# Properties

 Label 4840.2.a.bf.1.3 Level $4840$ Weight $2$ Character 4840.1 Self dual yes Analytic conductor $38.648$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4840 = 2^{3} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4840.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: $$38.6475945783$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.45753625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20$$ x^6 - x^5 - 13*x^4 + 11*x^3 + 41*x^2 - 30*x - 20 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.444728$$ of defining polynomial Character $$\chi$$ $$=$$ 4840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.719585 q^{3} -1.00000 q^{5} +4.18418 q^{7} -2.48220 q^{9} +O(q^{10})$$ $$q+0.719585 q^{3} -1.00000 q^{5} +4.18418 q^{7} -2.48220 q^{9} -6.05056 q^{13} -0.719585 q^{15} +6.52180 q^{17} +0.739455 q^{19} +3.01088 q^{21} -5.13093 q^{23} +1.00000 q^{25} -3.94491 q^{27} +2.08036 q^{29} -1.12698 q^{31} -4.18418 q^{35} +7.11676 q^{37} -4.35390 q^{39} -4.02170 q^{41} +6.29891 q^{43} +2.48220 q^{45} +9.24147 q^{47} +10.5074 q^{49} +4.69299 q^{51} +7.77015 q^{53} +0.532101 q^{57} +6.16416 q^{59} +14.6869 q^{61} -10.3860 q^{63} +6.05056 q^{65} -5.53268 q^{67} -3.69214 q^{69} +8.87252 q^{71} -8.09351 q^{73} +0.719585 q^{75} -0.570482 q^{79} +4.60789 q^{81} -15.2224 q^{83} -6.52180 q^{85} +1.49700 q^{87} -1.33867 q^{89} -25.3167 q^{91} -0.810957 q^{93} -0.739455 q^{95} +9.96643 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9}+O(q^{10})$$ 6 * q + 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 $$6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} - 6 q^{13} - 2 q^{15} + 11 q^{17} - 11 q^{19} + 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} - 6 q^{29} + q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} - 4 q^{41} + 3 q^{43} - 10 q^{45} + 14 q^{47} + 8 q^{49} + 31 q^{51} + 14 q^{53} - 5 q^{57} + 2 q^{59} - 4 q^{61} - 16 q^{63} + 6 q^{65} + 11 q^{67} + 8 q^{69} + 7 q^{71} - 9 q^{73} + 2 q^{75} + 36 q^{79} + 30 q^{81} - 45 q^{83} - 11 q^{85} + 25 q^{87} + q^{89} - 8 q^{91} + 55 q^{93} + 11 q^{95} + 20 q^{97}+O(q^{100})$$ 6 * q + 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 - 6 * q^13 - 2 * q^15 + 11 * q^17 - 11 * q^19 + 2 * q^21 + 18 * q^23 + 6 * q^25 - q^27 - 6 * q^29 + q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 - 4 * q^41 + 3 * q^43 - 10 * q^45 + 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 - 5 * q^57 + 2 * q^59 - 4 * q^61 - 16 * q^63 + 6 * q^65 + 11 * q^67 + 8 * q^69 + 7 * q^71 - 9 * q^73 + 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 - 11 * q^85 + 25 * q^87 + q^89 - 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * 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
$$3$$ 0.719585 0.415453 0.207726 0.978187i $$-0.433394\pi$$
0.207726 + 0.978187i $$0.433394\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.18418 1.58147 0.790736 0.612157i $$-0.209698\pi$$
0.790736 + 0.612157i $$0.209698\pi$$
$$8$$ 0 0
$$9$$ −2.48220 −0.827399
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −6.05056 −1.67812 −0.839062 0.544035i $$-0.816896\pi$$
−0.839062 + 0.544035i $$0.816896\pi$$
$$14$$ 0 0
$$15$$ −0.719585 −0.185796
$$16$$ 0 0
$$17$$ 6.52180 1.58177 0.790885 0.611965i $$-0.209621\pi$$
0.790885 + 0.611965i $$0.209621\pi$$
$$18$$ 0 0
$$19$$ 0.739455 0.169642 0.0848212 0.996396i $$-0.472968\pi$$
0.0848212 + 0.996396i $$0.472968\pi$$
$$20$$ 0 0
$$21$$ 3.01088 0.657027
$$22$$ 0 0
$$23$$ −5.13093 −1.06987 −0.534936 0.844893i $$-0.679664\pi$$
−0.534936 + 0.844893i $$0.679664\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −3.94491 −0.759198
$$28$$ 0 0
$$29$$ 2.08036 0.386313 0.193157 0.981168i $$-0.438127\pi$$
0.193157 + 0.981168i $$0.438127\pi$$
$$30$$ 0 0
$$31$$ −1.12698 −0.202411 −0.101206 0.994866i $$-0.532270\pi$$
−0.101206 + 0.994866i $$0.532270\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.18418 −0.707256
$$36$$ 0 0
$$37$$ 7.11676 1.16999 0.584994 0.811037i $$-0.301097\pi$$
0.584994 + 0.811037i $$0.301097\pi$$
$$38$$ 0 0
$$39$$ −4.35390 −0.697182
$$40$$ 0 0
$$41$$ −4.02170 −0.628085 −0.314042 0.949409i $$-0.601683\pi$$
−0.314042 + 0.949409i $$0.601683\pi$$
$$42$$ 0 0
$$43$$ 6.29891 0.960575 0.480288 0.877111i $$-0.340532\pi$$
0.480288 + 0.877111i $$0.340532\pi$$
$$44$$ 0 0
