LMFDB - Embedded newform 9200.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$73.4623698596$
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 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{3} -1.23607 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+2.23607 q^{3} -1.23607 q^{7} +2.00000 q^{9} +0.763932 q^{11} -3.00000 q^{13} -5.23607 q^{17} +2.00000 q^{19} -2.76393 q^{21} +1.00000 q^{23} -2.23607 q^{27} -3.00000 q^{29} +6.70820 q^{31} +1.70820 q^{33} -3.23607 q^{37} -6.70820 q^{39} +5.47214 q^{41} +2.23607 q^{47} -5.47214 q^{49} -11.7082 q^{51} +8.47214 q^{53} +4.47214 q^{57} +2.47214 q^{59} +10.9443 q^{61} -2.47214 q^{63} -7.23607 q^{67} +2.23607 q^{69} -7.76393 q^{71} -15.4721 q^{73} -0.944272 q^{77} -6.94427 q^{79} -11.0000 q^{81} -13.2361 q^{83} -6.70820 q^{87} -1.52786 q^{89} +3.70820 q^{91} +15.0000 q^{93} -4.29180 q^{97} +1.52786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^7 + 4 * q^9	$ 2 q + 2 q^{7} + 4 q^{9} + 6 q^{11} - 6 q^{13} - 6 q^{17} + 4 q^{19} - 10 q^{21} + 2 q^{23} - 6 q^{29} - 10 q^{33} - 2 q^{37} + 2 q^{41} - 2 q^{49} - 10 q^{51} + 8 q^{53} - 4 q^{59} + 4 q^{61} + 4 q^{63} - 10 q^{67} - 20 q^{71} - 22 q^{73} + 16 q^{77} + 4 q^{79} - 22 q^{81} - 22 q^{83} - 12 q^{89} - 6 q^{91} + 30 q^{93} - 22 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^7 + 4 * q^9 + 6 * q^11 - 6 * q^13 - 6 * q^17 + 4 * q^19 - 10 * q^21 + 2 * q^23 - 6 * q^29 - 10 * q^33 - 2 * q^37 + 2 * q^41 - 2 * q^49 - 10 * q^51 + 8 * q^53 - 4 * q^59 + 4 * q^61 + 4 * q^63 - 10 * q^67 - 20 * q^71 - 22 * q^73 + 16 * q^77 + 4 * q^79 - 22 * q^81 - 22 * q^83 - 12 * q^89 - 6 * q^91 + 30 * q^93 - 22 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

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	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.23607	−1.26993	−0.634967	−	0.772540i	$-0.718986\pi$
$17$	−5.23607	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−2.76393	−0.603139
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	6.70820	1.20483	0.602414	−	0.798183i	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	0	0
$33$	1.70820	0.297360
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.23607	−0.532006	−0.266003	−	0.963972i	$-0.585703\pi$
$37$	−3.23607	−0.532006	−0.266003	+	0.963972i	$0.585703\pi$
$38$	0	0
$39$	−6.70820	−1.07417
$40$	0	0
$41$	5.47214	0.854604	0.427302	−	0.904109i	$-0.359464\pi$
$41$	5.47214	0.854604	0.427302	+	0.904109i	$0.359464\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.23607	0.326164	0.163082	−	0.986613i	$-0.447856\pi$
$47$	2.23607	0.326164	0.163082	+	0.986613i	$0.447856\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−11.7082	−1.63948
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.47214	0.592349
$58$	0	0
$59$	2.47214	0.321845	0.160922	−	0.986967i	$-0.448553\pi$
$59$	2.47214	0.321845	0.160922	+	0.986967i	$0.448553\pi$
$60$	0	0
$61$	10.9443	1.40127	0.700635	−	0.713520i	$-0.252900\pi$
$61$	10.9443	1.40127	0.700635	+	0.713520i	$0.252900\pi$
$62$	0	0
$63$	−2.47214	−0.311460
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.23607	−0.884026	−0.442013	−	0.897009i	$-0.645736\pi$
$67$	−7.23607	−0.884026	−0.442013	+	0.897009i	$0.645736\pi$
$68$	0	0
$69$	2.23607	0.269191
$70$	0	0
$71$	−7.76393	−0.921409	−0.460705	−	0.887554i	$-0.652403\pi$
$71$	−7.76393	−0.921409	−0.460705	+	0.887554i	$0.652403\pi$
$72$	0	0
$73$	−15.4721	−1.81088	−0.905438	−	0.424478i	$-0.860458\pi$
$73$	−15.4721	−1.81088	−0.905438	+	0.424478i	$0.860458\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.944272	−0.107610
$78$	0	0
$79$	−6.94427	−0.781292	−0.390646	−	0.920541i	$-0.627748\pi$
$79$	−6.94427	−0.781292	−0.390646	+	0.920541i	$0.627748\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−13.2361	−1.45285	−0.726424	−	0.687247i	$-0.758819\pi$
$83$	−13.2361	−1.45285	−0.726424	+	0.687247i	$0.758819\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.70820	−0.719195
$88$	0	0
$89$	−1.52786	−0.161953	−0.0809766	−	0.996716i	$-0.525804\pi$
$89$	−1.52786	−0.161953	−0.0809766	+	0.996716i	$0.525804\pi$
$90$	0	0
$91$	3.70820	0.388725
$92$	0	0
$93$	15.0000	1.55543
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.29180	−0.435766	−0.217883	−	0.975975i	$-0.569915\pi$
$97$	−4.29180	−0.435766	−0.217883	+	0.975975i	$0.569915\pi$
$98$	0	0
$99$	1.52786	0.153556
$100$	0	0
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	0	0
$103$	18.1803	1.79136	0.895681	−	0.444697i	$-0.146689\pi$
