# Properties

 Label 273.2.a.e.1.4 Level $273$ Weight $2$ Character 273.1 Self dual yes Analytic conductor $2.180$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(1,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("273.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.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: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.17428.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 6x^{2} + 4x + 6$$ x^4 - x^3 - 6*x^2 + 4*x + 6 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.36865$$ of defining polynomial Character $$\chi$$ $$=$$ 273.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.61050 q^{2} +1.00000 q^{3} +4.81471 q^{4} -3.81471 q^{5} +2.61050 q^{6} +1.00000 q^{7} +7.34780 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.61050 q^{2} +1.00000 q^{3} +4.81471 q^{4} -3.81471 q^{5} +2.61050 q^{6} +1.00000 q^{7} +7.34780 q^{8} +1.00000 q^{9} -9.95830 q^{10} -4.73730 q^{11} +4.81471 q^{12} +1.00000 q^{13} +2.61050 q^{14} -3.81471 q^{15} +9.55201 q^{16} -5.22100 q^{17} +2.61050 q^{18} +2.92259 q^{19} -18.3667 q^{20} +1.00000 q^{21} -12.3667 q^{22} +3.33101 q^{23} +7.34780 q^{24} +9.55201 q^{25} +2.61050 q^{26} +1.00000 q^{27} +4.81471 q^{28} -0.922589 q^{29} -9.95830 q^{30} -7.51941 q^{31} +10.2399 q^{32} -4.73730 q^{33} -13.6294 q^{34} -3.81471 q^{35} +4.81471 q^{36} +0.154821 q^{37} +7.62942 q^{38} +1.00000 q^{39} -28.0297 q^{40} +6.36672 q^{41} +2.61050 q^{42} -6.55201 q^{43} -22.8087 q^{44} -3.81471 q^{45} +8.69560 q^{46} +9.03571 q^{47} +9.55201 q^{48} +1.00000 q^{49} +24.9355 q^{50} -5.22100 q^{51} +4.81471 q^{52} +8.55201 q^{53} +2.61050 q^{54} +18.0714 q^{55} +7.34780 q^{56} +2.92259 q^{57} -2.40842 q^{58} +3.95830 q^{59} -18.3667 q^{60} +12.4420 q^{61} -19.6294 q^{62} +1.00000 q^{63} +7.62729 q^{64} -3.81471 q^{65} -12.3667 q^{66} -10.6620 q^{67} -25.1376 q^{68} +3.33101 q^{69} -9.95830 q^{70} -6.58248 q^{71} +7.34780 q^{72} -7.73517 q^{73} +0.404161 q^{74} +9.55201 q^{75} +14.0714 q^{76} -4.73730 q^{77} +2.61050 q^{78} +13.3646 q^{79} -36.4381 q^{80} +1.00000 q^{81} +16.6203 q^{82} -1.40629 q^{83} +4.81471 q^{84} +19.9166 q^{85} -17.1040 q^{86} -0.922589 q^{87} -34.8087 q^{88} -1.96953 q^{89} -9.95830 q^{90} +1.00000 q^{91} +16.0378 q^{92} -7.51941 q^{93} +23.5877 q^{94} -11.1488 q^{95} +10.2399 q^{96} -2.11001 q^{97} +2.61050 q^{98} -4.73730 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + 4 q^{3} + 7 q^{4} - 3 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + q^2 + 4 * q^3 + 7 * q^4 - 3 * q^5 + q^6 + 4 * q^7 + 3 * q^8 + 4 * q^9 $$4 q + q^{2} + 4 q^{3} + 7 q^{4} - 3 q^{5} + q^{6} + 4 q^{7} + 3 q^{8} + 4 q^{9} - 4 q^{10} - 2 q^{11} + 7 q^{12} + 4 q^{13} + q^{14} - 3 q^{15} + 9 q^{16} - 2 q^{17} + q^{18} + 7 q^{19} - 32 q^{20} + 4 q^{21} - 8 q^{22} + 3 q^{23} + 3 q^{24} + 9 q^{25} + q^{26} + 4 q^{27} + 7 q^{28} + q^{29} - 4 q^{30} + 3 q^{31} + 7 q^{32} - 2 q^{33} - 30 q^{34} - 3 q^{35} + 7 q^{36} + 10 q^{37} + 6 q^{38} + 4 q^{39} - 14 q^{40} - 16 q^{41} + q^{42} + 3 q^{43} - 12 q^{44} - 3 q^{45} - 18 q^{46} + 5 q^{47} + 9 q^{48} + 4 q^{49} + 13 q^{50} - 2 q^{51} + 7 q^{52} + 5 q^{53} + q^{54} + 10 q^{55} + 3 q^{56} + 7 q^{57} - 4 q^{58} - 20 q^{59} - 32 q^{60} + 12 q^{61} - 54 q^{62} + 4 q^{63} + 5 q^{64} - 3 q^{65} - 8 q^{66} - 22 q^{67} - 10 q^{68} + 3 q^{69} - 4 q^{70} + 3 q^{72} - 13 q^{73} - 6 q^{74} + 9 q^{75} - 6 q^{76} - 2 q^{77} + q^{78} + 11 q^{79} - 42 q^{80} + 4 q^{81} + 10 q^{82} + q^{83} + 7 q^{84} + 8 q^{85} - 10 q^{86} + q^{87} - 60 q^{88} - 5 q^{89} - 4 q^{90} + 4 q^{91} + 34 q^{92} + 3 q^{93} + 34 q^{94} + 13 q^{95} + 7 q^{96} - 17 q^{97} + q^{98} - 2 q^{99}+O(q^{100})$$ 4 * q + q^2 + 4 * q^3 + 7 * q^4 - 3 * q^5 + q^6 + 4 * q^7 + 3 * q^8 + 4 * q^9 - 4 * q^10 - 2 * q^11 + 7 * q^12 + 4 * q^13 + q^14 - 3 * q^15 + 9 * q^16 - 2 * q^17 + q^18 + 7 * q^19 - 32 * q^20 + 4 * q^21 - 8 * q^22 + 3 * q^23 + 3 * q^24 + 9 * q^25 + q^26 + 4 * q^27 + 7 * q^28 + q^29 - 4 * q^30 + 3 * q^31 + 7 * q^32 - 2 * q^33 - 30 * q^34 - 3 * q^35 + 7 * q^36 + 10 * q^37 + 6 * q^38 + 4 * q^39 - 14 * q^40 - 16 * q^41 + q^42 + 3 * q^43 - 12 * q^44 - 3 * q^45 - 18 * q^46 + 5 * q^47 + 9 * q^48 + 4 * q^49 + 13 * q^50 - 2 * q^51 + 7 * q^52 + 5 * q^53 + q^54 + 10 * q^55 + 3 * q^56 + 7 * q^57 - 4 * q^58 - 20 * q^59 - 32 * q^60 + 12 * q^61 - 54 * q^62 + 4 * q^63 + 5 * q^64 - 3 * q^65 - 8 * q^66 - 22 * q^67 - 10 * q^68 + 3 * q^69 - 4 * q^70 + 3 * q^72 - 13 * q^73 - 6 * q^74 + 9 * q^75 - 6 * q^76 - 2 * q^77 + q^78 + 11 * q^79 - 42 * q^80 + 4 * q^81 + 10 * q^82 + q^83 + 7 * q^84 + 8 * q^85 - 10 * q^86 + q^87 - 60 * q^88 - 5 * q^89 - 4 * q^90 + 4 * q^91 + 34 * q^92 + 3 * q^93 + 34 * q^94 + 13 * q^95 + 7 * q^96 - 17 * q^97 + q^98 - 2 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.61050 1.84590 0.922951 0.384917i $$-0.125770\pi$$
0.922951 + 0.384917i $$0.125770\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 4.81471 2.40735
$$5$$ −3.81471 −1.70599 −0.852995 0.521919i $$-0.825216\pi$$
−0.852995 + 0.521919i $$0.825216\pi$$
$$6$$ 2.61050 1.06573
$$7$$ 1.00000 0.377964
$$8$$ 7.34780 2.59784
$$9$$ 1.00000 0.333333
$$10$$ −9.95830 −3.14909
$$11$$ −4.73730 −1.42835 −0.714175 0.699968i $$-0.753198\pi$$
−0.714175 + 0.699968i $$0.753198\pi$$
$$12$$ 4.81471 1.38989
$$13$$ 1.00000 0.277350
$$14$$ 2.61050 0.697685
$$15$$ −3.81471 −0.984954
$$16$$ 9.55201 2.38800
$$17$$ −5.22100 −1.26628 −0.633139 0.774038i $$-0.718234\pi$$
