LMFDB - Embedded newform 2160.3.e.b.271.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(271,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2160.271");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.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:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.4
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	2160.271
Dual form			2160.3.e.b.271.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{5} +7.74597i q^{7} +O(q^{10})$	$q+2.23607 q^{5} +7.74597i q^{7} -6.92820i q^{11} +22.0000 q^{13} -6.70820 q^{17} +27.1109i q^{19} +29.4449i q^{23} +5.00000 q^{25} -40.2492 q^{29} -19.3649i q^{31} +17.3205i q^{35} +2.00000 q^{37} +53.6656 q^{41} +15.4919i q^{43} -13.8564i q^{47} -11.0000 q^{49} -6.70820 q^{53} -15.4919i q^{55} +24.2487i q^{59} -31.0000 q^{61} +49.1935 q^{65} +23.2379i q^{67} -110.851i q^{71} +76.0000 q^{73} +53.6656 q^{77} +19.3649i q^{79} +129.904i q^{83} -15.0000 q^{85} -53.6656 q^{89} +170.411i q^{91} +60.6218i q^{95} -32.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 88 q^{13} + 20 q^{25} + 8 q^{37} - 44 q^{49} - 124 q^{61} + 304 q^{73} - 60 q^{85} - 128 q^{97}+O(q^{100}) $ 4 * q + 88 * q^13 + 20 * q^25 + 8 * q^37 - 44 * q^49 - 124 * q^61 + 304 * q^73 - 60 * q^85 - 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	7.74597i	1.10657i	0.832993	+	0.553283i	$0.186625\pi$
$7$	7.74597i	1.10657i	−0.832993	+	0.553283i	$0.813375\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 6.92820i	− 0.629837i	−0.949119	−	0.314918i	$-0.898023\pi$
$11$	− 6.92820i	− 0.629837i	0.949119	−	0.314918i	$-0.101977\pi$
$12$	0	0
$13$	22.0000	1.69231	0.846154	−	0.532939i	$-0.178912\pi$
$13$	22.0000	1.69231	0.846154	+	0.532939i	$0.178912\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.70820	−0.394600	−0.197300	−	0.980343i	$-0.563217\pi$
$17$	−6.70820	−0.394600	−0.197300	+	0.980343i	$0.563217\pi$
$18$	0	0
$19$	27.1109i	1.42689i	0.700712	+	0.713444i	$0.252866\pi$
$19$	27.1109i	1.42689i	−0.700712	+	0.713444i	$0.747134\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	29.4449i	1.28021i	0.768287	+	0.640106i	$0.221109\pi$
$23$	29.4449i	1.28021i	−0.768287	+	0.640106i	$0.778891\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−40.2492	−1.38790	−0.693952	−	0.720021i	$-0.744132\pi$
$29$	−40.2492	−1.38790	−0.693952	+	0.720021i	$0.744132\pi$
$30$	0	0
$31$	− 19.3649i	− 0.624675i	−0.949971	−	0.312337i	$-0.898888\pi$
$31$	− 19.3649i	− 0.624675i	0.949971	−	0.312337i	$-0.101112\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	17.3205i	0.494872i
$36$	0	0
$37$	2.00000	0.0540541	0.0270270	−	0.999635i	$-0.491396\pi$
$37$	2.00000	0.0540541	0.0270270	+	0.999635i	$0.491396\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	53.6656	1.30892	0.654459	−	0.756098i	$-0.272896\pi$
$41$	53.6656	1.30892	0.654459	+	0.756098i	$0.272896\pi$
$42$	0	0
$43$	15.4919i	0.360278i	0.983641	+	0.180139i	$0.0576547\pi$
$43$	15.4919i	0.360278i	−0.983641	+	0.180139i	$0.942345\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 13.8564i	− 0.294817i	−0.989076	−	0.147409i	$-0.952907\pi$
$47$	− 13.8564i	− 0.294817i	0.989076	−	0.147409i	$-0.0470932\pi$
$48$	0	0
$49$	−11.0000	−0.224490
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.70820	−0.126570	−0.0632849	−	0.997995i	$-0.520158\pi$
$53$	−6.70820	−0.126570	−0.0632849	+	0.997995i	$0.520158\pi$
$54$	0	0
$55$	− 15.4919i	− 0.281672i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	24.2487i	0.410995i	0.978658	+	0.205498i	$0.0658813\pi$
$59$	24.2487i	0.410995i	−0.978658	+	0.205498i	$0.934119\pi$
$60$	0	0
$61$	−31.0000	−0.508197	−0.254098	−	0.967178i	$-0.581779\pi$
$61$	−31.0000	−0.508197	−0.254098	+	0.967178i	$0.581779\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	49.1935	0.756823
$66$	0	0
$67$	23.2379i	0.346834i	0.984848	+	0.173417i	$0.0554809\pi$
$67$	23.2379i	0.346834i	−0.984848	+	0.173417i	$0.944519\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 110.851i	− 1.56129i	−0.624978	−	0.780643i	$-0.714892\pi$
$71$	− 110.851i	− 1.56129i	0.624978	−	0.780643i	$-0.285108\pi$
$72$	0	0
$73$	76.0000	1.04110	0.520548	−	0.853832i	$-0.325728\pi$
$73$	76.0000	1.04110	0.520548	+	0.853832i	$0.325728\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	53.6656	0.696956
$78$	0	0
$79$	19.3649i	0.245126i	0.992461	+	0.122563i	$0.0391113\pi$
$79$	19.3649i	0.245126i	−0.992461	+	0.122563i	$0.960889\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	129.904i	1.56511i	0.622584	+	0.782553i	$0.286083\pi$
