LMFDB - Embedded newform 2205.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.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:	$17.6070136457$
Analytic rank:	$0$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.00000 q^{5} +2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} -1.00000 q^{5} +2.82843 q^{8} +1.41421 q^{10} +5.82843 q^{11} +1.58579 q^{13} -4.00000 q^{16} +5.24264 q^{17} +6.00000 q^{19} -8.24264 q^{22} -4.58579 q^{23} +1.00000 q^{25} -2.24264 q^{26} -2.65685 q^{29} +1.75736 q^{31} -7.41421 q^{34} -6.24264 q^{37} -8.48528 q^{38} -2.82843 q^{40} -2.24264 q^{41} +2.00000 q^{43} +6.48528 q^{46} -1.24264 q^{47} -1.41421 q^{50} +4.24264 q^{53} -5.82843 q^{55} +3.75736 q^{58} -6.24264 q^{59} +2.82843 q^{61} -2.48528 q^{62} +8.00000 q^{64} -1.58579 q^{65} +0.242641 q^{67} +8.82843 q^{71} +8.48528 q^{73} +8.82843 q^{74} -15.4853 q^{79} +4.00000 q^{80} +3.17157 q^{82} -5.24264 q^{85} -2.82843 q^{86} +16.4853 q^{88} +8.00000 q^{89} +1.75736 q^{94} -6.00000 q^{95} +4.75736 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} + 6 q^{11} + 6 q^{13} - 8 q^{16} + 2 q^{17} + 12 q^{19} - 8 q^{22} - 12 q^{23} + 2 q^{25} + 4 q^{26} + 6 q^{29} + 12 q^{31} - 12 q^{34} - 4 q^{37} + 4 q^{41} + 4 q^{43} - 4 q^{46} + 6 q^{47} - 6 q^{55} + 16 q^{58} - 4 q^{59} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 8 q^{67} + 12 q^{71} + 12 q^{74} - 14 q^{79} + 8 q^{80} + 12 q^{82} - 2 q^{85} + 16 q^{88} + 16 q^{89} + 12 q^{94} - 12 q^{95} + 18 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 6 * q^11 + 6 * q^13 - 8 * q^16 + 2 * q^17 + 12 * q^19 - 8 * q^22 - 12 * q^23 + 2 * q^25 + 4 * q^26 + 6 * q^29 + 12 * q^31 - 12 * q^34 - 4 * q^37 + 4 * q^41 + 4 * q^43 - 4 * q^46 + 6 * q^47 - 6 * q^55 + 16 * q^58 - 4 * q^59 + 12 * q^62 + 16 * q^64 - 6 * q^65 - 8 * q^67 + 12 * q^71 + 12 * q^74 - 14 * q^79 + 8 * q^80 + 12 * q^82 - 2 * q^85 + 16 * q^88 + 16 * q^89 + 12 * q^94 - 12 * q^95 + 18 * q^97

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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	0	0
$10$	1.41421	0.447214
$11$	5.82843	1.75734	0.878668	−	0.477432i	$-0.158432\pi$
$11$	5.82843	1.75734	0.878668	+	0.477432i	$0.158432\pi$
$12$	0	0
$13$	1.58579	0.439818	0.219909	−	0.975520i	$-0.429424\pi$
$13$	1.58579	0.439818	0.219909	+	0.975520i	$0.429424\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	5.24264	1.27153	0.635764	−	0.771884i	$-0.280685\pi$
$17$	5.24264	1.27153	0.635764	+	0.771884i	$0.280685\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	−8.24264	−1.75734
$23$	−4.58579	−0.956203	−0.478101	−	0.878305i	$-0.658675\pi$
$23$	−4.58579	−0.956203	−0.478101	+	0.878305i	$0.658675\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.24264	−0.439818
$27$	0	0
$28$	0	0
$29$	−2.65685	−0.493365	−0.246683	−	0.969096i	$-0.579341\pi$
$29$	−2.65685	−0.493365	−0.246683	+	0.969096i	$0.579341\pi$
$30$	0	0
$31$	1.75736	0.315631	0.157816	−	0.987469i	$-0.449555\pi$
$31$	1.75736	0.315631	0.157816	+	0.987469i	$0.449555\pi$
$32$	0	0
$33$	0	0
$34$	−7.41421	−1.27153
$35$	0	0
$36$	0	0
$37$	−6.24264	−1.02628	−0.513142	−	0.858304i	$-0.671519\pi$
$37$	−6.24264	−1.02628	−0.513142	+	0.858304i	$0.671519\pi$
$38$	−8.48528	−1.37649
$39$	0	0
$40$	−2.82843	−0.447214
$41$	−2.24264	−0.350242	−0.175121	−	0.984547i	$-0.556032\pi$
$41$	−2.24264	−0.350242	−0.175121	+	0.984547i	$0.556032\pi$
$42$	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
$46$	6.48528	0.956203
$47$	−1.24264	−0.181258	−0.0906289	−	0.995885i	$-0.528888\pi$
$47$	−1.24264	−0.181258	−0.0906289	+	0.995885i	$0.528888\pi$
$48$	0	0
$49$	0	0
$50$	−1.41421	−0.200000
$51$	0	0
$52$	0	0
$53$	4.24264	0.582772	0.291386	−	0.956606i	$-0.405884\pi$
$53$	4.24264	0.582772	0.291386	+	0.956606i	$0.405884\pi$
$54$	0	0
$55$	−5.82843	−0.785905
$56$	0	0
$57$	0	0
$58$	3.75736	0.493365
$59$	−6.24264	−0.812723	−0.406361	−	0.913712i	$-0.633203\pi$
$59$	−6.24264	−0.812723	−0.406361	+	0.913712i	$0.633203\pi$
$60$	0	0
$61$	2.82843	0.362143	0.181071	−	0.983470i	$-0.442043\pi$
$61$	2.82843	0.362143	0.181071	+	0.983470i	$0.442043\pi$
$62$	−2.48528	−0.315631
$63$	0	0
$64$	8.00000	1.00000
$65$	−1.58579	−0.196693
$66$	0	0
$67$	0.242641	0.0296433	0.0148216	−	0.999890i	$-0.495282\pi$
$67$	0.242641	0.0296433	0.0148216	+	0.999890i	$0.495282\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.82843	1.04774	0.523871	−	0.851798i	$-0.324487\pi$
$71$	8.82843	1.04774	0.523871	+	0.851798i	$0.324487\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	8.82843	1.02628
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.4853	−1.74223	−0.871115	−	0.491079i	$-0.836603\pi$