−0.633139 + 0.774038i $$0.718234\pi$$
$$18$$ 2.61050 0.615301
$$19$$ 2.92259 0.670488 0.335244 0.942131i $$-0.391181\pi$$
0.335244 + 0.942131i $$0.391181\pi$$
$$20$$ −18.3667 −4.10692
$$21$$ 1.00000 0.218218
$$22$$ −12.3667 −2.63659
$$23$$ 3.33101 0.694563 0.347282 0.937761i $$-0.387105\pi$$
0.347282 + 0.937761i $$0.387105\pi$$
$$24$$ 7.34780 1.49986
$$25$$ 9.55201 1.91040
$$26$$ 2.61050 0.511961
$$27$$ 1.00000 0.192450
$$28$$ 4.81471 0.909895
$$29$$ −0.922589 −0.171321 −0.0856603 0.996324i $$-0.527300\pi$$
−0.0856603 + 0.996324i $$0.527300\pi$$
$$30$$ −9.95830 −1.81813
$$31$$ −7.51941 −1.35053 −0.675263 0.737577i $$-0.735970\pi$$
−0.675263 + 0.737577i $$0.735970\pi$$
$$32$$ 10.2399 1.81018
$$33$$ −4.73730 −0.824658
$$34$$ −13.6294 −2.33743
$$35$$ −3.81471 −0.644804
$$36$$ 4.81471 0.802452
$$37$$ 0.154821 0.0254525 0.0127262 0.999919i $$-0.495949\pi$$
0.0127262 + 0.999919i $$0.495949\pi$$
$$38$$ 7.62942 1.23766
$$39$$ 1.00000 0.160128
$$40$$ −28.0297 −4.43189
$$41$$ 6.36672 0.994314 0.497157 0.867661i $$-0.334377\pi$$
0.497157 + 0.867661i $$0.334377\pi$$
$$42$$ 2.61050 0.402809
$$43$$ −6.55201 −0.999172 −0.499586 0.866264i $$-0.666514\pi$$
−0.499586 + 0.866264i $$0.666514\pi$$
$$44$$ −22.8087 −3.43854
$$45$$ −3.81471 −0.568663
$$46$$ 8.69560 1.28210
$$47$$ 9.03571 1.31799 0.658997 0.752146i $$-0.270981\pi$$
0.658997 + 0.752146i $$0.270981\pi$$
$$48$$ 9.55201 1.37871
$$49$$ 1.00000 0.142857
$$50$$ 24.9355 3.52641
$$51$$ −5.22100 −0.731086
$$52$$ 4.81471 0.667680
$$53$$ 8.55201 1.17471 0.587354 0.809330i $$-0.300170\pi$$
0.587354 + 0.809330i $$0.300170\pi$$
$$54$$ 2.61050 0.355244
$$55$$ 18.0714 2.43675
$$56$$ 7.34780 0.981891
$$57$$ 2.92259 0.387106
$$58$$ −2.40842 −0.316241
$$59$$ 3.95830 0.515327 0.257663 0.966235i $$-0.417047\pi$$
0.257663 + 0.966235i $$0.417047\pi$$
$$60$$ −18.3667 −2.37113
$$61$$ 12.4420 1.59303 0.796517 0.604616i $$-0.206673\pi$$
0.796517 + 0.604616i $$0.206673\pi$$
$$62$$ −19.6294 −2.49294
$$63$$ 1.00000 0.125988
$$64$$ 7.62729 0.953411
$$65$$ −3.81471 −0.473156
$$66$$ −12.3667 −1.52224
$$67$$ −10.6620 −1.30257 −0.651286 0.758832i $$-0.725770\pi$$
−0.651286 + 0.758832i $$0.725770\pi$$
$$68$$ −25.1376 −3.04838
$$69$$ 3.33101 0.401006
$$70$$ −9.95830 −1.19024
$$71$$ −6.58248 −0.781196 −0.390598 0.920561i $$-0.627732\pi$$
−0.390598 + 0.920561i $$0.627732\pi$$
$$72$$ 7.34780 0.865946
$$73$$ −7.73517 −0.905333 −0.452667 0.891680i $$-0.649527\pi$$
−0.452667 + 0.891680i $$0.649527\pi$$
$$74$$ 0.404161 0.0469828
$$75$$ 9.55201 1.10297
$$76$$ 14.0714 1.61410
$$77$$ −4.73730 −0.539865
$$78$$ 2.61050 0.295581
$$79$$ 13.3646 1.50363 0.751817 0.659372i $$-0.229178\pi$$
0.751817 + 0.659372i $$0.229178\pi$$
$$80$$ −36.4381 −4.07391
$$81$$ 1.00000 0.111111
$$82$$ 16.6203 1.83541
$$83$$ −1.40629 −0.154360 −0.0771802 0.997017i $$-0.524592\pi$$
−0.0771802 + 0.997017i $$0.524592\pi$$
$$84$$ 4.81471 0.525328
$$85$$ 19.9166 2.16026
$$86$$ −17.1040 −1.84437
$$87$$ −0.922589 −0.0989120
$$88$$ −34.8087 −3.71062
$$89$$ −1.96953 −0.208770 −0.104385 0.994537i $$-0.533287\pi$$
−0.104385 + 0.994537i $$0.533287\pi$$
$$90$$ −9.95830 −1.04970
$$91$$ 1.00000 0.104828
$$92$$ 16.0378 1.67206
$$93$$ −7.51941 −0.779727
$$94$$ 23.5877 2.43289
$$95$$ −11.1488 −1.14385
$$96$$ 10.2399 1.04511
$$97$$ −2.11001 −0.214239 −0.107119 0.994246i $$-0.534163\pi$$
−0.107119 + 0.994246i $$0.534163\pi$$
$$98$$ 2.61050 0.263700
$$99$$ −4.73730 −0.476116
$$100$$ 45.9901 4.59901
$$101$$ 0.850419 0.0846198 0.0423099 0.999105i $$-0.486528\pi$$
0.0423099 + 0.999105i $$0.486528\pi$$
$$102$$ −13.6294 −1.34951
$$103$$ −1.47460 −0.145296 −0.0726482 0.997358i $$-0.523145\pi$$
−0.0726482 + 0.997358i $$0.523145\pi$$
$$104$$ 7.34780 0.720511
$$105$$ −3.81471 −0.372278
$$106$$ 22.3250 2.16840
$$107$$ 2.62418 0.253689 0.126844 0.991923i $$-0.459515\pi$$
0.126844 + 0.991923i $$0.459515\pi$$
$$108$$ 4.81471 0.463296
$$109$$ −17.9166 −1.71610 −0.858049 0.513567i $$-0.828324\pi$$
−0.858049 + 0.513567i $$0.828324\pi$$
$$110$$ 47.1754 4.49800
$$111$$ 0.154821 0.0146950
$$112$$ 9.55201 0.902580
$$113$$ 0.922589 0.0867899 0.0433950 0.999058i $$-0.486183\pi$$
0.0433950 + 0.999058i $$0.486183\pi$$
$$114$$ 7.62942 0.714561
$$115$$ −12.7068 −1.18492
$$116$$ −4.44200 −0.412429
$$117$$ 1.00000 0.0924500
$$118$$ 10.3331 0.951242
$$119$$ −5.22100 −0.478608
$$120$$ −28.0297 −2.55875
$$121$$ 11.4420 1.04018
$$122$$ 32.4798 2.94059
$$123$$ 6.36672 0.574068
$$124$$ −36.2038 −3.25119
$$125$$ −17.3646 −1.55314
$$126$$ 2.61050 0.232562
$$127$$ 17.4746 1.55062 0.775310 0.631581i $$-0.217594\pi$$
0.775310 + 0.631581i $$0.217594\pi$$
$$128$$ −0.568798 −0.0502751
$$129$$ −6.55201 −0.576872
$$130$$ −9.95830 −0.873401
$$131$$ 0.967402 0.0845223 0.0422611 0.999107i $$-0.486544\pi$$
0.0422611 + 0.999107i $$0.486544\pi$$
$$132$$ −22.8087 −1.98524
$$133$$ 2.92259 0.253421
$$134$$ −27.8332 −2.40442
$$135$$ −3.81471 −0.328318
$$136$$ −38.3629 −3.28959
$$137$$ 3.29628 0.281620 0.140810 0.990037i $$-0.455029\pi$$
0.140810 + 0.990037i $$0.455029\pi$$
$$138$$ 8.69560 0.740218
$$139$$ 0.370581 0.0314323 0.0157161 0.999876i $$-0.494997\pi$$
0.0157161 + 0.999876i $$0.494997\pi$$
$$140$$ −18.3667 −1.55227
$$141$$ 9.03571 0.760944
$$142$$ −17.1836 −1.44201
$$143$$ −4.73730 −0.396153
$$144$$ 9.55201 0.796001
$$145$$ 3.51941 0.292271
$$146$$ −20.1927 −1.67116
$$147$$ 1.00000 0.0824786
$$148$$ 0.745420 0.0612732
$$149$$ −15.7425 −1.28968 −0.644840 0.764318i $$-0.723076\pi$$
