# Properties

 Label 68.4.a.a.1.1 Level $68$ Weight $4$ Character 68.1 Self dual yes Analytic conductor $4.012$ 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] = [68,4,Mod(1,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("68.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 68.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: $$4.01212988039$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 68.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} -8.00000 q^{5} -12.0000 q^{7} -23.0000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} -8.00000 q^{5} -12.0000 q^{7} -23.0000 q^{9} -10.0000 q^{11} -38.0000 q^{13} +16.0000 q^{15} -17.0000 q^{17} +4.00000 q^{19} +24.0000 q^{21} +120.000 q^{23} -61.0000 q^{25} +100.000 q^{27} +56.0000 q^{29} +164.000 q^{31} +20.0000 q^{33} +96.0000 q^{35} -236.000 q^{37} +76.0000 q^{39} +70.0000 q^{41} -144.000 q^{43} +184.000 q^{45} +48.0000 q^{47} -199.000 q^{49} +34.0000 q^{51} -366.000 q^{53} +80.0000 q^{55} -8.00000 q^{57} -504.000 q^{59} -460.000 q^{61} +276.000 q^{63} +304.000 q^{65} -768.000 q^{67} -240.000 q^{69} +72.0000 q^{71} -734.000 q^{73} +122.000 q^{75} +120.000 q^{77} +736.000 q^{79} +421.000 q^{81} +856.000 q^{83} +136.000 q^{85} -112.000 q^{87} +906.000 q^{89} +456.000 q^{91} -328.000 q^{93} -32.0000 q^{95} +46.0000 q^{97} +230.000 q^{99} +O(q^{100})$$

## 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
$$3$$ −2.00000 −0.384900 −0.192450 0.981307i $$-0.561643\pi$$
−0.192450 + 0.981307i $$0.561643\pi$$
$$4$$ 0 0
$$5$$ −8.00000 −0.715542 −0.357771 0.933809i $$-0.616463\pi$$
−0.357771 + 0.933809i $$0.616463\pi$$
$$6$$ 0 0
$$7$$ −12.0000 −0.647939 −0.323970 0.946068i $$-0.605018\pi$$
−0.323970 + 0.946068i $$0.605018\pi$$
$$8$$ 0 0
$$9$$ −23.0000 −0.851852
$$10$$ 0 0
$$11$$ −10.0000 −0.274101 −0.137051 0.990564i $$-0.543762\pi$$
−0.137051 + 0.990564i $$0.543762\pi$$
$$12$$ 0 0
$$13$$ −38.0000 −0.810716 −0.405358 0.914158i $$-0.632853\pi$$
−0.405358 + 0.914158i $$0.632853\pi$$
$$14$$ 0 0
$$15$$ 16.0000 0.275412
$$16$$ 0 0
$$17$$ −17.0000 −0.242536
$$18$$ 0 0
$$19$$ 4.00000 0.0482980 0.0241490 0.999708i $$-0.492312\pi$$
0.0241490 + 0.999708i $$0.492312\pi$$
$$20$$ 0 0
$$21$$ 24.0000 0.249392
$$22$$ 0 0
$$23$$ 120.000 1.08790 0.543951 0.839117i $$-0.316928\pi$$
0.543951 + 0.839117i $$0.316928\pi$$
$$24$$ 0 0
$$25$$ −61.0000 −0.488000
$$26$$ 0 0
$$27$$ 100.000 0.712778
$$28$$ 0 0
$$29$$ 56.0000 0.358584 0.179292 0.983796i $$-0.442619\pi$$
0.179292 + 0.983796i $$0.442619\pi$$
$$30$$ 0 0
$$31$$ 164.000 0.950170 0.475085 0.879940i $$-0.342417\pi$$
0.475085 + 0.879940i $$0.342417\pi$$
$$32$$ 0 0
$$33$$ 20.0000 0.105502
$$34$$ 0 0
$$35$$ 96.0000 0.463627
$$36$$ 0 0
$$37$$ −236.000 −1.04860 −0.524299 0.851534i $$-0.675673\pi$$
−0.524299 + 0.851534i $$0.675673\pi$$
$$38$$ 0 0
$$39$$ 76.0000 0.312045
$$40$$ 0 0
$$41$$ 70.0000 0.266638 0.133319 0.991073i $$-0.457436\pi$$
0.133319 + 0.991073i $$0.457436\pi$$
$$42$$ 0 0
$$43$$ −144.000 −0.510693 −0.255346 0.966850i $$-0.582190\pi$$
−0.255346 + 0.966850i $$0.582190\pi$$
$$44$$ 0 0
$$45$$ 184.000 0.609536
$$46$$ 0 0
$$47$$ 48.0000 0.148969 0.0744843 0.997222i $$-0.476269\pi$$
0.0744843 + 0.997222i $$0.476269\pi$$
$$48$$ 0 0
$$49$$ −199.000 −0.580175
$$50$$ 0 0
$$51$$ 34.0000 0.0933520
$$52$$ 0 0
$$53$$ −366.000 −0.948565 −0.474283 0.880373i $$-0.657293\pi$$
−0.474283 + 0.880373i $$0.657293\pi$$
$$54$$ 0 0
$$55$$ 80.0000 0.196131
$$56$$ 0 0
$$57$$ −8.00000 −0.0185899
$$58$$ 0 0
$$59$$ −504.000 −1.11212 −0.556061 0.831141i $$-0.687688\pi$$
−0.556061 + 0.831141i $$0.687688\pi$$
$$60$$ 0 0
$$61$$ −460.000 −0.965524 −0.482762 0.875752i $$-0.660366\pi$$
−0.482762 + 0.875752i $$0.660366\pi$$
$$62$$ 0 0
$$63$$ 276.000 0.551948
$$64$$ 0 0
$$65$$ 304.000 0.580101
$$66$$ 0 0
$$67$$ −768.000 −1.40039 −0.700195 0.713952i $$-0.746904\pi$$
−0.700195 + 0.713952i $$0.746904\pi$$
$$68$$ 0 0
$$69$$ −240.000 −0.418733
$$70$$ 0 0
$$71$$ 72.0000 0.120350 0.0601748 0.998188i $$-0.480834\pi$$
0.0601748 + 0.998188i $$0.480834\pi$$
$$72$$ 0 0
$$73$$ −734.000 −1.17682 −0.588412 0.808561i $$-0.700247\pi$$
−0.588412 + 0.808561i $$0.700247\pi$$
$$74$$ 0 0
$$75$$ 122.000 0.187831
$$76$$ 0 0
$$77$$ 120.000 0.177601
