LMFDB - Embedded newform 1620.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{5} +0.732051 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +0.732051 q^{7} -1.73205 q^{11} -1.46410 q^{13} -1.26795 q^{17} +2.46410 q^{19} -3.46410 q^{23} +1.00000 q^{25} -4.26795 q^{29} -7.92820 q^{31} -0.732051 q^{35} +4.19615 q^{37} -0.803848 q^{41} +6.73205 q^{43} -4.73205 q^{47} -6.46410 q^{49} -10.7321 q^{53} +1.73205 q^{55} -4.26795 q^{59} -4.00000 q^{61} +1.46410 q^{65} -14.3923 q^{67} -0.803848 q^{71} +10.1962 q^{73} -1.26795 q^{77} +6.39230 q^{79} +9.12436 q^{83} +1.26795 q^{85} -5.19615 q^{89} -1.07180 q^{91} -2.46410 q^{95} -2.73205 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 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}+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

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
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.73205	−0.522233	−0.261116	−	0.965307i	$-0.584091\pi$
$11$	−1.73205	−0.522233	−0.261116	+	0.965307i	$0.584091\pi$
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.26795	−0.307523	−0.153761	−	0.988108i	$-0.549139\pi$
$17$	−1.26795	−0.307523	−0.153761	+	0.988108i	$0.549139\pi$
$18$	0	0
$19$	2.46410	0.565304	0.282652	−	0.959223i	$-0.408786\pi$
$19$	2.46410	0.565304	0.282652	+	0.959223i	$0.408786\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.26795	−0.792538	−0.396269	−	0.918134i	$-0.629695\pi$
$29$	−4.26795	−0.792538	−0.396269	+	0.918134i	$0.629695\pi$
$30$	0	0
$31$	−7.92820	−1.42395	−0.711974	−	0.702206i	$-0.752198\pi$
$31$	−7.92820	−1.42395	−0.711974	+	0.702206i	$0.752198\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.732051	−0.123739
$36$	0	0
$37$	4.19615	0.689843	0.344922	−	0.938631i	$-0.387905\pi$
$37$	4.19615	0.689843	0.344922	+	0.938631i	$0.387905\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.803848	−0.125540	−0.0627700	−	0.998028i	$-0.519993\pi$
$41$	−0.803848	−0.125540	−0.0627700	+	0.998028i	$0.519993\pi$
$42$	0	0
$43$	6.73205	1.02663	0.513314	−	0.858201i	$-0.328418\pi$
$43$	6.73205	1.02663	0.513314	+	0.858201i	$0.328418\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.73205	−0.690241	−0.345120	−	0.938558i	$-0.612162\pi$
$47$	−4.73205	−0.690241	−0.345120	+	0.938558i	$0.612162\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.7321	−1.47416	−0.737080	−	0.675805i	$-0.763796\pi$
$53$	−10.7321	−1.47416	−0.737080	+	0.675805i	$0.763796\pi$
$54$	0	0
$55$	1.73205	0.233550
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.26795	−0.555640	−0.277820	−	0.960633i	$-0.589612\pi$
$59$	−4.26795	−0.555640	−0.277820	+	0.960633i	$0.589612\pi$
$60$	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
$65$	1.46410	0.181599
$66$	0	0
$67$	−14.3923	−1.75830	−0.879150	−	0.476545i	$-0.841889\pi$
$67$	−14.3923	−1.75830	−0.879150	+	0.476545i	$0.841889\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.803848	−0.0953992	−0.0476996	−	0.998862i	$-0.515189\pi$
$71$	−0.803848	−0.0953992	−0.0476996	+	0.998862i	$0.515189\pi$
$72$	0	0
$73$	10.1962	1.19337	0.596685	−	0.802476i	$-0.296484\pi$
$73$	10.1962	1.19337	0.596685	+	0.802476i	$0.296484\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.26795	−0.144496
$78$	0	0
$79$	6.39230	0.719190	0.359595	−	0.933108i	$-0.382915\pi$
$79$	6.39230	0.719190	0.359595	+	0.933108i	$0.382915\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.12436	1.00153	0.500764	−	0.865584i	$-0.333052\pi$
$83$	9.12436	1.00153	0.500764	+	0.865584i	$0.333052\pi$
$84$	0	0
$85$	1.26795	0.137528
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.19615	−0.550791	−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615	−0.550791	−0.275396	+	0.961331i	$0.588809\pi$
$90$	0	0
$91$	−1.07180	−0.112355
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.46410	−0.252811
$96$	0	0
$97$	−2.73205	−0.277398	−0.138699	−	0.990335i	$-0.544292\pi$
$97$	−2.73205	−0.277398	−0.138699	+	0.990335i	$0.544292\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.1244	−1.80344	−0.901720	−	0.432320i	$-0.857695\pi$
$101$	−18.1244	−1.80344	−0.901720	+	0.432320i	$0.857695\pi$
$102$	0	0
$103$	−2.39230	−0.235721	−0.117860	−	0.993030i	$-0.537604\pi$
$103$	−2.39230	−0.235721	−0.117860	+	0.993030i	$0.537604\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.46410	−0.334887	−0.167444	−	0.985882i	$-0.553551\pi$
