LMFDB - Embedded newform 5200.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5200,2,Mod(1,5200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5200, 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("5200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5200 = 2^{4} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5200.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:	$41.5222090511$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	5200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} -0.828427 q^{7} -1.00000 q^{9} -0.585786 q^{11} +1.00000 q^{13} +4.82843 q^{17} -3.41421 q^{19} -1.17157 q^{21} -1.41421 q^{23} -5.65685 q^{27} +5.65685 q^{29} -10.2426 q^{31} -0.828427 q^{33} -8.48528 q^{37} +1.41421 q^{39} -8.82843 q^{41} +3.07107 q^{43} +0.828427 q^{47} -6.31371 q^{49} +6.82843 q^{51} +14.4853 q^{53} -4.82843 q^{57} -10.2426 q^{59} -8.00000 q^{61} +0.828427 q^{63} -2.00000 q^{67} -2.00000 q^{69} +7.89949 q^{71} +8.48528 q^{73} +0.485281 q^{77} -8.48528 q^{79} -5.00000 q^{81} -8.82843 q^{83} +8.00000 q^{87} +6.00000 q^{89} -0.828427 q^{91} -14.4853 q^{93} -3.65685 q^{97} +0.585786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} - 2 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{17} - 4 q^{19} - 8 q^{21} - 12 q^{31} + 4 q^{33} - 12 q^{41} - 8 q^{43} - 4 q^{47} + 10 q^{49} + 8 q^{51} + 12 q^{53} - 4 q^{57} - 12 q^{59} - 16 q^{61}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^7 - 2 * q^9 - 4 * q^11 + 2 * q^13 + 4 * q^17 - 4 * q^19 - 8 * q^21 - 12 * q^31 + 4 * q^33 - 12 * q^41 - 8 * q^43 - 4 * q^47 + 10 * q^49 + 8 * q^51 + 12 * q^53 - 4 * q^57 - 12 * q^59 - 16 * q^61 - 4 * q^63 - 4 * q^67 - 4 * q^69 - 4 * q^71 - 16 * q^77 - 10 * q^81 - 12 * q^83 + 16 * q^87 + 12 * q^89 + 4 * q^91 - 12 * q^93 + 4 * q^97 + 4 * q^99

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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.585786	−0.176621	−0.0883106	−	0.996093i	$-0.528147\pi$
$11$	−0.585786	−0.176621	−0.0883106	+	0.996093i	$0.528147\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	−3.41421	−0.783274	−0.391637	−	0.920120i	$-0.628091\pi$
$19$	−3.41421	−0.783274	−0.391637	+	0.920120i	$0.628091\pi$
$20$	0	0
$21$	−1.17157	−0.255658
$22$	0	0
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	5.65685	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$30$	0	0
$31$	−10.2426	−1.83963	−0.919816	−	0.392349i	$-0.871662\pi$
$31$	−10.2426	−1.83963	−0.919816	+	0.392349i	$0.871662\pi$
$32$	0	0
$33$	−0.828427	−0.144211
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	0	0
$39$	1.41421	0.226455
$40$	0	0
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0	0
$43$	3.07107	0.468333	0.234167	−	0.972196i	$-0.424764\pi$
$43$	3.07107	0.468333	0.234167	+	0.972196i	$0.424764\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.828427	0.120839	0.0604193	−	0.998173i	$-0.480756\pi$
$47$	0.828427	0.120839	0.0604193	+	0.998173i	$0.480756\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	0	0
$51$	6.82843	0.956171
$52$	0	0
$53$	14.4853	1.98971	0.994853	−	0.101327i	$-0.0323087\pi$
$53$	14.4853	1.98971	0.994853	+	0.101327i	$0.0323087\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.82843	−0.639541
$58$	0	0
$59$	−10.2426	−1.33348	−0.666739	−	0.745291i	$-0.732310\pi$
$59$	−10.2426	−1.33348	−0.666739	+	0.745291i	$0.732310\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	0.828427	0.104372
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	7.89949	0.937498	0.468749	−	0.883332i	$-0.344705\pi$
$71$	7.89949	0.937498	0.468749	+	0.883332i	$0.344705\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.485281	0.0553029
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−8.82843	−0.969046	−0.484523	−	0.874779i	$-0.661007\pi$
$83$	−8.82843	−0.969046	−0.484523	+	0.874779i	$0.661007\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	0	0
$93$	−14.4853	−1.50205
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	0	0
$99$	0.585786	0.0588738
$100$	0	0
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	0	0
$103$	17.4142	1.71587	0.857937	−	0.513755i	$-0.171746\pi$
$103$	17.4142	1.71587	0.857937	+	0.513755i	$0.171746\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.58579	−0.636672	−0.318336	−	0.947978i	$-0.603124\pi$
$107$	−6.58579	−0.636672	−0.318336	+	0.947978i	$0.603124\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	0	0
