# Properties

 Label 4080.2.a.bq.1.2 Level $4080$ Weight $2$ Character 4080.1 Self dual yes Analytic conductor $32.579$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4080,2,Mod(1,4080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4080.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: $$32.5789640247$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 510) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.44949$$ of defining polynomial Character $$\chi$$ $$=$$ 4080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} +4.89898 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} +4.89898 q^{7} +1.00000 q^{9} +6.89898 q^{13} +1.00000 q^{15} -1.00000 q^{17} -4.00000 q^{19} +4.89898 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -4.00000 q^{31} +4.89898 q^{35} +6.00000 q^{37} +6.89898 q^{39} -2.89898 q^{41} -8.89898 q^{43} +1.00000 q^{45} -9.79796 q^{47} +17.0000 q^{49} -1.00000 q^{51} -7.79796 q^{53} -4.00000 q^{57} -4.89898 q^{59} +11.7980 q^{61} +4.89898 q^{63} +6.89898 q^{65} -0.898979 q^{67} +4.00000 q^{69} +8.89898 q^{71} -10.8990 q^{73} +1.00000 q^{75} +5.79796 q^{79} +1.00000 q^{81} -13.7980 q^{83} -1.00000 q^{85} +6.00000 q^{87} -7.79796 q^{89} +33.7980 q^{91} -4.00000 q^{93} -4.00000 q^{95} -12.6969 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} - 8 q^{19} + 8 q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{29} - 8 q^{31} + 12 q^{37} + 4 q^{39} + 4 q^{41} - 8 q^{43} + 2 q^{45} + 34 q^{49} - 2 q^{51} + 4 q^{53} - 8 q^{57} + 4 q^{61} + 4 q^{65} + 8 q^{67} + 8 q^{69} + 8 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} - 8 q^{83} - 2 q^{85} + 12 q^{87} + 4 q^{89} + 48 q^{91} - 8 q^{93} - 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 2 * q^17 - 8 * q^19 + 8 * q^23 + 2 * q^25 + 2 * q^27 + 12 * q^29 - 8 * q^31 + 12 * q^37 + 4 * q^39 + 4 * q^41 - 8 * q^43 + 2 * q^45 + 34 * q^49 - 2 * q^51 + 4 * q^53 - 8 * q^57 + 4 * q^61 + 4 * q^65 + 8 * q^67 + 8 * q^69 + 8 * q^71 - 12 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 - 8 * q^83 - 2 * q^85 + 12 * q^87 + 4 * q^89 + 48 * q^91 - 8 * q^93 - 8 * q^95 + 4 * q^97

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.89898 1.85164 0.925820 0.377964i $$-0.123376\pi$$
0.925820 + 0.377964i $$0.123376\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 6.89898 1.91343 0.956716 0.291022i $$-0.0939953\pi$$
0.956716 + 0.291022i $$0.0939953\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 4.89898 1.06904
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 4.89898 0.828079
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 0 0
$$39$$ 6.89898 1.10472
$$40$$ 0 0
$$41$$ −2.89898 −0.452745 −0.226372 0.974041i $$-0.572687\pi$$
−0.226372 + 0.974041i $$0.572687\pi$$
$$42$$ 0 0
$$43$$ −8.89898 −1.35708 −0.678541 0.734563i $$-0.737387\pi$$
−0.678541 + 0.734563i $$0.737387\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ −9.79796 −1.42918 −0.714590 0.699544i $$-0.753387\pi$$
−0.714590 + 0.699544i $$0.753387\pi$$
$$48$$ 0 0
$$49$$ 17.0000 2.42857
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 0 0
$$53$$ −7.79796 −1.07113 −0.535566 0.844493i $$-0.679902\pi$$
−0.535566 + 0.844493i $$0.679902\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ −4.89898 −0.637793 −0.318896 0.947790i $$-0.603312\pi$$
−0.318896 + 0.947790i $$0.603312\pi$$
$$60$$ 0 0
$$61$$ 11.7980 1.51057 0.755287 0.655394i $$-0.227498\pi$$
0.755287 + 0.655394i $$0.227498\pi$$
$$62$$ 0 0
$$63$$ 4.89898 0.617213
$$64$$ 0 0
$$65$$ 6.89898 0.855713
$$66$$ 0 0
$$67$$ −0.898979 −0.109828 −0.0549139 0.998491i $$-0.517488\pi$$
−0.0549139 + 0.998491i $$0.517488\pi$$
$$68$$ 0 0
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 8.89898 1.05611 0.528057 0.849209i $$-0.322921\pi$$
0.528057 + 0.849209i $$0.322921\pi$$
