LMFDB - Embedded newform 216.3.e.a.161.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,3,Mod(161,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("216.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	216.e (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:	$5.88557371018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	216.161
Dual form			216.3.e.a.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{5} -3.00000 q^{7} +O(q^{10})$	$q+2.82843i q^{5} -3.00000 q^{7} +19.7990i q^{11} +7.00000 q^{13} +14.1421i q^{17} -19.0000 q^{19} +2.82843i q^{23} +17.0000 q^{25} +50.9117i q^{29} -10.0000 q^{31} -8.48528i q^{35} +63.0000 q^{37} -50.9117i q^{41} -50.0000 q^{43} +42.4264i q^{47} -40.0000 q^{49} -73.5391i q^{53} -56.0000 q^{55} -98.9949i q^{59} +79.0000 q^{61} +19.7990i q^{65} +77.0000 q^{67} -79.1960i q^{71} -17.0000 q^{73} -59.3970i q^{77} -11.0000 q^{79} +39.5980i q^{83} -40.0000 q^{85} -42.4264i q^{89} -21.0000 q^{91} -53.7401i q^{95} -97.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{7}+O(q^{10}) $ 2 * q - 6 * q^7	$ 2 q - 6 q^{7} + 14 q^{13} - 38 q^{19} + 34 q^{25} - 20 q^{31} + 126 q^{37} - 100 q^{43} - 80 q^{49} - 112 q^{55} + 158 q^{61} + 154 q^{67} - 34 q^{73} - 22 q^{79} - 80 q^{85} - 42 q^{91} - 194 q^{97}+O(q^{100}) $ 2 * q - 6 * q^7 + 14 * q^13 - 38 * q^19 + 34 * q^25 - 20 * q^31 + 126 * q^37 - 100 * q^43 - 80 * q^49 - 112 * q^55 + 158 * q^61 + 154 * q^67 - 34 * q^73 - 22 * q^79 - 80 * q^85 - 42 * q^91 - 194 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.82843i	0.565685i	0.959166	+	0.282843i	$0.0912774\pi$
$5$	2.82843i	0.565685i	−0.959166	+	0.282843i	$0.908723\pi$
$6$	0	0
$7$	−3.00000	−0.428571	−0.214286	−	0.976771i	$-0.568742\pi$
$7$	−3.00000	−0.428571	−0.214286	+	0.976771i	$0.568742\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	19.7990i	1.79991i	0.435985	+	0.899954i	$0.356400\pi$
$11$	19.7990i	1.79991i	−0.435985	+	0.899954i	$0.643600\pi$
$12$	0	0
$13$	7.00000	0.538462	0.269231	−	0.963076i	$-0.413231\pi$
$13$	7.00000	0.538462	0.269231	+	0.963076i	$0.413231\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.1421i	0.831890i	0.909390	+	0.415945i	$0.136549\pi$
$17$	14.1421i	0.831890i	−0.909390	+	0.415945i	$0.863451\pi$
$18$	0	0
$19$	−19.0000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$19$	−19.0000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843i	0.122975i	0.998108	+	0.0614875i	$0.0195844\pi$
$23$	2.82843i	0.122975i	−0.998108	+	0.0614875i	$0.980416\pi$
$24$	0	0
$25$	17.0000	0.680000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	50.9117i	1.75558i	0.479050	+	0.877788i	$0.340981\pi$
$29$	50.9117i	1.75558i	−0.479050	+	0.877788i	$0.659019\pi$
$30$	0	0
$31$	−10.0000	−0.322581	−0.161290	−	0.986907i	$-0.551566\pi$
$31$	−10.0000	−0.322581	−0.161290	+	0.986907i	$0.551566\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 8.48528i	− 0.242437i
$36$	0	0
$37$	63.0000	1.70270	0.851351	−	0.524596i	$-0.175784\pi$
$37$	63.0000	1.70270	0.851351	+	0.524596i	$0.175784\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 50.9117i	− 1.24175i	−0.783910	−	0.620874i	$-0.786778\pi$
$41$	− 50.9117i	− 1.24175i	0.783910	−	0.620874i	$-0.213222\pi$
$42$	0	0
$43$	−50.0000	−1.16279	−0.581395	−	0.813621i	$-0.697493\pi$
$43$	−50.0000	−1.16279	−0.581395	+	0.813621i	$0.697493\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	42.4264i	0.902690i	0.892350	+	0.451345i	$0.149056\pi$
$47$	42.4264i	0.902690i	−0.892350	+	0.451345i	$0.850944\pi$
$48$	0	0
$49$	−40.0000	−0.816327
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 73.5391i	− 1.38753i	−0.720201	−	0.693765i	$-0.755951\pi$
$53$	− 73.5391i	− 1.38753i	0.720201	−	0.693765i	$-0.244049\pi$
$54$	0	0
$55$	−56.0000	−1.01818
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 98.9949i	− 1.67788i	−0.544224	−	0.838940i	$-0.683176\pi$
$59$	− 98.9949i	− 1.67788i	0.544224	−	0.838940i	$-0.316824\pi$
$60$	0	0
$61$	79.0000	1.29508	0.647541	−	0.762031i	$-0.275797\pi$
$61$	79.0000	1.29508	0.647541	+	0.762031i	$0.275797\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	19.7990i	0.304600i
$66$	0	0
$67$	77.0000	1.14925	0.574627	−	0.818416i	$-0.305147\pi$
$67$	77.0000	1.14925	0.574627	+	0.818416i	$0.305147\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 79.1960i	− 1.11544i	−0.830030	−	0.557718i	$-0.811677\pi$
$71$	− 79.1960i	− 1.11544i	0.830030	−	0.557718i	$-0.188323\pi$
$72$	0	0
$73$	−17.0000	−0.232877	−0.116438	−	0.993198i	$-0.537148\pi$
$73$	−17.0000	−0.232877	−0.116438	+	0.993198i	$0.537148\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 59.3970i	− 0.771389i
$78$	0	0