$113$	3.17157	0.298356	0.149178	−	0.988810i	$-0.452337\pi$
$113$	3.17157	0.298356	0.149178	+	0.988810i	$0.452337\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−10.6569	−0.968805
$122$	0	0
$123$	−12.4853	−1.12576
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−9.41421	−0.835376	−0.417688	−	0.908590i	$-0.637160\pi$
$127$	−9.41421	−0.835376	−0.417688	+	0.908590i	$0.637160\pi$
$128$	0	0
$129$	4.34315	0.382393
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.31371	−0.453981	−0.226990	−	0.973897i	$-0.572889\pi$
$137$	−5.31371	−0.453981	−0.226990	+	0.973897i	$0.572889\pi$
$138$	0	0
$139$	12.4853	1.05899	0.529494	−	0.848314i	$-0.322382\pi$
$139$	12.4853	1.05899	0.529494	+	0.848314i	$0.322382\pi$
$140$	0	0
$141$	1.17157	0.0986642
$142$	0	0
$143$	−0.585786	−0.0489859
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.92893	−0.736446
$148$	0	0
$149$	−0.343146	−0.0281116	−0.0140558	−	0.999901i	$-0.504474\pi$
$149$	−0.343146	−0.0281116	−0.0140558	+	0.999901i	$0.504474\pi$
$150$	0	0
$151$	−18.2426	−1.48457	−0.742283	−	0.670087i	$-0.766257\pi$
$151$	−18.2426	−1.48457	−0.742283	+	0.670087i	$0.766257\pi$
$152$	0	0
$153$	−4.82843	−0.390355
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	20.4853	1.62459
$160$	0	0
$161$	1.17157	0.0923329
$162$	0	0
$163$	−14.9706	−1.17258	−0.586292	−	0.810099i	$-0.699413\pi$
$163$	−14.9706	−1.17258	−0.586292	+	0.810099i	$0.699413\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.82843	−0.683164	−0.341582	−	0.939852i	$-0.610963\pi$
$167$	−8.82843	−0.683164	−0.341582	+	0.939852i	$0.610963\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.41421	0.261091
$172$	0	0
$173$	−11.1716	−0.849359	−0.424679	−	0.905344i	$-0.639613\pi$
$173$	−11.1716	−0.849359	−0.424679	+	0.905344i	$0.639613\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.4853	−1.08878
$178$	0	0
$179$	5.65685	0.422813	0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685	0.422813	0.211407	+	0.977398i	$0.432196\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	−11.3137	−0.836333
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.82843	−0.206835
$188$	0	0
$189$	4.68629	0.340878
$190$	0	0
$191$	−13.6569	−0.988175	−0.494088	−	0.869412i	$-0.664498\pi$
$191$	−13.6569	−0.988175	−0.494088	+	0.869412i	$0.664498\pi$
$192$	0	0
$193$	−15.6569	−1.12701	−0.563503	−	0.826114i	$-0.690546\pi$
$193$	−15.6569	−1.12701	−0.563503	+	0.826114i	$0.690546\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.9706	1.63658	0.818292	−	0.574802i	$-0.194921\pi$
$197$	22.9706	1.63658	0.818292	+	0.574802i	$0.194921\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−2.82843	−0.199502
$202$	0	0
$203$	−4.68629	−0.328913
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.41421	0.0982946
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	19.3137	1.32961	0.664805	−	0.747017i	$-0.268515\pi$
$211$	19.3137	1.32961	0.664805	+	0.747017i	$0.268515\pi$
$212$	0	0
$213$	11.1716	0.765464
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.48528	0.576018
$218$	0	0
$219$	12.0000	0.810885
$220$	0	0
$221$	4.82843	0.324795
$222$	0	0
$223$	26.4853	1.77359	0.886793	−	0.462167i	$-0.152928\pi$
$223$	26.4853	1.77359	0.886793	+	0.462167i	$0.152928\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−27.6569	−1.83565	−0.917825	−	0.396985i	$-0.870056\pi$
$227$	−27.6569	−1.83565	−0.917825	+	0.396985i	$0.870056\pi$
$228$	0	0
$229$	0.828427	0.0547440	0.0273720	−	0.999625i	$-0.491286\pi$
$229$	0.828427	0.0547440	0.0273720	+	0.999625i	$0.491286\pi$
$230$	0	0
$231$	0.686292	0.0451547
$232$	0	0
$233$	−24.6274	−1.61340	−0.806698	−	0.590964i	$-0.798747\pi$
$233$	−24.6274	−1.61340	−0.806698	+	0.590964i	$0.798747\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	0.585786	0.0378914	0.0189457	−	0.999821i	$-0.493969\pi$
$239$	0.585786	0.0378914	0.0189457	+	0.999821i	$0.493969\pi$
$240$	0	0
$241$	2.48528	0.160091	0.0800455	−	0.996791i	$-0.474493\pi$
$241$	2.48528	0.160091	0.0800455	+	0.996791i	$0.474493\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.41421	−0.217241
$248$	0	0
$249$	−12.4853	−0.791223
$250$	0	0
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	0	0
$253$	0.828427	0.0520828
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.3431	−1.01946	−0.509729	−	0.860335i	$-0.670254\pi$
$257$	−16.3431	−1.01946	−0.509729	+	0.860335i	$0.670254\pi$
$258$	0	0
$259$	7.02944	0.436788
$260$	0	0
