LMFDB - Embedded newform 648.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(1,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, 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("648.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.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:	$5.17430605098$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
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	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.267949 q^{5} -3.46410 q^{7} +O(q^{10})$	$q+0.267949 q^{5} -3.46410 q^{7} +2.00000 q^{11} +4.46410 q^{13} +5.73205 q^{17} -0.535898 q^{19} +8.92820 q^{23} -4.92820 q^{25} +7.73205 q^{29} +2.92820 q^{31} -0.928203 q^{35} +6.46410 q^{37} -6.92820 q^{41} -11.4641 q^{43} -6.92820 q^{47} +5.00000 q^{49} +2.92820 q^{53} +0.535898 q^{55} +8.00000 q^{59} +3.53590 q^{61} +1.19615 q^{65} -7.46410 q^{67} -2.00000 q^{71} +1.00000 q^{73} -6.92820 q^{77} +7.46410 q^{79} +10.9282 q^{83} +1.53590 q^{85} +5.19615 q^{89} -15.4641 q^{91} -0.143594 q^{95} +15.8564 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5}+O(q^{10}) $ 2 * q + 4 * q^5	$ 2 q + 4 q^{5} + 4 q^{11} + 2 q^{13} + 8 q^{17} - 8 q^{19} + 4 q^{23} + 4 q^{25} + 12 q^{29} - 8 q^{31} + 12 q^{35} + 6 q^{37} - 16 q^{43} + 10 q^{49} - 8 q^{53} + 8 q^{55} + 16 q^{59} + 14 q^{61} - 8 q^{65} - 8 q^{67} - 4 q^{71} + 2 q^{73} + 8 q^{79} + 8 q^{83} + 10 q^{85} - 24 q^{91} - 28 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 + 4 * q^11 + 2 * q^13 + 8 * q^17 - 8 * q^19 + 4 * q^23 + 4 * q^25 + 12 * q^29 - 8 * q^31 + 12 * q^35 + 6 * q^37 - 16 * q^43 + 10 * q^49 - 8 * q^53 + 8 * q^55 + 16 * q^59 + 14 * q^61 - 8 * q^65 - 8 * q^67 - 4 * q^71 + 2 * q^73 + 8 * q^79 + 8 * q^83 + 10 * q^85 - 24 * q^91 - 28 * q^95 + 4 * 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
$5$	0.267949	0.119831	0.0599153	−	0.998203i	$-0.480917\pi$
$5$	0.267949	0.119831	0.0599153	+	0.998203i	$0.480917\pi$
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	4.46410	1.23812	0.619060	−	0.785344i	$-0.287514\pi$
$13$	4.46410	1.23812	0.619060	+	0.785344i	$0.287514\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.73205	1.39023	0.695113	−	0.718900i	$-0.255354\pi$
$17$	5.73205	1.39023	0.695113	+	0.718900i	$0.255354\pi$
$18$	0	0
$19$	−0.535898	−0.122944	−0.0614718	−	0.998109i	$-0.519579\pi$
$19$	−0.535898	−0.122944	−0.0614718	+	0.998109i	$0.519579\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.92820	1.86166	0.930830	−	0.365454i	$-0.119086\pi$
$23$	8.92820	1.86166	0.930830	+	0.365454i	$0.119086\pi$
$24$	0	0
$25$	−4.92820	−0.985641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.73205	1.43581	0.717903	−	0.696143i	$-0.245102\pi$
$29$	7.73205	1.43581	0.717903	+	0.696143i	$0.245102\pi$
$30$	0	0
$31$	2.92820	0.525921	0.262960	−	0.964807i	$-0.415301\pi$
$31$	2.92820	0.525921	0.262960	+	0.964807i	$0.415301\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.928203	−0.156895
$36$	0	0
$37$	6.46410	1.06269	0.531346	−	0.847155i	$-0.321686\pi$
$37$	6.46410	1.06269	0.531346	+	0.847155i	$0.321686\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	−11.4641	−1.74826	−0.874130	−	0.485693i	$-0.838567\pi$