$79$	−11.0000	−0.139241	−0.0696203	−	0.997574i	$-0.522179\pi$
$79$	−11.0000	−0.139241	−0.0696203	+	0.997574i	$0.522179\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	39.5980i	0.477084i	0.971132	+	0.238542i	$0.0766695\pi$
$83$	39.5980i	0.477084i	−0.971132	+	0.238542i	$0.923331\pi$
$84$	0	0
$85$	−40.0000	−0.470588
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 42.4264i	− 0.476701i	−0.971179	−	0.238351i	$-0.923393\pi$
$89$	− 42.4264i	− 0.476701i	0.971179	−	0.238351i	$-0.0766067\pi$
$90$	0	0
$91$	−21.0000	−0.230769
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 53.7401i	− 0.565685i
$96$	0	0
$97$	−97.0000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$97$	−97.0000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	133.000	1.29126	0.645631	−	0.763649i	$-0.276594\pi$
$103$	133.000	1.29126	0.645631	+	0.763649i	$0.276594\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	59.3970i	0.555112i	0.960709	+	0.277556i	$0.0895244\pi$
$107$	59.3970i	0.555112i	−0.960709	+	0.277556i	$0.910476\pi$
$108$	0	0
$109$	130.000	1.19266	0.596330	−	0.802739i	$-0.296625\pi$
$109$	130.000	1.19266	0.596330	+	0.802739i	$0.296625\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	59.3970i	0.525637i	0.964845	+	0.262818i	$0.0846520\pi$
$113$	59.3970i	0.525637i	−0.964845	+	0.262818i	$0.915348\pi$
$114$	0	0
$115$	−8.00000	−0.0695652
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 42.4264i	− 0.356524i
$120$	0	0
$121$	−271.000	−2.23967
$122$	0	0
$123$	0	0
$124$	0	0
$125$	118.794i	0.950352i
$126$	0	0
$127$	−50.0000	−0.393701	−0.196850	−	0.980434i	$-0.563071\pi$
$127$	−50.0000	−0.393701	−0.196850	+	0.980434i	$0.563071\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	141.421i	1.07955i	0.841809	+	0.539776i	$0.181491\pi$
$131$	141.421i	1.07955i	−0.841809	+	0.539776i	$0.818509\pi$
$132$	0	0
$133$	57.0000	0.428571
$134$	0	0
$135$	0	0
$136$	0	0
$137$	144.250i	1.05292i	0.850200	+	0.526459i	$0.176481\pi$
$137$	144.250i	1.05292i	−0.850200	+	0.526459i	$0.823519\pi$
$138$	0	0
$139$	189.000	1.35971	0.679856	−	0.733346i	$-0.262042\pi$
$139$	189.000	1.35971	0.679856	+	0.733346i	$0.262042\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	138.593i	0.969181i
$144$	0	0
$145$	−144.000	−0.993103
$146$	0	0
$147$	0	0
$148$	0	0
$149$	56.5685i	0.379655i	0.981817	+	0.189827i	$0.0607928\pi$
$149$	56.5685i	0.379655i	−0.981817	+	0.189827i	$0.939207\pi$
$150$	0	0
$151$	221.000	1.46358	0.731788	−	0.681532i	$-0.238686\pi$
$151$	221.000	1.46358	0.731788	+	0.681532i	$0.238686\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 28.2843i	− 0.182479i
$156$	0	0
$157$	−14.0000	−0.0891720	−0.0445860	−	0.999006i	$-0.514197\pi$
$157$	−14.0000	−0.0891720	−0.0445860	+	0.999006i	$0.514197\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 8.48528i	− 0.0527036i
$162$	0	0
$163$	93.0000	0.570552	0.285276	−	0.958445i	$-0.407915\pi$
$163$	93.0000	0.570552	0.285276	+	0.958445i	$0.407915\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	296.985i	1.77835i	0.457565	+	0.889176i	$0.348722\pi$
$167$	296.985i	1.77835i	−0.457565	+	0.889176i	$0.651278\pi$
$168$	0	0
$169$	−120.000	−0.710059
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 254.558i	− 1.47144i	−0.677288	−	0.735718i	$-0.736845\pi$
$173$	− 254.558i	− 1.47144i	0.677288	−	0.735718i	$-0.263155\pi$
$174$	0	0
$175$	−51.0000	−0.291429
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 33.9411i	− 0.189615i	−0.995496	−	0.0948076i	$-0.969776\pi$
$179$	− 33.9411i	− 0.189615i	0.995496	−	0.0948076i	$-0.0302236\pi$
$180$	0	0
$181$	111.000	0.613260	0.306630	−	0.951829i	$-0.400799\pi$
$181$	111.000	0.613260	0.306630	+	0.951829i	$0.400799\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	178.191i	0.963194i
$186$	0	0
$187$	−280.000	−1.49733
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 42.4264i	− 0.222128i	−0.993813	−	0.111064i	$-0.964574\pi$
$191$	− 42.4264i	− 0.222128i	0.993813	−	0.111064i	$-0.0354258\pi$
$192$	0	0
$193$	−217.000	−1.12435	−0.562176	−	0.827018i	$-0.690036\pi$
$193$	−217.000	−1.12435	−0.562176	+	0.827018i	$0.690036\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	296.985i	1.50754i	0.657140	+	0.753769i	$0.271766\pi$
$197$	296.985i	1.50754i	−0.657140	+	0.753769i	$0.728234\pi$
$198$	0	0
$199$	−11.0000	−0.0552764	−0.0276382	−	0.999618i	$-0.508799\pi$
$199$	−11.0000	−0.0552764	−0.0276382	+	0.999618i	$0.508799\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 152.735i	− 0.752389i
$204$	0	0
$205$	144.000	0.702439
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 376.181i	− 1.79991i
$210$	0	0
$211$	181.000	0.857820	0.428910	−	0.903347i	$-0.358898\pi$
