LMFDB - Embedded newform 1008.2.h.b.575.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,2,Mod(575,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.575"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1008.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$8.04892052375$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			575.7
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1008.575
Dual form			1008.2.h.b.575.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.86370i q^{5} -1.00000i q^{7} -5.27792 q^{11} +2.00000 q^{13} +1.03528i q^{17} +5.46410i q^{19} -0.378937 q^{23} -9.92820 q^{25} +6.31319i q^{29} -9.46410i q^{31} +3.86370 q^{35} -10.9282 q^{37} +5.93426i q^{41} -4.00000i q^{43} -8.48528 q^{47} -1.00000 q^{49} -7.07107i q^{53} -20.3923i q^{55} +2.82843 q^{59} -3.46410 q^{61} +7.72741i q^{65} +15.4641i q^{67} +4.52004 q^{71} +0.535898 q^{73} +5.27792i q^{77} +10.3923i q^{79} -11.8685 q^{83} -4.00000 q^{85} -3.86370i q^{89} -2.00000i q^{91} -21.1117 q^{95} +11.4641 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{13} - 24 q^{25} - 32 q^{37} - 8 q^{49} + 32 q^{73} - 32 q^{85} + 64 q^{97}+O(q^{100}) $ 8 * q + 16 * q^13 - 24 * q^25 - 32 * q^37 - 8 * q^49 + 32 * q^73 - 32 * q^85 + 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	3.86370i	1.72790i	0.503577	+	0.863950i	$0.332017\pi$
$5$	3.86370i	1.72790i	−0.503577	+	0.863950i	$0.667983\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.27792	−1.59135	−0.795676	−	0.605723i	$-0.792884\pi$
$11$	−5.27792	−1.59135	−0.795676	+	0.605723i	$0.792884\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.03528i	0.251091i	0.992088	+	0.125546i	$0.0400682\pi$
$17$	1.03528i	0.251091i	−0.992088	+	0.125546i	$0.959932\pi$
$18$	0	0
$19$	5.46410i	1.25355i	0.779200	+	0.626775i	$0.215626\pi$
$19$	5.46410i	1.25355i	−0.779200	+	0.626775i	$0.784374\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.378937	−0.0790139	−0.0395070	−	0.999219i	$-0.512579\pi$
$23$	−0.378937	−0.0790139	−0.0395070	+	0.999219i	$0.512579\pi$
$24$	0	0
$25$	−9.92820	−1.98564
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.31319i	1.17233i	0.810192	+	0.586165i	$0.199363\pi$
$29$	6.31319i	1.17233i	−0.810192	+	0.586165i	$0.800637\pi$
$30$	0	0
$31$	− 9.46410i	− 1.69980i	−0.526942	−	0.849901i	$-0.676661\pi$
$31$	− 9.46410i	− 1.69980i	0.526942	−	0.849901i	$-0.323339\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.86370	0.653085
$36$	0	0
$37$	−10.9282	−1.79659	−0.898293	−	0.439397i	$-0.855192\pi$
$37$	−10.9282	−1.79659	−0.898293	+	0.439397i	$0.855192\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.93426i	0.926775i	0.886156	+	0.463388i	$0.153366\pi$
$41$	5.93426i	0.926775i	−0.886156	+	0.463388i	$0.846634\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.07107i	− 0.971286i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	− 7.07107i	− 0.971286i	0.874157	−	0.485643i	$-0.161414\pi$
$54$	0	0
$55$	− 20.3923i	− 2.74970i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	−3.46410	−0.443533	−0.221766	−	0.975100i	$-0.571182\pi$
$61$	−3.46410	−0.443533	−0.221766	+	0.975100i	$0.571182\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.72741i	0.958467i
$66$	0	0
$67$	15.4641i	1.88924i	0.328165	+	0.944620i	$0.393570\pi$
$67$	15.4641i	1.88924i	−0.328165	+	0.944620i	$0.606430\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.52004	0.536430	0.268215	−	0.963359i	$-0.413566\pi$
$71$	4.52004	0.536430	0.268215	+	0.963359i	$0.413566\pi$
$72$	0	0
$73$	0.535898	0.0627222	0.0313611	−	0.999508i	$-0.490016\pi$
$73$	0.535898	0.0627222	0.0313611	+	0.999508i	$0.490016\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.27792i	0.601474i
$78$	0	0
$79$	10.3923i	1.16923i	0.811312	+	0.584613i	$0.198754\pi$
$79$	10.3923i	1.16923i	−0.811312	+	0.584613i	$0.801246\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−11.8685	−1.30274	−0.651369	−	0.758761i	$-0.725805\pi$
$83$	−11.8685	−1.30274	−0.651369	+	0.758761i	$0.725805\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.86370i	− 0.409552i	−0.978809	−	0.204776i	$-0.934353\pi$
$89$	− 3.86370i	− 0.409552i	0.978809	−	0.204776i	$-0.0656465\pi$
$90$	0	0
$91$	− 2.00000i	− 0.209657i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−21.1117	−2.16601
$96$	0	0
$97$	11.4641	1.16400	0.582002	−	0.813188i	$-0.302270\pi$
$97$	11.4641	1.16400	0.582002	+	0.813188i	$0.302270\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.4195i	1.43480i	0.696663	+	0.717399i	$0.254667\pi$
