LMFDB - Embedded newform 6592.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6592,2,Mod(1,6592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6592, 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("6592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6592 = 2^{6} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6592.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:	$52.6373850124$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	6592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79129 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.79129 q^{7} -3.00000 q^{9} -5.58258 q^{11} +2.79129 q^{13} -2.20871 q^{17} -4.79129 q^{19} -4.79129 q^{23} -5.00000 q^{25} -5.37386 q^{29} -3.58258 q^{31} +9.58258 q^{37} +2.79129 q^{41} +2.00000 q^{43} +2.00000 q^{47} -3.79129 q^{49} +9.58258 q^{53} +9.79129 q^{59} +9.79129 q^{61} +5.37386 q^{63} -2.00000 q^{67} -7.58258 q^{71} -7.16515 q^{73} +10.0000 q^{77} +8.79129 q^{79} +9.00000 q^{81} -1.37386 q^{83} +9.16515 q^{89} -5.00000 q^{91} -13.3739 q^{97} +16.7477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q + q^7 - 6 * q^9	$ 2 q + q^{7} - 6 q^{9} - 2 q^{11} + q^{13} - 9 q^{17} - 5 q^{19} - 5 q^{23} - 10 q^{25} + 3 q^{29} + 2 q^{31} + 10 q^{37} + q^{41} + 4 q^{43} + 4 q^{47} - 3 q^{49} + 10 q^{53} + 15 q^{59} + 15 q^{61} - 3 q^{63} - 4 q^{67} - 6 q^{71} + 4 q^{73} + 20 q^{77} + 13 q^{79} + 18 q^{81} + 11 q^{83} - 10 q^{91} - 13 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + q^7 - 6 * q^9 - 2 * q^11 + q^13 - 9 * q^17 - 5 * q^19 - 5 * q^23 - 10 * q^25 + 3 * q^29 + 2 * q^31 + 10 * q^37 + q^41 + 4 * q^43 + 4 * q^47 - 3 * q^49 + 10 * q^53 + 15 * q^59 + 15 * q^61 - 3 * q^63 - 4 * q^67 - 6 * q^71 + 4 * q^73 + 20 * q^77 + 13 * q^79 + 18 * q^81 + 11 * q^83 - 10 * q^91 - 13 * q^97 + 6 * 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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−1.79129	−0.677043	−0.338522	−	0.940959i	$-0.609927\pi$
$7$	−1.79129	−0.677043	−0.338522	+	0.940959i	$0.609927\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−5.58258	−1.68321	−0.841605	−	0.540094i	$-0.818389\pi$
$11$	−5.58258	−1.68321	−0.841605	+	0.540094i	$0.818389\pi$
$12$	0	0
$13$	2.79129	0.774164	0.387082	−	0.922045i	$-0.373483\pi$
$13$	2.79129	0.774164	0.387082	+	0.922045i	$0.373483\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.20871	−0.535691	−0.267846	−	0.963462i	$-0.586312\pi$
$17$	−2.20871	−0.535691	−0.267846	+	0.963462i	$0.586312\pi$
$18$	0	0
$19$	−4.79129	−1.09920	−0.549598	−	0.835429i	$-0.685219\pi$
$19$	−4.79129	−1.09920	−0.549598	+	0.835429i	$0.685219\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.79129	−0.999053	−0.499526	−	0.866299i	$-0.666493\pi$
$23$	−4.79129	−0.999053	−0.499526	+	0.866299i	$0.666493\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.37386	−0.997901	−0.498951	−	0.866630i	$-0.666281\pi$
$29$	−5.37386	−0.997901	−0.498951	+	0.866630i	$0.666281\pi$
$30$	0	0
$31$	−3.58258	−0.643450	−0.321725	−	0.946833i	$-0.604263\pi$
$31$	−3.58258	−0.643450	−0.321725	+	0.946833i	$0.604263\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.58258	1.57537	0.787683	−	0.616081i	$-0.211281\pi$
$37$	9.58258	1.57537	0.787683	+	0.616081i	$0.211281\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.79129	0.435926	0.217963	−	0.975957i	$-0.430059\pi$
$41$	2.79129	0.435926	0.217963	+	0.975957i	$0.430059\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−3.79129	−0.541613
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.58258	1.31627	0.658134	−	0.752901i	$-0.271346\pi$
$53$	9.58258	1.31627	0.658134	+	0.752901i	$0.271346\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.79129	1.27472	0.637359	−	0.770567i	$-0.280027\pi$
$59$	9.79129	1.27472	0.637359	+	0.770567i	$0.280027\pi$
$60$	0	0
$61$	9.79129	1.25365	0.626823	−	0.779162i	$-0.284355\pi$
$61$	9.79129	1.25365	0.626823	+	0.779162i	$0.284355\pi$
$62$	0	0
$63$	5.37386	0.677043
$64$	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
$69$	0	0
$70$	0	0
$71$	−7.58258	−0.899886	−0.449943	−	0.893057i	$-0.648556\pi$
$71$	−7.58258	−0.899886	−0.449943	+	0.893057i	$0.648556\pi$
$72$	0	0
