# Properties

 Label 156.4.a.a.1.1 Level $156$ Weight $4$ Character 156.1 Self dual yes Analytic conductor $9.204$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [156,4,Mod(1,156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(156, 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("156.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 156.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: $$9.20429796090$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 156.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -6.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -6.00000 q^{5} -4.00000 q^{7} +9.00000 q^{9} +36.0000 q^{11} +13.0000 q^{13} +18.0000 q^{15} +66.0000 q^{17} +56.0000 q^{19} +12.0000 q^{21} +96.0000 q^{23} -89.0000 q^{25} -27.0000 q^{27} +222.000 q^{29} +260.000 q^{31} -108.000 q^{33} +24.0000 q^{35} -106.000 q^{37} -39.0000 q^{39} -90.0000 q^{41} +44.0000 q^{43} -54.0000 q^{45} +168.000 q^{47} -327.000 q^{49} -198.000 q^{51} +30.0000 q^{53} -216.000 q^{55} -168.000 q^{57} +348.000 q^{59} -346.000 q^{61} -36.0000 q^{63} -78.0000 q^{65} -256.000 q^{67} -288.000 q^{69} -168.000 q^{71} -814.000 q^{73} +267.000 q^{75} -144.000 q^{77} +200.000 q^{79} +81.0000 q^{81} +1236.00 q^{83} -396.000 q^{85} -666.000 q^{87} +318.000 q^{89} -52.0000 q^{91} -780.000 q^{93} -336.000 q^{95} -502.000 q^{97} +324.000 q^{99} +O(q^{100})$$

## 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 $$-0.586471\pi$$
−0.268328 + 0.963328i $$0.586471\pi$$
$$6$$ 0 0
$$7$$ −4.00000 −0.215980 −0.107990 0.994152i $$-0.534441\pi$$
−0.107990 + 0.994152i $$0.534441\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 36.0000 0.986764 0.493382 0.869813i $$-0.335760\pi$$
0.493382 + 0.869813i $$0.335760\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ 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 $$-0.390220\pi$$
0.338086 + 0.941115i $$0.390220\pi$$
$$20$$ 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 $$-0.228517\pi$$
0.753184 + 0.657810i $$0.228517\pi$$
$$32$$ 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 $$-0.575670\pi$$
−0.235490 + 0.971877i $$0.575670\pi$$
$$38$$ 0 0
$$39$$ −39.0000 −0.160128
$$40$$ 0 0
$$41$$ −90.0000 −0.342820 −0.171410 0.985200i $$-0.554832\pi$$
−0.171410 + 0.985200i $$0.554832\pi$$
$$42$$ 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 $$-0.416048\pi$$
0.260695 + 0.965421i $$0.416048\pi$$
$$48$$ 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 $$-0.374565\pi$$
0.383947 + 0.923355i $$0.374565\pi$$
$$60$$ 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
$$65$$ −78.0000 −0.148842
$$66$$ 0 0
$$67$$ −256.000 −0.466797 −0.233398 0.972381i $$-0.574985\pi$$
−0.233398 + 0.972381i $$0.574985\pi$$
$$68$$ 0 0
$$69$$ −288.000 −0.502480
$$70$$ 0 0
$$71$$ −168.000 −0.280816 −0.140408 0.990094i $$-0.544841\pi$$
−0.140408 + 0.990094i $$0.544841\pi$$
$$72$$ 0 0
$$73$$ −814.000 −1.30509 −0.652544 0.757750i $$-0.726298\pi$$
−0.652544 + 0.757750i $$0.726298\pi$$
$$74$$ 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 $$-0.195480\pi$$
0.817281 + 0.576240i $$0.195480\pi$$
$$84$$ 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 $$-0.439355\pi$$
0.189370 + 0.981906i $$0.439355\pi$$
$$90$$ 0 0
$$91$$ −52.0000 −0.0599020
$$92$$ 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 $$-0.584624\pi$$
−0.262734 + 0.964868i $$0.584624\pi$$
$$98$$ 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 $$-0.292289\pi$$
0.607209 + 0.794542i $$0.292289\pi$$
$$110$$ 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
$$117$$ 117.000 0.0924500
$$118$$ 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 $$-0.470779\pi$$
0.0916720 + 0.995789i $$0.470779\pi$$
$$138$$ 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
$$143$$ 468.000 0.273679
