# Properties

 Label 4840.2.a.l.1.1 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ 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)

 $$f(q)$$ $$=$$ $$q-1.73205 q^{3} +1.00000 q^{5} +0.267949 q^{7} +O(q^{10})$$ $$q-1.73205 q^{3} +1.00000 q^{5} +0.267949 q^{7} +3.46410 q^{13} -1.73205 q^{15} +2.00000 q^{19} -0.464102 q^{21} +7.46410 q^{23} +1.00000 q^{25} +5.19615 q^{27} -4.92820 q^{29} -1.46410 q^{31} +0.267949 q^{35} -4.00000 q^{37} -6.00000 q^{39} -1.92820 q^{41} +1.73205 q^{43} +6.66025 q^{47} -6.92820 q^{49} -7.46410 q^{53} -3.46410 q^{57} -1.46410 q^{59} +1.53590 q^{61} +3.46410 q^{65} +5.73205 q^{67} -12.9282 q^{69} +2.53590 q^{71} +1.07180 q^{73} -1.73205 q^{75} +14.3923 q^{79} -9.00000 q^{81} -7.46410 q^{83} +8.53590 q^{87} +15.9282 q^{89} +0.928203 q^{91} +2.53590 q^{93} +2.00000 q^{95} -14.3923 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + 4 * q^7 $$2 q + 2 q^{5} + 4 q^{7} + 4 q^{19} + 6 q^{21} + 8 q^{23} + 2 q^{25} + 4 q^{29} + 4 q^{31} + 4 q^{35} - 8 q^{37} - 12 q^{39} + 10 q^{41} - 4 q^{47} - 8 q^{53} + 4 q^{59} + 10 q^{61} + 8 q^{67} - 12 q^{69} + 12 q^{71} + 16 q^{73} + 8 q^{79} - 18 q^{81} - 8 q^{83} + 24 q^{87} + 18 q^{89} - 12 q^{91} + 12 q^{93} + 4 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 4 * q^7 + 4 * q^19 + 6 * q^21 + 8 * q^23 + 2 * q^25 + 4 * q^29 + 4 * q^31 + 4 * q^35 - 8 * q^37 - 12 * q^39 + 10 * q^41 - 4 * q^47 - 8 * q^53 + 4 * q^59 + 10 * q^61 + 8 * q^67 - 12 * q^69 + 12 * q^71 + 16 * q^73 + 8 * q^79 - 18 * q^81 - 8 * q^83 + 24 * q^87 + 18 * q^89 - 12 * q^91 + 12 * q^93 + 4 * q^95 - 8 * 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
$$3$$ −1.73205 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0.267949 0.101275 0.0506376 0.998717i $$-0.483875\pi$$
0.0506376 + 0.998717i $$0.483875\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 3.46410 0.960769 0.480384 0.877058i $$-0.340497\pi$$
0.480384 + 0.877058i $$0.340497\pi$$
$$14$$ 0 0
$$15$$ −1.73205 −0.447214
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −0.464102 −0.101275
$$22$$ 0 0
$$23$$ 7.46410 1.55637 0.778186 0.628033i $$-0.216140\pi$$
0.778186 + 0.628033i $$0.216140\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.19615 1.00000
$$28$$ 0 0
$$29$$ −4.92820 −0.915144 −0.457572 0.889172i $$-0.651281\pi$$
−0.457572 + 0.889172i $$0.651281\pi$$
$$30$$ 0 0
$$31$$ −1.46410 −0.262960 −0.131480 0.991319i $$-0.541973\pi$$
−0.131480 + 0.991319i $$0.541973\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0.267949 0.0452917
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −1.92820 −0.301135 −0.150567 0.988600i $$-0.548110\pi$$
−0.150567 + 0.988600i $$0.548110\pi$$
$$42$$ 0 0
$$43$$ 1.73205 0.264135 0.132068 0.991241i $$-0.457838\pi$$
0.132068 + 0.991241i $$0.457838\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.66025 0.971498 0.485749 0.874098i $$-0.338547\pi$$
0.485749 + 0.874098i $$0.338547\pi$$
$$48$$ 0 0
$$49$$ −6.92820 −0.989743
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −7.46410 −1.02527 −0.512637 0.858606i $$-0.671331\pi$$
−0.512637 + 0.858606i $$0.671331\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −3.46410 −0.458831
$$58$$ 0 0
$$59$$ −1.46410 −0.190610 −0.0953049 0.995448i $$-0.530383\pi$$
−0.0953049 + 0.995448i $$0.530383\pi$$
$$60$$ 0 0
$$61$$ 1.53590 0.196652 0.0983258 0.995154i $$-0.468651\pi$$
0.0983258 + 0.995154i $$0.468651\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 3.46410 0.429669
$$66$$ 0 0
$$67$$ 5.73205 0.700281 0.350141 0.936697i $$-0.386134\pi$$
0.350141 + 0.936697i $$0.386134\pi$$
$$68$$ 0 0
$$69$$ −12.9282 −1.55637
$$70$$ 0 0
$$71$$ 2.53590 0.300956 0.150478 0.988613i $$-0.451919\pi$$
0.150478 + 0.988613i $$0.451919\pi$$
$$72$$ 0 0
$$73$$ 1.07180 0.125444 0.0627222 0.998031i $$-0.480022\pi$$
0.0627222 + 0.998031i $$0.480022\pi$$
$$74$$ 0 0
$$75$$ −1.73205 −0.200000
