LMFDB - Embedded newform 1728.3.g.h.703.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,3,Mod(703,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.703"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.0845896815$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 432)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.4
Root			$-0.309017 - 0.535233i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.h.703.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.70820 q^{5} +11.6190i q^{7} +15.5885i q^{11} +8.00000 q^{13} +26.8328 q^{17} +23.2379i q^{19} -31.1769i q^{23} +20.0000 q^{25} +13.4164 q^{29} -11.6190i q^{31} +77.9423i q^{35} -2.00000 q^{37} -40.2492 q^{41} +23.2379i q^{43} -31.1769i q^{47} -86.0000 q^{49} +20.1246 q^{53} +104.571i q^{55} +62.3538i q^{59} -104.000 q^{61} +53.6656 q^{65} -69.7137i q^{67} +62.3538i q^{71} +61.0000 q^{73} -181.122 q^{77} -92.9516i q^{79} +77.9423i q^{83} +180.000 q^{85} -147.580 q^{89} +92.9516i q^{91} +155.885i q^{95} +103.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{13} + 80 q^{25} - 8 q^{37} - 344 q^{49} - 416 q^{61} + 244 q^{73} + 720 q^{85} + 412 q^{97}+O(q^{100}) $ 4 * q + 32 * q^13 + 80 * q^25 - 8 * q^37 - 344 * q^49 - 416 * q^61 + 244 * q^73 + 720 * q^85 + 412 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	6.70820	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	6.70820	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	11.6190i	1.65985i	0.557875	+	0.829925i	$0.311617\pi$
$7$	11.6190i	1.65985i	−0.557875	+	0.829925i	$0.688383\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.5885i	1.41713i	0.705644	+	0.708566i	$0.250658\pi$
$11$	15.5885i	1.41713i	−0.705644	+	0.708566i	$0.749342\pi$
$12$	0	0
$13$	8.00000	0.615385	0.307692	−	0.951486i	$-0.400443\pi$
$13$	8.00000	0.615385	0.307692	+	0.951486i	$0.400443\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	26.8328	1.57840	0.789200	−	0.614136i	$-0.210495\pi$
$17$	26.8328	1.57840	0.789200	+	0.614136i	$0.210495\pi$
$18$	0	0
$19$	23.2379i	1.22305i	0.791226	+	0.611524i	$0.209443\pi$
$19$	23.2379i	1.22305i	−0.791226	+	0.611524i	$0.790557\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 31.1769i	− 1.35552i	−0.735284	−	0.677759i	$-0.762951\pi$
$23$	− 31.1769i	− 1.35552i	0.735284	−	0.677759i	$-0.237049\pi$
$24$	0	0
$25$	20.0000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	13.4164	0.462635	0.231317	−	0.972878i	$-0.425696\pi$
$29$	13.4164	0.462635	0.231317	+	0.972878i	$0.425696\pi$
$30$	0	0
$31$	− 11.6190i	− 0.374805i	−0.982283	−	0.187402i	$-0.939993\pi$
$31$	− 11.6190i	− 0.374805i	0.982283	−	0.187402i	$-0.0600068\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	77.9423i	2.22692i
$36$	0	0
$37$	−2.00000	−0.0540541	−0.0270270	−	0.999635i	$-0.508604\pi$
$37$	−2.00000	−0.0540541	−0.0270270	+	0.999635i	$0.508604\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−40.2492	−0.981688	−0.490844	−	0.871247i	$-0.663312\pi$
$41$	−40.2492	−0.981688	−0.490844	+	0.871247i	$0.663312\pi$
$42$	0	0
$43$	23.2379i	0.540416i	0.962802	+	0.270208i	$0.0870925\pi$
$43$	23.2379i	0.540416i	−0.962802	+	0.270208i	$0.912907\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 31.1769i	− 0.663339i	−0.943396	−	0.331669i	$-0.892388\pi$
$47$	− 31.1769i	− 0.663339i	0.943396	−	0.331669i	$-0.107612\pi$
$48$	0	0
$49$	−86.0000	−1.75510
$50$	0	0
$51$	0	0
$52$	0	0
$53$	20.1246	0.379710	0.189855	−	0.981812i	$-0.439198\pi$
$53$	20.1246	0.379710	0.189855	+	0.981812i	$0.439198\pi$
$54$	0	0
$55$	104.571i	1.90128i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	62.3538i	1.05684i	0.848982	+	0.528422i	$0.177216\pi$
$59$	62.3538i	1.05684i	−0.848982	+	0.528422i	$0.822784\pi$
$60$	0	0
$61$	−104.000	−1.70492	−0.852459	−	0.522794i	$-0.824890\pi$
$61$	−104.000	−1.70492	−0.852459	+	0.522794i	$0.824890\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	53.6656	0.825625
$66$	0	0
$67$	− 69.7137i	− 1.04050i	−0.854013	−	0.520251i	$-0.825838\pi$
$67$	− 69.7137i	− 1.04050i	0.854013	−	0.520251i	$-0.174162\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	62.3538i	0.878223i	0.898433	+	0.439111i	$0.144707\pi$
$71$	62.3538i	0.878223i	−0.898433	+	0.439111i	$0.855293\pi$
$72$	0	0
$73$	61.0000	0.835616	0.417808	−	0.908535i	$-0.362798\pi$
$73$	61.0000	0.835616	0.417808	+	0.908535i	$0.362798\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−181.122	−2.35223
$78$	0	0
$79$	− 92.9516i	− 1.17660i	−0.808642	−	0.588301i	$-0.799797\pi$
$79$	− 92.9516i	− 1.17660i	0.808642	−	0.588301i	$-0.200203\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	77.9423i	0.939064i	0.882916	+	0.469532i	$0.155577\pi$
