# Properties

 Label 5824.2.a.by.1.1 Level $5824$ Weight $2$ Character 5824.1 Self dual yes Analytic conductor $46.505$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5824,2,Mod(1,5824)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5824, 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("5824.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5824 = 2^{6} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5824.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: $$46.5048741372$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0.470683$$ of defining polynomial Character $$\chi$$ $$=$$ 5824.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.24914 q^{3} -0.529317 q^{5} -1.00000 q^{7} +2.05863 q^{9} +O(q^{10})$$ $$q-2.24914 q^{3} -0.529317 q^{5} -1.00000 q^{7} +2.05863 q^{9} +2.24914 q^{11} -1.00000 q^{13} +1.19051 q^{15} -1.30777 q^{17} +1.47068 q^{19} +2.24914 q^{21} +5.83709 q^{23} -4.71982 q^{25} +2.11727 q^{27} -5.22154 q^{29} -7.02760 q^{31} -5.05863 q^{33} +0.529317 q^{35} +2.36641 q^{37} +2.24914 q^{39} +6.49828 q^{41} -11.3940 q^{43} -1.08967 q^{45} +8.58451 q^{47} +1.00000 q^{49} +2.94137 q^{51} -11.2767 q^{53} -1.19051 q^{55} -3.30777 q^{57} +12.1725 q^{59} +2.00000 q^{61} -2.05863 q^{63} +0.529317 q^{65} +15.9379 q^{67} -13.1284 q^{69} +1.19051 q^{71} +7.64315 q^{73} +10.6155 q^{75} -2.24914 q^{77} -1.33881 q^{79} -10.9379 q^{81} +16.3500 q^{83} +0.692226 q^{85} +11.7440 q^{87} +6.91033 q^{89} +1.00000 q^{91} +15.8061 q^{93} -0.778457 q^{95} -3.47068 q^{97} +4.63016 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 7 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 7 * q^9 $$3 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 7 q^{9} - 2 q^{11} - 3 q^{13} - 6 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 10 q^{23} - 5 q^{25} + 8 q^{27} - 24 q^{29} - 4 q^{31} - 16 q^{33} + 2 q^{35} - 2 q^{39} + 2 q^{41} - 10 q^{43} - 22 q^{45} - 8 q^{47} + 3 q^{49} + 8 q^{51} - 8 q^{53} + 6 q^{55} - 2 q^{57} + 4 q^{59} + 6 q^{61} - 7 q^{63} + 2 q^{65} + 12 q^{67} + 6 q^{69} - 6 q^{71} - 10 q^{73} + 16 q^{75} + 2 q^{77} - 14 q^{79} + 3 q^{81} + 12 q^{83} + 10 q^{85} - 26 q^{87} + 2 q^{89} + 3 q^{91} + 22 q^{93} + 6 q^{95} - 10 q^{97} - 14 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 7 * q^9 - 2 * q^11 - 3 * q^13 - 6 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 10 * q^23 - 5 * q^25 + 8 * q^27 - 24 * q^29 - 4 * q^31 - 16 * q^33 + 2 * q^35 - 2 * q^39 + 2 * q^41 - 10 * q^43 - 22 * q^45 - 8 * q^47 + 3 * q^49 + 8 * q^51 - 8 * q^53 + 6 * q^55 - 2 * q^57 + 4 * q^59 + 6 * q^61 - 7 * q^63 + 2 * q^65 + 12 * q^67 + 6 * q^69 - 6 * q^71 - 10 * q^73 + 16 * q^75 + 2 * q^77 - 14 * q^79 + 3 * q^81 + 12 * q^83 + 10 * q^85 - 26 * q^87 + 2 * q^89 + 3 * q^91 + 22 * q^93 + 6 * q^95 - 10 * q^97 - 14 * 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
$$3$$ −2.24914 −1.29854 −0.649271 0.760557i $$-0.724926\pi$$
−0.649271 + 0.760557i $$0.724926\pi$$
$$4$$ 0 0
$$5$$ −0.529317 −0.236718 −0.118359 0.992971i $$-0.537763\pi$$
−0.118359 + 0.992971i $$0.537763\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 2.05863 0.686211
$$10$$ 0 0
$$11$$ 2.24914 0.678141 0.339071 0.940761i $$-0.389887\pi$$
0.339071 + 0.940761i $$0.389887\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 1.19051 0.307388
$$16$$ 0 0
$$17$$ −1.30777 −0.317182 −0.158591 0.987344i $$-0.550695\pi$$
−0.158591 + 0.987344i $$0.550695\pi$$
$$18$$ 0 0
$$19$$ 1.47068 0.337398 0.168699 0.985668i $$-0.446043\pi$$
0.168699 + 0.985668i $$0.446043\pi$$
$$20$$ 0 0
$$21$$ 2.24914 0.490803
$$22$$ 0 0
$$23$$ 5.83709 1.21712 0.608559 0.793509i $$-0.291748\pi$$
0.608559 + 0.793509i $$0.291748\pi$$
$$24$$ 0 0
$$25$$ −4.71982 −0.943965
$$26$$ 0 0
$$27$$ 2.11727 0.407468
$$28$$ 0 0
$$29$$ −5.22154 −0.969616 −0.484808 0.874621i $$-0.661111\pi$$
−0.484808 + 0.874621i $$0.661111\pi$$
$$30$$ 0 0
$$31$$ −7.02760 −1.26219 −0.631097 0.775704i $$-0.717395\pi$$
−0.631097 + 0.775704i $$0.717395\pi$$
$$32$$ 0 0
$$33$$ −5.05863 −0.880595
$$34$$ 0 0
$$35$$ 0.529317 0.0894708
$$36$$ 0 0
$$37$$ 2.36641 0.389035 0.194517 0.980899i $$-0.437686\pi$$
0.194517 + 0.980899i $$0.437686\pi$$
$$38$$ 0 0
$$39$$ 2.24914 0.360151
$$40$$ 0 0
$$41$$ 6.49828 1.01486 0.507431 0.861693i $$-0.330595\pi$$
0.507431 + 0.861693i $$0.330595\pi$$
$$42$$ 0 0
$$43$$ −11.3940 −1.73757 −0.868785 0.495190i $$-0.835098\pi$$