$43$	−11.4641	−1.74826	−0.874130	+	0.485693i	$0.838567\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.92820	0.402220	0.201110	−	0.979569i	$-0.435545\pi$
$53$	2.92820	0.402220	0.201110	+	0.979569i	$0.435545\pi$
$54$	0	0
$55$	0.535898	0.0722605
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	3.53590	0.452725	0.226363	−	0.974043i	$-0.427317\pi$
$61$	3.53590	0.452725	0.226363	+	0.974043i	$0.427317\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.19615	0.148364
$66$	0	0
$67$	−7.46410	−0.911885	−0.455943	−	0.890009i	$-0.650698\pi$
$67$	−7.46410	−0.911885	−0.455943	+	0.890009i	$0.650698\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.92820	−0.789542
$78$	0	0
$79$	7.46410	0.839777	0.419889	−	0.907576i	$-0.362069\pi$
$79$	7.46410	0.839777	0.419889	+	0.907576i	$0.362069\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.9282	1.19953	0.599763	−	0.800178i	$-0.295261\pi$
$83$	10.9282	1.19953	0.599763	+	0.800178i	$0.295261\pi$
$84$	0	0
$85$	1.53590	0.166592
$86$	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
$91$	−15.4641	−1.62108
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.143594	−0.0147324
$96$	0	0
$97$	15.8564	1.60997	0.804987	−	0.593292i	$-0.202172\pi$
$97$	15.8564	1.60997	0.804987	+	0.593292i	$0.202172\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−2.92820	−0.288524	−0.144262	−	0.989539i	$-0.546081\pi$
$103$	−2.92820	−0.288524	−0.144262	+	0.989539i	$0.546081\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.9282	−1.82986	−0.914929	−	0.403614i	$-0.867754\pi$
$107$	−18.9282	−1.82986	−0.914929	+	0.403614i	$0.867754\pi$
$108$	0	0
$109$	−2.46410	−0.236018	−0.118009	−	0.993013i	$-0.537651\pi$
$109$	−2.46410	−0.236018	−0.118009	+	0.993013i	$0.537651\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.73205	0.162938	0.0814688	−	0.996676i	$-0.474039\pi$
$113$	1.73205	0.162938	0.0814688	+	0.996676i	$0.474039\pi$
$114$	0	0
$115$	2.39230	0.223084
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−19.8564	−1.82023
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.66025	−0.237940
$126$	0	0
$127$	7.46410	0.662332	0.331166	−	0.943573i	$-0.392558\pi$
$127$	7.46410	0.662332	0.331166	+	0.943573i	$0.392558\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.9282	−1.82851	−0.914253	−	0.405144i	$-0.867221\pi$
$131$	−20.9282	−1.82851	−0.914253	+	0.405144i	$0.867221\pi$
$132$	0	0
$133$	1.85641	0.160971
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.33975	0.285334	0.142667	−	0.989771i	$-0.454432\pi$
$137$	3.33975	0.285334	0.142667	+	0.989771i	$0.454432\pi$
$138$	0	0
$139$	−18.9282	−1.60547	−0.802735	−	0.596336i	$-0.796622\pi$
$139$	−18.9282	−1.60547	−0.802735	+	0.596336i	$0.796622\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.92820	0.746614
$144$	0	0
$145$	2.07180	0.172053
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.2679	−1.00503	−0.502515	−	0.864569i	$-0.667592\pi$
$149$	−12.2679	−1.00503	−0.502515	+	0.864569i	$0.667592\pi$
$150$	0	0
$151$	14.9282	1.21484	0.607420	−	0.794381i	$-0.292205\pi$
$151$	14.9282	1.21484	0.607420	+	0.794381i	$0.292205\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.784610	0.0630214
$156$	0	0
$157$	−15.3923	−1.22844	−0.614220	−	0.789135i	$-0.710529\pi$
$157$	−15.3923	−1.22844	−0.614220	+	0.789135i	$0.710529\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−30.9282	−2.43748
