LMFDB - Embedded newform 1620.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1620,2,Mod(1,1620)] 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("1620.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1620, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$12.9357651274$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -2.73205 q^{7} +1.73205 q^{11} +5.46410 q^{13} -4.73205 q^{17} -4.46410 q^{19} +3.46410 q^{23} +1.00000 q^{25} -7.73205 q^{29} +5.92820 q^{31} +2.73205 q^{35} -6.19615 q^{37} -11.1962 q^{41} +3.26795 q^{43} -1.26795 q^{47} +0.464102 q^{49} -7.26795 q^{53} -1.73205 q^{55} -7.73205 q^{59} -4.00000 q^{61} -5.46410 q^{65} +6.39230 q^{67} -11.1962 q^{71} -0.196152 q^{73} -4.73205 q^{77} -14.3923 q^{79} -15.1244 q^{83} +4.73205 q^{85} +5.19615 q^{89} -14.9282 q^{91} +4.46410 q^{95} +0.732051 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 2 q^{19} + 2 q^{25} - 12 q^{29} - 2 q^{31} + 2 q^{35} - 2 q^{37} - 12 q^{41} + 10 q^{43} - 6 q^{47} - 6 q^{49} - 18 q^{53} - 12 q^{59} - 8 q^{61} - 4 q^{65}+ \cdots - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^13 - 6 * q^17 - 2 * q^19 + 2 * q^25 - 12 * q^29 - 2 * q^31 + 2 * q^35 - 2 * q^37 - 12 * q^41 + 10 * q^43 - 6 * q^47 - 6 * q^49 - 18 * q^53 - 12 * q^59 - 8 * q^61 - 4 * q^65 - 8 * q^67 - 12 * q^71 + 10 * q^73 - 6 * q^77 - 8 * q^79 - 6 * q^83 + 6 * q^85 - 16 * q^91 + 2 * q^95 - 2 * q^97

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
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.73205	−1.03262	−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205	−1.03262	−0.516309	+	0.856402i	$0.672694\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.73205	0.522233	0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205	0.522233	0.261116	+	0.965307i	$0.415909\pi$
$12$	0	0
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.73205	−1.14769	−0.573845	−	0.818964i	$-0.694549\pi$
$17$	−4.73205	−1.14769	−0.573845	+	0.818964i	$0.694549\pi$
$18$	0	0
$19$	−4.46410	−1.02414	−0.512068	−	0.858945i	$-0.671120\pi$
$19$	−4.46410	−1.02414	−0.512068	+	0.858945i	$0.671120\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.73205	−1.43581	−0.717903	−	0.696143i	$-0.754898\pi$
$29$	−7.73205	−1.43581	−0.717903	+	0.696143i	$0.754898\pi$
$30$	0	0
$31$	5.92820	1.06474	0.532368	−	0.846513i	$-0.321302\pi$
$31$	5.92820	1.06474	0.532368	+	0.846513i	$0.321302\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.73205	0.461801
$36$	0	0
$37$	−6.19615	−1.01864	−0.509321	−	0.860577i	$-0.670103\pi$
$37$	−6.19615	−1.01864	−0.509321	+	0.860577i	$0.670103\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.1962	−1.74855	−0.874273	−	0.485435i	$-0.838661\pi$
$41$	−11.1962	−1.74855	−0.874273	+	0.485435i	$0.838661\pi$
$42$	0	0
$43$	3.26795	0.498358	0.249179	−	0.968458i	$-0.419839\pi$
$43$	3.26795	0.498358	0.249179	+	0.968458i	$0.419839\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.26795	−0.184949	−0.0924747	−	0.995715i	$-0.529478\pi$
$47$	−1.26795	−0.184949	−0.0924747	+	0.995715i	$0.529478\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.26795	−0.998330	−0.499165	−	0.866507i	$-0.666360\pi$
$53$	−7.26795	−0.998330	−0.499165	+	0.866507i	$0.666360\pi$
$54$	0	0
$55$	−1.73205	−0.233550
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.73205	−1.00663	−0.503314	−	0.864104i	$-0.667886\pi$
$59$	−7.73205	−1.00663	−0.503314	+	0.864104i	$0.667886\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.46410	−0.677738
$66$	0	0
$67$	6.39230	0.780944	0.390472	−	0.920615i	$-0.372312\pi$
$67$	6.39230	0.780944	0.390472	+	0.920615i	$0.372312\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.1962	−1.32874	−0.664369	−	0.747404i	$-0.731300\pi$
$71$	−11.1962	−1.32874	−0.664369	+	0.747404i	$0.731300\pi$
$72$	0	0
$73$	−0.196152	−0.0229579	−0.0114790	−	0.999934i	$-0.503654\pi$
$73$	−0.196152	−0.0229579	−0.0114790	+	0.999934i	$0.503654\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.73205	−0.539267
$78$	0	0
$79$	−14.3923	−1.61926	−0.809630	−	0.586940i	$-0.800332\pi$
$79$	−14.3923	−1.61926	−0.809630	+	0.586940i	$0.800332\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.1244	−1.66011	−0.830057	−	0.557679i	$-0.811692\pi$
$83$	−15.1244	−1.66011	−0.830057	+	0.557679i	$0.811692\pi$
$84$	0	0
$85$	4.73205	0.513263
