# Properties

 Label 507.2.a.j.1.2 Level $507$ Weight $2$ Character 507.1 Self dual yes Analytic conductor $4.048$ 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] = [507,2,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$4.04841538248$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 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 1.2 Root $$0.445042$$ of defining polynomial Character $$\chi$$ $$=$$ 507.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.445042 q^{2} +1.00000 q^{3} -1.80194 q^{4} +0.246980 q^{5} -0.445042 q^{6} -1.75302 q^{7} +1.69202 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.445042 q^{2} +1.00000 q^{3} -1.80194 q^{4} +0.246980 q^{5} -0.445042 q^{6} -1.75302 q^{7} +1.69202 q^{8} +1.00000 q^{9} -0.109916 q^{10} -5.65279 q^{11} -1.80194 q^{12} +0.780167 q^{14} +0.246980 q^{15} +2.85086 q^{16} -3.80194 q^{17} -0.445042 q^{18} -5.58211 q^{19} -0.445042 q^{20} -1.75302 q^{21} +2.51573 q^{22} +8.34481 q^{23} +1.69202 q^{24} -4.93900 q^{25} +1.00000 q^{27} +3.15883 q^{28} -5.93900 q^{29} -0.109916 q^{30} -5.26875 q^{31} -4.65279 q^{32} -5.65279 q^{33} +1.69202 q^{34} -0.432960 q^{35} -1.80194 q^{36} -3.19806 q^{37} +2.48427 q^{38} +0.417895 q^{40} -0.445042 q^{41} +0.780167 q^{42} +1.71379 q^{43} +10.1860 q^{44} +0.246980 q^{45} -3.71379 q^{46} +6.73556 q^{47} +2.85086 q^{48} -3.92692 q^{49} +2.19806 q^{50} -3.80194 q^{51} -1.06100 q^{53} -0.445042 q^{54} -1.39612 q^{55} -2.96615 q^{56} -5.58211 q^{57} +2.64310 q^{58} +13.7017 q^{59} -0.445042 q^{60} -8.51573 q^{61} +2.34481 q^{62} -1.75302 q^{63} -3.63102 q^{64} +2.51573 q^{66} -5.96077 q^{67} +6.85086 q^{68} +8.34481 q^{69} +0.192685 q^{70} +5.71917 q^{71} +1.69202 q^{72} -7.35690 q^{73} +1.42327 q^{74} -4.93900 q^{75} +10.0586 q^{76} +9.90946 q^{77} +4.45473 q^{79} +0.704103 q^{80} +1.00000 q^{81} +0.198062 q^{82} +10.1860 q^{83} +3.15883 q^{84} -0.939001 q^{85} -0.762709 q^{86} -5.93900 q^{87} -9.56465 q^{88} -0.137063 q^{89} -0.109916 q^{90} -15.0368 q^{92} -5.26875 q^{93} -2.99761 q^{94} -1.37867 q^{95} -4.65279 q^{96} +13.6896 q^{97} +1.74764 q^{98} -5.65279 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 3 q^{3} - q^{4} - 4 q^{5} - q^{6} - 10 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 + 3 * q^3 - q^4 - 4 * q^5 - q^6 - 10 * q^7 + 3 * q^9 $$3 q - q^{2} + 3 q^{3} - q^{4} - 4 q^{5} - q^{6} - 10 q^{7} + 3 q^{9} - q^{10} + q^{11} - q^{12} + q^{14} - 4 q^{15} - 5 q^{16} - 7 q^{17} - q^{18} - 11 q^{19} - q^{20} - 10 q^{21} - 5 q^{22} + 2 q^{23} - 5 q^{25} + 3 q^{27} + q^{28} - 8 q^{29} - q^{30} - 8 q^{31} + 4 q^{32} + q^{33} + 18 q^{35} - q^{36} - 14 q^{37} + 20 q^{38} + 7 q^{40} - q^{41} + q^{42} - 3 q^{43} + 16 q^{44} - 4 q^{45} - 3 q^{46} + 9 q^{47} - 5 q^{48} + 17 q^{49} + 11 q^{50} - 7 q^{51} - 13 q^{53} - q^{54} - 13 q^{55} + 7 q^{56} - 11 q^{57} + 12 q^{58} + 14 q^{59} - q^{60} - 13 q^{61} - 16 q^{62} - 10 q^{63} + 4 q^{64} - 5 q^{66} - 5 q^{67} + 7 q^{68} + 2 q^{69} + 8 q^{70} + 6 q^{71} - 18 q^{73} + 7 q^{74} - 5 q^{75} - q^{76} - 15 q^{77} - 9 q^{79} + 16 q^{80} + 3 q^{81} + 5 q^{82} + 16 q^{83} + q^{84} + 7 q^{85} + 15 q^{86} - 8 q^{87} - 7 q^{88} + 5 q^{89} - q^{90} - 17 q^{92} - 8 q^{93} + 32 q^{94} + 3 q^{95} + 4 q^{96} - 5 q^{97} + 13 q^{98} + q^{99}+O(q^{100})$$ 3 * q - q^2 + 3 * q^3 - q^4 - 4 * q^5 - q^6 - 10 * q^7 + 3 * q^9 - q^10 + q^11 - q^12 + q^14 - 4 * q^15 - 5 * q^16 - 7 * q^17 - q^18 - 11 * q^19 - q^20 - 10 * q^21 - 5 * q^22 + 2 * q^23 - 5 * q^25 + 3 * q^27 + q^28 - 8 * q^29 - q^30 - 8 * q^31 + 4 * q^32 + q^33 + 18 * q^35 - q^36 - 14 * q^37 + 20 * q^38 + 7 * q^40 - q^41 + q^42 - 3 * q^43 + 16 * q^44 - 4 * q^45 - 3 * q^46 + 9 * q^47 - 5 * q^48 + 17 * q^49 + 11 * q^50 - 7 * q^51 - 13 * q^53 - q^54 - 13 * q^55 + 7 * q^56 - 11 * q^57 + 12 * q^58 + 14 * q^59 - q^60 - 13 * q^61 - 16 * q^62 - 10 * q^63 + 4 * q^64 - 5 * q^66 - 5 * q^67 + 7 * q^68 + 2 * q^69 + 8 * q^70 + 6 * q^71 - 18 * q^73 + 7 * q^74 - 5 * q^75 - q^76 - 15 * q^77 - 9 * q^79 + 16 * q^80 + 3 * q^81 + 5 * q^82 + 16 * q^83 + q^84 + 7 * q^85 + 15 * q^86 - 8 * q^87 - 7 * q^88 + 5 * q^89 - q^90 - 17 * q^92 - 8 * q^93 + 32 * q^94 + 3 * q^95 + 4 * q^96 - 5 * q^97 + 13 * q^98 + 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.445042 −0.314692 −0.157346 0.987544i $$-0.550294\pi$$
−0.157346 + 0.987544i $$0.550294\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.80194 −0.900969
$$5$$ 0.246980 0.110453 0.0552263 0.998474i $$-0.482412\pi$$
0.0552263 + 0.998474i $$0.482412\pi$$
$$6$$ −0.445042 −0.181688
$$7$$ −1.75302 −0.662579 −0.331290 0.943529i $$-0.607484\pi$$
−0.331290 + 0.943529i $$0.607484\pi$$
$$8$$ 1.69202 0.598220
$$9$$ 1.00000 0.333333
$$10$$ −0.109916 −0.0347586
$$11$$ −5.65279 −1.70438 −0.852191 0.523232i $$-0.824726\pi$$
−0.852191 + 0.523232i $$0.824726\pi$$
$$12$$ −1.80194 −0.520175
$$13$$ 0 0
$$14$$ 0.780167 0.208509
$$15$$ 0.246980 0.0637699
$$16$$ 2.85086 0.712714
$$17$$ −3.80194 −0.922105 −0.461053 0.887373i $$-0.652528\pi$$
−0.461053 + 0.887373i $$0.652528\pi$$
$$18$$ −0.445042 −0.104897
$$19$$ −5.58211 −1.28062 −0.640311 0.768115i $$-0.721195\pi$$
−0.640311 + 0.768115i $$0.721195\pi$$
$$20$$ −0.445042 −0.0995144
$$21$$ −1.75302 −0.382540
$$22$$ 2.51573 0.536355
$$23$$ 8.34481 1.74001 0.870007 0.493039i $$-0.164114\pi$$
0.870007 + 0.493039i $$0.164114\pi$$
$$24$$ 1.69202 0.345382
$$25$$ −4.93900 −0.987800