$107$	−3.46410	−0.334887	−0.167444	+	0.985882i	$0.553551\pi$
$108$	0	0
$109$	5.92820	0.567819	0.283909	−	0.958851i	$-0.408368\pi$
$109$	5.92820	0.567819	0.283909	+	0.958851i	$0.408368\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.6603	−1.66134	−0.830668	−	0.556767i	$-0.812041\pi$
$113$	−17.6603	−1.66134	−0.830668	+	0.556767i	$0.812041\pi$
$114$	0	0
$115$	3.46410	0.323029
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.928203	−0.0850883
$120$	0	0
$121$	−8.00000	−0.727273
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.19615	−0.549820	−0.274910	−	0.961470i	$-0.588648\pi$
$127$	−6.19615	−0.549820	−0.274910	+	0.961470i	$0.588648\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.7321	1.19977	0.599887	−	0.800084i	$-0.295212\pi$
$131$	13.7321	1.19977	0.599887	+	0.800084i	$0.295212\pi$
$132$	0	0
$133$	1.80385	0.156413
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.3923	−1.40049	−0.700245	−	0.713903i	$-0.746926\pi$
$137$	−16.3923	−1.40049	−0.700245	+	0.713903i	$0.746926\pi$
$138$	0	0
$139$	−5.39230	−0.457369	−0.228685	−	0.973501i	$-0.573442\pi$
$139$	−5.39230	−0.457369	−0.228685	+	0.973501i	$0.573442\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.53590	0.212062
$144$	0	0
$145$	4.26795	0.354434
$146$	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
$151$	3.39230	0.276062	0.138031	−	0.990428i	$-0.455923\pi$
$151$	3.39230	0.276062	0.138031	+	0.990428i	$0.455923\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.92820	0.636809
$156$	0	0
$157$	6.73205	0.537276	0.268638	−	0.963241i	$-0.413426\pi$
$157$	6.73205	0.537276	0.268638	+	0.963241i	$0.413426\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.53590	−0.199857
$162$	0	0
$163$	15.2679	1.19588	0.597939	−	0.801542i	$-0.295986\pi$
$163$	15.2679	1.19588	0.597939	+	0.801542i	$0.295986\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.12436	0.241770	0.120885	−	0.992667i	$-0.461427\pi$
$167$	3.12436	0.241770	0.120885	+	0.992667i	$0.461427\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.2487	1.84360	0.921798	−	0.387671i	$-0.126720\pi$
$173$	24.2487	1.84360	0.921798	+	0.387671i	$0.126720\pi$
$174$	0	0
$175$	0.732051	0.0553378
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.1244	0.906217	0.453108	−	0.891455i	$-0.350315\pi$
$179$	12.1244	0.906217	0.453108	+	0.891455i	$0.350315\pi$
$180$	0	0
$181$	−9.53590	−0.708798	−0.354399	−	0.935094i	$-0.615314\pi$
$181$	−9.53590	−0.708798	−0.354399	+	0.935094i	$0.615314\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.19615	−0.308507
$186$	0	0
$187$	2.19615	0.160599
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.0526	−1.37859	−0.689297	−	0.724479i	$-0.742081\pi$
$191$	−19.0526	−1.37859	−0.689297	+	0.724479i	$0.742081\pi$
$192$	0	0
$193$	20.5885	1.48199	0.740995	−	0.671511i	$-0.234354\pi$
$193$	20.5885	1.48199	0.740995	+	0.671511i	$0.234354\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.8564	0.987228	0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564	0.987228	0.493614	+	0.869681i	$0.335676\pi$
$198$	0	0
$199$	−11.8564	−0.840478	−0.420239	−	0.907413i	$-0.638054\pi$
$199$	−11.8564	−0.840478	−0.420239	+	0.907413i	$0.638054\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.12436	−0.219287
$204$	0	0
$205$	0.803848	0.0561432
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.26795	−0.295220
$210$	0	0
$211$	−6.07180	−0.418000	−0.209000	−	0.977916i	$-0.567021\pi$
$211$	−6.07180	−0.418000	−0.209000	+	0.977916i	$0.567021\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.73205	−0.459122
$216$	0	0
$217$	−5.80385	−0.393991
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.85641	0.124875
$222$	0	0
$223$	21.8564	1.46361	0.731807	−	0.681512i	$-0.238677\pi$
$223$	21.8564	1.46361	0.731807	+	0.681512i	$0.238677\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.8038	1.04894	0.524469	−	0.851429i	$-0.324264\pi$
$227$	15.8038	1.04894	0.524469	+	0.851429i	$0.324264\pi$
$228$	0	0
$229$	9.85641	0.651330	0.325665	−	0.945485i	$-0.394412\pi$
$229$	9.85641	0.651330	0.325665	+	0.945485i	$0.394412\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.58846	−0.431624	−0.215812	−	0.976435i	$-0.569240\pi$
