LMFDB - Embedded newform 3528.3.d.g.1961.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3528,3,Mod(1961,3528)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, 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("3528.1961"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3528.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,-12,0,0,0,0,0,12,0,0,0,0,0,-92] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$96.1310372663$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 17x^{6} + 96x^{5} - 108x^{4} - 1104x^{3} + 5366x^{2} - 8656x + 10256 $ x^8 - 2x^7 - 17x^6 + 96x^5 - 108x^4 - 1104x^3 + 5366x^2 - 8656*x + 10256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1961.8
Root			$0.725935 + 1.57128i$ of defining polynomial
Character	$\chi$	$=$	3528.1961
Dual form			3528.3.d.g.1961.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.95214i q^{5} -2.21465i q^{11} +15.7846 q^{13} -17.4519i q^{17} +3.58386 q^{19} -7.68853i q^{23} -55.1408 q^{25} +24.6165i q^{29} -45.9172 q^{31} +4.92662 q^{37} -28.9060i q^{41} -35.9761 q^{43} +47.5971i q^{47} +57.9526i q^{53} +19.8258 q^{55} -116.026i q^{59} -107.895 q^{61} +141.306i q^{65} -131.926 q^{67} +115.889i q^{71} +46.3746 q^{73} +20.1647 q^{79} -5.93622i q^{83} +156.232 q^{85} +77.9701i q^{89} +32.0832i q^{95} -87.3734 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{13} + 12 q^{19} - 92 q^{25} - 32 q^{31} + 132 q^{37} + 12 q^{43} - 220 q^{55} - 184 q^{61} - 332 q^{67} - 188 q^{73} + 112 q^{79} + 128 q^{85} - 516 q^{97}+O(q^{100}) $ 8 * q - 12 * q^13 + 12 * q^19 - 92 * q^25 - 32 * q^31 + 132 * q^37 + 12 * q^43 - 220 * q^55 - 184 * q^61 - 332 * q^67 - 188 * q^73 + 112 * q^79 + 128 * q^85 - 516 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	8.95214i	1.79043i	0.445637	+	0.895214i	$0.352977\pi$
$5$	8.95214i	1.79043i	−0.445637	+	0.895214i	$0.647023\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.21465i	− 0.201332i	−0.994920	−	0.100666i	$-0.967903\pi$
$11$	− 2.21465i	− 0.201332i	0.994920	−	0.100666i	$-0.0320973\pi$
$12$	0	0
$13$	15.7846	1.21420	0.607100	−	0.794626i	$-0.292333\pi$
$13$	15.7846	1.21420	0.607100	+	0.794626i	$0.292333\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 17.4519i	− 1.02658i	−0.858215	−	0.513291i	$-0.828426\pi$
$17$	− 17.4519i	− 1.02658i	0.858215	−	0.513291i	$-0.171574\pi$
$18$	0	0
$19$	3.58386	0.188624	0.0943120	−	0.995543i	$-0.469935\pi$
$19$	3.58386	0.188624	0.0943120	+	0.995543i	$0.469935\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 7.68853i	− 0.334284i	−0.985933	−	0.167142i	$-0.946546\pi$
$23$	− 7.68853i	− 0.334284i	0.985933	−	0.167142i	$-0.0534538\pi$
$24$	0	0
$25$	−55.1408	−2.20563
$26$	0	0
$27$	0	0
$28$	0	0
$29$	24.6165i	0.848845i	0.905464	+	0.424423i	$0.139523\pi$
$29$	24.6165i	0.848845i	−0.905464	+	0.424423i	$0.860477\pi$
$30$	0	0
$31$	−45.9172	−1.48120	−0.740600	−	0.671947i	$-0.765458\pi$
$31$	−45.9172	−1.48120	−0.740600	+	0.671947i	$0.765458\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.92662	0.133152	0.0665760	−	0.997781i	$-0.478793\pi$
$37$	4.92662	0.133152	0.0665760	+	0.997781i	$0.478793\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 28.9060i	− 0.705024i	−0.935807	−	0.352512i	$-0.885328\pi$
$41$	− 28.9060i	− 0.705024i	0.935807	−	0.352512i	$-0.114672\pi$
$42$	0	0
$43$	−35.9761	−0.836653	−0.418327	−	0.908297i	$-0.637383\pi$
$43$	−35.9761	−0.836653	−0.418327	+	0.908297i	$0.637383\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	47.5971i	1.01270i	0.862327	+	0.506352i	$0.169006\pi$
$47$	47.5971i	1.01270i	−0.862327	+	0.506352i	$0.830994\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	57.9526i	1.09345i	0.837314	+	0.546723i	$0.184125\pi$
$53$	57.9526i	1.09345i	−0.837314	+	0.546723i	$0.815875\pi$
$54$	0	0
$55$	19.8258	0.360470
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 116.026i	− 1.96654i	−0.182164	−	0.983268i	$-0.558310\pi$
$59$	− 116.026i	− 1.96654i	0.182164	−	0.983268i	$-0.441690\pi$
$60$	0	0
$61$	−107.895	−1.76877	−0.884387	−	0.466755i	$-0.845423\pi$
$61$	−107.895	−1.76877	−0.884387	+	0.466755i	$0.845423\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	141.306i	2.17394i
$66$	0	0
$67$	−131.926	−1.96905	−0.984524	−	0.175247i	$-0.943928\pi$
$67$	−131.926	−1.96905	−0.984524	+	0.175247i	$0.943928\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	115.889i	1.63224i	0.577883	+	0.816119i	$0.303879\pi$
$71$	115.889i	1.63224i	−0.577883	+	0.816119i	$0.696121\pi$
