# Properties

 Label 1216.4.a.c.1.1 Level $1216$ Weight $4$ Character 1216.1 Self dual yes Analytic conductor $71.746$ 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] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$71.7463225670$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1216.1

## $q$-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 $$-0.383537\pi$$
0.357771 + 0.933809i $$0.383537\pi$$
$$6$$ 0 0
$$7$$ 17.0000 0.917914 0.458957 0.888459i $$-0.348223\pi$$
0.458957 + 0.888459i $$0.348223\pi$$
$$8$$ 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 $$-0.274473\pi$$
0.650706 + 0.759330i $$0.274473\pi$$
$$14$$ 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 $$-0.674548\pi$$
−0.521286 + 0.853382i $$0.674548\pi$$
$$24$$ 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 $$-0.851587\pi$$
−0.893259 + 0.449543i $$0.851587\pi$$
$$30$$ 0 0
$$31$$ 72.0000 0.417148 0.208574 0.978007i $$-0.433118\pi$$
0.208574 + 0.978007i $$0.433118\pi$$
$$32$$ 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 $$-0.475934\pi$$
0.0755347 + 0.997143i $$0.475934\pi$$
$$38$$ 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 $$-0.291857\pi$$
0.608288 + 0.793716i $$0.291857\pi$$
$$48$$ 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 $$-0.554298\pi$$
−0.169757 + 0.985486i $$0.554298\pi$$
$$54$$ 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 $$-0.615426\pi$$
−0.354725 + 0.934971i $$0.615426\pi$$
$$62$$ 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 $$-0.715644\pi$$
−0.626821 + 0.779163i $$0.715644\pi$$
$$72$$ 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 $$-0.495013\pi$$
0.0156658 + 0.999877i $$0.495013\pi$$
$$80$$ 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 $$-0.292818\pi$$
0.605889 + 0.795549i $$0.292818\pi$$
$$102$$ 0 0
$$103$$ −310.000 −0.296555 −0.148278 0.988946i $$-0.547373\pi$$
−0.148278 + 0.988946i $$0.547373\pi$$
$$104$$ 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 $$-0.388805\pi$$
0.342269 + 0.939602i $$0.388805\pi$$
$$110$$ 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 $$-0.552052\pi$$
−0.162798 + 0.986659i $$0.552052\pi$$
$$128$$ 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 $$-0.917480\pi$$
−0.966584 + 0.256351i $$0.917480\pi$$
$$150$$ 0 0
$$151$$ 390.000 0.210184 0.105092 0.994463i $$-0.466486\pi$$
0.105092 + 0.994463i $$0.466486\pi$$
$$152$$ 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 $$-0.699342\pi$$
−0.586111 + 0.810231i $$0.699342\pi$$
$$158$$ 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 $$-0.634338\pi$$
−0.409617 + 0.912258i $$0.634338\pi$$
$$168$$ 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 $$-0.194675\pi$$
0.818736 + 0.574171i $$0.194675\pi$$
$$174$$ 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 $$-0.685324\pi$$
−0.549873 + 0.835248i $$0.685324\pi$$
$$182$$ 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 $$-0.588125\pi$$
−0.273329 + 0.961921i $$0.588125\pi$$
$$192$$ 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 $$-0.482724\pi$$
0.0542490 + 0.998527i $$0.482724\pi$$
$$198$$ 0 0
$$199$$ −3339.00 −1.18942 −0.594712 0.803939i $$-0.702734\pi$$
−0.594712 + 0.803939i $$0.702734\pi$$
$$200$$ 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 $$-0.695254\pi$$
−0.575658 + 0.817690i $$0.695254\pi$$
$$224$$ 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 $$-0.519761\pi$$
−0.0620419 + 0.998074i $$0.519761\pi$$
$$230$$ 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 $$-0.591676\pi$$
−0.284044 + 0.958811i $$0.591676\pi$$
$$240$$ 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 $$-0.698988\pi$$
−0.585209 + 0.810883i $$0.698988\pi$$
$$264$$ 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 $$-0.428331\pi$$
0.223258 + 0.974759i $$0.428331\pi$$
$$270$$ 0 0
$$271$$ 1861.00 0.417150 0.208575 0.978006i $$-0.433117\pi$$
0.208575 + 0.978006i $$0.433117\pi$$
$$272$$ 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 $$-0.456086\pi$$
0.137521 + 0.990499i $$0.456086\pi$$
$$278$$ 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 $$-0.317729\pi$$
0.541836 + 0.840484i $$0.317729\pi$$
$$294$$ 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 $$-0.416727\pi$$
0.258636 + 0.965975i $$0.416727\pi$$
$$312$$ 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 $$-0.850452\pi$$
−0.891651 + 0.452723i $$0.850452\pi$$
$$318$$ 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 $$-0.412944\pi$$
0.270098 + 0.962833i $$0.412944\pi$$
$$350$$ 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 $$-0.122382\pi$$
0.926996 + 0.375071i $$0.122382\pi$$
$$360$$ 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 $$-0.733427\pi$$
−0.669349 + 0.742948i $$0.733427\pi$$
$$368$$ 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 $$-0.650299\pi$$
−0.454828 + 0.890579i $$0.650299\pi$$
$$374$$ 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 $$-0.270330\pi$$
0.660533 + 0.750797i $$0.270330\pi$$
$$384$$ 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 $$-0.505352\pi$$
−0.0168138 + 0.999859i $$0.505352\pi$$
$$390$$ 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 $$-0.524652\pi$$
−0.0773688 + 0.997003i $$0.524652\pi$$
$$398$$ 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 $$-0.203975\pi$$
