LMFDB - Embedded newform 3528.3.d.k.1961.1

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 = [8,0,0,0,0,0,0,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,4] 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} - 38x^{6} - 60x^{5} + 385x^{4} + 1260x^{3} + 1230x^{2} + 540x + 594 $ x^8 - 38x^6 - 60x^5 + 385x^4 + 1260x^3 + 1230x^2 + 540x + 594
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1961.1
Root			$-1.87753 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3528.1961
Dual form			3528.3.d.k.1961.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.57392i q^{5} +14.5417i q^{11} +10.9055 q^{13} -26.8141i q^{17} -7.97599 q^{19} +26.8141i q^{23} -48.5120 q^{25} +35.7359i q^{29} +23.8447 q^{31} -38.3654 q^{37} +27.8548i q^{41} +68.4986 q^{43} +71.1239i q^{47} +63.4746i q^{53} +124.679 q^{55} -8.92188i q^{59} -5.77419 q^{61} -93.5030i q^{65} -34.6720 q^{67} +68.6199i q^{71} -131.264 q^{73} +30.0642 q^{79} -86.5943i q^{83} -229.901 q^{85} +128.558i q^{89} +68.3854i q^{95} +80.0096 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 20 q^{13} + 20 q^{19} + 4 q^{25} + 56 q^{31} - 76 q^{37} + 36 q^{43} + 324 q^{55} + 72 q^{61} + 156 q^{67} - 124 q^{73} - 184 q^{79} - 432 q^{85} + 556 q^{97}+O(q^{100}) $ 8 * q + 20 * q^13 + 20 * q^19 + 4 * q^25 + 56 * q^31 - 76 * q^37 + 36 * q^43 + 324 * q^55 + 72 * q^61 + 156 * q^67 - 124 * q^73 - 184 * q^79 - 432 * q^85 + 556 * 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
$5$	− 8.57392i	− 1.71478i	−0.514665	−	0.857392i	$-0.672084\pi$
$5$	− 8.57392i	− 1.71478i	0.514665	−	0.857392i	$-0.327916\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	14.5417i	1.32197i	0.750400	+	0.660984i	$0.229861\pi$
$11$	14.5417i	1.32197i	−0.750400	+	0.660984i	$0.770139\pi$
$12$	0	0
$13$	10.9055	0.838886	0.419443	−	0.907782i	$-0.362225\pi$
$13$	10.9055	0.838886	0.419443	+	0.907782i	$0.362225\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 26.8141i	− 1.57730i	−0.614844	−	0.788649i	$-0.710781\pi$
$17$	− 26.8141i	− 1.57730i	0.614844	−	0.788649i	$-0.289219\pi$
$18$	0	0
$19$	−7.97599	−0.419789	−0.209894	−	0.977724i	$-0.567312\pi$
$19$	−7.97599	−0.419789	−0.209894	+	0.977724i	$0.567312\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	26.8141i	1.16583i	0.812534	+	0.582914i	$0.198088\pi$
$23$	26.8141i	1.16583i	−0.812534	+	0.582914i	$0.801912\pi$
$24$	0	0
$25$	−48.5120	−1.94048
$26$	0	0
$27$	0	0
$28$	0	0
$29$	35.7359i	1.23227i	0.787639	+	0.616137i	$0.211303\pi$
$29$	35.7359i	1.23227i	−0.787639	+	0.616137i	$0.788697\pi$
$30$	0	0
$31$	23.8447	0.769182	0.384591	−	0.923087i	$-0.374342\pi$
$31$	23.8447	0.769182	0.384591	+	0.923087i	$0.374342\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−38.3654	−1.03690	−0.518451	−	0.855107i	$-0.673491\pi$
$37$	−38.3654	−1.03690	−0.518451	+	0.855107i	$0.673491\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	27.8548i	0.679386i	0.940536	+	0.339693i	$0.110323\pi$
$41$	27.8548i	0.679386i	−0.940536	+	0.339693i	$0.889677\pi$
$42$	0	0
$43$	68.4986	1.59299	0.796496	−	0.604644i	$-0.206685\pi$
$43$	68.4986	1.59299	0.796496	+	0.604644i	$0.206685\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	71.1239i	1.51327i	0.653835	+	0.756637i	$0.273159\pi$
$47$	71.1239i	1.51327i	−0.653835	+	0.756637i	$0.726841\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	63.4746i	1.19763i	0.800886	+	0.598817i	$0.204362\pi$
$53$	63.4746i	1.19763i	−0.800886	+	0.598817i	$0.795638\pi$
$54$	0	0
$55$	124.679	2.26689
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 8.92188i	− 0.151218i	−0.997138	−	0.0756091i	$-0.975910\pi$
$59$	− 8.92188i	− 0.151218i	0.997138	−	0.0756091i	$-0.0240901\pi$
$60$	0	0
$61$	−5.77419	−0.0946588	−0.0473294	−	0.998879i	$-0.515071\pi$
$61$	−5.77419	−0.0946588	−0.0473294	+	0.998879i	$0.515071\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 93.5030i	− 1.43851i
$66$	0	0
$67$	−34.6720	−0.517493	−0.258746	−	0.965945i	$-0.583309\pi$
$67$	−34.6720	−0.517493	−0.258746	+	0.965945i	$0.583309\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	68.6199i	0.966478i	0.875488	+	0.483239i	$0.160540\pi$
$71$	68.6199i	0.966478i	−0.875488	+	0.483239i	$0.839460\pi$
$72$	0	0
$73$	−131.264	−1.79814	−0.899069	−	0.437806i	$-0.855756\pi$