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 14.3923 1.61926 0.809630 0.586940i $$-0.199668\pi$$
0.809630 + 0.586940i $$0.199668\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ 0 0
$$83$$ −7.46410 −0.819292 −0.409646 0.912245i $$-0.634348\pi$$
−0.409646 + 0.912245i $$0.634348\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 8.53590 0.915144
$$88$$ 0 0
$$89$$ 15.9282 1.68839 0.844193 0.536039i $$-0.180080\pi$$
0.844193 + 0.536039i $$0.180080\pi$$
$$90$$ 0 0
$$91$$ 0.928203 0.0973021
$$92$$ 0 0
$$93$$ 2.53590 0.262960
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ −14.3923 −1.46132 −0.730659 0.682743i $$-0.760787\pi$$
−0.730659 + 0.682743i $$0.760787\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 11.3923 1.13358 0.566788 0.823863i $$-0.308186\pi$$
0.566788 + 0.823863i $$0.308186\pi$$
$$102$$ 0 0
$$103$$ −2.39230 −0.235721 −0.117860 0.993030i $$-0.537604\pi$$
−0.117860 + 0.993030i $$0.537604\pi$$
$$104$$ 0 0
$$105$$ −0.464102 −0.0452917
$$106$$ 0 0
$$107$$ 2.26795 0.219251 0.109625 0.993973i $$-0.465035\pi$$
0.109625 + 0.993973i $$0.465035\pi$$
$$108$$ 0 0
$$109$$ 7.53590 0.721808 0.360904 0.932603i $$-0.382468\pi$$
0.360904 + 0.932603i $$0.382468\pi$$
$$110$$ 0 0
$$111$$ 6.92820 0.657596
$$112$$ 0 0
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ 7.46410 0.696031
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 3.33975 0.301135
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −18.6603 −1.65583 −0.827915 0.560854i $$-0.810473\pi$$
−0.827915 + 0.560854i $$0.810473\pi$$
$$128$$ 0 0
$$129$$ −3.00000 −0.264135
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ 0.535898 0.0464683
$$134$$ 0 0
$$135$$ 5.19615 0.447214
$$136$$ 0 0
$$137$$ 10.3923 0.887875 0.443937 0.896058i $$-0.353581\pi$$
0.443937 + 0.896058i $$0.353581\pi$$
$$138$$ 0 0
$$139$$ 8.39230 0.711826 0.355913 0.934519i $$-0.384170\pi$$
0.355913 + 0.934519i $$0.384170\pi$$
$$140$$ 0 0
$$141$$ −11.5359 −0.971498
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −4.92820 −0.409265
$$146$$ 0 0
$$147$$ 12.0000 0.989743
$$148$$ 0 0
$$149$$ 6.46410 0.529560 0.264780 0.964309i $$-0.414701\pi$$
0.264780 + 0.964309i $$0.414701\pi$$
$$150$$ 0 0
$$151$$ 4.39230 0.357441 0.178720 0.983900i $$-0.442804\pi$$
0.178720 + 0.983900i $$0.442804\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1.46410 −0.117599
$$156$$ 0 0
$$157$$ −20.0000 −1.59617 −0.798087 0.602542i $$-0.794154\pi$$
−0.798087 + 0.602542i $$0.794154\pi$$
$$158$$ 0 0
$$159$$ 12.9282 1.02527
$$160$$ 0 0
$$161$$ 2.00000 0.157622
$$162$$ 0 0
$$163$$ 8.12436 0.636349 0.318174 0.948032i $$-0.396930\pi$$
0.318174 + 0.948032i $$0.396930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −20.5167 −1.58763 −0.793813 0.608161i $$-0.791907\pi$$
−0.793813 + 0.608161i $$0.791907\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 3.46410 0.263371 0.131685 0.991292i $$-0.457961\pi$$
0.131685 + 0.991292i $$0.457961\pi$$
$$174$$ 0 0
$$175$$ 0.267949 0.0202551
$$176$$ 0 0
$$177$$ 2.53590 0.190610
$$178$$ 0 0
$$179$$ 2.39230 0.178809 0.0894046 0.995995i $$-0.471504\pi$$
0.0894046 + 0.995995i $$0.471504\pi$$
$$180$$ 0 0
$$181$$ 17.3923 1.29276 0.646380 0.763016i $$-0.276282\pi$$
0.646380 + 0.763016i $$0.276282\pi$$
$$182$$ 0 0
$$183$$ −2.66025 −0.196652
$$184$$ 0 0
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 1.39230 0.101275
$$190$$ 0 0
$$191$$ −11.0718 −0.801127 −0.400564 0.916269i $$-0.631186\pi$$
−0.400564 + 0.916269i $$0.631186\pi$$
$$192$$ 0 0
$$193$$ 15.8564 1.14137 0.570685 0.821169i $$-0.306678\pi$$
0.570685 + 0.821169i $$0.306678\pi$$
$$194$$ 0 0
$$195$$ −6.00000 −0.429669
$$196$$ 0 0
$$197$$ 0.928203 0.0661317 0.0330659 0.999453i $$-0.489473\pi$$
0.0330659 + 0.999453i $$0.489473\pi$$
$$198$$ 0 0
$$199$$ 6.39230 0.453138 0.226569 0.973995i $$-0.427249\pi$$
0.226569 + 0.973995i $$0.427249\pi$$
$$200$$ 0 0
$$201$$ −9.92820 −0.700281