$79$	−15.4853	−1.74223	−0.871115	+	0.491079i	$0.836603\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	3.17157	0.350242
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−5.24264	−0.568644
$86$	−2.82843	−0.304997
$87$	0	0
$88$	16.4853	1.75734
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	1.75736	0.181258
$95$	−6.00000	−0.615587
$96$	0	0
$97$	4.75736	0.483037	0.241518	−	0.970396i	$-0.422355\pi$
$97$	4.75736	0.483037	0.241518	+	0.970396i	$0.422355\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.4853	1.44134	0.720670	−	0.693279i	$-0.243834\pi$
$101$	14.4853	1.44134	0.720670	+	0.693279i	$0.243834\pi$
$102$	0	0
$103$	10.7574	1.05995	0.529977	−	0.848012i	$-0.322201\pi$
$103$	10.7574	1.05995	0.529977	+	0.848012i	$0.322201\pi$
$104$	4.48528	0.439818
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−14.4853	−1.40035	−0.700173	−	0.713974i	$-0.746894\pi$
$107$	−14.4853	−1.40035	−0.700173	+	0.713974i	$0.746894\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	8.24264	0.785905
$111$	0	0
$112$	0	0
$113$	−1.07107	−0.100758	−0.0503788	−	0.998730i	$-0.516043\pi$
$113$	−1.07107	−0.100758	−0.0503788	+	0.998730i	$0.516043\pi$
$114$	0	0
$115$	4.58579	0.427627
$116$	0	0
$117$	0	0
$118$	8.82843	0.812723
$119$	0	0
$120$	0	0
$121$	22.9706	2.08823
$122$	−4.00000	−0.362143
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.242641	−0.0215309	−0.0107654	−	0.999942i	$-0.503427\pi$
$127$	−0.242641	−0.0215309	−0.0107654	+	0.999942i	$0.503427\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	2.24264	0.196693
$131$	−3.75736	−0.328282	−0.164141	−	0.986437i	$-0.552485\pi$
$131$	−3.75736	−0.328282	−0.164141	+	0.986437i	$0.552485\pi$
$132$	0	0
$133$	0	0
$134$	−0.343146	−0.0296433
$135$	0	0
$136$	14.8284	1.27153
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	16.2426	1.37768	0.688841	−	0.724912i	$-0.258120\pi$
$139$	16.2426	1.37768	0.688841	+	0.724912i	$0.258120\pi$
$140$	0	0
$141$	0	0
$142$	−12.4853	−1.04774
$143$	9.24264	0.772908
$144$	0	0
$145$	2.65685	0.220640
$146$	−12.0000	−0.993127
$147$	0	0
$148$	0	0
$149$	14.8284	1.21479	0.607396	−	0.794399i	$-0.292214\pi$
$149$	14.8284	1.21479	0.607396	+	0.794399i	$0.292214\pi$
$150$	0	0
$151$	9.48528	0.771901	0.385951	−	0.922519i	$-0.373874\pi$
$151$	9.48528	0.771901	0.385951	+	0.922519i	$0.373874\pi$
$152$	16.9706	1.37649
$153$	0	0
$154$	0	0
$155$	−1.75736	−0.141154
$156$	0	0
$157$	−20.8284	−1.66229	−0.831145	−	0.556056i	$-0.812314\pi$
$157$	−20.8284	−1.66229	−0.831145	+	0.556056i	$0.812314\pi$
$158$	21.8995	1.74223
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1.75736	−0.137647	−0.0688235	−	0.997629i	$-0.521925\pi$
$163$	−1.75736	−0.137647	−0.0688235	+	0.997629i	$0.521925\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.24264	−0.715217	−0.357609	−	0.933872i	$-0.616408\pi$
$167$	−9.24264	−0.715217	−0.357609	+	0.933872i	$0.616408\pi$
$168$	0	0
$169$	−10.4853	−0.806560
$170$	7.41421	0.568644
$171$	0	0
$172$	0	0
$173$	−1.24264	−0.0944762	−0.0472381	−	0.998884i	$-0.515042\pi$
$173$	−1.24264	−0.0944762	−0.0472381	+	0.998884i	$0.515042\pi$
$174$	0	0
$175$	0	0
$176$	−23.3137	−1.75734
$177$	0	0
$178$	−11.3137	−0.847998
$179$	−2.48528	−0.185759	−0.0928793	−	0.995677i	$-0.529607\pi$
$179$	−2.48528	−0.185759	−0.0928793	+	0.995677i	$0.529607\pi$
$180$	0	0
$181$	6.72792	0.500083	0.250041	−	0.968235i	$-0.419556\pi$
$181$	6.72792	0.500083	0.250041	+	0.968235i	$0.419556\pi$
$182$	0	0
$183$	0	0
$184$	−12.9706	−0.956203
$185$	6.24264	0.458968
$186$	0	0
$187$	30.5563	2.23450
$188$	0	0
$189$	0	0
$190$	8.48528	0.615587
$191$	19.9706	1.44502	0.722510	−	0.691361i	$-0.242989\pi$
$191$	19.9706	1.44502	0.722510	+	0.691361i	$0.242989\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−6.72792	−0.483037
$195$	0	0
$196$	0	0
$197$	−10.5858	−0.754206	−0.377103	−	0.926171i	$-0.623080\pi$
$197$	−10.5858	−0.754206	−0.377103	+	0.926171i	$0.623080\pi$
$198$	0	0
$199$	−4.58579	−0.325078	−0.162539	−	0.986702i	$-0.551968\pi$
$199$	−4.58579	−0.325078	−0.162539	+	0.986702i	$0.551968\pi$
$200$	2.82843	0.200000
$201$	0	0
$202$	−20.4853	−1.44134
$203$	0	0
$204$	0	0
$205$	2.24264	0.156633
$206$	−15.2132	−1.05995
$207$	0	0
$208$	−6.34315	−0.439818
$209$	34.9706	2.41896
$210$	0	0
$211$	9.00000	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$212$	0	0
$213$	0	0
$214$	20.4853	1.40035
$215$	−2.00000	−0.136399
$216$	0	0
$217$	0	0
$218$	−7.07107	−0.478913
$219$	0	0
$220$	0	0
$221$	8.31371	0.559241
$222$	0	0
$223$	18.2132	1.21965	0.609823	−	0.792538i	$-0.291240\pi$
$223$	18.2132	1.21965	0.609823	+	0.792538i	$0.291240\pi$
