# Properties

 Label 4840.2.a.bf.1.2 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.45753625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20$$ x^6 - x^5 - 13*x^4 + 11*x^3 + 41*x^2 - 30*x - 20 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.80540$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73383 q^{3} -1.00000 q^{5} -1.25225 q^{7} +0.00617996 q^{9} +O(q^{10})$$ $$q-1.73383 q^{3} -1.00000 q^{5} -1.25225 q^{7} +0.00617996 q^{9} -2.50777 q^{13} +1.73383 q^{15} -3.60412 q^{17} -7.05765 q^{19} +2.17119 q^{21} -2.20737 q^{23} +1.00000 q^{25} +5.19079 q^{27} +2.69960 q^{29} -2.83345 q^{31} +1.25225 q^{35} -8.07294 q^{37} +4.34805 q^{39} -10.3266 q^{41} +6.88581 q^{43} -0.00617996 q^{45} +1.59656 q^{47} -5.43188 q^{49} +6.24894 q^{51} +1.77393 q^{53} +12.2368 q^{57} -4.36887 q^{59} -8.33324 q^{61} -0.00773884 q^{63} +2.50777 q^{65} +5.43293 q^{67} +3.82721 q^{69} -4.63664 q^{71} -2.03474 q^{73} -1.73383 q^{75} +16.7625 q^{79} -9.01850 q^{81} -5.62561 q^{83} +3.60412 q^{85} -4.68066 q^{87} -4.72832 q^{89} +3.14034 q^{91} +4.91274 q^{93} +7.05765 q^{95} +19.0943 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9}+O(q^{10})$$ 6 * q + 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 $$6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} - 6 q^{13} - 2 q^{15} + 11 q^{17} - 11 q^{19} + 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} - 6 q^{29} + q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} - 4 q^{41} + 3 q^{43} - 10 q^{45} + 14 q^{47} + 8 q^{49} + 31 q^{51} + 14 q^{53} - 5 q^{57} + 2 q^{59} - 4 q^{61} - 16 q^{63} + 6 q^{65} + 11 q^{67} + 8 q^{69} + 7 q^{71} - 9 q^{73} + 2 q^{75} + 36 q^{79} + 30 q^{81} - 45 q^{83} - 11 q^{85} + 25 q^{87} + q^{89} - 8 q^{91} + 55 q^{93} + 11 q^{95} + 20 q^{97}+O(q^{100})$$ 6 * q + 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 - 6 * q^13 - 2 * q^15 + 11 * q^17 - 11 * q^19 + 2 * q^21 + 18 * q^23 + 6 * q^25 - q^27 - 6 * q^29 + q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 - 4 * q^41 + 3 * q^43 - 10 * q^45 + 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 - 5 * q^57 + 2 * q^59 - 4 * q^61 - 16 * q^63 + 6 * q^65 + 11 * q^67 + 8 * q^69 + 7 * q^71 - 9 * q^73 + 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 - 11 * q^85 + 25 * q^87 + q^89 - 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * q^97

## 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
$$3$$ −1.73383 −1.00103 −0.500515 0.865728i $$-0.666856\pi$$
−0.500515 + 0.865728i $$0.666856\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −1.25225 −0.473305 −0.236652 0.971594i $$-0.576050\pi$$
−0.236652 + 0.971594i $$0.576050\pi$$
$$8$$ 0 0
$$9$$ 0.00617996 0.00205999
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −2.50777 −0.695529 −0.347764 0.937582i $$-0.613059\pi$$
−0.347764 + 0.937582i $$0.613059\pi$$
$$14$$ 0 0
$$15$$ 1.73383 0.447674
$$16$$ 0 0
$$17$$ −3.60412 −0.874126 −0.437063 0.899431i $$-0.643981\pi$$
−0.437063 + 0.899431i $$0.643981\pi$$
$$18$$ 0 0
$$19$$ −7.05765 −1.61914 −0.809568 0.587026i $$-0.800299\pi$$
−0.809568 + 0.587026i $$0.800299\pi$$
$$20$$ 0 0
$$21$$ 2.17119 0.473792
$$22$$ 0 0
$$23$$ −2.20737 −0.460268 −0.230134 0.973159i $$-0.573916\pi$$
−0.230134 + 0.973159i $$0.573916\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.19079 0.998967
$$28$$ 0 0
$$29$$ 2.69960 0.501303 0.250652 0.968077i $$-0.419355\pi$$
0.250652 + 0.968077i $$0.419355\pi$$
$$30$$ 0 0
$$31$$ −2.83345 −0.508903 −0.254452 0.967085i $$-0.581895\pi$$
−0.254452 + 0.967085i $$0.581895\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 1.25225 0.211668
$$36$$ 0 0
$$37$$ −8.07294 −1.32718 −0.663592 0.748095i $$-0.730969\pi$$
−0.663592 + 0.748095i $$0.730969\pi$$
$$38$$ 0 0
$$39$$ 4.34805 0.696245
$$40$$ 0 0
$$41$$ −10.3266 −1.61274 −0.806371 0.591410i $$-0.798571\pi$$
−0.806371 + 0.591410i $$0.798571\pi$$
$$42$$ 0 0
$$43$$ 6.88581 1.05008 0.525038 0.851079i $$-0.324051\pi$$
0.525038 + 0.851079i $$0.324051\pi$$
$$44$$ 0 0
$$45$$ −0.00617996 −0.000921254 0
$$46$$ 0 0
$$47$$ 1.59656 0.232883 0.116441 0.993198i $$-0.462851\pi$$
0.116441 + 0.993198i $$0.462851\pi$$
$$48$$ 0 0
$$49$$ −5.43188 −0.775982
$$50$$ 0 0
$$51$$ 6.24894 0.875026
$$52$$ 0 0
$$53$$ 1.77393 0.243668 0.121834 0.992550i $$-0.461122\pi$$