$$202$$ 0 0
$$203$$ −1.32051 −0.0926815
$$204$$ 0 0
$$205$$ −1.92820 −0.134672
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 6.53590 0.449950 0.224975 0.974365i $$-0.427770\pi$$
0.224975 + 0.974365i $$0.427770\pi$$
$$212$$ 0 0
$$213$$ −4.39230 −0.300956
$$214$$ 0 0
$$215$$ 1.73205 0.118125
$$216$$ 0 0
$$217$$ −0.392305 −0.0266314
$$218$$ 0 0
$$219$$ −1.85641 −0.125444
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0.267949 0.0179432 0.00897160 0.999960i $$-0.497144\pi$$
0.00897160 + 0.999960i $$0.497144\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −6.80385 −0.451587 −0.225794 0.974175i $$-0.572497\pi$$
−0.225794 + 0.974175i $$0.572497\pi$$
$$228$$ 0 0
$$229$$ −4.60770 −0.304485 −0.152243 0.988343i $$-0.548649\pi$$
−0.152243 + 0.988343i $$0.548649\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.3923 0.942871 0.471436 0.881900i $$-0.343736\pi$$
0.471436 + 0.881900i $$0.343736\pi$$
$$234$$ 0 0
$$235$$ 6.66025 0.434467
$$236$$ 0 0
$$237$$ −24.9282 −1.61926
$$238$$ 0 0
$$239$$ 14.0000 0.905585 0.452792 0.891616i $$-0.350428\pi$$
0.452792 + 0.891616i $$0.350428\pi$$
$$240$$ 0 0
$$241$$ 21.0000 1.35273 0.676364 0.736567i $$-0.263554\pi$$
0.676364 + 0.736567i $$0.263554\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −6.92820 −0.442627
$$246$$ 0 0
$$247$$ 6.92820 0.440831
$$248$$ 0 0
$$249$$ 12.9282 0.819292
$$250$$ 0 0
$$251$$ 10.5359 0.665020 0.332510 0.943100i $$-0.392104\pi$$
0.332510 + 0.943100i $$0.392104\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 26.2487 1.63735 0.818675 0.574257i $$-0.194709\pi$$
0.818675 + 0.574257i $$0.194709\pi$$
$$258$$ 0 0
$$259$$ −1.07180 −0.0665982
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 26.3923 1.62742 0.813710 0.581272i $$-0.197445\pi$$
0.813710 + 0.581272i $$0.197445\pi$$
$$264$$ 0 0
$$265$$ −7.46410 −0.458516
$$266$$ 0 0
$$267$$ −27.5885 −1.68839
$$268$$ 0 0
$$269$$ −19.2487 −1.17361 −0.586807 0.809727i $$-0.699615\pi$$
−0.586807 + 0.809727i $$0.699615\pi$$
$$270$$ 0 0
$$271$$ 18.9282 1.14981 0.574903 0.818221i $$-0.305040\pi$$
0.574903 + 0.818221i $$0.305040\pi$$
$$272$$ 0 0
$$273$$ −1.60770 −0.0973021
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 31.8564 1.90039 0.950197 0.311650i $$-0.100882\pi$$
0.950197 + 0.311650i $$0.100882\pi$$
$$282$$ 0 0
$$283$$ 18.5167 1.10070 0.550351 0.834934i $$-0.314494\pi$$
0.550351 + 0.834934i $$0.314494\pi$$
$$284$$ 0 0
$$285$$ −3.46410 −0.205196
$$286$$ 0 0
$$287$$ −0.516660 −0.0304975
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 24.9282 1.46132
$$292$$ 0 0
$$293$$ 30.2487 1.76715 0.883574 0.468291i $$-0.155130\pi$$
0.883574 + 0.468291i $$0.155130\pi$$
$$294$$ 0 0
$$295$$ −1.46410 −0.0852433
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 25.8564 1.49531
$$300$$ 0 0
$$301$$ 0.464102 0.0267504
$$302$$ 0 0
$$303$$ −19.7321 −1.13358
$$304$$ 0 0
$$305$$ 1.53590 0.0879453
$$306$$ 0 0
$$307$$ 26.3923 1.50629 0.753144 0.657855i $$-0.228536\pi$$
0.753144 + 0.657855i $$0.228536\pi$$
$$308$$ 0 0
$$309$$ 4.14359 0.235721
$$310$$ 0 0
$$311$$ 11.0718 0.627824 0.313912 0.949452i $$-0.398360\pi$$
0.313912 + 0.949452i $$0.398360\pi$$
$$312$$ 0 0
$$313$$ 3.46410 0.195803 0.0979013 0.995196i $$-0.468787\pi$$
0.0979013 + 0.995196i $$0.468787\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −19.8564 −1.11525 −0.557623 0.830094i $$-0.688287\pi$$
−0.557623 + 0.830094i $$0.688287\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −3.92820 −0.219251
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 3.46410 0.192154
$$326$$ 0 0
$$327$$ −13.0526 −0.721808
$$328$$ 0 0
$$329$$ 1.78461 0.0983887
$$330$$ 0 0
$$331$$ −20.9282 −1.15032 −0.575159 0.818042i $$-0.695060\pi$$
−0.575159 + 0.818042i $$0.695060\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 5.73205 0.313175
$$336$$ 0 0
$$337$$ −1.32051 −0.0719327 −0.0359663 0.999353i $$-0.511451\pi$$