$233$	−6.58846	−0.431624	−0.215812	+	0.976435i	$0.569240\pi$
$234$	0	0
$235$	4.73205	0.308685
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.4641	1.00029	0.500145	−	0.865942i	$-0.333280\pi$
$239$	15.4641	1.00029	0.500145	+	0.865942i	$0.333280\pi$
$240$	0	0
$241$	−24.3205	−1.56662	−0.783311	−	0.621630i	$-0.786471\pi$
$241$	−24.3205	−1.56662	−0.783311	+	0.621630i	$0.786471\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.46410	0.412976
$246$	0	0
$247$	−3.60770	−0.229552
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.4641	1.73352	0.866759	−	0.498727i	$-0.166199\pi$
$251$	27.4641	1.73352	0.866759	+	0.498727i	$0.166199\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.53590	0.532455	0.266227	−	0.963910i	$-0.414223\pi$
$257$	8.53590	0.532455	0.266227	+	0.963910i	$0.414223\pi$
$258$	0	0
$259$	3.07180	0.190872
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.2487	1.49524	0.747620	−	0.664127i	$-0.231197\pi$
$263$	24.2487	1.49524	0.747620	+	0.664127i	$0.231197\pi$
$264$	0	0
$265$	10.7321	0.659265
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.7321	−1.20308	−0.601542	−	0.798841i	$-0.705447\pi$
$269$	−19.7321	−1.20308	−0.601542	+	0.798841i	$0.705447\pi$
$270$	0	0
$271$	−24.7846	−1.50556	−0.752779	−	0.658273i	$-0.771287\pi$
$271$	−24.7846	−1.50556	−0.752779	+	0.658273i	$0.771287\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.73205	−0.104447
$276$	0	0
$277$	−19.1244	−1.14907	−0.574536	−	0.818480i	$-0.694817\pi$
$277$	−19.1244	−1.14907	−0.574536	+	0.818480i	$0.694817\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.32051	−0.317395	−0.158697	−	0.987327i	$-0.550729\pi$
$281$	−5.32051	−0.317395	−0.158697	+	0.987327i	$0.550729\pi$
$282$	0	0
$283$	23.4641	1.39480	0.697398	−	0.716684i	$-0.254341\pi$
$283$	23.4641	1.39480	0.697398	+	0.716684i	$0.254341\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.588457	−0.0347355
$288$	0	0
$289$	−15.3923	−0.905430
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.58846	−0.384902	−0.192451	−	0.981307i	$-0.561644\pi$
$293$	−6.58846	−0.384902	−0.192451	+	0.981307i	$0.561644\pi$
$294$	0	0
$295$	4.26795	0.248490
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.07180	0.293310
$300$	0	0
$301$	4.92820	0.284057
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	−12.1962	−0.696071	−0.348036	−	0.937481i	$-0.613151\pi$
$307$	−12.1962	−0.696071	−0.348036	+	0.937481i	$0.613151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.5167	1.61703	0.808516	−	0.588475i	$-0.200271\pi$
$311$	28.5167	1.61703	0.808516	+	0.588475i	$0.200271\pi$
$312$	0	0
$313$	1.07180	0.0605815	0.0302908	−	0.999541i	$-0.490357\pi$
$313$	1.07180	0.0605815	0.0302908	+	0.999541i	$0.490357\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.1244	−1.18646	−0.593231	−	0.805032i	$-0.702148\pi$
$317$	−21.1244	−1.18646	−0.593231	+	0.805032i	$0.702148\pi$
$318$	0	0
$319$	7.39230	0.413890
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.12436	−0.173844
$324$	0	0
$325$	−1.46410	−0.0812137
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.46410	−0.190982
$330$	0	0
$331$	−8.60770	−0.473122	−0.236561	−	0.971617i	$-0.576020\pi$
$331$	−8.60770	−0.473122	−0.236561	+	0.971617i	$0.576020\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.3923	0.786336
$336$	0	0
$337$	14.2487	0.776177	0.388088	−	0.921622i	$-0.373136\pi$
$337$	14.2487	0.776177	0.388088	+	0.921622i	$0.373136\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13.7321	0.743632
$342$	0	0
$343$	−9.85641	−0.532196
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.1962	1.72838	0.864190	−	0.503166i	$-0.167831\pi$
$347$	32.1962	1.72838	0.864190	+	0.503166i	$0.167831\pi$
$348$	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
$353$	22.9808	1.22314	0.611571	−	0.791189i	$-0.290538\pi$
$353$	22.9808	1.22314	0.611571	+	0.791189i	$0.290538\pi$
$354$	0	0
$355$	0.803848	0.0426638
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.6603	1.40707	0.703537	−	0.710658i	$-0.251603\pi$
$359$	26.6603	1.40707	0.703537	+	0.710658i	$0.251603\pi$
$360$	0	0
$361$	−12.9282	−0.680432
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.1962	−0.533691
$366$	0	0
$367$	−5.60770	−0.292719	−0.146360	−	0.989231i	$-0.546756\pi$
