# Properties

 Label 4002.2.a.bb.1.1 Level $4002$ Weight $2$ Character 4002.1 Self dual yes Analytic conductor $31.956$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4002 = 2 \cdot 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4002.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: $$31.9561308889$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.23252.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 2$$ x^4 - x^3 - 6*x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0.565882$$ of defining polynomial Character $$\chi$$ $$=$$ 4002.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.96842 q^{5} -1.00000 q^{6} +2.40254 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.96842 q^{5} -1.00000 q^{6} +2.40254 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.96842 q^{10} -2.43412 q^{11} +1.00000 q^{12} +2.67978 q^{13} -2.40254 q^{14} -3.96842 q^{15} +1.00000 q^{16} -2.40254 q^{17} -1.00000 q^{18} -0.957013 q^{19} -3.96842 q^{20} +2.40254 q^{21} +2.43412 q^{22} +1.00000 q^{23} -1.00000 q^{24} +10.7484 q^{25} -2.67978 q^{26} +1.00000 q^{27} +2.40254 q^{28} -1.00000 q^{29} +3.96842 q^{30} -7.81154 q^{31} -1.00000 q^{32} -2.43412 q^{33} +2.40254 q^{34} -9.53431 q^{35} +1.00000 q^{36} +11.1890 q^{37} +0.957013 q^{38} +2.67978 q^{39} +3.96842 q^{40} +4.38883 q^{41} -2.40254 q^{42} -7.33214 q^{43} -2.43412 q^{44} -3.96842 q^{45} -1.00000 q^{46} -2.08878 q^{47} +1.00000 q^{48} -1.22779 q^{49} -10.7484 q^{50} -2.40254 q^{51} +2.67978 q^{52} +11.1574 q^{53} -1.00000 q^{54} +9.65961 q^{55} -2.40254 q^{56} -0.957013 q^{57} +1.00000 q^{58} -5.54801 q^{59} -3.96842 q^{60} -3.56588 q^{61} +7.81154 q^{62} +2.40254 q^{63} +1.00000 q^{64} -10.6345 q^{65} +2.43412 q^{66} +13.5284 q^{67} -2.40254 q^{68} +1.00000 q^{69} +9.53431 q^{70} -14.3257 q^{71} -1.00000 q^{72} -15.4712 q^{73} -11.1890 q^{74} +10.7484 q^{75} -0.957013 q^{76} -5.84807 q^{77} -2.67978 q^{78} -10.6661 q^{79} -3.96842 q^{80} +1.00000 q^{81} -4.38883 q^{82} -6.81800 q^{83} +2.40254 q^{84} +9.53431 q^{85} +7.33214 q^{86} -1.00000 q^{87} +2.43412 q^{88} +10.5169 q^{89} +3.96842 q^{90} +6.43828 q^{91} +1.00000 q^{92} -7.81154 q^{93} +2.08878 q^{94} +3.79783 q^{95} -1.00000 q^{96} +14.5169 q^{97} +1.22779 q^{98} -2.43412 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 $$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 3 q^{10} - 11 q^{11} + 4 q^{12} - q^{13} + 2 q^{14} - 3 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 8 q^{19} - 3 q^{20} - 2 q^{21} + 11 q^{22} + 4 q^{23} - 4 q^{24} + 3 q^{25} + q^{26} + 4 q^{27} - 2 q^{28} - 4 q^{29} + 3 q^{30} - 17 q^{31} - 4 q^{32} - 11 q^{33} - 2 q^{34} - 24 q^{35} + 4 q^{36} + 15 q^{37} - 8 q^{38} - q^{39} + 3 q^{40} + q^{41} + 2 q^{42} + 4 q^{43} - 11 q^{44} - 3 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} + 16 q^{49} - 3 q^{50} + 2 q^{51} - q^{52} + 2 q^{53} - 4 q^{54} + 13 q^{55} + 2 q^{56} + 8 q^{57} + 4 q^{58} - 13 q^{59} - 3 q^{60} - 13 q^{61} + 17 q^{62} - 2 q^{63} + 4 q^{64} - 13 q^{65} + 11 q^{66} - 13 q^{67} + 2 q^{68} + 4 q^{69} + 24 q^{70} - 15 q^{71} - 4 q^{72} - 22 q^{73} - 15 q^{74} + 3 q^{75} + 8 q^{76} - 12 q^{77} + q^{78} - 26 q^{79} - 3 q^{80} + 4 q^{81} - q^{82} - 22 q^{83} - 2 q^{84} + 24 q^{85} - 4 q^{86} - 4 q^{87} + 11 q^{88} - 24 q^{89} + 3 q^{90} + 30 q^{91} + 4 q^{92} - 17 q^{93} - 6 q^{94} - 4 q^{95} - 4 q^{96} - 8 q^{97} - 16 q^{98} - 11 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 + 3 * q^10 - 11 * q^11 + 4 * q^12 - q^13 + 2 * q^14 - 3 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 8 * q^19 - 3 * q^20 - 2 * q^21 + 11 * q^22 + 4 * q^23 - 4 * q^24 + 3 * q^25 + q^26 + 4 * q^27 - 2 * q^28 - 4 * q^29 + 3 * q^30 - 17 * q^31 - 4 * q^32 - 11 * q^33 - 2 * q^34 - 24 * q^35 + 4 * q^36 + 15 * q^37 - 8 * q^38 - q^39 + 3 * q^40 + q^41 + 2 * q^42 + 4 * q^43 - 11 * q^44 - 3 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 + 16 * q^49 - 3 * q^50 + 2 * q^51 - q^52 + 2 * q^53 - 4 * q^54 + 13 * q^55 + 2 * q^56 + 8 * q^57 + 4 * q^58 - 13 * q^59 - 3 * q^60 - 13 * q^61 + 17 * q^62 - 2 * q^63 + 4 * q^64 - 13 * q^65 + 11 * q^66 - 13 * q^67 + 2 * q^68 + 4 * q^69 + 24 * q^70 - 15 * q^71 - 4 * q^72 - 22 * q^73 - 15 * q^74 + 3 * q^75 + 8 * q^76 - 12 * q^77 + q^78 - 26 * q^79 - 3 * q^80 + 4 * q^81 - q^82 - 22 * q^83 - 2 * q^84 + 24 * q^85 - 4 * q^86 - 4 * q^87 + 11 * q^88 - 24 * q^89 + 3 * q^90 + 30 * q^91 + 4 * q^92 - 17 * q^93 - 6 * q^94 - 4 * q^95 - 4 * q^96 - 8 * q^97 - 16 * q^98 - 11 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −3.96842 −1.77473 −0.887367 0.461065i $$-0.847468\pi$$
−0.887367 + 0.461065i $$0.847468\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 2.40254 0.908076 0.454038 0.890982i $$-0.349983\pi$$
0.454038 + 0.890982i $$0.349983\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 3.96842 1.25493
$$11$$ −2.43412 −0.733914 −0.366957 0.930238i $$-0.619600\pi$$
−0.366957 + 0.930238i $$0.619600\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.67978 0.743237 0.371618 0.928386i $$-0.378803\pi$$
0.371618 + 0.928386i $$0.378803\pi$$
$$14$$ −2.40254 −0.642106
$$15$$ −3.96842 −1.02464
$$16$$ 1.00000 0.250000
$$17$$ −2.40254 −0.582702 −0.291351 0.956616i $$-0.594105\pi$$
−0.291351 + 0.956616i $$0.594105\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −0.957013 −0.219554 −0.109777 0.993956i $$-0.535014\pi$$
−0.109777 + 0.993956i $$0.535014\pi$$