$83$	77.9423i	0.939064i	−0.882916	+	0.469532i	$0.844423\pi$
$84$	0	0
$85$	180.000	2.11765
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−147.580	−1.65821	−0.829104	−	0.559095i	$-0.811149\pi$
$89$	−147.580	−1.65821	−0.829104	+	0.559095i	$0.811149\pi$
$90$	0	0
$91$	92.9516i	1.02145i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	155.885i	1.64089i
$96$	0	0
$97$	103.000	1.06186	0.530928	−	0.847417i	$-0.321844\pi$
$97$	103.000	1.06186	0.530928	+	0.847417i	$0.321844\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	100.623	0.996268	0.498134	−	0.867100i	$-0.334019\pi$
$101$	100.623	0.996268	0.498134	+	0.867100i	$0.334019\pi$
$102$	0	0
$103$	− 139.427i	− 1.35366i	−0.736137	−	0.676832i	$-0.763352\pi$
$103$	− 139.427i	− 1.35366i	0.736137	−	0.676832i	$-0.236648\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 46.7654i	− 0.437060i	−0.975830	−	0.218530i	$-0.929874\pi$
$107$	− 46.7654i	− 0.437060i	0.975830	−	0.218530i	$-0.0701261\pi$
$108$	0	0
$109$	38.0000	0.348624	0.174312	−	0.984690i	$-0.444230\pi$
$109$	38.0000	0.348624	0.174312	+	0.984690i	$0.444230\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−93.9149	−0.831105	−0.415552	−	0.909569i	$-0.636412\pi$
$113$	−93.9149	−0.831105	−0.415552	+	0.909569i	$0.636412\pi$
$114$	0	0
$115$	− 209.141i	− 1.81862i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	311.769i	2.61991i
$120$	0	0
$121$	−122.000	−1.00826
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−33.5410	−0.268328
$126$	0	0
$127$	− 151.046i	− 1.18934i	−0.803969	−	0.594671i	$-0.797282\pi$
$127$	− 151.046i	− 1.18934i	0.803969	−	0.594671i	$-0.202718\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.5885i	0.118996i	0.998228	+	0.0594979i	$0.0189500\pi$
$131$	15.5885i	0.118996i	−0.998228	+	0.0594979i	$0.981050\pi$
$132$	0	0
$133$	−270.000	−2.03008
$134$	0	0
$135$	0	0
$136$	0	0
$137$	174.413	1.27309	0.636545	−	0.771240i	$-0.280363\pi$
$137$	174.413	1.27309	0.636545	+	0.771240i	$0.280363\pi$
$138$	0	0
$139$	− 46.4758i	− 0.334358i	−0.985927	−	0.167179i	$-0.946534\pi$
$139$	− 46.4758i	− 0.334358i	0.985927	−	0.167179i	$-0.0534658\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	124.708i	0.872082i
$144$	0	0
$145$	90.0000	0.620690
$146$	0	0
$147$	0	0
$148$	0	0
$149$	114.039	0.765366	0.382683	−	0.923880i	$-0.375000\pi$
$149$	114.039	0.765366	0.382683	+	0.923880i	$0.375000\pi$
$150$	0	0
$151$	174.284i	1.15420i	0.816673	+	0.577100i	$0.195816\pi$
$151$	174.284i	1.15420i	−0.816673	+	0.577100i	$0.804184\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 77.9423i	− 0.502853i
$156$	0	0
$157$	244.000	1.55414	0.777070	−	0.629414i	$-0.216705\pi$
$157$	244.000	1.55414	0.777070	+	0.629414i	$0.216705\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	362.243	2.24996
$162$	0	0
$163$	− 185.903i	− 1.14051i	−0.821468	−	0.570255i	$-0.806844\pi$
$163$	− 185.903i	− 1.14051i	0.821468	−	0.570255i	$-0.193156\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	280.592i	1.68019i	0.542437	+	0.840096i	$0.317502\pi$
$167$	280.592i	1.68019i	−0.542437	+	0.840096i	$0.682498\pi$
$168$	0	0
$169$	−105.000	−0.621302
$170$	0	0
$171$	0	0
$172$	0	0
$173$	114.039	0.659188	0.329594	−	0.944123i	$-0.393088\pi$
$173$	114.039	0.659188	0.329594	+	0.944123i	$0.393088\pi$
$174$	0	0
$175$	232.379i	1.32788i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 15.5885i	− 0.0870864i	−0.999052	−	0.0435432i	$-0.986135\pi$
$179$	− 15.5885i	− 0.0870864i	0.999052	−	0.0435432i	$-0.0138646\pi$
$180$	0	0
$181$	−16.0000	−0.0883978	−0.0441989	−	0.999023i	$-0.514074\pi$
$181$	−16.0000	−0.0883978	−0.0441989	+	0.999023i	$0.514074\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−13.4164	−0.0725211
$186$	0	0
$187$	418.282i	2.23680i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−91.0000	−0.471503	−0.235751	−	0.971813i	$-0.575755\pi$
$193$	−91.0000	−0.471503	−0.235751	+	0.971813i	$0.575755\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	73.7902	0.374570	0.187285	−	0.982306i	$-0.440031\pi$
$197$	73.7902	0.374570	0.187285	+	0.982306i	$0.440031\pi$
$198$	0	0
$199$	− 11.6190i	− 0.0583867i	−0.999574	−	0.0291933i	$-0.990706\pi$
$199$	− 11.6190i	− 0.0583867i	0.999574	−	0.0291933i	$-0.00929385\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	155.885i	0.767904i
$204$	0	0
$205$	−270.000	−1.31707
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−362.243	−1.73322
$210$	0	0
$211$	395.044i	1.87225i	0.351670	+	0.936124i	$0.385614\pi$
$211$	395.044i	1.87225i	−0.351670	+	0.936124i	$0.614386\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	155.885i	0.725045i