$367$	−5.60770	−0.292719	−0.146360	+	0.989231i	$0.546756\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.85641	−0.407884
$372$	0	0
$373$	24.0526	1.24539	0.622697	−	0.782463i	$-0.286037\pi$
$373$	24.0526	1.24539	0.622697	+	0.782463i	$0.286037\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.24871	0.321825
$378$	0	0
$379$	11.4641	0.588871	0.294436	−	0.955671i	$-0.404868\pi$
$379$	11.4641	0.588871	0.294436	+	0.955671i	$0.404868\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−30.2487	−1.54564	−0.772818	−	0.634627i	$-0.781154\pi$
$383$	−30.2487	−1.54564	−0.772818	+	0.634627i	$0.781154\pi$
$384$	0	0
$385$	1.26795	0.0646207
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.5359	1.04121	0.520606	−	0.853797i	$-0.325706\pi$
$389$	20.5359	1.04121	0.520606	+	0.853797i	$0.325706\pi$
$390$	0	0
$391$	4.39230	0.222128
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.39230	−0.321632
$396$	0	0
$397$	2.67949	0.134480	0.0672399	−	0.997737i	$-0.478581\pi$
$397$	2.67949	0.134480	0.0672399	+	0.997737i	$0.478581\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.7846	1.33756	0.668780	−	0.743461i	$-0.266817\pi$
$401$	26.7846	1.33756	0.668780	+	0.743461i	$0.266817\pi$
$402$	0	0
$403$	11.6077	0.578220
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.26795	−0.360259
$408$	0	0
$409$	9.85641	0.487368	0.243684	−	0.969855i	$-0.421644\pi$
$409$	9.85641	0.487368	0.243684	+	0.969855i	$0.421644\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.12436	−0.153739
$414$	0	0
$415$	−9.12436	−0.447897
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.5359	0.710125	0.355063	−	0.934843i	$-0.384460\pi$
$419$	14.5359	0.710125	0.355063	+	0.934843i	$0.384460\pi$
$420$	0	0
$421$	−27.7846	−1.35414	−0.677070	−	0.735919i	$-0.736750\pi$
$421$	−27.7846	−1.35414	−0.677070	+	0.735919i	$0.736750\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.26795	−0.0615046
$426$	0	0
$427$	−2.92820	−0.141706
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.41154	−0.116160	−0.0580800	−	0.998312i	$-0.518498\pi$
$431$	−2.41154	−0.116160	−0.0580800	+	0.998312i	$0.518498\pi$
$432$	0	0
$433$	4.53590	0.217981	0.108991	−	0.994043i	$-0.465238\pi$
$433$	4.53590	0.217981	0.108991	+	0.994043i	$0.465238\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.53590	−0.408327
$438$	0	0
$439$	27.3923	1.30736	0.653682	−	0.756770i	$-0.273224\pi$
$439$	27.3923	1.30736	0.653682	+	0.756770i	$0.273224\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	40.7321	1.93524	0.967619	−	0.252415i	$-0.0812248\pi$
$443$	40.7321	1.93524	0.967619	+	0.252415i	$0.0812248\pi$
$444$	0	0
$445$	5.19615	0.246321
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−0.124356	−0.00586871	−0.00293435	−	0.999996i	$-0.500934\pi$
$449$	−0.124356	−0.00586871	−0.00293435	+	0.999996i	$0.500934\pi$
$450$	0	0
$451$	1.39230	0.0655611
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.07180	0.0502466
$456$	0	0
$457$	11.8038	0.552161	0.276080	−	0.961135i	$-0.410964\pi$
$457$	11.8038	0.552161	0.276080	+	0.961135i	$0.410964\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.5885	−1.00547	−0.502737	−	0.864439i	$-0.667674\pi$
$461$	−21.5885	−1.00547	−0.502737	+	0.864439i	$0.667674\pi$
$462$	0	0
$463$	18.3923	0.854763	0.427381	−	0.904071i	$-0.359436\pi$
$463$	18.3923	0.854763	0.427381	+	0.904071i	$0.359436\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.8038	0.731315	0.365657	−	0.930750i	$-0.380844\pi$
$467$	15.8038	0.731315	0.365657	+	0.930750i	$0.380844\pi$
$468$	0	0
$469$	−10.5359	−0.486503
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.6603	−0.536139
$474$	0	0
$475$	2.46410	0.113061
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.6603	−0.943991	−0.471996	−	0.881601i	$-0.656466\pi$
$479$	−20.6603	−0.943991	−0.471996	+	0.881601i	$0.656466\pi$
$480$	0	0
$481$	−6.14359	−0.280124
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.73205	0.124056
$486$	0	0
$487$	−37.4641	−1.69766	−0.848830	−	0.528666i	$-0.822693\pi$
$487$	−37.4641	−1.69766	−0.848830	+	0.528666i	$0.822693\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.73205	−0.348943	−0.174471	−	0.984662i	$-0.555822\pi$
$491$	−7.73205	−0.348943	−0.174471	+	0.984662i	$0.555822\pi$
$492$	0	0
$493$	5.41154	0.243724
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.588457	−0.0263959
$498$	0	0