−0.868785 + 0.495190i $$0.835098\pi$$
$$44$$ 0 0
$$45$$ −1.08967 −0.162438
$$46$$ 0 0
$$47$$ 8.58451 1.25218 0.626090 0.779751i $$-0.284654\pi$$
0.626090 + 0.779751i $$0.284654\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 2.94137 0.411874
$$52$$ 0 0
$$53$$ −11.2767 −1.54898 −0.774490 0.632587i $$-0.781993\pi$$
−0.774490 + 0.632587i $$0.781993\pi$$
$$54$$ 0 0
$$55$$ −1.19051 −0.160528
$$56$$ 0 0
$$57$$ −3.30777 −0.438125
$$58$$ 0 0
$$59$$ 12.1725 1.58472 0.792360 0.610054i $$-0.208852\pi$$
0.792360 + 0.610054i $$0.208852\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ −2.05863 −0.259363
$$64$$ 0 0
$$65$$ 0.529317 0.0656536
$$66$$ 0 0
$$67$$ 15.9379 1.94713 0.973564 0.228415i $$-0.0733542\pi$$
0.973564 + 0.228415i $$0.0733542\pi$$
$$68$$ 0 0
$$69$$ −13.1284 −1.58048
$$70$$ 0 0
$$71$$ 1.19051 0.141287 0.0706436 0.997502i $$-0.477495\pi$$
0.0706436 + 0.997502i $$0.477495\pi$$
$$72$$ 0 0
$$73$$ 7.64315 0.894562 0.447281 0.894393i $$-0.352392\pi$$
0.447281 + 0.894393i $$0.352392\pi$$
$$74$$ 0 0
$$75$$ 10.6155 1.22578
$$76$$ 0 0
$$77$$ −2.24914 −0.256313
$$78$$ 0 0
$$79$$ −1.33881 −0.150628 −0.0753139 0.997160i $$-0.523996\pi$$
−0.0753139 + 0.997160i $$0.523996\pi$$
$$80$$ 0 0
$$81$$ −10.9379 −1.21533
$$82$$ 0 0
$$83$$ 16.3500 1.79464 0.897322 0.441377i $$-0.145510\pi$$
0.897322 + 0.441377i $$0.145510\pi$$
$$84$$ 0 0
$$85$$ 0.692226 0.0750825
$$86$$ 0 0
$$87$$ 11.7440 1.25909
$$88$$ 0 0
$$89$$ 6.91033 0.732494 0.366247 0.930518i $$-0.380643\pi$$
0.366247 + 0.930518i $$0.380643\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 0 0
$$93$$ 15.8061 1.63901
$$94$$ 0 0
$$95$$ −0.778457 −0.0798680
$$96$$ 0 0
$$97$$ −3.47068 −0.352395 −0.176197 0.984355i $$-0.556380\pi$$
−0.176197 + 0.984355i $$0.556380\pi$$
$$98$$ 0 0
$$99$$ 4.63016 0.465348
$$100$$ 0 0
$$101$$ −7.75086 −0.771239 −0.385620 0.922658i $$-0.626012\pi$$
−0.385620 + 0.922658i $$0.626012\pi$$
$$102$$ 0 0
$$103$$ −16.9966 −1.67472 −0.837361 0.546651i $$-0.815902\pi$$
−0.837361 + 0.546651i $$0.815902\pi$$
$$104$$ 0 0
$$105$$ −1.19051 −0.116182
$$106$$ 0 0
$$107$$ 5.55691 0.537207 0.268604 0.963251i $$-0.413438\pi$$
0.268604 + 0.963251i $$0.413438\pi$$
$$108$$ 0 0
$$109$$ −7.92332 −0.758917 −0.379458 0.925209i $$-0.623890\pi$$
−0.379458 + 0.925209i $$0.623890\pi$$
$$110$$ 0 0
$$111$$ −5.32238 −0.505178
$$112$$ 0 0
$$113$$ −9.89229 −0.930588 −0.465294 0.885156i $$-0.654051\pi$$
−0.465294 + 0.885156i $$0.654051\pi$$
$$114$$ 0 0
$$115$$ −3.08967 −0.288113
$$116$$ 0 0
$$117$$ −2.05863 −0.190321
$$118$$ 0 0
$$119$$ 1.30777 0.119883
$$120$$ 0 0
$$121$$ −5.94137 −0.540124
$$122$$ 0 0
$$123$$ −14.6155 −1.31784
$$124$$ 0 0
$$125$$ 5.14486 0.460171
$$126$$ 0 0
$$127$$ 0.824101 0.0731271 0.0365635 0.999331i $$-0.488359\pi$$
0.0365635 + 0.999331i $$0.488359\pi$$
$$128$$ 0 0
$$129$$ 25.6267 2.25631
$$130$$ 0 0
$$131$$ −10.6155 −0.927485 −0.463742 0.885970i $$-0.653494\pi$$
−0.463742 + 0.885970i $$0.653494\pi$$
$$132$$ 0 0
$$133$$ −1.47068 −0.127524
$$134$$ 0 0
$$135$$ −1.12070 −0.0964549
$$136$$ 0 0
$$137$$ 11.3630 0.970804 0.485402 0.874291i $$-0.338673\pi$$
0.485402 + 0.874291i $$0.338673\pi$$
$$138$$ 0 0
$$139$$ 13.9233 1.18096 0.590480 0.807052i $$-0.298938\pi$$
0.590480 + 0.807052i $$0.298938\pi$$
$$140$$ 0 0
$$141$$ −19.3078 −1.62601
$$142$$ 0 0
$$143$$ −2.24914 −0.188083
$$144$$ 0 0
$$145$$ 2.76385 0.229525
$$146$$ 0 0
$$147$$ −2.24914 −0.185506
$$148$$ 0 0
$$149$$ −9.30777 −0.762523 −0.381261 0.924467i $$-0.624510\pi$$
−0.381261 + 0.924467i $$0.624510\pi$$
$$150$$ 0 0
$$151$$ 7.07324 0.575612 0.287806 0.957689i $$-0.407074\pi$$
0.287806 + 0.957689i $$0.407074\pi$$
$$152$$ 0 0
$$153$$ −2.69223 −0.217654
$$154$$ 0 0
$$155$$ 3.71982 0.298783
$$156$$ 0 0
$$157$$ 6.04059 0.482091 0.241046 0.970514i $$-0.422510\pi$$
0.241046 + 0.970514i $$0.422510\pi$$
$$158$$ 0 0
$$159$$ 25.3630 2.01141
$$160$$ 0 0
$$161$$ −5.83709 −0.460027
$$162$$ 0 0
$$163$$ −6.38101 −0.499800 −0.249900 0.968272i $$-0.580398\pi$$
−0.249900 + 0.968272i $$0.580398\pi$$
$$164$$ 0 0
$$165$$ 2.67762 0.208452
$$166$$ 0 0
$$167$$ −16.5845 −1.28335 −0.641674 0.766977i $$-0.721760\pi$$
−0.641674 + 0.766977i $$0.721760\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 3.02760 0.231526
$$172$$ 0 0
$$173$$ 23.3009 1.77153 0.885767 0.464130i $$-0.153633\pi$$
0.885767 + 0.464130i $$0.153633\pi$$
$$174$$ 0 0
$$175$$ 4.71982 0.356785
$$176$$ 0 0
$$177$$ −27.3776 −2.05782
$$178$$ 0 0