−0.0359663 + 0.999353i $$0.511451\pi$$
$$338$$ 0 0
$$339$$ 17.3205 0.940721
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −3.73205 −0.201512
$$344$$ 0 0
$$345$$ −12.9282 −0.696031
$$346$$ 0 0
$$347$$ −5.73205 −0.307713 −0.153856 0.988093i $$-0.549169\pi$$
−0.153856 + 0.988093i $$0.549169\pi$$
$$348$$ 0 0
$$349$$ 29.7128 1.59049 0.795245 0.606288i $$-0.207342\pi$$
0.795245 + 0.606288i $$0.207342\pi$$
$$350$$ 0 0
$$351$$ 18.0000 0.960769
$$352$$ 0 0
$$353$$ 33.4641 1.78111 0.890557 0.454871i $$-0.150315\pi$$
0.890557 + 0.454871i $$0.150315\pi$$
$$354$$ 0 0
$$355$$ 2.53590 0.134592
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 26.7846 1.41364 0.706819 0.707395i $$-0.250130\pi$$
0.706819 + 0.707395i $$0.250130\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1.07180 0.0561004
$$366$$ 0 0
$$367$$ 33.5885 1.75330 0.876652 0.481126i $$-0.159772\pi$$
0.876652 + 0.481126i $$0.159772\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.00000 −0.103835
$$372$$ 0 0
$$373$$ −12.3923 −0.641649 −0.320825 0.947139i $$-0.603960\pi$$
−0.320825 + 0.947139i $$0.603960\pi$$
$$374$$ 0 0
$$375$$ −1.73205 −0.0894427
$$376$$ 0 0
$$377$$ −17.0718 −0.879242
$$378$$ 0 0
$$379$$ −6.39230 −0.328351 −0.164175 0.986431i $$-0.552496\pi$$
−0.164175 + 0.986431i $$0.552496\pi$$
$$380$$ 0 0
$$381$$ 32.3205 1.65583
$$382$$ 0 0
$$383$$ 12.5359 0.640554 0.320277 0.947324i $$-0.396224\pi$$
0.320277 + 0.947324i $$0.396224\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −23.2487 −1.17876 −0.589378 0.807857i $$-0.700627\pi$$
−0.589378 + 0.807857i $$0.700627\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −24.2487 −1.22319
$$394$$ 0 0
$$395$$ 14.3923 0.724155
$$396$$ 0 0
$$397$$ −30.6410 −1.53783 −0.768914 0.639352i $$-0.779203\pi$$
−0.768914 + 0.639352i $$0.779203\pi$$
$$398$$ 0 0
$$399$$ −0.928203 −0.0464683
$$400$$ 0 0
$$401$$ −10.8564 −0.542143 −0.271072 0.962559i $$-0.587378\pi$$
−0.271072 + 0.962559i $$0.587378\pi$$
$$402$$ 0 0
$$403$$ −5.07180 −0.252644
$$404$$ 0 0
$$405$$ −9.00000 −0.447214
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.8564 −1.32796 −0.663982 0.747749i $$-0.731135\pi$$
−0.663982 + 0.747749i $$0.731135\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 0 0
$$413$$ −0.392305 −0.0193041
$$414$$ 0 0
$$415$$ −7.46410 −0.366398
$$416$$ 0 0
$$417$$ −14.5359 −0.711826
$$418$$ 0 0
$$419$$ 15.8564 0.774636 0.387318 0.921946i $$-0.373402\pi$$
0.387318 + 0.921946i $$0.373402\pi$$
$$420$$ 0 0
$$421$$ 18.4641 0.899885 0.449943 0.893057i $$-0.351444\pi$$
0.449943 + 0.893057i $$0.351444\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.411543 0.0199159
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.3923 1.46395 0.731973 0.681334i $$-0.238600\pi$$
0.731973 + 0.681334i $$0.238600\pi$$
$$432$$ 0 0
$$433$$ 29.1769 1.40215 0.701077 0.713086i $$-0.252703\pi$$
0.701077 + 0.713086i $$0.252703\pi$$
$$434$$ 0 0
$$435$$ 8.53590 0.409265
$$436$$ 0 0
$$437$$ 14.9282 0.714113
$$438$$ 0 0
$$439$$ 1.07180 0.0511541 0.0255770 0.999673i $$-0.491858\pi$$
0.0255770 + 0.999673i $$0.491858\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −20.6603 −0.981598 −0.490799 0.871273i $$-0.663295\pi$$
−0.490799 + 0.871273i $$0.663295\pi$$
$$444$$ 0 0
$$445$$ 15.9282 0.755069
$$446$$ 0 0
$$447$$ −11.1962 −0.529560
$$448$$ 0 0
$$449$$ 22.8564 1.07866 0.539330 0.842094i $$-0.318677\pi$$
0.539330 + 0.842094i $$0.318677\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −7.60770 −0.357441
$$454$$ 0 0
$$455$$ 0.928203 0.0435148
$$456$$ 0 0
$$457$$ 6.24871 0.292302 0.146151 0.989262i $$-0.453311\pi$$
0.146151 + 0.989262i $$0.453311\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −35.2487 −1.64170 −0.820848 0.571147i $$-0.806499\pi$$
−0.820848 + 0.571147i $$0.806499\pi$$
$$462$$ 0 0
$$463$$ −19.7321 −0.917026 −0.458513 0.888688i $$-0.651618\pi$$
−0.458513 + 0.888688i $$0.651618\pi$$
$$464$$ 0 0