$$45$$ 2.48220 0.370024
$$46$$ 0 0
$$47$$ 9.24147 1.34801 0.674003 0.738728i $$-0.264573\pi$$
0.674003 + 0.738728i $$0.264573\pi$$
$$48$$ 0 0
$$49$$ 10.5074 1.50106
$$50$$ 0 0
$$51$$ 4.69299 0.657151
$$52$$ 0 0
$$53$$ 7.77015 1.06731 0.533656 0.845702i $$-0.320818\pi$$
0.533656 + 0.845702i $$0.320818\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.532101 0.0704785
$$58$$ 0 0
$$59$$ 6.16416 0.802505 0.401253 0.915967i $$-0.368575\pi$$
0.401253 + 0.915967i $$0.368575\pi$$
$$60$$ 0 0
$$61$$ 14.6869 1.88046 0.940230 0.340541i $$-0.110610\pi$$
0.940230 + 0.340541i $$0.110610\pi$$
$$62$$ 0 0
$$63$$ −10.3860 −1.30851
$$64$$ 0 0
$$65$$ 6.05056 0.750480
$$66$$ 0 0
$$67$$ −5.53268 −0.675924 −0.337962 0.941160i $$-0.609738\pi$$
−0.337962 + 0.941160i $$0.609738\pi$$
$$68$$ 0 0
$$69$$ −3.69214 −0.444481
$$70$$ 0 0
$$71$$ 8.87252 1.05297 0.526487 0.850183i $$-0.323509\pi$$
0.526487 + 0.850183i $$0.323509\pi$$
$$72$$ 0 0
$$73$$ −8.09351 −0.947274 −0.473637 0.880720i $$-0.657059\pi$$
−0.473637 + 0.880720i $$0.657059\pi$$
$$74$$ 0 0
$$75$$ 0.719585 0.0830906
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −0.570482 −0.0641842 −0.0320921 0.999485i $$-0.510217\pi$$
−0.0320921 + 0.999485i $$0.510217\pi$$
$$80$$ 0 0
$$81$$ 4.60789 0.511988
$$82$$ 0 0
$$83$$ −15.2224 −1.67088 −0.835439 0.549584i $$-0.814786\pi$$
−0.835439 + 0.549584i $$0.814786\pi$$
$$84$$ 0 0
$$85$$ −6.52180 −0.707389
$$86$$ 0 0
$$87$$ 1.49700 0.160495
$$88$$ 0 0
$$89$$ −1.33867 −0.141899 −0.0709495 0.997480i $$-0.522603\pi$$
−0.0709495 + 0.997480i $$0.522603\pi$$
$$90$$ 0 0
$$91$$ −25.3167 −2.65391
$$92$$ 0 0
$$93$$ −0.810957 −0.0840923
$$94$$ 0 0
$$95$$ −0.739455 −0.0758664
$$96$$ 0 0
$$97$$ 9.96643 1.01194 0.505969 0.862552i $$-0.331135\pi$$
0.505969 + 0.862552i $$0.331135\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 11.8701 1.18112 0.590562 0.806992i $$-0.298906\pi$$
0.590562 + 0.806992i $$0.298906\pi$$
$$102$$ 0 0
$$103$$ −10.3935 −1.02410 −0.512050 0.858955i $$-0.671114\pi$$
−0.512050 + 0.858955i $$0.671114\pi$$
$$104$$ 0 0
$$105$$ −3.01088 −0.293832
$$106$$ 0 0
$$107$$ 8.34807 0.807039 0.403519 0.914971i $$-0.367787\pi$$
0.403519 + 0.914971i $$0.367787\pi$$
$$108$$ 0 0
$$109$$ −6.25046 −0.598686 −0.299343 0.954146i $$-0.596767\pi$$
−0.299343 + 0.954146i $$0.596767\pi$$
$$110$$ 0 0
$$111$$ 5.12112 0.486075
$$112$$ 0 0
$$113$$ 16.4937 1.55160 0.775798 0.630982i $$-0.217348\pi$$
0.775798 + 0.630982i $$0.217348\pi$$
$$114$$ 0 0
$$115$$ 5.13093 0.478461
$$116$$ 0 0
$$117$$ 15.0187 1.38848
$$118$$ 0 0
$$119$$ 27.2884 2.50152
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −2.89396 −0.260940
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 16.5082 1.46487 0.732434 0.680838i $$-0.238384\pi$$
0.732434 + 0.680838i $$0.238384\pi$$
$$128$$ 0 0
$$129$$ 4.53261 0.399074
$$130$$ 0 0
$$131$$ 12.1354 1.06027 0.530137 0.847912i $$-0.322141\pi$$
0.530137 + 0.847912i $$0.322141\pi$$
$$132$$ 0 0
$$133$$ 3.09401 0.268285
$$134$$ 0 0
$$135$$ 3.94491 0.339524
$$136$$ 0 0
$$137$$ 6.18352 0.528294 0.264147 0.964482i $$-0.414910\pi$$
0.264147 + 0.964482i $$0.414910\pi$$
$$138$$ 0 0
$$139$$ −5.20245 −0.441266 −0.220633 0.975357i $$-0.570812\pi$$
−0.220633 + 0.975357i $$0.570812\pi$$
$$140$$ 0 0
$$141$$ 6.65003 0.560033
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −2.08036 −0.172765
$$146$$ 0 0
$$147$$ 7.56096 0.623618
$$148$$ 0 0
$$149$$ −11.2912 −0.925013 −0.462506 0.886616i $$-0.653050\pi$$
−0.462506 + 0.886616i $$0.653050\pi$$
$$150$$ 0 0
$$151$$ −0.156619 −0.0127455 −0.00637275 0.999980i $$-0.502029\pi$$
−0.00637275 + 0.999980i $$0.502029\pi$$
$$152$$ 0 0
$$153$$ −16.1884 −1.30875
$$154$$ 0 0
$$155$$ 1.12698 0.0905210
$$156$$ 0 0
$$157$$ −13.6990 −1.09330 −0.546650 0.837361i $$-0.684097\pi$$
−0.546650 + 0.837361i $$0.684097\pi$$
$$158$$ 0 0
$$159$$ 5.59129 0.443418
$$160$$ 0 0
$$161$$ −21.4687 −1.69197
$$162$$ 0 0
$$163$$ 5.85320 0.458458 0.229229 0.973372i $$-0.426379\pi$$
0.229229 + 0.973372i $$0.426379\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −12.2892 −0.950965 −0.475483 0.879725i $$-0.657727\pi$$
−0.475483 + 0.879725i $$0.657727\pi$$
$$168$$ 0 0
$$169$$ 23.6093 1.81610
$$170$$ 0 0
$$171$$ −1.83547 −0.140362
$$172$$ 0 0
$$173$$ −19.0157 −1.44574 −0.722868 0.690987i $$-0.757176\pi$$
−0.722868 + 0.690987i $$0.757176\pi$$
