Properties

 Label 729.2.a.c.1.4 Level $729$ Weight $2$ Character 729.1 Self dual yes Analytic conductor $5.821$ Analytic rank $1$ Dimension $6$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.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: $$5.82109430735$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{36})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 6x^{4} + 9x^{2} - 3$$ x^6 - 6*x^4 + 9*x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$0.684040$$ of defining polynomial Character $$\chi$$ $$=$$ 729.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.684040 q^{2} -1.53209 q^{4} +1.04801 q^{5} -0.120615 q^{7} -2.41609 q^{8} +O(q^{10})$$ $$q+0.684040 q^{2} -1.53209 q^{4} +1.04801 q^{5} -0.120615 q^{7} -2.41609 q^{8} +0.716881 q^{10} -5.43372 q^{11} -4.57398 q^{13} -0.0825054 q^{14} +1.41147 q^{16} +4.77833 q^{17} -0.588526 q^{19} -1.60565 q^{20} -3.71688 q^{22} -7.79596 q^{23} -3.90167 q^{25} -3.12879 q^{26} +0.184793 q^{28} +5.06975 q^{29} -8.75877 q^{31} +5.79769 q^{32} +3.26857 q^{34} -0.126406 q^{35} -2.18479 q^{37} -0.402575 q^{38} -2.53209 q^{40} +7.55839 q^{41} -1.30541 q^{43} +8.32494 q^{44} -5.33275 q^{46} +2.41609 q^{47} -6.98545 q^{49} -2.66890 q^{50} +7.00774 q^{52} -3.04628 q^{53} -5.69459 q^{55} +0.291416 q^{56} +3.46791 q^{58} -0.0439002 q^{59} -10.2121 q^{61} -5.99135 q^{62} +1.14290 q^{64} -4.79358 q^{65} -1.85978 q^{67} -7.32083 q^{68} -0.0864665 q^{70} -6.51038 q^{71} +12.2344 q^{73} -1.49449 q^{74} +0.901674 q^{76} +0.655386 q^{77} +0.702333 q^{79} +1.47924 q^{80} +5.17024 q^{82} +6.77660 q^{83} +5.00774 q^{85} -0.892951 q^{86} +13.1284 q^{88} +6.85565 q^{89} +0.551689 q^{91} +11.9441 q^{92} +1.65270 q^{94} -0.616781 q^{95} -9.02734 q^{97} -4.77833 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{7}+O(q^{10})$$ 6 * q - 12 * q^7 $$6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64} + 6 q^{67} + 30 q^{70} + 12 q^{73} - 18 q^{76} - 48 q^{79} - 12 q^{82} - 18 q^{85} + 42 q^{88} + 12 q^{94} - 12 q^{97}+O(q^{100})$$ 6 * q - 12 * q^7 - 12 * q^10 - 12 * q^13 - 12 * q^16 - 24 * q^19 - 6 * q^22 - 6 * q^28 - 30 * q^31 - 6 * q^37 - 6 * q^40 - 12 * q^43 + 6 * q^46 - 6 * q^49 - 6 * q^52 - 30 * q^55 + 30 * q^58 - 12 * q^61 + 6 * q^64 + 6 * q^67 + 30 * q^70 + 12 * q^73 - 18 * q^76 - 48 * q^79 - 12 * q^82 - 18 * q^85 + 42 * q^88 + 12 * q^94 - 12 * q^97

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.684040 0.483690 0.241845 0.970315i $$-0.422248\pi$$
0.241845 + 0.970315i $$0.422248\pi$$
$$3$$ 0 0
$$4$$ −1.53209 −0.766044
$$5$$ 1.04801 0.468685 0.234342 0.972154i $$-0.424706\pi$$
0.234342 + 0.972154i $$0.424706\pi$$
$$6$$ 0 0
$$7$$ −0.120615 −0.0455881 −0.0227940 0.999740i $$-0.507256\pi$$
−0.0227940 + 0.999740i $$0.507256\pi$$
$$8$$ −2.41609 −0.854217
$$9$$ 0 0
$$10$$ 0.716881 0.226698
$$11$$ −5.43372 −1.63833 −0.819164 0.573560i $$-0.805562\pi$$
−0.819164 + 0.573560i $$0.805562\pi$$
$$12$$ 0 0
$$13$$ −4.57398 −1.26859 −0.634297 0.773090i $$-0.718710\pi$$
−0.634297 + 0.773090i $$0.718710\pi$$
$$14$$ −0.0825054 −0.0220505
$$15$$ 0 0
$$16$$ 1.41147 0.352869
$$17$$ 4.77833 1.15892 0.579458 0.815002i $$-0.303264\pi$$
0.579458 + 0.815002i $$0.303264\pi$$
$$18$$ 0 0
$$19$$ −0.588526 −0.135017 −0.0675085 0.997719i $$-0.521505\pi$$
−0.0675085 + 0.997719i $$0.521505\pi$$
$$20$$ −1.60565 −0.359033
$$21$$ 0 0
$$22$$ −3.71688 −0.792442
$$23$$ −7.79596 −1.62557 −0.812785 0.582564i $$-0.802049\pi$$
−0.812785 + 0.582564i $$0.802049\pi$$
$$24$$ 0 0
$$25$$ −3.90167 −0.780335
$$26$$ −3.12879 −0.613605
$$27$$ 0 0
$$28$$ 0.184793 0.0349225
$$29$$ 5.06975 0.941428 0.470714 0.882286i $$-0.343996\pi$$
0.470714 + 0.882286i $$0.343996\pi$$
$$30$$ 0 0
$$31$$ −8.75877 −1.57312 −0.786561 0.617513i $$-0.788140\pi$$
−0.786561 + 0.617513i $$0.788140\pi$$
$$32$$ 5.79769 1.02490
$$33$$ 0 0
$$34$$ 3.26857 0.560555
$$35$$ −0.126406 −0.0213664
$$36$$ 0 0
$$37$$ −2.18479 −0.359178 −0.179589 0.983742i $$-0.557477\pi$$
−0.179589 + 0.983742i $$0.557477\pi$$
$$38$$ −0.402575 −0.0653064
$$39$$ 0 0
$$40$$ −2.53209 −0.400358
$$41$$ 7.55839 1.18042 0.590211 0.807249i $$-0.299044\pi$$
0.590211 + 0.807249i $$0.299044\pi$$
$$42$$ 0 0
$$43$$ −1.30541 −0.199073 −0.0995364 0.995034i $$-0.531736\pi$$
−0.0995364 + 0.995034i $$0.531736\pi$$
$$44$$ 8.32494 1.25503
$$45$$ 0 0
$$46$$ −5.33275 −0.786271
$$47$$ 2.41609 0.352423 0.176212 0.984352i $$-0.443616\pi$$
0.176212 + 0.984352i $$0.443616\pi$$
$$48$$ 0 0
$$49$$ −6.98545 −0.997922
$$50$$ −2.66890 −0.377440
$$51$$ 0 0
$$52$$ 7.00774 0.971799
$$53$$ −3.04628 −0.418439 −0.209219 0.977869i $$-0.567092\pi$$
−0.209219 + 0.977869i $$0.567092\pi$$
$$54$$ 0 0
$$55$$ −5.69459 −0.767859
$$56$$ 0.291416 0.0389421
$$57$$ 0 0
$$58$$ 3.46791 0.455359