$261$	−5.65685	−0.350150
$262$	0	0
$263$	−13.4142	−0.827156	−0.413578	−	0.910469i	$-0.635721\pi$
$263$	−13.4142	−0.827156	−0.413578	+	0.910469i	$0.635721\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.48528	0.519291
$268$	0	0
$269$	−2.68629	−0.163786	−0.0818930	−	0.996641i	$-0.526097\pi$
$269$	−2.68629	−0.163786	−0.0818930	+	0.996641i	$0.526097\pi$
$270$	0	0
$271$	−1.27208	−0.0772732	−0.0386366	−	0.999253i	$-0.512301\pi$
$271$	−1.27208	−0.0772732	−0.0386366	+	0.999253i	$0.512301\pi$
$272$	0	0
$273$	−1.17157	−0.0709068
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.17157	0.430898	0.215449	−	0.976515i	$-0.430878\pi$
$277$	7.17157	0.430898	0.215449	+	0.976515i	$0.430878\pi$
$278$	0	0
$279$	10.2426	0.613211
$280$	0	0
$281$	−17.7990	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$281$	−17.7990	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$282$	0	0
$283$	8.72792	0.518821	0.259411	−	0.965767i	$-0.416472\pi$
$283$	8.72792	0.518821	0.259411	+	0.965767i	$0.416472\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.31371	0.431715
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	−5.17157	−0.303163
$292$	0	0
$293$	2.14214	0.125145	0.0625724	−	0.998040i	$-0.480070\pi$
$293$	2.14214	0.125145	0.0625724	+	0.998040i	$0.480070\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.31371	0.192281
$298$	0	0
$299$	−1.41421	−0.0817861
$300$	0	0
$301$	−2.54416	−0.146643
$302$	0	0
$303$	10.8284	0.622077
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.1716	1.09418	0.547090	−	0.837074i	$-0.315736\pi$
$307$	19.1716	1.09418	0.547090	+	0.837074i	$0.315736\pi$
$308$	0	0
$309$	24.6274	1.40100
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	−0.828427	−0.0468255	−0.0234127	−	0.999726i	$-0.507453\pi$
$313$	−0.828427	−0.0468255	−0.0234127	+	0.999726i	$0.507453\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.1421	−1.46829	−0.734144	−	0.678993i	$-0.762416\pi$
$317$	−26.1421	−1.46829	−0.734144	+	0.678993i	$0.762416\pi$
$318$	0	0
$319$	−3.31371	−0.185532
$320$	0	0
$321$	−9.31371	−0.519841
$322$	0	0
$323$	−16.4853	−0.917266
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.82843	−0.156412
$328$	0	0
$329$	−0.686292	−0.0378365
$330$	0	0
$331$	−22.0416	−1.21152	−0.605759	−	0.795648i	$-0.707130\pi$
$331$	−22.0416	−1.21152	−0.605759	+	0.795648i	$0.707130\pi$
$332$	0	0
$333$	8.48528	0.464991
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.17157	−0.390660	−0.195330	−	0.980738i	$-0.562578\pi$
$337$	−7.17157	−0.390660	−0.195330	+	0.980738i	$0.562578\pi$
$338$	0	0
$339$	4.48528	0.243607
$340$	0	0
$341$	6.00000	0.324918
$342$	0	0
$343$	11.0294	0.595534
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.24264	0.227757	0.113878	−	0.993495i	$-0.463673\pi$
$347$	4.24264	0.227757	0.113878	+	0.993495i	$0.463673\pi$
$348$	0	0
$349$	1.51472	0.0810810	0.0405405	−	0.999178i	$-0.487092\pi$
$349$	1.51472	0.0810810	0.0405405	+	0.999178i	$0.487092\pi$
$350$	0	0
$351$	−5.65685	−0.301941
$352$	0	0
$353$	−9.17157	−0.488154	−0.244077	−	0.969756i	$-0.578485\pi$
$353$	−9.17157	−0.488154	−0.244077	+	0.969756i	$0.578485\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.65685	−0.299392
$358$	0	0
$359$	27.8995	1.47248	0.736240	−	0.676721i	$-0.236600\pi$
$359$	27.8995	1.47248	0.736240	+	0.676721i	$0.236600\pi$
$360$	0	0
$361$	−7.34315	−0.386481
$362$	0	0
$363$	−15.0711	−0.791026
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.44365	0.231957	0.115978	−	0.993252i	$-0.463000\pi$
$367$	4.44365	0.231957	0.115978	+	0.993252i	$0.463000\pi$
$368$	0	0
$369$	8.82843	0.459590
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	25.3137	1.31069	0.655347	−	0.755328i	$-0.272522\pi$
$373$	25.3137	1.31069	0.655347	+	0.755328i	$0.272522\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.65685	0.291343
$378$	0	0
$379$	−14.9289	−0.766848	−0.383424	−	0.923572i	$-0.625255\pi$
$379$	−14.9289	−0.766848	−0.383424	+	0.923572i	$0.625255\pi$
$380$	0	0
$381$	−13.3137	−0.682082
$382$	0	0
$383$	33.1127	1.69198	0.845990	−	0.533199i	$-0.179010\pi$
$383$	33.1127	1.69198	0.845990	+	0.533199i	$0.179010\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.07107	−0.156111
$388$	0	0
$389$	−16.6274	−0.843044	−0.421522	−	0.906818i	$-0.638504\pi$
$389$	−16.6274	−0.843044	−0.421522	+	0.906818i	$0.638504\pi$
$390$	0	0
$391$	−6.82843	−0.345328
$392$	0	0
$393$	−24.0000	−1.21064
