# Properties

 Label 608.4.a.a.1.1 Level $608$ Weight $4$ Character 608.1 Self dual yes Analytic conductor $35.873$ 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] = [608,4,Mod(1,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, 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("608.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$608 = 2^{5} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 608.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: $$35.8731612835$$ 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 $$\chi$$ $$=$$ 608.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -8.00000 q^{5} -17.0000 q^{7} -26.0000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -8.00000 q^{5} -17.0000 q^{7} -26.0000 q^{9} -70.0000 q^{11} -61.0000 q^{13} +8.00000 q^{15} +83.0000 q^{17} +19.0000 q^{19} +17.0000 q^{21} +115.000 q^{23} -61.0000 q^{25} +53.0000 q^{27} +279.000 q^{29} -72.0000 q^{31} +70.0000 q^{33} +136.000 q^{35} -34.0000 q^{37} +61.0000 q^{39} +108.000 q^{41} -192.000 q^{43} +208.000 q^{45} -392.000 q^{47} -54.0000 q^{49} -83.0000 q^{51} +131.000 q^{53} +560.000 q^{55} -19.0000 q^{57} -609.000 q^{59} +338.000 q^{61} +442.000 q^{63} +488.000 q^{65} -461.000 q^{67} -115.000 q^{69} +750.000 q^{71} +1177.00 q^{73} +61.0000 q^{75} +1190.00 q^{77} -22.0000 q^{79} +649.000 q^{81} -810.000 q^{83} -664.000 q^{85} -279.000 q^{87} -476.000 q^{89} +1037.00 q^{91} +72.0000 q^{93} -152.000 q^{95} +1426.00 q^{97} +1820.00 q^{99} +O(q^{100})$$

## 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
$$3$$ −1.00000 −0.192450 −0.0962250 0.995360i $$-0.530677\pi$$
−0.0962250 + 0.995360i $$0.530677\pi$$
$$4$$ 0 0
$$5$$ −8.00000 −0.715542 −0.357771 0.933809i $$-0.616463\pi$$
−0.357771 + 0.933809i $$0.616463\pi$$
$$6$$ 0 0
$$7$$ −17.0000 −0.917914 −0.458957 0.888459i $$-0.651777\pi$$
−0.458957 + 0.888459i $$0.651777\pi$$
$$8$$ 0 0
$$9$$ −26.0000 −0.962963
$$10$$ 0 0
$$11$$ −70.0000 −1.91871 −0.959354 0.282204i $$-0.908934\pi$$
−0.959354 + 0.282204i $$0.908934\pi$$
$$12$$ 0 0
$$13$$ −61.0000 −1.30141 −0.650706 0.759330i $$-0.725527\pi$$
−0.650706 + 0.759330i $$0.725527\pi$$
$$14$$ 0 0
$$15$$ 8.00000 0.137706
$$16$$ 0 0
$$17$$ 83.0000 1.18414 0.592072 0.805885i $$-0.298310\pi$$
0.592072 + 0.805885i $$0.298310\pi$$
$$18$$ 0 0
$$19$$ 19.0000 0.229416
$$20$$ 0 0
$$21$$ 17.0000 0.176653
$$22$$ 0 0
$$23$$ 115.000 1.04257 0.521286 0.853382i $$-0.325452\pi$$
0.521286 + 0.853382i $$0.325452\pi$$
$$24$$ 0 0
$$25$$ −61.0000 −0.488000
$$26$$ 0 0
$$27$$ 53.0000 0.377772
$$28$$ 0 0
$$29$$ 279.000 1.78652 0.893259 0.449543i $$-0.148413\pi$$
0.893259 + 0.449543i $$0.148413\pi$$
$$30$$ 0 0
$$31$$ −72.0000 −0.417148 −0.208574 0.978007i $$-0.566882\pi$$
−0.208574 + 0.978007i $$0.566882\pi$$
$$32$$ 0 0
$$33$$ 70.0000 0.369256
$$34$$ 0 0
$$35$$ 136.000 0.656806
$$36$$ 0 0
$$37$$ −34.0000 −0.151069 −0.0755347 0.997143i $$-0.524066\pi$$
−0.0755347 + 0.997143i $$0.524066\pi$$
$$38$$ 0 0
$$39$$ 61.0000 0.250457
$$40$$ 0 0
$$41$$ 108.000 0.411385 0.205692 0.978617i $$-0.434055\pi$$
0.205692 + 0.978617i $$0.434055\pi$$
$$42$$ 0 0
$$43$$ −192.000 −0.680924 −0.340462 0.940258i $$-0.610583\pi$$
−0.340462 + 0.940258i $$0.610583\pi$$
$$44$$ 0 0
$$45$$ 208.000 0.689040
$$46$$ 0 0
$$47$$ −392.000 −1.21658 −0.608288 0.793716i $$-0.708143\pi$$
−0.608288 + 0.793716i $$0.708143\pi$$
$$48$$ 0 0
$$49$$ −54.0000 −0.157434
$$50$$ 0 0
$$51$$ −83.0000 −0.227889
$$52$$ 0 0
$$53$$ 131.000 0.339514 0.169757 0.985486i $$-0.445702\pi$$
0.169757 + 0.985486i $$0.445702\pi$$
$$54$$ 0 0
$$55$$ 560.000 1.37292
$$56$$ 0 0
$$57$$ −19.0000 −0.0441511
$$58$$ 0 0
$$59$$ −609.000 −1.34381 −0.671907 0.740635i $$-0.734525\pi$$
−0.671907 + 0.740635i $$0.734525\pi$$
$$60$$ 0 0
$$61$$ 338.000 0.709450 0.354725 0.934971i $$-0.384574\pi$$
0.354725 + 0.934971i $$0.384574\pi$$
$$62$$ 0 0
$$63$$ 442.000 0.883917
$$64$$ 0 0
$$65$$ 488.000 0.931215
$$66$$ 0 0
$$67$$ −461.000 −0.840599 −0.420299 0.907386i $$-0.638075\pi$$
−0.420299 + 0.907386i $$0.638075\pi$$
$$68$$ 0 0
$$69$$ −115.000 −0.200643
$$70$$ 0 0
$$71$$ 750.000 1.25364 0.626821 0.779163i $$-0.284356\pi$$
0.626821 + 0.779163i $$0.284356\pi$$