$$78$$ 0 0
$$79$$ 736.000 1.04818 0.524092 0.851662i $$-0.324405\pi$$
0.524092 + 0.851662i $$0.324405\pi$$
$$80$$ 0 0
$$81$$ 421.000 0.577503
$$82$$ 0 0
$$83$$ 856.000 1.13203 0.566013 0.824396i $$-0.308485\pi$$
0.566013 + 0.824396i $$0.308485\pi$$
$$84$$ 0 0
$$85$$ 136.000 0.173544
$$86$$ 0 0
$$87$$ −112.000 −0.138019
$$88$$ 0 0
$$89$$ 906.000 1.07905 0.539527 0.841968i $$-0.318603\pi$$
0.539527 + 0.841968i $$0.318603\pi$$
$$90$$ 0 0
$$91$$ 456.000 0.525294
$$92$$ 0 0
$$93$$ −328.000 −0.365721
$$94$$ 0 0
$$95$$ −32.0000 −0.0345593
$$96$$ 0 0
$$97$$ 46.0000 0.0481504 0.0240752 0.999710i $$-0.492336\pi$$
0.0240752 + 0.999710i $$0.492336\pi$$
$$98$$ 0 0
$$99$$ 230.000 0.233494
$$100$$ 0 0
$$101$$ 1178.00 1.16055 0.580274 0.814421i $$-0.302945\pi$$
0.580274 + 0.814421i $$0.302945\pi$$
$$102$$ 0 0
$$103$$ −608.000 −0.581631 −0.290816 0.956779i $$-0.593927\pi$$
−0.290816 + 0.956779i $$0.593927\pi$$
$$104$$ 0 0
$$105$$ −192.000 −0.178450
$$106$$ 0 0
$$107$$ −1798.00 −1.62448 −0.812239 0.583324i $$-0.801752\pi$$
−0.812239 + 0.583324i $$0.801752\pi$$
$$108$$ 0 0
$$109$$ 200.000 0.175748 0.0878740 0.996132i $$-0.471993\pi$$
0.0878740 + 0.996132i $$0.471993\pi$$
$$110$$ 0 0
$$111$$ 472.000 0.403606
$$112$$ 0 0
$$113$$ 2330.00 1.93972 0.969858 0.243670i $$-0.0783513\pi$$
0.969858 + 0.243670i $$0.0783513\pi$$
$$114$$ 0 0
$$115$$ −960.000 −0.778439
$$116$$ 0 0
$$117$$ 874.000 0.690610
$$118$$ 0 0
$$119$$ 204.000 0.157148
$$120$$ 0 0
$$121$$ −1231.00 −0.924869
$$122$$ 0 0
$$123$$ −140.000 −0.102629
$$124$$ 0 0
$$125$$ 1488.00 1.06473
$$126$$ 0 0
$$127$$ −2016.00 −1.40859 −0.704296 0.709907i $$-0.748737\pi$$
−0.704296 + 0.709907i $$0.748737\pi$$
$$128$$ 0 0
$$129$$ 288.000 0.196566
$$130$$ 0 0
$$131$$ −102.000 −0.0680289 −0.0340144 0.999421i $$-0.510829\pi$$
−0.0340144 + 0.999421i $$0.510829\pi$$
$$132$$ 0 0
$$133$$ −48.0000 −0.0312942
$$134$$ 0 0
$$135$$ −800.000 −0.510022
$$136$$ 0 0
$$137$$ −2186.00 −1.36323 −0.681615 0.731711i $$-0.738722\pi$$
−0.681615 + 0.731711i $$0.738722\pi$$
$$138$$ 0 0
$$139$$ −2590.00 −1.58044 −0.790219 0.612824i $$-0.790033\pi$$
−0.790219 + 0.612824i $$0.790033\pi$$
$$140$$ 0 0
$$141$$ −96.0000 −0.0573380
$$142$$ 0 0
$$143$$ 380.000 0.222218
$$144$$ 0 0
$$145$$ −448.000 −0.256582
$$146$$ 0 0
$$147$$ 398.000 0.223309
$$148$$ 0 0
$$149$$ −1282.00 −0.704869 −0.352435 0.935836i $$-0.614646\pi$$
−0.352435 + 0.935836i $$0.614646\pi$$
$$150$$ 0 0
$$151$$ 3312.00 1.78495 0.892473 0.451102i $$-0.148969\pi$$
0.892473 + 0.451102i $$0.148969\pi$$
$$152$$ 0 0
$$153$$ 391.000 0.206604
$$154$$ 0 0
$$155$$ −1312.00 −0.679886
$$156$$ 0 0
$$157$$ 310.000 0.157584 0.0787920 0.996891i $$-0.474894\pi$$
0.0787920 + 0.996891i $$0.474894\pi$$
$$158$$ 0 0
$$159$$ 732.000 0.365103
$$160$$ 0 0
$$161$$ −1440.00 −0.704894
$$162$$ 0 0
$$163$$ −1838.00 −0.883210 −0.441605 0.897210i $$-0.645591\pi$$
−0.441605 + 0.897210i $$0.645591\pi$$
$$164$$ 0 0
$$165$$ −160.000 −0.0754908
$$166$$ 0 0
$$167$$ 1684.00 0.780310 0.390155 0.920749i $$-0.372421\pi$$
0.390155 + 0.920749i $$0.372421\pi$$
$$168$$ 0 0
$$169$$ −753.000 −0.342740
$$170$$ 0 0
$$171$$ −92.0000 −0.0411428
$$172$$ 0 0
$$173$$ −1104.00 −0.485177 −0.242588 0.970129i $$-0.577996\pi$$
−0.242588 + 0.970129i $$0.577996\pi$$
$$174$$ 0 0
$$175$$ 732.000 0.316194
$$176$$ 0 0
$$177$$ 1008.00 0.428056
$$178$$ 0 0
$$179$$ −3660.00 −1.52828 −0.764138 0.645053i $$-0.776835\pi$$
−0.764138 + 0.645053i $$0.776835\pi$$
$$180$$ 0 0
$$181$$ 4724.00 1.93996 0.969978 0.243191i $$-0.0781943\pi$$
0.969978 + 0.243191i $$0.0781943\pi$$
$$182$$ 0 0
$$183$$ 920.000 0.371630
$$184$$ 0 0
$$185$$ 1888.00 0.750316
$$186$$ 0 0
$$187$$ 170.000 0.0664793
$$188$$ 0 0
$$189$$ −1200.00 −0.461837
$$190$$ 0 0
$$191$$ 4280.00 1.62141 0.810707 0.585453i $$-0.199083\pi$$
0.810707 + 0.585453i $$0.199083\pi$$
$$192$$ 0 0
$$193$$ 630.000 0.234966 0.117483 0.993075i $$-0.462517\pi$$
0.117483 + 0.993075i $$0.462517\pi$$
$$194$$ 0 0
$$195$$ −608.000 −0.223281
$$196$$ 0 0
$$197$$ −1204.00 −0.435439 −0.217719 0.976011i $$-0.569862\pi$$
−0.217719 + 0.976011i $$0.569862\pi$$
$$198$$ 0 0
$$199$$ 4632.00 1.65002 0.825009 0.565119i $$-0.191170\pi$$
0.825009 + 0.565119i $$0.191170\pi$$
$$200$$ 0 0
$$201$$ 1536.00 0.539010
