# Properties

 Label 4140.2.a.u.1.3 Level $4140$ Weight $2$ Character 4140.1 Self dual yes Analytic conductor $33.058$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4140.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: $$33.0580664368$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.14345904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12$$ x^5 - 2*x^4 - 13*x^3 + 34*x^2 - 11*x - 12 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.83694$$ of defining polynomial Character $$\chi$$ $$=$$ 4140.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -0.113901 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -0.113901 q^{7} +3.39991 q^{11} +6.27397 q^{13} +3.61662 q^{17} -1.39991 q^{19} +1.00000 q^{23} +1.00000 q^{25} -6.38787 q^{29} +9.45717 q^{31} -0.113901 q^{35} -7.78778 q^{37} +11.0563 q^{41} +1.55936 q^{43} +1.70196 q^{47} -6.98703 q^{49} -8.71849 q^{53} +3.39991 q^{55} -8.95988 q^{59} -1.83333 q^{61} +6.27397 q^{65} +9.21577 q^{67} -6.82852 q^{71} -2.27397 q^{73} -0.387254 q^{77} +11.7703 q^{79} +1.16101 q^{83} +3.61662 q^{85} -1.77220 q^{89} -0.714615 q^{91} -1.39991 q^{95} -0.131365 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{5} + 4 q^{7}+O(q^{10})$$ 5 * q + 5 * q^5 + 4 * q^7 $$5 q + 5 q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{23} + 5 q^{25} + 2 q^{29} + 8 q^{31} + 4 q^{35} + 8 q^{37} + 2 q^{41} + 18 q^{43} - 4 q^{47} + 13 q^{49} - 2 q^{53} + 4 q^{55} + 6 q^{59} + 10 q^{61} + 2 q^{65} + 16 q^{67} + 10 q^{71} + 18 q^{73} - 16 q^{77} + 14 q^{79} + 8 q^{83} + 2 q^{85} - 18 q^{89} + 36 q^{91} + 6 q^{95} + 6 q^{97}+O(q^{100})$$ 5 * q + 5 * q^5 + 4 * q^7 + 4 * q^11 + 2 * q^13 + 2 * q^17 + 6 * q^19 + 5 * q^23 + 5 * q^25 + 2 * q^29 + 8 * q^31 + 4 * q^35 + 8 * q^37 + 2 * q^41 + 18 * q^43 - 4 * q^47 + 13 * q^49 - 2 * q^53 + 4 * q^55 + 6 * q^59 + 10 * q^61 + 2 * q^65 + 16 * q^67 + 10 * q^71 + 18 * q^73 - 16 * q^77 + 14 * q^79 + 8 * q^83 + 2 * q^85 - 18 * q^89 + 36 * q^91 + 6 * q^95 + 6 * 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −0.113901 −0.0430507 −0.0215254 0.999768i $$-0.506852\pi$$
−0.0215254 + 0.999768i $$0.506852\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.39991 1.02511 0.512555 0.858654i $$-0.328699\pi$$
0.512555 + 0.858654i $$0.328699\pi$$
$$12$$ 0 0
$$13$$ 6.27397 1.74009 0.870043 0.492975i $$-0.164091\pi$$
0.870043 + 0.492975i $$0.164091\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.61662 0.877158 0.438579 0.898693i $$-0.355482\pi$$
0.438579 + 0.898693i $$0.355482\pi$$
$$18$$ 0 0
$$19$$ −1.39991 −0.321160 −0.160580 0.987023i $$-0.551337\pi$$
−0.160580 + 0.987023i $$0.551337\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.38787 −1.18620 −0.593099 0.805129i $$-0.702096\pi$$
−0.593099 + 0.805129i $$0.702096\pi$$
$$30$$ 0 0
$$31$$ 9.45717 1.69856 0.849279 0.527945i $$-0.177037\pi$$
0.849279 + 0.527945i $$0.177037\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −0.113901 −0.0192529
$$36$$ 0 0
$$37$$ −7.78778 −1.28030 −0.640151 0.768249i $$-0.721128\pi$$
−0.640151 + 0.768249i $$0.721128\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.0563 1.72671 0.863353 0.504600i $$-0.168360\pi$$
0.863353 + 0.504600i $$0.168360\pi$$
$$42$$ 0 0
$$43$$ 1.55936 0.237800 0.118900 0.992906i $$-0.462063\pi$$
0.118900 + 0.992906i $$0.462063\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.70196 0.248257 0.124128 0.992266i $$-0.460387\pi$$
0.124128 + 0.992266i $$0.460387\pi$$
$$48$$ 0 0
$$49$$ −6.98703 −0.998147
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −8.71849 −1.19758 −0.598788 0.800908i $$-0.704351\pi$$
−0.598788 + 0.800908i $$0.704351\pi$$
$$54$$ 0 0
$$55$$ 3.39991 0.458443
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −8.95988 −1.16648 −0.583239 0.812301i $$-0.698215\pi$$
−0.583239 + 0.812301i $$0.698215\pi$$
$$60$$ 0 0
$$61$$ −1.83333 −0.234734 −0.117367 0.993089i $$-0.537445\pi$$
−0.117367 + 0.993089i $$0.537445\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.27397 0.778190
$$66$$ 0 0