$211$	181.000	0.857820	0.428910	+	0.903347i	$0.358898\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 141.421i	− 0.657774i
$216$	0	0
$217$	30.0000	0.138249
$218$	0	0
$219$	0	0
$220$	0	0
$221$	98.9949i	0.447941i
$222$	0	0
$223$	70.0000	0.313901	0.156951	−	0.987606i	$-0.449834\pi$
$223$	70.0000	0.313901	0.156951	+	0.987606i	$0.449834\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 271.529i	− 1.19616i	−0.801435	−	0.598082i	$-0.795930\pi$
$227$	− 271.529i	− 1.19616i	0.801435	−	0.598082i	$-0.204070\pi$
$228$	0	0
$229$	250.000	1.09170	0.545852	−	0.837882i	$-0.316206\pi$
$229$	250.000	1.09170	0.545852	+	0.837882i	$0.316206\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 56.5685i	− 0.242783i	−0.992605	−	0.121392i	$-0.961264\pi$
$233$	− 56.5685i	− 0.242783i	0.992605	−	0.121392i	$-0.0387357\pi$
$234$	0	0
$235$	−120.000	−0.510638
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 147.078i	− 0.615390i	−0.951485	−	0.307695i	$-0.900442\pi$
$239$	− 147.078i	− 0.615390i	0.951485	−	0.307695i	$-0.0995576\pi$
$240$	0	0
$241$	39.0000	0.161826	0.0809129	−	0.996721i	$-0.474216\pi$
$241$	39.0000	0.161826	0.0809129	+	0.996721i	$0.474216\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 113.137i	− 0.461784i
$246$	0	0
$247$	−133.000	−0.538462
$248$	0	0
$249$	0	0
$250$	0	0
$251$	107.480i	0.428208i	0.976811	+	0.214104i	$0.0686832\pi$
$251$	107.480i	0.428208i	−0.976811	+	0.214104i	$0.931317\pi$
$252$	0	0
$253$	−56.0000	−0.221344
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 197.990i	− 0.770389i	−0.922835	−	0.385194i	$-0.874134\pi$
$257$	− 197.990i	− 0.770389i	0.922835	−	0.385194i	$-0.125866\pi$
$258$	0	0
$259$	−189.000	−0.729730
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 395.980i	− 1.50563i	−0.658234	−	0.752813i	$-0.728696\pi$
$263$	− 395.980i	− 1.50563i	0.658234	−	0.752813i	$-0.271304\pi$
$264$	0	0
$265$	208.000	0.784906
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 494.975i	− 1.84005i	−0.391854	−	0.920027i	$-0.628166\pi$
$269$	− 494.975i	− 1.84005i	0.391854	−	0.920027i	$-0.371834\pi$
$270$	0	0
$271$	−331.000	−1.22140	−0.610701	−	0.791861i	$-0.709112\pi$
$271$	−331.000	−1.22140	−0.610701	+	0.791861i	$0.709112\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	336.583i	1.22394i
$276$	0	0
$277$	130.000	0.469314	0.234657	−	0.972078i	$-0.424603\pi$
$277$	130.000	0.469314	0.234657	+	0.972078i	$0.424603\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 197.990i	− 0.704590i	−0.935889	−	0.352295i	$-0.885401\pi$
$281$	− 197.990i	− 0.704590i	0.935889	−	0.352295i	$-0.114599\pi$
$282$	0	0
$283$	−130.000	−0.459364	−0.229682	−	0.973266i	$-0.573769\pi$
$283$	−130.000	−0.459364	−0.229682	+	0.973266i	$0.573769\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	152.735i	0.532178i
$288$	0	0
$289$	89.0000	0.307958
$290$	0	0
$291$	0	0
$292$	0	0
$293$	59.3970i	0.202720i	0.994850	+	0.101360i	$0.0323194\pi$
$293$	59.3970i	0.202720i	−0.994850	+	0.101360i	$0.967681\pi$
$294$	0	0
$295$	280.000	0.949153
$296$	0	0
$297$	0	0
$298$	0	0
$299$	19.7990i	0.0662174i
$300$	0	0
$301$	150.000	0.498339
$302$	0	0
$303$	0	0
$304$	0	0
$305$	223.446i	0.732609i
$306$	0	0
$307$	414.000	1.34853	0.674267	−	0.738488i	$-0.264460\pi$
$307$	414.000	1.34853	0.674267	+	0.738488i	$0.264460\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 296.985i	− 0.954935i	−0.878649	−	0.477468i	$-0.841555\pi$
$311$	− 296.985i	− 0.954935i	0.878649	−	0.477468i	$-0.158445\pi$
$312$	0	0
$313$	−177.000	−0.565495	−0.282748	−	0.959194i	$-0.591246\pi$
$313$	−177.000	−0.565495	−0.282748	+	0.959194i	$0.591246\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	593.970i	1.87372i	0.349703	+	0.936861i	$0.386283\pi$
$317$	593.970i	1.87372i	−0.349703	+	0.936861i	$0.613717\pi$
$318$	0	0
$319$	−1008.00	−3.15987
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 268.701i	− 0.831890i
$324$	0	0
$325$	119.000	0.366154
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 127.279i	− 0.386867i
$330$	0	0
$331$	−499.000	−1.50755	−0.753776	−	0.657131i	$-0.771770\pi$
$331$	−499.000	−1.50755	−0.753776	+	0.657131i	$0.771770\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	217.789i	0.650116i
$336$	0	0
$337$	447.000	1.32641	0.663205	−	0.748438i	$-0.269196\pi$
$337$	447.000	1.32641	0.663205	+	0.748438i	$0.269196\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 197.990i	− 0.580616i
$342$	0	0
$343$	267.000	0.778426
$344$	0	0
$345$	0	0
$346$	0	0
$347$	237.588i	0.684691i	0.939574	+	0.342346i	$0.111221\pi$
$347$	237.588i	0.684691i	−0.939574	+	0.342346i	$0.888779\pi$
$348$	0	0
$349$	111.000	0.318052	0.159026	−	0.987274i	$-0.449165\pi$
