# Properties

 Label 920.2.a.g.1.2 Level $920$ Weight $2$ Character 920.1 Self dual yes Analytic conductor $7.346$ 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] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, 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("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.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: $$7.34623698596$$ 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.523976$$ of defining polynomial Character $$\chi$$ $$=$$ 920.1

## $q$-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 $$-0.548333\pi$$
−0.151259 + 0.988494i $$0.548333\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0.476024 0.179920 0.0899600 0.995945i $$-0.471326\pi$$
0.0899600 + 0.995945i $$0.471326\pi$$
$$8$$ 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 $$-0.378911\pi$$
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 $$-0.136633\pi$$
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
$$25$$ 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 $$-0.291864\pi$$
0.608269 + 0.793731i $$0.291864\pi$$
$$32$$ 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 $$-0.509781\pi$$
−0.0307242 + 0.999528i $$0.509781\pi$$
$$44$$ 0 0
$$45$$ 2.72545 0.406286
$$46$$ 0 0
$$47$$ 1.79853 0.262342 0.131171 0.991360i $$-0.458126\pi$$
0.131171 + 0.991360i $$0.458126\pi$$
$$48$$ 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 $$-0.289075\pi$$
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
$$65$$ −2.67750 −0.332102
$$66$$ 0 0
$$67$$ 14.7100 1.79711 0.898555 0.438860i $$-0.144618\pi$$
0.898555 + 0.438860i $$0.144618\pi$$
$$68$$ 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 $$-0.616303\pi$$
−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 $$-0.420586\pi$$
0.246906 + 0.969039i $$0.420586\pi$$
$$84$$ 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 $$-0.536501\pi$$
−0.114422 + 0.993432i $$0.536501\pi$$
$$104$$ 0 0
$$105$$ 0.249425 0.0243414
$$106$$ 0 0
$$107$$ −7.85384 −0.759260 −0.379630 0.925138i $$-0.623949\pi$$
−0.379630 + 0.925138i $$0.623949\pi$$
$$108$$ 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 $$-0.663964\pi$$
−0.492628 + 0.870240i $$0.663964\pi$$
$$128$$ 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
$$135$$ −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 $$-0.618900\pi$$
−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
$$145$$ 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 $$-0.512008\pi$$
−0.0377159 + 0.999289i $$0.512008\pi$$
$$152$$ 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 $$-0.600604\pi$$
−0.310821 + 0.950469i $$0.600604\pi$$
$$164$$ 0 0
$$165$$ −0.878968 −0.0684276
$$166$$ 0 0
$$167$$ 12.8059 0.990949 0.495475 0.868622i $$-0.334994\pi$$
0.495475 + 0.868622i $$0.334994\pi$$
$$168$$ 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 $$-0.662891\pi$$
−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
$$185$$ −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 $$-0.432072\pi$$
0.211785 + 0.977316i $$0.432072\pi$$
$$192$$ 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 $$-0.503464\pi$$
−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
$$205$$ −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 $$-0.296893\pi$$
0.595654 + 0.803241i $$0.296893\pi$$
$$212$$ 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 $$-0.520307\pi$$
−0.0637538 + 0.997966i $$0.520307\pi$$
$$224$$ 0 0
$$225$$ −2.72545 −0.181697
$$226$$ 0 0
$$227$$ 17.3550 1.15189 0.575946 0.817488i $$-0.304634\pi$$
0.575946 + 0.817488i $$0.304634\pi$$
$$228$$ 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 $$-0.434184\pi$$
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
$$245$$ 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 $$-0.866126\pi$$
