# Properties

 Label 209.2.a.c.1.1 Level $209$ Weight $2$ Character 209.1 Self dual yes Analytic conductor $1.669$ Analytic rank $0$ 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] = [209,2,Mod(1,209)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, 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("209.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$209 = 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 209.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: $$1.66887340224$$ Analytic rank: $$0$$ 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0.245526$$ of defining polynomial Character $$\chi$$ $$=$$ 209.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.18524 q^{2} +2.15766 q^{3} +2.77529 q^{4} -3.43077 q^{5} -4.71500 q^{6} +3.93972 q^{7} -1.69419 q^{8} +1.65548 q^{9} +O(q^{10})$$ $$q-2.18524 q^{2} +2.15766 q^{3} +2.77529 q^{4} -3.43077 q^{5} -4.71500 q^{6} +3.93972 q^{7} -1.69419 q^{8} +1.65548 q^{9} +7.49706 q^{10} +1.00000 q^{11} +5.98812 q^{12} +3.31182 q^{13} -8.60924 q^{14} -7.40242 q^{15} -1.84836 q^{16} +2.80637 q^{17} -3.61763 q^{18} -1.00000 q^{19} -9.52137 q^{20} +8.50056 q^{21} -2.18524 q^{22} +6.88998 q^{23} -3.65548 q^{24} +6.77018 q^{25} -7.23713 q^{26} -2.90101 q^{27} +10.9338 q^{28} +5.67979 q^{29} +16.1761 q^{30} +2.51864 q^{31} +7.42749 q^{32} +2.15766 q^{33} -6.13259 q^{34} -13.5163 q^{35} +4.59444 q^{36} -6.39893 q^{37} +2.18524 q^{38} +7.14577 q^{39} +5.81238 q^{40} +0.560629 q^{41} -18.5758 q^{42} -9.40080 q^{43} +2.77529 q^{44} -5.67958 q^{45} -15.0563 q^{46} -12.1742 q^{47} -3.98812 q^{48} +8.52137 q^{49} -14.7945 q^{50} +6.05517 q^{51} +9.19126 q^{52} +5.68316 q^{53} +6.33941 q^{54} -3.43077 q^{55} -6.67463 q^{56} -2.15766 q^{57} -12.4117 q^{58} +4.35730 q^{59} -20.5438 q^{60} -3.56412 q^{61} -5.50384 q^{62} +6.52213 q^{63} -12.5342 q^{64} -11.3621 q^{65} -4.71500 q^{66} -9.95563 q^{67} +7.78847 q^{68} +14.8662 q^{69} +29.5363 q^{70} -11.4671 q^{71} -2.80470 q^{72} -8.95834 q^{73} +13.9832 q^{74} +14.6077 q^{75} -2.77529 q^{76} +3.93972 q^{77} -15.6153 q^{78} +8.49105 q^{79} +6.34128 q^{80} -11.2258 q^{81} -1.22511 q^{82} -5.21960 q^{83} +23.5915 q^{84} -9.62799 q^{85} +20.5430 q^{86} +12.2550 q^{87} -1.69419 q^{88} -7.28423 q^{89} +12.4113 q^{90} +13.0476 q^{91} +19.1217 q^{92} +5.43436 q^{93} +26.6036 q^{94} +3.43077 q^{95} +16.0260 q^{96} +10.6574 q^{97} -18.6213 q^{98} +1.65548 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10})$$ 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 $$5 q + 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{8} + 4 q^{9} + 12 q^{10} + 5 q^{11} + 6 q^{12} + 4 q^{13} - 14 q^{14} + 3 q^{15} + 8 q^{16} - 4 q^{17} - 20 q^{18} - 5 q^{19} - 8 q^{20} + 10 q^{21} + 2 q^{22} + 3 q^{23} - 14 q^{24} + 6 q^{25} - 6 q^{26} - 11 q^{27} - 10 q^{28} + 10 q^{29} + 6 q^{30} + 11 q^{31} + 14 q^{32} + q^{33} - 4 q^{34} - 8 q^{35} - 26 q^{36} + q^{37} - 2 q^{38} + 2 q^{39} - 16 q^{40} + 2 q^{41} - 16 q^{42} + 20 q^{43} + 6 q^{44} - 28 q^{45} - 4 q^{46} - 20 q^{47} + 4 q^{48} + 3 q^{49} - 32 q^{50} + 24 q^{51} + 6 q^{52} - 14 q^{53} + 16 q^{54} - 5 q^{55} - 38 q^{56} - q^{57} - 6 q^{58} + 3 q^{59} - 40 q^{60} - 10 q^{61} - 6 q^{62} + 24 q^{63} - 2 q^{66} + 9 q^{67} + 24 q^{68} - 5 q^{69} + 50 q^{70} + 23 q^{71} - 12 q^{72} + 8 q^{74} - 18 q^{75} - 6 q^{76} + 6 q^{77} - 22 q^{78} + 44 q^{79} - 18 q^{80} + q^{81} - 30 q^{82} - 14 q^{83} + 14 q^{84} - 12 q^{85} + 52 q^{86} + 28 q^{87} + 6 q^{88} - 27 q^{89} + 26 q^{90} + 24 q^{91} + 58 q^{92} - 27 q^{93} - 8 q^{94} + 5 q^{95} + 50 q^{96} + 15 q^{97} - 10 q^{98} + 4 q^{99}+O(q^{100})$$ 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 + 5 * q^11 + 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 + 2 * q^22 + 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 + q^33 - 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 + 6 * q^44 - 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 - 5 * q^55 - 38 * q^56 - q^57 - 6 * q^58 + 3 * q^59 - 40 * q^60 - 10 * q^61 - 6 * q^62 + 24 * q^63 - 2 * q^66 + 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 + 6 * q^77 - 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 + 6 * q^88 - 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 + 4 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.18524 −1.54520 −0.772600 0.634893i $$-0.781044\pi$$
−0.772600 + 0.634893i $$0.781044\pi$$
$$3$$ 2.15766 1.24572 0.622862 0.782332i $$-0.285970\pi$$
0.622862 + 0.782332i $$0.285970\pi$$
$$4$$ 2.77529 1.38764
$$5$$ −3.43077 −1.53429 −0.767143 0.641476i $$-0.778322\pi$$
−0.767143 + 0.641476i $$0.778322\pi$$
$$6$$ −4.71500 −1.92489
$$7$$ 3.93972 1.48907 0.744537 0.667582i $$-0.232671\pi$$
0.744537 + 0.667582i $$0.232671\pi$$
$$8$$ −1.69419 −0.598987
$$9$$ 1.65548 0.551827
$$10$$ 7.49706 2.37078
$$11$$ 1.00000 0.301511
$$12$$ 5.98812 1.72862
$$13$$ 3.31182 0.918534 0.459267 0.888298i $$-0.348112\pi$$
0.459267 + 0.888298i $$0.348112\pi$$
$$14$$ −8.60924 −2.30092
$$15$$ −7.40242 −1.91130
$$16$$ −1.84836 −0.462089
$$17$$ 2.80637 0.680644 0.340322 0.940309i $$-0.389464\pi$$
0.340322 + 0.940309i $$0.389464\pi$$
$$18$$ −3.61763 −0.852684
$$19$$ −1.00000 −0.229416
$$20$$ −9.52137 −2.12904
$$21$$ 8.50056 1.85497
$$22$$ −2.18524 −0.465895
$$23$$ 6.88998 1.43666 0.718330 0.695702i $$-0.244907\pi$$
0.718330 + 0.695702i $$0.244907\pi$$
$$24$$ −3.65548 −0.746172
