# Properties

 Label 4600.2.a.y.1.2 Level $4600$ Weight $2$ Character 4600.1 Self dual yes Analytic conductor $36.731$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.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: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.523976$$ of defining polynomial Character $$\chi$$ $$=$$ 4600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.523976 q^{3} -0.476024 q^{7} -2.72545 q^{9} +O(q^{10})$$ $$q+0.523976 q^{3} -0.476024 q^{7} -2.72545 q^{9} -1.67750 q^{11} -2.67750 q^{13} -1.67750 q^{17} +7.92692 q^{19} -0.249425 q^{21} -1.00000 q^{23} -3.00000 q^{27} -2.20147 q^{29} +6.77340 q^{31} -0.878968 q^{33} -4.00000 q^{37} -1.40294 q^{39} +5.97487 q^{41} +0.402945 q^{43} -1.79853 q^{47} -6.77340 q^{49} -0.878968 q^{51} +10.8059 q^{53} +4.15352 q^{57} +9.45090 q^{59} +6.32250 q^{61} +1.29738 q^{63} -14.7100 q^{67} -0.523976 q^{69} +9.97487 q^{71} +6.10557 q^{73} +0.798528 q^{77} +6.60442 q^{81} -4.49885 q^{83} -1.15352 q^{87} -3.59706 q^{89} +1.27455 q^{91} +3.54910 q^{93} +6.08044 q^{97} +4.57193 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^7 + 3 * q^9 $$3 q - 3 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{17} + 3 q^{19} + 12 q^{21} - 3 q^{23} - 9 q^{27} + 3 q^{29} + 6 q^{31} + 15 q^{33} - 12 q^{37} + 15 q^{39} - 6 q^{41} - 18 q^{43} - 15 q^{47} - 6 q^{49} + 15 q^{51} - 6 q^{53} + 6 q^{57} + 6 q^{59} + 27 q^{61} - 12 q^{63} - 12 q^{67} + 6 q^{71} + 15 q^{73} + 12 q^{77} - 9 q^{81} + 12 q^{83} + 3 q^{87} - 30 q^{89} + 15 q^{91} + 33 q^{93} - 9 q^{97} + 9 q^{99}+O(q^{100})$$ 3 * q - 3 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^17 + 3 * q^19 + 12 * q^21 - 3 * q^23 - 9 * q^27 + 3 * q^29 + 6 * q^31 + 15 * q^33 - 12 * q^37 + 15 * q^39 - 6 * q^41 - 18 * q^43 - 15 * q^47 - 6 * q^49 + 15 * q^51 - 6 * q^53 + 6 * q^57 + 6 * q^59 + 27 * q^61 - 12 * q^63 - 12 * q^67 + 6 * q^71 + 15 * q^73 + 12 * q^77 - 9 * q^81 + 12 * q^83 + 3 * q^87 - 30 * q^89 + 15 * q^91 + 33 * q^93 - 9 * q^97 + 9 * q^99

## 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
$$3$$ 0.523976 0.302518 0.151259 0.988494i $$-0.451667\pi$$
0.151259 + 0.988494i $$0.451667\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.476024 −0.179920 −0.0899600 0.995945i $$-0.528674\pi$$
−0.0899600 + 0.995945i $$0.528674\pi$$
$$8$$ 0 0
$$9$$ −2.72545 −0.908483
$$10$$ 0 0
$$11$$ −1.67750 −0.505784 −0.252892 0.967495i $$-0.581382\pi$$
−0.252892 + 0.967495i $$0.581382\pi$$
$$12$$ 0 0
$$13$$ −2.67750 −0.742604 −0.371302 0.928512i $$-0.621089\pi$$
−0.371302 + 0.928512i $$0.621089\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.67750 −0.406853 −0.203426 0.979090i $$-0.565208\pi$$
−0.203426 + 0.979090i $$0.565208\pi$$
$$18$$ 0 0
$$19$$ 7.92692 1.81856 0.909280 0.416184i $$-0.136633\pi$$
0.909280 + 0.416184i $$0.136633\pi$$
$$20$$ 0 0
$$21$$ −0.249425 −0.0544290
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −3.00000 −0.577350
$$28$$ 0 0
$$29$$ −2.20147 −0.408803 −0.204402 0.978887i $$-0.565525\pi$$
−0.204402 + 0.978887i $$0.565525\pi$$
$$30$$ 0 0
$$31$$ 6.77340 1.21654 0.608269 0.793731i $$-0.291864\pi$$
0.608269 + 0.793731i $$0.291864\pi$$
$$32$$ 0 0
$$33$$ −0.878968 −0.153009
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ −1.40294 −0.224651
$$40$$ 0 0
$$41$$ 5.97487 0.933119 0.466559 0.884490i $$-0.345493\pi$$
0.466559 + 0.884490i $$0.345493\pi$$
$$42$$ 0 0
$$43$$ 0.402945 0.0614485 0.0307242 0.999528i $$-0.490219\pi$$
0.0307242 + 0.999528i $$0.490219\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −1.79853 −0.262342 −0.131171 0.991360i $$-0.541874\pi$$
−0.131171 + 0.991360i $$0.541874\pi$$
$$48$$ 0 0
$$49$$ −6.77340 −0.967629
$$50$$ 0 0
$$51$$ −0.878968 −0.123080
$$52$$ 0 0
$$53$$ 10.8059 1.48430 0.742152 0.670232i $$-0.233805\pi$$
0.742152 + 0.670232i $$0.233805\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.15352 0.550147
$$58$$ 0 0
$$59$$ 9.45090 1.23040 0.615201 0.788370i $$-0.289075\pi$$
0.615201 + 0.788370i $$0.289075\pi$$
$$60$$ 0 0
$$61$$ 6.32250 0.809514 0.404757 0.914424i $$-0.367356\pi$$
0.404757 + 0.914424i $$0.367356\pi$$
$$62$$ 0 0
$$63$$ 1.29738 0.163454