$73$	−7.16515	−0.838618	−0.419309	−	0.907844i	$-0.637728\pi$
$73$	−7.16515	−0.838618	−0.419309	+	0.907844i	$0.637728\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.0000	1.13961
$78$	0	0
$79$	8.79129	0.989097	0.494549	−	0.869150i	$-0.335333\pi$
$79$	8.79129	0.989097	0.494549	+	0.869150i	$0.335333\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−1.37386	−0.150801	−0.0754006	−	0.997153i	$-0.524024\pi$
$83$	−1.37386	−0.150801	−0.0754006	+	0.997153i	$0.524024\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.16515	0.971504	0.485752	−	0.874097i	$-0.338546\pi$
$89$	9.16515	0.971504	0.485752	+	0.874097i	$0.338546\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.3739	−1.35791	−0.678955	−	0.734180i	$-0.737567\pi$
$97$	−13.3739	−1.35791	−0.678955	+	0.734180i	$0.737567\pi$
$98$	0	0
$99$	16.7477	1.68321
$100$	0	0
$101$	4.41742	0.439550	0.219775	−	0.975551i	$-0.429468\pi$
$101$	4.41742	0.439550	0.219775	+	0.975551i	$0.429468\pi$
$102$	0	0
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.1652	1.46607	0.733035	−	0.680191i	$-0.238103\pi$
$107$	15.1652	1.46607	0.733035	+	0.680191i	$0.238103\pi$
$108$	0	0
$109$	−14.7477	−1.41258	−0.706288	−	0.707925i	$-0.749632\pi$
$109$	−14.7477	−1.41258	−0.706288	+	0.707925i	$0.749632\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.37386	−0.774164
$118$	0	0
$119$	3.95644	0.362686
$120$	0	0
$121$	20.1652	1.83320
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−13.1652	−1.16822	−0.584109	−	0.811675i	$-0.698556\pi$
$127$	−13.1652	−1.16822	−0.584109	+	0.811675i	$0.698556\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−21.5390	−1.88187	−0.940936	−	0.338584i	$-0.890052\pi$
$131$	−21.5390	−1.88187	−0.940936	+	0.338584i	$0.890052\pi$
$132$	0	0
$133$	8.58258	0.744204
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.95644	−0.167150	−0.0835749	−	0.996501i	$-0.526634\pi$
$137$	−1.95644	−0.167150	−0.0835749	+	0.996501i	$0.526634\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.5826	−1.30308
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.9564	1.14336	0.571678	−	0.820478i	$-0.306293\pi$
$149$	13.9564	1.14336	0.571678	+	0.820478i	$0.306293\pi$
$150$	0	0
$151$	8.74773	0.711880	0.355940	−	0.934509i	$-0.384161\pi$
$151$	8.74773	0.711880	0.355940	+	0.934509i	$0.384161\pi$
$152$	0	0
$153$	6.62614	0.535691
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.7477	−0.857762	−0.428881	−	0.903361i	$-0.641092\pi$
$157$	−10.7477	−0.857762	−0.428881	+	0.903361i	$0.641092\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.58258	0.676402
$162$	0	0
$163$	−9.37386	−0.734218	−0.367109	−	0.930178i	$-0.619652\pi$
$163$	−9.37386	−0.734218	−0.367109	+	0.930178i	$0.619652\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.16515	0.244927	0.122463	−	0.992473i	$-0.460921\pi$
$167$	3.16515	0.244927	0.122463	+	0.992473i	$0.460921\pi$
$168$	0	0
$169$	−5.20871	−0.400670
$170$	0	0
$171$	14.3739	1.09920
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	8.95644	0.677043
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.95644	0.370462	0.185231	−	0.982695i	$-0.440697\pi$
$179$	4.95644	0.370462	0.185231	+	0.982695i	$0.440697\pi$
$180$	0	0
$181$	9.16515	0.681240	0.340620	−	0.940201i	$-0.389363\pi$
$181$	9.16515	0.681240	0.340620	+	0.940201i	$0.389363\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.3303	0.901681
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.5826	0.982801	0.491400	−	0.870934i	$-0.336485\pi$
$191$	13.5826	0.982801	0.491400	+	0.870934i	$0.336485\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.3303	1.44847	0.724237	−	0.689551i	$-0.242192\pi$
$197$	20.3303	1.44847	0.724237	+	0.689551i	$0.242192\pi$
$198$	0	0
$199$	−17.1652	−1.21681	−0.608403	−	0.793629i	$-0.708189\pi$
$199$	−17.1652	−1.21681	−0.608403	+	0.793629i	$0.708189\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.62614	0.675622
$204$	0	0
$205$	0	0
$206$	0	0
$207$	14.3739	0.999053
$208$	0	0
$209$	26.7477	1.85018
$210$	0	0
$211$	−16.4174	−1.13022	−0.565111	−	0.825015i	$-0.691167\pi$
$211$	−16.4174	−1.13022	−0.565111	+	0.825015i	$0.691167\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.41742	0.435643