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.19615	0.550791	0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615	0.550791	0.275396	+	0.961331i	$0.411191\pi$
$90$	0	0
$91$	−14.9282	−1.56490
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.46410	0.458007
$96$	0	0
$97$	0.732051	0.0743285	0.0371642	−	0.999309i	$-0.488168\pi$
$97$	0.732051	0.0743285	0.0371642	+	0.999309i	$0.488168\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.12436	0.609396	0.304698	−	0.952449i	$-0.401444\pi$
$101$	6.12436	0.609396	0.304698	+	0.952449i	$0.401444\pi$
$102$	0	0
$103$	18.3923	1.81225	0.906124	−	0.423013i	$-0.139027\pi$
$103$	18.3923	1.81225	0.906124	+	0.423013i	$0.139027\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	0	0
$109$	−7.92820	−0.759384	−0.379692	−	0.925113i	$-0.623970\pi$
$109$	−7.92820	−0.759384	−0.379692	+	0.925113i	$0.623970\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.339746	−0.0319606	−0.0159803	−	0.999872i	$-0.505087\pi$
$113$	−0.339746	−0.0319606	−0.0159803	+	0.999872i	$0.505087\pi$
$114$	0	0
$115$	−3.46410	−0.323029
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.9282	1.18513
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.19615	0.372348	0.186174	−	0.982517i	$-0.440391\pi$
$127$	4.19615	0.372348	0.186174	+	0.982517i	$0.440391\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.2679	0.897115	0.448557	−	0.893754i	$-0.351938\pi$
$131$	10.2679	0.897115	0.448557	+	0.893754i	$0.351938\pi$
$132$	0	0
$133$	12.1962	1.05754
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.39230	0.375260	0.187630	−	0.982240i	$-0.439919\pi$
$137$	4.39230	0.375260	0.187630	+	0.982240i	$0.439919\pi$
$138$	0	0
$139$	15.3923	1.30556	0.652779	−	0.757548i	$-0.273603\pi$
$139$	15.3923	1.30556	0.652779	+	0.757548i	$0.273603\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.46410	0.791428
$144$	0	0
$145$	7.73205	0.642112
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−17.3923	−1.41537	−0.707683	−	0.706530i	$-0.750259\pi$
$151$	−17.3923	−1.41537	−0.707683	+	0.706530i	$0.750259\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.92820	−0.476165
$156$	0	0
$157$	3.26795	0.260811	0.130405	−	0.991461i	$-0.458372\pi$
$157$	3.26795	0.260811	0.130405	+	0.991461i	$0.458372\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.46410	−0.745876
$162$	0	0
$163$	18.7321	1.46721	0.733604	−	0.679577i	$-0.237837\pi$
$163$	18.7321	1.46721	0.733604	+	0.679577i	$0.237837\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.1244	−1.63465	−0.817326	−	0.576176i	$-0.804544\pi$
$167$	−21.1244	−1.63465	−0.817326	+	0.576176i	$0.804544\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−24.2487	−1.84360	−0.921798	−	0.387671i	$-0.873280\pi$
$173$	−24.2487	−1.84360	−0.921798	+	0.387671i	$0.873280\pi$
$174$	0	0
$175$	−2.73205	−0.206524
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.1244	−0.906217	−0.453108	−	0.891455i	$-0.649685\pi$
$179$	−12.1244	−0.906217	−0.453108	+	0.891455i	$0.649685\pi$
$180$	0	0
$181$	−16.4641	−1.22377	−0.611884	−	0.790948i	$-0.709588\pi$
$181$	−16.4641	−1.22377	−0.611884	+	0.790948i	$0.709588\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.19615	0.455550
$186$	0	0
$187$	−8.19615	−0.599362
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.0526	1.37859	0.689297	−	0.724479i	$-0.257919\pi$
$191$	19.0526	1.37859	0.689297	+	0.724479i	$0.257919\pi$
$192$	0	0
$193$	−10.5885	−0.762174	−0.381087	−	0.924539i	$-0.624450\pi$
$193$	−10.5885	−0.762174	−0.381087	+	0.924539i	$0.624450\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.8564	−0.987228	−0.493614	−	0.869681i	$-0.664324\pi$
$197$	−13.8564	−0.987228	−0.493614	+	0.869681i	$0.664324\pi$
$198$	0	0
$199$	15.8564	1.12403	0.562015	−	0.827127i	$-0.310026\pi$
$199$	15.8564	1.12403	0.562015	+	0.827127i	$0.310026\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	21.1244	1.48264
$204$	0	0
$205$	11.1962	0.781973
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.73205	−0.534837
$210$	0	0
$211$	−19.9282	−1.37191	−0.685957	−	0.727642i	$-0.740616\pi$
$211$	−19.9282	−1.37191	−0.685957	+	0.727642i	$0.740616\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.26795	−0.222872
$216$	0	0
$217$	−16.1962	−1.09947
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−25.8564	−1.73929
$222$	0	0
$223$	−5.85641	−0.392174	−0.196087	−	0.980587i	$-0.562823\pi$
