LMFDB - Embedded newform 81.3.b.a.80.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,3,Mod(80,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("81.80");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	81.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$2.20709014132$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			80.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	81.80
Dual form			81.3.b.a.80.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} +1.00000 q^{4} +3.46410i q^{5} +2.00000 q^{7} +8.66025i q^{8} +O(q^{10})$	\(q+1.73205i q^{2} +1.00000 q^{4} +3.46410i q^{5} +2.00000 q^{7} +8.66025i q^{8} -6.00000 q^{10} +1.73205i q^{11} -4.00000 q^{13} +3.46410i q^{14} -11.0000 q^{16} -15.5885i q^{17} +11.0000 q^{19} +3.46410i q^{20} -3.00000 q^{22} -27.7128i q^{23} +13.0000 q^{25} -6.92820i q^{26} +2.00000 q^{28} -45.0333i q^{29} +32.0000 q^{31} +15.5885i q^{32} +27.0000 q^{34} +6.92820i q^{35} -34.0000 q^{37} +19.0526i q^{38} -30.0000 q^{40} -12.1244i q^{41} -61.0000 q^{43} +1.73205i q^{44} +48.0000 q^{46} +48.4974i q^{47} -45.0000 q^{49} +22.5167i q^{50} -4.00000 q^{52} -6.00000 q^{55} +17.3205i q^{56} +78.0000 q^{58} +50.2295i q^{59} +56.0000 q^{61} +55.4256i q^{62} -71.0000 q^{64} -13.8564i q^{65} -31.0000 q^{67} -15.5885i q^{68} -12.0000 q^{70} +31.1769i q^{71} +65.0000 q^{73} -58.8897i q^{74} +11.0000 q^{76} +3.46410i q^{77} +38.0000 q^{79} -38.1051i q^{80} +21.0000 q^{82} +48.4974i q^{83} +54.0000 q^{85} -105.655i q^{86} -15.0000 q^{88} -124.708i q^{89} -8.00000 q^{91} -27.7128i q^{92} -84.0000 q^{94} +38.1051i q^{95} -115.000 q^{97} -77.9423i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 + 4 * q^7	$ 2 q + 2 q^{4} + 4 q^{7} - 12 q^{10} - 8 q^{13} - 22 q^{16} + 22 q^{19} - 6 q^{22} + 26 q^{25} + 4 q^{28} + 64 q^{31} + 54 q^{34} - 68 q^{37} - 60 q^{40} - 122 q^{43} + 96 q^{46} - 90 q^{49} - 8 q^{52} - 12 q^{55} + 156 q^{58} + 112 q^{61} - 142 q^{64} - 62 q^{67} - 24 q^{70} + 130 q^{73} + 22 q^{76} + 76 q^{79} + 42 q^{82} + 108 q^{85} - 30 q^{88} - 16 q^{91} - 168 q^{94} - 230 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 4 * q^7 - 12 * q^10 - 8 * q^13 - 22 * q^16 + 22 * q^19 - 6 * q^22 + 26 * q^25 + 4 * q^28 + 64 * q^31 + 54 * q^34 - 68 * q^37 - 60 * q^40 - 122 * q^43 + 96 * q^46 - 90 * q^49 - 8 * q^52 - 12 * q^55 + 156 * q^58 + 112 * q^61 - 142 * q^64 - 62 * q^67 - 24 * q^70 + 130 * q^73 + 22 * q^76 + 76 * q^79 + 42 * q^82 + 108 * q^85 - 30 * q^88 - 16 * q^91 - 168 * q^94 - 230 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.73205i	0.866025i	0.901388	+	0.433013i	$0.142549\pi$
$2$	1.73205i	0.866025i	−0.901388	+	0.433013i	$0.857451\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.250000
$5$	3.46410i	0.692820i	0.938083	+	0.346410i	$0.112599\pi$
$5$	3.46410i	0.692820i	−0.938083	+	0.346410i	$0.887401\pi$
$6$	0	0
$7$	2.00000	0.285714	0.142857	−	0.989743i	$-0.454371\pi$
$7$	2.00000	0.285714	0.142857	+	0.989743i	$0.454371\pi$
$8$	8.66025i	1.08253i
$9$	0	0
$10$	−6.00000	−0.600000
$11$	1.73205i	0.157459i	0.996896	+	0.0787296i	$0.0250864\pi$
$11$	1.73205i	0.157459i	−0.996896	+	0.0787296i	$0.974914\pi$
$12$	0	0
$13$	−4.00000	−0.307692	−0.153846	−	0.988095i	$-0.549166\pi$
$13$	−4.00000	−0.307692	−0.153846	+	0.988095i	$0.549166\pi$
$14$	3.46410i	0.247436i
$15$	0	0
$16$	−11.0000	−0.687500
$17$	− 15.5885i	− 0.916968i	−0.888703	−	0.458484i	$-0.848393\pi$
$17$	− 15.5885i	− 0.916968i	0.888703	−	0.458484i	$-0.151607\pi$
$18$	0	0
$19$	11.0000	0.578947	0.289474	−	0.957186i	$-0.406520\pi$
$19$	11.0000	0.578947	0.289474	+	0.957186i	$0.406520\pi$
$20$	3.46410i	0.173205i
$21$	0	0
$22$	−3.00000	−0.136364
$23$	− 27.7128i	− 1.20490i	−0.798155	−	0.602452i	$-0.794190\pi$
$23$	− 27.7128i	− 1.20490i	0.798155	−	0.602452i	$-0.205810\pi$
$24$	0	0
$25$	13.0000	0.520000
$26$	− 6.92820i	− 0.266469i
$27$	0	0
$28$	2.00000	0.0714286
$29$	− 45.0333i	− 1.55287i	−0.630195	−	0.776437i	$-0.717025\pi$
$29$	− 45.0333i	− 1.55287i	0.630195	−	0.776437i	$-0.282975\pi$
$30$	0	0
$31$	32.0000	1.03226	0.516129	−	0.856511i	$-0.327372\pi$
$31$	32.0000	1.03226	0.516129	+	0.856511i	$0.327372\pi$
$32$	15.5885i	0.487139i
$33$	0	0
$34$	27.0000	0.794118
$35$	6.92820i	0.197949i
$36$	0	0
$37$	−34.0000	−0.918919	−0.459459	−	0.888199i	$-0.651957\pi$
$37$	−34.0000	−0.918919	−0.459459	+	0.888199i	$0.651957\pi$
$38$	19.0526i	0.501383i
$39$	0	0
$40$	−30.0000	−0.750000
$41$	− 12.1244i	− 0.295716i	−0.989009	−	0.147858i	$-0.952762\pi$
$41$	− 12.1244i	− 0.295716i	0.989009	−	0.147858i	$-0.0472379\pi$
$42$	0	0
$43$	−61.0000	−1.41860	−0.709302	−	0.704904i	$-0.750990\pi$
$43$	−61.0000	−1.41860	−0.709302	+	0.704904i	$0.750990\pi$
$44$	1.73205i	0.0393648i
$45$	0	0
$46$	48.0000	1.04348
$47$	48.4974i	1.03186i	0.856631	+	0.515930i	$0.172554\pi$
$47$	48.4974i	1.03186i	−0.856631	+	0.515930i	$0.827446\pi$
$48$	0	0
$49$	−45.0000	−0.918367
$50$	22.5167i	0.450333i
$51$	0	0
$52$	−4.00000	−0.0769231
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−6.00000	−0.109091
$56$	17.3205i	0.309295i
$57$	0	0
$58$	78.0000	1.34483
$59$	50.2295i	0.851347i	0.904877	+	0.425674i	$0.139963\pi$
$59$	50.2295i	0.851347i	−0.904877	+	0.425674i	$0.860037\pi$
$60$	0	0
$61$	56.0000	0.918033	0.459016	−	0.888428i	$-0.348202\pi$
$61$	56.0000	0.918033	0.459016	+	0.888428i	$0.348202\pi$
$62$	55.4256i	0.893962i
$63$	0	0
$64$	−71.0000	−1.10938
$65$	− 13.8564i	− 0.213175i
$66$	0	0
$67$	−31.0000	−0.462687	−0.231343	−	0.972872i	$-0.574312\pi$
$67$	−31.0000	−0.462687	−0.231343	+	0.972872i	$0.574312\pi$