$$72$$ 0 0
$$73$$ −10.8990 −1.27563 −0.637815 0.770190i $$-0.720161\pi$$
−0.637815 + 0.770190i $$0.720161\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 5.79796 0.652321 0.326161 0.945314i $$-0.394245\pi$$
0.326161 + 0.945314i $$0.394245\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −13.7980 −1.51452 −0.757261 0.653112i $$-0.773463\pi$$
−0.757261 + 0.653112i $$0.773463\pi$$
$$84$$ 0 0
$$85$$ −1.00000 −0.108465
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ −7.79796 −0.826582 −0.413291 0.910599i $$-0.635621\pi$$
−0.413291 + 0.910599i $$0.635621\pi$$
$$90$$ 0 0
$$91$$ 33.7980 3.54299
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ −12.6969 −1.28918 −0.644589 0.764529i $$-0.722972\pi$$
−0.644589 + 0.764529i $$0.722972\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −18.8990 −1.88052 −0.940259 0.340459i $$-0.889418\pi$$
−0.940259 + 0.340459i $$0.889418\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 4.89898 0.478091
$$106$$ 0 0
$$107$$ −5.79796 −0.560510 −0.280255 0.959926i $$-0.590419\pi$$
−0.280255 + 0.959926i $$0.590419\pi$$
$$108$$ 0 0
$$109$$ 11.7980 1.13004 0.565020 0.825077i $$-0.308869\pi$$
0.565020 + 0.825077i $$0.308869\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ 7.79796 0.733570 0.366785 0.930306i $$-0.380458\pi$$
0.366785 + 0.930306i $$0.380458\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 0 0
$$117$$ 6.89898 0.637811
$$118$$ 0 0
$$119$$ −4.89898 −0.449089
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −2.89898 −0.261392
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ 0 0
$$129$$ −8.89898 −0.783511
$$130$$ 0 0
$$131$$ −9.79796 −0.856052 −0.428026 0.903767i $$-0.640791\pi$$
−0.428026 + 0.903767i $$0.640791\pi$$
$$132$$ 0 0
$$133$$ −19.5959 −1.69918
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ 0 0
$$139$$ −13.7980 −1.17033 −0.585164 0.810915i $$-0.698970\pi$$
−0.585164 + 0.810915i $$0.698970\pi$$
$$140$$ 0 0
$$141$$ −9.79796 −0.825137
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 0 0
$$147$$ 17.0000 1.40214
$$148$$ 0 0
$$149$$ −18.8990 −1.54826 −0.774132 0.633024i $$-0.781814\pi$$
−0.774132 + 0.633024i $$0.781814\pi$$
$$150$$ 0 0
$$151$$ 9.79796 0.797347 0.398673 0.917093i $$-0.369471\pi$$
0.398673 + 0.917093i $$0.369471\pi$$
$$152$$ 0 0
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ −1.10102 −0.0878710 −0.0439355 0.999034i $$-0.513990\pi$$
−0.0439355 + 0.999034i $$0.513990\pi$$
$$158$$ 0 0
$$159$$ −7.79796 −0.618418
$$160$$ 0 0
$$161$$ 19.5959 1.54437
$$162$$ 0 0
$$163$$ 2.20204 0.172477 0.0862386 0.996275i $$-0.472515\pi$$
0.0862386 + 0.996275i $$0.472515\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2.20204 −0.170399 −0.0851995 0.996364i $$-0.527153\pi$$
−0.0851995 + 0.996364i $$0.527153\pi$$
$$168$$ 0 0
$$169$$ 34.5959 2.66122
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 4.89898 0.370328
$$176$$ 0 0
$$177$$ −4.89898 −0.368230
$$178$$ 0 0
$$179$$ 4.89898 0.366167 0.183083 0.983097i $$-0.441392\pi$$
0.183083 + 0.983097i $$0.441392\pi$$
$$180$$ 0 0
$$181$$ −4.20204 −0.312335 −0.156168 0.987731i $$-0.549914\pi$$
−0.156168 + 0.987731i $$0.549914\pi$$
$$182$$ 0 0
$$183$$ 11.7980 0.872130
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4.89898 0.356348
$$190$$ 0 0
$$191$$ 9.79796 0.708955 0.354478 0.935064i $$-0.384659\pi$$
0.354478 + 0.935064i $$0.384659\pi$$
$$192$$ 0 0
$$193$$ −1.10102 −0.0792532 −0.0396266 0.999215i $$-0.512617\pi$$
−0.0396266 + 0.999215i $$0.512617\pi$$
$$194$$ 0 0
$$195$$ 6.89898 0.494046
$$196$$ 0 0
$$197$$ −13.5959 −0.968669 −0.484335 0.874883i $$-0.660938\pi$$
−0.484335 + 0.874883i $$0.660938\pi$$
$$198$$ 0 0
$$199$$ −15.5959 −1.10557 −0.552783 0.833325i $$-0.686434\pi$$
−0.552783 + 0.833325i $$0.686434\pi$$
$$200$$ 0 0
