LMFDB - Embedded newform 4032.3.d.o.449.6

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			449.6
Root			$-3.78561i$ of defining polynomial
Character	$\chi$	$=$	4032.449
Dual form			4032.3.d.o.449.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.54305i q^{5} -2.64575 q^{7} -5.28100i q^{11} +17.4394 q^{13} -7.16491i q^{17} +12.1575 q^{19} -41.2578i q^{23} +22.6190 q^{25} +38.3304i q^{29} +45.2159 q^{31} +4.08252i q^{35} -52.5559 q^{37} +64.3356i q^{41} +40.6940 q^{43} +79.0301i q^{47} +7.00000 q^{49} -88.4071i q^{53} -8.14883 q^{55} -39.6751i q^{59} -94.2190 q^{61} -26.9098i q^{65} -13.2802 q^{67} -62.1050i q^{71} +12.8299 q^{73} +13.9722i q^{77} -114.290 q^{79} +42.7913i q^{83} -11.0558 q^{85} +9.41534i q^{89} -46.1403 q^{91} -18.7595i q^{95} +63.2873 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 1.54305i	− 0.308609i	−0.988023	−	0.154305i	$-0.950686\pi$
$5$	− 1.54305i	− 0.308609i	0.988023	−	0.154305i	$-0.0493137\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.28100i	− 0.480091i	−0.970762	−	0.240045i	$-0.922838\pi$
$11$	− 5.28100i	− 0.480091i	0.970762	−	0.240045i	$-0.0771623\pi$
$12$	0	0
$13$	17.4394	1.34149	0.670746	−	0.741687i	$-0.265974\pi$
$13$	17.4394	1.34149	0.670746	+	0.741687i	$0.265974\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.16491i	− 0.421465i	−0.977544	−	0.210733i	$-0.932415\pi$
$17$	− 7.16491i	− 0.421465i	0.977544	−	0.210733i	$-0.0675849\pi$
$18$	0	0
$19$	12.1575	0.639866	0.319933	−	0.947440i	$-0.396340\pi$
$19$	12.1575	0.639866	0.319933	+	0.947440i	$0.396340\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 41.2578i	− 1.79382i	−0.442218	−	0.896908i	$-0.645808\pi$
$23$	− 41.2578i	− 1.79382i	0.442218	−	0.896908i	$-0.354192\pi$
$24$	0	0
$25$	22.6190	0.904760
$26$	0	0
$27$	0	0
$28$	0	0
$29$	38.3304i	1.32174i	0.750502	+	0.660868i	$0.229812\pi$
$29$	38.3304i	1.32174i	−0.750502	+	0.660868i	$0.770188\pi$
$30$	0	0
$31$	45.2159	1.45858	0.729288	−	0.684207i	$-0.239851\pi$
$31$	45.2159	1.45858	0.729288	+	0.684207i	$0.239851\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.08252i	0.116643i
$36$	0	0
$37$	−52.5559	−1.42043	−0.710215	−	0.703985i	$-0.751402\pi$
$37$	−52.5559	−1.42043	−0.710215	+	0.703985i	$0.751402\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	64.3356i	1.56916i	0.620026	+	0.784581i	$0.287122\pi$
$41$	64.3356i	1.56916i	−0.620026	+	0.784581i	$0.712878\pi$
$42$	0	0
$43$	40.6940	0.946372	0.473186	−	0.880962i	$-0.343104\pi$
$43$	40.6940	0.946372	0.473186	+	0.880962i	$0.343104\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	79.0301i	1.68149i	0.541430	+	0.840746i	$0.317883\pi$
$47$	79.0301i	1.68149i	−0.541430	+	0.840746i	$0.682117\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 88.4071i	− 1.66806i	−0.551720	−	0.834030i	$-0.686028\pi$
$53$	− 88.4071i	− 1.66806i	0.551720	−	0.834030i	$-0.313972\pi$
$54$	0	0
$55$	−8.14883	−0.148161
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 39.6751i	− 0.672459i	−0.941780	−	0.336229i	$-0.890848\pi$
$59$	− 39.6751i	− 0.672459i	0.941780	−	0.336229i	$-0.109152\pi$
$60$	0	0
$61$	−94.2190	−1.54457	−0.772287	−	0.635273i	$-0.780887\pi$
$61$	−94.2190	−1.54457	−0.772287	+	0.635273i	$0.780887\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 26.9098i	− 0.413997i
$66$	0	0
$67$	−13.2802	−0.198212	−0.0991062	−	0.995077i	$-0.531598\pi$
$67$	−13.2802	−0.198212	−0.0991062	+	0.995077i	$0.531598\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 62.1050i	− 0.874719i	−0.899287	−	0.437359i	$-0.855914\pi$
$71$	− 62.1050i	− 0.874719i	0.899287	−	0.437359i	$-0.144086\pi$
$72$	0	0
$73$	12.8299	0.175753	0.0878764	−	0.996131i	$-0.471992\pi$
$73$	12.8299	0.175753	0.0878764	+	0.996131i	$0.471992\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	13.9722i	0.181457i
$78$	0	0
$79$	−114.290	−1.44670	−0.723352	−	0.690479i	$-0.757400\pi$
$79$	−114.290	−1.44670	−0.723352	+	0.690479i	$0.757400\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	42.7913i	0.515557i	0.966204	+	0.257779i	$0.0829905\pi$
$83$	42.7913i	0.515557i	−0.966204	+	0.257779i	$0.917009\pi$
$84$	0	0
$85$	−11.0558	−0.130068
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.41534i	0.105790i	0.998600	+	0.0528952i	$0.0168449\pi$
$89$	9.41534i	0.105790i	−0.998600	+	0.0528952i	$0.983155\pi$
$90$	0	0
$91$	−46.1403	−0.507037
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 18.7595i	− 0.197469i
$96$	0	0
$97$	63.2873	0.652446	0.326223	−	0.945293i	$-0.394224\pi$