$$202$$ 0 0
$$203$$ −672.000 −0.232341
$$204$$ 0 0
$$205$$ −560.000 −0.190791
$$206$$ 0 0
$$207$$ −2760.00 −0.926731
$$208$$ 0 0
$$209$$ −40.0000 −0.0132386
$$210$$ 0 0
$$211$$ 3590.00 1.17131 0.585654 0.810561i $$-0.300838\pi$$
0.585654 + 0.810561i $$0.300838\pi$$
$$212$$ 0 0
$$213$$ −144.000 −0.0463226
$$214$$ 0 0
$$215$$ 1152.00 0.365422
$$216$$ 0 0
$$217$$ −1968.00 −0.615652
$$218$$ 0 0
$$219$$ 1468.00 0.452960
$$220$$ 0 0
$$221$$ 646.000 0.196627
$$222$$ 0 0
$$223$$ 2344.00 0.703883 0.351941 0.936022i $$-0.385522\pi$$
0.351941 + 0.936022i $$0.385522\pi$$
$$224$$ 0 0
$$225$$ 1403.00 0.415704
$$226$$ 0 0
$$227$$ 1258.00 0.367826 0.183913 0.982943i $$-0.441124\pi$$
0.183913 + 0.982943i $$0.441124\pi$$
$$228$$ 0 0
$$229$$ −5258.00 −1.51729 −0.758643 0.651507i $$-0.774137\pi$$
−0.758643 + 0.651507i $$0.774137\pi$$
$$230$$ 0 0
$$231$$ −240.000 −0.0683586
$$232$$ 0 0
$$233$$ −3694.00 −1.03864 −0.519318 0.854581i $$-0.673814\pi$$
−0.519318 + 0.854581i $$0.673814\pi$$
$$234$$ 0 0
$$235$$ −384.000 −0.106593
$$236$$ 0 0
$$237$$ −1472.00 −0.403446
$$238$$ 0 0
$$239$$ −1944.00 −0.526138 −0.263069 0.964777i $$-0.584735\pi$$
−0.263069 + 0.964777i $$0.584735\pi$$
$$240$$ 0 0
$$241$$ −1162.00 −0.310585 −0.155293 0.987869i $$-0.549632\pi$$
−0.155293 + 0.987869i $$0.549632\pi$$
$$242$$ 0 0
$$243$$ −3542.00 −0.935059
$$244$$ 0 0
$$245$$ 1592.00 0.415139
$$246$$ 0 0
$$247$$ −152.000 −0.0391560
$$248$$ 0 0
$$249$$ −1712.00 −0.435717
$$250$$ 0 0
$$251$$ −176.000 −0.0442590 −0.0221295 0.999755i $$-0.507045\pi$$
−0.0221295 + 0.999755i $$0.507045\pi$$
$$252$$ 0 0
$$253$$ −1200.00 −0.298195
$$254$$ 0 0
$$255$$ −272.000 −0.0667973
$$256$$ 0 0
$$257$$ −4474.00 −1.08592 −0.542958 0.839760i $$-0.682696\pi$$
−0.542958 + 0.839760i $$0.682696\pi$$
$$258$$ 0 0
$$259$$ 2832.00 0.679428
$$260$$ 0 0
$$261$$ −1288.00 −0.305461
$$262$$ 0 0
$$263$$ −2696.00 −0.632101 −0.316050 0.948742i $$-0.602357\pi$$
−0.316050 + 0.948742i $$0.602357\pi$$
$$264$$ 0 0
$$265$$ 2928.00 0.678738
$$266$$ 0 0
$$267$$ −1812.00 −0.415328
$$268$$ 0 0
$$269$$ 2304.00 0.522221 0.261110 0.965309i $$-0.415911\pi$$
0.261110 + 0.965309i $$0.415911\pi$$
$$270$$ 0 0
$$271$$ 1720.00 0.385544 0.192772 0.981244i $$-0.438252\pi$$
0.192772 + 0.981244i $$0.438252\pi$$
$$272$$ 0 0
$$273$$ −912.000 −0.202186
$$274$$ 0 0
$$275$$ 610.000 0.133761
$$276$$ 0 0
$$277$$ −4224.00 −0.916229 −0.458115 0.888893i $$-0.651475\pi$$
−0.458115 + 0.888893i $$0.651475\pi$$
$$278$$ 0 0
$$279$$ −3772.00 −0.809404
$$280$$ 0 0
$$281$$ 3126.00 0.663635 0.331818 0.943344i $$-0.392338\pi$$
0.331818 + 0.943344i $$0.392338\pi$$
$$282$$ 0 0
$$283$$ 5118.00 1.07503 0.537515 0.843254i $$-0.319363\pi$$
0.537515 + 0.843254i $$0.319363\pi$$
$$284$$ 0 0
$$285$$ 64.0000 0.0133019
$$286$$ 0 0
$$287$$ −840.000 −0.172765
$$288$$ 0 0
$$289$$ 289.000 0.0588235
$$290$$ 0 0
$$291$$ −92.0000 −0.0185331
$$292$$ 0 0
$$293$$ −4418.00 −0.880895 −0.440448 0.897778i $$-0.645180\pi$$
−0.440448 + 0.897778i $$0.645180\pi$$
$$294$$ 0 0
$$295$$ 4032.00 0.795770
$$296$$ 0 0
$$297$$ −1000.00 −0.195373
$$298$$ 0 0
$$299$$ −4560.00 −0.881979
$$300$$ 0 0
$$301$$ 1728.00 0.330898
$$302$$ 0 0
$$303$$ −2356.00 −0.446695
$$304$$ 0 0
$$305$$ 3680.00 0.690873
$$306$$ 0 0
$$307$$ 4476.00 0.832113 0.416057 0.909339i $$-0.363412\pi$$
0.416057 + 0.909339i $$0.363412\pi$$
$$308$$ 0 0
$$309$$ 1216.00 0.223870
$$310$$ 0 0
$$311$$ −300.000 −0.0546992 −0.0273496 0.999626i $$-0.508707\pi$$
−0.0273496 + 0.999626i $$0.508707\pi$$
$$312$$ 0 0
$$313$$ −1154.00 −0.208396 −0.104198 0.994557i $$-0.533228\pi$$
−0.104198 + 0.994557i $$0.533228\pi$$
$$314$$ 0 0
$$315$$ −2208.00 −0.394942
$$316$$ 0 0
$$317$$ 684.000 0.121190 0.0605951 0.998162i $$-0.480700\pi$$
0.0605951 + 0.998162i $$0.480700\pi$$
$$318$$ 0 0
$$319$$ −560.000 −0.0982883
$$320$$ 0 0
$$321$$ 3596.00 0.625262
$$322$$ 0 0
$$323$$ −68.0000 −0.0117140
$$324$$ 0 0
$$325$$ 2318.00 0.395629
$$326$$ 0 0
$$327$$ −400.000 −0.0676454
$$328$$ 0 0
$$329$$ −576.000 −0.0965225
$$330$$ 0 0
$$331$$ 9400.00 1.56094 0.780469 0.625194i $$-0.214980\pi$$
0.780469 + 0.625194i $$0.214980\pi$$
$$332$$ 0 0
$$333$$ 5428.00 0.893251
$$334$$ 0 0
$$335$$ 6144.00 1.00204
$$336$$ 0 0
$$337$$ −470.000 −0.0759719 −0.0379860 0.999278i $$-0.512094\pi$$