$72$	0	0
$73$	46.3746	0.635268	0.317634	−	0.948213i	$-0.397112\pi$
$73$	46.3746	0.635268	0.317634	+	0.948213i	$0.397112\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	20.1647	0.255249	0.127625	−	0.991823i	$-0.459265\pi$
$79$	20.1647	0.255249	0.127625	+	0.991823i	$0.459265\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5.93622i	− 0.0715208i	−0.999360	−	0.0357604i	$-0.988615\pi$
$83$	− 5.93622i	− 0.0715208i	0.999360	−	0.0357604i	$-0.0113853\pi$
$84$	0	0
$85$	156.232	1.83802
$86$	0	0
$87$	0	0
$88$	0	0
$89$	77.9701i	0.876068i	0.898958	+	0.438034i	$0.144325\pi$
$89$	77.9701i	0.876068i	−0.898958	+	0.438034i	$0.855675\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	32.0832i	0.337718i
$96$	0	0
$97$	−87.3734	−0.900757	−0.450378	−	0.892838i	$-0.648711\pi$
$97$	−87.3734	−0.900757	−0.450378	+	0.892838i	$0.648711\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	160.474i	1.58885i	0.607362	+	0.794425i	$0.292228\pi$
$101$	160.474i	1.58885i	−0.607362	+	0.794425i	$0.707772\pi$
$102$	0	0
$103$	16.1024	0.156334	0.0781669	−	0.996940i	$-0.475093\pi$
$103$	16.1024	0.156334	0.0781669	+	0.996940i	$0.475093\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 114.440i	− 1.06953i	−0.845001	−	0.534764i	$-0.820400\pi$
$107$	− 114.440i	− 1.06953i	0.845001	−	0.534764i	$-0.179600\pi$
$108$	0	0
$109$	−51.7946	−0.475180	−0.237590	−	0.971366i	$-0.576357\pi$
$109$	−51.7946	−0.475180	−0.237590	+	0.971366i	$0.576357\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	131.309i	1.16202i	0.813895	+	0.581012i	$0.197343\pi$
$113$	131.309i	1.16202i	−0.813895	+	0.581012i	$0.802657\pi$
$114$	0	0
$115$	68.8288	0.598511
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	116.095	0.959466
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 269.825i	− 2.15860i
$126$	0	0
$127$	−219.227	−1.72620	−0.863099	−	0.505035i	$-0.831480\pi$
$127$	−219.227	−1.72620	−0.863099	+	0.505035i	$0.831480\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 201.140i	− 1.53542i	−0.640798	−	0.767710i	$-0.721396\pi$
$131$	− 201.140i	− 1.53542i	0.640798	−	0.767710i	$-0.278604\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 72.1384i	− 0.526558i	−0.964720	−	0.263279i	$-0.915196\pi$
$137$	− 72.1384i	− 0.526558i	0.964720	−	0.263279i	$-0.0848039\pi$
$138$	0	0
$139$	−199.392	−1.43447	−0.717236	−	0.696830i	$-0.754593\pi$
$139$	−199.392	−1.43447	−0.717236	+	0.696830i	$0.754593\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 34.9573i	− 0.244457i
$144$	0	0
$145$	−220.370	−1.51980
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 56.1829i	− 0.377067i	−0.982067	−	0.188533i	$-0.939627\pi$
$149$	− 56.1829i	− 0.377067i	0.982067	−	0.188533i	$-0.0603734\pi$
$150$	0	0
$151$	107.907	0.714615	0.357307	−	0.933987i	$-0.383695\pi$
$151$	107.907	0.714615	0.357307	+	0.933987i	$0.383695\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 411.057i	− 2.65198i
$156$	0	0
$157$	7.46399	0.0475413	0.0237707	−	0.999717i	$-0.492433\pi$
$157$	7.46399	0.0475413	0.0237707	+	0.999717i	$0.492433\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−171.839	−1.05422	−0.527112	−	0.849796i	$-0.676725\pi$
$163$	−171.839	−1.05422	−0.527112	+	0.849796i	$0.676725\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.17115i	0.0249769i	0.999922	+	0.0124885i	$0.00397531\pi$
$167$	4.17115i	0.0249769i	−0.999922	+	0.0124885i	$0.996025\pi$
$168$	0	0
$169$	80.1534	0.474281
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 129.284i	− 0.747304i	−0.927569	−	0.373652i	$-0.878105\pi$
$173$	− 129.284i	− 0.747304i	0.927569	−	0.373652i	$-0.121895\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 269.141i	− 1.50358i	−0.659402	−	0.751791i	$-0.729190\pi$
$179$	− 269.141i	− 1.50358i	0.659402	−	0.751791i	$-0.270810\pi$
$180$	0	0
$181$	−218.196	−1.20550	−0.602752	−	0.797929i	$-0.705929\pi$
$181$	−218.196	−1.20550	−0.602752	+	0.797929i	$0.705929\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	44.1038i	0.238399i
$186$	0	0
$187$	−38.6498	−0.206683
$188$	0	0
$189$	0	0
$190$	0	0
$191$	243.372i	1.27420i	0.770782	+	0.637099i	$0.219866\pi$
$191$	243.372i	1.27420i	−0.770782	+	0.637099i	$0.780134\pi$
$192$	0	0
$193$	298.736	1.54785	0.773926	−	0.633276i	$-0.218290\pi$
$193$	298.736	1.54785	0.773926	+	0.633276i	$0.218290\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 47.0566i	− 0.238866i	−0.992842	−	0.119433i	$-0.961892\pi$
$197$	− 47.0566i	− 0.238866i	0.992842	−	0.119433i	$-0.0381077\pi$
$198$	0	0
