# Properties

 Label 2299.2.a.n.1.2 Level $2299$ Weight $2$ Character 2299.1 Self dual yes Analytic conductor $18.358$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2299,2,Mod(1,2299)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2299, 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("2299.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2299 = 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2299.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: $$18.3576074247$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.246832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2$$ x^5 - 2*x^4 - 5*x^3 + 6*x^2 + 7*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.51908$$ of defining polynomial Character $$\chi$$ $$=$$ 2299.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.82669 q^{2} -0.563416 q^{3} +1.33679 q^{4} +2.34577 q^{5} +1.02918 q^{6} -1.69239 q^{7} +1.21147 q^{8} -2.68256 q^{9} +O(q^{10})$$ $$q-1.82669 q^{2} -0.563416 q^{3} +1.33679 q^{4} +2.34577 q^{5} +1.02918 q^{6} -1.69239 q^{7} +1.21147 q^{8} -2.68256 q^{9} -4.28499 q^{10} -0.753170 q^{12} -4.11168 q^{13} +3.09147 q^{14} -1.32164 q^{15} -4.88657 q^{16} +6.16499 q^{17} +4.90021 q^{18} +1.00000 q^{19} +3.13581 q^{20} +0.953520 q^{21} +3.52199 q^{23} -0.682563 q^{24} +0.502638 q^{25} +7.51076 q^{26} +3.20164 q^{27} -2.26238 q^{28} +8.10336 q^{29} +2.41423 q^{30} -2.30144 q^{31} +6.50330 q^{32} -11.2615 q^{34} -3.96996 q^{35} -3.58603 q^{36} -6.56016 q^{37} -1.82669 q^{38} +2.31659 q^{39} +2.84184 q^{40} -7.75013 q^{41} -1.74178 q^{42} -7.75102 q^{43} -6.29268 q^{45} -6.43359 q^{46} -10.8969 q^{47} +2.75317 q^{48} -4.13581 q^{49} -0.918163 q^{50} -3.47345 q^{51} -5.49647 q^{52} +7.93511 q^{53} -5.84841 q^{54} -2.05029 q^{56} -0.563416 q^{57} -14.8023 q^{58} +10.9247 q^{59} -1.76676 q^{60} +4.51162 q^{61} +4.20401 q^{62} +4.53995 q^{63} -2.10636 q^{64} -9.64506 q^{65} +14.7201 q^{67} +8.24132 q^{68} -1.98435 q^{69} +7.25189 q^{70} +3.12026 q^{71} -3.24985 q^{72} -11.5827 q^{73} +11.9834 q^{74} -0.283194 q^{75} +1.33679 q^{76} -4.23168 q^{78} -4.96184 q^{79} -11.4628 q^{80} +6.24383 q^{81} +14.1571 q^{82} +1.82905 q^{83} +1.27466 q^{84} +14.4617 q^{85} +14.1587 q^{86} -4.56556 q^{87} -9.37496 q^{89} +11.4948 q^{90} +6.95858 q^{91} +4.70818 q^{92} +1.29666 q^{93} +19.9053 q^{94} +2.34577 q^{95} -3.66406 q^{96} -10.9937 q^{97} +7.55484 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10})$$ 5 * q - 2 * q^2 + q^3 + 6 * q^4 - 5 * q^5 + 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 $$5 q - 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} + 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} - 12 q^{10} + 6 q^{12} - 4 q^{13} - 14 q^{14} + 3 q^{15} + 8 q^{16} + 4 q^{17} + 20 q^{18} + 5 q^{19} - 8 q^{20} - 10 q^{21} + 3 q^{23} + 14 q^{24} + 6 q^{25} - 6 q^{26} - 11 q^{27} + 10 q^{28} - 10 q^{29} - 6 q^{30} + 11 q^{31} - 14 q^{32} - 4 q^{34} + 8 q^{35} - 26 q^{36} + q^{37} - 2 q^{38} - 2 q^{39} + 16 q^{40} - 2 q^{41} - 16 q^{42} - 20 q^{43} - 28 q^{45} + 4 q^{46} - 20 q^{47} + 4 q^{48} + 3 q^{49} + 32 q^{50} - 24 q^{51} - 6 q^{52} - 14 q^{53} - 16 q^{54} - 38 q^{56} + q^{57} - 6 q^{58} + 3 q^{59} - 40 q^{60} + 10 q^{61} + 6 q^{62} - 24 q^{63} + 9 q^{67} - 24 q^{68} - 5 q^{69} + 50 q^{70} + 23 q^{71} + 12 q^{72} - 8 q^{74} - 18 q^{75} + 6 q^{76} - 22 q^{78} - 44 q^{79} - 18 q^{80} + q^{81} - 30 q^{82} + 14 q^{83} - 14 q^{84} + 12 q^{85} + 52 q^{86} - 28 q^{87} - 27 q^{89} - 26 q^{90} + 24 q^{91} + 58 q^{92} - 27 q^{93} + 8 q^{94} - 5 q^{95} - 50 q^{96} + 15 q^{97} + 10 q^{98}+O(q^{100})$$ 5 * q - 2 * q^2 + q^3 + 6 * q^4 - 5 * q^5 + 2 * q^6 - 6 * q^7 - 6 * q^8 + 4 * q^9 - 12 * q^10 + 6 * q^12 - 4 * q^13 - 14 * q^14 + 3 * q^15 + 8 * q^16 + 4 * q^17 + 20 * q^18 + 5 * q^19 - 8 * q^20 - 10 * q^21 + 3 * q^23 + 14 * q^24 + 6 * q^25 - 6 * q^26 - 11 * q^27 + 10 * q^28 - 10 * q^29 - 6 * q^30 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 - 26 * q^36 + q^37 - 2 * q^38 - 2 * q^39 + 16 * q^40 - 2 * q^41 - 16 * q^42 - 20 * q^43 - 28 * q^45 + 4 * q^46 - 20 * q^47 + 4 * q^48 + 3 * q^49 + 32 * q^50 - 24 * q^51 - 6 * q^52 - 14 * q^53 - 16 * q^54 - 38 * q^56 + q^57 - 6 * q^58 + 3 * q^59 - 40 * q^60 + 10 * q^61 + 6 * q^62 - 24 * q^63 + 9 * q^67 - 24 * q^68 - 5 * q^69 + 50 * q^70 + 23 * q^71 + 12 * q^72 - 8 * q^74 - 18 * q^75 + 6 * q^76 - 22 * q^78 - 44 * q^79 - 18 * q^80 + q^81 - 30 * q^82 + 14 * q^83 - 14 * q^84 + 12 * q^85 + 52 * q^86 - 28 * q^87 - 27 * q^89 - 26 * q^90 + 24 * q^91 + 58 * q^92 - 27 * q^93 + 8 * q^94 - 5 * q^95 - 50 * q^96 + 15 * q^97 + 10 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.82669 −1.29166 −0.645832 0.763479i $$-0.723489\pi$$
−0.645832 + 0.763479i $$0.723489\pi$$
$$3$$ −0.563416 −0.325288 −0.162644 0.986685i $$-0.552002\pi$$
−0.162644 + 0.986685i $$0.552002\pi$$
$$4$$ 1.33679 0.668396
$$5$$ 2.34577 1.04906 0.524530 0.851392i $$-0.324241\pi$$
0.524530 + 0.851392i $$0.324241\pi$$
$$6$$ 1.02918 0.420163
$$7$$ −1.69239 −0.639664 −0.319832 0.947474i $$-0.603626\pi$$
−0.319832 + 0.947474i $$0.603626\pi$$
$$8$$ 1.21147 0.428321
$$9$$ −2.68256 −0.894188
$$10$$ −4.28499 −1.35503
$$11$$ 0 0
$$12$$ −0.753170 −0.217421
$$13$$ −4.11168 −1.14038 −0.570188 0.821514i $$-0.693130\pi$$
−0.570188 + 0.821514i $$0.693130\pi$$