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.16515	−0.414713
$222$	0	0
$223$	−19.1216	−1.28048	−0.640238	−	0.768176i	$-0.721164\pi$
$223$	−19.1216	−1.28048	−0.640238	+	0.768176i	$0.721164\pi$
$224$	0	0
$225$	15.0000	1.00000
$226$	0	0
$227$	12.4174	0.824173	0.412087	−	0.911145i	$-0.364800\pi$
$227$	12.4174	0.824173	0.412087	+	0.911145i	$0.364800\pi$
$228$	0	0
$229$	−5.79129	−0.382699	−0.191350	−	0.981522i	$-0.561286\pi$
$229$	−5.79129	−0.382699	−0.191350	+	0.981522i	$0.561286\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.5826	1.02085	0.510424	−	0.859923i	$-0.329488\pi$
$233$	15.5826	1.02085	0.510424	+	0.859923i	$0.329488\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.7913	−0.892084	−0.446042	−	0.895012i	$-0.647167\pi$
$239$	−13.7913	−0.892084	−0.446042	+	0.895012i	$0.647167\pi$
$240$	0	0
$241$	21.9129	1.41153	0.705766	−	0.708445i	$-0.250603\pi$
$241$	21.9129	1.41153	0.705766	+	0.708445i	$0.250603\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−13.3739	−0.850959
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	26.7477	1.68162
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.3303	−0.644387	−0.322193	−	0.946674i	$-0.604420\pi$
$257$	−10.3303	−0.644387	−0.322193	+	0.946674i	$0.604420\pi$
$258$	0	0
$259$	−17.1652	−1.06659
$260$	0	0
$261$	16.1216	0.997901
$262$	0	0
$263$	−8.33030	−0.513668	−0.256834	−	0.966455i	$-0.582679\pi$
$263$	−8.33030	−0.513668	−0.256834	+	0.966455i	$0.582679\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	29.1216	1.77557	0.887787	−	0.460254i	$-0.152242\pi$
$269$	29.1216	1.77557	0.887787	+	0.460254i	$0.152242\pi$
$270$	0	0
$271$	27.5826	1.67552	0.837761	−	0.546037i	$-0.183864\pi$
$271$	27.5826	1.67552	0.837761	+	0.546037i	$0.183864\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	27.9129	1.68321
$276$	0	0
$277$	−14.3303	−0.861024	−0.430512	−	0.902585i	$-0.641667\pi$
$277$	−14.3303	−0.861024	−0.430512	+	0.902585i	$0.641667\pi$
$278$	0	0
$279$	10.7477	0.643450
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	−21.5826	−1.28295	−0.641475	−	0.767144i	$-0.721677\pi$
$283$	−21.5826	−1.28295	−0.641475	+	0.767144i	$0.721677\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.00000	−0.295141
$288$	0	0
$289$	−12.1216	−0.713035
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.7477	−1.44578	−0.722889	−	0.690964i	$-0.757186\pi$
$293$	−24.7477	−1.44578	−0.722889	+	0.690964i	$0.757186\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−13.3739	−0.773430
$300$	0	0
$301$	−3.58258	−0.206496
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−33.1652	−1.89284	−0.946418	−	0.322945i	$-0.895327\pi$
$307$	−33.1652	−1.89284	−0.946418	+	0.322945i	$0.895327\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.3739	0.758362	0.379181	−	0.925322i	$-0.376206\pi$
$311$	13.3739	0.758362	0.379181	+	0.925322i	$0.376206\pi$
$312$	0	0
$313$	24.3739	1.37769	0.688846	−	0.724908i	$-0.258118\pi$
$313$	24.3739	1.37769	0.688846	+	0.724908i	$0.258118\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.53901	−0.479599	−0.239799	−	0.970822i	$-0.577082\pi$
$317$	−8.53901	−0.479599	−0.239799	+	0.970822i	$0.577082\pi$
$318$	0	0
$319$	30.0000	1.67968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.5826	0.588830
$324$	0	0
$325$	−13.9564	−0.774164
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.58258	−0.197514
$330$	0	0
$331$	4.41742	0.242804	0.121402	−	0.992603i	$-0.461261\pi$
$331$	4.41742	0.242804	0.121402	+	0.992603i	$0.461261\pi$
$332$	0	0
$333$	−28.7477	−1.57537
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.5390	0.846464	0.423232	−	0.906021i	$-0.360896\pi$
$337$	15.5390	0.846464	0.423232	+	0.906021i	$0.360896\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.0000	1.08306
$342$	0	0
$343$	19.3303	1.04374
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−36.7042	−1.97038	−0.985191	−	0.171458i	$-0.945152\pi$
$347$	−36.7042	−1.97038	−0.985191	+	0.171458i	$0.945152\pi$
$348$	0	0
$349$	22.3303	1.19531	0.597657	−	0.801752i	$-0.296099\pi$
$349$	22.3303	1.19531	0.597657	+	0.801752i	$0.296099\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.58258	0.403580	0.201790	−	0.979429i	$-0.435324\pi$