$$59$$ −0.0439002 −0.00571532 −0.00285766 0.999996i $$-0.500910\pi$$
−0.00285766 + 0.999996i $$0.500910\pi$$
$$60$$ 0 0
$$61$$ −10.2121 −1.30753 −0.653765 0.756698i $$-0.726811\pi$$
−0.653765 + 0.756698i $$0.726811\pi$$
$$62$$ −5.99135 −0.760902
$$63$$ 0 0
$$64$$ 1.14290 0.142863
$$65$$ −4.79358 −0.594570
$$66$$ 0 0
$$67$$ −1.85978 −0.227209 −0.113604 0.993526i $$-0.536240\pi$$
−0.113604 + 0.993526i $$0.536240\pi$$
$$68$$ −7.32083 −0.887781
$$69$$ 0 0
$$70$$ −0.0864665 −0.0103347
$$71$$ −6.51038 −0.772640 −0.386320 0.922365i $$-0.626254\pi$$
−0.386320 + 0.922365i $$0.626254\pi$$
$$72$$ 0 0
$$73$$ 12.2344 1.43193 0.715965 0.698136i $$-0.245987\pi$$
0.715965 + 0.698136i $$0.245987\pi$$
$$74$$ −1.49449 −0.173730
$$75$$ 0 0
$$76$$ 0.901674 0.103429
$$77$$ 0.655386 0.0746882
$$78$$ 0 0
$$79$$ 0.702333 0.0790187 0.0395093 0.999219i $$-0.487421\pi$$
0.0395093 + 0.999219i $$0.487421\pi$$
$$80$$ 1.47924 0.165384
$$81$$ 0 0
$$82$$ 5.17024 0.570958
$$83$$ 6.77660 0.743828 0.371914 0.928267i $$-0.378702\pi$$
0.371914 + 0.928267i $$0.378702\pi$$
$$84$$ 0 0
$$85$$ 5.00774 0.543166
$$86$$ −0.892951 −0.0962894
$$87$$ 0 0
$$88$$ 13.1284 1.39949
$$89$$ 6.85565 0.726697 0.363349 0.931653i $$-0.381633\pi$$
0.363349 + 0.931653i $$0.381633\pi$$
$$90$$ 0 0
$$91$$ 0.551689 0.0578327
$$92$$ 11.9441 1.24526
$$93$$ 0 0
$$94$$ 1.65270 0.170463
$$95$$ −0.616781 −0.0632804
$$96$$ 0 0
$$97$$ −9.02734 −0.916588 −0.458294 0.888801i $$-0.651539\pi$$
−0.458294 + 0.888801i $$0.651539\pi$$
$$98$$ −4.77833 −0.482684
$$99$$ 0 0
$$100$$ 5.97771 0.597771
$$101$$ 12.9921 1.29276 0.646382 0.763014i $$-0.276281\pi$$
0.646382 + 0.763014i $$0.276281\pi$$
$$102$$ 0 0
$$103$$ 9.07192 0.893883 0.446941 0.894563i $$-0.352513\pi$$
0.446941 + 0.894563i $$0.352513\pi$$
$$104$$ 11.0511 1.08365
$$105$$ 0 0
$$106$$ −2.08378 −0.202394
$$107$$ 11.3865 1.10077 0.550386 0.834911i $$-0.314481\pi$$
0.550386 + 0.834911i $$0.314481\pi$$
$$108$$ 0 0
$$109$$ 2.89899 0.277672 0.138836 0.990315i $$-0.455664\pi$$
0.138836 + 0.990315i $$0.455664\pi$$
$$110$$ −3.89533 −0.371405
$$111$$ 0 0
$$112$$ −0.170245 −0.0160866
$$113$$ −11.5801 −1.08937 −0.544683 0.838642i $$-0.683350\pi$$
−0.544683 + 0.838642i $$0.683350\pi$$
$$114$$ 0 0
$$115$$ −8.17024 −0.761879
$$116$$ −7.76730 −0.721176
$$117$$ 0 0
$$118$$ −0.0300295 −0.00276444
$$119$$ −0.576337 −0.0528327
$$120$$ 0 0
$$121$$ 18.5253 1.68412
$$122$$ −6.98551 −0.632438
$$123$$ 0 0
$$124$$ 13.4192 1.20508
$$125$$ −9.32905 −0.834415
$$126$$ 0 0
$$127$$ −15.4192 −1.36823 −0.684117 0.729372i $$-0.739812\pi$$
−0.684117 + 0.729372i $$0.739812\pi$$
$$128$$ −10.8136 −0.955795
$$129$$ 0 0
$$130$$ −3.27900 −0.287587
$$131$$ 13.3561 1.16693 0.583463 0.812140i $$-0.301697\pi$$
0.583463 + 0.812140i $$0.301697\pi$$
$$132$$ 0 0
$$133$$ 0.0709849 0.00615517
$$134$$ −1.27217 −0.109899
$$135$$ 0 0
$$136$$ −11.5449 −0.989965
$$137$$ −19.6236 −1.67656 −0.838279 0.545242i $$-0.816438\pi$$
−0.838279 + 0.545242i $$0.816438\pi$$
$$138$$ 0 0
$$139$$ 16.3063 1.38309 0.691543 0.722335i $$-0.256931\pi$$
0.691543 + 0.722335i $$0.256931\pi$$
$$140$$ 0.193665 0.0163676
$$141$$ 0 0
$$142$$ −4.45336 −0.373718
$$143$$ 24.8537 2.07837
$$144$$ 0 0
$$145$$ 5.31315 0.441233
$$146$$ 8.36884 0.692610
$$147$$ 0 0
$$148$$ 3.34730 0.275146
$$149$$ 11.0477 0.905062 0.452531 0.891749i $$-0.350521\pi$$
0.452531 + 0.891749i $$0.350521\pi$$
$$150$$ 0 0
$$151$$ −6.80066 −0.553430 −0.276715 0.960952i $$-0.589246\pi$$
−0.276715 + 0.960952i $$0.589246\pi$$
$$152$$ 1.42193 0.115334
$$153$$ 0 0
$$154$$ 0.448311 0.0361259
$$155$$ −9.17928 −0.737298
$$156$$ 0 0
$$157$$ 16.8871 1.34774 0.673870 0.738850i $$-0.264631\pi$$
0.673870 + 0.738850i $$0.264631\pi$$
$$158$$ 0.480424 0.0382205
$$159$$ 0 0
$$160$$ 6.07604 0.480353
$$161$$ 0.940307 0.0741066
$$162$$ 0 0
$$163$$ 8.41147 0.658838 0.329419 0.944184i $$-0.393147\pi$$
0.329419 + 0.944184i $$0.393147\pi$$
$$164$$ −11.5801 −0.904256
$$165$$ 0 0
$$166$$ 4.63547 0.359782
$$167$$ −2.62500 −0.203129 −0.101564 0.994829i $$-0.532385\pi$$
−0.101564 + 0.994829i $$0.532385\pi$$
$$168$$ 0 0
$$169$$ 7.92127 0.609329
$$170$$ 3.42550 0.262724
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ 13.1033 0.996223 0.498112 0.867113i $$-0.334027\pi$$
0.498112 + 0.867113i $$0.334027\pi$$
$$174$$ 0 0
$$175$$ 0.470599 0.0355740
$$176$$ −7.66955 −0.578114
$$177$$ 0 0
$$178$$ 4.68954 0.351496
$$179$$ −22.0988 −1.65174 −0.825872 0.563857i $$-0.809317\pi$$
−0.825872 + 0.563857i $$0.809317\pi$$
$$180$$ 0 0
$$181$$ −2.05913 −0.153054 −0.0765268 0.997068i $$-0.524383\pi$$
−0.0765268 + 0.997068i $$0.524383\pi$$
$$182$$ 0.377378 0.0279731
$$183$$ 0 0
$$184$$ 18.8357 1.38859
$$185$$ −2.28969 −0.168341
$$186$$ 0 0
$$187$$ −25.9641 −1.89868