$$20$$ −3.96842 −0.887367
$$21$$ 2.40254 0.524278
$$22$$ 2.43412 0.518956
$$23$$ 1.00000 0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 10.7484 2.14968
$$26$$ −2.67978 −0.525548
$$27$$ 1.00000 0.192450
$$28$$ 2.40254 0.454038
$$29$$ −1.00000 −0.185695
$$30$$ 3.96842 0.724532
$$31$$ −7.81154 −1.40299 −0.701497 0.712672i $$-0.747485\pi$$
−0.701497 + 0.712672i $$0.747485\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −2.43412 −0.423726
$$34$$ 2.40254 0.412033
$$35$$ −9.53431 −1.61159
$$36$$ 1.00000 0.166667
$$37$$ 11.1890 1.83945 0.919727 0.392558i $$-0.128410\pi$$
0.919727 + 0.392558i $$0.128410\pi$$
$$38$$ 0.957013 0.155248
$$39$$ 2.67978 0.429108
$$40$$ 3.96842 0.627463
$$41$$ 4.38883 0.685421 0.342710 0.939441i $$-0.388655\pi$$
0.342710 + 0.939441i $$0.388655\pi$$
$$42$$ −2.40254 −0.370720
$$43$$ −7.33214 −1.11814 −0.559070 0.829120i $$-0.688842\pi$$
−0.559070 + 0.829120i $$0.688842\pi$$
$$44$$ −2.43412 −0.366957
$$45$$ −3.96842 −0.591578
$$46$$ −1.00000 −0.147442
$$47$$ −2.08878 −0.304679 −0.152340 0.988328i $$-0.548681\pi$$
−0.152340 + 0.988328i $$0.548681\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −1.22779 −0.175399
$$50$$ −10.7484 −1.52005
$$51$$ −2.40254 −0.336423
$$52$$ 2.67978 0.371618
$$53$$ 11.1574 1.53259 0.766293 0.642492i $$-0.222099\pi$$
0.766293 + 0.642492i $$0.222099\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 9.65961 1.30250
$$56$$ −2.40254 −0.321053
$$57$$ −0.957013 −0.126759
$$58$$ 1.00000 0.131306
$$59$$ −5.54801 −0.722290 −0.361145 0.932510i $$-0.617614\pi$$
−0.361145 + 0.932510i $$0.617614\pi$$
$$60$$ −3.96842 −0.512321
$$61$$ −3.56588 −0.456564 −0.228282 0.973595i $$-0.573311\pi$$
−0.228282 + 0.973595i $$0.573311\pi$$
$$62$$ 7.81154 0.992067
$$63$$ 2.40254 0.302692
$$64$$ 1.00000 0.125000
$$65$$ −10.6345 −1.31905
$$66$$ 2.43412 0.299619
$$67$$ 13.5284 1.65275 0.826376 0.563119i $$-0.190399\pi$$
0.826376 + 0.563119i $$0.190399\pi$$
$$68$$ −2.40254 −0.291351
$$69$$ 1.00000 0.120386
$$70$$ 9.53431 1.13957
$$71$$ −14.3257 −1.70015 −0.850073 0.526665i $$-0.823442\pi$$
−0.850073 + 0.526665i $$0.823442\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −15.4712 −1.81076 −0.905381 0.424600i $$-0.860415\pi$$
−0.905381 + 0.424600i $$0.860415\pi$$
$$74$$ −11.1890 −1.30069
$$75$$ 10.7484 1.24112
$$76$$ −0.957013 −0.109777
$$77$$ −5.84807 −0.666450
$$78$$ −2.67978 −0.303425
$$79$$ −10.6661 −1.20003 −0.600013 0.799990i $$-0.704838\pi$$
−0.600013 + 0.799990i $$0.704838\pi$$
$$80$$ −3.96842 −0.443683
$$81$$ 1.00000 0.111111
$$82$$ −4.38883 −0.484666
$$83$$ −6.81800 −0.748373 −0.374186 0.927354i $$-0.622078\pi$$
−0.374186 + 0.927354i $$0.622078\pi$$
$$84$$ 2.40254 0.262139
$$85$$ 9.53431 1.03414
$$86$$ 7.33214 0.790645
$$87$$ −1.00000 −0.107211
$$88$$ 2.43412 0.259478
$$89$$ 10.5169 1.11479 0.557397 0.830246i $$-0.311800\pi$$
0.557397 + 0.830246i $$0.311800\pi$$
$$90$$ 3.96842 0.418309
$$91$$ 6.43828 0.674915
$$92$$ 1.00000 0.104257
$$93$$ −7.81154 −0.810019
$$94$$ 2.08878 0.215441
$$95$$ 3.79783 0.389650
$$96$$ −1.00000 −0.102062
$$97$$ 14.5169 1.47397 0.736986 0.675908i $$-0.236248\pi$$
0.736986 + 0.675908i $$0.236248\pi$$
$$98$$ 1.22779 0.124026
$$99$$ −2.43412 −0.244638
$$100$$ 10.7484 1.07484
$$101$$ 7.17110 0.713551 0.356775 0.934190i $$-0.383876\pi$$
0.356775 + 0.934190i $$0.383876\pi$$
$$102$$ 2.40254 0.237887
$$103$$ −3.87965 −0.382273 −0.191136 0.981563i $$-0.561217\pi$$
−0.191136 + 0.981563i $$0.561217\pi$$
$$104$$ −2.67978 −0.262774
$$105$$ −9.53431 −0.930453
$$106$$ −11.1574 −1.08370
$$107$$ −14.5801 −1.40951 −0.704756 0.709450i $$-0.748943\pi$$
−0.704756 + 0.709450i $$0.748943\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −10.1776 −0.974833 −0.487416 0.873170i $$-0.662061\pi$$
−0.487416 + 0.873170i $$0.662061\pi$$
$$110$$ −9.65961 −0.921008
$$111$$ 11.1890 1.06201
$$112$$ 2.40254 0.227019
$$113$$ 12.7419 1.19866 0.599330 0.800502i $$-0.295434\pi$$
0.599330 + 0.800502i $$0.295434\pi$$
$$114$$ 0.957013 0.0896325
$$115$$ −3.96842 −0.370057
$$116$$ −1.00000 −0.0928477
$$117$$ 2.67978 0.247746
$$118$$ 5.54801 0.510736
$$119$$ −5.77221 −0.529138
$$120$$ 3.96842 0.362266
$$121$$ −5.07507 −0.461370
$$122$$ 3.56588 0.322840
$$123$$ 4.38883 0.395728
$$124$$ −7.81154 −0.701497
$$125$$ −22.8120 −2.04037
$$126$$ −2.40254 −0.214035
$$127$$ 1.17110 0.103918 0.0519590 0.998649i $$-0.483453\pi$$
0.0519590 + 0.998649i $$0.483453\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −7.33214 −0.645559
$$130$$ 10.6345 0.932707
$$131$$ 2.33939 0.204393 0.102197 0.994764i $$-0.467413\pi$$
0.102197 + 0.994764i $$0.467413\pi$$
$$132$$ −2.43412 −0.211863
$$133$$ −2.29926 −0.199371
$$134$$ −13.5284 −1.16867
$$135$$ −3.96842 −0.341548
$$136$$ 2.40254 0.206016
$$137$$ 3.86099 0.329866 0.164933 0.986305i $$-0.447259\pi$$
0.164933 + 0.986305i $$0.447259\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ 4.66607 0.395771 0.197885 0.980225i $$-0.436593\pi$$
0.197885 + 0.980225i $$0.436593\pi$$
$$140$$ −9.53431 −0.805796
$$141$$ −2.08878 −0.175907
$$142$$ 14.3257 1.20218
$$143$$ −6.52290 −0.545472
$$144$$ 1.00000 0.0833333
$$145$$ 3.96842 0.329560
$$146$$ 15.4712 1.28040
$$147$$ −1.22779 −0.101267
$$148$$ 11.1890 0.919727
$$149$$ −6.43412 −0.527103 −0.263552 0.964645i $$-0.584894\pi$$
−0.263552 + 0.964645i $$0.584894\pi$$
$$150$$ −10.7484 −0.877602
$$151$$ −18.7191 −1.52334 −0.761670 0.647965i $$-0.775620\pi$$