−0.0379860 + 0.999278i $$0.512094\pi$$
$$338$$ 0 0
$$339$$ −4660.00 −0.746597
$$340$$ 0 0
$$341$$ −1640.00 −0.260443
$$342$$ 0 0
$$343$$ 6504.00 1.02386
$$344$$ 0 0
$$345$$ 1920.00 0.299621
$$346$$ 0 0
$$347$$ −6302.00 −0.974954 −0.487477 0.873136i $$-0.662083\pi$$
−0.487477 + 0.873136i $$0.662083\pi$$
$$348$$ 0 0
$$349$$ 10946.0 1.67887 0.839435 0.543459i $$-0.182886\pi$$
0.839435 + 0.543459i $$0.182886\pi$$
$$350$$ 0 0
$$351$$ −3800.00 −0.577860
$$352$$ 0 0
$$353$$ 7782.00 1.17335 0.586677 0.809821i $$-0.300436\pi$$
0.586677 + 0.809821i $$0.300436\pi$$
$$354$$ 0 0
$$355$$ −576.000 −0.0861152
$$356$$ 0 0
$$357$$ −408.000 −0.0604864
$$358$$ 0 0
$$359$$ −4920.00 −0.723308 −0.361654 0.932312i $$-0.617788\pi$$
−0.361654 + 0.932312i $$0.617788\pi$$
$$360$$ 0 0
$$361$$ −6843.00 −0.997667
$$362$$ 0 0
$$363$$ 2462.00 0.355982
$$364$$ 0 0
$$365$$ 5872.00 0.842067
$$366$$ 0 0
$$367$$ −5976.00 −0.849985 −0.424993 0.905197i $$-0.639723\pi$$
−0.424993 + 0.905197i $$0.639723\pi$$
$$368$$ 0 0
$$369$$ −1610.00 −0.227136
$$370$$ 0 0
$$371$$ 4392.00 0.614613
$$372$$ 0 0
$$373$$ −3150.00 −0.437268 −0.218634 0.975807i $$-0.570160\pi$$
−0.218634 + 0.975807i $$0.570160\pi$$
$$374$$ 0 0
$$375$$ −2976.00 −0.409813
$$376$$ 0 0
$$377$$ −2128.00 −0.290710
$$378$$ 0 0
$$379$$ 2414.00 0.327174 0.163587 0.986529i $$-0.447694\pi$$
0.163587 + 0.986529i $$0.447694\pi$$
$$380$$ 0 0
$$381$$ 4032.00 0.542167
$$382$$ 0 0
$$383$$ −10120.0 −1.35015 −0.675076 0.737749i $$-0.735889\pi$$
−0.675076 + 0.737749i $$0.735889\pi$$
$$384$$ 0 0
$$385$$ −960.000 −0.127081
$$386$$ 0 0
$$387$$ 3312.00 0.435035
$$388$$ 0 0
$$389$$ −11450.0 −1.49239 −0.746193 0.665730i $$-0.768120\pi$$
−0.746193 + 0.665730i $$0.768120\pi$$
$$390$$ 0 0
$$391$$ −2040.00 −0.263855
$$392$$ 0 0
$$393$$ 204.000 0.0261843
$$394$$ 0 0
$$395$$ −5888.00 −0.750019
$$396$$ 0 0
$$397$$ −6164.00 −0.779250 −0.389625 0.920974i $$-0.627395\pi$$
−0.389625 + 0.920974i $$0.627395\pi$$
$$398$$ 0 0
$$399$$ 96.0000 0.0120451
$$400$$ 0 0
$$401$$ 5758.00 0.717059 0.358530 0.933518i $$-0.383278\pi$$
0.358530 + 0.933518i $$0.383278\pi$$
$$402$$ 0 0
$$403$$ −6232.00 −0.770318
$$404$$ 0 0
$$405$$ −3368.00 −0.413228
$$406$$ 0 0
$$407$$ 2360.00 0.287422
$$408$$ 0 0
$$409$$ 3302.00 0.399201 0.199601 0.979877i $$-0.436035\pi$$
0.199601 + 0.979877i $$0.436035\pi$$
$$410$$ 0 0
$$411$$ 4372.00 0.524708
$$412$$ 0 0
$$413$$ 6048.00 0.720587
$$414$$ 0 0
$$415$$ −6848.00 −0.810012
$$416$$ 0 0
$$417$$ 5180.00 0.608311
$$418$$ 0 0
$$419$$ −9998.00 −1.16571 −0.582857 0.812575i $$-0.698065\pi$$
−0.582857 + 0.812575i $$0.698065\pi$$
$$420$$ 0 0
$$421$$ −5846.00 −0.676762 −0.338381 0.941009i $$-0.609879\pi$$
−0.338381 + 0.941009i $$0.609879\pi$$
$$422$$ 0 0
$$423$$ −1104.00 −0.126899
$$424$$ 0 0
$$425$$ 1037.00 0.118357
$$426$$ 0 0
$$427$$ 5520.00 0.625601
$$428$$ 0 0
$$429$$ −760.000 −0.0855318
$$430$$ 0 0
$$431$$ −13712.0 −1.53245 −0.766223 0.642575i $$-0.777866\pi$$
−0.766223 + 0.642575i $$0.777866\pi$$
$$432$$ 0 0
$$433$$ −62.0000 −0.00688113 −0.00344057 0.999994i $$-0.501095\pi$$
−0.00344057 + 0.999994i $$0.501095\pi$$
$$434$$ 0 0
$$435$$ 896.000 0.0987584
$$436$$ 0 0
$$437$$ 480.000 0.0525435
$$438$$ 0 0
$$439$$ −9392.00 −1.02108 −0.510542 0.859853i $$-0.670555\pi$$
−0.510542 + 0.859853i $$0.670555\pi$$
$$440$$ 0 0
$$441$$ 4577.00 0.494223
$$442$$ 0 0
$$443$$ 8528.00 0.914622 0.457311 0.889307i $$-0.348813\pi$$
0.457311 + 0.889307i $$0.348813\pi$$
$$444$$ 0 0
$$445$$ −7248.00 −0.772108
$$446$$ 0 0
$$447$$ 2564.00 0.271304
$$448$$ 0 0
$$449$$ 4238.00 0.445442 0.222721 0.974882i $$-0.428506\pi$$
0.222721 + 0.974882i $$0.428506\pi$$
$$450$$ 0 0
$$451$$ −700.000 −0.0730858
$$452$$ 0 0
$$453$$ −6624.00 −0.687026
$$454$$ 0 0
$$455$$ −3648.00 −0.375870
$$456$$ 0 0
$$457$$ −19382.0 −1.98392 −0.991960 0.126549i $$-0.959610\pi$$
−0.991960 + 0.126549i $$0.959610\pi$$
$$458$$ 0 0
$$459$$ −1700.00 −0.172874
$$460$$ 0 0
$$461$$ 18882.0 1.90764 0.953820 0.300377i $$-0.0971127\pi$$
0.953820 + 0.300377i $$0.0971127\pi$$
$$462$$ 0 0
$$463$$ 6224.00 0.624738 0.312369 0.949961i $$-0.398877\pi$$
0.312369 + 0.949961i $$0.398877\pi$$
$$464$$ 0 0
$$465$$ 2624.00 0.261688
$$466$$ 0 0