$103$	18.1803	1.79136	0.895681	+	0.444697i	$0.146689\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.4164	−1.29701	−0.648507	−	0.761209i	$-0.724606\pi$
$107$	−13.4164	−1.29701	−0.648507	+	0.761209i	$0.724606\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−7.23607	−0.686817
$112$	0	0
$113$	−13.2361	−1.24514	−0.622572	−	0.782562i	$-0.713912\pi$
$113$	−13.2361	−1.24514	−0.622572	+	0.782562i	$0.713912\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	6.47214	0.593300
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	12.2361	1.10329
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−20.7082	−1.83756	−0.918778	−	0.394775i	$-0.870823\pi$
$127$	−20.7082	−1.83756	−0.918778	+	0.394775i	$0.870823\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.29180	−0.462346	−0.231173	−	0.972913i	$-0.574256\pi$
$131$	−5.29180	−0.462346	−0.231173	+	0.972913i	$0.574256\pi$
$132$	0	0
$133$	−2.47214	−0.214361
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.8885	−1.18658	−0.593289	−	0.804989i	$-0.702171\pi$
$137$	−13.8885	−1.18658	−0.593289	+	0.804989i	$0.702171\pi$
$138$	0	0
$139$	−2.70820	−0.229707	−0.114853	−	0.993382i	$-0.536640\pi$
$139$	−2.70820	−0.229707	−0.114853	+	0.993382i	$0.536640\pi$
$140$	0	0
$141$	5.00000	0.421076
$142$	0	0
$143$	−2.29180	−0.191650
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−12.2361	−1.00921
$148$	0	0
$149$	−11.8885	−0.973947	−0.486974	−	0.873417i	$-0.661899\pi$
$149$	−11.8885	−0.973947	−0.486974	+	0.873417i	$0.661899\pi$
$150$	0	0
$151$	0.236068	0.0192109	0.00960547	−	0.999954i	$-0.496942\pi$
$151$	0.236068	0.0192109	0.00960547	+	0.999954i	$0.496942\pi$
$152$	0	0
$153$	−10.4721	−0.846622
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−15.4164	−1.23036	−0.615182	−	0.788385i	$-0.710917\pi$
$157$	−15.4164	−1.23036	−0.615182	+	0.788385i	$0.710917\pi$
$158$	0	0
$159$	18.9443	1.50238
$160$	0	0
$161$	−1.23607	−0.0974158
$162$	0	0
$163$	−10.2361	−0.801751	−0.400875	−	0.916133i	$-0.631294\pi$
$163$	−10.2361	−0.801751	−0.400875	+	0.916133i	$0.631294\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.4721	0.810358	0.405179	−	0.914237i	$-0.367209\pi$
$167$	10.4721	0.810358	0.405179	+	0.914237i	$0.367209\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	−5.05573	−0.384380	−0.192190	−	0.981358i	$-0.561559\pi$
$173$	−5.05573	−0.384380	−0.192190	+	0.981358i	$0.561559\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.52786	0.415500
$178$	0	0
$179$	12.7082	0.949856	0.474928	−	0.880025i	$-0.342474\pi$
$179$	12.7082	0.949856	0.474928	+	0.880025i	$0.342474\pi$
$180$	0	0
$181$	−14.6525	−1.08911	−0.544555	−	0.838725i	$-0.683301\pi$
$181$	−14.6525	−1.08911	−0.544555	+	0.838725i	$0.683301\pi$
$182$	0	0
$183$	24.4721	1.80903
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	2.76393	0.201046
$190$	0	0
$191$	3.81966	0.276381	0.138190	−	0.990406i	$-0.455871\pi$
$191$	3.81966	0.276381	0.138190	+	0.990406i	$0.455871\pi$
$192$	0	0
$193$	7.94427	0.571841	0.285921	−	0.958253i	$-0.407701\pi$
$193$	7.94427	0.571841	0.285921	+	0.958253i	$0.407701\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.47214	−0.532368	−0.266184	−	0.963922i	$-0.585763\pi$
$197$	−7.47214	−0.532368	−0.266184	+	0.963922i	$0.585763\pi$
$198$	0	0
$199$	25.7082	1.82241	0.911203	−	0.411957i	$-0.135155\pi$
$199$	25.7082	1.82241	0.911203	+	0.411957i	$0.135155\pi$
$200$	0	0
$201$	−16.1803	−1.14127
$202$	0	0
$203$	3.70820	0.260265
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	1.52786	0.105685
$210$	0	0
$211$	−3.41641	−0.235195	−0.117598	−	0.993061i	$-0.537519\pi$
$211$	−3.41641	−0.235195	−0.117598	+	0.993061i	$0.537519\pi$
$212$	0	0
$213$	−17.3607	−1.18953
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.29180	−0.562884
$218$	0	0
$219$	−34.5967	−2.33783
$220$	0	0
$221$	15.7082	1.05665
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.1803	0.675693	0.337846	−	0.941201i	$-0.390302\pi$
$227$	10.1803	0.675693	0.337846	+	0.941201i	$0.390302\pi$
$228$	0	0
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	0	0
$231$	−2.11146	−0.138924
$232$	0	0
$233$	15.4721	1.01361	0.506807	−	0.862060i	$-0.330826\pi$
$233$	15.4721	1.01361	0.506807	+	0.862060i	$0.330826\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−15.5279	−1.00864
$238$	0	0
$239$	−18.2361	−1.17959	−0.589797	−	0.807552i	$-0.700792\pi$
$239$	−18.2361	−1.17959	−0.589797	+	0.807552i	$0.700792\pi$
$240$	0	0
$241$	17.1246	1.10309	0.551547	−	0.834144i	$-0.314038\pi$
$241$	17.1246	1.10309	0.551547	+	0.834144i	$0.314038\pi$
$242$	0	0
$243$	−17.8885	−1.14755
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−29.5967	−1.87562