$$67$$ 9.21577 1.12589 0.562943 0.826496i $$-0.309669\pi$$
0.562943 + 0.826496i $$0.309669\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.82852 −0.810396 −0.405198 0.914229i $$-0.632797\pi$$
−0.405198 + 0.914229i $$0.632797\pi$$
$$72$$ 0 0
$$73$$ −2.27397 −0.266148 −0.133074 0.991106i $$-0.542485\pi$$
−0.133074 + 0.991106i $$0.542485\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −0.387254 −0.0441317
$$78$$ 0 0
$$79$$ 11.7703 1.32426 0.662132 0.749387i $$-0.269652\pi$$
0.662132 + 0.749387i $$0.269652\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.16101 0.127437 0.0637187 0.997968i $$-0.479704\pi$$
0.0637187 + 0.997968i $$0.479704\pi$$
$$84$$ 0 0
$$85$$ 3.61662 0.392277
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1.77220 −0.187853 −0.0939263 0.995579i $$-0.529942\pi$$
−0.0939263 + 0.995579i $$0.529942\pi$$
$$90$$ 0 0
$$91$$ −0.714615 −0.0749120
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.39991 −0.143627
$$96$$ 0 0
$$97$$ −0.131365 −0.0133380 −0.00666902 0.999978i $$-0.502123\pi$$
−0.00666902 + 0.999978i $$0.502123\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.40052 −0.935387 −0.467694 0.883891i $$-0.654915\pi$$
−0.467694 + 0.883891i $$0.654915\pi$$
$$102$$ 0 0
$$103$$ 9.24045 0.910489 0.455244 0.890366i $$-0.349552\pi$$
0.455244 + 0.890366i $$0.349552\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.28506 0.220905 0.110453 0.993881i $$-0.464770\pi$$
0.110453 + 0.993881i $$0.464770\pi$$
$$108$$ 0 0
$$109$$ −1.39991 −0.134087 −0.0670433 0.997750i $$-0.521357\pi$$
−0.0670433 + 0.997750i $$0.521357\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −9.59977 −0.903071 −0.451535 0.892253i $$-0.649124\pi$$
−0.451535 + 0.892253i $$0.649124\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −0.411938 −0.0377623
$$120$$ 0 0
$$121$$ 0.559357 0.0508506
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 6.73146 0.597320 0.298660 0.954360i $$-0.403460\pi$$
0.298660 + 0.954360i $$0.403460\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 4.22780 0.369385 0.184692 0.982796i $$-0.440871\pi$$
0.184692 + 0.982796i $$0.440871\pi$$
$$132$$ 0 0
$$133$$ 0.159451 0.0138262
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 14.4496 1.23451 0.617257 0.786761i $$-0.288244\pi$$
0.617257 + 0.786761i $$0.288244\pi$$
$$138$$ 0 0
$$139$$ 11.9870 1.01673 0.508363 0.861143i $$-0.330251\pi$$
0.508363 + 0.861143i $$0.330251\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 21.3309 1.78378
$$144$$ 0 0
$$145$$ −6.38787 −0.530484
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 4.19972 0.344054 0.172027 0.985092i $$-0.444968\pi$$
0.172027 + 0.985092i $$0.444968\pi$$
$$150$$ 0 0
$$151$$ 7.55936 0.615172 0.307586 0.951520i $$-0.400479\pi$$
0.307586 + 0.951520i $$0.400479\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 9.45717 0.759618
$$156$$ 0 0
$$157$$ 5.86203 0.467841 0.233921 0.972256i $$-0.424844\pi$$
0.233921 + 0.972256i $$0.424844\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −0.113901 −0.00897669
$$162$$ 0 0
$$163$$ 0.766767 0.0600578 0.0300289 0.999549i $$-0.490440\pi$$
0.0300289 + 0.999549i $$0.490440\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −14.0443 −1.08678 −0.543390 0.839481i $$-0.682859\pi$$
−0.543390 + 0.839481i $$0.682859\pi$$
$$168$$ 0 0
$$169$$ 26.3627 2.02790
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −2.13859 −0.162594 −0.0812968 0.996690i $$-0.525906\pi$$
−0.0812968 + 0.996690i $$0.525906\pi$$
$$174$$ 0 0
$$175$$ −0.113901 −0.00861014
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 8.93118 0.667547 0.333774 0.942653i $$-0.391678\pi$$
0.333774 + 0.942653i $$0.391678\pi$$
$$180$$ 0 0
$$181$$ 25.4623 1.89260 0.946298 0.323296i $$-0.104791\pi$$
0.946298 + 0.323296i $$0.104791\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −7.78778 −0.572569
$$186$$ 0 0
$$187$$ 12.2962 0.899184
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.3813 −0.895877 −0.447939 0.894064i $$-0.647842\pi$$
