LMFDB - Embedded newform 1600.2.d.b.801.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,2,Mod(801,1600)] 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("1600.801"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-12,0,-4,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			801.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1600.801
Dual form			1600.2.d.b.801.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051i q^{3} -1.26795 q^{7} +2.46410 q^{9} +3.46410i q^{11} +3.46410i q^{13} -3.46410 q^{17} +2.00000i q^{19} +0.928203i q^{21} -8.19615 q^{23} -4.00000i q^{27} -9.46410 q^{31} +2.53590 q^{33} -6.00000i q^{37} +2.53590 q^{39} +2.53590 q^{41} +10.1962i q^{43} +8.19615 q^{47} -5.39230 q^{49} +2.53590i q^{51} +10.3923i q^{53} +1.46410 q^{57} -6.00000i q^{59} +12.9282i q^{61} -3.12436 q^{63} +10.1962i q^{67} +6.00000i q^{69} -4.39230 q^{71} +14.3923 q^{73} -4.39230i q^{77} -12.0000 q^{79} +4.46410 q^{81} +4.73205i q^{83} +0.928203 q^{89} -4.39230i q^{91} +6.92820i q^{93} +6.39230 q^{97} +8.53590i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{7} - 4 q^{9} - 12 q^{23} - 24 q^{31} + 24 q^{33} + 24 q^{39} + 24 q^{41} + 12 q^{47} + 20 q^{49} - 8 q^{57} + 36 q^{63} + 24 q^{71} + 16 q^{73} - 48 q^{79} + 4 q^{81} - 24 q^{89} - 16 q^{97}+O(q^{100}) $ 4 * q - 12 * q^7 - 4 * q^9 - 12 * q^23 - 24 * q^31 + 24 * q^33 + 24 * q^39 + 24 * q^41 + 12 * q^47 + 20 * q^49 - 8 * q^57 + 36 * q^63 + 24 * q^71 + 16 * q^73 - 48 * q^79 + 4 * q^81 - 24 * q^89 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 0.732051i	− 0.422650i	−0.977416	−	0.211325i	$-0.932222\pi$
$3$	− 0.732051i	− 0.422650i	0.977416	−	0.211325i	$-0.0677778\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.26795	−0.479240	−0.239620	−	0.970867i	$-0.577023\pi$
$7$	−1.26795	−0.479240	−0.239620	+	0.970867i	$0.577023\pi$
$8$	0	0
$9$	2.46410	0.821367
$10$	0	0
$11$	3.46410i	1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$	3.46410i	1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	3.46410i	0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$	3.46410i	0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	2.00000i	0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$	2.00000i	0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$	0	0
$21$	0.928203i	0.202551i
$22$	0	0
$23$	−8.19615	−1.70902	−0.854508	−	0.519438i	$-0.826141\pi$
$23$	−8.19615	−1.70902	−0.854508	+	0.519438i	$0.826141\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−9.46410	−1.69980	−0.849901	−	0.526942i	$-0.823339\pi$
$31$	−9.46410	−1.69980	−0.849901	+	0.526942i	$0.823339\pi$
$32$	0	0
$33$	2.53590	0.441443
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	2.53590	0.406069
$40$	0	0
$41$	2.53590	0.396041	0.198020	−	0.980198i	$-0.436549\pi$
$41$	2.53590	0.396041	0.198020	+	0.980198i	$0.436549\pi$
$42$	0	0
$43$	10.1962i	1.55490i	0.628946	+	0.777449i	$0.283487\pi$
$43$	10.1962i	1.55490i	−0.628946	+	0.777449i	$0.716513\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.19615	1.19553	0.597766	−	0.801671i	$-0.296055\pi$
$47$	8.19615	1.19553	0.597766	+	0.801671i	$0.296055\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	0	0
$51$	2.53590i	0.355097i
$52$	0	0
$53$	10.3923i	1.42749i	0.700404	+	0.713746i	$0.253003\pi$
$53$	10.3923i	1.42749i	−0.700404	+	0.713746i	$0.746997\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.46410	0.193925
$58$	0	0
$59$	− 6.00000i	− 0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	− 6.00000i	− 0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$	0	0
$61$	12.9282i	1.65529i	0.561254	+	0.827643i	$0.310319\pi$
$61$	12.9282i	1.65529i	−0.561254	+	0.827643i	$0.689681\pi$
$62$	0	0
$63$	−3.12436	−0.393632
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.1962i	1.24566i	0.782358	+	0.622829i	$0.214017\pi$
$67$	10.1962i	1.24566i	−0.782358	+	0.622829i	$0.785983\pi$
$68$	0	0
$69$	6.00000i	0.722315i
$70$	0	0
$71$	−4.39230	−0.521271	−0.260635	−	0.965437i	$-0.583932\pi$
$71$	−4.39230	−0.521271	−0.260635	+	0.965437i	$0.583932\pi$
$72$	0	0
$73$	14.3923	1.68449	0.842246	−	0.539093i	$-0.181233\pi$
$73$	14.3923	1.68449	0.842246	+	0.539093i	$0.181233\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.39230i	− 0.500550i
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	4.73205i	0.519410i	0.965688	+	0.259705i	$0.0836253\pi$
$83$	4.73205i	0.519410i	−0.965688	+	0.259705i	$0.916375\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.928203	0.0983893	0.0491947	−	0.998789i	$-0.484335\pi$
$89$	0.928203	0.0983893	0.0491947	+	0.998789i	$0.484335\pi$
