LMFDB - Embedded newform 1648.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1648,2,Mod(1,1648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1648 = 2^{4} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1648.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:	$13.1593462531$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 824)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.618034 q^{5} +4.23607 q^{7} -2.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.618034 q^{5} +4.23607 q^{7} -2.00000 q^{9} +3.61803 q^{11} +1.85410 q^{13} -0.618034 q^{15} +0.618034 q^{17} +3.61803 q^{19} -4.23607 q^{21} +2.00000 q^{23} -4.61803 q^{25} +5.00000 q^{27} -5.70820 q^{29} -2.23607 q^{31} -3.61803 q^{33} +2.61803 q^{35} -1.47214 q^{37} -1.85410 q^{39} +8.94427 q^{41} -6.70820 q^{43} -1.23607 q^{45} +1.85410 q^{47} +10.9443 q^{49} -0.618034 q^{51} +2.38197 q^{53} +2.23607 q^{55} -3.61803 q^{57} +10.3262 q^{59} +6.85410 q^{61} -8.47214 q^{63} +1.14590 q^{65} +4.52786 q^{67} -2.00000 q^{69} -10.7984 q^{71} -14.7984 q^{73} +4.61803 q^{75} +15.3262 q^{77} +1.38197 q^{79} +1.00000 q^{81} +13.7984 q^{83} +0.381966 q^{85} +5.70820 q^{87} -7.70820 q^{89} +7.85410 q^{91} +2.23607 q^{93} +2.23607 q^{95} -1.23607 q^{97} -7.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^5 + 4 * q^7 - 4 * q^9	$ 2 q - 2 q^{3} - q^{5} + 4 q^{7} - 4 q^{9} + 5 q^{11} - 3 q^{13} + q^{15} - q^{17} + 5 q^{19} - 4 q^{21} + 4 q^{23} - 7 q^{25} + 10 q^{27} + 2 q^{29} - 5 q^{33} + 3 q^{35} + 6 q^{37} + 3 q^{39} + 2 q^{45} - 3 q^{47} + 4 q^{49} + q^{51} + 7 q^{53} - 5 q^{57} + 5 q^{59} + 7 q^{61} - 8 q^{63} + 9 q^{65} + 18 q^{67} - 4 q^{69} + 3 q^{71} - 5 q^{73} + 7 q^{75} + 15 q^{77} + 5 q^{79} + 2 q^{81} + 3 q^{83} + 3 q^{85} - 2 q^{87} - 2 q^{89} + 9 q^{91} + 2 q^{97} - 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 + 4 * q^7 - 4 * q^9 + 5 * q^11 - 3 * q^13 + q^15 - q^17 + 5 * q^19 - 4 * q^21 + 4 * q^23 - 7 * q^25 + 10 * q^27 + 2 * q^29 - 5 * q^33 + 3 * q^35 + 6 * q^37 + 3 * q^39 + 2 * q^45 - 3 * q^47 + 4 * q^49 + q^51 + 7 * q^53 - 5 * q^57 + 5 * q^59 + 7 * q^61 - 8 * q^63 + 9 * q^65 + 18 * q^67 - 4 * q^69 + 3 * q^71 - 5 * q^73 + 7 * q^75 + 15 * q^77 + 5 * q^79 + 2 * q^81 + 3 * q^83 + 3 * q^85 - 2 * q^87 - 2 * q^89 + 9 * q^91 + 2 * q^97 - 10 * 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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	0.618034	0.276393	0.138197	−	0.990405i	$-0.455869\pi$
$5$	0.618034	0.276393	0.138197	+	0.990405i	$0.455869\pi$
$6$	0	0
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.61803	1.09088	0.545439	−	0.838150i	$-0.316363\pi$
$11$	3.61803	1.09088	0.545439	+	0.838150i	$0.316363\pi$
$12$	0	0
$13$	1.85410	0.514235	0.257118	−	0.966380i	$-0.417227\pi$
$13$	1.85410	0.514235	0.257118	+	0.966380i	$0.417227\pi$
$14$	0	0
$15$	−0.618034	−0.159576
$16$	0	0
$17$	0.618034	0.149895	0.0749476	−	0.997187i	$-0.476121\pi$
$17$	0.618034	0.149895	0.0749476	+	0.997187i	$0.476121\pi$
$18$	0	0
$19$	3.61803	0.830034	0.415017	−	0.909814i	$-0.363776\pi$
$19$	3.61803	0.830034	0.415017	+	0.909814i	$0.363776\pi$
$20$	0	0
$21$	−4.23607	−0.924386
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−5.70820	−1.05999	−0.529993	−	0.848002i	$-0.677806\pi$
$29$	−5.70820	−1.05999	−0.529993	+	0.848002i	$0.677806\pi$
$30$	0	0
$31$	−2.23607	−0.401610	−0.200805	−	0.979631i	$-0.564356\pi$
$31$	−2.23607	−0.401610	−0.200805	+	0.979631i	$0.564356\pi$
$32$	0	0
$33$	−3.61803	−0.629819
$34$	0	0
$35$	2.61803	0.442529
$36$	0	0
$37$	−1.47214	−0.242018	−0.121009	−	0.992651i	$-0.538613\pi$
$37$	−1.47214	−0.242018	−0.121009	+	0.992651i	$0.538613\pi$
$38$	0	0
$39$	−1.85410	−0.296894
$40$	0	0
$41$	8.94427	1.39686	0.698430	−	0.715678i	$-0.253882\pi$
$41$	8.94427	1.39686	0.698430	+	0.715678i	$0.253882\pi$
$42$	0	0
$43$	−6.70820	−1.02299	−0.511496	−	0.859286i	$-0.670908\pi$
$43$	−6.70820	−1.02299	−0.511496	+	0.859286i	$0.670908\pi$
$44$	0	0
$45$	−1.23607	−0.184262
$46$	0	0
$47$	1.85410	0.270449	0.135224	−	0.990815i	$-0.456825\pi$
$47$	1.85410	0.270449	0.135224	+	0.990815i	$0.456825\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	−0.618034	−0.0865421
$52$	0	0
$53$	2.38197	0.327188	0.163594	−	0.986528i	$-0.447691\pi$
$53$	2.38197	0.327188	0.163594	+	0.986528i	$0.447691\pi$
$54$	0	0
$55$	2.23607	0.301511
$56$	0	0
$57$	−3.61803	−0.479220
$58$	0	0
$59$	10.3262	1.34436	0.672181	−	0.740387i	$-0.265358\pi$
$59$	10.3262	1.34436	0.672181	+	0.740387i	$0.265358\pi$
$60$	0	0
$61$	6.85410	0.877578	0.438789	−	0.898590i	$-0.355408\pi$
$61$	6.85410	0.877578	0.438789	+	0.898590i	$0.355408\pi$
$62$	0	0
$63$	−8.47214	−1.06739
$64$	0	0
$65$	1.14590	0.142131
