LMFDB - Embedded newform 9200.2.a.cf.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.43163$ 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)

$f(q)$	$=$	$q+1.43163 q^{3} +3.08719 q^{7} -0.950444 q^{9} +O(q^{10})$	$q+1.43163 q^{3} +3.08719 q^{7} -0.950444 q^{9} +6.46926 q^{11} -3.95044 q^{13} +3.43163 q^{17} -3.08719 q^{19} +4.41970 q^{21} -1.00000 q^{23} -5.65556 q^{27} +0.863254 q^{29} +5.95044 q^{31} +9.26157 q^{33} +7.03763 q^{37} -5.65556 q^{39} +5.60601 q^{41} +8.00000 q^{43} +3.90089 q^{47} +2.53074 q^{49} +4.91281 q^{51} +6.00000 q^{53} -4.41970 q^{57} -6.86325 q^{59} -13.5069 q^{61} -2.93420 q^{63} -10.0753 q^{67} -1.43163 q^{69} -2.56837 q^{71} -5.90089 q^{73} +19.9718 q^{77} -15.8018 q^{79} -5.24533 q^{81} +9.03763 q^{83} +1.23586 q^{87} +16.7641 q^{89} -12.1958 q^{91} +8.51882 q^{93} +14.2949 q^{97} -6.14867 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{7} + 10 q^{9}+O(q^{10}) $ 3 * q + q^3 + 3 * q^7 + 10 * q^9	$ 3 q + q^{3} + 3 q^{7} + 10 q^{9} - 3 q^{11} + q^{13} + 7 q^{17} - 3 q^{19} - 22 q^{21} - 3 q^{23} - 14 q^{27} - 4 q^{29} + 5 q^{31} + 9 q^{33} + 2 q^{37} - 14 q^{39} + q^{41} + 24 q^{43} - 14 q^{47} + 30 q^{49} + 21 q^{51} + 18 q^{53} + 22 q^{57} - 14 q^{59} + q^{61} + 8 q^{63} + 8 q^{67} - q^{69} - 11 q^{71} + 8 q^{73} + 24 q^{77} + 4 q^{79} + 7 q^{81} + 8 q^{83} + 36 q^{87} + 18 q^{89} - q^{91} + 16 q^{93} + 33 q^{97} - 57 q^{99}+O(q^{100}) $ 3 * q + q^3 + 3 * q^7 + 10 * q^9 - 3 * q^11 + q^13 + 7 * q^17 - 3 * q^19 - 22 * q^21 - 3 * q^23 - 14 * q^27 - 4 * q^29 + 5 * q^31 + 9 * q^33 + 2 * q^37 - 14 * q^39 + q^41 + 24 * q^43 - 14 * q^47 + 30 * q^49 + 21 * q^51 + 18 * q^53 + 22 * q^57 - 14 * q^59 + q^61 + 8 * q^63 + 8 * q^67 - q^69 - 11 * q^71 + 8 * q^73 + 24 * q^77 + 4 * q^79 + 7 * q^81 + 8 * q^83 + 36 * q^87 + 18 * q^89 - q^91 + 16 * q^93 + 33 * q^97 - 57 * 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
$3$	1.43163	0.826550	0.413275	−	0.910606i	$-0.364385\pi$
$3$	1.43163	0.826550	0.413275	+	0.910606i	$0.364385\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.08719	1.16685	0.583424	−	0.812168i	$-0.301713\pi$
$7$	3.08719	1.16685	0.583424	+	0.812168i	$0.301713\pi$
$8$	0	0
$9$	−0.950444	−0.316815
$10$	0	0
$11$	6.46926	1.95056	0.975278	−	0.220983i	$-0.0709265\pi$
$11$	6.46926	1.95056	0.975278	+	0.220983i	$0.0709265\pi$
$12$	0	0
$13$	−3.95044	−1.09566	−0.547828	−	0.836591i	$-0.684545\pi$
$13$	−3.95044	−1.09566	−0.547828	+	0.836591i	$0.684545\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.43163	0.832292	0.416146	−	0.909298i	$-0.363381\pi$
$17$	3.43163	0.832292	0.416146	+	0.909298i	$0.363381\pi$
$18$	0	0
$19$	−3.08719	−0.708250	−0.354125	−	0.935198i	$-0.615221\pi$
$19$	−3.08719	−0.708250	−0.354125	+	0.935198i	$0.615221\pi$
$20$	0	0
$21$	4.41970	0.964459
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.65556	−1.08841
$28$	0	0
$29$	0.863254	0.160302	0.0801511	−	0.996783i	$-0.474460\pi$
$29$	0.863254	0.160302	0.0801511	+	0.996783i	$0.474460\pi$
$30$	0	0
$31$	5.95044	1.06873	0.534366	−	0.845253i	$-0.320551\pi$
$31$	5.95044	1.06873	0.534366	+	0.845253i	$0.320551\pi$
$32$	0	0
$33$	9.26157	1.61223
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.03763	1.15698	0.578490	−	0.815690i	$-0.303642\pi$
$37$	7.03763	1.15698	0.578490	+	0.815690i	$0.303642\pi$
$38$	0	0
$39$	−5.65556	−0.905615
$40$	0	0
$41$	5.60601	0.875511	0.437756	−	0.899094i	$-0.355774\pi$
$41$	5.60601	0.875511	0.437756	+	0.899094i	$0.355774\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.90089	0.569003	0.284501	−	0.958676i	$-0.408172\pi$
$47$	3.90089	0.569003	0.284501	+	0.958676i	$0.408172\pi$
$48$	0	0
$49$	2.53074	0.361534
$50$	0	0
$51$	4.91281	0.687931
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.41970	−0.585404
$58$	0	0
$59$	−6.86325	−0.893520	−0.446760	−	0.894654i	$-0.647422\pi$
$59$	−6.86325	−0.893520	−0.446760	+	0.894654i	$0.647422\pi$
$60$	0	0
$61$	−13.5069	−1.72938	−0.864690	−	0.502305i	$-0.832485\pi$
$61$	−13.5069	−1.72938	−0.864690	+	0.502305i	$0.832485\pi$
$62$	0	0
$63$	−2.93420	−0.369674
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.0753	−1.23089	−0.615445	−	0.788180i	$-0.711024\pi$
$67$	−10.0753	−1.23089	−0.615445	+	0.788180i	$0.711024\pi$
$68$	0	0
$69$	−1.43163	−0.172348
$70$	0	0
$71$	−2.56837	−0.304810	−0.152405	−	0.988318i	$-0.548702\pi$
$71$	−2.56837	−0.304810	−0.152405	+	0.988318i	$0.548702\pi$
$72$	0	0
$73$	−5.90089	−0.690647	−0.345323	−	0.938484i	$-0.612231\pi$
$73$	−5.90089	−0.690647	−0.345323	+	0.938484i	$0.612231\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.9718	2.27600