$$174$$ 0 0
$$175$$ 4.18418 0.316294
$$176$$ 0 0
$$177$$ 4.43564 0.333403
$$178$$ 0 0
$$179$$ −0.166428 −0.0124394 −0.00621969 0.999981i $$-0.501980\pi$$
−0.00621969 + 0.999981i $$0.501980\pi$$
$$180$$ 0 0
$$181$$ 4.08469 0.303613 0.151806 0.988410i $$-0.451491\pi$$
0.151806 + 0.988410i $$0.451491\pi$$
$$182$$ 0 0
$$183$$ 10.5684 0.781242
$$184$$ 0 0
$$185$$ −7.11676 −0.523235
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −16.5062 −1.20065
$$190$$ 0 0
$$191$$ 14.7337 1.06609 0.533047 0.846086i $$-0.321047\pi$$
0.533047 + 0.846086i $$0.321047\pi$$
$$192$$ 0 0
$$193$$ 0.314976 0.0226725 0.0113362 0.999936i $$-0.496391\pi$$
0.0113362 + 0.999936i $$0.496391\pi$$
$$194$$ 0 0
$$195$$ 4.35390 0.311789
$$196$$ 0 0
$$197$$ 1.41533 0.100838 0.0504191 0.998728i $$-0.483944\pi$$
0.0504191 + 0.998728i $$0.483944\pi$$
$$198$$ 0 0
$$199$$ 22.3049 1.58115 0.790575 0.612365i $$-0.209782\pi$$
0.790575 + 0.612365i $$0.209782\pi$$
$$200$$ 0 0
$$201$$ −3.98124 −0.280815
$$202$$ 0 0
$$203$$ 8.70461 0.610944
$$204$$ 0 0
$$205$$ 4.02170 0.280888
$$206$$ 0 0
$$207$$ 12.7360 0.885211
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −24.3488 −1.67624 −0.838120 0.545486i $$-0.816345\pi$$
−0.838120 + 0.545486i $$0.816345\pi$$
$$212$$ 0 0
$$213$$ 6.38453 0.437461
$$214$$ 0 0
$$215$$ −6.29891 −0.429582
$$216$$ 0 0
$$217$$ −4.71548 −0.320108
$$218$$ 0 0
$$219$$ −5.82397 −0.393547
$$220$$ 0 0
$$221$$ −39.4606 −2.65441
$$222$$ 0 0
$$223$$ 6.22145 0.416619 0.208310 0.978063i $$-0.433204\pi$$
0.208310 + 0.978063i $$0.433204\pi$$
$$224$$ 0 0
$$225$$ −2.48220 −0.165480
$$226$$ 0 0
$$227$$ −16.0658 −1.06632 −0.533162 0.846013i $$-0.678996\pi$$
−0.533162 + 0.846013i $$0.678996\pi$$
$$228$$ 0 0
$$229$$ −15.6832 −1.03638 −0.518189 0.855266i $$-0.673394\pi$$
−0.518189 + 0.855266i $$0.673394\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −14.0236 −0.918718 −0.459359 0.888251i $$-0.651921\pi$$
−0.459359 + 0.888251i $$0.651921\pi$$
$$234$$ 0 0
$$235$$ −9.24147 −0.602847
$$236$$ 0 0
$$237$$ −0.410510 −0.0266655
$$238$$ 0 0
$$239$$ 10.8094 0.699200 0.349600 0.936899i $$-0.386317\pi$$
0.349600 + 0.936899i $$0.386317\pi$$
$$240$$ 0 0
$$241$$ 19.7912 1.27486 0.637431 0.770507i $$-0.279997\pi$$
0.637431 + 0.770507i $$0.279997\pi$$
$$242$$ 0 0
$$243$$ 15.1505 0.971905
$$244$$ 0 0
$$245$$ −10.5074 −0.671292
$$246$$ 0 0
$$247$$ −4.47412 −0.284681
$$248$$ 0 0
$$249$$ −10.9538 −0.694171
$$250$$ 0 0
$$251$$ 3.82947 0.241714 0.120857 0.992670i $$-0.461436\pi$$
0.120857 + 0.992670i $$0.461436\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −4.69299 −0.293887
$$256$$ 0 0
$$257$$ 19.0304 1.18709 0.593543 0.804802i $$-0.297729\pi$$
0.593543 + 0.804802i $$0.297729\pi$$
$$258$$ 0 0
$$259$$ 29.7778 1.85030
$$260$$ 0 0
$$261$$ −5.16387 −0.319635
$$262$$ 0 0
$$263$$ 26.6567 1.64372 0.821862 0.569686i $$-0.192935\pi$$
0.821862 + 0.569686i $$0.192935\pi$$
$$264$$ 0 0
$$265$$ −7.77015 −0.477317
$$266$$ 0 0
$$267$$ −0.963290 −0.0589524
$$268$$ 0 0
$$269$$ 0.454544 0.0277141 0.0138570 0.999904i $$-0.495589\pi$$
0.0138570 + 0.999904i $$0.495589\pi$$
$$270$$ 0 0
$$271$$ 16.2608 0.987771 0.493885 0.869527i $$-0.335576\pi$$
0.493885 + 0.869527i $$0.335576\pi$$
$$272$$ 0 0
$$273$$ −18.2175 −1.10257
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −11.4338 −0.686989 −0.343495 0.939155i $$-0.611611\pi$$
−0.343495 + 0.939155i $$0.611611\pi$$
$$278$$ 0 0
$$279$$ 2.79738 0.167475
$$280$$ 0 0
$$281$$ −5.69217 −0.339566 −0.169783 0.985481i $$-0.554307\pi$$
−0.169783 + 0.985481i $$0.554307\pi$$
$$282$$ 0 0
$$283$$ −0.580914 −0.0345318 −0.0172659 0.999851i $$-0.505496\pi$$
−0.0172659 + 0.999851i $$0.505496\pi$$
$$284$$ 0 0
$$285$$ −0.532101 −0.0315189
$$286$$ 0 0
$$287$$ −16.8275 −0.993298
$$288$$ 0 0
$$289$$ 25.5339 1.50199
$$290$$ 0 0
$$291$$ 7.17170 0.420413
$$292$$ 0 0
$$293$$ −1.17741 −0.0687850 −0.0343925 0.999408i $$-0.510950\pi$$
−0.0343925 + 0.999408i $$0.510950\pi$$
$$294$$ 0 0
$$295$$ −6.16416 −0.358891
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 31.0450 1.79538
$$300$$ 0 0
$$301$$ 26.3558 1.51912
$$302$$ 0 0
$$303$$ 8.54158 0.490701
$$304$$ 0 0
$$305$$ −14.6869 −0.840967
$$306$$ 0 0
$$307$$ −19.1615 −1.09360 −0.546802 0.837262i $$-0.684155\pi$$
−0.546802 + 0.837262i $$0.684155\pi$$
$$308$$ 0 0
