LMFDB - Embedded newform 4032.3.d.n.449.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4032,3,Mod(449,4032)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4032, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4032.449"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4032.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,-64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$109.864042590$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 44x^{10} + 719x^{8} + 5356x^{6} + 17809x^{4} + 20000x^{2} + 144 $ x^12 + 44x^10 + 719x^8 + 5356x^6 + 17809x^4 + 20000*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 2016)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.12
Root			$-1.48336i$ of defining polynomial
Character	$\chi$	$=$	4032.449
Dual form			4032.3.d.n.449.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.07031i q^{5} -2.64575 q^{7} -2.06952i q^{11} +4.46871 q^{13} -9.11630i q^{17} -5.96748 q^{19} -19.7029i q^{23} -57.2705 q^{25} +6.15244i q^{29} +53.8344 q^{31} -23.9978i q^{35} +36.4171 q^{37} -64.4821i q^{41} +73.4565 q^{43} +22.5045i q^{47} +7.00000 q^{49} -14.5690i q^{53} +18.7712 q^{55} +71.8103i q^{59} +38.8208 q^{61} +40.5325i q^{65} +92.4697 q^{67} -29.8169i q^{71} -62.3248 q^{73} +5.47543i q^{77} +120.133 q^{79} -110.484i q^{83} +82.6877 q^{85} +125.519i q^{89} -11.8231 q^{91} -54.1269i q^{95} -89.6087 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{13} - 64 q^{19} - 124 q^{25} + 160 q^{31} - 56 q^{37} + 64 q^{43} + 84 q^{49} - 160 q^{55} - 104 q^{61} + 64 q^{67} - 64 q^{73} + 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) $ 12 * q - 16 * q^13 - 64 * q^19 - 124 * q^25 + 160 * q^31 - 56 * q^37 + 64 * q^43 + 84 * q^49 - 160 * q^55 - 104 * q^61 + 64 * q^67 - 64 * q^73 + 32 * q^79 + 184 * q^85 - 96 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	9.07031i	1.81406i	0.421064	+	0.907031i	$0.361657\pi$
$5$	9.07031i	1.81406i	−0.421064	+	0.907031i	$0.638343\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.06952i	− 0.188138i	−0.995566	−	0.0940690i	$-0.970013\pi$
$11$	− 2.06952i	− 0.188138i	0.995566	−	0.0940690i	$-0.0299874\pi$
$12$	0	0
$13$	4.46871	0.343747	0.171873	−	0.985119i	$-0.445018\pi$
$13$	4.46871	0.343747	0.171873	+	0.985119i	$0.445018\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 9.11630i	− 0.536253i	−0.963384	−	0.268127i	$-0.913595\pi$
$17$	− 9.11630i	− 0.536253i	0.963384	−	0.268127i	$-0.0864045\pi$
$18$	0	0
$19$	−5.96748	−0.314078	−0.157039	−	0.987592i	$-0.550195\pi$
$19$	−5.96748	−0.314078	−0.157039	+	0.987592i	$0.550195\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 19.7029i	− 0.856649i	−0.903625	−	0.428325i	$-0.859104\pi$
$23$	− 19.7029i	− 0.856649i	0.903625	−	0.428325i	$-0.140896\pi$
$24$	0	0
$25$	−57.2705	−2.29082
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.15244i	0.212153i	0.994358	+	0.106077i	$0.0338289\pi$
$29$	6.15244i	0.212153i	−0.994358	+	0.106077i	$0.966171\pi$
$30$	0	0
$31$	53.8344	1.73659	0.868296	−	0.496046i	$-0.165215\pi$
$31$	53.8344	1.73659	0.868296	+	0.496046i	$0.165215\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 23.9978i	− 0.685651i
$36$	0	0
$37$	36.4171	0.984245	0.492122	−	0.870526i	$-0.336221\pi$
$37$	36.4171	0.984245	0.492122	+	0.870526i	$0.336221\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 64.4821i	− 1.57273i	−0.617759	−	0.786367i	$-0.711959\pi$
$41$	− 64.4821i	− 1.57273i	0.617759	−	0.786367i	$-0.288041\pi$
$42$	0	0
$43$	73.4565	1.70829	0.854146	−	0.520033i	$-0.174081\pi$
$43$	73.4565	1.70829	0.854146	+	0.520033i	$0.174081\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	22.5045i	0.478820i	0.970919	+	0.239410i	$0.0769540\pi$
$47$	22.5045i	0.478820i	−0.970919	+	0.239410i	$0.923046\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 14.5690i	− 0.274887i	−0.990510	−	0.137444i	$-0.956111\pi$
$53$	− 14.5690i	− 0.274887i	0.990510	−	0.137444i	$-0.0438886\pi$
$54$	0	0
$55$	18.7712	0.341294
$56$	0	0
$57$	0	0
$58$	0	0
$59$	71.8103i	1.21712i	0.793507	+	0.608562i	$0.208253\pi$
$59$	71.8103i	1.21712i	−0.793507	+	0.608562i	$0.791747\pi$
$60$	0	0
$61$	38.8208	0.636406	0.318203	−	0.948023i	$-0.396921\pi$
$61$	38.8208	0.636406	0.318203	+	0.948023i	$0.396921\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	40.5325i	0.623578i
$66$	0	0
$67$	92.4697	1.38014	0.690072	−	0.723741i	$-0.257579\pi$
$67$	92.4697	1.38014	0.690072	+	0.723741i	$0.257579\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 29.8169i	− 0.419956i	−0.977706	−	0.209978i	$-0.932661\pi$
$71$	− 29.8169i	− 0.419956i	0.977706	−	0.209978i	$-0.0673392\pi$