$78$	0	0
$79$	−15.8018	−1.77784	−0.888919	−	0.458064i	$-0.848543\pi$
$79$	−15.8018	−1.77784	−0.888919	+	0.458064i	$0.848543\pi$
$80$	0	0
$81$	−5.24533	−0.582814
$82$	0	0
$83$	9.03763	0.992009	0.496005	−	0.868320i	$-0.334800\pi$
$83$	9.03763	0.992009	0.496005	+	0.868320i	$0.334800\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.23586	0.132498
$88$	0	0
$89$	16.7641	1.77700	0.888498	−	0.458881i	$-0.151750\pi$
$89$	16.7641	1.77700	0.888498	+	0.458881i	$0.151750\pi$
$90$	0	0
$91$	−12.1958	−1.27846
$92$	0	0
$93$	8.51882	0.883360
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.2949	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$97$	14.2949	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$98$	0	0
$99$	−6.14867	−0.617964
$100$	0	0
$101$	0.863254	0.0858970	0.0429485	−	0.999077i	$-0.486325\pi$
$101$	0.863254	0.0858970	0.0429485	+	0.999077i	$0.486325\pi$
$102$	0	0
$103$	1.53074	0.150828	0.0754141	−	0.997152i	$-0.475972\pi$
$103$	1.53074	0.150828	0.0754141	+	0.997152i	$0.475972\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.6274	1.70410	0.852052	−	0.523457i	$-0.175358\pi$
$107$	17.6274	1.70410	0.852052	+	0.523457i	$0.175358\pi$
$108$	0	0
$109$	−2.91281	−0.278997	−0.139498	−	0.990222i	$-0.544549\pi$
$109$	−2.91281	−0.278997	−0.139498	+	0.990222i	$0.544549\pi$
$110$	0	0
$111$	10.0753	0.956302
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.75467	0.347120
$118$	0	0
$119$	10.5941	0.971158
$120$	0	0
$121$	30.8513	2.80467
$122$	0	0
$123$	8.02571	0.723654
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.9385	1.85799	0.928997	−	0.370088i	$-0.120673\pi$
$127$	20.9385	1.85799	0.928997	+	0.370088i	$0.120673\pi$
$128$	0	0
$129$	11.4530	1.00838
$130$	0	0
$131$	−10.7641	−0.940467	−0.470234	−	0.882542i	$-0.655830\pi$
$131$	−10.7641	−0.940467	−0.470234	+	0.882542i	$0.655830\pi$
$132$	0	0
$133$	−9.53074	−0.826420
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.26157	−0.278655	−0.139327	−	0.990246i	$-0.544494\pi$
$137$	−3.26157	−0.278655	−0.139327	+	0.990246i	$0.544494\pi$
$138$	0	0
$139$	16.9385	1.43671	0.718353	−	0.695678i	$-0.244896\pi$
$139$	16.9385	1.43671	0.718353	+	0.695678i	$0.244896\pi$
$140$	0	0
$141$	5.58462	0.470310
$142$	0	0
$143$	−25.5565	−2.13714
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.62308	0.298826
$148$	0	0
$149$	3.26157	0.267198	0.133599	−	0.991035i	$-0.457347\pi$
$149$	3.26157	0.267198	0.133599	+	0.991035i	$0.457347\pi$
$150$	0	0
$151$	−0.294881	−0.0239971	−0.0119986	−	0.999928i	$-0.503819\pi$
$151$	−0.294881	−0.0239971	−0.0119986	+	0.999928i	$0.503819\pi$
$152$	0	0
$153$	−3.26157	−0.263682
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.13675	−0.569574	−0.284787	−	0.958591i	$-0.591923\pi$
$157$	−7.13675	−0.569574	−0.284787	+	0.958591i	$0.591923\pi$
$158$	0	0
$159$	8.58976	0.681212
$160$	0	0
$161$	−3.08719	−0.243305
$162$	0	0
$163$	−8.12482	−0.636385	−0.318193	−	0.948026i	$-0.603076\pi$
$163$	−8.12482	−0.636385	−0.318193	+	0.948026i	$0.603076\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.80178	−0.603719	−0.301860	−	0.953352i	$-0.597607\pi$
$167$	−7.80178	−0.603719	−0.301860	+	0.953352i	$0.597607\pi$
$168$	0	0
$169$	2.60601	0.200462
$170$	0	0
$171$	2.93420	0.224384
$172$	0	0
$173$	19.3325	1.46982	0.734912	−	0.678163i	$-0.237223\pi$
$173$	19.3325	1.46982	0.734912	+	0.678163i	$0.237223\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.82562	−0.738539
$178$	0	0
$179$	−2.17438	−0.162521	−0.0812604	−	0.996693i	$-0.525895\pi$
$179$	−2.17438	−0.162521	−0.0812604	+	0.996693i	$0.525895\pi$
$180$	0	0
$181$	−7.26157	−0.539748	−0.269874	−	0.962896i	$-0.586982\pi$
$181$	−7.26157	−0.539748	−0.269874	+	0.962896i	$0.586982\pi$
$182$	0	0
$183$	−19.3368	−1.42942
$184$	0	0
$185$	0	0
$186$	0	0
$187$	22.2001	1.62343
$188$	0	0
$189$	−17.4598	−1.27001
$190$	0	0
$191$	−8.58976	−0.621533	−0.310767	−	0.950486i	$-0.600586\pi$
$191$	−8.58976	−0.621533	−0.310767	+	0.950486i	$0.600586\pi$
$192$	0	0
$193$	−2.44787	−0.176202	−0.0881008	−	0.996112i	$-0.528080\pi$
$193$	−2.44787	−0.176202	−0.0881008	+	0.996112i	$0.528080\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.7428	0.765389	0.382695	−	0.923875i	$-0.374996\pi$
$197$	10.7428	0.765389	0.382695	+	0.923875i	$0.374996\pi$
$198$	0	0
$199$	11.3111	0.801824	0.400912	−	0.916116i	$-0.368693\pi$
$199$	11.3111	0.801824	0.400912	+	0.916116i	$0.368693\pi$
$200$	0	0
$201$	−14.4240	−1.01739
$202$	0	0
$203$	2.66503	0.187048
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.950444	0.0660604
$208$	0	0
$209$	−19.9718	−1.38148
$210$	0	0
