LMFDB - Embedded newform 5200.2.a.br.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5200,2,Mod(1,5200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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:traces = [2,0,0,0,0,0,-2,0,10,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 325)
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)

$f(q)$	$=$	$q+2.82843 q^{3} -2.41421 q^{7} +5.00000 q^{9} -6.41421 q^{11} +1.00000 q^{13} -3.82843 q^{17} +3.65685 q^{19} -6.82843 q^{21} +3.17157 q^{23} +5.65685 q^{27} -0.171573 q^{29} -1.24264 q^{31} -18.1421 q^{33} -6.00000 q^{37} +2.82843 q^{39} -5.65685 q^{41} -0.828427 q^{43} -3.58579 q^{47} -1.17157 q^{49} -10.8284 q^{51} +3.00000 q^{53} +10.3431 q^{57} -13.2426 q^{59} +1.00000 q^{61} -12.0711 q^{63} +2.75736 q^{67} +8.97056 q^{69} +9.31371 q^{71} -6.00000 q^{73} +15.4853 q^{77} +6.00000 q^{79} +1.00000 q^{81} -6.89949 q^{83} -0.485281 q^{87} -8.48528 q^{89} -2.41421 q^{91} -3.51472 q^{93} -0.828427 q^{97} -32.0711 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} + 10 q^{9} - 10 q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} - 8 q^{21} + 12 q^{23} - 6 q^{29} + 6 q^{31} - 8 q^{33} - 12 q^{37} + 4 q^{43} - 10 q^{47} - 8 q^{49} - 16 q^{51} + 6 q^{53} + 32 q^{57}+ \cdots - 50 q^{99}+O(q^{100}) $ 2 * q - 2 * q^7 + 10 * q^9 - 10 * q^11 + 2 * q^13 - 2 * q^17 - 4 * q^19 - 8 * q^21 + 12 * q^23 - 6 * q^29 + 6 * q^31 - 8 * q^33 - 12 * q^37 + 4 * q^43 - 10 * q^47 - 8 * q^49 - 16 * q^51 + 6 * q^53 + 32 * q^57 - 18 * q^59 + 2 * q^61 - 10 * q^63 + 14 * q^67 - 16 * q^69 - 4 * q^71 - 12 * q^73 + 14 * q^77 + 12 * q^79 + 2 * q^81 + 6 * q^83 + 16 * q^87 - 2 * q^91 - 24 * q^93 + 4 * q^97 - 50 * 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
$3$	2.82843	1.63299	0.816497	−	0.577350i	$-0.195913\pi$
$3$	2.82843	1.63299	0.816497	+	0.577350i	$0.195913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.41421	−0.912487	−0.456243	−	0.889855i	$-0.650805\pi$
$7$	−2.41421	−0.912487	−0.456243	+	0.889855i	$0.650805\pi$
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	−6.41421	−1.93396	−0.966979	−	0.254856i	$-0.917972\pi$
$11$	−6.41421	−1.93396	−0.966979	+	0.254856i	$0.917972\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.82843	−0.928530	−0.464265	−	0.885696i	$-0.653681\pi$
$17$	−3.82843	−0.928530	−0.464265	+	0.885696i	$0.653681\pi$
$18$	0	0
$19$	3.65685	0.838940	0.419470	−	0.907769i	$-0.362216\pi$
$19$	3.65685	0.838940	0.419470	+	0.907769i	$0.362216\pi$
$20$	0	0
$21$	−6.82843	−1.49008
$22$	0	0
$23$	3.17157	0.661319	0.330659	−	0.943750i	$-0.392729\pi$
$23$	3.17157	0.661319	0.330659	+	0.943750i	$0.392729\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−0.171573	−0.0318603	−0.0159301	−	0.999873i	$-0.505071\pi$
$29$	−0.171573	−0.0318603	−0.0159301	+	0.999873i	$0.505071\pi$
$30$	0	0
$31$	−1.24264	−0.223185	−0.111592	−	0.993754i	$-0.535595\pi$
$31$	−1.24264	−0.223185	−0.111592	+	0.993754i	$0.535595\pi$
$32$	0	0
$33$	−18.1421	−3.15814
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	2.82843	0.452911
$40$	0	0
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	−0.828427	−0.126334	−0.0631670	−	0.998003i	$-0.520120\pi$
$43$	−0.828427	−0.126334	−0.0631670	+	0.998003i	$0.520120\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.58579	−0.523041	−0.261520	−	0.965198i	$-0.584224\pi$
$47$	−3.58579	−0.523041	−0.261520	+	0.965198i	$0.584224\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	−10.8284	−1.51628
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.3431	1.36998
$58$	0	0
$59$	−13.2426	−1.72404	−0.862022	−	0.506870i	$-0.830802\pi$
$59$	−13.2426	−1.72404	−0.862022	+	0.506870i	$0.830802\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	−12.0711	−1.52081
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.75736	0.336865	0.168433	−	0.985713i	$-0.446129\pi$
$67$	2.75736	0.336865	0.168433	+	0.985713i	$0.446129\pi$
$68$	0	0
$69$	8.97056	1.07993
$70$	0	0
$71$	9.31371	1.10533	0.552667	−	0.833402i	$-0.313610\pi$
$71$	9.31371	1.10533	0.552667	+	0.833402i	$0.313610\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.4853	1.76471
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.89949	−0.757318	−0.378659	−	0.925536i	$-0.623615\pi$
$83$	−6.89949	−0.757318	−0.378659	+	0.925536i	$0.623615\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.485281	−0.0520276