$216$	0	0
$217$	135.000	0.622120
$218$	0	0
$219$	0	0
$220$	0	0
$221$	214.663	0.971324
$222$	0	0
$223$	− 185.903i	− 0.833647i	−0.908988	−	0.416823i	$-0.863143\pi$
$223$	− 185.903i	− 0.833647i	0.908988	−	0.416823i	$-0.136857\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	436.477i	1.92281i	0.275145	+	0.961403i	$0.411274\pi$
$227$	436.477i	1.92281i	−0.275145	+	0.961403i	$0.588726\pi$
$228$	0	0
$229$	−154.000	−0.672489	−0.336245	−	0.941775i	$-0.609157\pi$
$229$	−154.000	−0.672489	−0.336245	+	0.941775i	$0.609157\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−67.0820	−0.287906	−0.143953	−	0.989585i	$-0.545981\pi$
$233$	−67.0820	−0.287906	−0.143953	+	0.989585i	$0.545981\pi$
$234$	0	0
$235$	− 209.141i	− 0.889962i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 93.5307i	− 0.391342i	−0.980670	−	0.195671i	$-0.937312\pi$
$239$	− 93.5307i	− 0.391342i	0.980670	−	0.195671i	$-0.0626885\pi$
$240$	0	0
$241$	−194.000	−0.804979	−0.402490	−	0.915425i	$-0.631855\pi$
$241$	−194.000	−0.804979	−0.402490	+	0.915425i	$0.631855\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−576.906	−2.35472
$246$	0	0
$247$	185.903i	0.752645i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	249.415i	0.993687i	0.867840	+	0.496843i	$0.165508\pi$
$251$	249.415i	0.993687i	−0.867840	+	0.496843i	$0.834492\pi$
$252$	0	0
$253$	486.000	1.92095
$254$	0	0
$255$	0	0
$256$	0	0
$257$	214.663	0.835263	0.417631	−	0.908617i	$-0.362860\pi$
$257$	214.663	0.835263	0.417631	+	0.908617i	$0.362860\pi$
$258$	0	0
$259$	− 23.2379i	− 0.0897216i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 31.1769i	− 0.118543i	−0.998242	−	0.0592717i	$-0.981122\pi$
$263$	− 31.1769i	− 0.118543i	0.998242	−	0.0592717i	$-0.0188778\pi$
$264$	0	0
$265$	135.000	0.509434
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−93.9149	−0.349126	−0.174563	−	0.984646i	$-0.555851\pi$
$269$	−93.9149	−0.349126	−0.174563	+	0.984646i	$0.555851\pi$
$270$	0	0
$271$	− 34.8569i	− 0.128623i	−0.997930	−	0.0643115i	$-0.979515\pi$
$271$	− 34.8569i	− 0.128623i	0.997930	−	0.0643115i	$-0.0204851\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	311.769i	1.13371i
$276$	0	0
$277$	176.000	0.635379	0.317690	−	0.948195i	$-0.397093\pi$
$277$	176.000	0.635379	0.317690	+	0.948195i	$0.397093\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.8328	0.0954904	0.0477452	−	0.998860i	$-0.484796\pi$
$281$	26.8328	0.0954904	0.0477452	+	0.998860i	$0.484796\pi$
$282$	0	0
$283$	− 116.190i	− 0.410564i	−0.978703	−	0.205282i	$-0.934189\pi$
$283$	− 116.190i	− 0.410564i	0.978703	−	0.205282i	$-0.0658111\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 467.654i	− 1.62946i
$288$	0	0
$289$	431.000	1.49135
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−335.410	−1.14474	−0.572372	−	0.819994i	$-0.693977\pi$
$293$	−335.410	−1.14474	−0.572372	+	0.819994i	$0.693977\pi$
$294$	0	0
$295$	418.282i	1.41791i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 249.415i	− 0.834165i
$300$	0	0
$301$	−270.000	−0.897010
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−697.653	−2.28739
$306$	0	0
$307$	139.427i	0.454161i	0.973876	+	0.227080i	$0.0729180\pi$
$307$	139.427i	0.454161i	−0.973876	+	0.227080i	$0.927082\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 405.300i	− 1.30322i	−0.758556	−	0.651608i	$-0.774095\pi$
$311$	− 405.300i	− 1.30322i	0.758556	−	0.651608i	$-0.225905\pi$
$312$	0	0
$313$	−391.000	−1.24920	−0.624601	−	0.780944i	$-0.714738\pi$
$313$	−391.000	−1.24920	−0.624601	+	0.780944i	$0.714738\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	288.453	0.909946	0.454973	−	0.890505i	$-0.349649\pi$
$317$	288.453	0.909946	0.454973	+	0.890505i	$0.349649\pi$
$318$	0	0
$319$	209.141i	0.655615i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	623.538i	1.93046i
$324$	0	0
$325$	160.000	0.492308
$326$	0	0
$327$	0	0
$328$	0	0
$329$	362.243	1.10104
$330$	0	0
$331$	− 441.520i	− 1.33390i	−0.745104	−	0.666949i	$-0.767600\pi$
$331$	− 441.520i	− 1.33390i	0.745104	−	0.666949i	$-0.232400\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 467.654i	− 1.39598i
$336$	0	0
$337$	514.000	1.52522	0.762611	−	0.646857i	$-0.223917\pi$
$337$	514.000	1.52522	0.762611	+	0.646857i	$0.223917\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	181.122	0.531148
$342$	0	0
$343$	− 429.901i	− 1.25336i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 389.711i	− 1.12309i	−0.827447	−	0.561544i	$-0.810208\pi$
$347$	− 389.711i	− 1.12309i	0.827447	−	0.561544i	$-0.189792\pi$
$348$	0	0