−0.761670 + 0.647965i $$0.775620\pi$$
$$152$$ 0.957013 0.0776240
$$153$$ −2.40254 −0.194234
$$154$$ 5.84807 0.471251
$$155$$ 30.9995 2.48994
$$156$$ 2.67978 0.214554
$$157$$ −4.60292 −0.367353 −0.183676 0.982987i $$-0.558800\pi$$
−0.183676 + 0.982987i $$0.558800\pi$$
$$158$$ 10.6661 0.848547
$$159$$ 11.1574 0.884839
$$160$$ 3.96842 0.313731
$$161$$ 2.40254 0.189347
$$162$$ −1.00000 −0.0785674
$$163$$ −11.5077 −0.901351 −0.450676 0.892688i $$-0.648817\pi$$
−0.450676 + 0.892688i $$0.648817\pi$$
$$164$$ 4.38883 0.342710
$$165$$ 9.65961 0.752000
$$166$$ 6.81800 0.529179
$$167$$ −5.03208 −0.389394 −0.194697 0.980863i $$-0.562372\pi$$
−0.194697 + 0.980863i $$0.562372\pi$$
$$168$$ −2.40254 −0.185360
$$169$$ −5.81879 −0.447599
$$170$$ −9.53431 −0.731248
$$171$$ −0.957013 −0.0731846
$$172$$ −7.33214 −0.559070
$$173$$ 17.3578 1.31969 0.659843 0.751403i $$-0.270623\pi$$
0.659843 + 0.751403i $$0.270623\pi$$
$$174$$ 1.00000 0.0758098
$$175$$ 25.8235 1.95207
$$176$$ −2.43412 −0.183479
$$177$$ −5.54801 −0.417014
$$178$$ −10.5169 −0.788278
$$179$$ −7.68623 −0.574496 −0.287248 0.957856i $$-0.592740\pi$$
−0.287248 + 0.957856i $$0.592740\pi$$
$$180$$ −3.96842 −0.295789
$$181$$ −13.4336 −0.998514 −0.499257 0.866454i $$-0.666394\pi$$
−0.499257 + 0.866454i $$0.666394\pi$$
$$182$$ −6.43828 −0.477237
$$183$$ −3.56588 −0.263598
$$184$$ −1.00000 −0.0737210
$$185$$ −44.4026 −3.26454
$$186$$ 7.81154 0.572770
$$187$$ 5.84807 0.427653
$$188$$ −2.08878 −0.152340
$$189$$ 2.40254 0.174759
$$190$$ −3.79783 −0.275524
$$191$$ 1.70490 0.123362 0.0616810 0.998096i $$-0.480354\pi$$
0.0616810 + 0.998096i $$0.480354\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −18.0484 −1.29916 −0.649578 0.760295i $$-0.725054\pi$$
−0.649578 + 0.760295i $$0.725054\pi$$
$$194$$ −14.5169 −1.04226
$$195$$ −10.6345 −0.761552
$$196$$ −1.22779 −0.0876994
$$197$$ −26.7145 −1.90333 −0.951665 0.307137i $$-0.900629\pi$$
−0.951665 + 0.307137i $$0.900629\pi$$
$$198$$ 2.43412 0.172985
$$199$$ −9.79367 −0.694255 −0.347128 0.937818i $$-0.612843\pi$$
−0.347128 + 0.937818i $$0.612843\pi$$
$$200$$ −10.7484 −0.760026
$$201$$ 13.5284 0.954217
$$202$$ −7.17110 −0.504557
$$203$$ −2.40254 −0.168625
$$204$$ −2.40254 −0.168212
$$205$$ −17.4168 −1.21644
$$206$$ 3.87965 0.270308
$$207$$ 1.00000 0.0695048
$$208$$ 2.67978 0.185809
$$209$$ 2.32948 0.161134
$$210$$ 9.53431 0.657930
$$211$$ −15.3083 −1.05387 −0.526934 0.849906i $$-0.676659\pi$$
−0.526934 + 0.849906i $$0.676659\pi$$
$$212$$ 11.1574 0.766293
$$213$$ −14.3257 −0.981580
$$214$$ 14.5801 0.996675
$$215$$ 29.0970 1.98440
$$216$$ −1.00000 −0.0680414
$$217$$ −18.7676 −1.27402
$$218$$ 10.1776 0.689311
$$219$$ −15.4712 −1.04544
$$220$$ 9.65961 0.651251
$$221$$ −6.43828 −0.433085
$$222$$ −11.1890 −0.750954
$$223$$ 14.2032 0.951115 0.475558 0.879685i $$-0.342246\pi$$
0.475558 + 0.879685i $$0.342246\pi$$
$$224$$ −2.40254 −0.160527
$$225$$ 10.7484 0.716559
$$226$$ −12.7419 −0.847581
$$227$$ −23.8006 −1.57970 −0.789852 0.613298i $$-0.789843\pi$$
−0.789852 + 0.613298i $$0.789843\pi$$
$$228$$ −0.957013 −0.0633797
$$229$$ 15.5055 1.02463 0.512317 0.858796i $$-0.328787\pi$$
0.512317 + 0.858796i $$0.328787\pi$$
$$230$$ 3.96842 0.261670
$$231$$ −5.84807 −0.384775
$$232$$ 1.00000 0.0656532
$$233$$ −11.5828 −0.758811 −0.379406 0.925230i $$-0.623872\pi$$
−0.379406 + 0.925230i $$0.623872\pi$$
$$234$$ −2.67978 −0.175183
$$235$$ 8.28915 0.540725
$$236$$ −5.54801 −0.361145
$$237$$ −10.6661 −0.692836
$$238$$ 5.77221 0.374157
$$239$$ −25.4602 −1.64689 −0.823443 0.567399i $$-0.807950\pi$$
−0.823443 + 0.567399i $$0.807950\pi$$
$$240$$ −3.96842 −0.256161
$$241$$ −14.8809 −0.958566 −0.479283 0.877660i $$-0.659103\pi$$
−0.479283 + 0.877660i $$0.659103\pi$$
$$242$$ 5.07507 0.326238
$$243$$ 1.00000 0.0641500
$$244$$ −3.56588 −0.228282
$$245$$ 4.87240 0.311286
$$246$$ −4.38883 −0.279822
$$247$$ −2.56458 −0.163180
$$248$$ 7.81154 0.496033
$$249$$ −6.81800 −0.432073
$$250$$ 22.8120 1.44276
$$251$$ −11.1002 −0.700638 −0.350319 0.936631i $$-0.613927\pi$$
−0.350319 + 0.936631i $$0.613927\pi$$
$$252$$ 2.40254 0.151346
$$253$$ −2.43412 −0.153032
$$254$$ −1.17110 −0.0734811
$$255$$ 9.53431 0.597061
$$256$$ 1.00000 0.0625000
$$257$$ 12.5169 0.780786 0.390393 0.920648i $$-0.372339\pi$$
0.390393 + 0.920648i $$0.372339\pi$$
$$258$$ 7.33214 0.456479
$$259$$ 26.8820 1.67036
$$260$$ −10.6345 −0.659523
$$261$$ −1.00000 −0.0618984
$$262$$ −2.33939 −0.144528
$$263$$ 9.18021 0.566076 0.283038 0.959109i $$-0.408658\pi$$
0.283038 + 0.959109i $$0.408658\pi$$
$$264$$ 2.43412 0.149810
$$265$$ −44.2773 −2.71993
$$266$$ 2.29926 0.140977
$$267$$ 10.5169 0.643627
$$268$$ 13.5284 0.826376
$$269$$ −0.841614 −0.0513141 −0.0256571 0.999671i $$-0.508168\pi$$
−0.0256571 + 0.999671i $$0.508168\pi$$
$$270$$ 3.96842 0.241511
$$271$$ 5.86178 0.356078 0.178039 0.984023i $$-0.443025\pi$$
0.178039 + 0.984023i $$0.443025\pi$$
$$272$$ −2.40254 −0.145676
$$273$$ 6.43828 0.389662
$$274$$ −3.86099 −0.233251
$$275$$ −26.1628 −1.57768
$$276$$ 1.00000 0.0601929
$$277$$ −10.0896 −0.606224 −0.303112 0.952955i $$-0.598026\pi$$
−0.303112 + 0.952955i $$0.598026\pi$$
$$278$$ −4.66607 −0.279852
$$279$$ −7.81154 −0.467665
$$280$$ 9.53431 0.569784
$$281$$ −6.17475 −0.368355 −0.184177 0.982893i $$-0.558962\pi$$
−0.184177 + 0.982893i $$0.558962\pi$$
$$282$$ 2.08878 0.124385
$$283$$ 22.0169 1.30877 0.654384 0.756163i $$-0.272928\pi$$
0.654384 + 0.756163i $$0.272928\pi$$