$101$	14.4195i	1.43480i	−0.696663	+	0.717399i	$0.745333\pi$
$102$	0	0
$103$	5.46410i	0.538394i	0.963085	+	0.269197i	$0.0867583\pi$
$103$	5.46410i	0.538394i	−0.963085	+	0.269197i	$0.913242\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.1769	0.983838	0.491919	−	0.870641i	$-0.336296\pi$
$107$	10.1769	0.983838	0.491919	+	0.870641i	$0.336296\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.89949i	0.931266i	0.884978	+	0.465633i	$0.154173\pi$
$113$	9.89949i	0.931266i	−0.884978	+	0.465633i	$0.845827\pi$
$114$	0	0
$115$	− 1.46410i	− 0.136528i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.03528	0.0949036
$120$	0	0
$121$	16.8564	1.53240
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 19.0411i	− 1.70309i
$126$	0	0
$127$	6.39230i	0.567225i	0.958939	+	0.283613i	$0.0915330\pi$
$127$	6.39230i	0.567225i	−0.958939	+	0.283613i	$0.908467\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	21.1117	1.84453	0.922267	−	0.386552i	$-0.126334\pi$
$131$	21.1117	1.84453	0.922267	+	0.386552i	$0.126334\pi$
$132$	0	0
$133$	5.46410	0.473798
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.48477i	− 0.297724i	−0.988858	−	0.148862i	$-0.952439\pi$
$137$	− 3.48477i	− 0.297724i	0.988858	−	0.148862i	$-0.0475610\pi$
$138$	0	0
$139$	− 5.85641i	− 0.496734i	−0.968666	−	0.248367i	$-0.920106\pi$
$139$	− 5.85641i	− 0.496734i	0.968666	−	0.248367i	$-0.0798939\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−10.5558	−0.882723
$144$	0	0
$145$	−24.3923	−2.02567
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2.72689i	− 0.223396i	−0.993742	−	0.111698i	$-0.964371\pi$
$149$	− 2.72689i	− 0.223396i	0.993742	−	0.111698i	$-0.0356289\pi$
$150$	0	0
$151$	− 10.9282i	− 0.889325i	−0.895698	−	0.444662i	$-0.853324\pi$
$151$	− 10.9282i	− 0.889325i	0.895698	−	0.444662i	$-0.146676\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	36.5665	2.93709
$156$	0	0
$157$	17.3205	1.38233	0.691164	−	0.722698i	$-0.257098\pi$
$157$	17.3205	1.38233	0.691164	+	0.722698i	$0.257098\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.378937i	0.0298644i
$162$	0	0
$163$	7.46410i	0.584634i	0.956322	+	0.292317i	$0.0944262\pi$
$163$	7.46410i	0.584634i	−0.956322	+	0.292317i	$0.905574\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.7675	−1.29751	−0.648754	−	0.760998i	$-0.724709\pi$
$167$	−16.7675	−1.29751	−0.648754	+	0.760998i	$0.724709\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.10583i	− 0.236132i	−0.993006	−	0.118066i	$-0.962331\pi$
$173$	− 3.10583i	− 0.236132i	0.993006	−	0.118066i	$-0.0376694\pi$
$174$	0	0
$175$	9.92820i	0.750502i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.89469	0.141616	0.0708078	−	0.997490i	$-0.477442\pi$
$179$	1.89469	0.141616	0.0708078	+	0.997490i	$0.477442\pi$
$180$	0	0
$181$	−0.928203	−0.0689928	−0.0344964	−	0.999405i	$-0.510983\pi$
$181$	−0.928203	−0.0689928	−0.0344964	+	0.999405i	$0.510983\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 42.2233i	− 3.10432i
$186$	0	0
$187$	− 5.46410i	− 0.399575i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.76217	0.272221	0.136110	−	0.990694i	$-0.456540\pi$
$191$	3.76217	0.272221	0.136110	+	0.990694i	$0.456540\pi$
$192$	0	0
$193$	13.8564	0.997406	0.498703	−	0.866773i	$-0.333810\pi$
$193$	13.8564	0.997406	0.498703	+	0.866773i	$0.333810\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 23.8386i	− 1.69843i	−0.528050	−	0.849213i	$-0.677077\pi$
$197$	− 23.8386i	− 1.69843i	0.528050	−	0.849213i	$-0.322923\pi$
$198$	0	0
$199$	4.00000i	0.283552i	0.989899	+	0.141776i	$0.0452813\pi$
$199$	4.00000i	0.283552i	−0.989899	+	0.141776i	$0.954719\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.31319	0.443099
$204$	0	0
$205$	−22.9282	−1.60138
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 28.8391i	− 1.99484i
$210$	0	0
$211$	22.9282i	1.57844i	0.614109	+	0.789221i	$0.289516\pi$
$211$	22.9282i	1.57844i	−0.614109	+	0.789221i	$0.710484\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.4548	1.05401
$216$	0	0
$217$	−9.46410	−0.642465
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.07055i	0.139280i
$222$	0	0
$223$	− 9.85641i	− 0.660034i	−0.943975	−	0.330017i	$-0.892946\pi$
$223$	− 9.85641i	− 0.660034i	0.943975	−	0.330017i	$-0.107054\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.34418	0.288333	0.144167	−	0.989553i	$-0.453950\pi$
$227$	4.34418	0.288333	0.144167	+	0.989553i	$0.453950\pi$