$223$	−5.85641	−0.392174	−0.196087	+	0.980587i	$0.562823\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.1962	1.73870	0.869350	−	0.494197i	$-0.164538\pi$
$227$	26.1962	1.73870	0.869350	+	0.494197i	$0.164538\pi$
$228$	0	0
$229$	−17.8564	−1.17998	−0.589992	−	0.807409i	$-0.700869\pi$
$229$	−17.8564	−1.17998	−0.589992	+	0.807409i	$0.700869\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.5885	1.61084	0.805422	−	0.592702i	$-0.201939\pi$
$233$	24.5885	1.61084	0.805422	+	0.592702i	$0.201939\pi$
$234$	0	0
$235$	1.26795	0.0827119
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.53590	0.552141	0.276071	−	0.961137i	$-0.410968\pi$
$239$	8.53590	0.552141	0.276071	+	0.961137i	$0.410968\pi$
$240$	0	0
$241$	10.3205	0.664802	0.332401	−	0.943138i	$-0.392141\pi$
$241$	10.3205	0.664802	0.332401	+	0.943138i	$0.392141\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.464102	−0.0296504
$246$	0	0
$247$	−24.3923	−1.55205
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.5359	1.29621	0.648107	−	0.761549i	$-0.275561\pi$
$251$	20.5359	1.29621	0.648107	+	0.761549i	$0.275561\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.4641	0.964624	0.482312	−	0.875999i	$-0.339797\pi$
$257$	15.4641	0.964624	0.482312	+	0.875999i	$0.339797\pi$
$258$	0	0
$259$	16.9282	1.05187
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.2487	−1.49524	−0.747620	−	0.664127i	$-0.768803\pi$
$263$	−24.2487	−1.49524	−0.747620	+	0.664127i	$0.768803\pi$
$264$	0	0
$265$	7.26795	0.446467
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−16.2679	−0.991874	−0.495937	−	0.868358i	$-0.665175\pi$
$269$	−16.2679	−0.991874	−0.495937	+	0.868358i	$0.665175\pi$
$270$	0	0
$271$	16.7846	1.01959	0.509796	−	0.860295i	$-0.329721\pi$
$271$	16.7846	1.01959	0.509796	+	0.860295i	$0.329721\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.73205	0.104447
$276$	0	0
$277$	5.12436	0.307893	0.153946	−	0.988079i	$-0.450802\pi$
$277$	5.12436	0.307893	0.153946	+	0.988079i	$0.450802\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	29.3205	1.74911	0.874557	−	0.484922i	$-0.161152\pi$
$281$	29.3205	1.74911	0.874557	+	0.484922i	$0.161152\pi$
$282$	0	0
$283$	16.5359	0.982957	0.491479	−	0.870890i	$-0.336457\pi$
$283$	16.5359	0.982957	0.491479	+	0.870890i	$0.336457\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.5885	1.80558
$288$	0	0
$289$	5.39230	0.317194
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.5885	1.43647	0.718237	−	0.695799i	$-0.244950\pi$
$293$	24.5885	1.43647	0.718237	+	0.695799i	$0.244950\pi$
$294$	0	0
$295$	7.73205	0.450177
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.9282	1.09465
$300$	0	0
$301$	−8.92820	−0.514613
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	−1.80385	−0.102951	−0.0514755	−	0.998674i	$-0.516392\pi$
$307$	−1.80385	−0.102951	−0.0514755	+	0.998674i	$0.516392\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.5167	−0.936574	−0.468287	−	0.883576i	$-0.655129\pi$
$311$	−16.5167	−0.936574	−0.468287	+	0.883576i	$0.655129\pi$
$312$	0	0
$313$	14.9282	0.843792	0.421896	−	0.906644i	$-0.361365\pi$
$313$	14.9282	0.843792	0.421896	+	0.906644i	$0.361365\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.12436	0.175481	0.0877406	−	0.996143i	$-0.472035\pi$
$317$	3.12436	0.175481	0.0877406	+	0.996143i	$0.472035\pi$
$318$	0	0
$319$	−13.3923	−0.749825
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.1244	1.17539
$324$	0	0
$325$	5.46410	0.303094
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.46410	0.190982
$330$	0	0
$331$	−29.3923	−1.61555	−0.807774	−	0.589493i	$-0.799328\pi$
$331$	−29.3923	−1.61555	−0.807774	+	0.589493i	$0.799328\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.39230	−0.349249
$336$	0	0
$337$	−34.2487	−1.86565	−0.932823	−	0.360335i	$-0.882662\pi$
$337$	−34.2487	−1.86565	−0.932823	+	0.360335i	$0.882662\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.2679	0.556041
$342$	0	0
$343$	17.8564	0.964155
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.8038	1.17049	0.585246	−	0.810856i	$-0.300998\pi$
$347$	21.8038	1.17049	0.585246	+	0.810856i	$0.300998\pi$
$348$	0	0
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−28.9808	−1.54249	−0.771245	−	0.636538i	$-0.780366\pi$
$353$	−28.9808	−1.54249	−0.771245	+	0.636538i	$0.780366\pi$
$354$	0	0
$355$	11.1962	0.594230
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.33975	0.492933	0.246466	−	0.969151i	$-0.420730\pi$