$88$	0	0
$89$	−8.48528	−0.899438	−0.449719	−	0.893170i	$-0.648476\pi$
$89$	−8.48528	−0.899438	−0.449719	+	0.893170i	$0.648476\pi$
$90$	0	0
$91$	−2.41421	−0.253078
$92$	0	0
$93$	−3.51472	−0.364459
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.828427	−0.0841140	−0.0420570	−	0.999115i	$-0.513391\pi$
$97$	−0.828427	−0.0841140	−0.0420570	+	0.999115i	$0.513391\pi$
$98$	0	0
$99$	−32.0711	−3.22326
$100$	0	0
$101$	−6.65685	−0.662382	−0.331191	−	0.943564i	$-0.607450\pi$
$101$	−6.65685	−0.662382	−0.331191	+	0.943564i	$0.607450\pi$
$102$	0	0
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.1421	−1.56052	−0.780260	−	0.625456i	$-0.784913\pi$
$107$	−16.1421	−1.56052	−0.780260	+	0.625456i	$0.784913\pi$
$108$	0	0
$109$	−16.4853	−1.57900	−0.789502	−	0.613748i	$-0.789661\pi$
$109$	−16.4853	−1.57900	−0.789502	+	0.613748i	$0.789661\pi$
$110$	0	0
$111$	−16.9706	−1.61077
$112$	0	0
$113$	−11.6569	−1.09658	−0.548292	−	0.836287i	$-0.684722\pi$
$113$	−11.6569	−1.09658	−0.548292	+	0.836287i	$0.684722\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.00000	0.462250
$118$	0	0
$119$	9.24264	0.847271
$120$	0	0
$121$	30.1421	2.74019
$122$	0	0
$123$	−16.0000	−1.44267
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.65685	0.324493	0.162247	−	0.986750i	$-0.448126\pi$
$127$	3.65685	0.324493	0.162247	+	0.986750i	$0.448126\pi$
$128$	0	0
$129$	−2.34315	−0.206302
$130$	0	0
$131$	−8.48528	−0.741362	−0.370681	−	0.928760i	$-0.620876\pi$
$131$	−8.48528	−0.741362	−0.370681	+	0.928760i	$0.620876\pi$
$132$	0	0
$133$	−8.82843	−0.765522
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.82843	0.241649	0.120824	−	0.992674i	$-0.461446\pi$
$137$	2.82843	0.241649	0.120824	+	0.992674i	$0.461446\pi$
$138$	0	0
$139$	18.4853	1.56790	0.783951	−	0.620823i	$-0.213202\pi$
$139$	18.4853	1.56790	0.783951	+	0.620823i	$0.213202\pi$
$140$	0	0
$141$	−10.1421	−0.854122
$142$	0	0
$143$	−6.41421	−0.536383
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.31371	−0.273310
$148$	0	0
$149$	8.82843	0.723253	0.361626	−	0.932323i	$-0.382222\pi$
$149$	8.82843	0.723253	0.361626	+	0.932323i	$0.382222\pi$
$150$	0	0
$151$	8.75736	0.712664	0.356332	−	0.934359i	$-0.384027\pi$
$151$	8.75736	0.712664	0.356332	+	0.934359i	$0.384027\pi$
$152$	0	0
$153$	−19.1421	−1.54755
$154$	0	0
$155$	0	0
$156$	0	0
$157$	23.4853	1.87433	0.937165	−	0.348887i	$-0.113440\pi$
$157$	23.4853	1.87433	0.937165	+	0.348887i	$0.113440\pi$
$158$	0	0
$159$	8.48528	0.672927
$160$	0	0
$161$	−7.65685	−0.603445
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.31371	0.411187	0.205594	−	0.978637i	$-0.434088\pi$
$167$	5.31371	0.411187	0.205594	+	0.978637i	$0.434088\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	18.2843	1.39823
$172$	0	0
$173$	18.6569	1.41845	0.709227	−	0.704980i	$-0.249044\pi$
$173$	18.6569	1.41845	0.709227	+	0.704980i	$0.249044\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−37.4558	−2.81535
$178$	0	0
$179$	−0.686292	−0.0512958	−0.0256479	−	0.999671i	$-0.508165\pi$
$179$	−0.686292	−0.0512958	−0.0256479	+	0.999671i	$0.508165\pi$
$180$	0	0
$181$	−17.4853	−1.29967	−0.649835	−	0.760075i	$-0.725162\pi$
$181$	−17.4853	−1.29967	−0.649835	+	0.760075i	$0.725162\pi$
$182$	0	0
$183$	2.82843	0.209083
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.5563	1.79574
$188$	0	0
$189$	−13.6569	−0.993390
$190$	0	0
$191$	−25.3137	−1.83164	−0.915818	−	0.401594i	$-0.868456\pi$
$191$	−25.3137	−1.83164	−0.915818	+	0.401594i	$0.868456\pi$
$192$	0	0
$193$	1.65685	0.119263	0.0596315	−	0.998220i	$-0.481007\pi$
$193$	1.65685	0.119263	0.0596315	+	0.998220i	$0.481007\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	7.79899	0.550098
$202$	0	0
$203$	0.414214	0.0290721
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.8579	1.10220
$208$	0	0
$209$	−23.4558	−1.62247
$210$	0	0
$211$	−17.7990	−1.22533	−0.612666	−	0.790342i	$-0.709903\pi$
$211$	−17.7990	−1.22533	−0.612666	+	0.790342i	$0.709903\pi$
$212$	0	0
$213$	26.3431	1.80500
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.00000	0.203653
$218$	0	0
$219$	−16.9706	−1.14676
$220$	0	0
$221$	−3.82843	−0.257528
$222$	0	0
$223$	10.9706	0.734643	0.367322	−	0.930094i	$-0.380275\pi$
$223$	10.9706	0.734643	0.367322	+	0.930094i	$0.380275\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.89949	0.590680	0.295340	−	0.955392i	$-0.404567\pi$