−0.644840 + 0.764318i $$0.723076\pi$$
$$150$$ 24.9355 2.03598
$$151$$ 10.2914 0.837505 0.418753 0.908100i $$-0.362467\pi$$
0.418753 + 0.908100i $$0.362467\pi$$
$$152$$ 21.4746 1.74182
$$153$$ −5.22100 −0.422093
$$154$$ −12.3667 −0.996539
$$155$$ 28.6844 2.30398
$$156$$ 4.81471 0.385485
$$157$$ −11.4137 −0.910909 −0.455455 0.890259i $$-0.650523\pi$$
−0.455455 + 0.890259i $$0.650523\pi$$
$$158$$ 34.8883 2.77556
$$159$$ 8.55201 0.678218
$$160$$ −39.0623 −3.08815
$$161$$ 3.33101 0.262520
$$162$$ 2.61050 0.205100
$$163$$ −13.4746 −1.05541 −0.527706 0.849427i $$-0.676948\pi$$
−0.527706 + 0.849427i $$0.676948\pi$$
$$164$$ 30.6539 2.39367
$$165$$ 18.0714 1.40686
$$166$$ −3.67112 −0.284934
$$167$$ 19.1905 1.48501 0.742504 0.669842i $$-0.233638\pi$$
0.742504 + 0.669842i $$0.233638\pi$$
$$168$$ 7.34780 0.566895
$$169$$ 1.00000 0.0769231
$$170$$ 51.9923 3.98763
$$171$$ 2.92259 0.223496
$$172$$ −31.5460 −2.40536
$$173$$ −19.5124 −1.48350 −0.741752 0.670675i $$-0.766005\pi$$
−0.741752 + 0.670675i $$0.766005\pi$$
$$174$$ −2.40842 −0.182582
$$175$$ 9.55201 0.722064
$$176$$ −45.2507 −3.41090
$$177$$ 3.95830 0.297524
$$178$$ −5.14146 −0.385369
$$179$$ −16.5856 −1.23967 −0.619833 0.784734i $$-0.712799\pi$$
−0.619833 + 0.784734i $$0.712799\pi$$
$$180$$ −18.3667 −1.36897
$$181$$ −2.81684 −0.209374 −0.104687 0.994505i $$-0.533384\pi$$
−0.104687 + 0.994505i $$0.533384\pi$$
$$182$$ 2.61050 0.193503
$$183$$ 12.4420 0.919739
$$184$$ 24.4756 1.80436
$$185$$ −0.590599 −0.0434217
$$186$$ −19.6294 −1.43930
$$187$$ 24.7334 1.80869
$$188$$ 43.5043 3.17288
$$189$$ 1.00000 0.0727393
$$190$$ −29.1040 −2.11143
$$191$$ 15.1601 1.09694 0.548472 0.836169i $$-0.315210\pi$$
0.548472 + 0.836169i $$0.315210\pi$$
$$192$$ 7.62729 0.550452
$$193$$ 0.0651962 0.00469293 0.00234646 0.999997i $$-0.499253\pi$$
0.00234646 + 0.999997i $$0.499253\pi$$
$$194$$ −5.50818 −0.395464
$$195$$ −3.81471 −0.273177
$$196$$ 4.81471 0.343908
$$197$$ −17.1415 −1.22128 −0.610639 0.791909i $$-0.709087\pi$$
−0.610639 + 0.791909i $$0.709087\pi$$
$$198$$ −12.3667 −0.878864
$$199$$ 6.44200 0.456661 0.228331 0.973584i $$-0.426673\pi$$
0.228331 + 0.973584i $$0.426673\pi$$
$$200$$ 70.1862 4.96292
$$201$$ −10.6620 −0.752041
$$202$$ 2.22002 0.156200
$$203$$ −0.922589 −0.0647531
$$204$$ −25.1376 −1.75998
$$205$$ −24.2872 −1.69629
$$206$$ −3.84944 −0.268203
$$207$$ 3.33101 0.231521
$$208$$ 9.55201 0.662313
$$209$$ −13.8452 −0.957691
$$210$$ −9.95830 −0.687188
$$211$$ −16.0266 −1.10332 −0.551659 0.834070i $$-0.686005\pi$$
−0.551659 + 0.834070i $$0.686005\pi$$
$$212$$ 41.1754 2.82794
$$213$$ −6.58248 −0.451024
$$214$$ 6.85042 0.468285
$$215$$ 24.9940 1.70458
$$216$$ 7.34780 0.499954
$$217$$ −7.51941 −0.510451
$$218$$ −46.7713 −3.16775
$$219$$ −7.73517 −0.522694
$$220$$ 87.0086 5.86612
$$221$$ −5.22100 −0.351202
$$222$$ 0.404161 0.0271255
$$223$$ 20.0266 1.34108 0.670540 0.741873i $$-0.266062\pi$$
0.670540 + 0.741873i $$0.266062\pi$$
$$224$$ 10.2399 0.684183
$$225$$ 9.55201 0.636801
$$226$$ 2.40842 0.160206
$$227$$ 19.2171 1.27549 0.637743 0.770249i $$-0.279868\pi$$
0.637743 + 0.770249i $$0.279868\pi$$
$$228$$ 14.0714 0.931902
$$229$$ 5.25884 0.347514 0.173757 0.984789i $$-0.444409\pi$$
0.173757 + 0.984789i $$0.444409\pi$$
$$230$$ −33.1712 −2.18724
$$231$$ −4.73730 −0.311691
$$232$$ −6.77900 −0.445063
$$233$$ −26.6234 −1.74416 −0.872079 0.489365i $$-0.837229\pi$$
−0.872079 + 0.489365i $$0.837229\pi$$
$$234$$ 2.61050 0.170654
$$235$$ −34.4686 −2.24848
$$236$$ 19.0581 1.24057
$$237$$ 13.3646 0.868123
$$238$$ −13.6294 −0.883464
$$239$$ 4.29104 0.277564 0.138782 0.990323i $$-0.455681\pi$$
0.138782 + 0.990323i $$0.455681\pi$$
$$240$$ −36.4381 −2.35207
$$241$$ 7.52367 0.484642 0.242321 0.970196i $$-0.422091\pi$$
0.242321 + 0.970196i $$0.422091\pi$$
$$242$$ 29.8693 1.92007
$$243$$ 1.00000 0.0641500
$$244$$ 59.9046 3.83500
$$245$$ −3.81471 −0.243713
$$246$$ 16.6203 1.05967
$$247$$ 2.92259 0.185960
$$248$$ −55.2511 −3.50845
$$249$$ −1.40629 −0.0891200
$$250$$ −45.3303 −2.86694
$$251$$ 11.3198 0.714498 0.357249 0.934009i $$-0.383715\pi$$
0.357249 + 0.934009i $$0.383715\pi$$
$$252$$ 4.81471 0.303298
$$253$$ −15.7800 −0.992079
$$254$$ 45.6174 2.86229
$$255$$ 19.9166 1.24723
$$256$$ −16.7394 −1.04621
$$257$$ −24.8504 −1.55013 −0.775063 0.631884i $$-0.782282\pi$$
−0.775063 + 0.631884i $$0.782282\pi$$
$$258$$ −17.1040 −1.06485
$$259$$ 0.154821 0.00962013
$$260$$ −18.3667 −1.13906
$$261$$ −0.922589 −0.0571068
$$262$$ 2.52540 0.156020
$$263$$ 17.1762 1.05913 0.529565 0.848270i $$-0.322355\pi$$
0.529565 + 0.848270i $$0.322355\pi$$
$$264$$ −34.8087 −2.14233
$$265$$ −32.6234 −2.00404
$$266$$ 7.62942 0.467790
$$267$$ −1.96953 −0.120533
$$268$$ −51.3345 −3.13575
$$269$$ −7.74640 −0.472306 −0.236153 0.971716i $$-0.575887\pi$$
−0.236153 + 0.971716i $$0.575887\pi$$
$$270$$ −9.95830 −0.606043
$$271$$ −29.7008 −1.80420 −0.902099 0.431530i $$-0.857974\pi$$
−0.902099 + 0.431530i $$0.857974\pi$$
$$272$$ −49.8710 −3.02388
$$273$$ 1.00000 0.0605228
$$274$$ 8.60494 0.519844
$$275$$ −45.2507 −2.72872
$$276$$ 16.0378 0.965364
$$277$$ 25.1488 1.51105 0.755523 0.655122i $$-0.227383\pi$$
0.755523 + 0.655122i $$0.227383\pi$$
$$278$$ 0.967402 0.0580209
$$279$$ −7.51941 −0.450175
$$280$$ −28.0297 −1.67510
$$281$$ 8.40030 0.501120 0.250560 0.968101i $$-0.419385\pi$$
0.250560 + 0.968101i $$0.419385\pi$$
$$282$$ 23.5877 1.40463
$$283$$ −21.3912 −1.27157 −0.635787 0.771864i $$-0.719324\pi$$