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	6.92820	0.532939
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.26795	−0.324486	−0.162243	−	0.986751i	$-0.551873\pi$
$173$	−4.26795	−0.324486	−0.162243	+	0.986751i	$0.551873\pi$
$174$	0	0
$175$	17.0718	1.29051
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.92820	0.517838	0.258919	−	0.965899i	$-0.416634\pi$
$179$	6.92820	0.517838	0.258919	+	0.965899i	$0.416634\pi$
$180$	0	0
$181$	−19.8564	−1.47592	−0.737958	−	0.674847i	$-0.764210\pi$
$181$	−19.8564	−1.47592	−0.737958	+	0.674847i	$0.764210\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.73205	0.127343
$186$	0	0
$187$	11.4641	0.838338
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−7.85641	−0.568470	−0.284235	−	0.958755i	$-0.591740\pi$
$191$	−7.85641	−0.568470	−0.284235	+	0.958755i	$0.591740\pi$
$192$	0	0
$193$	4.07180	0.293094	0.146547	−	0.989204i	$-0.453184\pi$
$193$	4.07180	0.293094	0.146547	+	0.989204i	$0.453184\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.6603	1.04450	0.522250	−	0.852792i	$-0.325093\pi$
$197$	14.6603	1.04450	0.522250	+	0.852792i	$0.325093\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−26.7846	−1.87991
$204$	0	0
$205$	−1.85641	−0.129657
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.07180	−0.0741377
$210$	0	0
$211$	9.32051	0.641650	0.320825	−	0.947138i	$-0.396040\pi$
$211$	9.32051	0.641650	0.320825	+	0.947138i	$0.396040\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.07180	−0.209495
$216$	0	0
$217$	−10.1436	−0.688592
$218$	0	0
$219$	0	0
$220$	0	0
$221$	25.5885	1.72127
$222$	0	0
$223$	7.46410	0.499833	0.249917	−	0.968267i	$-0.419597\pi$
$223$	7.46410	0.499833	0.249917	+	0.968267i	$0.419597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.7846	1.77776	0.888878	−	0.458143i	$-0.151485\pi$
$227$	26.7846	1.77776	0.888878	+	0.458143i	$0.151485\pi$
$228$	0	0
$229$	−14.4641	−0.955815	−0.477907	−	0.878410i	$-0.658605\pi$
$229$	−14.4641	−0.955815	−0.477907	+	0.878410i	$0.658605\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.26795	0.410627	0.205314	−	0.978696i	$-0.434179\pi$
$233$	6.26795	0.410627	0.205314	+	0.978696i	$0.434179\pi$
$234$	0	0
$235$	−1.85641	−0.121099
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.92820	−0.189410	−0.0947049	−	0.995505i	$-0.530191\pi$
$239$	−2.92820	−0.189410	−0.0947049	+	0.995505i	$0.530191\pi$
$240$	0	0
$241$	−21.9282	−1.41252	−0.706260	−	0.707953i	$-0.749619\pi$
$241$	−21.9282	−1.41252	−0.706260	+	0.707953i	$0.749619\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.33975	0.0855932
$246$	0	0
$247$	−2.39230	−0.152219
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	17.8564	1.12262
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.0526	0.689440	0.344720	−	0.938706i	$-0.387974\pi$
$257$	11.0526	0.689440	0.344720	+	0.938706i	$0.387974\pi$
$258$	0	0
$259$	−22.3923	−1.39139
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.7846	−0.788333	−0.394166	−	0.919039i	$-0.628967\pi$
$263$	−12.7846	−0.788333	−0.394166	+	0.919039i	$0.628967\pi$
$264$	0	0
$265$	0.784610	0.0481982
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−24.5167	−1.49481	−0.747404	−	0.664370i	$-0.768700\pi$
$269$	−24.5167	−1.49481	−0.747404	+	0.664370i	$0.768700\pi$