$224$	0	0
$225$	0	0
$226$	1.51472	0.100758
$227$	9.72792	0.645665	0.322832	−	0.946456i	$-0.395365\pi$
$227$	9.72792	0.645665	0.322832	+	0.946456i	$0.395365\pi$
$228$	0	0
$229$	−30.0416	−1.98521	−0.992603	−	0.121402i	$-0.961261\pi$
$229$	−30.0416	−1.98521	−0.992603	+	0.121402i	$0.961261\pi$
$230$	−6.48528	−0.427627
$231$	0	0
$232$	−7.51472	−0.493365
$233$	14.8284	0.971443	0.485721	−	0.874114i	$-0.338557\pi$
$233$	14.8284	0.971443	0.485721	+	0.874114i	$0.338557\pi$
$234$	0	0
$235$	1.24264	0.0810609
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.514719	0.0332944	0.0166472	−	0.999861i	$-0.494701\pi$
$239$	0.514719	0.0332944	0.0166472	+	0.999861i	$0.494701\pi$
$240$	0	0
$241$	24.7279	1.59287	0.796433	−	0.604727i	$-0.206718\pi$
$241$	24.7279	1.59287	0.796433	+	0.604727i	$0.206718\pi$
$242$	−32.4853	−2.08823
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9.51472	0.605407
$248$	4.97056	0.315631
$249$	0	0
$250$	1.41421	0.0894427
$251$	25.2132	1.59144	0.795722	−	0.605663i	$-0.207092\pi$
$251$	25.2132	1.59144	0.795722	+	0.605663i	$0.207092\pi$
$252$	0	0
$253$	−26.7279	−1.68037
$254$	0.343146	0.0215309
$255$	0	0
$256$	0	0
$257$	26.4853	1.65211	0.826053	−	0.563592i	$-0.190581\pi$
$257$	26.4853	1.65211	0.826053	+	0.563592i	$0.190581\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	5.31371	0.328282
$263$	−28.6274	−1.76524	−0.882621	−	0.470085i	$-0.844223\pi$
$263$	−28.6274	−1.76524	−0.882621	+	0.470085i	$0.844223\pi$
$264$	0	0
$265$	−4.24264	−0.260623
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.75736	0.472975	0.236487	−	0.971635i	$-0.424004\pi$
$269$	7.75736	0.472975	0.236487	+	0.971635i	$0.424004\pi$
$270$	0	0
$271$	−23.3137	−1.41621	−0.708103	−	0.706109i	$-0.750449\pi$
$271$	−23.3137	−1.41621	−0.708103	+	0.706109i	$0.750449\pi$
$272$	−20.9706	−1.27153
$273$	0	0
$274$	16.9706	1.02523
$275$	5.82843	0.351467
$276$	0	0
$277$	−27.2132	−1.63508	−0.817541	−	0.575870i	$-0.804664\pi$
$277$	−27.2132	−1.63508	−0.817541	+	0.575870i	$0.804664\pi$
$278$	−22.9706	−1.37768
$279$	0	0
$280$	0	0
$281$	20.3137	1.21181	0.605907	−	0.795535i	$-0.292810\pi$
$281$	20.3137	1.21181	0.605907	+	0.795535i	$0.292810\pi$
$282$	0	0
$283$	−6.55635	−0.389735	−0.194867	−	0.980830i	$-0.562428\pi$
$283$	−6.55635	−0.389735	−0.194867	+	0.980830i	$0.562428\pi$
$284$	0	0
$285$	0	0
$286$	−13.0711	−0.772908
$287$	0	0
$288$	0	0
$289$	10.4853	0.616781
$290$	−3.75736	−0.220640
$291$	0	0
$292$	0	0
$293$	−0.272078	−0.0158950	−0.00794748	−	0.999968i	$-0.502530\pi$
$293$	−0.272078	−0.0158950	−0.00794748	+	0.999968i	$0.502530\pi$
$294$	0	0
$295$	6.24264	0.363461
$296$	−17.6569	−1.02628
$297$	0	0
$298$	−20.9706	−1.21479
$299$	−7.27208	−0.420555
$300$	0	0
$301$	0	0
$302$	−13.4142	−0.771901
$303$	0	0
$304$	−24.0000	−1.37649
$305$	−2.82843	−0.161955
$306$	0	0
$307$	11.1005	0.633539	0.316770	−	0.948503i	$-0.397402\pi$
$307$	11.1005	0.633539	0.316770	+	0.948503i	$0.397402\pi$
$308$	0	0
$309$	0	0
$310$	2.48528	0.141154
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	24.2132	1.36861	0.684306	−	0.729195i	$-0.260105\pi$
$313$	24.2132	1.36861	0.684306	+	0.729195i	$0.260105\pi$
$314$	29.4558	1.66229
$315$	0	0
$316$	0	0
$317$	11.6569	0.654714	0.327357	−	0.944901i	$-0.393842\pi$
$317$	11.6569	0.654714	0.327357	+	0.944901i	$0.393842\pi$
$318$	0	0
$319$	−15.4853	−0.867009
$320$	−8.00000	−0.447214
$321$	0	0
$322$	0	0
$323$	31.4558	1.75025
$324$	0	0
$325$	1.58579	0.0879636
$326$	2.48528	0.137647
$327$	0	0
$328$	−6.34315	−0.350242
$329$	0	0
$330$	0	0
$331$	23.4558	1.28925	0.644625	−	0.764499i	$-0.277014\pi$
$331$	23.4558	1.28925	0.644625	+	0.764499i	$0.277014\pi$
$332$	0	0
$333$	0	0
$334$	13.0711	0.715217
$335$	−0.242641	−0.0132569
$336$	0	0
$337$	13.7574	0.749411	0.374706	−	0.927144i	$-0.377744\pi$
$337$	13.7574	0.749411	0.374706	+	0.927144i	$0.377744\pi$
$338$	14.8284	0.806560
$339$	0	0
$340$	0	0
$341$	10.2426	0.554670
$342$	0	0
$343$	0	0
$344$	5.65685	0.304997
$345$	0	0
$346$	1.75736	0.0944762
$347$	1.07107	0.0574979	0.0287490	−	0.999587i	$-0.490848\pi$
$347$	1.07107	0.0574979	0.0287490	+	0.999587i	$0.490848\pi$
$348$	0	0
$349$	22.9706	1.22959	0.614793	−	0.788688i	$-0.289240\pi$
$349$	22.9706	1.22959	0.614793	+	0.788688i	$0.289240\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−36.2132	−1.92743	−0.963717	−	0.266925i	$-0.913992\pi$
$353$	−36.2132	−1.92743	−0.963717	+	0.266925i	$0.913992\pi$
$354$	0	0
$355$	−8.82843	−0.468564
$356$	0	0
$357$	0	0
$358$	3.51472	0.185759
$359$	−11.3137	−0.597115	−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137	−0.597115	−0.298557	+	0.954392i	$0.596505\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−9.51472	−0.500083