$83$	129.904i	1.56511i	−0.622584	+	0.782553i	$0.713917\pi$
$84$	0	0
$85$	−15.0000	−0.176471
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−53.6656	−0.602985	−0.301492	−	0.953469i	$-0.597485\pi$
$89$	−53.6656	−0.602985	−0.301492	+	0.953469i	$0.597485\pi$
$90$	0	0
$91$	170.411i	1.87265i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	60.6218i	0.638124i
$96$	0	0
$97$	−32.0000	−0.329897	−0.164948	−	0.986302i	$-0.552746\pi$
$97$	−32.0000	−0.329897	−0.164948	+	0.986302i	$0.552746\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−26.8328	−0.265671	−0.132836	−	0.991138i	$-0.542408\pi$
$101$	−26.8328	−0.265671	−0.132836	+	0.991138i	$0.542408\pi$
$102$	0	0
$103$	201.395i	1.95529i	0.210256	+	0.977646i	$0.432570\pi$
$103$	201.395i	1.95529i	−0.210256	+	0.977646i	$0.567430\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 117.779i	− 1.10074i	−0.834920	−	0.550371i	$-0.814486\pi$
$107$	− 117.779i	− 1.10074i	0.834920	−	0.550371i	$-0.185514\pi$
$108$	0	0
$109$	−203.000	−1.86239	−0.931193	−	0.364527i	$-0.881231\pi$
$109$	−203.000	−1.86239	−0.931193	+	0.364527i	$0.881231\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−174.413	−1.54348	−0.771740	−	0.635938i	$-0.780613\pi$
$113$	−174.413	−1.54348	−0.771740	+	0.635938i	$0.780613\pi$
$114$	0	0
$115$	65.8407i	0.572528i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 51.9615i	− 0.436651i
$120$	0	0
$121$	73.0000	0.603306
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	116.190i	0.914878i	0.889241	+	0.457439i	$0.151233\pi$
$127$	116.190i	0.914878i	−0.889241	+	0.457439i	$0.848767\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 58.8897i	− 0.449540i	−0.974412	−	0.224770i	$-0.927837\pi$
$131$	− 58.8897i	− 0.449540i	0.974412	−	0.224770i	$-0.0721631\pi$
$132$	0	0
$133$	−210.000	−1.57895
$134$	0	0
$135$	0	0
$136$	0	0
$137$	248.204	1.81170	0.905852	−	0.423594i	$-0.139232\pi$
$137$	248.204	1.81170	0.905852	+	0.423594i	$0.139232\pi$
$138$	0	0
$139$	154.919i	1.11453i	0.830336	+	0.557264i	$0.188149\pi$
$139$	154.919i	1.11453i	−0.830336	+	0.557264i	$0.811851\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 152.420i	− 1.06588i
$144$	0	0
$145$	−90.0000	−0.620690
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−160.997	−1.08052	−0.540258	−	0.841499i	$-0.681673\pi$
$149$	−160.997	−1.08052	−0.540258	+	0.841499i	$0.681673\pi$
$150$	0	0
$151$	247.871i	1.64153i	0.571266	+	0.820765i	$0.306452\pi$
$151$	247.871i	1.64153i	−0.571266	+	0.820765i	$0.693548\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 43.3013i	− 0.279363i
$156$	0	0
$157$	56.0000	0.356688	0.178344	−	0.983968i	$-0.442926\pi$
$157$	56.0000	0.356688	0.178344	+	0.983968i	$0.442926\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−228.079	−1.41664
$162$	0	0
$163$	− 116.190i	− 0.712819i	−0.934330	−	0.356410i	$-0.884001\pi$
$163$	− 116.190i	− 0.712819i	0.934330	−	0.356410i	$-0.115999\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	168.009i	1.00604i	0.864274	+	0.503021i	$0.167778\pi$
$167$	168.009i	1.00604i	−0.864274	+	0.503021i	$0.832222\pi$
$168$	0	0
$169$	315.000	1.86391
$170$	0	0
$171$	0	0
$172$	0	0
$173$	221.371	1.27960	0.639800	−	0.768542i	$-0.279017\pi$
$173$	221.371	1.27960	0.639800	+	0.768542i	$0.279017\pi$
$174$	0	0
$175$	38.7298i	0.221313i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	284.056i	1.58691i	0.608631	+	0.793453i	$0.291719\pi$
$179$	284.056i	1.58691i	−0.608631	+	0.793453i	$0.708281\pi$
$180$	0	0
$181$	181.000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$181$	181.000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.47214	0.0241737
$186$	0	0
$187$	46.4758i	0.248534i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	51.9615i	0.272050i	0.990705	+	0.136025i	$0.0434327\pi$
$191$	51.9615i	0.272050i	−0.990705	+	0.136025i	$0.956567\pi$
$192$	0	0
$193$	74.0000	0.383420	0.191710	−	0.981452i	$-0.438597\pi$
$193$	74.0000	0.383420	0.191710	+	0.981452i	$0.438597\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	221.371	1.12371	0.561855	−	0.827236i	$-0.310088\pi$
$197$	221.371	1.12371	0.561855	+	0.827236i	$0.310088\pi$
$198$	0	0
$199$	185.903i	0.934187i	0.884208	+	0.467093i	$0.154699\pi$
$199$	185.903i	0.934187i	−0.884208	+	0.467093i	$0.845301\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 311.769i	− 1.53581i
$204$	0	0
$205$	120.000	0.585366
$206$	0	0
$207$	0	0
$208$	0	0
$209$	187.830	0.898707
$210$	0	0
$211$	− 290.474i	− 1.37665i	−0.725401	−	0.688326i	$-0.758346\pi$