$$179$$ −21.0422 −1.57277 −0.786384 0.617738i $$-0.788049\pi$$
−0.786384 + 0.617738i $$0.788049\pi$$
$$180$$ 0 0
$$181$$ −16.7474 −1.24483 −0.622413 0.782689i $$-0.713848\pi$$
−0.622413 + 0.782689i $$0.713848\pi$$
$$182$$ 0 0
$$183$$ −4.49828 −0.332523
$$184$$ 0 0
$$185$$ −1.25258 −0.0920914
$$186$$ 0 0
$$187$$ −2.94137 −0.215094
$$188$$ 0 0
$$189$$ −2.11727 −0.154008
$$190$$ 0 0
$$191$$ −7.43965 −0.538314 −0.269157 0.963096i $$-0.586745\pi$$
−0.269157 + 0.963096i $$0.586745\pi$$
$$192$$ 0 0
$$193$$ −1.50172 −0.108096 −0.0540480 0.998538i $$-0.517212\pi$$
−0.0540480 + 0.998538i $$0.517212\pi$$
$$194$$ 0 0
$$195$$ −1.19051 −0.0852540
$$196$$ 0 0
$$197$$ −23.9931 −1.70944 −0.854720 0.519090i $$-0.826271\pi$$
−0.854720 + 0.519090i $$0.826271\pi$$
$$198$$ 0 0
$$199$$ −2.01461 −0.142812 −0.0714059 0.997447i $$-0.522749\pi$$
−0.0714059 + 0.997447i $$0.522749\pi$$
$$200$$ 0 0
$$201$$ −35.8466 −2.52843
$$202$$ 0 0
$$203$$ 5.22154 0.366480
$$204$$ 0 0
$$205$$ −3.43965 −0.240235
$$206$$ 0 0
$$207$$ 12.0164 0.835199
$$208$$ 0 0
$$209$$ 3.30777 0.228803
$$210$$ 0 0
$$211$$ −10.1008 −0.695370 −0.347685 0.937611i $$-0.613032\pi$$
−0.347685 + 0.937611i $$0.613032\pi$$
$$212$$ 0 0
$$213$$ −2.67762 −0.183467
$$214$$ 0 0
$$215$$ 6.03104 0.411313
$$216$$ 0 0
$$217$$ 7.02760 0.477064
$$218$$ 0 0
$$219$$ −17.1905 −1.16163
$$220$$ 0 0
$$221$$ 1.30777 0.0879704
$$222$$ 0 0
$$223$$ 10.1414 0.679120 0.339560 0.940584i $$-0.389722\pi$$
0.339560 + 0.940584i $$0.389722\pi$$
$$224$$ 0 0
$$225$$ −9.71639 −0.647759
$$226$$ 0 0
$$227$$ 5.38445 0.357379 0.178689 0.983906i $$-0.442814\pi$$
0.178689 + 0.983906i $$0.442814\pi$$
$$228$$ 0 0
$$229$$ −3.32238 −0.219549 −0.109775 0.993957i $$-0.535013\pi$$
−0.109775 + 0.993957i $$0.535013\pi$$
$$230$$ 0 0
$$231$$ 5.05863 0.332834
$$232$$ 0 0
$$233$$ 13.7198 0.898816 0.449408 0.893327i $$-0.351635\pi$$
0.449408 + 0.893327i $$0.351635\pi$$
$$234$$ 0 0
$$235$$ −4.54392 −0.296413
$$236$$ 0 0
$$237$$ 3.01117 0.195597
$$238$$ 0 0
$$239$$ −3.50172 −0.226507 −0.113254 0.993566i $$-0.536127\pi$$
−0.113254 + 0.993566i $$0.536127\pi$$
$$240$$ 0 0
$$241$$ 1.58795 0.102289 0.0511444 0.998691i $$-0.483713\pi$$
0.0511444 + 0.998691i $$0.483713\pi$$
$$242$$ 0 0
$$243$$ 18.2491 1.17068
$$244$$ 0 0
$$245$$ −0.529317 −0.0338168
$$246$$ 0 0
$$247$$ −1.47068 −0.0935773
$$248$$ 0 0
$$249$$ −36.7734 −2.33042
$$250$$ 0 0
$$251$$ 4.92676 0.310974 0.155487 0.987838i $$-0.450305\pi$$
0.155487 + 0.987838i $$0.450305\pi$$
$$252$$ 0 0
$$253$$ 13.1284 0.825378
$$254$$ 0 0
$$255$$ −1.55691 −0.0974978
$$256$$ 0 0
$$257$$ −8.01461 −0.499938 −0.249969 0.968254i $$-0.580420\pi$$
−0.249969 + 0.968254i $$0.580420\pi$$
$$258$$ 0 0
$$259$$ −2.36641 −0.147041
$$260$$ 0 0
$$261$$ −10.7492 −0.665361
$$262$$ 0 0
$$263$$ 1.60256 0.0988179 0.0494090 0.998779i $$-0.484266\pi$$
0.0494090 + 0.998779i $$0.484266\pi$$
$$264$$ 0 0
$$265$$ 5.96896 0.366671
$$266$$ 0 0
$$267$$ −15.5423 −0.951174
$$268$$ 0 0
$$269$$ 11.8207 0.720719 0.360359 0.932814i $$-0.382654\pi$$
0.360359 + 0.932814i $$0.382654\pi$$
$$270$$ 0 0
$$271$$ 21.8827 1.32928 0.664641 0.747163i $$-0.268585\pi$$
0.664641 + 0.747163i $$0.268585\pi$$
$$272$$ 0 0
$$273$$ −2.24914 −0.136124
$$274$$ 0 0
$$275$$ −10.6155 −0.640142
$$276$$ 0 0
$$277$$ 10.2181 0.613946 0.306973 0.951718i $$-0.400684\pi$$
0.306973 + 0.951718i $$0.400684\pi$$
$$278$$ 0 0
$$279$$ −14.4672 −0.866131
$$280$$ 0 0
$$281$$ 1.54231 0.0920063 0.0460031 0.998941i $$-0.485352\pi$$
0.0460031 + 0.998941i $$0.485352\pi$$
$$282$$ 0 0
$$283$$ −15.8466 −0.941985 −0.470993 0.882137i $$-0.656104\pi$$
−0.470993 + 0.882137i $$0.656104\pi$$
$$284$$ 0 0
$$285$$ 1.75086 0.103712
$$286$$ 0 0
$$287$$ −6.49828 −0.383581
$$288$$ 0 0
$$289$$ −15.2897 −0.899396
$$290$$ 0 0
$$291$$ 7.80605 0.457599
$$292$$ 0 0
$$293$$ −11.0828 −0.647464 −0.323732 0.946149i $$-0.604938\pi$$
−0.323732 + 0.946149i $$0.604938\pi$$
$$294$$ 0 0
$$295$$ −6.44309 −0.375131
$$296$$ 0 0
$$297$$ 4.76203 0.276321
$$298$$ 0 0
$$299$$ −5.83709 −0.337568
$$300$$ 0 0
$$301$$ 11.3940 0.656740
$$302$$ 0 0
$$303$$ 17.4328 1.00149
$$304$$ 0 0
$$305$$ −1.05863 −0.0606172
$$306$$ 0 0
$$307$$ −20.4121 −1.16498 −0.582489 0.812839i $$-0.697921\pi$$
−0.582489 + 0.812839i $$0.697921\pi$$
$$308$$ 0 0
$$309$$ 38.2277 2.17470
$$310$$ 0 0
$$311$$ −1.92332 −0.109062 −0.0545308 0.998512i $$-0.517366\pi$$
−0.0545308 + 0.998512i $$0.517366\pi$$
$$312$$ 0 0