$$144$$ 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 $$-0.490023\pi$$
0.0313397 + 0.999509i $$0.490023\pi$$
$$150$$ 0 0
$$151$$ 2036.00 1.09727 0.548634 0.836063i $$-0.315148\pi$$
0.548634 + 0.836063i $$0.315148\pi$$
$$152$$ 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 $$-0.384934\pi$$
0.353669 + 0.935371i $$0.384934\pi$$
$$164$$ 0 0
$$165$$ 648.000 0.305738
$$166$$ 0 0
$$167$$ −240.000 −0.111208 −0.0556041 0.998453i $$-0.517708\pi$$
−0.0556041 + 0.998453i $$0.517708\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ 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 $$-0.625634\pi$$
−0.384523 + 0.923115i $$0.625634\pi$$
$$194$$ 0 0
$$195$$ 234.000 0.0859338
$$196$$ 0 0
$$197$$ −4374.00 −1.58190 −0.790951 0.611880i $$-0.790414\pi$$
−0.790951 + 0.611880i $$0.790414\pi$$
$$198$$ 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
$$221$$ 858.000 0.261155
$$222$$ 0 0
$$223$$ −2716.00 −0.815591 −0.407796 0.913073i $$-0.633702\pi$$
−0.407796 + 0.913073i $$0.633702\pi$$
$$224$$ 0 0
$$225$$ −801.000 −0.237333
$$226$$ 0 0
$$227$$ −4692.00 −1.37189 −0.685945 0.727653i $$-0.740611\pi$$
−0.685945 + 0.727653i $$0.740611\pi$$
$$228$$ 0 0
$$229$$ 6446.00 1.86010 0.930052 0.367429i $$-0.119762\pi$$
0.930052 + 0.367429i $$0.119762\pi$$
$$230$$ 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 $$-0.464779\pi$$
0.110424 + 0.993885i $$0.464779\pi$$
$$240$$ 0 0
$$241$$ 3818.00 1.02049 0.510247 0.860028i $$-0.329554\pi$$
0.510247 + 0.860028i $$0.329554\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 1962.00 0.511623
$$246$$ 0 0
$$247$$ 728.000 0.187537
$$248$$ 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 $$-0.385528\pi$$
0.351921 + 0.936030i $$0.385528\pi$$
$$272$$ 0 0
$$273$$ 156.000 0.0345844
$$274$$ 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 $$-0.367916\pi$$
0.403149 + 0.915134i $$0.367916\pi$$
$$282$$ 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 $$-0.248686\pi$$
0.710020 + 0.704182i $$0.248686\pi$$
$$294$$ 0 0
$$295$$ −2088.00 −0.412095
$$296$$ 0 0
$$297$$ −972.000 −0.189903
$$298$$ 0 0
$$299$$ 1248.00 0.241384
$$300$$ 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 $$-0.719942\pi$$
−0.637284 + 0.770629i $$0.719942\pi$$
$$308$$ 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 $$-0.144421\pi$$
0.898827 + 0.438304i $$0.144421\pi$$
$$318$$ 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
$$325$$ −1157.00 −0.197473
$$326$$ 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 $$-0.317411\pi$$
0.542675 + 0.839943i $$0.317411\pi$$
$$332$$ 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 $$-0.537238\pi$$
−0.116720 + 0.993165i $$0.537238\pi$$
$$350$$ 0 0
$$351$$ −351.000 −0.0533761
$$352$$ 0 0
$$353$$ −1362.00 −0.205360 −0.102680 0.994714i $$-0.532742\pi$$
−0.102680 + 0.994714i $$0.532742\pi$$
$$354$$ 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 $$-0.273619\pi$$
0.652742 + 0.757581i $$0.273619\pi$$
$$360$$ 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
$$377$$ 2886.00 0.394261
$$378$$ 0 0
$$379$$ −13168.0 −1.78468 −0.892341 0.451361i $$-0.850939\pi$$
−0.892341 + 0.451361i $$0.850939\pi$$
$$380$$ 0 0
$$381$$ −1392.00 −0.187177
$$382$$ 0 0
$$383$$ 4872.00 0.649994 0.324997 0.945715i $$-0.394637\pi$$
0.324997 + 0.945715i $$0.394637\pi$$
$$384$$ 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 $$-0.599857\pi$$
−0.308590 + 0.951195i $$0.599857\pi$$
$$398$$ 0 0
$$399$$ 672.000 0.0843160
$$400$$ 0 0
$$401$$ −90.0000 −0.0112079 −0.00560397 0.999984i $$-0.501784\pi$$
−0.00560397 + 0.999984i $$0.501784\pi$$
$$402$$ 0 0
$$403$$ 3380.00 0.417791
$$404$$ 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 $$-0.454552\pi$$
0.142296 + 0.989824i $$0.454552\pi$$
$$410$$ 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 $$-0.494436\pi$$
0.0174805 + 0.999847i $$0.494436\pi$$
