LMFDB - Embedded newform 3328.2.a.u.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1664)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{3} -1.00000 q^{5} -2.23607 q^{7} +2.00000 q^{9} +4.47214 q^{11} +1.00000 q^{13} +2.23607 q^{15} -7.00000 q^{17} +5.00000 q^{21} +4.47214 q^{23} -4.00000 q^{25} +2.23607 q^{27} -6.00000 q^{29} +8.94427 q^{31} -10.0000 q^{33} +2.23607 q^{35} +3.00000 q^{37} -2.23607 q^{39} +2.23607 q^{43} -2.00000 q^{45} -2.23607 q^{47} -2.00000 q^{49} +15.6525 q^{51} +14.0000 q^{53} -4.47214 q^{55} +8.94427 q^{59} +8.00000 q^{61} -4.47214 q^{63} -1.00000 q^{65} -4.47214 q^{67} -10.0000 q^{69} -6.70820 q^{71} +4.00000 q^{73} +8.94427 q^{75} -10.0000 q^{77} -4.47214 q^{79} -11.0000 q^{81} +4.47214 q^{83} +7.00000 q^{85} +13.4164 q^{87} -2.23607 q^{91} -20.0000 q^{93} +2.00000 q^{97} +8.94427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{9} + 2 q^{13} - 14 q^{17} + 10 q^{21} - 8 q^{25} - 12 q^{29} - 20 q^{33} + 6 q^{37} - 4 q^{45} - 4 q^{49} + 28 q^{53} + 16 q^{61} - 2 q^{65} - 20 q^{69} + 8 q^{73} - 20 q^{77} - 22 q^{81}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^9 + 2 * q^13 - 14 * q^17 + 10 * q^21 - 8 * q^25 - 12 * q^29 - 20 * q^33 + 6 * q^37 - 4 * q^45 - 4 * q^49 + 28 * q^53 + 16 * q^61 - 2 * q^65 - 20 * q^69 + 8 * q^73 - 20 * q^77 - 22 * q^81 + 14 * q^85 - 40 * q^93 + 4 * q^97

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
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−2.23607	−0.845154	−0.422577	−	0.906327i	$-0.638874\pi$
$7$	−2.23607	−0.845154	−0.422577	+	0.906327i	$0.638874\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.23607	0.577350
$16$	0	0
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	0	0
$23$	4.47214	0.932505	0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214	0.932505	0.466252	+	0.884652i	$0.345604\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	8.94427	1.60644	0.803219	−	0.595683i	$-0.203119\pi$
$31$	8.94427	1.60644	0.803219	+	0.595683i	$0.203119\pi$
$32$	0	0
$33$	−10.0000	−1.74078
$34$	0	0
$35$	2.23607	0.377964
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	−2.23607	−0.358057
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.23607	0.340997	0.170499	−	0.985358i	$-0.445462\pi$
$43$	2.23607	0.340997	0.170499	+	0.985358i	$0.445462\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−2.23607	−0.326164	−0.163082	−	0.986613i	$-0.552144\pi$
$47$	−2.23607	−0.326164	−0.163082	+	0.986613i	$0.552144\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	15.6525	2.19179
$52$	0	0
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	0	0
$55$	−4.47214	−0.603023
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	−4.47214	−0.563436
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−4.47214	−0.546358	−0.273179	−	0.961963i	$-0.588075\pi$
$67$	−4.47214	−0.546358	−0.273179	+	0.961963i	$0.588075\pi$
$68$	0	0
$69$	−10.0000	−1.20386
$70$	0	0
$71$	−6.70820	−0.796117	−0.398059	−	0.917360i	$-0.630316\pi$
$71$	−6.70820	−0.796117	−0.398059	+	0.917360i	$0.630316\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	8.94427	1.03280
$76$	0	0
$77$	−10.0000	−1.13961
$78$	0	0
$79$	−4.47214	−0.503155	−0.251577	−	0.967837i	$-0.580949\pi$
$79$	−4.47214	−0.503155	−0.251577	+	0.967837i	$0.580949\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	4.47214	0.490881	0.245440	−	0.969412i	$-0.421067\pi$
$83$	4.47214	0.490881	0.245440	+	0.969412i	$0.421067\pi$
$84$	0	0
$85$	7.00000	0.759257
$86$	0	0
$87$	13.4164	1.43839
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−2.23607	−0.234404