$72$	0	0
$73$	−62.3248	−0.853764	−0.426882	−	0.904307i	$-0.640388\pi$
$73$	−62.3248	−0.853764	−0.426882	+	0.904307i	$0.640388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.47543i	0.0711095i
$78$	0	0
$79$	120.133	1.52067	0.760335	−	0.649531i	$-0.225035\pi$
$79$	120.133	1.52067	0.760335	+	0.649531i	$0.225035\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 110.484i	− 1.33114i	−0.746336	−	0.665569i	$-0.768189\pi$
$83$	− 110.484i	− 1.33114i	0.746336	−	0.665569i	$-0.231811\pi$
$84$	0	0
$85$	82.6877	0.972796
$86$	0	0
$87$	0	0
$88$	0	0
$89$	125.519i	1.41032i	0.709047	+	0.705161i	$0.249125\pi$
$89$	125.519i	1.41032i	−0.709047	+	0.705161i	$0.750875\pi$
$90$	0	0
$91$	−11.8231	−0.129924
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 54.1269i	− 0.569757i
$96$	0	0
$97$	−89.6087	−0.923801	−0.461900	−	0.886932i	$-0.652832\pi$
$97$	−89.6087	−0.923801	−0.461900	+	0.886932i	$0.652832\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 85.3777i	− 0.845324i	−0.906287	−	0.422662i	$-0.861096\pi$
$101$	− 85.3777i	− 0.845324i	0.906287	−	0.422662i	$-0.138904\pi$
$102$	0	0
$103$	17.2854	0.167819	0.0839097	−	0.996473i	$-0.473259\pi$
$103$	17.2854	0.167819	0.0839097	+	0.996473i	$0.473259\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 68.5297i	− 0.640464i	−0.947339	−	0.320232i	$-0.896239\pi$
$107$	− 68.5297i	− 0.640464i	0.947339	−	0.320232i	$-0.103761\pi$
$108$	0	0
$109$	−47.5882	−0.436589	−0.218295	−	0.975883i	$-0.570049\pi$
$109$	−47.5882	−0.436589	−0.218295	+	0.975883i	$0.570049\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	147.922i	1.30904i	0.756044	+	0.654521i	$0.227130\pi$
$113$	147.922i	1.30904i	−0.756044	+	0.654521i	$0.772870\pi$
$114$	0	0
$115$	178.712	1.55401
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.1195i	0.202685i
$120$	0	0
$121$	116.717	0.964604
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 292.704i	− 2.34163i
$126$	0	0
$127$	−65.1644	−0.513105	−0.256553	−	0.966530i	$-0.582587\pi$
$127$	−65.1644	−0.513105	−0.256553	+	0.966530i	$0.582587\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 141.639i	− 1.08121i	−0.841276	−	0.540606i	$-0.818195\pi$
$131$	− 141.639i	− 1.08121i	0.841276	−	0.540606i	$-0.181805\pi$
$132$	0	0
$133$	15.7885	0.118710
$134$	0	0
$135$	0	0
$136$	0	0
$137$	51.7140i	0.377475i	0.982028	+	0.188737i	$0.0604395\pi$
$137$	51.7140i	0.377475i	−0.982028	+	0.188737i	$0.939561\pi$
$138$	0	0
$139$	−117.576	−0.845867	−0.422934	−	0.906161i	$-0.639000\pi$
$139$	−117.576	−0.845867	−0.422934	+	0.906161i	$0.639000\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 9.24806i	− 0.0646718i
$144$	0	0
$145$	−55.8045	−0.384859
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 122.781i	− 0.824033i	−0.911176	−	0.412017i	$-0.864825\pi$
$149$	− 122.781i	− 0.824033i	0.911176	−	0.412017i	$-0.135175\pi$
$150$	0	0
$151$	96.1236	0.636580	0.318290	−	0.947993i	$-0.396891\pi$
$151$	96.1236	0.636580	0.318290	+	0.947993i	$0.396891\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	488.294i	3.15029i
$156$	0	0
$157$	−27.4380	−0.174765	−0.0873823	−	0.996175i	$-0.527850\pi$
$157$	−27.4380	−0.174765	−0.0873823	+	0.996175i	$0.527850\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	52.1291i	0.323783i
$162$	0	0
$163$	−36.0228	−0.220999	−0.110499	−	0.993876i	$-0.535245\pi$
$163$	−36.0228	−0.220999	−0.110499	+	0.993876i	$0.535245\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	239.594i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$167$	239.594i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$168$	0	0
$169$	−149.031	−0.881838
$170$	0	0
$171$	0	0
$172$	0	0
$173$	38.2933i	0.221349i	0.993857	+	0.110674i	$0.0353010\pi$
$173$	38.2933i	0.221349i	−0.993857	+	0.110674i	$0.964699\pi$
$174$	0	0
$175$	151.524	0.865849
$176$	0	0
$177$	0	0
$178$	0	0
$179$	289.394i	1.61673i	0.588682	+	0.808364i	$0.299647\pi$
$179$	289.394i	1.61673i	−0.588682	+	0.808364i	$0.700353\pi$
$180$	0	0
$181$	124.035	0.685276	0.342638	−	0.939468i	$-0.388680\pi$
$181$	124.035	0.685276	0.342638	+	0.939468i	$0.388680\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	330.314i	1.78548i
$186$	0	0
$187$	−18.8663	−0.100890
$188$	0	0
$189$	0	0
$190$	0	0
$191$	177.779i	0.930781i	0.885105	+	0.465391i	$0.154086\pi$
$191$	177.779i	0.930781i	−0.885105	+	0.465391i	$0.845914\pi$
$192$	0	0
$193$	−69.0026	−0.357526	−0.178763	−	0.983892i	$-0.557210\pi$
$193$	−69.0026	−0.357526	−0.178763	+	0.983892i	$0.557210\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 46.2047i	− 0.234542i	−0.993100	−	0.117271i	$-0.962585\pi$
$197$	− 46.2047i	− 0.234542i	0.993100	−	0.117271i	$-0.0374146\pi$