$97$	63.2873	0.652446	0.326223	+	0.945293i	$0.394224\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 20.8527i	− 0.206462i	−0.994657	−	0.103231i	$-0.967082\pi$
$101$	− 20.8527i	− 0.206462i	0.994657	−	0.103231i	$-0.0329181\pi$
$102$	0	0
$103$	160.552	1.55876	0.779378	−	0.626554i	$-0.215535\pi$
$103$	160.552	1.55876	0.779378	+	0.626554i	$0.215535\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 65.0567i	− 0.608007i	−0.952671	−	0.304003i	$-0.901677\pi$
$107$	− 65.0567i	− 0.608007i	0.952671	−	0.304003i	$-0.0983234\pi$
$108$	0	0
$109$	71.1695	0.652932	0.326466	−	0.945209i	$-0.394142\pi$
$109$	71.1695	0.652932	0.326466	+	0.945209i	$0.394142\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	199.450i	1.76504i	0.470274	+	0.882520i	$0.344155\pi$
$113$	199.450i	1.76504i	−0.470274	+	0.882520i	$0.655845\pi$
$114$	0	0
$115$	−63.6627	−0.553588
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18.9566i	0.159299i
$120$	0	0
$121$	93.1110	0.769513
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 73.4784i	− 0.587827i
$126$	0	0
$127$	159.752	1.25789	0.628944	−	0.777451i	$-0.283488\pi$
$127$	159.752	1.25789	0.628944	+	0.777451i	$0.283488\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.89656i	0.0602791i	0.999546	+	0.0301396i	$0.00959517\pi$
$131$	7.89656i	0.0602791i	−0.999546	+	0.0301396i	$0.990405\pi$
$132$	0	0
$133$	−32.1656	−0.241847
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 79.5045i	− 0.580325i	−0.956977	−	0.290162i	$-0.906291\pi$
$137$	− 79.5045i	− 0.580325i	0.956977	−	0.290162i	$-0.0937093\pi$
$138$	0	0
$139$	205.295	1.47694	0.738471	−	0.674285i	$-0.235548\pi$
$139$	205.295	1.47694	0.738471	+	0.674285i	$0.235548\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 92.0975i	− 0.644038i
$144$	0	0
$145$	59.1455	0.407900
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 116.708i	− 0.783272i	−0.920120	−	0.391636i	$-0.871909\pi$
$149$	− 116.708i	− 0.783272i	0.920120	−	0.391636i	$-0.128091\pi$
$150$	0	0
$151$	−24.0881	−0.159524	−0.0797619	−	0.996814i	$-0.525416\pi$
$151$	−24.0881	−0.159524	−0.0797619	+	0.996814i	$0.525416\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 69.7702i	− 0.450130i
$156$	0	0
$157$	−294.586	−1.87634	−0.938170	−	0.346174i	$-0.887481\pi$
$157$	−294.586	−1.87634	−0.938170	+	0.346174i	$0.887481\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	109.158i	0.677998i
$162$	0	0
$163$	206.907	1.26937	0.634683	−	0.772772i	$-0.281131\pi$
$163$	206.907	1.26937	0.634683	+	0.772772i	$0.281131\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 20.1697i	− 0.120776i	−0.998175	−	0.0603882i	$-0.980766\pi$
$167$	− 20.1697i	− 0.120776i	0.998175	−	0.0603882i	$-0.0192339\pi$
$168$	0	0
$169$	135.133	0.799603
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 192.928i	− 1.11519i	−0.830113	−	0.557595i	$-0.811724\pi$
$173$	− 192.928i	− 1.11519i	0.830113	−	0.557595i	$-0.188276\pi$
$174$	0	0
$175$	−59.8443	−0.341967
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 162.749i	− 0.909215i	−0.890692	−	0.454607i	$-0.849780\pi$
$179$	− 162.749i	− 0.909215i	0.890692	−	0.454607i	$-0.150220\pi$
$180$	0	0
$181$	−328.731	−1.81620	−0.908098	−	0.418758i	$-0.862465\pi$
$181$	−328.731	−1.81620	−0.908098	+	0.418758i	$0.862465\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	81.0963i	0.438358i
$186$	0	0
$187$	−37.8379	−0.202342
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 246.878i	− 1.29255i	−0.763103	−	0.646277i	$-0.776325\pi$
$191$	− 246.878i	− 1.29255i	0.763103	−	0.646277i	$-0.223675\pi$
$192$	0	0
$193$	298.678	1.54755	0.773777	−	0.633458i	$-0.218365\pi$
$193$	298.678	1.54755	0.773777	+	0.633458i	$0.218365\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 93.8072i	− 0.476179i	−0.971243	−	0.238089i	$-0.923479\pi$
$197$	− 93.8072i	− 0.476179i	0.971243	−	0.238089i	$-0.0765211\pi$
$198$	0	0
$199$	−158.584	−0.796906	−0.398453	−	0.917189i	$-0.630453\pi$
$199$	−158.584	−0.796906	−0.398453	+	0.917189i	$0.630453\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 101.413i	− 0.499569i
$204$	0	0
$205$	99.2729	0.484258
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 64.2035i	− 0.307194i
$210$	0	0
$211$	−75.0497	−0.355686	−0.177843	−	0.984059i	$-0.556912\pi$
$211$	−75.0497	−0.355686	−0.177843	+	0.984059i	$0.556912\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 62.7928i	− 0.292059i
$216$	0	0
$217$	−119.630	−0.551290
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 124.952i	− 0.565392i
$222$	0	0