$$25$$ 6.77018 1.35404
$$26$$ −7.23713 −1.41932
$$27$$ −2.90101 −0.558299
$$28$$ 10.9338 2.06630
$$29$$ 5.67979 1.05471 0.527355 0.849645i $$-0.323184\pi$$
0.527355 + 0.849645i $$0.323184\pi$$
$$30$$ 16.1761 2.95334
$$31$$ 2.51864 0.452361 0.226180 0.974085i $$-0.427376\pi$$
0.226180 + 0.974085i $$0.427376\pi$$
$$32$$ 7.42749 1.31301
$$33$$ 2.15766 0.375600
$$34$$ −6.13259 −1.05173
$$35$$ −13.5163 −2.28466
$$36$$ 4.59444 0.765740
$$37$$ −6.39893 −1.05198 −0.525989 0.850491i $$-0.676305\pi$$
−0.525989 + 0.850491i $$0.676305\pi$$
$$38$$ 2.18524 0.354493
$$39$$ 7.14577 1.14424
$$40$$ 5.81238 0.919018
$$41$$ 0.560629 0.0875555 0.0437778 0.999041i $$-0.486061\pi$$
0.0437778 + 0.999041i $$0.486061\pi$$
$$42$$ −18.5758 −2.86631
$$43$$ −9.40080 −1.43361 −0.716805 0.697274i $$-0.754396\pi$$
−0.716805 + 0.697274i $$0.754396\pi$$
$$44$$ 2.77529 0.418390
$$45$$ −5.67958 −0.846661
$$46$$ −15.0563 −2.21993
$$47$$ −12.1742 −1.77579 −0.887896 0.460044i $$-0.847834\pi$$
−0.887896 + 0.460044i $$0.847834\pi$$
$$48$$ −3.98812 −0.575635
$$49$$ 8.52137 1.21734
$$50$$ −14.7945 −2.09226
$$51$$ 6.05517 0.847894
$$52$$ 9.19126 1.27460
$$53$$ 5.68316 0.780643 0.390321 0.920679i $$-0.372364\pi$$
0.390321 + 0.920679i $$0.372364\pi$$
$$54$$ 6.33941 0.862684
$$55$$ −3.43077 −0.462605
$$56$$ −6.67463 −0.891935
$$57$$ −2.15766 −0.285789
$$58$$ −12.4117 −1.62974
$$59$$ 4.35730 0.567273 0.283636 0.958932i $$-0.408459\pi$$
0.283636 + 0.958932i $$0.408459\pi$$
$$60$$ −20.5438 −2.65220
$$61$$ −3.56412 −0.456339 −0.228169 0.973621i $$-0.573274\pi$$
−0.228169 + 0.973621i $$0.573274\pi$$
$$62$$ −5.50384 −0.698988
$$63$$ 6.52213 0.821711
$$64$$ −12.5342 −1.56677
$$65$$ −11.3621 −1.40929
$$66$$ −4.71500 −0.580377
$$67$$ −9.95563 −1.21627 −0.608137 0.793832i $$-0.708083\pi$$
−0.608137 + 0.793832i $$0.708083\pi$$
$$68$$ 7.78847 0.944491
$$69$$ 14.8662 1.78968
$$70$$ 29.5363 3.53026
$$71$$ −11.4671 −1.36089 −0.680447 0.732797i $$-0.738214\pi$$
−0.680447 + 0.732797i $$0.738214\pi$$
$$72$$ −2.80470 −0.330537
$$73$$ −8.95834 −1.04849 −0.524247 0.851566i $$-0.675653\pi$$
−0.524247 + 0.851566i $$0.675653\pi$$
$$74$$ 13.9832 1.62552
$$75$$ 14.6077 1.68675
$$76$$ −2.77529 −0.318347
$$77$$ 3.93972 0.448972
$$78$$ −15.6153 −1.76808
$$79$$ 8.49105 0.955318 0.477659 0.878545i $$-0.341485\pi$$
0.477659 + 0.878545i $$0.341485\pi$$
$$80$$ 6.34128 0.708977
$$81$$ −11.2258 −1.24731
$$82$$ −1.22511 −0.135291
$$83$$ −5.21960 −0.572926 −0.286463 0.958091i $$-0.592480\pi$$
−0.286463 + 0.958091i $$0.592480\pi$$
$$84$$ 23.5915 2.57404
$$85$$ −9.62799 −1.04430
$$86$$ 20.5430 2.21521
$$87$$ 12.2550 1.31388
$$88$$ −1.69419 −0.180601
$$89$$ −7.28423 −0.772127 −0.386064 0.922472i $$-0.626166\pi$$
−0.386064 + 0.922472i $$0.626166\pi$$
$$90$$ 12.4113 1.30826
$$91$$ 13.0476 1.36776
$$92$$ 19.1217 1.99357
$$93$$ 5.43436 0.563517
$$94$$ 26.6036 2.74395
$$95$$ 3.43077 0.351989
$$96$$ 16.0260 1.63564
$$97$$ 10.6574 1.08209 0.541045 0.840993i $$-0.318029\pi$$
0.541045 + 0.840993i $$0.318029\pi$$
$$98$$ −18.6213 −1.88103
$$99$$ 1.65548 0.166382
$$100$$ 18.7892 1.87892
$$101$$ −11.4716 −1.14147 −0.570735 0.821134i $$-0.693342\pi$$
−0.570735 + 0.821134i $$0.693342\pi$$
$$102$$ −13.2320 −1.31017
$$103$$ 18.3034 1.80349 0.901745 0.432268i $$-0.142286\pi$$
0.901745 + 0.432268i $$0.142286\pi$$
$$104$$ −5.61086 −0.550190
$$105$$ −29.1634 −2.84606
$$106$$ −12.4191 −1.20625
$$107$$ 1.38838 0.134220 0.0671100 0.997746i $$-0.478622\pi$$
0.0671100 + 0.997746i $$0.478622\pi$$
$$108$$ −8.05113 −0.774720
$$109$$ −0.412113 −0.0394732 −0.0197366 0.999805i $$-0.506283\pi$$
−0.0197366 + 0.999805i $$0.506283\pi$$
$$110$$ 7.49706 0.714817
$$111$$ −13.8067 −1.31047
$$112$$ −7.28200 −0.688084
$$113$$ −6.54003 −0.615234 −0.307617 0.951510i $$-0.599532\pi$$
−0.307617 + 0.951510i $$0.599532\pi$$
$$114$$ 4.71500 0.441601
$$115$$ −23.6379 −2.20425
$$116$$ 15.7630 1.46356
$$117$$ 5.48266 0.506872
$$118$$ −9.52177 −0.876550
$$119$$ 11.0563 1.01353
$$120$$ 12.5411 1.14484
$$121$$ 1.00000 0.0909091
$$122$$ 7.78847 0.705135
$$123$$ 1.20964 0.109070
$$124$$ 6.98995 0.627716
$$125$$ −6.07307 −0.543192
$$126$$ −14.2524 −1.26971
$$127$$ 9.08005 0.805724 0.402862 0.915261i $$-0.368015\pi$$
0.402862 + 0.915261i $$0.368015\pi$$
$$128$$ 12.5352 1.10797
$$129$$ −20.2837 −1.78588
$$130$$ 24.8289 2.17764
$$131$$ −10.3876 −0.907571 −0.453785 0.891111i $$-0.649927\pi$$
−0.453785 + 0.891111i $$0.649927\pi$$
$$132$$ 5.98812 0.521199
$$133$$ −3.93972 −0.341617
$$134$$ 21.7555 1.87939
$$135$$ 9.95269 0.856591
$$136$$ −4.75452 −0.407697
$$137$$ −0.798293 −0.0682028 −0.0341014 0.999418i $$-0.510857\pi$$
−0.0341014 + 0.999418i $$0.510857\pi$$
$$138$$ −32.4863 −2.76542
$$139$$ 5.03184 0.426795 0.213398 0.976965i $$-0.431547\pi$$
0.213398 + 0.976965i $$0.431547\pi$$
$$140$$ −37.5115 −3.17030
$$141$$ −26.2678 −2.21215
$$142$$ 25.0584 2.10285
$$143$$ 3.31182 0.276948
$$144$$ −3.05992 −0.254993
$$145$$ −19.4860 −1.61823
$$146$$ 19.5761 1.62013
$$147$$ 18.3862 1.51647
$$148$$ −17.7589 −1.45977
$$149$$ 19.8351 1.62496 0.812479 0.582991i $$-0.198118\pi$$
0.812479 + 0.582991i $$0.198118\pi$$
$$150$$ −31.9214 −2.60637
$$151$$ 22.5447 1.83466 0.917331 0.398125i $$-0.130339\pi$$
0.917331 + 0.398125i $$0.130339\pi$$
$$152$$ 1.69419 0.137417
$$153$$ 4.64589 0.375598
$$154$$ −8.60924 −0.693752
$$155$$ −8.64087 −0.694051
$$156$$ 19.8316 1.58780