$92$	0	0
$93$	−20.0000	−2.07390
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	8.94427	0.898933
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−13.4164	−1.32196	−0.660979	−	0.750404i	$-0.729859\pi$
$103$	−13.4164	−1.32196	−0.660979	+	0.750404i	$0.729859\pi$
$104$	0	0
$105$	−5.00000	−0.487950
$106$	0	0
$107$	−17.8885	−1.72935	−0.864675	−	0.502331i	$-0.832476\pi$
$107$	−17.8885	−1.72935	−0.864675	+	0.502331i	$0.832476\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	−6.70820	−0.636715
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	−4.47214	−0.417029
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	15.6525	1.43486
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−13.4164	−1.19051	−0.595257	−	0.803535i	$-0.702950\pi$
$127$	−13.4164	−1.19051	−0.595257	+	0.803535i	$0.702950\pi$
$128$	0	0
$129$	−5.00000	−0.440225
$130$	0	0
$131$	2.23607	0.195366	0.0976831	−	0.995218i	$-0.468857\pi$
$131$	2.23607	0.195366	0.0976831	+	0.995218i	$0.468857\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.23607	−0.192450
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−6.70820	−0.568982	−0.284491	−	0.958679i	$-0.591825\pi$
$139$	−6.70820	−0.568982	−0.284491	+	0.958679i	$0.591825\pi$
$140$	0	0
$141$	5.00000	0.421076
$142$	0	0
$143$	4.47214	0.373979
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	4.47214	0.368856
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	20.1246	1.63772	0.818859	−	0.573995i	$-0.194607\pi$
$151$	20.1246	1.63772	0.818859	+	0.573995i	$0.194607\pi$
$152$	0	0
$153$	−14.0000	−1.13183
$154$	0	0
$155$	−8.94427	−0.718421
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	0	0
$159$	−31.3050	−2.48264
$160$	0	0
$161$	−10.0000	−0.788110
$162$	0	0
$163$	−13.4164	−1.05085	−0.525427	−	0.850839i	$-0.676094\pi$
$163$	−13.4164	−1.05085	−0.525427	+	0.850839i	$0.676094\pi$
$164$	0	0
$165$	10.0000	0.778499
$166$	0	0
$167$	8.94427	0.692129	0.346064	−	0.938211i	$-0.387518\pi$
$167$	8.94427	0.692129	0.346064	+	0.938211i	$0.387518\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	8.94427	0.676123
$176$	0	0
$177$	−20.0000	−1.50329
$178$	0	0
$179$	−24.5967	−1.83845	−0.919224	−	0.393736i	$-0.871182\pi$
$179$	−24.5967	−1.83845	−0.919224	+	0.393736i	$0.871182\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−17.8885	−1.32236
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	−31.3050	−2.28924
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	22.3607	1.61796	0.808981	−	0.587835i	$-0.200019\pi$
$191$	22.3607	1.61796	0.808981	+	0.587835i	$0.200019\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	0	0
$195$	2.23607	0.160128
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	−22.3607	−1.58511	−0.792553	−	0.609803i	$-0.791249\pi$
$199$	−22.3607	−1.58511	−0.792553	+	0.609803i	$0.791249\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	13.4164	0.941647
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.94427	0.621670
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−6.70820	−0.461812	−0.230906	−	0.972976i	$-0.574169\pi$
$211$	−6.70820	−0.461812	−0.230906	+	0.972976i	$0.574169\pi$
$212$	0	0
$213$	15.0000	1.02778
$214$	0	0
$215$	−2.23607	−0.152499
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	−8.94427	−0.604398
$220$	0	0
$221$	−7.00000	−0.470871
$222$	0	0
$223$	20.1246	1.34764	0.673822	−	0.738894i	$-0.264652\pi$
$223$	20.1246	1.34764	0.673822	+	0.738894i	$0.264652\pi$
$224$	0	0
$225$	−8.00000	−0.533333
$226$	0	0
$227$	−8.94427	−0.593652	−0.296826	−	0.954932i	$-0.595928\pi$