$499$	6.60770	0.295801	0.147901	−	0.989002i	$-0.452748\pi$
$499$	6.60770	0.295801	0.147901	+	0.989002i	$0.452748\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.679492	−0.0302970	−0.0151485	−	0.999885i	$-0.504822\pi$
$503$	−0.679492	−0.0302970	−0.0151485	+	0.999885i	$0.504822\pi$
$504$	0	0
$505$	18.1244	0.806523
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.7846	0.655316	0.327658	−	0.944796i	$-0.393741\pi$
$509$	14.7846	0.655316	0.327658	+	0.944796i	$0.393741\pi$
$510$	0	0
$511$	7.46410	0.330192
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.39230	0.105418
$516$	0	0
$517$	8.19615	0.360466
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.3923	0.455295	0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923	0.455295	0.227648	+	0.973744i	$0.426897\pi$
$522$	0	0
$523$	24.3923	1.06660	0.533301	−	0.845926i	$-0.320952\pi$
$523$	24.3923	1.06660	0.533301	+	0.845926i	$0.320952\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.0526	0.437896
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.17691	0.0509778
$534$	0	0
$535$	3.46410	0.149766
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11.1962	0.482252
$540$	0	0
$541$	−4.46410	−0.191927	−0.0959634	−	0.995385i	$-0.530593\pi$
$541$	−4.46410	−0.191927	−0.0959634	+	0.995385i	$0.530593\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.92820	−0.253936
$546$	0	0
$547$	−36.7846	−1.57280	−0.786398	−	0.617720i	$-0.788057\pi$
$547$	−36.7846	−1.57280	−0.786398	+	0.617720i	$0.788057\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.5167	−0.448025
$552$	0	0
$553$	4.67949	0.198992
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.3923	0.694564	0.347282	−	0.937761i	$-0.387105\pi$
$557$	16.3923	0.694564	0.347282	+	0.937761i	$0.387105\pi$
$558$	0	0
$559$	−9.85641	−0.416882
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.2679	0.812047	0.406024	−	0.913863i	$-0.366915\pi$
$563$	19.2679	0.812047	0.406024	+	0.913863i	$0.366915\pi$
$564$	0	0
$565$	17.6603	0.742972
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.94744	−0.207408	−0.103704	−	0.994608i	$-0.533069\pi$
$569$	−4.94744	−0.207408	−0.103704	+	0.994608i	$0.533069\pi$
$570$	0	0
$571$	−26.8564	−1.12391	−0.561953	−	0.827169i	$-0.689950\pi$
$571$	−26.8564	−1.12391	−0.561953	+	0.827169i	$0.689950\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.46410	−0.144463
$576$	0	0
$577$	22.1962	0.924038	0.462019	−	0.886870i	$-0.347125\pi$
$577$	22.1962	0.924038	0.462019	+	0.886870i	$0.347125\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.67949	0.277112
$582$	0	0
$583$	18.5885	0.769855
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.1962	0.585938	0.292969	−	0.956122i	$-0.405357\pi$
$587$	14.1962	0.585938	0.292969	+	0.956122i	$0.405357\pi$
$588$	0	0
$589$	−19.5359	−0.804963
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−36.9282	−1.51646	−0.758230	−	0.651987i	$-0.773935\pi$
$593$	−36.9282	−1.51646	−0.758230	+	0.651987i	$0.773935\pi$
$594$	0	0
$595$	0.928203	0.0380526
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.5885	−1.86269	−0.931347	−	0.364133i	$-0.881365\pi$
$599$	−45.5885	−1.86269	−0.931347	+	0.364133i	$0.881365\pi$
$600$	0	0
$601$	10.3205	0.420982	0.210491	−	0.977596i	$-0.432494\pi$
$601$	10.3205	0.420982	0.210491	+	0.977596i	$0.432494\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.00000	0.325246
$606$	0	0
$607$	−28.5885	−1.16037	−0.580185	−	0.814485i	$-0.697020\pi$
$607$	−28.5885	−1.16037	−0.580185	+	0.814485i	$0.697020\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.92820	0.280285
$612$	0	0
$613$	3.60770	0.145713	0.0728567	−	0.997342i	$-0.476788\pi$
$613$	3.60770	0.145713	0.0728567	+	0.997342i	$0.476788\pi$
$614$	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
$623$	−3.80385	−0.152398
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.32051	−0.212143
$630$	0	0
$631$	−13.9282	−0.554473	−0.277237	−	0.960802i	$-0.589419\pi$
$631$	−13.9282	−0.554473	−0.277237	+	0.960802i	$0.589419\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.19615	0.245887
$636$	0	0
$637$	9.46410	0.374981
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23.1962	−0.916193	−0.458096	−	0.888902i	$-0.651469\pi$
$641$	−23.1962	−0.916193	−0.458096	+	0.888902i	$0.651469\pi$
$642$	0	0
