# Properties

 Label 91.2.a.d.1.3 Level $91$ Weight $2$ Character 91.1 Self dual yes Analytic conductor $0.727$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,2,Mod(1,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("91.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.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: $$0.726638658394$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.34292$$ of defining polynomial Character $$\chi$$ $$=$$ 91.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.34292 q^{2} -1.14637 q^{3} +3.48929 q^{4} -1.34292 q^{5} -2.68585 q^{6} -1.00000 q^{7} +3.48929 q^{8} -1.68585 q^{9} +O(q^{10})$$ $$q+2.34292 q^{2} -1.14637 q^{3} +3.48929 q^{4} -1.34292 q^{5} -2.68585 q^{6} -1.00000 q^{7} +3.48929 q^{8} -1.68585 q^{9} -3.14637 q^{10} +1.14637 q^{11} -4.00000 q^{12} +1.00000 q^{13} -2.34292 q^{14} +1.53948 q^{15} +1.19656 q^{16} +5.83221 q^{17} -3.94981 q^{18} -3.34292 q^{19} -4.68585 q^{20} +1.14637 q^{21} +2.68585 q^{22} -3.17513 q^{23} -4.00000 q^{24} -3.19656 q^{25} +2.34292 q^{26} +5.37169 q^{27} -3.48929 q^{28} +10.4893 q^{29} +3.60688 q^{30} +1.63565 q^{31} -4.17513 q^{32} -1.31415 q^{33} +13.6644 q^{34} +1.34292 q^{35} -5.88240 q^{36} +8.51806 q^{37} -7.83221 q^{38} -1.14637 q^{39} -4.68585 q^{40} -0.292731 q^{41} +2.68585 q^{42} -8.15371 q^{43} +4.00000 q^{44} +2.26396 q^{45} -7.43910 q^{46} -10.6142 q^{47} -1.37169 q^{48} +1.00000 q^{49} -7.48929 q^{50} -6.68585 q^{51} +3.48929 q^{52} -0.782020 q^{53} +12.5855 q^{54} -1.53948 q^{55} -3.48929 q^{56} +3.83221 q^{57} +24.5756 q^{58} +12.6430 q^{59} +5.37169 q^{60} -2.00000 q^{61} +3.83221 q^{62} +1.68585 q^{63} -12.1751 q^{64} -1.34292 q^{65} -3.07896 q^{66} -6.10038 q^{67} +20.3503 q^{68} +3.63986 q^{69} +3.14637 q^{70} +1.53948 q^{71} -5.88240 q^{72} -15.3001 q^{73} +19.9572 q^{74} +3.66442 q^{75} -11.6644 q^{76} -1.14637 q^{77} -2.68585 q^{78} +0.882404 q^{79} -1.60688 q^{80} -1.10038 q^{81} -0.685846 q^{82} -12.1292 q^{83} +4.00000 q^{84} -7.83221 q^{85} -19.1035 q^{86} -12.0246 q^{87} +4.00000 q^{88} +5.73604 q^{89} +5.30429 q^{90} -1.00000 q^{91} -11.0790 q^{92} -1.87506 q^{93} -24.8683 q^{94} +4.48929 q^{95} +4.78623 q^{96} -5.34292 q^{97} +2.34292 q^{98} -1.93260 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9}+O(q^{10})$$ 3 * q + q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 3 * q^8 + 7 * q^9 $$3 q + q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} + 4 q^{6} - 3 q^{7} + 3 q^{8} + 7 q^{9} - 8 q^{10} + 2 q^{11} - 12 q^{12} + 3 q^{13} - q^{14} - 6 q^{15} - q^{16} + 4 q^{17} - 15 q^{18} - 4 q^{19} - 2 q^{20} + 2 q^{21} - 4 q^{22} + 10 q^{23} - 12 q^{24} - 5 q^{25} + q^{26} - 8 q^{27} - 3 q^{28} + 24 q^{29} + 20 q^{30} - 4 q^{31} + 7 q^{32} - 16 q^{33} + 14 q^{34} - 2 q^{35} - q^{36} - 10 q^{38} - 2 q^{39} - 2 q^{40} + 2 q^{41} - 4 q^{42} + 10 q^{43} + 12 q^{44} + 22 q^{45} - 18 q^{46} - 8 q^{47} + 20 q^{48} + 3 q^{49} - 15 q^{50} - 8 q^{51} + 3 q^{52} + 8 q^{53} + 32 q^{54} + 6 q^{55} - 3 q^{56} - 2 q^{57} + 12 q^{58} - 4 q^{59} - 8 q^{60} - 6 q^{61} - 2 q^{62} - 7 q^{63} - 17 q^{64} + 2 q^{65} + 12 q^{66} - 12 q^{67} + 22 q^{68} - 6 q^{69} + 8 q^{70} - 6 q^{71} - q^{72} - 10 q^{73} + 30 q^{74} - 16 q^{75} - 8 q^{76} - 2 q^{77} + 4 q^{78} - 14 q^{79} - 14 q^{80} + 3 q^{81} + 10 q^{82} - 12 q^{83} + 12 q^{84} - 10 q^{85} - 26 q^{86} - 26 q^{87} + 12 q^{88} + 2 q^{89} - 28 q^{90} - 3 q^{91} - 12 q^{92} - 22 q^{93} - 10 q^{94} + 6 q^{95} - 4 q^{96} - 10 q^{97} + q^{98} + 14 q^{99}+O(q^{100})$$ 3 * q + q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 + 4 * q^6 - 3 * q^7 + 3 * q^8 + 7 * q^9 - 8 * q^10 + 2 * q^11 - 12 * q^12 + 3 * q^13 - q^14 - 6 * q^15 - q^16 + 4 * q^17 - 15 * q^18 - 4 * q^19 - 2 * q^20 + 2 * q^21 - 4 * q^22 + 10 * q^23 - 12 * q^24 - 5 * q^25 + q^26 - 8 * q^27 - 3 * q^28 + 24 * q^29 + 20 * q^30 - 4 * q^31 + 7 * q^32 - 16 * q^33 + 14 * q^34 - 2 * q^35 - q^36 - 10 * q^38 - 2 * q^39 - 2 * q^40 + 2 * q^41 - 4 * q^42 + 10 * q^43 + 12 * q^44 + 22 * q^45 - 18 * q^46 - 8 * q^47 + 20 * q^48 + 3 * q^49 - 15 * q^50 - 8 * q^51 + 3 * q^52 + 8 * q^53 + 32 * q^54 + 6 * q^55 - 3 * q^56 - 2 * q^57 + 12 * q^58 - 4 * q^59 - 8 * q^60 - 6 * q^61 - 2 * q^62 - 7 * q^63 - 17 * q^64 + 2 * q^65 + 12 * q^66 - 12 * q^67 + 22 * q^68 - 6 * q^69 + 8 * q^70 - 6 * q^71 - q^72 - 10 * q^73 + 30 * q^74 - 16 * q^75 - 8 * q^76 - 2 * q^77 + 4 * q^78 - 14 * q^79 - 14 * q^80 + 3 * q^81 + 10 * q^82 - 12 * q^83 + 12 * q^84 - 10 * q^85 - 26 * q^86 - 26 * q^87 + 12 * q^88 + 2 * q^89 - 28 * q^90 - 3 * q^91 - 12 * q^92 - 22 * q^93 - 10 * q^94 + 6 * q^95 - 4 * q^96 - 10 * q^97 + q^98 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.34292 1.65670 0.828348 0.560213i $$-0.189281\pi$$
0.828348 + 0.560213i $$0.189281\pi$$
$$3$$ −1.14637 −0.661854 −0.330927 0.943656i $$-0.607361\pi$$
−0.330927 + 0.943656i $$0.607361\pi$$
$$4$$ 3.48929 1.74464
$$5$$ −1.34292 −0.600573 −0.300287 0.953849i $$-0.597082\pi$$
−0.300287 + 0.953849i $$0.597082\pi$$
$$6$$ −2.68585 −1.09649
$$7$$ −1.00000 −0.377964
$$8$$ 3.48929 1.23365
$$9$$ −1.68585 −0.561949
$$10$$ −3.14637 −0.994968
$$11$$ 1.14637 0.345642 0.172821 0.984953i $$-0.444712\pi$$
0.172821 + 0.984953i $$0.444712\pi$$
$$12$$ −4.00000 −1.15470
$$13$$ 1.00000 0.277350
$$14$$ −2.34292 −0.626173
$$15$$ 1.53948 0.397492
$$16$$ 1.19656 0.299139
$$17$$ 5.83221 1.41452 0.707260 0.706954i $$-0.249931\pi$$
0.707260 + 0.706954i $$0.249931\pi$$
$$18$$ −3.94981 −0.930979
$$19$$ −3.34292 −0.766919 −0.383460 0.923558i $$-0.625267\pi$$