$$14$$ 3.09147 0.826231
$$15$$ −1.32164 −0.341247
$$16$$ −4.88657 −1.22164
$$17$$ 6.16499 1.49523 0.747615 0.664132i $$-0.231199\pi$$
0.747615 + 0.664132i $$0.231199\pi$$
$$18$$ 4.90021 1.15499
$$19$$ 1.00000 0.229416
$$20$$ 3.13581 0.701188
$$21$$ 0.953520 0.208075
$$22$$ 0 0
$$23$$ 3.52199 0.734386 0.367193 0.930145i $$-0.380319\pi$$
0.367193 + 0.930145i $$0.380319\pi$$
$$24$$ −0.682563 −0.139328
$$25$$ 0.502638 0.100528
$$26$$ 7.51076 1.47298
$$27$$ 3.20164 0.616157
$$28$$ −2.26238 −0.427549
$$29$$ 8.10336 1.50476 0.752378 0.658731i $$-0.228906\pi$$
0.752378 + 0.658731i $$0.228906\pi$$
$$30$$ 2.41423 0.440776
$$31$$ −2.30144 −0.413350 −0.206675 0.978410i $$-0.566264\pi$$
−0.206675 + 0.978410i $$0.566264\pi$$
$$32$$ 6.50330 1.14963
$$33$$ 0 0
$$34$$ −11.2615 −1.93134
$$35$$ −3.96996 −0.671046
$$36$$ −3.58603 −0.597672
$$37$$ −6.56016 −1.07848 −0.539242 0.842151i $$-0.681289\pi$$
−0.539242 + 0.842151i $$0.681289\pi$$
$$38$$ −1.82669 −0.296328
$$39$$ 2.31659 0.370951
$$40$$ 2.84184 0.449334
$$41$$ −7.75013 −1.21037 −0.605183 0.796086i $$-0.706900\pi$$
−0.605183 + 0.796086i $$0.706900\pi$$
$$42$$ −1.74178 −0.268763
$$43$$ −7.75102 −1.18202 −0.591010 0.806664i $$-0.701271\pi$$
−0.591010 + 0.806664i $$0.701271\pi$$
$$44$$ 0 0
$$45$$ −6.29268 −0.938057
$$46$$ −6.43359 −0.948581
$$47$$ −10.8969 −1.58948 −0.794742 0.606948i $$-0.792394\pi$$
−0.794742 + 0.606948i $$0.792394\pi$$
$$48$$ 2.75317 0.397386
$$49$$ −4.13581 −0.590830
$$50$$ −0.918163 −0.129848
$$51$$ −3.47345 −0.486381
$$52$$ −5.49647 −0.762223
$$53$$ 7.93511 1.08997 0.544986 0.838445i $$-0.316535\pi$$
0.544986 + 0.838445i $$0.316535\pi$$
$$54$$ −5.84841 −0.795868
$$55$$ 0 0
$$56$$ −2.05029 −0.273981
$$57$$ −0.563416 −0.0746262
$$58$$ −14.8023 −1.94364
$$59$$ 10.9247 1.42228 0.711140 0.703051i $$-0.248179\pi$$
0.711140 + 0.703051i $$0.248179\pi$$
$$60$$ −1.76676 −0.228088
$$61$$ 4.51162 0.577653 0.288827 0.957381i $$-0.406735\pi$$
0.288827 + 0.957381i $$0.406735\pi$$
$$62$$ 4.20401 0.533910
$$63$$ 4.53995 0.571980
$$64$$ −2.10636 −0.263295
$$65$$ −9.64506 −1.19632
$$66$$ 0 0
$$67$$ 14.7201 1.79835 0.899173 0.437594i $$-0.144169\pi$$
0.899173 + 0.437594i $$0.144169\pi$$
$$68$$ 8.24132 0.999407
$$69$$ −1.98435 −0.238887
$$70$$ 7.25189 0.866766
$$71$$ 3.12026 0.370307 0.185153 0.982710i $$-0.440722\pi$$
0.185153 + 0.982710i $$0.440722\pi$$
$$72$$ −3.24985 −0.382999
$$73$$ −11.5827 −1.35565 −0.677824 0.735224i $$-0.737077\pi$$
−0.677824 + 0.735224i $$0.737077\pi$$
$$74$$ 11.9834 1.39304
$$75$$ −0.283194 −0.0327004
$$76$$ 1.33679 0.153341
$$77$$ 0 0
$$78$$ −4.23168 −0.479144
$$79$$ −4.96184 −0.558250 −0.279125 0.960255i $$-0.590044\pi$$
−0.279125 + 0.960255i $$0.590044\pi$$
$$80$$ −11.4628 −1.28158
$$81$$ 6.24383 0.693759
$$82$$ 14.1571 1.56339
$$83$$ 1.82905 0.200765 0.100382 0.994949i $$-0.467993\pi$$
0.100382 + 0.994949i $$0.467993\pi$$
$$84$$ 1.27466 0.139077
$$85$$ 14.4617 1.56859
$$86$$ 14.1587 1.52677
$$87$$ −4.56556 −0.489480
$$88$$ 0 0
$$89$$ −9.37496 −0.993743 −0.496872 0.867824i $$-0.665518\pi$$
−0.496872 + 0.867824i $$0.665518\pi$$
$$90$$ 11.4948 1.21165
$$91$$ 6.95858 0.729457
$$92$$ 4.70818 0.490861
$$93$$ 1.29666 0.134458
$$94$$ 19.9053 2.05308
$$95$$ 2.34577 0.240671
$$96$$ −3.66406 −0.373961
$$97$$ −10.9937 −1.11625 −0.558123 0.829758i $$-0.688478\pi$$
−0.558123 + 0.829758i $$0.688478\pi$$
$$98$$ 7.55484 0.763154
$$99$$ 0 0
$$100$$ 0.671923 0.0671923
$$101$$ −2.30621 −0.229476 −0.114738 0.993396i $$-0.536603\pi$$
−0.114738 + 0.993396i $$0.536603\pi$$
$$102$$ 6.34492 0.628241
$$103$$ 6.12000 0.603021 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$104$$ −4.98119 −0.488446
$$105$$ 2.23674 0.218283
$$106$$ −14.4950 −1.40788
$$107$$ −0.422947 −0.0408878 −0.0204439 0.999791i $$-0.506508\pi$$
−0.0204439 + 0.999791i $$0.506508\pi$$
$$108$$ 4.27993 0.411837
$$109$$ −14.0180 −1.34268 −0.671338 0.741151i $$-0.734280\pi$$
−0.671338 + 0.741151i $$0.734280\pi$$
$$110$$ 0 0
$$111$$ 3.69609 0.350818
$$112$$ 8.26999 0.781441
$$113$$ −2.53638 −0.238602 −0.119301 0.992858i $$-0.538065\pi$$
−0.119301 + 0.992858i $$0.538065\pi$$
$$114$$ 1.02918 0.0963920
$$115$$ 8.26179 0.770416
$$116$$ 10.8325 1.00577
$$117$$ 11.0298 1.01971
$$118$$ −19.9561 −1.83711
$$119$$ −10.4336 −0.956445
$$120$$ −1.60114 −0.146163
$$121$$ 0 0
$$122$$ −8.24132 −0.746134
$$123$$ 4.36654 0.393718
$$124$$ −3.07654 −0.276282
$$125$$ −10.5498 −0.943601
$$126$$ −8.29307 −0.738806
$$127$$ −1.07275 −0.0951914 −0.0475957 0.998867i $$-0.515156\pi$$
−0.0475957 + 0.998867i $$0.515156\pi$$
$$128$$ −9.15893 −0.809543
$$129$$ 4.36705 0.384497
$$130$$ 17.6185 1.54525
$$131$$ 7.82709 0.683856 0.341928 0.939726i $$-0.388920\pi$$
0.341928 + 0.939726i $$0.388920\pi$$
$$132$$ 0 0
$$133$$ −1.69239 −0.146749
$$134$$ −26.8890 −2.32286
$$135$$ 7.51032 0.646386
$$136$$ 7.46873 0.640438
$$137$$ −11.2473 −0.960919 −0.480460 0.877017i $$-0.659530\pi$$
−0.480460 + 0.877017i $$0.659530\pi$$
$$138$$ 3.62478 0.308562
$$139$$ 0.905926 0.0768397 0.0384198 0.999262i $$-0.487768\pi$$
0.0384198 + 0.999262i $$0.487768\pi$$
$$140$$ −5.30702 −0.448525
$$141$$ 6.13951 0.517040
$$142$$ −5.69975 −0.478312
$$143$$ 0 0
$$144$$ 13.1085 1.09238
$$145$$ 19.0086 1.57858
$$146$$ 21.1579 1.75104
$$147$$ 2.33018 0.192190
$$148$$ −8.76957 −0.720854