$90$	0	0
$91$	− 4.39230i	− 0.460439i
$92$	0	0
$93$	6.92820i	0.718421i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.39230	0.649040	0.324520	−	0.945879i	$-0.394797\pi$
$97$	6.39230	0.649040	0.324520	+	0.945879i	$0.394797\pi$
$98$	0	0
$99$	8.53590i	0.857890i
$100$	0	0
$101$	12.0000i	1.19404i	0.802225	+	0.597022i	$0.203650\pi$
$101$	12.0000i	1.19404i	−0.802225	+	0.597022i	$0.796350\pi$
$102$	0	0
$103$	−8.19615	−0.807591	−0.403795	−	0.914849i	$-0.632309\pi$
$103$	−8.19615	−0.807591	−0.403795	+	0.914849i	$0.632309\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.7321i	− 1.61755i	−0.588119	−	0.808774i	$-0.700131\pi$
$107$	− 16.7321i	− 1.61755i	0.588119	−	0.808774i	$-0.299869\pi$
$108$	0	0
$109$	0.928203i	0.0889057i	0.999011	+	0.0444529i	$0.0141545\pi$
$109$	0.928203i	0.0889057i	−0.999011	+	0.0444529i	$0.985846\pi$
$110$	0	0
$111$	−4.39230	−0.416899
$112$	0	0
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.53590i	0.789144i
$118$	0	0
$119$	4.39230	0.402642
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 1.85641i	− 0.167387i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.80385	0.337537	0.168768	−	0.985656i	$-0.446021\pi$
$127$	3.80385	0.337537	0.168768	+	0.985656i	$0.446021\pi$
$128$	0	0
$129$	7.46410	0.657178
$130$	0	0
$131$	10.3923i	0.907980i	0.891007	+	0.453990i	$0.150000\pi$
$131$	10.3923i	0.907980i	−0.891007	+	0.453990i	$0.850000\pi$
$132$	0	0
$133$	− 2.53590i	− 0.219890i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.9282	−1.10453	−0.552265	−	0.833668i	$-0.686237\pi$
$137$	−12.9282	−1.10453	−0.552265	+	0.833668i	$0.686237\pi$
$138$	0	0
$139$	− 10.0000i	− 0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$	− 10.0000i	− 0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$	0	0
$141$	− 6.00000i	− 0.505291i
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.94744i	0.325579i
$148$	0	0
$149$	− 18.0000i	− 1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$	− 18.0000i	− 1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$	0	0
$151$	2.53590	0.206368	0.103184	−	0.994662i	$-0.467097\pi$
$151$	2.53590	0.206368	0.103184	+	0.994662i	$0.467097\pi$
$152$	0	0
$153$	−8.53590	−0.690086
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.928203i	0.0740787i	0.999314	+	0.0370393i	$0.0117927\pi$
$157$	0.928203i	0.0740787i	−0.999314	+	0.0370393i	$0.988207\pi$
$158$	0	0
$159$	7.60770	0.603329
$160$	0	0
$161$	10.3923	0.819028
$162$	0	0
$163$	− 5.80385i	− 0.454592i	−0.973826	−	0.227296i	$-0.927011\pi$
$163$	− 5.80385i	− 0.454592i	0.973826	−	0.227296i	$-0.0729886\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.19615	0.634237	0.317119	−	0.948386i	$-0.397285\pi$
$167$	8.19615	0.634237	0.317119	+	0.948386i	$0.397285\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.92820i	0.376869i
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.39230	−0.330146
$178$	0	0
$179$	19.8564i	1.48414i	0.670324	+	0.742069i	$0.266155\pi$
$179$	19.8564i	1.48414i	−0.670324	+	0.742069i	$0.733845\pi$
$180$	0	0
$181$	− 6.92820i	− 0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	− 6.92820i	− 0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$	0	0
$183$	9.46410	0.699607
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	5.07180i	0.368919i
$190$	0	0
$191$	16.3923	1.18611	0.593053	−	0.805164i	$-0.297923\pi$
$191$	16.3923	1.18611	0.593053	+	0.805164i	$0.297923\pi$
$192$	0	0
$193$	−6.39230	−0.460128	−0.230064	−	0.973175i	$-0.573894\pi$
$193$	−6.39230	−0.460128	−0.230064	+	0.973175i	$0.573894\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 10.3923i	− 0.740421i	−0.928948	−	0.370211i	$-0.879286\pi$
$197$	− 10.3923i	− 0.740421i	0.928948	−	0.370211i	$-0.120714\pi$
$198$	0	0
$199$	−6.92820	−0.491127	−0.245564	−	0.969380i	$-0.578973\pi$
$199$	−6.92820	−0.491127	−0.245564	+	0.969380i	$0.578973\pi$
$200$	0	0
$201$	7.46410	0.526477
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−20.1962	−1.40373
$208$	0	0
$209$	−6.92820	−0.479234
$210$	0	0
$211$	− 6.39230i	− 0.440064i	−0.975493	−	0.220032i	$-0.929384\pi$
$211$	− 6.39230i	− 0.440064i	0.975493	−	0.220032i	$-0.0706162\pi$
$212$	0	0
$213$	3.21539i	0.220315i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	− 10.5359i	− 0.711950i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	15.1244	1.01280	0.506401	−	0.862298i	$-0.330976\pi$
$223$	15.1244	1.01280	0.506401	+	0.862298i	$0.330976\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.5885i	1.23376i	0.787058	+	0.616880i	$0.211603\pi$
$227$	18.5885i	1.23376i	−0.787058	+	0.616880i	$0.788397\pi$
$228$	0	0
