# Properties

 Label 5850.2.a.e.1.1 Level $5850$ Weight $2$ Character 5850.1 Self dual yes Analytic conductor $46.712$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(1,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5850.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: $$46.7124851824$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 5850.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{7} -1.00000 q^{8} +6.00000 q^{11} +1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +2.00000 q^{19} -6.00000 q^{22} +6.00000 q^{23} -1.00000 q^{26} -4.00000 q^{28} +10.0000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -6.00000 q^{37} -2.00000 q^{38} -10.0000 q^{41} +6.00000 q^{44} -6.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} +1.00000 q^{52} +6.00000 q^{53} +4.00000 q^{56} -10.0000 q^{58} +6.00000 q^{59} -6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -12.0000 q^{67} +4.00000 q^{68} +2.00000 q^{73} +6.00000 q^{74} +2.00000 q^{76} -24.0000 q^{77} -8.00000 q^{79} +10.0000 q^{82} +4.00000 q^{83} -6.00000 q^{88} -14.0000 q^{89} -4.00000 q^{91} +6.00000 q^{92} +8.00000 q^{94} +14.0000 q^{97} -9.00000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6.00000 1.80907 0.904534 0.426401i $$-0.140219\pi$$
0.904534 + 0.426401i $$0.140219\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 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 0
$$22$$ −6.00000 −1.27920
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ −4.00000 −0.755929
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ 0 0
$$58$$ −10.0000 −1.31306
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ −24.0000 −2.73505
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 10.0000 1.10432
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ −6.00000 −0.639602
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ 8.00000 0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ −9.00000 −0.909137
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ −2.00000 −0.197066 −0.0985329 0.995134i $$-0.531415\pi$$
−0.0985329 + 0.995134i $$0.531415\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −4.00000 −0.377964
$$113$$ 4.00000 0.376288 0.188144 0.982141i $$-0.439753\pi$$
0.188144 + 0.982141i $$0.439753\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 10.0000 0.928477
$$117$$ 0 0
$$118$$ −6.00000 −0.552345
$$119$$ −16.0000 −1.46672
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 6.00000 0.543214
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −10.0000 −0.887357 −0.443678 0.896186i $$-0.646327\pi$$
−0.443678 + 0.896186i $$0.646327\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ −8.00000 −0.693688
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 6.00000 0.501745
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 0 0
$$148$$ −6.00000 −0.493197
$$149$$ −16.0000 −1.31077 −0.655386 0.755295i $$-0.727494\pi$$
−0.655386 + 0.755295i $$0.727494\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 0 0
$$154$$ 24.0000 1.93398
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −24.0000 −1.89146
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −10.0000 −0.780869
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 22.0000 1.67263 0.836315 0.548250i $$-0.184706\pi$$
0.836315 + 0.548250i $$0.184706\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6.00000 0.452267
$$177$$ 0 0
$$178$$ 14.0000 1.04934
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 24.0000 1.75505
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 10.0000 0.703598
$$203$$ −40.0000 −2.80745
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2.00000 0.139347
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ −4.00000 −0.270914
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ −4.00000 −0.266076
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 0 0
$$229$$ 8.00000 0.528655 0.264327 0.964433i $$-0.414850\pi$$
0.264327 + 0.964433i $$0.414850\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −10.0000 −0.656532
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ 0 0
$$238$$ 16.0000 1.03713
$$239$$ 4.00000 0.258738 0.129369 0.991596i $$-0.458705\pi$$
0.129369 + 0.991596i $$0.458705\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ −25.0000 −1.60706
$$243$$ 0 0
$$244$$ −6.00000 −0.384111