0.121834 + 0.992550i $$0.461122\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 12.2368 1.62080
$$58$$ 0 0
$$59$$ −4.36887 −0.568779 −0.284389 0.958709i $$-0.591791\pi$$
−0.284389 + 0.958709i $$0.591791\pi$$
$$60$$ 0 0
$$61$$ −8.33324 −1.06696 −0.533481 0.845812i $$-0.679116\pi$$
−0.533481 + 0.845812i $$0.679116\pi$$
$$62$$ 0 0
$$63$$ −0.00773884 −0.000975002 0
$$64$$ 0 0
$$65$$ 2.50777 0.311050
$$66$$ 0 0
$$67$$ 5.43293 0.663738 0.331869 0.943326i $$-0.392321\pi$$
0.331869 + 0.943326i $$0.392321\pi$$
$$68$$ 0 0
$$69$$ 3.82721 0.460742
$$70$$ 0 0
$$71$$ −4.63664 −0.550267 −0.275134 0.961406i $$-0.588722\pi$$
−0.275134 + 0.961406i $$0.588722\pi$$
$$72$$ 0 0
$$73$$ −2.03474 −0.238148 −0.119074 0.992885i $$-0.537993\pi$$
−0.119074 + 0.992885i $$0.537993\pi$$
$$74$$ 0 0
$$75$$ −1.73383 −0.200206
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16.7625 1.88593 0.942967 0.332887i $$-0.108023\pi$$
0.942967 + 0.332887i $$0.108023\pi$$
$$80$$ 0 0
$$81$$ −9.01850 −1.00206
$$82$$ 0 0
$$83$$ −5.62561 −0.617491 −0.308746 0.951145i $$-0.599909\pi$$
−0.308746 + 0.951145i $$0.599909\pi$$
$$84$$ 0 0
$$85$$ 3.60412 0.390921
$$86$$ 0 0
$$87$$ −4.68066 −0.501820
$$88$$ 0 0
$$89$$ −4.72832 −0.501200 −0.250600 0.968091i $$-0.580628\pi$$
−0.250600 + 0.968091i $$0.580628\pi$$
$$90$$ 0 0
$$91$$ 3.14034 0.329197
$$92$$ 0 0
$$93$$ 4.91274 0.509427
$$94$$ 0 0
$$95$$ 7.05765 0.724100
$$96$$ 0 0
$$97$$ 19.0943 1.93873 0.969365 0.245625i $$-0.0789933\pi$$
0.969365 + 0.245625i $$0.0789933\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −14.2252 −1.41546 −0.707732 0.706481i $$-0.750281\pi$$
−0.707732 + 0.706481i $$0.750281\pi$$
$$102$$ 0 0
$$103$$ −15.8815 −1.56485 −0.782426 0.622744i $$-0.786018\pi$$
−0.782426 + 0.622744i $$0.786018\pi$$
$$104$$ 0 0
$$105$$ −2.17119 −0.211886
$$106$$ 0 0
$$107$$ −4.04935 −0.391465 −0.195733 0.980657i $$-0.562708\pi$$
−0.195733 + 0.980657i $$0.562708\pi$$
$$108$$ 0 0
$$109$$ 5.89844 0.564968 0.282484 0.959272i $$-0.408842\pi$$
0.282484 + 0.959272i $$0.408842\pi$$
$$110$$ 0 0
$$111$$ 13.9971 1.32855
$$112$$ 0 0
$$113$$ 3.41401 0.321163 0.160582 0.987023i $$-0.448663\pi$$
0.160582 + 0.987023i $$0.448663\pi$$
$$114$$ 0 0
$$115$$ 2.20737 0.205838
$$116$$ 0 0
$$117$$ −0.0154979 −0.00143278
$$118$$ 0 0
$$119$$ 4.51324 0.413728
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 17.9046 1.61440
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 6.19208 0.549458 0.274729 0.961522i $$-0.411412\pi$$
0.274729 + 0.961522i $$0.411412\pi$$
$$128$$ 0 0
$$129$$ −11.9389 −1.05116
$$130$$ 0 0
$$131$$ −7.39924 −0.646475 −0.323237 0.946318i $$-0.604771\pi$$
−0.323237 + 0.946318i $$0.604771\pi$$
$$132$$ 0 0
$$133$$ 8.83792 0.766345
$$134$$ 0 0
$$135$$ −5.19079 −0.446752
$$136$$ 0 0
$$137$$ 18.4830 1.57911 0.789553 0.613683i $$-0.210313\pi$$
0.789553 + 0.613683i $$0.210313\pi$$
$$138$$ 0 0
$$139$$ −17.8424 −1.51337 −0.756687 0.653778i $$-0.773183\pi$$
−0.756687 + 0.653778i $$0.773183\pi$$
$$140$$ 0 0
$$141$$ −2.76817 −0.233122
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −2.69960 −0.224190
$$146$$ 0 0
$$147$$ 9.41797 0.776781
$$148$$ 0 0
$$149$$ −7.39910 −0.606157 −0.303079 0.952966i $$-0.598015\pi$$
−0.303079 + 0.952966i $$0.598015\pi$$
$$150$$ 0 0
$$151$$ 4.41515 0.359300 0.179650 0.983731i $$-0.442504\pi$$
0.179650 + 0.983731i $$0.442504\pi$$
$$152$$ 0 0
$$153$$ −0.0222733 −0.00180069
$$154$$ 0 0
$$155$$ 2.83345 0.227589
$$156$$ 0 0
$$157$$ −2.33057 −0.186000 −0.0929998 0.995666i $$-0.529646\pi$$
−0.0929998 + 0.995666i $$0.529646\pi$$
$$158$$ 0 0
$$159$$ −3.07570 −0.243919
$$160$$ 0 0
$$161$$ 2.76417 0.217847
$$162$$ 0 0
$$163$$ 1.50615 0.117971 0.0589855 0.998259i $$-0.481213\pi$$
0.0589855 + 0.998259i $$0.481213\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 9.79732 0.758139 0.379070 0.925368i $$-0.376244\pi$$
0.379070 + 0.925368i $$0.376244\pi$$
$$168$$ 0 0
$$169$$ −6.71111 −0.516239
$$170$$ 0 0
$$171$$ −0.0436160 −0.00333540
$$172$$ 0 0
$$173$$ −20.0936 −1.52769 −0.763845 0.645400i $$-0.776691\pi$$
−0.763845 + 0.645400i $$0.776691\pi$$
$$174$$ 0 0
$$175$$ −1.25225 −0.0946610
$$176$$ 0 0
$$177$$ 7.57490 0.569364
$$178$$ 0 0
$$179$$ 16.2049 1.21121 0.605606 0.795764i $$-0.292931\pi$$
0.605606 + 0.795764i $$0.292931\pi$$
$$180$$ 0 0