$$149$$ −10.5174 −0.861622 −0.430811 0.902442i $$-0.641772\pi$$
−0.430811 + 0.902442i $$0.641772\pi$$
$$150$$ 0.517307 0.0422380
$$151$$ −13.2436 −1.07775 −0.538873 0.842387i $$-0.681150\pi$$
−0.538873 + 0.842387i $$0.681150\pi$$
$$152$$ 1.21147 0.0982635
$$153$$ −16.5380 −1.33702
$$154$$ 0 0
$$155$$ −5.39864 −0.433629
$$156$$ 3.09679 0.247942
$$157$$ 1.99915 0.159549 0.0797747 0.996813i $$-0.474580\pi$$
0.0797747 + 0.996813i $$0.474580\pi$$
$$158$$ 9.06373 0.721072
$$159$$ −4.47076 −0.354555
$$160$$ 15.2552 1.20603
$$161$$ −5.96059 −0.469761
$$162$$ −11.4055 −0.896104
$$163$$ −18.7557 −1.46906 −0.734531 0.678575i $$-0.762598\pi$$
−0.734531 + 0.678575i $$0.762598\pi$$
$$164$$ −10.3603 −0.809005
$$165$$ 0 0
$$166$$ −3.34111 −0.259320
$$167$$ 6.31203 0.488440 0.244220 0.969720i $$-0.421468\pi$$
0.244220 + 0.969720i $$0.421468\pi$$
$$168$$ 1.15516 0.0891229
$$169$$ 3.90593 0.300456
$$170$$ −26.4170 −2.02609
$$171$$ −2.68256 −0.205141
$$172$$ −10.3615 −0.790058
$$173$$ 21.5269 1.63666 0.818328 0.574751i $$-0.194901\pi$$
0.818328 + 0.574751i $$0.194901\pi$$
$$174$$ 8.33986 0.632243
$$175$$ −0.850661 −0.0643039
$$176$$ 0 0
$$177$$ −6.15516 −0.462650
$$178$$ 17.1251 1.28358
$$179$$ −7.97018 −0.595719 −0.297860 0.954610i $$-0.596273\pi$$
−0.297860 + 0.954610i $$0.596273\pi$$
$$180$$ −8.41200 −0.626994
$$181$$ −1.16955 −0.0869317 −0.0434659 0.999055i $$-0.513840\pi$$
−0.0434659 + 0.999055i $$0.513840\pi$$
$$182$$ −12.7112 −0.942214
$$183$$ −2.54191 −0.187904
$$184$$ 4.26680 0.314553
$$185$$ −15.3886 −1.13139
$$186$$ −2.36860 −0.173674
$$187$$ 0 0
$$188$$ −14.5670 −1.06240
$$189$$ −5.41844 −0.394133
$$190$$ −4.28499 −0.310866
$$191$$ −15.6673 −1.13365 −0.566824 0.823839i $$-0.691828\pi$$
−0.566824 + 0.823839i $$0.691828\pi$$
$$192$$ 1.18676 0.0856468
$$193$$ −21.6769 −1.56034 −0.780169 0.625568i $$-0.784867\pi$$
−0.780169 + 0.625568i $$0.784867\pi$$
$$194$$ 20.0821 1.44181
$$195$$ 5.43418 0.389149
$$196$$ −5.52872 −0.394908
$$197$$ −4.58794 −0.326877 −0.163439 0.986554i $$-0.552259\pi$$
−0.163439 + 0.986554i $$0.552259\pi$$
$$198$$ 0 0
$$199$$ 13.0619 0.925937 0.462968 0.886375i $$-0.346784\pi$$
0.462968 + 0.886375i $$0.346784\pi$$
$$200$$ 0.608933 0.0430580
$$201$$ −8.29353 −0.584980
$$202$$ 4.21272 0.296406
$$203$$ −13.7141 −0.962539
$$204$$ −4.64329 −0.325095
$$205$$ −18.1800 −1.26975
$$206$$ −11.1793 −0.778901
$$207$$ −9.44797 −0.656679
$$208$$ 20.0920 1.39313
$$209$$ 0 0
$$210$$ −4.08583 −0.281949
$$211$$ −2.91188 −0.200462 −0.100231 0.994964i $$-0.531958\pi$$
−0.100231 + 0.994964i $$0.531958\pi$$
$$212$$ 10.6076 0.728533
$$213$$ −1.75800 −0.120456
$$214$$ 0.772592 0.0528133
$$215$$ −18.1821 −1.24001
$$216$$ 3.87871 0.263913
$$217$$ 3.89493 0.264405
$$218$$ 25.6064 1.73429
$$219$$ 6.52585 0.440976
$$220$$ 0 0
$$221$$ −25.3485 −1.70512
$$222$$ −6.75161 −0.453139
$$223$$ 6.34161 0.424665 0.212333 0.977197i $$-0.431894\pi$$
0.212333 + 0.977197i $$0.431894\pi$$
$$224$$ −11.0061 −0.735378
$$225$$ −1.34836 −0.0898905
$$226$$ 4.63317 0.308194
$$227$$ 2.00429 0.133029 0.0665147 0.997785i $$-0.478812\pi$$
0.0665147 + 0.997785i $$0.478812\pi$$
$$228$$ −0.753170 −0.0498799
$$229$$ −20.9893 −1.38701 −0.693507 0.720450i $$-0.743935\pi$$
−0.693507 + 0.720450i $$0.743935\pi$$
$$230$$ −15.0917 −0.995118
$$231$$ 0 0
$$232$$ 9.81701 0.644518
$$233$$ 17.6006 1.15305 0.576527 0.817078i $$-0.304407\pi$$
0.576527 + 0.817078i $$0.304407\pi$$
$$234$$ −20.1481 −1.31712
$$235$$ −25.5617 −1.66746
$$236$$ 14.6041 0.950646
$$237$$ 2.79558 0.181592
$$238$$ 19.0589 1.23541
$$239$$ −26.6207 −1.72195 −0.860975 0.508647i $$-0.830146\pi$$
−0.860975 + 0.508647i $$0.830146\pi$$
$$240$$ 6.45830 0.416882
$$241$$ −13.1342 −0.846049 −0.423024 0.906118i $$-0.639032\pi$$
−0.423024 + 0.906118i $$0.639032\pi$$
$$242$$ 0 0
$$243$$ −13.1228 −0.841828
$$244$$ 6.03109 0.386101
$$245$$ −9.70166 −0.619816
$$246$$ −7.97632 −0.508551
$$247$$ −4.11168 −0.261620
$$248$$ −2.78813 −0.177046
$$249$$ −1.03052 −0.0653063
$$250$$ 19.2712 1.21882
$$251$$ −26.1636 −1.65143 −0.825715 0.564087i $$-0.809228\pi$$
−0.825715 + 0.564087i $$0.809228\pi$$
$$252$$ 6.06897 0.382309
$$253$$ 0 0
$$254$$ 1.95959 0.122955
$$255$$ −8.14792 −0.510243
$$256$$ 20.9432 1.30895
$$257$$ 12.6117 0.786697 0.393349 0.919389i $$-0.371317\pi$$
0.393349 + 0.919389i $$0.371317\pi$$
$$258$$ −7.97724 −0.496641
$$259$$ 11.1024 0.689867
$$260$$ −12.8934 −0.799618
$$261$$ −21.7378 −1.34554
$$262$$ −14.2977 −0.883312
$$263$$ 20.0550 1.23664 0.618322 0.785925i $$-0.287813\pi$$
0.618322 + 0.785925i $$0.287813\pi$$
$$264$$ 0 0
$$265$$ 18.6139 1.14345
$$266$$ 3.09147 0.189550
$$267$$ 5.28200 0.323253
$$268$$ 19.6777 1.20201
$$269$$ −20.1150 −1.22644 −0.613218 0.789914i $$-0.710125\pi$$
−0.613218 + 0.789914i $$0.710125\pi$$
$$270$$ −13.7190 −0.834913
$$271$$ −15.5586 −0.945117 −0.472558 0.881299i $$-0.656669\pi$$
−0.472558 + 0.881299i $$0.656669\pi$$
$$272$$ −30.1257 −1.82664
$$273$$ −3.92057 −0.237284
$$274$$ 20.5453 1.24119
$$275$$ 0 0
$$276$$ −2.65266 −0.159671
$$277$$ −1.89211 −0.113686 −0.0568429 0.998383i $$-0.518103\pi$$
−0.0568429 + 0.998383i $$0.518103\pi$$
$$278$$ −1.65485 −0.0992510
$$279$$ 6.17375 0.369613
$$280$$ −4.80951 −0.287423
$$281$$ −4.59299 −0.273995 −0.136997 0.990571i $$-0.543745\pi$$
−0.136997 + 0.990571i $$0.543745\pi$$