$211$	− 290.474i	− 1.37665i	0.725401	−	0.688326i	$-0.241654\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	34.6410i	0.161121i
$216$	0	0
$217$	150.000	0.691244
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−147.580	−0.667785
$222$	0	0
$223$	271.109i	1.21573i	0.794039	+	0.607867i	$0.207975\pi$
$223$	271.109i	1.21573i	−0.794039	+	0.607867i	$0.792025\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 168.009i	− 0.740127i	−0.929006	−	0.370064i	$-0.879336\pi$
$227$	− 168.009i	− 0.740127i	0.929006	−	0.370064i	$-0.120664\pi$
$228$	0	0
$229$	19.0000	0.0829694	0.0414847	−	0.999139i	$-0.486791\pi$
$229$	19.0000	0.0829694	0.0414847	+	0.999139i	$0.486791\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−281.745	−1.20920	−0.604602	−	0.796528i	$-0.706668\pi$
$233$	−281.745	−1.20920	−0.604602	+	0.796528i	$0.706668\pi$
$234$	0	0
$235$	− 30.9839i	− 0.131846i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 24.2487i	− 0.101459i	−0.998712	−	0.0507295i	$-0.983845\pi$
$239$	− 24.2487i	− 0.101459i	0.998712	−	0.0507295i	$-0.0161546\pi$
$240$	0	0
$241$	−209.000	−0.867220	−0.433610	−	0.901101i	$-0.642760\pi$
$241$	−209.000	−0.867220	−0.433610	+	0.901101i	$0.642760\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−24.5967	−0.100395
$246$	0	0
$247$	596.439i	2.41473i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	304.841i	1.21451i	0.794509	+	0.607253i	$0.207729\pi$
$251$	304.841i	1.21451i	−0.794509	+	0.607253i	$0.792271\pi$
$252$	0	0
$253$	204.000	0.806324
$254$	0	0
$255$	0	0
$256$	0	0
$257$	33.5410	0.130510	0.0652549	−	0.997869i	$-0.479214\pi$
$257$	33.5410	0.130510	0.0652549	+	0.997869i	$0.479214\pi$
$258$	0	0
$259$	15.4919i	0.0598144i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 464.190i	− 1.76498i	−0.470332	−	0.882490i	$-0.655866\pi$
$263$	− 464.190i	− 1.76498i	0.470332	−	0.882490i	$-0.344134\pi$
$264$	0	0
$265$	−15.0000	−0.0566038
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−362.243	−1.34663	−0.673314	−	0.739357i	$-0.735130\pi$
$269$	−362.243	−1.34663	−0.673314	+	0.739357i	$0.735130\pi$
$270$	0	0
$271$	274.982i	1.01469i	0.861742	+	0.507347i	$0.169374\pi$
$271$	274.982i	1.01469i	−0.861742	+	0.507347i	$0.830626\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 34.6410i	− 0.125967i
$276$	0	0
$277$	−116.000	−0.418773	−0.209386	−	0.977833i	$-0.567147\pi$
$277$	−116.000	−0.418773	−0.209386	+	0.977833i	$0.567147\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	469.574	1.67108	0.835541	−	0.549427i	$-0.185154\pi$
$281$	469.574	1.67108	0.835541	+	0.549427i	$0.185154\pi$
$282$	0	0
$283$	216.887i	0.766385i	0.923668	+	0.383193i	$0.125175\pi$
$283$	216.887i	0.766385i	−0.923668	+	0.383193i	$0.874825\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	415.692i	1.44840i
$288$	0	0
$289$	−244.000	−0.844291
$290$	0	0
$291$	0	0
$292$	0	0
$293$	328.702	1.12185	0.560925	−	0.827867i	$-0.310446\pi$
$293$	328.702	1.12185	0.560925	+	0.827867i	$0.310446\pi$
$294$	0	0
$295$	54.2218i	0.183803i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	647.787i	2.16651i
$300$	0	0
$301$	−120.000	−0.398671
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−69.3181	−0.227272
$306$	0	0
$307$	147.173i	0.479392i	0.970848	+	0.239696i	$0.0770478\pi$
$307$	147.173i	0.479392i	−0.970848	+	0.239696i	$0.922952\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	45.0333i	0.144802i	0.997376	+	0.0724008i	$0.0230661\pi$
$311$	45.0333i	0.144802i	−0.997376	+	0.0724008i	$0.976934\pi$
$312$	0	0
$313$	−256.000	−0.817891	−0.408946	−	0.912559i	$-0.634103\pi$
$313$	−256.000	−0.817891	−0.408946	+	0.912559i	$0.634103\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−140.872	−0.444392	−0.222196	−	0.975002i	$-0.571322\pi$
$317$	−140.872	−0.444392	−0.222196	+	0.975002i	$0.571322\pi$
$318$	0	0
$319$	278.855i	0.874153i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 181.865i	− 0.563051i
$324$	0	0
$325$	110.000	0.338462
$326$	0	0
$327$	0	0
$328$	0	0
$329$	107.331	0.326235
$330$	0	0
$331$	340.823i	1.02968i	0.857288	+	0.514838i	$0.172148\pi$
$331$	340.823i	1.02968i	−0.857288	+	0.514838i	$0.827852\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	51.9615i	0.155109i
$336$	0	0
$337$	−386.000	−1.14540	−0.572700	−	0.819765i	$-0.694104\pi$
$337$	−386.000	−1.14540	−0.572700	+	0.819765i	$0.694104\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−134.164	−0.393443
$342$	0	0