$73$	−131.264	−1.79814	−0.899069	+	0.437806i	$0.855756\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	30.0642	0.380559	0.190280	−	0.981730i	$-0.439061\pi$
$79$	30.0642	0.380559	0.190280	+	0.981730i	$0.439061\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 86.5943i	− 1.04331i	−0.853158	−	0.521653i	$-0.825316\pi$
$83$	− 86.5943i	− 1.04331i	0.853158	−	0.521653i	$-0.174684\pi$
$84$	0	0
$85$	−229.901	−2.70472
$86$	0	0
$87$	0	0
$88$	0	0
$89$	128.558i	1.44447i	0.691647	+	0.722236i	$0.256885\pi$
$89$	128.558i	1.44447i	−0.691647	+	0.722236i	$0.743115\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	68.3854i	0.719847i
$96$	0	0
$97$	80.0096	0.824841	0.412421	−	0.910994i	$-0.364683\pi$
$97$	80.0096	0.824841	0.412421	+	0.910994i	$0.364683\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 60.2984i	− 0.597013i	−0.954408	−	0.298507i	$-0.903511\pi$
$101$	− 60.2984i	− 0.597013i	0.954408	−	0.298507i	$-0.0964885\pi$
$102$	0	0
$103$	−136.616	−1.32637	−0.663184	−	0.748457i	$-0.730795\pi$
$103$	−136.616	−1.32637	−0.663184	+	0.748457i	$0.730795\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	88.5597i	0.827661i	0.910354	+	0.413830i	$0.135809\pi$
$107$	88.5597i	0.827661i	−0.910354	+	0.413830i	$0.864191\pi$
$108$	0	0
$109$	120.440	1.10496	0.552479	−	0.833527i	$-0.313682\pi$
$109$	120.440	1.10496	0.552479	+	0.833527i	$0.313682\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 6.02151i	− 0.0532877i	−0.999645	−	0.0266438i	$-0.991518\pi$
$113$	− 6.02151i	− 0.0532877i	0.999645	−	0.0266438i	$-0.00848200\pi$
$114$	0	0
$115$	229.901	1.99914
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−90.4596	−0.747600
$122$	0	0
$123$	0	0
$124$	0	0
$125$	201.590i	1.61272i
$126$	0	0
$127$	−63.3782	−0.499041	−0.249521	−	0.968369i	$-0.580273\pi$
$127$	−63.3782	−0.499041	−0.249521	+	0.968369i	$0.580273\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	37.8129i	0.288648i	0.989530	+	0.144324i	$0.0461007\pi$
$131$	37.8129i	0.288648i	−0.989530	+	0.144324i	$0.953899\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	64.9147i	0.473830i	0.971530	+	0.236915i	$0.0761363\pi$
$137$	64.9147i	0.473830i	−0.971530	+	0.236915i	$0.923864\pi$
$138$	0	0
$139$	−124.568	−0.896171	−0.448086	−	0.893991i	$-0.647894\pi$
$139$	−124.568	−0.896171	−0.448086	+	0.893991i	$0.647894\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	158.584i	1.10898i
$144$	0	0
$145$	306.397	2.11308
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 156.079i	− 1.04751i	−0.851868	−	0.523757i	$-0.824530\pi$
$149$	− 156.079i	− 1.04751i	0.851868	−	0.523757i	$-0.175470\pi$
$150$	0	0
$151$	176.449	1.16854	0.584270	−	0.811560i	$-0.301381\pi$
$151$	176.449	1.16854	0.584270	+	0.811560i	$0.301381\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 204.442i	− 1.31898i
$156$	0	0
$157$	43.4565	0.276793	0.138396	−	0.990377i	$-0.455805\pi$
$157$	43.4565	0.276793	0.138396	+	0.990377i	$0.455805\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−156.335	−0.959110	−0.479555	−	0.877512i	$-0.659202\pi$
$163$	−156.335	−0.959110	−0.479555	+	0.877512i	$0.659202\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	137.999i	0.826343i	0.910653	+	0.413172i	$0.135579\pi$
$167$	137.999i	0.826343i	−0.910653	+	0.413172i	$0.864421\pi$
$168$	0	0
$169$	−50.0696	−0.296270
$170$	0	0
$171$	0	0
$172$	0	0
$173$	61.8388i	0.357450i	0.983899	+	0.178725i	$0.0571972\pi$
$173$	61.8388i	0.357450i	−0.983899	+	0.178725i	$0.942803\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	194.288i	1.08541i	0.839925	+	0.542703i	$0.182599\pi$
$179$	194.288i	1.08541i	−0.839925	+	0.542703i	$0.817401\pi$
$180$	0	0
$181$	23.8714	0.131886	0.0659430	−	0.997823i	$-0.478994\pi$
$181$	23.8714	0.131886	0.0659430	+	0.997823i	$0.478994\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	328.942i	1.77806i
$186$	0	0
$187$	389.921	2.08514
$188$	0	0
$189$	0	0
$190$	0	0
$191$	164.171i	0.859532i	0.902940	+	0.429766i	$0.141404\pi$
$191$	164.171i	0.859532i	−0.902940	+	0.429766i	$0.858596\pi$
$192$	0	0
$193$	292.939	1.51782	0.758909	−	0.651197i	$-0.225733\pi$
$193$	292.939	1.51782	0.758909	+	0.651197i	$0.225733\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 232.573i	− 1.18058i	−0.807193	−	0.590288i	$-0.799014\pi$
$197$	− 232.573i	− 1.18058i	0.807193	−	0.590288i	$-0.200986\pi$
$198$	0	0
$199$	24.1124	0.121168	0.0605840	−	0.998163i	$-0.480704\pi$