$211$	−23.1129	−1.59116	−0.795579	−	0.605850i	$-0.792833\pi$
$211$	−23.1129	−1.59116	−0.795579	+	0.605850i	$0.792833\pi$
$212$	0	0
$213$	−3.67695	−0.251941
$214$	0	0
$215$	0	0
$216$	0	0
$217$	18.3701	1.24705
$218$	0	0
$219$	−8.44787	−0.570854
$220$	0	0
$221$	−13.5565	−0.911906
$222$	0	0
$223$	−5.72651	−0.383475	−0.191738	−	0.981446i	$-0.561412\pi$
$223$	−5.72651	−0.383475	−0.191738	+	0.981446i	$0.561412\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.6274	−1.16997	−0.584986	−	0.811044i	$-0.698900\pi$
$227$	−17.6274	−1.16997	−0.584986	+	0.811044i	$0.698900\pi$
$228$	0	0
$229$	−16.0753	−1.06228	−0.531142	−	0.847283i	$-0.678237\pi$
$229$	−16.0753	−1.06228	−0.531142	+	0.847283i	$0.678237\pi$
$230$	0	0
$231$	28.5922	1.88123
$232$	0	0
$233$	6.44787	0.422414	0.211207	−	0.977441i	$-0.432261\pi$
$233$	6.44787	0.422414	0.211207	+	0.977441i	$0.432261\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−22.6222	−1.46947
$238$	0	0
$239$	−4.34876	−0.281298	−0.140649	−	0.990060i	$-0.544919\pi$
$239$	−4.34876	−0.281298	−0.140649	+	0.990060i	$0.544919\pi$
$240$	0	0
$241$	0.764142	0.0492227	0.0246114	−	0.999697i	$-0.492165\pi$
$241$	0.764142	0.0492227	0.0246114	+	0.999697i	$0.492165\pi$
$242$	0	0
$243$	9.45734	0.606688
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.1958	0.775998
$248$	0	0
$249$	12.9385	0.819945
$250$	0	0
$251$	22.5402	1.42273	0.711363	−	0.702825i	$-0.248078\pi$
$251$	22.5402	1.42273	0.711363	+	0.702825i	$0.248078\pi$
$252$	0	0
$253$	−6.46926	−0.406719
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.90089	0.617600	0.308800	−	0.951127i	$-0.400073\pi$
$257$	9.90089	0.617600	0.308800	+	0.951127i	$0.400073\pi$
$258$	0	0
$259$	21.7265	1.35002
$260$	0	0
$261$	−0.820475	−0.0507861
$262$	0	0
$263$	−7.25725	−0.447501	−0.223751	−	0.974646i	$-0.571830\pi$
$263$	−7.25725	−0.447501	−0.223751	+	0.974646i	$0.571830\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	24.0000	1.46878
$268$	0	0
$269$	−6.93852	−0.423049	−0.211525	−	0.977373i	$-0.567843\pi$
$269$	−6.93852	−0.423049	−0.211525	+	0.977373i	$0.567843\pi$
$270$	0	0
$271$	10.2992	0.625632	0.312816	−	0.949814i	$-0.398728\pi$
$271$	10.2992	0.625632	0.312816	+	0.949814i	$0.398728\pi$
$272$	0	0
$273$	−17.4598	−1.05671
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.8018	1.06961	0.534803	−	0.844977i	$-0.320386\pi$
$277$	17.8018	1.06961	0.534803	+	0.844977i	$0.320386\pi$
$278$	0	0
$279$	−5.65556	−0.338590
$280$	0	0
$281$	−18.9385	−1.12978	−0.564889	−	0.825167i	$-0.691081\pi$
$281$	−18.9385	−1.12978	−0.564889	+	0.825167i	$0.691081\pi$
$282$	0	0
$283$	12.6889	0.754275	0.377138	−	0.926157i	$-0.376908\pi$
$283$	12.6889	0.754275	0.377138	+	0.926157i	$0.376908\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.3068	1.02159
$288$	0	0
$289$	−5.22394	−0.307290
$290$	0	0
$291$	20.4649	1.19968
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−36.5873	−2.12301
$298$	0	0
$299$	3.95044	0.228460
$300$	0	0
$301$	24.6975	1.42354
$302$	0	0
$303$	1.23586	0.0709982
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−22.9599	−1.31039	−0.655196	−	0.755459i	$-0.727414\pi$
$307$	−22.9599	−1.31039	−0.655196	+	0.755459i	$0.727414\pi$
$308$	0	0
$309$	2.19145	0.124667
$310$	0	0
$311$	−18.0753	−1.02495	−0.512477	−	0.858701i	$-0.671272\pi$
$311$	−18.0753	−1.02495	−0.512477	+	0.858701i	$0.671272\pi$
$312$	0	0
$313$	−15.1625	−0.857033	−0.428516	−	0.903534i	$-0.640964\pi$
$313$	−15.1625	−0.857033	−0.428516	+	0.903534i	$0.640964\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.0257	−0.787762	−0.393881	−	0.919161i	$-0.628868\pi$
$317$	−14.0257	−0.787762	−0.393881	+	0.919161i	$0.628868\pi$
$318$	0	0
$319$	5.58462	0.312678
$320$	0	0
$321$	25.2359	1.40853
$322$	0	0
$323$	−10.5941	−0.589471
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.17006	−0.230605
$328$	0	0
$329$	12.0428	0.663940
$330$	0	0
$331$	24.7403	1.35985	0.679925	−	0.733282i	$-0.262012\pi$
$331$	24.7403	1.35985	0.679925	+	0.733282i	$0.262012\pi$
$332$	0	0
$333$	−6.68888	−0.366548
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.7146	−0.801555	−0.400777	−	0.916176i	$-0.631260\pi$
$337$	−14.7146	−0.801555	−0.400777	+	0.916176i	$0.631260\pi$
$338$	0	0
$339$	8.58976	0.466532
$340$	0	0
$341$	38.4950	2.08462
$342$	0	0
$343$	−13.7975	−0.744993
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.43163	0.506316	0.253158	−	0.967425i	$-0.418531\pi$
$347$	9.43163	0.506316	0.253158	+	0.967425i	$0.418531\pi$
$348$	0	0
$349$	18.3488	0.982187	0.491093	−	0.871107i	$-0.336597\pi$
$349$	18.3488	0.982187	0.491093	+	0.871107i	$0.336597\pi$