$227$	8.89949	0.590680	0.295340	+	0.955392i	$0.404567\pi$
$228$	0	0
$229$	−4.82843	−0.319071	−0.159536	−	0.987192i	$-0.551000\pi$
$229$	−4.82843	−0.319071	−0.159536	+	0.987192i	$0.551000\pi$
$230$	0	0
$231$	43.7990	2.88176
$232$	0	0
$233$	20.6274	1.35135	0.675674	−	0.737201i	$-0.263853\pi$
$233$	20.6274	1.35135	0.675674	+	0.737201i	$0.263853\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.9706	1.10236
$238$	0	0
$239$	−23.5858	−1.52564	−0.762819	−	0.646612i	$-0.776185\pi$
$239$	−23.5858	−1.52564	−0.762819	+	0.646612i	$0.776185\pi$
$240$	0	0
$241$	4.97056	0.320182	0.160091	−	0.987102i	$-0.448821\pi$
$241$	4.97056	0.320182	0.160091	+	0.987102i	$0.448821\pi$
$242$	0	0
$243$	−14.1421	−0.907218
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.65685	0.232680
$248$	0	0
$249$	−19.5147	−1.23670
$250$	0	0
$251$	5.65685	0.357057	0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685	0.357057	0.178529	+	0.983935i	$0.442866\pi$
$252$	0	0
$253$	−20.3431	−1.27896
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.65685	−0.415243	−0.207622	−	0.978209i	$-0.566572\pi$
$257$	−6.65685	−0.415243	−0.207622	+	0.978209i	$0.566572\pi$
$258$	0	0
$259$	14.4853	0.900072
$260$	0	0
$261$	−0.857864	−0.0531005
$262$	0	0
$263$	26.1421	1.61199	0.805997	−	0.591920i	$-0.201630\pi$
$263$	26.1421	1.61199	0.805997	+	0.591920i	$0.201630\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−24.0000	−1.46878
$268$	0	0
$269$	15.1421	0.923232	0.461616	−	0.887080i	$-0.347270\pi$
$269$	15.1421	0.923232	0.461616	+	0.887080i	$0.347270\pi$
$270$	0	0
$271$	25.7279	1.56286	0.781430	−	0.623993i	$-0.214491\pi$
$271$	25.7279	1.56286	0.781430	+	0.623993i	$0.214491\pi$
$272$	0	0
$273$	−6.82843	−0.413275
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.31371	0.559607	0.279803	−	0.960057i	$-0.409731\pi$
$277$	9.31371	0.559607	0.279803	+	0.960057i	$0.409731\pi$
$278$	0	0
$279$	−6.21320	−0.371975
$280$	0	0
$281$	4.82843	0.288040	0.144020	−	0.989575i	$-0.453997\pi$
$281$	4.82843	0.288040	0.144020	+	0.989575i	$0.453997\pi$
$282$	0	0
$283$	22.4853	1.33661	0.668306	−	0.743887i	$-0.267020\pi$
$283$	22.4853	1.33661	0.668306	+	0.743887i	$0.267020\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.6569	0.806139
$288$	0	0
$289$	−2.34315	−0.137832
$290$	0	0
$291$	−2.34315	−0.137358
$292$	0	0
$293$	8.82843	0.515762	0.257881	−	0.966177i	$-0.416976\pi$
$293$	8.82843	0.515762	0.257881	+	0.966177i	$0.416976\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−36.2843	−2.10543
$298$	0	0
$299$	3.17157	0.183417
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	−18.8284	−1.08166
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.31371	0.531561	0.265781	−	0.964034i	$-0.414370\pi$
$307$	9.31371	0.531561	0.265781	+	0.964034i	$0.414370\pi$
$308$	0	0
$309$	−38.6274	−2.19744
$310$	0	0
$311$	−9.51472	−0.539530	−0.269765	−	0.962926i	$-0.586946\pi$
$311$	−9.51472	−0.539530	−0.269765	+	0.962926i	$0.586946\pi$
$312$	0	0
$313$	2.85786	0.161536	0.0807680	−	0.996733i	$-0.474263\pi$
$313$	2.85786	0.161536	0.0807680	+	0.996733i	$0.474263\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.1421	1.13130	0.565648	−	0.824647i	$-0.308626\pi$
$317$	20.1421	1.13130	0.565648	+	0.824647i	$0.308626\pi$
$318$	0	0
$319$	1.10051	0.0616165
$320$	0	0
$321$	−45.6569	−2.54832
$322$	0	0
$323$	−14.0000	−0.778981
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−46.6274	−2.57850
$328$	0	0
$329$	8.65685	0.477268
$330$	0	0
$331$	−27.6569	−1.52016	−0.760079	−	0.649831i	$-0.774840\pi$
$331$	−27.6569	−1.52016	−0.760079	+	0.649831i	$0.774840\pi$
$332$	0	0
$333$	−30.0000	−1.64399
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−20.7990	−1.13299	−0.566497	−	0.824064i	$-0.691702\pi$
$337$	−20.7990	−1.13299	−0.566497	+	0.824064i	$0.691702\pi$
$338$	0	0
$339$	−32.9706	−1.79072
$340$	0	0
$341$	7.97056	0.431630
$342$	0	0
$343$	19.7279	1.06521
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.4558	1.04444	0.522222	−	0.852809i	$-0.325103\pi$
$347$	19.4558	1.04444	0.522222	+	0.852809i	$0.325103\pi$
$348$	0	0
$349$	35.4558	1.89791	0.948954	−	0.315415i	$-0.102144\pi$
$349$	35.4558	1.89791	0.948954	+	0.315415i	$0.102144\pi$
$350$	0	0
$351$	5.65685	0.301941
$352$	0	0
$353$	26.1421	1.39141	0.695703	−	0.718330i	$-0.255093\pi$
$353$	26.1421	1.39141	0.695703	+	0.718330i	$0.255093\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	26.1421	1.38359