$199$	−274.647	−1.38014	−0.690069	−	0.723744i	$-0.742420\pi$
$199$	−274.647	−1.38014	−0.690069	+	0.723744i	$0.742420\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	258.770	1.26229
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 7.93698i	− 0.0379760i
$210$	0	0
$211$	166.122	0.787310	0.393655	−	0.919258i	$-0.371210\pi$
$211$	166.122	0.787310	0.393655	+	0.919258i	$0.371210\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 322.063i	− 1.49797i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 275.471i	− 1.24648i
$222$	0	0
$223$	−300.037	−1.34546	−0.672728	−	0.739890i	$-0.734877\pi$
$223$	−300.037	−1.34546	−0.672728	+	0.739890i	$0.734877\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.4578i	0.0504750i	0.999681	+	0.0252375i	$0.00803421\pi$
$227$	11.4578i	0.0504750i	−0.999681	+	0.0252375i	$0.991966\pi$
$228$	0	0
$229$	78.9688	0.344842	0.172421	−	0.985023i	$-0.444841\pi$
$229$	78.9688	0.344842	0.172421	+	0.985023i	$0.444841\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 213.389i	− 0.915831i	−0.888996	−	0.457915i	$-0.848596\pi$
$233$	− 213.389i	− 0.915831i	0.888996	−	0.457915i	$-0.151404\pi$
$234$	0	0
$235$	−426.095	−1.81317
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 8.32278i	− 0.0348234i	−0.999848	−	0.0174117i	$-0.994457\pi$
$239$	− 8.32278i	− 0.0348234i	0.999848	−	0.0174117i	$-0.00554259\pi$
$240$	0	0
$241$	117.399	0.487132	0.243566	−	0.969884i	$-0.421683\pi$
$241$	117.399	0.487132	0.243566	+	0.969884i	$0.421683\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	56.5697	0.229027
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 33.1840i	− 0.132207i	−0.997813	−	0.0661036i	$-0.978943\pi$
$251$	− 33.1840i	− 0.132207i	0.997813	−	0.0661036i	$-0.0210568\pi$
$252$	0	0
$253$	−17.0274	−0.0673020
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 298.619i	− 1.16194i	−0.813924	−	0.580971i	$-0.802673\pi$
$257$	− 298.619i	− 1.16194i	0.813924	−	0.580971i	$-0.197327\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.3518i	0.0773831i	0.999251	+	0.0386915i	$0.0123190\pi$
$263$	20.3518i	0.0773831i	−0.999251	+	0.0386915i	$0.987681\pi$
$264$	0	0
$265$	−518.800	−1.95774
$266$	0	0
$267$	0	0
$268$	0	0
$269$	228.218i	0.848395i	0.905570	+	0.424198i	$0.139444\pi$
$269$	228.218i	0.848395i	−0.905570	+	0.424198i	$0.860556\pi$
$270$	0	0
$271$	457.244	1.68725	0.843624	−	0.536935i	$-0.180418\pi$
$271$	457.244	1.68725	0.843624	+	0.536935i	$0.180418\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	122.117i	0.444064i
$276$	0	0
$277$	−389.955	−1.40778	−0.703890	−	0.710309i	$-0.748555\pi$
$277$	−389.955	−1.40778	−0.703890	+	0.710309i	$0.748555\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	301.317i	1.07230i	0.844122	+	0.536151i	$0.180122\pi$
$281$	301.317i	1.07230i	−0.844122	+	0.536151i	$0.819878\pi$
$282$	0	0
$283$	214.715	0.758710	0.379355	−	0.925251i	$-0.376146\pi$
$283$	214.715	0.758710	0.379355	+	0.925251i	$0.376146\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.5686	−0.0538706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	257.705i	0.879539i	0.898111	+	0.439770i	$0.144940\pi$
$293$	257.705i	0.879539i	−0.898111	+	0.439770i	$0.855060\pi$
$294$	0	0
$295$	1038.68	3.52094
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 121.360i	− 0.405888i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 965.893i	− 3.16686i
$306$	0	0
$307$	163.197	0.531586	0.265793	−	0.964030i	$-0.414366\pi$
$307$	163.197	0.531586	0.265793	+	0.964030i	$0.414366\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 300.468i	− 0.966136i	−0.875583	−	0.483068i	$-0.839522\pi$
$311$	− 300.468i	− 0.966136i	0.875583	−	0.483068i	$-0.160478\pi$
$312$	0	0
$313$	−466.435	−1.49021	−0.745103	−	0.666949i	$-0.767600\pi$
$313$	−466.435	−1.49021	−0.745103	+	0.666949i	$0.767600\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	254.635i	0.803265i	0.915801	+	0.401633i	$0.131557\pi$
$317$	254.635i	0.803265i	−0.915801	+	0.401633i	$0.868443\pi$
$318$	0	0
$319$	54.5169	0.170899
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 62.5451i	− 0.193638i
$324$	0	0
$325$	−870.375	−2.67808
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	576.952	1.74306	0.871529	−	0.490344i	$-0.163129\pi$
$331$	576.952	1.74306	0.871529	+	0.490344i	$0.163129\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1181.02i	− 3.52544i
$336$	0	0
$337$	−248.660	−0.737865	−0.368932	−	0.929456i	$-0.620277\pi$