$343$	294.347i	0.858154i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 103.923i	− 0.299490i	−0.988725	−	0.149745i	$-0.952155\pi$
$347$	− 103.923i	− 0.299490i	0.988725	−	0.149745i	$-0.0478453\pi$
$348$	0	0
$349$	413.000	1.18338	0.591691	−	0.806165i	$-0.298461\pi$
$349$	413.000	1.18338	0.591691	+	0.806165i	$0.298461\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−147.580	−0.418075	−0.209038	−	0.977908i	$-0.567033\pi$
$353$	−147.580	−0.418075	−0.209038	+	0.977908i	$0.567033\pi$
$354$	0	0
$355$	− 247.871i	− 0.698228i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	183.597i	0.511413i	0.966754	+	0.255707i	$0.0823081\pi$
$359$	183.597i	0.511413i	−0.966754	+	0.255707i	$0.917692\pi$
$360$	0	0
$361$	−374.000	−1.03601
$362$	0	0
$363$	0	0
$364$	0	0
$365$	169.941	0.465592
$366$	0	0
$367$	− 580.948i	− 1.58296i	−0.611193	−	0.791482i	$-0.709310\pi$
$367$	− 580.948i	− 1.58296i	0.611193	−	0.791482i	$-0.290690\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 51.9615i	− 0.140058i
$372$	0	0
$373$	−16.0000	−0.0428954	−0.0214477	−	0.999770i	$-0.506828\pi$
$373$	−16.0000	−0.0428954	−0.0214477	+	0.999770i	$0.506828\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−885.483	−2.34876
$378$	0	0
$379$	− 383.425i	− 1.01168i	−0.862628	−	0.505838i	$-0.831183\pi$
$379$	− 383.425i	− 1.01168i	0.862628	−	0.505838i	$-0.168817\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 497.099i	− 1.29791i	−0.760828	−	0.648954i	$-0.775207\pi$
$383$	− 497.099i	− 1.29791i	0.760828	−	0.648954i	$-0.224793\pi$
$384$	0	0
$385$	120.000	0.311688
$386$	0	0
$387$	0	0
$388$	0	0
$389$	563.489	1.44856	0.724279	−	0.689507i	$-0.242173\pi$
$389$	563.489	1.44856	0.724279	+	0.689507i	$0.242173\pi$
$390$	0	0
$391$	− 197.522i	− 0.505172i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	43.3013i	0.109623i
$396$	0	0
$397$	2.00000	0.00503778	0.00251889	−	0.999997i	$-0.499198\pi$
$397$	2.00000	0.00503778	0.00251889	+	0.999997i	$0.499198\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	147.580	0.368031	0.184016	−	0.982923i	$-0.441090\pi$
$401$	147.580	0.368031	0.184016	+	0.982923i	$0.441090\pi$
$402$	0	0
$403$	− 426.028i	− 1.05714i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 13.8564i	− 0.0340452i
$408$	0	0
$409$	49.0000	0.119804	0.0599022	−	0.998204i	$-0.480921\pi$
$409$	49.0000	0.119804	0.0599022	+	0.998204i	$0.480921\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−187.830	−0.454793
$414$	0	0
$415$	290.474i	0.699937i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	311.769i	0.744079i	0.928217	+	0.372040i	$0.121341\pi$
$419$	311.769i	0.744079i	−0.928217	+	0.372040i	$0.878659\pi$
$420$	0	0
$421$	31.0000	0.0736342	0.0368171	−	0.999322i	$-0.488278\pi$
$421$	31.0000	0.0736342	0.0368171	+	0.999322i	$0.488278\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−33.5410	−0.0789200
$426$	0	0
$427$	− 240.125i	− 0.562354i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 460.726i	− 1.06897i	−0.845178	−	0.534484i	$-0.820506\pi$
$431$	− 460.726i	− 1.06897i	0.845178	−	0.534484i	$-0.179494\pi$
$432$	0	0
$433$	−142.000	−0.327945	−0.163972	−	0.986465i	$-0.552431\pi$
$433$	−142.000	−0.327945	−0.163972	+	0.986465i	$0.552431\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−798.276	−1.82672
$438$	0	0
$439$	73.5867i	0.167623i	0.996482	+	0.0838117i	$0.0267094\pi$
$439$	73.5867i	0.167623i	−0.996482	+	0.0838117i	$0.973291\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 334.286i	− 0.754595i	−0.926092	−	0.377298i	$-0.876853\pi$
$443$	− 334.286i	− 0.754595i	0.926092	−	0.377298i	$-0.123147\pi$
$444$	0	0
$445$	−120.000	−0.269663
$446$	0	0
$447$	0	0
$448$	0	0
$449$	67.0820	0.149403	0.0747016	−	0.997206i	$-0.476200\pi$
$449$	67.0820	0.149403	0.0747016	+	0.997206i	$0.476200\pi$
$450$	0	0
$451$	− 371.806i	− 0.824404i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	381.051i	0.837475i
$456$	0	0
$457$	268.000	0.586433	0.293217	−	0.956046i	$-0.405274\pi$
$457$	268.000	0.586433	0.293217	+	0.956046i	$0.405274\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	630.571	1.36783	0.683917	−	0.729560i	$-0.260275\pi$
$461$	630.571	1.36783	0.683917	+	0.729560i	$0.260275\pi$
$462$	0	0
$463$	− 658.407i	− 1.42205i	−0.703169	−	0.711023i	$-0.748232\pi$
$463$	− 658.407i	− 1.42205i	0.703169	−	0.711023i	$-0.251768\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 687.624i	− 1.47243i	−0.676748	−	0.736214i	$-0.736612\pi$
$467$	− 687.624i	− 1.47243i	0.676748	−	0.736214i	$-0.263388\pi$
$468$	0	0
$469$	−180.000	−0.383795
$470$	0	0
$471$	0	0