$$72$$ 0 0
$$73$$ 1177.00 1.88709 0.943544 0.331247i $$-0.107469\pi$$
0.943544 + 0.331247i $$0.107469\pi$$
$$74$$ 0 0
$$75$$ 61.0000 0.0939156
$$76$$ 0 0
$$77$$ 1190.00 1.76121
$$78$$ 0 0
$$79$$ −22.0000 −0.0313316 −0.0156658 0.999877i $$-0.504987\pi$$
−0.0156658 + 0.999877i $$0.504987\pi$$
$$80$$ 0 0
$$81$$ 649.000 0.890261
$$82$$ 0 0
$$83$$ −810.000 −1.07119 −0.535597 0.844474i $$-0.679913\pi$$
−0.535597 + 0.844474i $$0.679913\pi$$
$$84$$ 0 0
$$85$$ −664.000 −0.847305
$$86$$ 0 0
$$87$$ −279.000 −0.343815
$$88$$ 0 0
$$89$$ −476.000 −0.566920 −0.283460 0.958984i $$-0.591482\pi$$
−0.283460 + 0.958984i $$0.591482\pi$$
$$90$$ 0 0
$$91$$ 1037.00 1.19458
$$92$$ 0 0
$$93$$ 72.0000 0.0802801
$$94$$ 0 0
$$95$$ −152.000 −0.164157
$$96$$ 0 0
$$97$$ 1426.00 1.49266 0.746332 0.665574i $$-0.231813\pi$$
0.746332 + 0.665574i $$0.231813\pi$$
$$98$$ 0 0
$$99$$ 1820.00 1.84765
$$100$$ 0 0
$$101$$ −1230.00 −1.21178 −0.605889 0.795549i $$-0.707182\pi$$
−0.605889 + 0.795549i $$0.707182\pi$$
$$102$$ 0 0
$$103$$ 310.000 0.296555 0.148278 0.988946i $$-0.452627\pi$$
0.148278 + 0.988946i $$0.452627\pi$$
$$104$$ 0 0
$$105$$ −136.000 −0.126402
$$106$$ 0 0
$$107$$ −1791.00 −1.61815 −0.809077 0.587702i $$-0.800033\pi$$
−0.809077 + 0.587702i $$0.800033\pi$$
$$108$$ 0 0
$$109$$ −779.000 −0.684538 −0.342269 0.939602i $$-0.611195\pi$$
−0.342269 + 0.939602i $$0.611195\pi$$
$$110$$ 0 0
$$111$$ 34.0000 0.0290733
$$112$$ 0 0
$$113$$ −1430.00 −1.19047 −0.595235 0.803552i $$-0.702941\pi$$
−0.595235 + 0.803552i $$0.702941\pi$$
$$114$$ 0 0
$$115$$ −920.000 −0.746004
$$116$$ 0 0
$$117$$ 1586.00 1.25321
$$118$$ 0 0
$$119$$ −1411.00 −1.08694
$$120$$ 0 0
$$121$$ 3569.00 2.68144
$$122$$ 0 0
$$123$$ −108.000 −0.0791710
$$124$$ 0 0
$$125$$ 1488.00 1.06473
$$126$$ 0 0
$$127$$ 466.000 0.325597 0.162798 0.986659i $$-0.447948\pi$$
0.162798 + 0.986659i $$0.447948\pi$$
$$128$$ 0 0
$$129$$ 192.000 0.131044
$$130$$ 0 0
$$131$$ 2240.00 1.49397 0.746984 0.664842i $$-0.231501\pi$$
0.746984 + 0.664842i $$0.231501\pi$$
$$132$$ 0 0
$$133$$ −323.000 −0.210584
$$134$$ 0 0
$$135$$ −424.000 −0.270312
$$136$$ 0 0
$$137$$ −1145.00 −0.714043 −0.357022 0.934096i $$-0.616208\pi$$
−0.357022 + 0.934096i $$0.616208\pi$$
$$138$$ 0 0
$$139$$ −2336.00 −1.42545 −0.712723 0.701446i $$-0.752538\pi$$
−0.712723 + 0.701446i $$0.752538\pi$$
$$140$$ 0 0
$$141$$ 392.000 0.234130
$$142$$ 0 0
$$143$$ 4270.00 2.49703
$$144$$ 0 0
$$145$$ −2232.00 −1.27833
$$146$$ 0 0
$$147$$ 54.0000 0.0302983
$$148$$ 0 0
$$149$$ 3516.00 1.93317 0.966584 0.256351i $$-0.0825204\pi$$
0.966584 + 0.256351i $$0.0825204\pi$$
$$150$$ 0 0
$$151$$ −390.000 −0.210184 −0.105092 0.994463i $$-0.533514\pi$$
−0.105092 + 0.994463i $$0.533514\pi$$
$$152$$ 0 0
$$153$$ −2158.00 −1.14029
$$154$$ 0 0
$$155$$ 576.000 0.298487
$$156$$ 0 0
$$157$$ 2306.00 1.17222 0.586111 0.810231i $$-0.300658\pi$$
0.586111 + 0.810231i $$0.300658\pi$$
$$158$$ 0 0
$$159$$ −131.000 −0.0653395
$$160$$ 0 0
$$161$$ −1955.00 −0.956991
$$162$$ 0 0
$$163$$ −1272.00 −0.611231 −0.305616 0.952155i $$-0.598862\pi$$
−0.305616 + 0.952155i $$0.598862\pi$$
$$164$$ 0 0
$$165$$ −560.000 −0.264218
$$166$$ 0 0
$$167$$ 1768.00 0.819233 0.409617 0.912258i $$-0.365662\pi$$
0.409617 + 0.912258i $$0.365662\pi$$
$$168$$ 0 0
$$169$$ 1524.00 0.693673
$$170$$ 0 0
$$171$$ −494.000 −0.220919
$$172$$ 0 0
$$173$$ −3726.00 −1.63747 −0.818736 0.574171i $$-0.805325\pi$$
−0.818736 + 0.574171i $$0.805325\pi$$
$$174$$ 0 0
$$175$$ 1037.00 0.447942
$$176$$ 0 0
$$177$$ 609.000 0.258617
$$178$$ 0 0
$$179$$ −1048.00 −0.437604 −0.218802 0.975769i $$-0.570215\pi$$
−0.218802 + 0.975769i $$0.570215\pi$$
$$180$$ 0 0
$$181$$ 2678.00 1.09975 0.549873 0.835248i $$-0.314676\pi$$
0.549873 + 0.835248i $$0.314676\pi$$
$$182$$ 0 0
$$183$$ −338.000 −0.136534
$$184$$ 0 0
$$185$$ 272.000 0.108096
$$186$$ 0 0
$$187$$ −5810.00 −2.27203
$$188$$ 0 0
$$189$$ −901.000 −0.346762
$$190$$ 0 0
$$191$$ 1443.00 0.546659 0.273329 0.961921i $$-0.411875\pi$$
0.273329 + 0.961921i $$0.411875\pi$$
$$192$$ 0 0
$$193$$ 890.000 0.331936 0.165968 0.986131i $$-0.446925\pi$$
0.165968 + 0.986131i $$0.446925\pi$$
$$194$$ 0 0
$$195$$ −488.000 −0.179212
$$196$$ 0 0