$250$	0	0
$251$	−15.7082	−0.991493	−0.495747	−	0.868467i	$-0.665105\pi$
$251$	−15.7082	−0.991493	−0.495747	+	0.868467i	$0.665105\pi$
$252$	0	0
$253$	0.763932	0.0480280
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.47214	−0.0918293	−0.0459147	−	0.998945i	$-0.514620\pi$
$257$	−1.47214	−0.0918293	−0.0459147	+	0.998945i	$0.514620\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−14.9443	−0.921503	−0.460752	−	0.887529i	$-0.652420\pi$
$263$	−14.9443	−0.921503	−0.460752	+	0.887529i	$0.652420\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.41641	−0.209081
$268$	0	0
$269$	9.94427	0.606313	0.303156	−	0.952941i	$-0.401960\pi$
$269$	9.94427	0.606313	0.303156	+	0.952941i	$0.401960\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	8.29180	0.501842
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.52786	−0.392221	−0.196111	−	0.980582i	$-0.562831\pi$
$277$	−6.52786	−0.392221	−0.196111	+	0.980582i	$0.562831\pi$
$278$	0	0
$279$	13.4164	0.803219
$280$	0	0
$281$	−13.2361	−0.789598	−0.394799	−	0.918768i	$-0.629186\pi$
$281$	−13.2361	−0.789598	−0.394799	+	0.918768i	$0.629186\pi$
$282$	0	0
$283$	14.2918	0.849559	0.424780	−	0.905297i	$-0.360352\pi$
$283$	14.2918	0.849559	0.424780	+	0.905297i	$0.360352\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.76393	−0.399262
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	−9.59675	−0.562571
$292$	0	0
$293$	10.4721	0.611789	0.305894	−	0.952065i	$-0.401045\pi$
$293$	10.4721	0.611789	0.305894	+	0.952065i	$0.401045\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.70820	−0.0991200
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.4721	1.05426	0.527130	−	0.849785i	$-0.323268\pi$
$307$	18.4721	1.05426	0.527130	+	0.849785i	$0.323268\pi$
$308$	0	0
$309$	40.6525	2.31264
$310$	0	0
$311$	9.18034	0.520569	0.260285	−	0.965532i	$-0.416184\pi$
$311$	9.18034	0.520569	0.260285	+	0.965532i	$0.416184\pi$
$312$	0	0
$313$	20.3607	1.15085	0.575427	−	0.817853i	$-0.304836\pi$
$313$	20.3607	1.15085	0.575427	+	0.817853i	$0.304836\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.41641	0.0795534	0.0397767	−	0.999209i	$-0.487335\pi$
$317$	1.41641	0.0795534	0.0397767	+	0.999209i	$0.487335\pi$
$318$	0	0
$319$	−2.29180	−0.128316
$320$	0	0
$321$	−30.0000	−1.67444
$322$	0	0
$323$	−10.4721	−0.582685
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.76393	−0.152381
$330$	0	0
$331$	−11.6525	−0.640478	−0.320239	−	0.947337i	$-0.603763\pi$
$331$	−11.6525	−0.640478	−0.320239	+	0.947337i	$0.603763\pi$
$332$	0	0
$333$	−6.47214	−0.354671
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.41641	0.186104	0.0930518	−	0.995661i	$-0.470338\pi$
$337$	3.41641	0.186104	0.0930518	+	0.995661i	$0.470338\pi$
$338$	0	0
$339$	−29.5967	−1.60747
$340$	0	0
$341$	5.12461	0.277513
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.8885	1.38977	0.694885	−	0.719121i	$-0.255455\pi$
$347$	25.8885	1.38977	0.694885	+	0.719121i	$0.255455\pi$
$348$	0	0
$349$	−2.41641	−0.129347	−0.0646737	−	0.997906i	$-0.520601\pi$
$349$	−2.41641	−0.129347	−0.0646737	+	0.997906i	$0.520601\pi$
$350$	0	0
$351$	6.70820	0.358057
$352$	0	0
$353$	35.3607	1.88206	0.941030	−	0.338324i	$-0.109860\pi$
$353$	35.3607	1.88206	0.941030	+	0.338324i	$0.109860\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	14.4721	0.765947
$358$	0	0
$359$	−15.8885	−0.838565	−0.419283	−	0.907856i	$-0.637718\pi$
$359$	−15.8885	−0.838565	−0.419283	+	0.907856i	$0.637718\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−23.2918	−1.22250
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.1803	0.949006	0.474503	−	0.880254i	$-0.342628\pi$
$367$	18.1803	0.949006	0.474503	+	0.880254i	$0.342628\pi$
$368$	0	0
$369$	10.9443	0.569736
$370$	0	0
$371$	−10.4721	−0.543686
$372$	0	0
$373$	5.70820	0.295560	0.147780	−	0.989020i	$-0.452787\pi$
$373$	5.70820	0.295560	0.147780	+	0.989020i	$0.452787\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.00000	0.463524
$378$	0	0
$379$	20.3607	1.04586	0.522929	−	0.852376i	$-0.324839\pi$
$379$	20.3607	1.04586	0.522929	+	0.852376i	$0.324839\pi$
$380$	0	0
$381$	−46.3050	−2.37227
$382$	0	0
$383$	24.9443	1.27459	0.637296	−	0.770619i	$-0.280053\pi$
$383$	24.9443	1.27459	0.637296	+	0.770619i	$0.280053\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	34.4721	1.74781	0.873903	−	0.486100i	$-0.161581\pi$
$389$	34.4721	1.74781	0.873903	+	0.486100i	$0.161581\pi$
$390$	0	0
$391$	−5.23607	−0.264799
$392$	0	0
$393$	−11.8328	−0.596887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.41641	−0.121276	−0.0606380	−	0.998160i	$-0.519314\pi$