$227$	−8.94427	−0.593652	−0.296826	+	0.954932i	$0.595928\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	0	0
$231$	22.3607	1.47122
$232$	0	0
$233$	−19.0000	−1.24473	−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000	−1.24473	−0.622366	+	0.782727i	$0.713828\pi$
$234$	0	0
$235$	2.23607	0.145865
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	24.5967	1.59103	0.795516	−	0.605933i	$-0.207200\pi$
$239$	24.5967	1.59103	0.795516	+	0.605933i	$0.207200\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	17.8885	1.14755
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	17.8885	1.12911	0.564557	−	0.825394i	$-0.309047\pi$
$251$	17.8885	1.12911	0.564557	+	0.825394i	$0.309047\pi$
$252$	0	0
$253$	20.0000	1.25739
$254$	0	0
$255$	−15.6525	−0.980196
$256$	0	0
$257$	17.0000	1.06043	0.530215	−	0.847863i	$-0.322111\pi$
$257$	17.0000	1.06043	0.530215	+	0.847863i	$0.322111\pi$
$258$	0	0
$259$	−6.70820	−0.416828
$260$	0	0
$261$	−12.0000	−0.742781
$262$	0	0
$263$	−26.8328	−1.65458	−0.827291	−	0.561773i	$-0.810119\pi$
$263$	−26.8328	−1.65458	−0.827291	+	0.561773i	$0.810119\pi$
$264$	0	0
$265$	−14.0000	−0.860013
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	15.6525	0.950820	0.475410	−	0.879764i	$-0.342300\pi$
$271$	15.6525	0.950820	0.475410	+	0.879764i	$0.342300\pi$
$272$	0	0
$273$	5.00000	0.302614
$274$	0	0
$275$	−17.8885	−1.07872
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	17.8885	1.07096
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	8.94427	0.531682	0.265841	−	0.964017i	$-0.414350\pi$
$283$	8.94427	0.531682	0.265841	+	0.964017i	$0.414350\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	−4.47214	−0.262161
$292$	0	0
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	0	0
$295$	−8.94427	−0.520756
$296$	0	0
$297$	10.0000	0.580259
$298$	0	0
$299$	4.47214	0.258630
$300$	0	0
$301$	−5.00000	−0.288195
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−26.8328	−1.53143	−0.765715	−	0.643180i	$-0.777615\pi$
$307$	−26.8328	−1.53143	−0.765715	+	0.643180i	$0.777615\pi$
$308$	0	0
$309$	30.0000	1.70664
$310$	0	0
$311$	−17.8885	−1.01437	−0.507183	−	0.861838i	$-0.669313\pi$
$311$	−17.8885	−1.01437	−0.507183	+	0.861838i	$0.669313\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	4.47214	0.251976
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	−26.8328	−1.50235
$320$	0	0
$321$	40.0000	2.23258
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	11.1803	0.618274
$328$	0	0
$329$	5.00000	0.275659
$330$	0	0
$331$	−4.47214	−0.245811	−0.122905	−	0.992418i	$-0.539221\pi$
$331$	−4.47214	−0.245811	−0.122905	+	0.992418i	$0.539221\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	4.47214	0.244339
$336$	0	0
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	0	0
$339$	31.3050	1.70025
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	20.1246	1.08663
$344$	0	0
$345$	10.0000	0.538382
$346$	0	0
$347$	−15.6525	−0.840269	−0.420134	−	0.907462i	$-0.638017\pi$
$347$	−15.6525	−0.840269	−0.420134	+	0.907462i	$0.638017\pi$
$348$	0	0
$349$	31.0000	1.65939	0.829696	−	0.558216i	$-0.188514\pi$
$349$	31.0000	1.65939	0.829696	+	0.558216i	$0.188514\pi$
$350$	0	0
$351$	2.23607	0.119352
$352$	0	0
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	6.70820	0.356034
$356$	0	0
$357$	−35.0000	−1.85240
$358$	0	0
$359$	8.94427	0.472061	0.236030	−	0.971746i	$-0.424154\pi$
$359$	8.94427	0.472061	0.236030	+	0.971746i	$0.424154\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−20.1246	−1.05627
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	4.47214	0.233444	0.116722	−	0.993165i	$-0.462761\pi$