$229$	18.9282i	1.25081i	0.780300	+	0.625405i	$0.215066\pi$
$229$	18.9282i	1.25081i	−0.780300	+	0.625405i	$0.784934\pi$
$230$	0	0
$231$	−3.21539	−0.211557
$232$	0	0
$233$	1.60770	0.105324	0.0526618	−	0.998612i	$-0.483229\pi$
$233$	1.60770	0.105324	0.0526618	+	0.998612i	$0.483229\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.78461i	0.570622i
$238$	0	0
$239$	20.7846	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$240$	0	0
$241$	−20.3923	−1.31358	−0.656792	−	0.754072i	$-0.728087\pi$
$241$	−20.3923	−1.31358	−0.656792	+	0.754072i	$0.728087\pi$
$242$	0	0
$243$	− 15.2679i	− 0.979439i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.92820	−0.440831
$248$	0	0
$249$	3.46410	0.219529
$250$	0	0
$251$	− 15.4641i	− 0.976085i	−0.872820	−	0.488043i	$-0.837711\pi$
$251$	− 15.4641i	− 0.976085i	0.872820	−	0.488043i	$-0.162289\pi$
$252$	0	0
$253$	− 28.3923i	− 1.78501i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	7.60770i	0.472719i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.1962	1.24535	0.622674	−	0.782481i	$-0.286046\pi$
$263$	20.1962	1.24535	0.622674	+	0.782481i	$0.286046\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 0.679492i	− 0.0415842i
$268$	0	0
$269$	− 14.7846i	− 0.901434i	−0.892667	−	0.450717i	$-0.851168\pi$
$269$	− 14.7846i	− 0.901434i	0.892667	−	0.450717i	$-0.148832\pi$
$270$	0	0
$271$	4.39230	0.266814	0.133407	−	0.991061i	$-0.457408\pi$
$271$	4.39230	0.266814	0.133407	+	0.991061i	$0.457408\pi$
$272$	0	0
$273$	−3.21539	−0.194604
$274$	0	0
$275$	0	0
$276$	0	0
$277$	19.8564i	1.19306i	0.802592	+	0.596528i	$0.203454\pi$
$277$	19.8564i	1.19306i	−0.802592	+	0.596528i	$0.796546\pi$
$278$	0	0
$279$	−23.3205	−1.39616
$280$	0	0
$281$	−28.3923	−1.69374	−0.846871	−	0.531798i	$-0.821517\pi$
$281$	−28.3923	−1.69374	−0.846871	+	0.531798i	$0.821517\pi$
$282$	0	0
$283$	10.5885i	0.629418i	0.949188	+	0.314709i	$0.101907\pi$
$283$	10.5885i	0.629418i	−0.949188	+	0.314709i	$0.898093\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.21539	−0.189798
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	− 4.67949i	− 0.274317i
$292$	0	0
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	13.8564	0.804030
$298$	0	0
$299$	− 28.3923i	− 1.64197i
$300$	0	0
$301$	− 12.9282i	− 0.745169i
$302$	0	0
$303$	8.78461	0.504663
$304$	0	0
$305$	0	0
$306$	0	0
$307$	34.1962i	1.95168i	0.218492	+	0.975839i	$0.429886\pi$
$307$	34.1962i	1.95168i	−0.218492	+	0.975839i	$0.570114\pi$
$308$	0	0
$309$	6.00000i	0.341328i
$310$	0	0
$311$	−7.60770	−0.431393	−0.215696	−	0.976460i	$-0.569202\pi$
$311$	−7.60770	−0.431393	−0.215696	+	0.976460i	$0.569202\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 22.3923i	− 1.25768i	−0.777536	−	0.628839i	$-0.783531\pi$
$317$	− 22.3923i	− 1.25768i	0.777536	−	0.628839i	$-0.216469\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.2487	−0.683656
$322$	0	0
$323$	− 6.92820i	− 0.385496i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.679492	0.0375760
$328$	0	0
$329$	−10.3923	−0.572946
$330$	0	0
$331$	− 5.60770i	− 0.308227i	−0.988053	−	0.154113i	$-0.950748\pi$
$331$	− 5.60770i	− 0.308227i	0.988053	−	0.154113i	$-0.0492521\pi$
$332$	0	0
$333$	− 14.7846i	− 0.810192i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	0.679492i	0.0369049i
$340$	0	0
$341$	− 32.7846i	− 1.77539i
$342$	0	0
$343$	15.7128	0.848412
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 28.0526i	− 1.50594i	−0.658055	−	0.752970i	$-0.728620\pi$
$347$	− 28.0526i	− 1.50594i	0.658055	−	0.752970i	$-0.271380\pi$
$348$	0	0
$349$	8.78461i	0.470229i	0.971968	+	0.235115i	$0.0755466\pi$
$349$	8.78461i	0.470229i	−0.971968	+	0.235115i	$0.924453\pi$
$350$	0	0
$351$	13.8564	0.739600
$352$	0	0
$353$	14.7846	0.786905	0.393453	−	0.919345i	$-0.371281\pi$
$353$	14.7846	0.786905	0.393453	+	0.919345i	$0.371281\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.21539i	− 0.170177i
$358$	0	0
$359$	8.78461	0.463634	0.231817	−	0.972759i	$-0.425533\pi$
$359$	8.78461	0.463634	0.231817	+	0.972759i	$0.425533\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	0.732051i	0.0384227i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.0526	−0.524739	−0.262370	−	0.964967i	$-0.584504\pi$
$367$	−10.0526	−0.524739	−0.262370	+	0.964967i	$0.584504\pi$
$368$	0	0
$369$	6.24871	0.325295
$370$	0	0
$371$	− 13.1769i	− 0.684111i
$372$	0	0
$373$	− 7.85641i	− 0.406789i	−0.979097	−	0.203395i	$-0.934803\pi$
$373$	− 7.85641i	− 0.406789i	0.979097	−	0.203395i	$-0.0651974\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 2.00000i	− 0.102733i	−0.998680	−	0.0513665i	$-0.983642\pi$