$223$	383.762	1.72090	0.860452	−	0.509531i	$-0.170181\pi$
$223$	383.762	1.72090	0.860452	+	0.509531i	$0.170181\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	35.1635i	0.154905i	0.996996	+	0.0774526i	$0.0246787\pi$
$227$	35.1635i	0.154905i	−0.996996	+	0.0774526i	$0.975321\pi$
$228$	0	0
$229$	219.763	0.959663	0.479832	−	0.877361i	$-0.340698\pi$
$229$	219.763	0.959663	0.479832	+	0.877361i	$0.340698\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	263.627i	1.13145i	0.824596	+	0.565723i	$0.191403\pi$
$233$	263.627i	1.13145i	−0.824596	+	0.565723i	$0.808597\pi$
$234$	0	0
$235$	121.947	0.518924
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 155.414i	− 0.650267i	−0.945668	−	0.325133i	$-0.894591\pi$
$239$	− 155.414i	− 0.650267i	0.945668	−	0.325133i	$-0.105409\pi$
$240$	0	0
$241$	166.284	0.689977	0.344988	−	0.938607i	$-0.387883\pi$
$241$	166.284	0.689977	0.344988	+	0.938607i	$0.387883\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 10.8013i	− 0.0440871i
$246$	0	0
$247$	212.019	0.858376
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 381.832i	− 1.52124i	−0.649196	−	0.760621i	$-0.724894\pi$
$251$	− 381.832i	− 1.52124i	0.649196	−	0.760621i	$-0.275106\pi$
$252$	0	0
$253$	−217.882	−0.861194
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 133.076i	− 0.517805i	−0.965903	−	0.258902i	$-0.916639\pi$
$257$	− 133.076i	− 0.517805i	0.965903	−	0.258902i	$-0.0833608\pi$
$258$	0	0
$259$	139.050	0.536872
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 360.430i	− 1.37045i	−0.728329	−	0.685227i	$-0.759703\pi$
$263$	− 360.430i	− 1.37045i	0.728329	−	0.685227i	$-0.240297\pi$
$264$	0	0
$265$	−136.416	−0.514779
$266$	0	0
$267$	0	0
$268$	0	0
$269$	62.5750i	0.232621i	0.993213	+	0.116310i	$0.0371067\pi$
$269$	62.5750i	0.232621i	−0.993213	+	0.116310i	$0.962893\pi$
$270$	0	0
$271$	−102.805	−0.379353	−0.189677	−	0.981847i	$-0.560744\pi$
$271$	−102.805	−0.379353	−0.189677	+	0.981847i	$0.560744\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 119.451i	− 0.434367i
$276$	0	0
$277$	−265.912	−0.959972	−0.479986	−	0.877276i	$-0.659358\pi$
$277$	−265.912	−0.959972	−0.479986	+	0.877276i	$0.659358\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.9943i	0.0391256i	0.999809	+	0.0195628i	$0.00622743\pi$
$281$	10.9943i	0.0391256i	−0.999809	+	0.0195628i	$0.993773\pi$
$282$	0	0
$283$	370.671	1.30979	0.654897	−	0.755719i	$-0.272712\pi$
$283$	370.671	1.30979	0.654897	+	0.755719i	$0.272712\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 170.216i	− 0.593088i
$288$	0	0
$289$	237.664	0.822367
$290$	0	0
$291$	0	0
$292$	0	0
$293$	400.085i	1.36548i	0.730663	+	0.682738i	$0.239211\pi$
$293$	400.085i	1.36548i	−0.730663	+	0.682738i	$0.760789\pi$
$294$	0	0
$295$	−61.2205	−0.207527
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 719.511i	− 2.40639i
$300$	0	0
$301$	−107.666	−0.357695
$302$	0	0
$303$	0	0
$304$	0	0
$305$	145.384i	0.476670i
$306$	0	0
$307$	−160.824	−0.523858	−0.261929	−	0.965087i	$-0.584359\pi$
$307$	−160.824	−0.523858	−0.261929	+	0.965087i	$0.584359\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 68.7757i	− 0.221144i	−0.993868	−	0.110572i	$-0.964732\pi$
$311$	− 68.7757i	− 0.221144i	0.993868	−	0.110572i	$-0.0352682\pi$
$312$	0	0
$313$	410.532	1.31160	0.655801	−	0.754934i	$-0.272331\pi$
$313$	410.532	1.31160	0.655801	+	0.754934i	$0.272331\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	102.907i	0.324627i	0.986739	+	0.162314i	$0.0518956\pi$
$317$	102.907i	0.324627i	−0.986739	+	0.162314i	$0.948104\pi$
$318$	0	0
$319$	202.423	0.634554
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 87.1070i	− 0.269681i
$324$	0	0
$325$	394.462	1.21373
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 209.094i	− 0.635544i
$330$	0	0
$331$	−161.830	−0.488911	−0.244456	−	0.969660i	$-0.578609\pi$
$331$	−161.830	−0.488911	−0.244456	+	0.969660i	$0.578609\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	20.4920i	0.0611702i
$336$	0	0
$337$	−194.690	−0.577715	−0.288858	−	0.957372i	$-0.593275\pi$
$337$	−194.690	−0.577715	−0.288858	+	0.957372i	$0.593275\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 238.785i	− 0.700249i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	605.635i	1.74534i	0.488306	+	0.872672i	$0.337615\pi$
$347$	605.635i	1.74534i	−0.488306	+	0.872672i	$0.662385\pi$
$348$	0	0
$349$	178.942	0.512727	0.256363	−	0.966580i	$-0.417476\pi$