$68$	− 15.5885i	− 0.229242i
$69$	0	0
$70$	−12.0000	−0.171429
$71$	31.1769i	0.439111i	0.975600	+	0.219556i	$0.0704608\pi$
$71$	31.1769i	0.439111i	−0.975600	+	0.219556i	$0.929539\pi$
$72$	0	0
$73$	65.0000	0.890411	0.445205	−	0.895428i	$-0.353131\pi$
$73$	65.0000	0.890411	0.445205	+	0.895428i	$0.353131\pi$
$74$	− 58.8897i	− 0.795807i
$75$	0	0
$76$	11.0000	0.144737
$77$	3.46410i	0.0449883i
$78$	0	0
$79$	38.0000	0.481013	0.240506	−	0.970648i	$-0.422687\pi$
$79$	38.0000	0.481013	0.240506	+	0.970648i	$0.422687\pi$
$80$	− 38.1051i	− 0.476314i
$81$	0	0
$82$	21.0000	0.256098
$83$	48.4974i	0.584306i	0.956372	+	0.292153i	$0.0943717\pi$
$83$	48.4974i	0.584306i	−0.956372	+	0.292153i	$0.905628\pi$
$84$	0	0
$85$	54.0000	0.635294
$86$	− 105.655i	− 1.22855i
$87$	0	0
$88$	−15.0000	−0.170455
$89$	− 124.708i	− 1.40121i	−0.713549	−	0.700605i	$-0.752914\pi$
$89$	− 124.708i	− 1.40121i	0.713549	−	0.700605i	$-0.247086\pi$
$90$	0	0
$91$	−8.00000	−0.0879121
$92$	− 27.7128i	− 0.301226i
$93$	0	0
$94$	−84.0000	−0.893617
$95$	38.1051i	0.401107i
$96$	0	0
$97$	−115.000	−1.18557	−0.592784	−	0.805362i	$-0.701971\pi$
$97$	−115.000	−1.18557	−0.592784	+	0.805362i	$0.701971\pi$
$98$	− 77.9423i	− 0.795329i
$99$	0	0
$100$	13.0000	0.130000
$101$	− 45.0333i	− 0.445874i	−0.974833	−	0.222937i	$-0.928436\pi$
$101$	− 45.0333i	− 0.445874i	0.974833	−	0.222937i	$-0.0715645\pi$
$102$	0	0
$103$	−40.0000	−0.388350	−0.194175	−	0.980967i	$-0.562203\pi$
$103$	−40.0000	−0.388350	−0.194175	+	0.980967i	$0.562203\pi$
$104$	− 34.6410i	− 0.333087i
$105$	0	0
$106$	0	0
$107$	140.296i	1.31118i	0.755118	+	0.655589i	$0.227580\pi$
$107$	140.296i	1.31118i	−0.755118	+	0.655589i	$0.772420\pi$
$108$	0	0
$109$	−52.0000	−0.477064	−0.238532	−	0.971135i	$-0.576666\pi$
$109$	−52.0000	−0.477064	−0.238532	+	0.971135i	$0.576666\pi$
$110$	− 10.3923i	− 0.0944755i
$111$	0	0
$112$	−22.0000	−0.196429
$113$	− 90.0666i	− 0.797050i	−0.917157	−	0.398525i	$-0.869522\pi$
$113$	− 90.0666i	− 0.797050i	0.917157	−	0.398525i	$-0.130478\pi$
$114$	0	0
$115$	96.0000	0.834783
$116$	− 45.0333i	− 0.388218i
$117$	0	0
$118$	−87.0000	−0.737288
$119$	− 31.1769i	− 0.261991i
$120$	0	0
$121$	118.000	0.975207
$122$	96.9948i	0.795040i
$123$	0	0
$124$	32.0000	0.258065
$125$	131.636i	1.05309i
$126$	0	0
$127$	−16.0000	−0.125984	−0.0629921	−	0.998014i	$-0.520064\pi$
$127$	−16.0000	−0.125984	−0.0629921	+	0.998014i	$0.520064\pi$
$128$	− 60.6218i	− 0.473608i
$129$	0	0
$130$	24.0000	0.184615
$131$	159.349i	1.21640i	0.793783	+	0.608201i	$0.208109\pi$
$131$	159.349i	1.21640i	−0.793783	+	0.608201i	$0.791891\pi$
$132$	0	0
$133$	22.0000	0.165414
$134$	− 53.6936i	− 0.400698i
$135$	0	0
$136$	135.000	0.992647
$137$	188.794i	1.37806i	0.724735	+	0.689028i	$0.241962\pi$
$137$	188.794i	1.37806i	−0.724735	+	0.689028i	$0.758038\pi$
$138$	0	0
$139$	5.00000	0.0359712	0.0179856	−	0.999838i	$-0.494275\pi$
$139$	5.00000	0.0359712	0.0179856	+	0.999838i	$0.494275\pi$
$140$	6.92820i	0.0494872i
$141$	0	0
$142$	−54.0000	−0.380282
$143$	− 6.92820i	− 0.0484490i
$144$	0	0
$145$	156.000	1.07586
$146$	112.583i	0.771119i
$147$	0	0
$148$	−34.0000	−0.229730
$149$	− 152.420i	− 1.02296i	−0.859296	−	0.511478i	$-0.829098\pi$
$149$	− 152.420i	− 1.02296i	0.859296	−	0.511478i	$-0.170902\pi$
$150$	0	0
$151$	20.0000	0.132450	0.0662252	−	0.997805i	$-0.478904\pi$
$151$	20.0000	0.132450	0.0662252	+	0.997805i	$0.478904\pi$
$152$	95.2628i	0.626729i
$153$	0	0
$154$	−6.00000	−0.0389610
$155$	110.851i	0.715169i
$156$	0	0
$157$	−40.0000	−0.254777	−0.127389	−	0.991853i	$-0.540660\pi$
$157$	−40.0000	−0.254777	−0.127389	+	0.991853i	$0.540660\pi$
$158$	65.8179i	0.416569i
$159$	0	0
$160$	−54.0000	−0.337500
$161$	− 55.4256i	− 0.344259i
$162$	0	0
$163$	−106.000	−0.650307	−0.325153	−	0.945661i	$-0.605416\pi$
$163$	−106.000	−0.650307	−0.325153	+	0.945661i	$0.605416\pi$
$164$	− 12.1244i	− 0.0739290i
$165$	0	0
$166$	−84.0000	−0.506024
$167$	190.526i	1.14087i	0.821342	+	0.570436i	$0.193226\pi$
$167$	190.526i	1.14087i	−0.821342	+	0.570436i	$0.806774\pi$
$168$	0	0
$169$	−153.000	−0.905325
$170$	93.5307i	0.550181i
$171$	0	0
$172$	−61.0000	−0.354651
$173$	− 232.095i	− 1.34159i	−0.741644	−	0.670794i	$-0.765953\pi$
$173$	− 232.095i	− 1.34159i	0.741644	−	0.670794i	$-0.234047\pi$
$174$	0	0
$175$	26.0000	0.148571
$176$	− 19.0526i	− 0.108253i
$177$	0	0
$178$	216.000	1.21348
$179$	− 62.3538i	− 0.348345i	−0.984715	−	0.174173i	$-0.944275\pi$
$179$	− 62.3538i	− 0.348345i	0.984715	−	0.174173i	$-0.0557251\pi$
$180$	0	0
$181$	−232.000	−1.28177	−0.640884	−	0.767638i	$-0.721432\pi$
$181$	−232.000	−1.28177	−0.640884	+	0.767638i	$0.721432\pi$
$182$	− 13.8564i	− 0.0761341i
$183$	0	0
$184$	240.000	1.30435
$185$	− 117.779i	− 0.636646i
$186$	0	0
$187$	27.0000	0.144385
$188$	48.4974i	0.257965i
$189$	0	0
$190$	−66.0000	−0.347368
$191$	− 232.095i	− 1.21516i	−0.794260	−	0.607578i	$-0.792141\pi$
$191$	− 232.095i	− 1.21516i	0.794260	−	0.607578i	$-0.207859\pi$
$192$	0	0
$193$	−265.000	−1.37306	−0.686528	−	0.727103i	$-0.740866\pi$
$193$	−265.000	−1.37306	−0.686528	+	0.727103i	$0.740866\pi$
$194$	− 199.186i	− 1.02673i
$195$	0	0
$196$	−45.0000	−0.229592
$197$	− 124.708i	− 0.633034i	−0.948587	−	0.316517i	$-0.897487\pi$
$197$	− 124.708i	− 0.633034i	0.948587	−	0.316517i	$-0.102513\pi$
$198$	0	0
$199$	290.000	1.45729	0.728643	−	0.684893i	$-0.240151\pi$
$199$	290.000	1.45729	0.728643	+	0.684893i	$0.240151\pi$
$200$	112.583i	0.562917i
$201$	0	0
$202$	78.0000	0.386139
$203$	− 90.0666i	− 0.443678i
$204$	0	0
$205$	42.0000	0.204878
$206$	− 69.2820i	− 0.336321i