−0.912853 + 0.408288i $$0.866126\pi$$
$$252$$ 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 $$-0.661659\pi$$
−0.486315 + 0.873784i $$0.661659\pi$$
$$264$$ 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 $$-0.783069\pi$$
−0.776622 + 0.629967i $$0.783069\pi$$
$$272$$ 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 $$-0.686611\pi$$
−0.553246 + 0.833018i $$0.686611\pi$$
$$284$$ 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 $$-0.513914\pi$$
−0.0436990 + 0.999045i $$0.513914\pi$$
$$308$$ 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 $$-0.520975\pi$$
−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 $$-0.283193\pi$$
0.629664 + 0.776868i $$0.283193\pi$$
$$332$$ 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 $$-0.528014\pi$$
−0.0878936 + 0.996130i $$0.528014\pi$$
$$348$$ 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 $$-0.298658\pi$$
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
$$365$$ 6.10557 0.319580
$$366$$ 0 0
$$367$$ 14.2877 0.745813 0.372906 0.927869i $$-0.378361\pi$$
0.372906 + 0.927869i $$0.378361\pi$$
$$368$$ 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 $$-0.464055\pi$$
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 $$-0.343307\pi$$
0.472624 + 0.881264i $$0.343307\pi$$
$$384$$ 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 $$-0.535801\pi$$
−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
$$425$$ 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 $$-0.626458\pi$$
−0.386911 + 0.922117i $$0.626458\pi$$
$$432$$ 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 $$-0.375444\pi$$
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 $$-0.527098\pi$$
−0.0850286 + 0.996379i $$0.527098\pi$$
$$444$$ 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 $$-0.783947\pi$$
−0.778358 + 0.627820i $$0.783947\pi$$
$$464$$ 0 0
$$465$$ 3.54910 0.164586
$$466$$ 0 0
$$467$$ −5.23976 −0.242467 −0.121234 0.992624i $$-0.538685\pi$$
−0.121234 + 0.992624i $$0.538685\pi$$
$$468$$ 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 $$-0.517483\pi$$
−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
$$485$$ 6.08044 0.276099
$$486$$ 0 0
$$487$$ 25.8944 1.17339 0.586694 0.809808i $$-0.300429\pi$$
0.586694 + 0.809808i $$0.300429\pi$$
$$488$$ 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
$$495$$ −4.57193 −0.205493
$$496$$ 0 0
$$497$$ 4.74828 0.212989
$$498$$ 0 0
$$499$$ −38.4583 −1.72163 −0.860814 0.508920i $$-0.830045\pi$$
−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 $$-0.179690\pi$$
0.844849 + 0.535004i $$0.179690\pi$$
$$504$$ 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 $$-0.410367\pi$$
0.277884 + 0.960615i $$0.410367\pi$$
$$524$$ 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 $$-0.325805\pi$$
0.520342 + 0.853958i $$0.325805\pi$$
$$548$$ 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 $$-0.634907\pi$$
−0.411248 + 0.911523i $$0.634907\pi$$
$$564$$ 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 $$-0.695007\pi$$
−0.575024 + 0.818137i $$0.695007\pi$$
$$572$$ 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 $$-0.750594\pi$$
−0.708425 + 0.705786i $$0.750594\pi$$
$$588$$ 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 $$-0.188732\pi$$
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
$$605$$ 8.18601 0.332809
$$606$$ 0 0
$$607$$ 16.9977 0.689915 0.344958 0.938618i $$-0.387893\pi$$
0.344958 + 0.938618i $$0.387893\pi$$
$$608$$ 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 $$-0.737098\pi$$
−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
$$625$$ 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 $$-0.469184\pi$$
0.0966600 + 0.995317i $$0.469184\pi$$
$$632$$ 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 $$-0.170120\pi$$