$349$	111.000	0.318052	0.159026	+	0.987274i	$0.449165\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	395.980i	1.12176i	0.827899	+	0.560878i	$0.189536\pi$
$353$	395.980i	1.12176i	−0.827899	+	0.560878i	$0.810464\pi$
$354$	0	0
$355$	224.000	0.630986
$356$	0	0
$357$	0	0
$358$	0	0
$359$	415.779i	1.15816i	0.815271	+	0.579079i	$0.196588\pi$
$359$	415.779i	1.15816i	−0.815271	+	0.579079i	$0.803412\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 48.0833i	− 0.131735i
$366$	0	0
$367$	493.000	1.34332	0.671662	−	0.740858i	$-0.265581\pi$
$367$	493.000	1.34332	0.671662	+	0.740858i	$0.265581\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	220.617i	0.594656i
$372$	0	0
$373$	47.0000	0.126005	0.0630027	−	0.998013i	$-0.479932\pi$
$373$	47.0000	0.126005	0.0630027	+	0.998013i	$0.479932\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	356.382i	0.945310i
$378$	0	0
$379$	−259.000	−0.683377	−0.341689	−	0.939813i	$-0.610999\pi$
$379$	−259.000	−0.683377	−0.341689	+	0.939813i	$0.610999\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	168.000	0.436364
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 183.848i	− 0.472616i	−0.971678	−	0.236308i	$-0.924063\pi$
$389$	− 183.848i	− 0.472616i	0.971678	−	0.236308i	$-0.0759375\pi$
$390$	0	0
$391$	−40.0000	−0.102302
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 31.1127i	− 0.0787663i
$396$	0	0
$397$	−670.000	−1.68766	−0.843829	−	0.536613i	$-0.819704\pi$
$397$	−670.000	−1.68766	−0.843829	+	0.536613i	$0.819704\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	475.176i	1.18498i	0.805579	+	0.592488i	$0.201854\pi$
$401$	475.176i	1.18498i	−0.805579	+	0.592488i	$0.798146\pi$
$402$	0	0
$403$	−70.0000	−0.173697
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1247.34i	3.06471i
$408$	0	0
$409$	231.000	0.564792	0.282396	−	0.959298i	$-0.408871\pi$
$409$	231.000	0.564792	0.282396	+	0.959298i	$0.408871\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	296.985i	0.719092i
$414$	0	0
$415$	−112.000	−0.269880
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 721.249i	− 1.72136i	−0.509148	−	0.860679i	$-0.670039\pi$
$419$	− 721.249i	− 1.72136i	0.509148	−	0.860679i	$-0.329961\pi$
$420$	0	0
$421$	−769.000	−1.82660	−0.913302	−	0.407284i	$-0.866476\pi$
$421$	−769.000	−1.82660	−0.913302	+	0.407284i	$0.866476\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	240.416i	0.565685i
$426$	0	0
$427$	−237.000	−0.555035
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 613.769i	− 1.42406i	−0.702150	−	0.712029i	$-0.747777\pi$
$431$	− 613.769i	− 1.42406i	0.702150	−	0.712029i	$-0.252223\pi$
$432$	0	0
$433$	−230.000	−0.531178	−0.265589	−	0.964086i	$-0.585566\pi$
$433$	−230.000	−0.531178	−0.265589	+	0.964086i	$0.585566\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 53.7401i	− 0.122975i
$438$	0	0
$439$	630.000	1.43508	0.717540	−	0.696517i	$-0.245268\pi$
$439$	630.000	1.43508	0.717540	+	0.696517i	$0.245268\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	141.421i	0.319236i	0.987179	+	0.159618i	$0.0510262\pi$
$443$	141.421i	0.319236i	−0.987179	+	0.159618i	$0.948974\pi$
$444$	0	0
$445$	120.000	0.269663
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 268.701i	− 0.598442i	−0.954184	−	0.299221i	$-0.903273\pi$
$449$	− 268.701i	− 0.598442i	0.954184	−	0.299221i	$-0.0967268\pi$
$450$	0	0
$451$	1008.00	2.23503
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 59.3970i	− 0.130543i
$456$	0	0
$457$	170.000	0.371991	0.185996	−	0.982551i	$-0.440449\pi$
$457$	170.000	0.371991	0.185996	+	0.982551i	$0.440449\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	494.975i	1.07370i	0.843678	+	0.536849i	$0.180386\pi$
$461$	494.975i	1.07370i	−0.843678	+	0.536849i	$0.819614\pi$
$462$	0	0
$463$	−43.0000	−0.0928726	−0.0464363	−	0.998921i	$-0.514786\pi$
$463$	−43.0000	−0.0928726	−0.0464363	+	0.998921i	$0.514786\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	296.985i	0.635942i	0.948100	+	0.317971i	$0.103001\pi$
$467$	296.985i	0.635942i	−0.948100	+	0.317971i	$0.896999\pi$
$468$	0	0
$469$	−231.000	−0.492537
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 989.949i	− 2.09292i
$474$	0	0
$475$	−323.000	−0.680000
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 401.637i	− 0.838490i	−0.907873	−	0.419245i	$-0.862295\pi$
$479$	− 401.637i	− 0.838490i	0.907873	−	0.419245i	$-0.137705\pi$
$480$	0	0
$481$	441.000	0.916840
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 274.357i	− 0.565685i
$486$	0	0
$487$	277.000	0.568789	0.284394	−	0.958707i	$-0.408208\pi$