$367$	4.47214	0.233444	0.116722	+	0.993165i	$0.462761\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−31.3050	−1.62527
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	−20.1246	−1.03923
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	31.3050	1.60803	0.804014	−	0.594611i	$-0.202694\pi$
$379$	31.3050	1.60803	0.804014	+	0.594611i	$0.202694\pi$
$380$	0	0
$381$	30.0000	1.53695
$382$	0	0
$383$	−24.5967	−1.25684	−0.628418	−	0.777876i	$-0.716297\pi$
$383$	−24.5967	−1.25684	−0.628418	+	0.777876i	$0.716297\pi$
$384$	0	0
$385$	10.0000	0.509647
$386$	0	0
$387$	4.47214	0.227331
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−31.3050	−1.58316
$392$	0	0
$393$	−5.00000	−0.252217
$394$	0	0
$395$	4.47214	0.225018
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	8.94427	0.445546
$404$	0	0
$405$	11.0000	0.546594
$406$	0	0
$407$	13.4164	0.665027
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	40.2492	1.98535
$412$	0	0
$413$	−20.0000	−0.984136
$414$	0	0
$415$	−4.47214	−0.219529
$416$	0	0
$417$	15.0000	0.734553
$418$	0	0
$419$	−2.23607	−0.109239	−0.0546195	−	0.998507i	$-0.517395\pi$
$419$	−2.23607	−0.109239	−0.0546195	+	0.998507i	$0.517395\pi$
$420$	0	0
$421$	−33.0000	−1.60832	−0.804161	−	0.594412i	$-0.797385\pi$
$421$	−33.0000	−1.60832	−0.804161	+	0.594412i	$0.797385\pi$
$422$	0	0
$423$	−4.47214	−0.217443
$424$	0	0
$425$	28.0000	1.35820
$426$	0	0
$427$	−17.8885	−0.865687
$428$	0	0
$429$	−10.0000	−0.482805
$430$	0	0
$431$	−2.23607	−0.107708	−0.0538538	−	0.998549i	$-0.517150\pi$
$431$	−2.23607	−0.107708	−0.0538538	+	0.998549i	$0.517150\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	−13.4164	−0.643268
$436$	0	0
$437$	0	0
$438$	0	0
$439$	35.7771	1.70755	0.853774	−	0.520644i	$-0.174308\pi$
$439$	35.7771	1.70755	0.853774	+	0.520644i	$0.174308\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	−38.0132	−1.80606	−0.903030	−	0.429578i	$-0.858662\pi$
$443$	−38.0132	−1.80606	−0.903030	+	0.429578i	$0.858662\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	22.3607	1.05762
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−45.0000	−2.11428
$454$	0	0
$455$	2.23607	0.104828
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	−15.6525	−0.730595
$460$	0	0
$461$	−25.0000	−1.16437	−0.582183	−	0.813058i	$-0.697801\pi$
$461$	−25.0000	−1.16437	−0.582183	+	0.813058i	$0.697801\pi$
$462$	0	0
$463$	−26.8328	−1.24703	−0.623513	−	0.781813i	$-0.714295\pi$
$463$	−26.8328	−1.24703	−0.623513	+	0.781813i	$0.714295\pi$
$464$	0	0
$465$	20.0000	0.927478
$466$	0	0
$467$	17.8885	0.827783	0.413892	−	0.910326i	$-0.364169\pi$
$467$	17.8885	0.827783	0.413892	+	0.910326i	$0.364169\pi$
$468$	0	0
$469$	10.0000	0.461757
$470$	0	0
$471$	−17.8885	−0.824261
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	0	0
$476$	0	0
$477$	28.0000	1.28203
$478$	0	0
$479$	33.5410	1.53253	0.766264	−	0.642526i	$-0.222113\pi$
$479$	33.5410	1.53253	0.766264	+	0.642526i	$0.222113\pi$
$480$	0	0
$481$	3.00000	0.136788
$482$	0	0
$483$	22.3607	1.01745
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−8.94427	−0.405304	−0.202652	−	0.979251i	$-0.564956\pi$
$487$	−8.94427	−0.405304	−0.202652	+	0.979251i	$0.564956\pi$
$488$	0	0
$489$	30.0000	1.35665
$490$	0	0
$491$	2.23607	0.100912	0.0504562	−	0.998726i	$-0.483932\pi$
$491$	2.23607	0.100912	0.0504562	+	0.998726i	$0.483932\pi$
$492$	0	0
$493$	42.0000	1.89158
$494$	0	0
$495$	−8.94427	−0.402015
$496$	0	0
$497$	15.0000	0.672842
$498$	0	0
$499$	8.94427	0.400401	0.200200	−	0.979755i	$-0.435841\pi$
$499$	8.94427	0.400401	0.200200	+	0.979755i	$0.435841\pi$