$207$	0	0
$208$	44.0000	0.211538
$209$	19.0526i	0.0911606i
$210$	0	0
$211$	−94.0000	−0.445498	−0.222749	−	0.974876i	$-0.571503\pi$
$211$	−94.0000	−0.445498	−0.222749	+	0.974876i	$0.571503\pi$
$212$	0	0
$213$	0	0
$214$	−243.000	−1.13551
$215$	− 211.310i	− 0.982838i
$216$	0	0
$217$	64.0000	0.294931
$218$	− 90.0666i	− 0.413150i
$219$	0	0
$220$	−6.00000	−0.0272727
$221$	62.3538i	0.282144i
$222$	0	0
$223$	−52.0000	−0.233184	−0.116592	−	0.993180i	$-0.537197\pi$
$223$	−52.0000	−0.233184	−0.116592	+	0.993180i	$0.537197\pi$
$224$	31.1769i	0.139183i
$225$	0	0
$226$	156.000	0.690265
$227$	188.794i	0.831690i	0.909436	+	0.415845i	$0.136514\pi$
$227$	188.794i	0.831690i	−0.909436	+	0.415845i	$0.863486\pi$
$228$	0	0
$229$	266.000	1.16157	0.580786	−	0.814056i	$-0.302745\pi$
$229$	266.000	1.16157	0.580786	+	0.814056i	$0.302745\pi$
$230$	166.277i	0.722943i
$231$	0	0
$232$	390.000	1.68103
$233$	− 202.650i	− 0.869742i	−0.900493	−	0.434871i	$-0.856794\pi$
$233$	− 202.650i	− 0.869742i	0.900493	−	0.434871i	$-0.143206\pi$
$234$	0	0
$235$	−168.000	−0.714894
$236$	50.2295i	0.212837i
$237$	0	0
$238$	54.0000	0.226891
$239$	− 401.836i	− 1.68132i	−0.541562	−	0.840661i	$-0.682167\pi$
$239$	− 401.836i	− 1.68132i	0.541562	−	0.840661i	$-0.317833\pi$
$240$	0	0
$241$	119.000	0.493776	0.246888	−	0.969044i	$-0.420592\pi$
$241$	119.000	0.493776	0.246888	+	0.969044i	$0.420592\pi$
$242$	204.382i	0.844554i
$243$	0	0
$244$	56.0000	0.229508
$245$	− 155.885i	− 0.636264i
$246$	0	0
$247$	−44.0000	−0.178138
$248$	277.128i	1.11745i
$249$	0	0
$250$	−228.000	−0.912000
$251$	389.711i	1.55264i	0.630342	+	0.776318i	$0.282915\pi$
$251$	389.711i	1.55264i	−0.630342	+	0.776318i	$0.717085\pi$
$252$	0	0
$253$	48.0000	0.189723
$254$	− 27.7128i	− 0.109106i
$255$	0	0
$256$	−179.000	−0.699219
$257$	174.937i	0.680689i	0.940301	+	0.340345i	$0.110544\pi$
$257$	174.937i	0.680689i	−0.940301	+	0.340345i	$0.889456\pi$
$258$	0	0
$259$	−68.0000	−0.262548
$260$	− 13.8564i	− 0.0532939i
$261$	0	0
$262$	−276.000	−1.05344
$263$	− 45.0333i	− 0.171229i	−0.996328	−	0.0856147i	$-0.972715\pi$
$263$	− 45.0333i	− 0.171229i	0.996328	−	0.0856147i	$-0.0272854\pi$
$264$	0	0
$265$	0	0
$266$	38.1051i	0.143252i
$267$	0	0
$268$	−31.0000	−0.115672
$269$	187.061i	0.695396i	0.937607	+	0.347698i	$0.113037\pi$
$269$	187.061i	0.695396i	−0.937607	+	0.347698i	$0.886963\pi$
$270$	0	0
$271$	−268.000	−0.988930	−0.494465	−	0.869198i	$-0.664636\pi$
$271$	−268.000	−0.988930	−0.494465	+	0.869198i	$0.664636\pi$
$272$	171.473i	0.630416i
$273$	0	0
$274$	−327.000	−1.19343
$275$	22.5167i	0.0818788i
$276$	0	0
$277$	56.0000	0.202166	0.101083	−	0.994878i	$-0.467769\pi$
$277$	56.0000	0.202166	0.101083	+	0.994878i	$0.467769\pi$
$278$	8.66025i	0.0311520i
$279$	0	0
$280$	−60.0000	−0.214286
$281$	48.4974i	0.172589i	0.996270	+	0.0862943i	$0.0275025\pi$
$281$	48.4974i	0.172589i	−0.996270	+	0.0862943i	$0.972497\pi$
$282$	0	0
$283$	374.000	1.32155	0.660777	−	0.750582i	$-0.270227\pi$
$283$	374.000	1.32155	0.660777	+	0.750582i	$0.270227\pi$
$284$	31.1769i	0.109778i
$285$	0	0
$286$	12.0000	0.0419580
$287$	− 24.2487i	− 0.0844903i
$288$	0	0
$289$	46.0000	0.159170
$290$	270.200i	0.931724i
$291$	0	0
$292$	65.0000	0.222603
$293$	252.879i	0.863070i	0.902096	+	0.431535i	$0.142028\pi$
$293$	252.879i	0.863070i	−0.902096	+	0.431535i	$0.857972\pi$
$294$	0	0
$295$	−174.000	−0.589831
$296$	− 294.449i	− 0.994759i
$297$	0	0
$298$	264.000	0.885906
$299$	110.851i	0.370740i
$300$	0	0
$301$	−122.000	−0.405316
$302$	34.6410i	0.114705i
$303$	0	0
$304$	−121.000	−0.398026
$305$	193.990i	0.636032i
$306$	0	0
$307$	533.000	1.73616	0.868078	−	0.496428i	$-0.165355\pi$
$307$	533.000	1.73616	0.868078	+	0.496428i	$0.165355\pi$
$308$	3.46410i	0.0112471i
$309$	0	0
$310$	−192.000	−0.619355
$311$	− 245.951i	− 0.790840i	−0.918500	−	0.395420i	$-0.870599\pi$
$311$	− 245.951i	− 0.790840i	0.918500	−	0.395420i	$-0.129401\pi$
$312$	0	0
$313$	155.000	0.495208	0.247604	−	0.968861i	$-0.420357\pi$
$313$	155.000	0.495208	0.247604	+	0.968861i	$0.420357\pi$
$314$	− 69.2820i	− 0.220643i
$315$	0	0
$316$	38.0000	0.120253
$317$	48.4974i	0.152989i	0.997070	+	0.0764944i	$0.0243727\pi$
$317$	48.4974i	0.152989i	−0.997070	+	0.0764944i	$0.975627\pi$
$318$	0	0
$319$	78.0000	0.244514
$320$	− 245.951i	− 0.768598i
$321$	0	0
$322$	96.0000	0.298137
$323$	− 171.473i	− 0.530876i
$324$	0	0
$325$	−52.0000	−0.160000
$326$	− 183.597i	− 0.563182i
$327$	0	0
$328$	105.000	0.320122
$329$	96.9948i	0.294817i
$330$	0	0
$331$	2.00000	0.00604230	0.00302115	−	0.999995i	$-0.499038\pi$
$331$	2.00000	0.00604230	0.00302115	+	0.999995i	$0.499038\pi$
$332$	48.4974i	0.146077i
$333$	0	0
$334$	−330.000	−0.988024
$335$	− 107.387i	− 0.320559i
$336$	0	0
$337$	77.0000	0.228487	0.114243	−	0.993453i	$-0.463556\pi$
$337$	77.0000	0.228487	0.114243	+	0.993453i	$0.463556\pi$
$338$	− 265.004i	− 0.784035i
$339$	0	0
$340$	54.0000	0.158824
$341$	55.4256i	0.162538i
$342$	0	0
$343$	−188.000	−0.548105
$344$	− 528.275i	− 1.53568i
$345$	0	0
$346$	402.000	1.16185
$347$	112.583i	0.324448i	0.986754	+	0.162224i	$0.0518666\pi$
$347$	112.583i	0.324448i	−0.986754	+	0.162224i	$0.948133\pi$
$348$	0	0
$349$	416.000	1.19198	0.595989	−	0.802993i	$-0.296760\pi$
$349$	416.000	1.19198	0.595989	+	0.802993i	$0.296760\pi$
$350$	45.0333i	0.128667i
$351$	0	0
$352$	−27.0000	−0.0767045
$353$	1.73205i	0.00490666i	0.999997	+	0.00245333i	$0.000780920\pi$
$353$	1.73205i	0.00490666i	−0.999997	+	0.00245333i	$0.999219\pi$
$354$	0	0
$355$	−108.000	−0.304225
$356$	− 124.708i	− 0.350302i
$357$	0	0