$228$	0	0
$229$	29.7128	1.96348	0.981739	−	0.190233i	$-0.0609243\pi$
$229$	29.7128	1.96348	0.981739	+	0.190233i	$0.0609243\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.4553i	1.34007i	0.742328	+	0.670037i	$0.233722\pi$
$233$	20.4553i	1.34007i	−0.742328	+	0.670037i	$0.766278\pi$
$234$	0	0
$235$	− 32.7846i	− 2.13863i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.2485	−1.43913	−0.719567	−	0.694423i	$-0.755660\pi$
$239$	−22.2485	−1.43913	−0.719567	+	0.694423i	$0.755660\pi$
$240$	0	0
$241$	−6.39230	−0.411765	−0.205882	−	0.978577i	$-0.566006\pi$
$241$	−6.39230	−0.411765	−0.205882	+	0.978577i	$0.566006\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.86370i	− 0.246843i
$246$	0	0
$247$	10.9282i	0.695345i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.4949	−1.54610	−0.773052	−	0.634343i	$-0.781271\pi$
$251$	−24.4949	−1.54610	−0.773052	+	0.634343i	$0.781271\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 0.277401i	− 0.0173038i	−0.999963	−	0.00865191i	$-0.997246\pi$
$257$	− 0.277401i	− 0.0173038i	0.999963	−	0.00865191i	$-0.00275402\pi$
$258$	0	0
$259$	10.9282i	0.679046i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.2485	1.37190	0.685950	−	0.727649i	$-0.259387\pi$
$263$	22.2485	1.37190	0.685950	+	0.727649i	$0.259387\pi$
$264$	0	0
$265$	27.3205	1.67829
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.1069i	0.799139i	0.916703	+	0.399570i	$0.130840\pi$
$269$	13.1069i	0.799139i	−0.916703	+	0.399570i	$0.869160\pi$
$270$	0	0
$271$	3.32051i	0.201707i	0.994901	+	0.100853i	$0.0321573\pi$
$271$	3.32051i	0.201707i	−0.994901	+	0.100853i	$0.967843\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	52.4002	3.15985
$276$	0	0
$277$	27.8564	1.67373	0.836865	−	0.547410i	$-0.184386\pi$
$277$	27.8564	1.67373	0.836865	+	0.547410i	$0.184386\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.4553i	1.22026i	0.792300	+	0.610131i	$0.208883\pi$
$281$	20.4553i	1.22026i	−0.792300	+	0.610131i	$0.791117\pi$
$282$	0	0
$283$	21.4641i	1.27591i	0.770074	+	0.637954i	$0.220219\pi$
$283$	21.4641i	1.27591i	−0.770074	+	0.637954i	$0.779781\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.93426	0.350288
$288$	0	0
$289$	15.9282	0.936953
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 18.0058i	− 1.05191i	−0.850512	−	0.525956i	$-0.823708\pi$
$293$	− 18.0058i	− 1.05191i	0.850512	−	0.525956i	$-0.176292\pi$
$294$	0	0
$295$	10.9282i	0.636265i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.757875	−0.0438290
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 13.3843i	− 0.766381i
$306$	0	0
$307$	14.5359i	0.829608i	0.909911	+	0.414804i	$0.136150\pi$
$307$	14.5359i	0.829608i	−0.909911	+	0.414804i	$0.863850\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.2832	−1.03675	−0.518374	−	0.855154i	$-0.673462\pi$
$311$	−18.2832	−1.03675	−0.518374	+	0.855154i	$0.673462\pi$
$312$	0	0
$313$	24.9282	1.40903	0.704513	−	0.709691i	$-0.251166\pi$
$313$	24.9282	1.40903	0.704513	+	0.709691i	$0.251166\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.3533i	− 0.862326i	−0.902274	−	0.431163i	$-0.858103\pi$
$317$	− 15.3533i	− 0.862326i	0.902274	−	0.431163i	$-0.141897\pi$
$318$	0	0
$319$	− 33.3205i	− 1.86559i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.65685	−0.314756
$324$	0	0
$325$	−19.8564	−1.10144
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.48528i	0.467809i
$330$	0	0
$331$	12.7846i	0.702706i	0.936243	+	0.351353i	$0.114278\pi$
$331$	12.7846i	0.702706i	−0.936243	+	0.351353i	$0.885722\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−59.7487	−3.26442
$336$	0	0
$337$	12.7846	0.696422	0.348211	−	0.937416i	$-0.386789\pi$
$337$	12.7846	0.696422	0.348211	+	0.937416i	$0.386789\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	49.9507i	2.70498i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.03579	0.324018	0.162009	−	0.986789i	$-0.448203\pi$
$347$	6.03579	0.324018	0.162009	+	0.986789i	$0.448203\pi$
$348$	0	0
$349$	−13.3205	−0.713030	−0.356515	−	0.934290i	$-0.616035\pi$
$349$	−13.3205	−0.713030	−0.356515	+	0.934290i	$0.616035\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 15.9353i	− 0.848150i	−0.905627	−	0.424075i	$-0.860599\pi$
$353$	− 15.9353i	− 0.848150i	0.905627	−	0.424075i	$-0.139401\pi$
$354$	0	0
$355$	17.4641i	0.926898i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.0759	0.795674	0.397837	−	0.917456i	$-0.369761\pi$