$487$	277.000	0.568789	0.284394	+	0.958707i	$0.408208\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	415.779i	0.846800i	0.905943	+	0.423400i	$0.139163\pi$
$491$	415.779i	0.846800i	−0.905943	+	0.423400i	$0.860837\pi$
$492$	0	0
$493$	−720.000	−1.46045
$494$	0	0
$495$	0	0
$496$	0	0
$497$	237.588i	0.478044i
$498$	0	0
$499$	198.000	0.396794	0.198397	−	0.980122i	$-0.436427\pi$
$499$	198.000	0.396794	0.198397	+	0.980122i	$0.436427\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 229.103i	− 0.455472i	−0.973723	−	0.227736i	$-0.926868\pi$
$503$	− 229.103i	− 0.455472i	0.973723	−	0.227736i	$-0.0731324\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 772.161i	− 1.51701i	−0.651664	−	0.758507i	$-0.725929\pi$
$509$	− 772.161i	− 1.51701i	0.651664	−	0.758507i	$-0.274071\pi$
$510$	0	0
$511$	51.0000	0.0998043
$512$	0	0
$513$	0	0
$514$	0	0
$515$	376.181i	0.730448i
$516$	0	0
$517$	−840.000	−1.62476
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 613.769i	− 1.17806i	−0.808112	−	0.589029i	$-0.799510\pi$
$521$	− 613.769i	− 1.17806i	0.808112	−	0.589029i	$-0.200490\pi$
$522$	0	0
$523$	277.000	0.529637	0.264818	−	0.964298i	$-0.414688\pi$
$523$	277.000	0.529637	0.264818	+	0.964298i	$0.414688\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 141.421i	− 0.268352i
$528$	0	0
$529$	521.000	0.984877
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 356.382i	− 0.668634i
$534$	0	0
$535$	−168.000	−0.314019
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 791.960i	− 1.46931i
$540$	0	0
$541$	−369.000	−0.682070	−0.341035	−	0.940051i	$-0.610778\pi$
$541$	−369.000	−0.682070	−0.341035	+	0.940051i	$0.610778\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	367.696i	0.674671i
$546$	0	0
$547$	197.000	0.360146	0.180073	−	0.983653i	$-0.442367\pi$
$547$	197.000	0.360146	0.180073	+	0.983653i	$0.442367\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 967.322i	− 1.75558i
$552$	0	0
$553$	33.0000	0.0596745
$554$	0	0
$555$	0	0
$556$	0	0
$557$	410.122i	0.736305i	0.929765	+	0.368153i	$0.120010\pi$
$557$	410.122i	0.736305i	−0.929765	+	0.368153i	$0.879990\pi$
$558$	0	0
$559$	−350.000	−0.626118
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 480.833i	− 0.854054i	−0.904239	−	0.427027i	$-0.859561\pi$
$563$	− 480.833i	− 0.854054i	0.904239	−	0.427027i	$-0.140439\pi$
$564$	0	0
$565$	−168.000	−0.297345
$566$	0	0
$567$	0	0
$568$	0	0
$569$	494.975i	0.869903i	0.900454	+	0.434951i	$0.143234\pi$
$569$	494.975i	0.869903i	−0.900454	+	0.434951i	$0.856766\pi$
$570$	0	0
$571$	381.000	0.667250	0.333625	−	0.942706i	$-0.391728\pi$
$571$	381.000	0.667250	0.333625	+	0.942706i	$0.391728\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	48.0833i	0.0836231i
$576$	0	0
$577$	−553.000	−0.958406	−0.479203	−	0.877704i	$-0.659074\pi$
$577$	−553.000	−0.958406	−0.479203	+	0.877704i	$0.659074\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 118.794i	− 0.204465i
$582$	0	0
$583$	1456.00	2.49743
$584$	0	0
$585$	0	0
$586$	0	0
$587$	87.6812i	0.149372i	0.997207	+	0.0746859i	$0.0237954\pi$
$587$	87.6812i	0.149372i	−0.997207	+	0.0746859i	$0.976205\pi$
$588$	0	0
$589$	190.000	0.322581
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 752.362i	− 1.26874i	−0.773030	−	0.634369i	$-0.781260\pi$
$593$	− 752.362i	− 1.26874i	0.773030	−	0.634369i	$-0.218740\pi$
$594$	0	0
$595$	120.000	0.201681
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 560.029i	− 0.934939i	−0.884009	−	0.467470i	$-0.845166\pi$
$599$	− 560.029i	− 0.934939i	0.884009	−	0.467470i	$-0.154834\pi$
$600$	0	0
$601$	−230.000	−0.382696	−0.191348	−	0.981522i	$-0.561286\pi$
$601$	−230.000	−0.382696	−0.191348	+	0.981522i	$0.561286\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 766.504i	− 1.26695i
$606$	0	0
$607$	−1027.00	−1.69193	−0.845964	−	0.533240i	$-0.820974\pi$
$607$	−1027.00	−1.69193	−0.845964	+	0.533240i	$0.820974\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	296.985i	0.486064i
$612$	0	0
$613$	−73.0000	−0.119086	−0.0595432	−	0.998226i	$-0.518964\pi$
$613$	−73.0000	−0.119086	−0.0595432	+	0.998226i	$0.518964\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	534.573i	0.866406i	0.901296	+	0.433203i	$0.142617\pi$
$617$	534.573i	0.866406i	−0.901296	+	0.433203i	$0.857383\pi$
$618$	0	0
$619$	−1211.00	−1.95638	−0.978191	−	0.207709i	$-0.933399\pi$
$619$	−1211.00	−1.95638	−0.978191	+	0.207709i	$0.933399\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	127.279i	0.204301i
$624$	0	0
$625$	89.0000	0.142400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	890.955i	1.41646i
$630$	0	0
$631$	349.000	0.553090	0.276545	−	0.961001i	$-0.410810\pi$