$$201$$ −0.898979 −0.0634091
$$202$$ 0 0
$$203$$ 29.3939 2.06305
$$204$$ 0 0
$$205$$ −2.89898 −0.202474
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ 8.89898 0.609748
$$214$$ 0 0
$$215$$ −8.89898 −0.606905
$$216$$ 0 0
$$217$$ −19.5959 −1.33026
$$218$$ 0 0
$$219$$ −10.8990 −0.736485
$$220$$ 0 0
$$221$$ −6.89898 −0.464076
$$222$$ 0 0
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 21.7980 1.44678 0.723391 0.690439i $$-0.242583\pi$$
0.723391 + 0.690439i $$0.242583\pi$$
$$228$$ 0 0
$$229$$ −13.5959 −0.898444 −0.449222 0.893420i $$-0.648299\pi$$
−0.449222 + 0.893420i $$0.648299\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ 0 0
$$235$$ −9.79796 −0.639148
$$236$$ 0 0
$$237$$ 5.79796 0.376618
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 13.5959 0.875790 0.437895 0.899026i $$-0.355724\pi$$
0.437895 + 0.899026i $$0.355724\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 17.0000 1.08609
$$246$$ 0 0
$$247$$ −27.5959 −1.75589
$$248$$ 0 0
$$249$$ −13.7980 −0.874410
$$250$$ 0 0
$$251$$ −4.89898 −0.309221 −0.154610 0.987976i $$-0.549412\pi$$
−0.154610 + 0.987976i $$0.549412\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −1.00000 −0.0626224
$$256$$ 0 0
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ 29.3939 1.82645
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ −7.79796 −0.479025
$$266$$ 0 0
$$267$$ −7.79796 −0.477227
$$268$$ 0 0
$$269$$ 9.59592 0.585073 0.292537 0.956254i $$-0.405501\pi$$
0.292537 + 0.956254i $$0.405501\pi$$
$$270$$ 0 0
$$271$$ 1.79796 0.109218 0.0546091 0.998508i $$-0.482609\pi$$
0.0546091 + 0.998508i $$0.482609\pi$$
$$272$$ 0 0
$$273$$ 33.7980 2.04555
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7.79796 0.468534 0.234267 0.972172i $$-0.424731\pi$$
0.234267 + 0.972172i $$0.424731\pi$$
$$278$$ 0 0
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ 27.7980 1.65829 0.829144 0.559036i $$-0.188829\pi$$
0.829144 + 0.559036i $$0.188829\pi$$
$$282$$ 0 0
$$283$$ −23.5959 −1.40263 −0.701316 0.712851i $$-0.747404\pi$$
−0.701316 + 0.712851i $$0.747404\pi$$
$$284$$ 0 0
$$285$$ −4.00000 −0.236940
$$286$$ 0 0
$$287$$ −14.2020 −0.838320
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ −12.6969 −0.744308
$$292$$ 0 0
$$293$$ 13.5959 0.794282 0.397141 0.917758i $$-0.370002\pi$$
0.397141 + 0.917758i $$0.370002\pi$$
$$294$$ 0 0
$$295$$ −4.89898 −0.285230
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 27.5959 1.59591
$$300$$ 0 0
$$301$$ −43.5959 −2.51283
$$302$$ 0 0
$$303$$ −18.8990 −1.08572
$$304$$ 0 0
$$305$$ 11.7980 0.675549
$$306$$ 0 0
$$307$$ −0.898979 −0.0513075 −0.0256537 0.999671i $$-0.508167\pi$$
−0.0256537 + 0.999671i $$0.508167\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 7.10102 0.402662 0.201331 0.979523i $$-0.435473\pi$$
0.201331 + 0.979523i $$0.435473\pi$$
$$312$$ 0 0
$$313$$ 6.89898 0.389953 0.194977 0.980808i $$-0.437537\pi$$
0.194977 + 0.980808i $$0.437537\pi$$
$$314$$ 0 0
$$315$$ 4.89898 0.276026
$$316$$ 0 0
$$317$$ 14.0000 0.786318 0.393159 0.919470i $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −5.79796 −0.323611
$$322$$ 0 0
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ 6.89898 0.382687
$$326$$ 0 0
$$327$$ 11.7980 0.652429
$$328$$ 0 0
$$329$$ −48.0000 −2.64633
$$330$$ 0 0
$$331$$ 5.79796 0.318685 0.159342 0.987223i $$-0.449063\pi$$
0.159342 + 0.987223i $$0.449063\pi$$
$$332$$ 0 0
$$333$$ 6.00000 0.328798
$$334$$ 0 0
$$335$$ −0.898979 −0.0491165
$$336$$ 0 0
$$337$$ 32.6969 1.78112 0.890558 0.454870i $$-0.150314\pi$$
0.890558 + 0.454870i $$0.150314\pi$$
$$338$$ 0 0
$$339$$ 7.79796 0.423527
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 48.9898 2.64520
$$344$$ 0 0
$$345$$ 4.00000 0.215353
$$346$$ 0 0
$$347$$ −21.7980 −1.17018 −0.585088 0.810970i $$-0.698940\pi$$
−0.585088 + 0.810970i $$0.698940\pi$$