$358$	108.000	0.301676
$359$	− 592.361i	− 1.65003i	−0.565110	−	0.825016i	$-0.691166\pi$
$359$	− 592.361i	− 1.65003i	0.565110	−	0.825016i	$-0.308834\pi$
$360$	0	0
$361$	−240.000	−0.664820
$362$	− 401.836i	− 1.11004i
$363$	0	0
$364$	−8.00000	−0.0219780
$365$	225.167i	0.616895i
$366$	0	0
$367$	−358.000	−0.975477	−0.487738	−	0.872990i	$-0.662178\pi$
$367$	−358.000	−0.975477	−0.487738	+	0.872990i	$0.662178\pi$
$368$	304.841i	0.828372i
$369$	0	0
$370$	204.000	0.551351
$371$	0	0
$372$	0	0
$373$	−580.000	−1.55496	−0.777480	−	0.628908i	$-0.783502\pi$
$373$	−580.000	−1.55496	−0.777480	+	0.628908i	$0.783502\pi$
$374$	46.7654i	0.125041i
$375$	0	0
$376$	−420.000	−1.11702
$377$	180.133i	0.477807i
$378$	0	0
$379$	83.0000	0.218997	0.109499	−	0.993987i	$-0.465075\pi$
$379$	83.0000	0.218997	0.109499	+	0.993987i	$0.465075\pi$
$380$	38.1051i	0.100277i
$381$	0	0
$382$	402.000	1.05236
$383$	− 557.720i	− 1.45619i	−0.685477	−	0.728094i	$-0.740406\pi$
$383$	− 557.720i	− 1.45619i	0.685477	−	0.728094i	$-0.259594\pi$
$384$	0	0
$385$	−12.0000	−0.0311688
$386$	− 458.993i	− 1.18910i
$387$	0	0
$388$	−115.000	−0.296392
$389$	516.151i	1.32687i	0.748235	+	0.663433i	$0.230901\pi$
$389$	516.151i	1.32687i	−0.748235	+	0.663433i	$0.769099\pi$
$390$	0	0
$391$	−432.000	−1.10486
$392$	− 389.711i	− 0.994162i
$393$	0	0
$394$	216.000	0.548223
$395$	131.636i	0.333255i
$396$	0	0
$397$	362.000	0.911839	0.455919	−	0.890021i	$-0.349311\pi$
$397$	362.000	0.911839	0.455919	+	0.890021i	$0.349311\pi$
$398$	502.295i	1.26205i
$399$	0	0
$400$	−143.000	−0.357500
$401$	393.176i	0.980488i	0.871585	+	0.490244i	$0.163092\pi$
$401$	393.176i	0.980488i	−0.871585	+	0.490244i	$0.836908\pi$
$402$	0	0
$403$	−128.000	−0.317618
$404$	− 45.0333i	− 0.111469i
$405$	0	0
$406$	156.000	0.384236
$407$	− 58.8897i	− 0.144692i
$408$	0	0
$409$	221.000	0.540342	0.270171	−	0.962812i	$-0.412920\pi$
$409$	221.000	0.540342	0.270171	+	0.962812i	$0.412920\pi$
$410$	72.7461i	0.177430i
$411$	0	0
$412$	−40.0000	−0.0970874
$413$	100.459i	0.243242i
$414$	0	0
$415$	−168.000	−0.404819
$416$	− 62.3538i	− 0.149889i
$417$	0	0
$418$	−33.0000	−0.0789474
$419$	782.887i	1.86847i	0.356664	+	0.934233i	$0.383914\pi$
$419$	782.887i	1.86847i	−0.356664	+	0.934233i	$0.616086\pi$
$420$	0	0
$421$	−682.000	−1.61995	−0.809976	−	0.586463i	$-0.800520\pi$
$421$	−682.000	−1.61995	−0.809976	+	0.586463i	$0.800520\pi$
$422$	− 162.813i	− 0.385812i
$423$	0	0
$424$	0	0
$425$	− 202.650i	− 0.476823i
$426$	0	0
$427$	112.000	0.262295
$428$	140.296i	0.327795i
$429$	0	0
$430$	366.000	0.851163
$431$	280.592i	0.651026i	0.945538	+	0.325513i	$0.105537\pi$
$431$	280.592i	0.651026i	−0.945538	+	0.325513i	$0.894463\pi$
$432$	0	0
$433$	−295.000	−0.681293	−0.340647	−	0.940191i	$-0.610646\pi$
$433$	−295.000	−0.681293	−0.340647	+	0.940191i	$0.610646\pi$
$434$	110.851i	0.255418i
$435$	0	0
$436$	−52.0000	−0.119266
$437$	− 304.841i	− 0.697577i
$438$	0	0
$439$	812.000	1.84966	0.924829	−	0.380383i	$-0.124208\pi$
$439$	812.000	1.84966	0.924829	+	0.380383i	$0.124208\pi$
$440$	− 51.9615i	− 0.118094i
$441$	0	0
$442$	−108.000	−0.244344
$443$	− 91.7987i	− 0.207221i	−0.994618	−	0.103610i	$-0.966961\pi$
$443$	− 91.7987i	− 0.207221i	0.994618	−	0.103610i	$-0.0330395\pi$
$444$	0	0
$445$	432.000	0.970787
$446$	− 90.0666i	− 0.201943i
$447$	0	0
$448$	−142.000	−0.316964
$449$	639.127i	1.42344i	0.702461	+	0.711722i	$0.252085\pi$
$449$	639.127i	1.42344i	−0.702461	+	0.711722i	$0.747915\pi$
$450$	0	0
$451$	21.0000	0.0465632
$452$	− 90.0666i	− 0.199262i
$453$	0	0
$454$	−327.000	−0.720264
$455$	− 27.7128i	− 0.0609073i
$456$	0	0
$457$	65.0000	0.142232	0.0711160	−	0.997468i	$-0.477344\pi$
$457$	65.0000	0.142232	0.0711160	+	0.997468i	$0.477344\pi$
$458$	460.726i	1.00595i
$459$	0	0
$460$	96.0000	0.208696
$461$	796.743i	1.72829i	0.503240	+	0.864147i	$0.332141\pi$
$461$	796.743i	1.72829i	−0.503240	+	0.864147i	$0.667859\pi$
$462$	0	0
$463$	734.000	1.58531	0.792657	−	0.609668i	$-0.208697\pi$
$463$	734.000	1.58531	0.792657	+	0.609668i	$0.208697\pi$
$464$	495.367i	1.06760i
$465$	0	0
$466$	351.000	0.753219
$467$	202.650i	0.433940i	0.976178	+	0.216970i	$0.0696174\pi$
$467$	202.650i	0.433940i	−0.976178	+	0.216970i	$0.930383\pi$
$468$	0	0
$469$	−62.0000	−0.132196
$470$	− 290.985i	− 0.619116i
$471$	0	0
$472$	−435.000	−0.921610
$473$	− 105.655i	− 0.223372i
$474$	0	0
$475$	143.000	0.301053
$476$	− 31.1769i	− 0.0654977i
$477$	0	0
$478$	696.000	1.45607
$479$	− 606.218i	− 1.26559i	−0.774319	−	0.632795i	$-0.781908\pi$
$479$	− 606.218i	− 1.26559i	0.774319	−	0.632795i	$-0.218092\pi$
$480$	0	0
$481$	136.000	0.282744
$482$	206.114i	0.427623i
$483$	0	0
$484$	118.000	0.243802
$485$	− 398.372i	− 0.821385i
$486$	0	0
$487$	−106.000	−0.217659	−0.108830	−	0.994060i	$-0.534710\pi$
$487$	−106.000	−0.217659	−0.108830	+	0.994060i	$0.534710\pi$
$488$	484.974i	0.993800i
$489$	0	0
$490$	270.000	0.551020
$491$	− 230.363i	− 0.469171i	−0.972096	−	0.234585i	$-0.924627\pi$
$491$	− 230.363i	− 0.469171i	0.972096	−	0.234585i	$-0.0753732\pi$
$492$	0	0
$493$	−702.000	−1.42394
$494$	− 76.2102i	− 0.154272i
$495$	0	0
$496$	−352.000	−0.709677
$497$	62.3538i	0.125460i
$498$	0	0
$499$	−787.000	−1.57715	−0.788577	−	0.614936i	$-0.789182\pi$
$499$	−787.000	−1.57715	−0.788577	+	0.614936i	$0.789182\pi$
$500$	131.636i	0.263272i
$501$	0	0
$502$	−675.000	−1.34462
$503$	− 623.538i	− 1.23964i	−0.784745	−	0.619819i	$-0.787206\pi$
$503$	− 623.538i	− 1.23964i	0.784745	−	0.619819i	$-0.212794\pi$
$504$	0	0
$505$	156.000	0.308911
$506$	83.1384i	0.164305i
$507$	0	0
$508$	−16.0000	−0.0314961