$$284$$ −14.3257 −0.850073
$$285$$ 3.79783 0.224964
$$286$$ 6.52290 0.385707
$$287$$ 10.5444 0.622414
$$288$$ −1.00000 −0.0589256
$$289$$ −11.2278 −0.660458
$$290$$ −3.96842 −0.233034
$$291$$ 14.5169 0.850998
$$292$$ −15.4712 −0.905381
$$293$$ −6.75485 −0.394622 −0.197311 0.980341i $$-0.563221\pi$$
−0.197311 + 0.980341i $$0.563221\pi$$
$$294$$ 1.22779 0.0716062
$$295$$ 22.0169 1.28187
$$296$$ −11.1890 −0.650345
$$297$$ −2.43412 −0.141242
$$298$$ 6.43412 0.372718
$$299$$ 2.67978 0.154976
$$300$$ 10.7484 0.620559
$$301$$ −17.6158 −1.01536
$$302$$ 18.7191 1.07716
$$303$$ 7.17110 0.411969
$$304$$ −0.957013 −0.0548885
$$305$$ 14.1509 0.810280
$$306$$ 2.40254 0.137344
$$307$$ 27.9990 1.59799 0.798994 0.601339i $$-0.205366\pi$$
0.798994 + 0.601339i $$0.205366\pi$$
$$308$$ −5.84807 −0.333225
$$309$$ −3.87965 −0.220705
$$310$$ −30.9995 −1.76065
$$311$$ 15.7248 0.891670 0.445835 0.895115i $$-0.352907\pi$$
0.445835 + 0.895115i $$0.352907\pi$$
$$312$$ −2.67978 −0.151713
$$313$$ −14.1390 −0.799184 −0.399592 0.916693i $$-0.630848\pi$$
−0.399592 + 0.916693i $$0.630848\pi$$
$$314$$ 4.60292 0.259758
$$315$$ −9.53431 −0.537197
$$316$$ −10.6661 −0.600013
$$317$$ 12.9889 0.729532 0.364766 0.931099i $$-0.381149\pi$$
0.364766 + 0.931099i $$0.381149\pi$$
$$318$$ −11.1574 −0.625675
$$319$$ 2.43412 0.136284
$$320$$ −3.96842 −0.221842
$$321$$ −14.5801 −0.813782
$$322$$ −2.40254 −0.133888
$$323$$ 2.29926 0.127934
$$324$$ 1.00000 0.0555556
$$325$$ 28.8033 1.59772
$$326$$ 11.5077 0.637352
$$327$$ −10.1776 −0.562820
$$328$$ −4.38883 −0.242333
$$329$$ −5.01837 −0.276672
$$330$$ −9.65961 −0.531744
$$331$$ 13.1188 0.721077 0.360539 0.932744i $$-0.382593\pi$$
0.360539 + 0.932744i $$0.382593\pi$$
$$332$$ −6.81800 −0.374186
$$333$$ 11.1890 0.613152
$$334$$ 5.03208 0.275343
$$335$$ −53.6863 −2.93319
$$336$$ 2.40254 0.131069
$$337$$ −23.8046 −1.29672 −0.648359 0.761334i $$-0.724545\pi$$
−0.648359 + 0.761334i $$0.724545\pi$$
$$338$$ 5.81879 0.316501
$$339$$ 12.7419 0.692047
$$340$$ 9.53431 0.517070
$$341$$ 19.0142 1.02968
$$342$$ 0.957013 0.0517493
$$343$$ −19.7676 −1.06735
$$344$$ 7.33214 0.395322
$$345$$ −3.96842 −0.213653
$$346$$ −17.3578 −0.933159
$$347$$ −20.9826 −1.12641 −0.563204 0.826318i $$-0.690431\pi$$
−0.563204 + 0.826318i $$0.690431\pi$$
$$348$$ −1.00000 −0.0536056
$$349$$ −17.3687 −0.929724 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$350$$ −25.8235 −1.38032
$$351$$ 2.67978 0.143036
$$352$$ 2.43412 0.129739
$$353$$ −0.957013 −0.0509367 −0.0254683 0.999676i $$-0.508108\pi$$
−0.0254683 + 0.999676i $$0.508108\pi$$
$$354$$ 5.54801 0.294874
$$355$$ 56.8504 3.01731
$$356$$ 10.5169 0.557397
$$357$$ −5.77221 −0.305498
$$358$$ 7.68623 0.406230
$$359$$ 34.0723 1.79827 0.899133 0.437675i $$-0.144198\pi$$
0.899133 + 0.437675i $$0.144198\pi$$
$$360$$ 3.96842 0.209154
$$361$$ −18.0841 −0.951796
$$362$$ 13.4336 0.706056
$$363$$ −5.07507 −0.266372
$$364$$ 6.43828 0.337457
$$365$$ 61.3961 3.21362
$$366$$ 3.56588 0.186392
$$367$$ 11.9140 0.621907 0.310954 0.950425i $$-0.399352\pi$$
0.310954 + 0.950425i $$0.399352\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 4.38883 0.228474
$$370$$ 44.4026 2.30838
$$371$$ 26.8061 1.39170
$$372$$ −7.81154 −0.405010
$$373$$ −8.58010 −0.444261 −0.222130 0.975017i $$-0.571301\pi$$
−0.222130 + 0.975017i $$0.571301\pi$$
$$374$$ −5.84807 −0.302397
$$375$$ −22.8120 −1.17801
$$376$$ 2.08878 0.107720
$$377$$ −2.67978 −0.138016
$$378$$ −2.40254 −0.123573
$$379$$ −28.1499 −1.44597 −0.722983 0.690866i $$-0.757229\pi$$
−0.722983 + 0.690866i $$0.757229\pi$$
$$380$$ 3.79783 0.194825
$$381$$ 1.17110 0.0599971
$$382$$ −1.70490 −0.0872301
$$383$$ −5.88381 −0.300649 −0.150324 0.988637i $$-0.548032\pi$$
−0.150324 + 0.988637i $$0.548032\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 23.2076 1.18277
$$386$$ 18.0484 0.918642
$$387$$ −7.33214 −0.372714
$$388$$ 14.5169 0.736986
$$389$$ 30.8132 1.56229 0.781146 0.624349i $$-0.214636\pi$$
0.781146 + 0.624349i $$0.214636\pi$$
$$390$$ 10.6345 0.538499
$$391$$ −2.40254 −0.121502
$$392$$ 1.22779 0.0620128
$$393$$ 2.33939 0.118007
$$394$$ 26.7145 1.34586
$$395$$ 42.3275 2.12973
$$396$$ −2.43412 −0.122319
$$397$$ 2.49132 0.125036 0.0625179 0.998044i $$-0.480087\pi$$
0.0625179 + 0.998044i $$0.480087\pi$$
$$398$$ 9.79367 0.490912
$$399$$ −2.29926 −0.115107
$$400$$ 10.7484 0.537419
$$401$$ 24.2302 1.21000 0.604998 0.796227i $$-0.293174\pi$$
0.604998 + 0.796227i $$0.293174\pi$$
$$402$$ −13.5284 −0.674733
$$403$$ −20.9332 −1.04276
$$404$$ 7.17110 0.356775
$$405$$ −3.96842 −0.197193
$$406$$ 2.40254 0.119236
$$407$$ −27.2353 −1.35000
$$408$$ 2.40254 0.118944
$$409$$ 19.4986 0.964142 0.482071 0.876132i $$-0.339885\pi$$
0.482071 + 0.876132i $$0.339885\pi$$
$$410$$ 17.4168 0.860152
$$411$$ 3.86099 0.190448
$$412$$ −3.87965 −0.191136
$$413$$ −13.3293 −0.655894
$$414$$ −1.00000 −0.0491473
$$415$$ 27.0567 1.32816
$$416$$ −2.67978 −0.131387
$$417$$ 4.66607 0.228498
$$418$$ −2.32948 −0.113939
$$419$$ 29.3202 1.43239 0.716194 0.697902i $$-0.245883\pi$$
0.716194 + 0.697902i $$0.245883\pi$$
$$420$$ −9.53431 −0.465226
$$421$$ −5.71781 −0.278669 −0.139335 0.990245i $$-0.544496\pi$$
−0.139335 + 0.990245i $$0.544496\pi$$
$$422$$ 15.3083 0.745197
$$423$$ −2.08878 −0.101560
$$424$$ −11.1574 −0.541851
$$425$$ −25.8235 −1.25262
$$426$$ 14.3257 0.694082
$$427$$ −8.56718 −0.414595
$$428$$ −14.5801 −0.704756
$$429$$ −6.52290 −0.314928
$$430$$ −29.0970 −1.40318