$349$	−158.000	−0.452722	−0.226361	−	0.974043i	$-0.572683\pi$
$349$	−158.000	−0.452722	−0.226361	+	0.974043i	$0.572683\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−281.745	−0.798143	−0.399072	−	0.916920i	$-0.630668\pi$
$353$	−281.745	−0.798143	−0.399072	+	0.916920i	$0.630668\pi$
$354$	0	0
$355$	418.282i	1.17826i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	530.008i	1.47634i	0.674612	+	0.738172i	$0.264311\pi$
$359$	530.008i	1.47634i	−0.674612	+	0.738172i	$0.735689\pi$
$360$	0	0
$361$	−179.000	−0.495845
$362$	0	0
$363$	0	0
$364$	0	0
$365$	409.200	1.12110
$366$	0	0
$367$	− 243.998i	− 0.664845i	−0.943131	−	0.332422i	$-0.892134\pi$
$367$	− 243.998i	− 0.664845i	0.943131	−	0.332422i	$-0.107866\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	233.827i	0.630261i
$372$	0	0
$373$	−44.0000	−0.117962	−0.0589812	−	0.998259i	$-0.518785\pi$
$373$	−44.0000	−0.117962	−0.0589812	+	0.998259i	$0.518785\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	107.331	0.284698
$378$	0	0
$379$	325.331i	0.858392i	0.903211	+	0.429196i	$0.141203\pi$
$379$	325.331i	0.858392i	−0.903211	+	0.429196i	$0.858797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 436.477i	− 1.13963i	−0.821774	−	0.569813i	$-0.807016\pi$
$383$	− 436.477i	− 1.13963i	0.821774	−	0.569813i	$-0.192984\pi$
$384$	0	0
$385$	−1215.00	−3.15584
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−422.617	−1.08642	−0.543209	−	0.839597i	$-0.682791\pi$
$389$	−422.617	−1.08642	−0.543209	+	0.839597i	$0.682791\pi$
$390$	0	0
$391$	− 836.564i	− 2.13955i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 623.538i	− 1.57858i
$396$	0	0
$397$	88.0000	0.221662	0.110831	−	0.993839i	$-0.464649\pi$
$397$	88.0000	0.221662	0.110831	+	0.993839i	$0.464649\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	134.164	0.334574	0.167287	−	0.985908i	$-0.446499\pi$
$401$	134.164	0.334574	0.167287	+	0.985908i	$0.446499\pi$
$402$	0	0
$403$	− 92.9516i	− 0.230649i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 31.1769i	− 0.0766018i
$408$	0	0
$409$	739.000	1.80685	0.903423	−	0.428750i	$-0.141046\pi$
$409$	739.000	1.80685	0.903423	+	0.428750i	$0.141046\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−724.486	−1.75420
$414$	0	0
$415$	522.853i	1.25989i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 311.769i	− 0.744079i	−0.928217	−	0.372040i	$-0.878659\pi$
$419$	− 311.769i	− 0.744079i	0.928217	−	0.372040i	$-0.121341\pi$
$420$	0	0
$421$	−256.000	−0.608076	−0.304038	−	0.952660i	$-0.598335\pi$
$421$	−256.000	−0.608076	−0.304038	+	0.952660i	$0.598335\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	536.656	1.26272
$426$	0	0
$427$	− 1208.37i	− 2.82991i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 218.238i	− 0.506354i	−0.967420	−	0.253177i	$-0.918525\pi$
$431$	− 218.238i	− 0.506354i	0.967420	−	0.253177i	$-0.0814754\pi$
$432$	0	0
$433$	173.000	0.399538	0.199769	−	0.979843i	$-0.435981\pi$
$433$	173.000	0.399538	0.199769	+	0.979843i	$0.435981\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	724.486	1.65786
$438$	0	0
$439$	− 801.708i	− 1.82621i	−0.407721	−	0.913107i	$-0.633676\pi$
$439$	− 801.708i	− 1.82621i	0.407721	−	0.913107i	$-0.366324\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 436.477i	− 0.985275i	−0.870235	−	0.492637i	$-0.836033\pi$
$443$	− 436.477i	− 0.985275i	0.870235	−	0.492637i	$-0.163967\pi$
$444$	0	0
$445$	−990.000	−2.22472
$446$	0	0
$447$	0	0
$448$	0	0
$449$	845.234	1.88248	0.941240	−	0.337738i	$-0.109662\pi$
$449$	845.234	1.88248	0.941240	+	0.337738i	$0.109662\pi$
$450$	0	0
$451$	− 627.423i	− 1.39118i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	623.538i	1.37041i
$456$	0	0
$457$	−437.000	−0.956236	−0.478118	−	0.878296i	$-0.658681\pi$
$457$	−437.000	−0.956236	−0.478118	+	0.878296i	$0.658681\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	623.863	1.35328	0.676641	−	0.736313i	$-0.263435\pi$
$461$	623.863	1.35328	0.676641	+	0.736313i	$0.263435\pi$
$462$	0	0
$463$	197.522i	0.426614i	0.976985	+	0.213307i	$0.0684234\pi$
$463$	197.522i	0.426614i	−0.976985	+	0.213307i	$0.931577\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 420.888i	− 0.901260i	−0.892711	−	0.450630i	$-0.851199\pi$
$467$	− 420.888i	− 0.901260i	0.892711	−	0.450630i	$-0.148801\pi$
$468$	0	0
$469$	810.000	1.72708
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−362.243	−0.765841
$474$	0	0
$475$	464.758i	0.978438i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	467.654i	0.976313i	0.872756	+	0.488156i	$0.162330\pi$
$479$	467.654i	0.976313i	−0.872756	+	0.488156i	$0.837670\pi$
$480$	0	0