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.7990	1.39519	0.697596	−	0.716492i	$-0.254253\pi$
$397$	27.7990	1.39519	0.697596	+	0.716492i	$0.254253\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	17.3137	0.864605	0.432303	−	0.901729i	$-0.357701\pi$
$401$	17.3137	0.864605	0.432303	+	0.901729i	$0.357701\pi$
$402$	0	0
$403$	−10.2426	−0.510222
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.97056	0.246382
$408$	0	0
$409$	12.8284	0.634325	0.317162	−	0.948371i	$-0.397270\pi$
$409$	12.8284	0.634325	0.317162	+	0.948371i	$0.397270\pi$
$410$	0	0
$411$	−7.51472	−0.370674
$412$	0	0
$413$	8.48528	0.417533
$414$	0	0
$415$	0	0
$416$	0	0
$417$	17.6569	0.864660
$418$	0	0
$419$	−5.17157	−0.252648	−0.126324	−	0.991989i	$-0.540318\pi$
$419$	−5.17157	−0.252648	−0.126324	+	0.991989i	$0.540318\pi$
$420$	0	0
$421$	−1.02944	−0.0501717	−0.0250859	−	0.999685i	$-0.507986\pi$
$421$	−1.02944	−0.0501717	−0.0250859	+	0.999685i	$0.507986\pi$
$422$	0	0
$423$	−0.828427	−0.0402795
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.62742	0.320723
$428$	0	0
$429$	−0.828427	−0.0399968
$430$	0	0
$431$	−3.61522	−0.174139	−0.0870696	−	0.996202i	$-0.527750\pi$
$431$	−3.61522	−0.174139	−0.0870696	+	0.996202i	$0.527750\pi$
$432$	0	0
$433$	3.65685	0.175737	0.0878686	−	0.996132i	$-0.471994\pi$
$433$	3.65685	0.175737	0.0878686	+	0.996132i	$0.471994\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.82843	0.230975
$438$	0	0
$439$	32.9706	1.57360	0.786800	−	0.617209i	$-0.211737\pi$
$439$	32.9706	1.57360	0.786800	+	0.617209i	$0.211737\pi$
$440$	0	0
$441$	6.31371	0.300653
$442$	0	0
$443$	−6.58579	−0.312900	−0.156450	−	0.987686i	$-0.550005\pi$
$443$	−6.58579	−0.312900	−0.156450	+	0.987686i	$0.550005\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.485281	−0.0229530
$448$	0	0
$449$	29.1127	1.37391	0.686957	−	0.726698i	$-0.258946\pi$
$449$	29.1127	1.37391	0.686957	+	0.726698i	$0.258946\pi$
$450$	0	0
$451$	5.17157	0.243520
$452$	0	0
$453$	−25.7990	−1.21214
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	−27.3137	−1.27489
$460$	0	0
$461$	26.4853	1.23354	0.616771	−	0.787142i	$-0.288440\pi$
$461$	26.4853	1.23354	0.616771	+	0.787142i	$0.288440\pi$
$462$	0	0
$463$	−15.6569	−0.727636	−0.363818	−	0.931470i	$-0.618527\pi$
$463$	−15.6569	−0.727636	−0.363818	+	0.931470i	$0.618527\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.5858	−0.489852	−0.244926	−	0.969542i	$-0.578764\pi$
$467$	−10.5858	−0.489852	−0.244926	+	0.969542i	$0.578764\pi$
$468$	0	0
$469$	1.65685	0.0765064
$470$	0	0
$471$	−25.4558	−1.17294
$472$	0	0
$473$	−1.79899	−0.0827176
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−14.4853	−0.663235
$478$	0	0
$479$	5.27208	0.240887	0.120444	−	0.992720i	$-0.461568\pi$
$479$	5.27208	0.240887	0.120444	+	0.992720i	$0.461568\pi$
$480$	0	0
$481$	−8.48528	−0.386896
$482$	0	0
$483$	1.65685	0.0753895
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.9706	1.04090	0.520448	−	0.853894i	$-0.325765\pi$
$487$	22.9706	1.04090	0.520448	+	0.853894i	$0.325765\pi$
$488$	0	0
$489$	−21.1716	−0.957412
$490$	0	0
$491$	−10.8284	−0.488680	−0.244340	−	0.969690i	$-0.578571\pi$
$491$	−10.8284	−0.488680	−0.244340	+	0.969690i	$0.578571\pi$
$492$	0	0
$493$	27.3137	1.23015
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.54416	−0.293546
$498$	0	0
$499$	−10.4437	−0.467522	−0.233761	−	0.972294i	$-0.575103\pi$
$499$	−10.4437	−0.467522	−0.233761	+	0.972294i	$0.575103\pi$
$500$	0	0
$501$	−12.4853	−0.557801
$502$	0	0
$503$	18.1005	0.807062	0.403531	−	0.914966i	$-0.367783\pi$
$503$	18.1005	0.807062	0.403531	+	0.914966i	$0.367783\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.41421	0.0628074
$508$	0	0
$509$	−21.1127	−0.935804	−0.467902	−	0.883780i	$-0.654990\pi$
$509$	−21.1127	−0.935804	−0.467902	+	0.883780i	$0.654990\pi$
$510$	0	0
$511$	−7.02944	−0.310964
$512$	0	0
$513$	19.3137	0.852721
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.485281	−0.0213427
$518$	0	0
$519$	−15.7990	−0.693499
$520$	0	0
$521$	−6.34315	−0.277898	−0.138949	−	0.990300i	$-0.544372\pi$
$521$	−6.34315	−0.277898	−0.138949	+	0.990300i	$0.544372\pi$
$522$	0	0
$523$	−28.2426	−1.23496	−0.617482	−	0.786585i	$-0.711847\pi$
$523$	−28.2426	−1.23496	−0.617482	+	0.786585i	$0.711847\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−49.4558	−2.15433
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	10.2426	0.444493
$532$	0	0
$533$	−8.82843	−0.382402