$$431$$ −18.7401 −0.902681 −0.451340 0.892352i $$-0.649054\pi$$
−0.451340 + 0.892352i $$0.649054\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 28.8710 1.38745 0.693727 0.720238i $$-0.255968\pi$$
0.693727 + 0.720238i $$0.255968\pi$$
$$434$$ 18.7676 0.900872
$$435$$ 3.96842 0.190271
$$436$$ −10.1776 −0.487416
$$437$$ −0.957013 −0.0457801
$$438$$ 15.4712 0.739240
$$439$$ −36.0238 −1.71932 −0.859662 0.510863i $$-0.829326\pi$$
−0.859662 + 0.510863i $$0.829326\pi$$
$$440$$ −9.65961 −0.460504
$$441$$ −1.22779 −0.0584663
$$442$$ 6.43828 0.306238
$$443$$ −1.56172 −0.0741996 −0.0370998 0.999312i $$-0.511812\pi$$
−0.0370998 + 0.999312i $$0.511812\pi$$
$$444$$ 11.1890 0.531005
$$445$$ −41.7357 −1.97846
$$446$$ −14.2032 −0.672540
$$447$$ −6.43412 −0.304323
$$448$$ 2.40254 0.113509
$$449$$ −36.4675 −1.72101 −0.860504 0.509444i $$-0.829851\pi$$
−0.860504 + 0.509444i $$0.829851\pi$$
$$450$$ −10.7484 −0.506684
$$451$$ −10.6829 −0.503040
$$452$$ 12.7419 0.599330
$$453$$ −18.7191 −0.879501
$$454$$ 23.8006 1.11702
$$455$$ −25.5498 −1.19779
$$456$$ 0.957013 0.0448162
$$457$$ 27.6697 1.29434 0.647168 0.762347i $$-0.275953\pi$$
0.647168 + 0.762347i $$0.275953\pi$$
$$458$$ −15.5055 −0.724526
$$459$$ −2.40254 −0.112141
$$460$$ −3.96842 −0.185029
$$461$$ −24.3229 −1.13283 −0.566415 0.824120i $$-0.691670\pi$$
−0.566415 + 0.824120i $$0.691670\pi$$
$$462$$ 5.84807 0.272077
$$463$$ −17.3981 −0.808558 −0.404279 0.914636i $$-0.632478\pi$$
−0.404279 + 0.914636i $$0.632478\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 30.9995 1.43757
$$466$$ 11.5828 0.536561
$$467$$ 16.4625 0.761796 0.380898 0.924617i $$-0.375615\pi$$
0.380898 + 0.924617i $$0.375615\pi$$
$$468$$ 2.67978 0.123873
$$469$$ 32.5024 1.50082
$$470$$ −8.28915 −0.382350
$$471$$ −4.60292 −0.212091
$$472$$ 5.54801 0.255368
$$473$$ 17.8473 0.820619
$$474$$ 10.6661 0.489909
$$475$$ −10.2863 −0.471970
$$476$$ −5.77221 −0.264569
$$477$$ 11.1574 0.510862
$$478$$ 25.4602 1.16452
$$479$$ −7.91998 −0.361873 −0.180936 0.983495i $$-0.557913\pi$$
−0.180936 + 0.983495i $$0.557913\pi$$
$$480$$ 3.96842 0.181133
$$481$$ 29.9839 1.36715
$$482$$ 14.8809 0.677809
$$483$$ 2.40254 0.109319
$$484$$ −5.07507 −0.230685
$$485$$ −57.6094 −2.61591
$$486$$ −1.00000 −0.0453609
$$487$$ −39.6715 −1.79769 −0.898844 0.438268i $$-0.855592\pi$$
−0.898844 + 0.438268i $$0.855592\pi$$
$$488$$ 3.56588 0.161420
$$489$$ −11.5077 −0.520395
$$490$$ −4.87240 −0.220112
$$491$$ −24.2260 −1.09330 −0.546652 0.837360i $$-0.684098\pi$$
−0.546652 + 0.837360i $$0.684098\pi$$
$$492$$ 4.38883 0.197864
$$493$$ 2.40254 0.108205
$$494$$ 2.56458 0.115386
$$495$$ 9.65961 0.434167
$$496$$ −7.81154 −0.350749
$$497$$ −34.4181 −1.54386
$$498$$ 6.81800 0.305522
$$499$$ −34.7429 −1.55531 −0.777654 0.628693i $$-0.783590\pi$$
−0.777654 + 0.628693i $$0.783590\pi$$
$$500$$ −22.8120 −1.02019
$$501$$ −5.03208 −0.224817
$$502$$ 11.1002 0.495426
$$503$$ −18.4667 −0.823390 −0.411695 0.911322i $$-0.635063\pi$$
−0.411695 + 0.911322i $$0.635063\pi$$
$$504$$ −2.40254 −0.107018
$$505$$ −28.4580 −1.26636
$$506$$ 2.43412 0.108210
$$507$$ −5.81879 −0.258422
$$508$$ 1.17110 0.0519590
$$509$$ −17.4483 −0.773384 −0.386692 0.922209i $$-0.626382\pi$$
−0.386692 + 0.922209i $$0.626382\pi$$
$$510$$ −9.53431 −0.422186
$$511$$ −37.1701 −1.64431
$$512$$ −1.00000 −0.0441942
$$513$$ −0.957013 −0.0422532
$$514$$ −12.5169 −0.552099
$$515$$ 15.3961 0.678433
$$516$$ −7.33214 −0.322779
$$517$$ 5.08433 0.223609
$$518$$ −26.8820 −1.18113
$$519$$ 17.3578 0.761921
$$520$$ 10.6345 0.466353
$$521$$ 2.17475 0.0952776 0.0476388 0.998865i $$-0.484830\pi$$
0.0476388 + 0.998865i $$0.484830\pi$$
$$522$$ 1.00000 0.0437688
$$523$$ −16.9739 −0.742216 −0.371108 0.928590i $$-0.621022\pi$$
−0.371108 + 0.928590i $$0.621022\pi$$
$$524$$ 2.33939 0.102197
$$525$$ 25.8235 1.12703
$$526$$ −9.18021 −0.400276
$$527$$ 18.7676 0.817528
$$528$$ −2.43412 −0.105931
$$529$$ 1.00000 0.0434783
$$530$$ 44.2773 1.92328
$$531$$ −5.54801 −0.240763
$$532$$ −2.29926 −0.0996857
$$533$$ 11.7611 0.509430
$$534$$ −10.5169 −0.455113
$$535$$ 57.8600 2.50151
$$536$$ −13.5284 −0.584336
$$537$$ −7.68623 −0.331686
$$538$$ 0.841614 0.0362846
$$539$$ 2.98859 0.128728
$$540$$ −3.96842 −0.170774
$$541$$ 14.5426 0.625234 0.312617 0.949879i $$-0.398794\pi$$
0.312617 + 0.949879i $$0.398794\pi$$
$$542$$ −5.86178 −0.251785
$$543$$ −13.4336 −0.576492
$$544$$ 2.40254 0.103008
$$545$$ 40.3889 1.73007
$$546$$ −6.43828 −0.275533
$$547$$ 40.5525 1.73390 0.866949 0.498396i $$-0.166078\pi$$
0.866949 + 0.498396i $$0.166078\pi$$
$$548$$ 3.86099 0.164933
$$549$$ −3.56588 −0.152188
$$550$$ 26.1628 1.11559
$$551$$ 0.957013 0.0407701
$$552$$ −1.00000 −0.0425628
$$553$$ −25.6257 −1.08971
$$554$$ 10.0896 0.428665
$$555$$ −44.4026 −1.88478
$$556$$ 4.66607 0.197885
$$557$$ −15.1458 −0.641749 −0.320875 0.947122i $$-0.603977\pi$$
−0.320875 + 0.947122i $$0.603977\pi$$
$$558$$ 7.81154 0.330689
$$559$$ −19.6485 −0.831043
$$560$$ −9.53431 −0.402898
$$561$$ 5.84807 0.246906
$$562$$ 6.17475 0.260466
$$563$$ 22.4468 0.946021 0.473011 0.881057i $$-0.343167\pi$$
0.473011 + 0.881057i $$0.343167\pi$$
$$564$$ −2.08878 −0.0879534
$$565$$ −50.5654 −2.12730
$$566$$ −22.0169 −0.925438
$$567$$ 2.40254 0.100897
$$568$$ 14.3257 0.601092
$$569$$ 19.3339 0.810521 0.405260 0.914201i $$-0.367181\pi$$
0.405260 + 0.914201i $$0.367181\pi$$
$$570$$ −3.79783 −0.159074
$$571$$ 33.7691 1.41319 0.706596 0.707618i $$-0.250230\pi$$