$$465$$ 2.53590 0.117599
$$466$$ 0 0
$$467$$ −15.3397 −0.709839 −0.354919 0.934897i $$-0.615492\pi$$
−0.354919 + 0.934897i $$0.615492\pi$$
$$468$$ 0 0
$$469$$ 1.53590 0.0709212
$$470$$ 0 0
$$471$$ 34.6410 1.59617
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −28.5359 −1.30384 −0.651919 0.758288i $$-0.726036\pi$$
−0.651919 + 0.758288i $$0.726036\pi$$
$$480$$ 0 0
$$481$$ −13.8564 −0.631798
$$482$$ 0 0
$$483$$ −3.46410 −0.157622
$$484$$ 0 0
$$485$$ −14.3923 −0.653521
$$486$$ 0 0
$$487$$ 6.67949 0.302677 0.151338 0.988482i $$-0.451642\pi$$
0.151338 + 0.988482i $$0.451642\pi$$
$$488$$ 0 0
$$489$$ −14.0718 −0.636349
$$490$$ 0 0
$$491$$ −32.2487 −1.45536 −0.727682 0.685915i $$-0.759402\pi$$
−0.727682 + 0.685915i $$0.759402\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0.679492 0.0304794
$$498$$ 0 0
$$499$$ 33.3205 1.49163 0.745815 0.666153i $$-0.232060\pi$$
0.745815 + 0.666153i $$0.232060\pi$$
$$500$$ 0 0
$$501$$ 35.5359 1.58763
$$502$$ 0 0
$$503$$ 38.6603 1.72378 0.861888 0.507099i $$-0.169282\pi$$
0.861888 + 0.507099i $$0.169282\pi$$
$$504$$ 0 0
$$505$$ 11.3923 0.506951
$$506$$ 0 0
$$507$$ 1.73205 0.0769231
$$508$$ 0 0
$$509$$ −10.4641 −0.463813 −0.231907 0.972738i $$-0.574496\pi$$
−0.231907 + 0.972738i $$0.574496\pi$$
$$510$$ 0 0
$$511$$ 0.287187 0.0127044
$$512$$ 0 0
$$513$$ 10.3923 0.458831
$$514$$ 0 0
$$515$$ −2.39230 −0.105418
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −17.0000 −0.744784 −0.372392 0.928076i $$-0.621462\pi$$
−0.372392 + 0.928076i $$0.621462\pi$$
$$522$$ 0 0
$$523$$ 9.32051 0.407557 0.203779 0.979017i $$-0.434678\pi$$
0.203779 + 0.979017i $$0.434678\pi$$
$$524$$ 0 0
$$525$$ −0.464102 −0.0202551
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 32.7128 1.42230
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −6.67949 −0.289321
$$534$$ 0 0
$$535$$ 2.26795 0.0980520
$$536$$ 0 0
$$537$$ −4.14359 −0.178809
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 12.4641 0.535874 0.267937 0.963436i $$-0.413658\pi$$
0.267937 + 0.963436i $$0.413658\pi$$
$$542$$ 0 0
$$543$$ −30.1244 −1.29276
$$544$$ 0 0
$$545$$ 7.53590 0.322802
$$546$$ 0 0
$$547$$ −22.3923 −0.957426 −0.478713 0.877971i $$-0.658897\pi$$
−0.478713 + 0.877971i $$0.658897\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −9.85641 −0.419897
$$552$$ 0 0
$$553$$ 3.85641 0.163991
$$554$$ 0 0
$$555$$ 6.92820 0.294086
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 6.00000 0.253773
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 32.6603 1.37647 0.688233 0.725490i $$-0.258387\pi$$
0.688233 + 0.725490i $$0.258387\pi$$
$$564$$ 0 0
$$565$$ −10.0000 −0.420703
$$566$$ 0 0
$$567$$ −2.41154 −0.101275
$$568$$ 0 0
$$569$$ 18.7128 0.784482 0.392241 0.919863i $$-0.371700\pi$$
0.392241 + 0.919863i $$0.371700\pi$$
$$570$$ 0 0
$$571$$ 16.2487 0.679987 0.339994 0.940428i $$-0.389575\pi$$
0.339994 + 0.940428i $$0.389575\pi$$
$$572$$ 0 0
$$573$$ 19.1769 0.801127
$$574$$ 0 0
$$575$$ 7.46410 0.311275
$$576$$ 0 0
$$577$$ 8.39230 0.349376 0.174688 0.984624i $$-0.444108\pi$$
0.174688 + 0.984624i $$0.444108\pi$$
$$578$$ 0 0
$$579$$ −27.4641 −1.14137
$$580$$ 0 0
$$581$$ −2.00000 −0.0829740
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 4.94744 0.204203 0.102101 0.994774i $$-0.467443\pi$$
0.102101 + 0.994774i $$0.467443\pi$$
$$588$$ 0 0
$$589$$ −2.92820 −0.120655
$$590$$ 0 0
$$591$$ −1.60770 −0.0661317
$$592$$ 0 0
$$593$$ −6.39230 −0.262500 −0.131250 0.991349i $$-0.541899\pi$$
−0.131250 + 0.991349i $$0.541899\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −11.0718 −0.453138
$$598$$ 0 0
$$599$$ −24.9282 −1.01854 −0.509269 0.860607i $$-0.670084\pi$$
−0.509269 + 0.860607i $$0.670084\pi$$
$$600$$ 0 0
$$601$$ 23.8564 0.973123 0.486562 0.873646i $$-0.338251\pi$$
0.486562 + 0.873646i $$0.338251\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −27.4641 −1.11473 −0.557367 0.830266i $$-0.688188\pi$$