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 36.0000 2.26330
$$254$$ 10.0000 0.627456
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −28.0000 −1.74659 −0.873296 0.487190i $$-0.838022\pi$$
−0.873296 + 0.487190i $$0.838022\pi$$
$$258$$ 0 0
$$259$$ 24.0000 1.49129
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −8.00000 −0.494242
$$263$$ 10.0000 0.616626 0.308313 0.951285i $$-0.400236\pi$$
0.308313 + 0.951285i $$0.400236\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −14.0000 −0.845771
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −14.0000 −0.841178 −0.420589 0.907251i $$-0.638177\pi$$
−0.420589 + 0.907251i $$0.638177\pi$$
$$278$$ −16.0000 −0.959616
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ 40.0000 2.36113
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ −18.0000 −1.05157 −0.525786 0.850617i $$-0.676229\pi$$
−0.525786 + 0.850617i $$0.676229\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ 16.0000 0.926855
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −16.0000 −0.920697
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ −24.0000 −1.36753
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ −8.00000 −0.452187 −0.226093 0.974106i $$-0.572595\pi$$
−0.226093 + 0.974106i $$0.572595\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 0 0
$$319$$ 60.0000 3.35936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 24.0000 1.33747
$$323$$ 8.00000 0.445132
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ 10.0000 0.552158
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 0 0
$$334$$ −16.0000 −0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 16.0000 0.871576 0.435788 0.900049i $$-0.356470\pi$$
0.435788 + 0.900049i $$0.356470\pi$$
$$338$$ −1.00000 −0.0543928
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −22.0000 −1.18273
$$347$$ −16.0000 −0.858925 −0.429463 0.903085i $$-0.641297\pi$$
−0.429463 + 0.903085i $$0.641297\pi$$
$$348$$ 0 0
$$349$$ −4.00000 −0.214115 −0.107058 0.994253i $$-0.534143\pi$$
−0.107058 + 0.994253i $$0.534143\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −6.00000 −0.319801
$$353$$ 26.0000 1.38384 0.691920 0.721974i $$-0.256765\pi$$
0.691920 + 0.721974i $$0.256765\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ −24.0000 −1.26844
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −6.00000 −0.315353
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −18.0000 −0.939592 −0.469796 0.882775i $$-0.655673\pi$$
−0.469796 + 0.882775i $$0.655673\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ 0 0
$$373$$ −18.0000 −0.932005 −0.466002 0.884783i $$-0.654306\pi$$
−0.466002 + 0.884783i $$0.654306\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 10.0000 0.515026
$$378$$ 0 0
$$379$$ 10.0000 0.513665 0.256833 0.966456i $$-0.417321\pi$$
0.256833 + 0.966456i $$0.417321\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −20.0000 −1.02195 −0.510976 0.859595i $$-0.670716\pi$$
−0.510976 + 0.859595i $$0.670716\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ 14.0000 0.710742
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ −9.00000 −0.454569
$$393$$ 0 0
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −10.0000 −0.501886 −0.250943 0.968002i $$-0.580741\pi$$
−0.250943 + 0.968002i $$0.580741\pi$$
$$398$$ 8.00000 0.401004
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 4.00000 0.199254
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 40.0000 1.98517
$$407$$ −36.0000 −1.78445
$$408$$ 0 0
$$409$$ −34.0000 −1.68119 −0.840596 0.541663i $$-0.817795\pi$$
−0.840596 + 0.541663i $$0.817795\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −2.00000 −0.0985329
$$413$$ −24.0000 −1.18096
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ −12.0000 −0.586939
$$419$$ −8.00000 −0.390826 −0.195413 0.980721i $$-0.562605\pi$$
−0.195413 + 0.980721i $$0.562605\pi$$
$$420$$ 0 0
$$421$$ 4.00000 0.194948 0.0974740 0.995238i $$-0.468924\pi$$
0.0974740 + 0.995238i $$0.468924\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 24.0000 1.16144
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ 4.00000 0.191565
$$437$$ 12.0000 0.574038
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −4.00000 −0.190261
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ 0 0
$$448$$ −4.00000 −0.188982
$$449$$ −38.0000 −1.79333 −0.896665 0.442709i $$-0.854018\pi$$