−0.635787 + 0.771864i $$0.719324\pi$$
$$284$$ −31.6927 −1.88062
$$285$$ −11.1488 −0.660400
$$286$$ −12.3667 −0.731259
$$287$$ 6.36672 0.375815
$$288$$ 10.2399 0.603393
$$289$$ 10.2588 0.603461
$$290$$ 9.18742 0.539504
$$291$$ −2.11001 −0.123691
$$292$$ −37.2426 −2.17946
$$293$$ −9.59895 −0.560777 −0.280388 0.959887i $$-0.590463\pi$$
−0.280388 + 0.959887i $$0.590463\pi$$
$$294$$ 2.61050 0.152247
$$295$$ −15.0998 −0.879142
$$296$$ 1.13760 0.0661215
$$297$$ −4.73730 −0.274886
$$298$$ −41.0959 −2.38062
$$299$$ 3.33101 0.192637
$$300$$ 45.9901 2.65524
$$301$$ −6.55201 −0.377651
$$302$$ 26.8658 1.54595
$$303$$ 0.850419 0.0488553
$$304$$ 27.9166 1.60113
$$305$$ −47.4626 −2.71770
$$306$$ −13.6294 −0.779142
$$307$$ −15.1488 −0.864589 −0.432295 0.901732i $$-0.642296\pi$$
−0.432295 + 0.901732i $$0.642296\pi$$
$$308$$ −22.8087 −1.29965
$$309$$ −1.47460 −0.0838869
$$310$$ 74.8805 4.25293
$$311$$ −4.37058 −0.247833 −0.123916 0.992293i $$-0.539545\pi$$
−0.123916 + 0.992293i $$0.539545\pi$$
$$312$$ 7.34780 0.415987
$$313$$ −1.49280 −0.0843783 −0.0421891 0.999110i $$-0.513433\pi$$
−0.0421891 + 0.999110i $$0.513433\pi$$
$$314$$ −29.7954 −1.68145
$$315$$ −3.81471 −0.214935
$$316$$ 64.3466 3.61978
$$317$$ 23.2129 1.30377 0.651883 0.758320i $$-0.273979\pi$$
0.651883 + 0.758320i $$0.273979\pi$$
$$318$$ 22.3250 1.25192
$$319$$ 4.37058 0.244706
$$320$$ −29.0959 −1.62651
$$321$$ 2.62418 0.146467
$$322$$ 8.69560 0.484587
$$323$$ −15.2588 −0.849024
$$324$$ 4.81471 0.267484
$$325$$ 9.55201 0.529850
$$326$$ −35.1754 −1.94819
$$327$$ −17.9166 −0.990790
$$328$$ 46.7814 2.58307
$$329$$ 9.03571 0.498155
$$330$$ 47.1754 2.59692
$$331$$ 1.10402 0.0606822 0.0303411 0.999540i $$-0.490341\pi$$
0.0303411 + 0.999540i $$0.490341\pi$$
$$332$$ −6.77088 −0.371600
$$333$$ 0.154821 0.00848416
$$334$$ 50.0969 2.74118
$$335$$ 40.6725 2.22218
$$336$$ 9.55201 0.521105
$$337$$ −4.24237 −0.231096 −0.115548 0.993302i $$-0.536862\pi$$
−0.115548 + 0.993302i $$0.536862\pi$$
$$338$$ 2.61050 0.141992
$$339$$ 0.922589 0.0501082
$$340$$ 95.8926 5.20051
$$341$$ 35.6217 1.92902
$$342$$ 7.62942 0.412552
$$343$$ 1.00000 0.0539949
$$344$$ −48.1428 −2.59569
$$345$$ −12.7068 −0.684113
$$346$$ −50.9372 −2.73840
$$347$$ 14.0336 0.753362 0.376681 0.926343i $$-0.377065\pi$$
0.376681 + 0.926343i $$0.377065\pi$$
$$348$$ −4.44200 −0.238116
$$349$$ 4.10575 0.219776 0.109888 0.993944i $$-0.464951\pi$$
0.109888 + 0.993944i $$0.464951\pi$$
$$350$$ 24.9355 1.33286
$$351$$ 1.00000 0.0533761
$$352$$ −48.5096 −2.58557
$$353$$ −16.0753 −0.855601 −0.427800 0.903873i $$-0.640711\pi$$
−0.427800 + 0.903873i $$0.640711\pi$$
$$354$$ 10.3331 0.549200
$$355$$ 25.1102 1.33271
$$356$$ −9.48272 −0.502583
$$357$$ −5.22100 −0.276325
$$358$$ −43.2967 −2.28830
$$359$$ −18.1510 −0.957971 −0.478985 0.877823i $$-0.658995\pi$$
−0.478985 + 0.877823i $$0.658995\pi$$
$$360$$ −28.0297 −1.47730
$$361$$ −10.4585 −0.550446
$$362$$ −7.35336 −0.386484
$$363$$ 11.4420 0.600549
$$364$$ 4.81471 0.252359
$$365$$ 29.5074 1.54449
$$366$$ 32.4798 1.69775
$$367$$ −23.3955 −1.22123 −0.610616 0.791927i $$-0.709078\pi$$
−0.610616 + 0.791927i $$0.709078\pi$$
$$368$$ 31.8178 1.65862
$$369$$ 6.36672 0.331438
$$370$$ −1.54176 −0.0801522
$$371$$ 8.55201 0.443998
$$372$$ −36.2038 −1.87708
$$373$$ −4.75164 −0.246031 −0.123015 0.992405i $$-0.539256\pi$$
−0.123015 + 0.992405i $$0.539256\pi$$
$$374$$ 64.5666 3.33866
$$375$$ −17.3646 −0.896704
$$376$$ 66.3926 3.42394
$$377$$ −0.922589 −0.0475158
$$378$$ 2.61050 0.134270
$$379$$ −6.81258 −0.349939 −0.174969 0.984574i $$-0.555983\pi$$
−0.174969 + 0.984574i $$0.555983\pi$$
$$380$$ −53.6784 −2.75364
$$381$$ 17.4746 0.895251
$$382$$ 39.5753 2.02485
$$383$$ 2.25746 0.115351 0.0576754 0.998335i $$-0.481631\pi$$
0.0576754 + 0.998335i $$0.481631\pi$$
$$384$$ −0.568798 −0.0290264
$$385$$ 18.0714 0.921005
$$386$$ 0.170195 0.00866268
$$387$$ −6.55201 −0.333057
$$388$$ −10.1591 −0.515749
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ −9.95830 −0.504258
$$391$$ −17.3912 −0.879511
$$392$$ 7.34780 0.371120
$$393$$ 0.967402 0.0487990
$$394$$ −44.7478 −2.25436
$$395$$ −50.9820 −2.56518
$$396$$ −22.8087 −1.14618
$$397$$ 24.9897 1.25420 0.627100 0.778939i $$-0.284242\pi$$
0.627100 + 0.778939i $$0.284242\pi$$
$$398$$ 16.8168 0.842952
$$399$$ 2.92259 0.146312
$$400$$ 91.2409 4.56204
$$401$$ 27.6529 1.38092 0.690460 0.723370i $$-0.257408\pi$$
0.690460 + 0.723370i $$0.257408\pi$$
$$402$$ −27.8332 −1.38819
$$403$$ −7.51941 −0.374568
$$404$$ 4.09452 0.203710
$$405$$ −3.81471 −0.189554
$$406$$ −2.40842 −0.119528
$$407$$ −0.733435 −0.0363550
$$408$$ −38.3629 −1.89924
$$409$$ 13.1488 0.650168 0.325084 0.945685i $$-0.394607\pi$$
0.325084 + 0.945685i $$0.394607\pi$$
$$410$$ −63.4017 −3.13119
$$411$$ 3.29628 0.162594
$$412$$ −7.09976 −0.349780
$$413$$ 3.95830 0.194775
$$414$$ 8.69560 0.427365
$$415$$ 5.36459 0.263337
$$416$$ 10.2399 0.502053
$$417$$ 0.370581 0.0181474
$$418$$ −36.1428 −1.76780
$$419$$ −5.69462 −0.278200 −0.139100 0.990278i $$-0.544421\pi$$
−0.139100 + 0.990278i $$0.544421\pi$$
$$420$$ −18.3667 −0.896204
$$421$$ −23.0206 −1.12196 −0.560978 0.827831i $$-0.689575\pi$$
−0.560978 + 0.827831i $$0.689575\pi$$
$$422$$ −41.8375 −2.03662
$$423$$ 9.03571 0.439331
$$424$$ 62.8384 3.05170
$$425$$ −49.8710 −2.41910
$$426$$ −17.1836 −0.832546
$$427$$ 12.4420 0.602111
$$428$$ 12.6347 0.610719
$$429$$ −4.73730 −0.228719
$$430$$ 65.2469 3.14648