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −14.7100 −1.79711 −0.898555 0.438860i $$-0.855382\pi$$
−0.898555 + 0.438860i $$0.855382\pi$$
$$68$$ 0 0
$$69$$ −0.523976 −0.0630793
$$70$$ 0 0
$$71$$ 9.97487 1.18380 0.591900 0.806012i $$-0.298378\pi$$
0.591900 + 0.806012i $$0.298378\pi$$
$$72$$ 0 0
$$73$$ 6.10557 0.714603 0.357301 0.933989i $$-0.383697\pi$$
0.357301 + 0.933989i $$0.383697\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0.798528 0.0910007
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 6.60442 0.733824
$$82$$ 0 0
$$83$$ −4.49885 −0.493813 −0.246906 0.969039i $$-0.579414\pi$$
−0.246906 + 0.969039i $$0.579414\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −1.15352 −0.123670
$$88$$ 0 0
$$89$$ −3.59706 −0.381287 −0.190644 0.981659i $$-0.561057\pi$$
−0.190644 + 0.981659i $$0.561057\pi$$
$$90$$ 0 0
$$91$$ 1.27455 0.133609
$$92$$ 0 0
$$93$$ 3.54910 0.368025
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.08044 0.617375 0.308688 0.951163i $$-0.400110\pi$$
0.308688 + 0.951163i $$0.400110\pi$$
$$98$$ 0 0
$$99$$ 4.57193 0.459496
$$100$$ 0 0
$$101$$ 11.4509 1.13941 0.569703 0.821850i $$-0.307058\pi$$
0.569703 + 0.821850i $$0.307058\pi$$
$$102$$ 0 0
$$103$$ 2.32250 0.228843 0.114422 0.993432i $$-0.463499\pi$$
0.114422 + 0.993432i $$0.463499\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.85384 0.759260 0.379630 0.925138i $$-0.376051\pi$$
0.379630 + 0.925138i $$0.376051\pi$$
$$108$$ 0 0
$$109$$ 13.9269 1.33396 0.666979 0.745077i $$-0.267587\pi$$
0.666979 + 0.745077i $$0.267587\pi$$
$$110$$ 0 0
$$111$$ −2.09591 −0.198935
$$112$$ 0 0
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 7.29738 0.674643
$$118$$ 0 0
$$119$$ 0.798528 0.0732009
$$120$$ 0 0
$$121$$ −8.18601 −0.744182
$$122$$ 0 0
$$123$$ 3.13069 0.282285
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 11.1033 0.985256 0.492628 0.870240i $$-0.336036\pi$$
0.492628 + 0.870240i $$0.336036\pi$$
$$128$$ 0 0
$$129$$ 0.211133 0.0185893
$$130$$ 0 0
$$131$$ 7.65237 0.668591 0.334295 0.942468i $$-0.391502\pi$$
0.334295 + 0.942468i $$0.391502\pi$$
$$132$$ 0 0
$$133$$ −3.77340 −0.327195
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 21.3778 1.82643 0.913215 0.407478i $$-0.133592\pi$$
0.913215 + 0.407478i $$0.133592\pi$$
$$138$$ 0 0
$$139$$ −8.60442 −0.729817 −0.364909 0.931043i $$-0.618900\pi$$
−0.364909 + 0.931043i $$0.618900\pi$$
$$140$$ 0 0
$$141$$ −0.942386 −0.0793632
$$142$$ 0 0
$$143$$ 4.49149 0.375597
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −3.54910 −0.292725
$$148$$ 0 0
$$149$$ 4.97487 0.407558 0.203779 0.979017i $$-0.434678\pi$$
0.203779 + 0.979017i $$0.434678\pi$$
$$150$$ 0 0
$$151$$ −0.926921 −0.0754318 −0.0377159 0.999289i $$-0.512008\pi$$
−0.0377159 + 0.999289i $$0.512008\pi$$
$$152$$ 0 0
$$153$$ 4.57193 0.369619
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −22.3527 −1.78394 −0.891970 0.452096i $$-0.850677\pi$$
−0.891970 + 0.452096i $$0.850677\pi$$
$$158$$ 0 0
$$159$$ 5.66203 0.449028
$$160$$ 0 0
$$161$$ 0.476024 0.0375159
$$162$$ 0 0
$$163$$ 7.93658 0.621641 0.310821 0.950469i $$-0.399396\pi$$
0.310821 + 0.950469i $$0.399396\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −12.8059 −0.990949 −0.495475 0.868622i $$-0.665006\pi$$
−0.495475 + 0.868622i $$0.665006\pi$$
$$168$$ 0 0
$$169$$ −5.83102 −0.448540
$$170$$ 0 0
$$171$$ −21.6044 −1.65213
$$172$$ 0 0
$$173$$ 24.6752 1.87602 0.938010 0.346608i $$-0.112666\pi$$
0.938010 + 0.346608i $$0.112666\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.95205 0.372219
$$178$$ 0 0
$$179$$ −13.1033 −0.979384 −0.489692 0.871895i $$-0.662891\pi$$
−0.489692 + 0.871895i $$0.662891\pi$$
$$180$$ 0 0
$$181$$ 9.52398 0.707912 0.353956 0.935262i $$-0.384836\pi$$
0.353956 + 0.935262i $$0.384836\pi$$
$$182$$ 0 0
$$183$$ 3.31284 0.244892
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.81399 0.205780
$$188$$ 0 0
$$189$$ 1.42807 0.103877
$$190$$ 0 0
$$191$$ 5.85384 0.423569 0.211785 0.977316i $$-0.432072\pi$$
0.211785 + 0.977316i $$0.432072\pi$$
$$192$$ 0 0
$$193$$ −4.41261 −0.317626 −0.158813 0.987309i $$-0.550767\pi$$