$379$	− 2.00000i	− 0.102733i	0.998680	−	0.0513665i	$-0.0163577\pi$
$380$	0	0
$381$	− 2.78461i	− 0.142660i
$382$	0	0
$383$	−3.80385	−0.194368	−0.0971838	−	0.995266i	$-0.530983\pi$
$383$	−3.80385	−0.194368	−0.0971838	+	0.995266i	$0.530983\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	25.1244i	1.27714i
$388$	0	0
$389$	− 26.7846i	− 1.35803i	−0.734123	−	0.679017i	$-0.762406\pi$
$389$	− 26.7846i	− 1.35803i	0.734123	−	0.679017i	$-0.237594\pi$
$390$	0	0
$391$	28.3923	1.43586
$392$	0	0
$393$	7.60770	0.383757
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.4641i	1.37838i	0.724579	+	0.689192i	$0.242034\pi$
$397$	27.4641i	1.37838i	−0.724579	+	0.689192i	$0.757966\pi$
$398$	0	0
$399$	−1.85641	−0.0929366
$400$	0	0
$401$	−4.14359	−0.206921	−0.103461	−	0.994634i	$-0.532992\pi$
$401$	−4.14359	−0.206921	−0.103461	+	0.994634i	$0.532992\pi$
$402$	0	0
$403$	− 32.7846i	− 1.63312i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.7846	1.03025
$408$	0	0
$409$	3.60770	0.178389	0.0891945	−	0.996014i	$-0.471571\pi$
$409$	3.60770	0.178389	0.0891945	+	0.996014i	$0.471571\pi$
$410$	0	0
$411$	9.46410i	0.466830i
$412$	0	0
$413$	7.60770i	0.374350i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7.32051	−0.358487
$418$	0	0
$419$	0.928203i	0.0453457i	0.999743	+	0.0226728i	$0.00721761\pi$
$419$	0.928203i	0.0453457i	−0.999743	+	0.0226728i	$0.992782\pi$
$420$	0	0
$421$	6.00000i	0.292422i	0.989253	+	0.146211i	$0.0467079\pi$
$421$	6.00000i	0.292422i	−0.989253	+	0.146211i	$0.953292\pi$
$422$	0	0
$423$	20.1962	0.981971
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 16.3923i	− 0.793279i
$428$	0	0
$429$	8.78461i	0.424125i
$430$	0	0
$431$	28.3923	1.36761	0.683805	−	0.729665i	$-0.260324\pi$
$431$	28.3923	1.36761	0.683805	+	0.729665i	$0.260324\pi$
$432$	0	0
$433$	−26.3923	−1.26833	−0.634167	−	0.773196i	$-0.718657\pi$
$433$	−26.3923	−1.26833	−0.634167	+	0.773196i	$0.718657\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 16.3923i	− 0.784150i
$438$	0	0
$439$	18.9282	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$439$	18.9282	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$440$	0	0
$441$	−13.2872	−0.632723
$442$	0	0
$443$	− 0.339746i	− 0.0161418i	−0.999967	−	0.00807091i	$-0.997431\pi$
$443$	− 0.339746i	− 0.0161418i	0.999967	−	0.00807091i	$-0.00256908\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.1769	−0.623247
$448$	0	0
$449$	−2.53590	−0.119676	−0.0598382	−	0.998208i	$-0.519058\pi$
$449$	−2.53590	−0.119676	−0.0598382	+	0.998208i	$0.519058\pi$
$450$	0	0
$451$	8.78461i	0.413651i
$452$	0	0
$453$	− 1.85641i	− 0.0872216i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.7846	−1.06582	−0.532910	−	0.846172i	$-0.678901\pi$
$457$	−22.7846	−1.06582	−0.532910	+	0.846172i	$0.678901\pi$
$458$	0	0
$459$	13.8564i	0.646762i
$460$	0	0
$461$	12.0000i	0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$	12.0000i	0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$	0	0
$463$	15.8038	0.734467	0.367234	−	0.930129i	$-0.380305\pi$
$463$	15.8038	0.734467	0.367234	+	0.930129i	$0.380305\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 22.9808i	− 1.06342i	−0.846926	−	0.531711i	$-0.821549\pi$
$467$	− 22.9808i	− 1.06342i	0.846926	−	0.531711i	$-0.178451\pi$
$468$	0	0
$469$	− 12.9282i	− 0.596969i
$470$	0	0
$471$	0.679492	0.0313093
$472$	0	0
$473$	−35.3205	−1.62404
$474$	0	0
$475$	0	0
$476$	0	0
$477$	25.6077i	1.17250i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	20.7846	0.947697
$482$	0	0
$483$	− 7.60770i	− 0.346162i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−39.1244	−1.77289	−0.886447	−	0.462830i	$-0.846834\pi$
$487$	−39.1244	−1.77289	−0.886447	+	0.462830i	$0.846834\pi$
$488$	0	0
$489$	−4.24871	−0.192133
$490$	0	0
$491$	− 22.3923i	− 1.01055i	−0.862958	−	0.505275i	$-0.831391\pi$
$491$	− 22.3923i	− 1.01055i	0.862958	−	0.505275i	$-0.168609\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.56922	0.249814
$498$	0	0
$499$	39.5692i	1.77136i	0.464295	+	0.885681i	$0.346308\pi$
$499$	39.5692i	1.77136i	−0.464295	+	0.885681i	$0.653692\pi$
$500$	0	0
$501$	− 6.00000i	− 0.268060i
$502$	0	0
$503$	−8.19615	−0.365448	−0.182724	−	0.983164i	$-0.558492\pi$
$503$	−8.19615	−0.365448	−0.182724	+	0.983164i	$0.558492\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.732051i	− 0.0325115i
$508$	0	0
$509$	− 8.78461i	− 0.389371i	−0.980866	−	0.194685i	$-0.937631\pi$
$509$	− 8.78461i	− 0.389371i	0.980866	−	0.194685i	$-0.0623686\pi$
$510$	0	0
$511$	−18.2487	−0.807275
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	0	0
$516$	0	0