$643$	−16.5885	−0.654185	−0.327092	−	0.944992i	$-0.606069\pi$
$643$	−16.5885	−0.654185	−0.327092	+	0.944992i	$0.606069\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−48.2487	−1.89685	−0.948426	−	0.316998i	$-0.897325\pi$
$647$	−48.2487	−1.89685	−0.948426	+	0.316998i	$0.897325\pi$
$648$	0	0
$649$	7.39230	0.290173
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.46410	−0.370359	−0.185179	−	0.982705i	$-0.559287\pi$
$653$	−9.46410	−0.370359	−0.185179	+	0.982705i	$0.559287\pi$
$654$	0	0
$655$	−13.7321	−0.536556
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.46410	−0.368669	−0.184335	−	0.982864i	$-0.559013\pi$
$659$	−9.46410	−0.368669	−0.184335	+	0.982864i	$0.559013\pi$
$660$	0	0
$661$	−5.39230	−0.209736	−0.104868	−	0.994486i	$-0.533442\pi$
$661$	−5.39230	−0.209736	−0.104868	+	0.994486i	$0.533442\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.80385	−0.0699502
$666$	0	0
$667$	14.7846	0.572462
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.92820	0.267460
$672$	0	0
$673$	−17.6077	−0.678727	−0.339363	−	0.940655i	$-0.610212\pi$
$673$	−17.6077	−0.678727	−0.339363	+	0.940655i	$0.610212\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.6410	−1.10076	−0.550382	−	0.834913i	$-0.685518\pi$
$677$	−28.6410	−1.10076	−0.550382	+	0.834913i	$0.685518\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.5359	0.556201	0.278100	−	0.960552i	$-0.410295\pi$
$683$	14.5359	0.556201	0.278100	+	0.960552i	$0.410295\pi$
$684$	0	0
$685$	16.3923	0.626318
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.7128	0.598610
$690$	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
$695$	5.39230	0.204542
$696$	0	0
$697$	1.01924	0.0386064
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−42.1244	−1.59101	−0.795507	−	0.605944i	$-0.792796\pi$
$701$	−42.1244	−1.59101	−0.795507	+	0.605944i	$0.792796\pi$
$702$	0	0
$703$	10.3397	0.389971
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13.2679	−0.498993
$708$	0	0
$709$	17.4641	0.655878	0.327939	−	0.944699i	$-0.393646\pi$
$709$	17.4641	0.655878	0.327939	+	0.944699i	$0.393646\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	27.4641	1.02854
$714$	0	0
$715$	−2.53590	−0.0948372
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.80385	−0.253741	−0.126870	−	0.991919i	$-0.540493\pi$
$719$	−6.80385	−0.253741	−0.126870	+	0.991919i	$0.540493\pi$
$720$	0	0
$721$	−1.75129	−0.0652214
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.26795	−0.158508
$726$	0	0
$727$	33.1769	1.23046	0.615232	−	0.788346i	$-0.289062\pi$
$727$	33.1769	1.23046	0.615232	+	0.788346i	$0.289062\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.53590	−0.315712
$732$	0	0
$733$	−39.5692	−1.46152	−0.730761	−	0.682633i	$-0.760835\pi$
$733$	−39.5692	−1.46152	−0.730761	+	0.682633i	$0.760835\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.9282	0.918242
$738$	0	0
$739$	36.1769	1.33079	0.665395	−	0.746492i	$-0.268263\pi$
$739$	36.1769	1.33079	0.665395	+	0.746492i	$0.268263\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.5885	−1.34230	−0.671150	−	0.741321i	$-0.734199\pi$
$743$	−36.5885	−1.34230	−0.671150	+	0.741321i	$0.734199\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.53590	−0.0926597
$750$	0	0
$751$	−0.784610	−0.0286308	−0.0143154	−	0.999898i	$-0.504557\pi$
$751$	−0.784610	−0.0286308	−0.0143154	+	0.999898i	$0.504557\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.39230	−0.123459
$756$	0	0
$757$	0.392305	0.0142586	0.00712928	−	0.999975i	$-0.497731\pi$
$757$	0.392305	0.0142586	0.00712928	+	0.999975i	$0.497731\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.87564	−0.212992	−0.106496	−	0.994313i	$-0.533963\pi$
$761$	−5.87564	−0.212992	−0.106496	+	0.994313i	$0.533963\pi$
$762$	0	0
$763$	4.33975	0.157109
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.24871	0.225628
$768$	0	0
$769$	−13.2487	−0.477761	−0.238880	−	0.971049i	$-0.576780\pi$
$769$	−13.2487	−0.477761	−0.238880	+	0.971049i	$0.576780\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.339746	0.0122198	0.00610991	−	0.999981i	$-0.498055\pi$
$773$	0.339746	0.0122198	0.00610991	+	0.999981i	$0.498055\pi$
$774$	0	0
$775$	−7.92820	−0.284789
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.98076	−0.0709682
$780$	0	0
$781$	1.39230	0.0498206