$350$	0	0
$351$	22.3420	1.19253
$352$	0	0
$353$	33.1129	1.76242	0.881211	−	0.472723i	$-0.156729\pi$
$353$	33.1129	1.76242	0.881211	+	0.472723i	$0.156729\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	15.1668	0.802711
$358$	0	0
$359$	−33.0376	−1.74366	−0.871830	−	0.489809i	$-0.837067\pi$
$359$	−33.0376	−1.74366	−0.871830	+	0.489809i	$0.837067\pi$
$360$	0	0
$361$	−9.46926	−0.498382
$362$	0	0
$363$	44.1676	2.31820
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.27349	−0.118675	−0.0593376	−	0.998238i	$-0.518899\pi$
$367$	−2.27349	−0.118675	−0.0593376	+	0.998238i	$0.518899\pi$
$368$	0	0
$369$	−5.32819	−0.277375
$370$	0	0
$371$	18.5231	0.961673
$372$	0	0
$373$	−23.9762	−1.24144	−0.620719	−	0.784033i	$-0.713159\pi$
$373$	−23.9762	−1.24144	−0.620719	+	0.784033i	$0.713159\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.41024	−0.175636
$378$	0	0
$379$	13.8795	0.712942	0.356471	−	0.934306i	$-0.383980\pi$
$379$	13.8795	0.712942	0.356471	+	0.934306i	$0.383980\pi$
$380$	0	0
$381$	29.9762	1.53572
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.60355	−0.386510
$388$	0	0
$389$	−26.6436	−1.35089	−0.675443	−	0.737412i	$-0.736048\pi$
$389$	−26.6436	−1.35089	−0.675443	+	0.737412i	$0.736048\pi$
$390$	0	0
$391$	−3.43163	−0.173545
$392$	0	0
$393$	−15.4102	−0.777344
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.66749	0.435009	0.217504	−	0.976059i	$-0.430208\pi$
$397$	8.66749	0.435009	0.217504	+	0.976059i	$0.430208\pi$
$398$	0	0
$399$	−13.6445	−0.683078
$400$	0	0
$401$	39.9762	1.99631	0.998157	−	0.0606854i	$-0.0193286\pi$
$401$	39.9762	1.99631	0.998157	+	0.0606854i	$0.0193286\pi$
$402$	0	0
$403$	−23.5069	−1.17096
$404$	0	0
$405$	0	0
$406$	0	0
$407$	45.5283	2.25675
$408$	0	0
$409$	30.1248	1.48958	0.744788	−	0.667301i	$-0.232550\pi$
$409$	30.1248	1.48958	0.744788	+	0.667301i	$0.232550\pi$
$410$	0	0
$411$	−4.66935	−0.230322
$412$	0	0
$413$	−21.1882	−1.04260
$414$	0	0
$415$	0	0
$416$	0	0
$417$	24.2496	1.18751
$418$	0	0
$419$	−25.8770	−1.26418	−0.632088	−	0.774897i	$-0.717802\pi$
$419$	−25.8770	−1.26418	−0.632088	+	0.774897i	$0.717802\pi$
$420$	0	0
$421$	−10.7146	−0.522197	−0.261098	−	0.965312i	$-0.584085\pi$
$421$	−10.7146	−0.522197	−0.261098	+	0.965312i	$0.584085\pi$
$422$	0	0
$423$	−3.70757	−0.180268
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−41.6983	−2.01792
$428$	0	0
$429$	−36.5873	−1.76645
$430$	0	0
$431$	−32.2496	−1.55341	−0.776705	−	0.629864i	$-0.783111\pi$
$431$	−32.2496	−1.55341	−0.776705	+	0.629864i	$0.783111\pi$
$432$	0	0
$433$	−20.6393	−0.991862	−0.495931	−	0.868362i	$-0.665173\pi$
$433$	−20.6393	−0.991862	−0.495931	+	0.868362i	$0.665173\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.08719	0.147680
$438$	0	0
$439$	16.2239	0.774326	0.387163	−	0.922011i	$-0.373455\pi$
$439$	16.2239	0.774326	0.387163	+	0.922011i	$0.373455\pi$
$440$	0	0
$441$	−2.40533	−0.114539
$442$	0	0
$443$	−15.6770	−0.744834	−0.372417	−	0.928065i	$-0.621471\pi$
$443$	−15.6770	−0.744834	−0.372417	+	0.928065i	$0.621471\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.66935	0.220853
$448$	0	0
$449$	22.5727	1.06527	0.532636	−	0.846345i	$-0.321202\pi$
$449$	22.5727	1.06527	0.532636	+	0.846345i	$0.321202\pi$
$450$	0	0
$451$	36.2667	1.70773
$452$	0	0
$453$	−0.422160	−0.0198348
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.45302	0.0679693	0.0339846	−	0.999422i	$-0.489180\pi$
$457$	1.45302	0.0679693	0.0339846	+	0.999422i	$0.489180\pi$
$458$	0	0
$459$	−19.4078	−0.905878
$460$	0	0
$461$	−29.7027	−1.38339	−0.691695	−	0.722189i	$-0.743136\pi$
$461$	−29.7027	−1.38339	−0.691695	+	0.722189i	$0.743136\pi$
$462$	0	0
$463$	22.9624	1.06715	0.533576	−	0.845752i	$-0.320848\pi$
$463$	22.9624	1.06715	0.533576	+	0.845752i	$0.320848\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.48550	0.438937	0.219468	−	0.975620i	$-0.429568\pi$
$467$	9.48550	0.438937	0.219468	+	0.975620i	$0.429568\pi$
$468$	0	0
$469$	−31.1043	−1.43626
$470$	0	0
$471$	−10.2172	−0.470782
$472$	0	0
$473$	51.7541	2.37966
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.70266	−0.261107
$478$	0	0
$479$	30.5659	1.39659	0.698296	−	0.715809i	$-0.253942\pi$
$479$	30.5659	1.39659	0.698296	+	0.715809i	$0.253942\pi$
$480$	0	0
$481$	−27.8018	−1.26765
$482$	0	0
$483$	−4.41970	−0.201104
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−21.1367	−0.957797	−0.478899	−	0.877870i	$-0.658964\pi$
$487$	−21.1367	−0.957797	−0.478899	+	0.877870i	$0.658964\pi$
$488$	0	0
$489$	−11.6317	−0.526004
$490$	0	0
$491$	28.0514	1.26594	0.632971	−	0.774175i	$-0.281835\pi$