$358$	0	0
$359$	−21.3848	−1.12865	−0.564323	−	0.825554i	$-0.690863\pi$
$359$	−21.3848	−1.12865	−0.564323	+	0.825554i	$0.690863\pi$
$360$	0	0
$361$	−5.62742	−0.296180
$362$	0	0
$363$	85.2548	4.47472
$364$	0	0
$365$	0	0
$366$	0	0
$367$	34.8284	1.81803	0.909015	−	0.416764i	$-0.136836\pi$
$367$	34.8284	1.81803	0.909015	+	0.416764i	$0.136836\pi$
$368$	0	0
$369$	−28.2843	−1.47242
$370$	0	0
$371$	−7.24264	−0.376019
$372$	0	0
$373$	1.14214	0.0591375	0.0295688	−	0.999563i	$-0.490587\pi$
$373$	1.14214	0.0591375	0.0295688	+	0.999563i	$0.490587\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.171573	−0.00883645
$378$	0	0
$379$	31.8701	1.63705	0.818527	−	0.574467i	$-0.194791\pi$
$379$	31.8701	1.63705	0.818527	+	0.574467i	$0.194791\pi$
$380$	0	0
$381$	10.3431	0.529895
$382$	0	0
$383$	−27.6569	−1.41320	−0.706600	−	0.707614i	$-0.749772\pi$
$383$	−27.6569	−1.41320	−0.706600	+	0.707614i	$0.749772\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.14214	−0.210557
$388$	0	0
$389$	28.6274	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$390$	0	0
$391$	−12.1421	−0.614054
$392$	0	0
$393$	−24.0000	−1.21064
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.65685	0.384286	0.192143	−	0.981367i	$-0.438456\pi$
$397$	7.65685	0.384286	0.192143	+	0.981367i	$0.438456\pi$
$398$	0	0
$399$	−24.9706	−1.25009
$400$	0	0
$401$	−3.85786	−0.192653	−0.0963263	−	0.995350i	$-0.530709\pi$
$401$	−3.85786	−0.192653	−0.0963263	+	0.995350i	$0.530709\pi$
$402$	0	0
$403$	−1.24264	−0.0619003
$404$	0	0
$405$	0	0
$406$	0	0
$407$	38.4853	1.90764
$408$	0	0
$409$	−10.8284	−0.535431	−0.267716	−	0.963498i	$-0.586269\pi$
$409$	−10.8284	−0.535431	−0.267716	+	0.963498i	$0.586269\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	31.9706	1.57317
$414$	0	0
$415$	0	0
$416$	0	0
$417$	52.2843	2.56037
$418$	0	0
$419$	−22.8284	−1.11524	−0.557621	−	0.830096i	$-0.688286\pi$
$419$	−22.8284	−1.11524	−0.557621	+	0.830096i	$0.688286\pi$
$420$	0	0
$421$	−16.9706	−0.827095	−0.413547	−	0.910483i	$-0.635710\pi$
$421$	−16.9706	−0.827095	−0.413547	+	0.910483i	$0.635710\pi$
$422$	0	0
$423$	−17.9289	−0.871735
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.41421	−0.116832
$428$	0	0
$429$	−18.1421	−0.875911
$430$	0	0
$431$	8.34315	0.401875	0.200938	−	0.979604i	$-0.435601\pi$
$431$	8.34315	0.401875	0.200938	+	0.979604i	$0.435601\pi$
$432$	0	0
$433$	16.3431	0.785401	0.392701	−	0.919666i	$-0.371541\pi$
$433$	16.3431	0.785401	0.392701	+	0.919666i	$0.371541\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.5980	0.554807
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	−5.85786	−0.278946
$442$	0	0
$443$	29.7990	1.41579	0.707896	−	0.706316i	$-0.249644\pi$
$443$	29.7990	1.41579	0.707896	+	0.706316i	$0.249644\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	24.9706	1.18107
$448$	0	0
$449$	−5.17157	−0.244062	−0.122031	−	0.992526i	$-0.538941\pi$
$449$	−5.17157	−0.244062	−0.122031	+	0.992526i	$0.538941\pi$
$450$	0	0
$451$	36.2843	1.70856
$452$	0	0
$453$	24.7696	1.16378
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.48528	0.396925	0.198462	−	0.980109i	$-0.436405\pi$
$457$	8.48528	0.396925	0.198462	+	0.980109i	$0.436405\pi$
$458$	0	0
$459$	−21.6569	−1.01086
$460$	0	0
$461$	−19.4558	−0.906149	−0.453074	−	0.891473i	$-0.649673\pi$
$461$	−19.4558	−0.906149	−0.453074	+	0.891473i	$0.649673\pi$
$462$	0	0
$463$	−28.5563	−1.32713	−0.663563	−	0.748120i	$-0.730957\pi$
$463$	−28.5563	−1.32713	−0.663563	+	0.748120i	$0.730957\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−31.1127	−1.43972	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	−31.1127	−1.43972	−0.719862	+	0.694117i	$0.755795\pi$
$468$	0	0
$469$	−6.65685	−0.307385
$470$	0	0
$471$	66.4264	3.06077
$472$	0	0
$473$	5.31371	0.244325
$474$	0	0
$475$	0	0
$476$	0	0
$477$	15.0000	0.686803
$478$	0	0
$479$	−14.2721	−0.652108	−0.326054	−	0.945351i	$-0.605719\pi$
$479$	−14.2721	−0.652108	−0.326054	+	0.945351i	$0.605719\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	−21.6569	−0.985421
$484$	0	0
$485$	0	0
$486$	0	0
$487$	42.2132	1.91286	0.956431	−	0.291957i	$-0.0943064\pi$
$487$	42.2132	1.91286	0.956431	+	0.291957i	$0.0943064\pi$
$488$	0	0
$489$	−28.2843	−1.27906
$490$	0	0
$491$	−11.1716	−0.504166	−0.252083	−	0.967706i	$-0.581116\pi$
$491$	−11.1716	−0.504166	−0.252083	+	0.967706i	$0.581116\pi$