$$197$$ −300.000 −0.108498 −0.0542490 0.998527i $$-0.517276\pi$$
−0.0542490 + 0.998527i $$0.517276\pi$$
$$198$$ 0 0
$$199$$ 3339.00 1.18942 0.594712 0.803939i $$-0.297266\pi$$
0.594712 + 0.803939i $$0.297266\pi$$
$$200$$ 0 0
$$201$$ 461.000 0.161773
$$202$$ 0 0
$$203$$ −4743.00 −1.63987
$$204$$ 0 0
$$205$$ −864.000 −0.294363
$$206$$ 0 0
$$207$$ −2990.00 −1.00396
$$208$$ 0 0
$$209$$ −1330.00 −0.440182
$$210$$ 0 0
$$211$$ −2149.00 −0.701153 −0.350576 0.936534i $$-0.614014\pi$$
−0.350576 + 0.936534i $$0.614014\pi$$
$$212$$ 0 0
$$213$$ −750.000 −0.241264
$$214$$ 0 0
$$215$$ 1536.00 0.487229
$$216$$ 0 0
$$217$$ 1224.00 0.382906
$$218$$ 0 0
$$219$$ −1177.00 −0.363170
$$220$$ 0 0
$$221$$ −5063.00 −1.54106
$$222$$ 0 0
$$223$$ 3834.00 1.15132 0.575658 0.817690i $$-0.304746\pi$$
0.575658 + 0.817690i $$0.304746\pi$$
$$224$$ 0 0
$$225$$ 1586.00 0.469926
$$226$$ 0 0
$$227$$ −329.000 −0.0961960 −0.0480980 0.998843i $$-0.515316\pi$$
−0.0480980 + 0.998843i $$0.515316\pi$$
$$228$$ 0 0
$$229$$ 430.000 0.124084 0.0620419 0.998074i $$-0.480239\pi$$
0.0620419 + 0.998074i $$0.480239\pi$$
$$230$$ 0 0
$$231$$ −1190.00 −0.338945
$$232$$ 0 0
$$233$$ −438.000 −0.123152 −0.0615758 0.998102i $$-0.519613\pi$$
−0.0615758 + 0.998102i $$0.519613\pi$$
$$234$$ 0 0
$$235$$ 3136.00 0.870511
$$236$$ 0 0
$$237$$ 22.0000 0.00602976
$$238$$ 0 0
$$239$$ 2099.00 0.568088 0.284044 0.958811i $$-0.408324\pi$$
0.284044 + 0.958811i $$0.408324\pi$$
$$240$$ 0 0
$$241$$ −2996.00 −0.800786 −0.400393 0.916344i $$-0.631126\pi$$
−0.400393 + 0.916344i $$0.631126\pi$$
$$242$$ 0 0
$$243$$ −2080.00 −0.549103
$$244$$ 0 0
$$245$$ 432.000 0.112651
$$246$$ 0 0
$$247$$ −1159.00 −0.298564
$$248$$ 0 0
$$249$$ 810.000 0.206151
$$250$$ 0 0
$$251$$ 5518.00 1.38762 0.693811 0.720157i $$-0.255930\pi$$
0.693811 + 0.720157i $$0.255930\pi$$
$$252$$ 0 0
$$253$$ −8050.00 −2.00039
$$254$$ 0 0
$$255$$ 664.000 0.163064
$$256$$ 0 0
$$257$$ −4068.00 −0.987373 −0.493687 0.869640i $$-0.664351\pi$$
−0.493687 + 0.869640i $$0.664351\pi$$
$$258$$ 0 0
$$259$$ 578.000 0.138669
$$260$$ 0 0
$$261$$ −7254.00 −1.72035
$$262$$ 0 0
$$263$$ 4992.00 1.17042 0.585209 0.810883i $$-0.301012\pi$$
0.585209 + 0.810883i $$0.301012\pi$$
$$264$$ 0 0
$$265$$ −1048.00 −0.242936
$$266$$ 0 0
$$267$$ 476.000 0.109104
$$268$$ 0 0
$$269$$ −1970.00 −0.446517 −0.223258 0.974759i $$-0.571669\pi$$
−0.223258 + 0.974759i $$0.571669\pi$$
$$270$$ 0 0
$$271$$ −1861.00 −0.417150 −0.208575 0.978006i $$-0.566883\pi$$
−0.208575 + 0.978006i $$0.566883\pi$$
$$272$$ 0 0
$$273$$ −1037.00 −0.229898
$$274$$ 0 0
$$275$$ 4270.00 0.936330
$$276$$ 0 0
$$277$$ −1268.00 −0.275042 −0.137521 0.990499i $$-0.543914\pi$$
−0.137521 + 0.990499i $$0.543914\pi$$
$$278$$ 0 0
$$279$$ 1872.00 0.401698
$$280$$ 0 0
$$281$$ −8912.00 −1.89198 −0.945988 0.324201i $$-0.894905\pi$$
−0.945988 + 0.324201i $$0.894905\pi$$
$$282$$ 0 0
$$283$$ 3302.00 0.693581 0.346791 0.937943i $$-0.387271\pi$$
0.346791 + 0.937943i $$0.387271\pi$$
$$284$$ 0 0
$$285$$ 152.000 0.0315919
$$286$$ 0 0
$$287$$ −1836.00 −0.377616
$$288$$ 0 0
$$289$$ 1976.00 0.402198
$$290$$ 0 0
$$291$$ −1426.00 −0.287263
$$292$$ 0 0
$$293$$ −5435.00 −1.08367 −0.541836 0.840484i $$-0.682271\pi$$
−0.541836 + 0.840484i $$0.682271\pi$$
$$294$$ 0 0
$$295$$ 4872.00 0.961555
$$296$$ 0 0
$$297$$ −3710.00 −0.724835
$$298$$ 0 0
$$299$$ −7015.00 −1.35682
$$300$$ 0 0
$$301$$ 3264.00 0.625029
$$302$$ 0 0
$$303$$ 1230.00 0.233207
$$304$$ 0 0
$$305$$ −2704.00 −0.507641
$$306$$ 0 0
$$307$$ 1740.00 0.323476 0.161738 0.986834i $$-0.448290\pi$$
0.161738 + 0.986834i $$0.448290\pi$$
$$308$$ 0 0
$$309$$ −310.000 −0.0570721
$$310$$ 0 0
$$311$$ −2837.00 −0.517272 −0.258636 0.965975i $$-0.583273\pi$$
−0.258636 + 0.965975i $$0.583273\pi$$
$$312$$ 0 0
$$313$$ −1579.00 −0.285145 −0.142572 0.989784i $$-0.545537\pi$$
−0.142572 + 0.989784i $$0.545537\pi$$
$$314$$ 0 0
$$315$$ −3536.00 −0.632479
$$316$$ 0 0
$$317$$ 10065.0 1.78330 0.891651 0.452723i $$-0.149548\pi$$
0.891651 + 0.452723i $$0.149548\pi$$
$$318$$ 0 0
$$319$$ −19530.0 −3.42781
$$320$$ 0 0
$$321$$ 1791.00 0.311414
$$322$$ 0 0
$$323$$ 1577.00 0.271661
$$324$$ 0 0
$$325$$ 3721.00 0.635089
$$326$$ 0 0
$$327$$ 779.000 0.131739