$66$	0	0
$67$	4.52786	0.553167	0.276583	−	0.960990i	$-0.410798\pi$
$67$	4.52786	0.553167	0.276583	+	0.960990i	$0.410798\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−10.7984	−1.28153	−0.640766	−	0.767737i	$-0.721383\pi$
$71$	−10.7984	−1.28153	−0.640766	+	0.767737i	$0.721383\pi$
$72$	0	0
$73$	−14.7984	−1.73202	−0.866009	−	0.500028i	$-0.833323\pi$
$73$	−14.7984	−1.73202	−0.866009	+	0.500028i	$0.833323\pi$
$74$	0	0
$75$	4.61803	0.533245
$76$	0	0
$77$	15.3262	1.74659
$78$	0	0
$79$	1.38197	0.155483	0.0777417	−	0.996974i	$-0.475229\pi$
$79$	1.38197	0.155483	0.0777417	+	0.996974i	$0.475229\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.7984	1.51457	0.757284	−	0.653086i	$-0.226526\pi$
$83$	13.7984	1.51457	0.757284	+	0.653086i	$0.226526\pi$
$84$	0	0
$85$	0.381966	0.0414300
$86$	0	0
$87$	5.70820	0.611984
$88$	0	0
$89$	−7.70820	−0.817068	−0.408534	−	0.912743i	$-0.633960\pi$
$89$	−7.70820	−0.817068	−0.408534	+	0.912743i	$0.633960\pi$
$90$	0	0
$91$	7.85410	0.823334
$92$	0	0
$93$	2.23607	0.231869
$94$	0	0
$95$	2.23607	0.229416
$96$	0	0
$97$	−1.23607	−0.125504	−0.0627518	−	0.998029i	$-0.519988\pi$
$97$	−1.23607	−0.125504	−0.0627518	+	0.998029i	$0.519988\pi$
$98$	0	0
$99$	−7.23607	−0.727252
$100$	0	0
$101$	11.3820	1.13255	0.566274	−	0.824217i	$-0.308384\pi$
$101$	11.3820	1.13255	0.566274	+	0.824217i	$0.308384\pi$
$102$	0	0
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	0	0
$105$	−2.61803	−0.255494
$106$	0	0
$107$	9.14590	0.884167	0.442084	−	0.896974i	$-0.354239\pi$
$107$	9.14590	0.884167	0.442084	+	0.896974i	$0.354239\pi$
$108$	0	0
$109$	2.09017	0.200202	0.100101	−	0.994977i	$-0.468083\pi$
$109$	2.09017	0.200202	0.100101	+	0.994977i	$0.468083\pi$
$110$	0	0
$111$	1.47214	0.139729
$112$	0	0
$113$	14.2361	1.33922	0.669608	−	0.742714i	$-0.266462\pi$
$113$	14.2361	1.33922	0.669608	+	0.742714i	$0.266462\pi$
$114$	0	0
$115$	1.23607	0.115264
$116$	0	0
$117$	−3.70820	−0.342824
$118$	0	0
$119$	2.61803	0.239995
$120$	0	0
$121$	2.09017	0.190015
$122$	0	0
$123$	−8.94427	−0.806478
$124$	0	0
$125$	−5.94427	−0.531672
$126$	0	0
$127$	−5.32624	−0.472627	−0.236314	−	0.971677i	$-0.575939\pi$
$127$	−5.32624	−0.472627	−0.236314	+	0.971677i	$0.575939\pi$
$128$	0	0
$129$	6.70820	0.590624
$130$	0	0
$131$	−5.47214	−0.478103	−0.239051	−	0.971007i	$-0.576836\pi$
$131$	−5.47214	−0.478103	−0.239051	+	0.971007i	$0.576836\pi$
$132$	0	0
$133$	15.3262	1.32895
$134$	0	0
$135$	3.09017	0.265959
$136$	0	0
$137$	−1.76393	−0.150703	−0.0753514	−	0.997157i	$-0.524008\pi$
$137$	−1.76393	−0.150703	−0.0753514	+	0.997157i	$0.524008\pi$
$138$	0	0
$139$	10.3262	0.875860	0.437930	−	0.899009i	$-0.355712\pi$
$139$	10.3262	0.875860	0.437930	+	0.899009i	$0.355712\pi$
$140$	0	0
$141$	−1.85410	−0.156144
$142$	0	0
$143$	6.70820	0.560968
$144$	0	0
$145$	−3.52786	−0.292973
$146$	0	0
$147$	−10.9443	−0.902668
$148$	0	0
$149$	17.4721	1.43137	0.715687	−	0.698422i	$-0.246114\pi$
$149$	17.4721	1.43137	0.715687	+	0.698422i	$0.246114\pi$
$150$	0	0
$151$	13.9443	1.13477	0.567384	−	0.823453i	$-0.307955\pi$
$151$	13.9443	1.13477	0.567384	+	0.823453i	$0.307955\pi$
$152$	0	0
$153$	−1.23607	−0.0999302
$154$	0	0
$155$	−1.38197	−0.111002
$156$	0	0
$157$	11.9443	0.953257	0.476628	−	0.879105i	$-0.341859\pi$
$157$	11.9443	0.953257	0.476628	+	0.879105i	$0.341859\pi$
$158$	0	0
$159$	−2.38197	−0.188902
$160$	0	0
$161$	8.47214	0.667698
$162$	0	0
$163$	−17.4721	−1.36852	−0.684262	−	0.729237i	$-0.739875\pi$
$163$	−17.4721	−1.36852	−0.684262	+	0.729237i	$0.739875\pi$
$164$	0	0
$165$	−2.23607	−0.174078
$166$	0	0
$167$	7.76393	0.600791	0.300396	−	0.953815i	$-0.402881\pi$
$167$	7.76393	0.600791	0.300396	+	0.953815i	$0.402881\pi$
$168$	0	0
$169$	−9.56231	−0.735562
$170$	0	0
$171$	−7.23607	−0.553356
$172$	0	0
$173$	4.38197	0.333155	0.166577	−	0.986028i	$-0.446728\pi$
$173$	4.38197	0.333155	0.166577	+	0.986028i	$0.446728\pi$
$174$	0	0
$175$	−19.5623	−1.47877
$176$	0	0
$177$	−10.3262	−0.776168
$178$	0	0
$179$	12.8541	0.960761	0.480380	−	0.877060i	$-0.340499\pi$
$179$	12.8541	0.960761	0.480380	+	0.877060i	$0.340499\pi$
$180$	0	0
$181$	−4.85410	−0.360803	−0.180401	−	0.983593i	$-0.557740\pi$
$181$	−4.85410	−0.360803	−0.180401	+	0.983593i	$0.557740\pi$
$182$	0	0
$183$	−6.85410	−0.506670
$184$	0	0
$185$	−0.909830	−0.0668920
$186$	0	0
$187$	2.23607	0.163517
$188$	0	0
$189$	21.1803	1.54064
$190$	0	0
$191$	−16.6180	−1.20244	−0.601219	−	0.799084i	$-0.705318\pi$
$191$	−16.6180	−1.20244	−0.601219	+	0.799084i	$0.705318\pi$
$192$	0	0
$193$	−13.0000	−0.935760	−0.467880	−	0.883792i	$-0.654982\pi$
$193$	−13.0000	−0.935760	−0.467880	+	0.883792i	$0.654982\pi$
$194$	0	0
$195$	−1.14590	−0.0820595
$196$	0	0