0.860550 + 0.509366i $$0.170120\pi$$
$$644$$ 0 0
$$645$$ −0.211133 −0.00831337
$$646$$ 0 0
$$647$$ −15.4606 −0.607817 −0.303909 0.952701i $$-0.598292\pi$$
−0.303909 + 0.952701i $$0.598292\pi$$
$$648$$ 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 $$-0.0499961\pi$$
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
$$665$$ −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 $$-0.441926\pi$$
0.181436 + 0.983403i $$0.441926\pi$$
$$684$$ 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 $$-0.649904\pi$$
−0.453721 + 0.891144i $$0.649904\pi$$
$$692$$ 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 $$-0.752599\pi$$
−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
$$725$$ −2.20147 −0.0817606
$$726$$ 0 0
$$727$$ −3.72315 −0.138084 −0.0690420 0.997614i $$-0.521994\pi$$
−0.0690420 + 0.997614i $$0.521994\pi$$
$$728$$ 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 $$-0.501474\pi$$
−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 $$-0.683605\pi$$
−0.545353 + 0.838206i $$0.683605\pi$$
$$744$$ 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 $$-0.437695\pi$$
0.194490 + 0.980905i $$0.437695\pi$$
$$752$$ 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 $$-0.540750\pi$$
−0.127670 + 0.991817i $$0.540750\pi$$
$$788$$ 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 $$-0.349676\pi$$
0.454897 + 0.890544i $$0.349676\pi$$
$$812$$ 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 $$-0.725011\pi$$
−0.649473 + 0.760384i $$0.725011\pi$$
$$824$$ 0 0
$$825$$ 0.878968 0.0306017
$$826$$ 0 0
$$827$$ −4.45320 −0.154853 −0.0774264 0.996998i $$-0.524670\pi$$
−0.0774264 + 0.996998i $$0.524670\pi$$
$$828$$ 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 $$-0.539435\pi$$
−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
$$845$$ 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 $$-0.820141\pi$$
−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 $$-0.384836\pi$$
0.353956 + 0.935262i $$0.384836\pi$$
$$864$$ 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 $$-0.0877624\pi$$
0.962231 + 0.272234i $$0.0877624\pi$$
$$884$$ 0 0
$$885$$ 4.95205 0.166461
$$886$$ 0 0
$$887$$ −15.5371 −0.521686 −0.260843 0.965381i $$-0.584001\pi$$
−0.260843 + 0.965381i $$0.584001\pi$$
$$888$$ 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 $$-0.683626\pi$$
−0.545409 + 0.838170i $$0.683626\pi$$
$$908$$ 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
$$915$$ 3.31284 0.109519
$$916$$ 0 0
$$917$$ 3.64271 0.120293
$$918$$ 0 0
$$919$$ −49.8229 −1.64351 −0.821753 0.569844i $$-0.807004\pi$$
−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
$$925$$ 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 $$-0.209911\pi$$
0.790326 + 0.612687i $$0.209911\pi$$
$$948$$ 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 $$-0.465123\pi$$
0.109350 + 0.994003i $$0.465123\pi$$
$$968$$ 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
$$975$$ −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 $$-0.421172\pi$$
0.245121 + 0.969493i $$0.421172\pi$$
$$984$$ 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 $$-0.809440\pi$$
−0.826090 + 0.563538i $$0.809440\pi$$
$$992$$ 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 920.2.a.g.1.2 3
3.2 odd 2 8280.2.a.bo.1.2 3
4.3 odd 2 1840.2.a.t.1.2 3
5.2 odd 4 4600.2.e.r.4049.4 6
5.3 odd 4 4600.2.e.r.4049.3 6
5.4 even 2 4600.2.a.y.1.2 3
8.3 odd 2 7360.2.a.ca.1.2 3
8.5 even 2 7360.2.a.cb.1.2 3
20.19 odd 2 9200.2.a.cd.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.2 3 1.1 even 1 trivial
1840.2.a.t.1.2 3 4.3 odd 2
4600.2.a.y.1.2 3 5.4 even 2
4600.2.e.r.4049.3 6 5.3 odd 4
4600.2.e.r.4049.4 6 5.2 odd 4
7360.2.a.ca.1.2 3 8.3 odd 2
7360.2.a.cb.1.2 3 8.5 even 2
8280.2.a.bo.1.2 3 3.2 odd 2
9200.2.a.cd.1.2 3 20.19 odd 2