$$313$$ −14.3664 −0.812037 −0.406019 0.913865i $$-0.633083\pi$$
−0.406019 + 0.913865i $$0.633083\pi$$
$$314$$ 0 0
$$315$$ 1.08967 0.0613959
$$316$$ 0 0
$$317$$ −15.5569 −0.873763 −0.436882 0.899519i $$-0.643917\pi$$
−0.436882 + 0.899519i $$0.643917\pi$$
$$318$$ 0 0
$$319$$ −11.7440 −0.657537
$$320$$ 0 0
$$321$$ −12.4983 −0.697586
$$322$$ 0 0
$$323$$ −1.92332 −0.107016
$$324$$ 0 0
$$325$$ 4.71982 0.261809
$$326$$ 0 0
$$327$$ 17.8207 0.985485
$$328$$ 0 0
$$329$$ −8.58451 −0.473279
$$330$$ 0 0
$$331$$ −31.5500 −1.73415 −0.867073 0.498180i $$-0.834002\pi$$
−0.867073 + 0.498180i $$0.834002\pi$$
$$332$$ 0 0
$$333$$ 4.87156 0.266960
$$334$$ 0 0
$$335$$ −8.43621 −0.460919
$$336$$ 0 0
$$337$$ −8.42666 −0.459029 −0.229515 0.973305i $$-0.573714\pi$$
−0.229515 + 0.973305i $$0.573714\pi$$
$$338$$ 0 0
$$339$$ 22.2491 1.20841
$$340$$ 0 0
$$341$$ −15.8061 −0.855946
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 6.94910 0.374127
$$346$$ 0 0
$$347$$ −19.3484 −1.03867 −0.519337 0.854569i $$-0.673821\pi$$
−0.519337 + 0.854569i $$0.673821\pi$$
$$348$$ 0 0
$$349$$ −27.2553 −1.45894 −0.729470 0.684013i $$-0.760233\pi$$
−0.729470 + 0.684013i $$0.760233\pi$$
$$350$$ 0 0
$$351$$ −2.11727 −0.113011
$$352$$ 0 0
$$353$$ −25.6742 −1.36650 −0.683249 0.730185i $$-0.739434\pi$$
−0.683249 + 0.730185i $$0.739434\pi$$
$$354$$ 0 0
$$355$$ −0.630155 −0.0334452
$$356$$ 0 0
$$357$$ −2.94137 −0.155674
$$358$$ 0 0
$$359$$ −23.4182 −1.23596 −0.617982 0.786192i $$-0.712049\pi$$
−0.617982 + 0.786192i $$0.712049\pi$$
$$360$$ 0 0
$$361$$ −16.8371 −0.886163
$$362$$ 0 0
$$363$$ 13.3630 0.701374
$$364$$ 0 0
$$365$$ −4.04564 −0.211759
$$366$$ 0 0
$$367$$ −14.6854 −0.766569 −0.383285 0.923630i $$-0.625207\pi$$
−0.383285 + 0.923630i $$0.625207\pi$$
$$368$$ 0 0
$$369$$ 13.3776 0.696409
$$370$$ 0 0
$$371$$ 11.2767 0.585459
$$372$$ 0 0
$$373$$ 23.6673 1.22545 0.612723 0.790298i $$-0.290074\pi$$
0.612723 + 0.790298i $$0.290074\pi$$
$$374$$ 0 0
$$375$$ −11.5715 −0.597551
$$376$$ 0 0
$$377$$ 5.22154 0.268923
$$378$$ 0 0
$$379$$ 32.7405 1.68177 0.840884 0.541215i $$-0.182035\pi$$
0.840884 + 0.541215i $$0.182035\pi$$
$$380$$ 0 0
$$381$$ −1.85352 −0.0949586
$$382$$ 0 0
$$383$$ −22.6155 −1.15560 −0.577800 0.816178i $$-0.696089\pi$$
−0.577800 + 0.816178i $$0.696089\pi$$
$$384$$ 0 0
$$385$$ 1.19051 0.0606739
$$386$$ 0 0
$$387$$ −23.4561 −1.19234
$$388$$ 0 0
$$389$$ −38.0483 −1.92913 −0.964563 0.263852i $$-0.915007\pi$$
−0.964563 + 0.263852i $$0.915007\pi$$
$$390$$ 0 0
$$391$$ −7.63359 −0.386047
$$392$$ 0 0
$$393$$ 23.8759 1.20438
$$394$$ 0 0
$$395$$ 0.708654 0.0356562
$$396$$ 0 0
$$397$$ 11.7052 0.587468 0.293734 0.955887i $$-0.405102\pi$$
0.293734 + 0.955887i $$0.405102\pi$$
$$398$$ 0 0
$$399$$ 3.30777 0.165596
$$400$$ 0 0
$$401$$ 3.55691 0.177624 0.0888119 0.996048i $$-0.471693\pi$$
0.0888119 + 0.996048i $$0.471693\pi$$
$$402$$ 0 0
$$403$$ 7.02760 0.350070
$$404$$ 0 0
$$405$$ 5.78963 0.287689
$$406$$ 0 0
$$407$$ 5.32238 0.263821
$$408$$ 0 0
$$409$$ 5.26213 0.260196 0.130098 0.991501i $$-0.458471\pi$$
0.130098 + 0.991501i $$0.458471\pi$$
$$410$$ 0 0
$$411$$ −25.5569 −1.26063
$$412$$ 0 0
$$413$$ −12.1725 −0.598968
$$414$$ 0 0
$$415$$ −8.65432 −0.424824
$$416$$ 0 0
$$417$$ −31.3155 −1.53353
$$418$$ 0 0
$$419$$ −26.0337 −1.27183 −0.635915 0.771759i $$-0.719377\pi$$
−0.635915 + 0.771759i $$0.719377\pi$$
$$420$$ 0 0
$$421$$ −22.2423 −1.08402 −0.542011 0.840372i $$-0.682337\pi$$
−0.542011 + 0.840372i $$0.682337\pi$$
$$422$$ 0 0
$$423$$ 17.6724 0.859260
$$424$$ 0 0
$$425$$ 6.17246 0.299408
$$426$$ 0 0
$$427$$ −2.00000 −0.0967868
$$428$$ 0 0
$$429$$ 5.05863 0.244233
$$430$$ 0 0
$$431$$ 27.6742 1.33302 0.666509 0.745497i $$-0.267788\pi$$
0.666509 + 0.745497i $$0.267788\pi$$
$$432$$ 0 0
$$433$$ −12.7880 −0.614552 −0.307276 0.951620i $$-0.599418\pi$$
−0.307276 + 0.951620i $$0.599418\pi$$
$$434$$ 0 0
$$435$$ −6.21629 −0.298048
$$436$$ 0 0
$$437$$ 8.58451 0.410653
$$438$$ 0 0
$$439$$ −18.1656 −0.866996 −0.433498 0.901155i $$-0.642721\pi$$
−0.433498 + 0.901155i $$0.642721\pi$$
$$440$$ 0 0
$$441$$ 2.05863 0.0980302
$$442$$ 0 0
$$443$$ −0.107714 −0.00511767 −0.00255883 0.999997i $$-0.500815\pi$$
−0.00255883 + 0.999997i $$0.500815\pi$$
$$444$$ 0 0
$$445$$ −3.65775 −0.173394
$$446$$ 0 0
$$447$$ 20.9345 0.990167
$$448$$ 0 0
$$449$$ −22.1725 −1.04638 −0.523192 0.852215i $$-0.675259\pi$$
−0.523192 + 0.852215i $$0.675259\pi$$
$$450$$ 0 0
$$451$$ 14.6155 0.688219