$$188$$ −3.70167 −0.269972
$$189$$ 0 0
$$190$$ −0.421903 −0.0306081
$$191$$ −3.94918 −0.285753 −0.142876 0.989741i $$-0.545635\pi$$
−0.142876 + 0.989741i $$0.545635\pi$$
$$192$$ 0 0
$$193$$ −8.44656 −0.607996 −0.303998 0.952673i $$-0.598322\pi$$
−0.303998 + 0.952673i $$0.598322\pi$$
$$194$$ −6.17507 −0.443344
$$195$$ 0 0
$$196$$ 10.7023 0.764452
$$197$$ −2.29498 −0.163511 −0.0817553 0.996652i $$-0.526053\pi$$
−0.0817553 + 0.996652i $$0.526053\pi$$
$$198$$ 0 0
$$199$$ −23.5030 −1.66608 −0.833042 0.553210i $$-0.813403\pi$$
−0.833042 + 0.553210i $$0.813403\pi$$
$$200$$ 9.42680 0.666575
$$201$$ 0 0
$$202$$ 8.88713 0.625296
$$203$$ −0.611486 −0.0429179
$$204$$ 0 0
$$205$$ 7.92127 0.553246
$$206$$ 6.20556 0.432362
$$207$$ 0 0
$$208$$ −6.45605 −0.447647
$$209$$ 3.19788 0.221202
$$210$$ 0 0
$$211$$ −1.66725 −0.114778 −0.0573892 0.998352i $$-0.518278\pi$$
−0.0573892 + 0.998352i $$0.518278\pi$$
$$212$$ 4.66717 0.320543
$$213$$ 0 0
$$214$$ 7.78880 0.532431
$$215$$ −1.36808 −0.0933023
$$216$$ 0 0
$$217$$ 1.05644 0.0717156
$$218$$ 1.98302 0.134307
$$219$$ 0 0
$$220$$ 8.72462 0.588214
$$221$$ −21.8560 −1.47019
$$222$$ 0 0
$$223$$ −9.80066 −0.656301 −0.328150 0.944626i $$-0.606425\pi$$
−0.328150 + 0.944626i $$0.606425\pi$$
$$224$$ −0.699287 −0.0467231
$$225$$ 0 0
$$226$$ −7.92127 −0.526915
$$227$$ −15.4808 −1.02749 −0.513747 0.857942i $$-0.671743\pi$$
−0.513747 + 0.857942i $$0.671743\pi$$
$$228$$ 0 0
$$229$$ −11.2540 −0.743687 −0.371843 0.928295i $$-0.621274\pi$$
−0.371843 + 0.928295i $$0.621274\pi$$
$$230$$ −5.58878 −0.368513
$$231$$ 0 0
$$232$$ −12.2490 −0.804184
$$233$$ −5.23476 −0.342940 −0.171470 0.985189i $$-0.554852\pi$$
−0.171470 + 0.985189i $$0.554852\pi$$
$$234$$ 0 0
$$235$$ 2.53209 0.165175
$$236$$ 0.0672590 0.00437819
$$237$$ 0 0
$$238$$ −0.394238 −0.0255546
$$239$$ 10.3098 0.666885 0.333443 0.942770i $$-0.391790\pi$$
0.333443 + 0.942770i $$0.391790\pi$$
$$240$$ 0 0
$$241$$ 10.7442 0.692096 0.346048 0.938217i $$-0.387523\pi$$
0.346048 + 0.938217i $$0.387523\pi$$
$$242$$ 12.6720 0.814590
$$243$$ 0 0
$$244$$ 15.6459 1.00163
$$245$$ −7.32083 −0.467710
$$246$$ 0 0
$$247$$ 2.69190 0.171282
$$248$$ 21.1620 1.34379
$$249$$ 0 0
$$250$$ −6.38144 −0.403598
$$251$$ −15.0729 −0.951392 −0.475696 0.879610i $$-0.657804\pi$$
−0.475696 + 0.879610i $$0.657804\pi$$
$$252$$ 0 0
$$253$$ 42.3610 2.66321
$$254$$ −10.5474 −0.661800
$$255$$ 0 0
$$256$$ −9.68273 −0.605171
$$257$$ 3.32245 0.207249 0.103624 0.994617i $$-0.466956\pi$$
0.103624 + 0.994617i $$0.466956\pi$$
$$258$$ 0 0
$$259$$ 0.263518 0.0163742
$$260$$ 7.34419 0.455467
$$261$$ 0 0
$$262$$ 9.13610 0.564430
$$263$$ −8.83867 −0.545016 −0.272508 0.962154i $$-0.587853\pi$$
−0.272508 + 0.962154i $$0.587853\pi$$
$$264$$ 0 0
$$265$$ −3.19253 −0.196116
$$266$$ 0.0485565 0.00297719
$$267$$ 0 0
$$268$$ 2.84936 0.174052
$$269$$ −8.09267 −0.493419 −0.246709 0.969090i $$-0.579349\pi$$
−0.246709 + 0.969090i $$0.579349\pi$$
$$270$$ 0 0
$$271$$ −19.0000 −1.15417 −0.577084 0.816685i $$-0.695809\pi$$
−0.577084 + 0.816685i $$0.695809\pi$$
$$272$$ 6.74449 0.408945
$$273$$ 0 0
$$274$$ −13.4233 −0.810933
$$275$$ 21.2006 1.27844
$$276$$ 0 0
$$277$$ −4.22937 −0.254118 −0.127059 0.991895i $$-0.540554\pi$$
−0.127059 + 0.991895i $$0.540554\pi$$
$$278$$ 11.1542 0.668984
$$279$$ 0 0
$$280$$ 0.305407 0.0182516
$$281$$ −7.30212 −0.435608 −0.217804 0.975993i $$-0.569889\pi$$
−0.217804 + 0.975993i $$0.569889\pi$$
$$282$$ 0 0
$$283$$ 16.0205 0.952322 0.476161 0.879358i $$-0.342028\pi$$
0.476161 + 0.879358i $$0.342028\pi$$
$$284$$ 9.97448 0.591877
$$285$$ 0 0
$$286$$ 17.0009 1.00529
$$287$$ −0.911654 −0.0538132
$$288$$ 0 0
$$289$$ 5.83244 0.343085
$$290$$ 3.63441 0.213420
$$291$$ 0 0
$$292$$ −18.7442 −1.09692
$$293$$ −18.0045 −1.05184 −0.525918 0.850535i $$-0.676278\pi$$
−0.525918 + 0.850535i $$0.676278\pi$$
$$294$$ 0 0
$$295$$ −0.0460079 −0.00267868
$$296$$ 5.27866 0.306816
$$297$$ 0 0
$$298$$ 7.55707 0.437769
$$299$$ 35.6585 2.06219
$$300$$ 0 0
$$301$$ 0.157451 0.00907535
$$302$$ −4.65193 −0.267688
$$303$$ 0 0
$$304$$ −0.830689 −0.0476433
$$305$$ −10.7024 −0.612819
$$306$$ 0 0
$$307$$ −13.5107 −0.771098 −0.385549 0.922687i $$-0.625988\pi$$
−0.385549 + 0.922687i $$0.625988\pi$$
$$308$$ −1.00411 −0.0572145
$$309$$ 0 0
$$310$$ −6.27900 −0.356623
$$311$$ −16.4223 −0.931221 −0.465611 0.884990i $$-0.654165\pi$$
−0.465611 + 0.884990i $$0.654165\pi$$
$$312$$ 0 0
$$313$$ −19.5648 −1.10587 −0.552934 0.833225i $$-0.686492\pi$$
−0.552934 + 0.833225i $$0.686492\pi$$
$$314$$ 11.5515 0.651887
$$315$$ 0 0
$$316$$ −1.07604 −0.0605318
$$317$$ 24.3786 1.36924 0.684619 0.728902i $$-0.259969\pi$$
0.684619 + 0.728902i $$0.259969\pi$$
$$318$$ 0 0
$$319$$ −27.5476 −1.54237
$$320$$ 1.19777 0.0669576
$$321$$ 0 0