$481$	−16.0000	−0.0332640
$482$	0	0
$483$	0	0
$484$	0	0
$485$	690.945	1.42463
$486$	0	0
$487$	371.806i	0.763463i	0.924273	+	0.381731i	$0.124672\pi$
$487$	371.806i	0.763463i	−0.924273	+	0.381731i	$0.875328\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 701.481i	− 1.42868i	−0.699800	−	0.714339i	$-0.746728\pi$
$491$	− 701.481i	− 1.42868i	0.699800	−	0.714339i	$-0.253272\pi$
$492$	0	0
$493$	360.000	0.730223
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−724.486	−1.45772
$498$	0	0
$499$	− 69.7137i	− 0.139707i	−0.997557	−	0.0698534i	$-0.977747\pi$
$499$	− 69.7137i	− 0.139707i	0.997557	−	0.0698534i	$-0.0222532\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 467.654i	− 0.929729i	−0.885382	−	0.464865i	$-0.846103\pi$
$503$	− 467.654i	− 0.929729i	0.885382	−	0.464865i	$-0.153897\pi$
$504$	0	0
$505$	675.000	1.33663
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−60.3738	−0.118613	−0.0593063	−	0.998240i	$-0.518889\pi$
$509$	−60.3738	−0.118613	−0.0593063	+	0.998240i	$0.518889\pi$
$510$	0	0
$511$	708.756i	1.38700i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 935.307i	− 1.81613i
$516$	0	0
$517$	486.000	0.940039
$518$	0	0
$519$	0	0
$520$	0	0
$521$	160.997	0.309015	0.154508	−	0.987992i	$-0.450621\pi$
$521$	160.997	0.309015	0.154508	+	0.987992i	$0.450621\pi$
$522$	0	0
$523$	139.427i	0.266592i	0.991076	+	0.133296i	$0.0425560\pi$
$523$	139.427i	0.266592i	−0.991076	+	0.133296i	$0.957444\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 311.769i	− 0.591592i
$528$	0	0
$529$	−443.000	−0.837429
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−321.994	−0.604116
$534$	0	0
$535$	− 313.712i	− 0.586377i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1340.61i	− 2.48721i
$540$	0	0
$541$	524.000	0.968577	0.484288	−	0.874908i	$-0.339079\pi$
$541$	524.000	0.968577	0.484288	+	0.874908i	$0.339079\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	254.912	0.467728
$546$	0	0
$547$	418.282i	0.764684i	0.924021	+	0.382342i	$0.124882\pi$
$547$	418.282i	0.764684i	−0.924021	+	0.382342i	$0.875118\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	311.769i	0.565824i
$552$	0	0
$553$	1080.00	1.95298
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−127.456	−0.228826	−0.114413	−	0.993433i	$-0.536499\pi$
$557$	−127.456	−0.228826	−0.114413	+	0.993433i	$0.536499\pi$
$558$	0	0
$559$	185.903i	0.332564i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	670.304i	1.19059i	0.803506	+	0.595296i	$0.202965\pi$
$563$	670.304i	1.19059i	−0.803506	+	0.595296i	$0.797035\pi$
$564$	0	0
$565$	−630.000	−1.11504
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.8328	0.0471578	0.0235789	−	0.999722i	$-0.492494\pi$
$569$	26.8328	0.0471578	0.0235789	+	0.999722i	$0.492494\pi$
$570$	0	0
$571$	743.613i	1.30230i	0.758949	+	0.651150i	$0.225713\pi$
$571$	743.613i	1.30230i	−0.758949	+	0.651150i	$0.774287\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 623.538i	− 1.08441i
$576$	0	0
$577$	542.000	0.939341	0.469671	−	0.882842i	$-0.344373\pi$
$577$	542.000	0.939341	0.469671	+	0.882842i	$0.344373\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−905.608	−1.55870
$582$	0	0
$583$	313.712i	0.538099i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 233.827i	− 0.398342i	−0.979965	−	0.199171i	$-0.936175\pi$
$587$	− 233.827i	− 0.398342i	0.979965	−	0.199171i	$-0.0638249\pi$
$588$	0	0
$589$	270.000	0.458404
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−228.079	−0.384619	−0.192309	−	0.981334i	$-0.561598\pi$
$593$	−228.079	−0.384619	−0.192309	+	0.981334i	$0.561598\pi$
$594$	0	0
$595$	2091.41i	3.51498i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1091.19i	1.82169i	0.412749	+	0.910845i	$0.364569\pi$
$599$	1091.19i	1.82169i	−0.412749	+	0.910845i	$0.635431\pi$
$600$	0	0
$601$	−841.000	−1.39933	−0.699667	−	0.714469i	$-0.746668\pi$
$601$	−841.000	−1.39933	−0.699667	+	0.714469i	$0.746668\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−818.401	−1.35273
$606$	0	0
$607$	− 371.806i	− 0.612531i	−0.951946	−	0.306266i	$-0.900920\pi$
$607$	− 371.806i	− 0.612531i	0.951946	−	0.306266i	$-0.0990796\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 249.415i	− 0.408208i
$612$	0	0
$613$	−578.000	−0.942904	−0.471452	−	0.881892i	$-0.656270\pi$
$613$	−578.000	−0.942904	−0.471452	+	0.881892i	$0.656270\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1033.06	−1.67433	−0.837166	−	0.546948i	$-0.815789\pi$
$617$	−1033.06	−1.67433	−0.837166	+	0.546948i	$0.815789\pi$
$618$	0	0
$619$	− 836.564i	− 1.35148i	−0.737141	−	0.675739i	$-0.763825\pi$