−0.557367 + 0.830266i $$0.688188\pi$$
$$608$$ 0 0
$$609$$ 2.28719 0.0926815
$$610$$ 0 0
$$611$$ 23.0718 0.933385
$$612$$ 0 0
$$613$$ 13.7128 0.553855 0.276928 0.960891i $$-0.410684\pi$$
0.276928 + 0.960891i $$0.410684\pi$$
$$614$$ 0 0
$$615$$ 3.33975 0.134672
$$616$$ 0 0
$$617$$ 11.3205 0.455746 0.227873 0.973691i $$-0.426823\pi$$
0.227873 + 0.973691i $$0.426823\pi$$
$$618$$ 0 0
$$619$$ −17.0718 −0.686173 −0.343087 0.939304i $$-0.611472\pi$$
−0.343087 + 0.939304i $$0.611472\pi$$
$$620$$ 0 0
$$621$$ 38.7846 1.55637
$$622$$ 0 0
$$623$$ 4.26795 0.170992
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −27.7128 −1.10323 −0.551615 0.834099i $$-0.685988\pi$$
−0.551615 + 0.834099i $$0.685988\pi$$
$$632$$ 0 0
$$633$$ −11.3205 −0.449950
$$634$$ 0 0
$$635$$ −18.6603 −0.740510
$$636$$ 0 0
$$637$$ −24.0000 −0.950915
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 38.7846 1.53190 0.765950 0.642900i $$-0.222269\pi$$
0.765950 + 0.642900i $$0.222269\pi$$
$$642$$ 0 0
$$643$$ −14.8038 −0.583807 −0.291903 0.956448i $$-0.594289\pi$$
−0.291903 + 0.956448i $$0.594289\pi$$
$$644$$ 0 0
$$645$$ −3.00000 −0.118125
$$646$$ 0 0
$$647$$ 5.87564 0.230995 0.115498 0.993308i $$-0.463154\pi$$
0.115498 + 0.993308i $$0.463154\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0.679492 0.0266314
$$652$$ 0 0
$$653$$ 3.85641 0.150913 0.0754564 0.997149i $$-0.475959\pi$$
0.0754564 + 0.997149i $$0.475959\pi$$
$$654$$ 0 0
$$655$$ 14.0000 0.547025
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −47.1769 −1.83775 −0.918876 0.394547i $$-0.870902\pi$$
−0.918876 + 0.394547i $$0.870902\pi$$
$$660$$ 0 0
$$661$$ 4.32051 0.168048 0.0840241 0.996464i $$-0.473223\pi$$
0.0840241 + 0.996464i $$0.473223\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0.535898 0.0207812
$$666$$ 0 0
$$667$$ −36.7846 −1.42431
$$668$$ 0 0
$$669$$ −0.464102 −0.0179432
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −29.4641 −1.13576 −0.567879 0.823112i $$-0.692236\pi$$
−0.567879 + 0.823112i $$0.692236\pi$$
$$674$$ 0 0
$$675$$ 5.19615 0.200000
$$676$$ 0 0
$$677$$ −39.7128 −1.52629 −0.763144 0.646229i $$-0.776345\pi$$
−0.763144 + 0.646229i $$0.776345\pi$$
$$678$$ 0 0
$$679$$ −3.85641 −0.147995
$$680$$ 0 0
$$681$$ 11.7846 0.451587
$$682$$ 0 0
$$683$$ −19.5885 −0.749531 −0.374766 0.927120i $$-0.622277\pi$$
−0.374766 + 0.927120i $$0.622277\pi$$
$$684$$ 0 0
$$685$$ 10.3923 0.397070
$$686$$ 0 0
$$687$$ 7.98076 0.304485
$$688$$ 0 0
$$689$$ −25.8564 −0.985051
$$690$$ 0 0
$$691$$ −43.5692 −1.65745 −0.828726 0.559655i $$-0.810934\pi$$
−0.828726 + 0.559655i $$0.810934\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 8.39230 0.318338
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ −24.9282 −0.942871
$$700$$ 0 0
$$701$$ 13.7128 0.517926 0.258963 0.965887i $$-0.416619\pi$$
0.258963 + 0.965887i $$0.416619\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 0 0
$$705$$ −11.5359 −0.434467
$$706$$ 0 0
$$707$$ 3.05256 0.114803
$$708$$ 0 0
$$709$$ −21.3923 −0.803405 −0.401702 0.915770i $$-0.631581\pi$$
−0.401702 + 0.915770i $$0.631581\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −10.9282 −0.409264
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −24.2487 −0.905585
$$718$$ 0 0
$$719$$ 32.6410 1.21730 0.608652 0.793437i $$-0.291710\pi$$
0.608652 + 0.793437i $$0.291710\pi$$
$$720$$ 0 0
$$721$$ −0.641016 −0.0238727
$$722$$ 0 0
$$723$$ −36.3731 −1.35273
$$724$$ 0 0
$$725$$ −4.92820 −0.183029
$$726$$ 0 0
$$727$$ 16.8038 0.623220 0.311610 0.950210i $$-0.399132\pi$$
0.311610 + 0.950210i $$0.399132\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −28.0000 −1.03420 −0.517102 0.855924i $$-0.672989\pi$$
−0.517102 + 0.855924i $$0.672989\pi$$
$$734$$ 0 0
$$735$$ 12.0000 0.442627
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 51.1769 1.88257 0.941287 0.337609i $$-0.109618\pi$$
0.941287 + 0.337609i $$0.109618\pi$$
$$740$$ 0 0