$270$	0	0
$271$	23.4641	1.42534	0.712671	−	0.701498i	$-0.247485\pi$
$271$	23.4641	1.42534	0.712671	+	0.701498i	$0.247485\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.85641	−0.594364
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.1244	−1.20052	−0.600259	−	0.799806i	$-0.704936\pi$
$281$	−20.1244	−1.20052	−0.600259	+	0.799806i	$0.704936\pi$
$282$	0	0
$283$	9.85641	0.585903	0.292951	−	0.956127i	$-0.405363\pi$
$283$	9.85641	0.585903	0.292951	+	0.956127i	$0.405363\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	15.8564	0.932730
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.1244	1.05884	0.529418	−	0.848361i	$-0.322410\pi$
$293$	18.1244	1.05884	0.529418	+	0.848361i	$0.322410\pi$
$294$	0	0
$295$	2.14359	0.124805
$296$	0	0
$297$	0	0
$298$	0	0
$299$	39.8564	2.30496
$300$	0	0
$301$	39.7128	2.28901
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.947441	0.0542503
$306$	0	0
$307$	−5.07180	−0.289463	−0.144731	−	0.989471i	$-0.546232\pi$
$307$	−5.07180	−0.289463	−0.144731	+	0.989471i	$0.546232\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−17.7846	−1.00525	−0.502623	−	0.864506i	$-0.667632\pi$
$313$	−17.7846	−1.00525	−0.502623	+	0.864506i	$0.667632\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.26795	0.239712	0.119856	−	0.992791i	$-0.461757\pi$
$317$	4.26795	0.239712	0.119856	+	0.992791i	$0.461757\pi$
$318$	0	0
$319$	15.4641	0.865823
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.07180	−0.170919
$324$	0	0
$325$	−22.0000	−1.22034
$326$	0	0
$327$	0	0
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−28.2487	−1.55269	−0.776345	−	0.630308i	$-0.782928\pi$
$331$	−28.2487	−1.55269	−0.776345	+	0.630308i	$0.782928\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−19.8564	−1.08165	−0.540824	−	0.841136i	$-0.681887\pi$
$337$	−19.8564	−1.08165	−0.540824	+	0.841136i	$0.681887\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.85641	0.317142
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.0718	0.594365	0.297183	−	0.954821i	$-0.403953\pi$
$347$	11.0718	0.594365	0.297183	+	0.954821i	$0.403953\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	29.8564	1.58910	0.794548	−	0.607201i	$-0.207708\pi$
$353$	29.8564	1.58910	0.794548	+	0.607201i	$0.207708\pi$
$354$	0	0
$355$	−0.535898	−0.0284425
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.7846	1.41364	0.706819	−	0.707395i	$-0.250130\pi$
$359$	26.7846	1.41364	0.706819	+	0.707395i	$0.250130\pi$
$360$	0	0
$361$	−18.7128	−0.984885
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.267949	0.0140251
$366$	0	0
$367$	9.07180	0.473544	0.236772	−	0.971565i	$-0.423911\pi$
$367$	9.07180	0.473544	0.236772	+	0.971565i	$0.423911\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.1436	−0.526629
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	34.5167	1.77770
$378$	0	0
$379$	16.7846	0.862167	0.431084	−	0.902312i	$-0.358131\pi$
$379$	16.7846	0.862167	0.431084	+	0.902312i	$0.358131\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	−1.85641	−0.0946112
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.0718	0.662766	0.331383	−	0.943496i	$-0.392485\pi$
$389$	13.0718	0.662766	0.331383	+	0.943496i	$0.392485\pi$
$390$	0	0
$391$	51.1769	2.58813