$517$	28.3923i	1.24869i
$518$	0	0
$519$	4.39230	0.192801
$520$	0	0
$521$	31.8564	1.39565	0.697827	−	0.716266i	$-0.254150\pi$
$521$	31.8564	1.39565	0.697827	+	0.716266i	$0.254150\pi$
$522$	0	0
$523$	− 14.9808i	− 0.655063i	−0.944840	−	0.327531i	$-0.893783\pi$
$523$	− 14.9808i	− 0.655063i	0.944840	−	0.327531i	$-0.106217\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.7846	1.42812
$528$	0	0
$529$	44.1769	1.92074
$530$	0	0
$531$	− 14.7846i	− 0.641597i
$532$	0	0
$533$	8.78461i	0.380504i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.5359	0.627270
$538$	0	0
$539$	− 18.6795i	− 0.804583i
$540$	0	0
$541$	− 15.7128i	− 0.675547i	−0.941227	−	0.337773i	$-0.890326\pi$
$541$	− 15.7128i	− 0.675547i	0.941227	−	0.337773i	$-0.109674\pi$
$542$	0	0
$543$	−5.07180	−0.217652
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.80385i	0.0771270i	0.999256	+	0.0385635i	$0.0122782\pi$
$547$	1.80385i	0.0771270i	−0.999256	+	0.0385635i	$0.987722\pi$
$548$	0	0
$549$	31.8564i	1.35960i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	15.2154	0.647024
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.7846i	− 0.626444i	−0.949680	−	0.313222i	$-0.898592\pi$
$557$	− 14.7846i	− 0.626444i	0.949680	−	0.313222i	$-0.101408\pi$
$558$	0	0
$559$	−35.3205	−1.49390
$560$	0	0
$561$	−8.78461	−0.370887
$562$	0	0
$563$	26.1962i	1.10404i	0.833832	+	0.552018i	$0.186142\pi$
$563$	26.1962i	1.10404i	−0.833832	+	0.552018i	$0.813858\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5.66025	−0.237708
$568$	0	0
$569$	−42.2487	−1.77116	−0.885579	−	0.464489i	$-0.846238\pi$
$569$	−42.2487	−1.77116	−0.885579	+	0.464489i	$0.846238\pi$
$570$	0	0
$571$	30.3923i	1.27188i	0.771739	+	0.635939i	$0.219387\pi$
$571$	30.3923i	1.27188i	−0.771739	+	0.635939i	$0.780613\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	4.67949i	0.194473i
$580$	0	0
$581$	− 6.00000i	− 0.248922i
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.5167i	0.557892i	0.960307	+	0.278946i	$0.0899851\pi$
$587$	13.5167i	0.557892i	−0.960307	+	0.278946i	$0.910015\pi$
$588$	0	0
$589$	− 18.9282i	− 0.779923i
$590$	0	0
$591$	−7.60770	−0.312939
$592$	0	0
$593$	0.928203	0.0381167	0.0190584	−	0.999818i	$-0.493933\pi$
$593$	0.928203	0.0381167	0.0190584	+	0.999818i	$0.493933\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.07180i	0.207575i
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−20.3923	−0.831819	−0.415910	−	0.909406i	$-0.636537\pi$
$601$	−20.3923	−0.831819	−0.415910	+	0.909406i	$0.636537\pi$
$602$	0	0
$603$	25.1244i	1.02314i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	8.19615	0.332672	0.166336	−	0.986069i	$-0.446806\pi$
$607$	8.19615	0.332672	0.166336	+	0.986069i	$0.446806\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	28.3923i	1.14863i
$612$	0	0
$613$	13.6077i	0.549610i	0.961500	+	0.274805i	$0.0886132\pi$
$613$	13.6077i	0.549610i	−0.961500	+	0.274805i	$0.911387\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.6077	−0.547825	−0.273913	−	0.961755i	$-0.588318\pi$
$617$	−13.6077	−0.547825	−0.273913	+	0.961755i	$0.588318\pi$
$618$	0	0
$619$	− 6.78461i	− 0.272696i	−0.990661	−	0.136348i	$-0.956463\pi$
$619$	− 6.78461i	− 0.272696i	0.990661	−	0.136348i	$-0.0435366\pi$
$620$	0	0
$621$	32.7846i	1.31560i
$622$	0	0
$623$	−1.17691	−0.0471521
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.07180i	0.202548i
$628$	0	0
$629$	20.7846i	0.828737i
$630$	0	0
$631$	21.4641	0.854472	0.427236	−	0.904140i	$-0.359487\pi$
$631$	21.4641	0.854472	0.427236	+	0.904140i	$0.359487\pi$
$632$	0	0
$633$	−4.67949	−0.185993
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.6795i	− 0.740108i
$638$	0	0
$639$	−10.8231	−0.428155
$640$	0	0
$641$	−4.39230	−0.173486	−0.0867428	−	0.996231i	$-0.527646\pi$
$641$	−4.39230	−0.173486	−0.0867428	+	0.996231i	$0.527646\pi$
$642$	0	0
$643$	10.5885i	0.417568i	0.977962	+	0.208784i	$0.0669506\pi$
$643$	10.5885i	0.417568i	−0.977962	+	0.208784i	$0.933049\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.5885	1.43844	0.719220	−	0.694782i	$-0.244499\pi$
$647$	36.5885	1.43844	0.719220	+	0.694782i	$0.244499\pi$
$648$	0	0
$649$	20.7846	0.815867
$650$	0	0
$651$	− 8.78461i	− 0.344296i
$652$	0	0
$653$	19.1769i	0.750451i	0.926934	+	0.375225i	$0.122435\pi$
$653$	19.1769i	0.750451i	−0.926934	+	0.375225i	$0.877565\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	35.4641	1.38359
$658$	0	0
$659$	− 40.6410i	− 1.58315i	−0.611073	−	0.791575i	$-0.709262\pi$
$659$	− 40.6410i	− 1.58315i	0.611073	−	0.791575i	$-0.290738\pi$
$660$	0	0
$661$	35.5692i	1.38348i	0.722146	+	0.691741i	$0.243156\pi$