$509$	− 214.774i	− 0.421953i	−0.977491	−	0.210977i	$-0.932336\pi$
$509$	− 214.774i	− 0.421953i	0.977491	−	0.210977i	$-0.0676644\pi$
$510$	0	0
$511$	130.000	0.254403
$512$	− 552.524i	− 1.07915i
$513$	0	0
$514$	−303.000	−0.589494
$515$	− 138.564i	− 0.269056i
$516$	0	0
$517$	−84.0000	−0.162476
$518$	− 117.779i	− 0.227373i
$519$	0	0
$520$	120.000	0.230769
$521$	202.650i	0.388963i	0.980906	+	0.194482i	$0.0623025\pi$
$521$	202.650i	0.388963i	−0.980906	+	0.194482i	$0.937698\pi$
$522$	0	0
$523$	−250.000	−0.478011	−0.239006	−	0.971018i	$-0.576821\pi$
$523$	−250.000	−0.478011	−0.239006	+	0.971018i	$0.576821\pi$
$524$	159.349i	0.304101i
$525$	0	0
$526$	78.0000	0.148289
$527$	− 498.831i	− 0.946548i
$528$	0	0
$529$	−239.000	−0.451796
$530$	0	0
$531$	0	0
$532$	22.0000	0.0413534
$533$	48.4974i	0.0909895i
$534$	0	0
$535$	−486.000	−0.908411
$536$	− 268.468i	− 0.500873i
$537$	0	0
$538$	−324.000	−0.602230
$539$	− 77.9423i	− 0.144605i
$540$	0	0
$541$	650.000	1.20148	0.600739	−	0.799445i	$-0.294873\pi$
$541$	650.000	1.20148	0.600739	+	0.799445i	$0.294873\pi$
$542$	− 464.190i	− 0.856438i
$543$	0	0
$544$	243.000	0.446691
$545$	− 180.133i	− 0.330520i
$546$	0	0
$547$	623.000	1.13894	0.569470	−	0.822012i	$-0.307149\pi$
$547$	623.000	1.13894	0.569470	+	0.822012i	$0.307149\pi$
$548$	188.794i	0.344514i
$549$	0	0
$550$	−39.0000	−0.0709091
$551$	− 495.367i	− 0.899032i
$552$	0	0
$553$	76.0000	0.137432
$554$	96.9948i	0.175081i
$555$	0	0
$556$	5.00000	0.00899281
$557$	− 530.008i	− 0.951540i	−0.879570	−	0.475770i	$-0.842170\pi$
$557$	− 530.008i	− 0.951540i	0.879570	−	0.475770i	$-0.157830\pi$
$558$	0	0
$559$	244.000	0.436494
$560$	− 76.2102i	− 0.136090i
$561$	0	0
$562$	−84.0000	−0.149466
$563$	112.583i	0.199970i	0.994989	+	0.0999852i	$0.0318795\pi$
$563$	112.583i	0.199970i	−0.994989	+	0.0999852i	$0.968120\pi$
$564$	0	0
$565$	312.000	0.552212
$566$	647.787i	1.14450i
$567$	0	0
$568$	−270.000	−0.475352
$569$	− 652.983i	− 1.14760i	−0.818996	−	0.573799i	$-0.805469\pi$
$569$	− 652.983i	− 1.14760i	0.818996	−	0.573799i	$-0.194531\pi$
$570$	0	0
$571$	545.000	0.954466	0.477233	−	0.878777i	$-0.341640\pi$
$571$	545.000	0.954466	0.477233	+	0.878777i	$0.341640\pi$
$572$	− 6.92820i	− 0.0121122i
$573$	0	0
$574$	42.0000	0.0731707
$575$	− 360.267i	− 0.626551i
$576$	0	0
$577$	−871.000	−1.50953	−0.754766	−	0.655994i	$-0.772250\pi$
$577$	−871.000	−1.50953	−0.754766	+	0.655994i	$0.772250\pi$
$578$	79.6743i	0.137845i
$579$	0	0
$580$	156.000	0.268966
$581$	96.9948i	0.166945i
$582$	0	0
$583$	0	0
$584$	562.917i	0.963898i
$585$	0	0
$586$	−438.000	−0.747440
$587$	1.73205i	0.00295068i	0.999999	+	0.00147534i	$0.000469616\pi$
$587$	1.73205i	0.00295068i	−0.999999	+	0.00147534i	$0.999530\pi$
$588$	0	0
$589$	352.000	0.597623
$590$	− 301.377i	− 0.510808i
$591$	0	0
$592$	374.000	0.631757
$593$	− 187.061i	− 0.315449i	−0.987483	−	0.157725i	$-0.949584\pi$
$593$	− 187.061i	− 0.315449i	0.987483	−	0.157725i	$-0.0504159\pi$
$594$	0	0
$595$	108.000	0.181513
$596$	− 152.420i	− 0.255739i
$597$	0	0
$598$	−192.000	−0.321070
$599$	564.649i	0.942652i	0.881959	+	0.471326i	$0.156224\pi$
$599$	564.649i	0.942652i	−0.881959	+	0.471326i	$0.843776\pi$
$600$	0	0
$601$	461.000	0.767055	0.383527	−	0.923529i	$-0.374709\pi$
$601$	461.000	0.767055	0.383527	+	0.923529i	$0.374709\pi$
$602$	− 211.310i	− 0.351014i
$603$	0	0
$604$	20.0000	0.0331126
$605$	408.764i	0.675643i
$606$	0	0
$607$	−112.000	−0.184514	−0.0922570	−	0.995735i	$-0.529408\pi$
$607$	−112.000	−0.184514	−0.0922570	+	0.995735i	$0.529408\pi$
$608$	171.473i	0.282028i
$609$	0	0
$610$	−336.000	−0.550820
$611$	− 193.990i	− 0.317495i
$612$	0	0
$613$	902.000	1.47145	0.735726	−	0.677279i	$-0.236841\pi$
$613$	902.000	1.47145	0.735726	+	0.677279i	$0.236841\pi$
$614$	923.183i	1.50356i
$615$	0	0
$616$	−30.0000	−0.0487013
$617$	− 355.070i	− 0.575479i	−0.957709	−	0.287739i	$-0.907096\pi$
$617$	− 355.070i	− 0.575479i	0.957709	−	0.287739i	$-0.0929037\pi$
$618$	0	0
$619$	−799.000	−1.29079	−0.645396	−	0.763848i	$-0.723308\pi$
$619$	−799.000	−1.29079	−0.645396	+	0.763848i	$0.723308\pi$
$620$	110.851i	0.178792i
$621$	0	0
$622$	426.000	0.684887
$623$	− 249.415i	− 0.400346i
$624$	0	0
$625$	−131.000	−0.209600
$626$	268.468i	0.428862i
$627$	0	0
$628$	−40.0000	−0.0636943
$629$	530.008i	0.842619i
$630$	0	0
$631$	830.000	1.31537	0.657686	−	0.753292i	$-0.271535\pi$
$631$	830.000	1.31537	0.657686	+	0.753292i	$0.271535\pi$
$632$	329.090i	0.520711i
$633$	0	0
$634$	−84.0000	−0.132492
$635$	− 55.4256i	− 0.0872845i
$636$	0	0
$637$	180.000	0.282575
$638$	135.100i	0.211755i
$639$	0	0
$640$	210.000	0.328125
$641$	375.855i	0.586357i	0.956058	+	0.293179i	$0.0947131\pi$
$641$	375.855i	0.586357i	−0.956058	+	0.293179i	$0.905287\pi$
$642$	0	0
$643$	−13.0000	−0.0202177	−0.0101089	−	0.999949i	$-0.503218\pi$
$643$	−13.0000	−0.0202177	−0.0101089	+	0.999949i	$0.503218\pi$
$644$	− 55.4256i	− 0.0860646i
$645$	0	0
$646$	297.000	0.459752
$647$	− 467.654i	− 0.722803i	−0.932410	−	0.361402i	$-0.882298\pi$
$647$	− 467.654i	− 0.722803i	0.932410	−	0.361402i	$-0.117702\pi$
$648$	0	0
$649$	−87.0000	−0.134052
$650$	− 90.0666i	− 0.138564i
$651$	0	0
$652$	−106.000	−0.162577
$653$	377.587i	0.578234i	0.957294	+	0.289117i	$0.0933617\pi$
$653$	377.587i	0.578234i	−0.957294	+	0.289117i	$0.906638\pi$
$654$	0	0
$655$	−552.000	−0.842748
$656$	133.368i	0.203305i
$657$	0	0
$658$	−168.000	−0.255319
$659$	983.805i	1.49288i	0.665455	+	0.746438i	$0.268237\pi$
$659$	983.805i	1.49288i	−0.665455	+	0.746438i	$0.731763\pi$
$660$	0	0
