# Properties

 Label 176.6.a.b.1.1 Level $176$ Weight $6$ Character 176.1 Self dual yes Analytic conductor $28.228$ 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] = [176,6,Mod(1,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("176.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 176.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: $$28.2275522871$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 176.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -51.0000 q^{5} +166.000 q^{7} -242.000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -51.0000 q^{5} +166.000 q^{7} -242.000 q^{9} +121.000 q^{11} +692.000 q^{13} +51.0000 q^{15} -738.000 q^{17} -1424.00 q^{19} -166.000 q^{21} +1779.00 q^{23} -524.000 q^{25} +485.000 q^{27} -2064.00 q^{29} -6245.00 q^{31} -121.000 q^{33} -8466.00 q^{35} -14785.0 q^{37} -692.000 q^{39} +5304.00 q^{41} -17798.0 q^{43} +12342.0 q^{45} +17184.0 q^{47} +10749.0 q^{49} +738.000 q^{51} -30726.0 q^{53} -6171.00 q^{55} +1424.00 q^{57} +34989.0 q^{59} -45940.0 q^{61} -40172.0 q^{63} -35292.0 q^{65} -25343.0 q^{67} -1779.00 q^{69} -13311.0 q^{71} -53260.0 q^{73} +524.000 q^{75} +20086.0 q^{77} -77234.0 q^{79} +58321.0 q^{81} -55014.0 q^{83} +37638.0 q^{85} +2064.00 q^{87} +125415. q^{89} +114872. q^{91} +6245.00 q^{93} +72624.0 q^{95} -88807.0 q^{97} -29282.0 q^{99} +O(q^{100})$$

## 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 −0.0641500 −0.0320750 0.999485i $$-0.510212\pi$$
−0.0320750 + 0.999485i $$0.510212\pi$$
$$4$$ 0 0
$$5$$ −51.0000 −0.912316 −0.456158 0.889899i $$-0.650775\pi$$
−0.456158 + 0.889899i $$0.650775\pi$$
$$6$$ 0 0
$$7$$ 166.000 1.28045 0.640226 0.768187i $$-0.278841\pi$$
0.640226 + 0.768187i $$0.278841\pi$$
$$8$$ 0 0
$$9$$ −242.000 −0.995885
$$10$$ 0 0
$$11$$ 121.000 0.301511
$$12$$ 0 0
$$13$$ 692.000 1.13566 0.567829 0.823146i $$-0.307783\pi$$
0.567829 + 0.823146i $$0.307783\pi$$
$$14$$ 0 0
$$15$$ 51.0000 0.0585251
$$16$$ 0 0
$$17$$ −738.000 −0.619347 −0.309674 0.950843i $$-0.600220\pi$$
−0.309674 + 0.950843i $$0.600220\pi$$
$$18$$ 0 0
$$19$$ −1424.00 −0.904953 −0.452476 0.891776i $$-0.649459\pi$$
−0.452476 + 0.891776i $$0.649459\pi$$
$$20$$ 0 0
$$21$$ −166.000 −0.0821410
$$22$$ 0 0
$$23$$ 1779.00 0.701223 0.350612 0.936521i $$-0.385974\pi$$
0.350612 + 0.936521i $$0.385974\pi$$
$$24$$ 0 0
$$25$$ −524.000 −0.167680
$$26$$ 0 0
$$27$$ 485.000 0.128036
$$28$$ 0 0
$$29$$ −2064.00 −0.455737 −0.227869 0.973692i $$-0.573176\pi$$
−0.227869 + 0.973692i $$0.573176\pi$$
$$30$$ 0 0
$$31$$ −6245.00 −1.16715 −0.583577 0.812058i $$-0.698347\pi$$
−0.583577 + 0.812058i $$0.698347\pi$$
$$32$$ 0 0
$$33$$ −121.000 −0.0193420
$$34$$ 0 0
$$35$$ −8466.00 −1.16818
$$36$$ 0 0
$$37$$ −14785.0 −1.77549 −0.887743 0.460340i $$-0.847727\pi$$
−0.887743 + 0.460340i $$0.847727\pi$$
$$38$$ 0 0
$$39$$ −692.000 −0.0728525
$$40$$ 0 0
$$41$$ 5304.00 0.492770 0.246385 0.969172i $$-0.420757\pi$$
0.246385 + 0.969172i $$0.420757\pi$$
$$42$$ 0 0
$$43$$ −17798.0 −1.46791 −0.733956 0.679197i $$-0.762328\pi$$
−0.733956 + 0.679197i $$0.762328\pi$$
$$44$$ 0 0
$$45$$ 12342.0 0.908561
$$46$$ 0 0
$$47$$ 17184.0 1.13470 0.567348 0.823478i $$-0.307969\pi$$
0.567348 + 0.823478i $$0.307969\pi$$
$$48$$ 0 0
$$49$$ 10749.0 0.639555
$$50$$ 0 0
$$51$$ 738.000 0.0397311
$$52$$ 0 0
$$53$$ −30726.0 −1.50251 −0.751253 0.660014i $$-0.770550\pi$$
−0.751253 + 0.660014i $$0.770550\pi$$
$$54$$ 0 0
$$55$$ −6171.00 −0.275074
$$56$$ 0 0
$$57$$ 1424.00 0.0580528
$$58$$ 0 0
$$59$$ 34989.0 1.30858 0.654292 0.756242i $$-0.272967\pi$$
0.654292 + 0.756242i $$0.272967\pi$$
$$60$$ 0 0
$$61$$ −45940.0 −1.58076 −0.790381 0.612616i $$-0.790117\pi$$
−0.790381 + 0.612616i $$0.790117\pi$$
$$62$$ 0 0
$$63$$ −40172.0 −1.27518
$$64$$ 0 0
$$65$$ −35292.0 −1.03608
$$66$$ 0 0
$$67$$ −25343.0 −0.689717 −0.344859 0.938655i $$-0.612073\pi$$
−0.344859 + 0.938655i $$0.612073\pi$$
$$68$$ 0 0
$$69$$ −1779.00 −0.0449835
$$70$$ 0 0
$$71$$ −13311.0 −0.313375 −0.156688 0.987648i $$-0.550082\pi$$
−0.156688 + 0.987648i $$0.550082\pi$$
$$72$$ 0 0
$$73$$ −53260.0 −1.16975 −0.584876 0.811123i $$-0.698857\pi$$
−0.584876 + 0.811123i $$0.698857\pi$$
$$74$$ 0 0
$$75$$ 524.000 0.0107567
$$76$$ 0 0
$$77$$ 20086.0 0.386071
$$78$$ 0 0
$$79$$ −77234.0 −1.39233 −0.696163 0.717884i $$-0.745111\pi$$