$397$	−2.41641	−0.121276	−0.0606380	+	0.998160i	$0.519314\pi$
$398$	0	0
$399$	−5.52786	−0.276739
$400$	0	0
$401$	8.18034	0.408507	0.204253	−	0.978918i	$-0.434523\pi$
$401$	8.18034	0.408507	0.204253	+	0.978918i	$0.434523\pi$
$402$	0	0
$403$	−20.1246	−1.00248
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.47214	−0.122539
$408$	0	0
$409$	−23.3607	−1.15511	−0.577556	−	0.816351i	$-0.695993\pi$
$409$	−23.3607	−1.15511	−0.577556	+	0.816351i	$0.695993\pi$
$410$	0	0
$411$	−31.0557	−1.53187
$412$	0	0
$413$	−3.05573	−0.150363
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.05573	−0.296550
$418$	0	0
$419$	31.4164	1.53479	0.767396	−	0.641173i	$-0.221552\pi$
$419$	31.4164	1.53479	0.767396	+	0.641173i	$0.221552\pi$
$420$	0	0
$421$	−23.7082	−1.15547	−0.577734	−	0.816225i	$-0.696063\pi$
$421$	−23.7082	−1.15547	−0.577734	+	0.816225i	$0.696063\pi$
$422$	0	0
$423$	4.47214	0.217443
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−13.5279	−0.654659
$428$	0	0
$429$	−5.12461	−0.247419
$430$	0	0
$431$	26.4721	1.27512	0.637559	−	0.770402i	$-0.279944\pi$
$431$	26.4721	1.27512	0.637559	+	0.770402i	$0.279944\pi$
$432$	0	0
$433$	−40.1803	−1.93094	−0.965472	−	0.260507i	$-0.916110\pi$
$433$	−40.1803	−1.93094	−0.965472	+	0.260507i	$0.916110\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	5.29180	0.252564	0.126282	−	0.991994i	$-0.459696\pi$
$439$	5.29180	0.252564	0.126282	+	0.991994i	$0.459696\pi$
$440$	0	0
$441$	−10.9443	−0.521156
$442$	0	0
$443$	−2.12461	−0.100943	−0.0504717	−	0.998725i	$-0.516072\pi$
$443$	−2.12461	−0.100943	−0.0504717	+	0.998725i	$0.516072\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−26.5836	−1.25736
$448$	0	0
$449$	2.94427	0.138949	0.0694744	−	0.997584i	$-0.477868\pi$
$449$	2.94427	0.138949	0.0694744	+	0.997584i	$0.477868\pi$
$450$	0	0
$451$	4.18034	0.196845
$452$	0	0
$453$	0.527864	0.0248012
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.1246	−1.64306	−0.821530	−	0.570165i	$-0.806879\pi$
$457$	−35.1246	−1.64306	−0.821530	+	0.570165i	$0.806879\pi$
$458$	0	0
$459$	11.7082	0.546492
$460$	0	0
$461$	7.47214	0.348012	0.174006	−	0.984745i	$-0.444329\pi$
$461$	7.47214	0.348012	0.174006	+	0.984745i	$0.444329\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.9443	−1.43193	−0.715965	−	0.698136i	$-0.754013\pi$
$467$	−30.9443	−1.43193	−0.715965	+	0.698136i	$0.754013\pi$
$468$	0	0
$469$	8.94427	0.413008
$470$	0	0
$471$	−34.4721	−1.58839
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	16.9443	0.775825
$478$	0	0
$479$	17.5967	0.804016	0.402008	−	0.915636i	$-0.368312\pi$
$479$	17.5967	0.804016	0.402008	+	0.915636i	$0.368312\pi$
$480$	0	0
$481$	9.70820	0.442656
$482$	0	0
$483$	−2.76393	−0.125763
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.29180	−0.0585369	−0.0292684	−	0.999572i	$-0.509318\pi$
$487$	−1.29180	−0.0585369	−0.0292684	+	0.999572i	$0.509318\pi$
$488$	0	0
$489$	−22.8885	−1.03506
$490$	0	0
$491$	−39.6525	−1.78949	−0.894746	−	0.446576i	$-0.852643\pi$
$491$	−39.6525	−1.78949	−0.894746	+	0.446576i	$0.852643\pi$
$492$	0	0
$493$	15.7082	0.707462
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.59675	0.430473
$498$	0	0
$499$	−32.7082	−1.46422	−0.732110	−	0.681186i	$-0.761464\pi$
$499$	−32.7082	−1.46422	−0.732110	+	0.681186i	$0.761464\pi$
$500$	0	0
$501$	23.4164	1.04617
$502$	0	0
$503$	−9.05573	−0.403775	−0.201887	−	0.979409i	$-0.564708\pi$
$503$	−9.05573	−0.403775	−0.201887	+	0.979409i	$0.564708\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8.94427	−0.397229
$508$	0	0
$509$	34.3050	1.52054	0.760270	−	0.649607i	$-0.225067\pi$
$509$	34.3050	1.52054	0.760270	+	0.649607i	$0.225067\pi$
$510$	0	0
$511$	19.1246	0.846023
$512$	0	0
$513$	−4.47214	−0.197450
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.70820	0.0751267
$518$	0	0
$519$	−11.3050	−0.496232
$520$	0	0
$521$	4.58359	0.200811	0.100405	−	0.994947i	$-0.467986\pi$
$521$	4.58359	0.200811	0.100405	+	0.994947i	$0.467986\pi$
$522$	0	0
$523$	0.875388	0.0382781	0.0191390	−	0.999817i	$-0.493907\pi$
$523$	0.875388	0.0382781	0.0191390	+	0.999817i	$0.493907\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−35.1246	−1.53005
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.94427	0.214563
$532$	0	0
$533$	−16.4164	−0.711074
$534$	0	0
$535$	0	0
$536$	0	0
$537$	28.4164	1.22626
$538$	0	0
$539$	−4.18034	−0.180060
$540$	0	0
$541$	−7.58359	−0.326044	−0.163022	−	0.986622i	$-0.552124\pi$
$541$	−7.58359	−0.326044	−0.163022	+	0.986622i	$0.552124\pi$
$542$	0	0