0.801614 + 0.597842i $$0.203975\pi$$
$$422$$ 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 $$-0.325592\pi$$
0.520911 + 0.853611i $$0.325592\pi$$
$$432$$ 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 $$-0.677605\pi$$
−0.529459 + 0.848336i $$0.677605\pi$$
$$440$$ 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 $$-0.301902\pi$$
0.582941 + 0.812515i $$0.301902\pi$$
$$462$$ 0 0
$$463$$ 13732.0 1.37836 0.689179 0.724591i $$-0.257971\pi$$
0.689179 + 0.724591i $$0.257971\pi$$
$$464$$ 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 $$-0.434253\pi$$
0.205086 + 0.978744i $$0.434253\pi$$
$$480$$ 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 $$-0.305603\pi$$
0.573454 + 0.819238i $$0.305603\pi$$
$$488$$ 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 $$-0.581866\pi$$
−0.254363 + 0.967109i $$0.581866\pi$$
$$504$$ 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 $$-0.619810\pi$$
−0.367569 + 0.929996i $$0.619810\pi$$
$$510$$ 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 $$-0.696720\pi$$
−0.579417 + 0.815031i $$0.696720\pi$$
$$542$$ 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 $$-0.650326\pi$$
−0.454903 + 0.890541i $$0.650326\pi$$
$$558$$ 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 $$-0.897591\pi$$
−0.948690 + 0.316207i $$0.897591\pi$$
$$600$$ 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 $$-0.704569\pi$$
−0.599336 + 0.800498i $$0.704569\pi$$
$$608$$ 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 $$-0.705901\pi$$
−0.602682 + 0.797982i $$0.705901\pi$$
$$614$$ 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 $$-0.371639\pi$$
0.392416 + 0.919788i $$0.371639\pi$$
$$632$$ 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 $$-0.529004\pi$$
−0.0909935 + 0.995851i $$0.529004\pi$$
$$648$$ 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 $$-0.644679\pi$$
−0.439033 + 0.898471i $$0.644679\pi$$
$$654$$ 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 $$-0.425573\pi$$
0.231696 + 0.972788i $$0.425573\pi$$
$$662$$ 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 $$-0.629699\pi$$
−0.396281 + 0.918129i $$0.629699\pi$$
$$678$$ 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 $$-0.290554\pi$$
0.611532 + 0.791220i $$0.290554\pi$$
$$702$$ 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 $$-0.604392\pi$$
−0.322111 + 0.946702i $$0.604392\pi$$
$$710$$ 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 $$-0.518163\pi$$
−0.0570298 + 0.998372i $$0.518163\pi$$
$$720$$ 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 $$-0.339274\pi$$
0.483750 + 0.875206i $$0.339274\pi$$
$$728$$ 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 $$-0.455328\pi$$
0.139883 + 0.990168i $$0.455328\pi$$
$$734$$ 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 $$-0.392541\pi$$
0.331215 + 0.943555i $$0.392541\pi$$
$$744$$ 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 $$-0.199371\pi$$
0.810177 + 0.586185i $$0.199371\pi$$
$$752$$ 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 $$-0.581948\pi$$
−0.254611 + 0.967043i $$0.581948\pi$$
$$758$$ 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 $$-0.173381\pi$$
0.855287 + 0.518155i $$0.173381\pi$$
$$774$$ 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 $$-0.407122\pi$$
0.287663 + 0.957732i $$0.407122\pi$$
$$798$$ 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 $$-0.488117\pi$$
0.0373233 + 0.999303i $$0.488117\pi$$
$$822$$ 0 0
$$823$$ −1721.00 −0.0728922 −0.0364461 0.999336i $$-0.511604\pi$$
−0.0364461 + 0.999336i $$0.511604\pi$$
$$824$$ 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 $$-0.320885\pi$$
0.533477 + 0.845814i $$0.320885\pi$$
$$830$$ 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 $$-0.519502\pi$$
−0.0612294 + 0.998124i $$0.519502\pi$$
$$840$$ 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 $$-0.513586\pi$$
−0.0426687 + 0.999089i $$0.513586\pi$$
$$854$$ 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 $$-0.706463\pi$$
−0.604089 + 0.796917i $$0.706463\pi$$
$$864$$ 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 $$-0.623272\pi$$
−0.377662 + 0.925944i $$0.623272\pi$$
$$878$$ 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 $$-0.513053\pi$$
−0.0409961 + 0.999159i $$0.513053\pi$$
$$888$$ 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 $$-0.703880\pi$$
−0.597603 + 0.801792i $$0.703880\pi$$
$$912$$ 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 $$-0.618241\pi$$
−0.362982 + 0.931796i $$0.618241\pi$$
$$920$$ 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 $$-0.0937653\pi$$
0.956926 + 0.290331i $$0.0937653\pi$$
$$942$$ 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 $$-0.609573\pi$$
−0.337474 + 0.941335i $$0.609573\pi$$
$$968$$ 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 $$-0.125076\pi$$
0.923788 + 0.382904i $$0.125076\pi$$
$$984$$ 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 $$-0.762161\pi$$
−0.733600 + 0.679581i $$0.762161\pi$$
$$992$$ 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 $$-0.370490\pi$$
0.395736 + 0.918364i $$0.370490\pi$$
$$998$$ 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 1216.4.a.c.1.1 1
4.3 odd 2 1216.4.a.d.1.1 1
8.3 odd 2 608.4.a.a.1.1 1
8.5 even 2 608.4.a.b.1.1 yes 1

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