0.706596 + 0.707618i $$0.250230\pi$$
$$572$$ −6.52290 −0.272736
$$573$$ 1.70490 0.0712231
$$574$$ −10.5444 −0.440113
$$575$$ 10.7484 0.448239
$$576$$ 1.00000 0.0416667
$$577$$ −14.6204 −0.608656 −0.304328 0.952567i $$-0.598432\pi$$
−0.304328 + 0.952567i $$0.598432\pi$$
$$578$$ 11.2278 0.467015
$$579$$ −18.0484 −0.750068
$$580$$ 3.96842 0.164780
$$581$$ −16.3805 −0.679579
$$582$$ −14.5169 −0.601747
$$583$$ −27.1584 −1.12479
$$584$$ 15.4712 0.640201
$$585$$ −10.6345 −0.439682
$$586$$ 6.75485 0.279040
$$587$$ 0.678781 0.0280163 0.0140081 0.999902i $$-0.495541\pi$$
0.0140081 + 0.999902i $$0.495541\pi$$
$$588$$ −1.22779 −0.0506333
$$589$$ 7.47575 0.308033
$$590$$ −22.0169 −0.906420
$$591$$ −26.7145 −1.09889
$$592$$ 11.1890 0.459864
$$593$$ 21.7200 0.891933 0.445966 0.895050i $$-0.352860\pi$$
0.445966 + 0.895050i $$0.352860\pi$$
$$594$$ 2.43412 0.0998731
$$595$$ 22.9066 0.939078
$$596$$ −6.43412 −0.263552
$$597$$ −9.79367 −0.400828
$$598$$ −2.67978 −0.109584
$$599$$ −18.9669 −0.774967 −0.387484 0.921877i $$-0.626656\pi$$
−0.387484 + 0.921877i $$0.626656\pi$$
$$600$$ −10.7484 −0.438801
$$601$$ 26.6085 1.08538 0.542692 0.839932i $$-0.317405\pi$$
0.542692 + 0.839932i $$0.317405\pi$$
$$602$$ 17.6158 0.717965
$$603$$ 13.5284 0.550917
$$604$$ −18.7191 −0.761670
$$605$$ 20.1400 0.818809
$$606$$ −7.17110 −0.291306
$$607$$ 11.6604 0.473281 0.236641 0.971597i $$-0.423954\pi$$
0.236641 + 0.971597i $$0.423954\pi$$
$$608$$ 0.957013 0.0388120
$$609$$ −2.40254 −0.0973559
$$610$$ −14.1509 −0.572954
$$611$$ −5.59746 −0.226449
$$612$$ −2.40254 −0.0971170
$$613$$ −14.5012 −0.585699 −0.292850 0.956159i $$-0.594603\pi$$
−0.292850 + 0.956159i $$0.594603\pi$$
$$614$$ −27.9990 −1.12995
$$615$$ −17.4168 −0.702311
$$616$$ 5.84807 0.235626
$$617$$ −33.5077 −1.34897 −0.674485 0.738289i $$-0.735634\pi$$
−0.674485 + 0.738289i $$0.735634\pi$$
$$618$$ 3.87965 0.156062
$$619$$ 16.4895 0.662770 0.331385 0.943496i $$-0.392484\pi$$
0.331385 + 0.943496i $$0.392484\pi$$
$$620$$ 30.9995 1.24497
$$621$$ 1.00000 0.0401286
$$622$$ −15.7248 −0.630506
$$623$$ 25.2674 1.01232
$$624$$ 2.67978 0.107277
$$625$$ 36.7859 1.47144
$$626$$ 14.1390 0.565109
$$627$$ 2.32948 0.0930306
$$628$$ −4.60292 −0.183676
$$629$$ −26.8820 −1.07185
$$630$$ 9.53431 0.379856
$$631$$ −12.9226 −0.514442 −0.257221 0.966353i $$-0.582807\pi$$
−0.257221 + 0.966353i $$0.582807\pi$$
$$632$$ 10.6661 0.424273
$$633$$ −15.3083 −0.608451
$$634$$ −12.9889 −0.515857
$$635$$ −4.64741 −0.184427
$$636$$ 11.1574 0.442419
$$637$$ −3.29021 −0.130363
$$638$$ −2.43412 −0.0963677
$$639$$ −14.3257 −0.566715
$$640$$ 3.96842 0.156866
$$641$$ 10.3997 0.410765 0.205382 0.978682i $$-0.434156\pi$$
0.205382 + 0.978682i $$0.434156\pi$$
$$642$$ 14.5801 0.575430
$$643$$ 30.2762 1.19398 0.596989 0.802249i $$-0.296364\pi$$
0.596989 + 0.802249i $$0.296364\pi$$
$$644$$ 2.40254 0.0946734
$$645$$ 29.0970 1.14569
$$646$$ −2.29926 −0.0904633
$$647$$ 0.236906 0.00931372 0.00465686 0.999989i $$-0.498518\pi$$
0.00465686 + 0.999989i $$0.498518\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 13.5045 0.530099
$$650$$ −28.8033 −1.12976
$$651$$ −18.7676 −0.735559
$$652$$ −11.5077 −0.450676
$$653$$ 17.3055 0.677217 0.338609 0.940927i $$-0.390044\pi$$
0.338609 + 0.940927i $$0.390044\pi$$
$$654$$ 10.1776 0.397974
$$655$$ −9.28369 −0.362744
$$656$$ 4.38883 0.171355
$$657$$ −15.4712 −0.603587
$$658$$ 5.01837 0.195637
$$659$$ −18.5771 −0.723661 −0.361830 0.932244i $$-0.617848\pi$$
−0.361830 + 0.932244i $$0.617848\pi$$
$$660$$ 9.65961 0.376000
$$661$$ 25.1374 0.977730 0.488865 0.872359i $$-0.337411\pi$$
0.488865 + 0.872359i $$0.337411\pi$$
$$662$$ −13.1188 −0.509879
$$663$$ −6.43828 −0.250042
$$664$$ 6.81800 0.264590
$$665$$ 9.12446 0.353831
$$666$$ −11.1890 −0.433564
$$667$$ −1.00000 −0.0387202
$$668$$ −5.03208 −0.194697
$$669$$ 14.2032 0.549127
$$670$$ 53.6863 2.07408
$$671$$ 8.67978 0.335079
$$672$$ −2.40254 −0.0926801
$$673$$ 33.3979 1.28739 0.643697 0.765280i $$-0.277400\pi$$
0.643697 + 0.765280i $$0.277400\pi$$
$$674$$ 23.8046 0.916919
$$675$$ 10.7484 0.413706
$$676$$ −5.81879 −0.223800
$$677$$ 22.2635 0.855657 0.427828 0.903860i $$-0.359279\pi$$
0.427828 + 0.903860i $$0.359279\pi$$
$$678$$ −12.7419 −0.489351
$$679$$ 34.8776 1.33848
$$680$$ −9.53431 −0.365624
$$681$$ −23.8006 −0.912042
$$682$$ −19.0142 −0.728092
$$683$$ −33.1893 −1.26995 −0.634977 0.772531i $$-0.718990\pi$$
−0.634977 + 0.772531i $$0.718990\pi$$
$$684$$ −0.957013 −0.0365923
$$685$$ −15.3220 −0.585425
$$686$$ 19.7676 0.754731
$$687$$ 15.5055 0.591573
$$688$$ −7.33214 −0.279535
$$689$$ 29.8993 1.13907
$$690$$ 3.96842 0.151075
$$691$$ 41.8965 1.59382 0.796909 0.604099i $$-0.206467\pi$$
0.796909 + 0.604099i $$0.206467\pi$$
$$692$$ 17.3578 0.659843
$$693$$ −5.84807 −0.222150
$$694$$ 20.9826 0.796490
$$695$$ −18.5169 −0.702388
$$696$$ 1.00000 0.0379049
$$697$$ −10.5444 −0.399396
$$698$$ 17.3687 0.657414
$$699$$ −11.5828 −0.438100
$$700$$ 25.8235 0.976035
$$701$$ 19.3024 0.729040 0.364520 0.931196i $$-0.381233\pi$$
0.364520 + 0.931196i $$0.381233\pi$$
$$702$$ −2.67978 −0.101142
$$703$$ −10.7080 −0.403859
$$704$$ −2.43412 −0.0917393
$$705$$ 8.28915 0.312188
$$706$$ 0.957013 0.0360177
$$707$$ 17.2289 0.647958
$$708$$ −5.54801 −0.208507
$$709$$ −12.9534 −0.486476 −0.243238 0.969967i $$-0.578210\pi$$
−0.243238 + 0.969967i $$0.578210\pi$$
$$710$$ −56.8504 −2.13356
$$711$$ −10.6661 −0.400009