$543$	−32.7639	−1.40603
$544$	0	0
$545$	0	0
$546$	0	0
$547$	37.5410	1.60514	0.802569	−	0.596559i	$-0.203466\pi$
$547$	37.5410	1.60514	0.802569	+	0.596559i	$0.203466\pi$
$548$	0	0
$549$	21.8885	0.934180
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	8.58359	0.365011
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.4164	−0.822700	−0.411350	−	0.911478i	$-0.634943\pi$
$557$	−19.4164	−0.822700	−0.411350	+	0.911478i	$0.634943\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.94427	−0.377627
$562$	0	0
$563$	−15.0557	−0.634523	−0.317262	−	0.948338i	$-0.602763\pi$
$563$	−15.0557	−0.634523	−0.317262	+	0.948338i	$0.602763\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	13.5967	0.571010
$568$	0	0
$569$	0.180340	0.00756024	0.00378012	−	0.999993i	$-0.498797\pi$
$569$	0.180340	0.00756024	0.00378012	+	0.999993i	$0.498797\pi$
$570$	0	0
$571$	27.7082	1.15955	0.579776	−	0.814776i	$-0.303140\pi$
$571$	27.7082	1.15955	0.579776	+	0.814776i	$0.303140\pi$
$572$	0	0
$573$	8.54102	0.356806
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.8885	0.536557	0.268279	−	0.963341i	$-0.413545\pi$
$577$	12.8885	0.536557	0.268279	+	0.963341i	$0.413545\pi$
$578$	0	0
$579$	17.7639	0.738244
$580$	0	0
$581$	16.3607	0.678755
$582$	0	0
$583$	6.47214	0.268048
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.2918	−0.466062	−0.233031	−	0.972469i	$-0.574864\pi$
$587$	−11.2918	−0.466062	−0.233031	+	0.972469i	$0.574864\pi$
$588$	0	0
$589$	13.4164	0.552813
$590$	0	0
$591$	−16.7082	−0.687284
$592$	0	0
$593$	−14.9443	−0.613688	−0.306844	−	0.951760i	$-0.599273\pi$
$593$	−14.9443	−0.613688	−0.306844	+	0.951760i	$0.599273\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	57.4853	2.35272
$598$	0	0
$599$	1.88854	0.0771638	0.0385819	−	0.999255i	$-0.487716\pi$
$599$	1.88854	0.0771638	0.0385819	+	0.999255i	$0.487716\pi$
$600$	0	0
$601$	11.1115	0.453246	0.226623	−	0.973983i	$-0.427232\pi$
$601$	11.1115	0.453246	0.226623	+	0.973983i	$0.427232\pi$
$602$	0	0
$603$	−14.4721	−0.589351
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.5279	0.711434	0.355717	−	0.934594i	$-0.384237\pi$
$607$	17.5279	0.711434	0.355717	+	0.934594i	$0.384237\pi$
$608$	0	0
$609$	8.29180	0.336001
$610$	0	0
$611$	−6.70820	−0.271385
$612$	0	0
$613$	7.70820	0.311331	0.155666	−	0.987810i	$-0.450248\pi$
$613$	7.70820	0.311331	0.155666	+	0.987810i	$0.450248\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.4721	0.663143	0.331572	−	0.943430i	$-0.392421\pi$
$617$	16.4721	0.663143	0.331572	+	0.943430i	$0.392421\pi$
$618$	0	0
$619$	7.41641	0.298091	0.149045	−	0.988830i	$-0.452380\pi$
$619$	7.41641	0.298091	0.149045	+	0.988830i	$0.452380\pi$
$620$	0	0
$621$	−2.23607	−0.0897303
$622$	0	0
$623$	1.88854	0.0756629
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.41641	0.136438
$628$	0	0
$629$	16.9443	0.675612
$630$	0	0
$631$	32.3607	1.28826	0.644129	−	0.764917i	$-0.277220\pi$
$631$	32.3607	1.28826	0.644129	+	0.764917i	$0.277220\pi$
$632$	0	0
$633$	−7.63932	−0.303636
$634$	0	0
$635$	0	0
$636$	0	0
$637$	16.4164	0.650442
$638$	0	0
$639$	−15.5279	−0.614273
$640$	0	0
$641$	45.3050	1.78944	0.894719	−	0.446629i	$-0.147376\pi$
$641$	45.3050	1.78944	0.894719	+	0.446629i	$0.147376\pi$
$642$	0	0
$643$	19.5967	0.772820	0.386410	−	0.922327i	$-0.373715\pi$
$643$	19.5967	0.772820	0.386410	+	0.922327i	$0.373715\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.70820	−0.263727	−0.131863	−	0.991268i	$-0.542096\pi$
$647$	−6.70820	−0.263727	−0.131863	+	0.991268i	$0.542096\pi$
$648$	0	0
$649$	1.88854	0.0741318
$650$	0	0
$651$	−18.5410	−0.726680
$652$	0	0
$653$	−24.3050	−0.951126	−0.475563	−	0.879682i	$-0.657756\pi$
$653$	−24.3050	−0.951126	−0.475563	+	0.879682i	$0.657756\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−30.9443	−1.20725
$658$	0	0
$659$	−20.6525	−0.804506	−0.402253	−	0.915528i	$-0.631773\pi$
$659$	−20.6525	−0.804506	−0.402253	+	0.915528i	$0.631773\pi$
$660$	0	0
$661$	−5.05573	−0.196645	−0.0983225	−	0.995155i	$-0.531348\pi$
$661$	−5.05573	−0.196645	−0.0983225	+	0.995155i	$0.531348\pi$
$662$	0	0
$663$	35.1246	1.36413
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.00000	−0.116160
$668$	0	0
$669$	8.94427	0.345806
$670$	0	0
$671$	8.36068	0.322760
$672$	0	0
$673$	−3.00000	−0.115642	−0.0578208	−	0.998327i	$-0.518415\pi$
$673$	−3.00000	−0.115642	−0.0578208	+	0.998327i	$0.518415\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	5.30495	0.203585
$680$	0	0
$681$	22.7639	0.872316
$682$	0	0
$683$	−22.5967	−0.864641	−0.432320	−	0.901720i	$-0.642305\pi$