$199$	24.1124	0.121168	0.0605840	+	0.998163i	$0.480704\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	238.825	1.16500
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 115.984i	− 0.554947i
$210$	0	0
$211$	393.303	1.86400	0.931998	−	0.362464i	$-0.118064\pi$
$211$	393.303	1.86400	0.931998	+	0.362464i	$0.118064\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 587.302i	− 2.73164i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 292.421i	− 1.32317i
$222$	0	0
$223$	198.609	0.890623	0.445311	−	0.895376i	$-0.353093\pi$
$223$	198.609	0.890623	0.445311	+	0.895376i	$0.353093\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 156.819i	− 0.690834i	−0.938449	−	0.345417i	$-0.887738\pi$
$227$	− 156.819i	− 0.690834i	0.938449	−	0.345417i	$-0.112262\pi$
$228$	0	0
$229$	−20.5714	−0.0898316	−0.0449158	−	0.998991i	$-0.514302\pi$
$229$	−20.5714	−0.0898316	−0.0449158	+	0.998991i	$0.514302\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 73.9599i	− 0.317424i	−0.987325	−	0.158712i	$-0.949266\pi$
$233$	− 73.9599i	− 0.317424i	0.987325	−	0.158712i	$-0.0507342\pi$
$234$	0	0
$235$	609.810	2.59494
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 238.052i	− 0.996035i	−0.867167	−	0.498017i	$-0.834062\pi$
$239$	− 238.052i	− 0.996035i	0.867167	−	0.498017i	$-0.165938\pi$
$240$	0	0
$241$	201.402	0.835695	0.417847	−	0.908517i	$-0.362785\pi$
$241$	201.402	0.835695	0.417847	+	0.908517i	$0.362785\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−86.9823	−0.352155
$248$	0	0
$249$	0	0
$250$	0	0
$251$	260.896i	1.03942i	0.854341	+	0.519712i	$0.173961\pi$
$251$	260.896i	1.03942i	−0.854341	+	0.519712i	$0.826039\pi$
$252$	0	0
$253$	−389.921	−1.54119
$254$	0	0
$255$	0	0
$256$	0	0
$257$	144.620i	0.562723i	0.959602	+	0.281361i	$0.0907859\pi$
$257$	144.620i	0.562723i	−0.959602	+	0.281361i	$0.909214\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 159.938i	− 0.608131i	−0.952651	−	0.304065i	$-0.901656\pi$
$263$	− 159.938i	− 0.608131i	0.952651	−	0.304065i	$-0.0983441\pi$
$264$	0	0
$265$	544.226	2.05368
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 119.728i	− 0.445086i	−0.974923	−	0.222543i	$-0.928564\pi$
$269$	− 119.728i	− 0.445086i	0.974923	−	0.222543i	$-0.0714358\pi$
$270$	0	0
$271$	505.421	1.86502	0.932511	−	0.361141i	$-0.117613\pi$
$271$	505.421	1.86502	0.932511	+	0.361141i	$0.117613\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 705.445i	− 2.56525i
$276$	0	0
$277$	291.401	1.05199	0.525995	−	0.850488i	$-0.323693\pi$
$277$	291.401	1.05199	0.525995	+	0.850488i	$0.323693\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 371.722i	− 1.32285i	−0.750010	−	0.661427i	$-0.769951\pi$
$281$	− 371.722i	− 1.32285i	0.750010	−	0.661427i	$-0.230049\pi$
$282$	0	0
$283$	−264.456	−0.934472	−0.467236	−	0.884133i	$-0.654750\pi$
$283$	−264.456	−0.934472	−0.467236	+	0.884133i	$0.654750\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−429.994	−1.48787
$290$	0	0
$291$	0	0
$292$	0	0
$293$	559.211i	1.90857i	0.298897	+	0.954285i	$0.403381\pi$
$293$	559.211i	1.90857i	−0.298897	+	0.954285i	$0.596619\pi$
$294$	0	0
$295$	−76.4954	−0.259306
$296$	0	0
$297$	0	0
$298$	0	0
$299$	292.421i	0.977998i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	49.5074i	0.162319i
$306$	0	0
$307$	230.693	0.751442	0.375721	−	0.926733i	$-0.377395\pi$
$307$	230.693	0.751442	0.375721	+	0.926733i	$0.377395\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 170.555i	− 0.548408i	−0.961672	−	0.274204i	$-0.911586\pi$
$311$	− 170.555i	− 0.548408i	0.961672	−	0.274204i	$-0.0884144\pi$
$312$	0	0
$313$	−262.708	−0.839323	−0.419662	−	0.907681i	$-0.637851\pi$
$313$	−262.708	−0.839323	−0.419662	+	0.907681i	$0.637851\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	204.187i	0.644123i	0.946719	+	0.322062i	$0.104376\pi$
$317$	204.187i	0.644123i	−0.946719	+	0.322062i	$0.895624\pi$
$318$	0	0
$319$	−519.659	−1.62903
$320$	0	0
$321$	0	0
$322$	0	0
$323$	213.869i	0.662132i
$324$	0	0
$325$	−529.049	−1.62784
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	163.406	0.493674	0.246837	−	0.969057i	$-0.420609\pi$
$331$	163.406	0.493674	0.246837	+	0.969057i	$0.420609\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	297.275i	0.887388i
$336$	0	0
$337$	588.142	1.74523	0.872614	−	0.488411i	$-0.162423\pi$