$198$	0	0
$199$	−290.833	−1.46147	−0.730735	−	0.682661i	$-0.760823\pi$
$199$	−290.833	−1.46147	−0.730735	+	0.682661i	$0.760823\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 16.2778i	− 0.0801863i
$204$	0	0
$205$	584.873	2.85304
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.3498i	0.0590900i
$210$	0	0
$211$	288.667	1.36809	0.684045	−	0.729440i	$-0.260219\pi$
$211$	288.667	1.36809	0.684045	+	0.729440i	$0.260219\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	666.274i	3.09895i
$216$	0	0
$217$	−142.432	−0.656370
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 40.7381i	− 0.184335i
$222$	0	0
$223$	244.448	1.09618	0.548089	−	0.836420i	$-0.315355\pi$
$223$	244.448	1.09618	0.548089	+	0.836420i	$0.315355\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	165.454i	0.728871i	0.931229	+	0.364436i	$0.118738\pi$
$227$	165.454i	0.728871i	−0.931229	+	0.364436i	$0.881262\pi$
$228$	0	0
$229$	25.3894	0.110871	0.0554354	−	0.998462i	$-0.482345\pi$
$229$	25.3894	0.110871	0.0554354	+	0.998462i	$0.482345\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 151.158i	− 0.648747i	−0.945929	−	0.324374i	$-0.894846\pi$
$233$	− 151.158i	− 0.648747i	0.945929	−	0.324374i	$-0.105154\pi$
$234$	0	0
$235$	−204.123	−0.868609
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 392.194i	− 1.64098i	−0.571662	−	0.820489i	$-0.693701\pi$
$239$	− 392.194i	− 1.64098i	0.571662	−	0.820489i	$-0.306299\pi$
$240$	0	0
$241$	−424.271	−1.76046	−0.880231	−	0.474545i	$-0.842612\pi$
$241$	−424.271	−1.76046	−0.880231	+	0.474545i	$0.842612\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	63.4922i	0.259152i
$246$	0	0
$247$	−26.6669	−0.107963
$248$	0	0
$249$	0	0
$250$	0	0
$251$	304.413i	1.21280i	0.795160	+	0.606400i	$0.207387\pi$
$251$	304.413i	1.21280i	−0.795160	+	0.606400i	$0.792613\pi$
$252$	0	0
$253$	−40.7756	−0.161168
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 34.8540i	− 0.135619i	−0.997698	−	0.0678093i	$-0.978399\pi$
$257$	− 34.8540i	− 0.135619i	0.997698	−	0.0678093i	$-0.0216009\pi$
$258$	0	0
$259$	−96.3505	−0.372010
$260$	0	0
$261$	0	0
$262$	0	0
$263$	158.888i	0.604137i	0.953286	+	0.302068i	$0.0976771\pi$
$263$	158.888i	0.604137i	−0.953286	+	0.302068i	$0.902323\pi$
$264$	0	0
$265$	132.146	0.498663
$266$	0	0
$267$	0	0
$268$	0	0
$269$	478.000i	1.77695i	0.458924	+	0.888476i	$0.348235\pi$
$269$	478.000i	1.77695i	−0.458924	+	0.888476i	$0.651765\pi$
$270$	0	0
$271$	190.210	0.701881	0.350941	−	0.936398i	$-0.385862\pi$
$271$	190.210	0.701881	0.350941	+	0.936398i	$0.385862\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	118.522i	0.430990i
$276$	0	0
$277$	377.904	1.36427	0.682137	−	0.731225i	$-0.261051\pi$
$277$	377.904	1.36427	0.682137	+	0.731225i	$0.261051\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 509.204i	− 1.81211i	−0.423156	−	0.906057i	$-0.639078\pi$
$281$	− 509.204i	− 1.81211i	0.423156	−	0.906057i	$-0.360922\pi$
$282$	0	0
$283$	−135.581	−0.479083	−0.239542	−	0.970886i	$-0.576997\pi$
$283$	−135.581	−0.479083	−0.239542	+	0.970886i	$0.576997\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	170.604i	0.594438i
$288$	0	0
$289$	205.893	0.712433
$290$	0	0
$291$	0	0
$292$	0	0
$293$	242.746i	0.828486i	0.910166	+	0.414243i	$0.135954\pi$
$293$	242.746i	0.828486i	−0.910166	+	0.414243i	$0.864046\pi$
$294$	0	0
$295$	−651.342	−2.20794
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 88.0466i	− 0.294470i
$300$	0	0
$301$	−194.348	−0.645674
$302$	0	0
$303$	0	0
$304$	0	0
$305$	352.116i	1.15448i
$306$	0	0
$307$	104.962	0.341897	0.170949	−	0.985280i	$-0.445317\pi$
$307$	104.962	0.341897	0.170949	+	0.985280i	$0.445317\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 354.169i	− 1.13881i	−0.822058	−	0.569403i	$-0.807174\pi$
$311$	− 354.169i	− 1.13881i	0.822058	−	0.569403i	$-0.192826\pi$
$312$	0	0
$313$	−592.147	−1.89184	−0.945922	−	0.324394i	$-0.894839\pi$
$313$	−592.147	−1.89184	−0.945922	+	0.324394i	$0.894839\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 330.539i	− 1.04271i	−0.853340	−	0.521354i	$-0.825427\pi$
$317$	− 330.539i	− 1.04271i	0.853340	−	0.521354i	$-0.174573\pi$
$318$	0	0
$319$	12.7326	0.0399140
$320$	0	0
$321$	0	0
$322$	0	0
$323$	54.4013i	0.168425i
$324$	0	0
$325$	−255.925	−0.787462
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 59.5414i	− 0.180977i
$330$	0	0
$331$	−313.072	−0.945838	−0.472919	−	0.881106i	$-0.656800\pi$
$331$	−313.072	−0.945838	−0.472919	+	0.881106i	$0.656800\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	838.729i	2.50367i
$336$	0	0
$337$	−7.27439	−0.0215857	−0.0107929	−	0.999942i	$-0.503436\pi$