$359$	15.0759	0.795674	0.397837	+	0.917456i	$0.369761\pi$
$360$	0	0
$361$	−10.8564	−0.571390
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.07055i	0.108378i
$366$	0	0
$367$	− 10.9282i	− 0.570448i	−0.958461	−	0.285224i	$-0.907932\pi$
$367$	− 10.9282i	− 0.570448i	0.958461	−	0.285224i	$-0.0920679\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.07107	−0.367112
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.6264i	0.650292i
$378$	0	0
$379$	20.7846i	1.06763i	0.845600	+	0.533817i	$0.179243\pi$
$379$	20.7846i	1.06763i	−0.845600	+	0.533817i	$0.820757\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	25.2528	1.29036	0.645178	−	0.764032i	$-0.276783\pi$
$383$	25.2528	1.29036	0.645178	+	0.764032i	$0.276783\pi$
$384$	0	0
$385$	−20.3923	−1.03929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 13.4858i	− 0.683757i	−0.939744	−	0.341879i	$-0.888937\pi$
$389$	− 13.4858i	− 0.683757i	0.939744	−	0.341879i	$-0.111063\pi$
$390$	0	0
$391$	− 0.392305i	− 0.0198397i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−40.1528	−2.02031
$396$	0	0
$397$	−11.4641	−0.575367	−0.287683	−	0.957726i	$-0.592885\pi$
$397$	−11.4641	−0.575367	−0.287683	+	0.957726i	$0.592885\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.79744i	0.239573i	0.992800	+	0.119786i	$0.0382210\pi$
$401$	4.79744i	0.239573i	−0.992800	+	0.119786i	$0.961779\pi$
$402$	0	0
$403$	− 18.9282i	− 0.942881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	57.6781	2.85900
$408$	0	0
$409$	−3.46410	−0.171289	−0.0856444	−	0.996326i	$-0.527295\pi$
$409$	−3.46410	−0.171289	−0.0856444	+	0.996326i	$0.527295\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.82843i	− 0.139178i
$414$	0	0
$415$	− 45.8564i	− 2.25100i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.38323	−0.165282	−0.0826408	−	0.996579i	$-0.526335\pi$
$419$	−3.38323	−0.165282	−0.0826408	+	0.996579i	$0.526335\pi$
$420$	0	0
$421$	−25.7128	−1.25317	−0.626583	−	0.779355i	$-0.715547\pi$
$421$	−25.7128	−1.25317	−0.626583	+	0.779355i	$0.715547\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 10.2784i	− 0.498577i
$426$	0	0
$427$	3.46410i	0.167640i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.4906	−1.03517	−0.517583	−	0.855633i	$-0.673168\pi$
$431$	−21.4906	−1.03517	−0.517583	+	0.855633i	$0.673168\pi$
$432$	0	0
$433$	−22.7846	−1.09496	−0.547479	−	0.836819i	$-0.684412\pi$
$433$	−22.7846	−1.09496	−0.547479	+	0.836819i	$0.684412\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.07055i	− 0.0990480i
$438$	0	0
$439$	− 13.8564i	− 0.661330i	−0.943748	−	0.330665i	$-0.892727\pi$
$439$	− 13.8564i	− 0.661330i	0.943748	−	0.330665i	$-0.107273\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−27.9053	−1.32582	−0.662911	−	0.748698i	$-0.730679\pi$
$443$	−27.9053	−1.32582	−0.662911	+	0.748698i	$0.730679\pi$
$444$	0	0
$445$	14.9282	0.707665
$446$	0	0
$447$	0	0
$448$	0	0
$449$	31.0112i	1.46351i	0.681569	+	0.731754i	$0.261298\pi$
$449$	31.0112i	1.46351i	−0.681569	+	0.731754i	$0.738702\pi$
$450$	0	0
$451$	− 31.3205i	− 1.47483i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.72741	0.362266
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 32.9058i	− 1.53258i	−0.642496	−	0.766289i	$-0.722101\pi$
$461$	− 32.9058i	− 1.53258i	0.642496	−	0.766289i	$-0.277899\pi$
$462$	0	0
$463$	− 0.535898i	− 0.0249053i	−0.999922	−	0.0124527i	$-0.996036\pi$
$463$	− 0.535898i	− 0.0249053i	0.999922	−	0.0124527i	$-0.00396391\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.5558	0.488466	0.244233	−	0.969717i	$-0.421464\pi$
$467$	10.5558	0.488466	0.244233	+	0.969717i	$0.421464\pi$
$468$	0	0
$469$	15.4641	0.714066
$470$	0	0
$471$	0	0
$472$	0	0
$473$	21.1117i	0.970716i
$474$	0	0
$475$	− 54.2487i	− 2.48910i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.3538	0.929989	0.464994	−	0.885314i	$-0.346056\pi$
$479$	20.3538	0.929989	0.464994	+	0.885314i	$0.346056\pi$
$480$	0	0
$481$	−21.8564	−0.996566
$482$	0	0
$483$	0	0
$484$	0	0
$485$	44.2939i	2.01128i
$486$	0	0
$487$	− 2.92820i	− 0.132690i	−0.997797	−	0.0663448i	$-0.978866\pi$
$487$	− 2.92820i	− 0.132690i	0.997797	−	0.0663448i	$-0.0211337\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−34.3201	−1.54884	−0.774421	−	0.632670i	$-0.781959\pi$
$491$	−34.3201	−1.54884	−0.774421	+	0.632670i	$0.781959\pi$
$492$	0	0
$493$	−6.53590	−0.294362
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.52004i	− 0.202752i
$498$	0	0
$499$	33.8564i	1.51562i	0.652475	+	0.757810i	$0.273731\pi$