$683$	−22.5967	−0.864641	−0.432320	+	0.901720i	$0.642305\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−26.8328	−1.02374
$688$	0	0
$689$	−25.4164	−0.968288
$690$	0	0
$691$	−24.9443	−0.948925	−0.474462	−	0.880276i	$-0.657358\pi$
$691$	−24.9443	−0.948925	−0.474462	+	0.880276i	$0.657358\pi$
$692$	0	0
$693$	−1.88854	−0.0717398
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−28.6525	−1.08529
$698$	0	0
$699$	34.5967	1.30857
$700$	0	0
$701$	−26.1803	−0.988818	−0.494409	−	0.869229i	$-0.664615\pi$
$701$	−26.1803	−0.988818	−0.494409	+	0.869229i	$0.664615\pi$
$702$	0	0
$703$	−6.47214	−0.244101
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.52786	0.207897
$708$	0	0
$709$	16.0689	0.603480	0.301740	−	0.953390i	$-0.402433\pi$
$709$	16.0689	0.603480	0.301740	+	0.953390i	$0.402433\pi$
$710$	0	0
$711$	−13.8885	−0.520861
$712$	0	0
$713$	6.70820	0.251224
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−40.7771	−1.52285
$718$	0	0
$719$	20.9443	0.781090	0.390545	−	0.920584i	$-0.372287\pi$
$719$	20.9443	0.781090	0.390545	+	0.920584i	$0.372287\pi$
$720$	0	0
$721$	−22.4721	−0.836906
$722$	0	0
$723$	38.2918	1.42409
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−14.2918	−0.530053	−0.265027	−	0.964241i	$-0.585381\pi$
$727$	−14.2918	−0.530053	−0.265027	+	0.964241i	$0.585381\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	0	0
$732$	0	0
$733$	26.7639	0.988548	0.494274	−	0.869306i	$-0.335434\pi$
$733$	26.7639	0.988548	0.494274	+	0.869306i	$0.335434\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.52786	−0.203621
$738$	0	0
$739$	−49.1803	−1.80913	−0.904564	−	0.426338i	$-0.859803\pi$
$739$	−49.1803	−1.80913	−0.904564	+	0.426338i	$0.859803\pi$
$740$	0	0
$741$	−13.4164	−0.492864
$742$	0	0
$743$	0.875388	0.0321149	0.0160574	−	0.999871i	$-0.494889\pi$
$743$	0.875388	0.0321149	0.0160574	+	0.999871i	$0.494889\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−26.4721	−0.968565
$748$	0	0
$749$	16.5836	0.605951
$750$	0	0
$751$	44.3607	1.61874	0.809372	−	0.587296i	$-0.199808\pi$
$751$	44.3607	1.61874	0.809372	+	0.587296i	$0.199808\pi$
$752$	0	0
$753$	−35.1246	−1.28001
$754$	0	0
$755$	0	0
$756$	0	0
$757$	47.5967	1.72993	0.864967	−	0.501829i	$-0.167339\pi$
$757$	47.5967	1.72993	0.864967	+	0.501829i	$0.167339\pi$
$758$	0	0
$759$	1.70820	0.0620039
$760$	0	0
$761$	−16.3050	−0.591054	−0.295527	−	0.955334i	$-0.595495\pi$
$761$	−16.3050	−0.591054	−0.295527	+	0.955334i	$0.595495\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.41641	−0.267791
$768$	0	0
$769$	17.1246	0.617529	0.308765	−	0.951138i	$-0.400084\pi$
$769$	17.1246	0.617529	0.308765	+	0.951138i	$0.400084\pi$
$770$	0	0
$771$	−3.29180	−0.118551
$772$	0	0
$773$	14.4721	0.520527	0.260263	−	0.965538i	$-0.416191\pi$
$773$	14.4721	0.520527	0.260263	+	0.965538i	$0.416191\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.94427	0.320874
$778$	0	0
$779$	10.9443	0.392119
$780$	0	0
$781$	−5.93112	−0.212232
$782$	0	0
$783$	6.70820	0.239732
$784$	0	0
$785$	0	0
$786$	0	0
$787$	51.4164	1.83280	0.916399	−	0.400267i	$-0.131083\pi$
$787$	51.4164	1.83280	0.916399	+	0.400267i	$0.131083\pi$
$788$	0	0
$789$	−33.4164	−1.18966
$790$	0	0
$791$	16.3607	0.581719
$792$	0	0
$793$	−32.8328	−1.16593
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.3607	−0.366994	−0.183497	−	0.983020i	$-0.558742\pi$
$797$	−10.3607	−0.366994	−0.183497	+	0.983020i	$0.558742\pi$
$798$	0	0
$799$	−11.7082	−0.414206
$800$	0	0
$801$	−3.05573	−0.107969
$802$	0	0
$803$	−11.8197	−0.417107
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.2361	0.782747
$808$	0	0
$809$	47.8885	1.68367	0.841836	−	0.539734i	$-0.181475\pi$
$809$	47.8885	1.68367	0.841836	+	0.539734i	$0.181475\pi$
$810$	0	0
$811$	55.6525	1.95422	0.977111	−	0.212728i	$-0.0682350\pi$
$811$	55.6525	1.95422	0.977111	+	0.212728i	$0.0682350\pi$
$812$	0	0
$813$	−17.8885	−0.627379
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	7.41641	0.259150
$820$	0	0
$821$	−21.0557	−0.734850	−0.367425	−	0.930053i	$-0.619761\pi$
$821$	−21.0557	−0.734850	−0.367425	+	0.930053i	$0.619761\pi$
$822$	0	0
$823$	27.5410	0.960020	0.480010	−	0.877263i	$-0.340633\pi$
$823$	27.5410	0.960020	0.480010	+	0.877263i	$0.340633\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.4721	0.364152	0.182076	−	0.983284i	$-0.441718\pi$
$827$	10.4721	0.364152	0.182076	+	0.983284i	$0.441718\pi$
$828$	0	0
$829$	−40.2492	−1.39791	−0.698957	−	0.715164i	$-0.746352\pi$
$829$	−40.2492	−1.39791	−0.698957	+	0.715164i	$0.746352\pi$
$830$	0	0
$831$	−14.5967	−0.506356
$832$	0	0