$$452$$ 0 0
$$453$$ −15.9087 −0.747457
$$454$$ 0 0
$$455$$ −0.529317 −0.0248147
$$456$$ 0 0
$$457$$ 4.35953 0.203930 0.101965 0.994788i $$-0.467487\pi$$
0.101965 + 0.994788i $$0.467487\pi$$
$$458$$ 0 0
$$459$$ −2.76891 −0.129241
$$460$$ 0 0
$$461$$ −32.3810 −1.50813 −0.754067 0.656797i $$-0.771911\pi$$
−0.754067 + 0.656797i $$0.771911\pi$$
$$462$$ 0 0
$$463$$ 8.36641 0.388820 0.194410 0.980920i $$-0.437721\pi$$
0.194410 + 0.980920i $$0.437721\pi$$
$$464$$ 0 0
$$465$$ −8.36641 −0.387983
$$466$$ 0 0
$$467$$ 19.5423 0.904310 0.452155 0.891939i $$-0.350655\pi$$
0.452155 + 0.891939i $$0.350655\pi$$
$$468$$ 0 0
$$469$$ −15.9379 −0.735945
$$470$$ 0 0
$$471$$ −13.5861 −0.626016
$$472$$ 0 0
$$473$$ −25.6267 −1.17832
$$474$$ 0 0
$$475$$ −6.94137 −0.318492
$$476$$ 0 0
$$477$$ −23.2147 −1.06293
$$478$$ 0 0
$$479$$ −28.5224 −1.30322 −0.651612 0.758553i $$-0.725907\pi$$
−0.651612 + 0.758553i $$0.725907\pi$$
$$480$$ 0 0
$$481$$ −2.36641 −0.107899
$$482$$ 0 0
$$483$$ 13.1284 0.597365
$$484$$ 0 0
$$485$$ 1.83709 0.0834180
$$486$$ 0 0
$$487$$ 24.8241 1.12489 0.562444 0.826836i $$-0.309861\pi$$
0.562444 + 0.826836i $$0.309861\pi$$
$$488$$ 0 0
$$489$$ 14.3518 0.649011
$$490$$ 0 0
$$491$$ −29.1690 −1.31638 −0.658190 0.752852i $$-0.728678\pi$$
−0.658190 + 0.752852i $$0.728678\pi$$
$$492$$ 0 0
$$493$$ 6.82860 0.307545
$$494$$ 0 0
$$495$$ −2.45082 −0.110156
$$496$$ 0 0
$$497$$ −1.19051 −0.0534016
$$498$$ 0 0
$$499$$ 33.3009 1.49075 0.745376 0.666644i $$-0.232270\pi$$
0.745376 + 0.666644i $$0.232270\pi$$
$$500$$ 0 0
$$501$$ 37.3009 1.66648
$$502$$ 0 0
$$503$$ −12.3258 −0.549581 −0.274791 0.961504i $$-0.588609\pi$$
−0.274791 + 0.961504i $$0.588609\pi$$
$$504$$ 0 0
$$505$$ 4.10266 0.182566
$$506$$ 0 0
$$507$$ −2.24914 −0.0998878
$$508$$ 0 0
$$509$$ −23.7052 −1.05072 −0.525358 0.850882i $$-0.676068\pi$$
−0.525358 + 0.850882i $$0.676068\pi$$
$$510$$ 0 0
$$511$$ −7.64315 −0.338113
$$512$$ 0 0
$$513$$ 3.11383 0.137479
$$514$$ 0 0
$$515$$ 8.99656 0.396436
$$516$$ 0 0
$$517$$ 19.3078 0.849155
$$518$$ 0 0
$$519$$ −52.4070 −2.30041
$$520$$ 0 0
$$521$$ 43.9018 1.92337 0.961687 0.274149i $$-0.0883962\pi$$
0.961687 + 0.274149i $$0.0883962\pi$$
$$522$$ 0 0
$$523$$ −37.4328 −1.63682 −0.818410 0.574634i $$-0.805144\pi$$
−0.818410 + 0.574634i $$0.805144\pi$$
$$524$$ 0 0
$$525$$ −10.6155 −0.463300
$$526$$ 0 0
$$527$$ 9.19051 0.400345
$$528$$ 0 0
$$529$$ 11.0716 0.481375
$$530$$ 0 0
$$531$$ 25.0586 1.08745
$$532$$ 0 0
$$533$$ −6.49828 −0.281472
$$534$$ 0 0
$$535$$ −2.94137 −0.127166
$$536$$ 0 0
$$537$$ 47.3269 2.04231
$$538$$ 0 0
$$539$$ 2.24914 0.0968773
$$540$$ 0 0
$$541$$ 34.9751 1.50370 0.751848 0.659336i $$-0.229163\pi$$
0.751848 + 0.659336i $$0.229163\pi$$
$$542$$ 0 0
$$543$$ 37.6673 1.61646
$$544$$ 0 0
$$545$$ 4.19395 0.179649
$$546$$ 0 0
$$547$$ −6.50783 −0.278255 −0.139127 0.990274i $$-0.544430\pi$$
−0.139127 + 0.990274i $$0.544430\pi$$
$$548$$ 0 0
$$549$$ 4.11727 0.175721
$$550$$ 0 0
$$551$$ −7.67924 −0.327146
$$552$$ 0 0
$$553$$ 1.33881 0.0569320
$$554$$ 0 0
$$555$$ 2.81722 0.119585
$$556$$ 0 0
$$557$$ 43.4328 1.84031 0.920153 0.391559i $$-0.128064\pi$$
0.920153 + 0.391559i $$0.128064\pi$$
$$558$$ 0 0
$$559$$ 11.3940 0.481915
$$560$$ 0 0
$$561$$ 6.61555 0.279309
$$562$$ 0 0
$$563$$ 33.8827 1.42799 0.713993 0.700152i $$-0.246885\pi$$
0.713993 + 0.700152i $$0.246885\pi$$
$$564$$ 0 0
$$565$$ 5.23615 0.220287
$$566$$ 0 0
$$567$$ 10.9379 0.459350
$$568$$ 0 0
$$569$$ 19.2147 0.805521 0.402760 0.915305i $$-0.368051\pi$$
0.402760 + 0.915305i $$0.368051\pi$$
$$570$$ 0 0
$$571$$ −20.8268 −0.871573 −0.435787 0.900050i $$-0.643530\pi$$
−0.435787 + 0.900050i $$0.643530\pi$$
$$572$$ 0 0
$$573$$ 16.7328 0.699023
$$574$$ 0 0
$$575$$ −27.5500 −1.14892
$$576$$ 0 0
$$577$$ 28.6448 1.19250 0.596249 0.802800i $$-0.296657\pi$$
0.596249 + 0.802800i $$0.296657\pi$$
$$578$$ 0 0
$$579$$ 3.37758 0.140367
$$580$$ 0 0
$$581$$ −16.3500 −0.678311
$$582$$ 0 0
$$583$$ −25.3630 −1.05043
$$584$$ 0 0
$$585$$ 1.08967 0.0450523
$$586$$ 0 0
$$587$$ 4.32076 0.178337 0.0891685 0.996017i $$-0.471579\pi$$
0.0891685 + 0.996017i $$0.471579\pi$$
$$588$$ 0 0
$$589$$ −10.3354 −0.425862
$$590$$ 0 0
$$591$$ 53.9639 2.21978
$$592$$ 0 0
$$593$$ −15.9690 −0.655767 −0.327883 0.944718i $$-0.606335\pi$$
−0.327883 + 0.944718i $$0.606335\pi$$
$$594$$ 0 0
$$595$$ −0.692226 −0.0283785
$$596$$ 0 0
$$597$$ 4.53114 0.185447
$$598$$ 0 0
$$599$$ −16.8697 −0.689279 −0.344640 0.938735i $$-0.611999\pi$$