$337$	−7.27439	−0.0215857	−0.0107929	+	0.999942i	$0.503436\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 111.411i	− 0.326719i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	283.015i	0.815606i	0.913070	+	0.407803i	$0.133705\pi$
$347$	283.015i	0.815606i	−0.913070	+	0.407803i	$0.866295\pi$
$348$	0	0
$349$	485.157	1.39014	0.695068	−	0.718944i	$-0.255374\pi$
$349$	485.157	1.39014	0.695068	+	0.718944i	$0.255374\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.63666i	0.0188007i	0.999956	+	0.00940037i	$0.00299228\pi$
$353$	6.63666i	0.0188007i	−0.999956	+	0.00940037i	$0.997008\pi$
$354$	0	0
$355$	270.448	0.761826
$356$	0	0
$357$	0	0
$358$	0	0
$359$	532.243i	1.48257i	0.671190	+	0.741286i	$0.265784\pi$
$359$	532.243i	1.48257i	−0.671190	+	0.741286i	$0.734216\pi$
$360$	0	0
$361$	−325.389	−0.901355
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 565.305i	− 1.54878i
$366$	0	0
$367$	444.300	1.21063	0.605314	−	0.795987i	$-0.293048\pi$
$367$	444.300	1.21063	0.605314	+	0.795987i	$0.293048\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	38.5460i	0.103898i
$372$	0	0
$373$	34.6623	0.0929285	0.0464642	−	0.998920i	$-0.485205\pi$
$373$	34.6623	0.0929285	0.0464642	+	0.998920i	$0.485205\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	27.4934i	0.0729269i
$378$	0	0
$379$	709.694	1.87254	0.936271	−	0.351278i	$-0.114253\pi$
$379$	709.694	1.87254	0.936271	+	0.351278i	$0.114253\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	282.603i	0.737866i	0.929456	+	0.368933i	$0.120277\pi$
$383$	282.603i	0.737866i	−0.929456	+	0.368933i	$0.879723\pi$
$384$	0	0
$385$	−49.6638	−0.128997
$386$	0	0
$387$	0	0
$388$	0	0
$389$	445.896i	1.14626i	0.819464	+	0.573131i	$0.194272\pi$
$389$	445.896i	1.14626i	−0.819464	+	0.573131i	$0.805728\pi$
$390$	0	0
$391$	−179.618	−0.459381
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1089.64i	2.75859i
$396$	0	0
$397$	−452.318	−1.13934	−0.569670	−	0.821874i	$-0.692929\pi$
$397$	−452.318	−1.13934	−0.569670	+	0.821874i	$0.692929\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 592.296i	− 1.47705i	−0.674227	−	0.738524i	$-0.735523\pi$
$401$	− 592.296i	− 1.47705i	0.674227	−	0.738524i	$-0.264477\pi$
$402$	0	0
$403$	240.570	0.596948
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 75.3657i	− 0.185174i
$408$	0	0
$409$	339.846	0.830920	0.415460	−	0.909612i	$-0.363621\pi$
$409$	339.846	0.830920	0.415460	+	0.909612i	$0.363621\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 189.992i	− 0.460029i
$414$	0	0
$415$	1002.13	2.41477
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 684.871i	− 1.63454i	−0.576257	−	0.817269i	$-0.695487\pi$
$419$	− 684.871i	− 1.63454i	0.576257	−	0.817269i	$-0.304513\pi$
$420$	0	0
$421$	164.455	0.390630	0.195315	−	0.980741i	$-0.437427\pi$
$421$	164.455	0.390630	0.195315	+	0.980741i	$0.437427\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	522.095i	1.22846i
$426$	0	0
$427$	−102.710	−0.240539
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 159.164i	− 0.369291i	−0.982805	−	0.184645i	$-0.940886\pi$
$431$	− 159.164i	− 0.369291i	0.982805	−	0.184645i	$-0.0591136\pi$
$432$	0	0
$433$	725.043	1.67447	0.837233	−	0.546847i	$-0.184172\pi$
$433$	725.043	1.67447	0.837233	+	0.546847i	$0.184172\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	117.577i	0.269055i
$438$	0	0
$439$	−47.8404	−0.108976	−0.0544880	−	0.998514i	$-0.517353\pi$
$439$	−47.8404	−0.108976	−0.0544880	+	0.998514i	$0.517353\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	518.646i	1.17076i	0.810759	+	0.585379i	$0.199054\pi$
$443$	518.646i	1.17076i	−0.810759	+	0.585379i	$0.800946\pi$
$444$	0	0
$445$	−1138.49	−2.55841
$446$	0	0
$447$	0	0
$448$	0	0
$449$	213.869i	0.476323i	0.971226	+	0.238161i	$0.0765447\pi$
$449$	213.869i	0.476323i	−0.971226	+	0.238161i	$0.923455\pi$
$450$	0	0
$451$	−133.447	−0.295891
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 107.239i	− 0.235690i
$456$	0	0
$457$	763.952	1.67167	0.835833	−	0.548983i	$-0.184985\pi$
$457$	763.952	1.67167	0.835833	+	0.548983i	$0.184985\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 169.162i	− 0.366945i	−0.983025	−	0.183473i	$-0.941266\pi$
$461$	− 169.162i	− 0.366945i	0.983025	−	0.183473i	$-0.0587339\pi$
$462$	0	0
$463$	183.784	0.396941	0.198471	−	0.980107i	$-0.436403\pi$
$463$	183.784	0.396941	0.198471	+	0.980107i	$0.436403\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 388.570i	− 0.832056i	−0.909352	−	0.416028i	$-0.863422\pi$
$467$	− 388.570i	− 0.832056i	0.909352	−	0.416028i	$-0.136578\pi$
$468$	0	0
$469$	−244.652	−0.521646