$619$	− 836.564i	− 1.35148i	0.737141	−	0.675739i	$-0.236175\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1714.73i	− 2.75238i
$624$	0	0
$625$	−725.000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−53.6656	−0.0853190
$630$	0	0
$631$	197.522i	0.313030i	0.987676	+	0.156515i	$0.0500260\pi$
$631$	197.522i	0.313030i	−0.987676	+	0.156515i	$0.949974\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1013.25i	− 1.59567i
$636$	0	0
$637$	−688.000	−1.08006
$638$	0	0
$639$	0	0
$640$	0	0
$641$	818.401	1.27676	0.638378	−	0.769723i	$-0.279606\pi$
$641$	818.401	1.27676	0.638378	+	0.769723i	$0.279606\pi$
$642$	0	0
$643$	− 697.137i	− 1.08419i	−0.840316	−	0.542097i	$-0.817630\pi$
$643$	− 697.137i	− 1.08419i	0.840316	−	0.542097i	$-0.182370\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	124.708i	0.192748i	0.995345	+	0.0963738i	$0.0307244\pi$
$647$	124.708i	0.192748i	−0.995345	+	0.0963738i	$0.969276\pi$
$648$	0	0
$649$	−972.000	−1.49769
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1227.60	−1.87994	−0.939970	−	0.341256i	$-0.889148\pi$
$653$	−1227.60	−1.87994	−0.939970	+	0.341256i	$0.889148\pi$
$654$	0	0
$655$	104.571i	0.159650i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 77.9423i	− 0.118274i	−0.998250	−	0.0591368i	$-0.981165\pi$
$659$	− 77.9423i	− 0.118274i	0.998250	−	0.0591368i	$-0.0188348\pi$
$660$	0	0
$661$	−494.000	−0.747352	−0.373676	−	0.927559i	$-0.621903\pi$
$661$	−494.000	−0.747352	−0.373676	+	0.927559i	$0.621903\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1811.22	−2.72363
$666$	0	0
$667$	− 418.282i	− 0.627110i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1621.20i	− 2.41609i
$672$	0	0
$673$	653.000	0.970282	0.485141	−	0.874436i	$-0.338768\pi$
$673$	653.000	0.970282	0.485141	+	0.874436i	$0.338768\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1328.22	1.96193	0.980963	−	0.194193i	$-0.0622088\pi$
$677$	1328.22	1.96193	0.980963	+	0.194193i	$0.0622088\pi$
$678$	0	0
$679$	1196.75i	1.76252i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	436.477i	0.639058i	0.947576	+	0.319529i	$0.103525\pi$
$683$	436.477i	0.639058i	−0.947576	+	0.319529i	$0.896475\pi$
$684$	0	0
$685$	1170.00	1.70803
$686$	0	0
$687$	0	0
$688$	0	0
$689$	160.997	0.233667
$690$	0	0
$691$	− 511.234i	− 0.739846i	−0.929062	−	0.369923i	$-0.879384\pi$
$691$	− 511.234i	− 0.739846i	0.929062	−	0.369923i	$-0.120616\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 311.769i	− 0.448589i
$696$	0	0
$697$	−1080.00	−1.54950
$698$	0	0
$699$	0	0
$700$	0	0
$701$	851.942	1.21532	0.607662	−	0.794196i	$-0.292108\pi$
$701$	851.942	1.21532	0.607662	+	0.794196i	$0.292108\pi$
$702$	0	0
$703$	− 46.4758i	− 0.0661107i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1169.13i	1.65366i
$708$	0	0
$709$	−4.00000	−0.00564175	−0.00282087	−	0.999996i	$-0.500898\pi$
$709$	−4.00000	−0.00564175	−0.00282087	+	0.999996i	$0.500898\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−362.243	−0.508055
$714$	0	0
$715$	836.564i	1.17002i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	62.3538i	0.0867230i	0.999059	+	0.0433615i	$0.0138067\pi$
$719$	62.3538i	0.0867230i	−0.999059	+	0.0433615i	$0.986193\pi$
$720$	0	0
$721$	1620.00	2.24688
$722$	0	0
$723$	0	0
$724$	0	0
$725$	268.328	0.370108
$726$	0	0
$727$	− 174.284i	− 0.239731i	−0.992790	−	0.119865i	$-0.961754\pi$
$727$	− 174.284i	− 0.239731i	0.992790	−	0.119865i	$-0.0382463\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	623.538i	0.852994i
$732$	0	0
$733$	278.000	0.379263	0.189632	−	0.981855i	$-0.439271\pi$
$733$	278.000	0.379263	0.189632	+	0.981855i	$0.439271\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1086.73	1.47453
$738$	0	0
$739$	− 1068.94i	− 1.44647i	−0.690601	−	0.723236i	$-0.742654\pi$
$739$	− 1068.94i	− 1.44647i	0.690601	−	0.723236i	$-0.257346\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 498.831i	− 0.671374i	−0.941974	−	0.335687i	$-0.891032\pi$
$743$	− 498.831i	− 0.671374i	0.941974	−	0.335687i	$-0.108968\pi$
$744$	0	0
$745$	765.000	1.02685
$746$	0	0
$747$	0	0
$748$	0	0
$749$	543.365	0.725453
$750$	0	0
$751$	− 429.901i	− 0.572438i	−0.958164	−	0.286219i	$-0.907601\pi$
$751$	− 429.901i	− 0.572438i	0.958164	−	0.286219i	$-0.0923985\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1169.13i	1.54852i
$756$	0	0
$757$	−1078.00	−1.42404	−0.712021	−	0.702158i	$-0.752220\pi$
$757$	−1078.00	−1.42404	−0.712021	+	0.702158i	$0.752220\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	697.653	0.916758	0.458379	−	0.888757i	$-0.348430\pi$
$761$	697.653	0.916758	0.458379	+	0.888757i	$0.348430\pi$
$762$	0	0