$492$	0	0
$493$	0.656854	0.0295832
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−22.4853	−1.00860
$498$	0	0
$499$	−2.55635	−0.114438	−0.0572190	−	0.998362i	$-0.518223\pi$
$499$	−2.55635	−0.114438	−0.0572190	+	0.998362i	$0.518223\pi$
$500$	0	0
$501$	15.0294	0.671466
$502$	0	0
$503$	14.6274	0.652204	0.326102	−	0.945335i	$-0.394265\pi$
$503$	14.6274	0.652204	0.326102	+	0.945335i	$0.394265\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.82843	0.125615
$508$	0	0
$509$	14.6274	0.648349	0.324174	−	0.945997i	$-0.394914\pi$
$509$	14.6274	0.648349	0.324174	+	0.945997i	$0.394914\pi$
$510$	0	0
$511$	14.4853	0.640791
$512$	0	0
$513$	20.6863	0.913322
$514$	0	0
$515$	0	0
$516$	0	0
$517$	23.0000	1.01154
$518$	0	0
$519$	52.7696	2.31633
$520$	0	0
$521$	0.343146	0.0150335	0.00751674	−	0.999972i	$-0.497607\pi$
$521$	0.343146	0.0150335	0.00751674	+	0.999972i	$0.497607\pi$
$522$	0	0
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.75736	0.207234
$528$	0	0
$529$	−12.9411	−0.562658
$530$	0	0
$531$	−66.2132	−2.87341
$532$	0	0
$533$	−5.65685	−0.245026
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.94113	−0.0837657
$538$	0	0
$539$	7.51472	0.323682
$540$	0	0
$541$	33.7990	1.45313	0.726566	−	0.687097i	$-0.241115\pi$
$541$	33.7990	1.45313	0.726566	+	0.687097i	$0.241115\pi$
$542$	0	0
$543$	−49.4558	−2.12235
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−28.4853	−1.21794	−0.608971	−	0.793192i	$-0.708418\pi$
$547$	−28.4853	−1.21794	−0.608971	+	0.793192i	$0.708418\pi$
$548$	0	0
$549$	5.00000	0.213395
$550$	0	0
$551$	−0.627417	−0.0267289
$552$	0	0
$553$	−14.4853	−0.615977
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.3137	1.15732	0.578659	−	0.815569i	$-0.303576\pi$
$557$	27.3137	1.15732	0.578659	+	0.815569i	$0.303576\pi$
$558$	0	0
$559$	−0.828427	−0.0350387
$560$	0	0
$561$	69.4558	2.93243
$562$	0	0
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.41421	−0.101387
$568$	0	0
$569$	−35.2843	−1.47919	−0.739597	−	0.673050i	$-0.764984\pi$
$569$	−35.2843	−1.47919	−0.739597	+	0.673050i	$0.764984\pi$
$570$	0	0
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	0	0
$573$	−71.5980	−2.99105
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.17157	−0.298556	−0.149278	−	0.988795i	$-0.547695\pi$
$577$	−7.17157	−0.298556	−0.149278	+	0.988795i	$0.547695\pi$
$578$	0	0
$579$	4.68629	0.194756
$580$	0	0
$581$	16.6569	0.691043
$582$	0	0
$583$	−19.2426	−0.796949
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.4142	−0.594938	−0.297469	−	0.954731i	$-0.596143\pi$
$587$	−14.4142	−0.594938	−0.297469	+	0.954731i	$0.596143\pi$
$588$	0	0
$589$	−4.54416	−0.187239
$590$	0	0
$591$	−33.9411	−1.39615
$592$	0	0
$593$	19.6569	0.807210	0.403605	−	0.914933i	$-0.367757\pi$
$593$	19.6569	0.807210	0.403605	+	0.914933i	$0.367757\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−28.2843	−1.15760
$598$	0	0
$599$	3.51472	0.143608	0.0718038	−	0.997419i	$-0.477124\pi$
$599$	3.51472	0.143608	0.0718038	+	0.997419i	$0.477124\pi$
$600$	0	0
$601$	13.8284	0.564073	0.282037	−	0.959404i	$-0.408990\pi$
$601$	13.8284	0.564073	0.282037	+	0.959404i	$0.408990\pi$
$602$	0	0
$603$	13.7868	0.561442
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21.5147	0.873255	0.436628	−	0.899642i	$-0.356173\pi$
$607$	21.5147	0.873255	0.436628	+	0.899642i	$0.356173\pi$
$608$	0	0
$609$	1.17157	0.0474745
$610$	0	0
$611$	−3.58579	−0.145065
$612$	0	0
$613$	−12.8284	−0.518135	−0.259068	−	0.965859i	$-0.583415\pi$
$613$	−12.8284	−0.518135	−0.259068	+	0.965859i	$0.583415\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−22.9706	−0.923265	−0.461632	−	0.887071i	$-0.652736\pi$
$619$	−22.9706	−0.923265	−0.461632	+	0.887071i	$0.652736\pi$
$620$	0	0
$621$	17.9411	0.719953
$622$	0	0
$623$	20.4853	0.820725
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−66.3431	−2.64949
$628$	0	0
$629$	22.9706	0.915896
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	−50.3431	−2.00096
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.17157	−0.0464194
$638$	0	0
$639$	46.5685	1.84222
$640$	0	0
$641$	28.1127	1.11038	0.555192	−	0.831722i	$-0.312645\pi$
$641$	28.1127	1.11038	0.555192	+	0.831722i	$0.312645\pi$
$642$	0	0
$643$	1.02944	0.0405970	0.0202985	−	0.999794i	$-0.493538\pi$
