# Properties

 Label 1840.2.a.t.1.2 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ Analytic rank $1$ 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] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$14.6924739719$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 1840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.523976 q^{3} -1.00000 q^{5} -0.476024 q^{7} -2.72545 q^{9} +O(q^{10})$$ $$q+0.523976 q^{3} -1.00000 q^{5} -0.476024 q^{7} -2.72545 q^{9} +1.67750 q^{11} +2.67750 q^{13} -0.523976 q^{15} +1.67750 q^{17} -7.92692 q^{19} -0.249425 q^{21} -1.00000 q^{23} +1.00000 q^{25} -3.00000 q^{27} -2.20147 q^{29} -6.77340 q^{31} +0.878968 q^{33} +0.476024 q^{35} +4.00000 q^{37} +1.40294 q^{39} +5.97487 q^{41} +0.402945 q^{43} +2.72545 q^{45} -1.79853 q^{47} -6.77340 q^{49} +0.878968 q^{51} -10.8059 q^{53} -1.67750 q^{55} -4.15352 q^{57} -9.45090 q^{59} +6.32250 q^{61} +1.29738 q^{63} -2.67750 q^{65} -14.7100 q^{67} -0.523976 q^{69} -9.97487 q^{71} -6.10557 q^{73} +0.523976 q^{75} -0.798528 q^{77} +6.60442 q^{81} -4.49885 q^{83} -1.67750 q^{85} -1.15352 q^{87} -3.59706 q^{89} -1.27455 q^{91} -3.54910 q^{93} +7.92692 q^{95} -6.08044 q^{97} -4.57193 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^5 - 3 * q^7 + 3 * q^9 $$3 q - 3 q^{5} - 3 q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{17} - 3 q^{19} + 12 q^{21} - 3 q^{23} + 3 q^{25} - 9 q^{27} + 3 q^{29} - 6 q^{31} - 15 q^{33} + 3 q^{35} + 12 q^{37} - 15 q^{39} - 6 q^{41} - 18 q^{43} - 3 q^{45} - 15 q^{47} - 6 q^{49} - 15 q^{51} + 6 q^{53} + 3 q^{55} - 6 q^{57} - 6 q^{59} + 27 q^{61} - 12 q^{63} - 12 q^{67} - 6 q^{71} - 15 q^{73} - 12 q^{77} - 9 q^{81} + 12 q^{83} + 3 q^{85} + 3 q^{87} - 30 q^{89} - 15 q^{91} - 33 q^{93} + 3 q^{95} + 9 q^{97} - 9 q^{99}+O(q^{100})$$ 3 * q - 3 * q^5 - 3 * q^7 + 3 * q^9 - 3 * q^11 - 3 * q^17 - 3 * q^19 + 12 * q^21 - 3 * q^23 + 3 * q^25 - 9 * q^27 + 3 * q^29 - 6 * q^31 - 15 * q^33 + 3 * q^35 + 12 * q^37 - 15 * q^39 - 6 * q^41 - 18 * q^43 - 3 * q^45 - 15 * q^47 - 6 * q^49 - 15 * q^51 + 6 * q^53 + 3 * q^55 - 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^85 + 3 * q^87 - 30 * q^89 - 15 * q^91 - 33 * q^93 + 3 * q^95 + 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
$$5$$ −1.00000 −0.447214
$$6$$ 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 $$-0.418618\pi$$
0.252892 + 0.967495i $$0.418618\pi$$
$$12$$ 0 0
$$13$$ 2.67750 0.742604 0.371302 0.928512i $$-0.378911\pi$$
0.371302 + 0.928512i $$0.378911\pi$$
$$14$$ 0 0
$$15$$ −0.523976 −0.135290
$$16$$ 0 0
$$17$$ 1.67750 0.406853 0.203426 0.979090i $$-0.434792\pi$$
0.203426 + 0.979090i $$0.434792\pi$$
$$18$$ 0 0
$$19$$ −7.92692 −1.81856 −0.909280 0.416184i $$-0.863367\pi$$
−0.909280 + 0.416184i $$0.863367\pi$$
$$20$$ 0 0
$$21$$ −0.249425 −0.0544290
$$22$$ 0 0
$$23$$ −1.00000 −0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 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 $$-0.708136\pi$$
−0.608269 + 0.793731i $$0.708136\pi$$