$$741$$ −12.0000 −0.440831
$$742$$ 0 0
$$743$$ −4.26795 −0.156576 −0.0782879 0.996931i $$-0.524945\pi$$
−0.0782879 + 0.996931i $$0.524945\pi$$
$$744$$ 0 0
$$745$$ 6.46410 0.236826
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0.607695 0.0222047
$$750$$ 0 0
$$751$$ 47.4641 1.73199 0.865995 0.500053i $$-0.166686\pi$$
0.865995 + 0.500053i $$0.166686\pi$$
$$752$$ 0 0
$$753$$ −18.2487 −0.665020
$$754$$ 0 0
$$755$$ 4.39230 0.159852
$$756$$ 0 0
$$757$$ −25.3205 −0.920290 −0.460145 0.887844i $$-0.652202\pi$$
−0.460145 + 0.887844i $$0.652202\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 4.14359 0.150205 0.0751026 0.997176i $$-0.476072\pi$$
0.0751026 + 0.997176i $$0.476072\pi$$
$$762$$ 0 0
$$763$$ 2.01924 0.0731013
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −5.07180 −0.183132
$$768$$ 0 0
$$769$$ 15.8564 0.571797 0.285898 0.958260i $$-0.407708\pi$$
0.285898 + 0.958260i $$0.407708\pi$$
$$770$$ 0 0
$$771$$ −45.4641 −1.63735
$$772$$ 0 0
$$773$$ 9.46410 0.340400 0.170200 0.985410i $$-0.445559\pi$$
0.170200 + 0.985410i $$0.445559\pi$$
$$774$$ 0 0
$$775$$ −1.46410 −0.0525921
$$776$$ 0 0
$$777$$ 1.85641 0.0665982
$$778$$ 0 0
$$779$$ −3.85641 −0.138170
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −25.6077 −0.915144
$$784$$ 0 0
$$785$$ −20.0000 −0.713831
$$786$$ 0 0
$$787$$ −27.5885 −0.983422 −0.491711 0.870758i $$-0.663628\pi$$
−0.491711 + 0.870758i $$0.663628\pi$$
$$788$$ 0 0
$$789$$ −45.7128 −1.62742
$$790$$ 0 0
$$791$$ −2.67949 −0.0952718
$$792$$ 0 0
$$793$$ 5.32051 0.188937
$$794$$ 0 0
$$795$$ 12.9282 0.458516
$$796$$ 0 0
$$797$$ −34.5359 −1.22332 −0.611662 0.791119i $$-0.709499\pi$$
−0.611662 + 0.791119i $$0.709499\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 2.00000 0.0704907
$$806$$ 0 0
$$807$$ 33.3397 1.17361
$$808$$ 0 0
$$809$$ 38.7846 1.36359 0.681797 0.731541i $$-0.261199\pi$$
0.681797 + 0.731541i $$0.261199\pi$$
$$810$$ 0 0
$$811$$ −44.3923 −1.55882 −0.779412 0.626511i $$-0.784482\pi$$
−0.779412 + 0.626511i $$0.784482\pi$$
$$812$$ 0 0
$$813$$ −32.7846 −1.14981
$$814$$ 0 0
$$815$$ 8.12436 0.284584
$$816$$ 0 0
$$817$$ 3.46410 0.121194
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25.1051 −0.876175 −0.438087 0.898932i $$-0.644344\pi$$
−0.438087 + 0.898932i $$0.644344\pi$$
$$822$$ 0 0
$$823$$ 26.1244 0.910638 0.455319 0.890328i $$-0.349525\pi$$
0.455319 + 0.890328i $$0.349525\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −2.80385 −0.0974993 −0.0487497 0.998811i $$-0.515524\pi$$
−0.0487497 + 0.998811i $$0.515524\pi$$
$$828$$ 0 0
$$829$$ −23.2487 −0.807461 −0.403731 0.914878i $$-0.632287\pi$$
−0.403731 + 0.914878i $$0.632287\pi$$
$$830$$ 0 0
$$831$$ 38.1051 1.32185
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −20.5167 −0.710008
$$836$$ 0 0
$$837$$ −7.60770 −0.262960
$$838$$ 0 0
$$839$$ 27.4641 0.948166 0.474083 0.880480i $$-0.342780\pi$$
0.474083 + 0.880480i $$0.342780\pi$$
$$840$$ 0 0
$$841$$ −4.71281 −0.162511
$$842$$ 0 0
$$843$$ −55.1769 −1.90039
$$844$$ 0 0
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −32.0718 −1.10070
$$850$$ 0 0
$$851$$ −29.8564 −1.02346
$$852$$ 0 0
$$853$$ 13.0718 0.447570 0.223785 0.974639i $$-0.428159\pi$$
0.223785 + 0.974639i $$0.428159\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −48.9282 −1.67136 −0.835678 0.549220i $$-0.814925\pi$$
−0.835678 + 0.549220i $$0.814925\pi$$
$$858$$ 0 0
$$859$$ −39.0718 −1.33311 −0.666556 0.745455i $$-0.732232\pi$$
−0.666556 + 0.745455i $$0.732232\pi$$
$$860$$ 0 0
$$861$$ 0.894882 0.0304975
$$862$$ 0 0
$$863$$ −32.5167 −1.10688 −0.553440 0.832889i $$-0.686685\pi$$
−0.553440 + 0.832889i $$0.686685\pi$$
$$864$$ 0 0
$$865$$ 3.46410 0.117783
$$866$$ 0 0
$$867$$ 29.4449 1.00000
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 19.8564 0.672809
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0.267949 0.00905834
$$876$$ 0 0