$$712$$ −10.5169 −0.394139
$$713$$ −7.81154 −0.292545
$$714$$ 5.77221 0.216019
$$715$$ 25.8856 0.968067
$$716$$ −7.68623 −0.287248
$$717$$ −25.4602 −0.950830
$$718$$ −34.0723 −1.27157
$$719$$ 19.4803 0.726491 0.363246 0.931693i $$-0.381669\pi$$
0.363246 + 0.931693i $$0.381669\pi$$
$$720$$ −3.96842 −0.147894
$$721$$ −9.32102 −0.347133
$$722$$ 18.0841 0.673021
$$723$$ −14.8809 −0.553428
$$724$$ −13.4336 −0.499257
$$725$$ −10.7484 −0.399185
$$726$$ 5.07507 0.188353
$$727$$ 10.2133 0.378790 0.189395 0.981901i $$-0.439347\pi$$
0.189395 + 0.981901i $$0.439347\pi$$
$$728$$ −6.43828 −0.238618
$$729$$ 1.00000 0.0370370
$$730$$ −61.3961 −2.27237
$$731$$ 17.6158 0.651543
$$732$$ −3.56588 −0.131799
$$733$$ 13.6821 0.505359 0.252679 0.967550i $$-0.418688\pi$$
0.252679 + 0.967550i $$0.418688\pi$$
$$734$$ −11.9140 −0.439755
$$735$$ 4.87240 0.179721
$$736$$ −1.00000 −0.0368605
$$737$$ −32.9296 −1.21298
$$738$$ −4.38883 −0.161555
$$739$$ 11.8517 0.435971 0.217985 0.975952i $$-0.430052\pi$$
0.217985 + 0.975952i $$0.430052\pi$$
$$740$$ −44.4026 −1.63227
$$741$$ −2.56458 −0.0942123
$$742$$ −26.8061 −0.984083
$$743$$ 3.44137 0.126252 0.0631258 0.998006i $$-0.479893\pi$$
0.0631258 + 0.998006i $$0.479893\pi$$
$$744$$ 7.81154 0.286385
$$745$$ 25.5333 0.935468
$$746$$ 8.58010 0.314140
$$747$$ −6.81800 −0.249458
$$748$$ 5.84807 0.213827
$$749$$ −35.0293 −1.27994
$$750$$ 22.8120 0.832978
$$751$$ −9.39350 −0.342774 −0.171387 0.985204i $$-0.554825\pi$$
−0.171387 + 0.985204i $$0.554825\pi$$
$$752$$ −2.08878 −0.0761699
$$753$$ −11.1002 −0.404513
$$754$$ 2.67978 0.0975917
$$755$$ 74.2854 2.70352
$$756$$ 2.40254 0.0873796
$$757$$ 5.37541 0.195373 0.0976864 0.995217i $$-0.468856\pi$$
0.0976864 + 0.995217i $$0.468856\pi$$
$$758$$ 28.1499 1.02245
$$759$$ −2.43412 −0.0883529
$$760$$ −3.79783 −0.137762
$$761$$ 0.227791 0.00825743 0.00412871 0.999991i $$-0.498686\pi$$
0.00412871 + 0.999991i $$0.498686\pi$$
$$762$$ −1.17110 −0.0424244
$$763$$ −24.4520 −0.885222
$$764$$ 1.70490 0.0616810
$$765$$ 9.53431 0.344714
$$766$$ 5.88381 0.212591
$$767$$ −14.8674 −0.536832
$$768$$ 1.00000 0.0360844
$$769$$ −16.4095 −0.591742 −0.295871 0.955228i $$-0.595610\pi$$
−0.295871 + 0.955228i $$0.595610\pi$$
$$770$$ −23.2076 −0.836345
$$771$$ 12.5169 0.450787
$$772$$ −18.0484 −0.649578
$$773$$ −8.79518 −0.316341 −0.158170 0.987412i $$-0.550559\pi$$
−0.158170 + 0.987412i $$0.550559\pi$$
$$774$$ 7.33214 0.263548
$$775$$ −83.9615 −3.01599
$$776$$ −14.5169 −0.521128
$$777$$ 26.8820 0.964385
$$778$$ −30.8132 −1.10471
$$779$$ −4.20017 −0.150487
$$780$$ −10.6345 −0.380776
$$781$$ 34.8704 1.24776
$$782$$ 2.40254 0.0859147
$$783$$ −1.00000 −0.0357371
$$784$$ −1.22779 −0.0438497
$$785$$ 18.2663 0.651953
$$786$$ −2.33939 −0.0834433
$$787$$ 15.9912 0.570026 0.285013 0.958524i $$-0.408002\pi$$
0.285013 + 0.958524i $$0.408002\pi$$
$$788$$ −26.7145 −0.951665
$$789$$ 9.18021 0.326824
$$790$$ −42.3275 −1.50594
$$791$$ 30.6130 1.08847
$$792$$ 2.43412 0.0864926
$$793$$ −9.55577 −0.339335
$$794$$ −2.49132 −0.0884136
$$795$$ −44.2773 −1.57035
$$796$$ −9.79367 −0.347128
$$797$$ −55.8590 −1.97863 −0.989313 0.145804i $$-0.953423\pi$$
−0.989313 + 0.145804i $$0.953423\pi$$
$$798$$ 2.29926 0.0813931
$$799$$ 5.01837 0.177537
$$800$$ −10.7484 −0.380013
$$801$$ 10.5169 0.371598
$$802$$ −24.2302 −0.855597
$$803$$ 37.6586 1.32894
$$804$$ 13.5284 0.477108
$$805$$ −9.53431 −0.336040
$$806$$ 20.9332 0.737340
$$807$$ −0.841614 −0.0296262
$$808$$ −7.17110 −0.252278
$$809$$ −34.1162 −1.19946 −0.599731 0.800202i $$-0.704726\pi$$
−0.599731 + 0.800202i $$0.704726\pi$$
$$810$$ 3.96842 0.139436
$$811$$ −12.3440 −0.433456 −0.216728 0.976232i $$-0.569538\pi$$
−0.216728 + 0.976232i $$0.569538\pi$$
$$812$$ −2.40254 −0.0843127
$$813$$ 5.86178 0.205582
$$814$$ 27.2353 0.954595
$$815$$ 45.6674 1.59966
$$816$$ −2.40254 −0.0841058
$$817$$ 7.01695 0.245492
$$818$$ −19.4986 −0.681751
$$819$$ 6.43828 0.224972
$$820$$ −17.4168 −0.608220
$$821$$ −44.9451 −1.56860 −0.784298 0.620385i $$-0.786976\pi$$
−0.784298 + 0.620385i $$0.786976\pi$$
$$822$$ −3.86099 −0.134667
$$823$$ 22.1748 0.772964 0.386482 0.922297i $$-0.373690\pi$$
0.386482 + 0.922297i $$0.373690\pi$$
$$824$$ 3.87965 0.135154
$$825$$ −26.1628 −0.910873
$$826$$ 13.3293 0.463787
$$827$$ 32.5483 1.13182 0.565908 0.824468i $$-0.308526\pi$$
0.565908 + 0.824468i $$0.308526\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ 46.2260 1.60550 0.802748 0.596319i $$-0.203371\pi$$
0.802748 + 0.596319i $$0.203371\pi$$
$$830$$ −27.0567 −0.939152
$$831$$ −10.0896 −0.350003
$$832$$ 2.67978 0.0929046
$$833$$ 2.94982 0.102205
$$834$$ −4.66607 −0.161573
$$835$$ 19.9694 0.691071
$$836$$ 2.32948 0.0805668
$$837$$ −7.81154 −0.270006
$$838$$ −29.3202 −1.01285
$$839$$ −7.00130 −0.241712 −0.120856 0.992670i $$-0.538564\pi$$
−0.120856 + 0.992670i $$0.538564\pi$$
$$840$$ 9.53431 0.328965
$$841$$ 1.00000 0.0344828
$$842$$ 5.71781 0.197049
$$843$$ −6.17475 −0.212670
$$844$$ −15.3083 −0.526934
$$845$$ 23.0914 0.794369
$$846$$ 2.08878 0.0718136
$$847$$ −12.1931 −0.418959
$$848$$ 11.1574 0.383146
$$849$$ 22.0169 0.755617
$$850$$ 25.8235 0.885737
$$851$$ 11.1890 0.383553
$$852$$ −14.3257 −0.490790
$$853$$ 11.8263 0.404924 0.202462 0.979290i $$-0.435106\pi$$
0.202462 + 0.979290i $$0.435106\pi$$
$$854$$ 8.56718 0.293163
$$855$$ 3.79783 0.129883
$$856$$ 14.5801 0.498337
$$857$$ −45.3788 −1.55011 −0.775055 0.631894i $$-0.782278\pi$$