$$309$$ −7.47900 −0.425466
$$310$$ 0 0
$$311$$ 18.6728 1.05884 0.529419 0.848361i $$-0.322410\pi$$
0.529419 + 0.848361i $$0.322410\pi$$
$$312$$ 0 0
$$313$$ −3.58974 −0.202904 −0.101452 0.994840i $$-0.532349\pi$$
−0.101452 + 0.994840i $$0.532349\pi$$
$$314$$ 0 0
$$315$$ 10.3860 0.585183
$$316$$ 0 0
$$317$$ 2.50027 0.140429 0.0702146 0.997532i $$-0.477632\pi$$
0.0702146 + 0.997532i $$0.477632\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 6.00715 0.335287
$$322$$ 0 0
$$323$$ 4.82258 0.268335
$$324$$ 0 0
$$325$$ −6.05056 −0.335625
$$326$$ 0 0
$$327$$ −4.49774 −0.248726
$$328$$ 0 0
$$329$$ 38.6680 2.13184
$$330$$ 0 0
$$331$$ −6.58629 −0.362015 −0.181008 0.983482i $$-0.557936\pi$$
−0.181008 + 0.983482i $$0.557936\pi$$
$$332$$ 0 0
$$333$$ −17.6652 −0.968047
$$334$$ 0 0
$$335$$ 5.53268 0.302283
$$336$$ 0 0
$$337$$ −15.2151 −0.828821 −0.414410 0.910090i $$-0.636012\pi$$
−0.414410 + 0.910090i $$0.636012\pi$$
$$338$$ 0 0
$$339$$ 11.8686 0.644615
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 14.6755 0.792405
$$344$$ 0 0
$$345$$ 3.69214 0.198778
$$346$$ 0 0
$$347$$ 7.66135 0.411283 0.205641 0.978627i $$-0.434072\pi$$
0.205641 + 0.978627i $$0.434072\pi$$
$$348$$ 0 0
$$349$$ −8.99401 −0.481438 −0.240719 0.970595i $$-0.577383\pi$$
−0.240719 + 0.970595i $$0.577383\pi$$
$$350$$ 0 0
$$351$$ 23.8689 1.27403
$$352$$ 0 0
$$353$$ 10.3310 0.549862 0.274931 0.961464i $$-0.411345\pi$$
0.274931 + 0.961464i $$0.411345\pi$$
$$354$$ 0 0
$$355$$ −8.87252 −0.470904
$$356$$ 0 0
$$357$$ 19.6363 1.03927
$$358$$ 0 0
$$359$$ 16.2627 0.858310 0.429155 0.903231i $$-0.358811\pi$$
0.429155 + 0.903231i $$0.358811\pi$$
$$360$$ 0 0
$$361$$ −18.4532 −0.971221
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 8.09351 0.423634
$$366$$ 0 0
$$367$$ −34.5427 −1.80311 −0.901557 0.432660i $$-0.857575\pi$$
−0.901557 + 0.432660i $$0.857575\pi$$
$$368$$ 0 0
$$369$$ 9.98266 0.519676
$$370$$ 0 0
$$371$$ 32.5117 1.68792
$$372$$ 0 0
$$373$$ 31.2168 1.61634 0.808172 0.588946i $$-0.200457\pi$$
0.808172 + 0.588946i $$0.200457\pi$$
$$374$$ 0 0
$$375$$ −0.719585 −0.0371592
$$376$$ 0 0
$$377$$ −12.5874 −0.648282
$$378$$ 0 0
$$379$$ −33.4663 −1.71905 −0.859525 0.511094i $$-0.829241\pi$$
−0.859525 + 0.511094i $$0.829241\pi$$
$$380$$ 0 0
$$381$$ 11.8791 0.608584
$$382$$ 0 0
$$383$$ −0.134288 −0.00686178 −0.00343089 0.999994i $$-0.501092\pi$$
−0.00343089 + 0.999994i $$0.501092\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −15.6351 −0.794779
$$388$$ 0 0
$$389$$ −8.53807 −0.432897 −0.216449 0.976294i $$-0.569447\pi$$
−0.216449 + 0.976294i $$0.569447\pi$$
$$390$$ 0 0
$$391$$ −33.4629 −1.69229
$$392$$ 0 0
$$393$$ 8.73245 0.440494
$$394$$ 0 0
$$395$$ 0.570482 0.0287040
$$396$$ 0 0
$$397$$ 38.0250 1.90842 0.954211 0.299135i $$-0.0966982\pi$$
0.954211 + 0.299135i $$0.0966982\pi$$
$$398$$ 0 0
$$399$$ 2.22641 0.111460
$$400$$ 0 0
$$401$$ −11.7343 −0.585982 −0.292991 0.956115i $$-0.594651\pi$$
−0.292991 + 0.956115i $$0.594651\pi$$
$$402$$ 0 0
$$403$$ 6.81885 0.339671
$$404$$ 0 0
$$405$$ −4.60789 −0.228968
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −28.9837 −1.43315 −0.716575 0.697510i $$-0.754291\pi$$
−0.716575 + 0.697510i $$0.754291\pi$$
$$410$$ 0 0
$$411$$ 4.44957 0.219481
$$412$$ 0 0
$$413$$ 25.7920 1.26914
$$414$$ 0 0
$$415$$ 15.2224 0.747239
$$416$$ 0 0
$$417$$ −3.74361 −0.183325
$$418$$ 0 0
$$419$$ 12.5318 0.612221 0.306110 0.951996i $$-0.400972\pi$$
0.306110 + 0.951996i $$0.400972\pi$$
$$420$$ 0 0
$$421$$ −11.4281 −0.556973 −0.278486 0.960440i $$-0.589833\pi$$
−0.278486 + 0.960440i $$0.589833\pi$$
$$422$$ 0 0
$$423$$ −22.9391 −1.11534
$$424$$ 0 0
$$425$$ 6.52180 0.316354
$$426$$ 0 0
$$427$$ 61.4525 2.97389
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 3.78747 0.182436 0.0912181 0.995831i $$-0.470924\pi$$
0.0912181 + 0.995831i $$0.470924\pi$$
$$432$$ 0 0
$$433$$ 25.6848 1.23433 0.617166 0.786833i $$-0.288281\pi$$
0.617166 + 0.786833i $$0.288281\pi$$
$$434$$ 0 0
$$435$$ −1.49700 −0.0717756
$$436$$ 0 0
$$437$$ −3.79409 −0.181496
$$438$$ 0 0
$$439$$ 29.1218 1.38991 0.694955 0.719054i $$-0.255424\pi$$
0.694955 + 0.719054i $$0.255424\pi$$
$$440$$ 0 0
$$441$$ −26.0814 −1.24197
$$442$$ 0 0
$$443$$ 12.1115 0.575437 0.287718 0.957715i $$-0.407103\pi$$
0.287718 + 0.957715i $$0.407103\pi$$
$$444$$ 0 0
$$445$$ 1.33867 0.0634592