$$282$$ −11.2150 −0.667842
$$283$$ 1.17798 0.0700234 0.0350117 0.999387i $$-0.488853\pi$$
0.0350117 + 0.999387i $$0.488853\pi$$
$$284$$ 4.17114 0.247512
$$285$$ −1.32164 −0.0782874
$$286$$ 0 0
$$287$$ 13.1163 0.774228
$$288$$ −17.4455 −1.02799
$$289$$ 21.0071 1.23571
$$290$$ −34.7229 −2.03900
$$291$$ 6.19405 0.363101
$$292$$ −15.4836 −0.906110
$$293$$ 11.2606 0.657851 0.328925 0.944356i $$-0.393314\pi$$
0.328925 + 0.944356i $$0.393314\pi$$
$$294$$ −4.25651 −0.248245
$$295$$ 25.6269 1.49206
$$296$$ −7.94745 −0.461936
$$297$$ 0 0
$$298$$ 19.2121 1.11293
$$299$$ −14.4813 −0.837476
$$300$$ −0.378572 −0.0218568
$$301$$ 13.1178 0.756096
$$302$$ 24.1919 1.39209
$$303$$ 1.29935 0.0746459
$$304$$ −4.88657 −0.280264
$$305$$ 10.5832 0.605993
$$306$$ 30.2098 1.72698
$$307$$ −4.35245 −0.248407 −0.124204 0.992257i $$-0.539638\pi$$
−0.124204 + 0.992257i $$0.539638\pi$$
$$308$$ 0 0
$$309$$ −3.44810 −0.196156
$$310$$ 9.86164 0.560103
$$311$$ 7.61725 0.431935 0.215967 0.976401i $$-0.430709\pi$$
0.215967 + 0.976401i $$0.430709\pi$$
$$312$$ 2.80648 0.158886
$$313$$ −17.5654 −0.992855 −0.496427 0.868078i $$-0.665355\pi$$
−0.496427 + 0.868078i $$0.665355\pi$$
$$314$$ −3.65182 −0.206084
$$315$$ 10.6497 0.600041
$$316$$ −6.63295 −0.373132
$$317$$ −1.94192 −0.109069 −0.0545345 0.998512i $$-0.517367\pi$$
−0.0545345 + 0.998512i $$0.517367\pi$$
$$318$$ 8.16670 0.457966
$$319$$ 0 0
$$320$$ −4.94104 −0.276213
$$321$$ 0.238295 0.0133003
$$322$$ 10.8882 0.606773
$$323$$ 6.16499 0.343029
$$324$$ 8.34671 0.463706
$$325$$ −2.06669 −0.114639
$$326$$ 34.2609 1.89754
$$327$$ 7.89793 0.436757
$$328$$ −9.38907 −0.518425
$$329$$ 18.4419 1.01674
$$330$$ 0 0
$$331$$ 13.3988 0.736462 0.368231 0.929734i $$-0.379964\pi$$
0.368231 + 0.929734i $$0.379964\pi$$
$$332$$ 2.44506 0.134190
$$333$$ 17.5980 0.964366
$$334$$ −11.5301 −0.630900
$$335$$ 34.5300 1.88657
$$336$$ −4.65944 −0.254193
$$337$$ −33.1631 −1.80651 −0.903254 0.429107i $$-0.858828\pi$$
−0.903254 + 0.429107i $$0.858828\pi$$
$$338$$ −7.13491 −0.388088
$$339$$ 1.42903 0.0776145
$$340$$ 19.3322 1.04844
$$341$$ 0 0
$$342$$ 4.90021 0.264973
$$343$$ 18.8462 1.01760
$$344$$ −9.39016 −0.506283
$$345$$ −4.65482 −0.250607
$$346$$ −39.3229 −2.11401
$$347$$ −18.0268 −0.967730 −0.483865 0.875143i $$-0.660767\pi$$
−0.483865 + 0.875143i $$0.660767\pi$$
$$348$$ −6.10321 −0.327166
$$349$$ −8.81411 −0.471808 −0.235904 0.971776i $$-0.575805\pi$$
−0.235904 + 0.971776i $$0.575805\pi$$
$$350$$ 1.55389 0.0830590
$$351$$ −13.1641 −0.702650
$$352$$ 0 0
$$353$$ −1.59249 −0.0847599 −0.0423799 0.999102i $$-0.513494\pi$$
−0.0423799 + 0.999102i $$0.513494\pi$$
$$354$$ 11.2436 0.597589
$$355$$ 7.31942 0.388474
$$356$$ −12.5324 −0.664214
$$357$$ 5.87844 0.311120
$$358$$ 14.5590 0.769469
$$359$$ −36.3774 −1.91993 −0.959963 0.280126i $$-0.909624\pi$$
−0.959963 + 0.280126i $$0.909624\pi$$
$$360$$ −7.62341 −0.401789
$$361$$ 1.00000 0.0526316
$$362$$ 2.13640 0.112287
$$363$$ 0 0
$$364$$ 9.30218 0.487567
$$365$$ −27.1703 −1.42216
$$366$$ 4.64329 0.242708
$$367$$ −5.00276 −0.261142 −0.130571 0.991439i $$-0.541681\pi$$
−0.130571 + 0.991439i $$0.541681\pi$$
$$368$$ −17.2105 −0.897158
$$369$$ 20.7902 1.08229
$$370$$ 28.1102 1.46138
$$371$$ −13.4293 −0.697216
$$372$$ 1.73337 0.0898712
$$373$$ 34.1313 1.76725 0.883625 0.468195i $$-0.155096\pi$$
0.883625 + 0.468195i $$0.155096\pi$$
$$374$$ 0 0
$$375$$ 5.94391 0.306942
$$376$$ −13.2014 −0.680808
$$377$$ −33.3185 −1.71599
$$378$$ 9.89780 0.509088
$$379$$ 15.7749 0.810302 0.405151 0.914250i $$-0.367219\pi$$
0.405151 + 0.914250i $$0.367219\pi$$
$$380$$ 3.13581 0.160864
$$381$$ 0.604405 0.0309646
$$382$$ 28.6193 1.46429
$$383$$ −22.4260 −1.14592 −0.572958 0.819585i $$-0.694204\pi$$
−0.572958 + 0.819585i $$0.694204\pi$$
$$384$$ 5.16028 0.263335
$$385$$ 0 0
$$386$$ 39.5970 2.01543
$$387$$ 20.7926 1.05695
$$388$$ −14.6964 −0.746094
$$389$$ −18.3488 −0.930321 −0.465160 0.885226i $$-0.654003\pi$$
−0.465160 + 0.885226i $$0.654003\pi$$
$$390$$ −9.92655 −0.502650
$$391$$ 21.7131 1.09808
$$392$$ −5.01042 −0.253065
$$393$$ −4.40990 −0.222450
$$394$$ 8.38074 0.422216
$$395$$ −11.6393 −0.585638
$$396$$ 0 0
$$397$$ −0.511483 −0.0256706 −0.0128353 0.999918i $$-0.504086\pi$$
−0.0128353 + 0.999918i $$0.504086\pi$$
$$398$$ −23.8601 −1.19600
$$399$$ 0.953520 0.0477357
$$400$$ −2.45618 −0.122809
$$401$$ 18.9824 0.947935 0.473967 0.880542i $$-0.342821\pi$$
0.473967 + 0.880542i $$0.342821\pi$$
$$402$$ 15.1497 0.755598
$$403$$ 9.46277 0.471374
$$404$$ −3.08292 −0.153381
$$405$$ 14.6466 0.727795
$$406$$ 25.0513 1.24328
$$407$$ 0 0
$$408$$ −4.20800 −0.208327
$$409$$ −1.10089 −0.0544355 −0.0272177 0.999630i $$-0.508665\pi$$
−0.0272177 + 0.999630i $$0.508665\pi$$
$$410$$ 33.2092 1.64009
$$411$$ 6.33689 0.312576
$$412$$ 8.18117 0.403057
$$413$$ −18.4889 −0.909781
$$414$$ 17.2585 0.848209
$$415$$ 4.29054 0.210614
$$416$$ −26.7395 −1.31101
$$417$$ −0.510413 −0.0249950
$$418$$ 0 0
$$419$$ −18.7929 −0.918094 −0.459047 0.888412i $$-0.651809\pi$$
−0.459047 + 0.888412i $$0.651809\pi$$
$$420$$ 2.99006 0.145900
$$421$$ −24.7618 −1.20682 −0.603408 0.797433i $$-0.706191\pi$$
−0.603408 + 0.797433i $$0.706191\pi$$
$$422$$ 5.31910 0.258930
$$423$$ 29.2318 1.42130
$$424$$ 9.61318 0.466857
$$425$$ 3.09876 0.150312
$$426$$ 3.21133 0.155589
$$427$$ −7.63542 −0.369504