$661$	35.5692i	1.38348i	−0.722146	+	0.691741i	$0.756844\pi$
$662$	0	0
$663$	−8.78461	−0.341166
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 11.0718i	− 0.428060i
$670$	0	0
$671$	−44.7846	−1.72889
$672$	0	0
$673$	39.1769	1.51016	0.755080	−	0.655633i	$-0.227598\pi$
$673$	39.1769	1.51016	0.755080	+	0.655633i	$0.227598\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 43.1769i	− 1.65942i	−0.558192	−	0.829712i	$-0.688505\pi$
$677$	− 43.1769i	− 1.65942i	0.558192	−	0.829712i	$-0.311495\pi$
$678$	0	0
$679$	−8.10512	−0.311046
$680$	0	0
$681$	13.6077	0.521448
$682$	0	0
$683$	− 11.6603i	− 0.446167i	−0.974799	−	0.223084i	$-0.928388\pi$
$683$	− 11.6603i	− 0.446167i	0.974799	−	0.223084i	$-0.0716123\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.8564	0.528655
$688$	0	0
$689$	−36.0000	−1.37149
$690$	0	0
$691$	− 5.60770i	− 0.213327i	−0.994295	−	0.106663i	$-0.965983\pi$
$691$	− 5.60770i	− 0.213327i	0.994295	−	0.106663i	$-0.0340167\pi$
$692$	0	0
$693$	− 10.8231i	− 0.411135i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−8.78461	−0.332741
$698$	0	0
$699$	− 1.17691i	− 0.0445150i
$700$	0	0
$701$	14.7846i	0.558407i	0.960232	+	0.279204i	$0.0900704\pi$
$701$	14.7846i	0.558407i	−0.960232	+	0.279204i	$0.909930\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 15.2154i	− 0.572234i
$708$	0	0
$709$	− 10.1436i	− 0.380951i	−0.981692	−	0.190475i	$-0.938997\pi$
$709$	− 10.1436i	− 0.380951i	0.981692	−	0.190475i	$-0.0610029\pi$
$710$	0	0
$711$	−29.5692	−1.10893
$712$	0	0
$713$	77.5692	2.90499
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 15.2154i	− 0.568229i
$718$	0	0
$719$	44.7846	1.67018	0.835092	−	0.550110i	$-0.185414\pi$
$719$	44.7846	1.67018	0.835092	+	0.550110i	$0.185414\pi$
$720$	0	0
$721$	10.3923	0.387030
$722$	0	0
$723$	14.9282i	0.555186i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.0526	−1.26294	−0.631470	−	0.775401i	$-0.717548\pi$
$727$	−34.0526	−1.26294	−0.631470	+	0.775401i	$0.717548\pi$
$728$	0	0
$729$	2.21539	0.0820515
$730$	0	0
$731$	− 35.3205i	− 1.30638i
$732$	0	0
$733$	38.7846i	1.43254i	0.697822	+	0.716271i	$0.254153\pi$
$733$	38.7846i	1.43254i	−0.697822	+	0.716271i	$0.745847\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35.3205	−1.30105
$738$	0	0
$739$	− 2.00000i	− 0.0735712i	−0.999323	−	0.0367856i	$-0.988288\pi$
$739$	− 2.00000i	− 0.0735712i	0.999323	−	0.0367856i	$-0.0117119\pi$
$740$	0	0
$741$	5.07180i	0.186317i
$742$	0	0
$743$	−36.5885	−1.34230	−0.671150	−	0.741321i	$-0.734199\pi$
$743$	−36.5885	−1.34230	−0.671150	+	0.741321i	$0.734199\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.6603i	0.426626i
$748$	0	0
$749$	21.2154i	0.775193i
$750$	0	0
$751$	−40.3923	−1.47394	−0.736968	−	0.675928i	$-0.763743\pi$
$751$	−40.3923	−1.47394	−0.736968	+	0.675928i	$0.763743\pi$
$752$	0	0
$753$	−11.3205	−0.412542
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 23.0718i	− 0.838559i	−0.907857	−	0.419279i	$-0.862283\pi$
$757$	− 23.0718i	− 0.838559i	0.907857	−	0.419279i	$-0.137717\pi$
$758$	0	0
$759$	−20.7846	−0.754434
$760$	0	0
$761$	−21.7128	−0.787089	−0.393544	−	0.919306i	$-0.628751\pi$
$761$	−21.7128	−0.787089	−0.393544	+	0.919306i	$0.628751\pi$
$762$	0	0
$763$	− 1.17691i	− 0.0426072i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.7846	0.750489
$768$	0	0
$769$	−6.78461	−0.244659	−0.122330	−	0.992490i	$-0.539037\pi$
$769$	−6.78461	−0.244659	−0.122330	+	0.992490i	$0.539037\pi$
$770$	0	0
$771$	4.39230i	0.158185i
$772$	0	0
$773$	46.3923i	1.66862i	0.551299	+	0.834308i	$0.314132\pi$
$773$	46.3923i	1.66862i	−0.551299	+	0.834308i	$0.685868\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.56922	0.199795
$778$	0	0
$779$	5.07180i	0.181716i
$780$	0	0
$781$	− 15.2154i	− 0.544449i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 5.80385i	− 0.206885i	−0.994635	−	0.103442i	$-0.967014\pi$
$787$	− 5.80385i	− 0.206885i	0.994635	−	0.103442i	$-0.0329857\pi$
$788$	0	0
$789$	− 14.7846i	− 0.526346i
$790$	0	0
$791$	1.17691	0.0418463
$792$	0	0
$793$	−44.7846	−1.59035
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.60770i	0.0569475i	0.999595	+	0.0284737i	$0.00906470\pi$
$797$	1.60770i	0.0569475i	−0.999595	+	0.0284737i	$0.990935\pi$
$798$	0	0
$799$	−28.3923	−1.00445
$800$	0	0
$801$	2.28719	0.0808138
$802$	0	0
$803$	49.8564i	1.75939i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.8231	−0.380991
$808$	0	0
$809$	35.5692	1.25055	0.625274	−	0.780406i	$-0.284987\pi$
$809$	35.5692	1.25055	0.625274	+	0.780406i	$0.284987\pi$
$810$	0	0
$811$	38.3923i	1.34814i	0.738669	+	0.674068i	$0.235455\pi$