$$181$$ −9.42873 −0.700832 −0.350416 0.936594i $$-0.613960\pi$$
−0.350416 + 0.936594i $$0.613960\pi$$
$$182$$ 0 0
$$183$$ 14.4485 1.06806
$$184$$ 0 0
$$185$$ 8.07294 0.593534
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −6.50015 −0.472816
$$190$$ 0 0
$$191$$ 8.17556 0.591563 0.295781 0.955256i $$-0.404420\pi$$
0.295781 + 0.955256i $$0.404420\pi$$
$$192$$ 0 0
$$193$$ −4.99518 −0.359561 −0.179781 0.983707i $$-0.557539\pi$$
−0.179781 + 0.983707i $$0.557539\pi$$
$$194$$ 0 0
$$195$$ −4.34805 −0.311370
$$196$$ 0 0
$$197$$ −7.00790 −0.499292 −0.249646 0.968337i $$-0.580314\pi$$
−0.249646 + 0.968337i $$0.580314\pi$$
$$198$$ 0 0
$$199$$ 18.8762 1.33810 0.669050 0.743217i $$-0.266701\pi$$
0.669050 + 0.743217i $$0.266701\pi$$
$$200$$ 0 0
$$201$$ −9.41979 −0.664421
$$202$$ 0 0
$$203$$ −3.38057 −0.237269
$$204$$ 0 0
$$205$$ 10.3266 0.721240
$$206$$ 0 0
$$207$$ −0.0136414 −0.000948145 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 13.7688 0.947881 0.473940 0.880557i $$-0.342831\pi$$
0.473940 + 0.880557i $$0.342831\pi$$
$$212$$ 0 0
$$213$$ 8.03916 0.550834
$$214$$ 0 0
$$215$$ −6.88581 −0.469608
$$216$$ 0 0
$$217$$ 3.54819 0.240867
$$218$$ 0 0
$$219$$ 3.52790 0.238394
$$220$$ 0 0
$$221$$ 9.03828 0.607980
$$222$$ 0 0
$$223$$ 28.7976 1.92843 0.964214 0.265125i $$-0.0854133\pi$$
0.964214 + 0.265125i $$0.0854133\pi$$
$$224$$ 0 0
$$225$$ 0.00617996 0.000411997 0
$$226$$ 0 0
$$227$$ 24.2769 1.61131 0.805656 0.592383i $$-0.201813\pi$$
0.805656 + 0.592383i $$0.201813\pi$$
$$228$$ 0 0
$$229$$ −16.3933 −1.08330 −0.541650 0.840604i $$-0.682200\pi$$
−0.541650 + 0.840604i $$0.682200\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.51844 −0.164988 −0.0824941 0.996592i $$-0.526289\pi$$
−0.0824941 + 0.996592i $$0.526289\pi$$
$$234$$ 0 0
$$235$$ −1.59656 −0.104148
$$236$$ 0 0
$$237$$ −29.0635 −1.88787
$$238$$ 0 0
$$239$$ 6.67824 0.431979 0.215990 0.976396i $$-0.430702\pi$$
0.215990 + 0.976396i $$0.430702\pi$$
$$240$$ 0 0
$$241$$ −2.14310 −0.138049 −0.0690247 0.997615i $$-0.521989\pi$$
−0.0690247 + 0.997615i $$0.521989\pi$$
$$242$$ 0 0
$$243$$ 0.0642239 0.00411997
$$244$$ 0 0
$$245$$ 5.43188 0.347030
$$246$$ 0 0
$$247$$ 17.6989 1.12616
$$248$$ 0 0
$$249$$ 9.75388 0.618127
$$250$$ 0 0
$$251$$ 14.3906 0.908325 0.454162 0.890919i $$-0.349939\pi$$
0.454162 + 0.890919i $$0.349939\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −6.24894 −0.391324
$$256$$ 0 0
$$257$$ 24.4034 1.52224 0.761120 0.648612i $$-0.224650\pi$$
0.761120 + 0.648612i $$0.224650\pi$$
$$258$$ 0 0
$$259$$ 10.1093 0.628162
$$260$$ 0 0
$$261$$ 0.0166834 0.00103268
$$262$$ 0 0
$$263$$ 0.992957 0.0612283 0.0306142 0.999531i $$-0.490254\pi$$
0.0306142 + 0.999531i $$0.490254\pi$$
$$264$$ 0 0
$$265$$ −1.77393 −0.108972
$$266$$ 0 0
$$267$$ 8.19811 0.501716
$$268$$ 0 0
$$269$$ 30.4230 1.85493 0.927463 0.373915i $$-0.121985\pi$$
0.927463 + 0.373915i $$0.121985\pi$$
$$270$$ 0 0
$$271$$ 25.9425 1.57589 0.787946 0.615744i $$-0.211144\pi$$
0.787946 + 0.615744i $$0.211144\pi$$
$$272$$ 0 0
$$273$$ −5.44483 −0.329536
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −5.03851 −0.302735 −0.151367 0.988478i $$-0.548368\pi$$
−0.151367 + 0.988478i $$0.548368\pi$$
$$278$$ 0 0
$$279$$ −0.0175106 −0.00104833
$$280$$ 0 0
$$281$$ 20.7739 1.23927 0.619634 0.784891i $$-0.287281\pi$$
0.619634 + 0.784891i $$0.287281\pi$$
$$282$$ 0 0
$$283$$ −4.55629 −0.270843 −0.135422 0.990788i $$-0.543239\pi$$
−0.135422 + 0.990788i $$0.543239\pi$$
$$284$$ 0 0
$$285$$ −12.2368 −0.724845
$$286$$ 0 0
$$287$$ 12.9314 0.763319
$$288$$ 0 0
$$289$$ −4.01035 −0.235903
$$290$$ 0 0
$$291$$ −33.1063 −1.94073
$$292$$ 0 0
$$293$$ −2.39092 −0.139679 −0.0698394 0.997558i $$-0.522249\pi$$
−0.0698394 + 0.997558i $$0.522249\pi$$
$$294$$ 0 0
$$295$$ 4.36887 0.254366
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 5.53556 0.320130
$$300$$ 0 0
$$301$$ −8.62274 −0.497006
$$302$$ 0 0
$$303$$ 24.6642 1.41692
$$304$$ 0 0
$$305$$ 8.33324 0.477160
$$306$$ 0 0
$$307$$ 3.26648 0.186428 0.0932138 0.995646i $$-0.470286\pi$$
0.0932138 + 0.995646i $$0.470286\pi$$
$$308$$ 0 0
$$309$$ 27.5359 1.56646
$$310$$ 0 0
$$311$$ 2.73567 0.155125 0.0775627 0.996987i $$-0.475286\pi$$
0.0775627 + 0.996987i $$0.475286\pi$$
$$312$$ 0 0
$$313$$ −10.3586 −0.585501 −0.292750 0.956189i $$-0.594570\pi$$