−0.383460 + 0.923558i $$0.625267\pi$$
$$20$$ −4.68585 −1.04779
$$21$$ 1.14637 0.250157
$$22$$ 2.68585 0.572624
$$23$$ −3.17513 −0.662061 −0.331031 0.943620i $$-0.607396\pi$$
−0.331031 + 0.943620i $$0.607396\pi$$
$$24$$ −4.00000 −0.816497
$$25$$ −3.19656 −0.639312
$$26$$ 2.34292 0.459485
$$27$$ 5.37169 1.03378
$$28$$ −3.48929 −0.659414
$$29$$ 10.4893 1.94781 0.973906 0.226952i $$-0.0728760\pi$$
0.973906 + 0.226952i $$0.0728760\pi$$
$$30$$ 3.60688 0.658524
$$31$$ 1.63565 0.293772 0.146886 0.989153i $$-0.453075\pi$$
0.146886 + 0.989153i $$0.453075\pi$$
$$32$$ −4.17513 −0.738067
$$33$$ −1.31415 −0.228765
$$34$$ 13.6644 2.34343
$$35$$ 1.34292 0.226995
$$36$$ −5.88240 −0.980401
$$37$$ 8.51806 1.40036 0.700180 0.713966i $$-0.253103\pi$$
0.700180 + 0.713966i $$0.253103\pi$$
$$38$$ −7.83221 −1.27055
$$39$$ −1.14637 −0.183565
$$40$$ −4.68585 −0.740897
$$41$$ −0.292731 −0.0457169 −0.0228584 0.999739i $$-0.507277\pi$$
−0.0228584 + 0.999739i $$0.507277\pi$$
$$42$$ 2.68585 0.414435
$$43$$ −8.15371 −1.24343 −0.621715 0.783244i $$-0.713564\pi$$
−0.621715 + 0.783244i $$0.713564\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 2.26396 0.337491
$$46$$ −7.43910 −1.09683
$$47$$ −10.6142 −1.54824 −0.774122 0.633036i $$-0.781809\pi$$
−0.774122 + 0.633036i $$0.781809\pi$$
$$48$$ −1.37169 −0.197987
$$49$$ 1.00000 0.142857
$$50$$ −7.48929 −1.05915
$$51$$ −6.68585 −0.936206
$$52$$ 3.48929 0.483877
$$53$$ −0.782020 −0.107419 −0.0537093 0.998557i $$-0.517104\pi$$
−0.0537093 + 0.998557i $$0.517104\pi$$
$$54$$ 12.5855 1.71266
$$55$$ −1.53948 −0.207584
$$56$$ −3.48929 −0.466276
$$57$$ 3.83221 0.507589
$$58$$ 24.5756 3.22693
$$59$$ 12.6430 1.64598 0.822989 0.568057i $$-0.192305\pi$$
0.822989 + 0.568057i $$0.192305\pi$$
$$60$$ 5.37169 0.693482
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 3.83221 0.486691
$$63$$ 1.68585 0.212397
$$64$$ −12.1751 −1.52189
$$65$$ −1.34292 −0.166569
$$66$$ −3.07896 −0.378994
$$67$$ −6.10038 −0.745281 −0.372640 0.927976i $$-0.621547\pi$$
−0.372640 + 0.927976i $$0.621547\pi$$
$$68$$ 20.3503 2.46783
$$69$$ 3.63986 0.438188
$$70$$ 3.14637 0.376063
$$71$$ 1.53948 0.182703 0.0913514 0.995819i $$-0.470881\pi$$
0.0913514 + 0.995819i $$0.470881\pi$$
$$72$$ −5.88240 −0.693248
$$73$$ −15.3001 −1.79074 −0.895369 0.445324i $$-0.853088\pi$$
−0.895369 + 0.445324i $$0.853088\pi$$
$$74$$ 19.9572 2.31997
$$75$$ 3.66442 0.423131
$$76$$ −11.6644 −1.33800
$$77$$ −1.14637 −0.130640
$$78$$ −2.68585 −0.304112
$$79$$ 0.882404 0.0992782 0.0496391 0.998767i $$-0.484193\pi$$
0.0496391 + 0.998767i $$0.484193\pi$$
$$80$$ −1.60688 −0.179655
$$81$$ −1.10038 −0.122265
$$82$$ −0.685846 −0.0757390
$$83$$ −12.1292 −1.33135 −0.665674 0.746243i $$-0.731856\pi$$
−0.665674 + 0.746243i $$0.731856\pi$$
$$84$$ 4.00000 0.436436
$$85$$ −7.83221 −0.849523
$$86$$ −19.1035 −2.05999
$$87$$ −12.0246 −1.28917
$$88$$ 4.00000 0.426401
$$89$$ 5.73604 0.608019 0.304009 0.952669i $$-0.401675\pi$$
0.304009 + 0.952669i $$0.401675\pi$$
$$90$$ 5.30429 0.559121
$$91$$ −1.00000 −0.104828
$$92$$ −11.0790 −1.15506
$$93$$ −1.87506 −0.194434
$$94$$ −24.8683 −2.56497
$$95$$ 4.48929 0.460591
$$96$$ 4.78623 0.488493
$$97$$ −5.34292 −0.542492 −0.271246 0.962510i $$-0.587436\pi$$
−0.271246 + 0.962510i $$0.587436\pi$$
$$98$$ 2.34292 0.236671
$$99$$ −1.93260 −0.194233
$$100$$ −11.1537 −1.11537
$$101$$ 11.1464 1.10910 0.554552 0.832149i $$-0.312889\pi$$
0.554552 + 0.832149i $$0.312889\pi$$
$$102$$ −15.6644 −1.55101
$$103$$ −3.41454 −0.336444 −0.168222 0.985749i $$-0.553803\pi$$
−0.168222 + 0.985749i $$0.553803\pi$$
$$104$$ 3.48929 0.342153
$$105$$ −1.53948 −0.150238
$$106$$ −1.83221 −0.177960
$$107$$ 4.97858 0.481297 0.240649 0.970612i $$-0.422640\pi$$
0.240649 + 0.970612i $$0.422640\pi$$
$$108$$ 18.7434 1.80358
$$109$$ −13.4966 −1.29274 −0.646372 0.763023i $$-0.723714\pi$$
−0.646372 + 0.763023i $$0.723714\pi$$
$$110$$ −3.60688 −0.343903
$$111$$ −9.76481 −0.926835
$$112$$ −1.19656 −0.113064
$$113$$ 16.4464 1.54715 0.773576 0.633704i $$-0.218466\pi$$
0.773576 + 0.633704i $$0.218466\pi$$
$$114$$ 8.97858 0.840921
$$115$$ 4.26396 0.397616
$$116$$ 36.6002 3.39824
$$117$$ −1.68585 −0.155857
$$118$$ 29.6216 2.72689
$$119$$ −5.83221 −0.534638
$$120$$ 5.37169 0.490366
$$121$$ −9.68585 −0.880531
$$122$$ −4.68585 −0.424237
$$123$$ 0.335577 0.0302579
$$124$$ 5.70727 0.512528
$$125$$ 11.0073 0.984527
$$126$$ 3.94981 0.351877
$$127$$ 12.0575 1.06993 0.534967 0.844873i $$-0.320324\pi$$
0.534967 + 0.844873i $$0.320324\pi$$
$$128$$ −20.1751 −1.78325
$$129$$ 9.34713 0.822969
$$130$$ −3.14637 −0.275955
$$131$$ −3.66442 −0.320162 −0.160081 0.987104i $$-0.551176\pi$$
−0.160081 + 0.987104i $$0.551176\pi$$
$$132$$ −4.58546 −0.399113
$$133$$ 3.34292 0.289868
$$134$$ −14.2927 −1.23470
$$135$$ −7.21377 −0.620862
$$136$$ 20.3503 1.74502
$$137$$ −13.1035 −1.11951 −0.559755 0.828658i $$-0.689105\pi$$
−0.559755 + 0.828658i $$0.689105\pi$$
$$138$$ 8.52792 0.725945
$$139$$ 7.49663 0.635856 0.317928 0.948115i $$-0.397013\pi$$
0.317928 + 0.948115i $$0.397013\pi$$
$$140$$ 4.68585 0.396026
$$141$$ 12.1678 1.02471
$$142$$ 3.60688 0.302683
$$143$$ 1.14637 0.0958639
$$144$$ −2.01721 −0.168101
$$145$$ −14.0863 −1.16980
$$146$$ −35.8469 −2.96671
$$147$$ −1.14637 −0.0945506
$$148$$ 29.7220 2.44313
$$149$$ 2.16779 0.177592 0.0887961 0.996050i $$-0.471698\pi$$
0.0887961 + 0.996050i $$0.471698\pi$$
$$150$$ 8.58546 0.701000
$$151$$ 14.9112 1.21345 0.606727 0.794910i $$-0.292482\pi$$
0.606727 + 0.794910i $$0.292482\pi$$
$$152$$ −11.6644 −0.946110