$363$	0	0
$364$	0	0
$365$	−8.48528	−0.444140
$366$	0	0
$367$	29.8701	1.55920	0.779602	−	0.626275i	$-0.215421\pi$
$367$	29.8701	1.55920	0.779602	+	0.626275i	$0.215421\pi$
$368$	18.3431	0.956203
$369$	0	0
$370$	−8.82843	−0.458968
$371$	0	0
$372$	0	0
$373$	0.485281	0.0251269	0.0125635	−	0.999921i	$-0.496001\pi$
$373$	0.485281	0.0251269	0.0125635	+	0.999921i	$0.496001\pi$
$374$	−43.2132	−2.23450
$375$	0	0
$376$	−3.51472	−0.181258
$377$	−4.21320	−0.216991
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	0	0
$382$	−28.2426	−1.44502
$383$	12.4853	0.637968	0.318984	−	0.947760i	$-0.396658\pi$
$383$	12.4853	0.637968	0.318984	+	0.947760i	$0.396658\pi$
$384$	0	0
$385$	0	0
$386$	22.6274	1.15171
$387$	0	0
$388$	0	0
$389$	35.1421	1.78178	0.890889	−	0.454222i	$-0.150083\pi$
$389$	35.1421	1.78178	0.890889	+	0.454222i	$0.150083\pi$
$390$	0	0
$391$	−24.0416	−1.21584
$392$	0	0
$393$	0	0
$394$	14.9706	0.754206
$395$	15.4853	0.779149
$396$	0	0
$397$	4.41421	0.221543	0.110772	−	0.993846i	$-0.464668\pi$
$397$	4.41421	0.221543	0.110772	+	0.993846i	$0.464668\pi$
$398$	6.48528	0.325078
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−6.17157	−0.308194	−0.154097	−	0.988056i	$-0.549247\pi$
$401$	−6.17157	−0.308194	−0.154097	+	0.988056i	$0.549247\pi$
$402$	0	0
$403$	2.78680	0.138820
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−36.3848	−1.80353
$408$	0	0
$409$	−14.4853	−0.716251	−0.358126	−	0.933673i	$-0.616584\pi$
$409$	−14.4853	−0.716251	−0.358126	+	0.933673i	$0.616584\pi$
$410$	−3.17157	−0.156633
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−49.4558	−2.41896
$419$	6.72792	0.328681	0.164340	−	0.986404i	$-0.447450\pi$
$419$	6.72792	0.328681	0.164340	+	0.986404i	$0.447450\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	−12.7279	−0.619586
$423$	0	0
$424$	12.0000	0.582772
$425$	5.24264	0.254305
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	2.82843	0.136399
$431$	−10.7990	−0.520169	−0.260085	−	0.965586i	$-0.583750\pi$
$431$	−10.7990	−0.520169	−0.260085	+	0.965586i	$0.583750\pi$
$432$	0	0
$433$	−22.9706	−1.10389	−0.551947	−	0.833879i	$-0.686115\pi$
$433$	−22.9706	−1.10389	−0.551947	+	0.833879i	$0.686115\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−27.5147	−1.31621
$438$	0	0
$439$	6.38478	0.304729	0.152364	−	0.988324i	$-0.451311\pi$
$439$	6.38478	0.304729	0.152364	+	0.988324i	$0.451311\pi$
$440$	−16.4853	−0.785905
$441$	0	0
$442$	−11.7574	−0.559241
$443$	−20.8284	−0.989588	−0.494794	−	0.869010i	$-0.664757\pi$
$443$	−20.8284	−0.989588	−0.494794	+	0.869010i	$0.664757\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	−25.7574	−1.21965
$447$	0	0
$448$	0	0
$449$	29.8284	1.40769	0.703845	−	0.710353i	$-0.251465\pi$
$449$	29.8284	1.40769	0.703845	+	0.710353i	$0.251465\pi$
$450$	0	0
$451$	−13.0711	−0.615493
$452$	0	0
$453$	0	0
$454$	−13.7574	−0.645665
$455$	0	0
$456$	0	0
$457$	−20.2426	−0.946911	−0.473455	−	0.880818i	$-0.656993\pi$
$457$	−20.2426	−0.946911	−0.473455	+	0.880818i	$0.656993\pi$
$458$	42.4853	1.98521
$459$	0	0
$460$	0	0
$461$	−36.9706	−1.72189	−0.860945	−	0.508697i	$-0.830127\pi$
$461$	−36.9706	−1.72189	−0.860945	+	0.508697i	$0.830127\pi$
$462$	0	0
$463$	29.4558	1.36893	0.684465	−	0.729046i	$-0.260036\pi$
$463$	29.4558	1.36893	0.684465	+	0.729046i	$0.260036\pi$
$464$	10.6274	0.493365
$465$	0	0
$466$	−20.9706	−0.971443
$467$	−19.7279	−0.912899	−0.456450	−	0.889749i	$-0.650879\pi$
$467$	−19.7279	−0.912899	−0.456450	+	0.889749i	$0.650879\pi$
$468$	0	0
$469$	0	0
$470$	−1.75736	−0.0810609
$471$	0	0
$472$	−17.6569	−0.812723
$473$	11.6569	0.535983
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	0	0
$478$	−0.727922	−0.0332944
$479$	−20.2426	−0.924910	−0.462455	−	0.886643i	$-0.653031\pi$
$479$	−20.2426	−0.924910	−0.462455	+	0.886643i	$0.653031\pi$
$480$	0	0
$481$	−9.89949	−0.451378
$482$	−34.9706	−1.59287
$483$	0	0
$484$	0	0
$485$	−4.75736	−0.216021
$486$	0	0
$487$	−31.6985	−1.43640	−0.718198	−	0.695839i	$-0.755033\pi$
$487$	−31.6985	−1.43640	−0.718198	+	0.695839i	$0.755033\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	0	0
$491$	−19.2843	−0.870287	−0.435143	−	0.900361i	$-0.643302\pi$
$491$	−19.2843	−0.870287	−0.435143	+	0.900361i	$0.643302\pi$
$492$	0	0
$493$	−13.9289	−0.627328
$494$	−13.4558	−0.605407
$495$	0	0
$496$	−7.02944	−0.315631
$497$	0	0
$498$	0	0
$499$	3.00000	0.134298	0.0671492	−	0.997743i	$-0.478610\pi$
$499$	3.00000	0.134298	0.0671492	+	0.997743i	$0.478610\pi$
$500$	0	0
$501$	0	0
$502$	−35.6569	−1.59144
$503$	32.7574	1.46058	0.730289	−	0.683138i	$-0.239385\pi$