$$431$$ 20.8045 1.00212 0.501058 0.865414i $$-0.332944\pi$$
0.501058 + 0.865414i $$0.332944\pi$$
$$432$$ 9.55201 0.459571
$$433$$ 31.4808 1.51287 0.756436 0.654068i $$-0.226939\pi$$
0.756436 + 0.654068i $$0.226939\pi$$
$$434$$ −19.6294 −0.942242
$$435$$ 3.51941 0.168743
$$436$$ −86.2632 −4.13126
$$437$$ 9.73517 0.465696
$$438$$ −20.1927 −0.964843
$$439$$ 22.3811 1.06819 0.534095 0.845425i $$-0.320653\pi$$
0.534095 + 0.845425i $$0.320653\pi$$
$$440$$ 132.785 6.33028
$$441$$ 1.00000 0.0476190
$$442$$ −13.6294 −0.648285
$$443$$ 8.52465 0.405018 0.202509 0.979280i $$-0.435090\pi$$
0.202509 + 0.979280i $$0.435090\pi$$
$$444$$ 0.745420 0.0353761
$$445$$ 7.51319 0.356159
$$446$$ 52.2795 2.47550
$$447$$ −15.7425 −0.744597
$$448$$ 7.62729 0.360356
$$449$$ 18.7142 0.883178 0.441589 0.897218i $$-0.354415\pi$$
0.441589 + 0.897218i $$0.354415\pi$$
$$450$$ 24.9355 1.17547
$$451$$ −30.1610 −1.42023
$$452$$ 4.44200 0.208934
$$453$$ 10.2914 0.483534
$$454$$ 50.1663 2.35442
$$455$$ −3.81471 −0.178836
$$456$$ 21.4746 1.00564
$$457$$ −14.1366 −0.661283 −0.330641 0.943756i $$-0.607265\pi$$
−0.330641 + 0.943756i $$0.607265\pi$$
$$458$$ 13.7282 0.641476
$$459$$ −5.22100 −0.243695
$$460$$ −61.1797 −2.85252
$$461$$ −2.67636 −0.124651 −0.0623253 0.998056i $$-0.519852\pi$$
−0.0623253 + 0.998056i $$0.519852\pi$$
$$462$$ −12.3667 −0.575352
$$463$$ −2.53162 −0.117655 −0.0588273 0.998268i $$-0.518736\pi$$
−0.0588273 + 0.998268i $$0.518736\pi$$
$$464$$ −8.81258 −0.409114
$$465$$ 28.6844 1.33021
$$466$$ −69.5005 −3.21955
$$467$$ 2.00426 0.0927460 0.0463730 0.998924i $$-0.485234\pi$$
0.0463730 + 0.998924i $$0.485234\pi$$
$$468$$ 4.81471 0.222560
$$469$$ −10.6620 −0.492326
$$470$$ −89.9803 −4.15048
$$471$$ −11.4137 −0.525914
$$472$$ 29.0848 1.33874
$$473$$ 31.0388 1.42717
$$474$$ 34.8883 1.60247
$$475$$ 27.9166 1.28090
$$476$$ −25.1376 −1.15218
$$477$$ 8.55201 0.391570
$$478$$ 11.2018 0.512357
$$479$$ −14.6609 −0.669872 −0.334936 0.942241i $$-0.608715\pi$$
−0.334936 + 0.942241i $$0.608715\pi$$
$$480$$ −39.0623 −1.78294
$$481$$ 0.154821 0.00705925
$$482$$ 19.6405 0.894602
$$483$$ 3.33101 0.151566
$$484$$ 55.0899 2.50409
$$485$$ 8.04907 0.365489
$$486$$ 2.61050 0.118415
$$487$$ 38.2143 1.73165 0.865827 0.500344i $$-0.166793\pi$$
0.865827 + 0.500344i $$0.166793\pi$$
$$488$$ 91.4213 4.13845
$$489$$ −13.4746 −0.609342
$$490$$ −9.95830 −0.449870
$$491$$ 21.3758 0.964677 0.482339 0.875985i $$-0.339787\pi$$
0.482339 + 0.875985i $$0.339787\pi$$
$$492$$ 30.6539 1.38198
$$493$$ 4.81684 0.216939
$$494$$ 7.62942 0.343264
$$495$$ 18.0714 0.812250
$$496$$ −71.8255 −3.22506
$$497$$ −6.58248 −0.295264
$$498$$ −3.67112 −0.164507
$$499$$ 27.0920 1.21281 0.606403 0.795158i $$-0.292612\pi$$
0.606403 + 0.795158i $$0.292612\pi$$
$$500$$ −83.6054 −3.73895
$$501$$ 19.1905 0.857370
$$502$$ 29.5503 1.31889
$$503$$ 24.8778 1.10925 0.554623 0.832102i $$-0.312863\pi$$
0.554623 + 0.832102i $$0.312863\pi$$
$$504$$ 7.34780 0.327297
$$505$$ −3.24410 −0.144361
$$506$$ −41.1936 −1.83128
$$507$$ 1.00000 0.0444116
$$508$$ 84.1351 3.73289
$$509$$ 2.70297 0.119807 0.0599034 0.998204i $$-0.480921\pi$$
0.0599034 + 0.998204i $$0.480921\pi$$
$$510$$ 51.9923 2.30226
$$511$$ −7.73517 −0.342184
$$512$$ −42.5607 −1.88093
$$513$$ 2.92259 0.129035
$$514$$ −64.8720 −2.86138
$$515$$ 5.62516 0.247874
$$516$$ −31.5460 −1.38874
$$517$$ −42.8049 −1.88256
$$518$$ 0.404161 0.0177578
$$519$$ −19.5124 −0.856501
$$520$$ −28.0297 −1.22918
$$521$$ −33.4472 −1.46535 −0.732675 0.680579i $$-0.761728\pi$$
−0.732675 + 0.680579i $$0.761728\pi$$
$$522$$ −2.40842 −0.105414
$$523$$ 19.3198 0.844795 0.422397 0.906411i $$-0.361189\pi$$
0.422397 + 0.906411i $$0.361189\pi$$
$$524$$ 4.65776 0.203475
$$525$$ 9.55201 0.416884
$$526$$ 44.8384 1.95505
$$527$$ 39.2588 1.71014
$$528$$ −45.2507 −1.96928
$$529$$ −11.9044 −0.517582
$$530$$ −85.1634 −3.69926
$$531$$ 3.95830 0.171776
$$532$$ 14.0714 0.610073
$$533$$ 6.36672 0.275773
$$534$$ −5.14146 −0.222493
$$535$$ −10.0105 −0.432791
$$536$$ −78.3424 −3.38387
$$537$$ −16.5856 −0.715721
$$538$$ −20.2220 −0.871832
$$539$$ −4.73730 −0.204050
$$540$$ −18.3667 −0.790378
$$541$$ −24.9554 −1.07292 −0.536459 0.843927i $$-0.680238\pi$$
−0.536459 + 0.843927i $$0.680238\pi$$
$$542$$ −77.5340 −3.33037
$$543$$ −2.81684 −0.120882
$$544$$ −53.4626 −2.29219
$$545$$ 68.3466 2.92765
$$546$$ 2.61050 0.111719
$$547$$ 3.80037 0.162492 0.0812460 0.996694i $$-0.474110\pi$$
0.0812460 + 0.996694i $$0.474110\pi$$
$$548$$ 15.8706 0.677960
$$549$$ 12.4420 0.531012
$$550$$ −118.127 −5.03695
$$551$$ −2.69635 −0.114868
$$552$$ 24.4756 1.04175
$$553$$ 13.3646 0.568320
$$554$$ 65.6510 2.78924
$$555$$ −0.590599 −0.0250695
$$556$$ 1.78424 0.0756686
$$557$$ 43.8792 1.85922 0.929610 0.368546i $$-0.120144\pi$$
0.929610 + 0.368546i $$0.120144\pi$$
$$558$$ −19.6294 −0.830980
$$559$$ −6.55201 −0.277120
$$560$$ −36.4381 −1.53979
$$561$$ 24.7334 1.04425
$$562$$ 21.9290 0.925018
$$563$$ 12.3768 0.521620 0.260810 0.965390i $$-0.416010\pi$$
0.260810 + 0.965390i $$0.416010\pi$$
$$564$$ 43.5043 1.83186
$$565$$ −3.51941 −0.148063
$$566$$ −55.8417 −2.34720
$$567$$ 1.00000 0.0419961
$$568$$ −48.3667 −2.02942
$$569$$ −26.6234 −1.11611 −0.558056 0.829803i $$-0.688453\pi$$
−0.558056 + 0.829803i $$0.688453\pi$$
$$570$$ −29.1040 −1.21903
$$571$$ 10.9983 0.460263 0.230132 0.973160i $$-0.426084\pi$$
0.230132 + 0.973160i $$0.426084\pi$$
$$572$$ −22.8087 −0.953680
$$573$$ 15.1601 0.633321
$$574$$ 16.6203 0.693719