$472$	0	0
$473$	107.331	0.226916
$474$	0	0
$475$	135.554i	0.285378i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	727.461i	1.51871i	0.650677	+	0.759354i	$0.274485\pi$
$479$	727.461i	1.51871i	−0.650677	+	0.759354i	$0.725515\pi$
$480$	0	0
$481$	44.0000	0.0914761
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−71.5542	−0.147534
$486$	0	0
$487$	− 464.758i	− 0.954329i	−0.878814	−	0.477164i	$-0.841665\pi$
$487$	− 464.758i	− 0.954329i	0.878814	−	0.477164i	$-0.158335\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 675.500i	− 1.37576i	−0.725823	−	0.687882i	$-0.758541\pi$
$491$	− 675.500i	− 1.37576i	0.725823	−	0.687882i	$-0.241459\pi$
$492$	0	0
$493$	270.000	0.547667
$494$	0	0
$495$	0	0
$496$	0	0
$497$	858.650	1.72767
$498$	0	0
$499$	− 143.300i	− 0.287175i	−0.989638	−	0.143588i	$-0.954136\pi$
$499$	− 143.300i	− 0.287175i	0.989638	−	0.143588i	$-0.0458639\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	285.788i	0.568168i	0.958799	+	0.284084i	$0.0916894\pi$
$503$	285.788i	0.568168i	−0.958799	+	0.284084i	$0.908311\pi$
$504$	0	0
$505$	−60.0000	−0.118812
$506$	0	0
$507$	0	0
$508$	0	0
$509$	187.830	0.369017	0.184509	−	0.982831i	$-0.440931\pi$
$509$	187.830	0.369017	0.184509	+	0.982831i	$0.440931\pi$
$510$	0	0
$511$	588.693i	1.15204i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	450.333i	0.874433i
$516$	0	0
$517$	−96.0000	−0.185687
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	131.681i	0.251781i	0.992044	+	0.125890i	$0.0401788\pi$
$523$	131.681i	0.251781i	−0.992044	+	0.125890i	$0.959821\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	129.904i	0.246497i
$528$	0	0
$529$	−338.000	−0.638941
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1180.64	2.21509
$534$	0	0
$535$	− 263.363i	− 0.492267i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	76.2102i	0.141392i
$540$	0	0
$541$	826.000	1.52680	0.763401	−	0.645925i	$-0.223528\pi$
$541$	826.000	1.52680	0.763401	+	0.645925i	$0.223528\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−453.922	−0.832884
$546$	0	0
$547$	− 449.266i	− 0.821327i	−0.911787	−	0.410664i	$-0.865297\pi$
$547$	− 449.266i	− 0.821327i	0.911787	−	0.410664i	$-0.134703\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1091.19i	− 1.98038i
$552$	0	0
$553$	−150.000	−0.271248
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1006.23	−1.80652	−0.903259	−	0.429096i	$-0.858832\pi$
$557$	−1006.23	−1.80652	−0.903259	+	0.429096i	$0.858832\pi$
$558$	0	0
$559$	340.823i	0.609700i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	256.344i	0.455317i	0.973741	+	0.227659i	$0.0731070\pi$
$563$	256.344i	0.455317i	−0.973741	+	0.227659i	$0.926893\pi$
$564$	0	0
$565$	−390.000	−0.690265
$566$	0	0
$567$	0	0
$568$	0	0
$569$	496.407	0.872420	0.436210	−	0.899845i	$-0.356320\pi$
$569$	496.407	0.872420	0.436210	+	0.899845i	$0.356320\pi$
$570$	0	0
$571$	− 577.075i	− 1.01064i	−0.862933	−	0.505319i	$-0.831375\pi$
$571$	− 577.075i	− 1.01064i	0.862933	−	0.505319i	$-0.168625\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	147.224i	0.256042i
$576$	0	0
$577$	992.000	1.71924	0.859619	−	0.510936i	$-0.170701\pi$
$577$	992.000	1.71924	0.859619	+	0.510936i	$0.170701\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1006.23	−1.73189
$582$	0	0
$583$	46.4758i	0.0797184i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 441.673i	− 0.752424i	−0.926534	−	0.376212i	$-0.877226\pi$
$587$	− 441.673i	− 0.752424i	0.926534	−	0.376212i	$-0.122774\pi$
$588$	0	0
$589$	525.000	0.891341
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−315.286	−0.531679	−0.265839	−	0.964017i	$-0.585649\pi$
$593$	−315.286	−0.531679	−0.265839	+	0.964017i	$0.585649\pi$
$594$	0	0
$595$	− 116.190i	− 0.195276i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	779.423i	1.30121i	0.759418	+	0.650603i	$0.225484\pi$
$599$	779.423i	1.30121i	−0.759418	+	0.650603i	$0.774516\pi$
$600$	0	0
$601$	869.000	1.44592	0.722962	−	0.690888i	$-0.242780\pi$
$601$	869.000	1.44592	0.722962	+	0.690888i	$0.242780\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	163.233	0.269807
$606$	0	0
$607$	1130.91i	1.86312i	0.363593	+	0.931558i	$0.381550\pi$
$607$	1130.91i	1.86312i	−0.363593	+	0.931558i	$0.618450\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 304.841i	− 0.498921i
$612$	0	0
$613$	−592.000	−0.965742	−0.482871	−	0.875691i	$-0.660406\pi$
$613$	−592.000	−0.965742	−0.482871	+	0.875691i	$0.660406\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	154.289	0.250063	0.125031	−	0.992153i	$-0.460097\pi$