−0.696163 + 0.717884i $$0.745111\pi$$
$$80$$ 0 0
$$81$$ 58321.0 0.987671
$$82$$ 0 0
$$83$$ −55014.0 −0.876553 −0.438276 0.898840i $$-0.644411\pi$$
−0.438276 + 0.898840i $$0.644411\pi$$
$$84$$ 0 0
$$85$$ 37638.0 0.565040
$$86$$ 0 0
$$87$$ 2064.00 0.0292356
$$88$$ 0 0
$$89$$ 125415. 1.67832 0.839159 0.543886i $$-0.183047\pi$$
0.839159 + 0.543886i $$0.183047\pi$$
$$90$$ 0 0
$$91$$ 114872. 1.45416
$$92$$ 0 0
$$93$$ 6245.00 0.0748730
$$94$$ 0 0
$$95$$ 72624.0 0.825603
$$96$$ 0 0
$$97$$ −88807.0 −0.958336 −0.479168 0.877723i $$-0.659062\pi$$
−0.479168 + 0.877723i $$0.659062\pi$$
$$98$$ 0 0
$$99$$ −29282.0 −0.300271
$$100$$ 0 0
$$101$$ 1482.00 0.0144559 0.00722794 0.999974i $$-0.497699\pi$$
0.00722794 + 0.999974i $$0.497699\pi$$
$$102$$ 0 0
$$103$$ 117496. 1.09126 0.545632 0.838025i $$-0.316290\pi$$
0.545632 + 0.838025i $$0.316290\pi$$
$$104$$ 0 0
$$105$$ 8466.00 0.0749385
$$106$$ 0 0
$$107$$ 79362.0 0.670121 0.335060 0.942197i $$-0.391243\pi$$
0.335060 + 0.942197i $$0.391243\pi$$
$$108$$ 0 0
$$109$$ 87842.0 0.708167 0.354084 0.935214i $$-0.384793\pi$$
0.354084 + 0.935214i $$0.384793\pi$$
$$110$$ 0 0
$$111$$ 14785.0 0.113897
$$112$$ 0 0
$$113$$ −47247.0 −0.348079 −0.174040 0.984739i $$-0.555682\pi$$
−0.174040 + 0.984739i $$0.555682\pi$$
$$114$$ 0 0
$$115$$ −90729.0 −0.639737
$$116$$ 0 0
$$117$$ −167464. −1.13098
$$118$$ 0 0
$$119$$ −122508. −0.793044
$$120$$ 0 0
$$121$$ 14641.0 0.0909091
$$122$$ 0 0
$$123$$ −5304.00 −0.0316112
$$124$$ 0 0
$$125$$ 186099. 1.06529
$$126$$ 0 0
$$127$$ 239416. 1.31718 0.658588 0.752504i $$-0.271154\pi$$
0.658588 + 0.752504i $$0.271154\pi$$
$$128$$ 0 0
$$129$$ 17798.0 0.0941666
$$130$$ 0 0
$$131$$ 98142.0 0.499662 0.249831 0.968289i $$-0.419625\pi$$
0.249831 + 0.968289i $$0.419625\pi$$
$$132$$ 0 0
$$133$$ −236384. −1.15875
$$134$$ 0 0
$$135$$ −24735.0 −0.116809
$$136$$ 0 0
$$137$$ 400137. 1.82141 0.910704 0.413059i $$-0.135540\pi$$
0.910704 + 0.413059i $$0.135540\pi$$
$$138$$ 0 0
$$139$$ −205766. −0.903310 −0.451655 0.892193i $$-0.649166\pi$$
−0.451655 + 0.892193i $$0.649166\pi$$
$$140$$ 0 0
$$141$$ −17184.0 −0.0727908
$$142$$ 0 0
$$143$$ 83732.0 0.342414
$$144$$ 0 0
$$145$$ 105264. 0.415776
$$146$$ 0 0
$$147$$ −10749.0 −0.0410275
$$148$$ 0 0
$$149$$ 87726.0 0.323715 0.161857 0.986814i $$-0.448252\pi$$
0.161857 + 0.986814i $$0.448252\pi$$
$$150$$ 0 0
$$151$$ 432778. 1.54462 0.772312 0.635243i $$-0.219100\pi$$
0.772312 + 0.635243i $$0.219100\pi$$
$$152$$ 0 0
$$153$$ 178596. 0.616798
$$154$$ 0 0
$$155$$ 318495. 1.06481
$$156$$ 0 0
$$157$$ −34075.0 −0.110328 −0.0551641 0.998477i $$-0.517568\pi$$
−0.0551641 + 0.998477i $$0.517568\pi$$
$$158$$ 0 0
$$159$$ 30726.0 0.0963858
$$160$$ 0 0
$$161$$ 295314. 0.897882
$$162$$ 0 0
$$163$$ −45020.0 −0.132720 −0.0663600 0.997796i $$-0.521139\pi$$
−0.0663600 + 0.997796i $$0.521139\pi$$
$$164$$ 0 0
$$165$$ 6171.00 0.0176460
$$166$$ 0 0
$$167$$ −482556. −1.33893 −0.669463 0.742845i $$-0.733476\pi$$
−0.669463 + 0.742845i $$0.733476\pi$$
$$168$$ 0 0
$$169$$ 107571. 0.289720
$$170$$ 0 0
$$171$$ 344608. 0.901229
$$172$$ 0 0
$$173$$ −766254. −1.94651 −0.973257 0.229719i $$-0.926219\pi$$
−0.973257 + 0.229719i $$0.926219\pi$$
$$174$$ 0 0
$$175$$ −86984.0 −0.214706
$$176$$ 0 0
$$177$$ −34989.0 −0.0839457
$$178$$ 0 0
$$179$$ −303399. −0.707753 −0.353876 0.935292i $$-0.615137\pi$$
−0.353876 + 0.935292i $$0.615137\pi$$
$$180$$ 0 0
$$181$$ −285181. −0.647030 −0.323515 0.946223i $$-0.604865\pi$$
−0.323515 + 0.946223i $$0.604865\pi$$
$$182$$ 0 0
$$183$$ 45940.0 0.101406
$$184$$ 0 0
$$185$$ 754035. 1.61980
$$186$$ 0 0
$$187$$ −89298.0 −0.186740
$$188$$ 0 0
$$189$$ 80510.0 0.163944
$$190$$ 0 0
$$191$$ −767067. −1.52142 −0.760711 0.649090i $$-0.775150\pi$$
−0.760711 + 0.649090i $$0.775150\pi$$
$$192$$ 0 0
$$193$$ 411668. 0.795525 0.397763 0.917488i $$-0.369787\pi$$
0.397763 + 0.917488i $$0.369787\pi$$
$$194$$ 0 0
$$195$$ 35292.0 0.0664645
$$196$$ 0 0
$$197$$ −759258. −1.39387 −0.696937 0.717132i $$-0.745455\pi$$
−0.696937 + 0.717132i $$0.745455\pi$$
$$198$$ 0 0
$$199$$ 46600.0 0.0834167 0.0417084 0.999130i $$-0.486720\pi$$
0.0417084 + 0.999130i $$0.486720\pi$$
$$200$$ 0 0
$$201$$ 25343.0 0.0442454
$$202$$ 0 0
$$203$$ −342624. −0.583549
$$204$$ 0 0
$$205$$ −270504. −0.449561
$$206$$ 0 0