$$153$$ −9.83221 −0.794887
$$154$$ −2.68585 −0.216432
$$155$$ −2.19656 −0.176432
$$156$$ −4.00000 −0.320256
$$157$$ 22.8683 1.82509 0.912546 0.408975i $$-0.134114\pi$$
0.912546 + 0.408975i $$0.134114\pi$$
$$158$$ 2.06740 0.164474
$$159$$ 0.896480 0.0710955
$$160$$ 5.60688 0.443263
$$161$$ 3.17513 0.250236
$$162$$ −2.57812 −0.202556
$$163$$ 7.07896 0.554467 0.277234 0.960803i $$-0.410582\pi$$
0.277234 + 0.960803i $$0.410582\pi$$
$$164$$ −1.02142 −0.0797597
$$165$$ 1.76481 0.137390
$$166$$ −28.4177 −2.20564
$$167$$ 2.61423 0.202295 0.101148 0.994871i $$-0.467749\pi$$
0.101148 + 0.994871i $$0.467749\pi$$
$$168$$ 4.00000 0.308607
$$169$$ 1.00000 0.0769231
$$170$$ −18.3503 −1.40740
$$171$$ 5.63565 0.430969
$$172$$ −28.4507 −2.16934
$$173$$ 11.0031 0.836553 0.418276 0.908320i $$-0.362634\pi$$
0.418276 + 0.908320i $$0.362634\pi$$
$$174$$ −28.1726 −2.13576
$$175$$ 3.19656 0.241637
$$176$$ 1.37169 0.103395
$$177$$ −14.4935 −1.08940
$$178$$ 13.4391 1.00730
$$179$$ 23.9614 1.79096 0.895478 0.445105i $$-0.146834\pi$$
0.895478 + 0.445105i $$0.146834\pi$$
$$180$$ 7.89962 0.588803
$$181$$ 6.56090 0.487668 0.243834 0.969817i $$-0.421595\pi$$
0.243834 + 0.969817i $$0.421595\pi$$
$$182$$ −2.34292 −0.173669
$$183$$ 2.29273 0.169484
$$184$$ −11.0790 −0.816752
$$185$$ −11.4391 −0.841019
$$186$$ −4.39312 −0.322119
$$187$$ 6.68585 0.488917
$$188$$ −37.0361 −2.70114
$$189$$ −5.37169 −0.390733
$$190$$ 10.5181 0.763060
$$191$$ −4.39312 −0.317875 −0.158937 0.987289i $$-0.550807\pi$$
−0.158937 + 0.987289i $$0.550807\pi$$
$$192$$ 13.9572 1.00727
$$193$$ −8.29273 −0.596924 −0.298462 0.954422i $$-0.596474\pi$$
−0.298462 + 0.954422i $$0.596474\pi$$
$$194$$ −12.5181 −0.898744
$$195$$ 1.53948 0.110245
$$196$$ 3.48929 0.249235
$$197$$ −3.17092 −0.225919 −0.112959 0.993600i $$-0.536033\pi$$
−0.112959 + 0.993600i $$0.536033\pi$$
$$198$$ −4.52792 −0.321786
$$199$$ −13.5970 −0.963867 −0.481934 0.876208i $$-0.660065\pi$$
−0.481934 + 0.876208i $$0.660065\pi$$
$$200$$ −11.1537 −0.788687
$$201$$ 6.99327 0.493267
$$202$$ 26.1151 1.83745
$$203$$ −10.4893 −0.736204
$$204$$ −23.3288 −1.63335
$$205$$ 0.393115 0.0274564
$$206$$ −8.00000 −0.557386
$$207$$ 5.35279 0.372045
$$208$$ 1.19656 0.0829663
$$209$$ −3.83221 −0.265080
$$210$$ −3.60688 −0.248899
$$211$$ 9.27552 0.638553 0.319277 0.947662i $$-0.396560\pi$$
0.319277 + 0.947662i $$0.396560\pi$$
$$212$$ −2.72869 −0.187407
$$213$$ −1.76481 −0.120923
$$214$$ 11.6644 0.797364
$$215$$ 10.9498 0.746771
$$216$$ 18.7434 1.27533
$$217$$ −1.63565 −0.111035
$$218$$ −31.6216 −2.14168
$$219$$ 17.5395 1.18521
$$220$$ −5.37169 −0.362159
$$221$$ 5.83221 0.392317
$$222$$ −22.8782 −1.53548
$$223$$ −19.5928 −1.31203 −0.656016 0.754747i $$-0.727760\pi$$
−0.656016 + 0.754747i $$0.727760\pi$$
$$224$$ 4.17513 0.278963
$$225$$ 5.38890 0.359260
$$226$$ 38.5328 2.56316
$$227$$ −19.6644 −1.30517 −0.652587 0.757714i $$-0.726316\pi$$
−0.652587 + 0.757714i $$0.726316\pi$$
$$228$$ 13.3717 0.885562
$$229$$ 7.76481 0.513113 0.256556 0.966529i $$-0.417412\pi$$
0.256556 + 0.966529i $$0.417412\pi$$
$$230$$ 9.99013 0.658730
$$231$$ 1.31415 0.0864650
$$232$$ 36.6002 2.40292
$$233$$ 12.1966 0.799023 0.399512 0.916728i $$-0.369180\pi$$
0.399512 + 0.916728i $$0.369180\pi$$
$$234$$ −3.94981 −0.258207
$$235$$ 14.2541 0.929835
$$236$$ 44.1151 2.87165
$$237$$ −1.01156 −0.0657077
$$238$$ −13.6644 −0.885733
$$239$$ −10.2927 −0.665781 −0.332891 0.942965i $$-0.608024\pi$$
−0.332891 + 0.942965i $$0.608024\pi$$
$$240$$ 1.84208 0.118906
$$241$$ −4.02877 −0.259516 −0.129758 0.991546i $$-0.541420\pi$$
−0.129758 + 0.991546i $$0.541420\pi$$
$$242$$ −22.6932 −1.45877
$$243$$ −14.8536 −0.952861
$$244$$ −6.97858 −0.446758
$$245$$ −1.34292 −0.0857962
$$246$$ 0.786230 0.0501282
$$247$$ −3.34292 −0.212705
$$248$$ 5.70727 0.362412
$$249$$ 13.9044 0.881158
$$250$$ 25.7894 1.63106
$$251$$ 2.91117 0.183752 0.0918758 0.995770i $$-0.470714\pi$$
0.0918758 + 0.995770i $$0.470714\pi$$
$$252$$ 5.88240 0.370557
$$253$$ −3.63986 −0.228836
$$254$$ 28.2499 1.77256
$$255$$ 8.97858 0.562260
$$256$$ −22.9185 −1.43241
$$257$$ −19.5970 −1.22243 −0.611214 0.791465i $$-0.709319\pi$$
−0.611214 + 0.791465i $$0.709319\pi$$
$$258$$ 21.8996 1.36341
$$259$$ −8.51806 −0.529286
$$260$$ −4.68585 −0.290604
$$261$$ −17.6833 −1.09457
$$262$$ −8.58546 −0.530412
$$263$$ 7.56825 0.466678 0.233339 0.972395i $$-0.425035\pi$$
0.233339 + 0.972395i $$0.425035\pi$$
$$264$$ −4.58546 −0.282216
$$265$$ 1.05019 0.0645128
$$266$$ 7.83221 0.480224
$$267$$ −6.57560 −0.402420
$$268$$ −21.2860 −1.30025
$$269$$ −9.47208 −0.577523 −0.288761 0.957401i $$-0.593243\pi$$
−0.288761 + 0.957401i $$0.593243\pi$$
$$270$$ −16.9013 −1.02858
$$271$$ 29.3717 1.78420 0.892102 0.451835i $$-0.149230\pi$$
0.892102 + 0.451835i $$0.149230\pi$$
$$272$$ 6.97858 0.423138
$$273$$ 1.14637 0.0693812
$$274$$ −30.7005 −1.85469
$$275$$ −3.66442 −0.220973
$$276$$ 12.7005 0.764483
$$277$$ −1.90383 −0.114390 −0.0571949 0.998363i $$-0.518216\pi$$
−0.0571949 + 0.998363i $$0.518216\pi$$
$$278$$ 17.5640 1.05342
$$279$$ −2.75746 −0.165085
$$280$$ 4.68585 0.280033
$$281$$ −20.5756 −1.22744 −0.613719 0.789525i $$-0.710327\pi$$
−0.613719 + 0.789525i $$0.710327\pi$$
$$282$$ 28.5082 1.69764
$$283$$ −26.9933 −1.60458 −0.802292 0.596932i $$-0.796386\pi$$
−0.802292 + 0.596932i $$0.796386\pi$$
$$284$$ 5.37169 0.318751
$$285$$ −5.14637 −0.304844
$$286$$ 2.68585 0.158817
$$287$$ 0.292731 0.0172794
$$288$$ 7.03863 0.414756
$$289$$ 17.0147 1.00086
$$290$$ −33.0031 −1.93801
$$291$$ 6.12494 0.359050
$$292$$ −53.3864 −3.12420