$499$	33.8564i	1.51562i	−0.652475	+	0.757810i	$0.726269\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.21166	−0.276964	−0.138482	−	0.990365i	$-0.544222\pi$
$503$	−6.21166	−0.276964	−0.138482	+	0.990365i	$0.544222\pi$
$504$	0	0
$505$	−55.7128	−2.47919
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.93426i	0.263031i	0.991314	+	0.131516i	$0.0419844\pi$
$509$	5.93426i	0.263031i	−0.991314	+	0.131516i	$0.958016\pi$
$510$	0	0
$511$	− 0.535898i	− 0.0237067i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−21.1117	−0.930291
$516$	0	0
$517$	44.7846	1.96962
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.66063i	0.160375i	0.996780	+	0.0801876i	$0.0255519\pi$
$521$	3.66063i	0.160375i	−0.996780	+	0.0801876i	$0.974448\pi$
$522$	0	0
$523$	11.7128i	0.512166i	0.966655	+	0.256083i	$0.0824320\pi$
$523$	11.7128i	0.512166i	−0.966655	+	0.256083i	$0.917568\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.79796	0.426806
$528$	0	0
$529$	−22.8564	−0.993757
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.8685i	0.514082i
$534$	0	0
$535$	39.3205i	1.69997i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.27792	0.227336
$540$	0	0
$541$	−29.7128	−1.27745	−0.638727	−	0.769434i	$-0.720539\pi$
$541$	−29.7128	−1.27745	−0.638727	+	0.769434i	$0.720539\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.9096i	1.32402i
$546$	0	0
$547$	− 4.53590i	− 0.193941i	−0.995287	−	0.0969705i	$-0.969085\pi$
$547$	− 4.53590i	− 0.193941i	0.995287	−	0.0969705i	$-0.0309153\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−34.4959	−1.46958
$552$	0	0
$553$	10.3923	0.441926
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.8690i	0.714764i	0.933958	+	0.357382i	$0.116331\pi$
$557$	16.8690i	0.714764i	−0.933958	+	0.357382i	$0.883669\pi$
$558$	0	0
$559$	− 8.00000i	− 0.338364i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.2832	0.770547	0.385273	−	0.922802i	$-0.374107\pi$
$563$	18.2832	0.770547	0.385273	+	0.922802i	$0.374107\pi$
$564$	0	0
$565$	−38.2487	−1.60914
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.14162i	0.383237i	0.981470	+	0.191618i	$0.0613736\pi$
$569$	9.14162i	0.383237i	−0.981470	+	0.191618i	$0.938626\pi$
$570$	0	0
$571$	18.3923i	0.769694i	0.922980	+	0.384847i	$0.125746\pi$
$571$	18.3923i	0.769694i	−0.922980	+	0.384847i	$0.874254\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.76217	0.156893
$576$	0	0
$577$	44.9282	1.87039	0.935193	−	0.354139i	$-0.115226\pi$
$577$	44.9282	1.87039	0.935193	+	0.354139i	$0.115226\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11.8685i	0.492389i
$582$	0	0
$583$	37.3205i	1.54566i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.7990	−0.817192	−0.408596	−	0.912715i	$-0.633981\pi$
$587$	−19.7990	−0.817192	−0.408596	+	0.912715i	$0.633981\pi$
$588$	0	0
$589$	51.7128	2.13079
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 14.4195i	− 0.592139i	−0.955166	−	0.296070i	$-0.904324\pi$
$593$	− 14.4195i	− 0.592139i	0.955166	−	0.296070i	$-0.0956761\pi$
$594$	0	0
$595$	4.00000i	0.163984i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−35.4297	−1.44762	−0.723809	−	0.690001i	$-0.757610\pi$
$599$	−35.4297	−1.44762	−0.723809	+	0.690001i	$0.757610\pi$
$600$	0	0
$601$	−4.92820	−0.201026	−0.100513	−	0.994936i	$-0.532048\pi$
$601$	−4.92820	−0.201026	−0.100513	+	0.994936i	$0.532048\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	65.1282i	2.64784i
$606$	0	0
$607$	− 24.7846i	− 1.00598i	−0.864293	−	0.502988i	$-0.832234\pi$
$607$	− 24.7846i	− 1.00598i	0.864293	−	0.502988i	$-0.167766\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.9706	−0.686555
$612$	0	0
$613$	−10.7846	−0.435586	−0.217793	−	0.975995i	$-0.569886\pi$
$613$	−10.7846	−0.435586	−0.217793	+	0.975995i	$0.569886\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 25.9091i	− 1.04306i	−0.853232	−	0.521531i	$-0.825361\pi$
$617$	− 25.9091i	− 1.04306i	0.853232	−	0.521531i	$-0.174639\pi$
$618$	0	0
$619$	− 14.1436i	− 0.568479i	−0.958753	−	0.284240i	$-0.908259\pi$
$619$	− 14.1436i	− 0.568479i	0.958753	−	0.284240i	$-0.0917411\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.86370	−0.154796
$624$	0	0
$625$	23.9282	0.957128
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 11.3137i	− 0.451107i
$630$	0	0
$631$	− 15.4641i	− 0.615616i	−0.951448	−	0.307808i	$-0.900405\pi$
$631$	− 15.4641i	− 0.615616i	0.951448	−	0.307808i	$-0.0995955\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−24.6980	−0.980109