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 152.020i	− 0.321394i
$474$	0	0
$475$	341.761	0.719496
$476$	0	0
$477$	0	0
$478$	0	0
$479$	281.584i	0.587857i	0.955827	+	0.293929i	$0.0949628\pi$
$479$	281.584i	0.587857i	−0.955827	+	0.293929i	$0.905037\pi$
$480$	0	0
$481$	162.737	0.338331
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 812.779i	− 1.67583i
$486$	0	0
$487$	159.831	0.328196	0.164098	−	0.986444i	$-0.447529\pi$
$487$	159.831	0.328196	0.164098	+	0.986444i	$0.447529\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.0430644i	0 8.77075e-5i	1.00000		4.38537e-5i	$1.39591e-5\pi$
$491$	0.0430644i	0 8.77075e-5i	−1.00000		4.38537e-5i	$0.999986\pi$
$492$	0	0
$493$	56.0875	0.113768
$494$	0	0
$495$	0	0
$496$	0	0
$497$	78.8880i	0.158728i
$498$	0	0
$499$	300.513	0.602231	0.301115	−	0.953588i	$-0.402641\pi$
$499$	300.513	0.602231	0.301115	+	0.953588i	$0.402641\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 624.071i	− 1.24070i	−0.784326	−	0.620349i	$-0.786991\pi$
$503$	− 624.071i	− 1.24070i	0.784326	−	0.620349i	$-0.213009\pi$
$504$	0	0
$505$	774.403	1.53347
$506$	0	0
$507$	0	0
$508$	0	0
$509$	833.447i	1.63742i	0.574207	+	0.818710i	$0.305310\pi$
$509$	833.447i	1.63742i	−0.574207	+	0.818710i	$0.694690\pi$
$510$	0	0
$511$	164.896	0.322693
$512$	0	0
$513$	0	0
$514$	0	0
$515$	156.784i	0.304435i
$516$	0	0
$517$	46.5735	0.0900842
$518$	0	0
$519$	0	0
$520$	0	0
$521$	869.182i	1.66830i	0.551541	+	0.834148i	$0.314040\pi$
$521$	869.182i	1.66830i	−0.551541	+	0.834148i	$0.685960\pi$
$522$	0	0
$523$	985.764	1.88483	0.942413	−	0.334453i	$-0.108551\pi$
$523$	985.764	1.88483	0.942413	+	0.334453i	$0.108551\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 490.770i	− 0.931253i
$528$	0	0
$529$	140.795	0.266152
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 288.152i	− 0.540622i
$534$	0	0
$535$	621.586	1.16184
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 14.4866i	− 0.0268768i
$540$	0	0
$541$	489.643	0.905069	0.452535	−	0.891747i	$-0.350520\pi$
$541$	489.643	0.905069	0.452535	+	0.891747i	$0.350520\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 431.640i	− 0.792000i
$546$	0	0
$547$	96.1073	0.175699	0.0878495	−	0.996134i	$-0.472001\pi$
$547$	96.1073	0.175699	0.0878495	+	0.996134i	$0.472001\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 36.7146i	− 0.0666326i
$552$	0	0
$553$	−317.842	−0.574759
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 180.363i	− 0.323811i	−0.986806	−	0.161905i	$-0.948236\pi$
$557$	− 180.363i	− 0.323811i	0.986806	−	0.161905i	$-0.0517640\pi$
$558$	0	0
$559$	328.256	0.587219
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 764.492i	− 1.35789i	−0.734189	−	0.678945i	$-0.762437\pi$
$563$	− 764.492i	− 1.35789i	0.734189	−	0.678945i	$-0.237563\pi$
$564$	0	0
$565$	−1341.70	−2.37468
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 421.556i	− 0.740872i	−0.928858	−	0.370436i	$-0.879208\pi$
$569$	− 421.556i	− 0.740872i	0.928858	−	0.370436i	$-0.120792\pi$
$570$	0	0
$571$	−550.600	−0.964273	−0.482136	−	0.876096i	$-0.660139\pi$
$571$	−550.600	−0.964273	−0.482136	+	0.876096i	$0.660139\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1128.40i	1.96243i
$576$	0	0
$577$	−532.473	−0.922829	−0.461415	−	0.887185i	$-0.652658\pi$
$577$	−532.473	−0.922829	−0.461415	+	0.887185i	$0.652658\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	292.314i	0.503123i
$582$	0	0
$583$	−30.1509	−0.0517167
$584$	0	0
$585$	0	0
$586$	0	0
$587$	242.765i	0.413569i	0.978386	+	0.206785i	$0.0663000\pi$
$587$	242.765i	0.413569i	−0.978386	+	0.206785i	$0.933700\pi$
$588$	0	0
$589$	−321.255	−0.545425
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 89.0422i	− 0.150155i	−0.997178	−	0.0750777i	$-0.976080\pi$
$593$	− 89.0422i	− 0.150155i	0.997178	−	0.0750777i	$-0.0239205\pi$
$594$	0	0
$595$	−218.771	−0.367682
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 628.172i	− 1.04870i	−0.851503	−	0.524350i	$-0.824308\pi$
$599$	− 628.172i	− 1.04870i	0.851503	−	0.524350i	$-0.175692\pi$
$600$	0	0
$601$	−24.3467	−0.0405102	−0.0202551	−	0.999795i	$-0.506448\pi$
$601$	−24.3467	−0.0405102	−0.0202551	+	0.999795i	$0.506448\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1058.66i	1.74985i
$606$	0	0
$607$	−547.391	−0.901798	−0.450899	−	0.892575i	$-0.648897\pi$
$607$	−547.391	−0.901798	−0.450899	+	0.892575i	$0.648897\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	100.566i	0.164593i
$612$	0	0
$613$	−521.556	−0.850825	−0.425413	−	0.905000i	$-0.639871\pi$
$613$	−521.556	−0.850825	−0.425413	+	0.905000i	$0.639871\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 986.523i	− 1.59890i	−0.600730	−	0.799452i	$-0.705123\pi$