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 3.15883 0.596963
$$29$$ −5.93900 −1.10284 −0.551422 0.834226i $$-0.685915\pi$$
−0.551422 + 0.834226i $$0.685915\pi$$
$$30$$ −0.109916 −0.0200679
$$31$$ −5.26875 −0.946295 −0.473148 0.880983i $$-0.656882\pi$$
−0.473148 + 0.880983i $$0.656882\pi$$
$$32$$ −4.65279 −0.822505
$$33$$ −5.65279 −0.984025
$$34$$ 1.69202 0.290179
$$35$$ −0.432960 −0.0731836
$$36$$ −1.80194 −0.300323
$$37$$ −3.19806 −0.525758 −0.262879 0.964829i $$-0.584672\pi$$
−0.262879 + 0.964829i $$0.584672\pi$$
$$38$$ 2.48427 0.403002
$$39$$ 0 0
$$40$$ 0.417895 0.0660750
$$41$$ −0.445042 −0.0695039 −0.0347519 0.999396i $$-0.511064\pi$$
−0.0347519 + 0.999396i $$0.511064\pi$$
$$42$$ 0.780167 0.120382
$$43$$ 1.71379 0.261351 0.130675 0.991425i $$-0.458285\pi$$
0.130675 + 0.991425i $$0.458285\pi$$
$$44$$ 10.1860 1.53559
$$45$$ 0.246980 0.0368175
$$46$$ −3.71379 −0.547569
$$47$$ 6.73556 0.982483 0.491241 0.871024i $$-0.336543\pi$$
0.491241 + 0.871024i $$0.336543\pi$$
$$48$$ 2.85086 0.411485
$$49$$ −3.92692 −0.560988
$$50$$ 2.19806 0.310853
$$51$$ −3.80194 −0.532378
$$52$$ 0 0
$$53$$ −1.06100 −0.145739 −0.0728697 0.997341i $$-0.523216\pi$$
−0.0728697 + 0.997341i $$0.523216\pi$$
$$54$$ −0.445042 −0.0605625
$$55$$ −1.39612 −0.188253
$$56$$ −2.96615 −0.396368
$$57$$ −5.58211 −0.739368
$$58$$ 2.64310 0.347057
$$59$$ 13.7017 1.78381 0.891905 0.452222i $$-0.149369\pi$$
0.891905 + 0.452222i $$0.149369\pi$$
$$60$$ −0.445042 −0.0574547
$$61$$ −8.51573 −1.09033 −0.545164 0.838330i $$-0.683533\pi$$
−0.545164 + 0.838330i $$0.683533\pi$$
$$62$$ 2.34481 0.297792
$$63$$ −1.75302 −0.220860
$$64$$ −3.63102 −0.453878
$$65$$ 0 0
$$66$$ 2.51573 0.309665
$$67$$ −5.96077 −0.728224 −0.364112 0.931355i $$-0.618627\pi$$
−0.364112 + 0.931355i $$0.618627\pi$$
$$68$$ 6.85086 0.830788
$$69$$ 8.34481 1.00460
$$70$$ 0.192685 0.0230303
$$71$$ 5.71917 0.678740 0.339370 0.940653i $$-0.389786\pi$$
0.339370 + 0.940653i $$0.389786\pi$$
$$72$$ 1.69202 0.199407
$$73$$ −7.35690 −0.861060 −0.430530 0.902576i $$-0.641673\pi$$
−0.430530 + 0.902576i $$0.641673\pi$$
$$74$$ 1.42327 0.165452
$$75$$ −4.93900 −0.570307
$$76$$ 10.0586 1.15380
$$77$$ 9.90946 1.12929
$$78$$ 0 0
$$79$$ 4.45473 0.501196 0.250598 0.968091i $$-0.419373\pi$$
0.250598 + 0.968091i $$0.419373\pi$$
$$80$$ 0.704103 0.0787211
$$81$$ 1.00000 0.111111
$$82$$ 0.198062 0.0218723
$$83$$ 10.1860 1.11806 0.559028 0.829149i $$-0.311174\pi$$
0.559028 + 0.829149i $$0.311174\pi$$
$$84$$ 3.15883 0.344657
$$85$$ −0.939001 −0.101849
$$86$$ −0.762709 −0.0822450
$$87$$ −5.93900 −0.636728
$$88$$ −9.56465 −1.01959
$$89$$ −0.137063 −0.0145287 −0.00726434 0.999974i $$-0.502312\pi$$
−0.00726434 + 0.999974i $$0.502312\pi$$
$$90$$ −0.109916 −0.0115862
$$91$$ 0 0
$$92$$ −15.0368 −1.56770
$$93$$ −5.26875 −0.546344
$$94$$ −2.99761 −0.309180
$$95$$ −1.37867 −0.141448
$$96$$ −4.65279 −0.474874
$$97$$ 13.6896 1.38997 0.694986 0.719024i $$-0.255411\pi$$
0.694986 + 0.719024i $$0.255411\pi$$
$$98$$ 1.74764 0.176539
$$99$$ −5.65279 −0.568127
$$100$$ 8.89977 0.889977
$$101$$ −5.41119 −0.538434 −0.269217 0.963080i $$-0.586765\pi$$
−0.269217 + 0.963080i $$0.586765\pi$$
$$102$$ 1.69202 0.167535
$$103$$ 13.7560 1.35542 0.677710 0.735330i $$-0.262973\pi$$
0.677710 + 0.735330i $$0.262973\pi$$
$$104$$ 0 0
$$105$$ −0.432960 −0.0422526
$$106$$ 0.472189 0.0458630
$$107$$ −12.8170 −1.23907 −0.619533 0.784970i $$-0.712678\pi$$
−0.619533 + 0.784970i $$0.712678\pi$$
$$108$$ −1.80194 −0.173392
$$109$$ −12.1468 −1.16345 −0.581724 0.813386i $$-0.697622\pi$$
−0.581724 + 0.813386i $$0.697622\pi$$
$$110$$ 0.621334 0.0592419
$$111$$ −3.19806 −0.303547
$$112$$ −4.99761 −0.472229
$$113$$ −1.63773 −0.154064 −0.0770322 0.997029i $$-0.524544\pi$$
−0.0770322 + 0.997029i $$0.524544\pi$$
$$114$$ 2.48427 0.232673
$$115$$ 2.06100 0.192189
$$116$$ 10.7017 0.993629
$$117$$ 0 0
$$118$$ −6.09783 −0.561351
$$119$$ 6.66487 0.610968
$$120$$ 0.417895 0.0381484
$$121$$ 20.9541 1.90492
$$122$$ 3.78986 0.343117
$$123$$ −0.445042 −0.0401281
$$124$$ 9.49396 0.852583
$$125$$ −2.45473 −0.219558
$$126$$ 0.780167 0.0695028
$$127$$ 10.7995 0.958305 0.479152 0.877732i $$-0.340944\pi$$
0.479152 + 0.877732i $$0.340944\pi$$
$$128$$ 10.9215 0.965337
$$129$$ 1.71379 0.150891
$$130$$ 0 0
$$131$$ 0.907542 0.0792923 0.0396462 0.999214i $$-0.487377\pi$$
0.0396462 + 0.999214i $$0.487377\pi$$
$$132$$ 10.1860 0.886576
$$133$$ 9.78554 0.848514
$$134$$ 2.65279 0.229166
$$135$$ 0.246980 0.0212566
$$136$$ −6.43296 −0.551622
$$137$$ 9.54825 0.815762 0.407881 0.913035i $$-0.366268\pi$$
0.407881 + 0.913035i $$0.366268\pi$$
$$138$$ −3.71379 −0.316139
$$139$$ −4.09246 −0.347118 −0.173559 0.984823i $$-0.555527\pi$$
−0.173559 + 0.984823i $$0.555527\pi$$
$$140$$ 0.780167 0.0659362
$$141$$ 6.73556 0.567237
$$142$$ −2.54527 −0.213594
$$143$$ 0 0
$$144$$ 2.85086 0.237571
$$145$$ −1.46681 −0.121812
$$146$$ 3.27413 0.270969
$$147$$ −3.92692 −0.323887
$$148$$ 5.76271 0.473692
$$149$$ 15.3884 1.26066 0.630332 0.776326i $$-0.282919\pi$$
0.630332 + 0.776326i $$0.282919\pi$$
$$150$$ 2.19806 0.179471
$$151$$ 3.67456 0.299032 0.149516 0.988759i $$-0.452228\pi$$
0.149516 + 0.988759i $$0.452228\pi$$
$$152$$ −9.44504 −0.766094
$$153$$ −3.80194 −0.307368
$$154$$ −4.41013 −0.355378
$$155$$ −1.30127 −0.104521
$$156$$ 0 0
$$157$$ −4.87800 −0.389307 −0.194653 0.980872i $$-0.562358\pi$$
−0.194653 + 0.980872i $$0.562358\pi$$
$$158$$ −1.98254 −0.157723
$$159$$ −1.06100 −0.0841427