$349$	178.942	0.512727	0.256363	+	0.966580i	$0.417476\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	171.611i	0.486150i	0.970007	+	0.243075i	$0.0781562\pi$
$353$	171.611i	0.486150i	−0.970007	+	0.243075i	$0.921844\pi$
$354$	0	0
$355$	−95.8310	−0.269946
$356$	0	0
$357$	0	0
$358$	0	0
$359$	167.294i	0.465999i	0.972477	+	0.232999i	$0.0748540\pi$
$359$	167.294i	0.465999i	−0.972477	+	0.232999i	$0.925146\pi$
$360$	0	0
$361$	−213.196	−0.590571
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 19.7972i	− 0.0542389i
$366$	0	0
$367$	382.371	1.04188	0.520942	−	0.853592i	$-0.325581\pi$
$367$	382.371	1.04188	0.520942	+	0.853592i	$0.325581\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	233.903i	0.630467i
$372$	0	0
$373$	−76.7573	−0.205784	−0.102892	−	0.994693i	$-0.532810\pi$
$373$	−76.7573	−0.205784	−0.102892	+	0.994693i	$0.532810\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	668.459i	1.77310i
$378$	0	0
$379$	−468.534	−1.23624	−0.618119	−	0.786085i	$-0.712105\pi$
$379$	−468.534	−1.23624	−0.618119	+	0.786085i	$0.712105\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 352.388i	− 0.920072i	−0.887900	−	0.460036i	$-0.847836\pi$
$383$	− 352.388i	− 0.920072i	0.887900	−	0.460036i	$-0.152164\pi$
$384$	0	0
$385$	21.5598	0.0559994
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 28.3218i	− 0.0728066i	−0.999337	−	0.0364033i	$-0.988410\pi$
$389$	− 28.3218i	− 0.0728066i	0.999337	−	0.0364033i	$-0.0115901\pi$
$390$	0	0
$391$	−295.608	−0.756031
$392$	0	0
$393$	0	0
$394$	0	0
$395$	176.354i	0.446467i
$396$	0	0
$397$	−394.854	−0.994593	−0.497297	−	0.867581i	$-0.665674\pi$
$397$	−394.854	−0.994593	−0.497297	+	0.867581i	$0.665674\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 458.679i	− 1.14384i	−0.820310	−	0.571919i	$-0.806199\pi$
$401$	− 458.679i	− 1.14384i	0.820310	−	0.571919i	$-0.193801\pi$
$402$	0	0
$403$	788.538	1.95667
$404$	0	0
$405$	0	0
$406$	0	0
$407$	277.548i	0.681936i
$408$	0	0
$409$	−200.052	−0.489126	−0.244563	−	0.969633i	$-0.578644\pi$
$409$	−200.052	−0.489126	−0.244563	+	0.969633i	$0.578644\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	104.970i	0.254166i
$414$	0	0
$415$	66.0289	0.159106
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 282.592i	− 0.674443i	−0.941425	−	0.337222i	$-0.890513\pi$
$419$	− 282.592i	− 0.674443i	0.941425	−	0.337222i	$-0.109487\pi$
$420$	0	0
$421$	−131.289	−0.311850	−0.155925	−	0.987769i	$-0.549836\pi$
$421$	−131.289	−0.311850	−0.155925	+	0.987769i	$0.549836\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 162.063i	− 0.381325i
$426$	0	0
$427$	249.280	0.583794
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 10.2557i	− 0.0237951i	−0.999929	−	0.0118976i	$-0.996213\pi$
$431$	− 10.2557i	− 0.0237951i	0.999929	−	0.0118976i	$-0.00378720\pi$
$432$	0	0
$433$	−2.74747	−0.00634520	−0.00317260	−	0.999995i	$-0.501010\pi$
$433$	−2.74747	−0.00634520	−0.00317260	+	0.999995i	$0.501010\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 501.589i	− 1.14780i
$438$	0	0
$439$	532.249	1.21241	0.606206	−	0.795307i	$-0.292691\pi$
$439$	532.249	1.21241	0.606206	+	0.795307i	$0.292691\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 419.913i	− 0.947885i	−0.880556	−	0.473943i	$-0.842830\pi$
$443$	− 419.913i	− 0.947885i	0.880556	−	0.473943i	$-0.157170\pi$
$444$	0	0
$445$	14.5283	0.0326479
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 417.221i	− 0.929223i	−0.885515	−	0.464611i	$-0.846194\pi$
$449$	− 417.221i	− 0.929223i	0.885515	−	0.464611i	$-0.153806\pi$
$450$	0	0
$451$	339.757	0.753340
$452$	0	0
$453$	0	0
$454$	0	0
$455$	71.1967i	0.156476i
$456$	0	0
$457$	−70.3057	−0.153842	−0.0769209	−	0.997037i	$-0.524509\pi$
$457$	−70.3057	−0.153842	−0.0769209	+	0.997037i	$0.524509\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 252.219i	− 0.547113i	−0.961856	−	0.273557i	$-0.911800\pi$
$461$	− 252.219i	− 0.547113i	0.961856	−	0.273557i	$-0.0882001\pi$
$462$	0	0
$463$	330.520	0.713865	0.356933	−	0.934130i	$-0.383823\pi$
$463$	330.520	0.713865	0.356933	+	0.934130i	$0.383823\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 398.164i	− 0.852599i	−0.904582	−	0.426299i	$-0.859817\pi$
$467$	− 398.164i	− 0.852599i	0.904582	−	0.426299i	$-0.140183\pi$
$468$	0	0
$469$	35.1362	0.0749173
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 214.905i	− 0.454345i
$474$	0	0
$475$	274.990	0.578925
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 823.897i	− 1.72003i	−0.510265	−	0.860017i	$-0.670453\pi$
$479$	− 823.897i	− 1.72003i	0.510265	−	0.860017i	$-0.329547\pi$