$$322$$ 0.643208 0.0358446
$$323$$ −2.81217 −0.156473
$$324$$ 0 0
$$325$$ 17.8462 0.989927
$$326$$ 5.75379 0.318673
$$327$$ 0 0
$$328$$ −18.2618 −1.00834
$$329$$ −0.291416 −0.0160663
$$330$$ 0 0
$$331$$ 28.4989 1.56644 0.783220 0.621745i $$-0.213576\pi$$
0.783220 + 0.621745i $$0.213576\pi$$
$$332$$ −10.3824 −0.569806
$$333$$ 0 0
$$334$$ −1.79561 −0.0982512
$$335$$ −1.94907 −0.106489
$$336$$ 0 0
$$337$$ −17.4466 −0.950374 −0.475187 0.879885i $$-0.657620\pi$$
−0.475187 + 0.879885i $$0.657620\pi$$
$$338$$ 5.41847 0.294726
$$339$$ 0 0
$$340$$ −7.67230 −0.416089
$$341$$ 47.5927 2.57729
$$342$$ 0 0
$$343$$ 1.68685 0.0910814
$$344$$ 3.15398 0.170051
$$345$$ 0 0
$$346$$ 8.96316 0.481863
$$347$$ −25.1890 −1.35222 −0.676109 0.736802i $$-0.736335\pi$$
−0.676109 + 0.736802i $$0.736335\pi$$
$$348$$ 0 0
$$349$$ −11.8520 −0.634425 −0.317213 0.948354i $$-0.602747\pi$$
−0.317213 + 0.948354i $$0.602747\pi$$
$$350$$ 0.321909 0.0172068
$$351$$ 0 0
$$352$$ −31.5030 −1.67912
$$353$$ 8.39749 0.446953 0.223477 0.974709i $$-0.428259\pi$$
0.223477 + 0.974709i $$0.428259\pi$$
$$354$$ 0 0
$$355$$ −6.82295 −0.362124
$$356$$ −10.5035 −0.556682
$$357$$ 0 0
$$358$$ −15.1165 −0.798932
$$359$$ −2.64025 −0.139347 −0.0696735 0.997570i $$-0.522196\pi$$
−0.0696735 + 0.997570i $$0.522196\pi$$
$$360$$ 0 0
$$361$$ −18.6536 −0.981770
$$362$$ −1.40852 −0.0740304
$$363$$ 0 0
$$364$$ −0.845237 −0.0443025
$$365$$ 12.8218 0.671124
$$366$$ 0 0
$$367$$ 9.19759 0.480110 0.240055 0.970759i $$-0.422835\pi$$
0.240055 + 0.970759i $$0.422835\pi$$
$$368$$ −11.0038 −0.573612
$$369$$ 0 0
$$370$$ −1.56624 −0.0814248
$$371$$ 0.367426 0.0190758
$$372$$ 0 0
$$373$$ −26.7912 −1.38719 −0.693597 0.720363i $$-0.743975\pi$$
−0.693597 + 0.720363i $$0.743975\pi$$
$$374$$ −17.7605 −0.918373
$$375$$ 0 0
$$376$$ −5.83750 −0.301046
$$377$$ −23.1889 −1.19429
$$378$$ 0 0
$$379$$ −27.1242 −1.39328 −0.696639 0.717422i $$-0.745322\pi$$
−0.696639 + 0.717422i $$0.745322\pi$$
$$380$$ 0.944964 0.0484756
$$381$$ 0 0
$$382$$ −2.70140 −0.138216
$$383$$ 34.4268 1.75913 0.879564 0.475781i $$-0.157834\pi$$
0.879564 + 0.475781i $$0.157834\pi$$
$$384$$ 0 0
$$385$$ 0.686852 0.0350052
$$386$$ −5.77778 −0.294081
$$387$$ 0 0
$$388$$ 13.8307 0.702147
$$389$$ 0.756594 0.0383609 0.0191804 0.999816i $$-0.493894\pi$$
0.0191804 + 0.999816i $$0.493894\pi$$
$$390$$ 0 0
$$391$$ −37.2517 −1.88390
$$392$$ 16.8775 0.852442
$$393$$ 0 0
$$394$$ −1.56986 −0.0790884
$$395$$ 0.736053 0.0370348
$$396$$ 0 0
$$397$$ 7.00774 0.351708 0.175854 0.984416i $$-0.443731\pi$$
0.175854 + 0.984416i $$0.443731\pi$$
$$398$$ −16.0770 −0.805867
$$399$$ 0 0
$$400$$ −5.50711 −0.275356
$$401$$ −9.45080 −0.471950 −0.235975 0.971759i $$-0.575828\pi$$
−0.235975 + 0.971759i $$0.575828\pi$$
$$402$$ 0 0
$$403$$ 40.0624 1.99565
$$404$$ −19.9051 −0.990314
$$405$$ 0 0
$$406$$ −0.418281 −0.0207590
$$407$$ 11.8715 0.588451
$$408$$ 0 0
$$409$$ 5.51754 0.272825 0.136412 0.990652i $$-0.456443\pi$$
0.136412 + 0.990652i $$0.456443\pi$$
$$410$$ 5.41847 0.267599
$$411$$ 0 0
$$412$$ −13.8990 −0.684754
$$413$$ 0.00529501 0.000260550 0
$$414$$ 0 0
$$415$$ 7.10195 0.348621
$$416$$ −26.5185 −1.30018
$$417$$ 0 0
$$418$$ 2.18748 0.106993
$$419$$ −0.466378 −0.0227841 −0.0113920 0.999935i $$-0.503626\pi$$
−0.0113920 + 0.999935i $$0.503626\pi$$
$$420$$ 0 0
$$421$$ −2.41416 −0.117659 −0.0588295 0.998268i $$-0.518737\pi$$
−0.0588295 + 0.998268i $$0.518737\pi$$
$$422$$ −1.14047 −0.0555171
$$423$$ 0 0
$$424$$ 7.36009 0.357438
$$425$$ −18.6435 −0.904342
$$426$$ 0 0
$$427$$ 1.23173 0.0596078
$$428$$ −17.4451 −0.843240
$$429$$ 0 0
$$430$$ −0.935822 −0.0451294
$$431$$ 12.0992 0.582796 0.291398 0.956602i $$-0.405880\pi$$
0.291398 + 0.956602i $$0.405880\pi$$
$$432$$ 0 0
$$433$$ 33.3509 1.60274 0.801371 0.598167i $$-0.204104\pi$$
0.801371 + 0.598167i $$0.204104\pi$$
$$434$$ 0.722645 0.0346881
$$435$$ 0 0
$$436$$ −4.44150 −0.212709
$$437$$ 4.58812 0.219480
$$438$$ 0 0
$$439$$ −9.04458 −0.431674 −0.215837 0.976429i $$-0.569248\pi$$
−0.215837 + 0.976429i $$0.569248\pi$$
$$440$$ 13.7587 0.655918
$$441$$ 0 0
$$442$$ −14.9504 −0.711117
$$443$$ 4.67712 0.222217 0.111108 0.993808i $$-0.464560\pi$$
0.111108 + 0.993808i $$0.464560\pi$$
$$444$$ 0 0
$$445$$ 7.18479 0.340592
$$446$$ −6.70405 −0.317446
$$447$$ 0 0
$$448$$ −0.137851 −0.00651285
$$449$$ 26.7069 1.26037 0.630187 0.776443i $$-0.282978\pi$$
0.630187 + 0.776443i $$0.282978\pi$$
$$450$$ 0 0
$$451$$ −41.0702 −1.93392
$$452$$ 17.7418 0.834503
$$453$$ 0 0
$$454$$ −10.5895 −0.496988
$$455$$ 0.578176 0.0271053
$$456$$ 0 0
$$457$$ −0.657756 −0.0307685 −0.0153843 0.999882i $$-0.504897\pi$$
−0.0153843 + 0.999882i $$0.504897\pi$$
$$458$$ −7.69820 −0.359713
$$459$$ 0 0
$$460$$ 12.5175 0.583633