$353$	7.58258	0.403580	0.201790	+	0.979429i	$0.435324\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.3303	−1.60077	−0.800386	−	0.599485i	$-0.795372\pi$
$359$	−30.3303	−1.60077	−0.800386	+	0.599485i	$0.795372\pi$
$360$	0	0
$361$	3.95644	0.208234
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	28.7913	1.50289	0.751446	−	0.659794i	$-0.229356\pi$
$367$	28.7913	1.50289	0.751446	+	0.659794i	$0.229356\pi$
$368$	0	0
$369$	−8.37386	−0.435926
$370$	0	0
$371$	−17.1652	−0.891170
$372$	0	0
$373$	−24.9564	−1.29220	−0.646098	−	0.763255i	$-0.723600\pi$
$373$	−24.9564	−1.29220	−0.646098	+	0.763255i	$0.723600\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15.0000	−0.772539
$378$	0	0
$379$	−25.4955	−1.30961	−0.654807	−	0.755796i	$-0.727250\pi$
$379$	−25.4955	−1.30961	−0.654807	+	0.755796i	$0.727250\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.00000	−0.304997
$388$	0	0
$389$	16.4174	0.832396	0.416198	−	0.909274i	$-0.363362\pi$
$389$	16.4174	0.832396	0.416198	+	0.909274i	$0.363362\pi$
$390$	0	0
$391$	10.5826	0.535184
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.7477	1.24205	0.621026	−	0.783790i	$-0.286716\pi$
$397$	24.7477	1.24205	0.621026	+	0.783790i	$0.286716\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−36.1216	−1.80383	−0.901913	−	0.431918i	$-0.857837\pi$
$401$	−36.1216	−1.80383	−0.901913	+	0.431918i	$0.857837\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−53.4955	−2.65167
$408$	0	0
$409$	−17.7913	−0.879723	−0.439861	−	0.898066i	$-0.644972\pi$
$409$	−17.7913	−0.879723	−0.439861	+	0.898066i	$0.644972\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−17.5390	−0.863039
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	39.8693	1.94774	0.973872	−	0.227098i	$-0.0729239\pi$
$419$	39.8693	1.94774	0.973872	+	0.227098i	$0.0729239\pi$
$420$	0	0
$421$	2.83485	0.138162	0.0690810	−	0.997611i	$-0.477993\pi$
$421$	2.83485	0.138162	0.0690810	+	0.997611i	$0.477993\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	11.0436	0.535691
$426$	0	0
$427$	−17.5390	−0.848772
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.0436	0.531950	0.265975	−	0.963980i	$-0.414306\pi$
$431$	11.0436	0.531950	0.265975	+	0.963980i	$0.414306\pi$
$432$	0	0
$433$	13.5826	0.652737	0.326368	−	0.945243i	$-0.394175\pi$
$433$	13.5826	0.652737	0.326368	+	0.945243i	$0.394175\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22.9564	1.09816
$438$	0	0
$439$	2.83485	0.135300	0.0676500	−	0.997709i	$-0.478450\pi$
$439$	2.83485	0.135300	0.0676500	+	0.997709i	$0.478450\pi$
$440$	0	0
$441$	11.3739	0.541613
$442$	0	0
$443$	21.9129	1.04111	0.520556	−	0.853827i	$-0.325725\pi$
$443$	21.9129	1.04111	0.520556	+	0.853827i	$0.325725\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.5826	−0.829773	−0.414887	−	0.909873i	$-0.636179\pi$
$449$	−17.5826	−0.829773	−0.414887	+	0.909873i	$0.636179\pi$
$450$	0	0
$451$	−15.5826	−0.733755
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.4955	−1.47330	−0.736648	−	0.676277i	$-0.763592\pi$
$457$	−31.4955	−1.47330	−0.736648	+	0.676277i	$0.763592\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.83485	−0.132032	−0.0660160	−	0.997819i	$-0.521029\pi$
$461$	−2.83485	−0.132032	−0.0660160	+	0.997819i	$0.521029\pi$
$462$	0	0
$463$	−0.747727	−0.0347498	−0.0173749	−	0.999849i	$-0.505531\pi$
$463$	−0.747727	−0.0347498	−0.0173749	+	0.999849i	$0.505531\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.46099	−0.113881	−0.0569404	−	0.998378i	$-0.518135\pi$
$467$	−2.46099	−0.113881	−0.0569404	+	0.998378i	$0.518135\pi$
$468$	0	0
$469$	3.58258	0.165428
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.1652	−0.513374
$474$	0	0
$475$	23.9564	1.09920
$476$	0	0
$477$	−28.7477	−1.31627
$478$	0	0
$479$	0.330303	0.0150919	0.00754596	−	0.999972i	$-0.497598\pi$
$479$	0.330303	0.0150919	0.00754596	+	0.999972i	$0.497598\pi$
$480$	0	0
$481$	26.7477	1.21959
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.16515	0.143427	0.0717134	−	0.997425i	$-0.477153\pi$
$487$	3.16515	0.143427	0.0717134	+	0.997425i	$0.477153\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.1216	−0.727557	−0.363779	−	0.931485i	$-0.618514\pi$