$$157$$ −11.8013 −0.941843 −0.470921 0.882175i $$-0.656078\pi$$
−0.470921 + 0.882175i $$0.656078\pi$$
$$158$$ −18.5550 −1.47616
$$159$$ 12.2623 0.972465
$$160$$ −25.4820 −2.01453
$$161$$ 27.1446 2.13929
$$162$$ 24.5312 1.92735
$$163$$ −24.8395 −1.94558 −0.972789 0.231691i $$-0.925574\pi$$
−0.972789 + 0.231691i $$0.925574\pi$$
$$164$$ 1.55591 0.121496
$$165$$ −7.40242 −0.576278
$$166$$ 11.4061 0.885285
$$167$$ −2.79938 −0.216623 −0.108311 0.994117i $$-0.534544\pi$$
−0.108311 + 0.994117i $$0.534544\pi$$
$$168$$ −14.4016 −1.11110
$$169$$ −2.03184 −0.156295
$$170$$ 21.0395 1.61366
$$171$$ −1.65548 −0.126598
$$172$$ −26.0899 −1.98934
$$173$$ −6.43926 −0.489568 −0.244784 0.969578i $$-0.578717\pi$$
−0.244784 + 0.969578i $$0.578717\pi$$
$$174$$ −26.7802 −2.03020
$$175$$ 26.6726 2.01626
$$176$$ −1.84836 −0.139325
$$177$$ 9.40156 0.706665
$$178$$ 15.9178 1.19309
$$179$$ 12.5241 0.936095 0.468048 0.883703i $$-0.344958\pi$$
0.468048 + 0.883703i $$0.344958\pi$$
$$180$$ −15.7625 −1.17486
$$181$$ 13.7515 1.02214 0.511071 0.859538i $$-0.329249\pi$$
0.511071 + 0.859538i $$0.329249\pi$$
$$182$$ −28.5123 −2.11347
$$183$$ −7.69015 −0.568472
$$184$$ −11.6729 −0.860541
$$185$$ 21.9533 1.61404
$$186$$ −11.8754 −0.870746
$$187$$ 2.80637 0.205222
$$188$$ −33.7869 −2.46417
$$189$$ −11.4292 −0.831348
$$190$$ −7.49706 −0.543894
$$191$$ 3.10678 0.224799 0.112399 0.993663i $$-0.464146\pi$$
0.112399 + 0.993663i $$0.464146\pi$$
$$192$$ −27.0444 −1.95176
$$193$$ −0.747815 −0.0538289 −0.0269144 0.999638i $$-0.508568\pi$$
−0.0269144 + 0.999638i $$0.508568\pi$$
$$194$$ −23.2889 −1.67205
$$195$$ −24.5155 −1.75559
$$196$$ 23.6492 1.68923
$$197$$ −3.41798 −0.243521 −0.121761 0.992559i $$-0.538854\pi$$
−0.121761 + 0.992559i $$0.538854\pi$$
$$198$$ −3.61763 −0.257094
$$199$$ 5.36785 0.380517 0.190258 0.981734i $$-0.439067\pi$$
0.190258 + 0.981734i $$0.439067\pi$$
$$200$$ −11.4700 −0.811049
$$201$$ −21.4808 −1.51514
$$202$$ 25.0683 1.76380
$$203$$ 22.3768 1.57054
$$204$$ 16.8048 1.17657
$$205$$ −1.92339 −0.134335
$$206$$ −39.9974 −2.78675
$$207$$ 11.4062 0.792789
$$208$$ −6.12142 −0.424444
$$209$$ −1.00000 −0.0691714
$$210$$ 63.7292 4.39773
$$211$$ 2.55492 0.175888 0.0879441 0.996125i $$-0.471970\pi$$
0.0879441 + 0.996125i $$0.471970\pi$$
$$212$$ 15.7724 1.08325
$$213$$ −24.7421 −1.69530
$$214$$ −3.03395 −0.207397
$$215$$ 32.2520 2.19957
$$216$$ 4.91486 0.334414
$$217$$ 9.92272 0.673598
$$218$$ 0.900566 0.0609940
$$219$$ −19.3290 −1.30613
$$220$$ −9.52137 −0.641931
$$221$$ 9.29418 0.625194
$$222$$ 30.1710 2.02494
$$223$$ −24.9404 −1.67013 −0.835066 0.550149i $$-0.814571\pi$$
−0.835066 + 0.550149i $$0.814571\pi$$
$$224$$ 29.2622 1.95516
$$225$$ 11.2079 0.747194
$$226$$ 14.2915 0.950660
$$227$$ −22.8254 −1.51497 −0.757487 0.652851i $$-0.773573\pi$$
−0.757487 + 0.652851i $$0.773573\pi$$
$$228$$ −5.98812 −0.396573
$$229$$ −0.603546 −0.0398834 −0.0199417 0.999801i $$-0.506348\pi$$
−0.0199417 + 0.999801i $$0.506348\pi$$
$$230$$ 51.6546 3.40601
$$231$$ 8.50056 0.559296
$$232$$ −9.62264 −0.631757
$$233$$ 17.3705 1.13798 0.568988 0.822346i $$-0.307335\pi$$
0.568988 + 0.822346i $$0.307335\pi$$
$$234$$ −11.9809 −0.783219
$$235$$ 41.7669 2.72457
$$236$$ 12.0928 0.787172
$$237$$ 18.3208 1.19006
$$238$$ −24.1607 −1.56610
$$239$$ 7.23486 0.467984 0.233992 0.972238i $$-0.424821\pi$$
0.233992 + 0.972238i $$0.424821\pi$$
$$240$$ 13.6823 0.883189
$$241$$ −12.2034 −0.786090 −0.393045 0.919519i $$-0.628578\pi$$
−0.393045 + 0.919519i $$0.628578\pi$$
$$242$$ −2.18524 −0.140473
$$243$$ −15.5185 −0.995509
$$244$$ −9.89146 −0.633236
$$245$$ −29.2349 −1.86775
$$246$$ −2.64337 −0.168535
$$247$$ −3.31182 −0.210726
$$248$$ −4.26705 −0.270958
$$249$$ −11.2621 −0.713707
$$250$$ 13.2711 0.839340
$$251$$ −14.0923 −0.889499 −0.444750 0.895655i $$-0.646707\pi$$
−0.444750 + 0.895655i $$0.646707\pi$$
$$252$$ 18.1008 1.14024
$$253$$ 6.88998 0.433169
$$254$$ −19.8421 −1.24501
$$255$$ −20.7739 −1.30091
$$256$$ −2.32415 −0.145259
$$257$$ 0.440920 0.0275038 0.0137519 0.999905i $$-0.495622\pi$$
0.0137519 + 0.999905i $$0.495622\pi$$
$$258$$ 44.3248 2.75954
$$259$$ −25.2100 −1.56647
$$260$$ −31.5331 −1.95560
$$261$$ 9.40279 0.582018
$$262$$ 22.6995 1.40238
$$263$$ −15.0661 −0.929016 −0.464508 0.885569i $$-0.653769\pi$$
−0.464508 + 0.885569i $$0.653769\pi$$
$$264$$ −3.65548 −0.224979
$$265$$ −19.4976 −1.19773
$$266$$ 8.60924 0.527866
$$267$$ −15.7169 −0.961857
$$268$$ −27.6297 −1.68775
$$269$$ −7.25751 −0.442498 −0.221249 0.975217i $$-0.571013\pi$$
−0.221249 + 0.975217i $$0.571013\pi$$
$$270$$ −21.7490 −1.32360
$$271$$ −16.8878 −1.02586 −0.512931 0.858430i $$-0.671440\pi$$
−0.512931 + 0.858430i $$0.671440\pi$$
$$272$$ −5.18716 −0.314518
$$273$$ 28.1523 1.70386
$$274$$ 1.74446 0.105387
$$275$$ 6.77018 0.408257
$$276$$ 41.2580 2.48344
$$277$$ −15.3818 −0.924204 −0.462102 0.886827i $$-0.652905\pi$$
−0.462102 + 0.886827i $$0.652905\pi$$
$$278$$ −10.9958 −0.659484
$$279$$ 4.16956 0.249625
$$280$$ 22.8991 1.36848
$$281$$ −25.8974 −1.54491 −0.772456 0.635069i $$-0.780972\pi$$
−0.772456 + 0.635069i $$0.780972\pi$$
$$282$$ 57.4015 3.41821
$$283$$ 18.6882 1.11090 0.555450 0.831550i $$-0.312546\pi$$
0.555450 + 0.831550i $$0.312546\pi$$
$$284$$ −31.8245 −1.88844
$$285$$ 7.40242 0.438482
$$286$$ −7.23713 −0.427941
$$287$$ 2.20872 0.130377
$$288$$ 12.2961 0.724553
$$289$$ −9.12431 −0.536724
$$290$$ 42.5817 2.50049