−0.292750 + 0.956189i $$0.594570\pi$$
$$314$$ 0 0
$$315$$ 0.00773884 0.000436034 0
$$316$$ 0 0
$$317$$ 31.2560 1.75551 0.877755 0.479110i $$-0.159041\pi$$
0.877755 + 0.479110i $$0.159041\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 7.02090 0.391868
$$322$$ 0 0
$$323$$ 25.4366 1.41533
$$324$$ 0 0
$$325$$ −2.50777 −0.139106
$$326$$ 0 0
$$327$$ −10.2269 −0.565550
$$328$$ 0 0
$$329$$ −1.99929 −0.110224
$$330$$ 0 0
$$331$$ −29.7878 −1.63729 −0.818644 0.574302i $$-0.805274\pi$$
−0.818644 + 0.574302i $$0.805274\pi$$
$$332$$ 0 0
$$333$$ −0.0498904 −0.00273398
$$334$$ 0 0
$$335$$ −5.43293 −0.296832
$$336$$ 0 0
$$337$$ −29.8351 −1.62522 −0.812610 0.582808i $$-0.801954\pi$$
−0.812610 + 0.582808i $$0.801954\pi$$
$$338$$ 0 0
$$339$$ −5.91933 −0.321494
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 15.5678 0.840581
$$344$$ 0 0
$$345$$ −3.82721 −0.206050
$$346$$ 0 0
$$347$$ −22.2170 −1.19267 −0.596336 0.802735i $$-0.703377\pi$$
−0.596336 + 0.802735i $$0.703377\pi$$
$$348$$ 0 0
$$349$$ −19.2134 −1.02847 −0.514234 0.857650i $$-0.671924\pi$$
−0.514234 + 0.857650i $$0.671924\pi$$
$$350$$ 0 0
$$351$$ −13.0173 −0.694811
$$352$$ 0 0
$$353$$ 31.0691 1.65364 0.826821 0.562466i $$-0.190147\pi$$
0.826821 + 0.562466i $$0.190147\pi$$
$$354$$ 0 0
$$355$$ 4.63664 0.246087
$$356$$ 0 0
$$357$$ −7.82521 −0.414154
$$358$$ 0 0
$$359$$ −27.5365 −1.45332 −0.726660 0.686998i $$-0.758928\pi$$
−0.726660 + 0.686998i $$0.758928\pi$$
$$360$$ 0 0
$$361$$ 30.8104 1.62160
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2.03474 0.106503
$$366$$ 0 0
$$367$$ 23.7891 1.24178 0.620890 0.783898i $$-0.286771\pi$$
0.620890 + 0.783898i $$0.286771\pi$$
$$368$$ 0 0
$$369$$ −0.0638179 −0.00332223
$$370$$ 0 0
$$371$$ −2.22140 −0.115329
$$372$$ 0 0
$$373$$ −15.5085 −0.803000 −0.401500 0.915859i $$-0.631511\pi$$
−0.401500 + 0.915859i $$0.631511\pi$$
$$374$$ 0 0
$$375$$ 1.73383 0.0895348
$$376$$ 0 0
$$377$$ −6.76997 −0.348671
$$378$$ 0 0
$$379$$ −19.0461 −0.978335 −0.489167 0.872190i $$-0.662699\pi$$
−0.489167 + 0.872190i $$0.662699\pi$$
$$380$$ 0 0
$$381$$ −10.7360 −0.550023
$$382$$ 0 0
$$383$$ −15.8224 −0.808488 −0.404244 0.914651i $$-0.632465\pi$$
−0.404244 + 0.914651i $$0.632465\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0.0425540 0.00216314
$$388$$ 0 0
$$389$$ −16.2272 −0.822752 −0.411376 0.911466i $$-0.634952\pi$$
−0.411376 + 0.911466i $$0.634952\pi$$
$$390$$ 0 0
$$391$$ 7.95560 0.402332
$$392$$ 0 0
$$393$$ 12.8291 0.647140
$$394$$ 0 0
$$395$$ −16.7625 −0.843415
$$396$$ 0 0
$$397$$ 31.3002 1.57091 0.785455 0.618919i $$-0.212429\pi$$
0.785455 + 0.618919i $$0.212429\pi$$
$$398$$ 0 0
$$399$$ −15.3235 −0.767134
$$400$$ 0 0
$$401$$ −6.86096 −0.342620 −0.171310 0.985217i $$-0.554800\pi$$
−0.171310 + 0.985217i $$0.554800\pi$$
$$402$$ 0 0
$$403$$ 7.10564 0.353957
$$404$$ 0 0
$$405$$ 9.01850 0.448133
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −22.8291 −1.12882 −0.564412 0.825493i $$-0.690897\pi$$
−0.564412 + 0.825493i $$0.690897\pi$$
$$410$$ 0 0
$$411$$ −32.0464 −1.58073
$$412$$ 0 0
$$413$$ 5.47091 0.269206
$$414$$ 0 0
$$415$$ 5.62561 0.276150
$$416$$ 0 0
$$417$$ 30.9358 1.51493
$$418$$ 0 0
$$419$$ 5.98106 0.292194 0.146097 0.989270i $$-0.453329\pi$$
0.146097 + 0.989270i $$0.453329\pi$$
$$420$$ 0 0
$$421$$ 37.8255 1.84350 0.921751 0.387781i $$-0.126758\pi$$
0.921751 + 0.387781i $$0.126758\pi$$
$$422$$ 0 0
$$423$$ 0.00986669 0.000479735 0
$$424$$ 0 0
$$425$$ −3.60412 −0.174825
$$426$$ 0 0
$$427$$ 10.4353 0.504999
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.51913 0.314015 0.157008 0.987597i $$-0.449815\pi$$
0.157008 + 0.987597i $$0.449815\pi$$
$$432$$ 0 0
$$433$$ −29.0198 −1.39460 −0.697301 0.716778i $$-0.745616\pi$$
−0.697301 + 0.716778i $$0.745616\pi$$
$$434$$ 0 0
$$435$$ 4.68066 0.224421
$$436$$ 0 0
$$437$$ 15.5788 0.745236
$$438$$ 0 0
$$439$$ 12.2777 0.585984 0.292992 0.956115i $$-0.405349\pi$$
0.292992 + 0.956115i $$0.405349\pi$$
$$440$$ 0 0
$$441$$ −0.0335688 −0.00159851
$$442$$ 0 0
$$443$$ 8.75445 0.415936 0.207968 0.978136i $$-0.433315\pi$$
0.207968 + 0.978136i $$0.433315\pi$$
$$444$$ 0 0
$$445$$ 4.72832 0.224144
$$446$$ 0 0
$$447$$ 12.8288 0.606782
$$448$$ 0 0
$$449$$ −21.0104 −0.991543 −0.495771 0.868453i $$-0.665115\pi$$