$$446$$ 0 0
$$447$$ −8.12500 −0.384299
$$448$$ 0 0
$$449$$ 29.6886 1.40109 0.700546 0.713607i $$-0.252940\pi$$
0.700546 + 0.713607i $$0.252940\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −0.112701 −0.00529516
$$454$$ 0 0
$$455$$ 25.3167 1.18686
$$456$$ 0 0
$$457$$ 0.751497 0.0351535 0.0175768 0.999846i $$-0.494405\pi$$
0.0175768 + 0.999846i $$0.494405\pi$$
$$458$$ 0 0
$$459$$ −25.7279 −1.20088
$$460$$ 0 0
$$461$$ −9.62920 −0.448477 −0.224238 0.974534i $$-0.571989\pi$$
−0.224238 + 0.974534i $$0.571989\pi$$
$$462$$ 0 0
$$463$$ −3.33661 −0.155065 −0.0775327 0.996990i $$-0.524704\pi$$
−0.0775327 + 0.996990i $$0.524704\pi$$
$$464$$ 0 0
$$465$$ 0.810957 0.0376072
$$466$$ 0 0
$$467$$ 1.69675 0.0785161 0.0392581 0.999229i $$-0.487501\pi$$
0.0392581 + 0.999229i $$0.487501\pi$$
$$468$$ 0 0
$$469$$ −23.1497 −1.06896
$$470$$ 0 0
$$471$$ −9.85761 −0.454214
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0.739455 0.0339285
$$476$$ 0 0
$$477$$ −19.2870 −0.883093
$$478$$ 0 0
$$479$$ 22.1315 1.01121 0.505607 0.862764i $$-0.331269\pi$$
0.505607 + 0.862764i $$0.331269\pi$$
$$480$$ 0 0
$$481$$ −43.0604 −1.96339
$$482$$ 0 0
$$483$$ −15.4486 −0.702935
$$484$$ 0 0
$$485$$ −9.96643 −0.452552
$$486$$ 0 0
$$487$$ −19.9414 −0.903631 −0.451815 0.892111i $$-0.649223\pi$$
−0.451815 + 0.892111i $$0.649223\pi$$
$$488$$ 0 0
$$489$$ 4.21188 0.190468
$$490$$ 0 0
$$491$$ 26.2451 1.18443 0.592213 0.805782i $$-0.298254\pi$$
0.592213 + 0.805782i $$0.298254\pi$$
$$492$$ 0 0
$$493$$ 13.5677 0.611059
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 37.1242 1.66525
$$498$$ 0 0
$$499$$ −8.34480 −0.373565 −0.186782 0.982401i $$-0.559806\pi$$
−0.186782 + 0.982401i $$0.559806\pi$$
$$500$$ 0 0
$$501$$ −8.84311 −0.395081
$$502$$ 0 0
$$503$$ 30.3824 1.35469 0.677343 0.735667i $$-0.263131\pi$$
0.677343 + 0.735667i $$0.263131\pi$$
$$504$$ 0 0
$$505$$ −11.8701 −0.528214
$$506$$ 0 0
$$507$$ 16.9889 0.754505
$$508$$ 0 0
$$509$$ 17.2197 0.763248 0.381624 0.924318i $$-0.375365\pi$$
0.381624 + 0.924318i $$0.375365\pi$$
$$510$$ 0 0
$$511$$ −33.8647 −1.49809
$$512$$ 0 0
$$513$$ −2.91708 −0.128792
$$514$$ 0 0
$$515$$ 10.3935 0.457992
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −13.6834 −0.600635
$$520$$ 0 0
$$521$$ −21.4975 −0.941822 −0.470911 0.882181i $$-0.656075\pi$$
−0.470911 + 0.882181i $$0.656075\pi$$
$$522$$ 0 0
$$523$$ −0.0139271 −0.000608991 0 −0.000304496 1.00000i $$-0.500097\pi$$
−0.000304496 1.00000i $$0.500097\pi$$
$$524$$ 0 0
$$525$$ 3.01088 0.131405
$$526$$ 0 0
$$527$$ −7.34993 −0.320168
$$528$$ 0 0
$$529$$ 3.32640 0.144626
$$530$$ 0 0
$$531$$ −15.3007 −0.663992
$$532$$ 0 0
$$533$$ 24.3336 1.05400
$$534$$ 0 0
$$535$$ −8.34807 −0.360919
$$536$$ 0 0
$$537$$ −0.119759 −0.00516798
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −42.7268 −1.83697 −0.918483 0.395459i $$-0.870585\pi$$
−0.918483 + 0.395459i $$0.870585\pi$$
$$542$$ 0 0
$$543$$ 2.93929 0.126137
$$544$$ 0 0
$$545$$ 6.25046 0.267740
$$546$$ 0 0
$$547$$ 17.8319 0.762438 0.381219 0.924485i $$-0.375504\pi$$
0.381219 + 0.924485i $$0.375504\pi$$
$$548$$ 0 0
$$549$$ −36.4557 −1.55589
$$550$$ 0 0
$$551$$ 1.53833 0.0655352
$$552$$ 0 0
$$553$$ −2.38700 −0.101506
$$554$$ 0 0
$$555$$ −5.12112 −0.217379
$$556$$ 0 0
$$557$$ −17.8786 −0.757541 −0.378771 0.925491i $$-0.623653\pi$$
−0.378771 + 0.925491i $$0.623653\pi$$
$$558$$ 0 0
$$559$$ −38.1120 −1.61196
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −10.6011 −0.446781 −0.223391 0.974729i $$-0.571713\pi$$
−0.223391 + 0.974729i $$0.571713\pi$$
$$564$$ 0 0
$$565$$ −16.4937 −0.693894
$$566$$ 0 0
$$567$$ 19.2803 0.809695
$$568$$ 0 0
$$569$$ −22.8718 −0.958836 −0.479418 0.877587i $$-0.659152\pi$$
−0.479418 + 0.877587i $$0.659152\pi$$
$$570$$ 0 0
$$571$$ −20.5139 −0.858480 −0.429240 0.903191i $$-0.641218\pi$$
−0.429240 + 0.903191i $$0.641218\pi$$
$$572$$ 0 0
$$573$$ 10.6022 0.442912
$$574$$ 0 0
$$575$$ −5.13093 −0.213974
$$576$$ 0 0
$$577$$ 24.2174 1.00819 0.504093 0.863650i $$-0.331827\pi$$
0.504093 + 0.863650i $$0.331827\pi$$
$$578$$ 0 0
$$579$$ 0.226652 0.00941935
$$580$$ 0 0
$$581$$ −63.6934 −2.64245
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −15.0187 −0.620946
$$586$$ 0 0
$$587$$ −31.9904 −1.32039 −0.660194 0.751095i $$-0.729526\pi$$
−0.660194 + 0.751095i $$0.729526\pi$$
$$588$$ 0 0