−0.447939 + 0.894064i $$0.647842\pi$$
$$192$$ 0 0
$$193$$ 3.31471 0.238598 0.119299 0.992858i $$-0.461935\pi$$
0.119299 + 0.992858i $$0.461935\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −16.3201 −1.16276 −0.581381 0.813632i $$-0.697487\pi$$
−0.581381 + 0.813632i $$0.697487\pi$$
$$198$$ 0 0
$$199$$ −9.80336 −0.694942 −0.347471 0.937691i $$-0.612959\pi$$
−0.347471 + 0.937691i $$0.612959\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0.727588 0.0510667
$$204$$ 0 0
$$205$$ 11.0563 0.772207
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4.75955 −0.329225
$$210$$ 0 0
$$211$$ 1.34987 0.0929286 0.0464643 0.998920i $$-0.485205\pi$$
0.0464643 + 0.998920i $$0.485205\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 1.55936 0.106347
$$216$$ 0 0
$$217$$ −1.07719 −0.0731241
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 22.6905 1.52633
$$222$$ 0 0
$$223$$ 15.2332 1.02009 0.510046 0.860147i $$-0.329628\pi$$
0.510046 + 0.860147i $$0.329628\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −18.8921 −1.25392 −0.626958 0.779053i $$-0.715700\pi$$
−0.626958 + 0.779053i $$0.715700\pi$$
$$228$$ 0 0
$$229$$ 7.57744 0.500731 0.250366 0.968151i $$-0.419449\pi$$
0.250366 + 0.968151i $$0.419449\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −17.0035 −1.11394 −0.556970 0.830533i $$-0.688036\pi$$
−0.556970 + 0.830533i $$0.688036\pi$$
$$234$$ 0 0
$$235$$ 1.70196 0.111024
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −22.9508 −1.48456 −0.742281 0.670089i $$-0.766256\pi$$
−0.742281 + 0.670089i $$0.766256\pi$$
$$240$$ 0 0
$$241$$ 4.51862 0.291070 0.145535 0.989353i $$-0.453510\pi$$
0.145535 + 0.989353i $$0.453510\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −6.98703 −0.446385
$$246$$ 0 0
$$247$$ −8.78297 −0.558847
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 25.0257 1.57961 0.789805 0.613358i $$-0.210182\pi$$
0.789805 + 0.613358i $$0.210182\pi$$
$$252$$ 0 0
$$253$$ 3.39991 0.213750
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 3.05160 0.190354 0.0951768 0.995460i $$-0.469658\pi$$
0.0951768 + 0.995460i $$0.469658\pi$$
$$258$$ 0 0
$$259$$ 0.887039 0.0551180
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −6.11437 −0.377028 −0.188514 0.982070i $$-0.560367\pi$$
−0.188514 + 0.982070i $$0.560367\pi$$
$$264$$ 0 0
$$265$$ −8.71849 −0.535572
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 10.7080 0.652879 0.326440 0.945218i $$-0.394151\pi$$
0.326440 + 0.945218i $$0.394151\pi$$
$$270$$ 0 0
$$271$$ 15.0980 0.917138 0.458569 0.888659i $$-0.348362\pi$$
0.458569 + 0.888659i $$0.348362\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 3.39991 0.205022
$$276$$ 0 0
$$277$$ −26.0886 −1.56751 −0.783755 0.621070i $$-0.786698\pi$$
−0.783755 + 0.621070i $$0.786698\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4.29082 0.255969 0.127984 0.991776i $$-0.459149\pi$$
0.127984 + 0.991776i $$0.459149\pi$$
$$282$$ 0 0
$$283$$ −20.6057 −1.22488 −0.612440 0.790517i $$-0.709812\pi$$
−0.612440 + 0.790517i $$0.709812\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1.25933 −0.0743360
$$288$$ 0 0
$$289$$ −3.92008 −0.230593
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 22.7238 1.32754 0.663770 0.747937i $$-0.268955\pi$$
0.663770 + 0.747937i $$0.268955\pi$$
$$294$$ 0 0
$$295$$ −8.95988 −0.521664
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 6.27397 0.362833
$$300$$ 0 0
$$301$$ −0.177613 −0.0102374
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −1.83333 −0.104976
$$306$$ 0 0
$$307$$ 20.7645 1.18509 0.592546 0.805536i $$-0.298123\pi$$
0.592546 + 0.805536i $$0.298123\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 33.3237 1.88961 0.944806 0.327629i $$-0.106250\pi$$
0.944806 + 0.327629i $$0.106250\pi$$
$$312$$ 0 0
$$313$$ −5.25070 −0.296787 −0.148393 0.988928i $$-0.547410\pi$$
−0.148393 + 0.988928i $$0.547410\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −23.8495 −1.33952 −0.669761 0.742576i $$-0.733604\pi$$
−0.669761 + 0.742576i $$0.733604\pi$$
$$318$$ 0 0