$$291$$ 22.9949 1.34799
$$292$$ −24.8620 −1.45494
$$293$$ 26.7471 1.56258 0.781291 0.624167i $$-0.214561\pi$$
0.781291 + 0.624167i $$0.214561\pi$$
$$294$$ −40.1783 −2.34325
$$295$$ −14.9489 −0.870359
$$296$$ 10.8410 0.630121
$$297$$ −2.90101 −0.168334
$$298$$ −43.3446 −2.51088
$$299$$ 22.8184 1.31962
$$300$$ 40.5406 2.34061
$$301$$ −37.0365 −2.13475
$$302$$ −49.2657 −2.83492
$$303$$ −24.7518 −1.42196
$$304$$ 1.84836 0.106010
$$305$$ 12.2277 0.700155
$$306$$ −10.1524 −0.580374
$$307$$ 22.6415 1.29222 0.646109 0.763245i $$-0.276395\pi$$
0.646109 + 0.763245i $$0.276395\pi$$
$$308$$ 10.9338 0.623014
$$309$$ 39.4925 2.24665
$$310$$ 18.8824 1.07245
$$311$$ −1.38723 −0.0786628 −0.0393314 0.999226i $$-0.512523\pi$$
−0.0393314 + 0.999226i $$0.512523\pi$$
$$312$$ −12.1063 −0.685384
$$313$$ 12.9018 0.729255 0.364627 0.931153i $$-0.381196\pi$$
0.364627 + 0.931153i $$0.381196\pi$$
$$314$$ 25.7886 1.45534
$$315$$ −22.3759 −1.26074
$$316$$ 23.5651 1.32564
$$317$$ −19.0712 −1.07114 −0.535572 0.844489i $$-0.679904\pi$$
−0.535572 + 0.844489i $$0.679904\pi$$
$$318$$ −26.7961 −1.50265
$$319$$ 5.67979 0.318007
$$320$$ 43.0018 2.40387
$$321$$ 2.99565 0.167201
$$322$$ −59.3175 −3.30564
$$323$$ −2.80637 −0.156150
$$324$$ −31.1549 −1.73083
$$325$$ 22.4216 1.24373
$$326$$ 54.2803 3.00631
$$327$$ −0.889197 −0.0491727
$$328$$ −0.949812 −0.0524446
$$329$$ −47.9630 −2.64428
$$330$$ 16.1761 0.890464
$$331$$ −12.4616 −0.684952 −0.342476 0.939527i $$-0.611265\pi$$
−0.342476 + 0.939527i $$0.611265\pi$$
$$332$$ −14.4859 −0.795017
$$333$$ −10.5933 −0.580510
$$334$$ 6.11733 0.334725
$$335$$ 34.1555 1.86611
$$336$$ −15.7121 −0.857163
$$337$$ −0.401035 −0.0218458 −0.0109229 0.999940i $$-0.503477\pi$$
−0.0109229 + 0.999940i $$0.503477\pi$$
$$338$$ 4.44006 0.241508
$$339$$ −14.1111 −0.766411
$$340$$ −26.7204 −1.44912
$$341$$ 2.51864 0.136392
$$342$$ 3.61763 0.195619
$$343$$ 5.99377 0.323633
$$344$$ 15.9268 0.858713
$$345$$ −51.0026 −2.74589
$$346$$ 14.0713 0.756480
$$347$$ 1.06608 0.0572304 0.0286152 0.999591i $$-0.490890\pi$$
0.0286152 + 0.999591i $$0.490890\pi$$
$$348$$ 34.0112 1.82319
$$349$$ 22.2695 1.19206 0.596029 0.802963i $$-0.296744\pi$$
0.596029 + 0.802963i $$0.296744\pi$$
$$350$$ −58.2861 −3.11552
$$351$$ −9.60762 −0.512817
$$352$$ 7.42749 0.395886
$$353$$ 12.3631 0.658023 0.329012 0.944326i $$-0.393284\pi$$
0.329012 + 0.944326i $$0.393284\pi$$
$$354$$ −20.5447 −1.09194
$$355$$ 39.3410 2.08800
$$356$$ −20.2158 −1.07144
$$357$$ 23.8557 1.26258
$$358$$ −27.3682 −1.44645
$$359$$ 32.1914 1.69899 0.849497 0.527593i $$-0.176905\pi$$
0.849497 + 0.527593i $$0.176905\pi$$
$$360$$ 9.62229 0.507139
$$361$$ 1.00000 0.0526316
$$362$$ −30.0504 −1.57941
$$363$$ 2.15766 0.113248
$$364$$ 36.2109 1.89797
$$365$$ 30.7340 1.60869
$$366$$ 16.8048 0.878403
$$367$$ −18.9777 −0.990628 −0.495314 0.868714i $$-0.664947\pi$$
−0.495314 + 0.868714i $$0.664947\pi$$
$$368$$ −12.7351 −0.663865
$$369$$ 0.928111 0.0483155
$$370$$ −47.9732 −2.49401
$$371$$ 22.3901 1.16243
$$372$$ 15.0819 0.781960
$$373$$ 1.15238 0.0596678 0.0298339 0.999555i $$-0.490502\pi$$
0.0298339 + 0.999555i $$0.490502\pi$$
$$374$$ −6.13259 −0.317109
$$375$$ −13.1036 −0.676667
$$376$$ 20.6254 1.06368
$$377$$ 18.8104 0.968787
$$378$$ 24.9755 1.28460
$$379$$ 13.5365 0.695325 0.347663 0.937620i $$-0.386975\pi$$
0.347663 + 0.937620i $$0.386975\pi$$
$$380$$ 9.52137 0.488436
$$381$$ 19.5916 1.00371
$$382$$ −6.78907 −0.347359
$$383$$ 13.1671 0.672809 0.336404 0.941718i $$-0.390789\pi$$
0.336404 + 0.941718i $$0.390789\pi$$
$$384$$ 27.0467 1.38022
$$385$$ −13.5163 −0.688852
$$386$$ 1.63416 0.0831764
$$387$$ −15.5629 −0.791105
$$388$$ 29.5772 1.50156
$$389$$ 0.223588 0.0113364 0.00566819 0.999984i $$-0.498196\pi$$
0.00566819 + 0.999984i $$0.498196\pi$$
$$390$$ 53.5723 2.71274
$$391$$ 19.3358 0.977854
$$392$$ −14.4368 −0.729170
$$393$$ −22.4129 −1.13058
$$394$$ 7.46913 0.376289
$$395$$ −29.1308 −1.46573
$$396$$ 4.59444 0.230879
$$397$$ −21.6504 −1.08660 −0.543301 0.839538i $$-0.682826\pi$$
−0.543301 + 0.839538i $$0.682826\pi$$
$$398$$ −11.7301 −0.587975
$$399$$ −8.50056 −0.425560
$$400$$ −12.5137 −0.625685
$$401$$ 1.30180 0.0650089 0.0325045 0.999472i $$-0.489652\pi$$
0.0325045 + 0.999472i $$0.489652\pi$$
$$402$$ 46.9408 2.34120
$$403$$ 8.34128 0.415509
$$404$$ −31.8371 −1.58395
$$405$$ 38.5132 1.91374
$$406$$ −48.8986 −2.42680
$$407$$ −6.39893 −0.317183
$$408$$ −10.2586 −0.507877
$$409$$ −2.15631 −0.106622 −0.0533112 0.998578i $$-0.516978\pi$$
−0.0533112 + 0.998578i $$0.516978\pi$$
$$410$$ 4.20307 0.207575
$$411$$ −1.72244 −0.0849618
$$412$$ 50.7973 2.50260
$$413$$ 17.1665 0.844710
$$414$$ −24.9254 −1.22502
$$415$$ 17.9073 0.879032
$$416$$ 24.5985 1.20604
$$417$$ 10.8570 0.531669
$$418$$ 2.18524 0.106884
$$419$$ −29.6335 −1.44769 −0.723844 0.689963i $$-0.757627\pi$$
−0.723844 + 0.689963i $$0.757627\pi$$
$$420$$ −80.9369 −3.94932
$$421$$ −5.25385 −0.256057 −0.128028 0.991770i $$-0.540865\pi$$
−0.128028 + 0.991770i $$0.540865\pi$$
$$422$$ −5.58313 −0.271782
$$423$$ −20.1542 −0.979931
$$424$$ −9.62837 −0.467595
$$425$$ 18.9996 0.921616
$$426$$ 54.0674 2.61957
$$427$$ −14.0416 −0.679522
$$428$$ 3.85316 0.186249
$$429$$ 7.14577 0.345001
$$430$$ −70.4784 −3.39877
$$431$$ 17.8489 0.859752 0.429876 0.902888i $$-0.358557\pi$$
0.429876 + 0.902888i $$0.358557\pi$$
$$432$$ 5.36209 0.257984
$$433$$ −0.696383 −0.0334660 −0.0167330 0.999860i $$-0.505327\pi$$