$833$	28.6525	0.992749
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−15.0000	−0.518476
$838$	0	0
$839$	0.875388	0.0302218	0.0151109	−	0.999886i	$-0.495190\pi$
$839$	0.875388	0.0302218	0.0151109	+	0.999886i	$0.495190\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−29.5967	−1.01937
$844$	0	0
$845$	0	0
$846$	0	0
$847$	12.8754	0.442404
$848$	0	0
$849$	31.9574	1.09678
$850$	0	0
$851$	−3.23607	−0.110931
$852$	0	0
$853$	37.4164	1.28111	0.640557	−	0.767911i	$-0.278704\pi$
$853$	37.4164	1.28111	0.640557	+	0.767911i	$0.278704\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.47214	0.255243	0.127622	−	0.991823i	$-0.459266\pi$
$857$	7.47214	0.255243	0.127622	+	0.991823i	$0.459266\pi$
$858$	0	0
$859$	3.29180	0.112315	0.0561573	−	0.998422i	$-0.482115\pi$
$859$	3.29180	0.112315	0.0561573	+	0.998422i	$0.482115\pi$
$860$	0	0
$861$	−15.1246	−0.515445
$862$	0	0
$863$	45.5410	1.55023	0.775117	−	0.631818i	$-0.217691\pi$
$863$	45.5410	1.55023	0.775117	+	0.631818i	$0.217691\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	23.2918	0.791031
$868$	0	0
$869$	−5.30495	−0.179958
$870$	0	0
$871$	21.7082	0.735554
$872$	0	0
$873$	−8.58359	−0.290511
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27.5279	0.929550	0.464775	−	0.885429i	$-0.346135\pi$
$877$	27.5279	0.929550	0.464775	+	0.885429i	$0.346135\pi$
$878$	0	0
$879$	23.4164	0.789816
$880$	0	0
$881$	21.8197	0.735123	0.367562	−	0.929999i	$-0.380193\pi$
$881$	21.8197	0.735123	0.367562	+	0.929999i	$0.380193\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35.0689	−1.17750	−0.588749	−	0.808316i	$-0.700379\pi$
$887$	−35.0689	−1.17750	−0.588749	+	0.808316i	$0.700379\pi$
$888$	0	0
$889$	25.5967	0.858487
$890$	0	0
$891$	−8.40325	−0.281520
$892$	0	0
$893$	4.47214	0.149654
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.70820	−0.223980
$898$	0	0
$899$	−20.1246	−0.671193
$900$	0	0
$901$	−44.3607	−1.47787
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.2492	1.33645	0.668227	−	0.743958i	$-0.267054\pi$
$907$	40.2492	1.33645	0.668227	+	0.743958i	$0.267054\pi$
$908$	0	0
$909$	−8.94427	−0.296663
$910$	0	0
$911$	31.3050	1.03718	0.518590	−	0.855023i	$-0.326457\pi$
$911$	31.3050	1.03718	0.518590	+	0.855023i	$0.326457\pi$
$912$	0	0
$913$	−10.1115	−0.334640
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.54102	0.216003
$918$	0	0
$919$	−0.875388	−0.0288764	−0.0144382	−	0.999896i	$-0.504596\pi$
$919$	−0.875388	−0.0288764	−0.0144382	+	0.999896i	$0.504596\pi$
$920$	0	0
$921$	41.3050	1.36104
$922$	0	0
$923$	23.2918	0.766659
$924$	0	0
$925$	0	0
$926$	0	0
$927$	36.3607	1.19424
$928$	0	0
$929$	−41.9443	−1.37615	−0.688073	−	0.725641i	$-0.741543\pi$
$929$	−41.9443	−1.37615	−0.688073	+	0.725641i	$0.741543\pi$
$930$	0	0
$931$	−10.9443	−0.358684
$932$	0	0
$933$	20.5279	0.672052
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−11.8197	−0.386131	−0.193066	−	0.981186i	$-0.561843\pi$
$937$	−11.8197	−0.386131	−0.193066	+	0.981186i	$0.561843\pi$
$938$	0	0
$939$	45.5279	1.48575
$940$	0	0
$941$	−24.6525	−0.803648	−0.401824	−	0.915717i	$-0.631624\pi$
$941$	−24.6525	−0.803648	−0.401824	+	0.915717i	$0.631624\pi$
$942$	0	0
$943$	5.47214	0.178197
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.1803	−1.07822	−0.539108	−	0.842237i	$-0.681239\pi$
$947$	−33.1803	−1.07822	−0.539108	+	0.842237i	$0.681239\pi$
$948$	0	0
$949$	46.4164	1.50674
$950$	0	0
$951$	3.16718	0.102703
$952$	0	0
$953$	−11.5279	−0.373424	−0.186712	−	0.982415i	$-0.559783\pi$
$953$	−11.5279	−0.373424	−0.186712	+	0.982415i	$0.559783\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.12461	−0.165655
$958$	0	0
$959$	17.1672	0.554357
$960$	0	0
$961$	14.0000	0.451613
$962$	0	0
$963$	−26.8328	−0.864675
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−39.5410	−1.27155	−0.635777	−	0.771873i	$-0.719320\pi$
$967$	−39.5410	−1.27155	−0.635777	+	0.771873i	$0.719320\pi$
$968$	0	0
$969$	−23.4164	−0.752243
$970$	0	0
$971$	−7.52786	−0.241581	−0.120790	−	0.992678i	$-0.538543\pi$
$971$	−7.52786	−0.241581	−0.120790	+	0.992678i	$0.538543\pi$
$972$	0	0
$973$	3.34752	0.107317
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.6525	1.74849	0.874244	−	0.485487i	$-0.161358\pi$
$977$	54.6525	1.74849	0.874244	+	0.485487i	$0.161358\pi$
$978$	0	0
$979$	−1.16718	−0.0373034
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−31.5279	−1.00558	−0.502791	−	0.864408i	$-0.667694\pi$
$983$	−31.5279	−1.00558	−0.502791	+	0.864408i	$0.667694\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.18034	−0.196722