$$207$$ −430518. −0.698338
$$208$$ 0 0
$$209$$ −172304. −0.272854
$$210$$ 0 0
$$211$$ 932428. 1.44181 0.720907 0.693032i $$-0.243726\pi$$
0.720907 + 0.693032i $$0.243726\pi$$
$$212$$ 0 0
$$213$$ 13311.0 0.0201030
$$214$$ 0 0
$$215$$ 907698. 1.33920
$$216$$ 0 0
$$217$$ −1.03667e6 −1.49448
$$218$$ 0 0
$$219$$ 53260.0 0.0750397
$$220$$ 0 0
$$221$$ −510696. −0.703367
$$222$$ 0 0
$$223$$ −169745. −0.228578 −0.114289 0.993448i $$-0.536459\pi$$
−0.114289 + 0.993448i $$0.536459\pi$$
$$224$$ 0 0
$$225$$ 126808. 0.166990
$$226$$ 0 0
$$227$$ −198078. −0.255136 −0.127568 0.991830i $$-0.540717\pi$$
−0.127568 + 0.991830i $$0.540717\pi$$
$$228$$ 0 0
$$229$$ −849997. −1.07110 −0.535548 0.844505i $$-0.679895\pi$$
−0.535548 + 0.844505i $$0.679895\pi$$
$$230$$ 0 0
$$231$$ −20086.0 −0.0247664
$$232$$ 0 0
$$233$$ −401832. −0.484903 −0.242451 0.970164i $$-0.577952\pi$$
−0.242451 + 0.970164i $$0.577952\pi$$
$$234$$ 0 0
$$235$$ −876384. −1.03520
$$236$$ 0 0
$$237$$ 77234.0 0.0893177
$$238$$ 0 0
$$239$$ −855174. −0.968411 −0.484206 0.874954i $$-0.660891\pi$$
−0.484206 + 0.874954i $$0.660891\pi$$
$$240$$ 0 0
$$241$$ 1.12546e6 1.24821 0.624107 0.781339i $$-0.285463\pi$$
0.624107 + 0.781339i $$0.285463\pi$$
$$242$$ 0 0
$$243$$ −176176. −0.191395
$$244$$ 0 0
$$245$$ −548199. −0.583476
$$246$$ 0 0
$$247$$ −985408. −1.02772
$$248$$ 0 0
$$249$$ 55014.0 0.0562309
$$250$$ 0 0
$$251$$ 1.19751e6 1.19976 0.599882 0.800088i $$-0.295214\pi$$
0.599882 + 0.800088i $$0.295214\pi$$
$$252$$ 0 0
$$253$$ 215259. 0.211427
$$254$$ 0 0
$$255$$ −37638.0 −0.0362473
$$256$$ 0 0
$$257$$ 37758.0 0.0356596 0.0178298 0.999841i $$-0.494324\pi$$
0.0178298 + 0.999841i $$0.494324\pi$$
$$258$$ 0 0
$$259$$ −2.45431e6 −2.27342
$$260$$ 0 0
$$261$$ 499488. 0.453862
$$262$$ 0 0
$$263$$ 631254. 0.562749 0.281375 0.959598i $$-0.409210\pi$$
0.281375 + 0.959598i $$0.409210\pi$$
$$264$$ 0 0
$$265$$ 1.56703e6 1.37076
$$266$$ 0 0
$$267$$ −125415. −0.107664
$$268$$ 0 0
$$269$$ −1.08034e6 −0.910292 −0.455146 0.890417i $$-0.650413\pi$$
−0.455146 + 0.890417i $$0.650413\pi$$
$$270$$ 0 0
$$271$$ 816100. 0.675025 0.337513 0.941321i $$-0.390414\pi$$
0.337513 + 0.941321i $$0.390414\pi$$
$$272$$ 0 0
$$273$$ −114872. −0.0932841
$$274$$ 0 0
$$275$$ −63404.0 −0.0505574
$$276$$ 0 0
$$277$$ 1.68820e6 1.32198 0.660989 0.750396i $$-0.270137\pi$$
0.660989 + 0.750396i $$0.270137\pi$$
$$278$$ 0 0
$$279$$ 1.51129e6 1.16235
$$280$$ 0 0
$$281$$ −879042. −0.664116 −0.332058 0.943259i $$-0.607743\pi$$
−0.332058 + 0.943259i $$0.607743\pi$$
$$282$$ 0 0
$$283$$ −1.54027e6 −1.14322 −0.571611 0.820525i $$-0.693681\pi$$
−0.571611 + 0.820525i $$0.693681\pi$$
$$284$$ 0 0
$$285$$ −72624.0 −0.0529624
$$286$$ 0 0
$$287$$ 880464. 0.630967
$$288$$ 0 0
$$289$$ −875213. −0.616409
$$290$$ 0 0
$$291$$ 88807.0 0.0614773
$$292$$ 0 0
$$293$$ 720840. 0.490535 0.245267 0.969455i $$-0.421124\pi$$
0.245267 + 0.969455i $$0.421124\pi$$
$$294$$ 0 0
$$295$$ −1.78444e6 −1.19384
$$296$$ 0 0
$$297$$ 58685.0 0.0386043
$$298$$ 0 0
$$299$$ 1.23107e6 0.796350
$$300$$ 0 0
$$301$$ −2.95447e6 −1.87959
$$302$$ 0 0
$$303$$ −1482.00 −0.000927346 0
$$304$$ 0 0
$$305$$ 2.34294e6 1.44215
$$306$$ 0 0
$$307$$ 1.03905e6 0.629201 0.314601 0.949224i $$-0.398129\pi$$
0.314601 + 0.949224i $$0.398129\pi$$
$$308$$ 0 0
$$309$$ −117496. −0.0700046
$$310$$ 0 0
$$311$$ 1.25135e6 0.733630 0.366815 0.930294i $$-0.380448\pi$$
0.366815 + 0.930294i $$0.380448\pi$$
$$312$$ 0 0
$$313$$ −1.44336e6 −0.832749 −0.416375 0.909193i $$-0.636699\pi$$
−0.416375 + 0.909193i $$0.636699\pi$$
$$314$$ 0 0
$$315$$ 2.04877e6 1.16337
$$316$$ 0 0
$$317$$ −2.01208e6 −1.12460 −0.562298 0.826934i $$-0.690083\pi$$
−0.562298 + 0.826934i $$0.690083\pi$$
$$318$$ 0 0
$$319$$ −249744. −0.137410
$$320$$ 0 0
$$321$$ −79362.0 −0.0429883
$$322$$ 0 0
$$323$$ 1.05091e6 0.560480
$$324$$ 0 0
$$325$$ −362608. −0.190427
$$326$$ 0 0
$$327$$ −87842.0 −0.0454290
$$328$$ 0 0
$$329$$ 2.85254e6 1.45292
$$330$$ 0 0
$$331$$ −2.01734e6 −1.01207 −0.506033 0.862514i $$-0.668888\pi$$
−0.506033 + 0.862514i $$0.668888\pi$$
$$332$$ 0 0
$$333$$ 3.57797e6 1.76818
$$334$$ 0 0
$$335$$ 1.29249e6 0.629240
$$336$$ 0 0
$$337$$ 264122. 0.126686 0.0633432 0.997992i $$-0.479824\pi$$
0.0633432 + 0.997992i $$0.479824\pi$$
$$338$$ 0 0
$$339$$ 47247.0 0.0223293
$$340$$ 0 0