$491$	28.0514	1.26594	0.632971	+	0.774175i	$0.281835\pi$
$492$	0	0
$493$	2.96237	0.133418
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.92905	−0.355667
$498$	0	0
$499$	−3.80178	−0.170191	−0.0850954	−	0.996373i	$-0.527120\pi$
$499$	−3.80178	−0.170191	−0.0850954	+	0.996373i	$0.527120\pi$
$500$	0	0
$501$	−11.1692	−0.499005
$502$	0	0
$503$	19.0872	0.851056	0.425528	−	0.904945i	$-0.360088\pi$
$503$	19.0872	0.851056	0.425528	+	0.904945i	$0.360088\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.73083	0.165692
$508$	0	0
$509$	−9.41024	−0.417101	−0.208551	−	0.978012i	$-0.566875\pi$
$509$	−9.41024	−0.417101	−0.208551	+	0.978012i	$0.566875\pi$
$510$	0	0
$511$	−18.2172	−0.805880
$512$	0	0
$513$	17.4598	0.770869
$514$	0	0
$515$	0	0
$516$	0	0
$517$	25.2359	1.10987
$518$	0	0
$519$	27.6770	1.21488
$520$	0	0
$521$	25.8018	1.13040	0.565198	−	0.824955i	$-0.308800\pi$
$521$	25.8018	1.13040	0.565198	+	0.824955i	$0.308800\pi$
$522$	0	0
$523$	−19.1129	−0.835749	−0.417874	−	0.908505i	$-0.637225\pi$
$523$	−19.1129	−0.835749	−0.417874	+	0.908505i	$0.637225\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.4197	0.889496
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.52314	0.283080
$532$	0	0
$533$	−22.1462	−0.959259
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.11290	−0.134332
$538$	0	0
$539$	16.3720	0.705193
$540$	0	0
$541$	30.8394	1.32589	0.662945	−	0.748668i	$-0.269306\pi$
$541$	30.8394	1.32589	0.662945	+	0.748668i	$0.269306\pi$
$542$	0	0
$543$	−10.3959	−0.446129
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13.8513	0.592240	0.296120	−	0.955151i	$-0.404307\pi$
$547$	13.8513	0.592240	0.296120	+	0.955151i	$0.404307\pi$
$548$	0	0
$549$	12.8375	0.547893
$550$	0	0
$551$	−2.66503	−0.113534
$552$	0	0
$553$	−48.7831	−2.07447
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.7641	1.21878	0.609388	−	0.792872i	$-0.291415\pi$
$557$	28.7641	1.21878	0.609388	+	0.792872i	$0.291415\pi$
$558$	0	0
$559$	−31.6036	−1.33669
$560$	0	0
$561$	31.7823	1.34185
$562$	0	0
$563$	36.1505	1.52356	0.761782	−	0.647834i	$-0.224325\pi$
$563$	36.1505	1.52356	0.761782	+	0.647834i	$0.224325\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−16.1933	−0.680055
$568$	0	0
$569$	−10.3488	−0.433843	−0.216921	−	0.976189i	$-0.569601\pi$
$569$	−10.3488	−0.433843	−0.216921	+	0.976189i	$0.569601\pi$
$570$	0	0
$571$	−19.6060	−0.820486	−0.410243	−	0.911976i	$-0.634556\pi$
$571$	−19.6060	−0.820486	−0.410243	+	0.911976i	$0.634556\pi$
$572$	0	0
$573$	−12.2973	−0.513729
$574$	0	0
$575$	0	0
$576$	0	0
$577$	34.1505	1.42171	0.710853	−	0.703341i	$-0.248309\pi$
$577$	34.1505	1.42171	0.710853	+	0.703341i	$0.248309\pi$
$578$	0	0
$579$	−3.50444	−0.145639
$580$	0	0
$581$	27.9009	1.15752
$582$	0	0
$583$	38.8156	1.60758
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.19062	−0.214240	−0.107120	−	0.994246i	$-0.534163\pi$
$587$	−5.19062	−0.214240	−0.107120	+	0.994246i	$0.534163\pi$
$588$	0	0
$589$	−18.3701	−0.756929
$590$	0	0
$591$	15.3796	0.632633
$592$	0	0
$593$	2.09911	0.0862002	0.0431001	−	0.999071i	$-0.486277\pi$
$593$	2.09911	0.0862002	0.0431001	+	0.999071i	$0.486277\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.1933	0.662748
$598$	0	0
$599$	−11.1581	−0.455909	−0.227955	−	0.973672i	$-0.573204\pi$
$599$	−11.1581	−0.455909	−0.227955	+	0.973672i	$0.573204\pi$
$600$	0	0
$601$	9.57784	0.390688	0.195344	−	0.980735i	$-0.437418\pi$
$601$	9.57784	0.390688	0.195344	+	0.980735i	$0.437418\pi$
$602$	0	0
$603$	9.57597	0.389964
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24.6975	−1.00244	−0.501221	−	0.865320i	$-0.667115\pi$
$607$	−24.6975	−1.00244	−0.501221	+	0.865320i	$0.667115\pi$
$608$	0	0
$609$	3.81533	0.154605
$610$	0	0
$611$	−15.4102	−0.623431
$612$	0	0
$613$	26.3488	1.06422	0.532108	−	0.846676i	$-0.321400\pi$
$613$	26.3488	1.06422	0.532108	+	0.846676i	$0.321400\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.94612	−0.239382	−0.119691	−	0.992811i	$-0.538190\pi$
$617$	−5.94612	−0.239382	−0.119691	+	0.992811i	$0.538190\pi$
$618$	0	0
$619$	−39.6856	−1.59510	−0.797549	−	0.603254i	$-0.793871\pi$
$619$	−39.6856	−1.59510	−0.797549	+	0.603254i	$0.793871\pi$
$620$	0	0
$621$	5.65556	0.226950
$622$	0	0
$623$	51.7541	2.07348
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−28.5922	−1.14186
$628$	0	0
$629$	24.1505	0.962945
$630$	0	0
$631$	23.0138	0.916164	0.458082	−	0.888910i	$-0.348537\pi$
$631$	23.0138	0.916164	0.458082	+	0.888910i	$0.348537\pi$
$632$	0	0
$633$	−33.0891	−1.31517
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9.99754	−0.396117
$638$	0	0