$500$	0	0
$501$	−20.0000	−0.893534
$502$	0	0
$503$	−35.7771	−1.59522	−0.797611	−	0.603173i	$-0.793903\pi$
$503$	−35.7771	−1.59522	−0.797611	+	0.603173i	$0.793903\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.23607	−0.0993073
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−8.94427	−0.395671
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.4164	0.591198
$516$	0	0
$517$	−10.0000	−0.439799
$518$	0	0
$519$	13.4164	0.588915
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	0	0
$523$	17.8885	0.782211	0.391106	−	0.920346i	$-0.372093\pi$
$523$	17.8885	0.782211	0.391106	+	0.920346i	$0.372093\pi$
$524$	0	0
$525$	−20.0000	−0.872872
$526$	0	0
$527$	−62.6099	−2.72733
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	17.8885	0.776297
$532$	0	0
$533$	0	0
$534$	0	0
$535$	17.8885	0.773389
$536$	0	0
$537$	55.0000	2.37343
$538$	0	0
$539$	−8.94427	−0.385257
$540$	0	0
$541$	45.0000	1.93470	0.967351	−	0.253442i	$-0.0815627\pi$
$541$	45.0000	1.93470	0.967351	+	0.253442i	$0.0815627\pi$
$542$	0	0
$543$	22.3607	0.959589
$544$	0	0
$545$	5.00000	0.214176
$546$	0	0
$547$	15.6525	0.669252	0.334626	−	0.942351i	$-0.391390\pi$
$547$	15.6525	0.669252	0.334626	+	0.942351i	$0.391390\pi$
$548$	0	0
$549$	16.0000	0.682863
$550$	0	0
$551$	0	0
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	6.70820	0.284747
$556$	0	0
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	0	0
$559$	2.23607	0.0945756
$560$	0	0
$561$	70.0000	2.95540
$562$	0	0
$563$	20.1246	0.848151	0.424076	−	0.905627i	$-0.360599\pi$
$563$	20.1246	0.848151	0.424076	+	0.905627i	$0.360599\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	24.5967	1.03297
$568$	0	0
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	0	0
$571$	33.5410	1.40365	0.701824	−	0.712350i	$-0.252369\pi$
$571$	33.5410	1.40365	0.701824	+	0.712350i	$0.252369\pi$
$572$	0	0
$573$	−50.0000	−2.08878
$574$	0	0
$575$	−17.8885	−0.746004
$576$	0	0
$577$	12.0000	0.499567	0.249783	−	0.968302i	$-0.419641\pi$
$577$	12.0000	0.499567	0.249783	+	0.968302i	$0.419641\pi$
$578$	0	0
$579$	58.1378	2.41612
$580$	0	0
$581$	−10.0000	−0.414870
$582$	0	0
$583$	62.6099	2.59304
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−31.3050	−1.29209	−0.646047	−	0.763298i	$-0.723579\pi$
$587$	−31.3050	−1.29209	−0.646047	+	0.763298i	$0.723579\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	6.70820	0.275939
$592$	0	0
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	−15.6525	−0.641689
$596$	0	0
$597$	50.0000	2.04636
$598$	0	0
$599$	−8.94427	−0.365453	−0.182727	−	0.983164i	$-0.558492\pi$
$599$	−8.94427	−0.365453	−0.182727	+	0.983164i	$0.558492\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	−8.94427	−0.364239
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	8.94427	0.363037	0.181518	−	0.983388i	$-0.441899\pi$
$607$	8.94427	0.363037	0.181518	+	0.983388i	$0.441899\pi$
$608$	0	0
$609$	−30.0000	−1.21566
$610$	0	0
$611$	−2.23607	−0.0904616
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	8.94427	0.359501	0.179750	−	0.983712i	$-0.442471\pi$
$619$	8.94427	0.359501	0.179750	+	0.983712i	$0.442471\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−21.0000	−0.837325
$630$	0	0
$631$	−42.4853	−1.69131	−0.845656	−	0.533728i	$-0.820791\pi$
$631$	−42.4853	−1.69131	−0.845656	+	0.533728i	$0.820791\pi$
$632$	0	0
$633$	15.0000	0.596196
$634$	0	0
$635$	13.4164	0.532414
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−13.4164	−0.530745
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−4.47214	−0.176364	−0.0881819	−	0.996104i	$-0.528106\pi$