$480$	0	0
$481$	−916.544	−1.90550
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 97.6553i	− 0.201351i
$486$	0	0
$487$	−421.546	−0.865597	−0.432799	−	0.901491i	$-0.642474\pi$
$487$	−421.546	−0.865597	−0.432799	+	0.901491i	$0.642474\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 670.944i	− 1.36648i	−0.730192	−	0.683242i	$-0.760569\pi$
$491$	− 670.944i	− 1.36648i	0.730192	−	0.683242i	$-0.239431\pi$
$492$	0	0
$493$	274.633	0.557066
$494$	0	0
$495$	0	0
$496$	0	0
$497$	164.314i	0.330613i
$498$	0	0
$499$	297.920	0.597033	0.298517	−	0.954404i	$-0.403508\pi$
$499$	297.920	0.597033	0.298517	+	0.954404i	$0.403508\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 272.635i	− 0.542017i	−0.962577	−	0.271008i	$-0.912643\pi$
$503$	− 272.635i	− 0.542017i	0.962577	−	0.271008i	$-0.0873571\pi$
$504$	0	0
$505$	−32.1767	−0.0637161
$506$	0	0
$507$	0	0
$508$	0	0
$509$	623.480i	1.22491i	0.790505	+	0.612456i	$0.209818\pi$
$509$	623.480i	1.22491i	−0.790505	+	0.612456i	$0.790182\pi$
$510$	0	0
$511$	−33.9449	−0.0664283
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 247.739i	− 0.481047i
$516$	0	0
$517$	417.358	0.807269
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 630.152i	− 1.20950i	−0.796414	−	0.604752i	$-0.793272\pi$
$521$	− 630.152i	− 1.20950i	0.796414	−	0.604752i	$-0.206728\pi$
$522$	0	0
$523$	−692.429	−1.32396	−0.661978	−	0.749524i	$-0.730283\pi$
$523$	−692.429	−1.32396	−0.661978	+	0.749524i	$0.730283\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 323.968i	− 0.614739i
$528$	0	0
$529$	−1173.20	−2.21777
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1121.98i	2.10502i
$534$	0	0
$535$	−100.386	−0.187637
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 36.9670i	− 0.0685844i
$540$	0	0
$541$	983.219	1.81741	0.908706	−	0.417438i	$-0.137072\pi$
$541$	983.219	1.81741	0.908706	+	0.417438i	$0.137072\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 109.818i	− 0.201501i
$546$	0	0
$547$	223.847	0.409226	0.204613	−	0.978843i	$-0.434406\pi$
$547$	223.847	0.409226	0.204613	+	0.978843i	$0.434406\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	466.000i	0.845734i
$552$	0	0
$553$	302.382	0.546803
$554$	0	0
$555$	0	0
$556$	0	0
$557$	734.687i	1.31901i	0.751701	+	0.659504i	$0.229234\pi$
$557$	734.687i	1.31901i	−0.751701	+	0.659504i	$0.770766\pi$
$558$	0	0
$559$	709.679	1.26955
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 489.532i	− 0.869506i	−0.900550	−	0.434753i	$-0.856836\pi$
$563$	− 489.532i	− 0.869506i	0.900550	−	0.434753i	$-0.143164\pi$
$564$	0	0
$565$	307.760	0.544708
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 12.4640i	− 0.0219051i	−0.999940	−	0.0109525i	$-0.996514\pi$
$569$	− 12.4640i	− 0.0219051i	0.999940	−	0.0109525i	$-0.00348637\pi$
$570$	0	0
$571$	625.809	1.09599	0.547994	−	0.836482i	$-0.315392\pi$
$571$	625.809	1.09599	0.547994	+	0.836482i	$0.315392\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 933.209i	− 1.62297i
$576$	0	0
$577$	51.9185	0.0899801	0.0449900	−	0.998987i	$-0.485674\pi$
$577$	51.9185	0.0899801	0.0449900	+	0.998987i	$0.485674\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 113.215i	− 0.194862i
$582$	0	0
$583$	−466.878	−0.800820
$584$	0	0
$585$	0	0
$586$	0	0
$587$	411.295i	0.700672i	0.936624	+	0.350336i	$0.113933\pi$
$587$	411.295i	0.700672i	−0.936624	+	0.350336i	$0.886067\pi$
$588$	0	0
$589$	549.710	0.933293
$590$	0	0
$591$	0	0
$592$	0	0
$593$	554.879i	0.935715i	0.883804	+	0.467858i	$0.154974\pi$
$593$	554.879i	0.935715i	−0.883804	+	0.467858i	$0.845026\pi$
$594$	0	0
$595$	29.2509	0.0491611
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 408.418i	− 0.681834i	−0.940093	−	0.340917i	$-0.889263\pi$
$599$	− 408.418i	− 0.681834i	0.940093	−	0.340917i	$-0.110737\pi$
$600$	0	0
$601$	105.962	0.176309	0.0881547	−	0.996107i	$-0.471903\pi$
$601$	105.962	0.176309	0.0881547	+	0.996107i	$0.471903\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 143.675i	− 0.237479i
$606$	0	0
$607$	831.924	1.37055	0.685275	−	0.728284i	$-0.259682\pi$
$607$	831.924	1.37055	0.685275	+	0.728284i	$0.259682\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1378.24i	2.25571i
$612$	0	0
$613$	−20.6017	−0.0336080	−0.0168040	−	0.999859i	$-0.505349\pi$
$613$	−20.6017	−0.0336080	−0.0168040	+	0.999859i	$0.505349\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	377.279i	0.611474i	0.952116	+	0.305737i	$0.0989027\pi$
$617$	377.279i	0.611474i	−0.952116	+	0.305737i	$0.901097\pi$
$618$	0	0