$$428$$ −0.565392 −0.0273293
$$429$$ 0 0
$$430$$ 33.2131 1.60168
$$431$$ 32.1985 1.55095 0.775474 0.631380i $$-0.217511\pi$$
0.775474 + 0.631380i $$0.217511\pi$$
$$432$$ −15.6451 −0.752723
$$433$$ −15.3042 −0.735475 −0.367737 0.929930i $$-0.619867\pi$$
−0.367737 + 0.929930i $$0.619867\pi$$
$$434$$ −7.11483 −0.341523
$$435$$ −10.7098 −0.513494
$$436$$ −18.7391 −0.897440
$$437$$ 3.52199 0.168480
$$438$$ −11.9207 −0.569593
$$439$$ −26.1812 −1.24956 −0.624781 0.780800i $$-0.714812\pi$$
−0.624781 + 0.780800i $$0.714812\pi$$
$$440$$ 0 0
$$441$$ 11.0946 0.528313
$$442$$ 46.3038 2.20245
$$443$$ 21.5579 1.02425 0.512123 0.858912i $$-0.328859\pi$$
0.512123 + 0.858912i $$0.328859\pi$$
$$444$$ 4.94091 0.234485
$$445$$ −21.9915 −1.04250
$$446$$ −11.5841 −0.548525
$$447$$ 5.92569 0.280275
$$448$$ 3.56479 0.168420
$$449$$ 11.7990 0.556829 0.278415 0.960461i $$-0.410191\pi$$
0.278415 + 0.960461i $$0.410191\pi$$
$$450$$ 2.46303 0.116108
$$451$$ 0 0
$$452$$ −3.39061 −0.159481
$$453$$ 7.46163 0.350578
$$454$$ −3.66122 −0.171829
$$455$$ 16.3232 0.765245
$$456$$ −0.682563 −0.0319639
$$457$$ 15.6863 0.733773 0.366886 0.930266i $$-0.380424\pi$$
0.366886 + 0.930266i $$0.380424\pi$$
$$458$$ 38.3410 1.79156
$$459$$ 19.7381 0.921296
$$460$$ 11.0443 0.514943
$$461$$ −27.2451 −1.26893 −0.634466 0.772951i $$-0.718780\pi$$
−0.634466 + 0.772951i $$0.718780\pi$$
$$462$$ 0 0
$$463$$ −17.9364 −0.833573 −0.416787 0.909004i $$-0.636844\pi$$
−0.416787 + 0.909004i $$0.636844\pi$$
$$464$$ −39.5977 −1.83828
$$465$$ 3.04168 0.141054
$$466$$ −32.1509 −1.48936
$$467$$ 29.0100 1.34242 0.671211 0.741266i $$-0.265774\pi$$
0.671211 + 0.741266i $$0.265774\pi$$
$$468$$ 14.7446 0.681570
$$469$$ −24.9122 −1.15034
$$470$$ 46.6933 2.15380
$$471$$ −1.12635 −0.0518995
$$472$$ 13.2350 0.609191
$$473$$ 0 0
$$474$$ −5.10665 −0.234556
$$475$$ 0.502638 0.0230626
$$476$$ −13.9475 −0.639285
$$477$$ −21.2864 −0.974639
$$478$$ 48.6277 2.22418
$$479$$ −34.0896 −1.55759 −0.778797 0.627276i $$-0.784170\pi$$
−0.778797 + 0.627276i $$0.784170\pi$$
$$480$$ −8.59504 −0.392308
$$481$$ 26.9733 1.22988
$$482$$ 23.9921 1.09281
$$483$$ 3.35829 0.152808
$$484$$ 0 0
$$485$$ −25.7888 −1.17101
$$486$$ 23.9713 1.08736
$$487$$ 6.38349 0.289264 0.144632 0.989486i $$-0.453800\pi$$
0.144632 + 0.989486i $$0.453800\pi$$
$$488$$ 5.46570 0.247421
$$489$$ 10.5673 0.477869
$$490$$ 17.7219 0.800594
$$491$$ 31.4552 1.41956 0.709778 0.704426i $$-0.248795\pi$$
0.709778 + 0.704426i $$0.248795\pi$$
$$492$$ 5.83716 0.263160
$$493$$ 49.9572 2.24996
$$494$$ 7.51076 0.337925
$$495$$ 0 0
$$496$$ 11.2461 0.504966
$$497$$ −5.28071 −0.236872
$$498$$ 1.88243 0.0843539
$$499$$ −31.9667 −1.43103 −0.715514 0.698599i $$-0.753807\pi$$
−0.715514 + 0.698599i $$0.753807\pi$$
$$500$$ −14.1029 −0.630699
$$501$$ −3.55630 −0.158884
$$502$$ 47.7927 2.13309
$$503$$ 34.0522 1.51831 0.759157 0.650907i $$-0.225611\pi$$
0.759157 + 0.650907i $$0.225611\pi$$
$$504$$ 5.50003 0.244991
$$505$$ −5.40983 −0.240734
$$506$$ 0 0
$$507$$ −2.20066 −0.0977347
$$508$$ −1.43405 −0.0636256
$$509$$ 25.7904 1.14314 0.571569 0.820554i $$-0.306335\pi$$
0.571569 + 0.820554i $$0.306335\pi$$
$$510$$ 14.8837 0.659062
$$511$$ 19.6024 0.867160
$$512$$ −19.9389 −0.881184
$$513$$ 3.20164 0.141356
$$514$$ −23.0377 −1.01615
$$515$$ 14.3561 0.632606
$$516$$ 5.83784 0.256996
$$517$$ 0 0
$$518$$ −20.2806 −0.891076
$$519$$ −12.1286 −0.532385
$$520$$ −11.6847 −0.512410
$$521$$ 31.3774 1.37467 0.687334 0.726342i $$-0.258781\pi$$
0.687334 + 0.726342i $$0.258781\pi$$
$$522$$ 39.7082 1.73798
$$523$$ 23.2388 1.01616 0.508082 0.861309i $$-0.330355\pi$$
0.508082 + 0.861309i $$0.330355\pi$$
$$524$$ 10.4632 0.457087
$$525$$ 0.479275 0.0209173
$$526$$ −36.6343 −1.59733
$$527$$ −14.1883 −0.618054
$$528$$ 0 0
$$529$$ −10.5956 −0.460677
$$530$$ −34.0019 −1.47695
$$531$$ −29.3063 −1.27178
$$532$$ −2.26238 −0.0980865
$$533$$ 31.8661 1.38027
$$534$$ −9.64856 −0.417534
$$535$$ −0.992136 −0.0428938
$$536$$ 17.8330 0.770268
$$537$$ 4.49052 0.193780
$$538$$ 36.7439 1.58414
$$539$$ 0 0
$$540$$ 10.0397 0.432042
$$541$$ −36.0085 −1.54812 −0.774062 0.633109i $$-0.781778\pi$$
−0.774062 + 0.633109i $$0.781778\pi$$
$$542$$ 28.4207 1.22077
$$543$$ 0.658941 0.0282779
$$544$$ 40.0928 1.71896
$$545$$ −32.8829 −1.40855
$$546$$ 7.16166 0.306491
$$547$$ 28.1855 1.20513 0.602563 0.798071i $$-0.294146\pi$$
0.602563 + 0.798071i $$0.294146\pi$$
$$548$$ −15.0353 −0.642275
$$549$$ −12.1027 −0.516530
$$550$$ 0 0
$$551$$ 8.10336 0.345215
$$552$$ −2.40398 −0.102320
$$553$$ 8.39738 0.357093
$$554$$ 3.45629 0.146844
$$555$$ 8.67019 0.368029
$$556$$ 1.21104 0.0513594
$$557$$ −21.6248 −0.916274 −0.458137 0.888882i $$-0.651483\pi$$
−0.458137 + 0.888882i $$0.651483\pi$$
$$558$$ −11.2775 −0.477415
$$559$$ 31.8697 1.34795
$$560$$ 19.3995 0.819779
$$561$$ 0 0
$$562$$ 8.38997 0.353909
$$563$$ −13.1791 −0.555434 −0.277717 0.960663i $$-0.589578\pi$$
−0.277717 + 0.960663i $$0.589578\pi$$
$$564$$ 8.20725 0.345588
$$565$$ −5.94976 −0.250308
$$566$$ −2.15179 −0.0904467
$$567$$ −10.5670 −0.443773
$$568$$ 3.78011 0.158610
$$569$$ −8.61143 −0.361010 −0.180505 0.983574i $$-0.557773\pi$$
−0.180505 + 0.983574i $$0.557773\pi$$
$$570$$ 2.41423 0.101121
$$571$$ 17.6546 0.738821 0.369410 0.929266i $$-0.379560\pi$$
0.369410 + 0.929266i $$0.379560\pi$$