−0.344640 + 0.938735i $$0.611999\pi$$
$$600$$ 0 0
$$601$$ −15.3415 −0.625792 −0.312896 0.949787i $$-0.601299\pi$$
−0.312896 + 0.949787i $$0.601299\pi$$
$$602$$ 0 0
$$603$$ 32.8103 1.33614
$$604$$ 0 0
$$605$$ 3.14486 0.127857
$$606$$ 0 0
$$607$$ 35.8353 1.45451 0.727254 0.686368i $$-0.240796\pi$$
0.727254 + 0.686368i $$0.240796\pi$$
$$608$$ 0 0
$$609$$ −11.7440 −0.475890
$$610$$ 0 0
$$611$$ −8.58451 −0.347292
$$612$$ 0 0
$$613$$ −19.6673 −0.794355 −0.397177 0.917742i $$-0.630010\pi$$
−0.397177 + 0.917742i $$0.630010\pi$$
$$614$$ 0 0
$$615$$ 7.73625 0.311956
$$616$$ 0 0
$$617$$ −41.4588 −1.66907 −0.834533 0.550958i $$-0.814263\pi$$
−0.834533 + 0.550958i $$0.814263\pi$$
$$618$$ 0 0
$$619$$ −10.8793 −0.437276 −0.218638 0.975806i $$-0.570161\pi$$
−0.218638 + 0.975806i $$0.570161\pi$$
$$620$$ 0 0
$$621$$ 12.3587 0.495937
$$622$$ 0 0
$$623$$ −6.91033 −0.276857
$$624$$ 0 0
$$625$$ 20.8759 0.835034
$$626$$ 0 0
$$627$$ −7.43965 −0.297111
$$628$$ 0 0
$$629$$ −3.09472 −0.123395
$$630$$ 0 0
$$631$$ −31.4396 −1.25159 −0.625796 0.779987i $$-0.715226\pi$$
−0.625796 + 0.779987i $$0.715226\pi$$
$$632$$ 0 0
$$633$$ 22.7182 0.902968
$$634$$ 0 0
$$635$$ −0.436210 −0.0173105
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ 2.45082 0.0969529
$$640$$ 0 0
$$641$$ 3.04221 0.120160 0.0600799 0.998194i $$-0.480864\pi$$
0.0600799 + 0.998194i $$0.480864\pi$$
$$642$$ 0 0
$$643$$ −8.02922 −0.316641 −0.158321 0.987388i $$-0.550608\pi$$
−0.158321 + 0.987388i $$0.550608\pi$$
$$644$$ 0 0
$$645$$ −13.5646 −0.534107
$$646$$ 0 0
$$647$$ 7.07324 0.278078 0.139039 0.990287i $$-0.455599\pi$$
0.139039 + 0.990287i $$0.455599\pi$$
$$648$$ 0 0
$$649$$ 27.3776 1.07466
$$650$$ 0 0
$$651$$ −15.8061 −0.619488
$$652$$ 0 0
$$653$$ 15.7586 0.616681 0.308341 0.951276i $$-0.400226\pi$$
0.308341 + 0.951276i $$0.400226\pi$$
$$654$$ 0 0
$$655$$ 5.61899 0.219552
$$656$$ 0 0
$$657$$ 15.7344 0.613859
$$658$$ 0 0
$$659$$ 12.2181 0.475950 0.237975 0.971271i $$-0.423516\pi$$
0.237975 + 0.971271i $$0.423516\pi$$
$$660$$ 0 0
$$661$$ −3.73443 −0.145253 −0.0726263 0.997359i $$-0.523138\pi$$
−0.0726263 + 0.997359i $$0.523138\pi$$
$$662$$ 0 0
$$663$$ −2.94137 −0.114233
$$664$$ 0 0
$$665$$ 0.778457 0.0301873
$$666$$ 0 0
$$667$$ −30.4786 −1.18014
$$668$$ 0 0
$$669$$ −22.8095 −0.881866
$$670$$ 0 0
$$671$$ 4.49828 0.173654
$$672$$ 0 0
$$673$$ −5.65775 −0.218090 −0.109045 0.994037i $$-0.534779\pi$$
−0.109045 + 0.994037i $$0.534779\pi$$
$$674$$ 0 0
$$675$$ −9.99312 −0.384636
$$676$$ 0 0
$$677$$ −9.39906 −0.361235 −0.180618 0.983553i $$-0.557810\pi$$
−0.180618 + 0.983553i $$0.557810\pi$$
$$678$$ 0 0
$$679$$ 3.47068 0.133193
$$680$$ 0 0
$$681$$ −12.1104 −0.464071
$$682$$ 0 0
$$683$$ 20.7328 0.793319 0.396660 0.917966i $$-0.370169\pi$$
0.396660 + 0.917966i $$0.370169\pi$$
$$684$$ 0 0
$$685$$ −6.01461 −0.229806
$$686$$ 0 0
$$687$$ 7.47250 0.285094
$$688$$ 0 0
$$689$$ 11.2767 0.429610
$$690$$ 0 0
$$691$$ −16.0862 −0.611949 −0.305975 0.952040i $$-0.598982\pi$$
−0.305975 + 0.952040i $$0.598982\pi$$
$$692$$ 0 0
$$693$$ −4.63016 −0.175885
$$694$$ 0 0
$$695$$ −7.36984 −0.279554
$$696$$ 0 0
$$697$$ −8.49828 −0.321895
$$698$$ 0 0
$$699$$ −30.8578 −1.16715
$$700$$ 0 0
$$701$$ 6.98013 0.263636 0.131818 0.991274i $$-0.457919\pi$$
0.131818 + 0.991274i $$0.457919\pi$$
$$702$$ 0 0
$$703$$ 3.48024 0.131260
$$704$$ 0 0
$$705$$ 10.2199 0.384905
$$706$$ 0 0
$$707$$ 7.75086 0.291501
$$708$$ 0 0
$$709$$ −8.39239 −0.315183 −0.157591 0.987504i $$-0.550373\pi$$
−0.157591 + 0.987504i $$0.550373\pi$$
$$710$$ 0 0
$$711$$ −2.75612 −0.103362
$$712$$ 0 0
$$713$$ −41.0207 −1.53624
$$714$$ 0 0
$$715$$ 1.19051 0.0445225
$$716$$ 0 0
$$717$$ 7.87586 0.294129
$$718$$ 0 0
$$719$$ −5.16129 −0.192484 −0.0962418 0.995358i $$-0.530682\pi$$
−0.0962418 + 0.995358i $$0.530682\pi$$
$$720$$ 0 0
$$721$$ 16.9966 0.632985
$$722$$ 0 0
$$723$$ −3.57152 −0.132826
$$724$$ 0 0
$$725$$ 24.6448 0.915284
$$726$$ 0 0
$$727$$ −40.4362 −1.49970 −0.749848 0.661610i $$-0.769873\pi$$
−0.749848 + 0.661610i $$0.769873\pi$$
$$728$$ 0 0
$$729$$ −8.23109 −0.304855
$$730$$ 0 0
$$731$$ 14.9008 0.551125
$$732$$ 0 0
$$733$$ 39.1311 1.44534 0.722670 0.691193i $$-0.242915\pi$$
0.722670 + 0.691193i $$0.242915\pi$$
$$734$$ 0 0
$$735$$ 1.19051 0.0439125
$$736$$ 0 0
$$737$$ 35.8466 1.32043
$$738$$ 0 0
$$739$$ 7.13531 0.262477 0.131238 0.991351i $$-0.458105\pi$$
0.131238 + 0.991351i $$0.458105\pi$$