−0.495771 + 0.868453i $$0.665115\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −7.65513 −0.359670
$$454$$ 0 0
$$455$$ −3.14034 −0.147222
$$456$$ 0 0
$$457$$ −13.8185 −0.646402 −0.323201 0.946330i $$-0.604759\pi$$
−0.323201 + 0.946330i $$0.604759\pi$$
$$458$$ 0 0
$$459$$ −18.7082 −0.873224
$$460$$ 0 0
$$461$$ −38.6522 −1.80021 −0.900106 0.435672i $$-0.856511\pi$$
−0.900106 + 0.435672i $$0.856511\pi$$
$$462$$ 0 0
$$463$$ −15.0206 −0.698066 −0.349033 0.937110i $$-0.613490\pi$$
−0.349033 + 0.937110i $$0.613490\pi$$
$$464$$ 0 0
$$465$$ −4.91274 −0.227823
$$466$$ 0 0
$$467$$ 11.7903 0.545591 0.272796 0.962072i $$-0.412052\pi$$
0.272796 + 0.962072i $$0.412052\pi$$
$$468$$ 0 0
$$469$$ −6.80337 −0.314150
$$470$$ 0 0
$$471$$ 4.04082 0.186191
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −7.05765 −0.323827
$$476$$ 0 0
$$477$$ 0.0109628 0.000501953 0
$$478$$ 0 0
$$479$$ 18.3651 0.839125 0.419562 0.907726i $$-0.362184\pi$$
0.419562 + 0.907726i $$0.362184\pi$$
$$480$$ 0 0
$$481$$ 20.2450 0.923094
$$482$$ 0 0
$$483$$ −4.79261 −0.218071
$$484$$ 0 0
$$485$$ −19.0943 −0.867026
$$486$$ 0 0
$$487$$ 22.3361 1.01214 0.506072 0.862492i $$-0.331097\pi$$
0.506072 + 0.862492i $$0.331097\pi$$
$$488$$ 0 0
$$489$$ −2.61142 −0.118092
$$490$$ 0 0
$$491$$ 8.71330 0.393226 0.196613 0.980481i $$-0.437006\pi$$
0.196613 + 0.980481i $$0.437006\pi$$
$$492$$ 0 0
$$493$$ −9.72968 −0.438203
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 5.80621 0.260444
$$498$$ 0 0
$$499$$ 38.9860 1.74525 0.872626 0.488390i $$-0.162415\pi$$
0.872626 + 0.488390i $$0.162415\pi$$
$$500$$ 0 0
$$501$$ −16.9869 −0.758920
$$502$$ 0 0
$$503$$ 12.7798 0.569821 0.284910 0.958554i $$-0.408036\pi$$
0.284910 + 0.958554i $$0.408036\pi$$
$$504$$ 0 0
$$505$$ 14.2252 0.633015
$$506$$ 0 0
$$507$$ 11.6360 0.516771
$$508$$ 0 0
$$509$$ 36.9761 1.63894 0.819469 0.573124i $$-0.194269\pi$$
0.819469 + 0.573124i $$0.194269\pi$$
$$510$$ 0 0
$$511$$ 2.54800 0.112717
$$512$$ 0 0
$$513$$ −36.6348 −1.61746
$$514$$ 0 0
$$515$$ 15.8815 0.699823
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 34.8390 1.52926
$$520$$ 0 0
$$521$$ −8.97119 −0.393035 −0.196518 0.980500i $$-0.562963\pi$$
−0.196518 + 0.980500i $$0.562963\pi$$
$$522$$ 0 0
$$523$$ 3.09464 0.135319 0.0676596 0.997708i $$-0.478447\pi$$
0.0676596 + 0.997708i $$0.478447\pi$$
$$524$$ 0 0
$$525$$ 2.17119 0.0947584
$$526$$ 0 0
$$527$$ 10.2121 0.444846
$$528$$ 0 0
$$529$$ −18.1275 −0.788154
$$530$$ 0 0
$$531$$ −0.0269994 −0.00117168
$$532$$ 0 0
$$533$$ 25.8967 1.12171
$$534$$ 0 0
$$535$$ 4.04935 0.175069
$$536$$ 0 0
$$537$$ −28.0966 −1.21246
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.8882 −0.812065 −0.406033 0.913859i $$-0.633088\pi$$
−0.406033 + 0.913859i $$0.633088\pi$$
$$542$$ 0 0
$$543$$ 16.3478 0.701553
$$544$$ 0 0
$$545$$ −5.89844 −0.252661
$$546$$ 0 0
$$547$$ −34.0440 −1.45562 −0.727808 0.685781i $$-0.759461\pi$$
−0.727808 + 0.685781i $$0.759461\pi$$
$$548$$ 0 0
$$549$$ −0.0514991 −0.00219793
$$550$$ 0 0
$$551$$ −19.0528 −0.811678
$$552$$ 0 0
$$553$$ −20.9908 −0.892622
$$554$$ 0 0
$$555$$ −13.9971 −0.594145
$$556$$ 0 0
$$557$$ 19.8622 0.841589 0.420795 0.907156i $$-0.361751\pi$$
0.420795 + 0.907156i $$0.361751\pi$$
$$558$$ 0 0
$$559$$ −17.2680 −0.730359
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −23.7516 −1.00101 −0.500505 0.865734i $$-0.666852\pi$$
−0.500505 + 0.865734i $$0.666852\pi$$
$$564$$ 0 0
$$565$$ −3.41401 −0.143628
$$566$$ 0 0
$$567$$ 11.2934 0.474278
$$568$$ 0 0
$$569$$ 36.4083 1.52632 0.763158 0.646211i $$-0.223648\pi$$
0.763158 + 0.646211i $$0.223648\pi$$
$$570$$ 0 0
$$571$$ −1.49673 −0.0626362 −0.0313181 0.999509i $$-0.509970\pi$$
−0.0313181 + 0.999509i $$0.509970\pi$$
$$572$$ 0 0
$$573$$ −14.1751 −0.592172
$$574$$ 0 0
$$575$$ −2.20737 −0.0920536
$$576$$ 0 0
$$577$$ 41.2565 1.71753 0.858766 0.512368i $$-0.171232\pi$$
0.858766 + 0.512368i $$0.171232\pi$$
$$578$$ 0 0
$$579$$ 8.66082 0.359931
$$580$$ 0 0
$$581$$ 7.04466 0.292262
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0.0154979 0.000640759 0
$$586$$ 0 0
$$587$$ 41.6959 1.72097 0.860486 0.509474i $$-0.170160\pi$$
0.860486 + 0.509474i $$0.170160\pi$$
$$588$$ 0 0
$$589$$ 19.9975 0.823984
$$590$$ 0 0
$$591$$ 12.1505 0.499806
$$592$$ 0 0
$$593$$ 16.0656 0.659736 0.329868 0.944027i $$-0.392996\pi$$