$617$	− 986.523i	− 1.59890i	0.600730	−	0.799452i	$-0.294877\pi$
$618$	0	0
$619$	768.839	1.24207	0.621033	−	0.783784i	$-0.286713\pi$
$619$	768.839	1.24207	0.621033	+	0.783784i	$0.286713\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 332.091i	− 0.533052i
$624$	0	0
$625$	1223.15	1.95704
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 331.989i	− 0.527804i
$630$	0	0
$631$	764.067	1.21088	0.605441	−	0.795890i	$-0.292997\pi$
$631$	764.067	1.21088	0.605441	+	0.795890i	$0.292997\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 591.061i	− 0.930805i
$636$	0	0
$637$	31.2809	0.0491067
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 577.215i	− 0.900492i	−0.892905	−	0.450246i	$-0.851336\pi$
$641$	− 577.215i	− 0.900492i	0.892905	−	0.450246i	$-0.148664\pi$
$642$	0	0
$643$	346.343	0.538636	0.269318	−	0.963051i	$-0.413202\pi$
$643$	346.343	0.538636	0.269318	+	0.963051i	$0.413202\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	513.567i	0.793767i	0.917869	+	0.396883i	$0.129908\pi$
$647$	513.567i	0.793767i	−0.917869	+	0.396883i	$0.870092\pi$
$648$	0	0
$649$	148.613	0.228987
$650$	0	0
$651$	0	0
$652$	0	0
$653$	946.639i	1.44968i	0.688919	+	0.724838i	$0.258085\pi$
$653$	946.639i	1.44968i	−0.688919	+	0.724838i	$0.741915\pi$
$654$	0	0
$655$	1284.71	1.96139
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 150.097i	− 0.227766i	−0.993494	−	0.113883i	$-0.963671\pi$
$659$	− 150.097i	− 0.227766i	0.993494	−	0.113883i	$-0.0363288\pi$
$660$	0	0
$661$	−409.569	−0.619621	−0.309810	−	0.950798i	$-0.600266\pi$
$661$	−409.569	−0.619621	−0.309810	+	0.950798i	$0.600266\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	143.206i	0.215348i
$666$	0	0
$667$	121.221	0.181741
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 80.3402i	− 0.119732i
$672$	0	0
$673$	−275.875	−0.409918	−0.204959	−	0.978771i	$-0.565706\pi$
$673$	−275.875	−0.409918	−0.204959	+	0.978771i	$0.565706\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1301.13i	1.92191i	0.276705	+	0.960955i	$0.410757\pi$
$677$	1301.13i	1.92191i	−0.276705	+	0.960955i	$0.589243\pi$
$678$	0	0
$679$	237.082	0.349164
$680$	0	0
$681$	0	0
$682$	0	0
$683$	767.412i	1.12359i	0.827276	+	0.561795i	$0.189889\pi$
$683$	767.412i	1.12359i	−0.827276	+	0.561795i	$0.810111\pi$
$684$	0	0
$685$	−469.062	−0.684763
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 65.1047i	− 0.0944916i
$690$	0	0
$691$	1146.24	1.65881	0.829405	−	0.558648i	$-0.188680\pi$
$691$	1146.24	1.65881	0.829405	+	0.558648i	$0.188680\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1066.45i	− 1.53446i
$696$	0	0
$697$	−587.838	−0.843384
$698$	0	0
$699$	0	0
$700$	0	0
$701$	945.712i	1.34909i	0.738234	+	0.674545i	$0.235660\pi$
$701$	945.712i	1.34909i	−0.738234	+	0.674545i	$0.764340\pi$
$702$	0	0
$703$	−217.318	−0.309130
$704$	0	0
$705$	0	0
$706$	0	0
$707$	225.888i	0.319502i
$708$	0	0
$709$	916.701	1.29295	0.646475	−	0.762935i	$-0.276243\pi$
$709$	916.701	1.29295	0.646475	+	0.762935i	$0.276243\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1060.69i	− 1.48765i
$714$	0	0
$715$	83.8828	0.117319
$716$	0	0
$717$	0	0
$718$	0	0
$719$	675.009i	0.938816i	0.882981	+	0.469408i	$0.155533\pi$
$719$	675.009i	0.938816i	−0.882981	+	0.469408i	$0.844467\pi$
$720$	0	0
$721$	−45.7329	−0.0634298
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 352.353i	− 0.486005i
$726$	0	0
$727$	105.334	0.144888	0.0724442	−	0.997372i	$-0.476920\pi$
$727$	105.334	0.144888	0.0724442	+	0.997372i	$0.476920\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 669.652i	− 0.916077i
$732$	0	0
$733$	1223.69	1.66943	0.834714	−	0.550683i	$-0.185633\pi$
$733$	1223.69	1.66943	0.834714	+	0.550683i	$0.185633\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 191.368i	− 0.259657i
$738$	0	0
$739$	−682.076	−0.922972	−0.461486	−	0.887148i	$-0.652683\pi$
$739$	−682.076	−0.922972	−0.461486	+	0.887148i	$0.652683\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1099.02i	1.47916i	0.673069	+	0.739579i	$0.264976\pi$
$743$	1099.02i	1.47916i	−0.673069	+	0.739579i	$0.735024\pi$
$744$	0	0
$745$	1113.66	1.49485
$746$	0	0
$747$	0	0
$748$	0	0
$749$	181.313i	0.242073i
$750$	0	0
$751$	−835.969	−1.11314	−0.556571	−	0.830800i	$-0.687883\pi$
$751$	−835.969	−1.11314	−0.556571	+	0.830800i	$0.687883\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	871.871i	1.15480i
$756$	0	0
$757$	849.591	1.12231	0.561156	−	0.827710i	$-0.310357\pi$
$757$	849.591	1.12231	0.561156	+	0.827710i	$0.310357\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	476.540i	0.626203i	0.949720	+	0.313101i	$0.101368\pi$