$$319$$ −21.7182 −1.21598
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −5.06292 −0.281708
$$324$$ 0 0
$$325$$ 6.27397 0.348017
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −0.193856 −0.0106876
$$330$$ 0 0
$$331$$ −20.8049 −1.14354 −0.571771 0.820413i $$-0.693743\pi$$
−0.571771 + 0.820413i $$0.693743\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 9.21577 0.503511
$$336$$ 0 0
$$337$$ 4.97050 0.270761 0.135380 0.990794i $$-0.456774\pi$$
0.135380 + 0.990794i $$0.456774\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 32.1535 1.74121
$$342$$ 0 0
$$343$$ 1.59314 0.0860216
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −27.3478 −1.46810 −0.734052 0.679093i $$-0.762373\pi$$
−0.734052 + 0.679093i $$0.762373\pi$$
$$348$$ 0 0
$$349$$ 24.7307 1.32380 0.661901 0.749591i $$-0.269750\pi$$
0.661901 + 0.749591i $$0.269750\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −25.5053 −1.35751 −0.678756 0.734364i $$-0.737480\pi$$
−0.678756 + 0.734364i $$0.737480\pi$$
$$354$$ 0 0
$$355$$ −6.82852 −0.362420
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 11.2644 0.594514 0.297257 0.954797i $$-0.403928\pi$$
0.297257 + 0.954797i $$0.403928\pi$$
$$360$$ 0 0
$$361$$ −17.0403 −0.896856
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −2.27397 −0.119025
$$366$$ 0 0
$$367$$ 36.6209 1.91160 0.955799 0.294022i $$-0.0949939\pi$$
0.955799 + 0.294022i $$0.0949939\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0.993048 0.0515565
$$372$$ 0 0
$$373$$ −38.4718 −1.99199 −0.995997 0.0893834i $$-0.971510\pi$$
−0.995997 + 0.0893834i $$0.971510\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −40.0773 −2.06409
$$378$$ 0 0
$$379$$ −8.63716 −0.443661 −0.221831 0.975085i $$-0.571203\pi$$
−0.221831 + 0.975085i $$0.571203\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −13.4684 −0.688203 −0.344102 0.938932i $$-0.611817\pi$$
−0.344102 + 0.938932i $$0.611817\pi$$
$$384$$ 0 0
$$385$$ −0.387254 −0.0197363
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −3.46292 −0.175577 −0.0877885 0.996139i $$-0.527980\pi$$
−0.0877885 + 0.996139i $$0.527980\pi$$
$$390$$ 0 0
$$391$$ 3.61662 0.182900
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 11.7703 0.592229
$$396$$ 0 0
$$397$$ 26.5791 1.33397 0.666984 0.745072i $$-0.267585\pi$$
0.666984 + 0.745072i $$0.267585\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 12.4575 0.622097 0.311049 0.950394i $$-0.399320\pi$$
0.311049 + 0.950394i $$0.399320\pi$$
$$402$$ 0 0
$$403$$ 59.3340 2.95564
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −26.4777 −1.31245
$$408$$ 0 0
$$409$$ 2.58310 0.127726 0.0638630 0.997959i $$-0.479658\pi$$
0.0638630 + 0.997959i $$0.479658\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1.02054 0.0502177
$$414$$ 0 0
$$415$$ 1.16101 0.0569918
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 36.6181 1.78891 0.894454 0.447159i $$-0.147564\pi$$
0.894454 + 0.447159i $$0.147564\pi$$
$$420$$ 0 0
$$421$$ −9.60553 −0.468145 −0.234072 0.972219i $$-0.575205\pi$$
−0.234072 + 0.972219i $$0.575205\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 3.61662 0.175432
$$426$$ 0 0
$$427$$ 0.208819 0.0101054
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −39.4934 −1.90233 −0.951166 0.308680i $$-0.900113\pi$$
−0.951166 + 0.308680i $$0.900113\pi$$
$$432$$ 0 0
$$433$$ 3.32495 0.159787 0.0798935 0.996803i $$-0.474542\pi$$
0.0798935 + 0.996803i $$0.474542\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1.39991 −0.0669666
$$438$$ 0 0
$$439$$ −14.5882 −0.696257 −0.348129 0.937447i $$-0.613183\pi$$
−0.348129 + 0.937447i $$0.613183\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 29.0497 1.38019 0.690097 0.723717i $$-0.257568\pi$$
0.690097 + 0.723717i $$0.257568\pi$$
$$444$$ 0 0
$$445$$ −1.77220 −0.0840102
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −3.07505 −0.145120 −0.0725602 0.997364i $$-0.523117\pi$$
−0.0725602 + 0.997364i $$0.523117\pi$$
$$450$$ 0 0
$$451$$ 37.5904 1.77006