−0.896665 + 0.442709i $$0.854018\pi$$
$$450$$ 0 0
$$451$$ −60.0000 −2.82529
$$452$$ 4.00000 0.188144
$$453$$ 0 0
$$454$$ −4.00000 −0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 14.0000 0.654892 0.327446 0.944870i $$-0.393812\pi$$
0.327446 + 0.944870i $$0.393812\pi$$
$$458$$ −8.00000 −0.373815
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 4.00000 0.186299 0.0931493 0.995652i $$-0.470307\pi$$
0.0931493 + 0.995652i $$0.470307\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 10.0000 0.464238
$$465$$ 0 0
$$466$$ −12.0000 −0.555889
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 0 0
$$469$$ 48.0000 2.21643
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −6.00000 −0.276172
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −16.0000 −0.733359
$$477$$ 0 0
$$478$$ −4.00000 −0.182956
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ −18.0000 −0.819878
$$483$$ 0 0
$$484$$ 25.0000 1.13636
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 40.0000 1.80151
$$494$$ −2.00000 −0.0899843
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −18.0000 −0.805791 −0.402895 0.915246i $$-0.631996\pi$$
−0.402895 + 0.915246i $$0.631996\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −20.0000 −0.892644
$$503$$ 14.0000 0.624229 0.312115 0.950044i $$-0.398963\pi$$
0.312115 + 0.950044i $$0.398963\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −36.0000 −1.60040
$$507$$ 0 0
$$508$$ −10.0000 −0.443678
$$509$$ −16.0000 −0.709188 −0.354594 0.935020i $$-0.615381\pi$$
−0.354594 + 0.935020i $$0.615381\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 28.0000 1.23503
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −48.0000 −2.11104
$$518$$ −24.0000 −1.05450
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ −10.0000 −0.436021
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −8.00000 −0.346844
$$533$$ −10.0000 −0.433148
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 10.0000 0.431131
$$539$$ 54.0000 2.32594
$$540$$ 0 0
$$541$$ −32.0000 −1.37579 −0.687894 0.725811i $$-0.741464\pi$$
−0.687894 + 0.725811i $$0.741464\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ 14.0000 0.598050
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 20.0000 0.852029
$$552$$ 0 0
$$553$$ 32.0000 1.36078
$$554$$ 14.0000 0.594803
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −18.0000 −0.759284
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 6.00000 0.250873
$$573$$ 0 0
$$574$$ −40.0000 −1.66957
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −16.0000 −0.663792
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −6.00000 −0.246598
$$593$$ −14.0000 −0.574911 −0.287456 0.957794i $$-0.592809\pi$$
−0.287456 + 0.957794i $$0.592809\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −16.0000 −0.655386
$$597$$ 0 0
$$598$$ −6.00000 −0.245358
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 42.0000 1.70473 0.852364 0.522949i $$-0.175168\pi$$
0.852364 + 0.522949i $$0.175168\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 0 0
$$613$$ −38.0000 −1.53481 −0.767403 0.641165i $$-0.778451\pi$$
−0.767403 + 0.641165i $$0.778451\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 24.0000 0.966988
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ −22.0000 −0.884255 −0.442127 0.896952i $$-0.645776\pi$$
−0.442127 + 0.896952i $$0.645776\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −8.00000 −0.320771
$$623$$ 56.0000 2.24359
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 8.00000 0.319744
$$627$$ 0 0
$$628$$ 6.00000 0.239426
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 8.00000 0.318223
$$633$$ 0 0
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 9.00000 0.356593
$$638$$ −60.0000 −2.37542
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ −20.0000 −0.788723 −0.394362 0.918955i $$-0.629034\pi$$
−0.394362 + 0.918955i $$0.629034\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 10.0000 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ 0 0
$$658$$ −32.0000 −1.24749
$$659$$ 32.0000 1.24654 0.623272 0.782006i $$-0.285803\pi$$
0.623272 + 0.782006i $$0.285803\pi$$
$$660$$ 0 0
$$661$$ −32.0000 −1.24466 −0.622328 0.782757i $$-0.713813\pi$$
−0.622328 + 0.782757i $$0.713813\pi$$
$$662$$ −2.00000 −0.0777322
$$663$$ 0 0