$$293$$ −14.9070 −0.870874 −0.435437 0.900219i $$-0.643406\pi$$
−0.435437 + 0.900219i $$0.643406\pi$$
$$294$$ −2.68585 −0.156642
$$295$$ −16.9786 −0.988531
$$296$$ 29.7220 1.72755
$$297$$ 6.15792 0.357319
$$298$$ 5.07896 0.294216
$$299$$ −3.17513 −0.183623
$$300$$ 12.7862 0.738213
$$301$$ 8.15371 0.469972
$$302$$ 34.9357 2.01033
$$303$$ −12.7778 −0.734066
$$304$$ −4.00000 −0.229416
$$305$$ 2.68585 0.153791
$$306$$ −23.0361 −1.31689
$$307$$ 26.0288 1.48554 0.742770 0.669546i $$-0.233512\pi$$
0.742770 + 0.669546i $$0.233512\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 3.91431 0.222677
$$310$$ −5.14637 −0.292294
$$311$$ 19.4966 1.10555 0.552776 0.833330i $$-0.313568\pi$$
0.552776 + 0.833330i $$0.313568\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ −3.48194 −0.196811 −0.0984055 0.995146i $$-0.531374\pi$$
−0.0984055 + 0.995146i $$0.531374\pi$$
$$314$$ 53.5787 3.02362
$$315$$ −2.26396 −0.127560
$$316$$ 3.07896 0.173205
$$317$$ 5.02142 0.282031 0.141016 0.990007i $$-0.454963\pi$$
0.141016 + 0.990007i $$0.454963\pi$$
$$318$$ 2.10038 0.117784
$$319$$ 12.0246 0.673246
$$320$$ 16.3503 0.914008
$$321$$ −5.70727 −0.318549
$$322$$ 7.43910 0.414565
$$323$$ −19.4966 −1.08482
$$324$$ −3.83956 −0.213309
$$325$$ −3.19656 −0.177313
$$326$$ 16.5855 0.918584
$$327$$ 15.4721 0.855608
$$328$$ −1.02142 −0.0563986
$$329$$ 10.6142 0.585182
$$330$$ 4.13481 0.227614
$$331$$ −6.14950 −0.338007 −0.169004 0.985615i $$-0.554055\pi$$
−0.169004 + 0.985615i $$0.554055\pi$$
$$332$$ −42.3221 −2.32273
$$333$$ −14.3601 −0.786931
$$334$$ 6.12494 0.335142
$$335$$ 8.19235 0.447596
$$336$$ 1.37169 0.0748320
$$337$$ −25.6258 −1.39593 −0.697963 0.716134i $$-0.745910\pi$$
−0.697963 + 0.716134i $$0.745910\pi$$
$$338$$ 2.34292 0.127438
$$339$$ −18.8536 −1.02399
$$340$$ −27.3288 −1.48211
$$341$$ 1.87506 0.101540
$$342$$ 13.2039 0.713985
$$343$$ −1.00000 −0.0539949
$$344$$ −28.4507 −1.53396
$$345$$ −4.88806 −0.263164
$$346$$ 25.7795 1.38591
$$347$$ −16.7005 −0.896532 −0.448266 0.893900i $$-0.647958\pi$$
−0.448266 + 0.893900i $$0.647958\pi$$
$$348$$ −41.9572 −2.24914
$$349$$ −23.5500 −1.26060 −0.630300 0.776351i $$-0.717068\pi$$
−0.630300 + 0.776351i $$0.717068\pi$$
$$350$$ 7.48929 0.400319
$$351$$ 5.37169 0.286720
$$352$$ −4.78623 −0.255107
$$353$$ −7.64973 −0.407154 −0.203577 0.979059i $$-0.565257\pi$$
−0.203577 + 0.979059i $$0.565257\pi$$
$$354$$ −33.9572 −1.80480
$$355$$ −2.06740 −0.109726
$$356$$ 20.0147 1.06078
$$357$$ 6.68585 0.353853
$$358$$ 56.1396 2.96707
$$359$$ 18.3748 0.969786 0.484893 0.874573i $$-0.338858\pi$$
0.484893 + 0.874573i $$0.338858\pi$$
$$360$$ 7.89962 0.416346
$$361$$ −7.82487 −0.411835
$$362$$ 15.3717 0.807918
$$363$$ 11.1035 0.582784
$$364$$ −3.48929 −0.182888
$$365$$ 20.5468 1.07547
$$366$$ 5.37169 0.280783
$$367$$ 5.33871 0.278679 0.139339 0.990245i $$-0.455502\pi$$
0.139339 + 0.990245i $$0.455502\pi$$
$$368$$ −3.79923 −0.198049
$$369$$ 0.493499 0.0256906
$$370$$ −26.8009 −1.39331
$$371$$ 0.782020 0.0406004
$$372$$ −6.54262 −0.339219
$$373$$ 21.5212 1.11433 0.557163 0.830403i $$-0.311890\pi$$
0.557163 + 0.830403i $$0.311890\pi$$
$$374$$ 15.6644 0.809988
$$375$$ −12.6184 −0.651614
$$376$$ −37.0361 −1.90999
$$377$$ 10.4893 0.540226
$$378$$ −12.5855 −0.647326
$$379$$ 4.61002 0.236801 0.118400 0.992966i $$-0.462223\pi$$
0.118400 + 0.992966i $$0.462223\pi$$
$$380$$ 15.6644 0.803568
$$381$$ −13.8223 −0.708140
$$382$$ −10.2927 −0.526622
$$383$$ −8.33558 −0.425928 −0.212964 0.977060i $$-0.568312\pi$$
−0.212964 + 0.977060i $$0.568312\pi$$
$$384$$ 23.1281 1.18025
$$385$$ 1.53948 0.0784592
$$386$$ −19.4292 −0.988922
$$387$$ 13.7459 0.698744
$$388$$ −18.6430 −0.946455
$$389$$ −6.44223 −0.326634 −0.163317 0.986574i $$-0.552219\pi$$
−0.163317 + 0.986574i $$0.552219\pi$$
$$390$$ 3.60688 0.182642
$$391$$ −18.5181 −0.936498
$$392$$ 3.48929 0.176236
$$393$$ 4.20077 0.211901
$$394$$ −7.42923 −0.374279
$$395$$ −1.18500 −0.0596238
$$396$$ −6.74338 −0.338868
$$397$$ 1.40046 0.0702872 0.0351436 0.999382i $$-0.488811\pi$$
0.0351436 + 0.999382i $$0.488811\pi$$
$$398$$ −31.8568 −1.59684
$$399$$ −3.83221 −0.191851
$$400$$ −3.82487 −0.191243
$$401$$ −6.97858 −0.348494 −0.174247 0.984702i $$-0.555749\pi$$
−0.174247 + 0.984702i $$0.555749\pi$$
$$402$$ 16.3847 0.817194
$$403$$ 1.63565 0.0814777
$$404$$ 38.8929 1.93499
$$405$$ 1.47773 0.0734291
$$406$$ −24.5756 −1.21967
$$407$$ 9.76481 0.484024
$$408$$ −23.3288 −1.15495
$$409$$ −18.3790 −0.908785 −0.454392 0.890802i $$-0.650144\pi$$
−0.454392 + 0.890802i $$0.650144\pi$$
$$410$$ 0.921039 0.0454869
$$411$$ 15.0214 0.740952
$$412$$ −11.9143 −0.586976
$$413$$ −12.6430 −0.622121
$$414$$ 12.5412 0.616365
$$415$$ 16.2885 0.799572
$$416$$ −4.17513 −0.204703
$$417$$ −8.59388 −0.420844
$$418$$ −8.97858 −0.439157
$$419$$ −30.0393 −1.46751 −0.733757 0.679412i $$-0.762235\pi$$
−0.733757 + 0.679412i $$0.762235\pi$$
$$420$$ −5.37169 −0.262112
$$421$$ −8.31729 −0.405360 −0.202680 0.979245i $$-0.564965\pi$$
−0.202680 + 0.979245i $$0.564965\pi$$
$$422$$ 21.7318 1.05789
$$423$$ 17.8940 0.870034
$$424$$ −2.72869 −0.132517
$$425$$ −18.6430 −0.904318
$$426$$ −4.13481 −0.200332
$$427$$ 2.00000 0.0967868
$$428$$ 17.3717 0.839692
$$429$$ −1.31415 −0.0634479
$$430$$ 25.6546 1.23717
$$431$$ 9.64973 0.464811 0.232406 0.972619i $$-0.425340\pi$$
0.232406 + 0.972619i $$0.425340\pi$$
$$432$$ 6.42754 0.309245
$$433$$ 26.3074 1.26425 0.632127 0.774865i $$-0.282182\pi$$
0.632127 + 0.774865i $$0.282182\pi$$
$$434$$ −3.83221 −0.183952
$$435$$ 16.1481 0.774240
$$436$$ −47.0937 −2.25538