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	3.69848	0.159305
$540$	0	0
$541$	−12.8284	−0.551537	−0.275769	−	0.961224i	$-0.588932\pi$
$541$	−12.8284	−0.551537	−0.275769	+	0.961224i	$0.588932\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−29.2132	−1.24907	−0.624533	−	0.780998i	$-0.714711\pi$
$547$	−29.2132	−1.24907	−0.624533	+	0.780998i	$0.714711\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	−19.3137	−0.822792
$552$	0	0
$553$	7.02944	0.298922
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.79899	−0.160968	−0.0804842	−	0.996756i	$-0.525647\pi$
$557$	−3.79899	−0.160968	−0.0804842	+	0.996756i	$0.525647\pi$
$558$	0	0
$559$	3.07107	0.129892
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−16.2426	−0.684546	−0.342273	−	0.939601i	$-0.611197\pi$
$563$	−16.2426	−0.684546	−0.342273	+	0.939601i	$0.611197\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.14214	0.173953
$568$	0	0
$569$	−21.6569	−0.907903	−0.453951	−	0.891027i	$-0.649986\pi$
$569$	−21.6569	−0.907903	−0.453951	+	0.891027i	$0.649986\pi$
$570$	0	0
$571$	28.4853	1.19207	0.596036	−	0.802958i	$-0.296742\pi$
$571$	28.4853	1.19207	0.596036	+	0.802958i	$0.296742\pi$
$572$	0	0
$573$	−19.3137	−0.806842
$574$	0	0
$575$	0	0
$576$	0	0
$577$	29.1716	1.21443	0.607214	−	0.794538i	$-0.292287\pi$
$577$	29.1716	1.21443	0.607214	+	0.794538i	$0.292287\pi$
$578$	0	0
$579$	−22.1421	−0.920196
$580$	0	0
$581$	7.31371	0.303424
$582$	0	0
$583$	−8.48528	−0.351424
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.6569	1.30662	0.653309	−	0.757091i	$-0.273380\pi$
$587$	31.6569	1.30662	0.653309	+	0.757091i	$0.273380\pi$
$588$	0	0
$589$	34.9706	1.44094
$590$	0	0
$591$	32.4853	1.33627
$592$	0	0
$593$	−20.6274	−0.847066	−0.423533	−	0.905881i	$-0.639210\pi$
$593$	−20.6274	−0.847066	−0.423533	+	0.905881i	$0.639210\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.65685	−0.231520
$598$	0	0
$599$	25.4558	1.04010	0.520049	−	0.854137i	$-0.325914\pi$
$599$	25.4558	1.04010	0.520049	+	0.854137i	$0.325914\pi$
$600$	0	0
$601$	0.627417	0.0255929	0.0127964	−	0.999918i	$-0.495927\pi$
$601$	0.627417	0.0255929	0.0127964	+	0.999918i	$0.495927\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.2426	1.63340	0.816699	−	0.577064i	$-0.195802\pi$
$607$	40.2426	1.63340	0.816699	+	0.577064i	$0.195802\pi$
$608$	0	0
$609$	−6.62742	−0.268556
$610$	0	0
$611$	0.828427	0.0335146
$612$	0	0
$613$	37.3137	1.50709	0.753543	−	0.657398i	$-0.228343\pi$
$613$	37.3137	1.50709	0.753543	+	0.657398i	$0.228343\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.9706	0.924760	0.462380	−	0.886682i	$-0.346996\pi$
$617$	22.9706	0.924760	0.462380	+	0.886682i	$0.346996\pi$
$618$	0	0
$619$	−10.2426	−0.411686	−0.205843	−	0.978585i	$-0.565994\pi$
$619$	−10.2426	−0.411686	−0.205843	+	0.978585i	$0.565994\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	−4.97056	−0.199141
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.82843	0.112956
$628$	0	0
$629$	−40.9706	−1.63360
$630$	0	0
$631$	18.2426	0.726228	0.363114	−	0.931745i	$-0.381714\pi$
$631$	18.2426	0.726228	0.363114	+	0.931745i	$0.381714\pi$
$632$	0	0
$633$	27.3137	1.08562
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.31371	−0.250158
$638$	0	0
$639$	−7.89949	−0.312499
$640$	0	0
$641$	36.3431	1.43547	0.717734	−	0.696317i	$-0.245179\pi$
$641$	36.3431	1.43547	0.717734	+	0.696317i	$0.245179\pi$
$642$	0	0
$643$	26.4853	1.04448	0.522239	−	0.852799i	$-0.325097\pi$
$643$	26.4853	1.04448	0.522239	+	0.852799i	$0.325097\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.58579	0.258914	0.129457	−	0.991585i	$-0.458677\pi$
$647$	6.58579	0.258914	0.129457	+	0.991585i	$0.458677\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	−13.0294	−0.509881	−0.254941	−	0.966957i	$-0.582056\pi$
$653$	−13.0294	−0.509881	−0.254941	+	0.966957i	$0.582056\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.48528	−0.331042
$658$	0	0
$659$	−46.1421	−1.79744	−0.898721	−	0.438520i	$-0.855503\pi$
$659$	−46.1421	−1.79744	−0.898721	+	0.438520i	$0.855503\pi$
$660$	0	0
$661$	−49.5980	−1.92914	−0.964569	−	0.263831i	$-0.915014\pi$
$661$	−49.5980	−1.92914	−0.964569	+	0.263831i	$0.915014\pi$
$662$	0	0
$663$	6.82843	0.265194
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	37.4558	1.44813
$670$	0	0
$671$	4.68629	0.180912
$672$	0	0
$673$	10.4853	0.404178	0.202089	−	0.979367i	$-0.435227\pi$