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −0.714615 −0.0335017
$$456$$ 0 0
$$457$$ 6.49525 0.303835 0.151918 0.988393i $$-0.451455\pi$$
0.151918 + 0.988393i $$0.451455\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 28.1367 1.31046 0.655228 0.755431i $$-0.272572\pi$$
0.655228 + 0.755431i $$0.272572\pi$$
$$462$$ 0 0
$$463$$ 22.7626 1.05787 0.528934 0.848663i $$-0.322592\pi$$
0.528934 + 0.848663i $$0.322592\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −11.0296 −0.510391 −0.255196 0.966889i $$-0.582140\pi$$
−0.255196 + 0.966889i $$0.582140\pi$$
$$468$$ 0 0
$$469$$ −1.04969 −0.0484702
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 5.30167 0.243771
$$474$$ 0 0
$$475$$ −1.39991 −0.0642321
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −10.9755 −0.501482 −0.250741 0.968054i $$-0.580674\pi$$
−0.250741 + 0.968054i $$0.580674\pi$$
$$480$$ 0 0
$$481$$ −48.8603 −2.22784
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −0.131365 −0.00596496
$$486$$ 0 0
$$487$$ 5.47416 0.248058 0.124029 0.992279i $$-0.460418\pi$$
0.124029 + 0.992279i $$0.460418\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20.5766 0.928611 0.464306 0.885675i $$-0.346304\pi$$
0.464306 + 0.885675i $$0.346304\pi$$
$$492$$ 0 0
$$493$$ −23.1025 −1.04048
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0.777778 0.0348881
$$498$$ 0 0
$$499$$ 21.2605 0.951752 0.475876 0.879512i $$-0.342131\pi$$
0.475876 + 0.879512i $$0.342131\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 21.7049 0.967773 0.483887 0.875131i $$-0.339225\pi$$
0.483887 + 0.875131i $$0.339225\pi$$
$$504$$ 0 0
$$505$$ −9.40052 −0.418318
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −15.6961 −0.695716 −0.347858 0.937547i $$-0.613091\pi$$
−0.347858 + 0.937547i $$0.613091\pi$$
$$510$$ 0 0
$$511$$ 0.259009 0.0114579
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 9.24045 0.407183
$$516$$ 0 0
$$517$$ 5.78651 0.254491
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −0.207458 −0.00908890 −0.00454445 0.999990i $$-0.501447\pi$$
−0.00454445 + 0.999990i $$0.501447\pi$$
$$522$$ 0 0
$$523$$ 26.1457 1.14327 0.571635 0.820508i $$-0.306309\pi$$
0.571635 + 0.820508i $$0.306309\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 34.2029 1.48990
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 69.3670 3.00462
$$534$$ 0 0
$$535$$ 2.28506 0.0987919
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −23.7552 −1.02321
$$540$$ 0 0
$$541$$ −36.9994 −1.59073 −0.795365 0.606130i $$-0.792721\pi$$
−0.795365 + 0.606130i $$0.792721\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −1.39991 −0.0599654
$$546$$ 0 0
$$547$$ 3.24644 0.138808 0.0694038 0.997589i $$-0.477890\pi$$
0.0694038 + 0.997589i $$0.477890\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8.94242 0.380960
$$552$$ 0 0
$$553$$ −1.34066 −0.0570105
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −45.8366 −1.94216 −0.971079 0.238760i $$-0.923259\pi$$
−0.971079 + 0.238760i $$0.923259\pi$$
$$558$$ 0 0
$$559$$ 9.78336 0.413792
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −37.4399 −1.57791 −0.788953 0.614454i $$-0.789376\pi$$
−0.788953 + 0.614454i $$0.789376\pi$$
$$564$$ 0 0
$$565$$ −9.59977 −0.403865
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −15.8277 −0.663533 −0.331766 0.943362i $$-0.607645\pi$$
−0.331766 + 0.943362i $$0.607645\pi$$
$$570$$ 0 0
$$571$$ −33.8423 −1.41626 −0.708128 0.706084i $$-0.750460\pi$$
−0.708128 + 0.706084i $$0.750460\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −37.3490 −1.55486 −0.777429 0.628970i $$-0.783477\pi$$
−0.777429 + 0.628970i $$0.783477\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −0.132241 −0.00548627
$$582$$ 0 0
$$583$$ −29.6420 −1.22765
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −7.28941 −0.300866 −0.150433 0.988620i $$-0.548067\pi$$
−0.150433 + 0.988620i $$0.548067\pi$$
$$588$$ 0 0
$$589$$ −13.2391 −0.545509
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −23.4510 −0.963018 −0.481509 0.876441i $$-0.659911\pi$$