−0.0167330 + 0.999860i $$0.505327\pi$$
$$434$$ −21.6836 −1.04084
$$435$$ −42.0442 −2.01586
$$436$$ −1.14373 −0.0547748
$$437$$ −6.88998 −0.329593
$$438$$ 42.2386 2.01824
$$439$$ −24.8210 −1.18464 −0.592321 0.805702i $$-0.701788\pi$$
−0.592321 + 0.805702i $$0.701788\pi$$
$$440$$ 5.81238 0.277094
$$441$$ 14.1070 0.671761
$$442$$ −20.3100 −0.966050
$$443$$ −3.93424 −0.186922 −0.0934608 0.995623i $$-0.529793\pi$$
−0.0934608 + 0.995623i $$0.529793\pi$$
$$444$$ −38.3175 −1.81847
$$445$$ 24.9905 1.18466
$$446$$ 54.5008 2.58069
$$447$$ 42.7974 2.02425
$$448$$ −49.3810 −2.33303
$$449$$ 6.13394 0.289479 0.144739 0.989470i $$-0.453766\pi$$
0.144739 + 0.989470i $$0.453766\pi$$
$$450$$ −24.4920 −1.15456
$$451$$ 0.560629 0.0263990
$$452$$ −18.1505 −0.853725
$$453$$ 48.6437 2.28548
$$454$$ 49.8790 2.34094
$$455$$ −44.7634 −2.09854
$$456$$ 3.65548 0.171184
$$457$$ 8.40189 0.393024 0.196512 0.980501i $$-0.437039\pi$$
0.196512 + 0.980501i $$0.437039\pi$$
$$458$$ 1.31889 0.0616279
$$459$$ −8.14129 −0.380003
$$460$$ −65.6021 −3.05871
$$461$$ 25.9903 1.21049 0.605246 0.796039i $$-0.293075\pi$$
0.605246 + 0.796039i $$0.293075\pi$$
$$462$$ −18.5758 −0.864224
$$463$$ 19.3724 0.900311 0.450155 0.892950i $$-0.351369\pi$$
0.450155 + 0.892950i $$0.351369\pi$$
$$464$$ −10.4983 −0.487370
$$465$$ −18.6440 −0.864596
$$466$$ −37.9587 −1.75840
$$467$$ −34.6720 −1.60443 −0.802214 0.597037i $$-0.796345\pi$$
−0.802214 + 0.597037i $$0.796345\pi$$
$$468$$ 15.2160 0.703358
$$469$$ −39.2224 −1.81112
$$470$$ −91.2709 −4.21001
$$471$$ −25.4631 −1.17328
$$472$$ −7.38210 −0.339789
$$473$$ −9.40080 −0.432249
$$474$$ −40.0353 −1.83888
$$475$$ −6.77018 −0.310637
$$476$$ 30.6844 1.40642
$$477$$ 9.40838 0.430780
$$478$$ −15.8099 −0.723130
$$479$$ 23.2352 1.06164 0.530821 0.847484i $$-0.321883\pi$$
0.530821 + 0.847484i $$0.321883\pi$$
$$480$$ −54.9814 −2.50955
$$481$$ −21.1921 −0.966277
$$482$$ 26.6674 1.21467
$$483$$ 58.5687 2.66497
$$484$$ 2.77529 0.126149
$$485$$ −36.5629 −1.66024
$$486$$ 33.9116 1.53826
$$487$$ 14.2680 0.646545 0.323272 0.946306i $$-0.395217\pi$$
0.323272 + 0.946306i $$0.395217\pi$$
$$488$$ 6.03830 0.273341
$$489$$ −53.5951 −2.42365
$$490$$ 63.8853 2.88604
$$491$$ 41.6168 1.87814 0.939071 0.343724i $$-0.111689\pi$$
0.939071 + 0.343724i $$0.111689\pi$$
$$492$$ 3.35711 0.151350
$$493$$ 15.9396 0.717882
$$494$$ 7.23713 0.325614
$$495$$ −5.67958 −0.255278
$$496$$ −4.65534 −0.209031
$$497$$ −45.1771 −2.02647
$$498$$ 24.6104 1.10282
$$499$$ 19.9664 0.893820 0.446910 0.894579i $$-0.352524\pi$$
0.446910 + 0.894579i $$0.352524\pi$$
$$500$$ −16.8545 −0.753757
$$501$$ −6.04010 −0.269852
$$502$$ 30.7951 1.37445
$$503$$ 2.31362 0.103159 0.0515797 0.998669i $$-0.483574\pi$$
0.0515797 + 0.998669i $$0.483574\pi$$
$$504$$ −11.0497 −0.492194
$$505$$ 39.3565 1.75134
$$506$$ −15.0563 −0.669334
$$507$$ −4.38401 −0.194701
$$508$$ 25.1998 1.11806
$$509$$ −13.2009 −0.585120 −0.292560 0.956247i $$-0.594507\pi$$
−0.292560 + 0.956247i $$0.594507\pi$$
$$510$$ 45.3960 2.01017
$$511$$ −35.2933 −1.56128
$$512$$ −19.9916 −0.883511
$$513$$ 2.90101 0.128083
$$514$$ −0.963517 −0.0424989
$$515$$ −62.7948 −2.76707
$$516$$ −56.2931 −2.47817
$$517$$ −12.1742 −0.535421
$$518$$ 55.0899 2.42051
$$519$$ −13.8937 −0.609866
$$520$$ 19.2496 0.844149
$$521$$ −15.6498 −0.685629 −0.342814 0.939403i $$-0.611380\pi$$
−0.342814 + 0.939403i $$0.611380\pi$$
$$522$$ −20.5474 −0.899334
$$523$$ −17.7350 −0.775498 −0.387749 0.921765i $$-0.626747\pi$$
−0.387749 + 0.921765i $$0.626747\pi$$
$$524$$ −28.8286 −1.25938
$$525$$ 57.5503 2.51170
$$526$$ 32.9231 1.43552
$$527$$ 7.06822 0.307897
$$528$$ −3.98812 −0.173560
$$529$$ 24.4719 1.06399
$$530$$ 42.6071 1.85073
$$531$$ 7.21344 0.313037
$$532$$ −10.9338 −0.474042
$$533$$ 1.85670 0.0804227
$$534$$ 34.3452 1.48626
$$535$$ −4.76322 −0.205932
$$536$$ 16.8667 0.728532
$$537$$ 27.0227 1.16612
$$538$$ 15.8594 0.683748
$$539$$ 8.52137 0.367041
$$540$$ 27.6216 1.18864
$$541$$ −34.5196 −1.48412 −0.742058 0.670336i $$-0.766150\pi$$
−0.742058 + 0.670336i $$0.766150\pi$$
$$542$$ 36.9040 1.58516
$$543$$ 29.6711 1.27331
$$544$$ 20.8442 0.893690
$$545$$ 1.41386 0.0605632
$$546$$ −61.5197 −2.63280
$$547$$ 1.99561 0.0853263 0.0426632 0.999090i $$-0.486416\pi$$
0.0426632 + 0.999090i $$0.486416\pi$$
$$548$$ −2.21549 −0.0946411
$$549$$ −5.90034 −0.251820
$$550$$ −14.7945 −0.630839
$$551$$ −5.67979 −0.241967
$$552$$ −25.1862 −1.07200
$$553$$ 33.4523 1.42254
$$554$$ 33.6130 1.42808
$$555$$ 47.3676 2.01064
$$556$$ 13.9648 0.592239
$$557$$ −23.5272 −0.996881 −0.498440 0.866924i $$-0.666094\pi$$
−0.498440 + 0.866924i $$0.666094\pi$$
$$558$$ −9.11150 −0.385721
$$559$$ −31.1338 −1.31682
$$560$$ 24.9829 1.05572
$$561$$ 6.05517 0.255650
$$562$$ 56.5922 2.38720
$$563$$ 14.2122 0.598972 0.299486 0.954101i $$-0.403185\pi$$
0.299486 + 0.954101i $$0.403185\pi$$
$$564$$ −72.9006 −3.06967
$$565$$ 22.4373 0.943945
$$566$$ −40.8383 −1.71656
$$567$$ −44.2266 −1.85734
$$568$$ 19.4275 0.815158
$$569$$ 22.1906 0.930277 0.465138 0.885238i $$-0.346005\pi$$
0.465138 + 0.885238i $$0.346005\pi$$
$$570$$ −16.1761 −0.677542
$$571$$ 8.67954 0.363227 0.181614 0.983370i $$-0.441868\pi$$
0.181614 + 0.983370i $$0.441868\pi$$
$$572$$ 9.19126 0.384306
$$573$$ 6.70337 0.280037
$$574$$ −4.82659 −0.201458
$$575$$ 46.6464 1.94529
$$576$$ −20.7501 −0.864586
$$577$$ 40.9150 1.70331 0.851657 0.524100i $$-0.175598\pi$$