$491$	−16.1216	−0.727557	−0.363779	+	0.931485i	$0.618514\pi$
$492$	0	0
$493$	11.8693	0.534567
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.5826	0.609262
$498$	0	0
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	12.8348	0.572278	0.286139	−	0.958188i	$-0.407628\pi$
$503$	12.8348	0.572278	0.286139	+	0.958188i	$0.407628\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.2087	0.762763	0.381381	−	0.924418i	$-0.375448\pi$
$509$	17.2087	0.762763	0.381381	+	0.924418i	$0.375448\pi$
$510$	0	0
$511$	12.8348	0.567780
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−11.1652	−0.491043
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.3303	1.06593	0.532965	−	0.846137i	$-0.321078\pi$
$521$	24.3303	1.06593	0.532965	+	0.846137i	$0.321078\pi$
$522$	0	0
$523$	8.53901	0.373385	0.186693	−	0.982418i	$-0.440223\pi$
$523$	8.53901	0.373385	0.186693	+	0.982418i	$0.440223\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.91288	0.344690
$528$	0	0
$529$	−0.0435608	−0.00189395
$530$	0	0
$531$	−29.3739	−1.27472
$532$	0	0
$533$	7.79129	0.337478
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	21.1652	0.911648
$540$	0	0
$541$	31.5390	1.35597	0.677984	−	0.735077i	$-0.262854\pi$
$541$	31.5390	1.35597	0.677984	+	0.735077i	$0.262854\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.62614	0.112285	0.0561427	−	0.998423i	$-0.482120\pi$
$547$	2.62614	0.112285	0.0561427	+	0.998423i	$0.482120\pi$
$548$	0	0
$549$	−29.3739	−1.25365
$550$	0	0
$551$	25.7477	1.09689
$552$	0	0
$553$	−15.7477	−0.669661
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.3303	1.79359	0.896796	−	0.442444i	$-0.145888\pi$
$557$	42.3303	1.79359	0.896796	+	0.442444i	$0.145888\pi$
$558$	0	0
$559$	5.58258	0.236118
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.4955	−0.821635	−0.410818	−	0.911718i	$-0.634757\pi$
$563$	−19.4955	−0.821635	−0.410818	+	0.911718i	$0.634757\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−16.1216	−0.677043
$568$	0	0
$569$	33.4955	1.40420	0.702101	−	0.712077i	$-0.252245\pi$
$569$	33.4955	1.40420	0.702101	+	0.712077i	$0.252245\pi$
$570$	0	0
$571$	24.1216	1.00946	0.504729	−	0.863278i	$-0.331593\pi$
$571$	24.1216	1.00946	0.504729	+	0.863278i	$0.331593\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	23.9564	0.999053
$576$	0	0
$577$	41.0780	1.71010	0.855050	−	0.518545i	$-0.173526\pi$
$577$	41.0780	1.71010	0.855050	+	0.518545i	$0.173526\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.46099	0.102099
$582$	0	0
$583$	−53.4955	−2.21556
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.83485	0.364653	0.182327	−	0.983238i	$-0.441637\pi$
$587$	8.83485	0.364653	0.182327	+	0.983238i	$0.441637\pi$
$588$	0	0
$589$	17.1652	0.707278
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.4174	0.838443	0.419222	−	0.907884i	$-0.362303\pi$
$593$	20.4174	0.838443	0.419222	+	0.907884i	$0.362303\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.41742	0.0987733	0.0493866	−	0.998780i	$-0.484273\pi$
$599$	2.41742	0.0987733	0.0493866	+	0.998780i	$0.484273\pi$
$600$	0	0
$601$	−13.1652	−0.537018	−0.268509	−	0.963277i	$-0.586531\pi$
$601$	−13.1652	−0.537018	−0.268509	+	0.963277i	$0.586531\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.2867	1.10753	0.553767	−	0.832671i	$-0.313190\pi$
$607$	27.2867	1.10753	0.553767	+	0.832671i	$0.313190\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.58258	0.225847
$612$	0	0
$613$	−1.37386	−0.0554898	−0.0277449	−	0.999615i	$-0.508833\pi$
$613$	−1.37386	−0.0554898	−0.0277449	+	0.999615i	$0.508833\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.5826	−0.788365	−0.394182	−	0.919032i	$-0.628972\pi$
$617$	−19.5826	−0.788365	−0.394182	+	0.919032i	$0.628972\pi$
$618$	0	0
$619$	−5.37386	−0.215994	−0.107997	−	0.994151i	$-0.534444\pi$
$619$	−5.37386	−0.215994	−0.107997	+	0.994151i	$0.534444\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.4174	−0.657750
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−21.1652	−0.843910
$630$	0	0
$631$	9.53901	0.379742	0.189871	−	0.981809i	$-0.439193\pi$
$631$	9.53901	0.379742	0.189871	+	0.981809i	$0.439193\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−10.5826	−0.419297