$$328$$ 0 0
$$329$$ 6664.00 1.11671
$$330$$ 0 0
$$331$$ 2953.00 0.490367 0.245184 0.969477i $$-0.421152\pi$$
0.245184 + 0.969477i $$0.421152\pi$$
$$332$$ 0 0
$$333$$ 884.000 0.145474
$$334$$ 0 0
$$335$$ 3688.00 0.601483
$$336$$ 0 0
$$337$$ −988.000 −0.159703 −0.0798513 0.996807i $$-0.525445\pi$$
−0.0798513 + 0.996807i $$0.525445\pi$$
$$338$$ 0 0
$$339$$ 1430.00 0.229106
$$340$$ 0 0
$$341$$ 5040.00 0.800385
$$342$$ 0 0
$$343$$ 6749.00 1.06242
$$344$$ 0 0
$$345$$ 920.000 0.143569
$$346$$ 0 0
$$347$$ −4458.00 −0.689677 −0.344839 0.938662i $$-0.612066\pi$$
−0.344839 + 0.938662i $$0.612066\pi$$
$$348$$ 0 0
$$349$$ −3522.00 −0.540196 −0.270098 0.962833i $$-0.587056\pi$$
−0.270098 + 0.962833i $$0.587056\pi$$
$$350$$ 0 0
$$351$$ −3233.00 −0.491638
$$352$$ 0 0
$$353$$ 8809.00 1.32820 0.664102 0.747642i $$-0.268814\pi$$
0.664102 + 0.747642i $$0.268814\pi$$
$$354$$ 0 0
$$355$$ −6000.00 −0.897034
$$356$$ 0 0
$$357$$ 1411.00 0.209182
$$358$$ 0 0
$$359$$ −12611.0 −1.85399 −0.926996 0.375071i $$-0.877618\pi$$
−0.926996 + 0.375071i $$0.877618\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 0 0
$$363$$ −3569.00 −0.516044
$$364$$ 0 0
$$365$$ −9416.00 −1.35029
$$366$$ 0 0
$$367$$ 9412.00 1.33870 0.669349 0.742948i $$-0.266573\pi$$
0.669349 + 0.742948i $$0.266573\pi$$
$$368$$ 0 0
$$369$$ −2808.00 −0.396148
$$370$$ 0 0
$$371$$ −2227.00 −0.311644
$$372$$ 0 0
$$373$$ 6553.00 0.909655 0.454828 0.890579i $$-0.349701\pi$$
0.454828 + 0.890579i $$0.349701\pi$$
$$374$$ 0 0
$$375$$ −1488.00 −0.204907
$$376$$ 0 0
$$377$$ −17019.0 −2.32499
$$378$$ 0 0
$$379$$ 14111.0 1.91249 0.956245 0.292568i $$-0.0945099\pi$$
0.956245 + 0.292568i $$0.0945099\pi$$
$$380$$ 0 0
$$381$$ −466.000 −0.0626612
$$382$$ 0 0
$$383$$ −9902.00 −1.32107 −0.660533 0.750797i $$-0.729670\pi$$
−0.660533 + 0.750797i $$0.729670\pi$$
$$384$$ 0 0
$$385$$ −9520.00 −1.26022
$$386$$ 0 0
$$387$$ 4992.00 0.655704
$$388$$ 0 0
$$389$$ 258.000 0.0336276 0.0168138 0.999859i $$-0.494648\pi$$
0.0168138 + 0.999859i $$0.494648\pi$$
$$390$$ 0 0
$$391$$ 9545.00 1.23456
$$392$$ 0 0
$$393$$ −2240.00 −0.287514
$$394$$ 0 0
$$395$$ 176.000 0.0224190
$$396$$ 0 0
$$397$$ 1224.00 0.154738 0.0773688 0.997003i $$-0.475348\pi$$
0.0773688 + 0.997003i $$0.475348\pi$$
$$398$$ 0 0
$$399$$ 323.000 0.0405269
$$400$$ 0 0
$$401$$ 12104.0 1.50734 0.753672 0.657251i $$-0.228281\pi$$
0.753672 + 0.657251i $$0.228281\pi$$
$$402$$ 0 0
$$403$$ 4392.00 0.542881
$$404$$ 0 0
$$405$$ −5192.00 −0.637019
$$406$$ 0 0
$$407$$ 2380.00 0.289858
$$408$$ 0 0
$$409$$ 6076.00 0.734569 0.367285 0.930109i $$-0.380287\pi$$
0.367285 + 0.930109i $$0.380287\pi$$
$$410$$ 0 0
$$411$$ 1145.00 0.137418
$$412$$ 0 0
$$413$$ 10353.0 1.23351
$$414$$ 0 0
$$415$$ 6480.00 0.766484
$$416$$ 0 0
$$417$$ 2336.00 0.274327
$$418$$ 0 0
$$419$$ 8148.00 0.950014 0.475007 0.879982i $$-0.342446\pi$$
0.475007 + 0.879982i $$0.342446\pi$$
$$420$$ 0 0
$$421$$ −13849.0 −1.60323 −0.801614 0.597842i $$-0.796025\pi$$
−0.801614 + 0.597842i $$0.796025\pi$$
$$422$$ 0 0
$$423$$ 10192.0 1.17152
$$424$$ 0 0
$$425$$ −5063.00 −0.577863
$$426$$ 0 0
$$427$$ −5746.00 −0.651214
$$428$$ 0 0
$$429$$ −4270.00 −0.480554
$$430$$ 0 0
$$431$$ −9322.00 −1.04182 −0.520911 0.853611i $$-0.674408\pi$$
−0.520911 + 0.853611i $$0.674408\pi$$
$$432$$ 0 0
$$433$$ 7090.00 0.786891 0.393445 0.919348i $$-0.371283\pi$$
0.393445 + 0.919348i $$0.371283\pi$$
$$434$$ 0 0
$$435$$ 2232.00 0.246014
$$436$$ 0 0
$$437$$ 2185.00 0.239182
$$438$$ 0 0
$$439$$ 9740.00 1.05892 0.529459 0.848336i $$-0.322395\pi$$
0.529459 + 0.848336i $$0.322395\pi$$
$$440$$ 0 0
$$441$$ 1404.00 0.151603
$$442$$ 0 0
$$443$$ −4658.00 −0.499567 −0.249784 0.968302i $$-0.580359\pi$$
−0.249784 + 0.968302i $$0.580359\pi$$
$$444$$ 0 0
$$445$$ 3808.00 0.405655
$$446$$ 0 0
$$447$$ −3516.00 −0.372038
$$448$$ 0 0
$$449$$ 14314.0 1.50450 0.752249 0.658879i $$-0.228969\pi$$
0.752249 + 0.658879i $$0.228969\pi$$
$$450$$ 0 0
$$451$$ −7560.00 −0.789327
$$452$$ 0 0
$$453$$ 390.000 0.0404499
$$454$$ 0 0
$$455$$ −8296.00 −0.854775
$$456$$ 0 0
$$457$$ 3641.00 0.372689 0.186344 0.982484i $$-0.440336\pi$$
0.186344 + 0.982484i $$0.440336\pi$$
$$458$$ 0 0
$$459$$ 4399.00 0.447337