$$422$$ 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
$$429$$ −1404.00 −0.158009
$$430$$ 0 0
$$431$$ 9816.00 1.09703 0.548515 0.836141i $$-0.315193\pi$$
0.548515 + 0.836141i $$0.315193\pi$$
$$432$$ 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 $$-0.558033\pi$$
−0.181309 + 0.983426i $$0.558033\pi$$
$$450$$ 0 0
$$451$$ −3240.00 −0.338283
$$452$$ 0 0
$$453$$ −6108.00 −0.633507
$$454$$ 0 0
$$455$$ 312.000 0.0321468
$$456$$ 0 0
$$457$$ −16654.0 −1.70469 −0.852343 0.522984i $$-0.824819\pi$$
−0.852343 + 0.522984i $$0.824819\pi$$
$$458$$ 0 0
$$459$$ −1782.00 −0.181213
$$460$$ 0 0
$$461$$ −14046.0 −1.41906 −0.709531 0.704674i $$-0.751093\pi$$
−0.709531 + 0.704674i $$0.751093\pi$$
$$462$$ 0 0
$$463$$ −4588.00 −0.460524 −0.230262 0.973129i $$-0.573958\pi$$
−0.230262 + 0.973129i $$0.573958\pi$$
$$464$$ 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 $$-0.289745\pi$$
0.613540 + 0.789664i $$0.289745\pi$$
$$480$$ 0 0
$$481$$ −1378.00 −0.130627
$$482$$ 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 $$-0.658667\pi$$
−0.478080 + 0.878316i $$0.658667\pi$$
$$488$$ 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 $$-0.218056\pi$$
0.774392 + 0.632707i $$0.218056\pi$$
$$500$$ 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
$$507$$ −507.000 −0.0444116
$$508$$ 0 0
$$509$$ 5010.00 0.436276 0.218138 0.975918i $$-0.430002\pi$$
0.218138 + 0.975918i $$0.430002\pi$$
$$510$$ 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
$$533$$ −1170.00 −0.0950813
$$534$$ 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 $$-0.421001\pi$$
0.245642 + 0.969361i $$0.421001\pi$$
$$542$$ 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 $$-0.661495\pi$$
−0.485863 + 0.874035i $$0.661495\pi$$
$$558$$ 0 0
$$559$$ 572.000 0.0432791
$$560$$ 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 $$-0.720605\pi$$
−0.638888 + 0.769300i $$0.720605\pi$$
$$578$$ 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
$$585$$ −702.000 −0.0496139
$$586$$ 0 0
$$587$$ 20028.0 1.40825 0.704126 0.710075i $$-0.251339\pi$$
0.704126 + 0.710075i $$0.251339\pi$$
$$588$$ 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 $$-0.257639\pi$$
0.689935 + 0.723871i $$0.257639\pi$$
$$594$$ 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
$$611$$ 2184.00 0.144608
$$612$$ 0 0
$$613$$ −12586.0 −0.829272 −0.414636 0.909987i $$-0.636091\pi$$
−0.414636 + 0.909987i $$0.636091\pi$$
$$614$$ 0 0
$$615$$ −1620.00 −0.106219
$$616$$ 0 0
$$617$$ −29610.0 −1.93202 −0.966008 0.258513i $$-0.916768\pi$$
−0.966008 + 0.258513i $$0.916768\pi$$
$$618$$ 0 0
$$619$$ −7120.00 −0.462321 −0.231161 0.972916i $$-0.574252\pi$$
−0.231161 + 0.972916i $$0.574252\pi$$
$$620$$ 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 $$-0.663539\pi$$
−0.491466 + 0.870897i $$0.663539\pi$$
$$632$$ 0 0
$$633$$ 13332.0 0.837124
$$634$$ 0 0
$$635$$ −2784.00 −0.173984
$$636$$ 0 0
$$637$$ −4251.00 −0.264412
$$638$$ 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 $$-0.236261\pi$$
0.736959 + 0.675937i $$0.236261\pi$$
$$644$$ 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 $$-0.445890\pi$$
0.169175 + 0.985586i $$0.445890\pi$$
$$662$$ 0 0
$$663$$ −2574.00 −0.150778
$$664$$ 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 $$-0.793221\pi$$
−0.796316 + 0.604881i $$0.793221\pi$$
$$684$$ 0 0
$$685$$ −1764.00 −0.0983927
$$686$$ 0 0
$$687$$ −19338.0 −1.07393
$$688$$ 0 0
$$689$$ 390.000 0.0215643
$$690$$ 0 0
$$691$$ 21680.0 1.19355 0.596777 0.802407i $$-0.296448\pi$$
0.596777 + 0.802407i $$0.296448\pi$$
$$692$$ 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 $$-0.480851\pi$$
0.0601210 + 0.998191i $$0.480851\pi$$
$$710$$ 0 0
$$711$$ 1800.00 0.0949441
$$712$$ 0 0
$$713$$ 24960.0 1.31102
$$714$$ 0 0
$$715$$ −2808.00 −0.146872