$$589$$ −0.833349 −0.0343375
$$590$$ 0 0
$$591$$ 1.01845 0.0418935
$$592$$ 0 0
$$593$$ −42.5206 −1.74611 −0.873056 0.487620i $$-0.837865\pi$$
−0.873056 + 0.487620i $$0.837865\pi$$
$$594$$ 0 0
$$595$$ −27.2884 −1.11872
$$596$$ 0 0
$$597$$ 16.0503 0.656894
$$598$$ 0 0
$$599$$ −4.29039 −0.175301 −0.0876503 0.996151i $$-0.527936\pi$$
−0.0876503 + 0.996151i $$0.527936\pi$$
$$600$$ 0 0
$$601$$ 19.9972 0.815704 0.407852 0.913048i $$-0.366278\pi$$
0.407852 + 0.913048i $$0.366278\pi$$
$$602$$ 0 0
$$603$$ 13.7332 0.559259
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −23.2114 −0.942121 −0.471060 0.882101i $$-0.656129\pi$$
−0.471060 + 0.882101i $$0.656129\pi$$
$$608$$ 0 0
$$609$$ 6.26371 0.253818
$$610$$ 0 0
$$611$$ −55.9161 −2.26212
$$612$$ 0 0
$$613$$ −5.94916 −0.240284 −0.120142 0.992757i $$-0.538335\pi$$
−0.120142 + 0.992757i $$0.538335\pi$$
$$614$$ 0 0
$$615$$ 2.89396 0.116696
$$616$$ 0 0
$$617$$ 39.7576 1.60058 0.800291 0.599612i $$-0.204678\pi$$
0.800291 + 0.599612i $$0.204678\pi$$
$$618$$ 0 0
$$619$$ −11.8287 −0.475436 −0.237718 0.971334i $$-0.576399\pi$$
−0.237718 + 0.971334i $$0.576399\pi$$
$$620$$ 0 0
$$621$$ 20.2410 0.812245
$$622$$ 0 0
$$623$$ −5.60125 −0.224410
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 46.4141 1.85065
$$630$$ 0 0
$$631$$ 14.8112 0.589623 0.294812 0.955555i $$-0.404743\pi$$
0.294812 + 0.955555i $$0.404743\pi$$
$$632$$ 0 0
$$633$$ −17.5210 −0.696398
$$634$$ 0 0
$$635$$ −16.5082 −0.655109
$$636$$ 0 0
$$637$$ −63.5756 −2.51896
$$638$$ 0 0
$$639$$ −22.0233 −0.871230
$$640$$ 0 0
$$641$$ −8.51035 −0.336139 −0.168069 0.985775i $$-0.553753\pi$$
−0.168069 + 0.985775i $$0.553753\pi$$
$$642$$ 0 0
$$643$$ 34.4340 1.35794 0.678971 0.734165i $$-0.262426\pi$$
0.678971 + 0.734165i $$0.262426\pi$$
$$644$$ 0 0
$$645$$ −4.53261 −0.178471
$$646$$ 0 0
$$647$$ 5.32491 0.209344 0.104672 0.994507i $$-0.466621\pi$$
0.104672 + 0.994507i $$0.466621\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −3.39319 −0.132990
$$652$$ 0 0
$$653$$ 28.2099 1.10394 0.551970 0.833864i $$-0.313876\pi$$
0.551970 + 0.833864i $$0.313876\pi$$
$$654$$ 0 0
$$655$$ −12.1354 −0.474169
$$656$$ 0 0
$$657$$ 20.0897 0.783773
$$658$$ 0 0
$$659$$ −19.9228 −0.776082 −0.388041 0.921642i $$-0.626848\pi$$
−0.388041 + 0.921642i $$0.626848\pi$$
$$660$$ 0 0
$$661$$ −16.6144 −0.646225 −0.323112 0.946361i $$-0.604729\pi$$
−0.323112 + 0.946361i $$0.604729\pi$$
$$662$$ 0 0
$$663$$ −28.3953 −1.10278
$$664$$ 0 0
$$665$$ −3.09401 −0.119981
$$666$$ 0 0
$$667$$ −10.6742 −0.413306
$$668$$ 0 0
$$669$$ 4.47687 0.173086
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 32.7246 1.26144 0.630720 0.776010i $$-0.282760\pi$$
0.630720 + 0.776010i $$0.282760\pi$$
$$674$$ 0 0
$$675$$ −3.94491 −0.151840
$$676$$ 0 0
$$677$$ 41.0308 1.57694 0.788472 0.615071i $$-0.210873\pi$$
0.788472 + 0.615071i $$0.210873\pi$$
$$678$$ 0 0
$$679$$ 41.7014 1.60035
$$680$$ 0 0
$$681$$ −11.5607 −0.443007
$$682$$ 0 0
$$683$$ 5.71534 0.218691 0.109346 0.994004i $$-0.465124\pi$$
0.109346 + 0.994004i $$0.465124\pi$$
$$684$$ 0 0
$$685$$ −6.18352 −0.236260
$$686$$ 0 0
$$687$$ −11.2854 −0.430566
$$688$$ 0 0
$$689$$ −47.0138 −1.79108
$$690$$ 0 0
$$691$$ −38.8709 −1.47872 −0.739358 0.673312i $$-0.764871\pi$$
−0.739358 + 0.673312i $$0.764871\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 5.20245 0.197340
$$696$$ 0 0
$$697$$ −26.2288 −0.993485
$$698$$ 0 0
$$699$$ −10.0912 −0.381684
$$700$$ 0 0
$$701$$ 3.37005 0.127285 0.0636426 0.997973i $$-0.479728\pi$$
0.0636426 + 0.997973i $$0.479728\pi$$
$$702$$ 0 0
$$703$$ 5.26252 0.198480
$$704$$ 0 0
$$705$$ −6.65003 −0.250455
$$706$$ 0 0
$$707$$ 49.6668 1.86791
$$708$$ 0 0
$$709$$ −22.6143 −0.849300 −0.424650 0.905358i $$-0.639603\pi$$
−0.424650 + 0.905358i $$0.639603\pi$$
$$710$$ 0 0
$$711$$ 1.41605 0.0531059
$$712$$ 0 0
$$713$$ 5.78244 0.216554
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 7.77827 0.290485
$$718$$ 0 0
$$719$$ −18.7071 −0.697655 −0.348828 0.937187i $$-0.613420\pi$$
−0.348828 + 0.937187i $$0.613420\pi$$
$$720$$ 0 0
$$721$$ −43.4883 −1.61959
$$722$$ 0 0
$$723$$ 14.2415 0.529645
$$724$$ 0 0
$$725$$ 2.08036 0.0772627
$$726$$ 0 0
$$727$$ −2.98028 −0.110532 −0.0552662 0.998472i $$-0.517601\pi$$
−0.0552662 + 0.998472i $$0.517601\pi$$
$$728$$ 0 0
$$729$$ −2.92160 −0.108207
$$730$$ 0 0
$$731$$ 41.0803 1.51941