$761$	476.540i	0.626203i	−0.949720	+	0.313101i	$0.898632\pi$
$762$	0	0
$763$	125.907	0.165015
$764$	0	0
$765$	0	0
$766$	0	0
$767$	320.899i	0.418382i
$768$	0	0
$769$	269.649	0.350649	0.175324	−	0.984511i	$-0.443903\pi$
$769$	269.649	0.350649	0.175324	+	0.984511i	$0.443903\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	642.838i	0.831614i	0.909453	+	0.415807i	$0.136501\pi$
$773$	642.838i	0.831614i	−0.909453	+	0.415807i	$0.863499\pi$
$774$	0	0
$775$	−3083.12	−3.97822
$776$	0	0
$777$	0	0
$778$	0	0
$779$	384.796i	0.493961i
$780$	0	0
$781$	−61.7065	−0.0790096
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 248.872i	− 0.317034i
$786$	0	0
$787$	−1328.18	−1.68765	−0.843825	−	0.536618i	$-0.819702\pi$
$787$	−1328.18	−1.68765	−0.843825	+	0.536618i	$0.819702\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 391.364i	− 0.494771i
$792$	0	0
$793$	173.479	0.218762
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 511.671i	− 0.641996i	−0.947080	−	0.320998i	$-0.895982\pi$
$797$	− 511.671i	− 0.641996i	0.947080	−	0.320998i	$-0.104018\pi$
$798$	0	0
$799$	205.158	0.256769
$800$	0	0
$801$	0	0
$802$	0	0
$803$	128.982i	0.160625i
$804$	0	0
$805$	−472.827	−0.587362
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 869.545i	− 1.07484i	−0.843315	−	0.537420i	$-0.819399\pi$
$809$	− 869.545i	− 1.07484i	0.843315	−	0.537420i	$-0.180601\pi$
$810$	0	0
$811$	184.029	0.226916	0.113458	−	0.993543i	$-0.463807\pi$
$811$	184.029	0.226916	0.113458	+	0.993543i	$0.463807\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 326.738i	− 0.400906i
$816$	0	0
$817$	−438.350	−0.536537
$818$	0	0
$819$	0	0
$820$	0	0
$821$	892.526i	1.08712i	0.839370	+	0.543560i	$0.182924\pi$
$821$	892.526i	1.08712i	−0.839370	+	0.543560i	$0.817076\pi$
$822$	0	0
$823$	1079.35	1.31148	0.655739	−	0.754988i	$-0.272357\pi$
$823$	1079.35	1.31148	0.655739	+	0.754988i	$0.272357\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1446.26i	− 1.74880i	−0.485203	−	0.874402i	$-0.661254\pi$
$827$	− 1446.26i	− 1.74880i	0.485203	−	0.874402i	$-0.338746\pi$
$828$	0	0
$829$	−891.157	−1.07498	−0.537489	−	0.843270i	$-0.680627\pi$
$829$	−891.157	−1.07498	−0.537489	+	0.843270i	$0.680627\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 63.8141i	− 0.0766076i
$834$	0	0
$835$	−2173.19	−2.60262
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 364.328i	− 0.434241i	−0.976145	−	0.217120i	$-0.930334\pi$
$839$	− 364.328i	− 0.434241i	0.976145	−	0.217120i	$-0.0696664\pi$
$840$	0	0
$841$	803.148	0.954991
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1351.75i	− 1.59971i
$846$	0	0
$847$	−308.804	−0.364586
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 717.523i	− 0.843152i
$852$	0	0
$853$	−1612.54	−1.89043	−0.945214	−	0.326450i	$-0.894148\pi$
$853$	−1612.54	−1.89043	−0.945214	+	0.326450i	$0.894148\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 221.077i	− 0.257966i	−0.991647	−	0.128983i	$-0.958829\pi$
$857$	− 221.077i	− 0.257966i	0.991647	−	0.128983i	$-0.0411713\pi$
$858$	0	0
$859$	297.734	0.346605	0.173302	−	0.984869i	$-0.444556\pi$
$859$	297.734	0.346605	0.173302	+	0.984869i	$0.444556\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 36.1731i	− 0.0419155i	−0.999780	−	0.0209578i	$-0.993328\pi$
$863$	− 36.1731i	− 0.0419155i	0.999780	−	0.0209578i	$-0.00667155\pi$
$864$	0	0
$865$	−347.332	−0.401540
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 248.617i	− 0.286096i
$870$	0	0
$871$	413.220	0.474420
$872$	0	0
$873$	0	0
$874$	0	0
$875$	774.421i	0.885053i
$876$	0	0
$877$	612.061	0.697904	0.348952	−	0.937141i	$-0.386538\pi$
$877$	612.061	0.697904	0.348952	+	0.937141i	$0.386538\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 463.592i	− 0.526211i	−0.964767	−	0.263105i	$-0.915253\pi$
$881$	− 463.592i	− 0.526211i	0.964767	−	0.263105i	$-0.0847467\pi$
$882$	0	0
$883$	−1618.82	−1.83332	−0.916658	−	0.399672i	$-0.869124\pi$
$883$	−1618.82	−1.83332	−0.916658	+	0.399672i	$0.869124\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1084.96i	− 1.22317i	−0.791177	−	0.611587i	$-0.790531\pi$
$887$	− 1084.96i	− 1.22317i	0.791177	−	0.611587i	$-0.209469\pi$
$888$	0	0
$889$	172.409	0.193936
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 134.295i	− 0.150387i
$894$	0	0
$895$	−2624.90	−2.93285
$896$	0	0
$897$	0	0
$898$	0	0
$899$	331.213i	0.368423i
$900$	0	0
$901$	−132.816	−0.147409
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1125.04i	1.24313i
$906$	0	0
$907$	697.474	0.768990	0.384495	−	0.923127i	$-0.374376\pi$