$$437$$ 10.6142 0.507748
$$438$$ 41.0937 1.96353
$$439$$ 33.8139 1.61385 0.806925 0.590654i $$-0.201130\pi$$
0.806925 + 0.590654i $$0.201130\pi$$
$$440$$ −5.37169 −0.256085
$$441$$ −1.68585 −0.0802784
$$442$$ 13.6644 0.649950
$$443$$ 26.4464 1.25651 0.628254 0.778008i $$-0.283770\pi$$
0.628254 + 0.778008i $$0.283770\pi$$
$$444$$ −34.0722 −1.61700
$$445$$ −7.70306 −0.365160
$$446$$ −45.9044 −2.17364
$$447$$ −2.48508 −0.117540
$$448$$ 12.1751 0.575221
$$449$$ 2.64300 0.124731 0.0623655 0.998053i $$-0.480136\pi$$
0.0623655 + 0.998053i $$0.480136\pi$$
$$450$$ 12.6258 0.595185
$$451$$ −0.335577 −0.0158017
$$452$$ 57.3864 2.69923
$$453$$ −17.0937 −0.803130
$$454$$ −46.0722 −2.16228
$$455$$ 1.34292 0.0629572
$$456$$ 13.3717 0.626187
$$457$$ −33.6890 −1.57590 −0.787952 0.615737i $$-0.788859\pi$$
−0.787952 + 0.615737i $$0.788859\pi$$
$$458$$ 18.1923 0.850073
$$459$$ 31.3288 1.46231
$$460$$ 14.8782 0.693699
$$461$$ 33.0790 1.54064 0.770320 0.637657i $$-0.220096\pi$$
0.770320 + 0.637657i $$0.220096\pi$$
$$462$$ 3.07896 0.143246
$$463$$ −2.51806 −0.117024 −0.0585120 0.998287i $$-0.518636\pi$$
−0.0585120 + 0.998287i $$0.518636\pi$$
$$464$$ 12.5510 0.582667
$$465$$ 2.51806 0.116772
$$466$$ 28.5756 1.32374
$$467$$ 2.57560 0.119184 0.0595922 0.998223i $$-0.481020\pi$$
0.0595922 + 0.998223i $$0.481020\pi$$
$$468$$ −5.88240 −0.271914
$$469$$ 6.10038 0.281690
$$470$$ 33.3963 1.54045
$$471$$ −26.2155 −1.20794
$$472$$ 44.1151 2.03056
$$473$$ −9.34713 −0.429782
$$474$$ −2.37000 −0.108858
$$475$$ 10.6858 0.490300
$$476$$ −20.3503 −0.932753
$$477$$ 1.31836 0.0603638
$$478$$ −24.1151 −1.10300
$$479$$ 0.513847 0.0234783 0.0117391 0.999931i $$-0.496263\pi$$
0.0117391 + 0.999931i $$0.496263\pi$$
$$480$$ −6.42754 −0.293376
$$481$$ 8.51806 0.388390
$$482$$ −9.43910 −0.429939
$$483$$ −3.63986 −0.165620
$$484$$ −33.7967 −1.53621
$$485$$ 7.17513 0.325806
$$486$$ −34.8009 −1.57860
$$487$$ 36.0575 1.63392 0.816962 0.576692i $$-0.195657\pi$$
0.816962 + 0.576692i $$0.195657\pi$$
$$488$$ −6.97858 −0.315905
$$489$$ −8.11508 −0.366976
$$490$$ −3.14637 −0.142138
$$491$$ −9.22846 −0.416475 −0.208237 0.978078i $$-0.566773\pi$$
−0.208237 + 0.978078i $$0.566773\pi$$
$$492$$ 1.17092 0.0527893
$$493$$ 61.1758 2.75522
$$494$$ −7.83221 −0.352388
$$495$$ 2.59533 0.116651
$$496$$ 1.95715 0.0878788
$$497$$ −1.53948 −0.0690551
$$498$$ 32.5770 1.45981
$$499$$ 1.00314 0.0449065 0.0224533 0.999748i $$-0.492852\pi$$
0.0224533 + 0.999748i $$0.492852\pi$$
$$500$$ 38.4078 1.71765
$$501$$ −2.99686 −0.133890
$$502$$ 6.82065 0.304421
$$503$$ −30.3503 −1.35325 −0.676626 0.736327i $$-0.736559\pi$$
−0.676626 + 0.736327i $$0.736559\pi$$
$$504$$ 5.88240 0.262023
$$505$$ −14.9687 −0.666099
$$506$$ −8.52792 −0.379112
$$507$$ −1.14637 −0.0509119
$$508$$ 42.0722 1.86665
$$509$$ 10.5995 0.469816 0.234908 0.972018i $$-0.424521\pi$$
0.234908 + 0.972018i $$0.424521\pi$$
$$510$$ 21.0361 0.931495
$$511$$ 15.3001 0.676836
$$512$$ −13.3461 −0.589818
$$513$$ −17.9572 −0.792828
$$514$$ −45.9143 −2.02519
$$515$$ 4.58546 0.202060
$$516$$ 32.6148 1.43579
$$517$$ −12.1678 −0.535139
$$518$$ −19.9572 −0.876867
$$519$$ −12.6136 −0.553676
$$520$$ −4.68585 −0.205488
$$521$$ −16.2646 −0.712564 −0.356282 0.934378i $$-0.615956\pi$$
−0.356282 + 0.934378i $$0.615956\pi$$
$$522$$ −41.4307 −1.81337
$$523$$ 7.22219 0.315804 0.157902 0.987455i $$-0.449527\pi$$
0.157902 + 0.987455i $$0.449527\pi$$
$$524$$ −12.7862 −0.558569
$$525$$ −3.66442 −0.159929
$$526$$ 17.7318 0.773144
$$527$$ 9.53948 0.415546
$$528$$ −1.57246 −0.0684326
$$529$$ −12.9185 −0.561675
$$530$$ 2.46052 0.106878
$$531$$ −21.3142 −0.924955
$$532$$ 11.6644 0.505717
$$533$$ −0.292731 −0.0126796
$$534$$ −15.4061 −0.666688
$$535$$ −6.68585 −0.289054
$$536$$ −21.2860 −0.919415
$$537$$ −27.4685 −1.18535
$$538$$ −22.1923 −0.956780
$$539$$ 1.14637 0.0493775
$$540$$ −25.1709 −1.08318
$$541$$ 17.3534 0.746081 0.373041 0.927815i $$-0.378315\pi$$
0.373041 + 0.927815i $$0.378315\pi$$
$$542$$ 68.8156 2.95588
$$543$$ −7.52119 −0.322765
$$544$$ −24.3503 −1.04401
$$545$$ 18.1249 0.776387
$$546$$ 2.68585 0.114944
$$547$$ −34.1109 −1.45848 −0.729238 0.684261i $$-0.760125\pi$$
−0.729238 + 0.684261i $$0.760125\pi$$
$$548$$ −45.7220 −1.95315
$$549$$ 3.37169 0.143900
$$550$$ −8.58546 −0.366085
$$551$$ −35.0649 −1.49381
$$552$$ 12.7005 0.540571
$$553$$ −0.882404 −0.0375236
$$554$$ −4.46052 −0.189509
$$555$$ 13.1134 0.556632
$$556$$ 26.1579 1.10934
$$557$$ −13.2222 −0.560242 −0.280121 0.959965i $$-0.590375\pi$$
−0.280121 + 0.959965i $$0.590375\pi$$
$$558$$ −6.46052 −0.273496
$$559$$ −8.15371 −0.344865
$$560$$ 1.60688 0.0679033
$$561$$ −7.66442 −0.323592
$$562$$ −48.2070 −2.03349
$$563$$ −41.3717 −1.74361 −0.871804 0.489854i $$-0.837050\pi$$
−0.871804 + 0.489854i $$0.837050\pi$$
$$564$$ 42.4569 1.78776
$$565$$ −22.0863 −0.929178
$$566$$ −63.2432 −2.65831
$$567$$ 1.10038 0.0462118
$$568$$ 5.37169 0.225391
$$569$$ −2.68164 −0.112420 −0.0562100 0.998419i $$-0.517902\pi$$
−0.0562100 + 0.998419i $$0.517902\pi$$
$$570$$ −12.0575 −0.505035
$$571$$ −28.9315 −1.21075 −0.605373 0.795942i $$-0.706976\pi$$
−0.605373 + 0.795942i $$0.706976\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 5.03612 0.210387
$$574$$ 0.685846 0.0286267
$$575$$ 10.1495 0.423263
$$576$$ 20.5254 0.855225
$$577$$ 37.5296 1.56238 0.781189 0.624294i $$-0.214613\pi$$
0.781189 + 0.624294i $$0.214613\pi$$
$$578$$ 39.8641 1.65813
$$579$$ 9.50650 0.395077
$$580$$ −49.1512 −2.04089
$$581$$ 12.1292 0.503202
$$582$$ 14.3503 0.594838
$$583$$ −0.896480 −0.0371284