$$160$$ −1.14914 −0.0908479
$$161$$ −14.6286 −1.15290
$$162$$ −0.445042 −0.0349658
$$163$$ −8.63102 −0.676034 −0.338017 0.941140i $$-0.609756\pi$$
−0.338017 + 0.941140i $$0.609756\pi$$
$$164$$ 0.801938 0.0626208
$$165$$ −1.39612 −0.108688
$$166$$ −4.53319 −0.351844
$$167$$ −9.46980 −0.732795 −0.366397 0.930458i $$-0.619409\pi$$
−0.366397 + 0.930458i $$0.619409\pi$$
$$168$$ −2.96615 −0.228843
$$169$$ 0 0
$$170$$ 0.417895 0.0320511
$$171$$ −5.58211 −0.426874
$$172$$ −3.08815 −0.235469
$$173$$ 4.77479 0.363021 0.181510 0.983389i $$-0.441901\pi$$
0.181510 + 0.983389i $$0.441901\pi$$
$$174$$ 2.64310 0.200373
$$175$$ 8.65817 0.654496
$$176$$ −16.1153 −1.21474
$$177$$ 13.7017 1.02988
$$178$$ 0.0609989 0.00457206
$$179$$ 3.43535 0.256770 0.128385 0.991724i $$-0.459021\pi$$
0.128385 + 0.991724i $$0.459021\pi$$
$$180$$ −0.445042 −0.0331715
$$181$$ −13.4862 −1.00242 −0.501210 0.865326i $$-0.667112\pi$$
−0.501210 + 0.865326i $$0.667112\pi$$
$$182$$ 0 0
$$183$$ −8.51573 −0.629501
$$184$$ 14.1196 1.04091
$$185$$ −0.789856 −0.0580714
$$186$$ 2.34481 0.171930
$$187$$ 21.4916 1.57162
$$188$$ −12.1371 −0.885186
$$189$$ −1.75302 −0.127513
$$190$$ 0.613564 0.0445126
$$191$$ −1.30127 −0.0941569 −0.0470784 0.998891i $$-0.514991\pi$$
−0.0470784 + 0.998891i $$0.514991\pi$$
$$192$$ −3.63102 −0.262046
$$193$$ 9.19567 0.661919 0.330959 0.943645i $$-0.392628\pi$$
0.330959 + 0.943645i $$0.392628\pi$$
$$194$$ −6.09246 −0.437413
$$195$$ 0 0
$$196$$ 7.07606 0.505433
$$197$$ −4.11231 −0.292990 −0.146495 0.989211i $$-0.546799\pi$$
−0.146495 + 0.989211i $$0.546799\pi$$
$$198$$ 2.51573 0.178785
$$199$$ −24.7724 −1.75607 −0.878034 0.478598i $$-0.841145\pi$$
−0.878034 + 0.478598i $$0.841145\pi$$
$$200$$ −8.35690 −0.590922
$$201$$ −5.96077 −0.420440
$$202$$ 2.40821 0.169441
$$203$$ 10.4112 0.730722
$$204$$ 6.85086 0.479656
$$205$$ −0.109916 −0.00767688
$$206$$ −6.12200 −0.426540
$$207$$ 8.34481 0.580005
$$208$$ 0 0
$$209$$ 31.5545 2.18267
$$210$$ 0.192685 0.0132966
$$211$$ −5.93900 −0.408858 −0.204429 0.978881i $$-0.565534\pi$$
−0.204429 + 0.978881i $$0.565534\pi$$
$$212$$ 1.91185 0.131307
$$213$$ 5.71917 0.391871
$$214$$ 5.70410 0.389924
$$215$$ 0.423272 0.0288669
$$216$$ 1.69202 0.115127
$$217$$ 9.23623 0.626996
$$218$$ 5.40581 0.366128
$$219$$ −7.35690 −0.497133
$$220$$ 2.51573 0.169610
$$221$$ 0 0
$$222$$ 1.42327 0.0955237
$$223$$ −14.2010 −0.950972 −0.475486 0.879723i $$-0.657728\pi$$
−0.475486 + 0.879723i $$0.657728\pi$$
$$224$$ 8.15644 0.544975
$$225$$ −4.93900 −0.329267
$$226$$ 0.728857 0.0484829
$$227$$ −16.0073 −1.06244 −0.531221 0.847233i $$-0.678267\pi$$
−0.531221 + 0.847233i $$0.678267\pi$$
$$228$$ 10.0586 0.666147
$$229$$ −1.84117 −0.121668 −0.0608339 0.998148i $$-0.519376\pi$$
−0.0608339 + 0.998148i $$0.519376\pi$$
$$230$$ −0.917231 −0.0604804
$$231$$ 9.90946 0.651995
$$232$$ −10.0489 −0.659744
$$233$$ −23.4252 −1.53464 −0.767318 0.641267i $$-0.778409\pi$$
−0.767318 + 0.641267i $$0.778409\pi$$
$$234$$ 0 0
$$235$$ 1.66355 0.108518
$$236$$ −24.6896 −1.60716
$$237$$ 4.45473 0.289366
$$238$$ −2.96615 −0.192267
$$239$$ −14.6015 −0.944491 −0.472246 0.881467i $$-0.656556\pi$$
−0.472246 + 0.881467i $$0.656556\pi$$
$$240$$ 0.704103 0.0454497
$$241$$ −8.63102 −0.555973 −0.277987 0.960585i $$-0.589667\pi$$
−0.277987 + 0.960585i $$0.589667\pi$$
$$242$$ −9.32544 −0.599462
$$243$$ 1.00000 0.0641500
$$244$$ 15.3448 0.982351
$$245$$ −0.969869 −0.0619627
$$246$$ 0.198062 0.0126280
$$247$$ 0 0
$$248$$ −8.91484 −0.566093
$$249$$ 10.1860 0.645510
$$250$$ 1.09246 0.0690931
$$251$$ −3.80194 −0.239976 −0.119988 0.992775i $$-0.538286\pi$$
−0.119988 + 0.992775i $$0.538286\pi$$
$$252$$ 3.15883 0.198988
$$253$$ −47.1715 −2.96565
$$254$$ −4.80625 −0.301571
$$255$$ −0.939001 −0.0588025
$$256$$ 2.40150 0.150094
$$257$$ −20.5961 −1.28475 −0.642375 0.766391i $$-0.722051\pi$$
−0.642375 + 0.766391i $$0.722051\pi$$
$$258$$ −0.762709 −0.0474842
$$259$$ 5.60627 0.348357
$$260$$ 0 0
$$261$$ −5.93900 −0.367615
$$262$$ −0.403894 −0.0249527
$$263$$ 0.332733 0.0205172 0.0102586 0.999947i $$-0.496735\pi$$
0.0102586 + 0.999947i $$0.496735\pi$$
$$264$$ −9.56465 −0.588663
$$265$$ −0.262045 −0.0160973
$$266$$ −4.35498 −0.267021
$$267$$ −0.137063 −0.00838814
$$268$$ 10.7409 0.656107
$$269$$ 27.3032 1.66471 0.832353 0.554247i $$-0.186994\pi$$
0.832353 + 0.554247i $$0.186994\pi$$
$$270$$ −0.109916 −0.00668929
$$271$$ −27.9855 −1.70000 −0.850000 0.526783i $$-0.823398\pi$$
−0.850000 + 0.526783i $$0.823398\pi$$
$$272$$ −10.8388 −0.657197
$$273$$ 0 0
$$274$$ −4.24937 −0.256714
$$275$$ 27.9191 1.68359
$$276$$ −15.0368 −0.905111
$$277$$ −2.10321 −0.126370 −0.0631849 0.998002i $$-0.520126\pi$$
−0.0631849 + 0.998002i $$0.520126\pi$$
$$278$$ 1.82132 0.109235
$$279$$ −5.26875 −0.315432
$$280$$ −0.732578 −0.0437799
$$281$$ −27.2349 −1.62470 −0.812349 0.583172i $$-0.801811\pi$$
−0.812349 + 0.583172i $$0.801811\pi$$
$$282$$ −2.99761 −0.178505
$$283$$ 5.28382 0.314090 0.157045 0.987591i $$-0.449803\pi$$
0.157045 + 0.987591i $$0.449803\pi$$
$$284$$ −10.3056 −0.611524
$$285$$ −1.37867 −0.0816651
$$286$$ 0 0
$$287$$ 0.780167 0.0460518
$$288$$ −4.65279 −0.274168
$$289$$ −2.54527 −0.149722
$$290$$ 0.652793 0.0383333
$$291$$ 13.6896 0.802500
$$292$$ 13.2567 0.775788
$$293$$ −32.6625 −1.90816 −0.954081 0.299548i $$-0.903164\pi$$
−0.954081 + 0.299548i $$0.903164\pi$$
$$294$$ 1.74764 0.101925
$$295$$ 3.38404 0.197027
$$296$$ −5.41119 −0.314519