$197$	9.76393	0.695651	0.347826	−	0.937559i	$-0.386920\pi$
$197$	9.76393	0.695651	0.347826	+	0.937559i	$0.386920\pi$
$198$	0	0
$199$	20.9443	1.48470	0.742350	−	0.670012i	$-0.233711\pi$
$199$	20.9443	1.48470	0.742350	+	0.670012i	$0.233711\pi$
$200$	0	0
$201$	−4.52786	−0.319371
$202$	0	0
$203$	−24.1803	−1.69713
$204$	0	0
$205$	5.52786	0.386083
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	13.0902	0.905466
$210$	0	0
$211$	2.03444	0.140057	0.0700284	−	0.997545i	$-0.477691\pi$
$211$	2.03444	0.140057	0.0700284	+	0.997545i	$0.477691\pi$
$212$	0	0
$213$	10.7984	0.739892
$214$	0	0
$215$	−4.14590	−0.282748
$216$	0	0
$217$	−9.47214	−0.643010
$218$	0	0
$219$	14.7984	0.999981
$220$	0	0
$221$	1.14590	0.0770814
$222$	0	0
$223$	−7.23607	−0.484563	−0.242281	−	0.970206i	$-0.577896\pi$
$223$	−7.23607	−0.484563	−0.242281	+	0.970206i	$0.577896\pi$
$224$	0	0
$225$	9.23607	0.615738
$226$	0	0
$227$	−11.8885	−0.789070	−0.394535	−	0.918881i	$-0.629094\pi$
$227$	−11.8885	−0.789070	−0.394535	+	0.918881i	$0.629094\pi$
$228$	0	0
$229$	−22.7082	−1.50060	−0.750300	−	0.661097i	$-0.770091\pi$
$229$	−22.7082	−1.50060	−0.750300	+	0.661097i	$0.770091\pi$
$230$	0	0
$231$	−15.3262	−1.00839
$232$	0	0
$233$	−21.6525	−1.41850	−0.709250	−	0.704957i	$-0.750966\pi$
$233$	−21.6525	−1.41850	−0.709250	+	0.704957i	$0.750966\pi$
$234$	0	0
$235$	1.14590	0.0747501
$236$	0	0
$237$	−1.38197	−0.0897683
$238$	0	0
$239$	−5.38197	−0.348130	−0.174065	−	0.984734i	$-0.555690\pi$
$239$	−5.38197	−0.348130	−0.174065	+	0.984734i	$0.555690\pi$
$240$	0	0
$241$	−5.38197	−0.346683	−0.173341	−	0.984862i	$-0.555456\pi$
$241$	−5.38197	−0.346683	−0.173341	+	0.984862i	$0.555456\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	6.76393	0.432132
$246$	0	0
$247$	6.70820	0.426833
$248$	0	0
$249$	−13.7984	−0.874436
$250$	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
$253$	7.23607	0.454928
$254$	0	0
$255$	−0.381966	−0.0239196
$256$	0	0
$257$	−4.23607	−0.264239	−0.132119	−	0.991234i	$-0.542178\pi$
$257$	−4.23607	−0.264239	−0.132119	+	0.991234i	$0.542178\pi$
$258$	0	0
$259$	−6.23607	−0.387490
$260$	0	0
$261$	11.4164	0.706658
$262$	0	0
$263$	−15.3262	−0.945056	−0.472528	−	0.881316i	$-0.656659\pi$
$263$	−15.3262	−0.945056	−0.472528	+	0.881316i	$0.656659\pi$
$264$	0	0
$265$	1.47214	0.0904326
$266$	0	0
$267$	7.70820	0.471734
$268$	0	0
$269$	−25.2148	−1.53737	−0.768686	−	0.639626i	$-0.779089\pi$
$269$	−25.2148	−1.53737	−0.768686	+	0.639626i	$0.779089\pi$
$270$	0	0
$271$	29.3607	1.78353	0.891767	−	0.452495i	$-0.149466\pi$
$271$	29.3607	1.78353	0.891767	+	0.452495i	$0.149466\pi$
$272$	0	0
$273$	−7.85410	−0.475352
$274$	0	0
$275$	−16.7082	−1.00754
$276$	0	0
$277$	−17.9443	−1.07817	−0.539083	−	0.842252i	$-0.681229\pi$
$277$	−17.9443	−1.07817	−0.539083	+	0.842252i	$0.681229\pi$
$278$	0	0
$279$	4.47214	0.267740
$280$	0	0
$281$	−9.29180	−0.554302	−0.277151	−	0.960826i	$-0.589390\pi$
$281$	−9.29180	−0.554302	−0.277151	+	0.960826i	$0.589390\pi$
$282$	0	0
$283$	−20.1803	−1.19960	−0.599798	−	0.800151i	$-0.704753\pi$
$283$	−20.1803	−1.19960	−0.599798	+	0.800151i	$0.704753\pi$
$284$	0	0
$285$	−2.23607	−0.132453
$286$	0	0
$287$	37.8885	2.23649
$288$	0	0
$289$	−16.6180	−0.977531
$290$	0	0
$291$	1.23607	0.0724596
$292$	0	0
$293$	17.0000	0.993151	0.496575	−	0.867994i	$-0.334591\pi$
$293$	17.0000	0.993151	0.496575	+	0.867994i	$0.334591\pi$
$294$	0	0
$295$	6.38197	0.371572
$296$	0	0
$297$	18.0902	1.04970
$298$	0	0
$299$	3.70820	0.214451
$300$	0	0
$301$	−28.4164	−1.63789
$302$	0	0
$303$	−11.3820	−0.653877
$304$	0	0
$305$	4.23607	0.242557
$306$	0	0
$307$	−30.9787	−1.76805	−0.884024	−	0.467441i	$-0.845176\pi$
$307$	−30.9787	−1.76805	−0.884024	+	0.467441i	$0.845176\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	−24.5967	−1.39475	−0.697377	−	0.716705i	$-0.745650\pi$
$311$	−24.5967	−1.39475	−0.697377	+	0.716705i	$0.745650\pi$
$312$	0	0
$313$	−16.7639	−0.947553	−0.473777	−	0.880645i	$-0.657110\pi$
$313$	−16.7639	−0.947553	−0.473777	+	0.880645i	$0.657110\pi$
$314$	0	0
$315$	−5.23607	−0.295019
$316$	0	0
$317$	25.3607	1.42440	0.712199	−	0.701978i	$-0.247699\pi$
$317$	25.3607	1.42440	0.712199	+	0.701978i	$0.247699\pi$
$318$	0	0
$319$	−20.6525	−1.15632
$320$	0	0
$321$	−9.14590	−0.510474
$322$	0	0
$323$	2.23607	0.124418
$324$	0	0
$325$	−8.56231	−0.474951
$326$	0	0
$327$	−2.09017	−0.115587
$328$	0	0
$329$	7.85410	0.433011
$330$	0	0
$331$	27.0344	1.48595	0.742974	−	0.669321i	$-0.233415\pi$
$331$	27.0344	1.48595	0.742974	+	0.669321i	$0.233415\pi$
$332$	0	0
$333$	2.94427	0.161345
$334$	0	0
$335$	2.79837	0.152891
$336$	0	0
$337$	10.3820	0.565542	0.282771	−	0.959187i	$-0.408746\pi$