$$584$$ −53.3864 −2.20914
$$585$$ 2.26396 0.0936033
$$586$$ −34.9259 −1.44277
$$587$$ 23.0649 0.951990 0.475995 0.879448i $$-0.342088\pi$$
0.475995 + 0.879448i $$0.342088\pi$$
$$588$$ −4.00000 −0.164957
$$589$$ −5.46787 −0.225299
$$590$$ −39.7795 −1.63770
$$591$$ 3.63504 0.149525
$$592$$ 10.1923 0.418903
$$593$$ −11.0502 −0.453777 −0.226889 0.973921i $$-0.572855\pi$$
−0.226889 + 0.973921i $$0.572855\pi$$
$$594$$ 14.4275 0.591969
$$595$$ 7.83221 0.321089
$$596$$ 7.56404 0.309835
$$597$$ 15.5872 0.637940
$$598$$ −7.43910 −0.304207
$$599$$ −44.6044 −1.82248 −0.911242 0.411870i $$-0.864876\pi$$
−0.911242 + 0.411870i $$0.864876\pi$$
$$600$$ 12.7862 0.521996
$$601$$ 47.8715 1.95272 0.976359 0.216156i $$-0.0693520\pi$$
0.976359 + 0.216156i $$0.0693520\pi$$
$$602$$ 19.1035 0.778601
$$603$$ 10.2843 0.418809
$$604$$ 52.0294 2.11705
$$605$$ 13.0073 0.528824
$$606$$ −29.9374 −1.21612
$$607$$ 45.0691 1.82930 0.914649 0.404249i $$-0.132467\pi$$
0.914649 + 0.404249i $$0.132467\pi$$
$$608$$ 13.9572 0.566037
$$609$$ 12.0246 0.487260
$$610$$ 6.29273 0.254785
$$611$$ −10.6142 −0.429406
$$612$$ −34.3074 −1.38680
$$613$$ −25.5212 −1.03079 −0.515396 0.856952i $$-0.672355\pi$$
−0.515396 + 0.856952i $$0.672355\pi$$
$$614$$ 60.9834 2.46109
$$615$$ −0.450654 −0.0181721
$$616$$ −4.00000 −0.161165
$$617$$ 29.2432 1.17729 0.588643 0.808393i $$-0.299663\pi$$
0.588643 + 0.808393i $$0.299663\pi$$
$$618$$ 9.17092 0.368909
$$619$$ 4.78623 0.192375 0.0961874 0.995363i $$-0.469335\pi$$
0.0961874 + 0.995363i $$0.469335\pi$$
$$620$$ −7.66442 −0.307811
$$621$$ −17.0558 −0.684428
$$622$$ 45.6791 1.83157
$$623$$ −5.73604 −0.229810
$$624$$ −1.37169 −0.0549116
$$625$$ 1.20077 0.0480307
$$626$$ −8.15792 −0.326056
$$627$$ 4.39312 0.175444
$$628$$ 79.7942 3.18413
$$629$$ 49.6791 1.98084
$$630$$ −5.30429 −0.211328
$$631$$ −28.3931 −1.13031 −0.565156 0.824984i $$-0.691184\pi$$
−0.565156 + 0.824984i $$0.691184\pi$$
$$632$$ 3.07896 0.122475
$$633$$ −10.6331 −0.422629
$$634$$ 11.7648 0.467240
$$635$$ −16.1923 −0.642574
$$636$$ 3.12808 0.124036
$$637$$ 1.00000 0.0396214
$$638$$ 28.1726 1.11536
$$639$$ −2.59533 −0.102670
$$640$$ 27.0937 1.07097
$$641$$ 5.96137 0.235460 0.117730 0.993046i $$-0.462438\pi$$
0.117730 + 0.993046i $$0.462438\pi$$
$$642$$ −13.3717 −0.527739
$$643$$ 31.1940 1.23017 0.615086 0.788460i $$-0.289121\pi$$
0.615086 + 0.788460i $$0.289121\pi$$
$$644$$ 11.0790 0.436572
$$645$$ −12.5525 −0.494253
$$646$$ −45.6791 −1.79722
$$647$$ 14.9112 0.586219 0.293109 0.956079i $$-0.405310\pi$$
0.293109 + 0.956079i $$0.405310\pi$$
$$648$$ −3.83956 −0.150832
$$649$$ 14.4935 0.568920
$$650$$ −7.48929 −0.293754
$$651$$ 1.87506 0.0734893
$$652$$ 24.7005 0.967348
$$653$$ −3.57246 −0.139801 −0.0699006 0.997554i $$-0.522268\pi$$
−0.0699006 + 0.997554i $$0.522268\pi$$
$$654$$ 36.2499 1.41748
$$655$$ 4.92104 0.192281
$$656$$ −0.350269 −0.0136757
$$657$$ 25.7936 1.00630
$$658$$ 24.8683 0.969468
$$659$$ −3.90383 −0.152071 −0.0760357 0.997105i $$-0.524226\pi$$
−0.0760357 + 0.997105i $$0.524226\pi$$
$$660$$ 6.15792 0.239697
$$661$$ 13.7936 0.536508 0.268254 0.963348i $$-0.413553\pi$$
0.268254 + 0.963348i $$0.413553\pi$$
$$662$$ −14.4078 −0.559975
$$663$$ −6.68585 −0.259657
$$664$$ −42.3221 −1.64242
$$665$$ −4.48929 −0.174087
$$666$$ −33.6447 −1.30371
$$667$$ −33.3049 −1.28957
$$668$$ 9.12181 0.352933
$$669$$ 22.4605 0.868374
$$670$$ 19.1940 0.741530
$$671$$ −2.29273 −0.0885099
$$672$$ −4.78623 −0.184633
$$673$$ 5.70306 0.219837 0.109918 0.993941i $$-0.464941\pi$$
0.109918 + 0.993941i $$0.464941\pi$$
$$674$$ −60.0393 −2.31263
$$675$$ −17.1709 −0.660909
$$676$$ 3.48929 0.134203
$$677$$ 35.2614 1.35521 0.677604 0.735427i $$-0.263019\pi$$
0.677604 + 0.735427i $$0.263019\pi$$
$$678$$ −44.1726 −1.69644
$$679$$ 5.34292 0.205043
$$680$$ −27.3288 −1.04801
$$681$$ 22.5426 0.863835
$$682$$ 4.39312 0.168221
$$683$$ 1.03612 0.0396459 0.0198229 0.999804i $$-0.493690\pi$$
0.0198229 + 0.999804i $$0.493690\pi$$
$$684$$ 19.6644 0.751888
$$685$$ 17.5970 0.672348
$$686$$ −2.34292 −0.0894532
$$687$$ −8.90131 −0.339606
$$688$$ −9.75639 −0.371959
$$689$$ −0.782020 −0.0297926
$$690$$ −11.4523 −0.435983
$$691$$ 3.67850 0.139937 0.0699684 0.997549i $$-0.477710\pi$$
0.0699684 + 0.997549i $$0.477710\pi$$
$$692$$ 38.3931 1.45949
$$693$$ 1.93260 0.0734132
$$694$$ −39.1281 −1.48528
$$695$$ −10.0674 −0.381878
$$696$$ −41.9572 −1.59038
$$697$$ −1.70727 −0.0646674
$$698$$ −55.1758 −2.08843
$$699$$ −13.9817 −0.528837
$$700$$ 11.1537 0.421571
$$701$$ −0.0617493 −0.00233224 −0.00116612 0.999999i $$-0.500371\pi$$
−0.00116612 + 0.999999i $$0.500371\pi$$
$$702$$ 12.5855 0.475008
$$703$$ −28.4752 −1.07396
$$704$$ −13.9572 −0.526030
$$705$$ −16.3404 −0.615415
$$706$$ −17.9227 −0.674531
$$707$$ −11.1464 −0.419202
$$708$$ −50.5720 −1.90061
$$709$$ −42.9834 −1.61428 −0.807138 0.590363i $$-0.798985\pi$$
−0.807138 + 0.590363i $$0.798985\pi$$
$$710$$ −4.84377 −0.181783
$$711$$ −1.48760 −0.0557892
$$712$$ 20.0147 0.750082
$$713$$ −5.19342 −0.194495
$$714$$ 15.6644 0.586226
$$715$$ −1.53948 −0.0575733
$$716$$ 83.6081 3.12458
$$717$$ 11.7992 0.440650
$$718$$ 43.0508 1.60664
$$719$$ 17.6546 0.658404 0.329202 0.944260i $$-0.393220\pi$$
0.329202 + 0.944260i $$0.393220\pi$$
$$720$$ 2.70896 0.100957
$$721$$ 3.41454 0.127164
$$722$$ −18.3331 −0.682286
$$723$$ 4.61844 0.171762
$$724$$ 22.8929 0.850807
$$725$$ −33.5296 −1.24526
$$726$$ 26.0147 0.965496
$$727$$ −23.8077 −0.882977 −0.441488 0.897267i $$-0.645549\pi$$
−0.441488 + 0.897267i $$0.645549\pi$$