$$297$$ −5.65279 −0.328008
$$298$$ −6.84846 −0.396721
$$299$$ 0 0
$$300$$ 8.89977 0.513829
$$301$$ −3.00431 −0.173166
$$302$$ −1.63533 −0.0941029
$$303$$ −5.41119 −0.310865
$$304$$ −15.9138 −0.912717
$$305$$ −2.10321 −0.120430
$$306$$ 1.69202 0.0967264
$$307$$ 20.7614 1.18491 0.592457 0.805602i $$-0.298158\pi$$
0.592457 + 0.805602i $$0.298158\pi$$
$$308$$ −17.8562 −1.01745
$$309$$ 13.7560 0.782552
$$310$$ 0.579121 0.0328919
$$311$$ −11.3013 −0.640836 −0.320418 0.947276i $$-0.603823\pi$$
−0.320418 + 0.947276i $$0.603823\pi$$
$$312$$ 0 0
$$313$$ −4.27173 −0.241453 −0.120726 0.992686i $$-0.538522\pi$$
−0.120726 + 0.992686i $$0.538522\pi$$
$$314$$ 2.17092 0.122512
$$315$$ −0.432960 −0.0243945
$$316$$ −8.02715 −0.451562
$$317$$ −15.4776 −0.869307 −0.434653 0.900598i $$-0.643129\pi$$
−0.434653 + 0.900598i $$0.643129\pi$$
$$318$$ 0.472189 0.0264790
$$319$$ 33.5719 1.87967
$$320$$ −0.896789 −0.0501320
$$321$$ −12.8170 −0.715375
$$322$$ 6.51035 0.362808
$$323$$ 21.2228 1.18087
$$324$$ −1.80194 −0.100108
$$325$$ 0 0
$$326$$ 3.84117 0.212743
$$327$$ −12.1468 −0.671717
$$328$$ −0.753020 −0.0415786
$$329$$ −11.8076 −0.650973
$$330$$ 0.621334 0.0342033
$$331$$ −6.06829 −0.333544 −0.166772 0.985996i $$-0.553334\pi$$
−0.166772 + 0.985996i $$0.553334\pi$$
$$332$$ −18.3545 −1.00733
$$333$$ −3.19806 −0.175253
$$334$$ 4.21446 0.230605
$$335$$ −1.47219 −0.0804343
$$336$$ −4.99761 −0.272642
$$337$$ 12.1239 0.660432 0.330216 0.943905i $$-0.392878\pi$$
0.330216 + 0.943905i $$0.392878\pi$$
$$338$$ 0 0
$$339$$ −1.63773 −0.0889491
$$340$$ 1.69202 0.0917627
$$341$$ 29.7832 1.61285
$$342$$ 2.48427 0.134334
$$343$$ 19.1551 1.03428
$$344$$ 2.89977 0.156345
$$345$$ 2.06100 0.110960
$$346$$ −2.12498 −0.114240
$$347$$ 23.1497 1.24274 0.621371 0.783516i $$-0.286576\pi$$
0.621371 + 0.783516i $$0.286576\pi$$
$$348$$ 10.7017 0.573672
$$349$$ 22.1957 1.18811 0.594053 0.804426i $$-0.297527\pi$$
0.594053 + 0.804426i $$0.297527\pi$$
$$350$$ −3.85325 −0.205965
$$351$$ 0 0
$$352$$ 26.3013 1.40186
$$353$$ 5.07069 0.269885 0.134943 0.990853i $$-0.456915\pi$$
0.134943 + 0.990853i $$0.456915\pi$$
$$354$$ −6.09783 −0.324096
$$355$$ 1.41252 0.0749687
$$356$$ 0.246980 0.0130899
$$357$$ 6.66487 0.352743
$$358$$ −1.52888 −0.0808036
$$359$$ 16.6746 0.880050 0.440025 0.897986i $$-0.354970\pi$$
0.440025 + 0.897986i $$0.354970\pi$$
$$360$$ 0.417895 0.0220250
$$361$$ 12.1599 0.639995
$$362$$ 6.00192 0.315454
$$363$$ 20.9541 1.09980
$$364$$ 0 0
$$365$$ −1.81700 −0.0951063
$$366$$ 3.78986 0.198099
$$367$$ −1.17928 −0.0615577 −0.0307789 0.999526i $$-0.509799\pi$$
−0.0307789 + 0.999526i $$0.509799\pi$$
$$368$$ 23.7899 1.24013
$$369$$ −0.445042 −0.0231680
$$370$$ 0.351519 0.0182746
$$371$$ 1.85995 0.0965639
$$372$$ 9.49396 0.492239
$$373$$ −30.0925 −1.55813 −0.779064 0.626944i $$-0.784305\pi$$
−0.779064 + 0.626944i $$0.784305\pi$$
$$374$$ −9.56465 −0.494576
$$375$$ −2.45473 −0.126762
$$376$$ 11.3967 0.587741
$$377$$ 0 0
$$378$$ 0.780167 0.0401275
$$379$$ 19.1631 0.984345 0.492172 0.870498i $$-0.336203\pi$$
0.492172 + 0.870498i $$0.336203\pi$$
$$380$$ 2.48427 0.127440
$$381$$ 10.7995 0.553277
$$382$$ 0.579121 0.0296304
$$383$$ 15.3884 0.786308 0.393154 0.919473i $$-0.371384\pi$$
0.393154 + 0.919473i $$0.371384\pi$$
$$384$$ 10.9215 0.557338
$$385$$ 2.44743 0.124733
$$386$$ −4.09246 −0.208301
$$387$$ 1.71379 0.0871169
$$388$$ −24.6679 −1.25232
$$389$$ −24.0315 −1.21844 −0.609222 0.793000i $$-0.708518\pi$$
−0.609222 + 0.793000i $$0.708518\pi$$
$$390$$ 0 0
$$391$$ −31.7265 −1.60448
$$392$$ −6.64443 −0.335594
$$393$$ 0.907542 0.0457794
$$394$$ 1.83015 0.0922016
$$395$$ 1.10023 0.0553585
$$396$$ 10.1860 0.511865
$$397$$ −29.6015 −1.48566 −0.742828 0.669482i $$-0.766516\pi$$
−0.742828 + 0.669482i $$0.766516\pi$$
$$398$$ 11.0248 0.552621
$$399$$ 9.78554 0.489890
$$400$$ −14.0804 −0.704019
$$401$$ 21.1032 1.05384 0.526922 0.849914i $$-0.323346\pi$$
0.526922 + 0.849914i $$0.323346\pi$$
$$402$$ 2.65279 0.132309
$$403$$ 0 0
$$404$$ 9.75063 0.485112
$$405$$ 0.246980 0.0122725
$$406$$ −4.63342 −0.229953
$$407$$ 18.0780 0.896092
$$408$$ −6.43296 −0.318479
$$409$$ 36.0224 1.78119 0.890596 0.454796i $$-0.150288\pi$$
0.890596 + 0.454796i $$0.150288\pi$$
$$410$$ 0.0489173 0.00241586
$$411$$ 9.54825 0.470981
$$412$$ −24.7875 −1.22119
$$413$$ −24.0194 −1.18192
$$414$$ −3.71379 −0.182523
$$415$$ 2.51573 0.123492
$$416$$ 0 0
$$417$$ −4.09246 −0.200409
$$418$$ −14.0431 −0.686869
$$419$$ 5.96854 0.291582 0.145791 0.989315i $$-0.453427\pi$$
0.145791 + 0.989315i $$0.453427\pi$$
$$420$$ 0.780167 0.0380683
$$421$$ 2.09544 0.102126 0.0510628 0.998695i $$-0.483739\pi$$
0.0510628 + 0.998695i $$0.483739\pi$$
$$422$$ 2.64310 0.128664
$$423$$ 6.73556 0.327494
$$424$$ −1.79523 −0.0871842
$$425$$ 18.7778 0.910856
$$426$$ −2.54527 −0.123319
$$427$$ 14.9282 0.722429
$$428$$ 23.0954 1.11636
$$429$$ 0 0
$$430$$ −0.188374 −0.00908418
$$431$$ 2.88577 0.139003 0.0695014 0.997582i $$-0.477859\pi$$
0.0695014 + 0.997582i $$0.477859\pi$$
$$432$$ 2.85086 0.137162
$$433$$ −12.6485 −0.607847 −0.303924 0.952696i $$-0.598297\pi$$
−0.303924 + 0.952696i $$0.598297\pi$$
$$434$$ −4.11051 −0.197311
$$435$$ −1.46681 −0.0703283
$$436$$ 21.8877 1.04823
$$437$$ −46.5816 −2.22830
$$438$$ 3.27413 0.156444
$$439$$ −10.9269 −0.521513 −0.260757 0.965405i $$-0.583972\pi$$
−0.260757 + 0.965405i $$0.583972\pi$$
$$440$$ −2.36227 −0.112617
$$441$$ −3.92692 −0.186996