$673$	10.4853	0.404178	0.202089	+	0.979367i	$0.435227\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.14214	−0.312928	−0.156464	−	0.987684i	$-0.550009\pi$
$677$	−8.14214	−0.312928	−0.156464	+	0.987684i	$0.550009\pi$
$678$	0	0
$679$	3.02944	0.116259
$680$	0	0
$681$	−39.1127	−1.49880
$682$	0	0
$683$	33.3137	1.27471	0.637357	−	0.770569i	$-0.280028\pi$
$683$	33.3137	1.27471	0.637357	+	0.770569i	$0.280028\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.17157	0.0446983
$688$	0	0
$689$	14.4853	0.551845
$690$	0	0
$691$	−21.0711	−0.801581	−0.400791	−	0.916170i	$-0.631265\pi$
$691$	−21.0711	−0.801581	−0.400791	+	0.916170i	$0.631265\pi$
$692$	0	0
$693$	−0.485281	−0.0184343
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−42.6274	−1.61463
$698$	0	0
$699$	−34.8284	−1.31733
$700$	0	0
$701$	37.3137	1.40932	0.704660	−	0.709545i	$-0.251100\pi$
$701$	37.3137	1.40932	0.704660	+	0.709545i	$0.251100\pi$
$702$	0	0
$703$	28.9706	1.09265
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.34315	−0.238559
$708$	0	0
$709$	−17.1127	−0.642681	−0.321340	−	0.946964i	$-0.604133\pi$
$709$	−17.1127	−0.642681	−0.321340	+	0.946964i	$0.604133\pi$
$710$	0	0
$711$	8.48528	0.318223
$712$	0	0
$713$	14.4853	0.542478
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.828427	0.0309382
$718$	0	0
$719$	4.97056	0.185371	0.0926854	−	0.995695i	$-0.470455\pi$
$719$	4.97056	0.185371	0.0926854	+	0.995695i	$0.470455\pi$
$720$	0	0
$721$	−14.4264	−0.537267
$722$	0	0
$723$	3.51472	0.130714
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19.3553	0.717850	0.358925	−	0.933366i	$-0.383143\pi$
$727$	19.3553	0.717850	0.358925	+	0.933366i	$0.383143\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	14.8284	0.548449
$732$	0	0
$733$	1.31371	0.0485229	0.0242615	−	0.999706i	$-0.492277\pi$
$733$	1.31371	0.0485229	0.0242615	+	0.999706i	$0.492277\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.17157	0.0431554
$738$	0	0
$739$	−30.7279	−1.13034	−0.565172	−	0.824973i	$-0.691190\pi$
$739$	−30.7279	−1.13034	−0.565172	+	0.824973i	$0.691190\pi$
$740$	0	0
$741$	−4.82843	−0.177377
$742$	0	0
$743$	−38.4853	−1.41189	−0.705944	−	0.708268i	$-0.749477\pi$
$743$	−38.4853	−1.41189	−0.705944	+	0.708268i	$0.749477\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.82843	0.323015
$748$	0	0
$749$	5.45584	0.199352
$750$	0	0
$751$	44.4853	1.62329	0.811645	−	0.584150i	$-0.198572\pi$
$751$	44.4853	1.62329	0.811645	+	0.584150i	$0.198572\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4.14214	−0.150548	−0.0752742	−	0.997163i	$-0.523983\pi$
$757$	−4.14214	−0.150548	−0.0752742	+	0.997163i	$0.523983\pi$
$758$	0	0
$759$	1.17157	0.0425254
$760$	0	0
$761$	−36.6274	−1.32774	−0.663871	−	0.747847i	$-0.731088\pi$
$761$	−36.6274	−1.32774	−0.663871	+	0.747847i	$0.731088\pi$
$762$	0	0
$763$	1.65685	0.0599822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.2426	−0.369840
$768$	0	0
$769$	10.9706	0.395609	0.197804	−	0.980242i	$-0.436619\pi$
$769$	10.9706	0.395609	0.197804	+	0.980242i	$0.436619\pi$
$770$	0	0
$771$	−23.1127	−0.832383
$772$	0	0
$773$	−6.14214	−0.220917	−0.110459	−	0.993881i	$-0.535232\pi$
$773$	−6.14214	−0.220917	−0.110459	+	0.993881i	$0.535232\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.94113	0.356636
$778$	0	0
$779$	30.1421	1.07995
$780$	0	0
$781$	−4.62742	−0.165582
$782$	0	0
$783$	−32.0000	−1.14359
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.51472	0.196578	0.0982892	−	0.995158i	$-0.468663\pi$
$787$	5.51472	0.196578	0.0982892	+	0.995158i	$0.468663\pi$
$788$	0	0
$789$	−18.9706	−0.675370
$790$	0	0
$791$	−2.62742	−0.0934202
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.9706	−0.388597	−0.194299	−	0.980942i	$-0.562243\pi$
$797$	−10.9706	−0.388597	−0.194299	+	0.980942i	$0.562243\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−4.97056	−0.175407
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.79899	−0.133731
$808$	0	0
$809$	−45.2548	−1.59108	−0.795538	−	0.605904i	$-0.792811\pi$
$809$	−45.2548	−1.59108	−0.795538	+	0.605904i	$0.792811\pi$
$810$	0	0
$811$	−8.38478	−0.294429	−0.147215	−	0.989105i	$-0.547031\pi$
$811$	−8.38478	−0.294429	−0.147215	+	0.989105i	$0.547031\pi$
$812$	0	0
$813$	−1.79899	−0.0630933
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−10.4853	−0.366834
$818$	0	0
$819$	0.828427	0.0289476
$820$	0	0
$821$	39.2548	1.37000	0.685002	−	0.728542i	$-0.259801\pi$