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	−26.0557	−0.826854
$994$	0	0
$995$	0	0
$996$	0	0
$997$	36.8328	1.16651	0.583253	−	0.812290i	$-0.301779\pi$
$997$	36.8328	1.16651	0.583253	+	0.812290i	$0.301779\pi$
$998$	0	0
$999$	7.23607	0.228939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.bt.1.2		2
4.3	odd	2		575.2.a.f.1.2		2
5.4	even	2		368.2.a.h.1.1		2
12.11	even	2		5175.2.a.be.1.1		2
15.14	odd	2		3312.2.a.ba.1.2		2
20.3	even	4		575.2.b.d.24.1		4
20.7	even	4		575.2.b.d.24.4		4
20.19	odd	2		23.2.a.a.1.1	✓	2
40.19	odd	2		1472.2.a.t.1.1		2
40.29	even	2		1472.2.a.s.1.2		2
60.59	even	2		207.2.a.d.1.2		2
115.114	odd	2		8464.2.a.bb.1.1		2
140.139	even	2		1127.2.a.c.1.1		2
220.219	even	2		2783.2.a.c.1.2		2
260.259	odd	2		3887.2.a.i.1.2		2
340.339	odd	2		6647.2.a.b.1.1		2
380.379	even	2		8303.2.a.e.1.2		2
460.19	even	22		529.2.c.n.177.2		20
460.39	odd	22		529.2.c.o.118.2		20
460.59	odd	22		529.2.c.o.399.2		20
460.79	even	22		529.2.c.n.399.2		20
460.99	even	22		529.2.c.n.118.2		20
460.119	odd	22		529.2.c.o.177.2		20
460.159	even	22		529.2.c.n.487.1		20
460.179	odd	22		529.2.c.o.255.1		20
460.199	even	22		529.2.c.n.501.2		20
460.219	odd	22		529.2.c.o.466.1		20
460.239	odd	22		529.2.c.o.334.1		20
460.259	odd	22		529.2.c.o.266.2		20
460.279	odd	22		529.2.c.o.170.2		20
460.319	even	22		529.2.c.n.170.2		20
460.339	even	22		529.2.c.n.266.2		20
460.359	even	22		529.2.c.n.334.1		20
460.379	even	22		529.2.c.n.466.1		20
460.399	odd	22		529.2.c.o.501.2		20
460.419	even	22		529.2.c.n.255.1		20
460.439	odd	22		529.2.c.o.487.1		20
460.459	even	2		529.2.a.a.1.1		2
1380.1379	odd	2		4761.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.2.a.a.1.1	✓	2	20.19	odd	2
207.2.a.d.1.2		2	60.59	even	2
368.2.a.h.1.1		2	5.4	even	2
529.2.a.a.1.1		2	460.459	even	2
529.2.c.n.118.2		20	460.99	even	22
529.2.c.n.170.2		20	460.319	even	22
529.2.c.n.177.2		20	460.19	even	22
529.2.c.n.255.1		20	460.419	even	22
529.2.c.n.266.2		20	460.339	even	22
529.2.c.n.334.1		20	460.359	even	22
529.2.c.n.399.2		20	460.79	even	22
529.2.c.n.466.1		20	460.379	even	22
529.2.c.n.487.1		20	460.159	even	22
529.2.c.n.501.2		20	460.199	even	22
529.2.c.o.118.2		20	460.39	odd	22
529.2.c.o.170.2		20	460.279	odd	22
529.2.c.o.177.2		20	460.119	odd	22
529.2.c.o.255.1		20	460.179	odd	22
529.2.c.o.266.2		20	460.259	odd	22
529.2.c.o.334.1		20	460.239	odd	22
529.2.c.o.399.2		20	460.59	odd	22
529.2.c.o.466.1		20	460.219	odd	22
529.2.c.o.487.1		20	460.439	odd	22
529.2.c.o.501.2		20	460.399	odd	22
575.2.a.f.1.2		2	4.3	odd	2
575.2.b.d.24.1		4	20.3	even	4
575.2.b.d.24.4		4	20.7	even	4
1127.2.a.c.1.1		2	140.139	even	2
1472.2.a.s.1.2		2	40.29	even	2
1472.2.a.t.1.1		2	40.19	odd	2
2783.2.a.c.1.2		2	220.219	even	2
3312.2.a.ba.1.2		2	15.14	odd	2
3887.2.a.i.1.2		2	260.259	odd	2
4761.2.a.w.1.2		2	1380.1379	odd	2
5175.2.a.be.1.1		2	12.11	even	2
6647.2.a.b.1.1		2	340.339	odd	2
8303.2.a.e.1.2		2	380.379	even	2
8464.2.a.bb.1.1		2	115.114	odd	2
9200.2.a.bt.1.2		2	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{3} -1.23607 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+2.23607 q^{3} -1.23607 q^{7} +2.00000 q^{9} +0.763932 q^{11} -3.00000 q^{13} -5.23607 q^{17} +2.00000 q^{19} -2.76393 q^{21} +1.00000 q^{23} -2.23607 q^{27} -3.00000 q^{29} +6.70820 q^{31} +1.70820 q^{33} -3.23607 q^{37} -6.70820 q^{39} +5.47214 q^{41} +2.23607 q^{47} -5.47214 q^{49} -11.7082 q^{51} +8.47214 q^{53} +4.47214 q^{57} +2.47214 q^{59} +10.9443 q^{61} -2.47214 q^{63} -7.23607 q^{67} +2.23607 q^{69} -7.76393 q^{71} -15.4721 q^{73} -0.944272 q^{77} -6.94427 q^{79} -11.0000 q^{81} -13.2361 q^{83} -6.70820 q^{87} -1.52786 q^{89} +3.70820 q^{91} +15.0000 q^{93} -4.29180 q^{97} +1.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^7 + 4 * q^9	\( 2 q + 2 q^{7} + 4 q^{9} + 6 q^{11} - 6 q^{13} - 6 q^{17} + 4 q^{19} - 10 q^{21} + 2 q^{23} - 6 q^{29} - 10 q^{33} - 2 q^{37} + 2 q^{41} - 2 q^{49} - 10 q^{51} + 8 q^{53} - 4 q^{59} + 4 q^{61} + 4 q^{63} - 10 q^{67} - 20 q^{71} - 22 q^{73} + 16 q^{77} + 4 q^{79} - 22 q^{81} - 22 q^{83} - 12 q^{89} - 6 q^{91} + 30 q^{93} - 22 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^7 + 4 * q^9 + 6 * q^11 - 6 * q^13 - 6 * q^17 + 4 * q^19 - 10 * q^21 + 2 * q^23 - 6 * q^29 - 10 * q^33 - 2 * q^37 + 2 * q^41 - 2 * q^49 - 10 * q^51 + 8 * q^53 - 4 * q^59 + 4 * q^61 + 4 * q^63 - 10 * q^67 - 20 * q^71 - 22 * q^73 + 16 * q^77 + 4 * q^79 - 22 * q^81 - 22 * q^83 - 12 * q^89 - 6 * q^91 + 30 * q^93 - 22 * q^97 + 12 * q^99

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