$907$	697.474	0.768990	0.384495	+	0.923127i	$0.374376\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1682.48i	1.84684i	0.383786	+	0.923422i	$0.374620\pi$
$911$	1682.48i	1.84684i	−0.383786	+	0.923422i	$0.625380\pi$
$912$	0	0
$913$	−228.649	−0.250437
$914$	0	0
$915$	0	0
$916$	0	0
$917$	374.741i	0.408660i
$918$	0	0
$919$	1617.18	1.75972	0.879861	−	0.475231i	$-0.157636\pi$
$919$	1617.18	1.75972	0.879861	+	0.475231i	$0.157636\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 133.243i	− 0.144358i
$924$	0	0
$925$	−2085.62	−2.25473
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 692.871i	− 0.745825i	−0.927867	−	0.372912i	$-0.878359\pi$
$929$	− 692.871i	− 0.745825i	0.927867	−	0.372912i	$-0.121641\pi$
$930$	0	0
$931$	−41.7724	−0.0448683
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 171.124i	− 0.183020i
$936$	0	0
$937$	−1134.83	−1.21113	−0.605565	−	0.795796i	$-0.707053\pi$
$937$	−1134.83	−1.21113	−0.605565	+	0.795796i	$0.707053\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	492.664i	0.523553i	0.965128	+	0.261777i	$0.0843084\pi$
$941$	492.664i	0.523553i	−0.965128	+	0.261777i	$0.915692\pi$
$942$	0	0
$943$	−1270.49	−1.34728
$944$	0	0
$945$	0	0
$946$	0	0
$947$	584.464i	0.617174i	0.951196	+	0.308587i	$0.0998562\pi$
$947$	584.464i	0.617174i	−0.951196	+	0.308587i	$0.900144\pi$
$948$	0	0
$949$	−278.511	−0.293479
$950$	0	0
$951$	0	0
$952$	0	0
$953$	476.167i	0.499651i	0.968291	+	0.249826i	$0.0803733\pi$
$953$	476.167i	0.499651i	−0.968291	+	0.249826i	$0.919627\pi$
$954$	0	0
$955$	−1612.51	−1.68850
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 136.822i	− 0.142672i
$960$	0	0
$961$	1937.14	2.01575
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 625.875i	− 0.648575i
$966$	0	0
$967$	−1579.38	−1.63328	−0.816638	−	0.577151i	$-0.804165\pi$
$967$	−1579.38	−1.63328	−0.816638	+	0.577151i	$0.804165\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	962.629i	0.991379i	0.868500	+	0.495690i	$0.165085\pi$
$971$	962.629i	0.991379i	−0.868500	+	0.495690i	$0.834915\pi$
$972$	0	0
$973$	311.076	0.319708
$974$	0	0
$975$	0	0
$976$	0	0
$977$	94.4955i	0.0967200i	0.998830	+	0.0483600i	$0.0153995\pi$
$977$	94.4955i	0.0967200i	−0.998830	+	0.0483600i	$0.984601\pi$
$978$	0	0
$979$	259.763	0.265335
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 983.668i	− 1.00068i	−0.865829	−	0.500340i	$-0.833208\pi$
$983$	− 983.668i	− 1.00068i	0.865829	−	0.500340i	$-0.166792\pi$
$984$	0	0
$985$	419.091	0.425473
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1447.31i	− 1.46341i
$990$	0	0
$991$	−636.396	−0.642175	−0.321088	−	0.947049i	$-0.604048\pi$
$991$	−636.396	−0.642175	−0.321088	+	0.947049i	$0.604048\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2637.94i	− 2.65120i
$996$	0	0
$997$	1147.65	1.15110	0.575550	−	0.817767i	$-0.304788\pi$
$997$	1147.65	1.15110	0.575550	+	0.817767i	$0.304788\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.3.d.n.449.12		12
3.2	odd	2	inner	4032.3.d.n.449.1		12
4.3	odd	2		4032.3.d.o.449.12		12
8.3	odd	2		2016.3.d.e.449.1	✓	12
8.5	even	2		2016.3.d.f.449.1	yes	12
12.11	even	2		4032.3.d.o.449.1		12
24.5	odd	2		2016.3.d.f.449.12	yes	12
24.11	even	2		2016.3.d.e.449.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.3.d.e.449.1	✓	12	8.3	odd	2
2016.3.d.e.449.12	yes	12	24.11	even	2
2016.3.d.f.449.1	yes	12	8.5	even	2
2016.3.d.f.449.12	yes	12	24.5	odd	2
4032.3.d.n.449.1		12	3.2	odd	2	inner
4032.3.d.n.449.12		12	1.1	even	1	trivial
4032.3.d.o.449.1		12	12.11	even	2
4032.3.d.o.449.12		12	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4032.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.07031i q^{5} -2.64575 q^{7} -2.06952i q^{11} +4.46871 q^{13} -9.11630i q^{17} -5.96748 q^{19} -19.7029i q^{23} -57.2705 q^{25} +6.15244i q^{29} +53.8344 q^{31} -23.9978i q^{35} +36.4171 q^{37} -64.4821i q^{41} +73.4565 q^{43} +22.5045i q^{47} +7.00000 q^{49} -14.5690i q^{53} +18.7712 q^{55} +71.8103i q^{59} +38.8208 q^{61} +40.5325i q^{65} +92.4697 q^{67} -29.8169i q^{71} -62.3248 q^{73} +5.47543i q^{77} +120.133 q^{79} -110.484i q^{83} +82.6877 q^{85} +125.519i q^{89} -11.8231 q^{91} -54.1269i q^{95} -89.6087 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{13} - 64 q^{19} - 124 q^{25} + 160 q^{31} - 56 q^{37} + 64 q^{43} + 84 q^{49} - 160 q^{55} - 104 q^{61} + 64 q^{67} - 64 q^{73} + 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) \) 12 * q - 16 * q^13 - 64 * q^19 - 124 * q^25 + 160 * q^31 - 56 * q^37 + 64 * q^43 + 84 * q^49 - 160 * q^55 - 104 * q^61 + 64 * q^67 - 64 * q^73 + 32 * q^79 + 184 * q^85 - 96 * q^97

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