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.3944i	1.35850i	0.733908	+	0.679248i	$0.237694\pi$
$641$	34.3944i	1.35850i	−0.733908	+	0.679248i	$0.762306\pi$
$642$	0	0
$643$	− 38.2487i	− 1.50838i	−0.656655	−	0.754191i	$-0.728029\pi$
$643$	− 38.2487i	− 1.50838i	0.656655	−	0.754191i	$-0.271971\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.203072	0.00798358	0.00399179	−	0.999992i	$-0.498729\pi$
$647$	0.203072	0.00798358	0.00399179	+	0.999992i	$0.498729\pi$
$648$	0	0
$649$	−14.9282	−0.585983
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.9406i	1.13253i	0.824222	+	0.566267i	$0.191613\pi$
$653$	28.9406i	1.13253i	−0.824222	+	0.566267i	$0.808387\pi$
$654$	0	0
$655$	81.5692i	3.18717i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	43.3601	1.68907	0.844536	−	0.535499i	$-0.179876\pi$
$659$	43.3601	1.68907	0.844536	+	0.535499i	$0.179876\pi$
$660$	0	0
$661$	−8.53590	−0.332008	−0.166004	−	0.986125i	$-0.553086\pi$
$661$	−8.53590	−0.332008	−0.166004	+	0.986125i	$0.553086\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	21.1117i	0.818675i
$666$	0	0
$667$	− 2.39230i	− 0.0926304i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	18.2832	0.705817
$672$	0	0
$673$	17.8564	0.688314	0.344157	−	0.938912i	$-0.388165\pi$
$673$	17.8564	0.688314	0.344157	+	0.938912i	$0.388165\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 22.7017i	− 0.872499i	−0.899826	−	0.436249i	$-0.856307\pi$
$677$	− 22.7017i	− 0.872499i	0.899826	−	0.436249i	$-0.143693\pi$
$678$	0	0
$679$	− 11.4641i	− 0.439952i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.6632	−1.09677	−0.548384	−	0.836227i	$-0.684757\pi$
$683$	−28.6632	−1.09677	−0.548384	+	0.836227i	$0.684757\pi$
$684$	0	0
$685$	13.4641	0.514437
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 14.1421i	− 0.538772i
$690$	0	0
$691$	− 25.8564i	− 0.983624i	−0.870701	−	0.491812i	$-0.836335\pi$
$691$	− 25.8564i	− 0.983624i	0.870701	−	0.491812i	$-0.163665\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22.6274	0.858307
$696$	0	0
$697$	−6.14359	−0.232705
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 12.1731i	− 0.459772i	−0.973218	−	0.229886i	$-0.926165\pi$
$701$	− 12.1731i	− 0.459772i	0.973218	−	0.229886i	$-0.0738354\pi$
$702$	0	0
$703$	− 59.7128i	− 2.25211i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.4195	0.542303
$708$	0	0
$709$	13.8564	0.520388	0.260194	−	0.965556i	$-0.416213\pi$
$709$	13.8564	0.520388	0.260194	+	0.965556i	$0.416213\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.58630i	0.134308i
$714$	0	0
$715$	− 40.7846i	− 1.52526i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.1117	0.787332	0.393666	−	0.919253i	$-0.371207\pi$
$719$	21.1117	0.787332	0.393666	+	0.919253i	$0.371207\pi$
$720$	0	0
$721$	5.46410	0.203494
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 62.6787i	− 2.32783i
$726$	0	0
$727$	29.4641i	1.09276i	0.837536	+	0.546382i	$0.183995\pi$
$727$	29.4641i	1.09276i	−0.837536	+	0.546382i	$0.816005\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.14110	0.153164
$732$	0	0
$733$	−8.92820	−0.329771	−0.164885	−	0.986313i	$-0.552725\pi$
$733$	−8.92820	−0.329771	−0.164885	+	0.986313i	$0.552725\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 81.6182i	− 3.00645i
$738$	0	0
$739$	2.67949i	0.0985667i	0.998785	+	0.0492834i	$0.0156937\pi$
$739$	2.67949i	0.0985667i	−0.998785	+	0.0492834i	$0.984306\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.00429	−0.110217	−0.0551084	−	0.998480i	$-0.517550\pi$
$743$	−3.00429	−0.110217	−0.0551084	+	0.998480i	$0.517550\pi$
$744$	0	0
$745$	10.5359	0.386005
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 10.1769i	− 0.371856i
$750$	0	0
$751$	21.8564i	0.797552i	0.917048	+	0.398776i	$0.130565\pi$
$751$	21.8564i	0.797552i	−0.917048	+	0.398776i	$0.869435\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	42.2233	1.53666
$756$	0	0
$757$	−37.8564	−1.37591	−0.687957	−	0.725751i	$-0.741492\pi$
$757$	−37.8564	−1.37591	−0.687957	+	0.725751i	$0.741492\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.1459i	0.440289i	0.975467	+	0.220144i	$0.0706529\pi$
$761$	12.1459i	0.440289i	−0.975467	+	0.220144i	$0.929347\pi$
$762$	0	0
$763$	− 8.00000i	− 0.289619i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.65685	0.204257
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 17.0449i	− 0.613062i	−0.951861	−	0.306531i	$-0.900832\pi$