$$728$$ −3.48929 −0.129322
$$729$$ 20.3288 0.752920
$$730$$ 48.1396 1.78173
$$731$$ −47.5542 −1.75885
$$732$$ 8.00000 0.295689
$$733$$ 31.3492 1.15791 0.578954 0.815360i $$-0.303461\pi$$
0.578954 + 0.815360i $$0.303461\pi$$
$$734$$ 12.5082 0.461686
$$735$$ 1.53948 0.0567846
$$736$$ 13.2566 0.488645
$$737$$ −6.99327 −0.257600
$$738$$ 1.15623 0.0425615
$$739$$ −24.8108 −0.912680 −0.456340 0.889806i $$-0.650840\pi$$
−0.456340 + 0.889806i $$0.650840\pi$$
$$740$$ −39.9143 −1.46728
$$741$$ 3.83221 0.140780
$$742$$ 1.83221 0.0672626
$$743$$ 21.3717 0.784051 0.392026 0.919954i $$-0.371774\pi$$
0.392026 + 0.919954i $$0.371774\pi$$
$$744$$ −6.54262 −0.239864
$$745$$ −2.91117 −0.106657
$$746$$ 50.4225 1.84610
$$747$$ 20.4479 0.748149
$$748$$ 23.3288 0.852987
$$749$$ −4.97858 −0.181913
$$750$$ −29.5640 −1.07953
$$751$$ 19.2243 0.701503 0.350751 0.936469i $$-0.385926\pi$$
0.350751 + 0.936469i $$0.385926\pi$$
$$752$$ −12.7005 −0.463141
$$753$$ −3.33727 −0.121617
$$754$$ 24.5756 0.894990
$$755$$ −20.0246 −0.728768
$$756$$ −18.7434 −0.681690
$$757$$ −19.8610 −0.721860 −0.360930 0.932593i $$-0.617541\pi$$
−0.360930 + 0.932593i $$0.617541\pi$$
$$758$$ 10.8009 0.392307
$$759$$ 4.17262 0.151456
$$760$$ 15.6644 0.568208
$$761$$ 6.12073 0.221876 0.110938 0.993827i $$-0.464614\pi$$
0.110938 + 0.993827i $$0.464614\pi$$
$$762$$ −32.3847 −1.17317
$$763$$ 13.4966 0.488611
$$764$$ −15.3288 −0.554578
$$765$$ 13.2039 0.477388
$$766$$ −19.5296 −0.705634
$$767$$ 12.6430 0.456512
$$768$$ 26.2730 0.948045
$$769$$ 3.82800 0.138041 0.0690206 0.997615i $$-0.478013\pi$$
0.0690206 + 0.997615i $$0.478013\pi$$
$$770$$ 3.60688 0.129983
$$771$$ 22.4653 0.809070
$$772$$ −28.9357 −1.04142
$$773$$ −6.53635 −0.235096 −0.117548 0.993067i $$-0.537503\pi$$
−0.117548 + 0.993067i $$0.537503\pi$$
$$774$$ 32.2056 1.15761
$$775$$ −5.22846 −0.187812
$$776$$ −18.6430 −0.669245
$$777$$ 9.76481 0.350311
$$778$$ −15.0937 −0.541134
$$779$$ 0.978577 0.0350612
$$780$$ 5.37169 0.192337
$$781$$ 1.76481 0.0631498
$$782$$ −43.3864 −1.55149
$$783$$ 56.3452 2.01361
$$784$$ 1.19656 0.0427342
$$785$$ −30.7104 −1.09610
$$786$$ 9.84208 0.351055
$$787$$ 30.8066 1.09814 0.549068 0.835778i $$-0.314983\pi$$
0.549068 + 0.835778i $$0.314983\pi$$
$$788$$ −11.0643 −0.394148
$$789$$ −8.67598 −0.308873
$$790$$ −2.77636 −0.0987786
$$791$$ −16.4464 −0.584768
$$792$$ −6.74338 −0.239616
$$793$$ −2.00000 −0.0710221
$$794$$ 3.28117 0.116444
$$795$$ −1.20390 −0.0426981
$$796$$ −47.4439 −1.68161
$$797$$ −38.8156 −1.37492 −0.687460 0.726222i $$-0.741274\pi$$
−0.687460 + 0.726222i $$0.741274\pi$$
$$798$$ −8.97858 −0.317838
$$799$$ −61.9044 −2.19002
$$800$$ 13.3461 0.471854
$$801$$ −9.67008 −0.341675
$$802$$ −16.3503 −0.577348
$$803$$ −17.5395 −0.618955
$$804$$ 24.4015 0.860576
$$805$$ −4.26396 −0.150285
$$806$$ 3.83221 0.134984
$$807$$ 10.8585 0.382236
$$808$$ 38.8929 1.36825
$$809$$ 1.04033 0.0365759 0.0182880 0.999833i $$-0.494178\pi$$
0.0182880 + 0.999833i $$0.494178\pi$$
$$810$$ 3.46221 0.121650
$$811$$ 12.6712 0.444944 0.222472 0.974939i $$-0.428587\pi$$
0.222472 + 0.974939i $$0.428587\pi$$
$$812$$ −36.6002 −1.28441
$$813$$ −33.6707 −1.18088
$$814$$ 22.8782 0.801880
$$815$$ −9.50650 −0.332998
$$816$$ −8.00000 −0.280056
$$817$$ 27.2572 0.953610
$$818$$ −43.0607 −1.50558
$$819$$ 1.68585 0.0589082
$$820$$ 1.37169 0.0479016
$$821$$ 27.0361 0.943567 0.471783 0.881714i $$-0.343610\pi$$
0.471783 + 0.881714i $$0.343610\pi$$
$$822$$ 35.1940 1.22753
$$823$$ −31.6363 −1.10277 −0.551386 0.834251i $$-0.685901\pi$$
−0.551386 + 0.834251i $$0.685901\pi$$
$$824$$ −11.9143 −0.415055
$$825$$ 4.20077 0.146252
$$826$$ −29.6216 −1.03067
$$827$$ 56.4800 1.96400 0.982002 0.188872i $$-0.0604832\pi$$
0.982002 + 0.188872i $$0.0604832\pi$$
$$828$$ 18.6774 0.649085
$$829$$ −42.6760 −1.48220 −0.741099 0.671396i $$-0.765695\pi$$
−0.741099 + 0.671396i $$0.765695\pi$$
$$830$$ 38.1627 1.32465
$$831$$ 2.18248 0.0757094
$$832$$ −12.1751 −0.422097
$$833$$ 5.83221 0.202074
$$834$$ −20.1348 −0.697211
$$835$$ −3.51071 −0.121493
$$836$$ −13.3717 −0.462470
$$837$$ 8.78623 0.303697
$$838$$ −70.3797 −2.43122
$$839$$ −40.1642 −1.38662 −0.693311 0.720639i $$-0.743849\pi$$
−0.693311 + 0.720639i $$0.743849\pi$$
$$840$$ −5.37169 −0.185341
$$841$$ 81.0252 2.79397
$$842$$ −19.4868 −0.671558
$$843$$ 23.5872 0.812385
$$844$$ 32.3650 1.11405
$$845$$ −1.34292 −0.0461980
$$846$$ 41.9242 1.44138
$$847$$ 9.68585 0.332810
$$848$$ −0.935731 −0.0321331
$$849$$ 30.9442 1.06200
$$850$$ −43.6791 −1.49818
$$851$$ −27.0460 −0.927124
$$852$$ −6.15792 −0.210967
$$853$$ 19.6932 0.674282 0.337141 0.941454i $$-0.390540\pi$$
0.337141 + 0.941454i $$0.390540\pi$$
$$854$$ 4.68585 0.160346
$$855$$ −7.56825 −0.258829
$$856$$ 17.3717 0.593752
$$857$$ 1.66442 0.0568556 0.0284278 0.999596i $$-0.490950\pi$$
0.0284278 + 0.999596i $$0.490950\pi$$
$$858$$ −3.07896 −0.105114
$$859$$ −41.2944 −1.40895 −0.704474 0.709730i $$-0.748817\pi$$
−0.704474 + 0.709730i $$0.748817\pi$$
$$860$$ 38.2070 1.30285
$$861$$ −0.335577 −0.0114364
$$862$$ 22.6086 0.770051
$$863$$ −31.3288 −1.06645 −0.533223 0.845975i $$-0.679019\pi$$
−0.533223 + 0.845975i $$0.679019\pi$$
$$864$$ −22.4275 −0.763000
$$865$$ −14.7764 −0.502411
$$866$$ 61.6363 2.09449
$$867$$ −19.5051 −0.662426
$$868$$ −5.70727 −0.193717
$$869$$ 1.01156 0.0343147
$$870$$ 37.8337 1.28268
$$871$$ −6.10038 −0.206704
$$872$$ −47.0937 −1.59479
$$873$$ 9.00735 0.304852
$$874$$ 24.8683 0.841184
$$875$$ −11.0073 −0.372116
$$876$$ 61.2003 2.06777
$$877$$ −53.8041 −1.81683 −0.908417 0.418065i $$-0.862708\pi$$