$$732$$ 0 0
$$733$$ −20.1444 −0.744050 −0.372025 0.928223i $$-0.621336\pi$$
−0.372025 + 0.928223i $$0.621336\pi$$
$$734$$ 0 0
$$735$$ −7.56096 −0.278890
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 23.2196 0.854146 0.427073 0.904217i $$-0.359545\pi$$
0.427073 + 0.904217i $$0.359545\pi$$
$$740$$ 0 0
$$741$$ −3.21951 −0.118272
$$742$$ 0 0
$$743$$ 31.4412 1.15346 0.576732 0.816933i $$-0.304328\pi$$
0.576732 + 0.816933i $$0.304328\pi$$
$$744$$ 0 0
$$745$$ 11.2912 0.413678
$$746$$ 0 0
$$747$$ 37.7850 1.38248
$$748$$ 0 0
$$749$$ 34.9299 1.27631
$$750$$ 0 0
$$751$$ 40.5769 1.48067 0.740337 0.672236i $$-0.234666\pi$$
0.740337 + 0.672236i $$0.234666\pi$$
$$752$$ 0 0
$$753$$ 2.75563 0.100421
$$754$$ 0 0
$$755$$ 0.156619 0.00569996
$$756$$ 0 0
$$757$$ −22.5716 −0.820378 −0.410189 0.912000i $$-0.634537\pi$$
−0.410189 + 0.912000i $$0.634537\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 21.9901 0.797139 0.398570 0.917138i $$-0.369507\pi$$
0.398570 + 0.917138i $$0.369507\pi$$
$$762$$ 0 0
$$763$$ −26.1531 −0.946805
$$764$$ 0 0
$$765$$ 16.1884 0.585293
$$766$$ 0 0
$$767$$ −37.2966 −1.34670
$$768$$ 0 0
$$769$$ −41.1834 −1.48511 −0.742556 0.669784i $$-0.766387\pi$$
−0.742556 + 0.669784i $$0.766387\pi$$
$$770$$ 0 0
$$771$$ 13.6940 0.493178
$$772$$ 0 0
$$773$$ −10.8352 −0.389716 −0.194858 0.980831i $$-0.562425\pi$$
−0.194858 + 0.980831i $$0.562425\pi$$
$$774$$ 0 0
$$775$$ −1.12698 −0.0404822
$$776$$ 0 0
$$777$$ 21.4277 0.768714
$$778$$ 0 0
$$779$$ −2.97387 −0.106550
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −8.20684 −0.293288
$$784$$ 0 0
$$785$$ 13.6990 0.488938
$$786$$ 0 0
$$787$$ −31.3348 −1.11696 −0.558482 0.829517i $$-0.688616\pi$$
−0.558482 + 0.829517i $$0.688616\pi$$
$$788$$ 0 0
$$789$$ 19.1818 0.682890
$$790$$ 0 0
$$791$$ 69.0126 2.45381
$$792$$ 0 0
$$793$$ −88.8638 −3.15565
$$794$$ 0 0
$$795$$ −5.59129 −0.198303
$$796$$ 0 0
$$797$$ −9.52077 −0.337243 −0.168621 0.985681i $$-0.553932\pi$$
−0.168621 + 0.985681i $$0.553932\pi$$
$$798$$ 0 0
$$799$$ 60.2710 2.13224
$$800$$ 0 0
$$801$$ 3.32285 0.117407
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 21.4687 0.756673
$$806$$ 0 0
$$807$$ 0.327084 0.0115139
$$808$$ 0 0
$$809$$ −29.0902 −1.02276 −0.511379 0.859355i $$-0.670865\pi$$
−0.511379 + 0.859355i $$0.670865\pi$$
$$810$$ 0 0
$$811$$ −43.6205 −1.53172 −0.765862 0.643005i $$-0.777687\pi$$
−0.765862 + 0.643005i $$0.777687\pi$$
$$812$$ 0 0
$$813$$ 11.7010 0.410372
$$814$$ 0 0
$$815$$ −5.85320 −0.205029
$$816$$ 0 0
$$817$$ 4.65776 0.162954
$$818$$ 0 0
$$819$$ 62.8410 2.19584
$$820$$ 0 0
$$821$$ 26.6375 0.929656 0.464828 0.885401i $$-0.346116\pi$$
0.464828 + 0.885401i $$0.346116\pi$$
$$822$$ 0 0
$$823$$ 19.9584 0.695706 0.347853 0.937549i $$-0.386911\pi$$
0.347853 + 0.937549i $$0.386911\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 26.6819 0.927819 0.463910 0.885883i $$-0.346446\pi$$
0.463910 + 0.885883i $$0.346446\pi$$
$$828$$ 0 0
$$829$$ −50.7077 −1.76115 −0.880576 0.473905i $$-0.842844\pi$$
−0.880576 + 0.473905i $$0.842844\pi$$
$$830$$ 0 0
$$831$$ −8.22758 −0.285412
$$832$$ 0 0
$$833$$ 68.5271 2.37432
$$834$$ 0 0
$$835$$ 12.2892 0.425285
$$836$$ 0 0
$$837$$ 4.44583 0.153670
$$838$$ 0 0
$$839$$ −2.69881 −0.0931733 −0.0465867 0.998914i $$-0.514834\pi$$
−0.0465867 + 0.998914i $$0.514834\pi$$
$$840$$ 0 0
$$841$$ −24.6721 −0.850762
$$842$$ 0 0
$$843$$ −4.09600 −0.141074
$$844$$ 0 0
$$845$$ −23.6093 −0.812186
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −0.418017 −0.0143463
$$850$$ 0 0
$$851$$ −36.5156 −1.25174
$$852$$ 0 0
$$853$$ −17.6391 −0.603951 −0.301976 0.953316i $$-0.597646\pi$$
−0.301976 + 0.953316i $$0.597646\pi$$
$$854$$ 0 0
$$855$$ 1.83547 0.0627718
$$856$$ 0 0
$$857$$ −8.72254 −0.297956 −0.148978 0.988840i $$-0.547598\pi$$
−0.148978 + 0.988840i $$0.547598\pi$$
$$858$$ 0 0
$$859$$ −34.2322 −1.16799 −0.583993 0.811759i $$-0.698510\pi$$
−0.583993 + 0.811759i $$0.698510\pi$$
$$860$$ 0 0
$$861$$ −12.1089 −0.412669
$$862$$ 0 0
$$863$$ −40.8286 −1.38982 −0.694911 0.719096i $$-0.744556\pi$$
−0.694911 + 0.719096i $$0.744556\pi$$
$$864$$ 0 0
$$865$$ 19.0157 0.646552
$$866$$ 0 0
$$867$$ 18.3738 0.624008
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 33.4758 1.13429
$$872$$ 0 0
$$873$$ −24.7387 −0.837276
$$874$$ 0 0
$$875$$ −4.18418 −0.141451
