996.00 −0.133717
$$964$$ 0 0
$$965$$ 12372.0 0.412714
$$966$$ 0 0
$$967$$ 24388.0 0.811029 0.405515 0.914089i $$-0.367092\pi$$
0.405515 + 0.914089i $$0.367092\pi$$
$$968$$ 0 0
$$969$$ 11088.0 0.367593
$$970$$ 0 0
$$971$$ −14100.0 −0.466005 −0.233002 0.972476i $$-0.574855\pi$$
−0.233002 + 0.972476i $$0.574855\pi$$
$$972$$ 0 0
$$973$$ 10256.0 0.337916
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −44838.0 −1.46826 −0.734132 0.679006i $$-0.762411\pi$$
−0.734132 + 0.679006i $$0.762411\pi$$
$$978$$ 0 0
$$979$$ 11448.0 0.373728
$$980$$ 0 0
$$981$$ −12438.0 −0.404806
$$982$$ 0 0
$$983$$ −13176.0 −0.427517 −0.213758 0.976887i $$-0.568571\pi$$
−0.213758 + 0.976887i $$0.568571\pi$$
$$984$$ 0 0
$$985$$ 26244.0 0.848937
$$986$$ 0 0
$$987$$ 2016.00 0.0650152
$$988$$ 0 0
$$989$$ 4224.00 0.135809
$$990$$ 0 0
$$991$$ −43648.0 −1.39912 −0.699558 0.714576i $$-0.746620\pi$$
−0.699558 + 0.714576i $$0.746620\pi$$
$$992$$ 0 0
$$993$$ 19608.0 0.626627
$$994$$ 0 0
$$995$$ −15216.0 −0.484804
$$996$$ 0 0
$$997$$ 62750.0 1.99329 0.996646 0.0818317i $$-0.0260770\pi$$
0.996646 + 0.0818317i $$0.0260770\pi$$
$$998$$ 0 0
$$999$$ −2862.00 −0.0906403
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 2028.4.a.a.1.1 1
13.5 odd 4 2028.4.b.a.337.1 2
13.8 odd 4 2028.4.b.a.337.2 2
13.12 even 2 156.4.a.a.1.1 1
39.38 odd 2 468.4.a.b.1.1 1
52.51 odd 2 624.4.a.h.1.1 1
104.51 odd 2 2496.4.a.e.1.1 1
104.77 even 2 2496.4.a.n.1.1 1
156.155 even 2 1872.4.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
156.4.a.a.1.1 1 13.12 even 2
468.4.a.b.1.1 1 39.38 odd 2
624.4.a.h.1.1 1 52.51 odd 2
1872.4.a.j.1.1 1 156.155 even 2
2028.4.a.a.1.1 1 1.1 even 1 trivial
2028.4.b.a.337.1 2 13.5 odd 4
2028.4.b.a.337.2 2 13.8 odd 4
2496.4.a.e.1.1 1 104.51 odd 2
2496.4.a.n.1.1 1 104.77 even 2