$638$	0	0
$639$	22.7477	0.899886
$640$	0	0
$641$	−20.2867	−0.801278	−0.400639	−	0.916236i	$-0.631212\pi$
$641$	−20.2867	−0.801278	−0.400639	+	0.916236i	$0.631212\pi$
$642$	0	0
$643$	24.7913	0.977673	0.488836	−	0.872375i	$-0.337421\pi$
$643$	24.7913	0.977673	0.488836	+	0.872375i	$0.337421\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.7042	1.12848	0.564239	−	0.825612i	$-0.309170\pi$
$647$	28.7042	1.12848	0.564239	+	0.825612i	$0.309170\pi$
$648$	0	0
$649$	−54.6606	−2.14562
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.66970	0.221872	0.110936	−	0.993828i	$-0.464615\pi$
$653$	5.66970	0.221872	0.110936	+	0.993828i	$0.464615\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	21.4955	0.838618
$658$	0	0
$659$	20.0436	0.780786	0.390393	−	0.920648i	$-0.372339\pi$
$659$	20.0436	0.780786	0.390393	+	0.920648i	$0.372339\pi$
$660$	0	0
$661$	−5.58258	−0.217137	−0.108569	−	0.994089i	$-0.534627\pi$
$661$	−5.58258	−0.217137	−0.108569	+	0.994089i	$0.534627\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.7477	0.996956
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−54.6606	−2.11015
$672$	0	0
$673$	−38.7042	−1.49194	−0.745968	−	0.665982i	$-0.768013\pi$
$673$	−38.7042	−1.49194	−0.745968	+	0.665982i	$0.768013\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.46099	0.171450	0.0857248	−	0.996319i	$-0.472679\pi$
$677$	4.46099	0.171450	0.0857248	+	0.996319i	$0.472679\pi$
$678$	0	0
$679$	23.9564	0.919364
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.8348	0.797223	0.398612	−	0.917120i	$-0.369492\pi$
$683$	20.8348	0.797223	0.398612	+	0.917120i	$0.369492\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	26.7477	1.01901
$690$	0	0
$691$	2.33030	0.0886489	0.0443244	−	0.999017i	$-0.485886\pi$
$691$	2.33030	0.0886489	0.0443244	+	0.999017i	$0.485886\pi$
$692$	0	0
$693$	−30.0000	−1.13961
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.16515	−0.233522
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.83485	−0.258149	−0.129074	−	0.991635i	$-0.541201\pi$
$701$	−6.83485	−0.258149	−0.129074	+	0.991635i	$0.541201\pi$
$702$	0	0
$703$	−45.9129	−1.73164
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.91288	−0.297594
$708$	0	0
$709$	−16.3739	−0.614933	−0.307467	−	0.951559i	$-0.599481\pi$
$709$	−16.3739	−0.614933	−0.307467	+	0.951559i	$0.599481\pi$
$710$	0	0
$711$	−26.3739	−0.989097
$712$	0	0
$713$	17.1652	0.642840
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.1652	−1.01309	−0.506545	−	0.862214i	$-0.669077\pi$
$719$	−27.1652	−1.01309	−0.506545	+	0.862214i	$0.669077\pi$
$720$	0	0
$721$	1.79129	0.0667110
$722$	0	0
$723$	0	0
$724$	0	0
$725$	26.8693	0.997901
$726$	0	0
$727$	−9.66970	−0.358629	−0.179315	−	0.983792i	$-0.557388\pi$
$727$	−9.66970	−0.358629	−0.179315	+	0.983792i	$0.557388\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−4.41742	−0.163384
$732$	0	0
$733$	−18.3303	−0.677045	−0.338523	−	0.940958i	$-0.609927\pi$
$733$	−18.3303	−0.677045	−0.338523	+	0.940958i	$0.609927\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.1652	0.411274
$738$	0	0
$739$	7.70417	0.283402	0.141701	−	0.989909i	$-0.454743\pi$
$739$	7.70417	0.283402	0.141701	+	0.989909i	$0.454743\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.0000	−1.24734	−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000	−1.24734	−0.623670	+	0.781688i	$0.714359\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.12159	0.150801
$748$	0	0
$749$	−27.1652	−0.992593
$750$	0	0
$751$	4.87841	0.178016	0.0890078	−	0.996031i	$-0.471630\pi$
$751$	4.87841	0.178016	0.0890078	+	0.996031i	$0.471630\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.12159	−0.331530	−0.165765	−	0.986165i	$-0.553009\pi$
$757$	−9.12159	−0.331530	−0.165765	+	0.986165i	$0.553009\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.7477	1.04210	0.521052	−	0.853525i	$-0.325540\pi$
$761$	28.7477	1.04210	0.521052	+	0.853525i	$0.325540\pi$
$762$	0	0
$763$	26.4174	0.956375
$764$	0	0
$765$	0	0
$766$	0	0
$767$	27.3303	0.986840
$768$	0	0
$769$	−3.25227	−0.117280	−0.0586400	−	0.998279i	$-0.518676\pi$