−0.481509 + 0.876441i $$0.659911\pi$$
$$594$$ 0 0
$$595$$ −0.411938 −0.0168878
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 13.6570 0.558011 0.279006 0.960289i $$-0.409995\pi$$
0.279006 + 0.960289i $$0.409995\pi$$
$$600$$ 0 0
$$601$$ 28.5752 1.16561 0.582804 0.812613i $$-0.301956\pi$$
0.582804 + 0.812613i $$0.301956\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0.559357 0.0227411
$$606$$ 0 0
$$607$$ −32.1335 −1.30426 −0.652129 0.758108i $$-0.726124\pi$$
−0.652129 + 0.758108i $$0.726124\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 10.6781 0.431988
$$612$$ 0 0
$$613$$ 21.0366 0.849660 0.424830 0.905273i $$-0.360334\pi$$
0.424830 + 0.905273i $$0.360334\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 38.7378 1.55953 0.779763 0.626075i $$-0.215340\pi$$
0.779763 + 0.626075i $$0.215340\pi$$
$$618$$ 0 0
$$619$$ 27.4941 1.10508 0.552540 0.833486i $$-0.313659\pi$$
0.552540 + 0.833486i $$0.313659\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0.201856 0.00808718
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −28.1654 −1.12303
$$630$$ 0 0
$$631$$ 6.22378 0.247765 0.123882 0.992297i $$-0.460465\pi$$
0.123882 + 0.992297i $$0.460465\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 6.73146 0.267130
$$636$$ 0 0
$$637$$ −43.8364 −1.73686
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 21.6929 0.856816 0.428408 0.903585i $$-0.359075\pi$$
0.428408 + 0.903585i $$0.359075\pi$$
$$642$$ 0 0
$$643$$ −23.2411 −0.916538 −0.458269 0.888813i $$-0.651530\pi$$
−0.458269 + 0.888813i $$0.651530\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.2830 0.482893 0.241446 0.970414i $$-0.422378\pi$$
0.241446 + 0.970414i $$0.422378\pi$$
$$648$$ 0 0
$$649$$ −30.4627 −1.19577
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 4.12546 0.161442 0.0807209 0.996737i $$-0.474278\pi$$
0.0807209 + 0.996737i $$0.474278\pi$$
$$654$$ 0 0
$$655$$ 4.22780 0.165194
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −3.05758 −0.119107 −0.0595533 0.998225i $$-0.518968\pi$$
−0.0595533 + 0.998225i $$0.518968\pi$$
$$660$$ 0 0
$$661$$ −41.6105 −1.61846 −0.809230 0.587492i $$-0.800115\pi$$
−0.809230 + 0.587492i $$0.800115\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0.159451 0.00618326
$$666$$ 0 0
$$667$$ −6.38787 −0.247339
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6.23314 −0.240628
$$672$$ 0 0
$$673$$ −21.8495 −0.842237 −0.421119 0.907006i $$-0.638362\pi$$
−0.421119 + 0.907006i $$0.638362\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 13.8625 0.532779 0.266390 0.963865i $$-0.414169\pi$$
0.266390 + 0.963865i $$0.414169\pi$$
$$678$$ 0 0
$$679$$ 0.0149626 0.000574212 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −27.3016 −1.04467 −0.522333 0.852742i $$-0.674938\pi$$
−0.522333 + 0.852742i $$0.674938\pi$$
$$684$$ 0 0
$$685$$ 14.4496 0.552092
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −54.6995 −2.08389
$$690$$ 0 0
$$691$$ 35.7403 1.35962 0.679812 0.733387i $$-0.262061\pi$$
0.679812 + 0.733387i $$0.262061\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 11.9870 0.454694
$$696$$ 0 0
$$697$$ 39.9865 1.51460
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.25046 0.0472292 0.0236146 0.999721i $$-0.492483\pi$$
0.0236146 + 0.999721i $$0.492483\pi$$
$$702$$ 0 0
$$703$$ 10.9022 0.411182
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1.07073 0.0402691
$$708$$ 0 0
$$709$$ −38.2680 −1.43718 −0.718592 0.695432i $$-0.755213\pi$$
−0.718592 + 0.695432i $$0.755213\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 9.45717 0.354174
$$714$$ 0 0
$$715$$ 21.3309 0.797731
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 21.1228 0.787749 0.393874 0.919164i $$-0.371135\pi$$
0.393874 + 0.919164i $$0.371135\pi$$
$$720$$ 0 0
$$721$$ −1.05250 −0.0391972
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −6.38787 −0.237240
$$726$$ 0 0
$$727$$ −29.9281 −1.10997 −0.554986 0.831859i $$-0.687277\pi$$
−0.554986 + 0.831859i $$0.687277\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 5.63960 0.208588