$643$	1.02944	0.0405970	0.0202985	+	0.999794i	$0.493538\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.8284	−0.740222	−0.370111	−	0.928988i	$-0.620680\pi$
$647$	−18.8284	−0.740222	−0.370111	+	0.928988i	$0.620680\pi$
$648$	0	0
$649$	84.9411	3.33423
$650$	0	0
$651$	8.48528	0.332564
$652$	0	0
$653$	−22.4558	−0.878765	−0.439383	−	0.898300i	$-0.644803\pi$
$653$	−22.4558	−0.878765	−0.439383	+	0.898300i	$0.644803\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−30.0000	−1.17041
$658$	0	0
$659$	8.62742	0.336076	0.168038	−	0.985780i	$-0.446257\pi$
$659$	8.62742	0.336076	0.168038	+	0.985780i	$0.446257\pi$
$660$	0	0
$661$	−6.82843	−0.265595	−0.132798	−	0.991143i	$-0.542396\pi$
$661$	−6.82843	−0.265595	−0.132798	+	0.991143i	$0.542396\pi$
$662$	0	0
$663$	−10.8284	−0.420541
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.544156	−0.0210698
$668$	0	0
$669$	31.0294	1.19967
$670$	0	0
$671$	−6.41421	−0.247618
$672$	0	0
$673$	24.4558	0.942704	0.471352	−	0.881945i	$-0.343766\pi$
$673$	24.4558	0.942704	0.471352	+	0.881945i	$0.343766\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−41.3137	−1.58781	−0.793907	−	0.608039i	$-0.791957\pi$
$677$	−41.3137	−1.58781	−0.793907	+	0.608039i	$0.791957\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	25.1716	0.964577
$682$	0	0
$683$	−21.5858	−0.825957	−0.412979	−	0.910741i	$-0.635512\pi$
$683$	−21.5858	−0.825957	−0.412979	+	0.910741i	$0.635512\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−13.6569	−0.521041
$688$	0	0
$689$	3.00000	0.114291
$690$	0	0
$691$	−8.89949	−0.338553	−0.169276	−	0.985569i	$-0.554143\pi$
$691$	−8.89949	−0.338553	−0.169276	+	0.985569i	$0.554143\pi$
$692$	0	0
$693$	77.4264	2.94119
$694$	0	0
$695$	0	0
$696$	0	0
$697$	21.6569	0.820312
$698$	0	0
$699$	58.3431	2.20674
$700$	0	0
$701$	−15.8284	−0.597831	−0.298916	−	0.954280i	$-0.596625\pi$
$701$	−15.8284	−0.597831	−0.298916	+	0.954280i	$0.596625\pi$
$702$	0	0
$703$	−21.9411	−0.827525
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.0711	0.604415
$708$	0	0
$709$	−49.2548	−1.84980	−0.924902	−	0.380205i	$-0.875853\pi$
$709$	−49.2548	−1.84980	−0.924902	+	0.380205i	$0.875853\pi$
$710$	0	0
$711$	30.0000	1.12509
$712$	0	0
$713$	−3.94113	−0.147596
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−66.7107	−2.49136
$718$	0	0
$719$	49.4558	1.84439	0.922196	−	0.386723i	$-0.126393\pi$
$719$	49.4558	1.84439	0.922196	+	0.386723i	$0.126393\pi$
$720$	0	0
$721$	32.9706	1.22789
$722$	0	0
$723$	14.0589	0.522855
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−16.6863	−0.618860	−0.309430	−	0.950922i	$-0.600138\pi$
$727$	−16.6863	−0.618860	−0.309430	+	0.950922i	$0.600138\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	3.17157	0.117305
$732$	0	0
$733$	−35.7990	−1.32227	−0.661133	−	0.750269i	$-0.729924\pi$
$733$	−35.7990	−1.32227	−0.661133	+	0.750269i	$0.729924\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.6863	−0.651483
$738$	0	0
$739$	27.7279	1.01999	0.509994	−	0.860178i	$-0.329648\pi$
$739$	27.7279	1.01999	0.509994	+	0.860178i	$0.329648\pi$
$740$	0	0
$741$	10.3431	0.379965
$742$	0	0
$743$	14.6985	0.539235	0.269618	−	0.962967i	$-0.413103\pi$
$743$	14.6985	0.539235	0.269618	+	0.962967i	$0.413103\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−34.4975	−1.26220
$748$	0	0
$749$	38.9706	1.42395
$750$	0	0
$751$	−31.4558	−1.14784	−0.573920	−	0.818911i	$-0.694578\pi$
$751$	−31.4558	−1.14784	−0.573920	+	0.818911i	$0.694578\pi$
$752$	0	0
$753$	16.0000	0.583072
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−16.3137	−0.592932	−0.296466	−	0.955043i	$-0.595808\pi$
$757$	−16.3137	−0.592932	−0.296466	+	0.955043i	$0.595808\pi$
$758$	0	0
$759$	−57.5391	−2.08854
$760$	0	0
$761$	−21.7990	−0.790213	−0.395106	−	0.918635i	$-0.629292\pi$
$761$	−21.7990	−0.790213	−0.395106	+	0.918635i	$0.629292\pi$
$762$	0	0
$763$	39.7990	1.44082
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.2426	−0.478164
$768$	0	0
$769$	4.97056	0.179243	0.0896215	−	0.995976i	$-0.471434\pi$
$769$	4.97056	0.179243	0.0896215	+	0.995976i	$0.471434\pi$
$770$	0	0
$771$	−18.8284	−0.678089
$772$	0	0
$773$	−5.79899	−0.208575	−0.104288	−	0.994547i	$-0.533256\pi$
$773$	−5.79899	−0.208575	−0.104288	+	0.994547i	$0.533256\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	40.9706	1.46981
$778$	0	0
$779$	−20.6863	−0.741163
$780$	0	0
$781$	−59.7401	−2.13767
$782$	0	0
$783$	−0.970563	−0.0346851