$$572$$ 0 0
$$573$$ 8.82722 0.368762
$$574$$ −23.9593 −1.00004
$$575$$ 1.77029 0.0738261
$$576$$ 5.65045 0.235435
$$577$$ −31.0907 −1.29432 −0.647162 0.762352i $$-0.724044\pi$$
−0.647162 + 0.762352i $$0.724044\pi$$
$$578$$ −38.3735 −1.59613
$$579$$ 12.2131 0.507560
$$580$$ 25.4106 1.05512
$$581$$ −3.09547 −0.128422
$$582$$ −11.3146 −0.469005
$$583$$ 0 0
$$584$$ −14.0321 −0.580652
$$585$$ 25.8735 1.06974
$$586$$ −20.5696 −0.849722
$$587$$ −26.7211 −1.10290 −0.551450 0.834208i $$-0.685925\pi$$
−0.551450 + 0.834208i $$0.685925\pi$$
$$588$$ 3.11497 0.128459
$$589$$ −2.30144 −0.0948290
$$590$$ −46.8124 −1.92724
$$591$$ 2.58492 0.106329
$$592$$ 32.0567 1.31752
$$593$$ 24.2460 0.995666 0.497833 0.867273i $$-0.334129\pi$$
0.497833 + 0.867273i $$0.334129\pi$$
$$594$$ 0 0
$$595$$ −24.4748 −1.00337
$$596$$ −14.0596 −0.575905
$$597$$ −7.35930 −0.301196
$$598$$ 26.4529 1.08174
$$599$$ −18.2095 −0.744019 −0.372009 0.928229i $$-0.621331\pi$$
−0.372009 + 0.928229i $$0.621331\pi$$
$$600$$ −0.343082 −0.0140063
$$601$$ −37.9824 −1.54934 −0.774668 0.632368i $$-0.782083\pi$$
−0.774668 + 0.632368i $$0.782083\pi$$
$$602$$ −23.9621 −0.976622
$$603$$ −39.4876 −1.60806
$$604$$ −17.7039 −0.720362
$$605$$ 0 0
$$606$$ −2.37351 −0.0964174
$$607$$ 20.0130 0.812301 0.406151 0.913806i $$-0.366871\pi$$
0.406151 + 0.913806i $$0.366871\pi$$
$$608$$ 6.50330 0.263744
$$609$$ 7.72672 0.313103
$$610$$ −19.3322 −0.782739
$$611$$ 44.8048 1.81261
$$612$$ −22.1079 −0.893657
$$613$$ 42.1414 1.70208 0.851038 0.525105i $$-0.175974\pi$$
0.851038 + 0.525105i $$0.175974\pi$$
$$614$$ 7.95057 0.320859
$$615$$ 10.2429 0.413034
$$616$$ 0 0
$$617$$ 16.9044 0.680545 0.340273 0.940327i $$-0.389481\pi$$
0.340273 + 0.940327i $$0.389481\pi$$
$$618$$ 6.29861 0.253367
$$619$$ 5.31177 0.213498 0.106749 0.994286i $$-0.465956\pi$$
0.106749 + 0.994286i $$0.465956\pi$$
$$620$$ −7.21686 −0.289836
$$621$$ 11.2762 0.452497
$$622$$ −13.9144 −0.557915
$$623$$ 15.8661 0.635662
$$624$$ −11.3202 −0.453169
$$625$$ −27.2605 −1.09042
$$626$$ 32.0865 1.28243
$$627$$ 0 0
$$628$$ 2.67245 0.106642
$$629$$ −40.4433 −1.61258
$$630$$ −19.4536 −0.775052
$$631$$ 34.0425 1.35521 0.677604 0.735427i $$-0.263018\pi$$
0.677604 + 0.735427i $$0.263018\pi$$
$$632$$ −6.01113 −0.239110
$$633$$ 1.64060 0.0652079
$$634$$ 3.54728 0.140880
$$635$$ −2.51643 −0.0998615
$$636$$ −5.97649 −0.236983
$$637$$ 17.0051 0.673768
$$638$$ 0 0
$$639$$ −8.37030 −0.331124
$$640$$ −21.4847 −0.849259
$$641$$ −15.0405 −0.594063 −0.297031 0.954868i $$-0.595997\pi$$
−0.297031 + 0.954868i $$0.595997\pi$$
$$642$$ −0.435291 −0.0171795
$$643$$ 32.3618 1.27622 0.638112 0.769943i $$-0.279715\pi$$
0.638112 + 0.769943i $$0.279715\pi$$
$$644$$ −7.96808 −0.313986
$$645$$ 10.2441 0.403361
$$646$$ −11.2615 −0.443079
$$647$$ −49.0042 −1.92655 −0.963276 0.268512i $$-0.913468\pi$$
−0.963276 + 0.268512i $$0.913468\pi$$
$$648$$ 7.56424 0.297151
$$649$$ 0 0
$$650$$ 3.77520 0.148075
$$651$$ −2.19447 −0.0860079
$$652$$ −25.0725 −0.981916
$$653$$ 31.1992 1.22092 0.610460 0.792047i $$-0.290984\pi$$
0.610460 + 0.792047i $$0.290984\pi$$
$$654$$ −14.4271 −0.564143
$$655$$ 18.3605 0.717406
$$656$$ 37.8715 1.47864
$$657$$ 31.0712 1.21220
$$658$$ −33.6876 −1.31328
$$659$$ 35.7237 1.39160 0.695799 0.718237i $$-0.255050\pi$$
0.695799 + 0.718237i $$0.255050\pi$$
$$660$$ 0 0
$$661$$ 33.8677 1.31730 0.658651 0.752449i $$-0.271128\pi$$
0.658651 + 0.752449i $$0.271128\pi$$
$$662$$ −24.4754 −0.951262
$$663$$ 14.2817 0.554657
$$664$$ 2.21585 0.0859916
$$665$$ −3.96996 −0.153949
$$666$$ −32.1461 −1.24564
$$667$$ 28.5400 1.10507
$$668$$ 8.43788 0.326471
$$669$$ −3.57296 −0.138139
$$670$$ −63.0755 −2.43682
$$671$$ 0 0
$$672$$ 6.20103 0.239210
$$673$$ −21.1648 −0.815842 −0.407921 0.913017i $$-0.633746\pi$$
−0.407921 + 0.913017i $$0.633746\pi$$
$$674$$ 60.5786 2.33340
$$675$$ 1.60927 0.0619408
$$676$$ 5.22141 0.200824
$$677$$ 34.5239 1.32686 0.663431 0.748238i $$-0.269100\pi$$
0.663431 + 0.748238i $$0.269100\pi$$
$$678$$ −2.61040 −0.100252
$$679$$ 18.6057 0.714022
$$680$$ 17.5199 0.671858
$$681$$ −1.12925 −0.0432729
$$682$$ 0 0
$$683$$ 12.8022 0.489862 0.244931 0.969540i $$-0.421235\pi$$
0.244931 + 0.969540i $$0.421235\pi$$
$$684$$ −3.58603 −0.137115
$$685$$ −26.3835 −1.00806
$$686$$ −34.4261 −1.31439
$$687$$ 11.8257 0.451179
$$688$$ 37.8759 1.44401
$$689$$ −32.6267 −1.24298
$$690$$ 8.50291 0.323700
$$691$$ 39.5406 1.50419 0.752097 0.659052i $$-0.229042\pi$$
0.752097 + 0.659052i $$0.229042\pi$$
$$692$$ 28.7769 1.09394
$$693$$ 0 0
$$694$$ 32.9294 1.24998
$$695$$ 2.12510 0.0806094
$$696$$ −5.53106 −0.209654
$$697$$ −47.7795 −1.80978
$$698$$ 16.1006 0.609418
$$699$$ −9.91646 −0.375075
$$700$$ −1.13716 −0.0429805
$$701$$ −22.4546 −0.848097 −0.424049 0.905639i $$-0.639391\pi$$
−0.424049 + 0.905639i $$0.639391\pi$$
$$702$$ 24.0468 0.907588
$$703$$ −6.56016 −0.247421
$$704$$ 0 0
$$705$$ 14.4019 0.542406
$$706$$ 2.90899 0.109481
$$707$$ 3.90301 0.146788
$$708$$ −8.22818 −0.309234
$$709$$ −1.41244 −0.0530451 −0.0265226 0.999648i $$-0.508443\pi$$
−0.0265226 + 0.999648i $$0.508443\pi$$
$$710$$ −13.3703 −0.501778
$$711$$ 13.3104 0.499181
$$712$$ −11.3575 −0.425641
$$713$$ −8.10564 −0.303559
$$714$$ −10.7381 −0.401863
$$715$$ 0 0
$$716$$ −10.6545 −0.398177
$$717$$ 14.9985 0.560130