$$732$$ 0 0
$$733$$ −39.5600 −1.46118 −0.730591 0.682816i $$-0.760755\pi$$
−0.730591 + 0.682816i $$0.760755\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 31.3327 1.15416
$$738$$ 0 0
$$739$$ −21.1460 −0.777868 −0.388934 0.921266i $$-0.627157\pi$$
−0.388934 + 0.921266i $$0.627157\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −41.0959 −1.50766 −0.753831 0.657068i $$-0.771796\pi$$
−0.753831 + 0.657068i $$0.771796\pi$$
$$744$$ 0 0
$$745$$ 4.19972 0.153866
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −0.260272 −0.00951013
$$750$$ 0 0
$$751$$ −14.3735 −0.524498 −0.262249 0.965000i $$-0.584464\pi$$
−0.262249 + 0.965000i $$0.584464\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 7.55936 0.275113
$$756$$ 0 0
$$757$$ −25.7135 −0.934574 −0.467287 0.884106i $$-0.654768\pi$$
−0.467287 + 0.884106i $$0.654768\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 29.4516 1.06762 0.533811 0.845604i $$-0.320759\pi$$
0.533811 + 0.845604i $$0.320759\pi$$
$$762$$ 0 0
$$763$$ 0.159451 0.00577252
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −56.2140 −2.02977
$$768$$ 0 0
$$769$$ 7.92378 0.285739 0.142869 0.989742i $$-0.454367\pi$$
0.142869 + 0.989742i $$0.454367\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −6.74082 −0.242450 −0.121225 0.992625i $$-0.538682\pi$$
−0.121225 + 0.992625i $$0.538682\pi$$
$$774$$ 0 0
$$775$$ 9.45717 0.339711
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −15.4778 −0.554550
$$780$$ 0 0
$$781$$ −23.2163 −0.830745
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 5.86203 0.209225
$$786$$ 0 0
$$787$$ −36.8407 −1.31323 −0.656614 0.754226i $$-0.728012\pi$$
−0.656614 + 0.754226i $$0.728012\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1.09343 0.0388778
$$792$$ 0 0
$$793$$ −11.5022 −0.408457
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −38.4548 −1.36214 −0.681070 0.732219i $$-0.738485\pi$$
−0.681070 + 0.732219i $$0.738485\pi$$
$$798$$ 0 0
$$799$$ 6.15535 0.217761
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −7.73129 −0.272831
$$804$$ 0 0
$$805$$ −0.113901 −0.00401450
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −22.2100 −0.780861 −0.390430 0.920632i $$-0.627674\pi$$
−0.390430 + 0.920632i $$0.627674\pi$$
$$810$$ 0 0
$$811$$ −18.7647 −0.658916 −0.329458 0.944170i $$-0.606866\pi$$
−0.329458 + 0.944170i $$0.606866\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0.766767 0.0268587
$$816$$ 0 0
$$817$$ −2.18295 −0.0763718
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 34.5882 1.20714 0.603568 0.797311i $$-0.293745\pi$$
0.603568 + 0.797311i $$0.293745\pi$$
$$822$$ 0 0
$$823$$ −46.0642 −1.60570 −0.802849 0.596183i $$-0.796683\pi$$
−0.802849 + 0.596183i $$0.796683\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −7.24847 −0.252054 −0.126027 0.992027i $$-0.540223\pi$$
−0.126027 + 0.992027i $$0.540223\pi$$
$$828$$ 0 0
$$829$$ −14.7688 −0.512940 −0.256470 0.966552i $$-0.582559\pi$$
−0.256470 + 0.966552i $$0.582559\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −25.2694 −0.875533
$$834$$ 0 0
$$835$$ −14.0443 −0.486023
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −13.3292 −0.460173 −0.230087 0.973170i $$-0.573901\pi$$
−0.230087 + 0.973170i $$0.573901\pi$$
$$840$$ 0 0
$$841$$ 11.8049 0.407066
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 26.3627 0.906905
$$846$$ 0 0
$$847$$ −0.0637116 −0.00218915
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −7.78778 −0.266962
$$852$$ 0 0
$$853$$ 11.0307 0.377685 0.188843 0.982007i $$-0.439526\pi$$
0.188843 + 0.982007i $$0.439526\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 12.1979 0.416672 0.208336 0.978057i $$-0.433195\pi$$
0.208336 + 0.978057i $$0.433195\pi$$
$$858$$ 0 0
$$859$$ −26.5718 −0.906617 −0.453309 0.891354i $$-0.649756\pi$$
−0.453309 + 0.891354i $$0.649756\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −23.0276 −0.783869 −0.391935 0.919993i $$-0.628194\pi$$
−0.391935 + 0.919993i $$0.628194\pi$$