$821$	39.2548	1.37000	0.685002	+	0.728542i	$0.259801\pi$
$822$	0	0
$823$	34.3848	1.19858	0.599289	−	0.800533i	$-0.295450\pi$
$823$	34.3848	1.19858	0.599289	+	0.800533i	$0.295450\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.8579	0.968713	0.484356	−	0.874871i	$-0.339054\pi$
$827$	27.8579	0.968713	0.484356	+	0.874871i	$0.339054\pi$
$828$	0	0
$829$	7.02944	0.244142	0.122071	−	0.992521i	$-0.461046\pi$
$829$	7.02944	0.244142	0.122071	+	0.992521i	$0.461046\pi$
$830$	0	0
$831$	10.1421	0.351827
$832$	0	0
$833$	−30.4853	−1.05625
$834$	0	0
$835$	0	0
$836$	0	0
$837$	57.9411	2.00274
$838$	0	0
$839$	18.7279	0.646560	0.323280	−	0.946303i	$-0.395214\pi$
$839$	18.7279	0.646560	0.323280	+	0.946303i	$0.395214\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	0	0
$843$	−25.1716	−0.866955
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.82843	0.303348
$848$	0	0
$849$	12.3431	0.423616
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	−37.4558	−1.28246	−0.641232	−	0.767347i	$-0.721576\pi$
$853$	−37.4558	−1.28246	−0.641232	+	0.767347i	$0.721576\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.343146	−0.0117216	−0.00586082	−	0.999983i	$-0.501866\pi$
$857$	−0.343146	−0.0117216	−0.00586082	+	0.999983i	$0.501866\pi$
$858$	0	0
$859$	−11.7990	−0.402576	−0.201288	−	0.979532i	$-0.564513\pi$
$859$	−11.7990	−0.402576	−0.201288	+	0.979532i	$0.564513\pi$
$860$	0	0
$861$	10.3431	0.352493
$862$	0	0
$863$	19.4558	0.662285	0.331142	−	0.943581i	$-0.392566\pi$
$863$	19.4558	0.662285	0.331142	+	0.943581i	$0.392566\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.92893	0.303242
$868$	0	0
$869$	4.97056	0.168615
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	3.65685	0.123766
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.68629	−0.0907096	−0.0453548	−	0.998971i	$-0.514442\pi$
$877$	−2.68629	−0.0907096	−0.0453548	+	0.998971i	$0.514442\pi$
$878$	0	0
$879$	3.02944	0.102180
$880$	0	0
$881$	52.9706	1.78462	0.892312	−	0.451420i	$-0.149082\pi$
$881$	52.9706	1.78462	0.892312	+	0.451420i	$0.149082\pi$
$882$	0	0
$883$	−32.2426	−1.08505	−0.542526	−	0.840039i	$-0.682532\pi$
$883$	−32.2426	−1.08505	−0.542526	+	0.840039i	$0.682532\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14.3848	0.482994	0.241497	−	0.970402i	$-0.422362\pi$
$887$	14.3848	0.482994	0.241497	+	0.970402i	$0.422362\pi$
$888$	0	0
$889$	7.79899	0.261570
$890$	0	0
$891$	2.92893	0.0981229
$892$	0	0
$893$	−2.82843	−0.0946497
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	−57.9411	−1.93244
$900$	0	0
$901$	69.9411	2.33008
$902$	0	0
$903$	−3.59798	−0.119733
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−33.2132	−1.10283	−0.551413	−	0.834232i	$-0.685911\pi$
$907$	−33.2132	−1.10283	−0.551413	+	0.834232i	$0.685911\pi$
$908$	0	0
$909$	−7.65685	−0.253962
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	5.17157	0.171154
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.0589	0.464265
$918$	0	0
$919$	16.4853	0.543799	0.271900	−	0.962326i	$-0.412348\pi$
$919$	16.4853	0.543799	0.271900	+	0.962326i	$0.412348\pi$
$920$	0	0
$921$	27.1127	0.893394
$922$	0	0
$923$	7.89949	0.260015
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.4142	−0.571958
$928$	0	0
$929$	11.1716	0.366527	0.183264	−	0.983064i	$-0.441334\pi$
$929$	11.1716	0.366527	0.183264	+	0.983064i	$0.441334\pi$
$930$	0	0
$931$	21.5563	0.706481
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−10.9706	−0.358393	−0.179196	−	0.983813i	$-0.557350\pi$
$937$	−10.9706	−0.358393	−0.179196	+	0.983813i	$0.557350\pi$
$938$	0	0
$939$	−1.17157	−0.0382328
$940$	0	0
$941$	−54.7696	−1.78544	−0.892718	−	0.450615i	$-0.851205\pi$
$941$	−54.7696	−1.78544	−0.892718	+	0.450615i	$0.851205\pi$
$942$	0	0
$943$	12.4853	0.406577
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45.1127	−1.46597	−0.732983	−	0.680247i	$-0.761872\pi$
$947$	−45.1127	−1.46597	−0.732983	+	0.680247i	$0.761872\pi$
$948$	0	0
$949$	8.48528	0.275444
$950$	0	0
$951$	−36.9706	−1.19885
$952$	0	0
$953$	55.2548	1.78988	0.894940	−	0.446187i	$-0.147218\pi$
$953$	55.2548	1.78988	0.894940	+	0.446187i	$0.147218\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.68629	−0.151486
$958$	0	0
$959$	4.40202	0.142149
$960$	0	0
$961$	73.9117	2.38425
$962$	0	0
$963$	6.58579	0.212224
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−19.9411	−0.641263	−0.320632	−	0.947204i	$-0.603895\pi$
$967$	−19.9411	−0.641263	−0.320632	+	0.947204i	$0.603895\pi$