$643$	−4.47214	−0.176364	−0.0881819	+	0.996104i	$0.528106\pi$
$644$	0	0
$645$	5.00000	0.196875
$646$	0	0
$647$	17.8885	0.703271	0.351636	−	0.936137i	$-0.385626\pi$
$647$	17.8885	0.703271	0.351636	+	0.936137i	$0.385626\pi$
$648$	0	0
$649$	40.0000	1.57014
$650$	0	0
$651$	44.7214	1.75277
$652$	0	0
$653$	−4.00000	−0.156532	−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000	−0.156532	−0.0782660	+	0.996933i	$0.524938\pi$
$654$	0	0
$655$	−2.23607	−0.0873704
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	15.6525	0.607892
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−26.8328	−1.03897
$668$	0	0
$669$	−45.0000	−1.73980
$670$	0	0
$671$	35.7771	1.38116
$672$	0	0
$673$	41.0000	1.58043	0.790217	−	0.612827i	$-0.209968\pi$
$673$	41.0000	1.58043	0.790217	+	0.612827i	$0.209968\pi$
$674$	0	0
$675$	−8.94427	−0.344265
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−4.47214	−0.171625
$680$	0	0
$681$	20.0000	0.766402
$682$	0	0
$683$	40.2492	1.54009	0.770047	−	0.637987i	$-0.220233\pi$
$683$	40.2492	1.54009	0.770047	+	0.637987i	$0.220233\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	24.5967	0.938424
$688$	0	0
$689$	14.0000	0.533358
$690$	0	0
$691$	35.7771	1.36102	0.680512	−	0.732737i	$-0.261757\pi$
$691$	35.7771	1.36102	0.680512	+	0.732737i	$0.261757\pi$
$692$	0	0
$693$	−20.0000	−0.759737
$694$	0	0
$695$	6.70820	0.254457
$696$	0	0
$697$	0	0
$698$	0	0
$699$	42.4853	1.60694
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−5.00000	−0.188311
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	0	0
$711$	−8.94427	−0.335436
$712$	0	0
$713$	40.0000	1.49801
$714$	0	0
$715$	−4.47214	−0.167248
$716$	0	0
$717$	−55.0000	−2.05401
$718$	0	0
$719$	−22.3607	−0.833913	−0.416956	−	0.908927i	$-0.636903\pi$
$719$	−22.3607	−0.833913	−0.416956	+	0.908927i	$0.636903\pi$
$720$	0	0
$721$	30.0000	1.11726
$722$	0	0
$723$	−44.7214	−1.66321
$724$	0	0
$725$	24.0000	0.891338
$726$	0	0
$727$	−13.4164	−0.497587	−0.248794	−	0.968557i	$-0.580034\pi$
$727$	−13.4164	−0.497587	−0.248794	+	0.968557i	$0.580034\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	−15.6525	−0.578928
$732$	0	0
$733$	−31.0000	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$734$	0	0
$735$	−4.47214	−0.164957
$736$	0	0
$737$	−20.0000	−0.736709
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.6525	0.574234	0.287117	−	0.957896i	$-0.407303\pi$
$743$	15.6525	0.574234	0.287117	+	0.957896i	$0.407303\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	8.94427	0.327254
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	17.8885	0.652762	0.326381	−	0.945238i	$-0.394171\pi$
$751$	17.8885	0.652762	0.326381	+	0.945238i	$0.394171\pi$
$752$	0	0
$753$	−40.0000	−1.45768
$754$	0	0
$755$	−20.1246	−0.732410
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	−44.7214	−1.62328
$760$	0	0
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	0	0
$763$	11.1803	0.404755
$764$	0	0
$765$	14.0000	0.506171
$766$	0	0
$767$	8.94427	0.322959
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	−38.0132	−1.36901
$772$	0	0
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	−35.7771	−1.28515
$776$	0	0
$777$	15.0000	0.538122
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−30.0000	−1.07348
$782$	0	0
$783$	−13.4164	−0.479463
$784$	0	0
$785$	−8.00000	−0.285532
$786$	0	0
$787$	−26.8328	−0.956487	−0.478243	−	0.878227i	$-0.658726\pi$
$787$	−26.8328	−0.956487	−0.478243	+	0.878227i	$0.658726\pi$
$788$	0	0
$789$	60.0000	2.13606
$790$	0	0
$791$	31.3050	1.11308
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	31.3050	1.11027