$$32$$ 0 0
$$33$$ 0.878968 0.153009
$$34$$ 0 0
$$35$$ 0.476024 0.0804627
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 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
$$45$$ 2.72545 0.406286
$$46$$ 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 $$-0.766195\pi$$
−0.742152 + 0.670232i $$0.766195\pi$$
$$54$$ 0 0
$$55$$ −1.67750 −0.226194
$$56$$ 0 0
$$57$$ −4.15352 −0.550147
$$58$$ 0 0
$$59$$ −9.45090 −1.23040 −0.615201 0.788370i $$-0.710925\pi$$
−0.615201 + 0.788370i $$0.710925\pi$$
$$60$$ 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
$$65$$ −2.67750 −0.332102
$$66$$ 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 $$-0.701622\pi$$
−0.591900 + 0.806012i $$0.701622\pi$$
$$72$$ 0 0
$$73$$ −6.10557 −0.714603 −0.357301 0.933989i $$-0.616303\pi$$
−0.357301 + 0.933989i $$0.616303\pi$$
$$74$$ 0 0
$$75$$ 0.523976 0.0605036
$$76$$ 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
$$85$$ −1.67750 −0.181950
$$86$$ 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
$$95$$ 7.92692 0.813285
$$96$$ 0 0
$$97$$ −6.08044 −0.617375 −0.308688 0.951163i $$-0.599890\pi$$
−0.308688 + 0.951163i $$0.599890\pi$$
$$98$$ 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
$$105$$ 0.249425 0.0243414
$$106$$ 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 $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 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
$$125$$ −1.00000 −0.0894427
$$126$$ 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 $$-0.608498\pi$$
−0.334295 + 0.942468i $$0.608498\pi$$
$$132$$ 0 0
$$133$$ 3.77340 0.327195
$$134$$ 0 0
$$135$$ 3.00000 0.258199
$$136$$ 0 0
$$137$$ −21.3778 −1.82643 −0.913215 0.407478i $$-0.866408\pi$$
−0.913215 + 0.407478i $$0.866408\pi$$
$$138$$ 0 0
$$139$$ 8.60442 0.729817 0.364909 0.931043i $$-0.381100\pi$$
0.364909 + 0.931043i $$0.381100\pi$$
$$140$$ 0 0
$$141$$ −0.942386 −0.0793632
$$142$$ 0 0
$$143$$ 4.49149 0.375597
$$144$$ 0 0
$$145$$ 2.20147 0.182822
$$146$$ 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 $$-0.487992\pi$$
0.0377159 + 0.999289i $$0.487992\pi$$
$$152$$ 0 0
$$153$$ −4.57193 −0.369619
$$154$$ 0 0
$$155$$ 6.77340 0.544053
$$156$$ 0 0
$$157$$ 22.3527 1.78394 0.891970 0.452096i $$-0.149323\pi$$
0.891970 + 0.452096i $$0.149323\pi$$
$$158$$ 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
$$165$$ −0.878968 −0.0684276
$$166$$ 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 $$-0.887334\pi$$
−0.938010 + 0.346608i $$0.887334\pi$$
$$174$$ 0 0
$$175$$ −0.476024 −0.0359840
$$176$$ 0 0
$$177$$ −4.95205 −0.372219
$$178$$ 0 0
$$179$$ 13.1033 0.979384 0.489692 0.871895i $$-0.337109\pi$$
0.489692 + 0.871895i $$0.337109\pi$$
$$180$$ 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
$$185$$ −4.00000 −0.294086
$$186$$ 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 $$-0.567928\pi$$
−0.211785 + 0.977316i $$0.567928\pi$$
$$192$$ 0 0
$$193$$ 4.41261 0.317626 0.158813 0.987309i $$-0.449233\pi$$