$639$	2.44109	0.0965682
$640$	0	0
$641$	25.4617	1.00568	0.502838	−	0.864381i	$-0.332289\pi$
$641$	25.4617	1.00568	0.502838	+	0.864381i	$0.332289\pi$
$642$	0	0
$643$	−16.4479	−0.648641	−0.324320	−	0.945947i	$-0.605136\pi$
$643$	−16.4479	−0.648641	−0.324320	+	0.945947i	$0.605136\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−34.9147	−1.37264	−0.686319	−	0.727301i	$-0.740774\pi$
$647$	−34.9147	−1.37264	−0.686319	+	0.727301i	$0.740774\pi$
$648$	0	0
$649$	−44.4002	−1.74286
$650$	0	0
$651$	26.2992	1.03075
$652$	0	0
$653$	34.2754	1.34130	0.670649	−	0.741775i	$-0.266016\pi$
$653$	34.2754	1.34130	0.670649	+	0.741775i	$0.266016\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.60846	0.218807
$658$	0	0
$659$	−35.2548	−1.37333	−0.686666	−	0.726973i	$-0.740926\pi$
$659$	−35.2548	−1.37333	−0.686666	+	0.726973i	$0.740926\pi$
$660$	0	0
$661$	28.5684	1.11118	0.555590	−	0.831456i	$-0.312492\pi$
$661$	28.5684	1.11118	0.555590	+	0.831456i	$0.312492\pi$
$662$	0	0
$663$	−19.4078	−0.753736
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.863254	−0.0334253
$668$	0	0
$669$	−8.19822	−0.316962
$670$	0	0
$671$	−87.3796	−3.37325
$672$	0	0
$673$	23.8770	0.920392	0.460196	−	0.887817i	$-0.347779\pi$
$673$	23.8770	0.920392	0.460196	+	0.887817i	$0.347779\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	44.1310	1.69359
$680$	0	0
$681$	−25.2359	−0.967040
$682$	0	0
$683$	34.6480	1.32577	0.662884	−	0.748722i	$-0.269332\pi$
$683$	34.6480	1.32577	0.662884	+	0.748722i	$0.269332\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−23.0138	−0.878031
$688$	0	0
$689$	−23.7027	−0.903000
$690$	0	0
$691$	18.1744	0.691386	0.345693	−	0.938348i	$-0.387644\pi$
$691$	18.1744	0.691386	0.345693	+	0.938348i	$0.387644\pi$
$692$	0	0
$693$	−18.9821	−0.721071
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.2377	0.728681
$698$	0	0
$699$	9.23095	0.349146
$700$	0	0
$701$	−40.6907	−1.53687	−0.768434	−	0.639929i	$-0.778964\pi$
$701$	−40.6907	−1.53687	−0.768434	+	0.639929i	$0.778964\pi$
$702$	0	0
$703$	−21.7265	−0.819431
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.66503	0.100229
$708$	0	0
$709$	−26.4454	−0.993178	−0.496589	−	0.867986i	$-0.665414\pi$
$709$	−26.4454	−0.993178	−0.496589	+	0.867986i	$0.665414\pi$
$710$	0	0
$711$	15.0187	0.563245
$712$	0	0
$713$	−5.95044	−0.222846
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.22580	−0.232507
$718$	0	0
$719$	2.24778	0.0838281	0.0419140	−	0.999121i	$-0.486654\pi$
$719$	2.24778	0.0838281	0.0419140	+	0.999121i	$0.486654\pi$
$720$	0	0
$721$	4.72568	0.175994
$722$	0	0
$723$	1.09397	0.0406850
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21.4830	0.796762	0.398381	−	0.917220i	$-0.369572\pi$
$727$	21.4830	0.796762	0.398381	+	0.917220i	$0.369572\pi$
$728$	0	0
$729$	29.2754	1.08427
$730$	0	0
$731$	27.4530	1.01539
$732$	0	0
$733$	−11.3778	−0.420247	−0.210123	−	0.977675i	$-0.567387\pi$
$733$	−11.3778	−0.420247	−0.210123	+	0.977675i	$0.567387\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−65.1795	−2.40092
$738$	0	0
$739$	−21.8770	−0.804760	−0.402380	−	0.915473i	$-0.631817\pi$
$739$	−21.8770	−0.804760	−0.402380	+	0.915473i	$0.631817\pi$
$740$	0	0
$741$	17.4598	0.641402
$742$	0	0
$743$	5.36068	0.196664	0.0983322	−	0.995154i	$-0.468649\pi$
$743$	5.36068	0.196664	0.0983322	+	0.995154i	$0.468649\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.58976	−0.314283
$748$	0	0
$749$	54.4191	1.98843
$750$	0	0
$751$	−24.3915	−0.890060	−0.445030	−	0.895516i	$-0.646807\pi$
$751$	−24.3915	−0.890060	−0.445030	+	0.895516i	$0.646807\pi$
$752$	0	0
$753$	32.2692	1.17595
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−41.9437	−1.52447	−0.762234	−	0.647301i	$-0.775898\pi$
$757$	−41.9437	−1.52447	−0.762234	+	0.647301i	$0.775898\pi$
$758$	0	0
$759$	−9.26157	−0.336174
$760$	0	0
$761$	−18.3745	−0.666074	−0.333037	−	0.942914i	$-0.608073\pi$
$761$	−18.3745	−0.666074	−0.333037	+	0.942914i	$0.608073\pi$
$762$	0	0
$763$	−8.99240	−0.325547
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.1129	0.978990
$768$	0	0
$769$	42.4993	1.53256	0.766282	−	0.642505i	$-0.222105\pi$
$769$	42.4993	1.53256	0.766282	+	0.642505i	$0.222105\pi$
$770$	0	0
$771$	14.1744	0.510478
$772$	0	0
$773$	15.0376	0.540866	0.270433	−	0.962739i	$-0.412833\pi$
$773$	15.0376	0.540866	0.270433	+	0.962739i	$0.412833\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	31.1043	1.11586
$778$	0	0
$779$	−17.3068	−0.620081
$780$	0	0
$781$	−16.6155	−0.594548
$782$	0	0
$783$	−4.88219	−0.174475
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−8.39154	−0.299126	−0.149563	−	0.988752i	$-0.547787\pi$