$$341$$ −755645. −0.351910
$$342$$ 0 0
$$343$$ −1.00563e6 −0.461532
$$344$$ 0 0
$$345$$ 90729.0 0.0410392
$$346$$ 0 0
$$347$$ 1.71049e6 0.762601 0.381300 0.924451i $$-0.375476\pi$$
0.381300 + 0.924451i $$0.375476\pi$$
$$348$$ 0 0
$$349$$ 218822. 0.0961673 0.0480836 0.998843i $$-0.484689\pi$$
0.0480836 + 0.998843i $$0.484689\pi$$
$$350$$ 0 0
$$351$$ 335620. 0.145405
$$352$$ 0 0
$$353$$ 3.68192e6 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$354$$ 0 0
$$355$$ 678861. 0.285897
$$356$$ 0 0
$$357$$ 122508. 0.0508738
$$358$$ 0 0
$$359$$ −1.88528e6 −0.772042 −0.386021 0.922490i $$-0.626151\pi$$
−0.386021 + 0.922490i $$0.626151\pi$$
$$360$$ 0 0
$$361$$ −448323. −0.181060
$$362$$ 0 0
$$363$$ −14641.0 −0.00583182
$$364$$ 0 0
$$365$$ 2.71626e6 1.06718
$$366$$ 0 0
$$367$$ 3.11666e6 1.20788 0.603940 0.797029i $$-0.293596\pi$$
0.603940 + 0.797029i $$0.293596\pi$$
$$368$$ 0 0
$$369$$ −1.28357e6 −0.490742
$$370$$ 0 0
$$371$$ −5.10052e6 −1.92389
$$372$$ 0 0
$$373$$ 1.39441e6 0.518943 0.259471 0.965751i $$-0.416452\pi$$
0.259471 + 0.965751i $$0.416452\pi$$
$$374$$ 0 0
$$375$$ −186099. −0.0683386
$$376$$ 0 0
$$377$$ −1.42829e6 −0.517562
$$378$$ 0 0
$$379$$ 4.26036e6 1.52352 0.761759 0.647860i $$-0.224336\pi$$
0.761759 + 0.647860i $$0.224336\pi$$
$$380$$ 0 0
$$381$$ −239416. −0.0844969
$$382$$ 0 0
$$383$$ −201765. −0.0702828 −0.0351414 0.999382i $$-0.511188\pi$$
−0.0351414 + 0.999382i $$0.511188\pi$$
$$384$$ 0 0
$$385$$ −1.02439e6 −0.352218
$$386$$ 0 0
$$387$$ 4.30712e6 1.46187
$$388$$ 0 0
$$389$$ 1.94882e6 0.652977 0.326489 0.945201i $$-0.394135\pi$$
0.326489 + 0.945201i $$0.394135\pi$$
$$390$$ 0 0
$$391$$ −1.31290e6 −0.434301
$$392$$ 0 0
$$393$$ −98142.0 −0.0320534
$$394$$ 0 0
$$395$$ 3.93893e6 1.27024
$$396$$ 0 0
$$397$$ −1.46826e6 −0.467548 −0.233774 0.972291i $$-0.575108\pi$$
−0.233774 + 0.972291i $$0.575108\pi$$
$$398$$ 0 0
$$399$$ 236384. 0.0743337
$$400$$ 0 0
$$401$$ 2.24618e6 0.697563 0.348781 0.937204i $$-0.386596\pi$$
0.348781 + 0.937204i $$0.386596\pi$$
$$402$$ 0 0
$$403$$ −4.32154e6 −1.32549
$$404$$ 0 0
$$405$$ −2.97437e6 −0.901068
$$406$$ 0 0
$$407$$ −1.78898e6 −0.535329
$$408$$ 0 0
$$409$$ −3.61488e6 −1.06853 −0.534263 0.845318i $$-0.679411\pi$$
−0.534263 + 0.845318i $$0.679411\pi$$
$$410$$ 0 0
$$411$$ −400137. −0.116843
$$412$$ 0 0
$$413$$ 5.80817e6 1.67558
$$414$$ 0 0
$$415$$ 2.80571e6 0.799693
$$416$$ 0 0
$$417$$ 205766. 0.0579473
$$418$$ 0 0
$$419$$ 3.81239e6 1.06087 0.530435 0.847726i $$-0.322029\pi$$
0.530435 + 0.847726i $$0.322029\pi$$
$$420$$ 0 0
$$421$$ 1.97346e6 0.542655 0.271327 0.962487i $$-0.412537\pi$$
0.271327 + 0.962487i $$0.412537\pi$$
$$422$$ 0 0
$$423$$ −4.15853e6 −1.13003
$$424$$ 0 0
$$425$$ 386712. 0.103852
$$426$$ 0 0
$$427$$ −7.62604e6 −2.02409
$$428$$ 0 0
$$429$$ −83732.0 −0.0219659
$$430$$ 0 0
$$431$$ 2.08359e6 0.540280 0.270140 0.962821i $$-0.412930\pi$$
0.270140 + 0.962821i $$0.412930\pi$$
$$432$$ 0 0
$$433$$ −72691.0 −0.0186321 −0.00931603 0.999957i $$-0.502965\pi$$
−0.00931603 + 0.999957i $$0.502965\pi$$
$$434$$ 0 0
$$435$$ −105264. −0.0266721
$$436$$ 0 0
$$437$$ −2.53330e6 −0.634574
$$438$$ 0 0
$$439$$ −594392. −0.147201 −0.0736007 0.997288i $$-0.523449\pi$$
−0.0736007 + 0.997288i $$0.523449\pi$$
$$440$$ 0 0
$$441$$ −2.60126e6 −0.636923
$$442$$ 0 0
$$443$$ −4.56651e6 −1.10554 −0.552770 0.833334i $$-0.686429\pi$$
−0.552770 + 0.833334i $$0.686429\pi$$
$$444$$ 0 0
$$445$$ −6.39616e6 −1.53116
$$446$$ 0 0
$$447$$ −87726.0 −0.0207663
$$448$$ 0 0
$$449$$ −5.44382e6 −1.27435 −0.637174 0.770720i $$-0.719897\pi$$
−0.637174 + 0.770720i $$0.719897\pi$$
$$450$$ 0 0
$$451$$ 641784. 0.148576
$$452$$ 0 0
$$453$$ −432778. −0.0990877
$$454$$ 0 0
$$455$$ −5.85847e6 −1.32665
$$456$$ 0 0
$$457$$ 6.70312e6 1.50137 0.750683 0.660662i $$-0.229724\pi$$
0.750683 + 0.660662i $$0.229724\pi$$
$$458$$ 0 0
$$459$$ −357930. −0.0792988
$$460$$ 0 0
$$461$$ −1.25994e6 −0.276120 −0.138060 0.990424i $$-0.544087\pi$$
−0.138060 + 0.990424i $$0.544087\pi$$
$$462$$ 0 0
$$463$$ 5.02308e6 1.08897 0.544487 0.838769i $$-0.316724\pi$$
0.544487 + 0.838769i $$0.316724\pi$$
$$464$$ 0 0
$$465$$ −318495. −0.0683078
$$466$$ 0 0
$$467$$ 2.35660e6 0.500028 0.250014 0.968242i $$-0.419565\pi$$
0.250014 + 0.968242i $$0.419565\pi$$
$$468$$ 0 0
$$469$$ −4.20694e6 −0.883149