$769$	−3.25227	−0.117280	−0.0586400	+	0.998279i	$0.518676\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28.3303	1.01897	0.509485	−	0.860479i	$-0.329836\pi$
$773$	28.3303	1.01897	0.509485	+	0.860479i	$0.329836\pi$
$774$	0	0
$775$	17.9129	0.643450
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.3739	−0.479168
$780$	0	0
$781$	42.3303	1.51470
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31.9564	1.13912	0.569562	−	0.821948i	$-0.307113\pi$
$787$	31.9564	1.13912	0.569562	+	0.821948i	$0.307113\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.7477	−0.382145
$792$	0	0
$793$	27.3303	0.970528
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.5390	0.444155	0.222077	−	0.975029i	$-0.428716\pi$
$797$	12.5390	0.444155	0.222077	+	0.975029i	$0.428716\pi$
$798$	0	0
$799$	−4.41742	−0.156277
$800$	0	0
$801$	−27.4955	−0.971504
$802$	0	0
$803$	40.0000	1.41157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−12.8348	−0.451249	−0.225625	−	0.974214i	$-0.572442\pi$
$809$	−12.8348	−0.451249	−0.225625	+	0.974214i	$0.572442\pi$
$810$	0	0
$811$	35.4955	1.24641	0.623207	−	0.782057i	$-0.285829\pi$
$811$	35.4955	1.24641	0.623207	+	0.782057i	$0.285829\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−9.58258	−0.335252
$818$	0	0
$819$	15.0000	0.524142
$820$	0	0
$821$	30.7913	1.07462	0.537312	−	0.843384i	$-0.319440\pi$
$821$	30.7913	1.07462	0.537312	+	0.843384i	$0.319440\pi$
$822$	0	0
$823$	−9.58258	−0.334028	−0.167014	−	0.985955i	$-0.553412\pi$
$823$	−9.58258	−0.334028	−0.167014	+	0.985955i	$0.553412\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.33030	0.220126	0.110063	−	0.993925i	$-0.464895\pi$
$827$	6.33030	0.220126	0.110063	+	0.993925i	$0.464895\pi$
$828$	0	0
$829$	20.7477	0.720598	0.360299	−	0.932837i	$-0.382675\pi$
$829$	20.7477	0.720598	0.360299	+	0.932837i	$0.382675\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.37386	0.290137
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.79129	−0.338033	−0.169016	−	0.985613i	$-0.554059\pi$
$839$	−9.79129	−0.338033	−0.169016	+	0.985613i	$0.554059\pi$
$840$	0	0
$841$	−0.121591	−0.00419278
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−36.1216	−1.24115
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−45.9129	−1.57387
$852$	0	0
$853$	4.33030	0.148267	0.0741334	−	0.997248i	$-0.476381\pi$
$853$	4.33030	0.148267	0.0741334	+	0.997248i	$0.476381\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.95644	0.203468	0.101734	−	0.994812i	$-0.467561\pi$
$857$	5.95644	0.203468	0.101734	+	0.994812i	$0.467561\pi$
$858$	0	0
$859$	−15.9129	−0.542940	−0.271470	−	0.962447i	$-0.587510\pi$
$859$	−15.9129	−0.542940	−0.271470	+	0.962447i	$0.587510\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.6606	0.975618	0.487809	−	0.872950i	$-0.337796\pi$
$863$	28.6606	0.975618	0.487809	+	0.872950i	$0.337796\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−49.0780	−1.66486
$870$	0	0
$871$	−5.58258	−0.189158
$872$	0	0
$873$	40.1216	1.35791
$874$	0	0
$875$	0	0
$876$	0	0
$877$	47.0780	1.58971	0.794856	−	0.606798i	$-0.207546\pi$
$877$	47.0780	1.58971	0.794856	+	0.606798i	$0.207546\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−36.7477	−1.23806	−0.619031	−	0.785366i	$-0.712475\pi$
$881$	−36.7477	−1.23806	−0.619031	+	0.785366i	$0.712475\pi$
$882$	0	0
$883$	−2.37386	−0.0798869	−0.0399434	−	0.999202i	$-0.512718\pi$
$883$	−2.37386	−0.0798869	−0.0399434	+	0.999202i	$0.512718\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.1652	−0.374889	−0.187445	−	0.982275i	$-0.560020\pi$
$887$	−11.1652	−0.374889	−0.187445	+	0.982275i	$0.560020\pi$
$888$	0	0
$889$	23.5826	0.790934
$890$	0	0
$891$	−50.2432	−1.68321
$892$	0	0
$893$	−9.58258	−0.320669
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.2523	0.642099
$900$	0	0
$901$	−21.1652	−0.705113
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	39.9564	1.32673	0.663366	−	0.748295i	$-0.269127\pi$
$907$	39.9564	1.32673	0.663366	+	0.748295i	$0.269127\pi$
$908$	0	0
$909$	−13.2523	−0.439550
$910$	0	0
$911$	−39.1652	−1.29760	−0.648800	−	0.760959i	$-0.724729\pi$
$911$	−39.1652	−1.29760	−0.648800	+	0.760959i	$0.724729\pi$
$912$	0	0
$913$	7.66970	0.253830
$914$	0	0
$915$	0	0
$916$	0	0
$917$	38.5826	1.27411