$$716$$ 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 $$-0.508165\pi$$
−0.0256485 + 0.999671i $$0.508165\pi$$
$$734$$ 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 $$-0.709420\pi$$
−0.611467 + 0.791270i $$0.709420\pi$$
$$740$$ 0 0
$$741$$ −2184.00 −0.108274
$$742$$ 0 0
$$743$$ 16968.0 0.837814 0.418907 0.908029i $$-0.362413\pi$$
0.418907 + 0.908029i $$0.362413\pi$$
$$744$$ 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 $$-0.812223\pi$$
−0.830987 + 0.556293i $$0.812223\pi$$
$$762$$ 0 0
$$763$$ −5528.00 −0.262290
$$764$$ 0 0
$$765$$ −3564.00 −0.168440
$$766$$ 0 0
$$767$$ 4524.00 0.212975
$$768$$ 0 0
$$769$$ 11522.0 0.540304 0.270152 0.962818i $$-0.412926\pi$$
0.270152 + 0.962818i $$0.412926\pi$$
$$770$$ 0 0
$$771$$ 14418.0 0.673478
$$772$$ 0 0
$$773$$ −28158.0 −1.31018 −0.655092 0.755549i $$-0.727370\pi$$
−0.655092 + 0.755549i $$0.727370\pi$$
$$774$$ 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 $$-0.393467\pi$$
0.328470 + 0.944514i $$0.393467\pi$$
$$788$$ 0 0
$$789$$ 13752.0 0.620512
$$790$$ 0 0
$$791$$ 3480.00 0.156428
$$792$$ 0 0
$$793$$ −4498.00 −0.201423
$$794$$ 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 $$-0.747145\pi$$
−0.700736 + 0.713420i $$0.747145\pi$$
$$812$$ 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
$$819$$ −468.000 −0.0199673
$$820$$ 0 0
$$821$$ 35778.0 1.52090 0.760451 0.649395i $$-0.224978\pi$$
0.760451 + 0.649395i $$0.224978\pi$$
$$822$$ 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 $$-0.611224\pi$$
−0.342352 + 0.939572i $$0.611224\pi$$
$$828$$ 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 $$-0.197147\pi$$
0.814252 + 0.580511i $$0.197147\pi$$
$$840$$ 0 0
$$841$$ 24895.0 1.02075
$$842$$ 0 0
$$843$$ −11394.0 −0.465516
$$844$$ 0 0
$$845$$ −1014.00 −0.0412813
$$846$$ 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 $$-0.544364\pi$$
−0.138924 + 0.990303i $$0.544364\pi$$
$$854$$ 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 $$-0.495932\pi$$
0.0127799 + 0.999918i $$0.495932\pi$$
$$864$$ 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
$$871$$ −3328.00 −0.129466
$$872$$ 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 $$-0.455946\pi$$
0.137958 + 0.990438i $$0.455946\pi$$
$$878$$ 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
$$897$$ −3744.00 −0.139363
$$898$$ 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
$$923$$ −2184.00 −0.0778843
$$924$$ 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 $$-0.587536\pi$$
−0.271548 + 0.962425i $$0.587536\pi$$
$$930$$ 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 $$-0.439864\pi$$
0.187800 + 0.982207i $$0.439864\pi$$
$$942$$ 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 $$-0.247718\pi$$
0.712159 + 0.702018i $$0.247718\pi$$
$$948$$ 0 0
$$949$$ −10582.0 −0.361967
$$950$$ 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 $$-0.632908\pi$$
−0.405515 + 0.914089i $$0.632908\pi$$
$$968$$ 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
$$975$$ 3471.00 0.114011
$$976$$ 0 0
$$977$$ 44838.0 1.46826 0.734132 0.679006i $$-0.237589\pi$$
0.734132 + 0.679006i $$0.237589\pi$$
$$978$$ 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 $$-0.431429\pi$$
0.213758 + 0.976887i $$0.431429\pi$$
$$984$$ 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 156.4.a.a.1.1 1
3.2 odd 2 468.4.a.b.1.1 1
4.3 odd 2 624.4.a.h.1.1 1
8.3 odd 2 2496.4.a.e.1.1 1
8.5 even 2 2496.4.a.n.1.1 1
12.11 even 2 1872.4.a.j.1.1 1
13.5 odd 4 2028.4.b.a.337.2 2
13.8 odd 4 2028.4.b.a.337.1 2
13.12 even 2 2028.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
156.4.a.a.1.1 1 1.1 even 1 trivial
468.4.a.b.1.1 1 3.2 odd 2
624.4.a.h.1.1 1 4.3 odd 2
1872.4.a.j.1.1 1 12.11 even 2
2028.4.a.a.1.1 1 13.12 even 2
2028.4.b.a.337.1 2 13.8 odd 4
2028.4.b.a.337.2 2 13.5 odd 4
2496.4.a.e.1.1 1 8.3 odd 2
2496.4.a.n.1.1 1 8.5 even 2