$787$	−8.39154	−0.299126	−0.149563	+	0.988752i	$0.547787\pi$
$788$	0	0
$789$	−10.3897	−0.369882
$790$	0	0
$791$	18.5231	0.658607
$792$	0	0
$793$	53.3582	1.89481
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.0514	0.781101	0.390551	−	0.920581i	$-0.372285\pi$
$797$	22.0514	0.781101	0.390551	+	0.920581i	$0.372285\pi$
$798$	0	0
$799$	13.3864	0.473576
$800$	0	0
$801$	−15.9334	−0.562978
$802$	0	0
$803$	−38.1744	−1.34714
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.93337	−0.349671
$808$	0	0
$809$	2.04524	0.0719066	0.0359533	−	0.999353i	$-0.488553\pi$
$809$	2.04524	0.0719066	0.0359533	+	0.999353i	$0.488553\pi$
$810$	0	0
$811$	−4.24965	−0.149225	−0.0746126	−	0.997213i	$-0.523772\pi$
$811$	−4.24965	−0.149225	−0.0746126	+	0.997213i	$0.523772\pi$
$812$	0	0
$813$	14.7446	0.517116
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.6975	−0.864057
$818$	0	0
$819$	11.5914	0.405036
$820$	0	0
$821$	−29.2548	−1.02100	−0.510500	−	0.859878i	$-0.670540\pi$
$821$	−29.2548	−1.02100	−0.510500	+	0.859878i	$0.670540\pi$
$822$	0	0
$823$	13.5846	0.473530	0.236765	−	0.971567i	$-0.423913\pi$
$823$	13.5846	0.473530	0.236765	+	0.971567i	$0.423913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.7880	0.861963	0.430981	−	0.902361i	$-0.358167\pi$
$827$	24.7880	0.861963	0.430981	+	0.902361i	$0.358167\pi$
$828$	0	0
$829$	26.1505	0.908246	0.454123	−	0.890939i	$-0.349953\pi$
$829$	26.1505	0.908246	0.454123	+	0.890939i	$0.349953\pi$
$830$	0	0
$831$	25.4855	0.884082
$832$	0	0
$833$	8.68455	0.300902
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−33.6531	−1.16322
$838$	0	0
$839$	−6.37260	−0.220007	−0.110003	−	0.993931i	$-0.535086\pi$
$839$	−6.37260	−0.220007	−0.110003	+	0.993931i	$0.535086\pi$
$840$	0	0
$841$	−28.2548	−0.974303
$842$	0	0
$843$	−27.1129	−0.933818
$844$	0	0
$845$	0	0
$846$	0	0
$847$	95.2439	3.27262
$848$	0	0
$849$	18.1657	0.623447
$850$	0	0
$851$	−7.03763	−0.241247
$852$	0	0
$853$	33.6293	1.15144	0.575722	−	0.817646i	$-0.304721\pi$
$853$	33.6293	1.15144	0.575722	+	0.817646i	$0.304721\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.1795	−0.381885	−0.190943	−	0.981601i	$-0.561154\pi$
$857$	−11.1795	−0.381885	−0.190943	+	0.981601i	$0.561154\pi$
$858$	0	0
$859$	−47.9009	−1.63436	−0.817179	−	0.576385i	$-0.804463\pi$
$859$	−47.9009	−1.63436	−0.817179	+	0.576385i	$0.804463\pi$
$860$	0	0
$861$	24.7769	0.844394
$862$	0	0
$863$	−30.1180	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$863$	−30.1180	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.47873	−0.253991
$868$	0	0
$869$	−102.226	−3.46777
$870$	0	0
$871$	39.8018	1.34863
$872$	0	0
$873$	−13.5865	−0.459833
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.3940	0.756191	0.378096	−	0.925767i	$-0.376579\pi$
$877$	22.3940	0.756191	0.378096	+	0.925767i	$0.376579\pi$
$878$	0	0
$879$	8.58976	0.289726
$880$	0	0
$881$	21.1129	0.711312	0.355656	−	0.934617i	$-0.384258\pi$
$881$	21.1129	0.711312	0.355656	+	0.934617i	$0.384258\pi$
$882$	0	0
$883$	49.1061	1.65255	0.826276	−	0.563265i	$-0.190455\pi$
$883$	49.1061	1.65255	0.826276	+	0.563265i	$0.190455\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.0566	−1.44570	−0.722849	−	0.691006i	$-0.757168\pi$
$887$	−43.0566	−1.44570	−0.722849	+	0.691006i	$0.757168\pi$
$888$	0	0
$889$	64.6412	2.16800
$890$	0	0
$891$	−33.9334	−1.13681
$892$	0	0
$893$	−12.0428	−0.402996
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.65556	0.188834
$898$	0	0
$899$	5.13675	0.171320
$900$	0	0
$901$	20.5898	0.685944
$902$	0	0
$903$	35.3576	1.17663
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−37.9762	−1.26098	−0.630489	−	0.776198i	$-0.717146\pi$
$907$	−37.9762	−1.26098	−0.630489	+	0.776198i	$0.717146\pi$
$908$	0	0
$909$	−0.820475	−0.0272134
$910$	0	0
$911$	15.7504	0.521833	0.260916	−	0.965361i	$-0.415975\pi$
$911$	15.7504	0.521833	0.260916	+	0.965361i	$0.415975\pi$
$912$	0	0
$913$	58.4668	1.93497
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−33.2309	−1.09738
$918$	0	0
$919$	−30.4240	−1.00360	−0.501798	−	0.864985i	$-0.667328\pi$
$919$	−30.4240	−1.00360	−0.501798	+	0.864985i	$0.667328\pi$
$920$	0	0
$921$	−32.8700	−1.08310
$922$	0	0
$923$	10.1462	0.333967
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.45488	−0.0477846
$928$	0	0
$929$	−37.8018	−1.24024	−0.620118	−	0.784509i	$-0.712915\pi$
$929$	−37.8018	−1.24024	−0.620118	+	0.784509i	$0.712915\pi$
$930$	0	0
$931$	−7.81287	−0.256057
$932$	0	0
$933$	−25.8770	−0.847176
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.6907	1.06796	0.533980	−	0.845497i	$-0.320696\pi$