−0.158813 + 0.987309i $$0.550767\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −11.4258 −0.814053 −0.407026 0.913416i $$-0.633434\pi$$
−0.407026 + 0.913416i $$0.633434\pi$$
$$198$$ 0 0
$$199$$ −0.307039 −0.0217654 −0.0108827 0.999941i $$-0.503464\pi$$
−0.0108827 + 0.999941i $$0.503464\pi$$
$$200$$ 0 0
$$201$$ −7.70768 −0.543658
$$202$$ 0 0
$$203$$ 1.04795 0.0735519
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 2.72545 0.189432
$$208$$ 0 0
$$209$$ −13.2974 −0.919799
$$210$$ 0 0
$$211$$ 17.3047 1.19131 0.595654 0.803241i $$-0.296893\pi$$
0.595654 + 0.803241i $$0.296893\pi$$
$$212$$ 0 0
$$213$$ 5.22660 0.358121
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.22430 −0.218880
$$218$$ 0 0
$$219$$ 3.19917 0.216180
$$220$$ 0 0
$$221$$ 4.49149 0.302130
$$222$$ 0 0
$$223$$ 1.90409 0.127508 0.0637538 0.997966i $$-0.479693\pi$$
0.0637538 + 0.997966i $$0.479693\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −17.3550 −1.15189 −0.575946 0.817488i $$-0.695366\pi$$
−0.575946 + 0.817488i $$0.695366\pi$$
$$228$$ 0 0
$$229$$ −6.19181 −0.409166 −0.204583 0.978849i $$-0.565584\pi$$
−0.204583 + 0.978849i $$0.565584\pi$$
$$230$$ 0 0
$$231$$ 0.418410 0.0275293
$$232$$ 0 0
$$233$$ 15.7985 1.03500 0.517498 0.855684i $$-0.326863\pi$$
0.517498 + 0.855684i $$0.326863\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6.34763 0.410594 0.205297 0.978700i $$-0.434184\pi$$
0.205297 + 0.978700i $$0.434184\pi$$
$$240$$ 0 0
$$241$$ 5.69296 0.366716 0.183358 0.983046i $$-0.441303\pi$$
0.183358 + 0.983046i $$0.441303\pi$$
$$242$$ 0 0
$$243$$ 12.4606 0.799345
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −21.2243 −1.35047
$$248$$ 0 0
$$249$$ −2.35729 −0.149387
$$250$$ 0 0
$$251$$ −28.9246 −1.82571 −0.912853 0.408288i $$-0.866126\pi$$
−0.912853 + 0.408288i $$0.866126\pi$$
$$252$$ 0 0
$$253$$ 1.67750 0.105463
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 14.5085 0.905016 0.452508 0.891760i $$-0.350529\pi$$
0.452508 + 0.891760i $$0.350529\pi$$
$$258$$ 0 0
$$259$$ 1.90409 0.118315
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 15.7734 0.972630 0.486315 0.873784i $$-0.338341\pi$$
0.486315 + 0.873784i $$0.338341\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −1.88477 −0.115346
$$268$$ 0 0
$$269$$ −6.20147 −0.378110 −0.189055 0.981966i $$-0.560543\pi$$
−0.189055 + 0.981966i $$0.560543\pi$$
$$270$$ 0 0
$$271$$ −25.5696 −1.55324 −0.776622 0.629967i $$-0.783069\pi$$
−0.776622 + 0.629967i $$0.783069\pi$$
$$272$$ 0 0
$$273$$ 0.667835 0.0404192
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −7.03829 −0.422890 −0.211445 0.977390i $$-0.567817\pi$$
−0.211445 + 0.977390i $$0.567817\pi$$
$$278$$ 0 0
$$279$$ −18.4606 −1.10520
$$280$$ 0 0
$$281$$ 29.0627 1.73373 0.866867 0.498540i $$-0.166130\pi$$
0.866867 + 0.498540i $$0.166130\pi$$
$$282$$ 0 0
$$283$$ 18.6141 1.10649 0.553246 0.833018i $$-0.313389\pi$$
0.553246 + 0.833018i $$0.313389\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −2.84418 −0.167887
$$288$$ 0 0
$$289$$ −14.1860 −0.834471
$$290$$ 0 0
$$291$$ 3.18601 0.186767
$$292$$ 0 0
$$293$$ −23.4006 −1.36708 −0.683540 0.729913i $$-0.739561\pi$$
−0.683540 + 0.729913i $$0.739561\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.03249 0.292015
$$298$$ 0 0
$$299$$ 2.67750 0.154844
$$300$$ 0 0
$$301$$ −0.191811 −0.0110558
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1.53134 0.0873981 0.0436990 0.999045i $$-0.486086\pi$$
0.0436990 + 0.999045i $$0.486086\pi$$
$$308$$ 0 0
$$309$$ 1.21694 0.0692291
$$310$$ 0 0
$$311$$ −7.55646 −0.428488 −0.214244 0.976780i $$-0.568729\pi$$
−0.214244 + 0.976780i $$0.568729\pi$$
$$312$$ 0 0
$$313$$ 2.32987 0.131692 0.0658459 0.997830i $$-0.479025\pi$$
0.0658459 + 0.997830i $$0.479025\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 15.3778 0.863704 0.431852 0.901944i $$-0.357860\pi$$
0.431852 + 0.901944i $$0.357860\pi$$
$$318$$ 0 0
$$319$$ 3.69296 0.206766
$$320$$ 0 0
$$321$$ 4.11523 0.229690
$$322$$ 0 0
$$323$$ −13.2974 −0.739886
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 7.29738 0.403546
$$328$$ 0 0
$$329$$ 0.856142 0.0472006
$$330$$ 0 0