0.158813 + 0.987309i $$0.449233\pi$$
$$194$$ 0 0
$$195$$ −1.40294 −0.100467
$$196$$ 0 0
$$197$$ 11.4258 0.814053 0.407026 0.913416i $$-0.366566\pi$$
0.407026 + 0.913416i $$0.366566\pi$$
$$198$$ 0 0
$$199$$ 0.307039 0.0217654 0.0108827 0.999941i $$-0.496536\pi$$
0.0108827 + 0.999941i $$0.496536\pi$$
$$200$$ 0 0
$$201$$ −7.70768 −0.543658
$$202$$ 0 0
$$203$$ 1.04795 0.0735519
$$204$$ 0 0
$$205$$ −5.97487 −0.417303
$$206$$ 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 $$-0.703107\pi$$
−0.595654 + 0.803241i $$0.703107\pi$$
$$212$$ 0 0
$$213$$ −5.22660 −0.358121
$$214$$ 0 0
$$215$$ −0.402945 −0.0274806
$$216$$ 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
$$225$$ −2.72545 −0.181697
$$226$$ 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 $$-0.673137\pi$$
−0.517498 + 0.855684i $$0.673137\pi$$
$$234$$ 0 0
$$235$$ 1.79853 0.117323
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −6.34763 −0.410594 −0.205297 0.978700i $$-0.565816\pi$$
−0.205297 + 0.978700i $$0.565816\pi$$
$$240$$ 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
$$245$$ 6.77340 0.432737
$$246$$ 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 $$-0.133874\pi$$
0.912853 + 0.408288i $$0.133874\pi$$
$$252$$ 0 0
$$253$$ −1.67750 −0.105463
$$254$$ 0 0
$$255$$ −0.878968 −0.0550431
$$256$$ 0 0
$$257$$ −14.5085 −0.905016 −0.452508 0.891760i $$-0.649471\pi$$
−0.452508 + 0.891760i $$0.649471\pi$$
$$258$$ 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
$$265$$ 10.8059 0.663801
$$266$$ 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 $$-0.216931\pi$$
0.776622 + 0.629967i $$0.216931\pi$$
$$272$$ 0 0
$$273$$ −0.667835 −0.0404192
$$274$$ 0 0
$$275$$ 1.67750 0.101157
$$276$$ 0 0
$$277$$ 7.03829 0.422890 0.211445 0.977390i $$-0.432183\pi$$
0.211445 + 0.977390i $$0.432183\pi$$
$$278$$ 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
$$285$$ 4.15352 0.246033
$$286$$ 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 $$-0.260439\pi$$
0.683540 + 0.729913i $$0.260439\pi$$
$$294$$ 0 0
$$295$$ 9.45090 0.550253
$$296$$ 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
$$305$$ −6.32250 −0.362026
$$306$$ 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 $$-0.431271\pi$$
0.214244 + 0.976780i $$0.431271\pi$$
$$312$$ 0 0
$$313$$ −2.32987 −0.131692 −0.0658459 0.997830i $$-0.520975\pi$$
−0.0658459 + 0.997830i $$0.520975\pi$$
$$314$$ 0 0
$$315$$ −1.29738 −0.0730990
$$316$$ 0 0
$$317$$ −15.3778 −0.863704 −0.431852 0.901944i $$-0.642140\pi$$
−0.431852 + 0.901944i $$0.642140\pi$$
$$318$$ 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
$$325$$ 2.67750 0.148521
$$326$$ 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 $$-0.716807\pi$$
−0.629664 + 0.776868i $$0.716807\pi$$
$$332$$ 0 0
$$333$$ −10.9018 −0.597415
$$334$$ 0 0
$$335$$ 14.7100 0.803692
$$336$$ 0 0
$$337$$ 26.5410 1.44578 0.722890 0.690963i $$-0.242813\pi$$
0.722890 + 0.690963i $$0.242813\pi$$
$$338$$ 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