$$740$$ 0 0
$$741$$ 3.30777 0.121514
$$742$$ 0 0
$$743$$ 13.8827 0.509308 0.254654 0.967032i $$-0.418038\pi$$
0.254654 + 0.967032i $$0.418038\pi$$
$$744$$ 0 0
$$745$$ 4.92676 0.180502
$$746$$ 0 0
$$747$$ 33.6586 1.23150
$$748$$ 0 0
$$749$$ −5.55691 −0.203045
$$750$$ 0 0
$$751$$ −37.3251 −1.36201 −0.681005 0.732278i $$-0.738457\pi$$
−0.681005 + 0.732278i $$0.738457\pi$$
$$752$$ 0 0
$$753$$ −11.0810 −0.403813
$$754$$ 0 0
$$755$$ −3.74398 −0.136258
$$756$$ 0 0
$$757$$ 7.10428 0.258209 0.129105 0.991631i $$-0.458790\pi$$
0.129105 + 0.991631i $$0.458790\pi$$
$$758$$ 0 0
$$759$$ −29.5277 −1.07179
$$760$$ 0 0
$$761$$ −25.9621 −0.941125 −0.470562 0.882367i $$-0.655949\pi$$
−0.470562 + 0.882367i $$0.655949\pi$$
$$762$$ 0 0
$$763$$ 7.92332 0.286843
$$764$$ 0 0
$$765$$ 1.42504 0.0515224
$$766$$ 0 0
$$767$$ −12.1725 −0.439522
$$768$$ 0 0
$$769$$ −21.4638 −0.774005 −0.387002 0.922079i $$-0.626489\pi$$
−0.387002 + 0.922079i $$0.626489\pi$$
$$770$$ 0 0
$$771$$ 18.0260 0.649190
$$772$$ 0 0
$$773$$ 40.4914 1.45637 0.728187 0.685378i $$-0.240363\pi$$
0.728187 + 0.685378i $$0.240363\pi$$
$$774$$ 0 0
$$775$$ 33.1690 1.19147
$$776$$ 0 0
$$777$$ 5.32238 0.190939
$$778$$ 0 0
$$779$$ 9.55691 0.342412
$$780$$ 0 0
$$781$$ 2.67762 0.0958127
$$782$$ 0 0
$$783$$ −11.0554 −0.395088
$$784$$ 0 0
$$785$$ −3.19738 −0.114119
$$786$$ 0 0
$$787$$ 5.02072 0.178969 0.0894847 0.995988i $$-0.471478\pi$$
0.0894847 + 0.995988i $$0.471478\pi$$
$$788$$ 0 0
$$789$$ −3.60438 −0.128319
$$790$$ 0 0
$$791$$ 9.89229 0.351729
$$792$$ 0 0
$$793$$ −2.00000 −0.0710221
$$794$$ 0 0
$$795$$ −13.4250 −0.476137
$$796$$ 0 0
$$797$$ −19.7002 −0.697815 −0.348908 0.937157i $$-0.613447\pi$$
−0.348908 + 0.937157i $$0.613447\pi$$
$$798$$ 0 0
$$799$$ −11.2266 −0.397169
$$800$$ 0 0
$$801$$ 14.2258 0.502645
$$802$$ 0 0
$$803$$ 17.1905 0.606640
$$804$$ 0 0
$$805$$ 3.08967 0.108897
$$806$$ 0 0
$$807$$ −26.5863 −0.935883
$$808$$ 0 0
$$809$$ −2.57678 −0.0905947 −0.0452974 0.998974i $$-0.514424\pi$$
−0.0452974 + 0.998974i $$0.514424\pi$$
$$810$$ 0 0
$$811$$ −41.2311 −1.44782 −0.723910 0.689895i $$-0.757657\pi$$
−0.723910 + 0.689895i $$0.757657\pi$$
$$812$$ 0 0
$$813$$ −49.2173 −1.72613
$$814$$ 0 0
$$815$$ 3.37758 0.118311
$$816$$ 0 0
$$817$$ −16.7570 −0.586252
$$818$$ 0 0
$$819$$ 2.05863 0.0719345
$$820$$ 0 0
$$821$$ −5.26719 −0.183826 −0.0919130 0.995767i $$-0.529298\pi$$
−0.0919130 + 0.995767i $$0.529298\pi$$
$$822$$ 0 0
$$823$$ 36.0191 1.25555 0.627774 0.778396i $$-0.283966\pi$$
0.627774 + 0.778396i $$0.283966\pi$$
$$824$$ 0 0
$$825$$ 23.8759 0.831251
$$826$$ 0 0
$$827$$ 16.3157 0.567353 0.283676 0.958920i $$-0.408446\pi$$
0.283676 + 0.958920i $$0.408446\pi$$
$$828$$ 0 0
$$829$$ 30.3956 1.05568 0.527842 0.849343i $$-0.323001\pi$$
0.527842 + 0.849343i $$0.323001\pi$$
$$830$$ 0 0
$$831$$ −22.9820 −0.797235
$$832$$ 0 0
$$833$$ −1.30777 −0.0453117
$$834$$ 0 0
$$835$$ 8.77846 0.303791
$$836$$ 0 0
$$837$$ −14.8793 −0.514304
$$838$$ 0 0
$$839$$ 29.8398 1.03018 0.515092 0.857135i $$-0.327758\pi$$
0.515092 + 0.857135i $$0.327758\pi$$
$$840$$ 0 0
$$841$$ −1.73549 −0.0598445
$$842$$ 0 0
$$843$$ −3.46886 −0.119474
$$844$$ 0 0
$$845$$ −0.529317 −0.0182090
$$846$$ 0 0
$$847$$ 5.94137 0.204148
$$848$$ 0 0
$$849$$ 35.6413 1.22321
$$850$$ 0 0
$$851$$ 13.8129 0.473501
$$852$$ 0 0
$$853$$ 0.203497 0.00696761 0.00348380 0.999994i $$-0.498891\pi$$
0.00348380 + 0.999994i $$0.498891\pi$$
$$854$$ 0 0
$$855$$ −1.60256 −0.0548063
$$856$$ 0 0
$$857$$ −12.6155 −0.430939 −0.215469 0.976511i $$-0.569128\pi$$
−0.215469 + 0.976511i $$0.569128\pi$$
$$858$$ 0 0
$$859$$ 27.9671 0.954227 0.477113 0.878842i $$-0.341683\pi$$
0.477113 + 0.878842i $$0.341683\pi$$
$$860$$ 0 0
$$861$$ 14.6155 0.498097
$$862$$ 0 0
$$863$$ −2.76891 −0.0942546 −0.0471273 0.998889i $$-0.515007\pi$$
−0.0471273 + 0.998889i $$0.515007\pi$$
$$864$$ 0 0
$$865$$ −12.3336 −0.419353
$$866$$ 0 0
$$867$$ 34.3887 1.16790
$$868$$ 0 0
$$869$$ −3.01117 −0.102147
$$870$$ 0 0
$$871$$ −15.9379 −0.540036
$$872$$ 0 0
$$873$$ −7.14486 −0.241817
$$874$$ 0 0
$$875$$ −5.14486 −0.173928
$$876$$ 0 0
$$877$$ −6.71133 −0.226626 −0.113313 0.993559i $$-0.536146\pi$$
−0.113313 + 0.993559i $$0.536146\pi$$
$$878$$ 0 0
$$879$$ 24.9268 0.840759
$$880$$ 0 0
$$881$$ −18.3741 −0.619040 −0.309520 0.950893i $$-0.600168\pi$$
−0.309520 + 0.950893i $$0.600168\pi$$
$$882$$ 0 0
$$883$$ −9.93105 −0.334207 −0.167103 0.985939i $$-0.553441\pi$$