$$348$$ 0 0
$$349$$ 4.20204 0.224930 0.112465 0.993656i $$-0.464125\pi$$
0.112465 + 0.993656i $$0.464125\pi$$
$$350$$ 0 0
$$351$$ 6.89898 0.368240
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ 0 0
$$355$$ 8.89898 0.472309
$$356$$ 0 0
$$357$$ −4.89898 −0.259281
$$358$$ 0 0
$$359$$ −37.3939 −1.97357 −0.986787 0.162025i $$-0.948198\pi$$
−0.986787 + 0.162025i $$0.948198\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ −11.0000 −0.577350
$$364$$ 0 0
$$365$$ −10.8990 −0.570479
$$366$$ 0 0
$$367$$ 27.1010 1.41466 0.707331 0.706883i $$-0.249899\pi$$
0.707331 + 0.706883i $$0.249899\pi$$
$$368$$ 0 0
$$369$$ −2.89898 −0.150915
$$370$$ 0 0
$$371$$ −38.2020 −1.98335
$$372$$ 0 0
$$373$$ 24.6969 1.27876 0.639380 0.768891i $$-0.279191\pi$$
0.639380 + 0.768891i $$0.279191\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 41.3939 2.13189
$$378$$ 0 0
$$379$$ 29.7980 1.53062 0.765309 0.643663i $$-0.222586\pi$$
0.765309 + 0.643663i $$0.222586\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ 0 0
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.89898 −0.452361
$$388$$ 0 0
$$389$$ −38.4949 −1.95177 −0.975884 0.218288i $$-0.929953\pi$$
−0.975884 + 0.218288i $$0.929953\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ −9.79796 −0.494242
$$394$$ 0 0
$$395$$ 5.79796 0.291727
$$396$$ 0 0
$$397$$ 1.59592 0.0800968 0.0400484 0.999198i $$-0.487249\pi$$
0.0400484 + 0.999198i $$0.487249\pi$$
$$398$$ 0 0
$$399$$ −19.5959 −0.981023
$$400$$ 0 0
$$401$$ 22.8990 1.14352 0.571760 0.820421i $$-0.306261\pi$$
0.571760 + 0.820421i $$0.306261\pi$$
$$402$$ 0 0
$$403$$ −27.5959 −1.37465
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −17.5959 −0.870062 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$410$$ 0 0
$$411$$ 14.0000 0.690569
$$412$$ 0 0
$$413$$ −24.0000 −1.18096
$$414$$ 0 0
$$415$$ −13.7980 −0.677315
$$416$$ 0 0
$$417$$ −13.7980 −0.675689
$$418$$ 0 0
$$419$$ 17.7980 0.869487 0.434744 0.900554i $$-0.356839\pi$$
0.434744 + 0.900554i $$0.356839\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 0 0
$$423$$ −9.79796 −0.476393
$$424$$ 0 0
$$425$$ −1.00000 −0.0485071
$$426$$ 0 0
$$427$$ 57.7980 2.79704
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 23.1010 1.11274 0.556369 0.830936i $$-0.312194\pi$$
0.556369 + 0.830936i $$0.312194\pi$$
$$432$$ 0 0
$$433$$ −35.3939 −1.70092 −0.850461 0.526039i $$-0.823677\pi$$
−0.850461 + 0.526039i $$0.823677\pi$$
$$434$$ 0 0
$$435$$ 6.00000 0.287678
$$436$$ 0 0
$$437$$ −16.0000 −0.765384
$$438$$ 0 0
$$439$$ −5.79796 −0.276721 −0.138361 0.990382i $$-0.544183\pi$$
−0.138361 + 0.990382i $$0.544183\pi$$
$$440$$ 0 0
$$441$$ 17.0000 0.809524
$$442$$ 0 0
$$443$$ −37.7980 −1.79584 −0.897918 0.440164i $$-0.854920\pi$$
−0.897918 + 0.440164i $$0.854920\pi$$
$$444$$ 0 0
$$445$$ −7.79796 −0.369659
$$446$$ 0 0
$$447$$ −18.8990 −0.893891
$$448$$ 0 0
$$449$$ 6.89898 0.325583 0.162791 0.986660i $$-0.447950\pi$$
0.162791 + 0.986660i $$0.447950\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 9.79796 0.460348
$$454$$ 0 0
$$455$$ 33.7980 1.58447
$$456$$ 0 0
$$457$$ 16.2020 0.757900 0.378950 0.925417i $$-0.376285\pi$$
0.378950 + 0.925417i $$0.376285\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −28.6969 −1.33655 −0.668275 0.743914i $$-0.732967\pi$$
−0.668275 + 0.743914i $$0.732967\pi$$
$$462$$ 0 0
$$463$$ −7.59592 −0.353012 −0.176506 0.984300i $$-0.556480\pi$$
−0.176506 + 0.984300i $$0.556480\pi$$
$$464$$ 0 0
$$465$$ −4.00000 −0.185496
$$466$$ 0 0
$$467$$ 7.59592 0.351497 0.175749 0.984435i $$-0.443765\pi$$
0.175749 + 0.984435i $$0.443765\pi$$
$$468$$ 0 0
$$469$$ −4.40408 −0.203362
$$470$$ 0 0
$$471$$ −1.10102 −0.0507323
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ −7.79796 −0.357044
$$478$$ 0 0