$503$	32.7574	1.46058	0.730289	+	0.683138i	$0.239385\pi$
$504$	0	0
$505$	−14.4853	−0.644587
$506$	37.7990	1.68037
$507$	0	0
$508$	0	0
$509$	−17.2132	−0.762962	−0.381481	−	0.924377i	$-0.624586\pi$
$509$	−17.2132	−0.762962	−0.381481	+	0.924377i	$0.624586\pi$
$510$	0	0
$511$	0	0
$512$	22.6274	1.00000
$513$	0	0
$514$	−37.4558	−1.65211
$515$	−10.7574	−0.474026
$516$	0	0
$517$	−7.24264	−0.318531
$518$	0	0
$519$	0	0
$520$	−4.48528	−0.196693
$521$	18.9706	0.831115	0.415558	−	0.909567i	$-0.363586\pi$
$521$	18.9706	0.831115	0.415558	+	0.909567i	$0.363586\pi$
$522$	0	0
$523$	15.5147	0.678411	0.339206	−	0.940712i	$-0.389842\pi$
$523$	15.5147	0.678411	0.339206	+	0.940712i	$0.389842\pi$
$524$	0	0
$525$	0	0
$526$	40.4853	1.76524
$527$	9.21320	0.401333
$528$	0	0
$529$	−1.97056	−0.0856766
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	−3.55635	−0.154043
$534$	0	0
$535$	14.4853	0.626253
$536$	0.686292	0.0296433
$537$	0	0
$538$	−10.9706	−0.472975
$539$	0	0
$540$	0	0
$541$	11.9706	0.514655	0.257327	−	0.966324i	$-0.417158\pi$
$541$	11.9706	0.514655	0.257327	+	0.966324i	$0.417158\pi$
$542$	32.9706	1.41621
$543$	0	0
$544$	0	0
$545$	−5.00000	−0.214176
$546$	0	0
$547$	−7.51472	−0.321306	−0.160653	−	0.987011i	$-0.551360\pi$
$547$	−7.51472	−0.321306	−0.160653	+	0.987011i	$0.551360\pi$
$548$	0	0
$549$	0	0
$550$	−8.24264	−0.351467
$551$	−15.9411	−0.679115
$552$	0	0
$553$	0	0
$554$	38.4853	1.63508
$555$	0	0
$556$	0	0
$557$	−31.7990	−1.34737	−0.673683	−	0.739020i	$-0.735289\pi$
$557$	−31.7990	−1.34737	−0.673683	+	0.739020i	$0.735289\pi$
$558$	0	0
$559$	3.17157	0.134143
$560$	0	0
$561$	0	0
$562$	−28.7279	−1.21181
$563$	−35.9411	−1.51474	−0.757369	−	0.652987i	$-0.773516\pi$
$563$	−35.9411	−1.51474	−0.757369	+	0.652987i	$0.773516\pi$
$564$	0	0
$565$	1.07107	0.0450602
$566$	9.27208	0.389735
$567$	0	0
$568$	24.9706	1.04774
$569$	2.14214	0.0898030	0.0449015	−	0.998991i	$-0.485703\pi$
$569$	2.14214	0.0898030	0.0449015	+	0.998991i	$0.485703\pi$
$570$	0	0
$571$	−34.4853	−1.44316	−0.721582	−	0.692329i	$-0.756585\pi$
$571$	−34.4853	−1.44316	−0.721582	+	0.692329i	$0.756585\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.58579	−0.191241
$576$	0	0
$577$	−9.72792	−0.404979	−0.202489	−	0.979284i	$-0.564903\pi$
$577$	−9.72792	−0.404979	−0.202489	+	0.979284i	$0.564903\pi$
$578$	−14.8284	−0.616781
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	24.7279	1.02413
$584$	24.0000	0.993127
$585$	0	0
$586$	0.384776	0.0158950
$587$	−13.4558	−0.555382	−0.277691	−	0.960670i	$-0.589569\pi$
$587$	−13.4558	−0.555382	−0.277691	+	0.960670i	$0.589569\pi$
$588$	0	0
$589$	10.5442	0.434464
$590$	−8.82843	−0.363461
$591$	0	0
$592$	24.9706	1.02628
$593$	−10.7574	−0.441752	−0.220876	−	0.975302i	$-0.570892\pi$
$593$	−10.7574	−0.441752	−0.220876	+	0.975302i	$0.570892\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	10.2843	0.420555
$599$	12.1716	0.497317	0.248658	−	0.968591i	$-0.420010\pi$
$599$	12.1716	0.497317	0.248658	+	0.968591i	$0.420010\pi$
$600$	0	0
$601$	22.9706	0.936989	0.468494	−	0.883466i	$-0.344797\pi$
$601$	22.9706	0.936989	0.468494	+	0.883466i	$0.344797\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−22.9706	−0.933886
$606$	0	0
$607$	−24.8995	−1.01064	−0.505320	−	0.862932i	$-0.668625\pi$
$607$	−24.8995	−1.01064	−0.505320	+	0.862932i	$0.668625\pi$
$608$	0	0
$609$	0	0
$610$	4.00000	0.161955
$611$	−1.97056	−0.0797204
$612$	0	0
$613$	43.9411	1.77477	0.887383	−	0.461034i	$-0.152521\pi$
$613$	43.9411	1.77477	0.887383	+	0.461034i	$0.152521\pi$
$614$	−15.6985	−0.633539
$615$	0	0
$616$	0	0
$617$	−7.41421	−0.298485	−0.149242	−	0.988801i	$-0.547684\pi$
$617$	−7.41421	−0.298485	−0.149242	+	0.988801i	$0.547684\pi$
$618$	0	0
$619$	31.0711	1.24885	0.624426	−	0.781084i	$-0.285333\pi$
$619$	31.0711	1.24885	0.624426	+	0.781084i	$0.285333\pi$
$620$	0	0
$621$	0	0
$622$	−14.1421	−0.567048
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−34.2426	−1.36861
$627$	0	0
$628$	0	0
$629$	−32.7279	−1.30495
$630$	0	0
$631$	8.45584	0.336622	0.168311	−	0.985734i	$-0.446169\pi$
$631$	8.45584	0.336622	0.168311	+	0.985734i	$0.446169\pi$
$632$	−43.7990	−1.74223
$633$	0	0
$634$	−16.4853	−0.654714
$635$	0.242641	0.00962890
$636$	0	0
$637$	0	0
$638$	21.8995	0.867009
$639$	0	0
$640$	11.3137	0.447214
$641$	0.686292	0.0271069	0.0135534	−	0.999908i	$-0.495686\pi$
$641$	0.686292	0.0271069	0.0135534	+	0.999908i	$0.495686\pi$
$642$	0	0
$643$	27.7279	1.09348	0.546741	−	0.837302i	$-0.315868\pi$
$643$	27.7279	1.09348	0.546741	+	0.837302i	$0.315868\pi$
$644$	0	0
$645$	0	0
$646$	−44.4853	−1.75025
$647$	28.4853	1.11987	0.559936	−	0.828536i	$-0.310826\pi$