$631$	349.000	0.553090	0.276545	+	0.961001i	$0.410810\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 141.421i	− 0.222711i
$636$	0	0
$637$	−280.000	−0.439560
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1074.80i	− 1.67676i	−0.545087	−	0.838379i	$-0.683503\pi$
$641$	− 1074.80i	− 1.67676i	0.545087	−	0.838379i	$-0.316497\pi$
$642$	0	0
$643$	750.000	1.16641	0.583204	−	0.812326i	$-0.301799\pi$
$643$	750.000	1.16641	0.583204	+	0.812326i	$0.301799\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	752.362i	1.16285i	0.813601	+	0.581423i	$0.197504\pi$
$647$	752.362i	1.16285i	−0.813601	+	0.581423i	$0.802496\pi$
$648$	0	0
$649$	1960.00	3.02003
$650$	0	0
$651$	0	0
$652$	0	0
$653$	933.381i	1.42937i	0.699445	+	0.714687i	$0.253431\pi$
$653$	933.381i	1.42937i	−0.699445	+	0.714687i	$0.746569\pi$
$654$	0	0
$655$	−400.000	−0.610687
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 424.264i	− 0.643800i	−0.946774	−	0.321900i	$-0.895679\pi$
$659$	− 424.264i	− 0.643800i	0.946774	−	0.321900i	$-0.104321\pi$
$660$	0	0
$661$	−361.000	−0.546142	−0.273071	−	0.961994i	$-0.588039\pi$
$661$	−361.000	−0.546142	−0.273071	+	0.961994i	$0.588039\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	161.220i	0.242437i
$666$	0	0
$667$	−144.000	−0.215892
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1564.12i	2.33103i
$672$	0	0
$673$	463.000	0.687964	0.343982	−	0.938976i	$-0.388224\pi$
$673$	463.000	0.687964	0.343982	+	0.938976i	$0.388224\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	494.975i	0.731130i	0.930786	+	0.365565i	$0.119124\pi$
$677$	494.975i	0.731130i	−0.930786	+	0.365565i	$0.880876\pi$
$678$	0	0
$679$	291.000	0.428571
$680$	0	0
$681$	0	0
$682$	0	0
$683$	554.372i	0.811672i	0.913946	+	0.405836i	$0.133020\pi$
$683$	554.372i	0.811672i	−0.913946	+	0.405836i	$0.866980\pi$
$684$	0	0
$685$	−408.000	−0.595620
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 514.774i	− 0.747132i
$690$	0	0
$691$	110.000	0.159190	0.0795948	−	0.996827i	$-0.474637\pi$
$691$	110.000	0.159190	0.0795948	+	0.996827i	$0.474637\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	534.573i	0.769169i
$696$	0	0
$697$	720.000	1.03300
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 585.484i	− 0.835213i	−0.908628	−	0.417607i	$-0.862869\pi$
$701$	− 585.484i	− 0.835213i	0.908628	−	0.417607i	$-0.137131\pi$
$702$	0	0
$703$	−1197.00	−1.70270
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	359.000	0.506347	0.253173	−	0.967421i	$-0.418526\pi$
$709$	359.000	0.506347	0.253173	+	0.967421i	$0.418526\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 28.2843i	− 0.0396694i
$714$	0	0
$715$	−392.000	−0.548252
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 277.186i	− 0.385516i	−0.981246	−	0.192758i	$-0.938257\pi$
$719$	− 277.186i	− 0.385516i	0.981246	−	0.192758i	$-0.0617432\pi$
$720$	0	0
$721$	−399.000	−0.553398
$722$	0	0
$723$	0	0
$724$	0	0
$725$	865.499i	1.19379i
$726$	0	0
$727$	−330.000	−0.453920	−0.226960	−	0.973904i	$-0.572879\pi$
$727$	−330.000	−0.453920	−0.226960	+	0.973904i	$0.572879\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 707.107i	− 0.967314i
$732$	0	0
$733$	−670.000	−0.914052	−0.457026	−	0.889453i	$-0.651085\pi$
$733$	−670.000	−0.914052	−0.457026	+	0.889453i	$0.651085\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1524.52i	2.06855i
$738$	0	0
$739$	−850.000	−1.15020	−0.575101	−	0.818082i	$-0.695037\pi$
$739$	−850.000	−1.15020	−0.575101	+	0.818082i	$0.695037\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	664.680i	0.894590i	0.894386	+	0.447295i	$0.147613\pi$
$743$	664.680i	0.894590i	−0.894386	+	0.447295i	$0.852387\pi$
$744$	0	0
$745$	−160.000	−0.214765
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 178.191i	− 0.237905i
$750$	0	0
$751$	−51.0000	−0.0679095	−0.0339547	−	0.999423i	$-0.510810\pi$
$751$	−51.0000	−0.0679095	−0.0339547	+	0.999423i	$0.510810\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	625.082i	0.827924i
$756$	0	0
$757$	743.000	0.981506	0.490753	−	0.871299i	$-0.336722\pi$
$757$	743.000	0.981506	0.490753	+	0.871299i	$0.336722\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1168.14i	1.53501i	0.641045	+	0.767504i	$0.278501\pi$
$761$	1168.14i	1.53501i	−0.641045	+	0.767504i	$0.721499\pi$
$762$	0	0
$763$	−390.000	−0.511140
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 692.965i	− 0.903474i
$768$	0	0
$769$	239.000	0.310793	0.155397	−	0.987852i	$-0.450334\pi$
$769$	239.000	0.310793	0.155397	+	0.987852i	$0.450334\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 322.441i	− 0.417129i	−0.978009	−	0.208564i	$-0.933121\pi$
$773$	− 322.441i	− 0.417129i	0.978009	−	0.208564i	$-0.0668791\pi$