$796$	0	0
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	15.6525	0.553745
$800$	0	0
$801$	0	0
$802$	0	0
$803$	17.8885	0.631273
$804$	0	0
$805$	10.0000	0.352454
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−25.0000	−0.878953	−0.439477	−	0.898254i	$-0.644836\pi$
$809$	−25.0000	−0.878953	−0.439477	+	0.898254i	$0.644836\pi$
$810$	0	0
$811$	26.8328	0.942228	0.471114	−	0.882072i	$-0.343852\pi$
$811$	26.8328	0.942228	0.471114	+	0.882072i	$0.343852\pi$
$812$	0	0
$813$	−35.0000	−1.22750
$814$	0	0
$815$	13.4164	0.469956
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−4.47214	−0.156269
$820$	0	0
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	0	0
$823$	−26.8328	−0.935333	−0.467667	−	0.883905i	$-0.654905\pi$
$823$	−26.8328	−0.935333	−0.467667	+	0.883905i	$0.654905\pi$
$824$	0	0
$825$	40.0000	1.39262
$826$	0	0
$827$	−44.7214	−1.55511	−0.777557	−	0.628812i	$-0.783541\pi$
$827$	−44.7214	−1.55511	−0.777557	+	0.628812i	$0.783541\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	62.6099	2.17191
$832$	0	0
$833$	14.0000	0.485071
$834$	0	0
$835$	−8.94427	−0.309529
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	8.94427	0.308791	0.154395	−	0.988009i	$-0.450657\pi$
$839$	8.94427	0.308791	0.154395	+	0.988009i	$0.450657\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−20.1246	−0.691490
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	13.4164	0.459909
$852$	0	0
$853$	41.0000	1.40381	0.701907	−	0.712269i	$-0.252332\pi$
$853$	41.0000	1.40381	0.701907	+	0.712269i	$0.252332\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.0132	−1.29398	−0.646991	−	0.762497i	$-0.723973\pi$
$863$	−38.0132	−1.29398	−0.646991	+	0.762497i	$0.723973\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	−71.5542	−2.43011
$868$	0	0
$869$	−20.0000	−0.678454
$870$	0	0
$871$	−4.47214	−0.151533
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	−20.1246	−0.680336
$876$	0	0
$877$	33.0000	1.11433	0.557165	−	0.830402i	$-0.311889\pi$
$877$	33.0000	1.11433	0.557165	+	0.830402i	$0.311889\pi$
$878$	0	0
$879$	46.9574	1.58383
$880$	0	0
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	0	0
$883$	42.4853	1.42974	0.714872	−	0.699255i	$-0.246485\pi$
$883$	42.4853	1.42974	0.714872	+	0.699255i	$0.246485\pi$
$884$	0	0
$885$	20.0000	0.672293
$886$	0	0
$887$	−8.94427	−0.300319	−0.150160	−	0.988662i	$-0.547979\pi$
$887$	−8.94427	−0.300319	−0.150160	+	0.988662i	$0.547979\pi$
$888$	0	0
$889$	30.0000	1.00617
$890$	0	0
$891$	−49.1935	−1.64804
$892$	0	0
$893$	0	0
$894$	0	0
$895$	24.5967	0.822179
$896$	0	0
$897$	−10.0000	−0.333890
$898$	0	0
$899$	−53.6656	−1.78985
$900$	0	0
$901$	−98.0000	−3.26485
$902$	0	0
$903$	11.1803	0.372058
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	−42.4853	−1.41070	−0.705350	−	0.708859i	$-0.749210\pi$
$907$	−42.4853	−1.41070	−0.705350	+	0.708859i	$0.749210\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.3050	−1.03718	−0.518590	−	0.855023i	$-0.673543\pi$
$911$	−31.3050	−1.03718	−0.518590	+	0.855023i	$0.673543\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	0	0
$915$	17.8885	0.591377
$916$	0	0
$917$	−5.00000	−0.165115
$918$	0	0
$919$	26.8328	0.885133	0.442566	−	0.896736i	$-0.354068\pi$
$919$	26.8328	0.885133	0.442566	+	0.896736i	$0.354068\pi$
$920$	0	0
$921$	60.0000	1.97707
$922$	0	0
$923$	−6.70820	−0.220803
$924$	0	0
$925$	−12.0000	−0.394558
$926$	0	0
$927$	−26.8328	−0.881305
$928$	0	0
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	40.0000	1.30954
$934$	0	0
$935$	31.3050	1.02378