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.73205	−0.240277
$786$	0	0
$787$	−25.8038	−0.919808	−0.459904	−	0.887969i	$-0.652116\pi$
$787$	−25.8038	−0.919808	−0.459904	+	0.887969i	$0.652116\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.9282	−0.459674
$792$	0	0
$793$	5.85641	0.207967
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.3205	−0.400993	−0.200496	−	0.979694i	$-0.564256\pi$
$797$	−11.3205	−0.400993	−0.200496	+	0.979694i	$0.564256\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17.6603	−0.623217
$804$	0	0
$805$	2.53590	0.0893787
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−39.5885	−1.39186	−0.695928	−	0.718112i	$-0.745007\pi$
$809$	−39.5885	−1.39186	−0.695928	+	0.718112i	$0.745007\pi$
$810$	0	0
$811$	23.2487	0.816373	0.408186	−	0.912899i	$-0.366161\pi$
$811$	23.2487	0.816373	0.408186	+	0.912899i	$0.366161\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15.2679	−0.534813
$816$	0	0
$817$	16.5885	0.580357
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.6603	0.721048	0.360524	−	0.932750i	$-0.382598\pi$
$821$	20.6603	0.721048	0.360524	+	0.932750i	$0.382598\pi$
$822$	0	0
$823$	−3.07180	−0.107076	−0.0535381	−	0.998566i	$-0.517050\pi$
$823$	−3.07180	−0.107076	−0.0535381	+	0.998566i	$0.517050\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38.5359	−1.34002	−0.670012	−	0.742350i	$-0.733711\pi$
$827$	−38.5359	−1.34002	−0.670012	+	0.742350i	$0.733711\pi$
$828$	0	0
$829$	−10.2154	−0.354795	−0.177398	−	0.984139i	$-0.556768\pi$
$829$	−10.2154	−0.354795	−0.177398	+	0.984139i	$0.556768\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.19615	0.283980
$834$	0	0
$835$	−3.12436	−0.108123
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.5885	−1.36675	−0.683373	−	0.730070i	$-0.739488\pi$
$839$	−39.5885	−1.36675	−0.683373	+	0.730070i	$0.739488\pi$
$840$	0	0
$841$	−10.7846	−0.371883
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.8564	0.373472
$846$	0	0
$847$	−5.85641	−0.201229
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.5359	−0.498284
$852$	0	0
$853$	10.1962	0.349110	0.174555	−	0.984647i	$-0.444151\pi$
$853$	10.1962	0.349110	0.174555	+	0.984647i	$0.444151\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.9282	−1.46640	−0.733200	−	0.680013i	$-0.761974\pi$
$857$	−42.9282	−1.46640	−0.733200	+	0.680013i	$0.761974\pi$
$858$	0	0
$859$	−33.7846	−1.15272	−0.576358	−	0.817197i	$-0.695527\pi$
$859$	−33.7846	−1.15272	−0.576358	+	0.817197i	$0.695527\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.48334	−0.152615	−0.0763073	−	0.997084i	$-0.524313\pi$
$863$	−4.48334	−0.152615	−0.0763073	+	0.997084i	$0.524313\pi$
$864$	0	0
$865$	−24.2487	−0.824481
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11.0718	−0.375585
$870$	0	0
$871$	21.0718	0.713991
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.732051	−0.0247478
$876$	0	0
$877$	−34.2487	−1.15650	−0.578248	−	0.815861i	$-0.696264\pi$
$877$	−34.2487	−1.15650	−0.578248	+	0.815861i	$0.696264\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	17.1962	0.579353	0.289677	−	0.957125i	$-0.406452\pi$
$881$	17.1962	0.579353	0.289677	+	0.957125i	$0.406452\pi$
$882$	0	0
$883$	32.8372	1.10506	0.552529	−	0.833493i	$-0.313663\pi$
$883$	32.8372	1.10506	0.552529	+	0.833493i	$0.313663\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.7321	0.561807	0.280904	−	0.959736i	$-0.409366\pi$
$887$	16.7321	0.561807	0.280904	+	0.959736i	$0.409366\pi$
$888$	0	0
$889$	−4.53590	−0.152129
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.6603	−0.390196
$894$	0	0
$895$	−12.1244	−0.405273
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.8372	1.12853
$900$	0	0
$901$	13.6077	0.453338
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.53590	0.316984
$906$	0	0
$907$	−42.7846	−1.42064	−0.710320	−	0.703879i	$-0.751450\pi$
$907$	−42.7846	−1.42064	−0.710320	+	0.703879i	$0.751450\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.1962	−0.569734	−0.284867	−	0.958567i	$-0.591949\pi$
$911$	−17.1962	−0.569734	−0.284867	+	0.958567i	$0.591949\pi$
$912$	0	0
$913$	−15.8038	−0.523031
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.0526	0.331965
$918$	0	0
$919$	30.6077	1.00965	0.504827	−	0.863220i	$-0.331556\pi$