$774$	0	0
$775$	−170.000	−0.219355
$776$	0	0
$777$	0	0
$778$	0	0
$779$	967.322i	1.24175i
$780$	0	0
$781$	1568.00	2.00768
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 39.5980i	− 0.0504433i
$786$	0	0
$787$	1453.00	1.84625	0.923126	−	0.384498i	$-0.125625\pi$
$787$	1453.00	1.84625	0.923126	+	0.384498i	$0.125625\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 178.191i	− 0.225273i
$792$	0	0
$793$	553.000	0.697352
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 395.980i	− 0.496838i	−0.968653	−	0.248419i	$-0.920089\pi$
$797$	− 395.980i	− 0.496838i	0.968653	−	0.248419i	$-0.0799110\pi$
$798$	0	0
$799$	−600.000	−0.750939
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 336.583i	− 0.419157i
$804$	0	0
$805$	24.0000	0.0298137
$806$	0	0
$807$	0	0
$808$	0	0
$809$	395.980i	0.489468i	0.969590	+	0.244734i	$0.0787007\pi$
$809$	395.980i	0.489468i	−0.969590	+	0.244734i	$0.921299\pi$
$810$	0	0
$811$	−770.000	−0.949445	−0.474723	−	0.880135i	$-0.657452\pi$
$811$	−770.000	−0.949445	−0.474723	+	0.880135i	$0.657452\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	263.044i	0.322753i
$816$	0	0
$817$	950.000	1.16279
$818$	0	0
$819$	0	0
$820$	0	0
$821$	296.985i	0.361736i	0.983507	+	0.180868i	$0.0578906\pi$
$821$	296.985i	0.361736i	−0.983507	+	0.180868i	$0.942109\pi$
$822$	0	0
$823$	437.000	0.530984	0.265492	−	0.964113i	$-0.414466\pi$
$823$	437.000	0.530984	0.265492	+	0.964113i	$0.414466\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	692.965i	0.837926i	0.908003	+	0.418963i	$0.137606\pi$
$827$	692.965i	0.837926i	−0.908003	+	0.418963i	$0.862394\pi$
$828$	0	0
$829$	199.000	0.240048	0.120024	−	0.992771i	$-0.461703\pi$
$829$	199.000	0.240048	0.120024	+	0.992771i	$0.461703\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 565.685i	− 0.679094i
$834$	0	0
$835$	−840.000	−1.00599
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 593.970i	− 0.707950i	−0.935255	−	0.353975i	$-0.884830\pi$
$839$	− 593.970i	− 0.707950i	0.935255	−	0.353975i	$-0.115170\pi$
$840$	0	0
$841$	−1751.00	−2.08205
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 339.411i	− 0.401670i
$846$	0	0
$847$	813.000	0.959858
$848$	0	0
$849$	0	0
$850$	0	0
$851$	178.191i	0.209390i
$852$	0	0
$853$	−1273.00	−1.49238	−0.746190	−	0.665733i	$-0.768119\pi$
$853$	−1273.00	−1.49238	−0.746190	+	0.665733i	$0.768119\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 950.352i	− 1.10893i	−0.832208	−	0.554464i	$-0.812923\pi$
$857$	− 950.352i	− 1.10893i	0.832208	−	0.554464i	$-0.187077\pi$
$858$	0	0
$859$	−1251.00	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$859$	−1251.00	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 353.553i	− 0.409679i	−0.978796	−	0.204840i	$-0.934333\pi$
$863$	− 353.553i	− 0.409679i	0.978796	−	0.204840i	$-0.0656673\pi$
$864$	0	0
$865$	720.000	0.832370
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 217.789i	− 0.250620i
$870$	0	0
$871$	539.000	0.618829
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 356.382i	− 0.407294i
$876$	0	0
$877$	327.000	0.372862	0.186431	−	0.982468i	$-0.440308\pi$
$877$	327.000	0.372862	0.186431	+	0.982468i	$0.440308\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	404.465i	0.459098i	0.973297	+	0.229549i	$0.0737251\pi$
$881$	404.465i	0.459098i	−0.973297	+	0.229549i	$0.926275\pi$
$882$	0	0
$883$	533.000	0.603624	0.301812	−	0.953367i	$-0.402409\pi$
$883$	533.000	0.603624	0.301812	+	0.953367i	$0.402409\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1385.93i	− 1.56249i	−0.624224	−	0.781245i	$-0.714585\pi$
$887$	− 1385.93i	− 1.56249i	0.624224	−	0.781245i	$-0.285415\pi$
$888$	0	0
$889$	150.000	0.168729
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 806.102i	− 0.902690i
$894$	0	0
$895$	96.0000	0.107263
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 509.117i	− 0.566315i
$900$	0	0
$901$	1040.00	1.15427
$902$	0	0
$903$	0	0
$904$	0	0
$905$	313.955i	0.346912i
$906$	0	0
$907$	−587.000	−0.647189	−0.323594	−	0.946196i	$-0.604891\pi$
$907$	−587.000	−0.647189	−0.323594	+	0.946196i	$0.604891\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28.2843i	0.0310475i	0.999879	+	0.0155237i	$0.00494156\pi$
$911$	28.2843i	0.0310475i	−0.999879	+	0.0155237i	$0.995058\pi$
$912$	0	0
$913$	−784.000	−0.858708
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 424.264i	− 0.462665i
$918$	0	0
$919$	−1370.00	−1.49075	−0.745375	−	0.666645i	$-0.767730\pi$
$919$	−1370.00	−1.49075	−0.745375	+	0.666645i	$0.767730\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 554.372i	− 0.600619i
$924$	0	0
$925$	1071.00	1.15784