$$461$$ −20.7037 −0.964268 −0.482134 0.876097i $$-0.660138\pi$$
−0.482134 + 0.876097i $$0.660138\pi$$
$$462$$ 0 0
$$463$$ −24.8239 −1.15366 −0.576832 0.816863i $$-0.695711\pi$$
−0.576832 + 0.816863i $$0.695711\pi$$
$$464$$ 7.15582 0.332200
$$465$$ 0 0
$$466$$ −3.58079 −0.165877
$$467$$ −26.6729 −1.23428 −0.617138 0.786855i $$-0.711708\pi$$
−0.617138 + 0.786855i $$0.711708\pi$$
$$468$$ 0 0
$$469$$ 0.224318 0.0103580
$$470$$ 1.73205 0.0798935
$$471$$ 0 0
$$472$$ 0.106067 0.00488212
$$473$$ 7.09321 0.326146
$$474$$ 0 0
$$475$$ 2.29624 0.105359
$$476$$ 0.883000 0.0404722
$$477$$ 0 0
$$478$$ 7.05232 0.322566
$$479$$ 3.22654 0.147424 0.0737121 0.997280i $$-0.476515\pi$$
0.0737121 + 0.997280i $$0.476515\pi$$
$$480$$ 0 0
$$481$$ 9.99319 0.455650
$$482$$ 7.34948 0.334760
$$483$$ 0 0
$$484$$ −28.3824 −1.29011
$$485$$ −9.46075 −0.429590
$$486$$ 0 0
$$487$$ −7.77930 −0.352514 −0.176257 0.984344i $$-0.556399\pi$$
−0.176257 + 0.984344i $$0.556399\pi$$
$$488$$ 24.6734 1.11691
$$489$$ 0 0
$$490$$ −5.00774 −0.226227
$$491$$ −17.0460 −0.769273 −0.384637 0.923068i $$-0.625673\pi$$
−0.384637 + 0.923068i $$0.625673\pi$$
$$492$$ 0 0
$$493$$ 24.2249 1.09104
$$494$$ 1.84137 0.0828472
$$495$$ 0 0
$$496$$ −12.3628 −0.555105
$$497$$ 0.785248 0.0352232
$$498$$ 0 0
$$499$$ 7.93077 0.355030 0.177515 0.984118i $$-0.443194\pi$$
0.177515 + 0.984118i $$0.443194\pi$$
$$500$$ 14.2929 0.639199
$$501$$ 0 0
$$502$$ −10.3105 −0.460178
$$503$$ −24.9496 −1.11245 −0.556224 0.831032i $$-0.687750\pi$$
−0.556224 + 0.831032i $$0.687750\pi$$
$$504$$ 0 0
$$505$$ 13.6159 0.605898
$$506$$ 28.9766 1.28817
$$507$$ 0 0
$$508$$ 23.6236 1.04813
$$509$$ 40.4690 1.79376 0.896878 0.442278i $$-0.145830\pi$$
0.896878 + 0.442278i $$0.145830\pi$$
$$510$$ 0 0
$$511$$ −1.47565 −0.0652790
$$512$$ 15.0038 0.663080
$$513$$ 0 0
$$514$$ 2.27269 0.100244
$$515$$ 9.50747 0.418949
$$516$$ 0 0
$$517$$ −13.1284 −0.577384
$$518$$ 0.180257 0.00792004
$$519$$ 0 0
$$520$$ 11.5817 0.507892
$$521$$ −25.3674 −1.11137 −0.555684 0.831394i $$-0.687543\pi$$
−0.555684 + 0.831394i $$0.687543\pi$$
$$522$$ 0 0
$$523$$ 12.7219 0.556291 0.278146 0.960539i $$-0.410280\pi$$
0.278146 + 0.960539i $$0.410280\pi$$
$$524$$ −20.4627 −0.893917
$$525$$ 0 0
$$526$$ −6.04601 −0.263618
$$527$$ −41.8523 −1.82311
$$528$$ 0 0
$$529$$ 37.7769 1.64248
$$530$$ −2.18382 −0.0948591
$$531$$ 0 0
$$532$$ −0.108755 −0.00471514
$$533$$ −34.5719 −1.49748
$$534$$ 0 0
$$535$$ 11.9331 0.515914
$$536$$ 4.49341 0.194086
$$537$$ 0 0
$$538$$ −5.53571 −0.238661
$$539$$ 37.9570 1.63492
$$540$$ 0 0
$$541$$ 2.09327 0.0899969 0.0449984 0.998987i $$-0.485672\pi$$
0.0449984 + 0.998987i $$0.485672\pi$$
$$542$$ −12.9968 −0.558259
$$543$$ 0 0
$$544$$ 27.7033 1.18777
$$545$$ 3.03817 0.130141
$$546$$ 0 0
$$547$$ 3.45100 0.147554 0.0737770 0.997275i $$-0.476495\pi$$
0.0737770 + 0.997275i $$0.476495\pi$$
$$548$$ 30.0651 1.28432
$$549$$ 0 0
$$550$$ 14.5021 0.618370
$$551$$ −2.98368 −0.127109
$$552$$ 0 0
$$553$$ −0.0847118 −0.00360231
$$554$$ −2.89306 −0.122914
$$555$$ 0 0
$$556$$ −24.9828 −1.05951
$$557$$ 11.1003 0.470337 0.235168 0.971955i $$-0.424436\pi$$
0.235168 + 0.971955i $$0.424436\pi$$
$$558$$ 0 0
$$559$$ 5.97090 0.252542
$$560$$ −0.178418 −0.00753954
$$561$$ 0 0
$$562$$ −4.99495 −0.210699
$$563$$ −24.3107 −1.02457 −0.512286 0.858815i $$-0.671201\pi$$
−0.512286 + 0.858815i $$0.671201\pi$$
$$564$$ 0 0
$$565$$ −12.1361 −0.510569
$$566$$ 10.9587 0.460628
$$567$$ 0 0
$$568$$ 15.7297 0.660002
$$569$$ −42.7640 −1.79276 −0.896379 0.443288i $$-0.853812\pi$$
−0.896379 + 0.443288i $$0.853812\pi$$
$$570$$ 0 0
$$571$$ 18.0283 0.754460 0.377230 0.926120i $$-0.376877\pi$$
0.377230 + 0.926120i $$0.376877\pi$$
$$572$$ −38.0781 −1.59212
$$573$$ 0 0
$$574$$ −0.623608 −0.0260289
$$575$$ 30.4173 1.26849
$$576$$ 0 0
$$577$$ 25.2763 1.05227 0.526133 0.850402i $$-0.323641\pi$$
0.526133 + 0.850402i $$0.323641\pi$$
$$578$$ 3.98963 0.165947
$$579$$ 0 0
$$580$$ −8.14022 −0.338004
$$581$$ −0.817358 −0.0339097
$$582$$ 0 0
$$583$$ 16.5526 0.685540
$$584$$ −29.5595 −1.22318
$$585$$ 0 0
$$586$$ −12.3158 −0.508763
$$587$$ −28.5653 −1.17902 −0.589508 0.807762i $$-0.700679\pi$$
−0.589508 + 0.807762i $$0.700679\pi$$
$$588$$ 0 0
$$589$$ 5.15476 0.212398
$$590$$ −0.0314712 −0.00129565
$$591$$ 0 0
$$592$$ −3.08378 −0.126743
$$593$$ −20.6009 −0.845977 −0.422989 0.906135i $$-0.639019\pi$$
−0.422989 + 0.906135i $$0.639019\pi$$
$$594$$ 0 0
$$595$$ −0.604007 −0.0247619
$$596$$ −16.9260 −0.693318
$$597$$ 0 0
$$598$$ 24.3919 0.997458
$$599$$ 14.4748 0.591424 0.295712 0.955277i $$-0.404443\pi$$
0.295712 + 0.955277i $$0.404443\pi$$
$$600$$ 0 0
$$601$$ −13.3550 −0.544763 −0.272382 0.962189i $$-0.587811\pi$$
−0.272382 + 0.962189i $$0.587811\pi$$
$$602$$ 0.107703 0.00438965
$$603$$ 0 0