$$470$$ 0 0
$$471$$ 34075.0 0.00707756
$$472$$ 0 0
$$473$$ −2.15356e6 −0.442592
$$474$$ 0 0
$$475$$ 746176. 0.151743
$$476$$ 0 0
$$477$$ 7.43569e6 1.49632
$$478$$ 0 0
$$479$$ 6.72258e6 1.33874 0.669371 0.742928i $$-0.266563\pi$$
0.669371 + 0.742928i $$0.266563\pi$$
$$480$$ 0 0
$$481$$ −1.02312e7 −2.01634
$$482$$ 0 0
$$483$$ −295314. −0.0575992
$$484$$ 0 0
$$485$$ 4.52916e6 0.874305
$$486$$ 0 0
$$487$$ −1.96001e6 −0.374487 −0.187243 0.982314i $$-0.559955\pi$$
−0.187243 + 0.982314i $$0.559955\pi$$
$$488$$ 0 0
$$489$$ 45020.0 0.00851399
$$490$$ 0 0
$$491$$ 579624. 0.108503 0.0542516 0.998527i $$-0.482723\pi$$
0.0542516 + 0.998527i $$0.482723\pi$$
$$492$$ 0 0
$$493$$ 1.52323e6 0.282260
$$494$$ 0 0
$$495$$ 1.49338e6 0.273942
$$496$$ 0 0
$$497$$ −2.20963e6 −0.401262
$$498$$ 0 0
$$499$$ −1.36905e6 −0.246132 −0.123066 0.992398i $$-0.539273\pi$$
−0.123066 + 0.992398i $$0.539273\pi$$
$$500$$ 0 0
$$501$$ 482556. 0.0858921
$$502$$ 0 0
$$503$$ −1.83343e6 −0.323105 −0.161552 0.986864i $$-0.551650\pi$$
−0.161552 + 0.986864i $$0.551650\pi$$
$$504$$ 0 0
$$505$$ −75582.0 −0.0131883
$$506$$ 0 0
$$507$$ −107571. −0.0185855
$$508$$ 0 0
$$509$$ −1.71266e6 −0.293006 −0.146503 0.989210i $$-0.546802\pi$$
−0.146503 + 0.989210i $$0.546802\pi$$
$$510$$ 0 0
$$511$$ −8.84116e6 −1.49781
$$512$$ 0 0
$$513$$ −690640. −0.115867
$$514$$ 0 0
$$515$$ −5.99230e6 −0.995578
$$516$$ 0 0
$$517$$ 2.07926e6 0.342124
$$518$$ 0 0
$$519$$ 766254. 0.124869
$$520$$ 0 0
$$521$$ −789435. −0.127415 −0.0637077 0.997969i $$-0.520293\pi$$
−0.0637077 + 0.997969i $$0.520293\pi$$
$$522$$ 0 0
$$523$$ −627392. −0.100296 −0.0501481 0.998742i $$-0.515969\pi$$
−0.0501481 + 0.998742i $$0.515969\pi$$
$$524$$ 0 0
$$525$$ 86984.0 0.0137734
$$526$$ 0 0
$$527$$ 4.60881e6 0.722873
$$528$$ 0 0
$$529$$ −3.27150e6 −0.508286
$$530$$ 0 0
$$531$$ −8.46734e6 −1.30320
$$532$$ 0 0
$$533$$ 3.67037e6 0.559618
$$534$$ 0 0
$$535$$ −4.04746e6 −0.611362
$$536$$ 0 0
$$537$$ 303399. 0.0454024
$$538$$ 0 0
$$539$$ 1.30063e6 0.192833
$$540$$ 0 0
$$541$$ 3.20895e6 0.471379 0.235689 0.971828i $$-0.424265\pi$$
0.235689 + 0.971828i $$0.424265\pi$$
$$542$$ 0 0
$$543$$ 285181. 0.0415070
$$544$$ 0 0
$$545$$ −4.47994e6 −0.646072
$$546$$ 0 0
$$547$$ −3.42658e6 −0.489658 −0.244829 0.969566i $$-0.578732\pi$$
−0.244829 + 0.969566i $$0.578732\pi$$
$$548$$ 0 0
$$549$$ 1.11175e7 1.57426
$$550$$ 0 0
$$551$$ 2.93914e6 0.412421
$$552$$ 0 0
$$553$$ −1.28208e7 −1.78280
$$554$$ 0 0
$$555$$ −754035. −0.103910
$$556$$ 0 0
$$557$$ 1.05198e7 1.43672 0.718358 0.695674i $$-0.244894\pi$$
0.718358 + 0.695674i $$0.244894\pi$$
$$558$$ 0 0
$$559$$ −1.23162e7 −1.66705
$$560$$ 0 0
$$561$$ 89298.0 0.0119794
$$562$$ 0 0
$$563$$ −5.47288e6 −0.727687 −0.363844 0.931460i $$-0.618536\pi$$
−0.363844 + 0.931460i $$0.618536\pi$$
$$564$$ 0 0
$$565$$ 2.40960e6 0.317558
$$566$$ 0 0
$$567$$ 9.68129e6 1.26466
$$568$$ 0 0
$$569$$ −1.17787e7 −1.52516 −0.762580 0.646893i $$-0.776068\pi$$
−0.762580 + 0.646893i $$0.776068\pi$$
$$570$$ 0 0
$$571$$ 8.35628e6 1.07256 0.536281 0.844039i $$-0.319829\pi$$
0.536281 + 0.844039i $$0.319829\pi$$
$$572$$ 0 0
$$573$$ 767067. 0.0975993
$$574$$ 0 0
$$575$$ −932196. −0.117581
$$576$$ 0 0
$$577$$ −1.37758e7 −1.72258 −0.861288 0.508117i $$-0.830342\pi$$
−0.861288 + 0.508117i $$0.830342\pi$$
$$578$$ 0 0
$$579$$ −411668. −0.0510330
$$580$$ 0 0
$$581$$ −9.13232e6 −1.12238
$$582$$ 0 0
$$583$$ −3.71785e6 −0.453023
$$584$$ 0 0
$$585$$ 8.54066e6 1.03182
$$586$$ 0 0
$$587$$ 1.27093e7 1.52239 0.761196 0.648522i $$-0.224612\pi$$
0.761196 + 0.648522i $$0.224612\pi$$
$$588$$ 0 0
$$589$$ 8.89288e6 1.05622
$$590$$ 0 0
$$591$$ 759258. 0.0894171
$$592$$ 0 0
$$593$$ 1.00825e6 0.117742 0.0588711 0.998266i $$-0.481250\pi$$
0.0588711 + 0.998266i $$0.481250\pi$$
$$594$$ 0 0
$$595$$ 6.24791e6 0.723506
$$596$$ 0 0
$$597$$ −46600.0 −0.00535119
$$598$$ 0 0
$$599$$ −1.05100e7 −1.19684 −0.598421 0.801182i $$-0.704205\pi$$
−0.598421 + 0.801182i $$0.704205\pi$$
$$600$$ 0 0
$$601$$ −199390. −0.0225173 −0.0112587 0.999937i $$-0.503584\pi$$
−0.0112587 + 0.999937i $$0.503584\pi$$
$$602$$ 0 0
$$603$$ 6.13301e6 0.686879
$$604$$ 0 0
$$605$$ −746691. −0.0829378
$$606$$ 0 0
$$607$$ −16190.0 −0.00178351 −0.000891754 1.00000i $$-0.500284\pi$$
−0.000891754 1.00000i $$0.500284\pi$$
$$608$$ 0 0