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 656.195i	− 0.706346i	−0.935558	−	0.353173i	$-0.885103\pi$
$929$	− 656.195i	− 0.706346i	0.935558	−	0.353173i	$-0.114897\pi$
$930$	0	0
$931$	760.000	0.816327
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 791.960i	− 0.847016i
$936$	0	0
$937$	−273.000	−0.291355	−0.145678	−	0.989332i	$-0.546536\pi$
$937$	−273.000	−0.291355	−0.145678	+	0.989332i	$0.546536\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	415.779i	0.441848i	0.975291	+	0.220924i	$0.0709072\pi$
$941$	415.779i	0.441848i	−0.975291	+	0.220924i	$0.929093\pi$
$942$	0	0
$943$	144.000	0.152704
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1145.51i	1.20962i	0.796369	+	0.604812i	$0.206752\pi$
$947$	1145.51i	1.20962i	−0.796369	+	0.604812i	$0.793248\pi$
$948$	0	0
$949$	−119.000	−0.125395
$950$	0	0
$951$	0	0
$952$	0	0
$953$	127.279i	0.133556i	0.997768	+	0.0667782i	$0.0212720\pi$
$953$	127.279i	0.133556i	−0.997768	+	0.0667782i	$0.978728\pi$
$954$	0	0
$955$	120.000	0.125654
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 432.749i	− 0.451251i
$960$	0	0
$961$	−861.000	−0.895942
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 613.769i	− 0.636030i
$966$	0	0
$967$	917.000	0.948294	0.474147	−	0.880446i	$-0.342757\pi$
$967$	917.000	0.948294	0.474147	+	0.880446i	$0.342757\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1824.34i	− 1.87882i	−0.342794	−	0.939411i	$-0.611373\pi$
$971$	− 1824.34i	− 1.87882i	0.342794	−	0.939411i	$-0.388627\pi$
$972$	0	0
$973$	−567.000	−0.582734
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 831.558i	− 0.851134i	−0.904927	−	0.425567i	$-0.860075\pi$
$977$	− 831.558i	− 0.851134i	0.904927	−	0.425567i	$-0.139925\pi$
$978$	0	0
$979$	840.000	0.858018
$980$	0	0
$981$	0	0
$982$	0	0
$983$	59.3970i	0.0604242i	0.999544	+	0.0302121i	$0.00961827\pi$
$983$	59.3970i	0.0604242i	−0.999544	+	0.0302121i	$0.990382\pi$
$984$	0	0
$985$	−840.000	−0.852792
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 141.421i	− 0.142994i
$990$	0	0
$991$	−1379.00	−1.39152	−0.695762	−	0.718273i	$-0.744933\pi$
$991$	−1379.00	−1.39152	−0.695762	+	0.718273i	$0.744933\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 31.1127i	− 0.0312690i
$996$	0	0
$997$	130.000	0.130391	0.0651956	−	0.997873i	$-0.479233\pi$
$997$	130.000	0.130391	0.0651956	+	0.997873i	$0.479233\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.e.a.161.2	yes	2
3.2	odd	2	inner	216.3.e.a.161.1	✓	2
4.3	odd	2		432.3.e.f.161.2		2
8.3	odd	2		1728.3.e.j.1025.1		2
8.5	even	2		1728.3.e.h.1025.1		2
9.2	odd	6		648.3.m.c.377.2		4
9.4	even	3		648.3.m.c.593.2		4
9.5	odd	6		648.3.m.c.593.1		4
9.7	even	3		648.3.m.c.377.1		4
12.11	even	2		432.3.e.f.161.1		2
24.5	odd	2		1728.3.e.h.1025.2		2
24.11	even	2		1728.3.e.j.1025.2		2
36.7	odd	6		1296.3.q.g.1025.1		4
36.11	even	6		1296.3.q.g.1025.2		4
36.23	even	6		1296.3.q.g.593.1		4
36.31	odd	6		1296.3.q.g.593.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.e.a.161.1	✓	2	3.2	odd	2	inner
216.3.e.a.161.2	yes	2	1.1	even	1	trivial
432.3.e.f.161.1		2	12.11	even	2
432.3.e.f.161.2		2	4.3	odd	2
648.3.m.c.377.1		4	9.7	even	3
648.3.m.c.377.2		4	9.2	odd	6
648.3.m.c.593.1		4	9.5	odd	6
648.3.m.c.593.2		4	9.4	even	3
1296.3.q.g.593.1		4	36.23	even	6
1296.3.q.g.593.2		4	36.31	odd	6
1296.3.q.g.1025.1		4	36.7	odd	6
1296.3.q.g.1025.2		4	36.11	even	6
1728.3.e.h.1025.1		2	8.5	even	2
1728.3.e.h.1025.2		2	24.5	odd	2
1728.3.e.j.1025.1		2	8.3	odd	2
1728.3.e.j.1025.2		2	24.11	even	2

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{5} -3.00000 q^{7} +O(q^{10})\)	\(q+2.82843i q^{5} -3.00000 q^{7} +19.7990i q^{11} +7.00000 q^{13} +14.1421i q^{17} -19.0000 q^{19} +2.82843i q^{23} +17.0000 q^{25} +50.9117i q^{29} -10.0000 q^{31} -8.48528i q^{35} +63.0000 q^{37} -50.9117i q^{41} -50.0000 q^{43} +42.4264i q^{47} -40.0000 q^{49} -73.5391i q^{53} -56.0000 q^{55} -98.9949i q^{59} +79.0000 q^{61} +19.7990i q^{65} +77.0000 q^{67} -79.1960i q^{71} -17.0000 q^{73} -59.3970i q^{77} -11.0000 q^{79} +39.5980i q^{83} -40.0000 q^{85} -42.4264i q^{89} -21.0000 q^{91} -53.7401i q^{95} -97.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{7}+O(q^{10}) \) 2 * q - 6 * q^7	\( 2 q - 6 q^{7} + 14 q^{13} - 38 q^{19} + 34 q^{25} - 20 q^{31} + 126 q^{37} - 100 q^{43} - 80 q^{49} - 112 q^{55} + 158 q^{61} + 154 q^{67} - 34 q^{73} - 22 q^{79} - 80 q^{85} - 42 q^{91} - 194 q^{97}+O(q^{100}) \) 2 * q - 6 * q^7 + 14 * q^13 - 38 * q^19 + 34 * q^25 - 20 * q^31 + 126 * q^37 - 100 * q^43 - 80 * q^49 - 112 * q^55 + 158 * q^61 + 154 * q^67 - 34 * q^73 - 22 * q^79 - 80 * q^85 - 42 * q^91 - 194 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more