$$604$$ 10.4192 0.423952
$$605$$ 19.4147 0.789319
$$606$$ 0 0
$$607$$ 31.2131 1.26690 0.633450 0.773784i $$-0.281638\pi$$
0.633450 + 0.773784i $$0.281638\pi$$
$$608$$ −3.41209 −0.138378
$$609$$ 0 0
$$610$$ −7.32089 −0.296414
$$611$$ −11.0511 −0.447082
$$612$$ 0 0
$$613$$ 30.0651 1.21432 0.607159 0.794580i $$-0.292309\pi$$
0.607159 + 0.794580i $$0.292309\pi$$
$$614$$ −9.24189 −0.372972
$$615$$ 0 0
$$616$$ −1.58347 −0.0638000
$$617$$ 42.1537 1.69704 0.848521 0.529161i $$-0.177493\pi$$
0.848521 + 0.529161i $$0.177493\pi$$
$$618$$ 0 0
$$619$$ 11.6108 0.466678 0.233339 0.972395i $$-0.425035\pi$$
0.233339 + 0.972395i $$0.425035\pi$$
$$620$$ 14.0635 0.564803
$$621$$ 0 0
$$622$$ −11.2335 −0.450422
$$623$$ −0.826892 −0.0331287
$$624$$ 0 0
$$625$$ 9.73143 0.389257
$$626$$ −13.3831 −0.534897
$$627$$ 0 0
$$628$$ −25.8726 −1.03243
$$629$$ −10.4397 −0.416257
$$630$$ 0 0
$$631$$ −29.3105 −1.16683 −0.583415 0.812174i $$-0.698284\pi$$
−0.583415 + 0.812174i $$0.698284\pi$$
$$632$$ −1.69690 −0.0674991
$$633$$ 0 0
$$634$$ 16.6759 0.662286
$$635$$ −16.1595 −0.641270
$$636$$ 0 0
$$637$$ 31.9513 1.26596
$$638$$ −18.8436 −0.746027
$$639$$ 0 0
$$640$$ −11.3327 −0.447966
$$641$$ 31.0979 1.22829 0.614146 0.789192i $$-0.289501\pi$$
0.614146 + 0.789192i $$0.289501\pi$$
$$642$$ 0 0
$$643$$ 42.0719 1.65915 0.829577 0.558392i $$-0.188581\pi$$
0.829577 + 0.558392i $$0.188581\pi$$
$$644$$ −1.44063 −0.0567690
$$645$$ 0 0
$$646$$ −1.92364 −0.0756845
$$647$$ −4.66717 −0.183485 −0.0917427 0.995783i $$-0.529244\pi$$
−0.0917427 + 0.995783i $$0.529244\pi$$
$$648$$ 0 0
$$649$$ 0.238541 0.00936356
$$650$$ 12.2075 0.478818
$$651$$ 0 0
$$652$$ −12.8871 −0.504699
$$653$$ −3.12413 −0.122257 −0.0611283 0.998130i $$-0.519470\pi$$
−0.0611283 + 0.998130i $$0.519470\pi$$
$$654$$ 0 0
$$655$$ 13.9973 0.546920
$$656$$ 10.6685 0.416534
$$657$$ 0 0
$$658$$ −0.199340 −0.00777110
$$659$$ 37.4292 1.45803 0.729017 0.684495i $$-0.239977\pi$$
0.729017 + 0.684495i $$0.239977\pi$$
$$660$$ 0 0
$$661$$ 26.5003 1.03074 0.515371 0.856967i $$-0.327654\pi$$
0.515371 + 0.856967i $$0.327654\pi$$
$$662$$ 19.4944 0.757671
$$663$$ 0 0
$$664$$ −16.3729 −0.635391
$$665$$ 0.0743929 0.00288483
$$666$$ 0 0
$$667$$ −39.5235 −1.53036
$$668$$ 4.02174 0.155606
$$669$$ 0 0
$$670$$ −1.33325 −0.0515078
$$671$$ 55.4898 2.14216
$$672$$ 0 0
$$673$$ 1.72100 0.0663397 0.0331698 0.999450i $$-0.489440\pi$$
0.0331698 + 0.999450i $$0.489440\pi$$
$$674$$ −11.9341 −0.459686
$$675$$ 0 0
$$676$$ −12.1361 −0.466773
$$677$$ −28.2792 −1.08686 −0.543429 0.839455i $$-0.682874\pi$$
−0.543429 + 0.839455i $$0.682874\pi$$
$$678$$ 0 0
$$679$$ 1.08883 0.0417855
$$680$$ −12.0992 −0.463982
$$681$$ 0 0
$$682$$ 32.5553 1.24661
$$683$$ 24.7139 0.945651 0.472825 0.881156i $$-0.343234\pi$$
0.472825 + 0.881156i $$0.343234\pi$$
$$684$$ 0 0
$$685$$ −20.5657 −0.785777
$$686$$ 1.15387 0.0440551
$$687$$ 0 0
$$688$$ −1.84255 −0.0702465
$$689$$ 13.9336 0.530829
$$690$$ 0 0
$$691$$ −43.7725 −1.66518 −0.832592 0.553887i $$-0.813144\pi$$
−0.832592 + 0.553887i $$0.813144\pi$$
$$692$$ −20.0754 −0.763151
$$693$$ 0 0
$$694$$ −17.2303 −0.654053
$$695$$ 17.0892 0.648231
$$696$$ 0 0
$$697$$ 36.1165 1.36801
$$698$$ −8.10728 −0.306865
$$699$$ 0 0
$$700$$ −0.721000 −0.0272512
$$701$$ 14.6504 0.553338 0.276669 0.960965i $$-0.410769\pi$$
0.276669 + 0.960965i $$0.410769\pi$$
$$702$$ 0 0
$$703$$ 1.28581 0.0484951
$$704$$ −6.21021 −0.234056
$$705$$ 0 0
$$706$$ 5.74422 0.216187
$$707$$ −1.56704 −0.0589346
$$708$$ 0 0
$$709$$ 5.07367 0.190546 0.0952729 0.995451i $$-0.469628\pi$$
0.0952729 + 0.995451i $$0.469628\pi$$
$$710$$ −4.66717 −0.175156
$$711$$ 0 0
$$712$$ −16.5639 −0.620757
$$713$$ 68.2830 2.55722
$$714$$ 0 0
$$715$$ 26.0469 0.974100
$$716$$ 33.8574 1.26531
$$717$$ 0 0
$$718$$ −1.80604 −0.0674007
$$719$$ 5.33717 0.199043 0.0995213 0.995035i $$-0.468269\pi$$
0.0995213 + 0.995035i $$0.468269\pi$$
$$720$$ 0 0
$$721$$ −1.09421 −0.0407504
$$722$$ −12.7598 −0.474872
$$723$$ 0 0
$$724$$ 3.15476 0.117246
$$725$$ −19.7805 −0.734629
$$726$$ 0 0
$$727$$ 37.1198 1.37670 0.688348 0.725380i $$-0.258336\pi$$
0.688348 + 0.725380i $$0.258336\pi$$
$$728$$ −1.33293 −0.0494017
$$729$$ 0 0
$$730$$ 8.77063 0.324616
$$731$$ −6.23767 −0.230708
$$732$$ 0 0
$$733$$ 29.9463 1.10609 0.553045 0.833151i $$-0.313466\pi$$
0.553045 + 0.833151i $$0.313466\pi$$
$$734$$ 6.29152 0.232224
$$735$$ 0 0
$$736$$ −45.1985 −1.66604
$$737$$ 10.1055 0.372243
$$738$$ 0 0
$$739$$ −28.6100 −1.05244 −0.526218 0.850350i $$-0.676390\pi$$
−0.526218 + 0.850350i $$0.676390\pi$$
$$740$$ 3.50800 0.128957
$$741$$ 0 0
$$742$$ 0.251334 0.00922678
$$743$$ −49.6436 −1.82125 −0.910624 0.413237i $$-0.864398\pi$$
−0.910624 + 0.413237i $$0.864398\pi$$
$$744$$ 0 0
$$745$$ 11.5781 0.424189