0.329868 + 0.944027i $$0.392996\pi$$
$$594$$ 0 0
$$595$$ −4.51324 −0.185025
$$596$$ 0 0
$$597$$ −32.7282 −1.33948
$$598$$ 0 0
$$599$$ −25.4316 −1.03911 −0.519553 0.854438i $$-0.673902\pi$$
−0.519553 + 0.854438i $$0.673902\pi$$
$$600$$ 0 0
$$601$$ 13.4615 0.549104 0.274552 0.961572i $$-0.411470\pi$$
0.274552 + 0.961572i $$0.411470\pi$$
$$602$$ 0 0
$$603$$ 0.0335753 0.00136729
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −2.05778 −0.0835226 −0.0417613 0.999128i $$-0.513297\pi$$
−0.0417613 + 0.999128i $$0.513297\pi$$
$$608$$ 0 0
$$609$$ 5.86134 0.237514
$$610$$ 0 0
$$611$$ −4.00380 −0.161977
$$612$$ 0 0
$$613$$ −38.3437 −1.54869 −0.774344 0.632765i $$-0.781920\pi$$
−0.774344 + 0.632765i $$0.781920\pi$$
$$614$$ 0 0
$$615$$ −17.9046 −0.721982
$$616$$ 0 0
$$617$$ 7.25724 0.292166 0.146083 0.989272i $$-0.453333\pi$$
0.146083 + 0.989272i $$0.453333\pi$$
$$618$$ 0 0
$$619$$ −17.6113 −0.707860 −0.353930 0.935272i $$-0.615155\pi$$
−0.353930 + 0.935272i $$0.615155\pi$$
$$620$$ 0 0
$$621$$ −11.4580 −0.459793
$$622$$ 0 0
$$623$$ 5.92102 0.237221
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 29.0958 1.16013
$$630$$ 0 0
$$631$$ −28.4353 −1.13199 −0.565996 0.824408i $$-0.691508\pi$$
−0.565996 + 0.824408i $$0.691508\pi$$
$$632$$ 0 0
$$633$$ −23.8727 −0.948857
$$634$$ 0 0
$$635$$ −6.19208 −0.245725
$$636$$ 0 0
$$637$$ 13.6219 0.539718
$$638$$ 0 0
$$639$$ −0.0286542 −0.00113354
$$640$$ 0 0
$$641$$ −42.5129 −1.67916 −0.839579 0.543237i $$-0.817199\pi$$
−0.839579 + 0.543237i $$0.817199\pi$$
$$642$$ 0 0
$$643$$ −22.9742 −0.906015 −0.453007 0.891507i $$-0.649649\pi$$
−0.453007 + 0.891507i $$0.649649\pi$$
$$644$$ 0 0
$$645$$ 11.9389 0.470092
$$646$$ 0 0
$$647$$ −14.2632 −0.560746 −0.280373 0.959891i $$-0.590458\pi$$
−0.280373 + 0.959891i $$0.590458\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −6.15196 −0.241115
$$652$$ 0 0
$$653$$ −26.8378 −1.05024 −0.525122 0.851027i $$-0.675980\pi$$
−0.525122 + 0.851027i $$0.675980\pi$$
$$654$$ 0 0
$$655$$ 7.39924 0.289112
$$656$$ 0 0
$$657$$ −0.0125746 −0.000490582 0
$$658$$ 0 0
$$659$$ 28.8747 1.12480 0.562400 0.826865i $$-0.309878\pi$$
0.562400 + 0.826865i $$0.309878\pi$$
$$660$$ 0 0
$$661$$ −23.6938 −0.921582 −0.460791 0.887509i $$-0.652434\pi$$
−0.460791 + 0.887509i $$0.652434\pi$$
$$662$$ 0 0
$$663$$ −15.6709 −0.608606
$$664$$ 0 0
$$665$$ −8.83792 −0.342720
$$666$$ 0 0
$$667$$ −5.95901 −0.230734
$$668$$ 0 0
$$669$$ −49.9302 −1.93041
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −5.90541 −0.227637 −0.113818 0.993502i $$-0.536308\pi$$
−0.113818 + 0.993502i $$0.536308\pi$$
$$674$$ 0 0
$$675$$ 5.19079 0.199793
$$676$$ 0 0
$$677$$ 30.4595 1.17065 0.585327 0.810797i $$-0.300966\pi$$
0.585327 + 0.810797i $$0.300966\pi$$
$$678$$ 0 0
$$679$$ −23.9108 −0.917610
$$680$$ 0 0
$$681$$ −42.0921 −1.61297
$$682$$ 0 0
$$683$$ −2.95068 −0.112905 −0.0564524 0.998405i $$-0.517979\pi$$
−0.0564524 + 0.998405i $$0.517979\pi$$
$$684$$ 0 0
$$685$$ −18.4830 −0.706198
$$686$$ 0 0
$$687$$ 28.4233 1.08442
$$688$$ 0 0
$$689$$ −4.44860 −0.169478
$$690$$ 0 0
$$691$$ −38.6147 −1.46897 −0.734487 0.678623i $$-0.762577\pi$$
−0.734487 + 0.678623i $$0.762577\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 17.8424 0.676801
$$696$$ 0 0
$$697$$ 37.2182 1.40974
$$698$$ 0 0
$$699$$ 4.36655 0.165158
$$700$$ 0 0
$$701$$ 33.1129 1.25066 0.625328 0.780362i $$-0.284965\pi$$
0.625328 + 0.780362i $$0.284965\pi$$
$$702$$ 0 0
$$703$$ 56.9760 2.14889
$$704$$ 0 0
$$705$$ 2.76817 0.104255
$$706$$ 0 0
$$707$$ 17.8135 0.669946
$$708$$ 0 0
$$709$$ 21.9947 0.826030 0.413015 0.910724i $$-0.364476\pi$$
0.413015 + 0.910724i $$0.364476\pi$$
$$710$$ 0 0
$$711$$ 0.103592 0.00388500
$$712$$ 0 0
$$713$$ 6.25447 0.234232
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −11.5790 −0.432424
$$718$$ 0 0
$$719$$ −5.99121 −0.223434 −0.111717 0.993740i $$-0.535635\pi$$
−0.111717 + 0.993740i $$0.535635\pi$$
$$720$$ 0 0
$$721$$ 19.8876 0.740652
$$722$$ 0 0
$$723$$ 3.71579 0.138192
$$724$$ 0 0
$$725$$ 2.69960 0.100261
$$726$$ 0 0
$$727$$ −4.06512 −0.150767 −0.0753834 0.997155i $$-0.524018\pi$$
−0.0753834 + 0.997155i $$0.524018\pi$$
$$728$$ 0 0
$$729$$ 26.9442 0.997932
$$730$$ 0 0
$$731$$ −24.8173 −0.917899
$$732$$ 0 0
$$733$$ −29.9274 −1.10539 −0.552697 0.833382i $$-0.686401\pi$$