$$345$$ 0.523976 0.0282099
$$346$$ 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 $$-0.404074\pi$$
0.296821 + 0.954933i $$0.404074\pi$$
$$354$$ 0 0
$$355$$ 9.97487 0.529411
$$356$$ 0 0
$$357$$ −0.418410 −0.0221446
$$358$$ 0 0
$$359$$ −22.4029 −1.18238 −0.591191 0.806532i $$-0.701342\pi$$
−0.591191 + 0.806532i $$0.701342\pi$$
$$360$$ 0 0
$$361$$ 43.8361 2.30716
$$362$$ 0 0
$$363$$ −4.28927 −0.225129
$$364$$ 0 0
$$365$$ 6.10557 0.319580
$$366$$ 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 $$-0.510218\pi$$
−0.0320963 + 0.999485i $$0.510218\pi$$
$$374$$ 0 0
$$375$$ −0.523976 −0.0270580
$$376$$ 0 0
$$377$$ −5.89443 −0.303579
$$378$$ 0 0
$$379$$ −4.38748 −0.225370 −0.112685 0.993631i $$-0.535945\pi$$
−0.112685 + 0.993631i $$0.535945\pi$$
$$380$$ 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
$$385$$ 0.798528 0.0406967
$$386$$ 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 $$-0.673722\pi$$
−0.519072 + 0.854731i $$0.673722\pi$$
$$398$$ 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
$$405$$ −6.60442 −0.328176
$$406$$ 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
$$415$$ 4.49885 0.220840
$$416$$ 0 0
$$417$$ 4.50851 0.220783
$$418$$ 0 0
$$419$$ 4.59476 0.224469 0.112234 0.993682i $$-0.464199\pi$$
0.112234 + 0.993682i $$0.464199\pi$$
$$420$$ 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
$$425$$ 1.67750 0.0813705
$$426$$ 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 $$-0.373542\pi$$
0.386911 + 0.922117i $$0.373542\pi$$
$$432$$ 0 0
$$433$$ −7.68486 −0.369311 −0.184655 0.982803i $$-0.559117\pi$$
−0.184655 + 0.982803i $$0.559117\pi$$
$$434$$ 0 0
$$435$$ 1.15352 0.0553070
$$436$$ 0 0
$$437$$ 7.92692 0.379196
$$438$$ 0 0
$$439$$ −15.9822 −0.762790 −0.381395 0.924412i $$-0.624556\pi$$
−0.381395 + 0.924412i $$0.624556\pi$$
$$440$$ 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
$$445$$ 3.59706 0.170517
$$446$$ 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
$$455$$ 1.27455 0.0597519
$$456$$ 0 0
$$457$$ −4.49885 −0.210447 −0.105224 0.994449i $$-0.533556\pi$$
−0.105224 + 0.994449i $$0.533556\pi$$
$$458$$ 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
$$465$$ 3.54910 0.164586
$$466$$ 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
$$475$$ −7.92692 −0.363712
$$476$$ 0 0
$$477$$ 29.4509 1.34846
$$478$$ 0 0
$$479$$ 2.40294 0.109793 0.0548967 0.998492i $$-0.482517\pi$$
0.0548967 + 0.998492i $$0.482517\pi$$
$$480$$ 0 0
$$481$$ 10.7100 0.488333
$$482$$ 0 0
$$483$$ 0.249425 0.0113492
$$484$$ 0 0
$$485$$ 6.08044 0.276099
$$486$$ 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 $$-0.496486\pi$$
0.0110384 + 0.999939i $$0.496486\pi$$
$$492$$ 0 0
$$493$$ −3.69296 −0.166323
$$494$$ 0 0
$$495$$ 4.57193 0.205493
$$496$$ 0 0
$$497$$ 4.74828 0.212989
$$498$$ 0 0
$$499$$ 38.4583 1.72163 0.860814 0.508920i $$-0.169955\pi$$
0.860814 + 0.508920i $$0.169955\pi$$
$$500$$ 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