$337$	−248.660	−0.737865	−0.368932	+	0.929456i	$0.620277\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	101.690i	0.298212i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 236.349i	− 0.681120i	−0.940223	−	0.340560i	$-0.889383\pi$
$347$	− 236.349i	− 0.681120i	0.940223	−	0.340560i	$-0.110617\pi$
$348$	0	0
$349$	−148.888	−0.426612	−0.213306	−	0.976985i	$-0.568423\pi$
$349$	−148.888	−0.426612	−0.213306	+	0.976985i	$0.568423\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 477.365i	− 1.35231i	−0.736759	−	0.676155i	$-0.763645\pi$
$353$	− 477.365i	− 1.35231i	0.736759	−	0.676155i	$-0.236355\pi$
$354$	0	0
$355$	−1037.45	−2.92241
$356$	0	0
$357$	0	0
$358$	0	0
$359$	367.084i	1.02252i	0.859426	+	0.511259i	$0.170821\pi$
$359$	367.084i	1.02252i	−0.859426	+	0.511259i	$0.829179\pi$
$360$	0	0
$361$	−348.156	−0.964421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	415.151i	1.13740i
$366$	0	0
$367$	104.045	0.283503	0.141751	−	0.989902i	$-0.454727\pi$
$367$	104.045	0.283503	0.141751	+	0.989902i	$0.454727\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	369.322	0.990139	0.495070	−	0.868853i	$-0.335143\pi$
$373$	369.322	0.990139	0.495070	+	0.868853i	$0.335143\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	388.562i	1.03067i
$378$	0	0
$379$	−91.9709	−0.242667	−0.121334	−	0.992612i	$-0.538717\pi$
$379$	−91.9709	−0.242667	−0.121334	+	0.992612i	$0.538717\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 272.142i	− 0.710554i	−0.934761	−	0.355277i	$-0.884387\pi$
$383$	− 272.142i	− 0.710554i	0.934761	−	0.355277i	$-0.115613\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	445.747i	1.14588i	0.819597	+	0.572940i	$0.194197\pi$
$389$	445.747i	1.14588i	−0.819597	+	0.572940i	$0.805803\pi$
$390$	0	0
$391$	−134.179	−0.343170
$392$	0	0
$393$	0	0
$394$	0	0
$395$	180.517i	0.457006i
$396$	0	0
$397$	−46.3238	−0.116685	−0.0583423	−	0.998297i	$-0.518581\pi$
$397$	−46.3238	−0.116685	−0.0583423	+	0.998297i	$0.518581\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 3.71258i	− 0.00925830i	−0.999989	−	0.00462915i	$-0.998526\pi$
$401$	− 3.71258i	− 0.00925830i	0.999989	−	0.00462915i	$-0.00147351\pi$
$402$	0	0
$403$	−724.784	−1.79847
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 10.9107i	− 0.0268077i
$408$	0	0
$409$	−284.499	−0.695595	−0.347798	−	0.937570i	$-0.613070\pi$
$409$	−284.499	−0.695595	−0.347798	+	0.937570i	$0.613070\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	53.1419	0.128053
$416$	0	0
$417$	0	0
$418$	0	0
$419$	129.962i	0.310172i	0.987901	+	0.155086i	$0.0495654\pi$
$419$	129.962i	0.310172i	−0.987901	+	0.155086i	$0.950435\pi$
$420$	0	0
$421$	−307.217	−0.729732	−0.364866	−	0.931060i	$-0.618885\pi$
$421$	−307.217	−0.729732	−0.364866	+	0.931060i	$0.618885\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	962.311i	2.26426i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	466.755i	1.08296i	0.840714	+	0.541479i	$0.182135\pi$
$431$	466.755i	1.08296i	−0.840714	+	0.541479i	$0.817865\pi$
$432$	0	0
$433$	1.18036	0.00272601	0.00136300	−	0.999999i	$-0.499566\pi$
$433$	1.18036	0.00272601	0.00136300	+	0.999999i	$0.499566\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 27.5546i	− 0.0630540i
$438$	0	0
$439$	369.550	0.841800	0.420900	−	0.907107i	$-0.361714\pi$
$439$	369.550	0.841800	0.420900	+	0.907107i	$0.361714\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 25.8549i	− 0.0583631i	−0.999574	−	0.0291816i	$-0.990710\pi$
$443$	− 25.8549i	− 0.0583631i	0.999574	−	0.0291816i	$-0.00929010\pi$
$444$	0	0
$445$	−697.999	−1.56854
$446$	0	0
$447$	0	0
$448$	0	0
$449$	690.627i	1.53814i	0.639162	+	0.769072i	$0.279281\pi$
$449$	690.627i	1.53814i	−0.639162	+	0.769072i	$0.720719\pi$
$450$	0	0
$451$	−64.0166	−0.141944
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−238.824	−0.522591	−0.261295	−	0.965259i	$-0.584150\pi$
$457$	−238.824	−0.522591	−0.261295	+	0.965259i	$0.584150\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 665.309i	− 1.44319i	−0.692318	−	0.721593i	$-0.743410\pi$
$461$	− 665.309i	− 1.44319i	0.692318	−	0.721593i	$-0.256590\pi$
$462$	0	0
$463$	852.096	1.84038	0.920190	−	0.391473i	$-0.128034\pi$
$463$	852.096	1.84038	0.920190	+	0.391473i	$0.128034\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 279.388i	− 0.598262i	−0.954212	−	0.299131i	$-0.903303\pi$
$467$	− 279.388i	− 0.598262i	0.954212	−	0.299131i	$-0.0966967\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	79.6744i	0.168445i
$474$	0	0
$475$	−197.617	−0.416035