$619$	770.486	1.24473	0.622364	−	0.782728i	$-0.286173\pi$
$619$	770.486	1.24473	0.622364	+	0.782728i	$0.286173\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 24.9107i	− 0.0399850i
$624$	0	0
$625$	452.095	0.723351
$626$	0	0
$627$	0	0
$628$	0	0
$629$	376.558i	0.598662i
$630$	0	0
$631$	741.949	1.17583	0.587915	−	0.808923i	$-0.299949\pi$
$631$	741.949	1.17583	0.587915	+	0.808923i	$0.299949\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 246.504i	− 0.388196i
$636$	0	0
$637$	122.076	0.191642
$638$	0	0
$639$	0	0
$640$	0	0
$641$	138.310i	0.215772i	0.994163	+	0.107886i	$0.0344082\pi$
$641$	138.310i	0.215772i	−0.994163	+	0.107886i	$0.965592\pi$
$642$	0	0
$643$	−1196.49	−1.86079	−0.930396	−	0.366556i	$-0.880537\pi$
$643$	−1196.49	−1.86079	−0.930396	+	0.366556i	$0.880537\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	318.477i	0.492237i	0.969240	+	0.246118i	$0.0791552\pi$
$647$	318.477i	0.492237i	−0.969240	+	0.246118i	$0.920845\pi$
$648$	0	0
$649$	−209.524	−0.322841
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 223.357i	− 0.342047i	−0.985267	−	0.171023i	$-0.945293\pi$
$653$	− 223.357i	− 0.342047i	0.985267	−	0.171023i	$-0.0547074\pi$
$654$	0	0
$655$	12.1848	0.0186027
$656$	0	0
$657$	0	0
$658$	0	0
$659$	218.539i	0.331622i	0.986158	+	0.165811i	$0.0530242\pi$
$659$	218.539i	0.331622i	−0.986158	+	0.165811i	$0.946976\pi$
$660$	0	0
$661$	136.291	0.206189	0.103095	−	0.994672i	$-0.467126\pi$
$661$	136.291	0.206189	0.103095	+	0.994672i	$0.467126\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	49.6330i	0.0746361i
$666$	0	0
$667$	1581.42	2.37095
$668$	0	0
$669$	0	0
$670$	0	0
$671$	497.571i	0.741536i
$672$	0	0
$673$	−486.602	−0.723034	−0.361517	−	0.932365i	$-0.617741\pi$
$673$	−486.602	−0.723034	−0.361517	+	0.932365i	$0.617741\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	164.519i	0.243012i	0.992591	+	0.121506i	$0.0387724\pi$
$677$	164.519i	0.243012i	−0.992591	+	0.121506i	$0.961228\pi$
$678$	0	0
$679$	−167.442	−0.246602
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1165.64i	− 1.70664i	−0.521385	−	0.853322i	$-0.674584\pi$
$683$	− 1165.64i	− 1.70664i	0.521385	−	0.853322i	$-0.325416\pi$
$684$	0	0
$685$	−122.679	−0.179094
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1541.77i	− 2.23769i
$690$	0	0
$691$	−972.997	−1.40810	−0.704050	−	0.710150i	$-0.748627\pi$
$691$	−972.997	−1.40810	−0.704050	+	0.710150i	$0.748627\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 316.780i	− 0.455798i
$696$	0	0
$697$	460.959	0.661347
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 912.244i	− 1.30135i	−0.759358	−	0.650674i	$-0.774487\pi$
$701$	− 912.244i	− 1.30135i	0.759358	−	0.650674i	$-0.225513\pi$
$702$	0	0
$703$	−638.946	−0.908885
$704$	0	0
$705$	0	0
$706$	0	0
$707$	55.1710i	0.0780353i
$708$	0	0
$709$	−20.6221	−0.0290862	−0.0145431	−	0.999894i	$-0.504629\pi$
$709$	−20.6221	−0.0290862	−0.0145431	+	0.999894i	$0.504629\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1865.50i	− 2.61642i
$714$	0	0
$715$	−142.111	−0.198756
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1172.99i	1.63142i	0.578461	+	0.815710i	$0.303653\pi$
$719$	1172.99i	1.63142i	−0.578461	+	0.815710i	$0.696347\pi$
$720$	0	0
$721$	−424.780	−0.589154
$722$	0	0
$723$	0	0
$724$	0	0
$725$	866.995i	1.19585i
$726$	0	0
$727$	−309.756	−0.426075	−0.213037	−	0.977044i	$-0.568336\pi$
$727$	−309.756	−0.426075	−0.213037	+	0.977044i	$0.568336\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 291.569i	− 0.398863i
$732$	0	0
$733$	525.414	0.716800	0.358400	−	0.933568i	$-0.383322\pi$
$733$	525.414	0.716800	0.358400	+	0.933568i	$0.383322\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	70.1329i	0.0951600i
$738$	0	0
$739$	−942.991	−1.27604	−0.638018	−	0.770021i	$-0.720246\pi$
$739$	−942.991	−1.27604	−0.638018	+	0.770021i	$0.720246\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1425.24i	1.91822i	0.283027	+	0.959112i	$0.408661\pi$
$743$	1425.24i	1.91822i	−0.283027	+	0.959112i	$0.591339\pi$
$744$	0	0
$745$	−180.085	−0.241725
$746$	0	0
$747$	0	0
$748$	0	0
$749$	172.124i	0.229805i
$750$	0	0
$751$	−1239.56	−1.65055	−0.825276	−	0.564730i	$-0.808980\pi$
$751$	−1239.56	−1.65055	−0.825276	+	0.564730i	$0.808980\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	37.1690i	0.0492305i
$756$	0	0
$757$	670.688	0.885981	0.442991	−	0.896526i	$-0.353917\pi$
$757$	670.688	0.885981	0.442991	+	0.896526i	$0.353917\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 705.573i	− 0.927166i	−0.886054	−	0.463583i	$-0.846564\pi$