$337$	588.142	1.74523	0.872614	+	0.488411i	$0.162423\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	346.741i	1.01683i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	555.009i	1.59945i	0.600366	+	0.799725i	$0.295021\pi$
$347$	555.009i	1.59945i	−0.600366	+	0.799725i	$0.704979\pi$
$348$	0	0
$349$	248.425	0.711819	0.355910	−	0.934520i	$-0.384171\pi$
$349$	248.425	0.711819	0.355910	+	0.934520i	$0.384171\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	580.389i	1.64416i	0.569370	+	0.822081i	$0.307187\pi$
$353$	580.389i	1.64416i	−0.569370	+	0.822081i	$0.692813\pi$
$354$	0	0
$355$	588.342	1.65730
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 193.354i	− 0.538592i	−0.963058	−	0.269296i	$-0.913209\pi$
$359$	− 193.354i	− 0.538592i	0.963058	−	0.269296i	$-0.0867909\pi$
$360$	0	0
$361$	−297.384	−0.823777
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1125.45i	3.08342i
$366$	0	0
$367$	332.929	0.907163	0.453582	−	0.891215i	$-0.350146\pi$
$367$	332.929	0.907163	0.453582	+	0.891215i	$0.350146\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−452.996	−1.21447	−0.607233	−	0.794524i	$-0.707721\pi$
$373$	−452.996	−1.21447	−0.607233	+	0.794524i	$0.707721\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	389.719i	1.03374i
$378$	0	0
$379$	−598.216	−1.57841	−0.789203	−	0.614133i	$-0.789506\pi$
$379$	−598.216	−1.57841	−0.789203	+	0.614133i	$0.789506\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 290.640i	− 0.758852i	−0.925222	−	0.379426i	$-0.876122\pi$
$383$	− 290.640i	− 0.758852i	0.925222	−	0.379426i	$-0.123878\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 93.1096i	− 0.239356i	−0.992813	−	0.119678i	$-0.961814\pi$
$389$	− 93.1096i	− 0.239356i	0.992813	−	0.119678i	$-0.0381862\pi$
$390$	0	0
$391$	718.994	1.83886
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 257.768i	− 0.652577i
$396$	0	0
$397$	−300.366	−0.756589	−0.378294	−	0.925685i	$-0.623489\pi$
$397$	−300.366	−0.756589	−0.378294	+	0.925685i	$0.623489\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	220.928i	0.550943i	0.961309	+	0.275471i	$0.0888339\pi$
$401$	220.928i	0.550943i	−0.961309	+	0.275471i	$0.911166\pi$
$402$	0	0
$403$	260.038	0.645257
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 557.896i	− 1.37075i
$408$	0	0
$409$	141.765	0.346614	0.173307	−	0.984868i	$-0.444555\pi$
$409$	141.765	0.346614	0.173307	+	0.984868i	$0.444555\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−742.452	−1.78904
$416$	0	0
$417$	0	0
$418$	0	0
$419$	455.434i	1.08695i	0.839424	+	0.543477i	$0.182893\pi$
$419$	455.434i	1.08695i	−0.839424	+	0.543477i	$0.817107\pi$
$420$	0	0
$421$	123.182	0.292593	0.146296	−	0.989241i	$-0.453265\pi$
$421$	123.182	0.292593	0.146296	+	0.989241i	$0.453265\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1300.80i	3.06072i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 104.545i	− 0.242564i	−0.992618	−	0.121282i	$-0.961299\pi$
$431$	− 104.545i	− 0.242564i	0.992618	−	0.121282i	$-0.0387005\pi$
$432$	0	0
$433$	249.952	0.577256	0.288628	−	0.957441i	$-0.406801\pi$
$433$	249.952	0.577256	0.288628	+	0.957441i	$0.406801\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 213.869i	− 0.489402i
$438$	0	0
$439$	−363.345	−0.827665	−0.413832	−	0.910353i	$-0.635810\pi$
$439$	−363.345	−0.827665	−0.413832	+	0.910353i	$0.635810\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	724.075i	1.63448i	0.576297	+	0.817240i	$0.304497\pi$
$443$	724.075i	1.63448i	−0.576297	+	0.817240i	$0.695503\pi$
$444$	0	0
$445$	1102.24	2.47695
$446$	0	0
$447$	0	0
$448$	0	0
$449$	498.027i	1.10919i	0.832120	+	0.554596i	$0.187127\pi$
$449$	498.027i	1.10919i	−0.832120	+	0.554596i	$0.812873\pi$
$450$	0	0
$451$	−405.055	−0.898126
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	636.745	1.39332	0.696658	−	0.717404i	$-0.254670\pi$
$457$	636.745	1.39332	0.696658	+	0.717404i	$0.254670\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 597.813i	− 1.29678i	−0.761310	−	0.648388i	$-0.775444\pi$
$461$	− 597.813i	− 1.29678i	0.761310	−	0.648388i	$-0.224556\pi$
$462$	0	0
$463$	−710.671	−1.53493	−0.767463	−	0.641093i	$-0.778481\pi$
$463$	−710.671	−1.53493	−0.767463	+	0.641093i	$0.778481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	208.489i	0.446444i	0.974768	+	0.223222i	$0.0716574\pi$