$811$	38.3923i	1.34814i	−0.738669	+	0.674068i	$0.764545\pi$
$812$	0	0
$813$	− 3.21539i	− 0.112769i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−20.3923	−0.713436
$818$	0	0
$819$	− 10.8231i	− 0.378189i
$820$	0	0
$821$	50.7846i	1.77240i	0.463308	+	0.886198i	$0.346663\pi$
$821$	50.7846i	1.77240i	−0.463308	+	0.886198i	$0.653337\pi$
$822$	0	0
$823$	−30.8372	−1.07492	−0.537458	−	0.843290i	$-0.680615\pi$
$823$	−30.8372	−1.07492	−0.537458	+	0.843290i	$0.680615\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.73205i	0.164550i	0.996610	+	0.0822748i	$0.0262185\pi$
$827$	4.73205i	0.164550i	−0.996610	+	0.0822748i	$0.973781\pi$
$828$	0	0
$829$	50.7846i	1.76382i	0.471416	+	0.881911i	$0.343743\pi$
$829$	50.7846i	1.76382i	−0.471416	+	0.881911i	$0.656257\pi$
$830$	0	0
$831$	14.5359	0.504245
$832$	0	0
$833$	18.6795	0.647206
$834$	0	0
$835$	0	0
$836$	0	0
$837$	37.8564i	1.30851i
$838$	0	0
$839$	8.78461	0.303278	0.151639	−	0.988436i	$-0.451545\pi$
$839$	8.78461	0.303278	0.151639	+	0.988436i	$0.451545\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	20.7846i	0.715860i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.26795	0.0435672
$848$	0	0
$849$	7.75129	0.266024
$850$	0	0
$851$	49.1769i	1.68576i
$852$	0	0
$853$	3.46410i	0.118609i	0.998240	+	0.0593043i	$0.0188882\pi$
$853$	3.46410i	0.118609i	−0.998240	+	0.0593043i	$0.981112\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.1436	0.551455	0.275727	−	0.961236i	$-0.411081\pi$
$857$	16.1436	0.551455	0.275727	+	0.961236i	$0.411081\pi$
$858$	0	0
$859$	51.5692i	1.75952i	0.475419	+	0.879760i	$0.342296\pi$
$859$	51.5692i	1.75952i	−0.475419	+	0.879760i	$0.657704\pi$
$860$	0	0
$861$	2.35383i	0.0802183i
$862$	0	0
$863$	−40.9808	−1.39500	−0.697501	−	0.716584i	$-0.745705\pi$
$863$	−40.9808	−1.39500	−0.697501	+	0.716584i	$0.745705\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.66025i	0.124309i
$868$	0	0
$869$	− 41.5692i	− 1.41014i
$870$	0	0
$871$	−35.3205	−1.19679
$872$	0	0
$873$	15.7513	0.533100
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.85641i	− 0.265292i	−0.991163	−	0.132646i	$-0.957653\pi$
$877$	− 7.85641i	− 0.265292i	0.991163	−	0.132646i	$-0.0423473\pi$
$878$	0	0
$879$	−21.9615	−0.740744
$880$	0	0
$881$	−7.60770	−0.256310	−0.128155	−	0.991754i	$-0.540905\pi$
$881$	−7.60770	−0.256310	−0.128155	+	0.991754i	$0.540905\pi$
$882$	0	0
$883$	− 5.80385i	− 0.195315i	−0.995220	−	0.0976575i	$-0.968865\pi$
$883$	− 5.80385i	− 0.195315i	0.995220	−	0.0976575i	$-0.0311350\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.3731	0.717637	0.358819	−	0.933407i	$-0.383180\pi$
$887$	21.3731	0.717637	0.358819	+	0.933407i	$0.383180\pi$
$888$	0	0
$889$	−4.82309	−0.161761
$890$	0	0
$891$	15.4641i	0.518067i
$892$	0	0
$893$	16.3923i	0.548548i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−20.7846	−0.693978
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 36.0000i	− 1.19933i
$902$	0	0
$903$	−9.46410	−0.314946
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.41154i	0.0468695i	0.999725	+	0.0234348i	$0.00746020\pi$
$907$	1.41154i	0.0468695i	−0.999725	+	0.0234348i	$0.992540\pi$
$908$	0	0
$909$	29.5692i	0.980749i
$910$	0	0
$911$	−37.1769	−1.23173	−0.615863	−	0.787853i	$-0.711193\pi$
$911$	−37.1769	−1.23173	−0.615863	+	0.787853i	$0.711193\pi$
$912$	0	0
$913$	−16.3923	−0.542506
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 13.1769i	− 0.435140i
$918$	0	0
$919$	39.7128	1.31000	0.655002	−	0.755627i	$-0.272668\pi$
$919$	39.7128	1.31000	0.655002	+	0.755627i	$0.272668\pi$
$920$	0	0
$921$	25.0333	0.824876
$922$	0	0
$923$	− 15.2154i	− 0.500821i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−20.1962	−0.663329
$928$	0	0
$929$	7.60770	0.249600	0.124800	−	0.992182i	$-0.460171\pi$
$929$	7.60770	0.249600	0.124800	+	0.992182i	$0.460171\pi$
$930$	0	0
$931$	− 10.7846i	− 0.353451i
$932$	0	0
$933$	5.56922i	0.182328i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.60770	0.313870	0.156935	−	0.987609i	$-0.449839\pi$
$937$	9.60770	0.313870	0.156935	+	0.987609i	$0.449839\pi$
$938$	0	0
$939$	16.1051i	0.525571i
$940$	0	0
$941$	− 20.7846i	− 0.677559i	−0.940866	−	0.338779i	$-0.889986\pi$
$941$	− 20.7846i	− 0.677559i	0.940866	−	0.338779i	$-0.110014\pi$
$942$	0	0
$943$	−20.7846	−0.676840
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 45.8038i	− 1.48843i	−0.667943	−	0.744213i	$-0.732825\pi$
$947$	− 45.8038i	− 1.48843i	0.667943	−	0.744213i	$-0.267175\pi$
$948$	0	0
$949$	49.8564i	1.61841i
$950$	0	0
$951$	−16.3923	−0.531557
$952$	0	0
$953$	−24.9282	−0.807504	−0.403752	−	0.914869i	$-0.632294\pi$