$$442$$ 0 0
$$443$$ −19.2403 −0.914133 −0.457067 0.889433i $$-0.651100\pi$$
−0.457067 + 0.889433i $$0.651100\pi$$
$$444$$ 5.76271 0.273486
$$445$$ −0.0338518 −0.00160473
$$446$$ 6.32006 0.299264
$$447$$ 15.3884 0.727844
$$448$$ 6.36526 0.300730
$$449$$ 28.8200 1.36010 0.680050 0.733166i $$-0.261958\pi$$
0.680050 + 0.733166i $$0.261958\pi$$
$$450$$ 2.19806 0.103618
$$451$$ 2.51573 0.118461
$$452$$ 2.95108 0.138807
$$453$$ 3.67456 0.172646
$$454$$ 7.12392 0.334342
$$455$$ 0 0
$$456$$ −9.44504 −0.442305
$$457$$ 18.0707 0.845311 0.422656 0.906290i $$-0.361098\pi$$
0.422656 + 0.906290i $$0.361098\pi$$
$$458$$ 0.819396 0.0382879
$$459$$ −3.80194 −0.177459
$$460$$ −3.71379 −0.173156
$$461$$ 7.56763 0.352460 0.176230 0.984349i $$-0.443610\pi$$
0.176230 + 0.984349i $$0.443610\pi$$
$$462$$ −4.41013 −0.205178
$$463$$ −35.3551 −1.64309 −0.821545 0.570143i $$-0.806888\pi$$
−0.821545 + 0.570143i $$0.806888\pi$$
$$464$$ −16.9312 −0.786013
$$465$$ −1.30127 −0.0603451
$$466$$ 10.4252 0.482938
$$467$$ 13.0000 0.601568 0.300784 0.953692i $$-0.402752\pi$$
0.300784 + 0.953692i $$0.402752\pi$$
$$468$$ 0 0
$$469$$ 10.4494 0.482506
$$470$$ −0.740348 −0.0341497
$$471$$ −4.87800 −0.224766
$$472$$ 23.1836 1.06711
$$473$$ −9.68771 −0.445441
$$474$$ −1.98254 −0.0910612
$$475$$ 27.5700 1.26500
$$476$$ −12.0097 −0.550463
$$477$$ −1.06100 −0.0485798
$$478$$ 6.49827 0.297224
$$479$$ 25.5265 1.16633 0.583167 0.812352i $$-0.301813\pi$$
0.583167 + 0.812352i $$0.301813\pi$$
$$480$$ −1.14914 −0.0524510
$$481$$ 0 0
$$482$$ 3.84117 0.174960
$$483$$ −14.6286 −0.665626
$$484$$ −37.7579 −1.71627
$$485$$ 3.38106 0.153526
$$486$$ −0.445042 −0.0201875
$$487$$ −16.0073 −0.725360 −0.362680 0.931914i $$-0.618138\pi$$
−0.362680 + 0.931914i $$0.618138\pi$$
$$488$$ −14.4088 −0.652256
$$489$$ −8.63102 −0.390308
$$490$$ 0.431632 0.0194992
$$491$$ 20.7385 0.935917 0.467959 0.883750i $$-0.344990\pi$$
0.467959 + 0.883750i $$0.344990\pi$$
$$492$$ 0.801938 0.0361541
$$493$$ 22.5797 1.01694
$$494$$ 0 0
$$495$$ −1.39612 −0.0627511
$$496$$ −15.0204 −0.674438
$$497$$ −10.0258 −0.449719
$$498$$ −4.53319 −0.203137
$$499$$ 8.06770 0.361160 0.180580 0.983560i $$-0.442203\pi$$
0.180580 + 0.983560i $$0.442203\pi$$
$$500$$ 4.42327 0.197815
$$501$$ −9.46980 −0.423079
$$502$$ 1.69202 0.0755186
$$503$$ −30.2422 −1.34843 −0.674216 0.738534i $$-0.735518\pi$$
−0.674216 + 0.738534i $$0.735518\pi$$
$$504$$ −2.96615 −0.132123
$$505$$ −1.33645 −0.0594714
$$506$$ 20.9933 0.933266
$$507$$ 0 0
$$508$$ −19.4601 −0.863403
$$509$$ −16.0495 −0.711382 −0.355691 0.934604i $$-0.615754\pi$$
−0.355691 + 0.934604i $$0.615754\pi$$
$$510$$ 0.417895 0.0185047
$$511$$ 12.8968 0.570520
$$512$$ −22.9119 −1.01257
$$513$$ −5.58211 −0.246456
$$514$$ 9.16613 0.404301
$$515$$ 3.39745 0.149710
$$516$$ −3.08815 −0.135948
$$517$$ −38.0747 −1.67452
$$518$$ −2.49502 −0.109625
$$519$$ 4.77479 0.209590
$$520$$ 0 0
$$521$$ −2.69309 −0.117986 −0.0589931 0.998258i $$-0.518789\pi$$
−0.0589931 + 0.998258i $$0.518789\pi$$
$$522$$ 2.64310 0.115686
$$523$$ 35.3957 1.54774 0.773872 0.633342i $$-0.218317\pi$$
0.773872 + 0.633342i $$0.218317\pi$$
$$524$$ −1.63533 −0.0714399
$$525$$ 8.65817 0.377874
$$526$$ −0.148080 −0.00645659
$$527$$ 20.0315 0.872584
$$528$$ −16.1153 −0.701328
$$529$$ 46.6359 2.02765
$$530$$ 0.116621 0.00506569
$$531$$ 13.7017 0.594604
$$532$$ −17.6329 −0.764485
$$533$$ 0 0
$$534$$ 0.0609989 0.00263968
$$535$$ −3.16554 −0.136858
$$536$$ −10.0858 −0.435638
$$537$$ 3.43535 0.148246
$$538$$ −12.1511 −0.523870
$$539$$ 22.1981 0.956138
$$540$$ −0.445042 −0.0191516
$$541$$ 34.7338 1.49332 0.746660 0.665205i $$-0.231656\pi$$
0.746660 + 0.665205i $$0.231656\pi$$
$$542$$ 12.4547 0.534976
$$543$$ −13.4862 −0.578748
$$544$$ 17.6896 0.758437
$$545$$ −3.00000 −0.128506
$$546$$ 0 0
$$547$$ −26.1183 −1.11674 −0.558368 0.829593i $$-0.688572\pi$$
−0.558368 + 0.829593i $$0.688572\pi$$
$$548$$ −17.2054 −0.734976
$$549$$ −8.51573 −0.363442
$$550$$ −12.4252 −0.529812
$$551$$ 33.1521 1.41233
$$552$$ 14.1196 0.600970
$$553$$ −7.80923 −0.332082
$$554$$ 0.936017 0.0397676
$$555$$ −0.789856 −0.0335275
$$556$$ 7.37435 0.312742
$$557$$ −24.7748 −1.04974 −0.524871 0.851182i $$-0.675886\pi$$
−0.524871 + 0.851182i $$0.675886\pi$$
$$558$$ 2.34481 0.0992639
$$559$$ 0 0
$$560$$ −1.23431 −0.0521590
$$561$$ 21.4916 0.907375
$$562$$ 12.1207 0.511280
$$563$$ 5.26098 0.221724 0.110862 0.993836i $$-0.464639\pi$$
0.110862 + 0.993836i $$0.464639\pi$$
$$564$$ −12.1371 −0.511063
$$565$$ −0.404485 −0.0170168
$$566$$ −2.35152 −0.0988417
$$567$$ −1.75302 −0.0736199
$$568$$ 9.67696 0.406036
$$569$$ −33.7458 −1.41470 −0.707350 0.706864i $$-0.750109\pi$$
−0.707350 + 0.706864i $$0.750109\pi$$
$$570$$ 0.613564 0.0256994
$$571$$ 23.0887 0.966234 0.483117 0.875556i $$-0.339505\pi$$
0.483117 + 0.875556i $$0.339505\pi$$
$$572$$ 0 0
$$573$$ −1.30127 −0.0543615
$$574$$ −0.347207 −0.0144921
$$575$$ −41.2150 −1.71879
$$576$$ −3.63102 −0.151293
$$577$$ −3.57002 −0.148622 −0.0743110 0.997235i $$-0.523676\pi$$
−0.0743110 + 0.997235i $$0.523676\pi$$
$$578$$ 1.13275 0.0471162
$$579$$ 9.19567 0.382159
$$580$$ 2.64310 0.109749
$$581$$ −17.8562 −0.740801
$$582$$ −6.09246 −0.252541
$$583$$ 5.99761 0.248396
$$584$$ −12.4480 −0.515103
$$585$$ 0 0
$$586$$ 14.5362 0.600484
$$587$$ 11.4625 0.473108 0.236554 0.971618i $$-0.423982\pi$$
0.236554 + 0.971618i $$0.423982\pi$$
$$588$$ 7.07606 0.291812
$$589$$ 29.4107 1.21185