0.851657 + 0.524100i $$0.175598\pi$$
$$578$$ 19.9388 0.829346
$$579$$ −1.61353 −0.0670559
$$580$$ −54.0794 −2.24552
$$581$$ −20.5638 −0.853128
$$582$$ −50.2495 −2.08291
$$583$$ 5.68316 0.235373
$$584$$ 15.1771 0.628034
$$585$$ −18.8097 −0.777687
$$586$$ −58.4489 −2.41450
$$587$$ −21.2684 −0.877840 −0.438920 0.898526i $$-0.644639\pi$$
−0.438920 + 0.898526i $$0.644639\pi$$
$$588$$ 51.0270 2.10432
$$589$$ −2.51864 −0.103779
$$590$$ 32.6670 1.34488
$$591$$ −7.37484 −0.303360
$$592$$ 11.8275 0.486107
$$593$$ −23.0212 −0.945368 −0.472684 0.881232i $$-0.656715\pi$$
−0.472684 + 0.881232i $$0.656715\pi$$
$$594$$ 6.33941 0.260109
$$595$$ −37.9316 −1.55504
$$596$$ 55.0482 2.25486
$$597$$ 11.5820 0.474019
$$598$$ −49.8637 −2.03908
$$599$$ −41.6249 −1.70075 −0.850374 0.526179i $$-0.823624\pi$$
−0.850374 + 0.526179i $$0.823624\pi$$
$$600$$ −24.7483 −1.01034
$$601$$ −20.8629 −0.851016 −0.425508 0.904955i $$-0.639905\pi$$
−0.425508 + 0.904955i $$0.639905\pi$$
$$602$$ 80.9338 3.29861
$$603$$ −16.4814 −0.671173
$$604$$ 62.5680 2.54586
$$605$$ −3.43077 −0.139481
$$606$$ 54.0888 2.19721
$$607$$ −0.146703 −0.00595450 −0.00297725 0.999996i $$-0.500948\pi$$
−0.00297725 + 0.999996i $$0.500948\pi$$
$$608$$ −7.42749 −0.301224
$$609$$ 48.2813 1.95646
$$610$$ −26.7204 −1.08188
$$611$$ −40.3188 −1.63113
$$612$$ 12.8937 0.521196
$$613$$ −18.3393 −0.740716 −0.370358 0.928889i $$-0.620765\pi$$
−0.370358 + 0.928889i $$0.620765\pi$$
$$614$$ −49.4772 −1.99674
$$615$$ −4.15001 −0.167345
$$616$$ −6.67463 −0.268929
$$617$$ 4.12050 0.165885 0.0829425 0.996554i $$-0.473568\pi$$
0.0829425 + 0.996554i $$0.473568\pi$$
$$618$$ −86.3007 −3.47153
$$619$$ 28.5699 1.14832 0.574161 0.818743i $$-0.305328\pi$$
0.574161 + 0.818743i $$0.305328\pi$$
$$620$$ −23.9809 −0.963096
$$621$$ −19.9879 −0.802087
$$622$$ 3.03144 0.121550
$$623$$ −28.6978 −1.14975
$$624$$ −13.2079 −0.528740
$$625$$ −13.0156 −0.520624
$$626$$ −28.1937 −1.12684
$$627$$ −2.15766 −0.0861685
$$628$$ −32.7519 −1.30694
$$629$$ −17.9577 −0.716022
$$630$$ 48.8968 1.94810
$$631$$ 33.9323 1.35082 0.675412 0.737441i $$-0.263966\pi$$
0.675412 + 0.737441i $$0.263966\pi$$
$$632$$ −14.3855 −0.572223
$$633$$ 5.51265 0.219108
$$634$$ 41.6752 1.65513
$$635$$ −31.1516 −1.23621
$$636$$ 34.0315 1.34943
$$637$$ 28.2213 1.11817
$$638$$ −12.4117 −0.491385
$$639$$ −18.9836 −0.750979
$$640$$ −43.0054 −1.69994
$$641$$ 19.7694 0.780844 0.390422 0.920636i $$-0.372329\pi$$
0.390422 + 0.920636i $$0.372329\pi$$
$$642$$ −6.54623 −0.258359
$$643$$ 36.7642 1.44984 0.724920 0.688833i $$-0.241877\pi$$
0.724920 + 0.688833i $$0.241877\pi$$
$$644$$ 75.3340 2.96858
$$645$$ 69.5887 2.74005
$$646$$ 6.13259 0.241284
$$647$$ 24.9430 0.980610 0.490305 0.871551i $$-0.336885\pi$$
0.490305 + 0.871551i $$0.336885\pi$$
$$648$$ 19.0187 0.747125
$$649$$ 4.35730 0.171039
$$650$$ −48.9967 −1.92181
$$651$$ 21.4098 0.839117
$$652$$ −68.9367 −2.69977
$$653$$ 8.60802 0.336858 0.168429 0.985714i $$-0.446131\pi$$
0.168429 + 0.985714i $$0.446131\pi$$
$$654$$ 1.94311 0.0759817
$$655$$ 35.6375 1.39247
$$656$$ −1.03624 −0.0404584
$$657$$ −14.8304 −0.578588
$$658$$ 104.811 4.08595
$$659$$ 27.7805 1.08217 0.541087 0.840967i $$-0.318013\pi$$
0.541087 + 0.840967i $$0.318013\pi$$
$$660$$ −20.5438 −0.799668
$$661$$ 30.8037 1.19812 0.599062 0.800702i $$-0.295540\pi$$
0.599062 + 0.800702i $$0.295540\pi$$
$$662$$ 27.2316 1.05839
$$663$$ 20.0537 0.778819
$$664$$ 8.84300 0.343175
$$665$$ 13.5163 0.524138
$$666$$ 23.1490 0.897004
$$667$$ 39.1336 1.51526
$$668$$ −7.76909 −0.300595
$$669$$ −53.8128 −2.08052
$$670$$ −74.6380 −2.88352
$$671$$ −3.56412 −0.137591
$$672$$ 63.1378 2.43559
$$673$$ 15.2015 0.585974 0.292987 0.956116i $$-0.405351\pi$$
0.292987 + 0.956116i $$0.405351\pi$$
$$674$$ 0.876359 0.0337561
$$675$$ −19.6403 −0.755957
$$676$$ −5.63894 −0.216882
$$677$$ −9.49027 −0.364741 −0.182370 0.983230i $$-0.558377\pi$$
−0.182370 + 0.983230i $$0.558377\pi$$
$$678$$ 30.8362 1.18426
$$679$$ 41.9870 1.61131
$$680$$ 16.3117 0.625523
$$681$$ −49.2493 −1.88724
$$682$$ −5.50384 −0.210753
$$683$$ 15.0960 0.577631 0.288816 0.957385i $$-0.406739\pi$$
0.288816 + 0.957385i $$0.406739\pi$$
$$684$$ −4.59444 −0.175673
$$685$$ 2.73876 0.104643
$$686$$ −13.0978 −0.500078
$$687$$ −1.30224 −0.0496837
$$688$$ 17.3760 0.662455
$$689$$ 18.8216 0.717047
$$690$$ 111.453 4.24294
$$691$$ 3.46673 0.131881 0.0659403 0.997824i $$-0.478995\pi$$
0.0659403 + 0.997824i $$0.478995\pi$$
$$692$$ −17.8708 −0.679345
$$693$$ 6.52213 0.247755
$$694$$ −2.32965 −0.0884325
$$695$$ −17.2631 −0.654826
$$696$$ −20.7624 −0.786995
$$697$$ 1.57333 0.0595941
$$698$$ −48.6643 −1.84197
$$699$$ 37.4795 1.41760
$$700$$ 74.0241 2.79785
$$701$$ −26.2612 −0.991872 −0.495936 0.868359i $$-0.665175\pi$$
−0.495936 + 0.868359i $$0.665175\pi$$
$$702$$ 20.9950 0.792405
$$703$$ 6.39893 0.241340
$$704$$ −12.5342 −0.472399
$$705$$ 90.1187 3.39407
$$706$$ −27.0165 −1.01678
$$707$$ −45.1950 −1.69973
$$708$$ 26.0920 0.980599
$$709$$ −20.1488 −0.756704 −0.378352 0.925662i $$-0.623509\pi$$
−0.378352 + 0.925662i $$0.623509\pi$$
$$710$$ −85.9696 −3.22638
$$711$$ 14.0568 0.527171
$$712$$ 12.3409 0.462494
$$713$$ 17.3534 0.649889
$$714$$ −52.1304 −1.95093
$$715$$ −11.3621 −0.424918
$$716$$ 34.7580 1.29897
$$717$$ 15.6104 0.582979
$$718$$ −70.3459 −2.62529
$$719$$ −43.4738 −1.62130 −0.810650 0.585532i $$-0.800886\pi$$