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.00000	0.100631
$396$	0	0
$397$	21.3923	1.07365	0.536825	−	0.843694i	$-0.319624\pi$
$397$	21.3923	1.07365	0.536825	+	0.843694i	$0.319624\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.80385	−0.339768	−0.169884	−	0.985464i	$-0.554339\pi$
$401$	−6.80385	−0.339768	−0.169884	+	0.985464i	$0.554339\pi$
$402$	0	0
$403$	13.0718	0.651153
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.9282	0.640827
$408$	0	0
$409$	2.07180	0.102444	0.0512219	−	0.998687i	$-0.483688\pi$
$409$	2.07180	0.102444	0.0512219	+	0.998687i	$0.483688\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−27.7128	−1.36366
$414$	0	0
$415$	2.92820	0.143740
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.85641	−0.0906914	−0.0453457	−	0.998971i	$-0.514439\pi$
$419$	−1.85641	−0.0906914	−0.0453457	+	0.998971i	$0.514439\pi$
$420$	0	0
$421$	−17.3923	−0.847649	−0.423825	−	0.905744i	$-0.639313\pi$
$421$	−17.3923	−0.847649	−0.423825	+	0.905744i	$0.639313\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−28.2487	−1.37026
$426$	0	0
$427$	−12.2487	−0.592757
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.0718	−0.629646	−0.314823	−	0.949150i	$-0.601945\pi$
$431$	−13.0718	−0.629646	−0.314823	+	0.949150i	$0.601945\pi$
$432$	0	0
$433$	−13.7846	−0.662446	−0.331223	−	0.943552i	$-0.607461\pi$
$433$	−13.7846	−0.662446	−0.331223	+	0.943552i	$0.607461\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.78461	−0.228879
$438$	0	0
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14.9282	−0.709260	−0.354630	−	0.935007i	$-0.615393\pi$
$443$	−14.9282	−0.709260	−0.354630	+	0.935007i	$0.615393\pi$
$444$	0	0
$445$	1.39230	0.0660016
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.7846	−0.603343	−0.301672	−	0.953412i	$-0.597545\pi$
$449$	−12.7846	−0.603343	−0.301672	+	0.953412i	$0.597545\pi$
$450$	0	0
$451$	−13.8564	−0.652473
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.14359	−0.194255
$456$	0	0
$457$	−24.0718	−1.12603	−0.563016	−	0.826446i	$-0.690359\pi$
$457$	−24.0718	−1.12603	−0.563016	+	0.826446i	$0.690359\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.7846	0.781737	0.390869	−	0.920446i	$-0.372175\pi$
$461$	16.7846	0.781737	0.390869	+	0.920446i	$0.372175\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.7128	−0.727102	−0.363551	−	0.931574i	$-0.618436\pi$
$467$	−15.7128	−0.727102	−0.363551	+	0.931574i	$0.618436\pi$
$468$	0	0
$469$	25.8564	1.19394
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−22.9282	−1.05424
$474$	0	0
$475$	2.64102	0.121178
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.8564	−0.724498	−0.362249	−	0.932081i	$-0.617991\pi$
$479$	−15.8564	−0.724498	−0.362249	+	0.932081i	$0.617991\pi$
$480$	0	0
$481$	28.8564	1.31574
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.24871	0.192924
$486$	0	0
$487$	21.0718	0.954854	0.477427	−	0.878671i	$-0.341569\pi$
$487$	21.0718	0.954854	0.477427	+	0.878671i	$0.341569\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.1436	−0.728550	−0.364275	−	0.931291i	$-0.618683\pi$
$491$	−16.1436	−0.728550	−0.364275	+	0.931291i	$0.618683\pi$
$492$	0	0
$493$	44.3205	1.99610
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.92820	0.310772
$498$	0	0