$$609$$ 342624. 0.0374347
$$610$$ 0 0
$$611$$ 1.18913e7 1.28863
$$612$$ 0 0
$$613$$ −1.15253e7 −1.23880 −0.619402 0.785074i $$-0.712625\pi$$
−0.619402 + 0.785074i $$0.712625\pi$$
$$614$$ 0 0
$$615$$ 270504. 0.0288394
$$616$$ 0 0
$$617$$ 1.69974e7 1.79750 0.898751 0.438459i $$-0.144476\pi$$
0.898751 + 0.438459i $$0.144476\pi$$
$$618$$ 0 0
$$619$$ 1.84875e7 1.93933 0.969663 0.244445i $$-0.0786058\pi$$
0.969663 + 0.244445i $$0.0786058\pi$$
$$620$$ 0 0
$$621$$ 862815. 0.0897819
$$622$$ 0 0
$$623$$ 2.08189e7 2.14901
$$624$$ 0 0
$$625$$ −7.85355e6 −0.804203
$$626$$ 0 0
$$627$$ 172304. 0.0175036
$$628$$ 0 0
$$629$$ 1.09113e7 1.09964
$$630$$ 0 0
$$631$$ 4.54281e6 0.454204 0.227102 0.973871i $$-0.427075\pi$$
0.227102 + 0.973871i $$0.427075\pi$$
$$632$$ 0 0
$$633$$ −932428. −0.0924924
$$634$$ 0 0
$$635$$ −1.22102e7 −1.20168
$$636$$ 0 0
$$637$$ 7.43831e6 0.726316
$$638$$ 0 0
$$639$$ 3.22126e6 0.312086
$$640$$ 0 0
$$641$$ 1.84286e7 1.77153 0.885764 0.464136i $$-0.153635\pi$$
0.885764 + 0.464136i $$0.153635\pi$$
$$642$$ 0 0
$$643$$ −9.66604e6 −0.921979 −0.460989 0.887406i $$-0.652505\pi$$
−0.460989 + 0.887406i $$0.652505\pi$$
$$644$$ 0 0
$$645$$ −907698. −0.0859097
$$646$$ 0 0
$$647$$ 4.51430e6 0.423965 0.211982 0.977273i $$-0.432008\pi$$
0.211982 + 0.977273i $$0.432008\pi$$
$$648$$ 0 0
$$649$$ 4.23367e6 0.394553
$$650$$ 0 0
$$651$$ 1.03667e6 0.0958712
$$652$$ 0 0
$$653$$ −5.37235e6 −0.493039 −0.246519 0.969138i $$-0.579287\pi$$
−0.246519 + 0.969138i $$0.579287\pi$$
$$654$$ 0 0
$$655$$ −5.00524e6 −0.455850
$$656$$ 0 0
$$657$$ 1.28889e7 1.16494
$$658$$ 0 0
$$659$$ −9.87956e6 −0.886184 −0.443092 0.896476i $$-0.646119\pi$$
−0.443092 + 0.896476i $$0.646119\pi$$
$$660$$ 0 0
$$661$$ 1.08052e7 0.961898 0.480949 0.876748i $$-0.340292\pi$$
0.480949 + 0.876748i $$0.340292\pi$$
$$662$$ 0 0
$$663$$ 510696. 0.0451210
$$664$$ 0 0
$$665$$ 1.20556e7 1.05714
$$666$$ 0 0
$$667$$ −3.67186e6 −0.319574
$$668$$ 0 0
$$669$$ 169745. 0.0146633
$$670$$ 0 0
$$671$$ −5.55874e6 −0.476618
$$672$$ 0 0
$$673$$ 1.13275e7 0.964042 0.482021 0.876160i $$-0.339903\pi$$
0.482021 + 0.876160i $$0.339903\pi$$
$$674$$ 0 0
$$675$$ −254140. −0.0214691
$$676$$ 0 0
$$677$$ −1.20595e7 −1.01125 −0.505624 0.862754i $$-0.668738\pi$$
−0.505624 + 0.862754i $$0.668738\pi$$
$$678$$ 0 0
$$679$$ −1.47420e7 −1.22710
$$680$$ 0 0
$$681$$ 198078. 0.0163670
$$682$$ 0 0
$$683$$ 5.14166e6 0.421747 0.210873 0.977513i $$-0.432369\pi$$
0.210873 + 0.977513i $$0.432369\pi$$
$$684$$ 0 0
$$685$$ −2.04070e7 −1.66170
$$686$$ 0 0
$$687$$ 849997. 0.0687109
$$688$$ 0 0
$$689$$ −2.12624e7 −1.70633
$$690$$ 0 0
$$691$$ −1.31243e7 −1.04563 −0.522817 0.852445i $$-0.675119\pi$$
−0.522817 + 0.852445i $$0.675119\pi$$
$$692$$ 0 0
$$693$$ −4.86081e6 −0.384482
$$694$$ 0 0
$$695$$ 1.04941e7 0.824104
$$696$$ 0 0
$$697$$ −3.91435e6 −0.305195
$$698$$ 0 0
$$699$$ 401832. 0.0311065
$$700$$ 0 0
$$701$$ 3.65956e6 0.281277 0.140638 0.990061i $$-0.455084\pi$$
0.140638 + 0.990061i $$0.455084\pi$$
$$702$$ 0 0
$$703$$ 2.10538e7 1.60673
$$704$$ 0 0
$$705$$ 876384. 0.0664082
$$706$$ 0 0
$$707$$ 246012. 0.0185101
$$708$$ 0 0
$$709$$ 1.02252e7 0.763935 0.381968 0.924176i $$-0.375247\pi$$
0.381968 + 0.924176i $$0.375247\pi$$
$$710$$ 0 0
$$711$$ 1.86906e7 1.38660
$$712$$ 0 0
$$713$$ −1.11099e7 −0.818436
$$714$$ 0 0
$$715$$ −4.27033e6 −0.312390
$$716$$ 0 0
$$717$$ 855174. 0.0621236
$$718$$ 0 0
$$719$$ −2.41683e7 −1.74351 −0.871753 0.489945i $$-0.837017\pi$$
−0.871753 + 0.489945i $$0.837017\pi$$
$$720$$ 0 0
$$721$$ 1.95043e7 1.39731
$$722$$ 0 0
$$723$$ −1.12546e6 −0.0800730
$$724$$ 0 0
$$725$$ 1.08154e6 0.0764181
$$726$$ 0 0
$$727$$ −1.68246e7 −1.18062 −0.590310 0.807177i $$-0.700994\pi$$
−0.590310 + 0.807177i $$0.700994\pi$$
$$728$$ 0 0
$$729$$ −1.39958e7 −0.975393
$$730$$ 0 0
$$731$$ 1.31349e7 0.909147
$$732$$ 0 0
$$733$$ −5.04168e6 −0.346590 −0.173295 0.984870i $$-0.555441\pi$$
−0.173295 + 0.984870i $$0.555441\pi$$
$$734$$ 0 0
$$735$$ 548199. 0.0374300
$$736$$ 0 0
$$737$$ −3.06650e6 −0.207958
$$738$$ 0 0
$$739$$ 6.26375e6 0.421913 0.210957 0.977495i $$-0.432342\pi$$
0.210957 + 0.977495i $$0.432342\pi$$
$$740$$ 0 0
$$741$$ 985408. 0.0659281
$$742$$ 0 0
$$743$$ −3.63976e6 −0.241880 −0.120940 0.992660i $$-0.538591\pi$$
−0.120940 + 0.992660i $$0.538591\pi$$