$$864$$ 0 0
$$865$$ −2.13859 −0.0727141
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 40.0180 1.35752
$$870$$ 0 0
$$871$$ 57.8195 1.95914
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −0.113901 −0.00385057
$$876$$ 0 0
$$877$$ −46.8332 −1.58144 −0.790722 0.612176i $$-0.790294\pi$$
−0.790722 + 0.612176i $$0.790294\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 17.0810 0.575474 0.287737 0.957710i $$-0.407097\pi$$
0.287737 + 0.957710i $$0.407097\pi$$
$$882$$ 0 0
$$883$$ 7.21425 0.242779 0.121389 0.992605i $$-0.461265\pi$$
0.121389 + 0.992605i $$0.461265\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 16.6625 0.559471 0.279735 0.960077i $$-0.409753\pi$$
0.279735 + 0.960077i $$0.409753\pi$$
$$888$$ 0 0
$$889$$ −0.766723 −0.0257151
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −2.38259 −0.0797303
$$894$$ 0 0
$$895$$ 8.93118 0.298536
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −60.4112 −2.01483
$$900$$ 0 0
$$901$$ −31.5314 −1.05046
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 25.4623 0.846395
$$906$$ 0 0
$$907$$ 28.0024 0.929804 0.464902 0.885362i $$-0.346090\pi$$
0.464902 + 0.885362i $$0.346090\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −1.45915 −0.0483439 −0.0241720 0.999708i $$-0.507695\pi$$
−0.0241720 + 0.999708i $$0.507695\pi$$
$$912$$ 0 0
$$913$$ 3.94733 0.130637
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −0.481553 −0.0159023
$$918$$ 0 0
$$919$$ 26.8825 0.886773 0.443386 0.896331i $$-0.353777\pi$$
0.443386 + 0.896331i $$0.353777\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −42.8419 −1.41016
$$924$$ 0 0
$$925$$ −7.78778 −0.256061
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −25.3843 −0.832833 −0.416416 0.909174i $$-0.636714\pi$$
−0.416416 + 0.909174i $$0.636714\pi$$
$$930$$ 0 0
$$931$$ 9.78118 0.320565
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 12.2962 0.402127
$$936$$ 0 0
$$937$$ 13.4201 0.438416 0.219208 0.975678i $$-0.429653\pi$$
0.219208 + 0.975678i $$0.429653\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 40.3482 1.31531 0.657657 0.753317i $$-0.271548\pi$$
0.657657 + 0.753317i $$0.271548\pi$$
$$942$$ 0 0
$$943$$ 11.0563 0.360043
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −40.6974 −1.32249 −0.661244 0.750171i $$-0.729971\pi$$
−0.661244 + 0.750171i $$0.729971\pi$$
$$948$$ 0 0
$$949$$ −14.2668 −0.463121
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 41.7552 1.35258 0.676292 0.736633i $$-0.263586\pi$$
0.676292 + 0.736633i $$0.263586\pi$$
$$954$$ 0 0
$$955$$ −12.3813 −0.400649
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −1.64583 −0.0531467
$$960$$ 0 0
$$961$$ 58.4380 1.88510
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 3.31471 0.106704
$$966$$ 0 0
$$967$$ −34.4343 −1.10733 −0.553666 0.832739i $$-0.686772\pi$$
−0.553666 + 0.832739i $$0.686772\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 6.96276 0.223446 0.111723 0.993739i $$-0.464363\pi$$
0.111723 + 0.993739i $$0.464363\pi$$
$$972$$ 0 0
$$973$$ −1.36534 −0.0437708
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −36.7569 −1.17596 −0.587978 0.808877i $$-0.700076\pi$$
−0.587978 + 0.808877i $$0.700076\pi$$
$$978$$ 0 0
$$979$$ −6.02530 −0.192569
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 6.56822 0.209494 0.104747 0.994499i $$-0.466597\pi$$
0.104747 + 0.994499i $$0.466597\pi$$
$$984$$ 0 0
$$985$$ −16.3201 −0.520003
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.55936 0.0495847
$$990$$ 0 0
$$991$$ 50.0562 1.59009 0.795044 0.606551i $$-0.207448\pi$$
0.795044 + 0.606551i $$0.207448\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −9.80336 −0.310787
$$996$$ 0 0
$$997$$ 14.7902 0.468410 0.234205 0.972187i $$-0.424751\pi$$
0.234205 + 0.972187i $$0.424751\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 4140.2.a.u.1.3 yes 5
3.2 odd 2 4140.2.a.t.1.3 5

By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.a.t.1.3 5 3.2 odd 2
4140.2.a.u.1.3 yes 5 1.1 even 1 trivial