$337$	10.3820	0.565542	0.282771	+	0.959187i	$0.408746\pi$
$338$	0	0
$339$	−14.2361	−0.773197
$340$	0	0
$341$	−8.09017	−0.438107
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	−1.23607	−0.0665477
$346$	0	0
$347$	8.70820	0.467481	0.233740	−	0.972299i	$-0.424903\pi$
$347$	8.70820	0.467481	0.233740	+	0.972299i	$0.424903\pi$
$348$	0	0
$349$	−15.0557	−0.805915	−0.402957	−	0.915219i	$-0.632018\pi$
$349$	−15.0557	−0.805915	−0.402957	+	0.915219i	$0.632018\pi$
$350$	0	0
$351$	9.27051	0.494823
$352$	0	0
$353$	14.3262	0.762509	0.381254	−	0.924470i	$-0.375492\pi$
$353$	14.3262	0.762509	0.381254	+	0.924470i	$0.375492\pi$
$354$	0	0
$355$	−6.67376	−0.354207
$356$	0	0
$357$	−2.61803	−0.138561
$358$	0	0
$359$	−14.2148	−0.750227	−0.375114	−	0.926979i	$-0.622396\pi$
$359$	−14.2148	−0.750227	−0.375114	+	0.926979i	$0.622396\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	0	0
$363$	−2.09017	−0.109705
$364$	0	0
$365$	−9.14590	−0.478718
$366$	0	0
$367$	28.0902	1.46629	0.733147	−	0.680070i	$-0.238050\pi$
$367$	28.0902	1.46629	0.733147	+	0.680070i	$0.238050\pi$
$368$	0	0
$369$	−17.8885	−0.931240
$370$	0	0
$371$	10.0902	0.523856
$372$	0	0
$373$	−14.0902	−0.729561	−0.364781	−	0.931093i	$-0.618856\pi$
$373$	−14.0902	−0.729561	−0.364781	+	0.931093i	$0.618856\pi$
$374$	0	0
$375$	5.94427	0.306961
$376$	0	0
$377$	−10.5836	−0.545083
$378$	0	0
$379$	−4.88854	−0.251108	−0.125554	−	0.992087i	$-0.540071\pi$
$379$	−4.88854	−0.251108	−0.125554	+	0.992087i	$0.540071\pi$
$380$	0	0
$381$	5.32624	0.272871
$382$	0	0
$383$	−15.6525	−0.799804	−0.399902	−	0.916558i	$-0.630956\pi$
$383$	−15.6525	−0.799804	−0.399902	+	0.916558i	$0.630956\pi$
$384$	0	0
$385$	9.47214	0.482745
$386$	0	0
$387$	13.4164	0.681994
$388$	0	0
$389$	29.3050	1.48582	0.742910	−	0.669391i	$-0.233445\pi$
$389$	29.3050	1.48582	0.742910	+	0.669391i	$0.233445\pi$
$390$	0	0
$391$	1.23607	0.0625106
$392$	0	0
$393$	5.47214	0.276033
$394$	0	0
$395$	0.854102	0.0429745
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	−15.3262	−0.767272
$400$	0	0
$401$	−26.9443	−1.34553	−0.672766	−	0.739855i	$-0.734894\pi$
$401$	−26.9443	−1.34553	−0.672766	+	0.739855i	$0.734894\pi$
$402$	0	0
$403$	−4.14590	−0.206522
$404$	0	0
$405$	0.618034	0.0307104
$406$	0	0
$407$	−5.32624	−0.264012
$408$	0	0
$409$	−17.2918	−0.855024	−0.427512	−	0.904010i	$-0.640610\pi$
$409$	−17.2918	−0.855024	−0.427512	+	0.904010i	$0.640610\pi$
$410$	0	0
$411$	1.76393	0.0870084
$412$	0	0
$413$	43.7426	2.15243
$414$	0	0
$415$	8.52786	0.418616
$416$	0	0
$417$	−10.3262	−0.505678
$418$	0	0
$419$	16.4377	0.803034	0.401517	−	0.915852i	$-0.368483\pi$
$419$	16.4377	0.803034	0.401517	+	0.915852i	$0.368483\pi$
$420$	0	0
$421$	−25.3607	−1.23600	−0.618002	−	0.786177i	$-0.712058\pi$
$421$	−25.3607	−1.23600	−0.618002	+	0.786177i	$0.712058\pi$
$422$	0	0
$423$	−3.70820	−0.180299
$424$	0	0
$425$	−2.85410	−0.138444
$426$	0	0
$427$	29.0344	1.40508
$428$	0	0
$429$	−6.70820	−0.323875
$430$	0	0
$431$	8.47214	0.408088	0.204044	−	0.978962i	$-0.434591\pi$
$431$	8.47214	0.408088	0.204044	+	0.978962i	$0.434591\pi$
$432$	0	0
$433$	−30.5967	−1.47039	−0.735193	−	0.677858i	$-0.762908\pi$
$433$	−30.5967	−1.47039	−0.735193	+	0.677858i	$0.762908\pi$
$434$	0	0
$435$	3.52786	0.169148
$436$	0	0
$437$	7.23607	0.346148
$438$	0	0
$439$	−6.56231	−0.313202	−0.156601	−	0.987662i	$-0.550054\pi$
$439$	−6.56231	−0.313202	−0.156601	+	0.987662i	$0.550054\pi$
$440$	0	0
$441$	−21.8885	−1.04231
$442$	0	0
$443$	20.2148	0.960433	0.480217	−	0.877150i	$-0.340558\pi$
$443$	20.2148	0.960433	0.480217	+	0.877150i	$0.340558\pi$
$444$	0	0
$445$	−4.76393	−0.225832
$446$	0	0
$447$	−17.4721	−0.826404
$448$	0	0
$449$	14.7082	0.694123	0.347062	−	0.937842i	$-0.387179\pi$
$449$	14.7082	0.694123	0.347062	+	0.937842i	$0.387179\pi$
$450$	0	0
$451$	32.3607	1.52380
$452$	0	0
$453$	−13.9443	−0.655159
$454$	0	0
$455$	4.85410	0.227564
$456$	0	0
$457$	30.2705	1.41599	0.707997	−	0.706215i	$-0.249599\pi$
$457$	30.2705	1.41599	0.707997	+	0.706215i	$0.249599\pi$
$458$	0	0
$459$	3.09017	0.144237
$460$	0	0
$461$	−20.7426	−0.966081	−0.483041	−	0.875598i	$-0.660468\pi$
$461$	−20.7426	−0.966081	−0.483041	+	0.875598i	$0.660468\pi$
$462$	0	0
$463$	37.7771	1.75565	0.877825	−	0.478981i	$-0.158994\pi$
$463$	37.7771	1.75565	0.877825	+	0.478981i	$0.158994\pi$
$464$	0	0
$465$	1.38197	0.0640871
$466$	0	0
$467$	29.8328	1.38050	0.690249	−	0.723572i	$-0.257501\pi$
$467$	29.8328	1.38050	0.690249	+	0.723572i	$0.257501\pi$
$468$	0	0
$469$	19.1803	0.885666
$470$	0	0
$471$	−11.9443	−0.550363
$472$	0	0
$473$	−24.2705	−1.11596
$474$	0	0
$475$	−16.7082	−0.766625
$476$	0	0
$477$	−4.76393	−0.218125
$478$	0	0
$479$	2.76393	0.126287	0.0631436	−	0.998004i	$-0.479887\pi$