$661$	−382.000	−0.577912	−0.288956	−	0.957342i	$-0.593308\pi$
$661$	−382.000	−0.577912	−0.288956	+	0.957342i	$0.593308\pi$
$662$	3.46410i	0.00523278i
$663$	0	0
$664$	−420.000	−0.632530
$665$	76.2102i	0.114602i
$666$	0	0
$667$	−1248.00	−1.87106
$668$	190.526i	0.285218i
$669$	0	0
$670$	186.000	0.277612
$671$	96.9948i	0.144553i
$672$	0	0
$673$	578.000	0.858841	0.429421	−	0.903105i	$-0.358718\pi$
$673$	578.000	0.858841	0.429421	+	0.903105i	$0.358718\pi$
$674$	133.368i	0.197875i
$675$	0	0
$676$	−153.000	−0.226331
$677$	− 699.749i	− 1.03360i	−0.856106	−	0.516801i	$-0.827123\pi$
$677$	− 699.749i	− 1.03360i	0.856106	−	0.516801i	$-0.172877\pi$
$678$	0	0
$679$	−230.000	−0.338733
$680$	467.654i	0.687726i
$681$	0	0
$682$	−96.0000	−0.140762
$683$	1044.43i	1.52918i	0.644520	+	0.764588i	$0.277057\pi$
$683$	1044.43i	1.52918i	−0.644520	+	0.764588i	$0.722943\pi$
$684$	0	0
$685$	−654.000	−0.954745
$686$	− 325.626i	− 0.474673i
$687$	0	0
$688$	671.000	0.975291
$689$	0	0
$690$	0	0
$691$	182.000	0.263386	0.131693	−	0.991291i	$-0.457959\pi$
$691$	182.000	0.263386	0.131693	+	0.991291i	$0.457959\pi$
$692$	− 232.095i	− 0.335397i
$693$	0	0
$694$	−195.000	−0.280980
$695$	17.3205i	0.0249216i
$696$	0	0
$697$	−189.000	−0.271162
$698$	720.533i	1.03228i
$699$	0	0
$700$	26.0000	0.0371429
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−374.000	−0.532006
$704$	− 122.976i	− 0.174681i
$705$	0	0
$706$	−3.00000	−0.00424929
$707$	− 90.0666i	− 0.127393i
$708$	0	0
$709$	−700.000	−0.987306	−0.493653	−	0.869659i	$-0.664339\pi$
$709$	−700.000	−0.987306	−0.493653	+	0.869659i	$0.664339\pi$
$710$	− 187.061i	− 0.263467i
$711$	0	0
$712$	1080.00	1.51685
$713$	− 886.810i	− 1.24377i
$714$	0	0
$715$	24.0000	0.0335664
$716$	− 62.3538i	− 0.0870864i
$717$	0	0
$718$	1026.00	1.42897
$719$	592.361i	0.823868i	0.911214	+	0.411934i	$0.135147\pi$
$719$	592.361i	0.823868i	−0.911214	+	0.411934i	$0.864853\pi$
$720$	0	0
$721$	−80.0000	−0.110957
$722$	− 415.692i	− 0.575751i
$723$	0	0
$724$	−232.000	−0.320442
$725$	− 585.433i	− 0.807494i
$726$	0	0
$727$	−664.000	−0.913343	−0.456671	−	0.889636i	$-0.650958\pi$
$727$	−664.000	−0.913343	−0.456671	+	0.889636i	$0.650958\pi$
$728$	− 69.2820i	− 0.0951676i
$729$	0	0
$730$	−390.000	−0.534247
$731$	950.896i	1.30082i
$732$	0	0
$733$	−670.000	−0.914052	−0.457026	−	0.889453i	$-0.651085\pi$
$733$	−670.000	−0.914052	−0.457026	+	0.889453i	$0.651085\pi$
$734$	− 620.074i	− 0.844788i
$735$	0	0
$736$	432.000	0.586957
$737$	− 53.6936i	− 0.0728542i
$738$	0	0
$739$	317.000	0.428958	0.214479	−	0.976729i	$-0.431195\pi$
$739$	317.000	0.428958	0.214479	+	0.976729i	$0.431195\pi$
$740$	− 117.779i	− 0.159161i
$741$	0	0
$742$	0	0
$743$	− 620.074i	− 0.834555i	−0.908779	−	0.417277i	$-0.862984\pi$
$743$	− 620.074i	− 0.834555i	0.908779	−	0.417277i	$-0.137016\pi$
$744$	0	0
$745$	528.000	0.708725
$746$	− 1004.59i	− 1.34663i
$747$	0	0
$748$	27.0000	0.0360963
$749$	280.592i	0.374622i
$750$	0	0
$751$	1310.00	1.74434	0.872170	−	0.489202i	$-0.162712\pi$
$751$	1310.00	1.74434	0.872170	+	0.489202i	$0.162712\pi$
$752$	− 533.472i	− 0.709404i
$753$	0	0
$754$	−312.000	−0.413793
$755$	69.2820i	0.0917643i
$756$	0	0
$757$	218.000	0.287979	0.143989	−	0.989579i	$-0.454007\pi$
$757$	218.000	0.287979	0.143989	+	0.989579i	$0.454007\pi$
$758$	143.760i	0.189657i
$759$	0	0
$760$	−330.000	−0.434211
$761$	658.179i	0.864887i	0.901661	+	0.432444i	$0.142349\pi$
$761$	658.179i	0.864887i	−0.901661	+	0.432444i	$0.857651\pi$
$762$	0	0
$763$	−104.000	−0.136304
$764$	− 232.095i	− 0.303789i
$765$	0	0
$766$	966.000	1.26110
$767$	− 200.918i	− 0.261953i
$768$	0	0
$769$	1022.00	1.32900	0.664499	−	0.747289i	$-0.268645\pi$
$769$	1022.00	1.32900	0.664499	+	0.747289i	$0.268645\pi$
$770$	− 20.7846i	− 0.0269930i
$771$	0	0
$772$	−265.000	−0.343264
$773$	− 1184.72i	− 1.53263i	−0.642465	−	0.766315i	$-0.722088\pi$
$773$	− 1184.72i	− 1.53263i	0.642465	−	0.766315i	$-0.277912\pi$
$774$	0	0
$775$	416.000	0.536774
$776$	− 995.929i	− 1.28341i
$777$	0	0
$778$	−894.000	−1.14910
$779$	− 133.368i	− 0.171204i
$780$	0	0
$781$	−54.0000	−0.0691421
$782$	− 748.246i	− 0.956836i
$783$	0	0
$784$	495.000	0.631378
$785$	− 138.564i	− 0.176515i
$786$	0	0
$787$	−130.000	−0.165184	−0.0825921	−	0.996583i	$-0.526320\pi$
$787$	−130.000	−0.165184	−0.0825921	+	0.996583i	$0.526320\pi$
$788$	− 124.708i	− 0.158258i
$789$	0	0
$790$	−228.000	−0.288608
$791$	− 180.133i	− 0.227729i
$792$	0	0
$793$	−224.000	−0.282472
$794$	627.002i	0.789676i
$795$	0	0
$796$	290.000	0.364322
$797$	315.233i	0.395525i	0.980250	+	0.197762i	$0.0633674\pi$
$797$	315.233i	0.395525i	−0.980250	+	0.197762i	$0.936633\pi$
$798$	0	0
$799$	756.000	0.946183
$800$	202.650i	0.253312i
$801$	0	0
$802$	−681.000	−0.849127
$803$	112.583i	0.140203i
$804$	0	0
$805$	192.000	0.238509
$806$	− 221.703i	− 0.275065i
$807$	0	0
$808$	390.000	0.482673
$809$	140.296i	0.173419i	0.996234	+	0.0867096i	$0.0276352\pi$
$809$	140.296i	0.173419i	−0.996234	+	0.0867096i	$0.972365\pi$
$810$	0	0
$811$	299.000	0.368681	0.184340	−	0.982862i	$-0.440985\pi$
$811$	299.000	0.368681	0.184340	+	0.982862i	$0.440985\pi$
$812$	− 90.0666i	− 0.110920i
$813$	0	0
$814$	102.000	0.125307
$815$	− 367.195i	− 0.450546i
$816$	0	0
$817$	−671.000	−0.821297
$818$	382.783i	0.467950i
$819$	0	0
$820$	42.0000	0.0512195
$821$	− 606.218i	− 0.738390i	−0.929352	−	0.369195i	$-0.879634\pi$
$821$	− 606.218i	− 0.738390i	0.929352	−	0.369195i	$-0.120366\pi$
$822$	0	0
$823$	−814.000	−0.989064	−0.494532	−	0.869159i	$-0.664661\pi$
$823$	−814.000	−0.989064	−0.494532	+	0.869159i	$0.664661\pi$