$$744$$ 0 0
$$745$$ −4.47403e6 −0.295330
$$746$$ 0 0
$$747$$ 1.33134e7 0.872945
$$748$$ 0 0
$$749$$ 1.31741e7 0.858057
$$750$$ 0 0
$$751$$ 1.87370e7 1.21227 0.606135 0.795362i $$-0.292719\pi$$
0.606135 + 0.795362i $$0.292719\pi$$
$$752$$ 0 0
$$753$$ −1.19751e6 −0.0769649
$$754$$ 0 0
$$755$$ −2.20717e7 −1.40918
$$756$$ 0 0
$$757$$ 489242. 0.0310302 0.0155151 0.999880i $$-0.495061\pi$$
0.0155151 + 0.999880i $$0.495061\pi$$
$$758$$ 0 0
$$759$$ −215259. −0.0135630
$$760$$ 0 0
$$761$$ 1.46969e7 0.919952 0.459976 0.887931i $$-0.347858\pi$$
0.459976 + 0.887931i $$0.347858\pi$$
$$762$$ 0 0
$$763$$ 1.45818e7 0.906774
$$764$$ 0 0
$$765$$ −9.10840e6 −0.562715
$$766$$ 0 0
$$767$$ 2.42124e7 1.48610
$$768$$ 0 0
$$769$$ 2.42072e7 1.47615 0.738073 0.674721i $$-0.235736\pi$$
0.738073 + 0.674721i $$0.235736\pi$$
$$770$$ 0 0
$$771$$ −37758.0 −0.00228756
$$772$$ 0 0
$$773$$ 1.35260e7 0.814181 0.407091 0.913388i $$-0.366543\pi$$
0.407091 + 0.913388i $$0.366543\pi$$
$$774$$ 0 0
$$775$$ 3.27238e6 0.195708
$$776$$ 0 0
$$777$$ 2.45431e6 0.145840
$$778$$ 0 0
$$779$$ −7.55290e6 −0.445933
$$780$$ 0 0
$$781$$ −1.61063e6 −0.0944862
$$782$$ 0 0
$$783$$ −1.00104e6 −0.0583508
$$784$$ 0 0
$$785$$ 1.73782e6 0.100654
$$786$$ 0 0
$$787$$ −1.42094e7 −0.817786 −0.408893 0.912582i $$-0.634085\pi$$
−0.408893 + 0.912582i $$0.634085\pi$$
$$788$$ 0 0
$$789$$ −631254. −0.0361004
$$790$$ 0 0
$$791$$ −7.84300e6 −0.445698
$$792$$ 0 0
$$793$$ −3.17905e7 −1.79521
$$794$$ 0 0
$$795$$ −1.56703e6 −0.0879343
$$796$$ 0 0
$$797$$ −7.93333e6 −0.442395 −0.221197 0.975229i $$-0.570997\pi$$
−0.221197 + 0.975229i $$0.570997\pi$$
$$798$$ 0 0
$$799$$ −1.26818e7 −0.702771
$$800$$ 0 0
$$801$$ −3.03504e7 −1.67141
$$802$$ 0 0
$$803$$ −6.44446e6 −0.352694
$$804$$ 0 0
$$805$$ −1.50610e7 −0.819152
$$806$$ 0 0
$$807$$ 1.08034e6 0.0583952
$$808$$ 0 0
$$809$$ −1.04685e7 −0.562359 −0.281180 0.959655i $$-0.590726\pi$$
−0.281180 + 0.959655i $$0.590726\pi$$
$$810$$ 0 0
$$811$$ −1.19147e7 −0.636110 −0.318055 0.948072i $$-0.603030\pi$$
−0.318055 + 0.948072i $$0.603030\pi$$
$$812$$ 0 0
$$813$$ −816100. −0.0433029
$$814$$ 0 0
$$815$$ 2.29602e6 0.121083
$$816$$ 0 0
$$817$$ 2.53444e7 1.32839
$$818$$ 0 0
$$819$$ −2.77990e7 −1.44817
$$820$$ 0 0
$$821$$ −1.86112e6 −0.0963645 −0.0481822 0.998839i $$-0.515343\pi$$
−0.0481822 + 0.998839i $$0.515343\pi$$
$$822$$ 0 0
$$823$$ −2.30153e7 −1.18445 −0.592225 0.805773i $$-0.701750\pi$$
−0.592225 + 0.805773i $$0.701750\pi$$
$$824$$ 0 0
$$825$$ 63404.0 0.00324326
$$826$$ 0 0
$$827$$ 1.68351e7 0.855959 0.427980 0.903788i $$-0.359225\pi$$
0.427980 + 0.903788i $$0.359225\pi$$
$$828$$ 0 0
$$829$$ −2.35299e7 −1.18914 −0.594570 0.804044i $$-0.702678\pi$$
−0.594570 + 0.804044i $$0.702678\pi$$
$$830$$ 0 0
$$831$$ −1.68820e6 −0.0848049
$$832$$ 0 0
$$833$$ −7.93276e6 −0.396106
$$834$$ 0 0
$$835$$ 2.46104e7 1.22152
$$836$$ 0 0
$$837$$ −3.02882e6 −0.149438
$$838$$ 0 0
$$839$$ −2.91549e7 −1.42990 −0.714952 0.699173i $$-0.753552\pi$$
−0.714952 + 0.699173i $$0.753552\pi$$
$$840$$ 0 0
$$841$$ −1.62511e7 −0.792303
$$842$$ 0 0
$$843$$ 879042. 0.0426030
$$844$$ 0 0
$$845$$ −5.48612e6 −0.264316
$$846$$ 0 0
$$847$$ 2.43041e6 0.116405
$$848$$ 0 0
$$849$$ 1.54027e6 0.0733377
$$850$$ 0 0
$$851$$ −2.63025e7 −1.24501
$$852$$ 0 0
$$853$$ −9.49052e6 −0.446599 −0.223299 0.974750i $$-0.571683\pi$$
−0.223299 + 0.974750i $$0.571683\pi$$
$$854$$ 0 0
$$855$$ −1.75750e7 −0.822205
$$856$$ 0 0
$$857$$ −1.81553e6 −0.0844405 −0.0422203 0.999108i $$-0.513443\pi$$
−0.0422203 + 0.999108i $$0.513443\pi$$
$$858$$ 0 0
$$859$$ 1.07812e7 0.498522 0.249261 0.968436i $$-0.419812\pi$$
0.249261 + 0.968436i $$0.419812\pi$$
$$860$$ 0 0
$$861$$ −880464. −0.0404766
$$862$$ 0 0
$$863$$ 2.83355e7 1.29510 0.647550 0.762023i $$-0.275794\pi$$
0.647550 + 0.762023i $$0.275794\pi$$
$$864$$ 0 0
$$865$$ 3.90790e7 1.77584
$$866$$ 0 0
$$867$$ 875213. 0.0395427
$$868$$ 0 0
$$869$$ −9.34531e6 −0.419802
$$870$$ 0 0
$$871$$ −1.75374e7 −0.783283
$$872$$ 0 0
$$873$$ 2.14913e7 0.954392
$$874$$ 0 0
$$875$$ 3.08924e7 1.36406
$$876$$ 0 0
$$877$$ −2.68919e7 −1.18065 −0.590326 0.807165i $$-0.701001\pi$$
−0.590326 + 0.807165i $$0.701001\pi$$
$$878$$ 0 0
$$879$$ −720840. −0.0314678
$$880$$ 0 0
$$881$$ −1.92132e7 −0.833989 −0.416995 0.908909i $$-0.636917\pi$$