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 60.0000 2.32321
$$668$$ 16.0000 0.619059
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −36.0000 −1.38976
$$672$$ 0 0
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ −16.0000 −0.616297
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ −56.0000 −2.14908
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −24.0000 −0.919007
$$683$$ 20.0000 0.765279 0.382639 0.923898i $$-0.375015\pi$$
0.382639 + 0.923898i $$0.375015\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ 18.0000 0.684752 0.342376 0.939563i $$-0.388768\pi$$
0.342376 + 0.939563i $$0.388768\pi$$
$$692$$ 22.0000 0.836315
$$693$$ 0 0
$$694$$ 16.0000 0.607352
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −40.0000 −1.51511
$$698$$ 4.00000 0.151402
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.00000 0.226617 0.113308 0.993560i $$-0.463855\pi$$
0.113308 + 0.993560i $$0.463855\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ 6.00000 0.226134
$$705$$ 0 0
$$706$$ −26.0000 −0.978523
$$707$$ 40.0000 1.50435
$$708$$ 0 0
$$709$$ −8.00000 −0.300446 −0.150223 0.988652i $$-0.547999\pi$$
−0.150223 + 0.988652i $$0.547999\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 14.0000 0.524672
$$713$$ 24.0000 0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 24.0000 0.896922
$$717$$ 0 0
$$718$$ −20.0000 −0.746393
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 15.0000 0.558242
$$723$$ 0 0
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 38.0000 1.40934 0.704671 0.709534i $$-0.251095\pi$$
0.704671 + 0.709534i $$0.251095\pi$$
$$728$$ 4.00000 0.148250
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −18.0000 −0.664845 −0.332423 0.943131i $$-0.607866\pi$$
−0.332423 + 0.943131i $$0.607866\pi$$
$$734$$ 18.0000 0.664392
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ −72.0000 −2.65215
$$738$$ 0 0
$$739$$ −2.00000 −0.0735712 −0.0367856 0.999323i $$-0.511712\pi$$
−0.0367856 + 0.999323i $$0.511712\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 24.0000 0.881068
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 18.0000 0.659027
$$747$$ 0 0
$$748$$ 24.0000 0.877527
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −48.0000 −1.75154 −0.875772 0.482724i $$-0.839647\pi$$
−0.875772 + 0.482724i $$0.839647\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ −10.0000 −0.364179
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ −10.0000 −0.363216
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ −16.0000 −0.579239
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 20.0000 0.722629
$$767$$ 6.00000 0.216647
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −2.00000 −0.0719816
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ −20.0000 −0.716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −24.0000 −0.858238
$$783$$ 0 0
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −12.0000 −0.427754 −0.213877 0.976861i $$-0.568609\pi$$
−0.213877 + 0.976861i $$0.568609\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −16.0000 −0.568895
$$792$$ 0 0
$$793$$ −6.00000 −0.213066
$$794$$ 10.0000 0.354887
$$795$$ 0 0
$$796$$ −8.00000 −0.283552
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 0 0
$$799$$ −32.0000 −1.13208
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 30.0000 1.05934
$$803$$ 12.0000 0.423471
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 10.0000 0.351799
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 0 0
$$811$$ −50.0000 −1.75574 −0.877869 0.478901i $$-0.841035\pi$$
−0.877869 + 0.478901i $$0.841035\pi$$
$$812$$ −40.0000 −1.40372
$$813$$ 0 0
$$814$$ 36.0000 1.26180
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 34.0000 1.18878
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 28.0000 0.977207 0.488603 0.872506i $$-0.337507\pi$$
0.488603 + 0.872506i $$0.337507\pi$$
$$822$$ 0 0
$$823$$ −14.0000 −0.488009 −0.244005 0.969774i $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ 2.00000 0.0696733
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 44.0000 1.53003 0.765015 0.644013i $$-0.222732\pi$$
0.765015 + 0.644013i $$0.222732\pi$$
$$828$$ 0 0
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 36.0000 1.24733
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 12.0000 0.415029
$$837$$ 0 0
$$838$$ 8.00000 0.276355