$$590$$ −1.50604 −0.0620027
$$591$$ −4.11231 −0.169158
$$592$$ −9.11721 −0.374715
$$593$$ 21.8538 0.897430 0.448715 0.893675i $$-0.351882\pi$$
0.448715 + 0.893675i $$0.351882\pi$$
$$594$$ 2.51573 0.103222
$$595$$ 1.64609 0.0674830
$$596$$ −27.7289 −1.13582
$$597$$ −24.7724 −1.01387
$$598$$ 0 0
$$599$$ 27.0573 1.10553 0.552765 0.833337i $$-0.313573\pi$$
0.552765 + 0.833337i $$0.313573\pi$$
$$600$$ −8.35690 −0.341169
$$601$$ 10.8780 0.443723 0.221861 0.975078i $$-0.428787\pi$$
0.221861 + 0.975078i $$0.428787\pi$$
$$602$$ 1.33704 0.0544939
$$603$$ −5.96077 −0.242741
$$604$$ −6.62133 −0.269418
$$605$$ 5.17523 0.210403
$$606$$ 2.40821 0.0978267
$$607$$ −29.6359 −1.20289 −0.601443 0.798916i $$-0.705407\pi$$
−0.601443 + 0.798916i $$0.705407\pi$$
$$608$$ 25.9724 1.05332
$$609$$ 10.4112 0.421883
$$610$$ 0.936017 0.0378982
$$611$$ 0 0
$$612$$ 6.85086 0.276929
$$613$$ −10.2343 −0.413360 −0.206680 0.978409i $$-0.566266\pi$$
−0.206680 + 0.978409i $$0.566266\pi$$
$$614$$ −9.23968 −0.372883
$$615$$ −0.109916 −0.00443225
$$616$$ 16.7670 0.675563
$$617$$ −26.2828 −1.05810 −0.529052 0.848590i $$-0.677452\pi$$
−0.529052 + 0.848590i $$0.677452\pi$$
$$618$$ −6.12200 −0.246263
$$619$$ −29.0834 −1.16896 −0.584479 0.811408i $$-0.698701\pi$$
−0.584479 + 0.811408i $$0.698701\pi$$
$$620$$ 2.34481 0.0941700
$$621$$ 8.34481 0.334866
$$622$$ 5.02954 0.201666
$$623$$ 0.240275 0.00962641
$$624$$ 0 0
$$625$$ 24.0887 0.963549
$$626$$ 1.90110 0.0759833
$$627$$ 31.5545 1.26016
$$628$$ 8.78986 0.350753
$$629$$ 12.1588 0.484804
$$630$$ 0.192685 0.00767677
$$631$$ −25.4480 −1.01307 −0.506535 0.862219i $$-0.669074\pi$$
−0.506535 + 0.862219i $$0.669074\pi$$
$$632$$ 7.53750 0.299826
$$633$$ −5.93900 −0.236054
$$634$$ 6.88816 0.273564
$$635$$ 2.66727 0.105847
$$636$$ 1.91185 0.0758099
$$637$$ 0 0
$$638$$ −14.9409 −0.591517
$$639$$ 5.71917 0.226247
$$640$$ 2.69740 0.106624
$$641$$ −26.7409 −1.05620 −0.528102 0.849181i $$-0.677096\pi$$
−0.528102 + 0.849181i $$0.677096\pi$$
$$642$$ 5.70410 0.225123
$$643$$ 32.9614 1.29987 0.649935 0.759990i $$-0.274796\pi$$
0.649935 + 0.759990i $$0.274796\pi$$
$$644$$ 26.3599 1.03872
$$645$$ 0.423272 0.0166663
$$646$$ −9.44504 −0.371610
$$647$$ 34.4946 1.35612 0.678060 0.735006i $$-0.262821\pi$$
0.678060 + 0.735006i $$0.262821\pi$$
$$648$$ 1.69202 0.0664689
$$649$$ −77.4529 −3.04029
$$650$$ 0 0
$$651$$ 9.23623 0.361996
$$652$$ 15.5526 0.609085
$$653$$ 36.1517 1.41472 0.707362 0.706852i $$-0.249885\pi$$
0.707362 + 0.706852i $$0.249885\pi$$
$$654$$ 5.40581 0.211384
$$655$$ 0.224144 0.00875805
$$656$$ −1.26875 −0.0495364
$$657$$ −7.35690 −0.287020
$$658$$ 5.25487 0.204856
$$659$$ −6.81700 −0.265553 −0.132776 0.991146i $$-0.542389\pi$$
−0.132776 + 0.991146i $$0.542389\pi$$
$$660$$ 2.51573 0.0979246
$$661$$ 10.8944 0.423743 0.211871 0.977298i $$-0.432044\pi$$
0.211871 + 0.977298i $$0.432044\pi$$
$$662$$ 2.70065 0.104964
$$663$$ 0 0
$$664$$ 17.2349 0.668844
$$665$$ 2.41683 0.0937206
$$666$$ 1.42327 0.0551507
$$667$$ −49.5599 −1.91897
$$668$$ 17.0640 0.660225
$$669$$ −14.2010 −0.549044
$$670$$ 0.655186 0.0253120
$$671$$ 48.1377 1.85833
$$672$$ 8.15644 0.314642
$$673$$ 20.7385 0.799412 0.399706 0.916643i $$-0.369112\pi$$
0.399706 + 0.916643i $$0.369112\pi$$
$$674$$ −5.39565 −0.207833
$$675$$ −4.93900 −0.190102
$$676$$ 0 0
$$677$$ −25.5786 −0.983067 −0.491534 0.870859i $$-0.663564\pi$$
−0.491534 + 0.870859i $$0.663564\pi$$
$$678$$ 0.728857 0.0279916
$$679$$ −23.9982 −0.920966
$$680$$ −1.58881 −0.0609281
$$681$$ −16.0073 −0.613401
$$682$$ −13.2547 −0.507551
$$683$$ −21.6310 −0.827688 −0.413844 0.910348i $$-0.635814\pi$$
−0.413844 + 0.910348i $$0.635814\pi$$
$$684$$ 10.0586 0.384600
$$685$$ 2.35822 0.0901031
$$686$$ −8.52483 −0.325479
$$687$$ −1.84117 −0.0702449
$$688$$ 4.88577 0.186268
$$689$$ 0 0
$$690$$ −0.917231 −0.0349184
$$691$$ −2.62996 −0.100048 −0.0500242 0.998748i $$-0.515930\pi$$
−0.0500242 + 0.998748i $$0.515930\pi$$
$$692$$ −8.60388 −0.327070
$$693$$ 9.90946 0.376429
$$694$$ −10.3026 −0.391081
$$695$$ −1.01075 −0.0383401
$$696$$ −10.0489 −0.380903
$$697$$ 1.69202 0.0640899
$$698$$ −9.87800 −0.373888
$$699$$ −23.4252 −0.886022
$$700$$ −15.6015 −0.589681
$$701$$ 40.0925 1.51427 0.757136 0.653258i $$-0.226598\pi$$
0.757136 + 0.653258i $$0.226598\pi$$
$$702$$ 0 0
$$703$$ 17.8519 0.673298
$$704$$ 20.5254 0.773581
$$705$$ 1.66355 0.0626528
$$706$$ −2.25667 −0.0849308
$$707$$ 9.48593 0.356755
$$708$$ −24.6896 −0.927893
$$709$$ −23.2097 −0.871657 −0.435829 0.900030i $$-0.643545\pi$$
−0.435829 + 0.900030i $$0.643545\pi$$
$$710$$ −0.628630 −0.0235921
$$711$$ 4.45473 0.167065
$$712$$ −0.231914 −0.00869135
$$713$$ −43.9667 −1.64657
$$714$$ −2.96615 −0.111005
$$715$$ 0 0
$$716$$ −6.19029 −0.231342
$$717$$ −14.6015 −0.545302
$$718$$ −7.42088 −0.276945
$$719$$ −26.0146 −0.970181 −0.485090 0.874464i $$-0.661213\pi$$
−0.485090 + 0.874464i $$0.661213\pi$$
$$720$$ 0.704103 0.0262404
$$721$$ −24.1146 −0.898073
$$722$$ −5.41166 −0.201401
$$723$$ −8.63102 −0.320991
$$724$$ 24.3013 0.903150
$$725$$ 29.3327 1.08939
$$726$$ −9.32544 −0.346099
$$727$$ −16.5472 −0.613701 −0.306851 0.951758i $$-0.599275\pi$$
−0.306851 + 0.951758i $$0.599275\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0.808643 0.0299292
$$731$$ −6.51573 −0.240993
$$732$$ 15.3448 0.567161
$$733$$ −18.8750 −0.697165 −0.348582 0.937278i $$-0.613337\pi$$
−0.348582 + 0.937278i $$0.613337\pi$$
$$734$$ 0.524827 0.0193717