$479$	2.76393	0.126287	0.0631436	+	0.998004i	$0.479887\pi$
$480$	0	0
$481$	−2.72949	−0.124454
$482$	0	0
$483$	−8.47214	−0.385496
$484$	0	0
$485$	−0.763932	−0.0346884
$486$	0	0
$487$	35.4721	1.60740	0.803698	−	0.595037i	$-0.202863\pi$
$487$	35.4721	1.60740	0.803698	+	0.595037i	$0.202863\pi$
$488$	0	0
$489$	17.4721	0.790117
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−3.52786	−0.158887
$494$	0	0
$495$	−4.47214	−0.201008
$496$	0	0
$497$	−45.7426	−2.05184
$498$	0	0
$499$	−27.6869	−1.23944	−0.619718	−	0.784824i	$-0.712753\pi$
$499$	−27.6869	−1.23944	−0.619718	+	0.784824i	$0.712753\pi$
$500$	0	0
$501$	−7.76393	−0.346867
$502$	0	0
$503$	3.76393	0.167825	0.0839127	−	0.996473i	$-0.473258\pi$
$503$	3.76393	0.167825	0.0839127	+	0.996473i	$0.473258\pi$
$504$	0	0
$505$	7.03444	0.313029
$506$	0	0
$507$	9.56231	0.424677
$508$	0	0
$509$	−33.5066	−1.48515	−0.742576	−	0.669761i	$-0.766396\pi$
$509$	−33.5066	−1.48515	−0.742576	+	0.669761i	$0.766396\pi$
$510$	0	0
$511$	−62.6869	−2.77311
$512$	0	0
$513$	18.0902	0.798701
$514$	0	0
$515$	0.618034	0.0272338
$516$	0	0
$517$	6.70820	0.295026
$518$	0	0
$519$	−4.38197	−0.192347
$520$	0	0
$521$	−35.3607	−1.54918	−0.774590	−	0.632464i	$-0.782044\pi$
$521$	−35.3607	−1.54918	−0.774590	+	0.632464i	$0.782044\pi$
$522$	0	0
$523$	−32.4721	−1.41991	−0.709954	−	0.704248i	$-0.751284\pi$
$523$	−32.4721	−1.41991	−0.709954	+	0.704248i	$0.751284\pi$
$524$	0	0
$525$	19.5623	0.853769
$526$	0	0
$527$	−1.38197	−0.0601994
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−20.6525	−0.896241
$532$	0	0
$533$	16.5836	0.718315
$534$	0	0
$535$	5.65248	0.244378
$536$	0	0
$537$	−12.8541	−0.554695
$538$	0	0
$539$	39.5967	1.70555
$540$	0	0
$541$	14.3262	0.615933	0.307967	−	0.951397i	$-0.400352\pi$
$541$	14.3262	0.615933	0.307967	+	0.951397i	$0.400352\pi$
$542$	0	0
$543$	4.85410	0.208309
$544$	0	0
$545$	1.29180	0.0553345
$546$	0	0
$547$	−25.2705	−1.08049	−0.540244	−	0.841508i	$-0.681668\pi$
$547$	−25.2705	−1.08049	−0.540244	+	0.841508i	$0.681668\pi$
$548$	0	0
$549$	−13.7082	−0.585052
$550$	0	0
$551$	−20.6525	−0.879825
$552$	0	0
$553$	5.85410	0.248942
$554$	0	0
$555$	0.909830	0.0386201
$556$	0	0
$557$	−20.7984	−0.881255	−0.440628	−	0.897690i	$-0.645244\pi$
$557$	−20.7984	−0.881255	−0.440628	+	0.897690i	$0.645244\pi$
$558$	0	0
$559$	−12.4377	−0.526058
$560$	0	0
$561$	−2.23607	−0.0944069
$562$	0	0
$563$	−25.7984	−1.08727	−0.543636	−	0.839321i	$-0.682953\pi$
$563$	−25.7984	−1.08727	−0.543636	+	0.839321i	$0.682953\pi$
$564$	0	0
$565$	8.79837	0.370150
$566$	0	0
$567$	4.23607	0.177898
$568$	0	0
$569$	24.3820	1.02215	0.511073	−	0.859538i	$-0.329248\pi$
$569$	24.3820	1.02215	0.511073	+	0.859538i	$0.329248\pi$
$570$	0	0
$571$	−17.4508	−0.730295	−0.365148	−	0.930950i	$-0.618982\pi$
$571$	−17.4508	−0.730295	−0.365148	+	0.930950i	$0.618982\pi$
$572$	0	0
$573$	16.6180	0.694228
$574$	0	0
$575$	−9.23607	−0.385171
$576$	0	0
$577$	34.3607	1.43045	0.715227	−	0.698892i	$-0.246323\pi$
$577$	34.3607	1.43045	0.715227	+	0.698892i	$0.246323\pi$
$578$	0	0
$579$	13.0000	0.540262
$580$	0	0
$581$	58.4508	2.42495
$582$	0	0
$583$	8.61803	0.356922
$584$	0	0
$585$	−2.29180	−0.0947541
$586$	0	0
$587$	18.4164	0.760127	0.380063	−	0.924960i	$-0.375902\pi$
$587$	18.4164	0.760127	0.380063	+	0.924960i	$0.375902\pi$
$588$	0	0
$589$	−8.09017	−0.333350
$590$	0	0
$591$	−9.76393	−0.401634
$592$	0	0
$593$	−2.29180	−0.0941128	−0.0470564	−	0.998892i	$-0.514984\pi$
$593$	−2.29180	−0.0941128	−0.0470564	+	0.998892i	$0.514984\pi$
$594$	0	0
$595$	1.61803	0.0663329
$596$	0	0
$597$	−20.9443	−0.857192
$598$	0	0
$599$	−3.32624	−0.135906	−0.0679532	−	0.997689i	$-0.521647\pi$
$599$	−3.32624	−0.135906	−0.0679532	+	0.997689i	$0.521647\pi$
$600$	0	0
$601$	−22.5623	−0.920336	−0.460168	−	0.887832i	$-0.652211\pi$
$601$	−22.5623	−0.920336	−0.460168	+	0.887832i	$0.652211\pi$
$602$	0	0
$603$	−9.05573	−0.368778
$604$	0	0
$605$	1.29180	0.0525190
$606$	0	0
$607$	−30.5410	−1.23962	−0.619811	−	0.784751i	$-0.712791\pi$
$607$	−30.5410	−1.23962	−0.619811	+	0.784751i	$0.712791\pi$
$608$	0	0
$609$	24.1803	0.979837
$610$	0	0
$611$	3.43769	0.139074
$612$	0	0
$613$	−15.0000	−0.605844	−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000	−0.605844	−0.302922	+	0.953015i	$0.597962\pi$
$614$	0	0
$615$	−5.52786	−0.222905
$616$	0	0
$617$	34.5623	1.39143	0.695713	−	0.718320i	$-0.255089\pi$
$617$	34.5623	1.39143	0.695713	+	0.718320i	$0.255089\pi$
$618$	0	0
$619$	14.8541	0.597037	0.298518	−	0.954404i	$-0.403508\pi$
$619$	14.8541	0.597037	0.298518	+	0.954404i	$0.403508\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	0	0
$623$	−32.6525	−1.30819
$624$	0	0