$$460$$ 0 0
$$461$$ −11540.0 −1.16588 −0.582941 0.812515i $$-0.698098\pi$$
−0.582941 + 0.812515i $$0.698098\pi$$
$$462$$ 0 0
$$463$$ −13732.0 −1.37836 −0.689179 0.724591i $$-0.742029\pi$$
−0.689179 + 0.724591i $$0.742029\pi$$
$$464$$ 0 0
$$465$$ −576.000 −0.0574438
$$466$$ 0 0
$$467$$ 15978.0 1.58324 0.791621 0.611013i $$-0.209238\pi$$
0.791621 + 0.611013i $$0.209238\pi$$
$$468$$ 0 0
$$469$$ 7837.00 0.771597
$$470$$ 0 0
$$471$$ −2306.00 −0.225594
$$472$$ 0 0
$$473$$ 13440.0 1.30649
$$474$$ 0 0
$$475$$ −1159.00 −0.111955
$$476$$ 0 0
$$477$$ −3406.00 −0.326939
$$478$$ 0 0
$$479$$ −4300.00 −0.410171 −0.205086 0.978744i $$-0.565747\pi$$
−0.205086 + 0.978744i $$0.565747\pi$$
$$480$$ 0 0
$$481$$ 2074.00 0.196603
$$482$$ 0 0
$$483$$ 1955.00 0.184173
$$484$$ 0 0
$$485$$ −11408.0 −1.06806
$$486$$ 0 0
$$487$$ −12326.0 −1.14691 −0.573454 0.819238i $$-0.694397\pi$$
−0.573454 + 0.819238i $$0.694397\pi$$
$$488$$ 0 0
$$489$$ 1272.00 0.117632
$$490$$ 0 0
$$491$$ −7236.00 −0.665084 −0.332542 0.943088i $$-0.607906\pi$$
−0.332542 + 0.943088i $$0.607906\pi$$
$$492$$ 0 0
$$493$$ 23157.0 2.11549
$$494$$ 0 0
$$495$$ −14560.0 −1.32207
$$496$$ 0 0
$$497$$ −12750.0 −1.15074
$$498$$ 0 0
$$499$$ −1148.00 −0.102989 −0.0514945 0.998673i $$-0.516398\pi$$
−0.0514945 + 0.998673i $$0.516398\pi$$
$$500$$ 0 0
$$501$$ −1768.00 −0.157662
$$502$$ 0 0
$$503$$ 5739.00 0.508726 0.254363 0.967109i $$-0.418134\pi$$
0.254363 + 0.967109i $$0.418134\pi$$
$$504$$ 0 0
$$505$$ 9840.00 0.867078
$$506$$ 0 0
$$507$$ −1524.00 −0.133497
$$508$$ 0 0
$$509$$ 8442.00 0.735138 0.367569 0.929996i $$-0.380190\pi$$
0.367569 + 0.929996i $$0.380190\pi$$
$$510$$ 0 0
$$511$$ −20009.0 −1.73218
$$512$$ 0 0
$$513$$ 1007.00 0.0866669
$$514$$ 0 0
$$515$$ −2480.00 −0.212198
$$516$$ 0 0
$$517$$ 27440.0 2.33425
$$518$$ 0 0
$$519$$ 3726.00 0.315131
$$520$$ 0 0
$$521$$ 744.000 0.0625628 0.0312814 0.999511i $$-0.490041\pi$$
0.0312814 + 0.999511i $$0.490041\pi$$
$$522$$ 0 0
$$523$$ 10013.0 0.837166 0.418583 0.908179i $$-0.362527\pi$$
0.418583 + 0.908179i $$0.362527\pi$$
$$524$$ 0 0
$$525$$ −1037.00 −0.0862065
$$526$$ 0 0
$$527$$ −5976.00 −0.493963
$$528$$ 0 0
$$529$$ 1058.00 0.0869565
$$530$$ 0 0
$$531$$ 15834.0 1.29404
$$532$$ 0 0
$$533$$ −6588.00 −0.535381
$$534$$ 0 0
$$535$$ 14328.0 1.15786
$$536$$ 0 0
$$537$$ 1048.00 0.0842170
$$538$$ 0 0
$$539$$ 3780.00 0.302071
$$540$$ 0 0
$$541$$ 14582.0 1.15883 0.579417 0.815031i $$-0.303280\pi$$
0.579417 + 0.815031i $$0.303280\pi$$
$$542$$ 0 0
$$543$$ −2678.00 −0.211646
$$544$$ 0 0
$$545$$ 6232.00 0.489816
$$546$$ 0 0
$$547$$ 17924.0 1.40105 0.700526 0.713627i $$-0.252949\pi$$
0.700526 + 0.713627i $$0.252949\pi$$
$$548$$ 0 0
$$549$$ −8788.00 −0.683174
$$550$$ 0 0
$$551$$ 5301.00 0.409855
$$552$$ 0 0
$$553$$ 374.000 0.0287597
$$554$$ 0 0
$$555$$ −272.000 −0.0208032
$$556$$ 0 0
$$557$$ 11960.0 0.909805 0.454903 0.890541i $$-0.349674\pi$$
0.454903 + 0.890541i $$0.349674\pi$$
$$558$$ 0 0
$$559$$ 11712.0 0.886162
$$560$$ 0 0
$$561$$ 5810.00 0.437252
$$562$$ 0 0
$$563$$ 1348.00 0.100908 0.0504542 0.998726i $$-0.483933\pi$$
0.0504542 + 0.998726i $$0.483933\pi$$
$$564$$ 0 0
$$565$$ 11440.0 0.851831
$$566$$ 0 0
$$567$$ −11033.0 −0.817182
$$568$$ 0 0
$$569$$ −9820.00 −0.723508 −0.361754 0.932274i $$-0.617822\pi$$
−0.361754 + 0.932274i $$0.617822\pi$$
$$570$$ 0 0
$$571$$ −20904.0 −1.53206 −0.766029 0.642806i $$-0.777770\pi$$
−0.766029 + 0.642806i $$0.777770\pi$$
$$572$$ 0 0
$$573$$ −1443.00 −0.105205
$$574$$ 0 0
$$575$$ −7015.00 −0.508775
$$576$$ 0 0
$$577$$ −15093.0 −1.08896 −0.544480 0.838774i $$-0.683273\pi$$
−0.544480 + 0.838774i $$0.683273\pi$$
$$578$$ 0 0
$$579$$ −890.000 −0.0638811
$$580$$ 0 0
$$581$$ 13770.0 0.983263
$$582$$ 0 0
$$583$$ −9170.00 −0.651428
$$584$$ 0 0
$$585$$ −12688.0 −0.896725
$$586$$ 0 0
$$587$$ −16740.0 −1.17706 −0.588530 0.808476i $$-0.700293\pi$$
−0.588530 + 0.808476i $$0.700293\pi$$
$$588$$ 0 0
$$589$$ −1368.00 −0.0957003
$$590$$ 0 0
$$591$$ 300.000 0.0208805
$$592$$ 0 0
$$593$$ 3042.00 0.210658 0.105329 0.994437i $$-0.466410\pi$$
0.105329 + 0.994437i $$0.466410\pi$$
$$594$$ 0 0
$$595$$ 11288.0 0.777753
$$596$$ 0 0
$$597$$ −3339.00 −0.228905
$$598$$ 0 0