−0.552697 + 0.833382i $$0.686401\pi$$
$$734$$ 0 0
$$735$$ −9.41797 −0.347387
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −19.6749 −0.723755 −0.361877 0.932226i $$-0.617864\pi$$
−0.361877 + 0.932226i $$0.617864\pi$$
$$740$$ 0 0
$$741$$ −30.6870 −1.12732
$$742$$ 0 0
$$743$$ −26.6733 −0.978548 −0.489274 0.872130i $$-0.662738\pi$$
−0.489274 + 0.872130i $$0.662738\pi$$
$$744$$ 0 0
$$745$$ 7.39910 0.271082
$$746$$ 0 0
$$747$$ −0.0347660 −0.00127202
$$748$$ 0 0
$$749$$ 5.07079 0.185282
$$750$$ 0 0
$$751$$ 7.86537 0.287011 0.143506 0.989649i $$-0.454162\pi$$
0.143506 + 0.989649i $$0.454162\pi$$
$$752$$ 0 0
$$753$$ −24.9509 −0.909260
$$754$$ 0 0
$$755$$ −4.41515 −0.160684
$$756$$ 0 0
$$757$$ −24.4559 −0.888866 −0.444433 0.895812i $$-0.646595\pi$$
−0.444433 + 0.895812i $$0.646595\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −52.2028 −1.89235 −0.946174 0.323658i $$-0.895087\pi$$
−0.946174 + 0.323658i $$0.895087\pi$$
$$762$$ 0 0
$$763$$ −7.38631 −0.267402
$$764$$ 0 0
$$765$$ 0.0222733 0.000805292 0
$$766$$ 0 0
$$767$$ 10.9561 0.395602
$$768$$ 0 0
$$769$$ 26.7480 0.964558 0.482279 0.876018i $$-0.339809\pi$$
0.482279 + 0.876018i $$0.339809\pi$$
$$770$$ 0 0
$$771$$ −42.3114 −1.52381
$$772$$ 0 0
$$773$$ −50.2424 −1.80709 −0.903546 0.428490i $$-0.859046\pi$$
−0.903546 + 0.428490i $$0.859046\pi$$
$$774$$ 0 0
$$775$$ −2.83345 −0.101781
$$776$$ 0 0
$$777$$ −17.5279 −0.628809
$$778$$ 0 0
$$779$$ 72.8814 2.61125
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 14.0131 0.500786
$$784$$ 0 0
$$785$$ 2.33057 0.0831816
$$786$$ 0 0
$$787$$ −32.3414 −1.15285 −0.576423 0.817152i $$-0.695552\pi$$
−0.576423 + 0.817152i $$0.695552\pi$$
$$788$$ 0 0
$$789$$ −1.72162 −0.0612914
$$790$$ 0 0
$$791$$ −4.27518 −0.152008
$$792$$ 0 0
$$793$$ 20.8978 0.742103
$$794$$ 0 0
$$795$$ 3.07570 0.109084
$$796$$ 0 0
$$797$$ 38.6252 1.36817 0.684087 0.729400i $$-0.260201\pi$$
0.684087 + 0.729400i $$0.260201\pi$$
$$798$$ 0 0
$$799$$ −5.75420 −0.203569
$$800$$ 0 0
$$801$$ −0.0292208 −0.00103247
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −2.76417 −0.0974242
$$806$$ 0 0
$$807$$ −52.7485 −1.85684
$$808$$ 0 0
$$809$$ −29.6195 −1.04137 −0.520684 0.853750i $$-0.674323\pi$$
−0.520684 + 0.853750i $$0.674323\pi$$
$$810$$ 0 0
$$811$$ −14.6182 −0.513315 −0.256657 0.966502i $$-0.582621\pi$$
−0.256657 + 0.966502i $$0.582621\pi$$
$$812$$ 0 0
$$813$$ −44.9799 −1.57751
$$814$$ 0 0
$$815$$ −1.50615 −0.0527583
$$816$$ 0 0
$$817$$ −48.5976 −1.70022
$$818$$ 0 0
$$819$$ 0.0194072 0.000678142 0
$$820$$ 0 0
$$821$$ 27.9890 0.976822 0.488411 0.872614i $$-0.337577\pi$$
0.488411 + 0.872614i $$0.337577\pi$$
$$822$$ 0 0
$$823$$ 17.6590 0.615556 0.307778 0.951458i $$-0.400415\pi$$
0.307778 + 0.951458i $$0.400415\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −3.87712 −0.134821 −0.0674103 0.997725i $$-0.521474\pi$$
−0.0674103 + 0.997725i $$0.521474\pi$$
$$828$$ 0 0
$$829$$ −20.8274 −0.723366 −0.361683 0.932301i $$-0.617798\pi$$
−0.361683 + 0.932301i $$0.617798\pi$$
$$830$$ 0 0
$$831$$ 8.73593 0.303046
$$832$$ 0 0
$$833$$ 19.5771 0.678307
$$834$$ 0 0
$$835$$ −9.79732 −0.339050
$$836$$ 0 0
$$837$$ −14.7079 −0.508378
$$838$$ 0 0
$$839$$ −29.2063 −1.00831 −0.504157 0.863612i $$-0.668197\pi$$
−0.504157 + 0.863612i $$0.668197\pi$$
$$840$$ 0 0
$$841$$ −21.7122 −0.748695
$$842$$ 0 0
$$843$$ −36.0185 −1.24054
$$844$$ 0 0
$$845$$ 6.71111 0.230869
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 7.89985 0.271122
$$850$$ 0 0
$$851$$ 17.8199 0.610860
$$852$$ 0 0
$$853$$ 57.3270 1.96284 0.981420 0.191870i $$-0.0614553\pi$$
0.981420 + 0.191870i $$0.0614553\pi$$
$$854$$ 0 0
$$855$$ 0.0436160 0.00149163
$$856$$ 0 0
$$857$$ 43.7938 1.49597 0.747984 0.663717i $$-0.231022\pi$$
0.747984 + 0.663717i $$0.231022\pi$$
$$858$$ 0 0
$$859$$ −32.0382 −1.09313 −0.546564 0.837417i $$-0.684065\pi$$
−0.546564 + 0.837417i $$0.684065\pi$$
$$860$$ 0 0
$$861$$ −22.4210 −0.764105
$$862$$ 0 0
$$863$$ 50.5619 1.72115 0.860574 0.509325i $$-0.170105\pi$$
0.860574 + 0.509325i $$0.170105\pi$$
$$864$$ 0 0
$$865$$ 20.0936 0.683204
$$866$$ 0 0
$$867$$ 6.95329 0.236146
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −13.6245 −0.461649
$$872$$ 0 0
$$873$$ 0.118002 0.00399376
$$874$$ 0 0
$$875$$ 1.25225 0.0423337
$$876$$ 0 0