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ −4.00000 −0.137849
$$843$$ 0 0
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −100.000 −3.43604
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −36.0000 −1.23406
$$852$$ 0 0
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ −24.0000 −0.821263
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −16.0000 −0.544962
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −16.0000 −0.543702
$$867$$ 0 0
$$868$$ −16.0000 −0.543075
$$869$$ −48.0000 −1.62829
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ −4.00000 −0.135457
$$873$$ 0 0
$$874$$ −12.0000 −0.405906
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ 32.0000 1.07995
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ 40.0000 1.34611 0.673054 0.739594i $$-0.264982\pi$$
0.673054 + 0.739594i $$0.264982\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 6.00000 0.201460 0.100730 0.994914i $$-0.467882\pi$$
0.100730 + 0.994914i $$0.467882\pi$$
$$888$$ 0 0
$$889$$ 40.0000 1.34156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ −16.0000 −0.535420
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ 38.0000 1.26808
$$899$$ 40.0000 1.33407
$$900$$ 0 0
$$901$$ 24.0000 0.799556
$$902$$ 60.0000 1.99778
$$903$$ 0 0
$$904$$ −4.00000 −0.133038
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 40.0000 1.32818 0.664089 0.747653i $$-0.268820\pi$$
0.664089 + 0.747653i $$0.268820\pi$$
$$908$$ 4.00000 0.132745
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 0 0
$$913$$ 24.0000 0.794284
$$914$$ −14.0000 −0.463079
$$915$$ 0 0
$$916$$ 8.00000 0.264327
$$917$$ −32.0000 −1.05673
$$918$$ 0 0
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −4.00000 −0.131733
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ −10.0000 −0.328266
$$929$$ −46.0000 −1.50921 −0.754606 0.656179i $$-0.772172\pi$$
−0.754606 + 0.656179i $$0.772172\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ 12.0000 0.393073
$$933$$ 0 0
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 48.0000 1.56809 0.784046 0.620703i $$-0.213153\pi$$
0.784046 + 0.620703i $$0.213153\pi$$
$$938$$ −48.0000 −1.56726
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 8.00000 0.260793 0.130396 0.991462i $$-0.458375\pi$$
0.130396 + 0.991462i $$0.458375\pi$$
$$942$$ 0 0
$$943$$ −60.0000 −1.95387
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ 2.00000 0.0649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 16.0000 0.518563
$$953$$ 24.0000 0.777436 0.388718 0.921357i $$-0.372918\pi$$
0.388718 + 0.921357i $$0.372918\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 4.00000 0.129369
$$957$$ 0 0
$$958$$ −20.0000 −0.646171
$$959$$ −56.0000 −1.80833
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 6.00000 0.193448
$$963$$ 0 0
$$964$$ 18.0000 0.579741
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ −25.0000 −0.803530
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ −64.0000 −2.05175
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ −58.0000 −1.85558 −0.927792 0.373097i $$-0.878296\pi$$
−0.927792 + 0.373097i $$0.878296\pi$$
$$978$$ 0 0
$$979$$ −84.0000 −2.68465
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 28.0000 0.893061 0.446531 0.894768i $$-0.352659\pi$$
0.446531 + 0.894768i $$0.352659\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −40.0000 −1.27386
$$987$$ 0 0
$$988$$ 2.00000 0.0636285
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −32.0000 −1.01651 −0.508257 0.861206i $$-0.669710\pi$$
−0.508257 + 0.861206i $$0.669710\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 18.0000 0.569780
$$999$$ 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 5850.2.a.e.1.1 1
3.2 odd 2 1950.2.a.v.1.1 1
5.2 odd 4 1170.2.e.c.469.1 2
5.3 odd 4 1170.2.e.c.469.2 2
5.4 even 2 5850.2.a.cc.1.1 1
15.2 even 4 390.2.e.d.79.2 yes 2
15.8 even 4 390.2.e.d.79.1 2
15.14 odd 2 1950.2.a.d.1.1 1
60.23 odd 4 3120.2.l.i.1249.1 2
60.47 odd 4 3120.2.l.i.1249.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.e.d.79.1 2 15.8 even 4
390.2.e.d.79.2 yes 2 15.2 even 4
1170.2.e.c.469.1 2 5.2 odd 4
1170.2.e.c.469.2 2 5.3 odd 4
1950.2.a.d.1.1 1 15.14 odd 2
1950.2.a.v.1.1 1 3.2 odd 2
3120.2.l.i.1249.1 2 60.23 odd 4
3120.2.l.i.1249.2 2 60.47 odd 4
5850.2.a.e.1.1 1 1.1 even 1 trivial
5850.2.a.cc.1.1 1 5.4 even 2