$$331$$ 22.9115 1.25933 0.629664 0.776868i $$-0.283193\pi$$
0.629664 + 0.776868i $$0.283193\pi$$
$$332$$ 0 0
$$333$$ 10.9018 0.597415
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −26.5410 −1.44578 −0.722890 0.690963i $$-0.757187\pi$$
−0.722890 + 0.690963i $$0.757187\pi$$
$$338$$ 0 0
$$339$$ 5.23976 0.284585
$$340$$ 0 0
$$341$$ −11.3624 −0.615306
$$342$$ 0 0
$$343$$ 6.55646 0.354016
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 3.27455 0.175787 0.0878936 0.996130i $$-0.471986\pi$$
0.0878936 + 0.996130i $$0.471986\pi$$
$$348$$ 0 0
$$349$$ 32.0553 1.71588 0.857941 0.513749i $$-0.171744\pi$$
0.857941 + 0.513749i $$0.171744\pi$$
$$350$$ 0 0
$$351$$ 8.03249 0.428742
$$352$$ 0 0
$$353$$ −11.1535 −0.593642 −0.296821 0.954933i $$-0.595926\pi$$
−0.296821 + 0.954933i $$0.595926\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0.418410 0.0221446
$$358$$ 0 0
$$359$$ 22.4029 1.18238 0.591191 0.806532i $$-0.298658\pi$$
0.591191 + 0.806532i $$0.298658\pi$$
$$360$$ 0 0
$$361$$ 43.8361 2.30716
$$362$$ 0 0
$$363$$ −4.28927 −0.225129
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −14.2877 −0.745813 −0.372906 0.927869i $$-0.621639\pi$$
−0.372906 + 0.927869i $$0.621639\pi$$
$$368$$ 0 0
$$369$$ −16.2842 −0.847722
$$370$$ 0 0
$$371$$ −5.14386 −0.267056
$$372$$ 0 0
$$373$$ 1.23976 0.0641925 0.0320963 0.999485i $$-0.489782\pi$$
0.0320963 + 0.999485i $$0.489782\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 5.89443 0.303579
$$378$$ 0 0
$$379$$ 4.38748 0.225370 0.112685 0.993631i $$-0.464055\pi$$
0.112685 + 0.993631i $$0.464055\pi$$
$$380$$ 0 0
$$381$$ 5.81785 0.298057
$$382$$ 0 0
$$383$$ −18.4989 −0.945247 −0.472624 0.881264i $$-0.656693\pi$$
−0.472624 + 0.881264i $$0.656693\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.09821 −0.0558249
$$388$$ 0 0
$$389$$ 21.6775 1.09909 0.549546 0.835463i $$-0.314801\pi$$
0.549546 + 0.835463i $$0.314801\pi$$
$$390$$ 0 0
$$391$$ 1.67750 0.0848346
$$392$$ 0 0
$$393$$ 4.00966 0.202261
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 20.6849 1.03814 0.519072 0.854731i $$-0.326278\pi$$
0.519072 + 0.854731i $$0.326278\pi$$
$$398$$ 0 0
$$399$$ −1.97717 −0.0989825
$$400$$ 0 0
$$401$$ −28.5638 −1.42641 −0.713205 0.700956i $$-0.752757\pi$$
−0.713205 + 0.700956i $$0.752757\pi$$
$$402$$ 0 0
$$403$$ −18.1358 −0.903406
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 6.70998 0.332602
$$408$$ 0 0
$$409$$ 16.7734 0.829391 0.414696 0.909960i $$-0.363888\pi$$
0.414696 + 0.909960i $$0.363888\pi$$
$$410$$ 0 0
$$411$$ 11.2015 0.552528
$$412$$ 0 0
$$413$$ −4.49885 −0.221374
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −4.50851 −0.220783
$$418$$ 0 0
$$419$$ −4.59476 −0.224469 −0.112234 0.993682i $$-0.535801\pi$$
−0.112234 + 0.993682i $$0.535801\pi$$
$$420$$ 0 0
$$421$$ 31.8310 1.55135 0.775674 0.631133i $$-0.217410\pi$$
0.775674 + 0.631133i $$0.217410\pi$$
$$422$$ 0 0
$$423$$ 4.90179 0.238333
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −3.00966 −0.145648
$$428$$ 0 0
$$429$$ 2.35343 0.113625
$$430$$ 0 0
$$431$$ −16.0650 −0.773823 −0.386911 0.922117i $$-0.626458\pi$$
−0.386911 + 0.922117i $$0.626458\pi$$
$$432$$ 0 0
$$433$$ 7.68486 0.369311 0.184655 0.982803i $$-0.440883\pi$$
0.184655 + 0.982803i $$0.440883\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −7.92692 −0.379196
$$438$$ 0 0
$$439$$ 15.9822 0.762790 0.381395 0.924412i $$-0.375444\pi$$
0.381395 + 0.924412i $$0.375444\pi$$
$$440$$ 0 0
$$441$$ 18.4606 0.879074
$$442$$ 0 0
$$443$$ 3.57929 0.170057 0.0850286 0.996379i $$-0.472902\pi$$
0.0850286 + 0.996379i $$0.472902\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 2.60672 0.123293
$$448$$ 0 0
$$449$$ −28.5217 −1.34602 −0.673011 0.739633i $$-0.734999\pi$$
−0.673011 + 0.739633i $$0.734999\pi$$
$$450$$ 0 0
$$451$$ −10.0228 −0.471956
$$452$$ 0 0
$$453$$ −0.485685 −0.0228195
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4.49885 0.210447 0.105224 0.994449i $$-0.466444\pi$$
0.105224 + 0.994449i $$0.466444\pi$$
$$458$$ 0 0
$$459$$ 5.03249 0.234896
$$460$$ 0 0
$$461$$ 2.29738 0.107000 0.0534998 0.998568i $$-0.482962\pi$$
0.0534998 + 0.998568i $$0.482962\pi$$