$625$	19.4164	0.776656
$626$	0	0
$627$	−13.0902	−0.522771
$628$	0	0
$629$	−0.909830	−0.0362773
$630$	0	0
$631$	48.3262	1.92384	0.961919	−	0.273336i	$-0.0881271\pi$
$631$	48.3262	1.92384	0.961919	+	0.273336i	$0.0881271\pi$
$632$	0	0
$633$	−2.03444	−0.0808618
$634$	0	0
$635$	−3.29180	−0.130631
$636$	0	0
$637$	20.2918	0.803990
$638$	0	0
$639$	21.5967	0.854354
$640$	0	0
$641$	−47.3607	−1.87063	−0.935317	−	0.353810i	$-0.884886\pi$
$641$	−47.3607	−1.87063	−0.935317	+	0.353810i	$0.884886\pi$
$642$	0	0
$643$	35.5410	1.40160	0.700800	−	0.713357i	$-0.252826\pi$
$643$	35.5410	1.40160	0.700800	+	0.713357i	$0.252826\pi$
$644$	0	0
$645$	4.14590	0.163245
$646$	0	0
$647$	34.0344	1.33803	0.669016	−	0.743248i	$-0.266716\pi$
$647$	34.0344	1.33803	0.669016	+	0.743248i	$0.266716\pi$
$648$	0	0
$649$	37.3607	1.46653
$650$	0	0
$651$	9.47214	0.371242
$652$	0	0
$653$	−16.1803	−0.633186	−0.316593	−	0.948562i	$-0.602539\pi$
$653$	−16.1803	−0.633186	−0.316593	+	0.948562i	$0.602539\pi$
$654$	0	0
$655$	−3.38197	−0.132144
$656$	0	0
$657$	29.5967	1.15468
$658$	0	0
$659$	−47.9443	−1.86764	−0.933822	−	0.357738i	$-0.883548\pi$
$659$	−47.9443	−1.86764	−0.933822	+	0.357738i	$0.883548\pi$
$660$	0	0
$661$	30.5623	1.18874	0.594368	−	0.804193i	$-0.297402\pi$
$661$	30.5623	1.18874	0.594368	+	0.804193i	$0.297402\pi$
$662$	0	0
$663$	−1.14590	−0.0445030
$664$	0	0
$665$	9.47214	0.367314
$666$	0	0
$667$	−11.4164	−0.442045
$668$	0	0
$669$	7.23607	0.279763
$670$	0	0
$671$	24.7984	0.957331
$672$	0	0
$673$	−10.0689	−0.388127	−0.194063	−	0.980989i	$-0.562167\pi$
$673$	−10.0689	−0.388127	−0.194063	+	0.980989i	$0.562167\pi$
$674$	0	0
$675$	−23.0902	−0.888741
$676$	0	0
$677$	9.43769	0.362720	0.181360	−	0.983417i	$-0.441950\pi$
$677$	9.43769	0.362720	0.181360	+	0.983417i	$0.441950\pi$
$678$	0	0
$679$	−5.23607	−0.200942
$680$	0	0
$681$	11.8885	0.455570
$682$	0	0
$683$	15.3475	0.587257	0.293628	−	0.955920i	$-0.405137\pi$
$683$	15.3475	0.587257	0.293628	+	0.955920i	$0.405137\pi$
$684$	0	0
$685$	−1.09017	−0.0416533
$686$	0	0
$687$	22.7082	0.866372
$688$	0	0
$689$	4.41641	0.168252
$690$	0	0
$691$	13.5066	0.513814	0.256907	−	0.966436i	$-0.417297\pi$
$691$	13.5066	0.513814	0.256907	+	0.966436i	$0.417297\pi$
$692$	0	0
$693$	−30.6525	−1.16439
$694$	0	0
$695$	6.38197	0.242082
$696$	0	0
$697$	5.52786	0.209383
$698$	0	0
$699$	21.6525	0.818972
$700$	0	0
$701$	14.7984	0.558927	0.279463	−	0.960156i	$-0.409843\pi$
$701$	14.7984	0.558927	0.279463	+	0.960156i	$0.409843\pi$
$702$	0	0
$703$	−5.32624	−0.200883
$704$	0	0
$705$	−1.14590	−0.0431570
$706$	0	0
$707$	48.2148	1.81330
$708$	0	0
$709$	16.9656	0.637155	0.318577	−	0.947897i	$-0.396795\pi$
$709$	16.9656	0.637155	0.318577	+	0.947897i	$0.396795\pi$
$710$	0	0
$711$	−2.76393	−0.103656
$712$	0	0
$713$	−4.47214	−0.167483
$714$	0	0
$715$	4.14590	0.155048
$716$	0	0
$717$	5.38197	0.200993
$718$	0	0
$719$	−13.4934	−0.503220	−0.251610	−	0.967829i	$-0.580960\pi$
$719$	−13.4934	−0.503220	−0.251610	+	0.967829i	$0.580960\pi$
$720$	0	0
$721$	4.23607	0.157759
$722$	0	0
$723$	5.38197	0.200157
$724$	0	0
$725$	26.3607	0.979011
$726$	0	0
$727$	−10.2705	−0.380912	−0.190456	−	0.981696i	$-0.560997\pi$
$727$	−10.2705	−0.380912	−0.190456	+	0.981696i	$0.560997\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−4.14590	−0.153342
$732$	0	0
$733$	−11.9443	−0.441172	−0.220586	−	0.975368i	$-0.570797\pi$
$733$	−11.9443	−0.441172	−0.220586	+	0.975368i	$0.570797\pi$
$734$	0	0
$735$	−6.76393	−0.249491
$736$	0	0
$737$	16.3820	0.603437
$738$	0	0
$739$	4.47214	0.164510	0.0822551	−	0.996611i	$-0.473788\pi$
$739$	4.47214	0.164510	0.0822551	+	0.996611i	$0.473788\pi$
$740$	0	0
$741$	−6.70820	−0.246432
$742$	0	0
$743$	8.70820	0.319473	0.159737	−	0.987160i	$-0.448936\pi$
$743$	8.70820	0.319473	0.159737	+	0.987160i	$0.448936\pi$
$744$	0	0
$745$	10.7984	0.395622
$746$	0	0
$747$	−27.5967	−1.00971
$748$	0	0
$749$	38.7426	1.41563
$750$	0	0
$751$	−16.3050	−0.594976	−0.297488	−	0.954726i	$-0.596149\pi$
$751$	−16.3050	−0.594976	−0.297488	+	0.954726i	$0.596149\pi$
$752$	0	0
$753$	15.7082	0.572439
$754$	0	0
$755$	8.61803	0.313642
$756$	0	0
$757$	−17.1803	−0.624430	−0.312215	−	0.950011i	$-0.601071\pi$
$757$	−17.1803	−0.624430	−0.312215	+	0.950011i	$0.601071\pi$
$758$	0	0
$759$	−7.23607	−0.262653
$760$	0	0
$761$	−15.7639	−0.571442	−0.285721	−	0.958313i	$-0.592233\pi$
$761$	−15.7639	−0.571442	−0.285721	+	0.958313i	$0.592233\pi$
$762$	0	0
$763$	8.85410	0.320540
$764$	0	0
$765$	−0.763932	−0.0276200
$766$	0	0
$767$	19.1459	0.691318
$768$	0	0
$769$	−41.6312	−1.50126	−0.750630	−	0.660723i	$-0.770250\pi$
$769$	−41.6312	−1.50126	−0.750630	+	0.660723i	$0.770250\pi$
$770$	0	0
$771$	4.23607	0.152558