$824$	− 346.410i	− 0.420401i
$825$	0	0
$826$	−174.000	−0.210654
$827$	1434.14i	1.73415i	0.498182	+	0.867073i	$0.334001\pi$
$827$	1434.14i	1.73415i	−0.498182	+	0.867073i	$0.665999\pi$
$828$	0	0
$829$	−718.000	−0.866104	−0.433052	−	0.901369i	$-0.642563\pi$
$829$	−718.000	−0.866104	−0.433052	+	0.901369i	$0.642563\pi$
$830$	− 290.985i	− 0.350584i
$831$	0	0
$832$	284.000	0.341346
$833$	701.481i	0.842114i
$834$	0	0
$835$	−660.000	−0.790419
$836$	19.0526i	0.0227901i
$837$	0	0
$838$	−1356.00	−1.61814
$839$	796.743i	0.949635i	0.880084	+	0.474817i	$0.157486\pi$
$839$	796.743i	0.949635i	−0.880084	+	0.474817i	$0.842514\pi$
$840$	0	0
$841$	−1187.00	−1.41141
$842$	− 1181.26i	− 1.40292i
$843$	0	0
$844$	−94.0000	−0.111374
$845$	− 530.008i	− 0.627228i
$846$	0	0
$847$	236.000	0.278630
$848$	0	0
$849$	0	0
$850$	351.000	0.412941
$851$	942.236i	1.10721i
$852$	0	0
$853$	1424.00	1.66940	0.834701	−	0.550703i	$-0.185640\pi$
$853$	1424.00	1.66940	0.834701	+	0.550703i	$0.185640\pi$
$854$	193.990i	0.227154i
$855$	0	0
$856$	−1215.00	−1.41939
$857$	− 699.749i	− 0.816509i	−0.912868	−	0.408255i	$-0.866138\pi$
$857$	− 699.749i	− 0.816509i	0.912868	−	0.408255i	$-0.133862\pi$
$858$	0	0
$859$	311.000	0.362049	0.181024	−	0.983479i	$-0.442059\pi$
$859$	311.000	0.362049	0.181024	+	0.983479i	$0.442059\pi$
$860$	− 211.310i	− 0.245710i
$861$	0	0
$862$	−486.000	−0.563805
$863$	1028.84i	1.19216i	0.802923	+	0.596082i	$0.203277\pi$
$863$	1028.84i	1.19216i	−0.802923	+	0.596082i	$0.796723\pi$
$864$	0	0
$865$	804.000	0.929480
$866$	− 510.955i	− 0.590017i
$867$	0	0
$868$	64.0000	0.0737327
$869$	65.8179i	0.0757399i
$870$	0	0
$871$	124.000	0.142365
$872$	− 450.333i	− 0.516437i
$873$	0	0
$874$	528.000	0.604119
$875$	263.272i	0.300882i
$876$	0	0
$877$	104.000	0.118586	0.0592930	−	0.998241i	$-0.481115\pi$
$877$	104.000	0.118586	0.0592930	+	0.998241i	$0.481115\pi$
$878$	1406.43i	1.60185i
$879$	0	0
$880$	66.0000	0.0750000
$881$	− 62.3538i	− 0.0707762i	−0.999374	−	0.0353881i	$-0.988733\pi$
$881$	− 62.3538i	− 0.0707762i	0.999374	−	0.0353881i	$-0.0112667\pi$
$882$	0	0
$883$	119.000	0.134768	0.0673839	−	0.997727i	$-0.478535\pi$
$883$	119.000	0.134768	0.0673839	+	0.997727i	$0.478535\pi$
$884$	62.3538i	0.0705360i
$885$	0	0
$886$	159.000	0.179458
$887$	1188.19i	1.33956i	0.742561	+	0.669778i	$0.233611\pi$
$887$	1188.19i	1.33956i	−0.742561	+	0.669778i	$0.766389\pi$
$888$	0	0
$889$	−32.0000	−0.0359955
$890$	748.246i	0.840726i
$891$	0	0
$892$	−52.0000	−0.0582960
$893$	533.472i	0.597393i
$894$	0	0
$895$	216.000	0.241341
$896$	− 121.244i	− 0.135316i
$897$	0	0
$898$	−1107.00	−1.23274
$899$	− 1441.07i	− 1.60297i
$900$	0	0
$901$	0	0
$902$	36.3731i	0.0403249i
$903$	0	0
$904$	780.000	0.862832
$905$	− 803.672i	− 0.888035i
$906$	0	0
$907$	695.000	0.766262	0.383131	−	0.923694i	$-0.374846\pi$
$907$	695.000	0.766262	0.383131	+	0.923694i	$0.374846\pi$
$908$	188.794i	0.207922i
$909$	0	0
$910$	48.0000	0.0527473
$911$	1732.05i	1.90126i	0.310322	+	0.950632i	$0.399563\pi$
$911$	1732.05i	1.90126i	−0.310322	+	0.950632i	$0.600437\pi$
$912$	0	0
$913$	−84.0000	−0.0920044
$914$	112.583i	0.123176i
$915$	0	0
$916$	266.000	0.290393
$917$	318.697i	0.347543i
$918$	0	0
$919$	56.0000	0.0609358	0.0304679	−	0.999536i	$-0.490300\pi$
$919$	56.0000	0.0609358	0.0304679	+	0.999536i	$0.490300\pi$
$920$	831.384i	0.903679i
$921$	0	0
$922$	−1380.00	−1.49675
$923$	− 124.708i	− 0.135111i
$924$	0	0
$925$	−442.000	−0.477838
$926$	1271.33i	1.37292i
$927$	0	0
$928$	702.000	0.756466
$929$	796.743i	0.857635i	0.903391	+	0.428818i	$0.141070\pi$
$929$	796.743i	0.857635i	−0.903391	+	0.428818i	$0.858930\pi$
$930$	0	0
$931$	−495.000	−0.531686
$932$	− 202.650i	− 0.217436i
$933$	0	0
$934$	−351.000	−0.375803
$935$	93.5307i	0.100033i
$936$	0	0
$937$	470.000	0.501601	0.250800	−	0.968039i	$-0.419306\pi$
$937$	470.000	0.501601	0.250800	+	0.968039i	$0.419306\pi$
$938$	− 107.387i	− 0.114485i
$939$	0	0
$940$	−168.000	−0.178723
$941$	− 401.836i	− 0.427031i	−0.976940	−	0.213515i	$-0.931509\pi$
$941$	− 401.836i	− 0.427031i	0.976940	−	0.213515i	$-0.0684913\pi$
$942$	0	0
$943$	−336.000	−0.356310
$944$	− 552.524i	− 0.585301i
$945$	0	0
$946$	183.000	0.193446
$947$	1.73205i	0.00182899i	1.00000		0.000914494i	$0.000291092\pi$
$947$	1.73205i	0.00182899i	−1.00000		0.000914494i	$0.999709\pi$
$948$	0	0
$949$	−260.000	−0.273973
$950$	247.683i	0.260719i
$951$	0	0
$952$	270.000	0.283613
$953$	− 826.188i	− 0.866934i	−0.901169	−	0.433467i	$-0.857290\pi$
$953$	− 826.188i	− 0.866934i	0.901169	−	0.433467i	$-0.142710\pi$
$954$	0	0
$955$	804.000	0.841885
$956$	− 401.836i	− 0.420330i
$957$	0	0
$958$	1050.00	1.09603
$959$	377.587i	0.393730i
$960$	0	0
$961$	63.0000	0.0655567
$962$	235.559i	0.244864i
$963$	0	0
$964$	119.000	0.123444
$965$	− 917.987i	− 0.951282i
$966$	0	0
$967$	1202.00	1.24302	0.621510	−	0.783406i	$-0.286520\pi$
$967$	1202.00	1.24302	0.621510	+	0.783406i	$0.286520\pi$
$968$	1021.91i	1.05569i
$969$	0	0
$970$	690.000	0.711340
$971$	187.061i	0.192648i	0.995350	+	0.0963241i	$0.0307085\pi$
$971$	187.061i	0.192648i	−0.995350	+	0.0963241i	$0.969291\pi$
$972$	0	0
$973$	10.0000	0.0102775
$974$	− 183.597i	− 0.188498i
$975$	0	0
$976$	−616.000	−0.631148
$977$	− 417.424i	− 0.427251i	−0.976916	−	0.213626i	$-0.931473\pi$
$977$	− 417.424i	− 0.427251i	0.976916	−	0.213626i	$-0.0685272\pi$
$978$	0	0
$979$	216.000	0.220633
$980$	− 155.885i	− 0.159066i
$981$	0	0
$982$	399.000	0.406314
$983$	− 1167.40i	− 1.18759i	−0.804616	−	0.593796i	$-0.797629\pi$
$983$	− 1167.40i	− 1.18759i	0.804616	−	0.593796i	$-0.202371\pi$