$$462$$ 0 0
$$463$$ 33.4966 1.55672 0.778358 0.627820i $$-0.216053\pi$$
0.778358 + 0.627820i $$0.216053\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 5.23976 0.242467 0.121234 0.992624i $$-0.461315\pi$$
0.121234 + 0.992624i $$0.461315\pi$$
$$468$$ 0 0
$$469$$ 7.00230 0.323336
$$470$$ 0 0
$$471$$ −11.7123 −0.539674
$$472$$ 0 0
$$473$$ −0.675938 −0.0310797
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −29.4509 −1.34846
$$478$$ 0 0
$$479$$ −2.40294 −0.109793 −0.0548967 0.998492i $$-0.517483\pi$$
−0.0548967 + 0.998492i $$0.517483\pi$$
$$480$$ 0 0
$$481$$ 10.7100 0.488333
$$482$$ 0 0
$$483$$ 0.249425 0.0113492
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −25.8944 −1.17339 −0.586694 0.809808i $$-0.699571\pi$$
−0.586694 + 0.809808i $$0.699571\pi$$
$$488$$ 0 0
$$489$$ 4.15858 0.188058
$$490$$ 0 0
$$491$$ −0.489189 −0.0220768 −0.0110384 0.999939i $$-0.503514\pi$$
−0.0110384 + 0.999939i $$0.503514\pi$$
$$492$$ 0 0
$$493$$ 3.69296 0.166323
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −4.74828 −0.212989
$$498$$ 0 0
$$499$$ −38.4583 −1.72163 −0.860814 0.508920i $$-0.830045\pi$$
−0.860814 + 0.508920i $$0.830045\pi$$
$$500$$ 0 0
$$501$$ −6.70998 −0.299780
$$502$$ 0 0
$$503$$ −37.8960 −1.68970 −0.844849 0.535004i $$-0.820310\pi$$
−0.844849 + 0.535004i $$0.820310\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −3.05531 −0.135691
$$508$$ 0 0
$$509$$ 8.70032 0.385635 0.192818 0.981235i $$-0.438237\pi$$
0.192818 + 0.981235i $$0.438237\pi$$
$$510$$ 0 0
$$511$$ −2.90639 −0.128571
$$512$$ 0 0
$$513$$ −23.7808 −1.04995
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.01702 0.132689
$$518$$ 0 0
$$519$$ 12.9292 0.567530
$$520$$ 0 0
$$521$$ 25.1129 1.10022 0.550109 0.835093i $$-0.314586\pi$$
0.550109 + 0.835093i $$0.314586\pi$$
$$522$$ 0 0
$$523$$ −12.7100 −0.555769 −0.277884 0.960615i $$-0.589633\pi$$
−0.277884 + 0.960615i $$0.589633\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −11.3624 −0.494952
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −25.7579 −1.11780
$$532$$ 0 0
$$533$$ −15.9977 −0.692937
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −6.86580 −0.296281
$$538$$ 0 0
$$539$$ 11.3624 0.489411
$$540$$ 0 0
$$541$$ −0.251725 −0.0108225 −0.00541124 0.999985i $$-0.501722\pi$$
−0.00541124 + 0.999985i $$0.501722\pi$$
$$542$$ 0 0
$$543$$ 4.99034 0.214156
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −24.3395 −1.04068 −0.520342 0.853958i $$-0.674195\pi$$
−0.520342 + 0.853958i $$0.674195\pi$$
$$548$$ 0 0
$$549$$ −17.2317 −0.735429
$$550$$ 0 0
$$551$$ −17.4509 −0.743433
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 23.4966 0.995581 0.497790 0.867297i $$-0.334145\pi$$
0.497790 + 0.867297i $$0.334145\pi$$
$$558$$ 0 0
$$559$$ −1.07888 −0.0456319
$$560$$ 0 0
$$561$$ 1.47447 0.0622520
$$562$$ 0 0
$$563$$ 19.5159 0.822496 0.411248 0.911523i $$-0.365093\pi$$
0.411248 + 0.911523i $$0.365093\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −3.14386 −0.132030
$$568$$ 0 0
$$569$$ −8.51817 −0.357100 −0.178550 0.983931i $$-0.557141\pi$$
−0.178550 + 0.983931i $$0.557141\pi$$
$$570$$ 0 0
$$571$$ −27.4811 −1.15005 −0.575024 0.818137i $$-0.695007\pi$$
−0.575024 + 0.818137i $$0.695007\pi$$
$$572$$ 0 0
$$573$$ 3.06728 0.128137
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 26.1512 1.08869 0.544345 0.838862i $$-0.316778\pi$$
0.544345 + 0.838862i $$0.316778\pi$$
$$578$$ 0 0
$$579$$ −2.31210 −0.0960877
$$580$$ 0 0
$$581$$ 2.14156 0.0888468
$$582$$ 0 0
$$583$$ −18.1268 −0.750737
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 34.3276 1.41685 0.708425 0.705786i $$-0.249406\pi$$
0.708425 + 0.705786i $$0.249406\pi$$
$$588$$ 0 0
$$589$$ 53.6922 2.21235
$$590$$ 0 0
$$591$$ −5.98683 −0.246265
$$592$$ 0 0
$$593$$ −15.9497 −0.654978 −0.327489 0.944855i $$-0.606202\pi$$
−0.327489 + 0.944855i $$0.606202\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −0.160881 −0.00658443
$$598$$ 0 0
$$599$$ 40.5940 1.65863 0.829313 0.558784i $$-0.188732\pi$$
0.829313 + 0.558784i $$0.188732\pi$$
$$600$$ 0 0
$$601$$ −29.0302 −1.18417 −0.592083 0.805877i $$-0.701694\pi$$
−0.592083 + 0.805877i $$0.701694\pi$$