$918$	0	0
$919$	29.0780	0.959196	0.479598	−	0.877488i	$-0.340783\pi$
$919$	29.0780	0.959196	0.479598	+	0.877488i	$0.340783\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−21.1652	−0.696659
$924$	0	0
$925$	−47.9129	−1.57537
$926$	0	0
$927$	3.00000	0.0985329
$928$	0	0
$929$	−35.7042	−1.17142	−0.585708	−	0.810522i	$-0.699183\pi$
$929$	−35.7042	−1.17142	−0.585708	+	0.810522i	$0.699183\pi$
$930$	0	0
$931$	18.1652	0.595339
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.6606	0.674953	0.337476	−	0.941334i	$-0.390427\pi$
$937$	20.6606	0.674953	0.337476	+	0.941334i	$0.390427\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	25.1652	0.820360	0.410180	−	0.912004i	$-0.365466\pi$
$941$	25.1652	0.820360	0.410180	+	0.912004i	$0.365466\pi$
$942$	0	0
$943$	−13.3739	−0.435513
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−37.5826	−1.22127	−0.610635	−	0.791912i	$-0.709086\pi$
$947$	−37.5826	−1.22127	−0.610635	+	0.791912i	$0.709086\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−56.1216	−1.81796	−0.908978	−	0.416843i	$-0.863136\pi$
$953$	−56.1216	−1.81796	−0.908978	+	0.416843i	$0.863136\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.50455	0.113168
$960$	0	0
$961$	−18.1652	−0.585973
$962$	0	0
$963$	−45.4955	−1.46607
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11.2523	−0.361849	−0.180924	−	0.983497i	$-0.557909\pi$
$967$	−11.2523	−0.361849	−0.180924	+	0.983497i	$0.557909\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	23.4955	0.754005	0.377003	−	0.926212i	$-0.376955\pi$
$971$	23.4955	0.754005	0.377003	+	0.926212i	$0.376955\pi$
$972$	0	0
$973$	−28.6606	−0.918817
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.1216	0.419797	0.209898	−	0.977723i	$-0.432687\pi$
$977$	13.1216	0.419797	0.209898	+	0.977723i	$0.432687\pi$
$978$	0	0
$979$	−51.1652	−1.63525
$980$	0	0
$981$	44.2432	1.41258
$982$	0	0
$983$	−56.5390	−1.80331	−0.901657	−	0.432451i	$-0.857649\pi$
$983$	−56.5390	−1.80331	−0.901657	+	0.432451i	$0.857649\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.58258	−0.304708
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−54.7477	−1.73388	−0.866939	−	0.498414i	$-0.833916\pi$
$997$	−54.7477	−1.73388	−0.866939	+	0.498414i	$0.833916\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6592.2.a.l.1.1		2
4.3	odd	2		6592.2.a.k.1.2		2
8.3	odd	2		824.2.a.c.1.2	✓	2
8.5	even	2		1648.2.a.e.1.1		2
24.11	even	2		7416.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
824.2.a.c.1.2	✓	2	8.3	odd	2
1648.2.a.e.1.1		2	8.5	even	2
6592.2.a.k.1.2		2	4.3	odd	2
6592.2.a.l.1.1		2	1.1	even	1	trivial
7416.2.a.f.1.2		2	24.11	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:	\( N \)	\(=\)	\( 6592 = 2^{6} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6592.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79129 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.79129 q^{7} -3.00000 q^{9} -5.58258 q^{11} +2.79129 q^{13} -2.20871 q^{17} -4.79129 q^{19} -4.79129 q^{23} -5.00000 q^{25} -5.37386 q^{29} -3.58258 q^{31} +9.58258 q^{37} +2.79129 q^{41} +2.00000 q^{43} +2.00000 q^{47} -3.79129 q^{49} +9.58258 q^{53} +9.79129 q^{59} +9.79129 q^{61} +5.37386 q^{63} -2.00000 q^{67} -7.58258 q^{71} -7.16515 q^{73} +10.0000 q^{77} +8.79129 q^{79} +9.00000 q^{81} -1.37386 q^{83} +9.16515 q^{89} -5.00000 q^{91} -13.3739 q^{97} +16.7477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q + q^7 - 6 * q^9	\( 2 q + q^{7} - 6 q^{9} - 2 q^{11} + q^{13} - 9 q^{17} - 5 q^{19} - 5 q^{23} - 10 q^{25} + 3 q^{29} + 2 q^{31} + 10 q^{37} + q^{41} + 4 q^{43} + 4 q^{47} - 3 q^{49} + 10 q^{53} + 15 q^{59} + 15 q^{61} - 3 q^{63} - 4 q^{67} - 6 q^{71} + 4 q^{73} + 20 q^{77} + 13 q^{79} + 18 q^{81} + 11 q^{83} - 10 q^{91} - 13 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + q^7 - 6 * q^9 - 2 * q^11 + q^13 - 9 * q^17 - 5 * q^19 - 5 * q^23 - 10 * q^25 + 3 * q^29 + 2 * q^31 + 10 * q^37 + q^41 + 4 * q^43 + 4 * q^47 - 3 * q^49 + 10 * q^53 + 15 * q^59 + 15 * q^61 - 3 * q^63 - 4 * q^67 - 6 * q^71 + 4 * q^73 + 20 * q^77 + 13 * q^79 + 18 * q^81 + 11 * q^83 - 10 * q^91 - 13 * q^97 + 6 * q^99

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