$919$	30.6077	1.00965	0.504827	+	0.863220i	$0.331556\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.17691	0.0387386
$924$	0	0
$925$	4.19615	0.137969
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.5167	1.52616	0.763081	−	0.646303i	$-0.223686\pi$
$929$	46.5167	1.52616	0.763081	+	0.646303i	$0.223686\pi$
$930$	0	0
$931$	−15.9282	−0.522026
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.19615	−0.0718219
$936$	0	0
$937$	32.9282	1.07572	0.537859	−	0.843035i	$-0.319233\pi$
$937$	32.9282	1.07572	0.537859	+	0.843035i	$0.319233\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−15.4641	−0.504115	−0.252058	−	0.967712i	$-0.581107\pi$
$941$	−15.4641	−0.504115	−0.252058	+	0.967712i	$0.581107\pi$
$942$	0	0
$943$	2.78461	0.0906794
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.6410	0.540760	0.270380	−	0.962754i	$-0.412851\pi$
$947$	16.6410	0.540760	0.270380	+	0.962754i	$0.412851\pi$
$948$	0	0
$949$	−14.9282	−0.484590
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−9.46410	−0.306572	−0.153286	−	0.988182i	$-0.548986\pi$
$953$	−9.46410	−0.306572	−0.153286	+	0.988182i	$0.548986\pi$
$954$	0	0
$955$	19.0526	0.616526
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	31.8564	1.02763
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20.5885	−0.662766
$966$	0	0
$967$	38.5885	1.24092	0.620461	−	0.784238i	$-0.286946\pi$
$967$	38.5885	1.24092	0.620461	+	0.784238i	$0.286946\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.7321	0.825781	0.412890	−	0.910781i	$-0.364519\pi$
$971$	25.7321	0.825781	0.412890	+	0.910781i	$0.364519\pi$
$972$	0	0
$973$	−3.94744	−0.126549
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.3923	−0.524436	−0.262218	−	0.965009i	$-0.584454\pi$
$977$	−16.3923	−0.524436	−0.262218	+	0.965009i	$0.584454\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−49.2679	−1.57140	−0.785702	−	0.618605i	$-0.787698\pi$
$983$	−49.2679	−1.57140	−0.785702	+	0.618605i	$0.787698\pi$
$984$	0	0
$985$	−13.8564	−0.441502
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−23.3205	−0.741549
$990$	0	0
$991$	14.2154	0.451567	0.225783	−	0.974178i	$-0.427506\pi$
$991$	14.2154	0.451567	0.225783	+	0.974178i	$0.427506\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.8564	0.375873
$996$	0	0
$997$	41.8038	1.32394	0.661971	−	0.749530i	$-0.269720\pi$
$997$	41.8038	1.32394	0.661971	+	0.749530i	$0.269720\pi$
$998$	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.1.2	✓	2
3.2	odd	2		1620.2.a.h.1.2	yes	2
4.3	odd	2		6480.2.a.bh.1.1		2
5.2	odd	4		8100.2.d.m.649.3		4
5.3	odd	4		8100.2.d.m.649.2		4
5.4	even	2		8100.2.a.t.1.1		2
9.2	odd	6		1620.2.i.m.1081.1		4
9.4	even	3		1620.2.i.n.541.1		4
9.5	odd	6		1620.2.i.m.541.1		4
9.7	even	3		1620.2.i.n.1081.1		4
12.11	even	2		6480.2.a.bp.1.1		2
15.2	even	4		8100.2.d.l.649.3		4
15.8	even	4		8100.2.d.l.649.2		4
15.14	odd	2		8100.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.2.a.g.1.2	✓	2	1.1	even	1	trivial
1620.2.a.h.1.2	yes	2	3.2	odd	2
1620.2.i.m.541.1		4	9.5	odd	6
1620.2.i.m.1081.1		4	9.2	odd	6
1620.2.i.n.541.1		4	9.4	even	3
1620.2.i.n.1081.1		4	9.7	even	3
6480.2.a.bh.1.1		2	4.3	odd	2
6480.2.a.bp.1.1		2	12.11	even	2
8100.2.a.s.1.1		2	15.14	odd	2
8100.2.a.t.1.1		2	5.4	even	2
8100.2.d.l.649.2		4	15.8	even	4
8100.2.d.l.649.3		4	15.2	even	4
8100.2.d.m.649.2		4	5.3	odd	4
8100.2.d.m.649.3		4	5.2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +0.732051 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +0.732051 q^{7} -1.73205 q^{11} -1.46410 q^{13} -1.26795 q^{17} +2.46410 q^{19} -3.46410 q^{23} +1.00000 q^{25} -4.26795 q^{29} -7.92820 q^{31} -0.732051 q^{35} +4.19615 q^{37} -0.803848 q^{41} +6.73205 q^{43} -4.73205 q^{47} -6.46410 q^{49} -10.7321 q^{53} +1.73205 q^{55} -4.26795 q^{59} -4.00000 q^{61} +1.46410 q^{65} -14.3923 q^{67} -0.803848 q^{71} +10.1962 q^{73} -1.26795 q^{77} +6.39230 q^{79} +9.12436 q^{83} +1.26795 q^{85} -5.19615 q^{89} -1.07180 q^{91} -2.46410 q^{95} -2.73205 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 2 q^{19} + 2 q^{25} - 12 q^{29} - 2 q^{31} + 2 q^{35} - 2 q^{37} - 12 q^{41} + 10 q^{43} - 6 q^{47} - 6 q^{49} - 18 q^{53} - 12 q^{59} - 8 q^{61} - 4 q^{65} - 8 q^{67} - 12 q^{71} + 10 q^{73} - 6 q^{77} - 8 q^{79} - 6 q^{83} + 6 q^{85} - 16 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more