$$467$$ 10068.0 0.997626 0.498813 0.866710i $$-0.333769\pi$$
0.498813 + 0.866710i $$0.333769\pi$$
$$468$$ 0 0
$$469$$ 9216.00 0.907367
$$470$$ 0 0
$$471$$ −620.000 −0.0606541
$$472$$ 0 0
$$473$$ 1440.00 0.139982
$$474$$ 0 0
$$475$$ −244.000 −0.0235694
$$476$$ 0 0
$$477$$ 8418.00 0.808037
$$478$$ 0 0
$$479$$ 20064.0 1.91388 0.956939 0.290289i $$-0.0937515\pi$$
0.956939 + 0.290289i $$0.0937515\pi$$
$$480$$ 0 0
$$481$$ 8968.00 0.850116
$$482$$ 0 0
$$483$$ 2880.00 0.271314
$$484$$ 0 0
$$485$$ −368.000 −0.0344536
$$486$$ 0 0
$$487$$ −15616.0 −1.45304 −0.726518 0.687147i $$-0.758863\pi$$
−0.726518 + 0.687147i $$0.758863\pi$$
$$488$$ 0 0
$$489$$ 3676.00 0.339948
$$490$$ 0 0
$$491$$ 5868.00 0.539347 0.269673 0.962952i $$-0.413084\pi$$
0.269673 + 0.962952i $$0.413084\pi$$
$$492$$ 0 0
$$493$$ −952.000 −0.0869694
$$494$$ 0 0
$$495$$ −1840.00 −0.167074
$$496$$ 0 0
$$497$$ −864.000 −0.0779793
$$498$$ 0 0
$$499$$ −962.000 −0.0863027 −0.0431513 0.999069i $$-0.513740\pi$$
−0.0431513 + 0.999069i $$0.513740\pi$$
$$500$$ 0 0
$$501$$ −3368.00 −0.300342
$$502$$ 0 0
$$503$$ −4128.00 −0.365921 −0.182961 0.983120i $$-0.558568\pi$$
−0.182961 + 0.983120i $$0.558568\pi$$
$$504$$ 0 0
$$505$$ −9424.00 −0.830421
$$506$$ 0 0
$$507$$ 1506.00 0.131921
$$508$$ 0 0
$$509$$ 12006.0 1.04549 0.522747 0.852488i $$-0.324907\pi$$
0.522747 + 0.852488i $$0.324907\pi$$
$$510$$ 0 0
$$511$$ 8808.00 0.762511
$$512$$ 0 0
$$513$$ 400.000 0.0344258
$$514$$ 0 0
$$515$$ 4864.00 0.416181
$$516$$ 0 0
$$517$$ −480.000 −0.0408324
$$518$$ 0 0
$$519$$ 2208.00 0.186745
$$520$$ 0 0
$$521$$ −20146.0 −1.69407 −0.847037 0.531534i $$-0.821616\pi$$
−0.847037 + 0.531534i $$0.821616\pi$$
$$522$$ 0 0
$$523$$ 1496.00 0.125077 0.0625387 0.998043i $$-0.480080\pi$$
0.0625387 + 0.998043i $$0.480080\pi$$
$$524$$ 0 0
$$525$$ −1464.00 −0.121703
$$526$$ 0 0
$$527$$ −2788.00 −0.230450
$$528$$ 0 0
$$529$$ 2233.00 0.183529
$$530$$ 0 0
$$531$$ 11592.0 0.947363
$$532$$ 0 0
$$533$$ −2660.00 −0.216168
$$534$$ 0 0
$$535$$ 14384.0 1.16238
$$536$$ 0 0
$$537$$ 7320.00 0.588233
$$538$$ 0 0
$$539$$ 1990.00 0.159027
$$540$$ 0 0
$$541$$ −16756.0 −1.33160 −0.665801 0.746129i $$-0.731910\pi$$
−0.665801 + 0.746129i $$0.731910\pi$$
$$542$$ 0 0
$$543$$ −9448.00 −0.746690
$$544$$ 0 0
$$545$$ −1600.00 −0.125755
$$546$$ 0 0
$$547$$ 11010.0 0.860610 0.430305 0.902684i $$-0.358406\pi$$
0.430305 + 0.902684i $$0.358406\pi$$
$$548$$ 0 0
$$549$$ 10580.0 0.822483
$$550$$ 0 0
$$551$$ 224.000 0.0173189
$$552$$ 0 0
$$553$$ −8832.00 −0.679159
$$554$$ 0 0
$$555$$ −3776.00 −0.288797
$$556$$ 0 0
$$557$$ −1606.00 −0.122169 −0.0610847 0.998133i $$-0.519456\pi$$
−0.0610847 + 0.998133i $$0.519456\pi$$
$$558$$ 0 0
$$559$$ 5472.00 0.414027
$$560$$ 0 0
$$561$$ −340.000 −0.0255879
$$562$$ 0 0
$$563$$ 24480.0 1.83252 0.916260 0.400584i $$-0.131193\pi$$
0.916260 + 0.400584i $$0.131193\pi$$
$$564$$ 0 0
$$565$$ −18640.0 −1.38795
$$566$$ 0 0
$$567$$ −5052.00 −0.374187
$$568$$ 0 0
$$569$$ 7026.00 0.517654 0.258827 0.965924i $$-0.416664\pi$$
0.258827 + 0.965924i $$0.416664\pi$$
$$570$$ 0 0
$$571$$ −15542.0 −1.13908 −0.569538 0.821965i $$-0.692878\pi$$
−0.569538 + 0.821965i $$0.692878\pi$$
$$572$$ 0 0
$$573$$ −8560.00 −0.624082
$$574$$ 0 0
$$575$$ −7320.00 −0.530896
$$576$$ 0 0
$$577$$ 7886.00 0.568975 0.284487 0.958680i $$-0.408177\pi$$
0.284487 + 0.958680i $$0.408177\pi$$
$$578$$ 0 0
$$579$$ −1260.00 −0.0904384
$$580$$ 0 0
$$581$$ −10272.0 −0.733484
$$582$$ 0 0
$$583$$ 3660.00 0.260003
$$584$$ 0 0
$$585$$ −6992.00 −0.494160
$$586$$ 0 0
$$587$$ 1732.00 0.121784 0.0608921 0.998144i $$-0.480605\pi$$
0.0608921 + 0.998144i $$0.480605\pi$$
$$588$$ 0 0
$$589$$ 656.000 0.0458914
$$590$$ 0 0
$$591$$ 2408.00 0.167600
$$592$$ 0 0
$$593$$ −21182.0 −1.46685 −0.733424 0.679772i $$-0.762079\pi$$
−0.733424 + 0.679772i $$0.762079\pi$$
$$594$$ 0 0
$$595$$ −1632.00 −0.112446
$$596$$ 0 0
$$597$$ −9264.00 −0.635093
$$598$$ 0 0
$$599$$ −22280.0 −1.51976 −0.759880 0.650063i $$-0.774742\pi$$
−0.759880 + 0.650063i $$0.774742\pi$$
$$600$$ 0 0
$$601$$ −26190.0 −1.77756 −0.888779 0.458336i $$-0.848446\pi$$
−0.888779 + 0.458336i $$0.848446\pi$$
$$602$$ 0 0
$$603$$ 17664.0 1.19292
$$604$$ 0 0
$$605$$ 9848.00 0.661782
$$606$$ 0 0
$$607$$ 12444.0 0.832103 0.416051 0.909341i $$-0.363414\pi$$