$467$	208.489i	0.446444i	−0.974768	+	0.223222i	$0.928343\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	996.083i	2.10588i
$474$	0	0
$475$	386.931	0.814592
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 311.954i	− 0.651260i	−0.945497	−	0.325630i	$-0.894424\pi$
$479$	− 311.954i	− 0.651260i	0.945497	−	0.325630i	$-0.105576\pi$
$480$	0	0
$481$	−418.395	−0.869844
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 685.996i	− 1.41442i
$486$	0	0
$487$	−955.175	−1.96135	−0.980673	−	0.195654i	$-0.937317\pi$
$487$	−955.175	−1.96135	−0.980673	+	0.195654i	$0.937317\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 131.161i	− 0.267131i	−0.991040	−	0.133565i	$-0.957357\pi$
$491$	− 131.161i	− 0.267131i	0.991040	−	0.133565i	$-0.0426426\pi$
$492$	0	0
$493$	958.226	1.94366
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	348.305	0.698006	0.349003	−	0.937122i	$-0.386520\pi$
$499$	348.305	0.698006	0.349003	+	0.937122i	$0.386520\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 602.739i	− 1.19829i	−0.800641	−	0.599144i	$-0.795508\pi$
$503$	− 602.739i	− 1.19829i	0.800641	−	0.599144i	$-0.204492\pi$
$504$	0	0
$505$	−516.993	−1.02375
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.0888i	0.0276793i	0.999904	+	0.0138397i	$0.00440544\pi$
$509$	14.0888i	0.0276793i	−0.999904	+	0.0138397i	$0.995595\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1171.33i	2.27443i
$516$	0	0
$517$	−1034.26	−2.00050
$518$	0	0
$519$	0	0
$520$	0	0
$521$	47.8232i	0.0917911i	0.998946	+	0.0458956i	$0.0146141\pi$
$521$	47.8232i	0.0917911i	−0.998946	+	0.0458956i	$0.985386\pi$
$522$	0	0
$523$	384.974	0.736088	0.368044	−	0.929808i	$-0.380028\pi$
$523$	384.974	0.736088	0.368044	+	0.929808i	$0.380028\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 639.372i	− 1.21323i
$528$	0	0
$529$	−189.994	−0.359157
$530$	0	0
$531$	0	0
$532$	0	0
$533$	303.771i	0.569927i
$534$	0	0
$535$	759.303	1.41926
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	401.183	0.741558	0.370779	−	0.928721i	$-0.379091\pi$
$541$	401.183	0.741558	0.370779	+	0.928721i	$0.379091\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1032.65i	− 1.89476i
$546$	0	0
$547$	826.934	1.51176	0.755881	−	0.654709i	$-0.227209\pi$
$547$	826.934	1.51176	0.755881	+	0.654709i	$0.227209\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 285.029i	− 0.517295i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	716.450i	1.28627i	0.765754	+	0.643133i	$0.222366\pi$
$557$	716.450i	1.28627i	−0.765754	+	0.643133i	$0.777634\pi$
$558$	0	0
$559$	747.014	1.33634
$560$	0	0
$561$	0	0
$562$	0	0
$563$	248.777i	0.441877i	0.975288	+	0.220938i	$0.0709120\pi$
$563$	248.777i	0.441877i	−0.975288	+	0.220938i	$0.929088\pi$
$564$	0	0
$565$	−51.6279	−0.0913768
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 54.3872i	− 0.0955838i	−0.998857	−	0.0477919i	$-0.984782\pi$
$569$	− 54.3872i	− 0.0955838i	0.998857	−	0.0477919i	$-0.0152184\pi$
$570$	0	0
$571$	420.729	0.736828	0.368414	−	0.929662i	$-0.379901\pi$
$571$	420.729	0.736828	0.368414	+	0.929662i	$0.379901\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1300.80i	− 2.26227i
$576$	0	0
$577$	−712.798	−1.23535	−0.617676	−	0.786433i	$-0.711926\pi$
$577$	−712.798	−1.23535	−0.617676	+	0.786433i	$0.711926\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−923.025	−1.58323
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.9882i	0.0613088i	0.999530	+	0.0306544i	$0.00975912\pi$
$587$	35.9882i	0.0613088i	−0.999530	+	0.0306544i	$0.990241\pi$
$588$	0	0
$589$	−190.185	−0.322894
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 410.930i	− 0.692968i	−0.938056	−	0.346484i	$-0.887375\pi$
$593$	− 410.930i	− 0.692968i	0.938056	−	0.346484i	$-0.112625\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 499.293i	− 0.833545i	−0.909011	−	0.416773i	$-0.863161\pi$
$599$	− 499.293i	− 0.833545i	0.909011	−	0.416773i	$-0.136839\pi$
$600$	0	0
$601$	70.6569	0.117566	0.0587828	−	0.998271i	$-0.481278\pi$
$601$	70.6569	0.117566	0.0587828	+	0.998271i	$0.481278\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	775.593i	1.28197i
$606$	0	0
$607$	830.528	1.36825	0.684125	−	0.729364i	$-0.260184\pi$
$607$	830.528	1.36825	0.684125	+	0.729364i	$0.260184\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	775.643i	1.26947i
$612$	0	0