$499$	−24.2487	−1.08552	−0.542761	−	0.839887i	$-0.682621\pi$
$499$	−24.2487	−1.08552	−0.542761	+	0.839887i	$0.682621\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.85641	−0.0827731	−0.0413865	−	0.999143i	$-0.513178\pi$
$503$	−1.85641	−0.0827731	−0.0413865	+	0.999143i	$0.513178\pi$
$504$	0	0
$505$	−3.21539	−0.143083
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.78461	0.389371	0.194685	−	0.980866i	$-0.437631\pi$
$509$	8.78461	0.389371	0.194685	+	0.980866i	$0.437631\pi$
$510$	0	0
$511$	−3.46410	−0.153243
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.784610	−0.0345740
$516$	0	0
$517$	−13.8564	−0.609404
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.7846	0.560104	0.280052	−	0.959985i	$-0.409648\pi$
$521$	12.7846	0.560104	0.280052	+	0.959985i	$0.409648\pi$
$522$	0	0
$523$	−28.5359	−1.24779	−0.623894	−	0.781509i	$-0.714450\pi$
$523$	−28.5359	−1.24779	−0.623894	+	0.781509i	$0.714450\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.7846	0.731149
$528$	0	0
$529$	56.7128	2.46577
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−30.9282	−1.33965
$534$	0	0
$535$	−5.07180	−0.219273
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.0000	0.430730
$540$	0	0
$541$	−13.3923	−0.575780	−0.287890	−	0.957663i	$-0.592954\pi$
$541$	−13.3923	−0.575780	−0.287890	+	0.957663i	$0.592954\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.660254	−0.0282822
$546$	0	0
$547$	9.85641	0.421430	0.210715	−	0.977548i	$-0.432421\pi$
$547$	9.85641	0.421430	0.210715	+	0.977548i	$0.432421\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.14359	−0.176523
$552$	0	0
$553$	−25.8564	−1.09953
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.5885	0.745247	0.372623	−	0.927983i	$-0.378458\pi$
$557$	17.5885	0.745247	0.372623	+	0.927983i	$0.378458\pi$
$558$	0	0
$559$	−51.1769	−2.16455
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.9282	0.629149	0.314574	−	0.949233i	$-0.398138\pi$
$563$	14.9282	0.629149	0.314574	+	0.949233i	$0.398138\pi$
$564$	0	0
$565$	0.464102	0.0195249
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.4115	−0.520319	−0.260159	−	0.965566i	$-0.583775\pi$
$569$	−12.4115	−0.520319	−0.260159	+	0.965566i	$0.583775\pi$
$570$	0	0
$571$	13.8564	0.579873	0.289936	−	0.957046i	$-0.406366\pi$
$571$	13.8564	0.579873	0.289936	+	0.957046i	$0.406366\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−44.0000	−1.83493
$576$	0	0
$577$	24.8564	1.03479	0.517393	−	0.855748i	$-0.326903\pi$
$577$	24.8564	1.03479	0.517393	+	0.855748i	$0.326903\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−37.8564	−1.57055
$582$	0	0
$583$	5.85641	0.242548
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.0000	1.07313	0.536567	−	0.843857i	$-0.319721\pi$
$587$	26.0000	1.07313	0.536567	+	0.843857i	$0.319721\pi$
$588$	0	0
$589$	−1.56922	−0.0646586
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−31.5885	−1.29718	−0.648591	−	0.761137i	$-0.724642\pi$
$593$	−31.5885	−1.29718	−0.648591	+	0.761137i	$0.724642\pi$
$594$	0	0
$595$	−5.32051	−0.218120
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.07180	−0.207228	−0.103614	−	0.994618i	$-0.533041\pi$
$599$	−5.07180	−0.207228	−0.103614	+	0.994618i	$0.533041\pi$
$600$	0	0
$601$	20.8564	0.850751	0.425375	−	0.905017i	$-0.360142\pi$