$$479$$ 32.8990 1.50319 0.751596 0.659623i $$-0.229284\pi$$
0.751596 + 0.659623i $$0.229284\pi$$
$$480$$ 0 0
$$481$$ 41.3939 1.88740
$$482$$ 0 0
$$483$$ 19.5959 0.891645
$$484$$ 0 0
$$485$$ −12.6969 −0.576538
$$486$$ 0 0
$$487$$ 20.8990 0.947023 0.473512 0.880788i $$-0.342986\pi$$
0.473512 + 0.880788i $$0.342986\pi$$
$$488$$ 0 0
$$489$$ 2.20204 0.0995797
$$490$$ 0 0
$$491$$ 1.30306 0.0588063 0.0294032 0.999568i $$-0.490639\pi$$
0.0294032 + 0.999568i $$0.490639\pi$$
$$492$$ 0 0
$$493$$ −6.00000 −0.270226
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 43.5959 1.95554
$$498$$ 0 0
$$499$$ 39.5959 1.77256 0.886278 0.463153i $$-0.153282\pi$$
0.886278 + 0.463153i $$0.153282\pi$$
$$500$$ 0 0
$$501$$ −2.20204 −0.0983799
$$502$$ 0 0
$$503$$ −4.00000 −0.178351 −0.0891756 0.996016i $$-0.528423\pi$$
−0.0891756 + 0.996016i $$0.528423\pi$$
$$504$$ 0 0
$$505$$ −18.8990 −0.840994
$$506$$ 0 0
$$507$$ 34.5959 1.53646
$$508$$ 0 0
$$509$$ −15.3031 −0.678296 −0.339148 0.940733i $$-0.610139\pi$$
−0.339148 + 0.940733i $$0.610139\pi$$
$$510$$ 0 0
$$511$$ −53.3939 −2.36201
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 6.00000 0.263371
$$520$$ 0 0
$$521$$ −40.2929 −1.76526 −0.882631 0.470066i $$-0.844230\pi$$
−0.882631 + 0.470066i $$0.844230\pi$$
$$522$$ 0 0
$$523$$ −18.6969 −0.817560 −0.408780 0.912633i $$-0.634046\pi$$
−0.408780 + 0.912633i $$0.634046\pi$$
$$524$$ 0 0
$$525$$ 4.89898 0.213809
$$526$$ 0 0
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −4.89898 −0.212598
$$532$$ 0 0
$$533$$ −20.0000 −0.866296
$$534$$ 0 0
$$535$$ −5.79796 −0.250668
$$536$$ 0 0
$$537$$ 4.89898 0.211407
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −7.79796 −0.335260 −0.167630 0.985850i $$-0.553611\pi$$
−0.167630 + 0.985850i $$0.553611\pi$$
$$542$$ 0 0
$$543$$ −4.20204 −0.180327
$$544$$ 0 0
$$545$$ 11.7980 0.505369
$$546$$ 0 0
$$547$$ −0.404082 −0.0172773 −0.00863865 0.999963i $$-0.502750\pi$$
−0.00863865 + 0.999963i $$0.502750\pi$$
$$548$$ 0 0
$$549$$ 11.7980 0.503525
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 28.4041 1.20786
$$554$$ 0 0
$$555$$ 6.00000 0.254686
$$556$$ 0 0
$$557$$ 19.7980 0.838866 0.419433 0.907786i $$-0.362229\pi$$
0.419433 + 0.907786i $$0.362229\pi$$
$$558$$ 0 0
$$559$$ −61.3939 −2.59668
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −21.7980 −0.918674 −0.459337 0.888262i $$-0.651913\pi$$
−0.459337 + 0.888262i $$0.651913\pi$$
$$564$$ 0 0
$$565$$ 7.79796 0.328063
$$566$$ 0 0
$$567$$ 4.89898 0.205738
$$568$$ 0 0
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 0 0
$$571$$ −29.7980 −1.24701 −0.623503 0.781821i $$-0.714291\pi$$
−0.623503 + 0.781821i $$0.714291\pi$$
$$572$$ 0 0
$$573$$ 9.79796 0.409316
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ −27.3939 −1.14042 −0.570211 0.821498i $$-0.693139\pi$$
−0.570211 + 0.821498i $$0.693139\pi$$
$$578$$ 0 0
$$579$$ −1.10102 −0.0457569
$$580$$ 0 0
$$581$$ −67.5959 −2.80435
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 6.89898 0.285238
$$586$$ 0 0
$$587$$ 41.3939 1.70851 0.854254 0.519856i $$-0.174014\pi$$
0.854254 + 0.519856i $$0.174014\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ −13.5959 −0.559261
$$592$$ 0 0
$$593$$ 1.59592 0.0655365 0.0327682 0.999463i $$-0.489568\pi$$
0.0327682 + 0.999463i $$0.489568\pi$$
$$594$$ 0 0
$$595$$ −4.89898 −0.200839
$$596$$ 0 0
$$597$$ −15.5959 −0.638298
$$598$$ 0 0
$$599$$ −21.3939 −0.874130 −0.437065 0.899430i $$-0.643982\pi$$
−0.437065 + 0.899430i $$0.643982\pi$$
$$600$$ 0 0
$$601$$ 16.2020 0.660895 0.330448 0.943824i $$-0.392800\pi$$
0.330448 + 0.943824i $$0.392800\pi$$
$$602$$ 0 0
$$603$$ −0.898979 −0.0366093
$$604$$ 0 0
$$605$$ −11.0000 −0.447214
$$606$$ 0 0
$$607$$ 32.4949 1.31893 0.659464 0.751736i $$-0.270783\pi$$
0.659464 + 0.751736i $$0.270783\pi$$
$$608$$ 0 0