$968$	0	0
$969$	−23.3137	−0.748944
$970$	0	0
$971$	12.2843	0.394221	0.197111	−	0.980381i	$-0.436844\pi$
$971$	12.2843	0.394221	0.197111	+	0.980381i	$0.436844\pi$
$972$	0	0
$973$	−10.3431	−0.331586
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56.4853	1.80712	0.903562	−	0.428457i	$-0.140943\pi$
$977$	56.4853	1.80712	0.903562	+	0.428457i	$0.140943\pi$
$978$	0	0
$979$	−3.51472	−0.112331
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	−34.9706	−1.11539	−0.557694	−	0.830047i	$-0.688314\pi$
$983$	−34.9706	−1.11539	−0.557694	+	0.830047i	$0.688314\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.970563	−0.0308934
$988$	0	0
$989$	−4.34315	−0.138104
$990$	0	0
$991$	−15.0294	−0.477426	−0.238713	−	0.971090i	$-0.576725\pi$
$991$	−15.0294	−0.477426	−0.238713	+	0.971090i	$0.576725\pi$
$992$	0	0
$993$	−31.1716	−0.989200
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−23.1716	−0.733851	−0.366926	−	0.930250i	$-0.619590\pi$
$997$	−23.1716	−0.733851	−0.366926	+	0.930250i	$0.619590\pi$
$998$	0	0
$999$	48.0000	1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.bu.1.2		2
4.3	odd	2		325.2.a.i.1.1		2
5.4	even	2		1040.2.a.j.1.1		2
12.11	even	2		2925.2.a.u.1.2		2
15.14	odd	2		9360.2.a.cd.1.2		2
20.3	even	4		325.2.b.f.274.3		4
20.7	even	4		325.2.b.f.274.2		4
20.19	odd	2		65.2.a.b.1.2	✓	2
40.19	odd	2		4160.2.a.bf.1.1		2
40.29	even	2		4160.2.a.z.1.2		2
52.51	odd	2		4225.2.a.r.1.2		2
60.23	odd	4		2925.2.c.r.2224.2		4
60.47	odd	4		2925.2.c.r.2224.3		4
60.59	even	2		585.2.a.m.1.1		2
140.139	even	2		3185.2.a.j.1.2		2
220.219	even	2		7865.2.a.j.1.1		2
260.19	even	12		845.2.m.f.361.2		8
260.59	even	12		845.2.m.f.361.3		8
260.99	even	4		845.2.c.b.506.3		4
260.119	even	12		845.2.m.f.316.3		8
260.139	odd	6		845.2.e.h.146.1		4
260.159	odd	6		845.2.e.h.191.1		4
260.179	odd	6		845.2.e.c.191.2		4
260.199	odd	6		845.2.e.c.146.2		4
260.219	even	12		845.2.m.f.316.2		8
260.239	even	4		845.2.c.b.506.2		4
260.259	odd	2		845.2.a.g.1.1		2
780.779	even	2		7605.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.2	✓	2	20.19	odd	2
325.2.a.i.1.1		2	4.3	odd	2
325.2.b.f.274.2		4	20.7	even	4
325.2.b.f.274.3		4	20.3	even	4
585.2.a.m.1.1		2	60.59	even	2
845.2.a.g.1.1		2	260.259	odd	2
845.2.c.b.506.2		4	260.239	even	4
845.2.c.b.506.3		4	260.99	even	4
845.2.e.c.146.2		4	260.199	odd	6
845.2.e.c.191.2		4	260.179	odd	6
845.2.e.h.146.1		4	260.139	odd	6
845.2.e.h.191.1		4	260.159	odd	6
845.2.m.f.316.2		8	260.219	even	12
845.2.m.f.316.3		8	260.119	even	12
845.2.m.f.361.2		8	260.19	even	12
845.2.m.f.361.3		8	260.59	even	12
1040.2.a.j.1.1		2	5.4	even	2
2925.2.a.u.1.2		2	12.11	even	2
2925.2.c.r.2224.2		4	60.23	odd	4
2925.2.c.r.2224.3		4	60.47	odd	4
3185.2.a.j.1.2		2	140.139	even	2
4160.2.a.z.1.2		2	40.29	even	2
4160.2.a.bf.1.1		2	40.19	odd	2
4225.2.a.r.1.2		2	52.51	odd	2
5200.2.a.bu.1.2		2	1.1	even	1	trivial
7605.2.a.x.1.2		2	780.779	even	2
7865.2.a.j.1.1		2	220.219	even	2
9360.2.a.cd.1.2		2	15.14	odd	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:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} -0.828427 q^{7} -1.00000 q^{9} -0.585786 q^{11} +1.00000 q^{13} +4.82843 q^{17} -3.41421 q^{19} -1.17157 q^{21} -1.41421 q^{23} -5.65685 q^{27} +5.65685 q^{29} -10.2426 q^{31} -0.828427 q^{33} -8.48528 q^{37} +1.41421 q^{39} -8.82843 q^{41} +3.07107 q^{43} +0.828427 q^{47} -6.31371 q^{49} +6.82843 q^{51} +14.4853 q^{53} -4.82843 q^{57} -10.2426 q^{59} -8.00000 q^{61} +0.828427 q^{63} -2.00000 q^{67} -2.00000 q^{69} +7.89949 q^{71} +8.48528 q^{73} +0.485281 q^{77} -8.48528 q^{79} -5.00000 q^{81} -8.82843 q^{83} +8.00000 q^{87} +6.00000 q^{89} -0.828427 q^{91} -14.4853 q^{93} -3.65685 q^{97} +0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} - 2 q^{9} - 4 q^{11} + 2 q^{13} + 4 q^{17} - 4 q^{19} - 8 q^{21} - 12 q^{31} + 4 q^{33} - 12 q^{41} - 8 q^{43} - 4 q^{47} + 10 q^{49} + 8 q^{51} + 12 q^{53} - 4 q^{57} - 12 q^{59} - 16 q^{61}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^7 - 2 * q^9 - 4 * q^11 + 2 * q^13 + 4 * q^17 - 4 * q^19 - 8 * q^21 - 12 * q^31 + 4 * q^33 - 12 * q^41 - 8 * q^43 - 4 * q^47 + 10 * q^49 + 8 * q^51 + 12 * q^53 - 4 * q^57 - 12 * q^59 - 16 * q^61 - 4 * q^63 - 4 * q^67 - 4 * q^69 - 4 * q^71 - 16 * q^77 - 10 * q^81 - 12 * q^83 + 16 * q^87 + 12 * q^89 + 4 * q^91 - 12 * q^93 + 4 * q^97 + 4 * q^99

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