−0.810650 + 0.585532i $$0.800886\pi$$
$$720$$ 10.4979 0.391233
$$721$$ 72.1103 2.68553
$$722$$ −2.18524 −0.0813263
$$723$$ −26.3307 −0.979250
$$724$$ 38.1644 1.41837
$$725$$ 38.4532 1.42811
$$726$$ −4.71500 −0.174990
$$727$$ 9.55640 0.354427 0.177214 0.984172i $$-0.443292\pi$$
0.177214 + 0.984172i $$0.443292\pi$$
$$728$$ −22.1052 −0.819273
$$729$$ 0.193986 0.00718467
$$730$$ −67.1612 −2.48575
$$731$$ −26.3821 −0.975777
$$732$$ −21.3424 −0.788837
$$733$$ −20.3072 −0.750065 −0.375032 0.927012i $$-0.622368\pi$$
−0.375032 + 0.927012i $$0.622368\pi$$
$$734$$ 41.4709 1.53072
$$735$$ −63.0788 −2.32670
$$736$$ 51.1753 1.88635
$$737$$ −9.95563 −0.366720
$$738$$ −2.02815 −0.0746572
$$739$$ −6.60855 −0.243099 −0.121550 0.992585i $$-0.538786\pi$$
−0.121550 + 0.992585i $$0.538786\pi$$
$$740$$ 60.9266 2.23971
$$741$$ −7.14577 −0.262507
$$742$$ −48.9277 −1.79619
$$743$$ −39.1431 −1.43602 −0.718010 0.696033i $$-0.754947\pi$$
−0.718010 + 0.696033i $$0.754947\pi$$
$$744$$ −9.20684 −0.337539
$$745$$ −68.0498 −2.49315
$$746$$ −2.51822 −0.0921988
$$747$$ −8.64096 −0.316156
$$748$$ 7.78847 0.284775
$$749$$ 5.46983 0.199863
$$750$$ 28.6345 1.04559
$$751$$ 43.2173 1.57702 0.788511 0.615020i $$-0.210852\pi$$
0.788511 + 0.615020i $$0.210852\pi$$
$$752$$ 22.5023 0.820574
$$753$$ −30.4064 −1.10807
$$754$$ −41.1054 −1.49697
$$755$$ −77.3457 −2.81490
$$756$$ −31.7192 −1.15362
$$757$$ 9.93714 0.361171 0.180586 0.983559i $$-0.442201\pi$$
0.180586 + 0.983559i $$0.442201\pi$$
$$758$$ −29.5806 −1.07442
$$759$$ 14.8662 0.539609
$$760$$ −5.81238 −0.210837
$$761$$ 14.5829 0.528630 0.264315 0.964436i $$-0.414854\pi$$
0.264315 + 0.964436i $$0.414854\pi$$
$$762$$ −42.8125 −1.55093
$$763$$ −1.62361 −0.0587785
$$764$$ 8.62221 0.311941
$$765$$ −15.9390 −0.576275
$$766$$ −28.7734 −1.03962
$$767$$ 14.4306 0.521059
$$768$$ −5.01471 −0.180953
$$769$$ −11.8924 −0.428853 −0.214426 0.976740i $$-0.568788\pi$$
−0.214426 + 0.976740i $$0.568788\pi$$
$$770$$ 29.5363 1.06441
$$771$$ 0.951354 0.0342622
$$772$$ −2.07540 −0.0746953
$$773$$ 15.0935 0.542876 0.271438 0.962456i $$-0.412501\pi$$
0.271438 + 0.962456i $$0.412501\pi$$
$$774$$ 34.0086 1.22242
$$775$$ 17.0516 0.612513
$$776$$ −18.0556 −0.648158
$$777$$ −54.3945 −1.95139
$$778$$ −0.488595 −0.0175170
$$779$$ −0.560629 −0.0200866
$$780$$ −68.0375 −2.43613
$$781$$ −11.4671 −0.410325
$$782$$ −42.2534 −1.51098
$$783$$ −16.4771 −0.588844
$$784$$ −15.7505 −0.562519
$$785$$ 40.4874 1.44506
$$786$$ 48.9777 1.74698
$$787$$ −5.98214 −0.213240 −0.106620 0.994300i $$-0.534003\pi$$
−0.106620 + 0.994300i $$0.534003\pi$$
$$788$$ −9.48589 −0.337921
$$789$$ −32.5075 −1.15730
$$790$$ 63.6580 2.26485
$$791$$ −25.7659 −0.916128
$$792$$ −2.80470 −0.0996608
$$793$$ −11.8037 −0.419163
$$794$$ 47.3114 1.67902
$$795$$ −42.0692 −1.49204
$$796$$ 14.8973 0.528022
$$797$$ 24.2718 0.859751 0.429876 0.902888i $$-0.358557\pi$$
0.429876 + 0.902888i $$0.358557\pi$$
$$798$$ 18.5758 0.657576
$$799$$ −34.1653 −1.20868
$$800$$ 50.2854 1.77786
$$801$$ −12.0589 −0.426081
$$802$$ −2.84476 −0.100452
$$803$$ −8.95834 −0.316133
$$804$$ −59.6155 −2.10248
$$805$$ −93.1268 −3.28229
$$806$$ −18.2277 −0.642044
$$807$$ −15.6592 −0.551230
$$808$$ 19.4351 0.683726
$$809$$ −17.9557 −0.631289 −0.315644 0.948878i $$-0.602221\pi$$
−0.315644 + 0.948878i $$0.602221\pi$$
$$810$$ −84.1607 −2.95711
$$811$$ 5.48300 0.192534 0.0962671 0.995356i $$-0.469310\pi$$
0.0962671 + 0.995356i $$0.469310\pi$$
$$812$$ 62.1019 2.17935
$$813$$ −36.4381 −1.27794
$$814$$ 13.9832 0.490111
$$815$$ 85.2185 2.98508
$$816$$ −11.1921 −0.391802
$$817$$ 9.40080 0.328892
$$818$$ 4.71205 0.164753
$$819$$ 21.6001 0.754770
$$820$$ −5.33795 −0.186409
$$821$$ 52.6324 1.83688 0.918441 0.395558i $$-0.129449\pi$$
0.918441 + 0.395558i $$0.129449\pi$$
$$822$$ 3.76395 0.131283
$$823$$ −20.4784 −0.713831 −0.356916 0.934137i $$-0.616172\pi$$
−0.356916 + 0.934137i $$0.616172\pi$$
$$824$$ −31.0095 −1.08027
$$825$$ 14.6077 0.508575
$$826$$ −37.5131 −1.30525
$$827$$ −0.359755 −0.0125099 −0.00625496 0.999980i $$-0.501991\pi$$
−0.00625496 + 0.999980i $$0.501991\pi$$
$$828$$ 31.6556 1.10011
$$829$$ 22.7422 0.789870 0.394935 0.918709i $$-0.370767\pi$$
0.394935 + 0.918709i $$0.370767\pi$$
$$830$$ −39.1317 −1.35828
$$831$$ −33.1887 −1.15130
$$832$$ −41.5109 −1.43913
$$833$$ 23.9141 0.828574
$$834$$ −23.7251 −0.821535
$$835$$ 9.60403 0.332361
$$836$$ −2.77529 −0.0959853
$$837$$ −7.30659 −0.252553
$$838$$ 64.7563 2.23697
$$839$$ 13.1616 0.454388 0.227194 0.973849i $$-0.427045\pi$$
0.227194 + 0.973849i $$0.427045\pi$$
$$840$$ 49.4084 1.70475
$$841$$ 3.25998 0.112413
$$842$$ 11.4809 0.395659
$$843$$ −55.8778 −1.92453
$$844$$ 7.09065 0.244070
$$845$$ 6.97077 0.239802
$$846$$ 44.0418 1.51419
$$847$$ 3.93972 0.135370
$$848$$ −10.5045 −0.360726
$$849$$ 40.3228 1.38387
$$850$$ −41.5187 −1.42408
$$851$$ −44.0885 −1.51133
$$852$$ −68.6663 −2.35247
$$853$$ −36.9102 −1.26378 −0.631891 0.775057i $$-0.717721\pi$$
−0.631891 + 0.775057i $$0.717721\pi$$
$$854$$ 30.6844 1.05000
$$855$$ 5.67958 0.194237
$$856$$ −2.35218 −0.0803960
$$857$$ 24.3615 0.832174 0.416087 0.909325i $$-0.363401\pi$$
0.416087 + 0.909325i $$0.363401\pi$$
$$858$$ −15.6153 −0.533096
$$859$$ −7.69750 −0.262635 −0.131318 0.991340i $$-0.541921\pi$$
−0.131318 + 0.991340i $$0.541921\pi$$
$$860$$ 89.5085 3.05222
$$861$$ 4.76566 0.162413
$$862$$ −39.0042 −1.32849