$$609$$ 29.3939 1.19110
$$610$$ 0 0
$$611$$ −67.5959 −2.73464
$$612$$ 0 0
$$613$$ −14.4949 −0.585443 −0.292722 0.956198i $$-0.594561\pi$$
−0.292722 + 0.956198i $$0.594561\pi$$
$$614$$ 0 0
$$615$$ −2.89898 −0.116898
$$616$$ 0 0
$$617$$ −37.5959 −1.51355 −0.756777 0.653673i $$-0.773227\pi$$
−0.756777 + 0.653673i $$0.773227\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 0 0
$$623$$ −38.2020 −1.53053
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 0 0
$$633$$ 12.0000 0.476957
$$634$$ 0 0
$$635$$ 12.0000 0.476205
$$636$$ 0 0
$$637$$ 117.283 4.64691
$$638$$ 0 0
$$639$$ 8.89898 0.352038
$$640$$ 0 0
$$641$$ −20.6969 −0.817480 −0.408740 0.912651i $$-0.634032\pi$$
−0.408740 + 0.912651i $$0.634032\pi$$
$$642$$ 0 0
$$643$$ −29.7980 −1.17512 −0.587558 0.809182i $$-0.699911\pi$$
−0.587558 + 0.809182i $$0.699911\pi$$
$$644$$ 0 0
$$645$$ −8.89898 −0.350397
$$646$$ 0 0
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −19.5959 −0.768025
$$652$$ 0 0
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 0 0
$$655$$ −9.79796 −0.382838
$$656$$ 0 0
$$657$$ −10.8990 −0.425210
$$658$$ 0 0
$$659$$ −30.6969 −1.19578 −0.597891 0.801577i $$-0.703995\pi$$
−0.597891 + 0.801577i $$0.703995\pi$$
$$660$$ 0 0
$$661$$ −19.7980 −0.770051 −0.385026 0.922906i $$-0.625807\pi$$
−0.385026 + 0.922906i $$0.625807\pi$$
$$662$$ 0 0
$$663$$ −6.89898 −0.267934
$$664$$ 0 0
$$665$$ −19.5959 −0.759897
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 0 0
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −33.1010 −1.27595 −0.637975 0.770057i $$-0.720228\pi$$
−0.637975 + 0.770057i $$0.720228\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −13.5959 −0.522534 −0.261267 0.965267i $$-0.584140\pi$$
−0.261267 + 0.965267i $$0.584140\pi$$
$$678$$ 0 0
$$679$$ −62.2020 −2.38710
$$680$$ 0 0
$$681$$ 21.7980 0.835300
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 14.0000 0.534913
$$686$$ 0 0
$$687$$ −13.5959 −0.518717
$$688$$ 0 0
$$689$$ −53.7980 −2.04954
$$690$$ 0 0
$$691$$ 27.1918 1.03443 0.517213 0.855857i $$-0.326969\pi$$
0.517213 + 0.855857i $$0.326969\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −13.7980 −0.523386
$$696$$ 0 0
$$697$$ 2.89898 0.109807
$$698$$ 0 0
$$699$$ 14.0000 0.529529
$$700$$ 0 0
$$701$$ 14.8990 0.562727 0.281363 0.959601i $$-0.409213\pi$$
0.281363 + 0.959601i $$0.409213\pi$$
$$702$$ 0 0
$$703$$ −24.0000 −0.905177
$$704$$ 0 0
$$705$$ −9.79796 −0.369012
$$706$$ 0 0
$$707$$ −92.5857 −3.48204
$$708$$ 0 0
$$709$$ −31.7980 −1.19420 −0.597099 0.802168i $$-0.703680\pi$$
−0.597099 + 0.802168i $$0.703680\pi$$
$$710$$ 0 0
$$711$$ 5.79796 0.217440
$$712$$ 0 0
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 12.4949 0.465981 0.232991 0.972479i $$-0.425149\pi$$
0.232991 + 0.972479i $$0.425149\pi$$
$$720$$ 0 0
$$721$$ −19.5959 −0.729790
$$722$$ 0 0
$$723$$ 13.5959 0.505638
$$724$$ 0 0
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ −0.404082 −0.0149866 −0.00749329 0.999972i $$-0.502385\pi$$
−0.00749329 + 0.999972i $$0.502385\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 8.89898 0.329141
$$732$$ 0 0
$$733$$ −46.4949 −1.71733 −0.858664 0.512539i $$-0.828705\pi$$
−0.858664 + 0.512539i $$0.828705\pi$$
$$734$$ 0 0
$$735$$ 17.0000 0.627054
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 9.39388 0.345559 0.172780 0.984960i $$-0.444725\pi$$
0.172780 + 0.984960i $$0.444725\pi$$
$$740$$ 0 0
$$741$$ −27.5959 −1.01376
$$742$$ 0 0
$$743$$ 9.39388 0.344628 0.172314 0.985042i $$-0.444876\pi$$
0.172314 + 0.985042i $$0.444876\pi$$
$$744$$ 0 0
$$745$$ −18.8990 −0.692405
$$746$$ 0 0
$$747$$ −13.7980 −0.504841
$$748$$ 0 0
$$749$$ −28.4041 −1.03786
$$750$$ 0 0
$$751$$ −21.7980 −0.795419 −0.397709 0.917511i $$-0.630195\pi$$
−0.397709 + 0.917511i $$0.630195\pi$$
$$752$$ 0 0
$$753$$ −4.89898 −0.178529