$617$	154.289	0.250063	0.125031	+	0.992153i	$0.460097\pi$
$618$	0	0
$619$	− 1192.88i	− 1.92711i	−0.267517	−	0.963553i	$-0.586203\pi$
$619$	− 1192.88i	− 1.92711i	0.267517	−	0.963553i	$-0.413797\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 415.692i	− 0.667243i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.4164	−0.0213297
$630$	0	0
$631$	− 987.611i	− 1.56515i	−0.622555	−	0.782576i	$-0.713905\pi$
$631$	− 987.611i	− 1.56515i	0.622555	−	0.782576i	$-0.286095\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	259.808i	0.409146i
$636$	0	0
$637$	−242.000	−0.379906
$638$	0	0
$639$	0	0
$640$	0	0
$641$	160.997	0.251165	0.125583	−	0.992083i	$-0.459920\pi$
$641$	160.997	0.251165	0.125583	+	0.992083i	$0.459920\pi$
$642$	0	0
$643$	− 441.520i	− 0.686656i	−0.939215	−	0.343328i	$-0.888446\pi$
$643$	− 441.520i	− 0.686656i	0.939215	−	0.343328i	$-0.111554\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 247.683i	− 0.382818i	−0.981510	−	0.191409i	$-0.938694\pi$
$647$	− 247.683i	− 0.382818i	0.981510	−	0.191409i	$-0.0613057\pi$
$648$	0	0
$649$	168.000	0.258860
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−784.860	−1.20193	−0.600965	−	0.799276i	$-0.705217\pi$
$653$	−784.860	−1.20193	−0.600965	+	0.799276i	$0.705217\pi$
$654$	0	0
$655$	− 131.681i	− 0.201040i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	571.577i	0.867340i	0.901072	+	0.433670i	$0.142782\pi$
$659$	571.577i	0.867340i	−0.901072	+	0.433670i	$0.857218\pi$
$660$	0	0
$661$	−46.0000	−0.0695915	−0.0347958	−	0.999394i	$-0.511078\pi$
$661$	−46.0000	−0.0695915	−0.0347958	+	0.999394i	$0.511078\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−469.574	−0.706127
$666$	0	0
$667$	− 1185.13i	− 1.77681i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	214.774i	0.320081i
$672$	0	0
$673$	878.000	1.30461	0.652303	−	0.757958i	$-0.273803\pi$
$673$	878.000	1.30461	0.652303	+	0.757958i	$0.273803\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	603.738	0.891785	0.445892	−	0.895087i	$-0.352886\pi$
$677$	603.738	0.891785	0.445892	+	0.895087i	$0.352886\pi$
$678$	0	0
$679$	− 247.871i	− 0.365053i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1172.60i	− 1.71684i	−0.512952	−	0.858418i	$-0.671448\pi$
$683$	− 1172.60i	− 1.71684i	0.512952	−	0.858418i	$-0.328552\pi$
$684$	0	0
$685$	555.000	0.810219
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−147.580	−0.214195
$690$	0	0
$691$	− 1080.56i	− 1.56377i	−0.623425	−	0.781883i	$-0.714259\pi$
$691$	− 1080.56i	− 1.56377i	0.623425	−	0.781883i	$-0.285741\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	346.410i	0.498432i
$696$	0	0
$697$	−360.000	−0.516499
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−952.565	−1.35887	−0.679433	−	0.733738i	$-0.737774\pi$
$701$	−952.565	−1.35887	−0.679433	+	0.733738i	$0.737774\pi$
$702$	0	0
$703$	54.2218i	0.0771291i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 207.846i	− 0.293983i
$708$	0	0
$709$	−1346.00	−1.89845	−0.949224	−	0.314600i	$-0.898130\pi$
$709$	−1346.00	−1.89845	−0.949224	+	0.314600i	$0.898130\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	570.197	0.799716
$714$	0	0
$715$	− 340.823i	− 0.476675i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1118.90i	1.55620i	0.628143	+	0.778098i	$0.283815\pi$
$719$	1118.90i	1.55620i	−0.628143	+	0.778098i	$0.716185\pi$
$720$	0	0
$721$	−1560.00	−2.16366
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−201.246	−0.277581
$726$	0	0
$727$	340.823i	0.468807i	0.972139	+	0.234403i	$0.0753137\pi$
$727$	340.823i	0.468807i	−0.972139	+	0.234403i	$0.924686\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 103.923i	− 0.142166i
$732$	0	0
$733$	472.000	0.643929	0.321965	−	0.946752i	$-0.395657\pi$
$733$	472.000	0.643929	0.321965	+	0.946752i	$0.395657\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	160.997	0.218449
$738$	0	0
$739$	− 1359.42i	− 1.83954i	−0.392462	−	0.919768i	$-0.628377\pi$
$739$	− 1359.42i	− 1.83954i	0.392462	−	0.919768i	$-0.371623\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 48.4974i	− 0.0652724i	−0.999467	−	0.0326362i	$-0.989610\pi$
$743$	− 48.4974i	− 0.0652724i	0.999467	−	0.0326362i	$-0.0103903\pi$
$744$	0	0
$745$	−360.000	−0.483221
$746$	0	0
$747$	0	0
$748$	0	0
$749$	912.316	1.21805
$750$	0	0
$751$	− 27.1109i	− 0.0360997i	−0.999837	−	0.0180499i	$-0.994254\pi$
$751$	− 27.1109i	− 0.0360997i	0.999837	−	0.0180499i	$-0.00574576\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	554.256i	0.734114i