$359$	9.33975	0.492933	0.246466	+	0.969151i	$0.420730\pi$
$360$	0	0
$361$	0.928203	0.0488528
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.196152	0.0102671
$366$	0	0
$367$	−26.3923	−1.37767	−0.688834	−	0.724920i	$-0.741877\pi$
$367$	−26.3923	−1.37767	−0.688834	+	0.724920i	$0.741877\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	19.8564	1.03089
$372$	0	0
$373$	−14.0526	−0.727614	−0.363807	−	0.931474i	$-0.618523\pi$
$373$	−14.0526	−0.727614	−0.363807	+	0.931474i	$0.618523\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−42.2487	−2.17592
$378$	0	0
$379$	4.53590	0.232993	0.116497	−	0.993191i	$-0.462834\pi$
$379$	4.53590	0.232993	0.116497	+	0.993191i	$0.462834\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.2487	0.932466	0.466233	−	0.884662i	$-0.345611\pi$
$383$	18.2487	0.932466	0.466233	+	0.884662i	$0.345611\pi$
$384$	0	0
$385$	4.73205	0.241168
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.4641	1.39249	0.696243	−	0.717807i	$-0.254854\pi$
$389$	27.4641	1.39249	0.696243	+	0.717807i	$0.254854\pi$
$390$	0	0
$391$	−16.3923	−0.828994
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.3923	0.724155
$396$	0	0
$397$	37.3205	1.87306	0.936531	−	0.350584i	$-0.114017\pi$
$397$	37.3205	1.87306	0.936531	+	0.350584i	$0.114017\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.7846	−0.738308	−0.369154	−	0.929368i	$-0.620353\pi$
$401$	−14.7846	−0.738308	−0.369154	+	0.929368i	$0.620353\pi$
$402$	0	0
$403$	32.3923	1.61358
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.7321	−0.531968
$408$	0	0
$409$	−17.8564	−0.882942	−0.441471	−	0.897275i	$-0.645543\pi$
$409$	−17.8564	−0.882942	−0.441471	+	0.897275i	$0.645543\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	21.1244	1.03946
$414$	0	0
$415$	15.1244	0.742425
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.4641	1.04859	0.524295	−	0.851537i	$-0.324329\pi$
$419$	21.4641	1.04859	0.524295	+	0.851537i	$0.324329\pi$
$420$	0	0
$421$	13.7846	0.671821	0.335910	−	0.941894i	$-0.390956\pi$
$421$	13.7846	0.671821	0.335910	+	0.941894i	$0.390956\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.73205	−0.229538
$426$	0	0
$427$	10.9282	0.528853
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−33.5885	−1.61790	−0.808950	−	0.587878i	$-0.799963\pi$
$431$	−33.5885	−1.61790	−0.808950	+	0.587878i	$0.799963\pi$
$432$	0	0
$433$	11.4641	0.550930	0.275465	−	0.961311i	$-0.411168\pi$
$433$	11.4641	0.550930	0.275465	+	0.961311i	$0.411168\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.4641	−0.739748
$438$	0	0
$439$	6.60770	0.315368	0.157684	−	0.987490i	$-0.449597\pi$
$439$	6.60770	0.315368	0.157684	+	0.987490i	$0.449597\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.2679	1.77065	0.885327	−	0.464969i	$-0.153935\pi$
$443$	37.2679	1.77065	0.885327	+	0.464969i	$0.153935\pi$
$444$	0	0
$445$	−5.19615	−0.246321
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.1244	1.13850	0.569249	−	0.822165i	$-0.307234\pi$
$449$	24.1244	1.13850	0.569249	+	0.822165i	$0.307234\pi$
$450$	0	0
$451$	−19.3923	−0.913148
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.9282	0.699845
$456$	0	0
$457$	22.1962	1.03829	0.519146	−	0.854686i	$-0.326250\pi$
$457$	22.1962	1.03829	0.519146	+	0.854686i	$0.326250\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.58846	0.446579	0.223289	−	0.974752i	$-0.428320\pi$
$461$	9.58846	0.446579	0.223289	+	0.974752i	$0.428320\pi$
$462$	0	0
$463$	−2.39230	−0.111180	−0.0555899	−	0.998454i	$-0.517704\pi$
$463$	−2.39230	−0.111180	−0.0555899	+	0.998454i	$0.517704\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.1962	1.21221	0.606107	−	0.795383i	$-0.292730\pi$
$467$	26.1962	1.21221	0.606107	+	0.795383i	$0.292730\pi$
$468$	0	0
$469$	−17.4641	−0.806417
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.66025	0.260259
$474$	0	0
$475$	−4.46410	−0.204827
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.33975	−0.152597	−0.0762984	−	0.997085i	$-0.524310\pi$
$479$	−3.33975	−0.152597	−0.0762984	+	0.997085i	$0.524310\pi$
$480$	0	0
$481$	−33.8564	−1.54372
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.732051	−0.0332407
$486$	0	0
$487$	−30.5359	−1.38371	−0.691857	−	0.722035i	$-0.743207\pi$
$487$	−30.5359	−1.38371	−0.691857	+	0.722035i	$0.743207\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.26795	−0.192610	−0.0963049	−	0.995352i	$-0.530702\pi$