$984$	0	0
$985$	432.000	0.438579
$986$	− 1215.90i	− 1.23316i
$987$	0	0
$988$	−44.0000	−0.0445344
$989$	1690.48i	1.70928i
$990$	0	0
$991$	−1420.00	−1.43290	−0.716448	−	0.697640i	$-0.754233\pi$
$991$	−1420.00	−1.43290	−0.716448	+	0.697640i	$0.754233\pi$
$992$	498.831i	0.502853i
$993$	0	0
$994$	−108.000	−0.108652
$995$	1004.59i	1.00964i
$996$	0	0
$997$	524.000	0.525577	0.262788	−	0.964853i	$-0.415358\pi$
$997$	524.000	0.525577	0.262788	+	0.964853i	$0.415358\pi$
$998$	− 1363.12i	− 1.36586i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.3.b.a.80.2		2
3.2	odd	2	inner	81.3.b.a.80.1		2
4.3	odd	2		1296.3.e.a.161.2		2
9.2	odd	6		9.3.d.a.5.1	yes	2
9.4	even	3		9.3.d.a.2.1	✓	2
9.5	odd	6		27.3.d.a.8.1		2
9.7	even	3		27.3.d.a.17.1		2
12.11	even	2		1296.3.e.a.161.1		2
36.7	odd	6		432.3.q.a.17.1		2
36.11	even	6		144.3.q.a.113.1		2
36.23	even	6		432.3.q.a.305.1		2
36.31	odd	6		144.3.q.a.65.1		2
45.2	even	12		225.3.i.a.149.1		4
45.4	even	6		225.3.j.a.101.1		2
45.7	odd	12		675.3.i.a.449.2		4
45.13	odd	12		225.3.i.a.74.1		4
45.14	odd	6		675.3.j.a.251.1		2
45.22	odd	12		225.3.i.a.74.2		4
45.23	even	12		675.3.i.a.224.2		4
45.29	odd	6		225.3.j.a.176.1		2
45.32	even	12		675.3.i.a.224.1		4
45.34	even	6		675.3.j.a.476.1		2
45.38	even	12		225.3.i.a.149.2		4
45.43	odd	12		675.3.i.a.449.1		4
63.2	odd	6		441.3.n.b.410.1		2
63.4	even	3		441.3.n.b.128.1		2
63.11	odd	6		441.3.j.a.275.1		2
63.13	odd	6		441.3.r.a.344.1		2
63.20	even	6		441.3.r.a.50.1		2
63.31	odd	6		441.3.n.a.128.1		2
63.38	even	6		441.3.j.b.275.1		2
63.40	odd	6		441.3.j.b.263.1		2
63.47	even	6		441.3.n.a.410.1		2
63.58	even	3		441.3.j.a.263.1		2
72.5	odd	6		1728.3.q.a.1601.1		2
72.11	even	6		576.3.q.a.257.1		2
72.13	even	6		576.3.q.b.65.1		2
72.29	odd	6		576.3.q.b.257.1		2
72.43	odd	6		1728.3.q.b.449.1		2
72.59	even	6		1728.3.q.b.1601.1		2
72.61	even	6		1728.3.q.a.449.1		2
72.67	odd	6		576.3.q.a.65.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.3.d.a.2.1	✓	2	9.4	even	3
9.3.d.a.5.1	yes	2	9.2	odd	6
27.3.d.a.8.1		2	9.5	odd	6
27.3.d.a.17.1		2	9.7	even	3
81.3.b.a.80.1		2	3.2	odd	2	inner
81.3.b.a.80.2		2	1.1	even	1	trivial
144.3.q.a.65.1		2	36.31	odd	6
144.3.q.a.113.1		2	36.11	even	6
225.3.i.a.74.1		4	45.13	odd	12
225.3.i.a.74.2		4	45.22	odd	12
225.3.i.a.149.1		4	45.2	even	12
225.3.i.a.149.2		4	45.38	even	12
225.3.j.a.101.1		2	45.4	even	6
225.3.j.a.176.1		2	45.29	odd	6
432.3.q.a.17.1		2	36.7	odd	6
432.3.q.a.305.1		2	36.23	even	6
441.3.j.a.263.1		2	63.58	even	3
441.3.j.a.275.1		2	63.11	odd	6
441.3.j.b.263.1		2	63.40	odd	6
441.3.j.b.275.1		2	63.38	even	6
441.3.n.a.128.1		2	63.31	odd	6
441.3.n.a.410.1		2	63.47	even	6
441.3.n.b.128.1		2	63.4	even	3
441.3.n.b.410.1		2	63.2	odd	6
441.3.r.a.50.1		2	63.20	even	6
441.3.r.a.344.1		2	63.13	odd	6
576.3.q.a.65.1		2	72.67	odd	6
576.3.q.a.257.1		2	72.11	even	6
576.3.q.b.65.1		2	72.13	even	6
576.3.q.b.257.1		2	72.29	odd	6
675.3.i.a.224.1		4	45.32	even	12
675.3.i.a.224.2		4	45.23	even	12
675.3.i.a.449.1		4	45.43	odd	12
675.3.i.a.449.2		4	45.7	odd	12
675.3.j.a.251.1		2	45.14	odd	6
675.3.j.a.476.1		2	45.34	even	6
1296.3.e.a.161.1		2	12.11	even	2
1296.3.e.a.161.2		2	4.3	odd	2
1728.3.q.a.449.1		2	72.61	even	6
1728.3.q.a.1601.1		2	72.5	odd	6
1728.3.q.b.449.1		2	72.43	odd	6
1728.3.q.b.1601.1		2	72.59	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	81.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} +1.00000 q^{4} +3.46410i q^{5} +2.00000 q^{7} +8.66025i q^{8} +O(q^{10})\)	\(q+1.73205i q^{2} +1.00000 q^{4} +3.46410i q^{5} +2.00000 q^{7} +8.66025i q^{8} -6.00000 q^{10} +1.73205i q^{11} -4.00000 q^{13} +3.46410i q^{14} -11.0000 q^{16} -15.5885i q^{17} +11.0000 q^{19} +3.46410i q^{20} -3.00000 q^{22} -27.7128i q^{23} +13.0000 q^{25} -6.92820i q^{26} +2.00000 q^{28} -45.0333i q^{29} +32.0000 q^{31} +15.5885i q^{32} +27.0000 q^{34} +6.92820i q^{35} -34.0000 q^{37} +19.0526i q^{38} -30.0000 q^{40} -12.1244i q^{41} -61.0000 q^{43} +1.73205i q^{44} +48.0000 q^{46} +48.4974i q^{47} -45.0000 q^{49} +22.5167i q^{50} -4.00000 q^{52} -6.00000 q^{55} +17.3205i q^{56} +78.0000 q^{58} +50.2295i q^{59} +56.0000 q^{61} +55.4256i q^{62} -71.0000 q^{64} -13.8564i q^{65} -31.0000 q^{67} -15.5885i q^{68} -12.0000 q^{70} +31.1769i q^{71} +65.0000 q^{73} -58.8897i q^{74} +11.0000 q^{76} +3.46410i q^{77} +38.0000 q^{79} -38.1051i q^{80} +21.0000 q^{82} +48.4974i q^{83} +54.0000 q^{85} -105.655i q^{86} -15.0000 q^{88} -124.708i q^{89} -8.00000 q^{91} -27.7128i q^{92} -84.0000 q^{94} +38.1051i q^{95} -115.000 q^{97} -77.9423i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 + 4 * q^7	\( 2 q + 2 q^{4} + 4 q^{7} - 12 q^{10} - 8 q^{13} - 22 q^{16} + 22 q^{19} - 6 q^{22} + 26 q^{25} + 4 q^{28} + 64 q^{31} + 54 q^{34} - 68 q^{37} - 60 q^{40} - 122 q^{43} + 96 q^{46} - 90 q^{49} - 8 q^{52} - 12 q^{55} + 156 q^{58} + 112 q^{61} - 142 q^{64} - 62 q^{67} - 24 q^{70} + 130 q^{73} + 22 q^{76} + 76 q^{79} + 42 q^{82} + 108 q^{85} - 30 q^{88} - 16 q^{91} - 168 q^{94} - 230 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 4 * q^7 - 12 * q^10 - 8 * q^13 - 22 * q^16 + 22 * q^19 - 6 * q^22 + 26 * q^25 + 4 * q^28 + 64 * q^31 + 54 * q^34 - 68 * q^37 - 60 * q^40 - 122 * q^43 + 96 * q^46 - 90 * q^49 - 8 * q^52 - 12 * q^55 + 156 * q^58 + 112 * q^61 - 142 * q^64 - 62 * q^67 - 24 * q^70 + 130 * q^73 + 22 * q^76 + 76 * q^79 + 42 * q^82 + 108 * q^85 - 30 * q^88 - 16 * q^91 - 168 * q^94 - 230 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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