$756$	0	0
$757$	−542.000	−0.715984	−0.357992	−	0.933725i	$-0.616539\pi$
$757$	−542.000	−0.715984	−0.357992	+	0.933725i	$0.616539\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	952.565	1.25173	0.625864	−	0.779932i	$-0.284747\pi$
$761$	952.565	1.25173	0.625864	+	0.779932i	$0.284747\pi$
$762$	0	0
$763$	− 1572.43i	− 2.06085i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	533.472i	0.695530i
$768$	0	0
$769$	−127.000	−0.165150	−0.0825748	−	0.996585i	$-0.526314\pi$
$769$	−127.000	−0.165150	−0.0825748	+	0.996585i	$0.526314\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	73.7902	0.0954596	0.0477298	−	0.998860i	$-0.484801\pi$
$773$	73.7902	0.0954596	0.0477298	+	0.998860i	$0.484801\pi$
$774$	0	0
$775$	− 96.8246i	− 0.124935i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1454.92i	1.86768i
$780$	0	0
$781$	−768.000	−0.983355
$782$	0	0
$783$	0	0
$784$	0	0
$785$	125.220	0.159516
$786$	0	0
$787$	− 960.500i	− 1.22046i	−0.792225	−	0.610229i	$-0.791078\pi$
$787$	− 960.500i	− 1.22046i	0.792225	−	0.610229i	$-0.208922\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1351.00i	− 1.70796i
$792$	0	0
$793$	−682.000	−0.860025
$794$	0	0
$795$	0	0
$796$	0	0
$797$	731.194	0.917433	0.458717	−	0.888583i	$-0.348309\pi$
$797$	731.194	0.917433	0.458717	+	0.888583i	$0.348309\pi$
$798$	0	0
$799$	92.9516i	0.116335i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 526.543i	− 0.655720i
$804$	0	0
$805$	−510.000	−0.633540
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1234.31	−1.52572	−0.762861	−	0.646562i	$-0.776206\pi$
$809$	−1234.31	−1.52572	−0.762861	+	0.646562i	$0.776206\pi$
$810$	0	0
$811$	− 247.871i	− 0.305636i	−0.988254	−	0.152818i	$-0.951165\pi$
$811$	− 247.871i	− 0.305636i	0.988254	−	0.152818i	$-0.0488349\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 259.808i	− 0.318782i
$816$	0	0
$817$	−420.000	−0.514076
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−576.906	−0.702686	−0.351343	−	0.936247i	$-0.614275\pi$
$821$	−576.906	−0.702686	−0.351343	+	0.936247i	$0.614275\pi$
$822$	0	0
$823$	− 139.427i	− 0.169414i	−0.996406	−	0.0847068i	$-0.973005\pi$
$823$	− 139.427i	− 0.169414i	0.996406	−	0.0847068i	$-0.0269954\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	715.337i	0.864978i	0.901639	+	0.432489i	$0.142365\pi$
$827$	715.337i	0.864978i	−0.901639	+	0.432489i	$0.857635\pi$
$828$	0	0
$829$	34.0000	0.0410133	0.0205066	−	0.999790i	$-0.493472\pi$
$829$	34.0000	0.0410133	0.0205066	+	0.999790i	$0.493472\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	73.7902	0.0885837
$834$	0	0
$835$	375.679i	0.449915i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	651.251i	0.776223i	0.921612	+	0.388112i	$0.126872\pi$
$839$	651.251i	0.776223i	−0.921612	+	0.388112i	$0.873128\pi$
$840$	0	0
$841$	779.000	0.926278
$842$	0	0
$843$	0	0
$844$	0	0
$845$	704.361	0.833564
$846$	0	0
$847$	565.456i	0.667598i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	58.8897i	0.0692006i
$852$	0	0
$853$	1028.00	1.20516	0.602579	−	0.798059i	$-0.294140\pi$
$853$	1028.00	1.20516	0.602579	+	0.798059i	$0.294140\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−677.529	−0.790582	−0.395291	−	0.918556i	$-0.629356\pi$
$857$	−677.529	−0.790582	−0.395291	+	0.918556i	$0.629356\pi$
$858$	0	0
$859$	143.300i	0.166822i	0.996515	+	0.0834112i	$0.0265815\pi$
$859$	143.300i	0.166822i	−0.996515	+	0.0834112i	$0.973419\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1688.75i	− 1.95684i	−0.206635	−	0.978418i	$-0.566251\pi$
$863$	− 1688.75i	− 1.95684i	0.206635	−	0.978418i	$-0.433749\pi$
$864$	0	0
$865$	495.000	0.572254
$866$	0	0
$867$	0	0
$868$	0	0
$869$	134.164	0.154389
$870$	0	0
$871$	511.234i	0.586950i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	86.6025i	0.0989743i
$876$	0	0
$877$	−568.000	−0.647662	−0.323831	−	0.946115i	$-0.604971\pi$
$877$	−568.000	−0.647662	−0.323831	+	0.946115i	$0.604971\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−281.745	−0.319801	−0.159900	−	0.987133i	$-0.551117\pi$
$881$	−281.745	−0.319801	−0.159900	+	0.987133i	$0.551117\pi$
$882$	0	0
$883$	− 642.915i	− 0.728103i	−0.931379	−	0.364052i	$-0.881393\pi$
$883$	− 642.915i	− 0.728103i	0.931379	−	0.364052i	$-0.118607\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1155.28i	− 1.30246i	−0.758882	−	0.651228i	$-0.774254\pi$
$887$	− 1155.28i	− 1.30246i	0.758882	−	0.651228i	$-0.225746\pi$
$888$	0	0
$889$	−900.000	−1.01237
$890$	0	0
$891$	0	0
$892$	0	0