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	2.23607	0.0729713
$940$	0	0
$941$	5.00000	0.162995	0.0814977	−	0.996674i	$-0.474030\pi$
$941$	5.00000	0.162995	0.0814977	+	0.996674i	$0.474030\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	−31.3050	−1.01727	−0.508637	−	0.860981i	$-0.669851\pi$
$947$	−31.3050	−1.01727	−0.508637	+	0.860981i	$0.669851\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	−49.1935	−1.59521
$952$	0	0
$953$	39.0000	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$954$	0	0
$955$	−22.3607	−0.723575
$956$	0	0
$957$	60.0000	1.93952
$958$	0	0
$959$	40.2492	1.29972
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	−35.7771	−1.15290
$964$	0	0
$965$	26.0000	0.836970
$966$	0	0
$967$	46.9574	1.51005	0.755025	−	0.655697i	$-0.227625\pi$
$967$	46.9574	1.51005	0.755025	+	0.655697i	$0.227625\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.6525	0.502312	0.251156	−	0.967947i	$-0.419189\pi$
$971$	15.6525	0.502312	0.251156	+	0.967947i	$0.419189\pi$
$972$	0	0
$973$	15.0000	0.480878
$974$	0	0
$975$	8.94427	0.286446
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	−55.9017	−1.78299	−0.891494	−	0.453033i	$-0.850342\pi$
$983$	−55.9017	−1.78299	−0.891494	+	0.453033i	$0.850342\pi$
$984$	0	0
$985$	3.00000	0.0955879
$986$	0	0
$987$	−11.1803	−0.355874
$988$	0	0
$989$	10.0000	0.317982
$990$	0	0
$991$	35.7771	1.13650	0.568248	−	0.822857i	$-0.307621\pi$
$991$	35.7771	1.13650	0.568248	+	0.822857i	$0.307621\pi$
$992$	0	0
$993$	10.0000	0.317340
$994$	0	0
$995$	22.3607	0.708881
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	0	0
$999$	6.70820	0.212238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.u.1.1		2
4.3	odd	2	inner	3328.2.a.u.1.2		2
8.3	odd	2		3328.2.a.v.1.1		2
8.5	even	2		3328.2.a.v.1.2		2
16.3	odd	4		1664.2.b.g.833.4	yes	4
16.5	even	4		1664.2.b.g.833.3	yes	4
16.11	odd	4		1664.2.b.g.833.1	✓	4
16.13	even	4		1664.2.b.g.833.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.g.833.1	✓	4	16.11	odd	4
1664.2.b.g.833.2	yes	4	16.13	even	4
1664.2.b.g.833.3	yes	4	16.5	even	4
1664.2.b.g.833.4	yes	4	16.3	odd	4
3328.2.a.u.1.1		2	1.1	even	1	trivial
3328.2.a.u.1.2		2	4.3	odd	2	inner
3328.2.a.v.1.1		2	8.3	odd	2
3328.2.a.v.1.2		2	8.5	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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{3} -1.00000 q^{5} -2.23607 q^{7} +2.00000 q^{9} +4.47214 q^{11} +1.00000 q^{13} +2.23607 q^{15} -7.00000 q^{17} +5.00000 q^{21} +4.47214 q^{23} -4.00000 q^{25} +2.23607 q^{27} -6.00000 q^{29} +8.94427 q^{31} -10.0000 q^{33} +2.23607 q^{35} +3.00000 q^{37} -2.23607 q^{39} +2.23607 q^{43} -2.00000 q^{45} -2.23607 q^{47} -2.00000 q^{49} +15.6525 q^{51} +14.0000 q^{53} -4.47214 q^{55} +8.94427 q^{59} +8.00000 q^{61} -4.47214 q^{63} -1.00000 q^{65} -4.47214 q^{67} -10.0000 q^{69} -6.70820 q^{71} +4.00000 q^{73} +8.94427 q^{75} -10.0000 q^{77} -4.47214 q^{79} -11.0000 q^{81} +4.47214 q^{83} +7.00000 q^{85} +13.4164 q^{87} -2.23607 q^{91} -20.0000 q^{93} +2.00000 q^{97} +8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{9} + 2 q^{13} - 14 q^{17} + 10 q^{21} - 8 q^{25} - 12 q^{29} - 20 q^{33} + 6 q^{37} - 4 q^{45} - 4 q^{49} + 28 q^{53} + 16 q^{61} - 2 q^{65} - 20 q^{69} + 8 q^{73} - 20 q^{77} - 22 q^{81}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^9 + 2 * q^13 - 14 * q^17 + 10 * q^21 - 8 * q^25 - 12 * q^29 - 20 * q^33 + 6 * q^37 - 4 * q^45 - 4 * q^49 + 28 * q^53 + 16 * q^61 - 2 * q^65 - 20 * q^69 + 8 * q^73 - 20 * q^77 - 22 * q^81 + 14 * q^85 - 40 * q^93 + 4 * q^97

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