$$746$$ −18.3262 −0.670971
$$747$$ 0 0
$$748$$ 39.7793 1.45448
$$749$$ −1.37338 −0.0501821
$$750$$ 0 0
$$751$$ −31.6236 −1.15396 −0.576981 0.816758i $$-0.695769\pi$$
−0.576981 + 0.816758i $$0.695769\pi$$
$$752$$ 3.41025 0.124359
$$753$$ 0 0
$$754$$ −15.8621 −0.577665
$$755$$ −7.12716 −0.259384
$$756$$ 0 0
$$757$$ 3.04189 0.110559 0.0552797 0.998471i $$-0.482395\pi$$
0.0552797 + 0.998471i $$0.482395\pi$$
$$758$$ −18.5541 −0.673914
$$759$$ 0 0
$$760$$ 1.49020 0.0540552
$$761$$ −37.9283 −1.37490 −0.687450 0.726232i $$-0.741270\pi$$
−0.687450 + 0.726232i $$0.741270\pi$$
$$762$$ 0 0
$$763$$ −0.349660 −0.0126586
$$764$$ 6.05050 0.218899
$$765$$ 0 0
$$766$$ 23.5493 0.850872
$$767$$ 0.200798 0.00725041
$$768$$ 0 0
$$769$$ −20.3878 −0.735201 −0.367601 0.929984i $$-0.619821\pi$$
−0.367601 + 0.929984i $$0.619821\pi$$
$$770$$ 0.469834 0.0169317
$$771$$ 0 0
$$772$$ 12.9409 0.465752
$$773$$ −24.4664 −0.879994 −0.439997 0.897999i $$-0.645021\pi$$
−0.439997 + 0.897999i $$0.645021\pi$$
$$774$$ 0 0
$$775$$ 34.1739 1.22756
$$776$$ 21.8109 0.782965
$$777$$ 0 0
$$778$$ 0.517541 0.0185547
$$779$$ −4.44831 −0.159377
$$780$$ 0 0
$$781$$ 35.3756 1.26584
$$782$$ −25.4816 −0.911221
$$783$$ 0 0
$$784$$ −9.85978 −0.352135
$$785$$ 17.6979 0.631665
$$786$$ 0 0
$$787$$ 37.5749 1.33940 0.669700 0.742631i $$-0.266422\pi$$
0.669700 + 0.742631i $$0.266422\pi$$
$$788$$ 3.51611 0.125256
$$789$$ 0 0
$$790$$ 0.503490 0.0179134
$$791$$ 1.39673 0.0496622
$$792$$ 0 0
$$793$$ 46.7101 1.65872
$$794$$ 4.79358 0.170118
$$795$$ 0 0
$$796$$ 36.0087 1.27629
$$797$$ −2.30493 −0.0816449 −0.0408224 0.999166i $$-0.512998\pi$$
−0.0408224 + 0.999166i $$0.512998\pi$$
$$798$$ 0 0
$$799$$ 11.5449 0.408429
$$800$$ −22.6207 −0.799762
$$801$$ 0 0
$$802$$ −6.46473 −0.228277
$$803$$ −66.4784 −2.34597
$$804$$ 0 0
$$805$$ 0.985452 0.0347326
$$806$$ 27.4043 0.965276
$$807$$ 0 0
$$808$$ −31.3901 −1.10430
$$809$$ 28.8614 1.01471 0.507356 0.861736i $$-0.330623\pi$$
0.507356 + 0.861736i $$0.330623\pi$$
$$810$$ 0 0
$$811$$ −51.8631 −1.82116 −0.910580 0.413334i $$-0.864364\pi$$
−0.910580 + 0.413334i $$0.864364\pi$$
$$812$$ 0.936851 0.0328770
$$813$$ 0 0
$$814$$ 8.12061 0.284627
$$815$$ 8.81531 0.308787
$$816$$ 0 0
$$817$$ 0.768266 0.0268782
$$818$$ 3.77422 0.131963
$$819$$ 0 0
$$820$$ −12.1361 −0.423811
$$821$$ −18.4411 −0.643598 −0.321799 0.946808i $$-0.604288\pi$$
−0.321799 + 0.946808i $$0.604288\pi$$
$$822$$ 0 0
$$823$$ −34.7665 −1.21188 −0.605942 0.795509i $$-0.707204\pi$$
−0.605942 + 0.795509i $$0.707204\pi$$
$$824$$ −21.9186 −0.763570
$$825$$ 0 0
$$826$$ 0.00362200 0.000126025 0
$$827$$ 32.7773 1.13978 0.569889 0.821722i $$-0.306986\pi$$
0.569889 + 0.821722i $$0.306986\pi$$
$$828$$ 0 0
$$829$$ 5.35267 0.185906 0.0929530 0.995670i $$-0.470369\pi$$
0.0929530 + 0.995670i $$0.470369\pi$$
$$830$$ 4.85802 0.168624
$$831$$ 0 0
$$832$$ −5.22762 −0.181235
$$833$$ −33.3788 −1.15651
$$834$$ 0 0
$$835$$ −2.75103 −0.0952033
$$836$$ −4.89944 −0.169451
$$837$$ 0 0
$$838$$ −0.319022 −0.0110204
$$839$$ −5.43837 −0.187754 −0.0938768 0.995584i $$-0.529926\pi$$
−0.0938768 + 0.995584i $$0.529926\pi$$
$$840$$ 0 0
$$841$$ −3.29767 −0.113713
$$842$$ −1.65138 −0.0569105
$$843$$ 0 0
$$844$$ 2.55438 0.0879253
$$845$$ 8.30158 0.285583
$$846$$ 0 0
$$847$$ −2.23442 −0.0767757
$$848$$ −4.29975 −0.147654
$$849$$ 0 0
$$850$$ −12.7529 −0.437421
$$851$$ 17.0325 0.583868
$$852$$ 0 0
$$853$$ −47.1171 −1.61326 −0.806629 0.591057i $$-0.798711\pi$$
−0.806629 + 0.591057i $$0.798711\pi$$
$$854$$ 0.842556 0.0288317
$$855$$ 0 0
$$856$$ −27.5107 −0.940298
$$857$$ 46.9560 1.60399 0.801993 0.597333i $$-0.203773\pi$$
0.801993 + 0.597333i $$0.203773\pi$$
$$858$$ 0 0
$$859$$ 7.16344 0.244413 0.122207 0.992505i $$-0.461003\pi$$
0.122207 + 0.992505i $$0.461003\pi$$
$$860$$ 2.09602 0.0714737
$$861$$ 0 0
$$862$$ 8.27631 0.281892
$$863$$ 35.4309 1.20608 0.603041 0.797710i $$-0.293955\pi$$
0.603041 + 0.797710i $$0.293955\pi$$
$$864$$ 0 0
$$865$$ 13.7324 0.466914
$$866$$ 22.8134 0.775230
$$867$$ 0 0
$$868$$ −1.61856 −0.0549373
$$869$$ −3.81628 −0.129458
$$870$$ 0 0
$$871$$ 8.50662 0.288236
$$872$$ −7.00421 −0.237193
$$873$$ 0 0
$$874$$ 3.13846 0.106160
$$875$$ 1.12522 0.0380394
$$876$$ 0 0
$$877$$ 8.17293 0.275980 0.137990 0.990434i $$-0.455936\pi$$
0.137990 + 0.990434i $$0.455936\pi$$
$$878$$ −6.18686 −0.208796
$$879$$ 0 0
$$880$$ −8.03777 −0.270953
$$881$$ 33.2307 1.11957 0.559785 0.828638i $$-0.310884\pi$$
0.559785 + 0.828638i $$0.310884\pi$$
$$882$$ 0 0
$$883$$ 33.0479 1.11215 0.556075 0.831132i $$-0.312307\pi$$
0.556075 + 0.831132i $$0.312307\pi$$
$$884$$ 33.4853 1.12623
$$885$$ 0 0
$$886$$ 3.19934 0.107484
$$887$$ 49.7472 1.67035 0.835174 0.549986i $$-0.185367\pi$$
0.835174 + 0.549986i $$0.185367\pi$$