$$754$$ 0 0
$$755$$ 9.79796 0.356584
$$756$$ 0 0
$$757$$ −4.69694 −0.170713 −0.0853566 0.996350i $$-0.527203\pi$$
−0.0853566 + 0.996350i $$0.527203\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −1.59592 −0.0578520 −0.0289260 0.999582i $$-0.509209\pi$$
−0.0289260 + 0.999582i $$0.509209\pi$$
$$762$$ 0 0
$$763$$ 57.7980 2.09243
$$764$$ 0 0
$$765$$ −1.00000 −0.0361551
$$766$$ 0 0
$$767$$ −33.7980 −1.22037
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ 0 0
$$773$$ −35.3939 −1.27303 −0.636515 0.771265i $$-0.719625\pi$$
−0.636515 + 0.771265i $$0.719625\pi$$
$$774$$ 0 0
$$775$$ −4.00000 −0.143684
$$776$$ 0 0
$$777$$ 29.3939 1.05450
$$778$$ 0 0
$$779$$ 11.5959 0.415467
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ −1.10102 −0.0392971
$$786$$ 0 0
$$787$$ 45.7980 1.63252 0.816260 0.577684i $$-0.196043\pi$$
0.816260 + 0.577684i $$0.196043\pi$$
$$788$$ 0 0
$$789$$ −8.00000 −0.284808
$$790$$ 0 0
$$791$$ 38.2020 1.35831
$$792$$ 0 0
$$793$$ 81.3939 2.89038
$$794$$ 0 0
$$795$$ −7.79796 −0.276565
$$796$$ 0 0
$$797$$ 26.0000 0.920967 0.460484 0.887668i $$-0.347676\pi$$
0.460484 + 0.887668i $$0.347676\pi$$
$$798$$ 0 0
$$799$$ 9.79796 0.346627
$$800$$ 0 0
$$801$$ −7.79796 −0.275527
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 19.5959 0.690665
$$806$$ 0 0
$$807$$ 9.59592 0.337792
$$808$$ 0 0
$$809$$ 32.6969 1.14956 0.574782 0.818307i $$-0.305087\pi$$
0.574782 + 0.818307i $$0.305087\pi$$
$$810$$ 0 0
$$811$$ −25.3939 −0.891700 −0.445850 0.895108i $$-0.647098\pi$$
−0.445850 + 0.895108i $$0.647098\pi$$
$$812$$ 0 0
$$813$$ 1.79796 0.0630572
$$814$$ 0 0
$$815$$ 2.20204 0.0771341
$$816$$ 0 0
$$817$$ 35.5959 1.24534
$$818$$ 0 0
$$819$$ 33.7980 1.18100
$$820$$ 0 0
$$821$$ 15.7980 0.551353 0.275676 0.961251i $$-0.411098\pi$$
0.275676 + 0.961251i $$0.411098\pi$$
$$822$$ 0 0
$$823$$ −20.8990 −0.728493 −0.364246 0.931303i $$-0.618673\pi$$
−0.364246 + 0.931303i $$0.618673\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 4.00000 0.139094 0.0695468 0.997579i $$-0.477845\pi$$
0.0695468 + 0.997579i $$0.477845\pi$$
$$828$$ 0 0
$$829$$ 3.39388 0.117874 0.0589371 0.998262i $$-0.481229\pi$$
0.0589371 + 0.998262i $$0.481229\pi$$
$$830$$ 0 0
$$831$$ 7.79796 0.270508
$$832$$ 0 0
$$833$$ −17.0000 −0.589015
$$834$$ 0 0
$$835$$ −2.20204 −0.0762048
$$836$$ 0 0
$$837$$ −4.00000 −0.138260
$$838$$ 0 0
$$839$$ 7.10102 0.245154 0.122577 0.992459i $$-0.460884\pi$$
0.122577 + 0.992459i $$0.460884\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 27.7980 0.957413
$$844$$ 0 0
$$845$$ 34.5959 1.19014
$$846$$ 0 0
$$847$$ −53.8888 −1.85164
$$848$$ 0 0
$$849$$ −23.5959 −0.809810
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ 11.3939 0.390119 0.195059 0.980791i $$-0.437510\pi$$
0.195059 + 0.980791i $$0.437510\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ 0 0
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 0 0
$$859$$ −5.79796 −0.197824 −0.0989119 0.995096i $$-0.531536\pi$$
−0.0989119 + 0.995096i $$0.531536\pi$$
$$860$$ 0 0
$$861$$ −14.2020 −0.484004
$$862$$ 0 0
$$863$$ 45.3939 1.54523 0.772613 0.634878i $$-0.218949\pi$$
0.772613 + 0.634878i $$0.218949\pi$$
$$864$$ 0 0
$$865$$ 6.00000 0.204006
$$866$$ 0 0
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −6.20204 −0.210148
$$872$$ 0 0
$$873$$ −12.6969 −0.429726
$$874$$ 0 0
$$875$$ 4.89898 0.165616
$$876$$ 0 0
$$877$$ −31.3939 −1.06010 −0.530048 0.847968i $$-0.677826\pi$$
−0.530048 + 0.847968i $$0.677826\pi$$
$$878$$ 0 0
$$879$$ 13.5959 0.458579
$$880$$ 0 0
$$881$$ 34.4949 1.16216 0.581081 0.813846i $$-0.302630\pi$$
0.581081 + 0.813846i $$0.302630\pi$$
$$882$$ 0 0
$$883$$ −26.6969 −0.898424 −0.449212 0.893425i $$-0.648295\pi$$
−0.449212 + 0.893425i $$0.648295\pi$$
$$884$$ 0 0
$$885$$ −4.89898 −0.164677
$$886$$ 0 0