$$599$$ 27816.0 1.89738 0.948690 0.316207i $$-0.102409\pi$$
0.948690 + 0.316207i $$0.102409\pi$$
$$600$$ 0 0
$$601$$ 19320.0 1.31128 0.655640 0.755074i $$-0.272399\pi$$
0.655640 + 0.755074i $$0.272399\pi$$
$$602$$ 0 0
$$603$$ 11986.0 0.809465
$$604$$ 0 0
$$605$$ −28552.0 −1.91868
$$606$$ 0 0
$$607$$ 17926.0 1.19867 0.599336 0.800498i $$-0.295431\pi$$
0.599336 + 0.800498i $$0.295431\pi$$
$$608$$ 0 0
$$609$$ 4743.00 0.315593
$$610$$ 0 0
$$611$$ 23912.0 1.58327
$$612$$ 0 0
$$613$$ 18294.0 1.20536 0.602682 0.797982i $$-0.294099\pi$$
0.602682 + 0.797982i $$0.294099\pi$$
$$614$$ 0 0
$$615$$ 864.000 0.0566502
$$616$$ 0 0
$$617$$ 22538.0 1.47058 0.735288 0.677755i $$-0.237047\pi$$
0.735288 + 0.677755i $$0.237047\pi$$
$$618$$ 0 0
$$619$$ −17558.0 −1.14009 −0.570045 0.821614i $$-0.693074\pi$$
−0.570045 + 0.821614i $$0.693074\pi$$
$$620$$ 0 0
$$621$$ 6095.00 0.393855
$$622$$ 0 0
$$623$$ 8092.00 0.520384
$$624$$ 0 0
$$625$$ −4279.00 −0.273856
$$626$$ 0 0
$$627$$ 1330.00 0.0847131
$$628$$ 0 0
$$629$$ −2822.00 −0.178888
$$630$$ 0 0
$$631$$ −12440.0 −0.784831 −0.392416 0.919788i $$-0.628361\pi$$
−0.392416 + 0.919788i $$0.628361\pi$$
$$632$$ 0 0
$$633$$ 2149.00 0.134937
$$634$$ 0 0
$$635$$ −3728.00 −0.232978
$$636$$ 0 0
$$637$$ 3294.00 0.204887
$$638$$ 0 0
$$639$$ −19500.0 −1.20721
$$640$$ 0 0
$$641$$ −10842.0 −0.668071 −0.334035 0.942561i $$-0.608410\pi$$
−0.334035 + 0.942561i $$0.608410\pi$$
$$642$$ 0 0
$$643$$ 322.000 0.0197487 0.00987437 0.999951i $$-0.496857\pi$$
0.00987437 + 0.999951i $$0.496857\pi$$
$$644$$ 0 0
$$645$$ −1536.00 −0.0937674
$$646$$ 0 0
$$647$$ 2995.00 0.181987 0.0909935 0.995851i $$-0.470996\pi$$
0.0909935 + 0.995851i $$0.470996\pi$$
$$648$$ 0 0
$$649$$ 42630.0 2.57839
$$650$$ 0 0
$$651$$ −1224.00 −0.0736902
$$652$$ 0 0
$$653$$ 14652.0 0.878066 0.439033 0.898471i $$-0.355321\pi$$
0.439033 + 0.898471i $$0.355321\pi$$
$$654$$ 0 0
$$655$$ −17920.0 −1.06900
$$656$$ 0 0
$$657$$ −30602.0 −1.81720
$$658$$ 0 0
$$659$$ 20613.0 1.21847 0.609233 0.792992i $$-0.291478\pi$$
0.609233 + 0.792992i $$0.291478\pi$$
$$660$$ 0 0
$$661$$ −7875.00 −0.463392 −0.231696 0.972788i $$-0.574427\pi$$
−0.231696 + 0.972788i $$0.574427\pi$$
$$662$$ 0 0
$$663$$ 5063.00 0.296577
$$664$$ 0 0
$$665$$ 2584.00 0.150682
$$666$$ 0 0
$$667$$ 32085.0 1.86257
$$668$$ 0 0
$$669$$ −3834.00 −0.221571
$$670$$ 0 0
$$671$$ −23660.0 −1.36123
$$672$$ 0 0
$$673$$ 6216.00 0.356031 0.178016 0.984028i $$-0.443032\pi$$
0.178016 + 0.984028i $$0.443032\pi$$
$$674$$ 0 0
$$675$$ −3233.00 −0.184353
$$676$$ 0 0
$$677$$ 13961.0 0.792563 0.396281 0.918129i $$-0.370301\pi$$
0.396281 + 0.918129i $$0.370301\pi$$
$$678$$ 0 0
$$679$$ −24242.0 −1.37014
$$680$$ 0 0
$$681$$ 329.000 0.0185129
$$682$$ 0 0
$$683$$ −7788.00 −0.436310 −0.218155 0.975914i $$-0.570004\pi$$
−0.218155 + 0.975914i $$0.570004\pi$$
$$684$$ 0 0
$$685$$ 9160.00 0.510928
$$686$$ 0 0
$$687$$ −430.000 −0.0238799
$$688$$ 0 0
$$689$$ −7991.00 −0.441847
$$690$$ 0 0
$$691$$ −11782.0 −0.648637 −0.324319 0.945948i $$-0.605135\pi$$
−0.324319 + 0.945948i $$0.605135\pi$$
$$692$$ 0 0
$$693$$ −30940.0 −1.69598
$$694$$ 0 0
$$695$$ 18688.0 1.01997
$$696$$ 0 0
$$697$$ 8964.00 0.487139
$$698$$ 0 0
$$699$$ 438.000 0.0237005
$$700$$ 0 0
$$701$$ −22700.0 −1.22306 −0.611532 0.791220i $$-0.709446\pi$$
−0.611532 + 0.791220i $$0.709446\pi$$
$$702$$ 0 0
$$703$$ −646.000 −0.0346577
$$704$$ 0 0
$$705$$ −3136.00 −0.167530
$$706$$ 0 0
$$707$$ 20910.0 1.11231
$$708$$ 0 0
$$709$$ 12162.0 0.644222 0.322111 0.946702i $$-0.395608\pi$$
0.322111 + 0.946702i $$0.395608\pi$$
$$710$$ 0 0
$$711$$ 572.000 0.0301711
$$712$$ 0 0
$$713$$ −8280.00 −0.434907
$$714$$ 0 0
$$715$$ −34160.0 −1.78673
$$716$$ 0 0
$$717$$ −2099.00 −0.109329
$$718$$ 0 0
$$719$$ 2199.00 0.114060 0.0570298 0.998372i $$-0.481837\pi$$
0.0570298 + 0.998372i $$0.481837\pi$$
$$720$$ 0 0
$$721$$ −5270.00 −0.272212
$$722$$ 0 0
$$723$$ 2996.00 0.154111
$$724$$ 0 0
$$725$$ −17019.0 −0.871820
$$726$$ 0 0
$$727$$ −18965.0 −0.967501 −0.483750 0.875206i $$-0.660726\pi$$
−0.483750 + 0.875206i $$0.660726\pi$$
$$728$$ 0 0
$$729$$ −15443.0 −0.784586
$$730$$ 0 0
$$731$$ −15936.0 −0.806312
$$732$$ 0 0