$761$	− 705.573i	− 0.927166i	0.886054	−	0.463583i	$-0.153436\pi$
$762$	0	0
$763$	−188.297	−0.246785
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 691.910i	− 0.902099i
$768$	0	0
$769$	230.155	0.299292	0.149646	−	0.988740i	$-0.452187\pi$
$769$	230.155	0.299292	0.149646	+	0.988740i	$0.452187\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1448.15i	− 1.87342i	−0.350113	−	0.936708i	$-0.613857\pi$
$773$	− 1448.15i	− 1.87342i	0.350113	−	0.936708i	$-0.386143\pi$
$774$	0	0
$775$	1022.74	1.31966
$776$	0	0
$777$	0	0
$778$	0	0
$779$	782.158i	1.00405i
$780$	0	0
$781$	−327.977	−0.419944
$782$	0	0
$783$	0	0
$784$	0	0
$785$	454.559i	0.579056i
$786$	0	0
$787$	−1302.64	−1.65520	−0.827600	−	0.561318i	$-0.810294\pi$
$787$	−1302.64	−1.65520	−0.827600	+	0.561318i	$0.810294\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 527.694i	− 0.667123i
$792$	0	0
$793$	−1643.12	−2.07204
$794$	0	0
$795$	0	0
$796$	0	0
$797$	322.552i	0.404707i	0.979313	+	0.202354i	$0.0648590\pi$
$797$	322.552i	0.404707i	−0.979313	+	0.202354i	$0.935141\pi$
$798$	0	0
$799$	566.243	0.708690
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 67.7549i	− 0.0843773i
$804$	0	0
$805$	168.436	0.209237
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1351.66i	1.67078i	0.549658	+	0.835390i	$0.314758\pi$
$809$	1351.66i	1.67078i	−0.549658	+	0.835390i	$0.685242\pi$
$810$	0	0
$811$	780.010	0.961789	0.480894	−	0.876779i	$-0.340312\pi$
$811$	780.010	0.961789	0.480894	+	0.876779i	$0.340312\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 319.267i	− 0.391738i
$816$	0	0
$817$	494.736	0.605552
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1622.00i	1.97563i	0.155621	+	0.987817i	$0.450262\pi$
$821$	1622.00i	1.97563i	−0.155621	+	0.987817i	$0.549738\pi$
$822$	0	0
$823$	−1349.34	−1.63954	−0.819770	−	0.572693i	$-0.805899\pi$
$823$	−1349.34	−1.63954	−0.819770	+	0.572693i	$0.805899\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 922.170i	− 1.11508i	−0.830150	−	0.557539i	$-0.811746\pi$
$827$	− 922.170i	− 1.11508i	0.830150	−	0.557539i	$-0.188254\pi$
$828$	0	0
$829$	408.532	0.492801	0.246401	−	0.969168i	$-0.420752\pi$
$829$	408.532	0.492801	0.246401	+	0.969168i	$0.420752\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 50.1544i	− 0.0602093i
$834$	0	0
$835$	−31.1228	−0.0372728
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 39.9837i	− 0.0476564i	−0.999716	−	0.0238282i	$-0.992415\pi$
$839$	− 39.9837i	− 0.0476564i	0.999716	−	0.0238282i	$-0.00758547\pi$
$840$	0	0
$841$	−628.216	−0.746987
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 208.516i	− 0.246765i
$846$	0	0
$847$	−246.349	−0.290848
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2168.34i	2.54799i
$852$	0	0
$853$	825.840	0.968159	0.484080	−	0.875024i	$-0.339154\pi$
$853$	825.840	0.968159	0.484080	+	0.875024i	$0.339154\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1679.49i	− 1.95973i	−0.199653	−	0.979867i	$-0.563981\pi$
$857$	− 1679.49i	− 1.95973i	0.199653	−	0.979867i	$-0.436019\pi$
$858$	0	0
$859$	−131.762	−0.153390	−0.0766949	−	0.997055i	$-0.524437\pi$
$859$	−131.762	−0.153390	−0.0766949	+	0.997055i	$0.524437\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 905.936i	− 1.04975i	−0.851179	−	0.524876i	$-0.824112\pi$
$863$	− 905.936i	− 1.04975i	0.851179	−	0.524876i	$-0.175888\pi$
$864$	0	0
$865$	−297.697	−0.344158
$866$	0	0
$867$	0	0
$868$	0	0
$869$	603.564i	0.694550i
$870$	0	0
$871$	−231.599	−0.265901
$872$	0	0
$873$	0	0
$874$	0	0
$875$	194.405i	0.222178i
$876$	0	0
$877$	−238.592	−0.272055	−0.136028	−	0.990705i	$-0.543434\pi$
$877$	−238.592	−0.272055	−0.136028	+	0.990705i	$0.543434\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	511.426i	0.580506i	0.956950	+	0.290253i	$0.0937394\pi$
$881$	511.426i	0.580506i	−0.956950	+	0.290253i	$0.906261\pi$
$882$	0	0
$883$	735.153	0.832563	0.416281	−	0.909236i	$-0.363333\pi$
$883$	735.153	0.832563	0.416281	+	0.909236i	$0.363333\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1593.66i	1.79669i	0.439295	+	0.898343i	$0.355228\pi$
$887$	1593.66i	1.79669i	−0.439295	+	0.898343i	$0.644772\pi$
$888$	0	0
$889$	−422.663	−0.475437
$890$	0	0
$891$	0	0
$892$	0	0
$893$	960.805i	1.07593i
$894$	0	0
$895$	−251.130	−0.280592
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1733.14i	1.92785i
$900$	0	0
$901$	−633.429	−0.703029
$902$	0	0
$903$	0	0
$904$	0	0
$905$	507.248i	0.560495i
$906$	0	0