$$602$$ 0 0
$$603$$ 40.0913 1.63264
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −16.9977 −0.689915 −0.344958 0.938618i $$-0.612107\pi$$
−0.344958 + 0.938618i $$0.612107\pi$$
$$608$$ 0 0
$$609$$ 0.549103 0.0222508
$$610$$ 0 0
$$611$$ 4.81555 0.194816
$$612$$ 0 0
$$613$$ −30.3024 −1.22390 −0.611952 0.790895i $$-0.709615\pi$$
−0.611952 + 0.790895i $$0.709615\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.13069 0.246812 0.123406 0.992356i $$-0.460618\pi$$
0.123406 + 0.992356i $$0.460618\pi$$
$$618$$ 0 0
$$619$$ −33.7305 −1.35574 −0.677872 0.735180i $$-0.737098\pi$$
−0.677872 + 0.735180i $$0.737098\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 0 0
$$623$$ 1.71228 0.0686012
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −6.96751 −0.278256
$$628$$ 0 0
$$629$$ 6.70998 0.267545
$$630$$ 0 0
$$631$$ 4.85614 0.193320 0.0966600 0.995317i $$-0.469184\pi$$
0.0966600 + 0.995317i $$0.469184\pi$$
$$632$$ 0 0
$$633$$ 9.06728 0.360392
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 18.1358 0.718565
$$638$$ 0 0
$$639$$ −27.1860 −1.07546
$$640$$ 0 0
$$641$$ 7.78887 0.307642 0.153821 0.988099i $$-0.450842\pi$$
0.153821 + 0.988099i $$0.450842\pi$$
$$642$$ 0 0
$$643$$ −43.6427 −1.72110 −0.860550 0.509366i $$-0.829880\pi$$
−0.860550 + 0.509366i $$0.829880\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 15.4606 0.607817 0.303909 0.952701i $$-0.401708\pi$$
0.303909 + 0.952701i $$0.401708\pi$$
$$648$$ 0 0
$$649$$ −15.8538 −0.622318
$$650$$ 0 0
$$651$$ −1.68946 −0.0662150
$$652$$ 0 0
$$653$$ 9.70613 0.379830 0.189915 0.981801i $$-0.439179\pi$$
0.189915 + 0.981801i $$0.439179\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −16.6404 −0.649204
$$658$$ 0 0
$$659$$ 50.7100 1.97538 0.987690 0.156422i $$-0.0499961\pi$$
0.987690 + 0.156422i $$0.0499961\pi$$
$$660$$ 0 0
$$661$$ 33.0132 1.28406 0.642032 0.766678i $$-0.278092\pi$$
0.642032 + 0.766678i $$0.278092\pi$$
$$662$$ 0 0
$$663$$ 2.35343 0.0913998
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.20147 0.0852413
$$668$$ 0 0
$$669$$ 0.997701 0.0385733
$$670$$ 0 0
$$671$$ −10.6060 −0.409439
$$672$$ 0 0
$$673$$ −19.2494 −0.742011 −0.371005 0.928631i $$-0.620987\pi$$
−0.371005 + 0.928631i $$0.620987\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 25.1941 0.968288 0.484144 0.874988i $$-0.339131\pi$$
0.484144 + 0.874988i $$0.339131\pi$$
$$678$$ 0 0
$$679$$ −2.89443 −0.111078
$$680$$ 0 0
$$681$$ −9.09361 −0.348468
$$682$$ 0 0
$$683$$ −9.48339 −0.362872 −0.181436 0.983403i $$-0.558074\pi$$
−0.181436 + 0.983403i $$0.558074\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −3.24436 −0.123780
$$688$$ 0 0
$$689$$ −28.9327 −1.10225
$$690$$ 0 0
$$691$$ −23.8538 −0.907443 −0.453721 0.891144i $$-0.649904\pi$$
−0.453721 + 0.891144i $$0.649904\pi$$
$$692$$ 0 0
$$693$$ −2.17635 −0.0826726
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −10.0228 −0.379642
$$698$$ 0 0
$$699$$ 8.27806 0.313105
$$700$$ 0 0
$$701$$ 19.8767 0.750731 0.375366 0.926877i $$-0.377517\pi$$
0.375366 + 0.926877i $$0.377517\pi$$
$$702$$ 0 0
$$703$$ −31.7077 −1.19588
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −5.45090 −0.205002
$$708$$ 0 0
$$709$$ −6.72545 −0.252580 −0.126290 0.991993i $$-0.540307\pi$$
−0.126290 + 0.991993i $$0.540307\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −6.77340 −0.253666
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 3.32601 0.124212
$$718$$ 0 0
$$719$$ −38.2294 −1.42571 −0.712857 0.701309i $$-0.752599\pi$$
−0.712857 + 0.701309i $$0.752599\pi$$
$$720$$ 0 0
$$721$$ −1.10557 −0.0411735
$$722$$ 0 0
$$723$$ 2.98298 0.110938
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 3.72315 0.138084 0.0690420 0.997614i $$-0.478006\pi$$
0.0690420 + 0.997614i $$0.478006\pi$$
$$728$$ 0 0
$$729$$ −13.2842 −0.492008
$$730$$ 0 0
$$731$$ −0.675938 −0.0250005
$$732$$ 0 0
$$733$$ 8.49885 0.313912 0.156956 0.987606i $$-0.449832\pi$$
0.156956 + 0.987606i $$0.449832\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 24.6759 0.908950
$$738$$ 0 0
$$739$$ −0.251725 −0.00925984 −0.00462992 0.999989i $$-0.501474\pi$$