$647$	28.4853	1.11987	0.559936	+	0.828536i	$0.310826\pi$
$648$	0	0
$649$	−36.3848	−1.42823
$650$	−2.24264	−0.0879636
$651$	0	0
$652$	0	0
$653$	−1.02944	−0.0402850	−0.0201425	−	0.999797i	$-0.506412\pi$
$653$	−1.02944	−0.0402850	−0.0201425	+	0.999797i	$0.506412\pi$
$654$	0	0
$655$	3.75736	0.146812
$656$	8.97056	0.350242
$657$	0	0
$658$	0	0
$659$	13.9706	0.544216	0.272108	−	0.962267i	$-0.412279\pi$
$659$	13.9706	0.544216	0.272108	+	0.962267i	$0.412279\pi$
$660$	0	0
$661$	−37.4558	−1.45686	−0.728432	−	0.685118i	$-0.759750\pi$
$661$	−37.4558	−1.45686	−0.728432	+	0.685118i	$0.759750\pi$
$662$	−33.1716	−1.28925
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.1838	0.471757
$668$	0	0
$669$	0	0
$670$	0.343146	0.0132569
$671$	16.4853	0.636407
$672$	0	0
$673$	20.4853	0.789650	0.394825	−	0.918756i	$-0.370805\pi$
$673$	20.4853	0.789650	0.394825	+	0.918756i	$0.370805\pi$
$674$	−19.4558	−0.749411
$675$	0	0
$676$	0	0
$677$	−1.78680	−0.0686722	−0.0343361	−	0.999410i	$-0.510932\pi$
$677$	−1.78680	−0.0686722	−0.0343361	+	0.999410i	$0.510932\pi$
$678$	0	0
$679$	0	0
$680$	−14.8284	−0.568644
$681$	0	0
$682$	−14.4853	−0.554670
$683$	7.79899	0.298420	0.149210	−	0.988806i	$-0.452327\pi$
$683$	7.79899	0.298420	0.149210	+	0.988806i	$0.452327\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	6.72792	0.256313
$690$	0	0
$691$	9.17157	0.348903	0.174452	−	0.984666i	$-0.444185\pi$
$691$	9.17157	0.348903	0.174452	+	0.984666i	$0.444185\pi$
$692$	0	0
$693$	0	0
$694$	−1.51472	−0.0574979
$695$	−16.2426	−0.616118
$696$	0	0
$697$	−11.7574	−0.445342
$698$	−32.4853	−1.22959
$699$	0	0
$700$	0	0
$701$	4.45584	0.168295	0.0841475	−	0.996453i	$-0.473183\pi$
$701$	4.45584	0.168295	0.0841475	+	0.996453i	$0.473183\pi$
$702$	0	0
$703$	−37.4558	−1.41267
$704$	46.6274	1.75734
$705$	0	0
$706$	51.2132	1.92743
$707$	0	0
$708$	0	0
$709$	17.0000	0.638448	0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000	0.638448	0.319224	+	0.947679i	$0.396578\pi$
$710$	12.4853	0.468564
$711$	0	0
$712$	22.6274	0.847998
$713$	−8.05887	−0.301807
$714$	0	0
$715$	−9.24264	−0.345655
$716$	0	0
$717$	0	0
$718$	16.0000	0.597115
$719$	−17.2132	−0.641944	−0.320972	−	0.947089i	$-0.604010\pi$
$719$	−17.2132	−0.641944	−0.320972	+	0.947089i	$0.604010\pi$
$720$	0	0
$721$	0	0
$722$	−24.0416	−0.894737
$723$	0	0
$724$	0	0
$725$	−2.65685	−0.0986731
$726$	0	0
$727$	−29.3137	−1.08719	−0.543593	−	0.839349i	$-0.682936\pi$
$727$	−29.3137	−1.08719	−0.543593	+	0.839349i	$0.682936\pi$
$728$	0	0
$729$	0	0
$730$	12.0000	0.444140
$731$	10.4853	0.387812
$732$	0	0
$733$	−44.6985	−1.65098	−0.825488	−	0.564420i	$-0.809100\pi$
$733$	−44.6985	−1.65098	−0.825488	+	0.564420i	$0.809100\pi$
$734$	−42.2426	−1.55920
$735$	0	0
$736$	0	0
$737$	1.41421	0.0520932
$738$	0	0
$739$	−3.97056	−0.146060	−0.0730298	−	0.997330i	$-0.523267\pi$
$739$	−3.97056	−0.146060	−0.0730298	+	0.997330i	$0.523267\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.7279	−1.34742	−0.673708	−	0.738997i	$-0.735300\pi$
$743$	−36.7279	−1.34742	−0.673708	+	0.738997i	$0.735300\pi$
$744$	0	0
$745$	−14.8284	−0.543272
$746$	−0.686292	−0.0251269
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.5147	0.456669	0.228334	−	0.973583i	$-0.426672\pi$
$751$	12.5147	0.456669	0.228334	+	0.973583i	$0.426672\pi$
$752$	4.97056	0.181258
$753$	0	0
$754$	5.95837	0.216991
$755$	−9.48528	−0.345205
$756$	0	0
$757$	−16.4853	−0.599168	−0.299584	−	0.954070i	$-0.596848\pi$
$757$	−16.4853	−0.599168	−0.299584	+	0.954070i	$0.596848\pi$
$758$	−2.82843	−0.102733
$759$	0	0
$760$	−16.9706	−0.615587
$761$	−12.7279	−0.461387	−0.230693	−	0.973026i	$-0.574099\pi$
$761$	−12.7279	−0.461387	−0.230693	+	0.973026i	$0.574099\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−17.6569	−0.637968
$767$	−9.89949	−0.357450
$768$	0	0
$769$	−3.17157	−0.114370	−0.0571849	−	0.998364i	$-0.518212\pi$
$769$	−3.17157	−0.114370	−0.0571849	+	0.998364i	$0.518212\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−36.2132	−1.30250	−0.651249	−	0.758864i	$-0.725755\pi$
$773$	−36.2132	−1.30250	−0.651249	+	0.758864i	$0.725755\pi$
$774$	0	0
$775$	1.75736	0.0631262
$776$	13.4558	0.483037
$777$	0	0
$778$	−49.6985	−1.78178
$779$	−13.4558	−0.482106
$780$	0	0
$781$	51.4558	1.84123
$782$	34.0000	1.21584
$783$	0	0
$784$	0	0
$785$	20.8284	0.743398
$786$	0	0
$787$	45.7279	1.63002	0.815012	−	0.579444i	$-0.196730\pi$
$787$	45.7279	1.63002	0.815012	+	0.579444i	$0.196730\pi$
$788$	0	0
$789$	0	0
$790$	−21.8995	−0.779149
$791$	0	0
$792$	0	0
$793$	4.48528	0.159277
$794$	−6.24264	−0.221543
$795$	0	0
$796$	0	0