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 259.912i	− 0.542614i	−0.962493	−	0.271307i	$-0.912544\pi$
$479$	− 259.912i	− 0.542614i	0.962493	−	0.271307i	$-0.0874558\pi$
$480$	0	0
$481$	77.7647	0.161673
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 782.179i	− 1.61274i
$486$	0	0
$487$	−306.709	−0.629792	−0.314896	−	0.949126i	$-0.601970\pi$
$487$	−306.709	−0.629792	−0.314896	+	0.949126i	$0.601970\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	107.088i	0.218102i	0.994036	+	0.109051i	$0.0347811\pi$
$491$	107.088i	0.218102i	−0.994036	+	0.109051i	$0.965219\pi$
$492$	0	0
$493$	429.605	0.871409
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−406.030	−0.813687	−0.406843	−	0.913498i	$-0.633370\pi$
$499$	−406.030	−0.813687	−0.406843	+	0.913498i	$0.633370\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 300.937i	− 0.598283i	−0.954209	−	0.299142i	$-0.903300\pi$
$503$	− 300.937i	− 0.598283i	0.954209	−	0.299142i	$-0.0967003\pi$
$504$	0	0
$505$	−1436.58	−2.84472
$506$	0	0
$507$	0	0
$508$	0	0
$509$	453.146i	0.890266i	0.895464	+	0.445133i	$0.146844\pi$
$509$	453.146i	0.890266i	−0.895464	+	0.445133i	$0.853156\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	144.151i	0.279904i
$516$	0	0
$517$	105.411	0.203889
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 390.313i	− 0.749160i	−0.927195	−	0.374580i	$-0.877787\pi$
$521$	− 390.313i	− 0.749160i	0.927195	−	0.374580i	$-0.122213\pi$
$522$	0	0
$523$	−341.009	−0.652024	−0.326012	−	0.945366i	$-0.605705\pi$
$523$	−341.009	−0.652024	−0.326012	+	0.945366i	$0.605705\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	801.342i	1.52057i
$528$	0	0
$529$	469.886	0.888254
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 456.269i	− 0.856040i
$534$	0	0
$535$	1024.48	1.91491
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−910.408	−1.68282	−0.841412	−	0.540394i	$-0.818275\pi$
$541$	−910.408	−1.68282	−0.841412	+	0.540394i	$0.818275\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 463.673i	− 0.850776i
$546$	0	0
$547$	−93.6660	−0.171236	−0.0856179	−	0.996328i	$-0.527286\pi$
$547$	−93.6660	−0.171236	−0.0856179	+	0.996328i	$0.527286\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	88.2220i	0.160113i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	388.970i	0.698330i	0.937061	+	0.349165i	$0.113535\pi$
$557$	388.970i	0.698330i	−0.937061	+	0.349165i	$0.886465\pi$
$558$	0	0
$559$	−567.868	−1.01586
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 232.914i	− 0.413701i	−0.978373	−	0.206850i	$-0.933679\pi$
$563$	− 232.914i	− 0.413701i	0.978373	−	0.206850i	$-0.0663213\pi$
$564$	0	0
$565$	−1175.49	−2.08052
$566$	0	0
$567$	0	0
$568$	0	0
$569$	487.506i	0.856777i	0.903595	+	0.428388i	$0.140918\pi$
$569$	487.506i	0.856777i	−0.903595	+	0.428388i	$0.859082\pi$
$570$	0	0
$571$	−760.019	−1.33103	−0.665515	−	0.746384i	$-0.731788\pi$
$571$	−760.019	−1.33103	−0.665515	+	0.746384i	$0.731788\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	423.952i	0.737307i
$576$	0	0
$577$	−449.306	−0.778693	−0.389346	−	0.921091i	$-0.627299\pi$
$577$	−449.306	−0.778693	−0.389346	+	0.921091i	$0.627299\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	128.345	0.220145
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 39.3268i	− 0.0669962i	−0.999439	−	0.0334981i	$-0.989335\pi$
$587$	− 39.3268i	− 0.0669962i	0.999439	−	0.0334981i	$-0.0106648\pi$
$588$	0	0
$589$	−164.560	−0.279390
$590$	0	0
$591$	0	0
$592$	0	0
$593$	420.981i	0.709918i	0.934882	+	0.354959i	$0.115505\pi$
$593$	420.981i	0.709918i	−0.934882	+	0.354959i	$0.884495\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	962.589i	1.60699i	0.595309	+	0.803497i	$0.297030\pi$
$599$	962.589i	1.60699i	−0.595309	+	0.803497i	$0.702970\pi$
$600$	0	0
$601$	−313.194	−0.521121	−0.260560	−	0.965458i	$-0.583907\pi$
$601$	−313.194	−0.521121	−0.260560	+	0.965458i	$0.583907\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1039.30i	1.71785i
$606$	0	0
$607$	18.4202	0.0303463	0.0151731	−	0.999885i	$-0.495170\pi$
$607$	18.4202	0.0303463	0.0151731	+	0.999885i	$0.495170\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	751.300i	1.22962i
$612$	0	0
$613$	995.607	1.62416	0.812078	−	0.583549i	$-0.198336\pi$
$613$	995.607	1.62416	0.812078	+	0.583549i	$0.198336\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 425.645i	− 0.689863i	−0.938628	−	0.344931i	$-0.887902\pi$
$617$	− 425.645i	− 0.689863i	0.938628	−	0.344931i	$-0.112098\pi$
$618$	0	0