$491$	−4.26795	−0.192610	−0.0963049	+	0.995352i	$0.530702\pi$
$492$	0	0
$493$	36.5885	1.64786
$494$	0	0
$495$	0	0
$496$	0	0
$497$	30.5885	1.37208
$498$	0	0
$499$	27.3923	1.22625	0.613124	−	0.789987i	$-0.289913\pi$
$499$	27.3923	1.22625	0.613124	+	0.789987i	$0.289913\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−35.3205	−1.57486	−0.787432	−	0.616402i	$-0.788590\pi$
$503$	−35.3205	−1.57486	−0.787432	+	0.616402i	$0.788590\pi$
$504$	0	0
$505$	−6.12436	−0.272530
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.7846	−1.18721	−0.593603	−	0.804758i	$-0.702295\pi$
$509$	−26.7846	−1.18721	−0.593603	+	0.804758i	$0.702295\pi$
$510$	0	0
$511$	0.535898	0.0237067
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−18.3923	−0.810462
$516$	0	0
$517$	−2.19615	−0.0965867
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.3923	−0.455295	−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923	−0.455295	−0.227648	+	0.973744i	$0.573103\pi$
$522$	0	0
$523$	3.60770	0.157753	0.0788767	−	0.996884i	$-0.474867\pi$
$523$	3.60770	0.157753	0.0788767	+	0.996884i	$0.474867\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.0526	−1.22199
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−61.1769	−2.64987
$534$	0	0
$535$	−3.46410	−0.149766
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.803848	0.0346242
$540$	0	0
$541$	2.46410	0.105940	0.0529700	−	0.998596i	$-0.483131\pi$
$541$	2.46410	0.105940	0.0529700	+	0.998596i	$0.483131\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.92820	0.339607
$546$	0	0
$547$	4.78461	0.204575	0.102288	−	0.994755i	$-0.467384\pi$
$547$	4.78461	0.204575	0.102288	+	0.994755i	$0.467384\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	34.5167	1.47046
$552$	0	0
$553$	39.3205	1.67208
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.39230	−0.186108	−0.0930540	−	0.995661i	$-0.529663\pi$
$557$	−4.39230	−0.186108	−0.0930540	+	0.995661i	$0.529663\pi$
$558$	0	0
$559$	17.8564	0.755246
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.7321	0.958042	0.479021	−	0.877804i	$-0.340992\pi$
$563$	22.7321	0.958042	0.479021	+	0.877804i	$0.340992\pi$
$564$	0	0
$565$	0.339746	0.0142932
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−43.0526	−1.80486	−0.902429	−	0.430839i	$-0.858218\pi$
$569$	−43.0526	−1.80486	−0.902429	+	0.430839i	$0.858218\pi$
$570$	0	0
$571$	0.856406	0.0358395	0.0179197	−	0.999839i	$-0.494296\pi$
$571$	0.856406	0.0358395	0.0179197	+	0.999839i	$0.494296\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.46410	0.144463
$576$	0	0
$577$	11.8038	0.491401	0.245700	−	0.969346i	$-0.420982\pi$
$577$	11.8038	0.491401	0.245700	+	0.969346i	$0.420982\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	41.3205	1.71426
$582$	0	0
$583$	−12.5885	−0.521361
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.80385	0.157002	0.0785008	−	0.996914i	$-0.474987\pi$
$587$	3.80385	0.157002	0.0785008	+	0.996914i	$0.474987\pi$
$588$	0	0
$589$	−26.4641	−1.09043
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23.0718	−0.947445	−0.473723	−	0.880674i	$-0.657090\pi$
$593$	−23.0718	−0.947445	−0.473723	+	0.880674i	$0.657090\pi$
$594$	0	0
$595$	−12.9282	−0.530005
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−14.4115	−0.588840	−0.294420	−	0.955676i	$-0.595126\pi$
$599$	−14.4115	−0.588840	−0.294420	+	0.955676i	$0.595126\pi$
$600$	0	0
$601$	−24.3205	−0.992054	−0.496027	−	0.868307i	$-0.665208\pi$
$601$	−24.3205	−0.992054	−0.496027	+	0.868307i	$0.665208\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.00000	0.325246
$606$	0	0
$607$	2.58846	0.105062	0.0525311	−	0.998619i	$-0.483271\pi$
$607$	2.58846	0.105062	0.0525311	+	0.998619i	$0.483271\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.92820	−0.280285
$612$	0	0
$613$	24.3923	0.985196	0.492598	−	0.870257i	$-0.336047\pi$
$613$	24.3923	0.985196	0.492598	+	0.870257i	$0.336047\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.1962	−0.568757
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	29.3205	1.16909
$630$	0	0
$631$	−0.0717968	−0.00285818	−0.00142909	−	0.999999i	$-0.500455\pi$
$631$	−0.0717968	−0.00285818	−0.00142909	+	0.999999i	$0.500455\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.19615	−0.166519
$636$	0	0
$637$	2.53590	0.100476
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.8038	−0.505722	−0.252861	−	0.967503i	$-0.581371\pi$