$$718$$ 66.4502 2.47990
$$719$$ −39.0879 −1.45773 −0.728867 0.684655i $$-0.759953\pi$$
−0.728867 + 0.684655i $$0.759953\pi$$
$$720$$ 30.7496 1.14597
$$721$$ −10.3574 −0.385731
$$722$$ −1.82669 −0.0679823
$$723$$ 7.40002 0.275210
$$724$$ −1.56344 −0.0581049
$$725$$ 4.07306 0.151270
$$726$$ 0 0
$$727$$ 9.42098 0.349405 0.174702 0.984621i $$-0.444104\pi$$
0.174702 + 0.984621i $$0.444104\pi$$
$$728$$ 8.43013 0.312442
$$729$$ −11.3379 −0.419922
$$730$$ 49.6316 1.83695
$$731$$ −47.7850 −1.76739
$$732$$ −3.39801 −0.125594
$$733$$ −4.63361 −0.171146 −0.0855731 0.996332i $$-0.527272\pi$$
−0.0855731 + 0.996332i $$0.527272\pi$$
$$734$$ 9.13849 0.337308
$$735$$ 5.46606 0.201619
$$736$$ 22.9046 0.844274
$$737$$ 0 0
$$738$$ −37.9772 −1.39796
$$739$$ 13.1654 0.484298 0.242149 0.970239i $$-0.422148\pi$$
0.242149 + 0.970239i $$0.422148\pi$$
$$740$$ −20.5714 −0.756219
$$741$$ 2.31659 0.0851019
$$742$$ 24.5312 0.900568
$$743$$ 14.7689 0.541817 0.270909 0.962605i $$-0.412676\pi$$
0.270909 + 0.962605i $$0.412676\pi$$
$$744$$ 1.57087 0.0575911
$$745$$ −24.6715 −0.903894
$$746$$ −62.3472 −2.28269
$$747$$ −4.90655 −0.179521
$$748$$ 0 0
$$749$$ 0.715792 0.0261545
$$750$$ −10.8577 −0.396466
$$751$$ 35.0415 1.27868 0.639341 0.768924i $$-0.279207\pi$$
0.639341 + 0.768924i $$0.279207\pi$$
$$752$$ 53.2487 1.94178
$$753$$ 14.7410 0.537191
$$754$$ 60.8625 2.21648
$$755$$ −31.0664 −1.13062
$$756$$ −7.24333 −0.263437
$$757$$ −23.8005 −0.865044 −0.432522 0.901623i $$-0.642376\pi$$
−0.432522 + 0.901623i $$0.642376\pi$$
$$758$$ −28.8158 −1.04664
$$759$$ 0 0
$$760$$ 2.84184 0.103084
$$761$$ 39.0282 1.41477 0.707386 0.706827i $$-0.249874\pi$$
0.707386 + 0.706827i $$0.249874\pi$$
$$762$$ −1.10406 −0.0399959
$$763$$ 23.7239 0.858862
$$764$$ −20.9440 −0.757726
$$765$$ −38.7943 −1.40261
$$766$$ 40.9654 1.48014
$$767$$ −44.9190 −1.62193
$$768$$ −11.7997 −0.425787
$$769$$ −9.84757 −0.355113 −0.177556 0.984111i $$-0.556819\pi$$
−0.177556 + 0.984111i $$0.556819\pi$$
$$770$$ 0 0
$$771$$ −7.10564 −0.255903
$$772$$ −28.9776 −1.04292
$$773$$ −42.9837 −1.54602 −0.773008 0.634396i $$-0.781249\pi$$
−0.773008 + 0.634396i $$0.781249\pi$$
$$774$$ −37.9816 −1.36522
$$775$$ −1.15679 −0.0415531
$$776$$ −13.3186 −0.478111
$$777$$ −6.25524 −0.224405
$$778$$ 33.5175 1.20166
$$779$$ −7.75013 −0.277677
$$780$$ 7.26437 0.260106
$$781$$ 0 0
$$782$$ −39.6630 −1.41835
$$783$$ 25.9441 0.927166
$$784$$ 20.2099 0.721783
$$785$$ 4.68954 0.167377
$$786$$ 8.05552 0.287331
$$787$$ 25.8709 0.922199 0.461099 0.887349i $$-0.347455\pi$$
0.461099 + 0.887349i $$0.347455\pi$$
$$788$$ −6.13313 −0.218484
$$789$$ −11.2993 −0.402266
$$790$$ 21.2614 0.756448
$$791$$ 4.29254 0.152625
$$792$$ 0 0
$$793$$ −18.5503 −0.658741
$$794$$ 0.934320 0.0331578
$$795$$ −10.4874 −0.371949
$$796$$ 17.4611 0.618893
$$797$$ 3.62989 0.128577 0.0642886 0.997931i $$-0.479522\pi$$
0.0642886 + 0.997931i $$0.479522\pi$$
$$798$$ −1.74178 −0.0616585
$$799$$ −67.1796 −2.37664
$$800$$ 3.26880 0.115570
$$801$$ 25.1489 0.888593
$$802$$ −34.6749 −1.22441
$$803$$ 0 0
$$804$$ −11.0867 −0.390999
$$805$$ −13.9822 −0.492807
$$806$$ −17.2855 −0.608857
$$807$$ 11.3331 0.398945
$$808$$ −2.79391 −0.0982894
$$809$$ 13.1327 0.461720 0.230860 0.972987i $$-0.425846\pi$$
0.230860 + 0.972987i $$0.425846\pi$$
$$810$$ −26.7548 −0.940067
$$811$$ −9.40230 −0.330160 −0.165080 0.986280i $$-0.552788\pi$$
−0.165080 + 0.986280i $$0.552788\pi$$
$$812$$ −18.3329 −0.643358
$$813$$ 8.76595 0.307435
$$814$$ 0 0
$$815$$ −43.9966 −1.54114
$$816$$ 16.9733 0.594183
$$817$$ −7.75102 −0.271174
$$818$$ 2.01098 0.0703124
$$819$$ −18.6668 −0.652272
$$820$$ −24.3029 −0.848695
$$821$$ 21.1701 0.738843 0.369422 0.929262i $$-0.379556\pi$$
0.369422 + 0.929262i $$0.379556\pi$$
$$822$$ −11.5755 −0.403743
$$823$$ 25.2277 0.879384 0.439692 0.898149i $$-0.355088\pi$$
0.439692 + 0.898149i $$0.355088\pi$$
$$824$$ 7.41422 0.258286
$$825$$ 0 0
$$826$$ 33.7735 1.17513
$$827$$ 39.4460 1.37167 0.685836 0.727756i $$-0.259437\pi$$
0.685836 + 0.727756i $$0.259437\pi$$
$$828$$ −12.6300 −0.438922
$$829$$ −36.5068 −1.26793 −0.633967 0.773360i $$-0.718574\pi$$
−0.633967 + 0.773360i $$0.718574\pi$$
$$830$$ −7.83748 −0.272043
$$831$$ 1.06604 0.0369806
$$832$$ 8.66069 0.300255
$$833$$ −25.4972 −0.883427
$$834$$ 0.932366 0.0322852
$$835$$ 14.8066 0.512403
$$836$$ 0 0
$$837$$ −7.36838 −0.254688
$$838$$ 34.3288 1.18587
$$839$$ 42.4670 1.46612 0.733061 0.680163i $$-0.238091\pi$$
0.733061 + 0.680163i $$0.238091\pi$$
$$840$$ 2.70975 0.0934953
$$841$$ 36.6645 1.26429
$$842$$ 45.2321 1.55880
$$843$$ 2.58776 0.0891273
$$844$$ −3.89258 −0.133988
$$845$$ 9.16241 0.315196
$$846$$ −53.3973 −1.83584
$$847$$ 0 0
$$848$$ −38.7755 −1.33156
$$849$$ −0.663690 −0.0227778
$$850$$ −5.66047 −0.194153
$$851$$ −23.1048 −0.792023
$$852$$ −2.35009 −0.0805126
$$853$$ −21.5088 −0.736446 −0.368223 0.929738i $$-0.620034\pi$$
−0.368223 + 0.929738i $$0.620034\pi$$
$$854$$ 13.9475 0.477275
$$855$$ −6.29268 −0.215205
$$856$$ −0.512389 −0.0175131
$$857$$ 12.6370 0.431670 0.215835 0.976430i $$-0.430753\pi$$
0.215835 + 0.976430i $$0.430753\pi$$
$$858$$ 0 0
$$859$$ −3.39720 −0.115911 −0.0579555 0.998319i $$-0.518458\pi$$
−0.0579555 + 0.998319i $$0.518458\pi$$
$$860$$ −24.3057 −0.828818
$$861$$ −7.38990 −0.251847
$$862$$ −58.8167 −2.00330