$953$	−24.9282	−0.807504	−0.403752	+	0.914869i	$0.632294\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16.3923	0.529335
$960$	0	0
$961$	58.5692	1.88933
$962$	0	0
$963$	− 41.2295i	− 1.32860i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.19615	0.263570	0.131785	−	0.991278i	$-0.457929\pi$
$967$	8.19615	0.263570	0.131785	+	0.991278i	$0.457929\pi$
$968$	0	0
$969$	−5.07180	−0.162930
$970$	0	0
$971$	33.0333i	1.06009i	0.847970	+	0.530045i	$0.177825\pi$
$971$	33.0333i	1.06009i	−0.847970	+	0.530045i	$0.822175\pi$
$972$	0	0
$973$	12.6795i	0.406486i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.32051	−0.170218	−0.0851091	−	0.996372i	$-0.527124\pi$
$977$	−5.32051	−0.170218	−0.0851091	+	0.996372i	$0.527124\pi$
$978$	0	0
$979$	3.21539i	0.102764i
$980$	0	0
$981$	2.28719i	0.0730243i
$982$	0	0
$983$	11.4115	0.363972	0.181986	−	0.983301i	$-0.441747\pi$
$983$	11.4115	0.363972	0.181986	+	0.983301i	$0.441747\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	7.60770i	0.242156i
$988$	0	0
$989$	− 83.5692i	− 2.65735i
$990$	0	0
$991$	44.1051	1.40105	0.700523	−	0.713630i	$-0.252950\pi$
$991$	44.1051	1.40105	0.700523	+	0.713630i	$0.252950\pi$
$992$	0	0
$993$	−4.10512	−0.130272
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.6077i	0.811004i	0.914094	+	0.405502i	$0.132903\pi$
$997$	25.6077i	0.811004i	−0.914094	+	0.405502i	$0.867097\pi$
$998$	0	0
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.d.b.801.2		4
4.3	odd	2		1600.2.d.h.801.3		4
5.2	odd	4		1600.2.f.h.1249.1		4
5.3	odd	4		1600.2.f.d.1249.4		4
5.4	even	2		320.2.d.b.161.3	yes	4
8.3	odd	2		1600.2.d.h.801.2		4
8.5	even	2	inner	1600.2.d.b.801.3		4
15.14	odd	2		2880.2.k.l.1441.3		4
16.3	odd	4		6400.2.a.cd.1.1		2
16.5	even	4		6400.2.a.ck.1.1		2
16.11	odd	4		6400.2.a.y.1.2		2
16.13	even	4		6400.2.a.bf.1.2		2
20.3	even	4		1600.2.f.i.1249.1		4
20.7	even	4		1600.2.f.e.1249.4		4
20.19	odd	2		320.2.d.a.161.2	✓	4
40.3	even	4		1600.2.f.e.1249.3		4
40.13	odd	4		1600.2.f.h.1249.2		4
40.19	odd	2		320.2.d.a.161.3	yes	4
40.27	even	4		1600.2.f.i.1249.2		4
40.29	even	2		320.2.d.b.161.2	yes	4
40.37	odd	4		1600.2.f.d.1249.3		4
60.59	even	2		2880.2.k.e.1441.4		4
80.19	odd	4		1280.2.a.b.1.2		2
80.29	even	4		1280.2.a.m.1.1		2
80.59	odd	4		1280.2.a.p.1.1		2
80.69	even	4		1280.2.a.c.1.2		2
120.29	odd	2		2880.2.k.l.1441.1		4
120.59	even	2		2880.2.k.e.1441.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.d.a.161.2	✓	4	20.19	odd	2
320.2.d.a.161.3	yes	4	40.19	odd	2
320.2.d.b.161.2	yes	4	40.29	even	2
320.2.d.b.161.3	yes	4	5.4	even	2
1280.2.a.b.1.2		2	80.19	odd	4
1280.2.a.c.1.2		2	80.69	even	4
1280.2.a.m.1.1		2	80.29	even	4
1280.2.a.p.1.1		2	80.59	odd	4
1600.2.d.b.801.2		4	1.1	even	1	trivial
1600.2.d.b.801.3		4	8.5	even	2	inner
1600.2.d.h.801.2		4	8.3	odd	2
1600.2.d.h.801.3		4	4.3	odd	2
1600.2.f.d.1249.3		4	40.37	odd	4
1600.2.f.d.1249.4		4	5.3	odd	4
1600.2.f.e.1249.3		4	40.3	even	4
1600.2.f.e.1249.4		4	20.7	even	4
1600.2.f.h.1249.1		4	5.2	odd	4
1600.2.f.h.1249.2		4	40.13	odd	4
1600.2.f.i.1249.1		4	20.3	even	4
1600.2.f.i.1249.2		4	40.27	even	4
2880.2.k.e.1441.2		4	120.59	even	2
2880.2.k.e.1441.4		4	60.59	even	2
2880.2.k.l.1441.1		4	120.29	odd	2
2880.2.k.l.1441.3		4	15.14	odd	2
6400.2.a.y.1.2		2	16.11	odd	4
6400.2.a.bf.1.2		2	16.13	even	4
6400.2.a.cd.1.1		2	16.3	odd	4
6400.2.a.ck.1.1		2	16.5	even	4

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.732051i q^{3} -1.26795 q^{7} +2.46410 q^{9} +3.46410i q^{11} +3.46410i q^{13} -3.46410 q^{17} +2.00000i q^{19} +0.928203i q^{21} -8.19615 q^{23} -4.00000i q^{27} -9.46410 q^{31} +2.53590 q^{33} -6.00000i q^{37} +2.53590 q^{39} +2.53590 q^{41} +10.1962i q^{43} +8.19615 q^{47} -5.39230 q^{49} +2.53590i q^{51} +10.3923i q^{53} +1.46410 q^{57} -6.00000i q^{59} +12.9282i q^{61} -3.12436 q^{63} +10.1962i q^{67} +6.00000i q^{69} -4.39230 q^{71} +14.3923 q^{73} -4.39230i q^{77} -12.0000 q^{79} +4.46410 q^{81} +4.73205i q^{83} +0.928203 q^{89} -4.39230i q^{91} +6.92820i q^{93} +6.39230 q^{97} +8.53590i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{7} - 4 q^{9} - 12 q^{23} - 24 q^{31} + 24 q^{33} + 24 q^{39} + 24 q^{41} + 12 q^{47} + 20 q^{49} - 8 q^{57} + 36 q^{63} + 24 q^{71} + 16 q^{73} - 48 q^{79} + 4 q^{81} - 24 q^{89} - 16 q^{97}+O(q^{100}) \) 4 * q - 12 * q^7 - 4 * q^9 - 12 * q^23 - 24 * q^31 + 24 * q^33 + 24 * q^39 + 24 * q^41 + 12 * q^47 + 20 * q^49 - 8 * q^57 + 36 * q^63 + 24 * q^71 + 16 * q^73 - 48 * q^79 + 4 * q^81 - 24 * q^89 - 16 * q^97

Introduction

L-functions

Modular forms

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