$$863$$ −29.0225 −0.987937 −0.493969 0.869480i $$-0.664454\pi$$
−0.493969 + 0.869480i $$0.664454\pi$$
$$864$$ −21.5472 −0.733051
$$865$$ 22.0916 0.751137
$$866$$ 1.52177 0.0517117
$$867$$ −19.6871 −0.668610
$$868$$ 27.5384 0.934714
$$869$$ 8.49105 0.288039
$$870$$ 91.8768 3.11491
$$871$$ −32.9713 −1.11719
$$872$$ 0.698197 0.0236439
$$873$$ 17.6431 0.597127
$$874$$ 15.0563 0.509286
$$875$$ −23.9262 −0.808852
$$876$$ −53.6436 −1.81245
$$877$$ −36.0010 −1.21567 −0.607834 0.794064i $$-0.707961\pi$$
−0.607834 + 0.794064i $$0.707961\pi$$
$$878$$ 54.2399 1.83051
$$879$$ 57.7111 1.94655
$$880$$ 6.34128 0.213765
$$881$$ −15.3587 −0.517448 −0.258724 0.965951i $$-0.583302\pi$$
−0.258724 + 0.965951i $$0.583302\pi$$
$$882$$ −30.8272 −1.03800
$$883$$ 40.2870 1.35577 0.677883 0.735170i $$-0.262898\pi$$
0.677883 + 0.735170i $$0.262898\pi$$
$$884$$ 25.7940 0.867547
$$885$$ −32.2546 −1.08423
$$886$$ 8.59728 0.288831
$$887$$ −7.09855 −0.238346 −0.119173 0.992874i $$-0.538024\pi$$
−0.119173 + 0.992874i $$0.538024\pi$$
$$888$$ 23.3912 0.784956
$$889$$ 35.7728 1.19978
$$890$$ −54.6104 −1.83054
$$891$$ −11.2258 −0.376079
$$892$$ −69.2168 −2.31755
$$893$$ 12.1742 0.407395
$$894$$ −93.5228 −3.12787
$$895$$ −42.9673 −1.43624
$$896$$ 49.3851 1.64984
$$897$$ 49.2342 1.64388
$$898$$ −13.4042 −0.447302
$$899$$ 14.3053 0.477110
$$900$$ 31.1052 1.03684
$$901$$ 15.9490 0.531339
$$902$$ −1.22511 −0.0407917
$$903$$ −79.9121 −2.65931
$$904$$ 11.0801 0.368517
$$905$$ −47.1783 −1.56826
$$906$$ −106.298 −3.53153
$$907$$ 28.4435 0.944452 0.472226 0.881478i $$-0.343451\pi$$
0.472226 + 0.881478i $$0.343451\pi$$
$$908$$ −63.3470 −2.10224
$$909$$ −18.9911 −0.629895
$$910$$ 97.8190 3.24267
$$911$$ 49.5740 1.64246 0.821230 0.570598i $$-0.193288\pi$$
0.821230 + 0.570598i $$0.193288\pi$$
$$912$$ 3.98812 0.132060
$$913$$ −5.21960 −0.172744
$$914$$ −18.3602 −0.607301
$$915$$ 26.3831 0.872199
$$916$$ −1.67501 −0.0553440
$$917$$ −40.9243 −1.35144
$$918$$ 17.7907 0.587180
$$919$$ 17.8421 0.588556 0.294278 0.955720i $$-0.404921\pi$$
0.294278 + 0.955720i $$0.404921\pi$$
$$920$$ 40.0472 1.32032
$$921$$ 48.8526 1.60975
$$922$$ −56.7952 −1.87045
$$923$$ −37.9770 −1.25003
$$924$$ 23.5915 0.776103
$$925$$ −43.3219 −1.42441
$$926$$ −42.3334 −1.39116
$$927$$ 30.3010 0.995215
$$928$$ 42.1865 1.38484
$$929$$ 36.0244 1.18192 0.590961 0.806700i $$-0.298749\pi$$
0.590961 + 0.806700i $$0.298749\pi$$
$$930$$ 40.7417 1.33597
$$931$$ −8.52137 −0.279277
$$932$$ 48.2080 1.57911
$$933$$ −2.99317 −0.0979921
$$934$$ 75.7667 2.47916
$$935$$ −9.62799 −0.314869
$$936$$ −9.28867 −0.303610
$$937$$ −3.50371 −0.114461 −0.0572307 0.998361i $$-0.518227\pi$$
−0.0572307 + 0.998361i $$0.518227\pi$$
$$938$$ 85.7104 2.79854
$$939$$ 27.8377 0.908450
$$940$$ 115.915 3.78074
$$941$$ 44.2074 1.44112 0.720560 0.693392i $$-0.243885\pi$$
0.720560 + 0.693392i $$0.243885\pi$$
$$942$$ 55.6430 1.81295
$$943$$ 3.86272 0.125788
$$944$$ −8.05385 −0.262130
$$945$$ 39.2108 1.27553
$$946$$ 20.5430 0.667912
$$947$$ 37.9515 1.23326 0.616629 0.787254i $$-0.288498\pi$$
0.616629 + 0.787254i $$0.288498\pi$$
$$948$$ 50.8454 1.65138
$$949$$ −29.6684 −0.963077
$$950$$ 14.7945 0.479996
$$951$$ −41.1491 −1.33435
$$952$$ −18.7315 −0.607090
$$953$$ 13.0303 0.422094 0.211047 0.977476i $$-0.432313\pi$$
0.211047 + 0.977476i $$0.432313\pi$$
$$954$$ −20.5596 −0.665641
$$955$$ −10.6586 −0.344906
$$956$$ 20.0788 0.649396
$$957$$ 12.2550 0.396149
$$958$$ −50.7745 −1.64045
$$959$$ −3.14505 −0.101559
$$960$$ 92.7831 2.99456
$$961$$ −24.6565 −0.795370
$$962$$ 46.3099 1.49309
$$963$$ 2.29844 0.0740662
$$964$$ −33.8679 −1.09081
$$965$$ 2.56558 0.0825889
$$966$$ −127.987 −4.11791
$$967$$ 24.0504 0.773409 0.386705 0.922204i $$-0.373613\pi$$
0.386705 + 0.922204i $$0.373613\pi$$
$$968$$ −1.69419 −0.0544534
$$969$$ −6.05517 −0.194520
$$970$$ 79.8989 2.56540
$$971$$ 54.1335 1.73723 0.868614 0.495490i $$-0.165012\pi$$
0.868614 + 0.495490i $$0.165012\pi$$
$$972$$ −43.0682 −1.38141
$$973$$ 19.8240 0.635529
$$974$$ −31.1791 −0.999041
$$975$$ 48.3781 1.54934
$$976$$ 6.58776 0.210869
$$977$$ −50.2507 −1.60766 −0.803832 0.594857i $$-0.797209\pi$$
−0.803832 + 0.594857i $$0.797209\pi$$
$$978$$ 117.118 3.74503
$$979$$ −7.28423 −0.232805
$$980$$ −81.1351 −2.59177
$$981$$ −0.682245 −0.0217824
$$982$$ −90.9429 −2.90210
$$983$$ 6.64633 0.211985 0.105993 0.994367i $$-0.466198\pi$$
0.105993 + 0.994367i $$0.466198\pi$$
$$984$$ −2.04937 −0.0653315
$$985$$ 11.7263 0.373631
$$986$$ −34.8318 −1.10927
$$987$$ −103.488 −3.29405
$$988$$ −9.19126 −0.292413
$$989$$ −64.7714 −2.05961
$$990$$ 12.4113 0.394456
$$991$$ 51.7993 1.64546 0.822729 0.568434i $$-0.192451\pi$$
0.822729 + 0.568434i $$0.192451\pi$$
$$992$$ 18.7072 0.593953
$$993$$ −26.8879 −0.853260
$$994$$ 98.7230 3.13130
$$995$$ −18.4159 −0.583822
$$996$$ −31.2556 −0.990371
$$997$$ −14.2042 −0.449851 −0.224925 0.974376i $$-0.572214\pi$$
−0.224925 + 0.974376i $$0.572214\pi$$
$$998$$ −43.6315 −1.38113
$$999$$ 18.5633 0.587318
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 209.2.a.c.1.1 5
3.2 odd 2 1881.2.a.k.1.5 5
4.3 odd 2 3344.2.a.t.1.2 5
5.4 even 2 5225.2.a.h.1.5 5
11.10 odd 2 2299.2.a.n.1.5 5
19.18 odd 2 3971.2.a.h.1.5 5

By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.1 5 1.1 even 1 trivial
1881.2.a.k.1.5 5 3.2 odd 2
2299.2.a.n.1.5 5 11.10 odd 2
3344.2.a.t.1.2 5 4.3 odd 2
3971.2.a.h.1.5 5 19.18 odd 2
5225.2.a.h.1.5 5 5.4 even 2