−0.416995 + 0.908909i $$0.636917\pi$$
$$882$$ 0 0
$$883$$ −1.15931e7 −0.500378 −0.250189 0.968197i $$-0.580493\pi$$
−0.250189 + 0.968197i $$0.580493\pi$$
$$884$$ 0 0
$$885$$ 1.78444e6 0.0765850
$$886$$ 0 0
$$887$$ −1.31857e7 −0.562721 −0.281361 0.959602i $$-0.590786\pi$$
−0.281361 + 0.959602i $$0.590786\pi$$
$$888$$ 0 0
$$889$$ 3.97431e7 1.68658
$$890$$ 0 0
$$891$$ 7.05684e6 0.297794
$$892$$ 0 0
$$893$$ −2.44700e7 −1.02685
$$894$$ 0 0
$$895$$ 1.54733e7 0.645694
$$896$$ 0 0
$$897$$ −1.23107e6 −0.0510859
$$898$$ 0 0
$$899$$ 1.28897e7 0.531916
$$900$$ 0 0
$$901$$ 2.26758e7 0.930573
$$902$$ 0 0
$$903$$ 2.95447e6 0.120576
$$904$$ 0 0
$$905$$ 1.45442e7 0.590295
$$906$$ 0 0
$$907$$ −2.98195e6 −0.120360 −0.0601800 0.998188i $$-0.519167\pi$$
−0.0601800 + 0.998188i $$0.519167\pi$$
$$908$$ 0 0
$$909$$ −358644. −0.0143964
$$910$$ 0 0
$$911$$ −2.96579e7 −1.18398 −0.591989 0.805946i $$-0.701657\pi$$
−0.591989 + 0.805946i $$0.701657\pi$$
$$912$$ 0 0
$$913$$ −6.65669e6 −0.264291
$$914$$ 0 0
$$915$$ −2.34294e6 −0.0925142
$$916$$ 0 0
$$917$$ 1.62916e7 0.639793
$$918$$ 0 0
$$919$$ −3.18057e7 −1.24227 −0.621135 0.783704i $$-0.713328\pi$$
−0.621135 + 0.783704i $$0.713328\pi$$
$$920$$ 0 0
$$921$$ −1.03905e6 −0.0403633
$$922$$ 0 0
$$923$$ −9.21121e6 −0.355887
$$924$$ 0 0
$$925$$ 7.74734e6 0.297713
$$926$$ 0 0
$$927$$ −2.84340e7 −1.08677
$$928$$ 0 0
$$929$$ −2.33444e7 −0.887451 −0.443725 0.896163i $$-0.646343\pi$$
−0.443725 + 0.896163i $$0.646343\pi$$
$$930$$ 0 0
$$931$$ −1.53066e7 −0.578767
$$932$$ 0 0
$$933$$ −1.25135e6 −0.0470624
$$934$$ 0 0
$$935$$ 4.55420e6 0.170366
$$936$$ 0 0
$$937$$ 2.07372e7 0.771616 0.385808 0.922579i $$-0.373923\pi$$
0.385808 + 0.922579i $$0.373923\pi$$
$$938$$ 0 0
$$939$$ 1.44336e6 0.0534209
$$940$$ 0 0
$$941$$ 2.69193e7 0.991036 0.495518 0.868598i $$-0.334978\pi$$
0.495518 + 0.868598i $$0.334978\pi$$
$$942$$ 0 0
$$943$$ 9.43582e6 0.345542
$$944$$ 0 0
$$945$$ −4.10601e6 −0.149569
$$946$$ 0 0
$$947$$ −1.01896e7 −0.369216 −0.184608 0.982812i $$-0.559102\pi$$
−0.184608 + 0.982812i $$0.559102\pi$$
$$948$$ 0 0
$$949$$ −3.68559e7 −1.32844
$$950$$ 0 0
$$951$$ 2.01208e6 0.0721429
$$952$$ 0 0
$$953$$ 1.03924e7 0.370665 0.185333 0.982676i $$-0.440664\pi$$
0.185333 + 0.982676i $$0.440664\pi$$
$$954$$ 0 0
$$955$$ 3.91204e7 1.38802
$$956$$ 0 0
$$957$$ 249744. 0.00881486
$$958$$ 0 0
$$959$$ 6.64227e7 2.33222
$$960$$ 0 0
$$961$$ 1.03709e7 0.362249
$$962$$ 0 0
$$963$$ −1.92056e7 −0.667363
$$964$$ 0 0
$$965$$ −2.09951e7 −0.725770
$$966$$ 0 0
$$967$$ 8.18877e6 0.281613 0.140806 0.990037i $$-0.455030\pi$$
0.140806 + 0.990037i $$0.455030\pi$$
$$968$$ 0 0
$$969$$ −1.05091e6 −0.0359548
$$970$$ 0 0
$$971$$ 1.73274e7 0.589775 0.294887 0.955532i $$-0.404718\pi$$
0.294887 + 0.955532i $$0.404718\pi$$
$$972$$ 0 0
$$973$$ −3.41572e7 −1.15664
$$974$$ 0 0
$$975$$ 362608. 0.0122159
$$976$$ 0 0
$$977$$ −438963. −0.0147127 −0.00735634 0.999973i $$-0.502342\pi$$
−0.00735634 + 0.999973i $$0.502342\pi$$
$$978$$ 0 0
$$979$$ 1.51752e7 0.506032
$$980$$ 0 0
$$981$$ −2.12578e7 −0.705253
$$982$$ 0 0
$$983$$ 2.79124e7 0.921326 0.460663 0.887575i $$-0.347612\pi$$
0.460663 + 0.887575i $$0.347612\pi$$
$$984$$ 0 0
$$985$$ 3.87222e7 1.27165
$$986$$ 0 0
$$987$$ −2.85254e6 −0.0932051
$$988$$ 0 0
$$989$$ −3.16626e7 −1.02933
$$990$$ 0 0
$$991$$ 4.26846e7 1.38066 0.690331 0.723494i $$-0.257465\pi$$
0.690331 + 0.723494i $$0.257465\pi$$
$$992$$ 0 0
$$993$$ 2.01734e6 0.0649240
$$994$$ 0 0
$$995$$ −2.37660e6 −0.0761024
$$996$$ 0 0
$$997$$ −2.21044e7 −0.704273 −0.352137 0.935949i $$-0.614545\pi$$
−0.352137 + 0.935949i $$0.614545\pi$$
$$998$$ 0 0
$$999$$ −7.17072e6 −0.227326
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 176.6.a.b.1.1 1
4.3 odd 2 22.6.a.b.1.1 1
8.3 odd 2 704.6.a.e.1.1 1
8.5 even 2 704.6.a.f.1.1 1
12.11 even 2 198.6.a.i.1.1 1
20.3 even 4 550.6.b.f.199.2 2
20.7 even 4 550.6.b.f.199.1 2
20.19 odd 2 550.6.a.f.1.1 1
28.27 even 2 1078.6.a.a.1.1 1
44.43 even 2 242.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
22.6.a.b.1.1 1 4.3 odd 2
176.6.a.b.1.1 1 1.1 even 1 trivial
198.6.a.i.1.1 1 12.11 even 2
242.6.a.d.1.1 1 44.43 even 2
550.6.a.f.1.1 1 20.19 odd 2
550.6.b.f.199.1 2 20.7 even 4
550.6.b.f.199.2 2 20.3 even 4
704.6.a.e.1.1 1 8.3 odd 2
704.6.a.f.1.1 1 8.5 even 2
1078.6.a.a.1.1 1 28.27 even 2