$$505$$ −11.4509 −0.509558
$$506$$ 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
$$515$$ −2.32250 −0.102342
$$516$$ 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
$$525$$ −0.249425 −0.0108858
$$526$$ 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
$$535$$ −7.85384 −0.339551
$$536$$ 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
$$545$$ −13.9269 −0.596564
$$546$$ 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
$$555$$ −2.09591 −0.0889662
$$556$$ 0 0
$$557$$ −23.4966 −0.995581 −0.497790 0.867297i $$-0.665855\pi$$
−0.497790 + 0.867297i $$0.665855\pi$$
$$558$$ 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
$$565$$ 10.0000 0.420703
$$566$$ 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 $$-0.304993\pi$$
0.575024 + 0.818137i $$0.304993\pi$$
$$572$$ 0 0
$$573$$ −3.06728 −0.128137
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −26.1512 −1.08869 −0.544345 0.838862i $$-0.683222\pi$$
−0.544345 + 0.838862i $$0.683222\pi$$
$$578$$ 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
$$585$$ 7.29738 0.301709
$$586$$ 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 $$-0.393798\pi$$
0.327489 + 0.944855i $$0.393798\pi$$
$$594$$ 0 0
$$595$$ 0.798528 0.0327364
$$596$$ 0 0
$$597$$ 0.160881 0.00658443
$$598$$ 0 0
$$599$$ −40.5940 −1.65863 −0.829313 0.558784i $$-0.811268\pi$$
−0.829313 + 0.558784i $$0.811268\pi$$
$$600$$ 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
$$605$$ 8.18601 0.332809
$$606$$ 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 $$-0.290385\pi$$
0.611952 + 0.790895i $$0.290385\pi$$
$$614$$ 0 0
$$615$$ −3.13069 −0.126242
$$616$$ 0 0
$$617$$ −6.13069 −0.246812 −0.123406 0.992356i $$-0.539382\pi$$
−0.123406 + 0.992356i $$0.539382\pi$$
$$618$$ 0 0
$$619$$ 33.7305 1.35574 0.677872 0.735180i $$-0.262902\pi$$
0.677872 + 0.735180i $$0.262902\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 0 0
$$623$$ 1.71228 0.0686012
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 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 $$-0.530816\pi$$
−0.0966600 + 0.995317i $$0.530816\pi$$
$$632$$ 0 0
$$633$$ −9.06728 −0.360392
$$634$$ 0 0
$$635$$ −11.1033 −0.440620
$$636$$ 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
$$645$$ −0.211133 −0.00831337
$$646$$ 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 $$-0.560821\pi$$
−0.189915 + 0.981801i $$0.560821\pi$$
$$654$$ 0 0
$$655$$ 7.65237 0.299003
$$656$$ 0 0
$$657$$ 16.6404 0.649204
$$658$$ 0 0
$$659$$ −50.7100 −1.97538 −0.987690 0.156422i $$-0.950004\pi$$
−0.987690 + 0.156422i $$0.950004\pi$$
$$660$$ 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
$$665$$ −3.77340 −0.146326
$$666$$ 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 $$-0.379013\pi$$
0.371005 + 0.928631i $$0.379013\pi$$
$$674$$ 0 0
$$675$$ −3.00000 −0.115470
$$676$$ 0 0
$$677$$ −25.1941 −0.968288 −0.484144 0.874988i $$-0.660869\pi$$
−0.484144 + 0.874988i $$0.660869\pi$$
$$678$$ 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