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−38.7574	−1.38155	−0.690775	−	0.723069i	$-0.742731\pi$
$787$	−38.7574	−1.38155	−0.690775	+	0.723069i	$0.742731\pi$
$788$	0	0
$789$	73.9411	2.63237
$790$	0	0
$791$	28.1421	1.00062
$792$	0	0
$793$	1.00000	0.0355110
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.4853	−0.406830	−0.203415	−	0.979093i	$-0.565204\pi$
$797$	−11.4853	−0.406830	−0.203415	+	0.979093i	$0.565204\pi$
$798$	0	0
$799$	13.7279	0.485659
$800$	0	0
$801$	−42.4264	−1.49906
$802$	0	0
$803$	38.4853	1.35812
$804$	0	0
$805$	0	0
$806$	0	0
$807$	42.8284	1.50763
$808$	0	0
$809$	17.3137	0.608718	0.304359	−	0.952557i	$-0.401558\pi$
$809$	17.3137	0.608718	0.304359	+	0.952557i	$0.401558\pi$
$810$	0	0
$811$	−9.58579	−0.336602	−0.168301	−	0.985736i	$-0.553828\pi$
$811$	−9.58579	−0.336602	−0.168301	+	0.985736i	$0.553828\pi$
$812$	0	0
$813$	72.7696	2.55214
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.02944	−0.105987
$818$	0	0
$819$	−12.0711	−0.421797
$820$	0	0
$821$	−20.8284	−0.726917	−0.363459	−	0.931610i	$-0.618404\pi$
$821$	−20.8284	−0.726917	−0.363459	+	0.931610i	$0.618404\pi$
$822$	0	0
$823$	20.2843	0.707065	0.353533	−	0.935422i	$-0.384980\pi$
$823$	20.2843	0.707065	0.353533	+	0.935422i	$0.384980\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.3553	1.12511	0.562553	−	0.826761i	$-0.309819\pi$
$827$	32.3553	1.12511	0.562553	+	0.826761i	$0.309819\pi$
$828$	0	0
$829$	7.97056	0.276829	0.138415	−	0.990374i	$-0.455799\pi$
$829$	7.97056	0.276829	0.138415	+	0.990374i	$0.455799\pi$
$830$	0	0
$831$	26.3431	0.913834
$832$	0	0
$833$	4.48528	0.155406
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.02944	−0.242973
$838$	0	0
$839$	1.02944	0.0355401	0.0177701	−	0.999842i	$-0.494343\pi$
$839$	1.02944	0.0355401	0.0177701	+	0.999842i	$0.494343\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	0	0
$843$	13.6569	0.470367
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−72.7696	−2.50039
$848$	0	0
$849$	63.5980	2.18268
$850$	0	0
$851$	−19.0294	−0.652321
$852$	0	0
$853$	−42.4264	−1.45265	−0.726326	−	0.687350i	$-0.758774\pi$
$853$	−42.4264	−1.45265	−0.726326	+	0.687350i	$0.758774\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.62742	−0.158070	−0.0790348	−	0.996872i	$-0.525184\pi$
$857$	−4.62742	−0.158070	−0.0790348	+	0.996872i	$0.525184\pi$
$858$	0	0
$859$	−24.1421	−0.823719	−0.411860	−	0.911247i	$-0.635121\pi$
$859$	−24.1421	−0.823719	−0.411860	+	0.911247i	$0.635121\pi$
$860$	0	0
$861$	38.6274	1.31642
$862$	0	0
$863$	−29.1838	−0.993427	−0.496713	−	0.867915i	$-0.665460\pi$
$863$	−29.1838	−0.993427	−0.496713	+	0.867915i	$0.665460\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−6.62742	−0.225079
$868$	0	0
$869$	−38.4853	−1.30552
$870$	0	0
$871$	2.75736	0.0934296
$872$	0	0
$873$	−4.14214	−0.140190
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−15.7990	−0.533494	−0.266747	−	0.963767i	$-0.585949\pi$
$877$	−15.7990	−0.533494	−0.266747	+	0.963767i	$0.585949\pi$
$878$	0	0
$879$	24.9706	0.842236
$880$	0	0
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	0	0
$883$	−32.9706	−1.10955	−0.554774	−	0.832001i	$-0.687195\pi$
$883$	−32.9706	−1.10955	−0.554774	+	0.832001i	$0.687195\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.62742	−0.289680	−0.144840	−	0.989455i	$-0.546267\pi$
$887$	−8.62742	−0.289680	−0.144840	+	0.989455i	$0.546267\pi$
$888$	0	0
$889$	−8.82843	−0.296096
$890$	0	0
$891$	−6.41421	−0.214884
$892$	0	0
$893$	−13.1127	−0.438800
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.97056	0.299518
$898$	0	0
$899$	0.213203	0.00711073
$900$	0	0
$901$	−11.4853	−0.382630
$902$	0	0
$903$	5.65685	0.188248
$904$	0	0
$905$	0	0
$906$	0	0
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	0	0
$909$	−33.2843	−1.10397
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	44.2548	1.46462
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20.4853	0.676484
$918$	0	0
$919$	18.9706	0.625781	0.312891	−	0.949789i	$-0.398703\pi$
$919$	18.9706	0.625781	0.312891	+	0.949789i	$0.398703\pi$
$920$	0	0
$921$	26.3431	0.868036
$922$	0	0
$923$	9.31371	0.306564
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−68.2843	−2.24275
$928$	0	0
$929$	30.2843	0.993595	0.496797	−	0.867867i	$-0.334509\pi$
$929$	30.2843	0.993595	0.496797	+	0.867867i	$0.334509\pi$
$930$	0	0
$931$	−4.28427	−0.140411
$932$	0	0
$933$	−26.9117	−0.881049