$772$	0	0
$773$	21.0000	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$774$	0	0
$775$	10.3262	0.370929
$776$	0	0
$777$	6.23607	0.223718
$778$	0	0
$779$	32.3607	1.15944
$780$	0	0
$781$	−39.0689	−1.39799
$782$	0	0
$783$	−28.5410	−1.01997
$784$	0	0
$785$	7.38197	0.263474
$786$	0	0
$787$	−50.4721	−1.79914	−0.899569	−	0.436779i	$-0.856119\pi$
$787$	−50.4721	−1.79914	−0.899569	+	0.436779i	$0.856119\pi$
$788$	0	0
$789$	15.3262	0.545629
$790$	0	0
$791$	60.3050	2.14420
$792$	0	0
$793$	12.7082	0.451282
$794$	0	0
$795$	−1.47214	−0.0522113
$796$	0	0
$797$	19.2918	0.683350	0.341675	−	0.939818i	$-0.389006\pi$
$797$	19.2918	0.683350	0.341675	+	0.939818i	$0.389006\pi$
$798$	0	0
$799$	1.14590	0.0405390
$800$	0	0
$801$	15.4164	0.544712
$802$	0	0
$803$	−53.5410	−1.88942
$804$	0	0
$805$	5.23607	0.184547
$806$	0	0
$807$	25.2148	0.887602
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−32.5967	−1.14463	−0.572313	−	0.820035i	$-0.693954\pi$
$811$	−32.5967	−1.14463	−0.572313	+	0.820035i	$0.693954\pi$
$812$	0	0
$813$	−29.3607	−1.02972
$814$	0	0
$815$	−10.7984	−0.378251
$816$	0	0
$817$	−24.2705	−0.849118
$818$	0	0
$819$	−15.7082	−0.548889
$820$	0	0
$821$	1.94427	0.0678556	0.0339278	−	0.999424i	$-0.489198\pi$
$821$	1.94427	0.0678556	0.0339278	+	0.999424i	$0.489198\pi$
$822$	0	0
$823$	−3.85410	−0.134346	−0.0671728	−	0.997741i	$-0.521398\pi$
$823$	−3.85410	−0.134346	−0.0671728	+	0.997741i	$0.521398\pi$
$824$	0	0
$825$	16.7082	0.581705
$826$	0	0
$827$	−11.0689	−0.384903	−0.192451	−	0.981307i	$-0.561644\pi$
$827$	−11.0689	−0.384903	−0.192451	+	0.981307i	$0.561644\pi$
$828$	0	0
$829$	10.4377	0.362516	0.181258	−	0.983436i	$-0.441983\pi$
$829$	10.4377	0.362516	0.181258	+	0.983436i	$0.441983\pi$
$830$	0	0
$831$	17.9443	0.622480
$832$	0	0
$833$	6.76393	0.234356
$834$	0	0
$835$	4.79837	0.166055
$836$	0	0
$837$	−11.1803	−0.386449
$838$	0	0
$839$	−15.3820	−0.531044	−0.265522	−	0.964105i	$-0.585544\pi$
$839$	−15.3820	−0.531044	−0.265522	+	0.964105i	$0.585544\pi$
$840$	0	0
$841$	3.58359	0.123572
$842$	0	0
$843$	9.29180	0.320026
$844$	0	0
$845$	−5.90983	−0.203304
$846$	0	0
$847$	8.85410	0.304231
$848$	0	0
$849$	20.1803	0.692587
$850$	0	0
$851$	−2.94427	−0.100928
$852$	0	0
$853$	6.43769	0.220422	0.110211	−	0.993908i	$-0.464847\pi$
$853$	6.43769	0.220422	0.110211	+	0.993908i	$0.464847\pi$
$854$	0	0
$855$	−4.47214	−0.152944
$856$	0	0
$857$	23.1803	0.791825	0.395913	−	0.918288i	$-0.370428\pi$
$857$	23.1803	0.791825	0.395913	+	0.918288i	$0.370428\pi$
$858$	0	0
$859$	−19.0902	−0.651348	−0.325674	−	0.945482i	$-0.605591\pi$
$859$	−19.0902	−0.651348	−0.325674	+	0.945482i	$0.605591\pi$
$860$	0	0
$861$	−37.8885	−1.29124
$862$	0	0
$863$	−27.1246	−0.923333	−0.461666	−	0.887054i	$-0.652748\pi$
$863$	−27.1246	−0.923333	−0.461666	+	0.887054i	$0.652748\pi$
$864$	0	0
$865$	2.70820	0.0920817
$866$	0	0
$867$	16.6180	0.564378
$868$	0	0
$869$	5.00000	0.169613
$870$	0	0
$871$	8.39512	0.284458
$872$	0	0
$873$	2.47214	0.0836691
$874$	0	0
$875$	−25.1803	−0.851251
$876$	0	0
$877$	−44.7082	−1.50969	−0.754844	−	0.655904i	$-0.772288\pi$
$877$	−44.7082	−1.50969	−0.754844	+	0.655904i	$0.772288\pi$
$878$	0	0
$879$	−17.0000	−0.573396
$880$	0	0
$881$	−8.00000	−0.269527	−0.134763	−	0.990878i	$-0.543027\pi$
$881$	−8.00000	−0.269527	−0.134763	+	0.990878i	$0.543027\pi$
$882$	0	0
$883$	1.83282	0.0616792	0.0308396	−	0.999524i	$-0.490182\pi$
$883$	1.83282	0.0616792	0.0308396	+	0.999524i	$0.490182\pi$
$884$	0	0
$885$	−6.38197	−0.214527
$886$	0	0
$887$	15.5410	0.521816	0.260908	−	0.965364i	$-0.415978\pi$
$887$	15.5410	0.521816	0.260908	+	0.965364i	$0.415978\pi$
$888$	0	0
$889$	−22.5623	−0.756715
$890$	0	0
$891$	3.61803	0.121209
$892$	0	0
$893$	6.70820	0.224481
$894$	0	0
$895$	7.94427	0.265548
$896$	0	0
$897$	−3.70820	−0.123813
$898$	0	0
$899$	12.7639	0.425701
$900$	0	0
$901$	1.47214	0.0490440
$902$	0	0
$903$	28.4164	0.945639
$904$	0	0
$905$	−3.00000	−0.0997234
$906$	0	0
$907$	24.7639	0.822273	0.411136	−	0.911574i	$-0.365132\pi$
$907$	24.7639	0.822273	0.411136	+	0.911574i	$0.365132\pi$
$908$	0	0
$909$	−22.7639	−0.755032
$910$	0	0
$911$	−4.96556	−0.164516	−0.0822581	−	0.996611i	$-0.526213\pi$
$911$	−4.96556	−0.164516	−0.0822581	+	0.996611i	$0.526213\pi$
$912$	0	0
$913$	49.9230	1.65221
$914$	0	0
$915$	−4.23607	−0.140040
$916$	0	0
$917$	−23.1803	−0.765482
$918$	0	0
$919$	−50.1033	−1.65276	−0.826378	−	0.563116i	$-0.809603\pi$
$919$	−50.1033	−1.65276	−0.826378	+	0.563116i	$0.809603\pi$
$920$	0	0
$921$	30.9787	1.02078
$922$	0	0
$923$	−20.0213	−0.659009
$924$	0	0
$925$	6.79837	0.223529
$926$	0	0
$927$	−2.00000	−0.0656886
$928$	0	0
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	39.5967	1.29773