$797$	−55.1838	−1.95471	−0.977355	−	0.211608i	$-0.932130\pi$
$797$	−55.1838	−1.95471	−0.977355	+	0.211608i	$0.932130\pi$
$798$	0	0
$799$	−6.51472	−0.230474
$800$	0	0
$801$	0	0
$802$	8.72792	0.308194
$803$	49.4558	1.74526
$804$	0	0
$805$	0	0
$806$	−3.94113	−0.138820
$807$	0	0
$808$	40.9706	1.44134
$809$	−30.5980	−1.07577	−0.537884	−	0.843019i	$-0.680776\pi$
$809$	−30.5980	−1.07577	−0.537884	+	0.843019i	$0.680776\pi$
$810$	0	0
$811$	−23.3553	−0.820117	−0.410058	−	0.912059i	$-0.634492\pi$
$811$	−23.3553	−0.820117	−0.410058	+	0.912059i	$0.634492\pi$
$812$	0	0
$813$	0	0
$814$	51.4558	1.80353
$815$	1.75736	0.0615576
$816$	0	0
$817$	12.0000	0.419827
$818$	20.4853	0.716251
$819$	0	0
$820$	0	0
$821$	6.51472	0.227365	0.113683	−	0.993517i	$-0.463735\pi$
$821$	6.51472	0.227365	0.113683	+	0.993517i	$0.463735\pi$
$822$	0	0
$823$	−8.72792	−0.304236	−0.152118	−	0.988362i	$-0.548609\pi$
$823$	−8.72792	−0.304236	−0.152118	+	0.988362i	$0.548609\pi$
$824$	30.4264	1.05995
$825$	0	0
$826$	0	0
$827$	6.04163	0.210088	0.105044	−	0.994468i	$-0.466502\pi$
$827$	6.04163	0.210088	0.105044	+	0.994468i	$0.466502\pi$
$828$	0	0
$829$	54.0416	1.87694	0.938472	−	0.345356i	$-0.112242\pi$
$829$	54.0416	1.87694	0.938472	+	0.345356i	$0.112242\pi$
$830$	0	0
$831$	0	0
$832$	12.6863	0.439818
$833$	0	0
$834$	0	0
$835$	9.24264	0.319855
$836$	0	0
$837$	0	0
$838$	−9.51472	−0.328681
$839$	48.7279	1.68227	0.841137	−	0.540822i	$-0.181887\pi$
$839$	48.7279	1.68227	0.841137	+	0.540822i	$0.181887\pi$
$840$	0	0
$841$	−21.9411	−0.756591
$842$	26.8701	0.926003
$843$	0	0
$844$	0	0
$845$	10.4853	0.360705
$846$	0	0
$847$	0	0
$848$	−16.9706	−0.582772
$849$	0	0
$850$	−7.41421	−0.254305
$851$	28.6274	0.981335
$852$	0	0
$853$	−22.9706	−0.786497	−0.393249	−	0.919432i	$-0.628649\pi$
$853$	−22.9706	−0.786497	−0.393249	+	0.919432i	$0.628649\pi$
$854$	0	0
$855$	0	0
$856$	−40.9706	−1.40035
$857$	5.51472	0.188379	0.0941896	−	0.995554i	$-0.469974\pi$
$857$	5.51472	0.188379	0.0941896	+	0.995554i	$0.469974\pi$
$858$	0	0
$859$	48.7696	1.66400	0.831998	−	0.554779i	$-0.187197\pi$
$859$	48.7696	1.66400	0.831998	+	0.554779i	$0.187197\pi$
$860$	0	0
$861$	0	0
$862$	15.2721	0.520169
$863$	18.3848	0.625825	0.312913	−	0.949782i	$-0.398695\pi$
$863$	18.3848	0.625825	0.312913	+	0.949782i	$0.398695\pi$
$864$	0	0
$865$	1.24264	0.0422511
$866$	32.4853	1.10389
$867$	0	0
$868$	0	0
$869$	−90.2548	−3.06169
$870$	0	0
$871$	0.384776	0.0130376
$872$	14.1421	0.478913
$873$	0	0
$874$	38.9117	1.31621
$875$	0	0
$876$	0	0
$877$	46.9706	1.58608	0.793042	−	0.609167i	$-0.208496\pi$
$877$	46.9706	1.58608	0.793042	+	0.609167i	$0.208496\pi$
$878$	−9.02944	−0.304729
$879$	0	0
$880$	23.3137	0.785905
$881$	19.0294	0.641118	0.320559	−	0.947229i	$-0.396129\pi$
$881$	19.0294	0.641118	0.320559	+	0.947229i	$0.396129\pi$
$882$	0	0
$883$	48.4853	1.63166	0.815830	−	0.578292i	$-0.196281\pi$
$883$	48.4853	1.63166	0.815830	+	0.578292i	$0.196281\pi$
$884$	0	0
$885$	0	0
$886$	29.4558	0.989588
$887$	30.9706	1.03989	0.519945	−	0.854200i	$-0.325952\pi$
$887$	30.9706	1.03989	0.519945	+	0.854200i	$0.325952\pi$
$888$	0	0
$889$	0	0
$890$	11.3137	0.379236
$891$	0	0
$892$	0	0
$893$	−7.45584	−0.249500
$894$	0	0
$895$	2.48528	0.0830738
$896$	0	0
$897$	0	0
$898$	−42.1838	−1.40769
$899$	−4.66905	−0.155721
$900$	0	0
$901$	22.2426	0.741010
$902$	18.4853	0.615493
$903$	0	0
$904$	−3.02944	−0.100758
$905$	−6.72792	−0.223644
$906$	0	0
$907$	32.1838	1.06864	0.534322	−	0.845281i	$-0.320567\pi$
$907$	32.1838	1.06864	0.534322	+	0.845281i	$0.320567\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.65685	0.187420	0.0937100	−	0.995600i	$-0.470127\pi$
$911$	5.65685	0.187420	0.0937100	+	0.995600i	$0.470127\pi$
$912$	0	0
$913$	0	0
$914$	28.6274	0.946911
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−38.4558	−1.26854	−0.634271	−	0.773111i	$-0.718699\pi$
$919$	−38.4558	−1.26854	−0.634271	+	0.773111i	$0.718699\pi$
$920$	12.9706	0.427627
$921$	0	0
$922$	52.2843	1.72189
$923$	14.0000	0.460816
$924$	0	0
$925$	−6.24264	−0.205257
$926$	−41.6569	−1.36893
$927$	0	0
$928$	0	0
$929$	−54.7279	−1.79556	−0.897782	−	0.440439i	$-0.854823\pi$
$929$	−54.7279	−1.79556	−0.897782	+	0.440439i	$0.854823\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	27.8995	0.912899
$935$	−30.5563	−0.999299
$936$	0	0
$937$	−17.4437	−0.569859	−0.284930	−	0.958548i	$-0.591970\pi$
$937$	−17.4437	−0.569859	−0.284930	+	0.958548i	$0.591970\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.97056	0.162036	0.0810179	−	0.996713i	$-0.474183\pi$
$941$	4.97056	0.162036	0.0810179	+	0.996713i	$0.474183\pi$
$942$	0	0
$943$	10.2843	0.334902
$944$	24.9706	0.812723
$945$	0	0