0.416051 + 0.909341i $$0.363414\pi$$
$$608$$ 0 0
$$609$$ 1344.00 0.0894280
$$610$$ 0 0
$$611$$ −1824.00 −0.120771
$$612$$ 0 0
$$613$$ −3418.00 −0.225207 −0.112603 0.993640i $$-0.535919\pi$$
−0.112603 + 0.993640i $$0.535919\pi$$
$$614$$ 0 0
$$615$$ 1120.00 0.0734354
$$616$$ 0 0
$$617$$ 5346.00 0.348820 0.174410 0.984673i $$-0.444198\pi$$
0.174410 + 0.984673i $$0.444198\pi$$
$$618$$ 0 0
$$619$$ −17846.0 −1.15879 −0.579395 0.815047i $$-0.696711\pi$$
−0.579395 + 0.815047i $$0.696711\pi$$
$$620$$ 0 0
$$621$$ 12000.0 0.775432
$$622$$ 0 0
$$623$$ −10872.0 −0.699161
$$624$$ 0 0
$$625$$ −4279.00 −0.273856
$$626$$ 0 0
$$627$$ 80.0000 0.00509552
$$628$$ 0 0
$$629$$ 4012.00 0.254323
$$630$$ 0 0
$$631$$ −13416.0 −0.846407 −0.423203 0.906035i $$-0.639094\pi$$
−0.423203 + 0.906035i $$0.639094\pi$$
$$632$$ 0 0
$$633$$ −7180.00 −0.450836
$$634$$ 0 0
$$635$$ 16128.0 1.00791
$$636$$ 0 0
$$637$$ 7562.00 0.470357
$$638$$ 0 0
$$639$$ −1656.00 −0.102520
$$640$$ 0 0
$$641$$ −7762.00 −0.478285 −0.239142 0.970985i $$-0.576866\pi$$
−0.239142 + 0.970985i $$0.576866\pi$$
$$642$$ 0 0
$$643$$ −9106.00 −0.558485 −0.279242 0.960221i $$-0.590083\pi$$
−0.279242 + 0.960221i $$0.590083\pi$$
$$644$$ 0 0
$$645$$ −2304.00 −0.140651
$$646$$ 0 0
$$647$$ −5488.00 −0.333471 −0.166735 0.986002i $$-0.553323\pi$$
−0.166735 + 0.986002i $$0.553323\pi$$
$$648$$ 0 0
$$649$$ 5040.00 0.304834
$$650$$ 0 0
$$651$$ 3936.00 0.236965
$$652$$ 0 0
$$653$$ 2640.00 0.158210 0.0791050 0.996866i $$-0.474794\pi$$
0.0791050 + 0.996866i $$0.474794\pi$$
$$654$$ 0 0
$$655$$ 816.000 0.0486775
$$656$$ 0 0
$$657$$ 16882.0 1.00248
$$658$$ 0 0
$$659$$ 7548.00 0.446173 0.223087 0.974799i $$-0.428387\pi$$
0.223087 + 0.974799i $$0.428387\pi$$
$$660$$ 0 0
$$661$$ −5718.00 −0.336467 −0.168233 0.985747i $$-0.553806\pi$$
−0.168233 + 0.985747i $$0.553806\pi$$
$$662$$ 0 0
$$663$$ −1292.00 −0.0756819
$$664$$ 0 0
$$665$$ 384.000 0.0223923
$$666$$ 0 0
$$667$$ 6720.00 0.390104
$$668$$ 0 0
$$669$$ −4688.00 −0.270925
$$670$$ 0 0
$$671$$ 4600.00 0.264651
$$672$$ 0 0
$$673$$ 26150.0 1.49778 0.748892 0.662692i $$-0.230586\pi$$
0.748892 + 0.662692i $$0.230586\pi$$
$$674$$ 0 0
$$675$$ −6100.00 −0.347836
$$676$$ 0 0
$$677$$ 19276.0 1.09429 0.547147 0.837037i $$-0.315714\pi$$
0.547147 + 0.837037i $$0.315714\pi$$
$$678$$ 0 0
$$679$$ −552.000 −0.0311986
$$680$$ 0 0
$$681$$ −2516.00 −0.141576
$$682$$ 0 0
$$683$$ 17758.0 0.994862 0.497431 0.867503i $$-0.334277\pi$$
0.497431 + 0.867503i $$0.334277\pi$$
$$684$$ 0 0
$$685$$ 17488.0 0.975448
$$686$$ 0 0
$$687$$ 10516.0 0.584004
$$688$$ 0 0
$$689$$ 13908.0 0.769017
$$690$$ 0 0
$$691$$ −3994.00 −0.219883 −0.109941 0.993938i $$-0.535066\pi$$
−0.109941 + 0.993938i $$0.535066\pi$$
$$692$$ 0 0
$$693$$ −2760.00 −0.151290
$$694$$ 0 0
$$695$$ 20720.0 1.13087
$$696$$ 0 0
$$697$$ −1190.00 −0.0646692
$$698$$ 0 0
$$699$$ 7388.00 0.399771
$$700$$ 0 0
$$701$$ −14486.0 −0.780497 −0.390249 0.920709i $$-0.627611\pi$$
−0.390249 + 0.920709i $$0.627611\pi$$
$$702$$ 0 0
$$703$$ −944.000 −0.0506453
$$704$$ 0 0
$$705$$ 768.000 0.0410277
$$706$$ 0 0
$$707$$ −14136.0 −0.751965
$$708$$ 0 0
$$709$$ −7256.00 −0.384351 −0.192175 0.981361i $$-0.561554\pi$$
−0.192175 + 0.981361i $$0.561554\pi$$
$$710$$ 0 0
$$711$$ −16928.0 −0.892897
$$712$$ 0 0
$$713$$ 19680.0 1.03369
$$714$$ 0 0
$$715$$ −3040.00 −0.159006
$$716$$ 0 0
$$717$$ 3888.00 0.202510
$$718$$ 0 0
$$719$$ −37448.0 −1.94238 −0.971192 0.238297i $$-0.923411\pi$$
−0.971192 + 0.238297i $$0.923411\pi$$
$$720$$ 0 0
$$721$$ 7296.00 0.376862
$$722$$ 0 0
$$723$$ 2324.00 0.119544
$$724$$ 0 0
$$725$$ −3416.00 −0.174989
$$726$$ 0 0
$$727$$ 30856.0 1.57412 0.787060 0.616876i $$-0.211602\pi$$
0.787060 + 0.616876i $$0.211602\pi$$
$$728$$ 0 0
$$729$$ −4283.00 −0.217599
$$730$$ 0 0
$$731$$ 2448.00 0.123861
$$732$$ 0 0
$$733$$ −5294.00 −0.266764 −0.133382 0.991065i $$-0.542584\pi$$
−0.133382 + 0.991065i $$0.542584\pi$$
$$734$$ 0 0
$$735$$ −3184.00 −0.159787
$$736$$ 0 0
$$737$$ 7680.00 0.383849
$$738$$ 0 0
$$739$$ −836.000 −0.0416140 −0.0208070 0.999784i $$-0.506624\pi$$
−0.0208070 + 0.999784i $$0.506624\pi$$
$$740$$ 0 0
$$741$$ 304.000 0.0150711
$$742$$ 0 0
$$743$$ 24452.0 1.20734 0.603672 0.797233i $$-0.293704\pi$$