$932$	0	0
$933$	24.5967	0.805261
$934$	0	0
$935$	1.38197	0.0451951
$936$	0	0
$937$	30.8197	1.00683	0.503417	−	0.864043i	$-0.332076\pi$
$937$	30.8197	1.00683	0.503417	+	0.864043i	$0.332076\pi$
$938$	0	0
$939$	16.7639	0.547070
$940$	0	0
$941$	22.9787	0.749085	0.374542	−	0.927210i	$-0.377800\pi$
$941$	22.9787	0.749085	0.374542	+	0.927210i	$0.377800\pi$
$942$	0	0
$943$	17.8885	0.582531
$944$	0	0
$945$	13.0902	0.425823
$946$	0	0
$947$	−52.5410	−1.70735	−0.853677	−	0.520803i	$-0.825633\pi$
$947$	−52.5410	−1.70735	−0.853677	+	0.520803i	$0.825633\pi$
$948$	0	0
$949$	−27.4377	−0.890665
$950$	0	0
$951$	−25.3607	−0.822376
$952$	0	0
$953$	54.8885	1.77801	0.889007	−	0.457893i	$-0.151396\pi$
$953$	54.8885	1.77801	0.889007	+	0.457893i	$0.151396\pi$
$954$	0	0
$955$	−10.2705	−0.332346
$956$	0	0
$957$	20.6525	0.667600
$958$	0	0
$959$	−7.47214	−0.241288
$960$	0	0
$961$	−26.0000	−0.838710
$962$	0	0
$963$	−18.2918	−0.589445
$964$	0	0
$965$	−8.03444	−0.258638
$966$	0	0
$967$	9.94427	0.319786	0.159893	−	0.987134i	$-0.448885\pi$
$967$	9.94427	0.319786	0.159893	+	0.987134i	$0.448885\pi$
$968$	0	0
$969$	−2.23607	−0.0718329
$970$	0	0
$971$	−14.6525	−0.470220	−0.235110	−	0.971969i	$-0.575545\pi$
$971$	−14.6525	−0.470220	−0.235110	+	0.971969i	$0.575545\pi$
$972$	0	0
$973$	43.7426	1.40232
$974$	0	0
$975$	8.56231	0.274213
$976$	0	0
$977$	−48.1033	−1.53896	−0.769481	−	0.638670i	$-0.779485\pi$
$977$	−48.1033	−1.53896	−0.769481	+	0.638670i	$0.779485\pi$
$978$	0	0
$979$	−27.8885	−0.891322
$980$	0	0
$981$	−4.18034	−0.133468
$982$	0	0
$983$	19.7082	0.628594	0.314297	−	0.949325i	$-0.398231\pi$
$983$	19.7082	0.628594	0.314297	+	0.949325i	$0.398231\pi$
$984$	0	0
$985$	6.03444	0.192273
$986$	0	0
$987$	−7.85410	−0.249999
$988$	0	0
$989$	−13.4164	−0.426617
$990$	0	0
$991$	47.2705	1.50160	0.750799	−	0.660531i	$-0.229669\pi$
$991$	47.2705	1.50160	0.750799	+	0.660531i	$0.229669\pi$
$992$	0	0
$993$	−27.0344	−0.857912
$994$	0	0
$995$	12.9443	0.410361
$996$	0	0
$997$	−29.3607	−0.929862	−0.464931	−	0.885347i	$-0.653921\pi$
$997$	−29.3607	−0.929862	−0.464931	+	0.885347i	$0.653921\pi$
$998$	0	0
$999$	−7.36068	−0.232882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1648.2.a.b.1.2		2
4.3	odd	2		824.2.a.d.1.2	✓	2
8.3	odd	2		6592.2.a.e.1.1		2
8.5	even	2		6592.2.a.r.1.1		2
12.11	even	2		7416.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
824.2.a.d.1.2	✓	2	4.3	odd	2
1648.2.a.b.1.2		2	1.1	even	1	trivial
6592.2.a.e.1.1		2	8.3	odd	2
6592.2.a.r.1.1		2	8.5	even	2
7416.2.a.g.1.1		2	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1648 = 2^{4} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1648.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.618034 q^{5} +4.23607 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.618034 q^{5} +4.23607 q^{7} -2.00000 q^{9} +3.61803 q^{11} +1.85410 q^{13} -0.618034 q^{15} +0.618034 q^{17} +3.61803 q^{19} -4.23607 q^{21} +2.00000 q^{23} -4.61803 q^{25} +5.00000 q^{27} -5.70820 q^{29} -2.23607 q^{31} -3.61803 q^{33} +2.61803 q^{35} -1.47214 q^{37} -1.85410 q^{39} +8.94427 q^{41} -6.70820 q^{43} -1.23607 q^{45} +1.85410 q^{47} +10.9443 q^{49} -0.618034 q^{51} +2.38197 q^{53} +2.23607 q^{55} -3.61803 q^{57} +10.3262 q^{59} +6.85410 q^{61} -8.47214 q^{63} +1.14590 q^{65} +4.52786 q^{67} -2.00000 q^{69} -10.7984 q^{71} -14.7984 q^{73} +4.61803 q^{75} +15.3262 q^{77} +1.38197 q^{79} +1.00000 q^{81} +13.7984 q^{83} +0.381966 q^{85} +5.70820 q^{87} -7.70820 q^{89} +7.85410 q^{91} +2.23607 q^{93} +2.23607 q^{95} -1.23607 q^{97} -7.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^5 + 4 * q^7 - 4 * q^9	\( 2 q - 2 q^{3} - q^{5} + 4 q^{7} - 4 q^{9} + 5 q^{11} - 3 q^{13} + q^{15} - q^{17} + 5 q^{19} - 4 q^{21} + 4 q^{23} - 7 q^{25} + 10 q^{27} + 2 q^{29} - 5 q^{33} + 3 q^{35} + 6 q^{37} + 3 q^{39} + 2 q^{45} - 3 q^{47} + 4 q^{49} + q^{51} + 7 q^{53} - 5 q^{57} + 5 q^{59} + 7 q^{61} - 8 q^{63} + 9 q^{65} + 18 q^{67} - 4 q^{69} + 3 q^{71} - 5 q^{73} + 7 q^{75} + 15 q^{77} + 5 q^{79} + 2 q^{81} + 3 q^{83} + 3 q^{85} - 2 q^{87} - 2 q^{89} + 9 q^{91} + 2 q^{97} - 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 + 4 * q^7 - 4 * q^9 + 5 * q^11 - 3 * q^13 + q^15 - q^17 + 5 * q^19 - 4 * q^21 + 4 * q^23 - 7 * q^25 + 10 * q^27 + 2 * q^29 - 5 * q^33 + 3 * q^35 + 6 * q^37 + 3 * q^39 + 2 * q^45 - 3 * q^47 + 4 * q^49 + q^51 + 7 * q^53 - 5 * q^57 + 5 * q^59 + 7 * q^61 - 8 * q^63 + 9 * q^65 + 18 * q^67 - 4 * q^69 + 3 * q^71 - 5 * q^73 + 7 * q^75 + 15 * q^77 + 5 * q^79 + 2 * q^81 + 3 * q^83 + 3 * q^85 - 2 * q^87 - 2 * q^89 + 9 * q^91 + 2 * q^97 - 10 * q^99

Introduction

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