$$685$$ 21.3778 0.816804
$$686$$ 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 $$-0.350096\pi$$
0.453721 + 0.891144i $$0.350096\pi$$
$$692$$ 0 0
$$693$$ 2.17635 0.0826726
$$694$$ 0 0
$$695$$ −8.60442 −0.326384
$$696$$ 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
$$705$$ 0.942386 0.0354923
$$706$$ 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
$$715$$ −4.49149 −0.167972
$$716$$ 0 0
$$717$$ −3.32601 −0.124212
$$718$$ 0 0
$$719$$ 38.2294 1.42571 0.712857 0.701309i $$-0.247401\pi$$
0.712857 + 0.701309i $$0.247401\pi$$
$$720$$ 0 0
$$721$$ −1.10557 −0.0411735
$$722$$ 0 0
$$723$$ 2.98298 0.110938
$$724$$ 0 0
$$725$$ −2.20147 −0.0817606
$$726$$ 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 $$-0.550168\pi$$
−0.156956 + 0.987606i $$0.550168\pi$$
$$734$$ 0 0
$$735$$ 3.54910 0.130911
$$736$$ 0 0
$$737$$ −24.6759 −0.908950
$$738$$ 0 0
$$739$$ 0.251725 0.00925984 0.00462992 0.999989i $$-0.498526\pi$$
0.00462992 + 0.999989i $$0.498526\pi$$
$$740$$ 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
$$745$$ −4.97487 −0.182265
$$746$$ 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 $$-0.562305\pi$$
−0.194490 + 0.980905i $$0.562305\pi$$
$$752$$ 0 0
$$753$$ 15.1558 0.552309
$$754$$ 0 0
$$755$$ −0.926921 −0.0337341
$$756$$ 0 0
$$757$$ −16.1918 −0.588501 −0.294251 0.955728i $$-0.595070\pi$$
−0.294251 + 0.955728i $$0.595070\pi$$
$$758$$ 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
$$765$$ 4.57193 0.165298
$$766$$ 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 $$-0.628375\pi$$
−0.392458 + 0.919770i $$0.628375\pi$$
$$774$$ 0 0
$$775$$ −6.77340 −0.243308
$$776$$ 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
$$785$$ −22.3527 −0.797802
$$786$$ 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
$$795$$ 5.66203 0.200812
$$796$$ 0 0
$$797$$ −48.0604 −1.70239 −0.851193 0.524853i $$-0.824120\pi$$
−0.851193 + 0.524853i $$0.824120\pi$$
$$798$$ 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
$$805$$ −0.476024 −0.0167776
$$806$$ 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 $$-0.650324\pi$$
−0.454897 + 0.890544i $$0.650324\pi$$
$$812$$ 0 0
$$813$$ 13.3979 0.469884
$$814$$ 0 0
$$815$$ −7.93658 −0.278006
$$816$$ 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
$$825$$ 0.878968 0.0306017
$$826$$ 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
$$835$$ 12.8059 0.443166
$$836$$ 0 0
$$837$$ 20.3202 0.702369
$$838$$ 0 0
$$839$$ 7.15858 0.247142 0.123571 0.992336i $$-0.460565\pi$$
0.123571 + 0.992336i $$0.460565\pi$$
$$840$$ 0 0
$$841$$ −24.1535 −0.832880
$$842$$ 0 0
$$843$$ 15.2282 0.524486
$$844$$ 0 0
$$845$$ 5.83102 0.200593
$$846$$ 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 $$-0.823082\pi$$
−0.849476 + 0.527627i $$0.823082\pi$$
$$854$$ 0 0
$$855$$ −21.6044 −0.738855
$$856$$ 0 0
$$857$$ −44.0360 −1.50424 −0.752120 0.659026i $$-0.770969\pi$$
−0.752120 + 0.659026i $$0.770969\pi$$
$$858$$ 0 0