$$876$$ 0 0
$$877$$ −4.01395 −0.135541 −0.0677707 0.997701i $$-0.521589\pi$$
−0.0677707 + 0.997701i $$0.521589\pi$$
$$878$$ 0 0
$$879$$ −0.847247 −0.0285769
$$880$$ 0 0
$$881$$ 21.1713 0.713278 0.356639 0.934242i $$-0.383923\pi$$
0.356639 + 0.934242i $$0.383923\pi$$
$$882$$ 0 0
$$883$$ 6.66990 0.224460 0.112230 0.993682i $$-0.464201\pi$$
0.112230 + 0.993682i $$0.464201\pi$$
$$884$$ 0 0
$$885$$ −4.43564 −0.149102
$$886$$ 0 0
$$887$$ 2.53548 0.0851332 0.0425666 0.999094i $$-0.486447\pi$$
0.0425666 + 0.999094i $$0.486447\pi$$
$$888$$ 0 0
$$889$$ 69.0735 2.31665
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 6.83365 0.228679
$$894$$ 0 0
$$895$$ 0.166428 0.00556306
$$896$$ 0 0
$$897$$ 22.3395 0.745895
$$898$$ 0 0
$$899$$ −2.34452 −0.0781942
$$900$$ 0 0
$$901$$ 50.6754 1.68824
$$902$$ 0 0
$$903$$ 18.9653 0.631124
$$904$$ 0 0
$$905$$ −4.08469 −0.135780
$$906$$ 0 0
$$907$$ −36.3808 −1.20801 −0.604003 0.796982i $$-0.706428\pi$$
−0.604003 + 0.796982i $$0.706428\pi$$
$$908$$ 0 0
$$909$$ −29.4640 −0.977260
$$910$$ 0 0
$$911$$ −58.3264 −1.93244 −0.966220 0.257719i $$-0.917029\pi$$
−0.966220 + 0.257719i $$0.917029\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −10.5684 −0.349382
$$916$$ 0 0
$$917$$ 50.7767 1.67679
$$918$$ 0 0
$$919$$ 29.7560 0.981560 0.490780 0.871283i $$-0.336712\pi$$
0.490780 + 0.871283i $$0.336712\pi$$
$$920$$ 0 0
$$921$$ −13.7883 −0.454341
$$922$$ 0 0
$$923$$ −53.6837 −1.76702
$$924$$ 0 0
$$925$$ 7.11676 0.233998
$$926$$ 0 0
$$927$$ 25.7987 0.847340
$$928$$ 0 0
$$929$$ −26.0033 −0.853140 −0.426570 0.904455i $$-0.640278\pi$$
−0.426570 + 0.904455i $$0.640278\pi$$
$$930$$ 0 0
$$931$$ 7.76973 0.254643
$$932$$ 0 0
$$933$$ 13.4367 0.439897
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −6.46058 −0.211058 −0.105529 0.994416i $$-0.533654\pi$$
−0.105529 + 0.994416i $$0.533654\pi$$
$$938$$ 0 0
$$939$$ −2.58312 −0.0842970
$$940$$ 0 0
$$941$$ −29.8484 −0.973030 −0.486515 0.873672i $$-0.661732\pi$$
−0.486515 + 0.873672i $$0.661732\pi$$
$$942$$ 0 0
$$943$$ 20.6351 0.671970
$$944$$ 0 0
$$945$$ 16.5062 0.536947
$$946$$ 0 0
$$947$$ 36.1069 1.17332 0.586658 0.809834i $$-0.300443\pi$$
0.586658 + 0.809834i $$0.300443\pi$$
$$948$$ 0 0
$$949$$ 48.9703 1.58964
$$950$$ 0 0
$$951$$ 1.79916 0.0583417
$$952$$ 0 0
$$953$$ −34.5190 −1.11818 −0.559091 0.829107i $$-0.688850\pi$$
−0.559091 + 0.829107i $$0.688850\pi$$
$$954$$ 0 0
$$955$$ −14.7337 −0.476772
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 25.8730 0.835482
$$960$$ 0 0
$$961$$ −29.7299 −0.959030
$$962$$ 0 0
$$963$$ −20.7216 −0.667743
$$964$$ 0 0
$$965$$ −0.314976 −0.0101394
$$966$$ 0 0
$$967$$ −28.4086 −0.913559 −0.456780 0.889580i $$-0.650997\pi$$
−0.456780 + 0.889580i $$0.650997\pi$$
$$968$$ 0 0
$$969$$ 3.47026 0.111481
$$970$$ 0 0
$$971$$ −30.9718 −0.993932 −0.496966 0.867770i $$-0.665553\pi$$
−0.496966 + 0.867770i $$0.665553\pi$$
$$972$$ 0 0
$$973$$ −21.7680 −0.697850
$$974$$ 0 0
$$975$$ −4.35390 −0.139436
$$976$$ 0 0
$$977$$ −38.7235 −1.23887 −0.619437 0.785046i $$-0.712639\pi$$
−0.619437 + 0.785046i $$0.712639\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 15.5149 0.495352
$$982$$ 0 0
$$983$$ 11.1197 0.354663 0.177331 0.984151i $$-0.443254\pi$$
0.177331 + 0.984151i $$0.443254\pi$$
$$984$$ 0 0
$$985$$ −1.41533 −0.0450962
$$986$$ 0 0
$$987$$ 27.8249 0.885677
$$988$$ 0 0
$$989$$ −32.3193 −1.02769
$$990$$ 0 0
$$991$$ −11.5682 −0.367475 −0.183737 0.982975i $$-0.558820\pi$$
−0.183737 + 0.982975i $$0.558820\pi$$
$$992$$ 0 0
$$993$$ −4.73940 −0.150400
$$994$$ 0 0
$$995$$ −22.3049 −0.707112
$$996$$ 0 0
$$997$$ 28.1683 0.892097 0.446049 0.895009i $$-0.352831\pi$$
0.446049 + 0.895009i $$0.352831\pi$$
$$998$$ 0 0
$$999$$ −28.0750 −0.888253
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 4840.2.a.bf.1.3 6
4.3 odd 2 9680.2.a.cx.1.4 6
11.3 even 5 440.2.y.b.361.2 12
11.4 even 5 440.2.y.b.401.2 yes 12
11.10 odd 2 4840.2.a.be.1.3 6
44.3 odd 10 880.2.bo.j.801.2 12
44.15 odd 10 880.2.bo.j.401.2 12
44.43 even 2 9680.2.a.cy.1.4 6

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.361.2 12 11.3 even 5
440.2.y.b.401.2 yes 12 11.4 even 5
880.2.bo.j.401.2 12 44.15 odd 10
880.2.bo.j.801.2 12 44.3 odd 10
4840.2.a.be.1.3 6 11.10 odd 2
4840.2.a.bf.1.3 6 1.1 even 1 trivial
9680.2.a.cx.1.4 6 4.3 odd 2
9680.2.a.cy.1.4 6 44.43 even 2