0.603672 + 0.797233i $$0.293704\pi$$
$$744$$ 0 0
$$745$$ 10256.0 0.504363
$$746$$ 0 0
$$747$$ −19688.0 −0.964319
$$748$$ 0 0
$$749$$ 21576.0 1.05256
$$750$$ 0 0
$$751$$ −5680.00 −0.275987 −0.137993 0.990433i $$-0.544065\pi$$
−0.137993 + 0.990433i $$0.544065\pi$$
$$752$$ 0 0
$$753$$ 352.000 0.0170353
$$754$$ 0 0
$$755$$ −26496.0 −1.27720
$$756$$ 0 0
$$757$$ 10622.0 0.509991 0.254995 0.966942i $$-0.417926\pi$$
0.254995 + 0.966942i $$0.417926\pi$$
$$758$$ 0 0
$$759$$ 2400.00 0.114775
$$760$$ 0 0
$$761$$ −28842.0 −1.37388 −0.686939 0.726715i $$-0.741046\pi$$
−0.686939 + 0.726715i $$0.741046\pi$$
$$762$$ 0 0
$$763$$ −2400.00 −0.113874
$$764$$ 0 0
$$765$$ −3128.00 −0.147834
$$766$$ 0 0
$$767$$ 19152.0 0.901615
$$768$$ 0 0
$$769$$ 35598.0 1.66931 0.834653 0.550776i $$-0.185668\pi$$
0.834653 + 0.550776i $$0.185668\pi$$
$$770$$ 0 0
$$771$$ 8948.00 0.417969
$$772$$ 0 0
$$773$$ −32962.0 −1.53371 −0.766857 0.641818i $$-0.778180\pi$$
−0.766857 + 0.641818i $$0.778180\pi$$
$$774$$ 0 0
$$775$$ −10004.0 −0.463683
$$776$$ 0 0
$$777$$ −5664.00 −0.261512
$$778$$ 0 0
$$779$$ 280.000 0.0128781
$$780$$ 0 0
$$781$$ −720.000 −0.0329880
$$782$$ 0 0
$$783$$ 5600.00 0.255591
$$784$$ 0 0
$$785$$ −2480.00 −0.112758
$$786$$ 0 0
$$787$$ −26318.0 −1.19204 −0.596020 0.802970i $$-0.703252\pi$$
−0.596020 + 0.802970i $$0.703252\pi$$
$$788$$ 0 0
$$789$$ 5392.00 0.243296
$$790$$ 0 0
$$791$$ −27960.0 −1.25682
$$792$$ 0 0
$$793$$ 17480.0 0.782765
$$794$$ 0 0
$$795$$ −5856.00 −0.261246
$$796$$ 0 0
$$797$$ −17134.0 −0.761502 −0.380751 0.924678i $$-0.624335\pi$$
−0.380751 + 0.924678i $$0.624335\pi$$
$$798$$ 0 0
$$799$$ −816.000 −0.0361302
$$800$$ 0 0
$$801$$ −20838.0 −0.919194
$$802$$ 0 0
$$803$$ 7340.00 0.322569
$$804$$ 0 0
$$805$$ 11520.0 0.504381
$$806$$ 0 0
$$807$$ −4608.00 −0.201003
$$808$$ 0 0
$$809$$ −21826.0 −0.948531 −0.474265 0.880382i $$-0.657286\pi$$
−0.474265 + 0.880382i $$0.657286\pi$$
$$810$$ 0 0
$$811$$ −3386.00 −0.146607 −0.0733037 0.997310i $$-0.523354\pi$$
−0.0733037 + 0.997310i $$0.523354\pi$$
$$812$$ 0 0
$$813$$ −3440.00 −0.148396
$$814$$ 0 0
$$815$$ 14704.0 0.631974
$$816$$ 0 0
$$817$$ −576.000 −0.0246655
$$818$$ 0 0
$$819$$ −10488.0 −0.447473
$$820$$ 0 0
$$821$$ −17284.0 −0.734733 −0.367366 0.930076i $$-0.619741\pi$$
−0.367366 + 0.930076i $$0.619741\pi$$
$$822$$ 0 0
$$823$$ −5656.00 −0.239557 −0.119779 0.992801i $$-0.538219\pi$$
−0.119779 + 0.992801i $$0.538219\pi$$
$$824$$ 0 0
$$825$$ −1220.00 −0.0514848
$$826$$ 0 0
$$827$$ −8714.00 −0.366403 −0.183202 0.983075i $$-0.558646\pi$$
−0.183202 + 0.983075i $$0.558646\pi$$
$$828$$ 0 0
$$829$$ 11198.0 0.469147 0.234573 0.972098i $$-0.424631\pi$$
0.234573 + 0.972098i $$0.424631\pi$$
$$830$$ 0 0
$$831$$ 8448.00 0.352657
$$832$$ 0 0
$$833$$ 3383.00 0.140713
$$834$$ 0 0
$$835$$ −13472.0 −0.558345
$$836$$ 0 0
$$837$$ 16400.0 0.677260
$$838$$ 0 0
$$839$$ −6636.00 −0.273063 −0.136532 0.990636i $$-0.543596\pi$$
−0.136532 + 0.990636i $$0.543596\pi$$
$$840$$ 0 0
$$841$$ −21253.0 −0.871417
$$842$$ 0 0
$$843$$ −6252.00 −0.255433
$$844$$ 0 0
$$845$$ 6024.00 0.245245
$$846$$ 0 0
$$847$$ 14772.0 0.599258
$$848$$ 0 0
$$849$$ −10236.0 −0.413779
$$850$$ 0 0
$$851$$ −28320.0 −1.14077
$$852$$ 0 0
$$853$$ −16112.0 −0.646734 −0.323367 0.946274i $$-0.604815\pi$$
−0.323367 + 0.946274i $$0.604815\pi$$
$$854$$ 0 0
$$855$$ 736.000 0.0294394
$$856$$ 0 0
$$857$$ 41742.0 1.66380 0.831902 0.554923i $$-0.187252\pi$$
0.831902 + 0.554923i $$0.187252\pi$$
$$858$$ 0 0
$$859$$ −22248.0 −0.883693 −0.441846 0.897091i $$-0.645676\pi$$
−0.441846 + 0.897091i $$0.645676\pi$$
$$860$$ 0 0
$$861$$ 1680.00 0.0664974
$$862$$ 0 0
$$863$$ −11888.0 −0.468913 −0.234457 0.972127i $$-0.575331\pi$$
−0.234457 + 0.972127i $$0.575331\pi$$
$$864$$ 0 0
$$865$$ 8832.00 0.347164
$$866$$ 0 0
$$867$$ −578.000 −0.0226412
$$868$$ 0 0
$$869$$ −7360.00 −0.287308
$$870$$ 0 0
$$871$$ 29184.0 1.13532
$$872$$ 0 0
$$873$$ −1058.00 −0.0410170
$$874$$ 0 0
$$875$$ −17856.0 −0.689878
$$876$$ 0 0
$$877$$ 40024.0 1.54107 0.770533 0.637400i $$-0.219990\pi$$
0.770533 + 0.637400i $$0.219990\pi$$
$$878$$ 0 0
$$879$$ 8836.00 0.339057
$$880$$ 0 0
$$881$$ 36818.0 1.40798 0.703990 0.710210i $$-0.251400\pi$$
0.703990 + 0.710210i $$0.251400\pi$$
$$882$$ 0 0