$613$	−226.918	−0.370176	−0.185088	−	0.982722i	$-0.559257\pi$
$613$	−226.918	−0.370176	−0.185088	+	0.982722i	$0.559257\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	418.262i	0.677895i	0.940805	+	0.338948i	$0.110071\pi$
$617$	418.262i	0.677895i	−0.940805	+	0.338948i	$0.889929\pi$
$618$	0	0
$619$	513.201	0.829080	0.414540	−	0.910031i	$-0.363942\pi$
$619$	513.201	0.829080	0.414540	+	0.910031i	$0.363942\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	515.616	0.824986
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1028.73i	1.63550i
$630$	0	0
$631$	−976.542	−1.54761	−0.773805	−	0.633424i	$-0.781649\pi$
$631$	−976.542	−1.54761	−0.773805	+	0.633424i	$0.781649\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	543.400i	0.855748i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 952.274i	− 1.48561i	−0.669510	−	0.742803i	$-0.733496\pi$
$641$	− 952.274i	− 1.48561i	0.669510	−	0.742803i	$-0.266504\pi$
$642$	0	0
$643$	−910.267	−1.41566	−0.707828	−	0.706385i	$-0.750325\pi$
$643$	−910.267	−1.41566	−0.707828	+	0.706385i	$0.750325\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 775.131i	− 1.19804i	−0.800735	−	0.599019i	$-0.795557\pi$
$647$	− 775.131i	− 1.19804i	0.800735	−	0.599019i	$-0.204443\pi$
$648$	0	0
$649$	129.739	0.199906
$650$	0	0
$651$	0	0
$652$	0	0
$653$	231.119i	0.353934i	0.984217	+	0.176967i	$0.0566285\pi$
$653$	231.119i	0.353934i	−0.984217	+	0.176967i	$0.943371\pi$
$654$	0	0
$655$	324.204	0.494968
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 731.379i	− 1.10983i	−0.831907	−	0.554915i	$-0.812751\pi$
$659$	− 731.379i	− 1.10983i	0.831907	−	0.554915i	$-0.187249\pi$
$660$	0	0
$661$	627.701	0.949623	0.474811	−	0.880088i	$-0.342516\pi$
$661$	627.701	0.949623	0.474811	+	0.880088i	$0.342516\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−958.226	−1.43662
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 83.9662i	− 0.125136i
$672$	0	0
$673$	−536.968	−0.797872	−0.398936	−	0.916979i	$-0.630620\pi$
$673$	−536.968	−0.797872	−0.398936	+	0.916979i	$0.630620\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	416.461i	0.615157i	0.951523	+	0.307579i	$0.0995187\pi$
$677$	416.461i	0.615157i	−0.951523	+	0.307579i	$0.900481\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	838.290i	1.22736i	0.789553	+	0.613682i	$0.210312\pi$
$683$	838.290i	1.22736i	−0.789553	+	0.613682i	$0.789688\pi$
$684$	0	0
$685$	556.573	0.812515
$686$	0	0
$687$	0	0
$688$	0	0
$689$	692.224i	1.00468i
$690$	0	0
$691$	412.350	0.596744	0.298372	−	0.954450i	$-0.403556\pi$
$691$	412.350	0.596744	0.298372	+	0.954450i	$0.403556\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1068.03i	1.53674i
$696$	0	0
$697$	746.901	1.07159
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.5457i	0.0278825i	0.999903	+	0.0139413i	$0.00443779\pi$
$701$	19.5457i	0.0278825i	−0.999903	+	0.0139413i	$0.995562\pi$
$702$	0	0
$703$	306.002	0.435280
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−120.712	−0.170257	−0.0851287	−	0.996370i	$-0.527130\pi$
$709$	−120.712	−0.170257	−0.0851287	+	0.996370i	$0.527130\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	639.372i	0.896735i
$714$	0	0
$715$	1359.69	1.90166
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 884.044i	− 1.22955i	−0.788704	−	0.614773i	$-0.789247\pi$
$719$	− 884.044i	− 1.22955i	0.788704	−	0.614773i	$-0.210753\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1733.62i	− 2.39120i
$726$	0	0
$727$	−1322.62	−1.81929	−0.909643	−	0.415391i	$-0.863645\pi$
$727$	−1322.62	−1.81929	−0.909643	+	0.415391i	$0.863645\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1836.73i	− 2.51262i
$732$	0	0
$733$	−324.784	−0.443088	−0.221544	−	0.975150i	$-0.571110\pi$
$733$	−324.784	−0.443088	−0.221544	+	0.975150i	$0.571110\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 504.188i	− 0.684109i
$738$	0	0
$739$	−164.499	−0.222597	−0.111298	−	0.993787i	$-0.535501\pi$
$739$	−164.499	−0.222597	−0.111298	+	0.993787i	$0.535501\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	247.995i	0.333775i	0.985976	+	0.166888i	$0.0533717\pi$
$743$	247.995i	0.333775i	−0.985976	+	0.166888i	$0.946628\pi$
$744$	0	0
$745$	−1338.21	−1.79626
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	624.280	0.831264	0.415632	−	0.909533i	$-0.363560\pi$