$$887$$ 12.0000 0.402921 0.201460 0.979497i $$-0.435431\pi$$
0.201460 + 0.979497i $$0.435431\pi$$
$$888$$ 0 0
$$889$$ 58.7878 1.97168
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 39.1918 1.31150
$$894$$ 0 0
$$895$$ 4.89898 0.163755
$$896$$ 0 0
$$897$$ 27.5959 0.921401
$$898$$ 0 0
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 7.79796 0.259788
$$902$$ 0 0
$$903$$ −43.5959 −1.45078
$$904$$ 0 0
$$905$$ −4.20204 −0.139681
$$906$$ 0 0
$$907$$ 33.3939 1.10883 0.554413 0.832242i $$-0.312943\pi$$
0.554413 + 0.832242i $$0.312943\pi$$
$$908$$ 0 0
$$909$$ −18.8990 −0.626840
$$910$$ 0 0
$$911$$ −23.1010 −0.765371 −0.382685 0.923879i $$-0.625001\pi$$
−0.382685 + 0.923879i $$0.625001\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 11.7980 0.390028
$$916$$ 0 0
$$917$$ −48.0000 −1.58510
$$918$$ 0 0
$$919$$ −1.79796 −0.0593092 −0.0296546 0.999560i $$-0.509441\pi$$
−0.0296546 + 0.999560i $$0.509441\pi$$
$$920$$ 0 0
$$921$$ −0.898979 −0.0296224
$$922$$ 0 0
$$923$$ 61.3939 2.02080
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ 36.2929 1.19073 0.595365 0.803455i $$-0.297007\pi$$
0.595365 + 0.803455i $$0.297007\pi$$
$$930$$ 0 0
$$931$$ −68.0000 −2.22861
$$932$$ 0 0
$$933$$ 7.10102 0.232477
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 39.3939 1.28694 0.643471 0.765471i $$-0.277494\pi$$
0.643471 + 0.765471i $$0.277494\pi$$
$$938$$ 0 0
$$939$$ 6.89898 0.225140
$$940$$ 0 0
$$941$$ −16.2020 −0.528171 −0.264086 0.964499i $$-0.585070\pi$$
−0.264086 + 0.964499i $$0.585070\pi$$
$$942$$ 0 0
$$943$$ −11.5959 −0.377615
$$944$$ 0 0
$$945$$ 4.89898 0.159364
$$946$$ 0 0
$$947$$ −21.7980 −0.708338 −0.354169 0.935181i $$-0.615236\pi$$
−0.354169 + 0.935181i $$0.615236\pi$$
$$948$$ 0 0
$$949$$ −75.1918 −2.44083
$$950$$ 0 0
$$951$$ 14.0000 0.453981
$$952$$ 0 0
$$953$$ −41.1918 −1.33433 −0.667167 0.744908i $$-0.732493\pi$$
−0.667167 + 0.744908i $$0.732493\pi$$
$$954$$ 0 0
$$955$$ 9.79796 0.317055
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 68.5857 2.21475
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −5.79796 −0.186837
$$964$$ 0 0
$$965$$ −1.10102 −0.0354431
$$966$$ 0 0
$$967$$ 41.3939 1.33114 0.665569 0.746337i $$-0.268189\pi$$
0.665569 + 0.746337i $$0.268189\pi$$
$$968$$ 0 0
$$969$$ 4.00000 0.128499
$$970$$ 0 0
$$971$$ −38.6969 −1.24184 −0.620922 0.783872i $$-0.713242\pi$$
−0.620922 + 0.783872i $$0.713242\pi$$
$$972$$ 0 0
$$973$$ −67.5959 −2.16703
$$974$$ 0 0
$$975$$ 6.89898 0.220944
$$976$$ 0 0
$$977$$ −29.5959 −0.946857 −0.473429 0.880832i $$-0.656984\pi$$
−0.473429 + 0.880832i $$0.656984\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 11.7980 0.376680
$$982$$ 0 0
$$983$$ −9.39388 −0.299618 −0.149809 0.988715i $$-0.547866\pi$$
−0.149809 + 0.988715i $$0.547866\pi$$
$$984$$ 0 0
$$985$$ −13.5959 −0.433202
$$986$$ 0 0
$$987$$ −48.0000 −1.52786
$$988$$ 0 0
$$989$$ −35.5959 −1.13188
$$990$$ 0 0
$$991$$ 15.5959 0.495421 0.247710 0.968834i $$-0.420322\pi$$
0.247710 + 0.968834i $$0.420322\pi$$
$$992$$ 0 0
$$993$$ 5.79796 0.183993
$$994$$ 0 0
$$995$$ −15.5959 −0.494424
$$996$$ 0 0
$$997$$ −21.5959 −0.683950 −0.341975 0.939709i $$-0.611096\pi$$
−0.341975 + 0.939709i $$0.611096\pi$$
$$998$$ 0 0
$$999$$ 6.00000 0.189832
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 4080.2.a.bq.1.2 2
4.3 odd 2 510.2.a.h.1.1 2
12.11 even 2 1530.2.a.s.1.1 2
20.3 even 4 2550.2.d.u.2449.4 4
20.7 even 4 2550.2.d.u.2449.1 4
20.19 odd 2 2550.2.a.bl.1.2 2
60.59 even 2 7650.2.a.cu.1.2 2
68.67 odd 2 8670.2.a.be.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.h.1.1 2 4.3 odd 2
1530.2.a.s.1.1 2 12.11 even 2
2550.2.a.bl.1.2 2 20.19 odd 2
2550.2.d.u.2449.1 4 20.7 even 4
2550.2.d.u.2449.4 4 20.3 even 4
4080.2.a.bq.1.2 2 1.1 even 1 trivial
7650.2.a.cu.1.2 2 60.59 even 2
8670.2.a.be.1.2 2 68.67 odd 2