$619$	867.073	1.40076	0.700382	−	0.713768i	$-0.253013\pi$
$619$	867.073	1.40076	0.700382	+	0.713768i	$0.253013\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1036.99	1.65918
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 85.9789i	− 0.136691i
$630$	0	0
$631$	−657.164	−1.04146	−0.520732	−	0.853720i	$-0.674341\pi$
$631$	−657.164	−1.04146	−0.520732	+	0.853720i	$0.674341\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1962.55i	− 3.09063i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 147.733i	− 0.230472i	−0.993338	−	0.115236i	$-0.963238\pi$
$641$	− 147.733i	− 0.230472i	0.993338	−	0.115236i	$-0.0367625\pi$
$642$	0	0
$643$	85.1602	0.132442	0.0662210	−	0.997805i	$-0.478906\pi$
$643$	85.1602	0.132442	0.0662210	+	0.997805i	$0.478906\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 351.987i	− 0.544029i	−0.962293	−	0.272015i	$-0.912310\pi$
$647$	− 351.987i	− 0.544029i	0.962293	−	0.272015i	$-0.0876899\pi$
$648$	0	0
$649$	−256.956	−0.395926
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 890.233i	− 1.36330i	−0.731680	−	0.681648i	$-0.761263\pi$
$653$	− 890.233i	− 1.36330i	0.731680	−	0.681648i	$-0.238737\pi$
$654$	0	0
$655$	1800.63	2.74906
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 598.195i	− 0.907732i	−0.891070	−	0.453866i	$-0.850044\pi$
$659$	− 598.195i	− 0.907732i	0.891070	−	0.453866i	$-0.149956\pi$
$660$	0	0
$661$	−747.356	−1.13064	−0.565322	−	0.824870i	$-0.691248\pi$
$661$	−747.356	−1.13064	−0.565322	+	0.824870i	$0.691248\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	189.265	0.283755
$668$	0	0
$669$	0	0
$670$	0	0
$671$	238.950i	0.356110i
$672$	0	0
$673$	1200.21	1.78337	0.891687	−	0.452652i	$-0.149522\pi$
$673$	1200.21	1.78337	0.891687	+	0.452652i	$0.149522\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	181.902i	0.268688i	0.990935	+	0.134344i	$0.0428927\pi$
$677$	181.902i	0.268688i	−0.990935	+	0.134344i	$0.957107\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	601.353i	0.880458i	0.897885	+	0.440229i	$0.145103\pi$
$683$	601.353i	0.880458i	−0.897885	+	0.440229i	$0.854897\pi$
$684$	0	0
$685$	645.793	0.942764
$686$	0	0
$687$	0	0
$688$	0	0
$689$	914.759i	1.32766i
$690$	0	0
$691$	626.970	0.907337	0.453669	−	0.891170i	$-0.350115\pi$
$691$	626.970	0.907337	0.453669	+	0.891170i	$0.350115\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1784.98i	− 2.56832i
$696$	0	0
$697$	−504.464	−0.723765
$698$	0	0
$699$	0	0
$700$	0	0
$701$	277.434i	0.395769i	0.980225	+	0.197885i	$0.0634071\pi$
$701$	277.434i	0.395769i	−0.980225	+	0.197885i	$0.936593\pi$
$702$	0	0
$703$	17.6563	0.0251156
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	640.571	0.903485	0.451743	−	0.892148i	$-0.350802\pi$
$709$	640.571	0.903485	0.451743	+	0.892148i	$0.350802\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	353.036i	0.495141i
$714$	0	0
$715$	312.943	0.437682
$716$	0	0
$717$	0	0
$718$	0	0
$719$	118.213i	0.164414i	0.996615	+	0.0822068i	$0.0261968\pi$
$719$	118.213i	0.164414i	−0.996615	+	0.0822068i	$0.973803\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1357.37i	− 1.87224i
$726$	0	0
$727$	−623.825	−0.858081	−0.429041	−	0.903285i	$-0.641148\pi$
$727$	−623.825	−0.858081	−0.429041	+	0.903285i	$0.641148\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	627.851i	0.858893i
$732$	0	0
$733$	−152.293	−0.207767	−0.103884	−	0.994589i	$-0.533127\pi$
$733$	−152.293	−0.207767	−0.103884	+	0.994589i	$0.533127\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	292.170i	0.396432i
$738$	0	0
$739$	−441.575	−0.597531	−0.298765	−	0.954327i	$-0.596575\pi$
$739$	−441.575	−0.597531	−0.298765	+	0.954327i	$0.596575\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 82.3447i	− 0.110827i	−0.998463	−	0.0554137i	$-0.982352\pi$
$743$	− 82.3447i	− 0.110827i	0.998463	−	0.0554137i	$-0.0176478\pi$
$744$	0	0
$745$	502.958	0.675111
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−410.932	−0.547179	−0.273590	−	0.961847i	$-0.588211\pi$
$751$	−410.932	−0.547179	−0.273590	+	0.961847i	$0.588211\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	965.997i	1.27947i
$756$	0	0
$757$	219.158	0.289508	0.144754	−	0.989468i	$-0.453761\pi$
$757$	219.158	0.289508	0.144754	+	0.989468i	$0.453761\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	474.786i	0.623898i	0.950099	+	0.311949i	$0.100982\pi$
$761$	474.786i	0.623898i	−0.950099	+	0.311949i	$0.899018\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1831.42i	− 2.38777i
$768$	0	0