$751$	624.280	0.831264	0.415632	+	0.909533i	$0.363560\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1512.86i	− 2.00379i
$756$	0	0
$757$	−1091.35	−1.44167	−0.720836	−	0.693105i	$-0.756242\pi$
$757$	−1091.35	−1.44167	−0.720836	+	0.693105i	$0.756242\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 483.228i	− 0.634991i	−0.948260	−	0.317496i	$-0.897158\pi$
$761$	− 483.228i	− 0.634991i	0.948260	−	0.317496i	$-0.102842\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 97.2977i	− 0.126855i
$768$	0	0
$769$	−1025.78	−1.33391	−0.666957	−	0.745096i	$-0.732404\pi$
$769$	−1025.78	−1.33391	−0.666957	+	0.745096i	$0.732404\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1191.73i	1.54170i	0.637016	+	0.770851i	$0.280168\pi$
$773$	1191.73i	1.54170i	−0.637016	+	0.770851i	$0.719832\pi$
$774$	0	0
$775$	−1156.75	−1.49258
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 222.170i	− 0.285199i
$780$	0	0
$781$	−997.847	−1.27765
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 372.592i	− 0.474640i
$786$	0	0
$787$	−1411.25	−1.79320	−0.896602	−	0.442837i	$-0.853972\pi$
$787$	−1411.25	−1.79320	−0.896602	+	0.442837i	$0.853972\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−62.9705	−0.0794080
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1495.22i	1.87607i	0.346547	+	0.938033i	$0.387354\pi$
$797$	1495.22i	1.87607i	−0.346547	+	0.938033i	$0.612646\pi$
$798$	0	0
$799$	1907.12	2.38688
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1908.80i	− 2.37708i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 623.142i	− 0.770263i	−0.922862	−	0.385131i	$-0.874156\pi$
$809$	− 623.142i	− 0.770263i	0.922862	−	0.385131i	$-0.125844\pi$
$810$	0	0
$811$	−215.591	−0.265833	−0.132917	−	0.991127i	$-0.542434\pi$
$811$	−215.591	−0.265833	−0.132917	+	0.991127i	$0.542434\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1340.40i	1.64467i
$816$	0	0
$817$	−546.344	−0.668720
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1334.59i	− 1.62556i	−0.582569	−	0.812781i	$-0.697952\pi$
$821$	− 1334.59i	− 1.62556i	0.582569	−	0.812781i	$-0.302048\pi$
$822$	0	0
$823$	168.907	0.205233	0.102617	−	0.994721i	$-0.467278\pi$
$823$	168.907	0.205233	0.102617	+	0.994721i	$0.467278\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	336.132i	0.406447i	0.979132	+	0.203224i	$0.0651419\pi$
$827$	336.132i	0.406447i	−0.979132	+	0.203224i	$0.934858\pi$
$828$	0	0
$829$	−1250.09	−1.50795	−0.753976	−	0.656902i	$-0.771867\pi$
$829$	−1250.09	−1.50795	−0.753976	+	0.656902i	$0.771867\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1183.19	1.41700
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 173.924i	− 0.207299i	−0.994614	−	0.103649i	$-0.966948\pi$
$839$	− 173.924i	− 0.207299i	0.994614	−	0.103649i	$-0.0330520\pi$
$840$	0	0
$841$	−436.057	−0.518498
$842$	0	0
$843$	0	0
$844$	0	0
$845$	429.292i	0.508038i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1028.73i	− 1.20885i
$852$	0	0
$853$	−74.2360	−0.0870293	−0.0435147	−	0.999053i	$-0.513856\pi$
$853$	−74.2360	−0.0870293	−0.0435147	+	0.999053i	$0.513856\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 888.921i	− 1.03725i	−0.855003	−	0.518624i	$-0.826445\pi$
$857$	− 888.921i	− 1.03725i	0.855003	−	0.518624i	$-0.173555\pi$
$858$	0	0
$859$	−62.0393	−0.0722227	−0.0361114	−	0.999348i	$-0.511497\pi$
$859$	−62.0393	−0.0722227	−0.0361114	+	0.999348i	$0.511497\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 655.596i	− 0.759671i	−0.925054	−	0.379835i	$-0.875981\pi$
$863$	− 655.596i	− 0.759671i	0.925054	−	0.379835i	$-0.124019\pi$
$864$	0	0
$865$	530.201	0.612949
$866$	0	0
$867$	0	0
$868$	0	0
$869$	437.183i	0.503087i
$870$	0	0
$871$	−378.116	−0.434118
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1267.24	1.44498	0.722488	−	0.691384i	$-0.242999\pi$
$877$	1267.24	1.44498	0.722488	+	0.691384i	$0.242999\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.20047i	0.00703800i	0.999994	+	0.00351900i	$0.00112013\pi$
$881$	6.20047i	0.00703800i	−0.999994	+	0.00351900i	$0.998880\pi$
$882$	0	0
$883$	1122.22	1.27092	0.635459	−	0.772135i	$-0.280811\pi$
$883$	1122.22	1.27092	0.635459	+	0.772135i	$0.280811\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	480.508i	0.541723i	0.962618	+	0.270861i	$0.0873085\pi$