$$859$$ 49.5062 1.68913 0.844565 0.535453i $$-0.179859\pi$$
0.844565 + 0.535453i $$0.179859\pi$$
$$860$$ 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
$$865$$ 24.6752 0.838982
$$866$$ 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
$$875$$ 0.476024 0.0160925
$$876$$ 0 0
$$877$$ −5.42071 −0.183044 −0.0915222 0.995803i $$-0.529173\pi$$
−0.0915222 + 0.995803i $$0.529173\pi$$
$$878$$ 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
$$885$$ 4.95205 0.166461
$$886$$ 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
$$895$$ −13.1033 −0.437994
$$896$$ 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
$$905$$ −9.52398 −0.316588
$$906$$ 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 $$-0.547379\pi$$
−0.148297 + 0.988943i $$0.547379\pi$$
$$912$$ 0 0
$$913$$ −7.54680 −0.249763
$$914$$ 0 0
$$915$$ −3.31284 −0.109519
$$916$$ 0 0
$$917$$ 3.64271 0.120293
$$918$$ 0 0
$$919$$ 49.8229 1.64351 0.821753 0.569844i $$-0.192996\pi$$
0.821753 + 0.569844i $$0.192996\pi$$
$$920$$ 0 0
$$921$$ 0.802385 0.0264395
$$922$$ 0 0
$$923$$ −26.7077 −0.879094
$$924$$ 0 0
$$925$$ 4.00000 0.131519
$$926$$ 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
$$935$$ −2.81399 −0.0920274
$$936$$ 0 0
$$937$$ 21.3322 0.696891 0.348446 0.937329i $$-0.386710\pi$$
0.348446 + 0.937329i $$0.386710\pi$$
$$938$$ 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
$$945$$ −1.42807 −0.0464551
$$946$$ 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 $$-0.581107\pi$$
−0.252056 + 0.967713i $$0.581107\pi$$
$$954$$ 0 0
$$955$$ 5.85384 0.189426
$$956$$ 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
$$965$$ −4.41261 −0.142047
$$966$$ 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 $$-0.533075\pi$$
−0.103721 + 0.994606i $$0.533075\pi$$
$$972$$ 0 0
$$973$$ −4.09591 −0.131309
$$974$$ 0 0
$$975$$ 1.40294 0.0449302
$$976$$ 0 0
$$977$$ 9.18601 0.293886 0.146943 0.989145i $$-0.453057\pi$$
0.146943 + 0.989145i $$0.453057\pi$$
$$978$$ 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
$$985$$ −11.4258 −0.364055
$$986$$ 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 $$-0.190560\pi$$
0.826090 + 0.563538i $$0.190560\pi$$
$$992$$ 0 0
$$993$$ −12.0051 −0.380969
$$994$$ 0 0
$$995$$ −0.307039 −0.00973379
$$996$$ 0 0
$$997$$ 15.7123 0.497613 0.248807 0.968553i $$-0.419962\pi$$
0.248807 + 0.968553i $$0.419962\pi$$
$$998$$ 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 1840.2.a.t.1.2 3
4.3 odd 2 920.2.a.g.1.2 3
5.4 even 2 9200.2.a.cd.1.2 3
8.3 odd 2 7360.2.a.cb.1.2 3
8.5 even 2 7360.2.a.ca.1.2 3
12.11 even 2 8280.2.a.bo.1.2 3
20.3 even 4 4600.2.e.r.4049.3 6
20.7 even 4 4600.2.e.r.4049.4 6
20.19 odd 2 4600.2.a.y.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.2 3 4.3 odd 2
1840.2.a.t.1.2 3 1.1 even 1 trivial
4600.2.a.y.1.2 3 20.19 odd 2
4600.2.e.r.4049.3 6 20.3 even 4
4600.2.e.r.4049.4 6 20.7 even 4
7360.2.a.ca.1.2 3 8.5 even 2
7360.2.a.cb.1.2 3 8.3 odd 2
8280.2.a.bo.1.2 3 12.11 even 2
9200.2.a.cd.1.2 3 5.4 even 2