−0.908417 + 0.418065i $$0.862708\pi$$
$$878$$ 79.2234 2.67366
$$879$$ 17.0888 0.576392
$$880$$ −1.84208 −0.0620964
$$881$$ 8.09196 0.272625 0.136313 0.990666i $$-0.456475\pi$$
0.136313 + 0.990666i $$0.456475\pi$$
$$882$$ −3.94981 −0.132997
$$883$$ −27.0705 −0.910996 −0.455498 0.890237i $$-0.650539\pi$$
−0.455498 + 0.890237i $$0.650539\pi$$
$$884$$ 20.3503 0.684454
$$885$$ 19.4637 0.654264
$$886$$ 61.9620 2.08165
$$887$$ −38.4935 −1.29249 −0.646243 0.763132i $$-0.723661\pi$$
−0.646243 + 0.763132i $$0.723661\pi$$
$$888$$ −34.0722 −1.14339
$$889$$ −12.0575 −0.404397
$$890$$ −18.0477 −0.604959
$$891$$ −1.26144 −0.0422599
$$892$$ −68.3650 −2.28903
$$893$$ 35.4826 1.18738
$$894$$ −5.82235 −0.194728
$$895$$ −32.1783 −1.07560
$$896$$ 20.1751 0.674004
$$897$$ 3.63986 0.121532
$$898$$ 6.19235 0.206641
$$899$$ 17.1568 0.572213
$$900$$ 18.8034 0.626781
$$901$$ −4.56090 −0.151946
$$902$$ −0.786230 −0.0261786
$$903$$ −9.34713 −0.311053
$$904$$ 57.3864 1.90864
$$905$$ −8.81079 −0.292881
$$906$$ −40.0491 −1.33054
$$907$$ 15.7031 0.521411 0.260706 0.965418i $$-0.416045\pi$$
0.260706 + 0.965418i $$0.416045\pi$$
$$908$$ −68.6148 −2.27706
$$909$$ −18.7911 −0.623260
$$910$$ 3.14637 0.104301
$$911$$ 44.9399 1.48893 0.744463 0.667663i $$-0.232705\pi$$
0.744463 + 0.667663i $$0.232705\pi$$
$$912$$ 4.58546 0.151840
$$913$$ −13.9044 −0.460170
$$914$$ −78.9307 −2.61080
$$915$$ −3.07896 −0.101787
$$916$$ 27.0937 0.895200
$$917$$ 3.66442 0.121010
$$918$$ 73.4011 2.42260
$$919$$ −27.2432 −0.898669 −0.449334 0.893364i $$-0.648339\pi$$
−0.449334 + 0.893364i $$0.648339\pi$$
$$920$$ 14.8782 0.490519
$$921$$ −29.8385 −0.983211
$$922$$ 77.5015 2.55237
$$923$$ 1.53948 0.0506726
$$924$$ 4.58546 0.150851
$$925$$ −27.2285 −0.895266
$$926$$ −5.89962 −0.193873
$$927$$ 5.75639 0.189065
$$928$$ −43.7942 −1.43761
$$929$$ −17.1422 −0.562416 −0.281208 0.959647i $$-0.590735\pi$$
−0.281208 + 0.959647i $$0.590735\pi$$
$$930$$ 5.89962 0.193456
$$931$$ −3.34292 −0.109560
$$932$$ 42.5573 1.39401
$$933$$ −22.3503 −0.731715
$$934$$ 6.03442 0.197452
$$935$$ −8.97858 −0.293631
$$936$$ −5.88240 −0.192272
$$937$$ −51.5197 −1.68308 −0.841538 0.540197i $$-0.818350\pi$$
−0.841538 + 0.540197i $$0.818350\pi$$
$$938$$ 14.2927 0.466674
$$939$$ 3.99158 0.130260
$$940$$ 49.7367 1.62223
$$941$$ 32.7575 1.06786 0.533931 0.845528i $$-0.320714\pi$$
0.533931 + 0.845528i $$0.320714\pi$$
$$942$$ −61.4208 −2.00120
$$943$$ 0.929460 0.0302674
$$944$$ 15.1281 0.492377
$$945$$ 7.21377 0.234664
$$946$$ −21.8996 −0.712018
$$947$$ 20.9295 0.680116 0.340058 0.940404i $$-0.389553\pi$$
0.340058 + 0.940404i $$0.389553\pi$$
$$948$$ −3.52962 −0.114637
$$949$$ −15.3001 −0.496662
$$950$$ 25.0361 0.812279
$$951$$ −5.75639 −0.186664
$$952$$ −20.3503 −0.659556
$$953$$ 46.4120 1.50343 0.751716 0.659487i $$-0.229226\pi$$
0.751716 + 0.659487i $$0.229226\pi$$
$$954$$ 3.08883 0.100004
$$955$$ 5.89962 0.190907
$$956$$ −35.9143 −1.16155
$$957$$ −13.7845 −0.445591
$$958$$ 1.20390 0.0388964
$$959$$ 13.1035 0.423135
$$960$$ −18.7434 −0.604940
$$961$$ −28.3246 −0.913698
$$962$$ 19.9572 0.643444
$$963$$ −8.39312 −0.270464
$$964$$ −14.0575 −0.452763
$$965$$ 11.1365 0.358497
$$966$$ −8.52792 −0.274381
$$967$$ −23.2186 −0.746660 −0.373330 0.927699i $$-0.621784\pi$$
−0.373330 + 0.927699i $$0.621784\pi$$
$$968$$ −33.7967 −1.08627
$$969$$ 22.3503 0.717994
$$970$$ 16.8108 0.539762
$$971$$ −14.1004 −0.452503 −0.226251 0.974069i $$-0.572647\pi$$
−0.226251 + 0.974069i $$0.572647\pi$$
$$972$$ −51.8286 −1.66240
$$973$$ −7.49663 −0.240331
$$974$$ 84.4800 2.70692
$$975$$ 3.66442 0.117355
$$976$$ −2.39312 −0.0766018
$$977$$ −2.95402 −0.0945074 −0.0472537 0.998883i $$-0.515047\pi$$
−0.0472537 + 0.998883i $$0.515047\pi$$
$$978$$ −19.0130 −0.607969
$$979$$ 6.57560 0.210157
$$980$$ −4.68585 −0.149684
$$981$$ 22.7533 0.726455
$$982$$ −21.6216 −0.689972
$$983$$ −35.0367 −1.11750 −0.558749 0.829337i $$-0.688719\pi$$
−0.558749 + 0.829337i $$0.688719\pi$$
$$984$$ 1.17092 0.0373277
$$985$$ 4.25831 0.135681
$$986$$ 143.330 4.56456
$$987$$ −12.1678 −0.387305
$$988$$ −11.6644 −0.371095
$$989$$ 25.8891 0.823227
$$990$$ 6.08065 0.193256
$$991$$ 47.9718 1.52388 0.761938 0.647650i $$-0.224248\pi$$
0.761938 + 0.647650i $$0.224248\pi$$
$$992$$ −6.82908 −0.216823
$$993$$ 7.04958 0.223712
$$994$$ −3.60688 −0.114403
$$995$$ 18.2598 0.578873
$$996$$ 48.5166 1.53731
$$997$$ −38.4422 −1.21748 −0.608739 0.793371i $$-0.708324\pi$$
−0.608739 + 0.793371i $$0.708324\pi$$
$$998$$ 2.35027 0.0743965
$$999$$ 45.7564 1.44767
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 91.2.a.d.1.3 3
3.2 odd 2 819.2.a.i.1.1 3
4.3 odd 2 1456.2.a.t.1.2 3
5.4 even 2 2275.2.a.m.1.1 3
7.2 even 3 637.2.e.j.508.1 6
7.3 odd 6 637.2.e.i.79.1 6
7.4 even 3 637.2.e.j.79.1 6
7.5 odd 6 637.2.e.i.508.1 6
7.6 odd 2 637.2.a.j.1.3 3
8.3 odd 2 5824.2.a.bs.1.2 3
8.5 even 2 5824.2.a.by.1.2 3
13.5 odd 4 1183.2.c.f.337.1 6
13.8 odd 4 1183.2.c.f.337.6 6
13.12 even 2 1183.2.a.i.1.1 3
21.20 even 2 5733.2.a.x.1.1 3
91.90 odd 2 8281.2.a.bg.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.3 3 1.1 even 1 trivial
637.2.a.j.1.3 3 7.6 odd 2
637.2.e.i.79.1 6 7.3 odd 6
637.2.e.i.508.1 6 7.5 odd 6
637.2.e.j.79.1 6 7.4 even 3
637.2.e.j.508.1 6 7.2 even 3
819.2.a.i.1.1 3 3.2 odd 2
1183.2.a.i.1.1 3 13.12 even 2
1183.2.c.f.337.1 6 13.5 odd 4
1183.2.c.f.337.6 6 13.8 odd 4
1456.2.a.t.1.2 3 4.3 odd 2
2275.2.a.m.1.1 3 5.4 even 2
5733.2.a.x.1.1 3 21.20 even 2
5824.2.a.bs.1.2 3 8.3 odd 2
5824.2.a.by.1.2 3 8.5 even 2
8281.2.a.bg.1.1 3 91.90 odd 2