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.97056	0.0643755	0.0321877	−	0.999482i	$-0.489753\pi$
$937$	1.97056	0.0643755	0.0321877	+	0.999482i	$0.489753\pi$
$938$	0	0
$939$	8.08326	0.263787
$940$	0	0
$941$	−35.6569	−1.16238	−0.581190	−	0.813768i	$-0.697413\pi$
$941$	−35.6569	−1.16238	−0.581190	+	0.813768i	$0.697413\pi$
$942$	0	0
$943$	−17.9411	−0.584243
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.4142	0.858347	0.429173	−	0.903222i	$-0.358805\pi$
$947$	26.4142	0.858347	0.429173	+	0.903222i	$0.358805\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	56.9706	1.84740
$952$	0	0
$953$	−47.3431	−1.53359	−0.766797	−	0.641889i	$-0.778151\pi$
$953$	−47.3431	−1.53359	−0.766797	+	0.641889i	$0.778151\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.11270	0.100619
$958$	0	0
$959$	−6.82843	−0.220501
$960$	0	0
$961$	−29.4558	−0.950189
$962$	0	0
$963$	−80.7107	−2.60087
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−43.7279	−1.40620	−0.703098	−	0.711093i	$-0.748200\pi$
$967$	−43.7279	−1.40620	−0.703098	+	0.711093i	$0.748200\pi$
$968$	0	0
$969$	−39.5980	−1.27207
$970$	0	0
$971$	−36.8284	−1.18188	−0.590940	−	0.806715i	$-0.701243\pi$
$971$	−36.8284	−1.18188	−0.590940	+	0.806715i	$0.701243\pi$
$972$	0	0
$973$	−44.6274	−1.43069
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.48528	−0.271468	−0.135734	−	0.990745i	$-0.543339\pi$
$977$	−8.48528	−0.271468	−0.135734	+	0.990745i	$0.543339\pi$
$978$	0	0
$979$	54.4264	1.73948
$980$	0	0
$981$	−82.4264	−2.63167
$982$	0	0
$983$	−26.6985	−0.851549	−0.425775	−	0.904829i	$-0.639998\pi$
$983$	−26.6985	−0.851549	−0.425775	+	0.904829i	$0.639998\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.4853	0.779375
$988$	0	0
$989$	−2.62742	−0.0835470
$990$	0	0
$991$	13.5147	0.429309	0.214655	−	0.976690i	$-0.431137\pi$
$991$	13.5147	0.429309	0.214655	+	0.976690i	$0.431137\pi$
$992$	0	0
$993$	−78.2254	−2.48241
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.6569	0.400847	0.200423	−	0.979709i	$-0.435768\pi$
$997$	12.6569	0.400847	0.200423	+	0.979709i	$0.435768\pi$
$998$	0	0
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.br.1.2		2
4.3	odd	2		325.2.a.h.1.2	yes	2
5.4	even	2		5200.2.a.bt.1.1		2
12.11	even	2		2925.2.a.w.1.1		2
20.3	even	4		325.2.b.d.274.1		4
20.7	even	4		325.2.b.d.274.4		4
20.19	odd	2		325.2.a.f.1.1	✓	2
52.51	odd	2		4225.2.a.s.1.1		2
60.23	odd	4		2925.2.c.q.2224.4		4
60.47	odd	4		2925.2.c.q.2224.1		4
60.59	even	2		2925.2.a.bd.1.2		2
260.259	odd	2		4225.2.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.a.f.1.1	✓	2	20.19	odd	2
325.2.a.h.1.2	yes	2	4.3	odd	2
325.2.b.d.274.1		4	20.3	even	4
325.2.b.d.274.4		4	20.7	even	4
2925.2.a.w.1.1		2	12.11	even	2
2925.2.a.bd.1.2		2	60.59	even	2
2925.2.c.q.2224.1		4	60.47	odd	4
2925.2.c.q.2224.4		4	60.23	odd	4
4225.2.a.s.1.1		2	52.51	odd	2
4225.2.a.z.1.2		2	260.259	odd	2
5200.2.a.br.1.2		2	1.1	even	1	trivial
5200.2.a.bt.1.1		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82843 q^{3} -2.41421 q^{7} +5.00000 q^{9} -6.41421 q^{11} +1.00000 q^{13} -3.82843 q^{17} +3.65685 q^{19} -6.82843 q^{21} +3.17157 q^{23} +5.65685 q^{27} -0.171573 q^{29} -1.24264 q^{31} -18.1421 q^{33} -6.00000 q^{37} +2.82843 q^{39} -5.65685 q^{41} -0.828427 q^{43} -3.58579 q^{47} -1.17157 q^{49} -10.8284 q^{51} +3.00000 q^{53} +10.3431 q^{57} -13.2426 q^{59} +1.00000 q^{61} -12.0711 q^{63} +2.75736 q^{67} +8.97056 q^{69} +9.31371 q^{71} -6.00000 q^{73} +15.4853 q^{77} +6.00000 q^{79} +1.00000 q^{81} -6.89949 q^{83} -0.485281 q^{87} -8.48528 q^{89} -2.41421 q^{91} -3.51472 q^{93} -0.828427 q^{97} -32.0711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} + 10 q^{9} - 10 q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} - 8 q^{21} + 12 q^{23} - 6 q^{29} + 6 q^{31} - 8 q^{33} - 12 q^{37} + 4 q^{43} - 10 q^{47} - 8 q^{49} - 16 q^{51} + 6 q^{53} + 32 q^{57}+ \cdots - 50 q^{99}+O(q^{100}) \) 2 * q - 2 * q^7 + 10 * q^9 - 10 * q^11 + 2 * q^13 - 2 * q^17 - 4 * q^19 - 8 * q^21 + 12 * q^23 - 6 * q^29 + 6 * q^31 - 8 * q^33 - 12 * q^37 + 4 * q^43 - 10 * q^47 - 8 * q^49 - 16 * q^51 + 6 * q^53 + 32 * q^57 - 18 * q^59 + 2 * q^61 - 10 * q^63 + 14 * q^67 - 16 * q^69 - 4 * q^71 - 12 * q^73 + 14 * q^77 + 12 * q^79 + 2 * q^81 + 6 * q^83 + 16 * q^87 - 2 * q^91 - 24 * q^93 + 4 * q^97 - 50 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more