$763$	441.520i	0.578663i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	498.831i	0.650366i
$768$	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
$773$	−469.574	−0.607470	−0.303735	−	0.952757i	$-0.598234\pi$
$773$	−469.574	−0.607470	−0.303735	+	0.952757i	$0.598234\pi$
$774$	0	0
$775$	− 232.379i	− 0.299844i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 935.307i	− 1.20065i
$780$	0	0
$781$	−972.000	−1.24456
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1636.80	2.08510
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1091.19i	− 1.37951i
$792$	0	0
$793$	−832.000	−1.04918
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−690.945	−0.866932	−0.433466	−	0.901170i	$-0.642710\pi$
$797$	−690.945	−0.866932	−0.433466	+	0.901170i	$0.642710\pi$
$798$	0	0
$799$	− 836.564i	− 1.04701i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	950.896i	1.18418i
$804$	0	0
$805$	2430.00	3.01863
$806$	0	0
$807$	0	0
$808$	0	0
$809$	898.899	1.11112	0.555562	−	0.831475i	$-0.312503\pi$
$809$	898.899	1.11112	0.555562	+	0.831475i	$0.312503\pi$
$810$	0	0
$811$	− 1092.18i	− 1.34671i	−0.739320	−	0.673355i	$-0.764853\pi$
$811$	− 1092.18i	− 1.34671i	0.739320	−	0.673355i	$-0.235147\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1247.08i	− 1.53016i
$816$	0	0
$817$	−540.000	−0.660955
$818$	0	0
$819$	0	0
$820$	0	0
$821$	550.073	0.670003	0.335002	−	0.942218i	$-0.391263\pi$
$821$	550.073	0.670003	0.335002	+	0.942218i	$0.391263\pi$
$822$	0	0
$823$	639.042i	0.776479i	0.921559	+	0.388240i	$0.126917\pi$
$823$	639.042i	0.776479i	−0.921559	+	0.388240i	$0.873083\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	748.246i	0.904771i	0.891822	+	0.452386i	$0.149427\pi$
$827$	748.246i	0.904771i	−0.891822	+	0.452386i	$0.850573\pi$
$828$	0	0
$829$	146.000	0.176116	0.0880579	−	0.996115i	$-0.471934\pi$
$829$	146.000	0.176116	0.0880579	+	0.996115i	$0.471934\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2307.62	−2.77025
$834$	0	0
$835$	1882.27i	2.25422i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	685.892i	0.817511i	0.912644	+	0.408756i	$0.134037\pi$
$839$	685.892i	0.817511i	−0.912644	+	0.408756i	$0.865963\pi$
$840$	0	0
$841$	−661.000	−0.785969
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−704.361	−0.833564
$846$	0	0
$847$	− 1417.51i	− 1.67357i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	62.3538i	0.0732712i
$852$	0	0
$853$	−1058.00	−1.24033	−0.620164	−	0.784472i	$-0.712934\pi$
$853$	−1058.00	−1.24033	−0.620164	+	0.784472i	$0.712934\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	939.149	1.09586	0.547928	−	0.836526i	$-0.315417\pi$
$857$	939.149	1.09586	0.547928	+	0.836526i	$0.315417\pi$
$858$	0	0
$859$	929.516i	1.08209i	0.840993	+	0.541045i	$0.181971\pi$
$859$	929.516i	1.08209i	−0.840993	+	0.541045i	$0.818029\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 311.769i	− 0.361262i	−0.983551	−	0.180631i	$-0.942186\pi$
$863$	− 311.769i	− 0.361262i	0.983551	−	0.180631i	$-0.0578140\pi$
$864$	0	0
$865$	765.000	0.884393
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1448.97	1.66740
$870$	0	0
$871$	− 557.710i	− 0.640310i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 389.711i	− 0.445384i
$876$	0	0
$877$	−62.0000	−0.0706956	−0.0353478	−	0.999375i	$-0.511254\pi$
$877$	−62.0000	−0.0706956	−0.0353478	+	0.999375i	$0.511254\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	160.997	0.182743	0.0913717	−	0.995817i	$-0.470875\pi$
$881$	160.997	0.182743	0.0913717	+	0.995817i	$0.470875\pi$
$882$	0	0
$883$	− 999.230i	− 1.13163i	−0.824532	−	0.565815i	$-0.808562\pi$
$883$	− 999.230i	− 1.13163i	0.824532	−	0.565815i	$-0.191438\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 124.708i	− 0.140595i	−0.997526	−	0.0702974i	$-0.977605\pi$
$887$	− 124.708i	− 0.140595i	0.997526	−	0.0702974i	$-0.0223948\pi$
$888$	0	0
$889$	1755.00	1.97413
$890$	0	0
$891$	0	0
$892$	0	0
$893$	724.486	0.811295
$894$	0	0
$895$	− 104.571i	− 0.116839i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 155.885i	− 0.173398i
$900$	0	0
$901$	540.000	0.599334
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−107.331	−0.118598
$906$	0	0
$907$	− 813.327i	− 0.896722i	−0.893853	−	0.448361i	$-0.852008\pi$
$907$	− 813.327i	− 0.896722i	0.893853	−	0.448361i	$-0.147992\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1153.55i	− 1.26624i	−0.774053	−	0.633121i	$-0.781774\pi$
$911$	− 1153.55i	− 1.26624i	0.774053	−	0.633121i	$-0.218226\pi$
$912$	0	0
$913$	−1215.00	−1.33078
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−181.122	−0.197515
$918$	0	0