$$863$$ −10.9308 −0.372088 −0.186044 0.982541i $$-0.559567\pi$$
−0.186044 + 0.982541i $$0.559567\pi$$
$$864$$ 20.8212 0.708353
$$865$$ 50.4971 1.71695
$$866$$ 27.9561 0.949987
$$867$$ −11.8358 −0.401963
$$868$$ 5.20672 0.176727
$$869$$ 0 0
$$870$$ 19.5634 0.663261
$$871$$ −60.5243 −2.05079
$$872$$ −16.9824 −0.575096
$$873$$ 29.4914 0.998133
$$874$$ −6.43359 −0.217619
$$875$$ 17.8544 0.603588
$$876$$ 8.72371 0.294747
$$877$$ −31.7230 −1.07121 −0.535605 0.844468i $$-0.679917\pi$$
−0.535605 + 0.844468i $$0.679917\pi$$
$$878$$ 47.8250 1.61402
$$879$$ −6.34439 −0.213991
$$880$$ 0 0
$$881$$ −6.45839 −0.217589 −0.108794 0.994064i $$-0.534699\pi$$
−0.108794 + 0.994064i $$0.534699\pi$$
$$882$$ −20.2663 −0.682403
$$883$$ −48.5242 −1.63297 −0.816485 0.577367i $$-0.804080\pi$$
−0.816485 + 0.577367i $$0.804080\pi$$
$$884$$ −33.8857 −1.13970
$$885$$ −14.4386 −0.485348
$$886$$ −39.3796 −1.32298
$$887$$ 44.0097 1.47770 0.738851 0.673869i $$-0.235369\pi$$
0.738851 + 0.673869i $$0.235369\pi$$
$$888$$ 4.47772 0.150262
$$889$$ 1.81552 0.0608905
$$890$$ 40.1716 1.34656
$$891$$ 0 0
$$892$$ 8.47742 0.283845
$$893$$ −10.8969 −0.364652
$$894$$ −10.8244 −0.362022
$$895$$ −18.6962 −0.624946
$$896$$ 15.5005 0.517835
$$897$$ 8.15900 0.272421
$$898$$ −21.5531 −0.719236
$$899$$ −18.6494 −0.621991
$$900$$ −1.80248 −0.0600825
$$901$$ 48.9199 1.62976
$$902$$ 0 0
$$903$$ −7.39076 −0.245949
$$904$$ −3.07275 −0.102198
$$905$$ −2.74349 −0.0911966
$$906$$ −13.6301 −0.452829
$$907$$ −2.05084 −0.0680971 −0.0340485 0.999420i $$-0.510840\pi$$
−0.0340485 + 0.999420i $$0.510840\pi$$
$$908$$ 2.67932 0.0889164
$$909$$ 6.18655 0.205195
$$910$$ −29.8175 −0.988439
$$911$$ −16.8417 −0.557990 −0.278995 0.960293i $$-0.590001\pi$$
−0.278995 + 0.960293i $$0.590001\pi$$
$$912$$ 2.75317 0.0911666
$$913$$ 0 0
$$914$$ −28.6539 −0.947788
$$915$$ −5.96275 −0.197122
$$916$$ −28.0584 −0.927075
$$917$$ −13.2465 −0.437438
$$918$$ −36.0554 −1.19001
$$919$$ −8.01221 −0.264299 −0.132149 0.991230i $$-0.542188\pi$$
−0.132149 + 0.991230i $$0.542188\pi$$
$$920$$ 10.0089 0.329985
$$921$$ 2.45224 0.0808039
$$922$$ 49.7683 1.63903
$$923$$ −12.8295 −0.422289
$$924$$ 0 0
$$925$$ −3.29738 −0.108417
$$926$$ 32.7641 1.07670
$$927$$ −16.4173 −0.539214
$$928$$ 52.6986 1.72992
$$929$$ 16.7897 0.550853 0.275426 0.961322i $$-0.411181\pi$$
0.275426 + 0.961322i $$0.411181\pi$$
$$930$$ −5.55620 −0.182195
$$931$$ −4.13581 −0.135546
$$932$$ 23.5284 0.770698
$$933$$ −4.29168 −0.140503
$$934$$ −52.9922 −1.73396
$$935$$ 0 0
$$936$$ 13.3624 0.436763
$$937$$ −21.8471 −0.713713 −0.356856 0.934159i $$-0.616151\pi$$
−0.356856 + 0.934159i $$0.616151\pi$$
$$938$$ 45.5068 1.48585
$$939$$ 9.89661 0.322964
$$940$$ −34.1707 −1.11453
$$941$$ −17.8421 −0.581635 −0.290817 0.956779i $$-0.593927\pi$$
−0.290817 + 0.956779i $$0.593927\pi$$
$$942$$ 2.05749 0.0670368
$$943$$ −27.2959 −0.888877
$$944$$ −53.3845 −1.73752
$$945$$ −12.7104 −0.413470
$$946$$ 0 0
$$947$$ 41.9736 1.36396 0.681980 0.731371i $$-0.261119\pi$$
0.681980 + 0.731371i $$0.261119\pi$$
$$948$$ 3.73711 0.121376
$$949$$ 47.6242 1.54595
$$950$$ −0.918163 −0.0297892
$$951$$ 1.09411 0.0354788
$$952$$ −12.6400 −0.409665
$$953$$ 8.06945 0.261395 0.130697 0.991422i $$-0.458278\pi$$
0.130697 + 0.991422i $$0.458278\pi$$
$$954$$ 38.8837 1.25891
$$955$$ −36.7520 −1.18927
$$956$$ −35.5864 −1.15094
$$957$$ 0 0
$$958$$ 62.2711 2.01189
$$959$$ 19.0348 0.614666
$$960$$ 2.78386 0.0898487
$$961$$ −25.7034 −0.829142
$$962$$ −49.2718 −1.58859
$$963$$ 1.13458 0.0365614
$$964$$ −17.5577 −0.565496
$$965$$ −50.8491 −1.63689
$$966$$ −6.13455 −0.197376
$$967$$ 3.86360 0.124245 0.0621225 0.998069i $$-0.480213\pi$$
0.0621225 + 0.998069i $$0.480213\pi$$
$$968$$ 0 0
$$969$$ −3.47345 −0.111583
$$970$$ 47.1081 1.51255
$$971$$ 42.7787 1.37283 0.686417 0.727208i $$-0.259182\pi$$
0.686417 + 0.727208i $$0.259182\pi$$
$$972$$ −17.5425 −0.562675
$$973$$ −1.53318 −0.0491516
$$974$$ −11.6607 −0.373631
$$975$$ 1.16440 0.0372908
$$976$$ −22.0463 −0.705686
$$977$$ 54.5146 1.74408 0.872039 0.489437i $$-0.162798\pi$$
0.872039 + 0.489437i $$0.162798\pi$$
$$978$$ −19.3031 −0.617246
$$979$$ 0 0
$$980$$ −12.9691 −0.414283
$$981$$ 37.6040 1.20060
$$982$$ −57.4590 −1.83359
$$983$$ 33.5422 1.06983 0.534916 0.844905i $$-0.320343\pi$$
0.534916 + 0.844905i $$0.320343\pi$$
$$984$$ 5.28995 0.168637
$$985$$ −10.7623 −0.342914
$$986$$ −91.2562 −2.90619
$$987$$ −10.3905 −0.330732
$$988$$ −5.49647 −0.174866
$$989$$ −27.2991 −0.868060
$$990$$ 0 0
$$991$$ 31.7767 1.00942 0.504710 0.863289i $$-0.331600\pi$$
0.504710 + 0.863289i $$0.331600\pi$$
$$992$$ −14.9669 −0.475200
$$993$$ −7.54907 −0.239562
$$994$$ 9.64621 0.305959
$$995$$ 30.6403 0.971363
$$996$$ −1.37759 −0.0436505
$$997$$ 47.5668 1.50646 0.753228 0.657759i $$-0.228496\pi$$
0.753228 + 0.657759i $$0.228496\pi$$
$$998$$ 58.3933 1.84841
$$999$$ −21.0033 −0.664515
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 2299.2.a.n.1.2 5
11.10 odd 2 209.2.a.c.1.4 5
33.32 even 2 1881.2.a.k.1.2 5
44.43 even 2 3344.2.a.t.1.4 5
55.54 odd 2 5225.2.a.h.1.2 5
209.208 even 2 3971.2.a.h.1.2 5

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.4 5 11.10 odd 2
1881.2.a.k.1.2 5 33.32 even 2
2299.2.a.n.1.2 5 1.1 even 1 trivial
3344.2.a.t.1.4 5 44.43 even 2
3971.2.a.h.1.2 5 209.208 even 2
5225.2.a.h.1.2 5 55.54 odd 2