$641$	−12.8038	−0.505722	−0.252861	+	0.967503i	$0.581371\pi$
$642$	0	0
$643$	14.5885	0.575313	0.287656	−	0.957734i	$-0.407124\pi$
$643$	14.5885	0.575313	0.287656	+	0.957734i	$0.407124\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.248711	0.00977785	0.00488893	−	0.999988i	$-0.498444\pi$
$647$	0.248711	0.00977785	0.00488893	+	0.999988i	$0.498444\pi$
$648$	0	0
$649$	−13.3923	−0.525694
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.53590	−0.0992374	−0.0496187	−	0.998768i	$-0.515801\pi$
$653$	−2.53590	−0.0992374	−0.0496187	+	0.998768i	$0.515801\pi$
$654$	0	0
$655$	−10.2679	−0.401202
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.53590	−0.0987846	−0.0493923	−	0.998779i	$-0.515728\pi$
$659$	−2.53590	−0.0987846	−0.0493923	+	0.998779i	$0.515728\pi$
$660$	0	0
$661$	15.3923	0.598691	0.299346	−	0.954145i	$-0.403232\pi$
$661$	15.3923	0.598691	0.299346	+	0.954145i	$0.403232\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.1962	−0.472947
$666$	0	0
$667$	−26.7846	−1.03710
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.92820	−0.267460
$672$	0	0
$673$	−38.3923	−1.47991	−0.739957	−	0.672654i	$-0.765154\pi$
$673$	−38.3923	−1.47991	−0.739957	+	0.672654i	$0.765154\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.6410	1.56196	0.780981	−	0.624555i	$-0.214720\pi$
$677$	40.6410	1.56196	0.780981	+	0.624555i	$0.214720\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21.4641	0.821301	0.410651	−	0.911793i	$-0.365302\pi$
$683$	21.4641	0.821301	0.410651	+	0.911793i	$0.365302\pi$
$684$	0	0
$685$	−4.39230	−0.167821
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−39.7128	−1.51294
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.3923	−0.583863
$696$	0	0
$697$	52.9808	2.00679
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17.8756	−0.675154	−0.337577	−	0.941298i	$-0.609607\pi$
$701$	−17.8756	−0.675154	−0.337577	+	0.941298i	$0.609607\pi$
$702$	0	0
$703$	27.6603	1.04323
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16.7321	−0.629274
$708$	0	0
$709$	10.5359	0.395684	0.197842	−	0.980234i	$-0.436607\pi$
$709$	10.5359	0.395684	0.197842	+	0.980234i	$0.436607\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.5359	0.769075
$714$	0	0
$715$	−9.46410	−0.353937
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17.1962	−0.641308	−0.320654	−	0.947196i	$-0.603903\pi$
$719$	−17.1962	−0.641308	−0.320654	+	0.947196i	$0.603903\pi$
$720$	0	0
$721$	−50.2487	−1.87136
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.73205	−0.287161
$726$	0	0
$727$	−29.1769	−1.08211	−0.541056	−	0.840987i	$-0.681975\pi$
$727$	−29.1769	−1.08211	−0.541056	+	0.840987i	$0.681975\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−15.4641	−0.571960
$732$	0	0
$733$	43.5692	1.60927	0.804633	−	0.593773i	$-0.202362\pi$
$733$	43.5692	1.60927	0.804633	+	0.593773i	$0.202362\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.0718	0.407835
$738$	0	0
$739$	−26.1769	−0.962933	−0.481467	−	0.876464i	$-0.659896\pi$
$739$	−26.1769	−0.962933	−0.481467	+	0.876464i	$0.659896\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.41154	−0.198530	−0.0992651	−	0.995061i	$-0.531649\pi$
$743$	−5.41154	−0.198530	−0.0992651	+	0.995061i	$0.531649\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.46410	−0.345811
$750$	0	0
$751$	40.7846	1.48825	0.744126	−	0.668040i	$-0.232866\pi$
$751$	40.7846	1.48825	0.744126	+	0.668040i	$0.232866\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.3923	0.632971
$756$	0	0
$757$	−20.3923	−0.741171	−0.370585	−	0.928798i	$-0.620843\pi$
$757$	−20.3923	−0.741171	−0.370585	+	0.928798i	$0.620843\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.1244	−1.09201	−0.546004	−	0.837783i	$-0.683851\pi$
$761$	−30.1244	−1.09201	−0.546004	+	0.837783i	$0.683851\pi$
$762$	0	0
$763$	21.6603	0.784154
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−42.2487	−1.52551
$768$	0	0
$769$	35.2487	1.27110	0.635551	−	0.772059i	$-0.280773\pi$
$769$	35.2487	1.27110	0.635551	+	0.772059i	$0.280773\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	17.6603	0.635195	0.317598	−	0.948226i	$-0.397124\pi$
$773$	17.6603	0.635195	0.317598	+	0.948226i	$0.397124\pi$
$774$	0	0
$775$	5.92820	0.212947
$776$	0	0
$777$	0	0
$778$	0	0
$779$	49.9808	1.79075
$780$	0	0
$781$	−19.3923	−0.693911
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.26795	−0.116638