$907$	421.538	0.464761	0.232380	−	0.972625i	$-0.425349\pi$
$907$	421.538	0.464761	0.232380	+	0.972625i	$0.425349\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1025.61i	1.12581i	0.826522	+	0.562904i	$0.190316\pi$
$911$	1025.61i	1.12581i	−0.826522	+	0.562904i	$0.809684\pi$
$912$	0	0
$913$	225.981	0.247514
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 20.8923i	− 0.0227834i
$918$	0	0
$919$	−601.821	−0.654865	−0.327432	−	0.944875i	$-0.606183\pi$
$919$	−601.821	−0.654865	−0.327432	+	0.944875i	$0.606183\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1083.07i	− 1.17343i
$924$	0	0
$925$	−1188.76	−1.28515
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 291.928i	− 0.314239i	−0.987580	−	0.157120i	$-0.949779\pi$
$929$	− 291.928i	− 0.314239i	0.987580	−	0.157120i	$-0.0502208\pi$
$930$	0	0
$931$	85.1022	0.0914094
$932$	0	0
$933$	0	0
$934$	0	0
$935$	58.3856i	0.0624445i
$936$	0	0
$937$	−831.922	−0.887857	−0.443929	−	0.896062i	$-0.646416\pi$
$937$	−831.922	−0.887857	−0.443929	+	0.896062i	$0.646416\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	537.447i	0.571144i	0.958357	+	0.285572i	$0.0921836\pi$
$941$	537.447i	0.571144i	−0.958357	+	0.285572i	$0.907816\pi$
$942$	0	0
$943$	2654.34	2.81479
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 461.676i	− 0.487514i	−0.969836	−	0.243757i	$-0.921620\pi$
$947$	− 461.676i	− 0.487514i	0.969836	−	0.243757i	$-0.0783800\pi$
$948$	0	0
$949$	223.747	0.235771
$950$	0	0
$951$	0	0
$952$	0	0
$953$	379.663i	0.398387i	0.979960	+	0.199193i	$0.0638322\pi$
$953$	379.663i	0.398387i	−0.979960	+	0.199193i	$0.936168\pi$
$954$	0	0
$955$	−380.944	−0.398894
$956$	0	0
$957$	0	0
$958$	0	0
$959$	210.349i	0.219342i
$960$	0	0
$961$	1083.47	1.12744
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 460.874i	− 0.477590i
$966$	0	0
$967$	−382.730	−0.395791	−0.197896	−	0.980223i	$-0.563411\pi$
$967$	−382.730	−0.395791	−0.197896	+	0.980223i	$0.563411\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 812.948i	− 0.837227i	−0.908164	−	0.418614i	$-0.862516\pi$
$971$	− 812.948i	− 0.837227i	0.908164	−	0.418614i	$-0.137484\pi$
$972$	0	0
$973$	−543.159	−0.558232
$974$	0	0
$975$	0	0
$976$	0	0
$977$	810.148i	0.829220i	0.909999	+	0.414610i	$0.136082\pi$
$977$	810.148i	0.829220i	−0.909999	+	0.414610i	$0.863918\pi$
$978$	0	0
$979$	49.7224	0.0507890
$980$	0	0
$981$	0	0
$982$	0	0
$983$	519.397i	0.528379i	0.964471	+	0.264190i	$0.0851045\pi$
$983$	519.397i	0.528379i	−0.964471	+	0.264190i	$0.914896\pi$
$984$	0	0
$985$	−144.749	−0.146953
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1678.94i	− 1.69762i
$990$	0	0
$991$	−1240.08	−1.25134	−0.625669	−	0.780089i	$-0.715174\pi$
$991$	−1240.08	−1.25134	−0.625669	+	0.780089i	$0.715174\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	244.703i	0.245933i
$996$	0	0
$997$	647.369	0.649317	0.324658	−	0.945831i	$-0.394751\pi$
$997$	647.369	0.649317	0.324658	+	0.945831i	$0.394751\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.3.d.o.449.6		12
3.2	odd	2	inner	4032.3.d.o.449.7		12
4.3	odd	2		4032.3.d.n.449.6		12
8.3	odd	2		2016.3.d.f.449.7	yes	12
8.5	even	2		2016.3.d.e.449.7	yes	12
12.11	even	2		4032.3.d.n.449.7		12
24.5	odd	2		2016.3.d.e.449.6	✓	12
24.11	even	2		2016.3.d.f.449.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.3.d.e.449.6	✓	12	24.5	odd	2
2016.3.d.e.449.7	yes	12	8.5	even	2
2016.3.d.f.449.6	yes	12	24.11	even	2
2016.3.d.f.449.7	yes	12	8.3	odd	2
4032.3.d.n.449.6		12	4.3	odd	2
4032.3.d.n.449.7		12	12.11	even	2
4032.3.d.o.449.6		12	1.1	even	1	trivial
4032.3.d.o.449.7		12	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.54305i q^{5} -2.64575 q^{7} -5.28100i q^{11} +17.4394 q^{13} -7.16491i q^{17} +12.1575 q^{19} -41.2578i q^{23} +22.6190 q^{25} +38.3304i q^{29} +45.2159 q^{31} +4.08252i q^{35} -52.5559 q^{37} +64.3356i q^{41} +40.6940 q^{43} +79.0301i q^{47} +7.00000 q^{49} -88.4071i q^{53} -8.14883 q^{55} -39.6751i q^{59} -94.2190 q^{61} -26.9098i q^{65} -13.2802 q^{67} -62.1050i q^{71} +12.8299 q^{73} +13.9722i q^{77} -114.290 q^{79} +42.7913i q^{83} -11.0558 q^{85} +9.41534i q^{89} -46.1403 q^{91} -18.7595i q^{95} +63.2873 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{13} + 64 q^{19} - 124 q^{25} - 160 q^{31} - 56 q^{37} - 64 q^{43} + 84 q^{49} + 160 q^{55} - 104 q^{61} - 64 q^{67} - 64 q^{73} - 32 q^{79} + 184 q^{85} - 96 q^{97}+O(q^{100}) \) 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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