$887$	480.508i	0.541723i	−0.962618	+	0.270861i	$0.912692\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 567.283i	− 0.635256i
$894$	0	0
$895$	1665.80	1.86123
$896$	0	0
$897$	0	0
$898$	0	0
$899$	852.111i	0.947843i
$900$	0	0
$901$	1702.01	1.88903
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 204.671i	− 0.226156i
$906$	0	0
$907$	−170.449	−0.187926	−0.0939630	−	0.995576i	$-0.529954\pi$
$907$	−170.449	−0.187926	−0.0939630	+	0.995576i	$0.529954\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1180.21i	1.29552i	0.761847	+	0.647758i	$0.224293\pi$
$911$	1180.21i	1.29552i	−0.761847	+	0.647758i	$0.775707\pi$
$912$	0	0
$913$	1259.22	1.37922
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	535.681	0.582895	0.291448	−	0.956587i	$-0.405863\pi$
$919$	535.681	0.582895	0.291448	+	0.956587i	$0.405863\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	748.336i	0.810765i
$924$	0	0
$925$	1861.18	2.01209
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 493.610i	− 0.531335i	−0.964065	−	0.265667i	$-0.914408\pi$
$929$	− 493.610i	− 0.531335i	0.964065	−	0.265667i	$-0.0855923\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 3343.15i	− 3.57556i
$936$	0	0
$937$	−196.477	−0.209687	−0.104844	−	0.994489i	$-0.533434\pi$
$937$	−196.477	−0.209687	−0.104844	+	0.994489i	$0.533434\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 680.985i	− 0.723682i	−0.932240	−	0.361841i	$-0.882148\pi$
$941$	− 680.985i	− 0.723682i	0.932240	−	0.361841i	$-0.117852\pi$
$942$	0	0
$943$	−746.901	−0.792048
$944$	0	0
$945$	0	0
$946$	0	0
$947$	652.717i	0.689247i	0.938741	+	0.344623i	$0.111993\pi$
$947$	652.717i	0.689247i	−0.938741	+	0.344623i	$0.888007\pi$
$948$	0	0
$949$	−1431.50	−1.50843
$950$	0	0
$951$	0	0
$952$	0	0
$953$	153.185i	0.160740i	0.996765	+	0.0803701i	$0.0256102\pi$
$953$	153.185i	0.160740i	−0.996765	+	0.0803701i	$0.974390\pi$
$954$	0	0
$955$	1407.58	1.47391
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−392.432	−0.408358
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 2511.63i	− 2.60273i
$966$	0	0
$967$	1697.56	1.75550	0.877748	−	0.479122i	$-0.159045\pi$
$967$	1697.56	1.75550	0.877748	+	0.479122i	$0.159045\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 508.067i	− 0.523241i	−0.965171	−	0.261620i	$-0.915743\pi$
$971$	− 508.067i	− 0.523241i	0.965171	−	0.261620i	$-0.0842568\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 630.303i	− 0.645141i	−0.946545	−	0.322571i	$-0.895453\pi$
$977$	− 630.303i	− 0.645141i	0.946545	−	0.322571i	$-0.104547\pi$
$978$	0	0
$979$	−1869.44	−1.90954
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 281.461i	− 0.286328i	−0.989699	−	0.143164i	$-0.954272\pi$
$983$	− 281.461i	− 0.286328i	0.989699	−	0.143164i	$-0.0457277\pi$
$984$	0	0
$985$	−1994.06	−2.02443
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1836.73i	1.85716i
$990$	0	0
$991$	−241.021	−0.243210	−0.121605	−	0.992579i	$-0.538804\pi$
$991$	−241.021	−0.243210	−0.121605	+	0.992579i	$0.538804\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 206.738i	− 0.207777i
$996$	0	0
$997$	−1425.37	−1.42966	−0.714830	−	0.699299i	$-0.753496\pi$
$997$	−1425.37	−1.42966	−0.714830	+	0.699299i	$0.753496\pi$
$998$	0	0
$999$	0	0

Twists

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

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

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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)

\(f(q)\)	\(=\)	\(q-8.57392i q^{5} +14.5417i q^{11} +10.9055 q^{13} -26.8141i q^{17} -7.97599 q^{19} +26.8141i q^{23} -48.5120 q^{25} +35.7359i q^{29} +23.8447 q^{31} -38.3654 q^{37} +27.8548i q^{41} +68.4986 q^{43} +71.1239i q^{47} +63.4746i q^{53} +124.679 q^{55} -8.92188i q^{59} -5.77419 q^{61} -93.5030i q^{65} -34.6720 q^{67} +68.6199i q^{71} -131.264 q^{73} +30.0642 q^{79} -86.5943i q^{83} -229.901 q^{85} +128.558i q^{89} +68.3854i q^{95} +80.0096 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{13} + 20 q^{19} + 4 q^{25} + 56 q^{31} - 76 q^{37} + 36 q^{43} + 324 q^{55} + 72 q^{61} + 156 q^{67} - 124 q^{73} - 184 q^{79} - 432 q^{85} + 556 q^{97}+O(q^{100}) \) 8 * q + 20 * q^13 + 20 * q^19 + 4 * q^25 + 56 * q^31 - 76 * q^37 + 36 * q^43 + 324 * q^55 + 72 * q^61 + 156 * q^67 - 124 * q^73 - 184 * q^79 - 432 * q^85 + 556 * q^97

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