$769$	109.532	0.142434	0.0712172	−	0.997461i	$-0.477312\pi$
$769$	109.532	0.142434	0.0712172	+	0.997461i	$0.477312\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	709.918i	0.918393i	0.888335	+	0.459196i	$0.151863\pi$
$773$	709.918i	0.918393i	−0.888335	+	0.459196i	$0.848137\pi$
$774$	0	0
$775$	2531.91	3.26698
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 103.595i	− 0.132984i
$780$	0	0
$781$	256.653	0.328621
$782$	0	0
$783$	0	0
$784$	0	0
$785$	66.8186i	0.0851193i
$786$	0	0
$787$	335.213	0.425938	0.212969	−	0.977059i	$-0.431687\pi$
$787$	335.213	0.425938	0.212969	+	0.977059i	$0.431687\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1703.08	−2.14764
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 342.591i	− 0.429851i	−0.976630	−	0.214925i	$-0.931049\pi$
$797$	− 342.591i	− 0.429851i	0.976630	−	0.214925i	$-0.0689509\pi$
$798$	0	0
$799$	830.659	1.03962
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 102.703i	− 0.127900i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 353.604i	− 0.437088i	−0.975827	−	0.218544i	$-0.929869\pi$
$809$	− 353.604i	− 0.437088i	0.975827	−	0.218544i	$-0.0701308\pi$
$810$	0	0
$811$	588.302	0.725404	0.362702	−	0.931905i	$-0.381854\pi$
$811$	588.302	0.725404	0.362702	+	0.931905i	$0.381854\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1538.32i	− 1.88751i
$816$	0	0
$817$	−128.933	−0.157813
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1437.17i	1.75052i	0.483655	+	0.875259i	$0.339309\pi$
$821$	1437.17i	1.75052i	−0.483655	+	0.875259i	$0.660691\pi$
$822$	0	0
$823$	1380.24	1.67708	0.838540	−	0.544840i	$-0.183410\pi$
$823$	1380.24	1.67708	0.838540	+	0.544840i	$0.183410\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 713.927i	− 0.863273i	−0.902048	−	0.431637i	$-0.857936\pi$
$827$	− 713.927i	− 0.863273i	0.902048	−	0.431637i	$-0.142064\pi$
$828$	0	0
$829$	−266.457	−0.321420	−0.160710	−	0.987002i	$-0.551378\pi$
$829$	−266.457	−0.321420	−0.160710	+	0.987002i	$0.551378\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−37.3407	−0.0447194
$836$	0	0
$837$	0	0
$838$	0	0
$839$	258.193i	0.307739i	0.988091	+	0.153869i	$0.0491735\pi$
$839$	258.193i	0.307739i	−0.988091	+	0.153869i	$0.950827\pi$
$840$	0	0
$841$	235.028	0.279462
$842$	0	0
$843$	0	0
$844$	0	0
$845$	717.545i	0.849165i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 37.8785i	− 0.0445106i
$852$	0	0
$853$	276.361	0.323987	0.161993	−	0.986792i	$-0.448208\pi$
$853$	276.361	0.323987	0.161993	+	0.986792i	$0.448208\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	587.874i	0.685968i	0.939341	+	0.342984i	$0.111438\pi$
$857$	587.874i	0.685968i	−0.939341	+	0.342984i	$0.888562\pi$
$858$	0	0
$859$	815.995	0.949936	0.474968	−	0.880003i	$-0.342460\pi$
$859$	815.995	0.949936	0.474968	+	0.880003i	$0.342460\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	428.043i	0.495995i	0.968761	+	0.247997i	$0.0797724\pi$
$863$	428.043i	0.495995i	−0.968761	+	0.247997i	$0.920228\pi$
$864$	0	0
$865$	1157.37	1.33799
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 44.6577i	− 0.0513898i
$870$	0	0
$871$	−2082.40	−2.39082
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	762.296	0.869208	0.434604	−	0.900622i	$-0.356888\pi$
$877$	762.296	0.869208	0.434604	+	0.900622i	$0.356888\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	995.350i	1.12980i	0.825161	+	0.564898i	$0.191085\pi$
$881$	995.350i	1.12980i	−0.825161	+	0.564898i	$0.808915\pi$
$882$	0	0
$883$	−874.928	−0.990859	−0.495429	−	0.868648i	$-0.664989\pi$
$883$	−874.928	−0.990859	−0.495429	+	0.868648i	$0.664989\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 837.525i	− 0.944222i	−0.881539	−	0.472111i	$-0.843492\pi$
$887$	− 837.525i	− 0.944222i	0.881539	−	0.472111i	$-0.156508\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	170.581i	0.191020i
$894$	0	0
$895$	2409.39	2.69205
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1130.32i	− 1.25731i
$900$	0	0
$901$	1011.38	1.12251
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1953.32i	− 2.15837i
$906$	0	0
$907$	−1267.95	−1.39796	−0.698982	−	0.715139i	$-0.746363\pi$
$907$	−1267.95	−1.39796	−0.698982	+	0.715139i	$0.746363\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1262.85i	− 1.38623i	−0.720829	−	0.693113i	$-0.756239\pi$
$911$	− 1262.85i	− 1.38623i	0.720829	−	0.693113i	$-0.243761\pi$
$912$	0	0
$913$	−13.1466	−0.0143994
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	65.7968	0.0715961	0.0357980	−	0.999359i	$-0.488603\pi$