$$735$$ −0.969869 −0.0357742
$$736$$ −38.8267 −1.43117
$$737$$ 33.6950 1.24117
$$738$$ 0.198062 0.00729077
$$739$$ −47.3239 −1.74084 −0.870419 0.492312i $$-0.836152\pi$$
−0.870419 + 0.492312i $$0.836152\pi$$
$$740$$ 1.42327 0.0523205
$$741$$ 0 0
$$742$$ −0.827757 −0.0303879
$$743$$ 8.88769 0.326058 0.163029 0.986621i $$-0.447874\pi$$
0.163029 + 0.986621i $$0.447874\pi$$
$$744$$ −8.91484 −0.326834
$$745$$ 3.80061 0.139244
$$746$$ 13.3924 0.490331
$$747$$ 10.1860 0.372686
$$748$$ −38.7265 −1.41598
$$749$$ 22.4685 0.820980
$$750$$ 1.09246 0.0398909
$$751$$ 0.710808 0.0259377 0.0129689 0.999916i $$-0.495872\pi$$
0.0129689 + 0.999916i $$0.495872\pi$$
$$752$$ 19.2021 0.700229
$$753$$ −3.80194 −0.138550
$$754$$ 0 0
$$755$$ 0.907542 0.0330288
$$756$$ 3.15883 0.114886
$$757$$ 9.78554 0.355662 0.177831 0.984061i $$-0.443092\pi$$
0.177831 + 0.984061i $$0.443092\pi$$
$$758$$ −8.52840 −0.309766
$$759$$ −47.1715 −1.71222
$$760$$ −2.33273 −0.0846171
$$761$$ 18.8810 0.684435 0.342218 0.939621i $$-0.388822\pi$$
0.342218 + 0.939621i $$0.388822\pi$$
$$762$$ −4.80625 −0.174112
$$763$$ 21.2935 0.770877
$$764$$ 2.34481 0.0848324
$$765$$ −0.939001 −0.0339497
$$766$$ −6.84846 −0.247445
$$767$$ 0 0
$$768$$ 2.40150 0.0866567
$$769$$ 12.4349 0.448413 0.224207 0.974542i $$-0.428021\pi$$
0.224207 + 0.974542i $$0.428021\pi$$
$$770$$ −1.08921 −0.0392524
$$771$$ −20.5961 −0.741751
$$772$$ −16.5700 −0.596368
$$773$$ −45.6746 −1.64280 −0.821400 0.570353i $$-0.806807\pi$$
−0.821400 + 0.570353i $$0.806807\pi$$
$$774$$ −0.762709 −0.0274150
$$775$$ 26.0224 0.934751
$$776$$ 23.1631 0.831508
$$777$$ 5.60627 0.201124
$$778$$ 10.6950 0.383435
$$779$$ 2.48427 0.0890082
$$780$$ 0 0
$$781$$ −32.3293 −1.15683
$$782$$ 14.1196 0.504916
$$783$$ −5.93900 −0.212243
$$784$$ −11.1951 −0.399824
$$785$$ −1.20477 −0.0430000
$$786$$ −0.403894 −0.0144064
$$787$$ 4.51871 0.161075 0.0805374 0.996752i $$-0.474336\pi$$
0.0805374 + 0.996752i $$0.474336\pi$$
$$788$$ 7.41013 0.263975
$$789$$ 0.332733 0.0118456
$$790$$ −0.489647 −0.0174209
$$791$$ 2.87097 0.102080
$$792$$ −9.56465 −0.339865
$$793$$ 0 0
$$794$$ 13.1739 0.467524
$$795$$ −0.262045 −0.00929378
$$796$$ 44.6383 1.58216
$$797$$ −28.7391 −1.01799 −0.508996 0.860769i $$-0.669983\pi$$
−0.508996 + 0.860769i $$0.669983\pi$$
$$798$$ −4.35498 −0.154165
$$799$$ −25.6082 −0.905953
$$800$$ 22.9801 0.812471
$$801$$ −0.137063 −0.00484289
$$802$$ −9.39181 −0.331636
$$803$$ 41.5870 1.46757
$$804$$ 10.7409 0.378804
$$805$$ −3.61297 −0.127341
$$806$$ 0 0
$$807$$ 27.3032 0.961118
$$808$$ −9.15585 −0.322102
$$809$$ 5.42891 0.190870 0.0954352 0.995436i $$-0.469576\pi$$
0.0954352 + 0.995436i $$0.469576\pi$$
$$810$$ −0.109916 −0.00386206
$$811$$ −0.629104 −0.0220908 −0.0110454 0.999939i $$-0.503516\pi$$
−0.0110454 + 0.999939i $$0.503516\pi$$
$$812$$ −18.7603 −0.658358
$$813$$ −27.9855 −0.981495
$$814$$ −8.04546 −0.281993
$$815$$ −2.13169 −0.0746697
$$816$$ −10.8388 −0.379433
$$817$$ −9.56657 −0.334692
$$818$$ −16.0315 −0.560527
$$819$$ 0 0
$$820$$ 0.198062 0.00691663
$$821$$ −36.2640 −1.26562 −0.632811 0.774307i $$-0.718099\pi$$
−0.632811 + 0.774307i $$0.718099\pi$$
$$822$$ −4.24937 −0.148214
$$823$$ 41.7396 1.45495 0.727476 0.686133i $$-0.240693\pi$$
0.727476 + 0.686133i $$0.240693\pi$$
$$824$$ 23.2755 0.810839
$$825$$ 27.9191 0.972020
$$826$$ 10.6896 0.371940
$$827$$ −38.1997 −1.32833 −0.664167 0.747584i $$-0.731214\pi$$
−0.664167 + 0.747584i $$0.731214\pi$$
$$828$$ −15.0368 −0.522566
$$829$$ 15.9788 0.554967 0.277484 0.960730i $$-0.410500\pi$$
0.277484 + 0.960730i $$0.410500\pi$$
$$830$$ −1.11960 −0.0388621
$$831$$ −2.10321 −0.0729596
$$832$$ 0 0
$$833$$ 14.9299 0.517290
$$834$$ 1.82132 0.0630670
$$835$$ −2.33885 −0.0809391
$$836$$ −56.8592 −1.96652
$$837$$ −5.26875 −0.182115
$$838$$ −2.65625 −0.0917587
$$839$$ 4.63879 0.160149 0.0800744 0.996789i $$-0.474484\pi$$
0.0800744 + 0.996789i $$0.474484\pi$$
$$840$$ −0.732578 −0.0252763
$$841$$ 6.27173 0.216267
$$842$$ −0.932559 −0.0321381
$$843$$ −27.2349 −0.938020
$$844$$ 10.7017 0.368368
$$845$$ 0 0
$$846$$ −2.99761 −0.103060
$$847$$ −36.7329 −1.26216
$$848$$ −3.02475 −0.103870
$$849$$ 5.28382 0.181340
$$850$$ −8.35690 −0.286639
$$851$$ −26.6872 −0.914827
$$852$$ −10.3056 −0.353064
$$853$$ −18.3884 −0.629605 −0.314803 0.949157i $$-0.601938\pi$$
−0.314803 + 0.949157i $$0.601938\pi$$
$$854$$ −6.64370 −0.227343
$$855$$ −1.37867 −0.0471494
$$856$$ −21.6866 −0.741234
$$857$$ 28.1849 0.962778 0.481389 0.876507i $$-0.340132\pi$$
0.481389 + 0.876507i $$0.340132\pi$$
$$858$$ 0 0
$$859$$ 33.3957 1.13944 0.569722 0.821837i $$-0.307051\pi$$
0.569722 + 0.821837i $$0.307051\pi$$
$$860$$ −0.762709 −0.0260082
$$861$$ 0.780167 0.0265880
$$862$$ −1.28429 −0.0437431
$$863$$ 43.0640 1.46592 0.732958 0.680274i $$-0.238139\pi$$
0.732958 + 0.680274i $$0.238139\pi$$
$$864$$ −4.65279 −0.158291
$$865$$ 1.17928 0.0400966
$$866$$ 5.62910 0.191285
$$867$$ −2.54527 −0.0864419
$$868$$ −16.6431 −0.564904
$$869$$ −25.1817 −0.854230
$$870$$ 0.652793 0.0221317
$$871$$ 0 0
$$872$$ −20.5526 −0.695998
$$873$$ 13.6896 0.463324
$$874$$ 20.7308 0.701229
$$875$$ 4.30319 0.145474
$$876$$ 13.2567 0.447901
$$877$$ 30.1702 1.01877 0.509387 0.860537i $$-0.329872\pi$$
0.509387 + 0.860537i $$0.329872\pi$$
$$878$$ 4.86294 0.164116
$$879$$ −32.6625 −1.10168
$$880$$ −3.98015 −0.134171
$$881$$ −3.56273 −0.120031 −0.0600157 0.998197i $$-0.519115\pi$$
−0.0600157 + 0.998197i $$0.519115\pi$$