$$883$$ −38424.0 −1.46441 −0.732203 0.681086i $$-0.761508\pi$$
−0.732203 + 0.681086i $$0.761508\pi$$
$$884$$ 0 0
$$885$$ −8064.00 −0.306292
$$886$$ 0 0
$$887$$ −13248.0 −0.501493 −0.250747 0.968053i $$-0.580676\pi$$
−0.250747 + 0.968053i $$0.580676\pi$$
$$888$$ 0 0
$$889$$ 24192.0 0.912681
$$890$$ 0 0
$$891$$ −4210.00 −0.158294
$$892$$ 0 0
$$893$$ 192.000 0.00719489
$$894$$ 0 0
$$895$$ 29280.0 1.09354
$$896$$ 0 0
$$897$$ 9120.00 0.339474
$$898$$ 0 0
$$899$$ 9184.00 0.340716
$$900$$ 0 0
$$901$$ 6222.00 0.230061
$$902$$ 0 0
$$903$$ −3456.00 −0.127363
$$904$$ 0 0
$$905$$ −37792.0 −1.38812
$$906$$ 0 0
$$907$$ 29014.0 1.06218 0.531088 0.847317i $$-0.321783\pi$$
0.531088 + 0.847317i $$0.321783\pi$$
$$908$$ 0 0
$$909$$ −27094.0 −0.988615
$$910$$ 0 0
$$911$$ −16156.0 −0.587565 −0.293783 0.955872i $$-0.594914\pi$$
−0.293783 + 0.955872i $$0.594914\pi$$
$$912$$ 0 0
$$913$$ −8560.00 −0.310290
$$914$$ 0 0
$$915$$ −7360.00 −0.265917
$$916$$ 0 0
$$917$$ 1224.00 0.0440786
$$918$$ 0 0
$$919$$ 46584.0 1.67210 0.836052 0.548650i $$-0.184858\pi$$
0.836052 + 0.548650i $$0.184858\pi$$
$$920$$ 0 0
$$921$$ −8952.00 −0.320281
$$922$$ 0 0
$$923$$ −2736.00 −0.0975694
$$924$$ 0 0
$$925$$ 14396.0 0.511716
$$926$$ 0 0
$$927$$ 13984.0 0.495464
$$928$$ 0 0
$$929$$ 34766.0 1.22781 0.613905 0.789380i $$-0.289598\pi$$
0.613905 + 0.789380i $$0.289598\pi$$
$$930$$ 0 0
$$931$$ −796.000 −0.0280213
$$932$$ 0 0
$$933$$ 600.000 0.0210537
$$934$$ 0 0
$$935$$ −1360.00 −0.0475687
$$936$$ 0 0
$$937$$ −19690.0 −0.686493 −0.343247 0.939245i $$-0.611527\pi$$
−0.343247 + 0.939245i $$0.611527\pi$$
$$938$$ 0 0
$$939$$ 2308.00 0.0802116
$$940$$ 0 0
$$941$$ 8564.00 0.296683 0.148341 0.988936i $$-0.452607\pi$$
0.148341 + 0.988936i $$0.452607\pi$$
$$942$$ 0 0
$$943$$ 8400.00 0.290076
$$944$$ 0 0
$$945$$ 9600.00 0.330464
$$946$$ 0 0
$$947$$ 42578.0 1.46103 0.730517 0.682895i $$-0.239279\pi$$
0.730517 + 0.682895i $$0.239279\pi$$
$$948$$ 0 0
$$949$$ 27892.0 0.954070
$$950$$ 0 0
$$951$$ −1368.00 −0.0466461
$$952$$ 0 0
$$953$$ −47406.0 −1.61137 −0.805683 0.592348i $$-0.798201\pi$$
−0.805683 + 0.592348i $$0.798201\pi$$
$$954$$ 0 0
$$955$$ −34240.0 −1.16019
$$956$$ 0 0
$$957$$ 1120.00 0.0378312
$$958$$ 0 0
$$959$$ 26232.0 0.883290
$$960$$ 0 0
$$961$$ −2895.00 −0.0971770
$$962$$ 0 0
$$963$$ 41354.0 1.38382
$$964$$ 0 0
$$965$$ −5040.00 −0.168128
$$966$$ 0 0
$$967$$ 51400.0 1.70932 0.854660 0.519188i $$-0.173766\pi$$
0.854660 + 0.519188i $$0.173766\pi$$
$$968$$ 0 0
$$969$$ 136.000 0.00450872
$$970$$ 0 0
$$971$$ 5840.00 0.193012 0.0965059 0.995332i $$-0.469233\pi$$
0.0965059 + 0.995332i $$0.469233\pi$$
$$972$$ 0 0
$$973$$ 31080.0 1.02403
$$974$$ 0 0
$$975$$ −4636.00 −0.152278
$$976$$ 0 0
$$977$$ −18358.0 −0.601151 −0.300575 0.953758i $$-0.597179\pi$$
−0.300575 + 0.953758i $$0.597179\pi$$
$$978$$ 0 0
$$979$$ −9060.00 −0.295770
$$980$$ 0 0
$$981$$ −4600.00 −0.149711
$$982$$ 0 0
$$983$$ 54888.0 1.78093 0.890466 0.455051i $$-0.150379\pi$$
0.890466 + 0.455051i $$0.150379\pi$$
$$984$$ 0 0
$$985$$ 9632.00 0.311575
$$986$$ 0 0
$$987$$ 1152.00 0.0371515
$$988$$ 0 0
$$989$$ −17280.0 −0.555583
$$990$$ 0 0
$$991$$ −6472.00 −0.207457 −0.103728 0.994606i $$-0.533077\pi$$
−0.103728 + 0.994606i $$0.533077\pi$$
$$992$$ 0 0
$$993$$ −18800.0 −0.600806
$$994$$ 0 0
$$995$$ −37056.0 −1.18066
$$996$$ 0 0
$$997$$ 16444.0 0.522354 0.261177 0.965291i $$-0.415889\pi$$
0.261177 + 0.965291i $$0.415889\pi$$
$$998$$ 0 0
$$999$$ −23600.0 −0.747418
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 68.4.a.a.1.1 1
3.2 odd 2 612.4.a.c.1.1 1
4.3 odd 2 272.4.a.b.1.1 1
5.2 odd 4 1700.4.e.c.749.2 2
5.3 odd 4 1700.4.e.c.749.1 2
5.4 even 2 1700.4.a.b.1.1 1
8.3 odd 2 1088.4.a.e.1.1 1
8.5 even 2 1088.4.a.h.1.1 1
12.11 even 2 2448.4.a.l.1.1 1
17.4 even 4 1156.4.b.b.577.2 2
17.13 even 4 1156.4.b.b.577.1 2
17.16 even 2 1156.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
68.4.a.a.1.1 1 1.1 even 1 trivial
272.4.a.b.1.1 1 4.3 odd 2
612.4.a.c.1.1 1 3.2 odd 2
1088.4.a.e.1.1 1 8.3 odd 2
1088.4.a.h.1.1 1 8.5 even 2
1156.4.a.a.1.1 1 17.16 even 2
1156.4.b.b.577.1 2 17.13 even 4
1156.4.b.b.577.2 2 17.4 even 4
1700.4.a.b.1.1 1 5.4 even 2
1700.4.e.c.749.1 2 5.3 odd 4
1700.4.e.c.749.2 2 5.2 odd 4
2448.4.a.l.1.1 1 12.11 even 2