$786$	0	0
$787$	−36.1962	−1.29025	−0.645127	−	0.764076i	$-0.723195\pi$
$787$	−36.1962	−1.29025	−0.645127	+	0.764076i	$0.723195\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.928203	0.0330031
$792$	0	0
$793$	−21.8564	−0.776144
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.3205	0.826055	0.413027	−	0.910719i	$-0.364471\pi$
$797$	23.3205	0.826055	0.413027	+	0.910719i	$0.364471\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.339746	−0.0119894
$804$	0	0
$805$	9.46410	0.333566
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.41154	−0.295734	−0.147867	−	0.989007i	$-0.547241\pi$
$809$	−8.41154	−0.295734	−0.147867	+	0.989007i	$0.547241\pi$
$810$	0	0
$811$	−25.2487	−0.886602	−0.443301	−	0.896373i	$-0.646193\pi$
$811$	−25.2487	−0.886602	−0.443301	+	0.896373i	$0.646193\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.7321	−0.656155
$816$	0	0
$817$	−14.5885	−0.510386
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.33975	0.116558	0.0582790	−	0.998300i	$-0.481439\pi$
$821$	3.33975	0.116558	0.0582790	+	0.998300i	$0.481439\pi$
$822$	0	0
$823$	−16.9282	−0.590080	−0.295040	−	0.955485i	$-0.595333\pi$
$823$	−16.9282	−0.590080	−0.295040	+	0.955485i	$0.595333\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−45.4641	−1.58094	−0.790471	−	0.612500i	$-0.790164\pi$
$827$	−45.4641	−1.58094	−0.790471	+	0.612500i	$0.790164\pi$
$828$	0	0
$829$	−51.7846	−1.79855	−0.899277	−	0.437380i	$-0.855907\pi$
$829$	−51.7846	−1.79855	−0.899277	+	0.437380i	$0.855907\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.19615	−0.0760922
$834$	0	0
$835$	21.1244	0.731038
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.41154	−0.290399	−0.145199	−	0.989402i	$-0.546382\pi$
$839$	−8.41154	−0.290399	−0.145199	+	0.989402i	$0.546382\pi$
$840$	0	0
$841$	30.7846	1.06154
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−16.8564	−0.579878
$846$	0	0
$847$	21.8564	0.750995
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−21.4641	−0.735780
$852$	0	0
$853$	−0.196152	−0.00671613	−0.00335807	−	0.999994i	$-0.501069\pi$
$853$	−0.196152	−0.00671613	−0.00335807	+	0.999994i	$0.501069\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29.0718	−0.993074	−0.496537	−	0.868016i	$-0.665395\pi$
$857$	−29.0718	−0.993074	−0.496537	+	0.868016i	$0.665395\pi$
$858$	0	0
$859$	7.78461	0.265607	0.132804	−	0.991142i	$-0.457602\pi$
$859$	7.78461	0.265607	0.132804	+	0.991142i	$0.457602\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−49.5167	−1.68557	−0.842783	−	0.538253i	$-0.819085\pi$
$863$	−49.5167	−1.68557	−0.842783	+	0.538253i	$0.819085\pi$
$864$	0	0
$865$	24.2487	0.824481
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−24.9282	−0.845631
$870$	0	0
$871$	34.9282	1.18350
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.73205	0.0923602
$876$	0	0
$877$	14.2487	0.481145	0.240572	−	0.970631i	$-0.422665\pi$
$877$	14.2487	0.481145	0.240572	+	0.970631i	$0.422665\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.80385	0.229227	0.114614	−	0.993410i	$-0.463437\pi$
$881$	6.80385	0.229227	0.114614	+	0.993410i	$0.463437\pi$
$882$	0	0
$883$	−46.8372	−1.57620	−0.788098	−	0.615550i	$-0.788934\pi$
$883$	−46.8372	−1.57620	−0.788098	+	0.615550i	$0.788934\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.2679	0.445494	0.222747	−	0.974876i	$-0.428498\pi$
$887$	13.2679	0.445494	0.222747	+	0.974876i	$0.428498\pi$
$888$	0	0
$889$	−11.4641	−0.384494
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.66025	0.189413
$894$	0	0
$895$	12.1244	0.405273
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−45.8372	−1.52876
$900$	0	0
$901$	34.3923	1.14577
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.4641	0.547285
$906$	0	0
$907$	−1.21539	−0.0403564	−0.0201782	−	0.999796i	$-0.506423\pi$
$907$	−1.21539	−0.0403564	−0.0201782	+	0.999796i	$0.506423\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.80385	−0.225422	−0.112711	−	0.993628i	$-0.535953\pi$
$911$	−6.80385	−0.225422	−0.112711	+	0.993628i	$0.535953\pi$
$912$	0	0
$913$	−26.1962	−0.866966
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−28.0526	−0.926377
$918$	0	0
$919$	51.3923	1.69528	0.847638	−	0.530575i	$-0.178024\pi$
$919$	51.3923	1.69528	0.847638	+	0.530575i	$0.178024\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−61.1769	−2.01366
$924$	0	0