$919$	65.7968	0.0715961	0.0357980	+	0.999359i	$0.488603\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1829.26i	1.98186i
$924$	0	0
$925$	−271.658	−0.293684
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1111.92i	− 1.19690i	−0.801160	−	0.598450i	$-0.795783\pi$
$929$	− 1111.92i	− 1.19690i	0.801160	−	0.598450i	$-0.204217\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 345.998i	− 0.370052i
$936$	0	0
$937$	636.217	0.678993	0.339497	−	0.940607i	$-0.389743\pi$
$937$	636.217	0.678993	0.339497	+	0.940607i	$0.389743\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1650.38i	− 1.75385i	−0.480623	−	0.876927i	$-0.659589\pi$
$941$	− 1650.38i	− 1.75385i	0.480623	−	0.876927i	$-0.340411\pi$
$942$	0	0
$943$	−222.245	−0.235678
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 710.808i	− 0.750589i	−0.926906	−	0.375294i	$-0.877542\pi$
$947$	− 710.808i	− 0.750589i	0.926906	−	0.375294i	$-0.122458\pi$
$948$	0	0
$949$	732.004	0.771342
$950$	0	0
$951$	0	0
$952$	0	0
$953$	839.240i	0.880629i	0.897844	+	0.440315i	$0.145133\pi$
$953$	839.240i	0.880629i	−0.897844	+	0.440315i	$0.854867\pi$
$954$	0	0
$955$	−2178.70	−2.28136
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1147.39	1.19395
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2674.32i	2.77132i
$966$	0	0
$967$	−848.150	−0.877094	−0.438547	−	0.898708i	$-0.644507\pi$
$967$	−848.150	−0.877094	−0.438547	+	0.898708i	$0.644507\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	558.752i	0.575440i	0.957715	+	0.287720i	$0.0928972\pi$
$971$	558.752i	0.575440i	−0.957715	+	0.287720i	$0.907103\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1215.22i	1.24383i	0.783086	+	0.621913i	$0.213644\pi$
$977$	1215.22i	1.24383i	−0.783086	+	0.621913i	$0.786356\pi$
$978$	0	0
$979$	172.676	0.176380
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1083.10i	1.10183i	0.834560	+	0.550917i	$0.185722\pi$
$983$	1083.10i	1.10183i	−0.834560	+	0.550917i	$0.814278\pi$
$984$	0	0
$985$	421.257	0.427673
$986$	0	0
$987$	0	0
$988$	0	0
$989$	276.603i	0.279680i
$990$	0	0
$991$	−700.907	−0.707272	−0.353636	−	0.935383i	$-0.615055\pi$
$991$	−700.907	−0.707272	−0.353636	+	0.935383i	$0.615055\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2458.68i	− 2.47104i
$996$	0	0
$997$	530.354	0.531949	0.265975	−	0.963980i	$-0.414306\pi$
$997$	530.354	0.531949	0.265975	+	0.963980i	$0.414306\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.3.d.g.1961.8		8
3.2	odd	2	inner	3528.3.d.g.1961.1		8
7.2	even	3		504.3.cu.a.305.8	yes	16
7.4	even	3		504.3.cu.a.233.1	✓	16
7.6	odd	2		3528.3.d.j.1961.1		8
21.2	odd	6		504.3.cu.a.305.1	yes	16
21.11	odd	6		504.3.cu.a.233.8	yes	16
21.20	even	2		3528.3.d.j.1961.8		8
28.11	odd	6		1008.3.dc.g.737.1		16
28.23	odd	6		1008.3.dc.g.305.8		16
84.11	even	6		1008.3.dc.g.737.8		16
84.23	even	6		1008.3.dc.g.305.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.cu.a.233.1	✓	16	7.4	even	3
504.3.cu.a.233.8	yes	16	21.11	odd	6
504.3.cu.a.305.1	yes	16	21.2	odd	6
504.3.cu.a.305.8	yes	16	7.2	even	3
1008.3.dc.g.305.1		16	84.23	even	6
1008.3.dc.g.305.8		16	28.23	odd	6
1008.3.dc.g.737.1		16	28.11	odd	6
1008.3.dc.g.737.8		16	84.11	even	6
3528.3.d.g.1961.1		8	3.2	odd	2	inner
3528.3.d.g.1961.8		8	1.1	even	1	trivial
3528.3.d.j.1961.1		8	7.6	odd	2
3528.3.d.j.1961.8		8	21.20	even	2

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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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3528.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.95214i q^{5} -2.21465i q^{11} +15.7846 q^{13} -17.4519i q^{17} +3.58386 q^{19} -7.68853i q^{23} -55.1408 q^{25} +24.6165i q^{29} -45.9172 q^{31} +4.92662 q^{37} -28.9060i q^{41} -35.9761 q^{43} +47.5971i q^{47} +57.9526i q^{53} +19.8258 q^{55} -116.026i q^{59} -107.895 q^{61} +141.306i q^{65} -131.926 q^{67} +115.889i q^{71} +46.3746 q^{73} +20.1647 q^{79} -5.93622i q^{83} +156.232 q^{85} +77.9701i q^{89} +32.0832i q^{95} -87.3734 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{13} + 12 q^{19} - 92 q^{25} - 32 q^{31} + 132 q^{37} + 12 q^{43} - 220 q^{55} - 184 q^{61} - 332 q^{67} - 188 q^{73} + 112 q^{79} + 128 q^{85} - 516 q^{97}+O(q^{100}) \) 8 * q - 12 * q^13 + 12 * q^19 - 92 * q^25 - 32 * q^31 + 132 * q^37 + 12 * q^43 - 220 * q^55 - 184 * q^61 - 332 * q^67 - 188 * q^73 + 112 * q^79 + 128 * q^85 - 516 * q^97

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