$773$	− 17.0449i	− 0.613062i	0.951861	−	0.306531i	$-0.0991683\pi$
$774$	0	0
$775$	93.9615i	3.37520i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.4254	−1.16176
$780$	0	0
$781$	−23.8564	−0.853649
$782$	0	0
$783$	0	0
$784$	0	0
$785$	66.9213i	2.38852i
$786$	0	0
$787$	− 20.7846i	− 0.740891i	−0.928854	−	0.370446i	$-0.879205\pi$
$787$	− 20.7846i	− 0.740891i	0.928854	−	0.370446i	$-0.120795\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.89949	0.351986
$792$	0	0
$793$	−6.92820	−0.246028
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.20788i	0.290738i	0.989378	+	0.145369i	$0.0464369\pi$
$797$	8.20788i	0.290738i	−0.989378	+	0.145369i	$0.953563\pi$
$798$	0	0
$799$	− 8.78461i	− 0.310777i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.82843	−0.0998130
$804$	0	0
$805$	−1.46410	−0.0516028
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 47.5756i	− 1.67267i	−0.548220	−	0.836334i	$-0.684694\pi$
$809$	− 47.5756i	− 1.67267i	0.548220	−	0.836334i	$-0.315306\pi$
$810$	0	0
$811$	− 22.1436i	− 0.777567i	−0.921329	−	0.388783i	$-0.872895\pi$
$811$	− 22.1436i	− 0.777567i	0.921329	−	0.388783i	$-0.127105\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−28.8391	−1.01019
$816$	0	0
$817$	21.8564	0.764659
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.41421i	0.0493564i	0.999695	+	0.0246782i	$0.00785611\pi$
$821$	1.41421i	0.0493564i	−0.999695	+	0.0246782i	$0.992144\pi$
$822$	0	0
$823$	34.3923i	1.19884i	0.800435	+	0.599420i	$0.204602\pi$
$823$	34.3923i	1.19884i	−0.800435	+	0.599420i	$0.795398\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.6317	−0.891302	−0.445651	−	0.895207i	$-0.647028\pi$
$827$	−25.6317	−0.891302	−0.445651	+	0.895207i	$0.647028\pi$
$828$	0	0
$829$	−27.0718	−0.940242	−0.470121	−	0.882602i	$-0.655790\pi$
$829$	−27.0718	−0.940242	−0.470121	+	0.882602i	$0.655790\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.03528i	− 0.0358702i
$834$	0	0
$835$	− 64.7846i	− 2.24196i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37.3244	1.28858	0.644290	−	0.764781i	$-0.277153\pi$
$839$	37.3244	1.28858	0.644290	+	0.764781i	$0.277153\pi$
$840$	0	0
$841$	−10.8564	−0.374359
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 34.7733i	− 1.19624i
$846$	0	0
$847$	− 16.8564i	− 0.579193i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.14110	0.141955
$852$	0	0
$853$	21.3205	0.730000	0.365000	−	0.931007i	$-0.381069\pi$
$853$	21.3205	0.730000	0.365000	+	0.931007i	$0.381069\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 44.0165i	− 1.50357i	−0.659406	−	0.751787i	$-0.729192\pi$
$857$	− 44.0165i	− 1.50357i	0.659406	−	0.751787i	$-0.270808\pi$
$858$	0	0
$859$	22.5359i	0.768915i	0.923143	+	0.384457i	$0.125611\pi$
$859$	22.5359i	0.768915i	−0.923143	+	0.384457i	$0.874389\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.5916	0.564785	0.282393	−	0.959299i	$-0.408872\pi$
$863$	16.5916	0.564785	0.282393	+	0.959299i	$0.408872\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 54.8497i	− 1.86065i
$870$	0	0
$871$	30.9282i	1.04796i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−19.0411	−0.643707
$876$	0	0
$877$	−5.21539	−0.176111	−0.0880556	−	0.996116i	$-0.528065\pi$
$877$	−5.21539	−0.176111	−0.0880556	+	0.996116i	$0.528065\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23.6627i	0.797216i	0.917121	+	0.398608i	$0.130507\pi$
$881$	23.6627i	0.797216i	−0.917121	+	0.398608i	$0.869493\pi$
$882$	0	0
$883$	25.8564i	0.870137i	0.900397	+	0.435069i	$0.143276\pi$
$883$	25.8564i	0.870137i	−0.900397	+	0.435069i	$0.856724\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.0833	1.61448	0.807239	−	0.590225i	$-0.200961\pi$
$887$	48.0833	1.61448	0.807239	+	0.590225i	$0.200961\pi$
$888$	0	0
$889$	6.39230	0.214391
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 46.3644i	− 1.55153i
$894$	0	0
$895$	7.32051i	0.244698i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	59.7487	1.99273
$900$	0	0
$901$	7.32051	0.243881
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 3.58630i	− 0.119213i
$906$	0	0
$907$	7.71281i	0.256100i	0.991768	+	0.128050i	$0.0408718\pi$
$907$	7.71281i	0.256100i	−0.991768	+	0.128050i	$0.959128\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.89469	0.0627738	0.0313869	−	0.999507i	$-0.490008\pi$
$911$	1.89469	0.0627738	0.0313869	+	0.999507i	$0.490008\pi$
$912$	0	0
$913$	62.6410	2.07312
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 21.1117i	− 0.697169i