−0.00462992 + 0.999989i $$0.501474\pi$$
$$740$$ 0 0
$$741$$ −11.1210 −0.408541
$$742$$ 0 0
$$743$$ 29.7305 1.09071 0.545353 0.838206i $$-0.316395\pi$$
0.545353 + 0.838206i $$0.316395\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.2614 0.448621
$$748$$ 0 0
$$749$$ −3.73861 −0.136606
$$750$$ 0 0
$$751$$ 10.6597 0.388979 0.194490 0.980905i $$-0.437695\pi$$
0.194490 + 0.980905i $$0.437695\pi$$
$$752$$ 0 0
$$753$$ −15.1558 −0.552309
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 16.1918 0.588501 0.294251 0.955728i $$-0.404930\pi$$
0.294251 + 0.955728i $$0.404930\pi$$
$$758$$ 0 0
$$759$$ 0.878968 0.0319045
$$760$$ 0 0
$$761$$ −46.9343 −1.70137 −0.850683 0.525679i $$-0.823811\pi$$
−0.850683 + 0.525679i $$0.823811\pi$$
$$762$$ 0 0
$$763$$ −6.62954 −0.240006
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −25.3047 −0.913701
$$768$$ 0 0
$$769$$ −35.9954 −1.29803 −0.649014 0.760777i $$-0.724818\pi$$
−0.649014 + 0.760777i $$0.724818\pi$$
$$770$$ 0 0
$$771$$ 7.60212 0.273784
$$772$$ 0 0
$$773$$ 21.8229 0.784916 0.392458 0.919770i $$-0.371625\pi$$
0.392458 + 0.919770i $$0.371625\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0.997701 0.0357923
$$778$$ 0 0
$$779$$ 47.3624 1.69693
$$780$$ 0 0
$$781$$ −16.7328 −0.598747
$$782$$ 0 0
$$783$$ 6.60442 0.236023
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 7.16318 0.255340 0.127670 0.991817i $$-0.459250\pi$$
0.127670 + 0.991817i $$0.459250\pi$$
$$788$$ 0 0
$$789$$ 8.26489 0.294238
$$790$$ 0 0
$$791$$ −4.76024 −0.169255
$$792$$ 0 0
$$793$$ −16.9285 −0.601148
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 48.0604 1.70239 0.851193 0.524853i $$-0.175880\pi$$
0.851193 + 0.524853i $$0.175880\pi$$
$$798$$ 0 0
$$799$$ 3.01702 0.106735
$$800$$ 0 0
$$801$$ 9.80359 0.346393
$$802$$ 0 0
$$803$$ −10.2421 −0.361435
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −3.24943 −0.114385
$$808$$ 0 0
$$809$$ 0.0611183 0.00214880 0.00107440 0.999999i $$-0.499658\pi$$
0.00107440 + 0.999999i $$0.499658\pi$$
$$810$$ 0 0
$$811$$ 25.9092 0.909794 0.454897 0.890544i $$-0.349676\pi$$
0.454897 + 0.890544i $$0.349676\pi$$
$$812$$ 0 0
$$813$$ −13.3979 −0.469884
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 3.19411 0.111748
$$818$$ 0 0
$$819$$ −3.47372 −0.121382
$$820$$ 0 0
$$821$$ −21.0936 −0.736172 −0.368086 0.929792i $$-0.619987\pi$$
−0.368086 + 0.929792i $$0.619987\pi$$
$$822$$ 0 0
$$823$$ 37.2641 1.29895 0.649473 0.760384i $$-0.274989\pi$$
0.649473 + 0.760384i $$0.274989\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 4.45320 0.154853 0.0774264 0.996998i $$-0.475330\pi$$
0.0774264 + 0.996998i $$0.475330\pi$$
$$828$$ 0 0
$$829$$ −38.1300 −1.32431 −0.662154 0.749368i $$-0.730358\pi$$
−0.662154 + 0.749368i $$0.730358\pi$$
$$830$$ 0 0
$$831$$ −3.68790 −0.127932
$$832$$ 0 0
$$833$$ 11.3624 0.393682
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −20.3202 −0.702369
$$838$$ 0 0
$$839$$ −7.15858 −0.247142 −0.123571 0.992336i $$-0.539435\pi$$
−0.123571 + 0.992336i $$0.539435\pi$$
$$840$$ 0 0
$$841$$ −24.1535 −0.832880
$$842$$ 0 0
$$843$$ 15.2282 0.524486
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 3.89673 0.133893
$$848$$ 0 0
$$849$$ 9.75334 0.334734
$$850$$ 0 0
$$851$$ 4.00000 0.137118
$$852$$ 0 0
$$853$$ 49.6199 1.69895 0.849476 0.527627i $$-0.176918\pi$$
0.849476 + 0.527627i $$0.176918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 44.0360 1.50424 0.752120 0.659026i $$-0.229031\pi$$
0.752120 + 0.659026i $$0.229031\pi$$
$$858$$ 0 0
$$859$$ −49.5062 −1.68913 −0.844565 0.535453i $$-0.820141\pi$$
−0.844565 + 0.535453i $$0.820141\pi$$
$$860$$ 0 0
$$861$$ −1.49028 −0.0507887
$$862$$ 0 0
$$863$$ −20.7962 −0.707912 −0.353956 0.935262i $$-0.615164\pi$$
−0.353956 + 0.935262i $$0.615164\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −7.43313 −0.252442
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 39.3859 1.33454
$$872$$ 0 0
$$873$$ −16.5719 −0.560875
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 5.42071 0.183044 0.0915222 0.995803i $$-0.470827\pi$$
0.0915222 + 0.995803i $$0.470827\pi$$
$$878$$ 0 0
$$879$$ −12.2614 −0.413566
$$880$$ 0 0