$$888$$ 0 0
$$889$$ 1.85978 0.0623752
$$890$$ 4.91469 0.164741
$$891$$ 0 0
$$892$$ 15.0155 0.502756
$$893$$ −1.42193 −0.0475831
$$894$$ 0 0
$$895$$ −23.1598 −0.774147
$$896$$ 1.30428 0.0435729
$$897$$ 0 0
$$898$$ 18.2686 0.609630
$$899$$ −44.4047 −1.48098
$$900$$ 0 0
$$901$$ −14.5561 −0.484935
$$902$$ −28.0936 −0.935416
$$903$$ 0 0
$$904$$ 27.9786 0.930556
$$905$$ −2.15799 −0.0717338
$$906$$ 0 0
$$907$$ 34.5485 1.14716 0.573582 0.819148i $$-0.305553\pi$$
0.573582 + 0.819148i $$0.305553\pi$$
$$908$$ 23.7179 0.787106
$$909$$ 0 0
$$910$$ 0.395496 0.0131106
$$911$$ 13.6369 0.451811 0.225905 0.974149i $$-0.427466\pi$$
0.225905 + 0.974149i $$0.427466\pi$$
$$912$$ 0 0
$$913$$ −36.8221 −1.21863
$$914$$ −0.449932 −0.0148824
$$915$$ 0 0
$$916$$ 17.2422 0.569697
$$917$$ −1.61094 −0.0531979
$$918$$ 0 0
$$919$$ 32.7701 1.08099 0.540493 0.841348i $$-0.318238\pi$$
0.540493 + 0.841348i $$0.318238\pi$$
$$920$$ 19.7401 0.650810
$$921$$ 0 0
$$922$$ −14.1622 −0.466407
$$923$$ 29.7783 0.980166
$$924$$ 0 0
$$925$$ 8.52435 0.280279
$$926$$ −16.9805 −0.558015
$$927$$ 0 0
$$928$$ 29.3928 0.964866
$$929$$ −5.53147 −0.181482 −0.0907408 0.995875i $$-0.528923\pi$$
−0.0907408 + 0.995875i $$0.528923\pi$$
$$930$$ 0 0
$$931$$ 4.11112 0.134736
$$932$$ 8.02011 0.262708
$$933$$ 0 0
$$934$$ −18.2453 −0.597006
$$935$$ −27.2106 −0.889883
$$936$$ 0 0
$$937$$ −0.994014 −0.0324730 −0.0162365 0.999868i $$-0.505168\pi$$
−0.0162365 + 0.999868i $$0.505168\pi$$
$$938$$ 0.153442 0.00501007
$$939$$ 0 0
$$940$$ −3.87939 −0.126532
$$941$$ −11.4572 −0.373493 −0.186747 0.982408i $$-0.559794\pi$$
−0.186747 + 0.982408i $$0.559794\pi$$
$$942$$ 0 0
$$943$$ −58.9249 −1.91886
$$944$$ −0.0619640 −0.00201676
$$945$$ 0 0
$$946$$ 4.85204 0.157754
$$947$$ 45.6850 1.48456 0.742282 0.670088i $$-0.233743\pi$$
0.742282 + 0.670088i $$0.233743\pi$$
$$948$$ 0 0
$$949$$ −55.9600 −1.81654
$$950$$ 1.57072 0.0509608
$$951$$ 0 0
$$952$$ 1.39248 0.0451306
$$953$$ 14.5053 0.469873 0.234936 0.972011i $$-0.424512\pi$$
0.234936 + 0.972011i $$0.424512\pi$$
$$954$$ 0 0
$$955$$ −4.13878 −0.133928
$$956$$ −15.7955 −0.510864
$$957$$ 0 0
$$958$$ 2.20708 0.0713076
$$959$$ 2.36690 0.0764311
$$960$$ 0 0
$$961$$ 45.7161 1.47471
$$962$$ 6.83575 0.220393
$$963$$ 0 0
$$964$$ −16.4611 −0.530176
$$965$$ −8.85208 −0.284959
$$966$$ 0 0
$$967$$ −25.0368 −0.805130 −0.402565 0.915391i $$-0.631881\pi$$
−0.402565 + 0.915391i $$0.631881\pi$$
$$968$$ −44.7588 −1.43860
$$969$$ 0 0
$$970$$ −6.47153 −0.207788
$$971$$ −27.0907 −0.869383 −0.434692 0.900579i $$-0.643143\pi$$
−0.434692 + 0.900579i $$0.643143\pi$$
$$972$$ 0 0
$$973$$ −1.96679 −0.0630522
$$974$$ −5.32136 −0.170507
$$975$$ 0 0
$$976$$ −14.4142 −0.461386
$$977$$ 2.47340 0.0791310 0.0395655 0.999217i $$-0.487403\pi$$
0.0395655 + 0.999217i $$0.487403\pi$$
$$978$$ 0 0
$$979$$ −37.2517 −1.19057
$$980$$ 11.2162 0.358287
$$981$$ 0 0
$$982$$ −11.6601 −0.372089
$$983$$ −19.1080 −0.609451 −0.304726 0.952440i $$-0.598565\pi$$
−0.304726 + 0.952440i $$0.598565\pi$$
$$984$$ 0 0
$$985$$ −2.40516 −0.0766349
$$986$$ 16.5708 0.527723
$$987$$ 0 0
$$988$$ −4.12424 −0.131209
$$989$$ 10.1769 0.323607
$$990$$ 0 0
$$991$$ −38.3164 −1.21716 −0.608581 0.793492i $$-0.708261\pi$$
−0.608581 + 0.793492i $$0.708261\pi$$
$$992$$ −50.7806 −1.61229
$$993$$ 0 0
$$994$$ 0.537141 0.0170371
$$995$$ −24.6314 −0.780867
$$996$$ 0 0
$$997$$ 41.3746 1.31035 0.655174 0.755478i $$-0.272595\pi$$
0.655174 + 0.755478i $$0.272595\pi$$
$$998$$ 5.42497 0.171724
$$999$$ 0 0
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 729.2.a.c.1.4 yes 6
3.2 odd 2 inner 729.2.a.c.1.3 6
9.2 odd 6 729.2.c.c.244.4 12
9.4 even 3 729.2.c.c.487.3 12
9.5 odd 6 729.2.c.c.487.4 12
9.7 even 3 729.2.c.c.244.3 12
27.2 odd 18 729.2.e.m.568.2 12
27.4 even 9 729.2.e.r.406.2 12
27.5 odd 18 729.2.e.q.649.1 12
27.7 even 9 729.2.e.r.325.2 12
27.11 odd 18 729.2.e.q.82.1 12
27.13 even 9 729.2.e.m.163.1 12
27.14 odd 18 729.2.e.m.163.2 12
27.16 even 9 729.2.e.q.82.2 12
27.20 odd 18 729.2.e.r.325.1 12
27.22 even 9 729.2.e.q.649.2 12
27.23 odd 18 729.2.e.r.406.1 12
27.25 even 9 729.2.e.m.568.1 12

By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.c.1.3 6 3.2 odd 2 inner
729.2.a.c.1.4 yes 6 1.1 even 1 trivial
729.2.c.c.244.3 12 9.7 even 3
729.2.c.c.244.4 12 9.2 odd 6
729.2.c.c.487.3 12 9.4 even 3
729.2.c.c.487.4 12 9.5 odd 6
729.2.e.m.163.1 12 27.13 even 9
729.2.e.m.163.2 12 27.14 odd 18
729.2.e.m.568.1 12 27.25 even 9
729.2.e.m.568.2 12 27.2 odd 18
729.2.e.q.82.1 12 27.11 odd 18
729.2.e.q.82.2 12 27.16 even 9
729.2.e.q.649.1 12 27.5 odd 18
729.2.e.q.649.2 12 27.22 even 9
729.2.e.r.325.1 12 27.20 odd 18
729.2.e.r.325.2 12 27.7 even 9
729.2.e.r.406.1 12 27.23 odd 18
729.2.e.r.406.2 12 27.4 even 9