$893$	375.659	0.420671
$894$	0	0
$895$	635.169i	0.709686i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	779.423i	0.866989i
$900$	0	0
$901$	45.0000	0.0499445
$902$	0	0
$903$	0	0
$904$	0	0
$905$	404.728	0.447214
$906$	0	0
$907$	− 666.153i	− 0.734458i	−0.930131	−	0.367229i	$-0.880307\pi$
$907$	− 666.153i	− 0.734458i	0.930131	−	0.367229i	$-0.119693\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1617.74i	1.77578i	0.460056	+	0.887890i	$0.347829\pi$
$911$	1617.74i	1.77578i	−0.460056	+	0.887890i	$0.652171\pi$
$912$	0	0
$913$	900.000	0.985761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	456.158	0.497446
$918$	0	0
$919$	588.693i	0.640580i	0.947319	+	0.320290i	$0.103780\pi$
$919$	588.693i	0.640580i	−0.947319	+	0.320290i	$0.896220\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 2438.73i	− 2.64218i
$924$	0	0
$925$	10.0000	0.0108108
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−643.988	−0.693205	−0.346603	−	0.938012i	$-0.612665\pi$
$929$	−643.988	−0.693205	−0.346603	+	0.938012i	$0.612665\pi$
$930$	0	0
$931$	− 298.220i	− 0.320322i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	103.923i	0.111148i
$936$	0	0
$937$	1712.00	1.82711	0.913554	−	0.406718i	$-0.133327\pi$
$937$	1712.00	1.82711	0.913554	+	0.406718i	$0.133327\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	630.571	0.670108	0.335054	−	0.942199i	$-0.391246\pi$
$941$	630.571	0.670108	0.335054	+	0.942199i	$0.391246\pi$
$942$	0	0
$943$	1580.18i	1.67569i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	857.365i	0.905349i	0.891676	+	0.452674i	$0.149530\pi$
$947$	857.365i	0.905349i	−0.891676	+	0.452674i	$0.850470\pi$
$948$	0	0
$949$	1672.00	1.76185
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1838.05	−1.92870	−0.964348	−	0.264636i	$-0.914748\pi$
$953$	−1838.05	−1.92870	−0.964348	+	0.264636i	$0.914748\pi$
$954$	0	0
$955$	116.190i	0.121664i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1922.58i	2.00477i
$960$	0	0
$961$	586.000	0.609781
$962$	0	0
$963$	0	0
$964$	0	0
$965$	165.469	0.171470
$966$	0	0
$967$	− 1386.53i	− 1.43384i	−0.697153	−	0.716922i	$-0.745550\pi$
$967$	− 1386.53i	− 1.43384i	0.697153	−	0.716922i	$-0.254450\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1811.73i	1.86583i	0.360091	+	0.932917i	$0.382746\pi$
$971$	1811.73i	1.86583i	−0.360091	+	0.932917i	$0.617254\pi$
$972$	0	0
$973$	−1200.00	−1.23330
$974$	0	0
$975$	0	0
$976$	0	0
$977$	764.735	0.782738	0.391369	−	0.920234i	$-0.372002\pi$
$977$	764.735	0.782738	0.391369	+	0.920234i	$0.372002\pi$
$978$	0	0
$979$	371.806i	0.379782i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1120.64i	− 1.14002i	−0.821639	−	0.570009i	$-0.806940\pi$
$983$	− 1120.64i	− 1.14002i	0.821639	−	0.570009i	$-0.193060\pi$
$984$	0	0
$985$	495.000	0.502538
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−456.158	−0.461231
$990$	0	0
$991$	1738.97i	1.75476i	0.479794	+	0.877381i	$0.340711\pi$
$991$	1738.97i	1.75476i	−0.479794	+	0.877381i	$0.659289\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	415.692i	0.417781i
$996$	0	0
$997$	632.000	0.633902	0.316951	−	0.948442i	$-0.397341\pi$
$997$	632.000	0.633902	0.316951	+	0.948442i	$0.397341\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.e.b.271.4	yes	4
3.2	odd	2	inner	2160.3.e.b.271.2	yes	4
4.3	odd	2	inner	2160.3.e.b.271.3	yes	4
12.11	even	2	inner	2160.3.e.b.271.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.3.e.b.271.1	✓	4	12.11	even	2	inner
2160.3.e.b.271.2	yes	4	3.2	odd	2	inner
2160.3.e.b.271.3	yes	4	4.3	odd	2	inner
2160.3.e.b.271.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23607 q^{5} +7.74597i q^{7} +O(q^{10})\)	\(q+2.23607 q^{5} +7.74597i q^{7} -6.92820i q^{11} +22.0000 q^{13} -6.70820 q^{17} +27.1109i q^{19} +29.4449i q^{23} +5.00000 q^{25} -40.2492 q^{29} -19.3649i q^{31} +17.3205i q^{35} +2.00000 q^{37} +53.6656 q^{41} +15.4919i q^{43} -13.8564i q^{47} -11.0000 q^{49} -6.70820 q^{53} -15.4919i q^{55} +24.2487i q^{59} -31.0000 q^{61} +49.1935 q^{65} +23.2379i q^{67} -110.851i q^{71} +76.0000 q^{73} +53.6656 q^{77} +19.3649i q^{79} +129.904i q^{83} -15.0000 q^{85} -53.6656 q^{89} +170.411i q^{91} +60.6218i q^{95} -32.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 88 q^{13} + 20 q^{25} + 8 q^{37} - 44 q^{49} - 124 q^{61} + 304 q^{73} - 60 q^{85} - 128 q^{97}+O(q^{100}) \) 4 * q + 88 * q^13 + 20 * q^25 + 8 * q^37 - 44 * q^49 - 124 * q^61 + 304 * q^73 - 60 * q^85 - 128 * q^97

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