−0.775055 + 0.631894i $$0.782278\pi$$
$$858$$ 6.52290 0.222688
$$859$$ 38.8061 1.32405 0.662023 0.749483i $$-0.269698\pi$$
0.662023 + 0.749483i $$0.269698\pi$$
$$860$$ 29.0970 0.992201
$$861$$ 10.5444 0.359351
$$862$$ 18.7401 0.638292
$$863$$ 29.6882 1.01060 0.505300 0.862944i $$-0.331382\pi$$
0.505300 + 0.862944i $$0.331382\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −68.8830 −2.34209
$$866$$ −28.8710 −0.981078
$$867$$ −11.2278 −0.381316
$$868$$ −18.7676 −0.637012
$$869$$ 25.9625 0.880717
$$870$$ −3.96842 −0.134542
$$871$$ 36.2530 1.22839
$$872$$ 10.1776 0.344655
$$873$$ 14.5169 0.491324
$$874$$ 0.957013 0.0323715
$$875$$ −54.8069 −1.85281
$$876$$ −15.4712 −0.522722
$$877$$ 46.7757 1.57950 0.789751 0.613427i $$-0.210210\pi$$
0.789751 + 0.613427i $$0.210210\pi$$
$$878$$ 36.0238 1.21575
$$879$$ −6.75485 −0.227835
$$880$$ 9.65961 0.325625
$$881$$ 29.0886 0.980021 0.490010 0.871717i $$-0.336993\pi$$
0.490010 + 0.871717i $$0.336993\pi$$
$$882$$ 1.22779 0.0413419
$$883$$ 51.3220 1.72712 0.863562 0.504243i $$-0.168228\pi$$
0.863562 + 0.504243i $$0.168228\pi$$
$$884$$ −6.43828 −0.216543
$$885$$ 22.0169 0.740089
$$886$$ 1.56172 0.0524671
$$887$$ −24.4365 −0.820497 −0.410248 0.911974i $$-0.634558\pi$$
−0.410248 + 0.911974i $$0.634558\pi$$
$$888$$ −11.1890 −0.375477
$$889$$ 2.81361 0.0943654
$$890$$ 41.7357 1.39898
$$891$$ −2.43412 −0.0815460
$$892$$ 14.2032 0.475558
$$893$$ 1.99899 0.0668935
$$894$$ 6.43412 0.215189
$$895$$ 30.5022 1.01958
$$896$$ −2.40254 −0.0802633
$$897$$ 2.67978 0.0894752
$$898$$ 36.4675 1.21694
$$899$$ 7.81154 0.260529
$$900$$ 10.7484 0.358280
$$901$$ −26.8061 −0.893041
$$902$$ 10.6829 0.355703
$$903$$ −17.6158 −0.586216
$$904$$ −12.7419 −0.423790
$$905$$ 53.3103 1.77210
$$906$$ 18.7191 0.621901
$$907$$ 22.4655 0.745954 0.372977 0.927841i $$-0.378337\pi$$
0.372977 + 0.927841i $$0.378337\pi$$
$$908$$ −23.8006 −0.789852
$$909$$ 7.17110 0.237850
$$910$$ 25.5498 0.846968
$$911$$ −36.6941 −1.21573 −0.607864 0.794041i $$-0.707974\pi$$
−0.607864 + 0.794041i $$0.707974\pi$$
$$912$$ −0.957013 −0.0316899
$$913$$ 16.5958 0.549241
$$914$$ −27.6697 −0.915234
$$915$$ 14.1509 0.467815
$$916$$ 15.5055 0.512317
$$917$$ 5.62048 0.185605
$$918$$ 2.40254 0.0792957
$$919$$ −16.2931 −0.537460 −0.268730 0.963216i $$-0.586604\pi$$
−0.268730 + 0.963216i $$0.586604\pi$$
$$920$$ 3.96842 0.130835
$$921$$ 27.9990 0.922599
$$922$$ 24.3229 0.801031
$$923$$ −38.3896 −1.26361
$$924$$ −5.84807 −0.192387
$$925$$ 120.263 3.95423
$$926$$ 17.3981 0.571737
$$927$$ −3.87965 −0.127424
$$928$$ 1.00000 0.0328266
$$929$$ 54.6598 1.79333 0.896665 0.442711i $$-0.145983\pi$$
0.896665 + 0.442711i $$0.145983\pi$$
$$930$$ −30.9995 −1.01651
$$931$$ 1.17501 0.0385095
$$932$$ −11.5828 −0.379406
$$933$$ 15.7248 0.514806
$$934$$ −16.4625 −0.538671
$$935$$ −23.2076 −0.758971
$$936$$ −2.67978 −0.0875913
$$937$$ 7.19313 0.234989 0.117495 0.993074i $$-0.462514\pi$$
0.117495 + 0.993074i $$0.462514\pi$$
$$938$$ −32.5024 −1.06124
$$939$$ −14.1390 −0.461409
$$940$$ 8.28915 0.270362
$$941$$ 11.9152 0.388424 0.194212 0.980960i $$-0.437785\pi$$
0.194212 + 0.980960i $$0.437785\pi$$
$$942$$ 4.60292 0.149971
$$943$$ 4.38883 0.142920
$$944$$ −5.54801 −0.180572
$$945$$ −9.53431 −0.310151
$$946$$ −17.8473 −0.580266
$$947$$ 26.2032 0.851489 0.425744 0.904843i $$-0.360012\pi$$
0.425744 + 0.904843i $$0.360012\pi$$
$$948$$ −10.6661 −0.346418
$$949$$ −41.4593 −1.34582
$$950$$ 10.2863 0.333733
$$951$$ 12.9889 0.421196
$$952$$ 5.77221 0.187078
$$953$$ −50.2514 −1.62780 −0.813902 0.581003i $$-0.802661\pi$$
−0.813902 + 0.581003i $$0.802661\pi$$
$$954$$ −11.1574 −0.361234
$$955$$ −6.76575 −0.218935
$$956$$ −25.4602 −0.823443
$$957$$ 2.43412 0.0786839
$$958$$ 7.91998 0.255883
$$959$$ 9.27618 0.299543
$$960$$ −3.96842 −0.128080
$$961$$ 30.0202 0.968393
$$962$$ −29.9839 −0.966721
$$963$$ −14.5801 −0.469837
$$964$$ −14.8809 −0.479283
$$965$$ 71.6239 2.30565
$$966$$ −2.40254 −0.0773005
$$967$$ −21.4191 −0.688793 −0.344396 0.938824i $$-0.611916\pi$$
−0.344396 + 0.938824i $$0.611916\pi$$
$$968$$ 5.07507 0.163119
$$969$$ 2.29926 0.0738630
$$970$$ 57.6094 1.84973
$$971$$ 7.41395 0.237925 0.118963 0.992899i $$-0.462043\pi$$
0.118963 + 0.992899i $$0.462043\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 11.2104 0.359390
$$974$$ 39.6715 1.27116
$$975$$ 28.8033 0.922444
$$976$$ −3.56588 −0.114141
$$977$$ 51.0784 1.63414 0.817071 0.576537i $$-0.195596\pi$$
0.817071 + 0.576537i $$0.195596\pi$$
$$978$$ 11.5077 0.367975
$$979$$ −25.5995 −0.818163
$$980$$ 4.87240 0.155643
$$981$$ −10.1776 −0.324944
$$982$$ 24.2260 0.773083
$$983$$ −57.0261 −1.81885 −0.909426 0.415866i $$-0.863478\pi$$
−0.909426 + 0.415866i $$0.863478\pi$$
$$984$$ −4.38883 −0.139911
$$985$$ 106.015 3.37790
$$986$$ −2.40254 −0.0765125
$$987$$ −5.01837 −0.159737
$$988$$ −2.56458 −0.0815902
$$989$$ −7.33214 −0.233148
$$990$$ −9.65961 −0.307003
$$991$$ −49.3420 −1.56740 −0.783701 0.621138i $$-0.786670\pi$$
−0.783701 + 0.621138i $$0.786670\pi$$
$$992$$ 7.81154 0.248017
$$993$$ 13.1188 0.416314
$$994$$ 34.4181 1.09167
$$995$$ 38.8654 1.23212
$$996$$ −6.81800 −0.216037
$$997$$ −21.4582 −0.679589 −0.339795 0.940500i $$-0.610358\pi$$
−0.339795 + 0.940500i $$0.610358\pi$$
$$998$$ 34.7429 1.09977
$$999$$ 11.1890 0.354003
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 4002.2.a.bb.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.bb.1.1 4 1.1 even 1 trivial