$$575$$ 31.8178 1.32689
$$576$$ 7.62729 0.317804
$$577$$ 30.4238 1.26656 0.633280 0.773923i $$-0.281708\pi$$
0.633280 + 0.773923i $$0.281708\pi$$
$$578$$ 26.7807 1.11393
$$579$$ 0.0651962 0.00270946
$$580$$ 16.9449 0.703600
$$581$$ −1.40629 −0.0583428
$$582$$ −5.50818 −0.228321
$$583$$ −40.5134 −1.67789
$$584$$ −56.8365 −2.35191
$$585$$ −3.81471 −0.157719
$$586$$ −25.0581 −1.03514
$$587$$ 3.84207 0.158579 0.0792895 0.996852i $$-0.474735\pi$$
0.0792895 + 0.996852i $$0.474735\pi$$
$$588$$ 4.81471 0.198555
$$589$$ −21.9761 −0.905511
$$590$$ −39.4179 −1.62281
$$591$$ −17.1415 −0.705105
$$592$$ 1.47886 0.0607806
$$593$$ −23.8904 −0.981061 −0.490530 0.871424i $$-0.663197\pi$$
−0.490530 + 0.871424i $$0.663197\pi$$
$$594$$ −12.3667 −0.507413
$$595$$ 19.9166 0.816501
$$596$$ −75.7958 −3.10471
$$597$$ 6.44200 0.263653
$$598$$ 8.69560 0.355589
$$599$$ −15.4206 −0.630070 −0.315035 0.949080i $$-0.602016\pi$$
−0.315035 + 0.949080i $$0.602016\pi$$
$$600$$ 70.1862 2.86534
$$601$$ −1.49280 −0.0608928 −0.0304464 0.999536i $$-0.509693\pi$$
−0.0304464 + 0.999536i $$0.509693\pi$$
$$602$$ −17.1040 −0.697108
$$603$$ −10.6620 −0.434191
$$604$$ 49.5503 2.01617
$$605$$ −43.6479 −1.77454
$$606$$ 2.22002 0.0901821
$$607$$ −4.44626 −0.180468 −0.0902340 0.995921i $$-0.528761\pi$$
−0.0902340 + 0.995921i $$0.528761\pi$$
$$608$$ 29.9271 1.21370
$$609$$ −0.922589 −0.0373852
$$610$$ −123.901 −5.01661
$$611$$ 9.03571 0.365546
$$612$$ −25.1376 −1.01613
$$613$$ 26.6640 1.07695 0.538474 0.842642i $$-0.319001\pi$$
0.538474 + 0.842642i $$0.319001\pi$$
$$614$$ −39.5460 −1.59595
$$615$$ −24.2872 −0.979354
$$616$$ −34.8087 −1.40248
$$617$$ 27.3495 1.10105 0.550525 0.834819i $$-0.314428\pi$$
0.550525 + 0.834819i $$0.314428\pi$$
$$618$$ −3.84944 −0.154847
$$619$$ −9.24836 −0.371723 −0.185861 0.982576i $$-0.559508\pi$$
−0.185861 + 0.982576i $$0.559508\pi$$
$$620$$ 138.107 5.54651
$$621$$ 3.33101 0.133669
$$622$$ −11.4094 −0.457475
$$623$$ −1.96953 −0.0789076
$$624$$ 9.55201 0.382386
$$625$$ 18.4808 0.739233
$$626$$ −3.89697 −0.155754
$$627$$ −13.8452 −0.552923
$$628$$ −54.9535 −2.19288
$$629$$ −0.808323 −0.0322299
$$630$$ −9.95830 −0.396748
$$631$$ 24.5134 0.975864 0.487932 0.872882i $$-0.337751\pi$$
0.487932 + 0.872882i $$0.337751\pi$$
$$632$$ 98.2003 3.90620
$$633$$ −16.0266 −0.637000
$$634$$ 60.5972 2.40662
$$635$$ −66.6605 −2.64534
$$636$$ 41.1754 1.63271
$$637$$ 1.00000 0.0396214
$$638$$ 11.4094 0.451703
$$639$$ −6.58248 −0.260399
$$640$$ 2.16980 0.0857689
$$641$$ 34.3972 1.35861 0.679304 0.733857i $$-0.262282\pi$$
0.679304 + 0.733857i $$0.262282\pi$$
$$642$$ 6.85042 0.270364
$$643$$ −7.69036 −0.303278 −0.151639 0.988436i $$-0.548455\pi$$
−0.151639 + 0.988436i $$0.548455\pi$$
$$644$$ 16.0378 0.631979
$$645$$ 24.9940 0.984138
$$646$$ −39.8332 −1.56722
$$647$$ 41.9123 1.64774 0.823872 0.566776i $$-0.191809\pi$$
0.823872 + 0.566776i $$0.191809\pi$$
$$648$$ 7.34780 0.288649
$$649$$ −18.7516 −0.736066
$$650$$ 24.9355 0.978051
$$651$$ −7.51941 −0.294709
$$652$$ −64.8763 −2.54075
$$653$$ 34.2262 1.33938 0.669688 0.742642i $$-0.266428\pi$$
0.669688 + 0.742642i $$0.266428\pi$$
$$654$$ −46.7713 −1.82890
$$655$$ −3.69036 −0.144194
$$656$$ 60.8149 2.37442
$$657$$ −7.73517 −0.301778
$$658$$ 23.5877 0.919545
$$659$$ 45.1685 1.75951 0.879757 0.475424i $$-0.157705\pi$$
0.879757 + 0.475424i $$0.157705\pi$$
$$660$$ 87.0086 3.38681
$$661$$ 11.9839 0.466119 0.233059 0.972463i $$-0.425126\pi$$
0.233059 + 0.972463i $$0.425126\pi$$
$$662$$ 2.88203 0.112013
$$663$$ −5.22100 −0.202767
$$664$$ −10.3331 −0.401004
$$665$$ −11.1488 −0.432333
$$666$$ 0.404161 0.0156609
$$667$$ −3.07315 −0.118993
$$668$$ 92.3968 3.57494
$$669$$ 20.0266 0.774273
$$670$$ 106.176 4.10192
$$671$$ −58.9415 −2.27541
$$672$$ 10.2399 0.395013
$$673$$ −29.5117 −1.13759 −0.568796 0.822479i $$-0.692591\pi$$
−0.568796 + 0.822479i $$0.692591\pi$$
$$674$$ −11.0747 −0.426581
$$675$$ 9.55201 0.367657
$$676$$ 4.81471 0.185181
$$677$$ −31.0662 −1.19397 −0.596985 0.802252i $$-0.703635\pi$$
−0.596985 + 0.802252i $$0.703635\pi$$
$$678$$ 2.40842 0.0924948
$$679$$ −2.11001 −0.0809747
$$680$$ 146.343 5.61200
$$681$$ 19.2171 0.736402
$$682$$ 92.9904 3.56079
$$683$$ 5.79433 0.221714 0.110857 0.993836i $$-0.464640\pi$$
0.110857 + 0.993836i $$0.464640\pi$$
$$684$$ 14.0714 0.538034
$$685$$ −12.5744 −0.480441
$$686$$ 2.61050 0.0996693
$$687$$ 5.25884 0.200637
$$688$$ −62.5848 −2.38602
$$689$$ 8.55201 0.325806
$$690$$ −33.1712 −1.26281
$$691$$ 18.8617 0.717531 0.358766 0.933428i $$-0.383198\pi$$
0.358766 + 0.933428i $$0.383198\pi$$
$$692$$ −93.9467 −3.57132
$$693$$ −4.73730 −0.179955
$$694$$ 36.6347 1.39063
$$695$$ −1.41366 −0.0536232
$$696$$ −6.77900 −0.256957
$$697$$ −33.2406 −1.25908
$$698$$ 10.7181 0.405685
$$699$$ −26.6234 −1.00699
$$700$$ 45.9901 1.73826
$$701$$ −32.3180 −1.22064 −0.610318 0.792157i $$-0.708958\pi$$
−0.610318 + 0.792157i $$0.708958\pi$$
$$702$$ 2.61050 0.0985270
$$703$$ 0.452479 0.0170656
$$704$$ −36.1328 −1.36180
$$705$$ −34.4686 −1.29816
$$706$$ −41.9645 −1.57936
$$707$$ 0.850419 0.0319833
$$708$$ 19.0581 0.716246
$$709$$ 4.72296 0.177374 0.0886872 0.996060i $$-0.471733\pi$$
0.0886872 + 0.996060i $$0.471733\pi$$
$$710$$ 65.5503 2.46006
$$711$$ 13.3646 0.501211
$$712$$ −14.4717 −0.542350
$$713$$ −25.0472 −0.938026
$$714$$ −13.6294 −0.510068
$$715$$ 18.0714 0.675833
$$716$$ −79.8548 −2.98431
$$717$$ 4.29104 0.160252
$$718$$ −47.3831 −1.76832