$$882$$ 1.74764 0.0588462
$$883$$ −10.2088 −0.343554 −0.171777 0.985136i $$-0.554951\pi$$
−0.171777 + 0.985136i $$0.554951\pi$$
$$884$$ 0 0
$$885$$ 3.38404 0.113753
$$886$$ 8.56273 0.287670
$$887$$ −9.33645 −0.313487 −0.156744 0.987639i $$-0.550100\pi$$
−0.156744 + 0.987639i $$0.550100\pi$$
$$888$$ −5.41119 −0.181588
$$889$$ −18.9318 −0.634953
$$890$$ 0.0150655 0.000504996 0
$$891$$ −5.65279 −0.189376
$$892$$ 25.5894 0.856797
$$893$$ −37.5986 −1.25819
$$894$$ −6.84846 −0.229047
$$895$$ 0.848462 0.0283610
$$896$$ −19.1457 −0.639613
$$897$$ 0 0
$$898$$ −12.8261 −0.428013
$$899$$ 31.2911 1.04362
$$900$$ 8.89977 0.296659
$$901$$ 4.03385 0.134387
$$902$$ −1.11960 −0.0372788
$$903$$ −3.00431 −0.0999772
$$904$$ −2.77107 −0.0921644
$$905$$ −3.33081 −0.110720
$$906$$ −1.63533 −0.0543303
$$907$$ −41.0804 −1.36405 −0.682026 0.731328i $$-0.738901\pi$$
−0.682026 + 0.731328i $$0.738901\pi$$
$$908$$ 28.8442 0.957227
$$909$$ −5.41119 −0.179478
$$910$$ 0 0
$$911$$ −18.9705 −0.628519 −0.314260 0.949337i $$-0.601756\pi$$
−0.314260 + 0.949337i $$0.601756\pi$$
$$912$$ −15.9138 −0.526958
$$913$$ −57.5792 −1.90559
$$914$$ −8.04221 −0.266013
$$915$$ −2.10321 −0.0695300
$$916$$ 3.31767 0.109619
$$917$$ −1.59094 −0.0525375
$$918$$ 1.69202 0.0558450
$$919$$ 29.0019 0.956685 0.478343 0.878173i $$-0.341238\pi$$
0.478343 + 0.878173i $$0.341238\pi$$
$$920$$ 3.48725 0.114971
$$921$$ 20.7614 0.684111
$$922$$ −3.36791 −0.110916
$$923$$ 0 0
$$924$$ −17.8562 −0.587427
$$925$$ 15.7952 0.519344
$$926$$ 15.7345 0.517068
$$927$$ 13.7560 0.451806
$$928$$ 27.6329 0.907096
$$929$$ 50.7211 1.66410 0.832052 0.554697i $$-0.187166\pi$$
0.832052 + 0.554697i $$0.187166\pi$$
$$930$$ 0.579121 0.0189901
$$931$$ 21.9205 0.718415
$$932$$ 42.2107 1.38266
$$933$$ −11.3013 −0.369987
$$934$$ −5.78554 −0.189309
$$935$$ 5.30798 0.173589
$$936$$ 0 0
$$937$$ 51.3051 1.67606 0.838032 0.545620i $$-0.183706\pi$$
0.838032 + 0.545620i $$0.183706\pi$$
$$938$$ −4.65040 −0.151841
$$939$$ −4.27173 −0.139403
$$940$$ −2.99761 −0.0977712
$$941$$ 34.7036 1.13131 0.565653 0.824643i $$-0.308624\pi$$
0.565653 + 0.824643i $$0.308624\pi$$
$$942$$ 2.17092 0.0707322
$$943$$ −3.71379 −0.120938
$$944$$ 39.0616 1.27135
$$945$$ −0.432960 −0.0140842
$$946$$ 4.31144 0.140177
$$947$$ 13.0127 0.422855 0.211428 0.977394i $$-0.432189\pi$$
0.211428 + 0.977394i $$0.432189\pi$$
$$948$$ −8.02715 −0.260710
$$949$$ 0 0
$$950$$ −12.2698 −0.398085
$$951$$ −15.4776 −0.501894
$$952$$ 11.2771 0.365493
$$953$$ 26.0151 0.842711 0.421355 0.906896i $$-0.361555\pi$$
0.421355 + 0.906896i $$0.361555\pi$$
$$954$$ 0.472189 0.0152877
$$955$$ −0.321388 −0.0103999
$$956$$ 26.3110 0.850957
$$957$$ 33.5719 1.08523
$$958$$ −11.3604 −0.367036
$$959$$ −16.7383 −0.540507
$$960$$ −0.896789 −0.0289437
$$961$$ −3.24027 −0.104525
$$962$$ 0 0
$$963$$ −12.8170 −0.413022
$$964$$ 15.5526 0.500914
$$965$$ 2.27114 0.0731107
$$966$$ 6.51035 0.209467
$$967$$ −36.6644 −1.17905 −0.589524 0.807751i $$-0.700685\pi$$
−0.589524 + 0.807751i $$0.700685\pi$$
$$968$$ 35.4547 1.13956
$$969$$ 21.2228 0.681775
$$970$$ −1.50471 −0.0483134
$$971$$ −37.8465 −1.21455 −0.607277 0.794490i $$-0.707738\pi$$
−0.607277 + 0.794490i $$0.707738\pi$$
$$972$$ −1.80194 −0.0577972
$$973$$ 7.17416 0.229993
$$974$$ 7.12392 0.228265
$$975$$ 0 0
$$976$$ −24.2771 −0.777091
$$977$$ −28.8998 −0.924586 −0.462293 0.886727i $$-0.652973\pi$$
−0.462293 + 0.886727i $$0.652973\pi$$
$$978$$ 3.84117 0.122827
$$979$$ 0.774791 0.0247624
$$980$$ 1.74764 0.0558264
$$981$$ −12.1468 −0.387816
$$982$$ −9.22952 −0.294526
$$983$$ 19.3991 0.618735 0.309368 0.950942i $$-0.399883\pi$$
0.309368 + 0.950942i $$0.399883\pi$$
$$984$$ −0.753020 −0.0240054
$$985$$ −1.01566 −0.0323615
$$986$$ −10.0489 −0.320023
$$987$$ −11.8076 −0.375839
$$988$$ 0 0
$$989$$ 14.3013 0.454754
$$990$$ 0.621334 0.0197473
$$991$$ −5.18300 −0.164643 −0.0823217 0.996606i $$-0.526233\pi$$
−0.0823217 + 0.996606i $$0.526233\pi$$
$$992$$ 24.5144 0.778333
$$993$$ −6.06829 −0.192572
$$994$$ 4.46191 0.141523
$$995$$ −6.11828 −0.193962
$$996$$ −18.3545 −0.581585
$$997$$ −49.3642 −1.56338 −0.781690 0.623667i $$-0.785642\pi$$
−0.781690 + 0.623667i $$0.785642\pi$$
$$998$$ −3.59047 −0.113654
$$999$$ −3.19806 −0.101182
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 507.2.a.j.1.2 3
3.2 odd 2 1521.2.a.q.1.2 3
4.3 odd 2 8112.2.a.by.1.3 3
13.2 odd 12 507.2.j.h.316.3 12
13.3 even 3 507.2.e.k.22.2 6
13.4 even 6 507.2.e.j.484.2 6
13.5 odd 4 507.2.b.g.337.4 6
13.6 odd 12 507.2.j.h.361.4 12
13.7 odd 12 507.2.j.h.361.3 12
13.8 odd 4 507.2.b.g.337.3 6
13.9 even 3 507.2.e.k.484.2 6
13.10 even 6 507.2.e.j.22.2 6
13.11 odd 12 507.2.j.h.316.4 12
13.12 even 2 507.2.a.k.1.2 yes 3
39.5 even 4 1521.2.b.m.1351.3 6
39.8 even 4 1521.2.b.m.1351.4 6
39.38 odd 2 1521.2.a.p.1.2 3
52.51 odd 2 8112.2.a.cf.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.2 3 1.1 even 1 trivial
507.2.a.k.1.2 yes 3 13.12 even 2
507.2.b.g.337.3 6 13.8 odd 4
507.2.b.g.337.4 6 13.5 odd 4
507.2.e.j.22.2 6 13.10 even 6
507.2.e.j.484.2 6 13.4 even 6
507.2.e.k.22.2 6 13.3 even 3
507.2.e.k.484.2 6 13.9 even 3
507.2.j.h.316.3 12 13.2 odd 12
507.2.j.h.316.4 12 13.11 odd 12
507.2.j.h.361.3 12 13.7 odd 12
507.2.j.h.361.4 12 13.6 odd 12
1521.2.a.p.1.2 3 39.38 odd 2
1521.2.a.q.1.2 3 3.2 odd 2
1521.2.b.m.1351.3 6 39.5 even 4
1521.2.b.m.1351.4 6 39.8 even 4
8112.2.a.by.1.3 3 4.3 odd 2
8112.2.a.cf.1.1 3 52.51 odd 2