$$733$$ −5552.00 −0.279765 −0.139883 0.990168i $$-0.544672\pi$$
−0.139883 + 0.990168i $$0.544672\pi$$
$$734$$ 0 0
$$735$$ −432.000 −0.0216797
$$736$$ 0 0
$$737$$ 32270.0 1.61286
$$738$$ 0 0
$$739$$ 13136.0 0.653878 0.326939 0.945046i $$-0.393983\pi$$
0.326939 + 0.945046i $$0.393983\pi$$
$$740$$ 0 0
$$741$$ 1159.00 0.0574587
$$742$$ 0 0
$$743$$ −13416.0 −0.662430 −0.331215 0.943555i $$-0.607459\pi$$
−0.331215 + 0.943555i $$0.607459\pi$$
$$744$$ 0 0
$$745$$ −28128.0 −1.38326
$$746$$ 0 0
$$747$$ 21060.0 1.03152
$$748$$ 0 0
$$749$$ 30447.0 1.48533
$$750$$ 0 0
$$751$$ −33348.0 −1.62035 −0.810177 0.586185i $$-0.800629\pi$$
−0.810177 + 0.586185i $$0.800629\pi$$
$$752$$ 0 0
$$753$$ −5518.00 −0.267048
$$754$$ 0 0
$$755$$ 3120.00 0.150395
$$756$$ 0 0
$$757$$ 10606.0 0.509223 0.254611 0.967043i $$-0.418052\pi$$
0.254611 + 0.967043i $$0.418052\pi$$
$$758$$ 0 0
$$759$$ 8050.00 0.384976
$$760$$ 0 0
$$761$$ −29061.0 −1.38431 −0.692155 0.721749i $$-0.743339\pi$$
−0.692155 + 0.721749i $$0.743339\pi$$
$$762$$ 0 0
$$763$$ 13243.0 0.628347
$$764$$ 0 0
$$765$$ 17264.0 0.815923
$$766$$ 0 0
$$767$$ 37149.0 1.74886
$$768$$ 0 0
$$769$$ −15955.0 −0.748182 −0.374091 0.927392i $$-0.622045\pi$$
−0.374091 + 0.927392i $$0.622045\pi$$
$$770$$ 0 0
$$771$$ 4068.00 0.190020
$$772$$ 0 0
$$773$$ −36763.0 −1.71057 −0.855287 0.518155i $$-0.826619\pi$$
−0.855287 + 0.518155i $$0.826619\pi$$
$$774$$ 0 0
$$775$$ 4392.00 0.203568
$$776$$ 0 0
$$777$$ −578.000 −0.0266868
$$778$$ 0 0
$$779$$ 2052.00 0.0943781
$$780$$ 0 0
$$781$$ −52500.0 −2.40537
$$782$$ 0 0
$$783$$ 14787.0 0.674897
$$784$$ 0 0
$$785$$ −18448.0 −0.838774
$$786$$ 0 0
$$787$$ 31655.0 1.43377 0.716886 0.697190i $$-0.245567\pi$$
0.716886 + 0.697190i $$0.245567\pi$$
$$788$$ 0 0
$$789$$ −4992.00 −0.225247
$$790$$ 0 0
$$791$$ 24310.0 1.09275
$$792$$ 0 0
$$793$$ −20618.0 −0.923287
$$794$$ 0 0
$$795$$ 1048.00 0.0467531
$$796$$ 0 0
$$797$$ −12945.0 −0.575327 −0.287663 0.957732i $$-0.592878\pi$$
−0.287663 + 0.957732i $$0.592878\pi$$
$$798$$ 0 0
$$799$$ −32536.0 −1.44060
$$800$$ 0 0
$$801$$ 12376.0 0.545923
$$802$$ 0 0
$$803$$ −82390.0 −3.62077
$$804$$ 0 0
$$805$$ 15640.0 0.684767
$$806$$ 0 0
$$807$$ 1970.00 0.0859322
$$808$$ 0 0
$$809$$ −20263.0 −0.880605 −0.440302 0.897850i $$-0.645129\pi$$
−0.440302 + 0.897850i $$0.645129\pi$$
$$810$$ 0 0
$$811$$ 29477.0 1.27630 0.638149 0.769913i $$-0.279700\pi$$
0.638149 + 0.769913i $$0.279700\pi$$
$$812$$ 0 0
$$813$$ 1861.00 0.0802806
$$814$$ 0 0
$$815$$ 10176.0 0.437362
$$816$$ 0 0
$$817$$ −3648.00 −0.156215
$$818$$ 0 0
$$819$$ −26962.0 −1.15034
$$820$$ 0 0
$$821$$ −1756.00 −0.0746466 −0.0373233 0.999303i $$-0.511883\pi$$
−0.0373233 + 0.999303i $$0.511883\pi$$
$$822$$ 0 0
$$823$$ 1721.00 0.0728922 0.0364461 0.999336i $$-0.488396\pi$$
0.0364461 + 0.999336i $$0.488396\pi$$
$$824$$ 0 0
$$825$$ −4270.00 −0.180197
$$826$$ 0 0
$$827$$ 3063.00 0.128792 0.0643960 0.997924i $$-0.479488\pi$$
0.0643960 + 0.997924i $$0.479488\pi$$
$$828$$ 0 0
$$829$$ −25467.0 −1.06695 −0.533477 0.845814i $$-0.679115\pi$$
−0.533477 + 0.845814i $$0.679115\pi$$
$$830$$ 0 0
$$831$$ 1268.00 0.0529319
$$832$$ 0 0
$$833$$ −4482.00 −0.186425
$$834$$ 0 0
$$835$$ −14144.0 −0.586196
$$836$$ 0 0
$$837$$ −3816.00 −0.157587
$$838$$ 0 0
$$839$$ 2976.00 0.122459 0.0612294 0.998124i $$-0.480498\pi$$
0.0612294 + 0.998124i $$0.480498\pi$$
$$840$$ 0 0
$$841$$ 53452.0 2.19164
$$842$$ 0 0
$$843$$ 8912.00 0.364111
$$844$$ 0 0
$$845$$ −12192.0 −0.496352
$$846$$ 0 0
$$847$$ −60673.0 −2.46133
$$848$$ 0 0
$$849$$ −3302.00 −0.133480
$$850$$ 0 0
$$851$$ −3910.00 −0.157501
$$852$$ 0 0
$$853$$ 2126.00 0.0853375 0.0426687 0.999089i $$-0.486414\pi$$
0.0426687 + 0.999089i $$0.486414\pi$$
$$854$$ 0 0
$$855$$ 3952.00 0.158077
$$856$$ 0 0
$$857$$ −4684.00 −0.186701 −0.0933503 0.995633i $$-0.529758\pi$$
−0.0933503 + 0.995633i $$0.529758\pi$$
$$858$$ 0 0
$$859$$ 1370.00 0.0544165 0.0272083 0.999630i $$-0.491338\pi$$
0.0272083 + 0.999630i $$0.491338\pi$$
$$860$$ 0 0
$$861$$ 1836.00 0.0726721
$$862$$ 0 0
$$863$$ 30630.0 1.20818 0.604089 0.796917i $$-0.293537\pi$$
0.604089 + 0.796917i $$0.293537\pi$$
$$864$$ 0 0
$$865$$ 29808.0 1.17168
$$866$$ 0 0
$$867$$ −1976.00 −0.0774031
$$868$$ 0 0