$946$	−16.4853	−0.535983
$947$	43.7574	1.42192	0.710962	−	0.703231i	$-0.248260\pi$
$947$	43.7574	1.42192	0.710962	+	0.703231i	$0.248260\pi$
$948$	0	0
$949$	13.4558	0.436795
$950$	−8.48528	−0.275299
$951$	0	0
$952$	0	0
$953$	29.0122	0.939797	0.469899	−	0.882720i	$-0.344290\pi$
$953$	29.0122	0.939797	0.469899	+	0.882720i	$0.344290\pi$
$954$	0	0
$955$	−19.9706	−0.646232
$956$	0	0
$957$	0	0
$958$	28.6274	0.924910
$959$	0	0
$960$	0	0
$961$	−27.9117	−0.900377
$962$	14.0000	0.451378
$963$	0	0
$964$	0	0
$965$	16.0000	0.515058
$966$	0	0
$967$	24.4264	0.785500	0.392750	−	0.919645i	$-0.371524\pi$
$967$	24.4264	0.785500	0.392750	+	0.919645i	$0.371524\pi$
$968$	64.9706	2.08823
$969$	0	0
$970$	6.72792	0.216021
$971$	42.7279	1.37120	0.685602	−	0.727976i	$-0.259539\pi$
$971$	42.7279	1.37120	0.685602	+	0.727976i	$0.259539\pi$
$972$	0	0
$973$	0	0
$974$	44.8284	1.43640
$975$	0	0
$976$	−11.3137	−0.362143
$977$	30.7696	0.984405	0.492203	−	0.870481i	$-0.336192\pi$
$977$	30.7696	0.984405	0.492203	+	0.870481i	$0.336192\pi$
$978$	0	0
$979$	46.6274	1.49022
$980$	0	0
$981$	0	0
$982$	27.2721	0.870287
$983$	42.2132	1.34639	0.673196	−	0.739464i	$-0.264921\pi$
$983$	42.2132	1.34639	0.673196	+	0.739464i	$0.264921\pi$
$984$	0	0
$985$	10.5858	0.337291
$986$	19.6985	0.627328
$987$	0	0
$988$	0	0
$989$	−9.17157	−0.291639
$990$	0	0
$991$	−47.9411	−1.52290	−0.761450	−	0.648224i	$-0.775512\pi$
$991$	−47.9411	−1.52290	−0.761450	+	0.648224i	$0.775512\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.58579	0.145379
$996$	0	0
$997$	3.72792	0.118064	0.0590322	−	0.998256i	$-0.481199\pi$
$997$	3.72792	0.118064	0.0590322	+	0.998256i	$0.481199\pi$
$998$	−4.24264	−0.134298
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.a.t.1.1		2
3.2	odd	2		245.2.a.f.1.2	yes	2
7.6	odd	2		2205.2.a.v.1.1		2
12.11	even	2		3920.2.a.br.1.1		2
15.2	even	4		1225.2.b.j.99.3		4
15.8	even	4		1225.2.b.j.99.2		4
15.14	odd	2		1225.2.a.p.1.1		2
21.2	odd	6		245.2.e.f.116.1		4
21.5	even	6		245.2.e.g.116.1		4
21.11	odd	6		245.2.e.f.226.1		4
21.17	even	6		245.2.e.g.226.1		4
21.20	even	2		245.2.a.e.1.2	✓	2
84.83	odd	2		3920.2.a.bw.1.2		2
105.62	odd	4		1225.2.b.i.99.4		4
105.83	odd	4		1225.2.b.i.99.1		4
105.104	even	2		1225.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.a.e.1.2	✓	2	21.20	even	2
245.2.a.f.1.2	yes	2	3.2	odd	2
245.2.e.f.116.1		4	21.2	odd	6
245.2.e.f.226.1		4	21.11	odd	6
245.2.e.g.116.1		4	21.5	even	6
245.2.e.g.226.1		4	21.17	even	6
1225.2.a.p.1.1		2	15.14	odd	2
1225.2.a.r.1.1		2	105.104	even	2
1225.2.b.i.99.1		4	105.83	odd	4
1225.2.b.i.99.4		4	105.62	odd	4
1225.2.b.j.99.2		4	15.8	even	4
1225.2.b.j.99.3		4	15.2	even	4
2205.2.a.t.1.1		2	1.1	even	1	trivial
2205.2.a.v.1.1		2	7.6	odd	2
3920.2.a.br.1.1		2	12.11	even	2
3920.2.a.bw.1.2		2	84.83	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.00000 q^{5} +2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} -1.00000 q^{5} +2.82843 q^{8} +1.41421 q^{10} +5.82843 q^{11} +1.58579 q^{13} -4.00000 q^{16} +5.24264 q^{17} +6.00000 q^{19} -8.24264 q^{22} -4.58579 q^{23} +1.00000 q^{25} -2.24264 q^{26} -2.65685 q^{29} +1.75736 q^{31} -7.41421 q^{34} -6.24264 q^{37} -8.48528 q^{38} -2.82843 q^{40} -2.24264 q^{41} +2.00000 q^{43} +6.48528 q^{46} -1.24264 q^{47} -1.41421 q^{50} +4.24264 q^{53} -5.82843 q^{55} +3.75736 q^{58} -6.24264 q^{59} +2.82843 q^{61} -2.48528 q^{62} +8.00000 q^{64} -1.58579 q^{65} +0.242641 q^{67} +8.82843 q^{71} +8.48528 q^{73} +8.82843 q^{74} -15.4853 q^{79} +4.00000 q^{80} +3.17157 q^{82} -5.24264 q^{85} -2.82843 q^{86} +16.4853 q^{88} +8.00000 q^{89} +1.75736 q^{94} -6.00000 q^{95} +4.75736 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} + 6 q^{11} + 6 q^{13} - 8 q^{16} + 2 q^{17} + 12 q^{19} - 8 q^{22} - 12 q^{23} + 2 q^{25} + 4 q^{26} + 6 q^{29} + 12 q^{31} - 12 q^{34} - 4 q^{37} + 4 q^{41} + 4 q^{43} - 4 q^{46} + 6 q^{47} - 6 q^{55} + 16 q^{58} - 4 q^{59} + 12 q^{62} + 16 q^{64} - 6 q^{65} - 8 q^{67} + 12 q^{71} + 12 q^{74} - 14 q^{79} + 8 q^{80} + 12 q^{82} - 2 q^{85} + 16 q^{88} + 16 q^{89} + 12 q^{94} - 12 q^{95} + 18 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 6 * q^11 + 6 * q^13 - 8 * q^16 + 2 * q^17 + 12 * q^19 - 8 * q^22 - 12 * q^23 + 2 * q^25 + 4 * q^26 + 6 * q^29 + 12 * q^31 - 12 * q^34 - 4 * q^37 + 4 * q^41 + 4 * q^43 - 4 * q^46 + 6 * q^47 - 6 * q^55 + 16 * q^58 - 4 * q^59 + 12 * q^62 + 16 * q^64 - 6 * q^65 - 8 * q^67 + 12 * q^71 + 12 * q^74 - 14 * q^79 + 8 * q^80 + 12 * q^82 - 2 * q^85 + 16 * q^88 + 16 * q^89 + 12 * q^94 - 12 * q^95 + 18 * q^97

Introduction

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