$$719$$ −40.8902 −1.52495 −0.762474 0.647019i $$-0.776015\pi$$
−0.762474 + 0.647019i $$0.776015\pi$$
$$720$$ −36.4381 −1.35797
$$721$$ −1.47460 −0.0549169
$$722$$ −27.3018 −1.01607
$$723$$ 7.52367 0.279808
$$724$$ −13.5623 −0.504037
$$725$$ −8.81258 −0.327291
$$726$$ 29.8693 1.10856
$$727$$ 18.7292 0.694627 0.347313 0.937749i $$-0.387094\pi$$
0.347313 + 0.937749i $$0.387094\pi$$
$$728$$ 7.34780 0.272328
$$729$$ 1.00000 0.0370370
$$730$$ 77.0291 2.85098
$$731$$ 34.2080 1.26523
$$732$$ 59.9046 2.21414
$$733$$ −14.1100 −0.521165 −0.260583 0.965452i $$-0.583915\pi$$
−0.260583 + 0.965452i $$0.583915\pi$$
$$734$$ −61.0738 −2.25428
$$735$$ −3.81471 −0.140708
$$736$$ 34.1093 1.25728
$$737$$ 50.5092 1.86053
$$738$$ 16.6203 0.611802
$$739$$ −49.9209 −1.83637 −0.918184 0.396154i $$-0.870345\pi$$
−0.918184 + 0.396154i $$0.870345\pi$$
$$740$$ −2.84356 −0.104531
$$741$$ 2.92259 0.107364
$$742$$ 22.3250 0.819577
$$743$$ 2.15868 0.0791945 0.0395972 0.999216i $$-0.487393\pi$$
0.0395972 + 0.999216i $$0.487393\pi$$
$$744$$ −55.2511 −2.02560
$$745$$ 60.0532 2.20018
$$746$$ −12.4042 −0.454149
$$747$$ −1.40629 −0.0514535
$$748$$ 119.084 4.35415
$$749$$ 2.62418 0.0958854
$$750$$ −45.3303 −1.65523
$$751$$ 16.8372 0.614399 0.307199 0.951645i $$-0.400608\pi$$
0.307199 + 0.951645i $$0.400608\pi$$
$$752$$ 86.3092 3.14737
$$753$$ 11.3198 0.412516
$$754$$ −2.40842 −0.0877095
$$755$$ −39.2588 −1.42878
$$756$$ 4.81471 0.175109
$$757$$ −17.6028 −0.639785 −0.319893 0.947454i $$-0.603647\pi$$
−0.319893 + 0.947454i $$0.603647\pi$$
$$758$$ −17.7842 −0.645953
$$759$$ −15.7800 −0.572777
$$760$$ −81.9193 −2.97153
$$761$$ 19.2179 0.696648 0.348324 0.937374i $$-0.386751\pi$$
0.348324 + 0.937374i $$0.386751\pi$$
$$762$$ 45.6174 1.65255
$$763$$ −17.9166 −0.648624
$$764$$ 72.9913 2.64073
$$765$$ 19.9166 0.720086
$$766$$ 5.89310 0.212926
$$767$$ 3.95830 0.142926
$$768$$ −16.7394 −0.604032
$$769$$ −14.0406 −0.506315 −0.253158 0.967425i $$-0.581469\pi$$
−0.253158 + 0.967425i $$0.581469\pi$$
$$770$$ 47.1754 1.70008
$$771$$ −24.8504 −0.894966
$$772$$ 0.313901 0.0112975
$$773$$ 22.4339 0.806891 0.403445 0.915004i $$-0.367813\pi$$
0.403445 + 0.915004i $$0.367813\pi$$
$$774$$ −17.1040 −0.614791
$$775$$ −71.8255 −2.58005
$$776$$ −15.5039 −0.556558
$$777$$ 0.154821 0.00555419
$$778$$ 15.6630 0.561546
$$779$$ 18.6073 0.666676
$$780$$ −18.3667 −0.657634
$$781$$ 31.1832 1.11582
$$782$$ −45.3997 −1.62349
$$783$$ −0.922589 −0.0329707
$$784$$ 9.55201 0.341143
$$785$$ 43.5398 1.55400
$$786$$ 2.52540 0.0900781
$$787$$ −0.926847 −0.0330385 −0.0165193 0.999864i $$-0.505258\pi$$
−0.0165193 + 0.999864i $$0.505258\pi$$
$$788$$ −82.5311 −2.94005
$$789$$ 17.1762 0.611488
$$790$$ −133.089 −4.73508
$$791$$ 0.922589 0.0328035
$$792$$ −34.8087 −1.23687
$$793$$ 12.4420 0.441828
$$794$$ 65.2357 2.31513
$$795$$ −32.6234 −1.15703
$$796$$ 31.0164 1.09935
$$797$$ −26.0154 −0.921512 −0.460756 0.887527i $$-0.652422\pi$$
−0.460756 + 0.887527i $$0.652422\pi$$
$$798$$ 7.62942 0.270079
$$799$$ −47.1754 −1.66895
$$800$$ 97.8118 3.45817
$$801$$ −1.96953 −0.0695900
$$802$$ 72.1879 2.54904
$$803$$ 36.6438 1.29313
$$804$$ −51.3345 −1.81043
$$805$$ −12.7068 −0.447857
$$806$$ −19.6294 −0.691417
$$807$$ −7.74640 −0.272686
$$808$$ 6.24870 0.219829
$$809$$ −44.8539 −1.57698 −0.788490 0.615048i $$-0.789137\pi$$
−0.788490 + 0.615048i $$0.789137\pi$$
$$810$$ −9.95830 −0.349899
$$811$$ 27.2511 0.956916 0.478458 0.878110i $$-0.341196\pi$$
0.478458 + 0.878110i $$0.341196\pi$$
$$812$$ −4.44200 −0.155884
$$813$$ −29.7008 −1.04165
$$814$$ −1.91463 −0.0671078
$$815$$ 51.4017 1.80052
$$816$$ −49.8710 −1.74584
$$817$$ −19.1488 −0.669933
$$818$$ 34.3250 1.20015
$$819$$ 1.00000 0.0349428
$$820$$ −116.936 −4.08357
$$821$$ 20.4003 0.711975 0.355988 0.934491i $$-0.384145\pi$$
0.355988 + 0.934491i $$0.384145\pi$$
$$822$$ 8.60494 0.300132
$$823$$ 54.3509 1.89455 0.947276 0.320418i $$-0.103824\pi$$
0.947276 + 0.320418i $$0.103824\pi$$
$$824$$ −10.8350 −0.377457
$$825$$ −45.2507 −1.57543
$$826$$ 10.3331 0.359536
$$827$$ 3.48272 0.121106 0.0605530 0.998165i $$-0.480714\pi$$
0.0605530 + 0.998165i $$0.480714\pi$$
$$828$$ 16.0378 0.557353
$$829$$ −29.8417 −1.03645 −0.518223 0.855246i $$-0.673406\pi$$
−0.518223 + 0.855246i $$0.673406\pi$$
$$830$$ 14.0043 0.486095
$$831$$ 25.1488 0.872403
$$832$$ 7.62729 0.264429
$$833$$ −5.22100 −0.180897
$$834$$ 0.967402 0.0334984
$$835$$ −73.2063 −2.53341
$$836$$ −66.6605 −2.30550
$$837$$ −7.51941 −0.259909
$$838$$ −14.8658 −0.513530
$$839$$ −43.0033 −1.48464 −0.742320 0.670045i $$-0.766275\pi$$
−0.742320 + 0.670045i $$0.766275\pi$$
$$840$$ −28.0297 −0.967117
$$841$$ −28.1488 −0.970649
$$842$$ −60.0953 −2.07102
$$843$$ 8.40030 0.289322
$$844$$ −77.1634 −2.65608
$$845$$ −3.81471 −0.131230
$$846$$ 23.5877 0.810962
$$847$$ 11.4420 0.393152
$$848$$ 81.6889 2.80521
$$849$$ −21.3912 −0.734144
$$850$$ −130.188 −4.46542
$$851$$ 0.515711 0.0176784
$$852$$ −31.6927 −1.08577
$$853$$ 34.1023 1.16764 0.583820 0.811883i $$-0.301557\pi$$
0.583820 + 0.811883i $$0.301557\pi$$
$$854$$ 32.4798 1.11144
$$855$$ −11.1488 −0.381282
$$856$$ 19.2819 0.659043
$$857$$ −45.7344 −1.56226 −0.781129 0.624370i $$-0.785356\pi$$
−0.781129 + 0.624370i $$0.785356\pi$$
$$858$$ −12.3667 −0.422193
$$859$$ 15.9861 0.545437 0.272719 0.962094i $$-0.412077\pi$$
0.272719 + 0.962094i $$0.412077\pi$$
$$860$$ 120.339 4.10352
$$861$$ 6.36672 0.216977
$$862$$ 54.3100 1.84981
$$863$$ −24.5096 −0.834315 −0.417157 0.908834i $$-0.636974\pi$$