$$877$$ 19.9976 0.675270 0.337635 0.941277i $$-0.390373\pi$$
0.337635 + 0.941277i $$0.390373\pi$$
$$878$$ 0 0
$$879$$ 4.14545 0.139823
$$880$$ 0 0
$$881$$ −19.7712 −0.666108 −0.333054 0.942908i $$-0.608079\pi$$
−0.333054 + 0.942908i $$0.608079\pi$$
$$882$$ 0 0
$$883$$ −8.72958 −0.293774 −0.146887 0.989153i $$-0.546925\pi$$
−0.146887 + 0.989153i $$0.546925\pi$$
$$884$$ 0 0
$$885$$ −7.57490 −0.254627
$$886$$ 0 0
$$887$$ −50.1444 −1.68368 −0.841842 0.539724i $$-0.818529\pi$$
−0.841842 + 0.539724i $$0.818529\pi$$
$$888$$ 0 0
$$889$$ −7.75401 −0.260061
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −11.2680 −0.377069
$$894$$ 0 0
$$895$$ −16.2049 −0.541671
$$896$$ 0 0
$$897$$ −9.59774 −0.320459
$$898$$ 0 0
$$899$$ −7.64920 −0.255115
$$900$$ 0 0
$$901$$ −6.39345 −0.212997
$$902$$ 0 0
$$903$$ 14.9504 0.497518
$$904$$ 0 0
$$905$$ 9.42873 0.313422
$$906$$ 0 0
$$907$$ 21.5643 0.716031 0.358015 0.933716i $$-0.383453\pi$$
0.358015 + 0.933716i $$0.383453\pi$$
$$908$$ 0 0
$$909$$ −0.0879113 −0.00291584
$$910$$ 0 0
$$911$$ 25.9189 0.858731 0.429365 0.903131i $$-0.358737\pi$$
0.429365 + 0.903131i $$0.358737\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −14.4485 −0.477651
$$916$$ 0 0
$$917$$ 9.26568 0.305980
$$918$$ 0 0
$$919$$ 13.1654 0.434286 0.217143 0.976140i $$-0.430326\pi$$
0.217143 + 0.976140i $$0.430326\pi$$
$$920$$ 0 0
$$921$$ −5.66353 −0.186620
$$922$$ 0 0
$$923$$ 11.6276 0.382727
$$924$$ 0 0
$$925$$ −8.07294 −0.265437
$$926$$ 0 0
$$927$$ −0.0981471 −0.00322357
$$928$$ 0 0
$$929$$ 40.2771 1.32145 0.660725 0.750628i $$-0.270249\pi$$
0.660725 + 0.750628i $$0.270249\pi$$
$$930$$ 0 0
$$931$$ 38.3363 1.25642
$$932$$ 0 0
$$933$$ −4.74319 −0.155285
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 37.3771 1.22106 0.610528 0.791995i $$-0.290957\pi$$
0.610528 + 0.791995i $$0.290957\pi$$
$$938$$ 0 0
$$939$$ 17.9600 0.586103
$$940$$ 0 0
$$941$$ −38.2224 −1.24601 −0.623007 0.782216i $$-0.714089\pi$$
−0.623007 + 0.782216i $$0.714089\pi$$
$$942$$ 0 0
$$943$$ 22.7946 0.742293
$$944$$ 0 0
$$945$$ 6.50015 0.211450
$$946$$ 0 0
$$947$$ −58.9505 −1.91564 −0.957818 0.287377i $$-0.907217\pi$$
−0.957818 + 0.287377i $$0.907217\pi$$
$$948$$ 0 0
$$949$$ 5.10265 0.165639
$$950$$ 0 0
$$951$$ −54.1926 −1.75732
$$952$$ 0 0
$$953$$ 20.6255 0.668126 0.334063 0.942551i $$-0.391580\pi$$
0.334063 + 0.942551i $$0.391580\pi$$
$$954$$ 0 0
$$955$$ −8.17556 −0.264555
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −23.1452 −0.747399
$$960$$ 0 0
$$961$$ −22.9715 −0.741017
$$962$$ 0 0
$$963$$ −0.0250248 −0.000806413 0
$$964$$ 0 0
$$965$$ 4.99518 0.160801
$$966$$ 0 0
$$967$$ −35.8174 −1.15181 −0.575904 0.817517i $$-0.695350\pi$$
−0.575904 + 0.817517i $$0.695350\pi$$
$$968$$ 0 0
$$969$$ −44.1028 −1.41679
$$970$$ 0 0
$$971$$ 11.3094 0.362937 0.181469 0.983397i $$-0.441915\pi$$
0.181469 + 0.983397i $$0.441915\pi$$
$$972$$ 0 0
$$973$$ 22.3431 0.716287
$$974$$ 0 0
$$975$$ 4.34805 0.139249
$$976$$ 0 0
$$977$$ 23.2137 0.742671 0.371336 0.928499i $$-0.378900\pi$$
0.371336 + 0.928499i $$0.378900\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0.0364521 0.00116383
$$982$$ 0 0
$$983$$ 34.1621 1.08960 0.544801 0.838565i $$-0.316605\pi$$
0.544801 + 0.838565i $$0.316605\pi$$
$$984$$ 0 0
$$985$$ 7.00790 0.223290
$$986$$ 0 0
$$987$$ 3.46644 0.110338
$$988$$ 0 0
$$989$$ −15.1995 −0.483316
$$990$$ 0 0
$$991$$ −27.7516 −0.881560 −0.440780 0.897615i $$-0.645298\pi$$
−0.440780 + 0.897615i $$0.645298\pi$$
$$992$$ 0 0
$$993$$ 51.6472 1.63897
$$994$$ 0 0
$$995$$ −18.8762 −0.598416
$$996$$ 0 0
$$997$$ −43.1978 −1.36809 −0.684044 0.729440i $$-0.739781\pi$$
−0.684044 + 0.729440i $$0.739781\pi$$
$$998$$ 0 0
$$999$$ −41.9049 −1.32581
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 4840.2.a.bf.1.2 6
4.3 odd 2 9680.2.a.cx.1.5 6
11.5 even 5 440.2.y.b.201.1 yes 12
11.9 even 5 440.2.y.b.81.1 12
11.10 odd 2 4840.2.a.be.1.2 6
44.27 odd 10 880.2.bo.j.641.3 12
44.31 odd 10 880.2.bo.j.81.3 12
44.43 even 2 9680.2.a.cy.1.5 6

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.81.1 12 11.9 even 5
440.2.y.b.201.1 yes 12 11.5 even 5
880.2.bo.j.81.3 12 44.31 odd 10
880.2.bo.j.641.3 12 44.27 odd 10
4840.2.a.be.1.2 6 11.10 odd 2
4840.2.a.bf.1.2 6 1.1 even 1 trivial
9680.2.a.cx.1.5 6 4.3 odd 2
9680.2.a.cy.1.5 6 44.43 even 2