$937$	32.6907	1.06796	0.533980	+	0.845497i	$0.320696\pi$
$938$	0	0
$939$	−21.7070	−0.708381
$940$	0	0
$941$	−46.4882	−1.51547	−0.757736	−	0.652561i	$-0.773694\pi$
$941$	−46.4882	−1.51547	−0.757736	+	0.652561i	$0.773694\pi$
$942$	0	0
$943$	−5.60601	−0.182557
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−37.2052	−1.20901	−0.604504	−	0.796602i	$-0.706629\pi$
$947$	−37.2052	−1.20901	−0.604504	+	0.796602i	$0.706629\pi$
$948$	0	0
$949$	23.3111	0.756711
$950$	0	0
$951$	−20.0796	−0.651125
$952$	0	0
$953$	−49.1104	−1.59084	−0.795422	−	0.606056i	$-0.792751\pi$
$953$	−49.1104	−1.59084	−0.795422	+	0.606056i	$0.792751\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.99509	0.258445
$958$	0	0
$959$	−10.0691	−0.325148
$960$	0	0
$961$	4.40778	0.142187
$962$	0	0
$963$	−16.7538	−0.539885
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.96751	0.0632709	0.0316355	−	0.999499i	$-0.489928\pi$
$967$	1.96751	0.0632709	0.0316355	+	0.999499i	$0.489928\pi$
$968$	0	0
$969$	−15.1668	−0.487227
$970$	0	0
$971$	−26.1205	−0.838247	−0.419123	−	0.907929i	$-0.637663\pi$
$971$	−26.1205	−0.838247	−0.419123	+	0.907929i	$0.637663\pi$
$972$	0	0
$973$	52.2924	1.67642
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.639319	0.0204536	0.0102268	−	0.999948i	$-0.496745\pi$
$977$	0.639319	0.0204536	0.0102268	+	0.999948i	$0.496745\pi$
$978$	0	0
$979$	108.452	3.46613
$980$	0	0
$981$	2.76846	0.0883902
$982$	0	0
$983$	−6.46926	−0.206337	−0.103169	−	0.994664i	$-0.532898\pi$
$983$	−6.46926	−0.206337	−0.103169	+	0.994664i	$0.532898\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	17.2408	0.548780
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−37.2334	−1.18276	−0.591379	−	0.806394i	$-0.701416\pi$
$991$	−37.2334	−1.18276	−0.591379	+	0.806394i	$0.701416\pi$
$992$	0	0
$993$	35.4189	1.12398
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−9.65124	−0.305658	−0.152829	−	0.988253i	$-0.548838\pi$
$997$	−9.65124	−0.305658	−0.152829	+	0.988253i	$0.548838\pi$
$998$	0	0
$999$	−39.8018	−1.25927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cf.1.2		3
4.3	odd	2		1150.2.a.q.1.2		3
5.4	even	2		1840.2.a.r.1.2		3
20.3	even	4		1150.2.b.j.599.5		6
20.7	even	4		1150.2.b.j.599.2		6
20.19	odd	2		230.2.a.d.1.2	✓	3
40.19	odd	2		7360.2.a.bz.1.2		3
40.29	even	2		7360.2.a.ce.1.2		3
60.59	even	2		2070.2.a.z.1.2		3
460.459	even	2		5290.2.a.r.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.d.1.2	✓	3	20.19	odd	2
1150.2.a.q.1.2		3	4.3	odd	2
1150.2.b.j.599.2		6	20.7	even	4
1150.2.b.j.599.5		6	20.3	even	4
1840.2.a.r.1.2		3	5.4	even	2
2070.2.a.z.1.2		3	60.59	even	2
5290.2.a.r.1.2		3	460.459	even	2
7360.2.a.bz.1.2		3	40.19	odd	2
7360.2.a.ce.1.2		3	40.29	even	2
9200.2.a.cf.1.2		3	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)

\(f(q)\)	\(=\)	\(q+1.43163 q^{3} +3.08719 q^{7} -0.950444 q^{9} +O(q^{10})\)	\(q+1.43163 q^{3} +3.08719 q^{7} -0.950444 q^{9} +6.46926 q^{11} -3.95044 q^{13} +3.43163 q^{17} -3.08719 q^{19} +4.41970 q^{21} -1.00000 q^{23} -5.65556 q^{27} +0.863254 q^{29} +5.95044 q^{31} +9.26157 q^{33} +7.03763 q^{37} -5.65556 q^{39} +5.60601 q^{41} +8.00000 q^{43} +3.90089 q^{47} +2.53074 q^{49} +4.91281 q^{51} +6.00000 q^{53} -4.41970 q^{57} -6.86325 q^{59} -13.5069 q^{61} -2.93420 q^{63} -10.0753 q^{67} -1.43163 q^{69} -2.56837 q^{71} -5.90089 q^{73} +19.9718 q^{77} -15.8018 q^{79} -5.24533 q^{81} +9.03763 q^{83} +1.23586 q^{87} +16.7641 q^{89} -12.1958 q^{91} +8.51882 q^{93} +14.2949 q^{97} -6.14867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{7} + 10 q^{9}+O(q^{10}) \) 3 * q + q^3 + 3 * q^7 + 10 * q^9	\( 3 q + q^{3} + 3 q^{7} + 10 q^{9} - 3 q^{11} + q^{13} + 7 q^{17} - 3 q^{19} - 22 q^{21} - 3 q^{23} - 14 q^{27} - 4 q^{29} + 5 q^{31} + 9 q^{33} + 2 q^{37} - 14 q^{39} + q^{41} + 24 q^{43} - 14 q^{47} + 30 q^{49} + 21 q^{51} + 18 q^{53} + 22 q^{57} - 14 q^{59} + q^{61} + 8 q^{63} + 8 q^{67} - q^{69} - 11 q^{71} + 8 q^{73} + 24 q^{77} + 4 q^{79} + 7 q^{81} + 8 q^{83} + 36 q^{87} + 18 q^{89} - q^{91} + 16 q^{93} + 33 q^{97} - 57 q^{99}+O(q^{100}) \) 3 * q + q^3 + 3 * q^7 + 10 * q^9 - 3 * q^11 + q^13 + 7 * q^17 - 3 * q^19 - 22 * q^21 - 3 * q^23 - 14 * q^27 - 4 * q^29 + 5 * q^31 + 9 * q^33 + 2 * q^37 - 14 * q^39 + q^41 + 24 * q^43 - 14 * q^47 + 30 * q^49 + 21 * q^51 + 18 * q^53 + 22 * q^57 - 14 * q^59 + q^61 + 8 * q^63 + 8 * q^67 - q^69 - 11 * q^71 + 8 * q^73 + 24 * q^77 + 4 * q^79 + 7 * q^81 + 8 * q^83 + 36 * q^87 + 18 * q^89 - q^91 + 16 * q^93 + 33 * q^97 - 57 * q^99

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