−0.417157 + 0.908834i $$0.636974\pi$$
$$864$$ 10.2399 0.348369
$$865$$ 74.4343 2.53084
$$866$$ 82.1807 2.79261
$$867$$ 10.2588 0.348408
$$868$$ −36.2038 −1.22884
$$869$$ −63.3120 −2.14771
$$870$$ 9.18742 0.311483
$$871$$ −10.6620 −0.361269
$$872$$ −131.648 −4.45815
$$873$$ −2.11001 −0.0714130
$$874$$ 25.4137 0.859630
$$875$$ −17.3646 −0.587030
$$876$$ −37.2426 −1.25831
$$877$$ 53.4913 1.80627 0.903136 0.429354i $$-0.141259\pi$$
0.903136 + 0.429354i $$0.141259\pi$$
$$878$$ 58.4258 1.97177
$$879$$ −9.59895 −0.323765
$$880$$ 172.618 5.81896
$$881$$ 26.1745 0.881840 0.440920 0.897546i $$-0.354652\pi$$
0.440920 + 0.897546i $$0.354652\pi$$
$$882$$ 2.61050 0.0879001
$$883$$ 23.1840 0.780202 0.390101 0.920772i $$-0.372440\pi$$
0.390101 + 0.920772i $$0.372440\pi$$
$$884$$ −25.1376 −0.845469
$$885$$ −15.0998 −0.507573
$$886$$ 22.2536 0.747624
$$887$$ −13.7008 −0.460029 −0.230015 0.973187i $$-0.573877\pi$$
−0.230015 + 0.973187i $$0.573877\pi$$
$$888$$ 1.13760 0.0381752
$$889$$ 17.4746 0.586079
$$890$$ 19.6132 0.657435
$$891$$ −4.73730 −0.158705
$$892$$ 96.4223 3.22846
$$893$$ 26.4077 0.883699
$$894$$ −41.0959 −1.37445
$$895$$ 63.2692 2.11486
$$896$$ −0.568798 −0.0190022
$$897$$ 3.33101 0.111219
$$898$$ 48.8534 1.63026
$$899$$ 6.93733 0.231373
$$900$$ 45.9901 1.53300
$$901$$ −44.6500 −1.48751
$$902$$ −78.7354 −2.62160
$$903$$ −6.55201 −0.218037
$$904$$ 6.77900 0.225466
$$905$$ 10.7454 0.357190
$$906$$ 26.8658 0.892556
$$907$$ 25.5621 0.848777 0.424388 0.905480i $$-0.360489\pi$$
0.424388 + 0.905480i $$0.360489\pi$$
$$908$$ 92.5249 3.07055
$$909$$ 0.850419 0.0282066
$$910$$ −9.95830 −0.330114
$$911$$ 2.07643 0.0687951 0.0343976 0.999408i $$-0.489049\pi$$
0.0343976 + 0.999408i $$0.489049\pi$$
$$912$$ 27.9166 0.924411
$$913$$ 6.66202 0.220481
$$914$$ −36.9036 −1.22066
$$915$$ −47.4626 −1.56907
$$916$$ 25.3198 0.836589
$$917$$ 0.967402 0.0319464
$$918$$ −13.6294 −0.449838
$$919$$ 17.8514 0.588863 0.294432 0.955673i $$-0.404870\pi$$
0.294432 + 0.955673i $$0.404870\pi$$
$$920$$ −93.3672 −3.07823
$$921$$ −15.1488 −0.499171
$$922$$ −6.98664 −0.230093
$$923$$ −6.58248 −0.216665
$$924$$ −22.8087 −0.750352
$$925$$ 1.47886 0.0486245
$$926$$ −6.60881 −0.217179
$$927$$ −1.47460 −0.0484321
$$928$$ −9.44724 −0.310121
$$929$$ −37.0553 −1.21575 −0.607873 0.794034i $$-0.707977\pi$$
−0.607873 + 0.794034i $$0.707977\pi$$
$$930$$ 74.8805 2.45543
$$931$$ 2.92259 0.0957840
$$932$$ −128.184 −4.19881
$$933$$ −4.37058 −0.143086
$$934$$ 5.23212 0.171200
$$935$$ −94.3509 −3.08560
$$936$$ 7.34780 0.240170
$$937$$ 39.7540 1.29871 0.649354 0.760486i $$-0.275039\pi$$
0.649354 + 0.760486i $$0.275039\pi$$
$$938$$ −27.8332 −0.908786
$$939$$ −1.49280 −0.0487158
$$940$$ −165.956 −5.41290
$$941$$ −47.3530 −1.54366 −0.771832 0.635827i $$-0.780659\pi$$
−0.771832 + 0.635827i $$0.780659\pi$$
$$942$$ −29.7954 −0.970785
$$943$$ 21.2076 0.690614
$$944$$ 37.8097 1.23060
$$945$$ −3.81471 −0.124093
$$946$$ 81.0268 2.63441
$$947$$ 8.80446 0.286106 0.143053 0.989715i $$-0.454308\pi$$
0.143053 + 0.989715i $$0.454308\pi$$
$$948$$ 64.3466 2.08988
$$949$$ −7.73517 −0.251094
$$950$$ 72.8763 2.36442
$$951$$ 23.2129 0.752729
$$952$$ −38.3629 −1.24335
$$953$$ 7.72895 0.250365 0.125183 0.992134i $$-0.460048\pi$$
0.125183 + 0.992134i $$0.460048\pi$$
$$954$$ 22.3250 0.722799
$$955$$ −57.8312 −1.87137
$$956$$ 20.6601 0.668196
$$957$$ 4.37058 0.141281
$$958$$ −38.2722 −1.23652
$$959$$ 3.29628 0.106442
$$960$$ −29.0959 −0.939066
$$961$$ 25.5415 0.823920
$$962$$ 0.404161 0.0130307
$$963$$ 2.62418 0.0845630
$$964$$ 36.2243 1.16671
$$965$$ −0.248705 −0.00800608
$$966$$ 8.69560 0.279776
$$967$$ 14.3054 0.460030 0.230015 0.973187i $$-0.426122\pi$$
0.230015 + 0.973187i $$0.426122\pi$$
$$968$$ 84.0735 2.70222
$$969$$ −15.2588 −0.490184
$$970$$ 21.0121 0.674658
$$971$$ −15.1692 −0.486803 −0.243402 0.969926i $$-0.578263\pi$$
−0.243402 + 0.969926i $$0.578263\pi$$
$$972$$ 4.81471 0.154432
$$973$$ 0.370581 0.0118803
$$974$$ 99.7583 3.19646
$$975$$ 9.55201 0.305909
$$976$$ 118.846 3.80417
$$977$$ −8.25746 −0.264180 −0.132090 0.991238i $$-0.542169\pi$$
−0.132090 + 0.991238i $$0.542169\pi$$
$$978$$ −35.1754 −1.12479
$$979$$ 9.33026 0.298196
$$980$$ −18.3667 −0.586703
$$981$$ −17.9166 −0.572033
$$982$$ 55.8016 1.78070
$$983$$ −25.8525 −0.824568 −0.412284 0.911055i $$-0.635269\pi$$
−0.412284 + 0.911055i $$0.635269\pi$$
$$984$$ 46.7814 1.49134
$$985$$ 65.3897 2.08349
$$986$$ 12.5744 0.400449
$$987$$ 9.03571 0.287610
$$988$$ 14.0714 0.447671
$$989$$ −21.8248 −0.693988
$$990$$ 47.1754 1.49933
$$991$$ −56.2100 −1.78557 −0.892785 0.450484i $$-0.851252\pi$$
−0.892785 + 0.450484i $$0.851252\pi$$
$$992$$ −76.9981 −2.44469
$$993$$ 1.10402 0.0350349
$$994$$ −17.1836 −0.545029
$$995$$ −24.5744 −0.779059
$$996$$ −6.77088 −0.214544
$$997$$ 37.4789 1.18697 0.593484 0.804846i $$-0.297752\pi$$
0.593484 + 0.804846i $$0.297752\pi$$
$$998$$ 70.7237 2.23872
$$999$$ 0.154821 0.00489833
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 273.2.a.e.1.4 4
3.2 odd 2 819.2.a.k.1.1 4
4.3 odd 2 4368.2.a.br.1.1 4
5.4 even 2 6825.2.a.bg.1.1 4
7.6 odd 2 1911.2.a.s.1.4 4
13.12 even 2 3549.2.a.w.1.1 4
21.20 even 2 5733.2.a.bf.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.4 4 1.1 even 1 trivial
819.2.a.k.1.1 4 3.2 odd 2
1911.2.a.s.1.4 4 7.6 odd 2
3549.2.a.w.1.1 4 13.12 even 2
4368.2.a.br.1.1 4 4.3 odd 2
5733.2.a.bf.1.1 4 21.20 even 2
6825.2.a.bg.1.1 4 5.4 even 2