$$869$$ 1540.00 0.0601161
$$870$$ 0 0
$$871$$ 28121.0 1.09397
$$872$$ 0 0
$$873$$ −37076.0 −1.43738
$$874$$ 0 0
$$875$$ −25296.0 −0.977327
$$876$$ 0 0
$$877$$ 19617.0 0.755324 0.377662 0.925944i $$-0.376728\pi$$
0.377662 + 0.925944i $$0.376728\pi$$
$$878$$ 0 0
$$879$$ 5435.00 0.208553
$$880$$ 0 0
$$881$$ 1614.00 0.0617220 0.0308610 0.999524i $$-0.490175\pi$$
0.0308610 + 0.999524i $$0.490175\pi$$
$$882$$ 0 0
$$883$$ 35258.0 1.34374 0.671872 0.740667i $$-0.265490\pi$$
0.671872 + 0.740667i $$0.265490\pi$$
$$884$$ 0 0
$$885$$ −4872.00 −0.185051
$$886$$ 0 0
$$887$$ 2166.00 0.0819923 0.0409961 0.999159i $$-0.486947\pi$$
0.0409961 + 0.999159i $$0.486947\pi$$
$$888$$ 0 0
$$889$$ −7922.00 −0.298870
$$890$$ 0 0
$$891$$ −45430.0 −1.70815
$$892$$ 0 0
$$893$$ −7448.00 −0.279102
$$894$$ 0 0
$$895$$ 8384.00 0.313124
$$896$$ 0 0
$$897$$ 7015.00 0.261119
$$898$$ 0 0
$$899$$ −20088.0 −0.745242
$$900$$ 0 0
$$901$$ 10873.0 0.402033
$$902$$ 0 0
$$903$$ −3264.00 −0.120287
$$904$$ 0 0
$$905$$ −21424.0 −0.786915
$$906$$ 0 0
$$907$$ −30419.0 −1.11361 −0.556806 0.830642i $$-0.687973\pi$$
−0.556806 + 0.830642i $$0.687973\pi$$
$$908$$ 0 0
$$909$$ 31980.0 1.16690
$$910$$ 0 0
$$911$$ 32864.0 1.19521 0.597603 0.801792i $$-0.296120\pi$$
0.597603 + 0.801792i $$0.296120\pi$$
$$912$$ 0 0
$$913$$ 56700.0 2.05531
$$914$$ 0 0
$$915$$ 2704.00 0.0976956
$$916$$ 0 0
$$917$$ −38080.0 −1.37133
$$918$$ 0 0
$$919$$ 20225.0 0.725964 0.362982 0.931796i $$-0.381759\pi$$
0.362982 + 0.931796i $$0.381759\pi$$
$$920$$ 0 0
$$921$$ −1740.00 −0.0622529
$$922$$ 0 0
$$923$$ −45750.0 −1.63151
$$924$$ 0 0
$$925$$ 2074.00 0.0737218
$$926$$ 0 0
$$927$$ −8060.00 −0.285572
$$928$$ 0 0
$$929$$ −29415.0 −1.03883 −0.519416 0.854522i $$-0.673850\pi$$
−0.519416 + 0.854522i $$0.673850\pi$$
$$930$$ 0 0
$$931$$ −1026.00 −0.0361179
$$932$$ 0 0
$$933$$ 2837.00 0.0995490
$$934$$ 0 0
$$935$$ 46480.0 1.62573
$$936$$ 0 0
$$937$$ 3697.00 0.128896 0.0644481 0.997921i $$-0.479471\pi$$
0.0644481 + 0.997921i $$0.479471\pi$$
$$938$$ 0 0
$$939$$ 1579.00 0.0548762
$$940$$ 0 0
$$941$$ −55245.0 −1.91385 −0.956926 0.290331i $$-0.906235\pi$$
−0.956926 + 0.290331i $$0.906235\pi$$
$$942$$ 0 0
$$943$$ 12420.0 0.428898
$$944$$ 0 0
$$945$$ 7208.00 0.248123
$$946$$ 0 0
$$947$$ 54296.0 1.86313 0.931564 0.363576i $$-0.118444\pi$$
0.931564 + 0.363576i $$0.118444\pi$$
$$948$$ 0 0
$$949$$ −71797.0 −2.45588
$$950$$ 0 0
$$951$$ −10065.0 −0.343197
$$952$$ 0 0
$$953$$ −25770.0 −0.875941 −0.437971 0.898989i $$-0.644303\pi$$
−0.437971 + 0.898989i $$0.644303\pi$$
$$954$$ 0 0
$$955$$ −11544.0 −0.391157
$$956$$ 0 0
$$957$$ 19530.0 0.659682
$$958$$ 0 0
$$959$$ 19465.0 0.655430
$$960$$ 0 0
$$961$$ −24607.0 −0.825988
$$962$$ 0 0
$$963$$ 46566.0 1.55822
$$964$$ 0 0
$$965$$ −7120.00 −0.237514
$$966$$ 0 0
$$967$$ 20296.0 0.674949 0.337474 0.941335i $$-0.390427\pi$$
0.337474 + 0.941335i $$0.390427\pi$$
$$968$$ 0 0
$$969$$ −1577.00 −0.0522813
$$970$$ 0 0
$$971$$ 34476.0 1.13943 0.569715 0.821842i $$-0.307053\pi$$
0.569715 + 0.821842i $$0.307053\pi$$
$$972$$ 0 0
$$973$$ 39712.0 1.30844
$$974$$ 0 0
$$975$$ −3721.00 −0.122223
$$976$$ 0 0
$$977$$ 39952.0 1.30827 0.654134 0.756379i $$-0.273033\pi$$
0.654134 + 0.756379i $$0.273033\pi$$
$$978$$ 0 0
$$979$$ 33320.0 1.08775
$$980$$ 0 0
$$981$$ 20254.0 0.659185
$$982$$ 0 0
$$983$$ −56942.0 −1.84758 −0.923788 0.382904i $$-0.874924\pi$$
−0.923788 + 0.382904i $$0.874924\pi$$
$$984$$ 0 0
$$985$$ 2400.00 0.0776349
$$986$$ 0 0
$$987$$ −6664.00 −0.214911
$$988$$ 0 0
$$989$$ −22080.0 −0.709912
$$990$$ 0 0
$$991$$ 45772.0 1.46720 0.733600 0.679581i $$-0.237839\pi$$
0.733600 + 0.679581i $$0.237839\pi$$
$$992$$ 0 0
$$993$$ −2953.00 −0.0943712
$$994$$ 0 0
$$995$$ −26712.0 −0.851083
$$996$$ 0 0
$$997$$ −24916.0 −0.791472 −0.395736 0.918364i $$-0.629510\pi$$
−0.395736 + 0.918364i $$0.629510\pi$$
$$998$$ 0 0
$$999$$ −1802.00 −0.0570698
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 608.4.a.a.1.1 1
4.3 odd 2 608.4.a.b.1.1 yes 1
8.3 odd 2 1216.4.a.c.1.1 1
8.5 even 2 1216.4.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.a.1.1 1 1.1 even 1 trivial
608.4.a.b.1.1 yes 1 4.3 odd 2
1216.4.a.c.1.1 1 8.3 odd 2
1216.4.a.d.1.1 1 8.5 even 2