$918$	0	0
$919$	21.0718i	0.695094i	0.937662	+	0.347547i	$0.112985\pi$
$919$	21.0718i	0.695094i	−0.937662	+	0.347547i	$0.887015\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.04008	0.297558
$924$	0	0
$925$	108.497	3.56737
$926$	0	0
$927$	0	0
$928$	0	0
$929$	11.0363i	0.362089i	0.983475	+	0.181045i	$0.0579479\pi$
$929$	11.0363i	0.362089i	−0.983475	+	0.181045i	$0.942052\pi$
$930$	0	0
$931$	− 5.46410i	− 0.179079i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	21.1117	0.690425
$936$	0	0
$937$	43.5692	1.42334	0.711672	−	0.702512i	$-0.247938\pi$
$937$	43.5692	1.42334	0.711672	+	0.702512i	$0.247938\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	58.9165i	1.92062i	0.278930	+	0.960311i	$0.410020\pi$
$941$	58.9165i	1.92062i	−0.278930	+	0.960311i	$0.589980\pi$
$942$	0	0
$943$	− 2.24871i	− 0.0732281i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.3581	−0.759036	−0.379518	−	0.925184i	$-0.623910\pi$
$947$	−23.3581	−0.759036	−0.379518	+	0.925184i	$0.623910\pi$
$948$	0	0
$949$	1.07180	0.0347920
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 33.6365i	− 1.08959i	−0.838568	−	0.544797i	$-0.816607\pi$
$953$	− 33.6365i	− 1.08959i	0.838568	−	0.544797i	$-0.183393\pi$
$954$	0	0
$955$	14.5359i	0.470371i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.48477	−0.112529
$960$	0	0
$961$	−58.5692	−1.88933
$962$	0	0
$963$	0	0
$964$	0	0
$965$	53.5370i	1.72342i
$966$	0	0
$967$	23.4641i	0.754555i	0.926100	+	0.377277i	$0.123140\pi$
$967$	23.4641i	0.754555i	−0.926100	+	0.377277i	$0.876860\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−36.0117	−1.15567	−0.577835	−	0.816154i	$-0.696102\pi$
$971$	−36.0117	−1.15567	−0.577835	+	0.816154i	$0.696102\pi$
$972$	0	0
$973$	−5.85641	−0.187748
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.8300i	0.570431i	0.958463	+	0.285216i	$0.0920652\pi$
$977$	17.8300i	0.570431i	−0.958463	+	0.285216i	$0.907935\pi$
$978$	0	0
$979$	20.3923i	0.651741i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	61.2645	1.95403	0.977016	−	0.213165i	$-0.0683771\pi$
$983$	61.2645	1.95403	0.977016	+	0.213165i	$0.0683771\pi$
$984$	0	0
$985$	92.1051	2.93471
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.51575i	0.0481980i
$990$	0	0
$991$	− 54.6410i	− 1.73573i	−0.496801	−	0.867865i	$-0.665492\pi$
$991$	− 54.6410i	− 1.73573i	0.496801	−	0.867865i	$-0.334508\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15.4548	−0.489951
$996$	0	0
$997$	−28.5359	−0.903741	−0.451870	−	0.892084i	$-0.649243\pi$
$997$	−28.5359	−0.903741	−0.451870	+	0.892084i	$0.649243\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.2.h.b.575.7	yes	8
3.2	odd	2	inner	1008.2.h.b.575.1	✓	8
4.3	odd	2	inner	1008.2.h.b.575.8	yes	8
7.6	odd	2		7056.2.h.k.4607.1		8
8.3	odd	2		4032.2.h.f.575.2		8
8.5	even	2		4032.2.h.f.575.1		8
12.11	even	2	inner	1008.2.h.b.575.2	yes	8
21.20	even	2		7056.2.h.k.4607.8		8
24.5	odd	2		4032.2.h.f.575.7		8
24.11	even	2		4032.2.h.f.575.8		8
28.27	even	2		7056.2.h.k.4607.2		8
84.83	odd	2		7056.2.h.k.4607.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.2.h.b.575.1	✓	8	3.2	odd	2	inner
1008.2.h.b.575.2	yes	8	12.11	even	2	inner
1008.2.h.b.575.7	yes	8	1.1	even	1	trivial
1008.2.h.b.575.8	yes	8	4.3	odd	2	inner
4032.2.h.f.575.1		8	8.5	even	2
4032.2.h.f.575.2		8	8.3	odd	2
4032.2.h.f.575.7		8	24.5	odd	2
4032.2.h.f.575.8		8	24.11	even	2
7056.2.h.k.4607.1		8	7.6	odd	2
7056.2.h.k.4607.2		8	28.27	even	2
7056.2.h.k.4607.7		8	84.83	odd	2
7056.2.h.k.4607.8		8	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1008.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.86370i q^{5} -1.00000i q^{7} -5.27792 q^{11} +2.00000 q^{13} +1.03528i q^{17} +5.46410i q^{19} -0.378937 q^{23} -9.92820 q^{25} +6.31319i q^{29} -9.46410i q^{31} +3.86370 q^{35} -10.9282 q^{37} +5.93426i q^{41} -4.00000i q^{43} -8.48528 q^{47} -1.00000 q^{49} -7.07107i q^{53} -20.3923i q^{55} +2.82843 q^{59} -3.46410 q^{61} +7.72741i q^{65} +15.4641i q^{67} +4.52004 q^{71} +0.535898 q^{73} +5.27792i q^{77} +10.3923i q^{79} -11.8685 q^{83} -4.00000 q^{85} -3.86370i q^{89} -2.00000i q^{91} -21.1117 q^{95} +11.4641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{13} - 24 q^{25} - 32 q^{37} - 8 q^{49} + 32 q^{73} - 32 q^{85} + 64 q^{97}+O(q^{100}) \) 8 * q + 16 * q^13 - 24 * q^25 - 32 * q^37 - 8 * q^49 + 32 * q^73 - 32 * q^85 + 64 * q^97

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