$$881$$ −45.8036 −1.54316 −0.771581 0.636131i $$-0.780534\pi$$
−0.771581 + 0.636131i $$0.780534\pi$$
$$882$$ 0 0
$$883$$ −57.1860 −1.92446 −0.962231 0.272234i $$-0.912238\pi$$
−0.962231 + 0.272234i $$0.912238\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 15.5371 0.521686 0.260843 0.965381i $$-0.415999\pi$$
0.260843 + 0.965381i $$0.415999\pi$$
$$888$$ 0 0
$$889$$ −5.28542 −0.177267
$$890$$ 0 0
$$891$$ −11.0789 −0.371157
$$892$$ 0 0
$$893$$ −14.2568 −0.477085
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.40294 0.0468430
$$898$$ 0 0
$$899$$ −14.9115 −0.497325
$$900$$ 0 0
$$901$$ −18.1268 −0.603892
$$902$$ 0 0
$$903$$ −0.100505 −0.00334458
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 32.8515 1.09082 0.545409 0.838170i $$-0.316374\pi$$
0.545409 + 0.838170i $$0.316374\pi$$
$$908$$ 0 0
$$909$$ −31.2088 −1.03513
$$910$$ 0 0
$$911$$ 8.95205 0.296595 0.148297 0.988943i $$-0.452621\pi$$
0.148297 + 0.988943i $$0.452621\pi$$
$$912$$ 0 0
$$913$$ 7.54680 0.249763
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −3.64271 −0.120293
$$918$$ 0 0
$$919$$ −49.8229 −1.64351 −0.821753 0.569844i $$-0.807004\pi$$
−0.821753 + 0.569844i $$0.807004\pi$$
$$920$$ 0 0
$$921$$ 0.802385 0.0264395
$$922$$ 0 0
$$923$$ −26.7077 −0.879094
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −6.32987 −0.207900
$$928$$ 0 0
$$929$$ 25.0723 0.822597 0.411298 0.911501i $$-0.365075\pi$$
0.411298 + 0.911501i $$0.365075\pi$$
$$930$$ 0 0
$$931$$ −53.6922 −1.75969
$$932$$ 0 0
$$933$$ −3.95941 −0.129625
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −21.3322 −0.696891 −0.348446 0.937329i $$-0.613290\pi$$
−0.348446 + 0.937329i $$0.613290\pi$$
$$938$$ 0 0
$$939$$ 1.22079 0.0398391
$$940$$ 0 0
$$941$$ 3.93428 0.128254 0.0641270 0.997942i $$-0.479574\pi$$
0.0641270 + 0.997942i $$0.479574\pi$$
$$942$$ 0 0
$$943$$ −5.97487 −0.194569
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −48.6420 −1.58065 −0.790326 0.612687i $$-0.790089\pi$$
−0.790326 + 0.612687i $$0.790089\pi$$
$$948$$ 0 0
$$949$$ −16.3476 −0.530667
$$950$$ 0 0
$$951$$ 8.05761 0.261286
$$952$$ 0 0
$$953$$ 15.5623 0.504111 0.252056 0.967713i $$-0.418893\pi$$
0.252056 + 0.967713i $$0.418893\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 1.93502 0.0625505
$$958$$ 0 0
$$959$$ −10.1763 −0.328611
$$960$$ 0 0
$$961$$ 14.8790 0.479967
$$962$$ 0 0
$$963$$ −21.4052 −0.689774
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −6.80083 −0.218700 −0.109350 0.994003i $$-0.534877\pi$$
−0.109350 + 0.994003i $$0.534877\pi$$
$$968$$ 0 0
$$969$$ −6.96751 −0.223829
$$970$$ 0 0
$$971$$ 6.46406 0.207442 0.103721 0.994606i $$-0.466925\pi$$
0.103721 + 0.994606i $$0.466925\pi$$
$$972$$ 0 0
$$973$$ 4.09591 0.131309
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −9.18601 −0.293886 −0.146943 0.989145i $$-0.546943\pi$$
−0.146943 + 0.989145i $$0.546943\pi$$
$$978$$ 0 0
$$979$$ 6.03405 0.192849
$$980$$ 0 0
$$981$$ −37.9571 −1.21188
$$982$$ 0 0
$$983$$ −15.3705 −0.490241 −0.245121 0.969493i $$-0.578828\pi$$
−0.245121 + 0.969493i $$0.578828\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0.448598 0.0142790
$$988$$ 0 0
$$989$$ −0.402945 −0.0128129
$$990$$ 0 0
$$991$$ −52.0109 −1.65218 −0.826090 0.563538i $$-0.809440\pi$$
−0.826090 + 0.563538i $$0.809440\pi$$
$$992$$ 0 0
$$993$$ 12.0051 0.380969
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −15.7123 −0.497613 −0.248807 0.968553i $$-0.580038\pi$$
−0.248807 + 0.968553i $$0.580038\pi$$
$$998$$ 0 0
$$999$$ 12.0000 0.379663
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 4600.2.a.y.1.2 3
4.3 odd 2 9200.2.a.cd.1.2 3
5.2 odd 4 4600.2.e.r.4049.3 6
5.3 odd 4 4600.2.e.r.4049.4 6
5.4 even 2 920.2.a.g.1.2 3
15.14 odd 2 8280.2.a.bo.1.2 3
20.19 odd 2 1840.2.a.t.1.2 3
40.19 odd 2 7360.2.a.ca.1.2 3
40.29 even 2 7360.2.a.cb.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.2 3 5.4 even 2
1840.2.a.t.1.2 3 20.19 odd 2
4600.2.a.y.1.2 3 1.1 even 1 trivial
4600.2.e.r.4049.3 6 5.2 odd 4
4600.2.e.r.4049.4 6 5.3 odd 4
7360.2.a.ca.1.2 3 40.19 odd 2
7360.2.a.cb.1.2 3 40.29 even 2
8280.2.a.bo.1.2 3 15.14 odd 2
9200.2.a.cd.1.2 3 4.3 odd 2