$$877$$ 35.6077 1.20239 0.601193 0.799104i $$-0.294692\pi$$
0.601193 + 0.799104i $$0.294692\pi$$
$$878$$ 0 0
$$879$$ −52.3923 −1.76715
$$880$$ 0 0
$$881$$ 0.856406 0.0288531 0.0144265 0.999896i $$-0.495408\pi$$
0.0144265 + 0.999896i $$0.495408\pi$$
$$882$$ 0 0
$$883$$ 11.4641 0.385798 0.192899 0.981219i $$-0.438211\pi$$
0.192899 + 0.981219i $$0.438211\pi$$
$$884$$ 0 0
$$885$$ 2.53590 0.0852433
$$886$$ 0 0
$$887$$ −12.5167 −0.420268 −0.210134 0.977673i $$-0.567390\pi$$
−0.210134 + 0.977673i $$0.567390\pi$$
$$888$$ 0 0
$$889$$ −5.00000 −0.167695
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 13.3205 0.445754
$$894$$ 0 0
$$895$$ 2.39230 0.0799659
$$896$$ 0 0
$$897$$ −44.7846 −1.49531
$$898$$ 0 0
$$899$$ 7.21539 0.240647
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ −0.803848 −0.0267504
$$904$$ 0 0
$$905$$ 17.3923 0.578140
$$906$$ 0 0
$$907$$ −52.9090 −1.75681 −0.878407 0.477914i $$-0.841393\pi$$
−0.878407 + 0.477914i $$0.841393\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −12.2487 −0.405818 −0.202909 0.979198i $$-0.565040\pi$$
−0.202909 + 0.979198i $$0.565040\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −2.66025 −0.0879453
$$916$$ 0 0
$$917$$ 3.75129 0.123878
$$918$$ 0 0
$$919$$ −42.7846 −1.41133 −0.705667 0.708544i $$-0.749353\pi$$
−0.705667 + 0.708544i $$0.749353\pi$$
$$920$$ 0 0
$$921$$ −45.7128 −1.50629
$$922$$ 0 0
$$923$$ 8.78461 0.289149
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −13.2154 −0.433583 −0.216791 0.976218i $$-0.569559\pi$$
−0.216791 + 0.976218i $$0.569559\pi$$
$$930$$ 0 0
$$931$$ −13.8564 −0.454125
$$932$$ 0 0
$$933$$ −19.1769 −0.627824
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −39.0718 −1.27642 −0.638210 0.769862i $$-0.720325\pi$$
−0.638210 + 0.769862i $$0.720325\pi$$
$$938$$ 0 0
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ 20.4641 0.667111 0.333555 0.942731i $$-0.391752\pi$$
0.333555 + 0.942731i $$0.391752\pi$$
$$942$$ 0 0
$$943$$ −14.3923 −0.468678
$$944$$ 0 0
$$945$$ 1.39230 0.0452917
$$946$$ 0 0
$$947$$ 49.3205 1.60270 0.801351 0.598195i $$-0.204115\pi$$
0.801351 + 0.598195i $$0.204115\pi$$
$$948$$ 0 0
$$949$$ 3.71281 0.120523
$$950$$ 0 0
$$951$$ 34.3923 1.11525
$$952$$ 0 0
$$953$$ 4.00000 0.129573 0.0647864 0.997899i $$-0.479363\pi$$
0.0647864 + 0.997899i $$0.479363\pi$$
$$954$$ 0 0
$$955$$ −11.0718 −0.358275
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 2.78461 0.0899197
$$960$$ 0 0
$$961$$ −28.8564 −0.930852
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 15.8564 0.510436
$$966$$ 0 0
$$967$$ 3.17691 0.102163 0.0510813 0.998694i $$-0.483733\pi$$
0.0510813 + 0.998694i $$0.483733\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −0.784610 −0.0251793 −0.0125897 0.999921i $$-0.504008\pi$$
−0.0125897 + 0.999921i $$0.504008\pi$$
$$972$$ 0 0
$$973$$ 2.24871 0.0720904
$$974$$ 0 0
$$975$$ −6.00000 −0.192154
$$976$$ 0 0
$$977$$ 22.9282 0.733538 0.366769 0.930312i $$-0.380464\pi$$
0.366769 + 0.930312i $$0.380464\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −39.7321 −1.26726 −0.633628 0.773638i $$-0.718435\pi$$
−0.633628 + 0.773638i $$0.718435\pi$$
$$984$$ 0 0
$$985$$ 0.928203 0.0295750
$$986$$ 0 0
$$987$$ −3.09103 −0.0983887
$$988$$ 0 0
$$989$$ 12.9282 0.411093
$$990$$ 0 0
$$991$$ 6.78461 0.215520 0.107760 0.994177i $$-0.465632\pi$$
0.107760 + 0.994177i $$0.465632\pi$$
$$992$$ 0 0
$$993$$ 36.2487 1.15032
$$994$$ 0 0
$$995$$ 6.39230 0.202650
$$996$$ 0 0
$$997$$ −2.67949 −0.0848604 −0.0424302 0.999099i $$-0.513510\pi$$
−0.0424302 + 0.999099i $$0.513510\pi$$
$$998$$ 0 0
$$999$$ −20.7846 −0.657596
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.l.1.1 yes 2
4.3 odd 2 9680.2.a.bq.1.2 2
11.10 odd 2 4840.2.a.k.1.1 2
44.43 even 2 9680.2.a.br.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.k.1.1 2 11.10 odd 2
4840.2.a.l.1.1 yes 2 1.1 even 1 trivial
9680.2.a.bq.1.2 2 4.3 odd 2
9680.2.a.br.1.2 2 44.43 even 2