$925$	−6.19615	−0.203728
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.48334	0.0486668	0.0243334	−	0.999704i	$-0.492254\pi$
$929$	1.48334	0.0486668	0.0243334	+	0.999704i	$0.492254\pi$
$930$	0	0
$931$	−2.07180	−0.0679004
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.19615	0.268043
$936$	0	0
$937$	19.0718	0.623048	0.311524	−	0.950238i	$-0.399160\pi$
$937$	19.0718	0.623048	0.311524	+	0.950238i	$0.399160\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.53590	−0.278262	−0.139131	−	0.990274i	$-0.544431\pi$
$941$	−8.53590	−0.278262	−0.139131	+	0.990274i	$0.544431\pi$
$942$	0	0
$943$	−38.7846	−1.26300
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−52.6410	−1.71060	−0.855302	−	0.518130i	$-0.826628\pi$
$947$	−52.6410	−1.71060	−0.855302	+	0.518130i	$0.826628\pi$
$948$	0	0
$949$	−1.07180	−0.0347920
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.53590	−0.0821458	−0.0410729	−	0.999156i	$-0.513078\pi$
$953$	−2.53590	−0.0821458	−0.0410729	+	0.999156i	$0.513078\pi$
$954$	0	0
$955$	−19.0526	−0.616526
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	4.14359	0.133664
$962$	0	0
$963$	0	0
$964$	0	0
$965$	10.5885	0.340854
$966$	0	0
$967$	7.41154	0.238339	0.119170	−	0.992874i	$-0.461977\pi$
$967$	7.41154	0.238339	0.119170	+	0.992874i	$0.461977\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.2679	0.714612	0.357306	−	0.933987i	$-0.383695\pi$
$971$	22.2679	0.714612	0.357306	+	0.933987i	$0.383695\pi$
$972$	0	0
$973$	−42.0526	−1.34814
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.39230	0.140522	0.0702611	−	0.997529i	$-0.477617\pi$
$977$	4.39230	0.140522	0.0702611	+	0.997529i	$0.477617\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−52.7321	−1.68189	−0.840946	−	0.541120i	$-0.818001\pi$
$983$	−52.7321	−1.68189	−0.840946	+	0.541120i	$0.818001\pi$
$984$	0	0
$985$	13.8564	0.441502
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.3205	0.359971
$990$	0	0
$991$	55.7846	1.77206	0.886028	−	0.463631i	$-0.153454\pi$
$991$	55.7846	1.77206	0.886028	+	0.463631i	$0.153454\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15.8564	−0.502682
$996$	0	0
$997$	52.1962	1.65307	0.826534	−	0.562886i	$-0.190309\pi$
$997$	52.1962	1.65307	0.826534	+	0.562886i	$0.190309\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.a.g.1.1	✓	2
3.2	odd	2		1620.2.a.h.1.1	yes	2
4.3	odd	2		6480.2.a.bh.1.2		2
5.2	odd	4		8100.2.d.m.649.1		4
5.3	odd	4		8100.2.d.m.649.4		4
5.4	even	2		8100.2.a.t.1.2		2
9.2	odd	6		1620.2.i.m.1081.2		4
9.4	even	3		1620.2.i.n.541.2		4
9.5	odd	6		1620.2.i.m.541.2		4
9.7	even	3		1620.2.i.n.1081.2		4
12.11	even	2		6480.2.a.bp.1.2		2
15.2	even	4		8100.2.d.l.649.1		4
15.8	even	4		8100.2.d.l.649.4		4
15.14	odd	2		8100.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.2.a.g.1.1	✓	2	1.1	even	1	trivial
1620.2.a.h.1.1	yes	2	3.2	odd	2
1620.2.i.m.541.2		4	9.5	odd	6
1620.2.i.m.1081.2		4	9.2	odd	6
1620.2.i.n.541.2		4	9.4	even	3
1620.2.i.n.1081.2		4	9.7	even	3
6480.2.a.bh.1.2		2	4.3	odd	2
6480.2.a.bp.1.2		2	12.11	even	2
8100.2.a.s.1.2		2	15.14	odd	2
8100.2.a.t.1.2		2	5.4	even	2
8100.2.d.l.649.1		4	15.2	even	4
8100.2.d.l.649.4		4	15.8	even	4
8100.2.d.m.649.1		4	5.2	odd	4
8100.2.d.m.649.4		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -2.73205 q^{7} +1.73205 q^{11} +5.46410 q^{13} -4.73205 q^{17} -4.46410 q^{19} +3.46410 q^{23} +1.00000 q^{25} -7.73205 q^{29} +5.92820 q^{31} +2.73205 q^{35} -6.19615 q^{37} -11.1962 q^{41} +3.26795 q^{43} -1.26795 q^{47} +0.464102 q^{49} -7.26795 q^{53} -1.73205 q^{55} -7.73205 q^{59} -4.00000 q^{61} -5.46410 q^{65} +6.39230 q^{67} -11.1962 q^{71} -0.196152 q^{73} -4.73205 q^{77} -14.3923 q^{79} -15.1244 q^{83} +4.73205 q^{85} +5.19615 q^{89} -14.9282 q^{91} +4.46410 q^{95} +0.732051 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 2 q^{19} + 2 q^{25} - 12 q^{29} - 2 q^{31} + 2 q^{35} - 2 q^{37} - 12 q^{41} + 10 q^{43} - 6 q^{47} - 6 q^{49} - 18 q^{53} - 12 q^{59} - 8 q^{61} - 4 q^{65}+ \cdots - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^13 - 6 * q^17 - 2 * q^19 + 2 * q^25 - 12 * q^29 - 2 * q^31 + 2 * q^35 - 2 * q^37 - 12 * q^41 + 10 * q^43 - 6 * q^47 - 6 * q^49 - 18 * q^53 - 12 * q^59 - 8 * q^61 - 4 * q^65 - 8 * q^67 - 12 * q^71 + 10 * q^73 - 6 * q^77 - 8 * q^79 - 6 * q^83 + 6 * q^85 - 16 * q^91 + 2 * q^95 - 2 * q^97

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