$919$	34.8569i	0.0379291i	0.999820	+	0.0189646i	$0.00603697\pi$
$919$	34.8569i	0.0379291i	−0.999820	+	0.0189646i	$0.993963\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	498.831i	0.540445i
$924$	0	0
$925$	−40.0000	−0.0432432
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1583.14	−1.70413	−0.852065	−	0.523437i	$-0.824650\pi$
$929$	−1583.14	−1.70413	−0.852065	+	0.523437i	$0.824650\pi$
$930$	0	0
$931$	− 1998.46i	− 2.14657i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2805.92i	3.00099i
$936$	0	0
$937$	1127.00	1.20277	0.601387	−	0.798958i	$-0.294615\pi$
$937$	1127.00	1.20277	0.601387	+	0.798958i	$0.294615\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	758.027	0.805555	0.402777	−	0.915298i	$-0.368045\pi$
$941$	758.027	0.805555	0.402777	+	0.915298i	$0.368045\pi$
$942$	0	0
$943$	1254.85i	1.33070i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	545.596i	0.576131i	0.957611	+	0.288065i	$0.0930121\pi$
$947$	545.596i	0.576131i	−0.957611	+	0.288065i	$0.906988\pi$
$948$	0	0
$949$	488.000	0.514226
$950$	0	0
$951$	0	0
$952$	0	0
$953$	617.155	0.647592	0.323796	−	0.946127i	$-0.395041\pi$
$953$	617.155	0.647592	0.323796	+	0.946127i	$0.395041\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2026.50i	2.11314i
$960$	0	0
$961$	826.000	0.859521
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−610.447	−0.632587
$966$	0	0
$967$	104.571i	0.108139i	0.998537	+	0.0540696i	$0.0172193\pi$
$967$	104.571i	0.108139i	−0.998537	+	0.0540696i	$0.982781\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 607.950i	− 0.626107i	−0.949736	−	0.313053i	$-0.898648\pi$
$971$	− 607.950i	− 0.626107i	0.949736	−	0.313053i	$-0.101352\pi$
$972$	0	0
$973$	540.000	0.554985
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1569.72	1.60667	0.803337	−	0.595525i	$-0.203056\pi$
$977$	1569.72	1.60667	0.803337	+	0.595525i	$0.203056\pi$
$978$	0	0
$979$	− 2300.55i	− 2.34990i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 904.131i	− 0.919767i	−0.887979	−	0.459883i	$-0.847891\pi$
$983$	− 904.131i	− 0.919767i	0.887979	−	0.459883i	$-0.152109\pi$
$984$	0	0
$985$	495.000	0.502538
$986$	0	0
$987$	0	0
$988$	0	0
$989$	724.486	0.732544
$990$	0	0
$991$	− 1707.99i	− 1.72350i	−0.507336	−	0.861749i	$-0.669370\pi$
$991$	− 1707.99i	− 1.72350i	0.507336	−	0.861749i	$-0.330630\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 77.9423i	− 0.0783340i
$996$	0	0
$997$	−32.0000	−0.0320963	−0.0160481	−	0.999871i	$-0.505109\pi$
$997$	−32.0000	−0.0320963	−0.0160481	+	0.999871i	$0.505109\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.h.703.4		4
3.2	odd	2	inner	1728.3.g.h.703.2		4
4.3	odd	2	inner	1728.3.g.h.703.3		4
8.3	odd	2		432.3.g.e.271.1	✓	4
8.5	even	2		432.3.g.e.271.2	yes	4
12.11	even	2	inner	1728.3.g.h.703.1		4
24.5	odd	2		432.3.g.e.271.4	yes	4
24.11	even	2		432.3.g.e.271.3	yes	4
72.5	odd	6		1296.3.o.x.703.1		4
72.11	even	6		1296.3.o.x.271.1		4
72.13	even	6		1296.3.o.y.703.2		4
72.29	odd	6		1296.3.o.y.271.1		4
72.43	odd	6		1296.3.o.y.271.2		4
72.59	even	6		1296.3.o.y.703.1		4
72.61	even	6		1296.3.o.x.271.2		4
72.67	odd	6		1296.3.o.x.703.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.3.g.e.271.1	✓	4	8.3	odd	2
432.3.g.e.271.2	yes	4	8.5	even	2
432.3.g.e.271.3	yes	4	24.11	even	2
432.3.g.e.271.4	yes	4	24.5	odd	2
1296.3.o.x.271.1		4	72.11	even	6
1296.3.o.x.271.2		4	72.61	even	6
1296.3.o.x.703.1		4	72.5	odd	6
1296.3.o.x.703.2		4	72.67	odd	6
1296.3.o.y.271.1		4	72.29	odd	6
1296.3.o.y.271.2		4	72.43	odd	6
1296.3.o.y.703.1		4	72.59	even	6
1296.3.o.y.703.2		4	72.13	even	6
1728.3.g.h.703.1		4	12.11	even	2	inner
1728.3.g.h.703.2		4	3.2	odd	2	inner
1728.3.g.h.703.3		4	4.3	odd	2	inner
1728.3.g.h.703.4		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.70820 q^{5} +11.6190i q^{7} +15.5885i q^{11} +8.00000 q^{13} +26.8328 q^{17} +23.2379i q^{19} -31.1769i q^{23} +20.0000 q^{25} +13.4164 q^{29} -11.6190i q^{31} +77.9423i q^{35} -2.00000 q^{37} -40.2492 q^{41} +23.2379i q^{43} -31.1769i q^{47} -86.0000 q^{49} +20.1246 q^{53} +104.571i q^{55} +62.3538i q^{59} -104.000 q^{61} +53.6656 q^{65} -69.7137i q^{67} +62.3538i q^{71} +61.0000 q^{73} -181.122 q^{77} -92.9516i q^{79} +77.9423i q^{83} +180.000 q^{85} -147.580 q^{89} +92.9516i q^{91} +155.885i q^{95} +103.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{13} + 80 q^{25} - 8 q^{37} - 344 q^{49} - 416 q^{61} + 244 q^{73} + 720 q^{85} + 412 q^{97}+O(q^{100}) \) 4 * q + 32 * q^13 + 80 * q^25 - 8 * q^37 - 344 * q^49 - 416 * q^61 + 244 * q^73 + 720 * q^85 + 412 * q^97

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