−0.167103 + 0.985939i $$0.553441\pi$$
$$884$$ 0 0
$$885$$ 14.4914 0.487123
$$886$$ 0 0
$$887$$ −51.3776 −1.72509 −0.862545 0.505980i $$-0.831131\pi$$
−0.862545 + 0.505980i $$0.831131\pi$$
$$888$$ 0 0
$$889$$ −0.824101 −0.0276394
$$890$$ 0 0
$$891$$ −24.6009 −0.824162
$$892$$ 0 0
$$893$$ 12.6251 0.422483
$$894$$ 0 0
$$895$$ 11.1380 0.372302
$$896$$ 0 0
$$897$$ 13.1284 0.438346
$$898$$ 0 0
$$899$$ 36.6949 1.22384
$$900$$ 0 0
$$901$$ 14.7474 0.491308
$$902$$ 0 0
$$903$$ −25.6267 −0.852804
$$904$$ 0 0
$$905$$ 8.86469 0.294672
$$906$$ 0 0
$$907$$ −4.34225 −0.144182 −0.0720910 0.997398i $$-0.522967\pi$$
−0.0720910 + 0.997398i $$0.522967\pi$$
$$908$$ 0 0
$$909$$ −15.9562 −0.529233
$$910$$ 0 0
$$911$$ 31.4853 1.04315 0.521577 0.853204i $$-0.325344\pi$$
0.521577 + 0.853204i $$0.325344\pi$$
$$912$$ 0 0
$$913$$ 36.7734 1.21702
$$914$$ 0 0
$$915$$ 2.38101 0.0787139
$$916$$ 0 0
$$917$$ 10.6155 0.350556
$$918$$ 0 0
$$919$$ 43.4588 1.43357 0.716786 0.697293i $$-0.245612\pi$$
0.716786 + 0.697293i $$0.245612\pi$$
$$920$$ 0 0
$$921$$ 45.9096 1.51277
$$922$$ 0 0
$$923$$ −1.19051 −0.0391860
$$924$$ 0 0
$$925$$ −11.1690 −0.367235
$$926$$ 0 0
$$927$$ −34.9897 −1.14921
$$928$$ 0 0
$$929$$ 4.40517 0.144529 0.0722645 0.997385i $$-0.476977\pi$$
0.0722645 + 0.997385i $$0.476977\pi$$
$$930$$ 0 0
$$931$$ 1.47068 0.0481997
$$932$$ 0 0
$$933$$ 4.32582 0.141621
$$934$$ 0 0
$$935$$ 1.55691 0.0509165
$$936$$ 0 0
$$937$$ −34.0990 −1.11397 −0.556983 0.830524i $$-0.688041\pi$$
−0.556983 + 0.830524i $$0.688041\pi$$
$$938$$ 0 0
$$939$$ 32.3121 1.05446
$$940$$ 0 0
$$941$$ −44.4672 −1.44959 −0.724795 0.688964i $$-0.758066\pi$$
−0.724795 + 0.688964i $$0.758066\pi$$
$$942$$ 0 0
$$943$$ 37.9311 1.23521
$$944$$ 0 0
$$945$$ 1.12070 0.0364565
$$946$$ 0 0
$$947$$ −57.9311 −1.88251 −0.941253 0.337702i $$-0.890350\pi$$
−0.941253 + 0.337702i $$0.890350\pi$$
$$948$$ 0 0
$$949$$ −7.64315 −0.248107
$$950$$ 0 0
$$951$$ 34.9897 1.13462
$$952$$ 0 0
$$953$$ 35.3060 1.14367 0.571836 0.820368i $$-0.306231\pi$$
0.571836 + 0.820368i $$0.306231\pi$$
$$954$$ 0 0
$$955$$ 3.93793 0.127428
$$956$$ 0 0
$$957$$ 26.4139 0.853839
$$958$$ 0 0
$$959$$ −11.3630 −0.366929
$$960$$ 0 0
$$961$$ 18.3871 0.593133
$$962$$ 0 0
$$963$$ 11.4396 0.368638
$$964$$ 0 0
$$965$$ 0.794885 0.0255882
$$966$$ 0 0
$$967$$ 23.7148 0.762616 0.381308 0.924448i $$-0.375474\pi$$
0.381308 + 0.924448i $$0.375474\pi$$
$$968$$ 0 0
$$969$$ 4.32582 0.138965
$$970$$ 0 0
$$971$$ 23.9379 0.768205 0.384102 0.923291i $$-0.374511\pi$$
0.384102 + 0.923291i $$0.374511\pi$$
$$972$$ 0 0
$$973$$ −13.9233 −0.446361
$$974$$ 0 0
$$975$$ −10.6155 −0.339970
$$976$$ 0 0
$$977$$ −16.1871 −0.517870 −0.258935 0.965895i $$-0.583372\pi$$
−0.258935 + 0.965895i $$0.583372\pi$$
$$978$$ 0 0
$$979$$ 15.5423 0.496734
$$980$$ 0 0
$$981$$ −16.3112 −0.520777
$$982$$ 0 0
$$983$$ 45.7243 1.45838 0.729190 0.684312i $$-0.239897\pi$$
0.729190 + 0.684312i $$0.239897\pi$$
$$984$$ 0 0
$$985$$ 12.7000 0.404654
$$986$$ 0 0
$$987$$ 19.3078 0.614573
$$988$$ 0 0
$$989$$ −66.5078 −2.11483
$$990$$ 0 0
$$991$$ −5.40356 −0.171650 −0.0858248 0.996310i $$-0.527353\pi$$
−0.0858248 + 0.996310i $$0.527353\pi$$
$$992$$ 0 0
$$993$$ 70.9605 2.25186
$$994$$ 0 0
$$995$$ 1.06637 0.0338061
$$996$$ 0 0
$$997$$ −6.04832 −0.191552 −0.0957761 0.995403i $$-0.530533\pi$$
−0.0957761 + 0.995403i $$0.530533\pi$$
$$998$$ 0 0
$$999$$ 5.01031 0.158519
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 5824.2.a.by.1.1 3
4.3 odd 2 5824.2.a.bs.1.3 3
8.3 odd 2 1456.2.a.t.1.1 3
8.5 even 2 91.2.a.d.1.2 3
24.5 odd 2 819.2.a.i.1.2 3
40.29 even 2 2275.2.a.m.1.2 3
56.5 odd 6 637.2.e.i.508.2 6
56.13 odd 2 637.2.a.j.1.2 3
56.37 even 6 637.2.e.j.508.2 6
56.45 odd 6 637.2.e.i.79.2 6
56.53 even 6 637.2.e.j.79.2 6
104.5 odd 4 1183.2.c.f.337.3 6
104.21 odd 4 1183.2.c.f.337.4 6
104.77 even 2 1183.2.a.i.1.2 3
168.125 even 2 5733.2.a.x.1.2 3
728.181 odd 2 8281.2.a.bg.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.2 3 8.5 even 2
637.2.a.j.1.2 3 56.13 odd 2
637.2.e.i.79.2 6 56.45 odd 6
637.2.e.i.508.2 6 56.5 odd 6
637.2.e.j.79.2 6 56.53 even 6
637.2.e.j.508.2 6 56.37 even 6
819.2.a.i.1.2 3 24.5 odd 2
1183.2.a.i.1.2 3 104.77 even 2
1183.2.c.f.337.3 6 104.5 odd 4
1183.2.c.f.337.4 6 104.21 odd 4
1456.2.a.t.1.1 3 8.3 odd 2
2275.2.a.m.1.2 3 40.29 even 2
5733.2.a.x.1.2 3 168.125 even 2
5824.2.a.bs.1.3 3 4.3 odd 2
5824.2.a.by.1.1 3 1.1 even 1 trivial
8281.2.a.bg.1.2 3 728.181 odd 2