$601$	20.8564	0.850751	0.425375	+	0.905017i	$0.360142\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.87564	−0.0762558
$606$	0	0
$607$	−23.1769	−0.940722	−0.470361	−	0.882474i	$-0.655876\pi$
$607$	−23.1769	−0.940722	−0.470361	+	0.882474i	$0.655876\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−30.9282	−1.25122
$612$	0	0
$613$	−23.8564	−0.963551	−0.481776	−	0.876295i	$-0.660008\pi$
$613$	−23.8564	−0.963551	−0.481776	+	0.876295i	$0.660008\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.4449	−1.18541	−0.592703	−	0.805421i	$-0.701939\pi$
$617$	−29.4449	−1.18541	−0.592703	+	0.805421i	$0.701939\pi$
$618$	0	0
$619$	36.7846	1.47850	0.739249	−	0.673432i	$-0.235181\pi$
$619$	36.7846	1.47850	0.739249	+	0.673432i	$0.235181\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−18.0000	−0.721155
$624$	0	0
$625$	23.9282	0.957128
$626$	0	0
$627$	0	0
$628$	0	0
$629$	37.0526	1.47738
$630$	0	0
$631$	−28.7846	−1.14590	−0.572949	−	0.819591i	$-0.694201\pi$
$631$	−28.7846	−1.14590	−0.572949	+	0.819591i	$0.694201\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.00000	0.0793676
$636$	0	0
$637$	22.3205	0.884371
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.9808	0.868188	0.434094	−	0.900868i	$-0.357069\pi$
$641$	21.9808	0.868188	0.434094	+	0.900868i	$0.357069\pi$
$642$	0	0
$643$	−8.78461	−0.346431	−0.173216	−	0.984884i	$-0.555416\pi$
$643$	−8.78461	−0.346431	−0.173216	+	0.984884i	$0.555416\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.7128	−0.696363	−0.348181	−	0.937427i	$-0.613201\pi$
$647$	−17.7128	−0.696363	−0.348181	+	0.937427i	$0.613201\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.7846	−1.28296	−0.641480	−	0.767139i	$-0.721679\pi$
$653$	−32.7846	−1.28296	−0.641480	+	0.767139i	$0.721679\pi$
$654$	0	0
$655$	−5.60770	−0.219111
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.7846	−1.51083	−0.755417	−	0.655244i	$-0.772566\pi$
$659$	−38.7846	−1.51083	−0.755417	+	0.655244i	$0.772566\pi$
$660$	0	0
$661$	4.32051	0.168048	0.0840241	−	0.996464i	$-0.473223\pi$
$661$	4.32051	0.168048	0.0840241	+	0.996464i	$0.473223\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.497423	0.0192892
$666$	0	0
$667$	69.0333	2.67298
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.07180	0.273004
$672$	0	0
$673$	12.0718	0.465334	0.232667	−	0.972557i	$-0.425255\pi$
$673$	12.0718	0.465334	0.232667	+	0.972557i	$0.425255\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.6410	−1.48509	−0.742547	−	0.669793i	$-0.766383\pi$
$677$	−38.6410	−1.48509	−0.742547	+	0.669793i	$0.766383\pi$
$678$	0	0
$679$	−54.9282	−2.10795
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.9282	0.571212	0.285606	−	0.958347i	$-0.407805\pi$
$683$	14.9282	0.571212	0.285606	+	0.958347i	$0.407805\pi$
$684$	0	0
$685$	0.894882	0.0341917
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.0718	0.497996
$690$	0	0
$691$	−38.3923	−1.46051	−0.730256	−	0.683174i	$-0.760599\pi$
$691$	−38.3923	−1.46051	−0.730256	+	0.683174i	$0.760599\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.07180	−0.192384
$696$	0	0
$697$	−39.7128	−1.50423
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−23.4449	−0.885500	−0.442750	−	0.896645i	$-0.645997\pi$
$701$	−23.4449	−0.885500	−0.442750	+	0.896645i	$0.645997\pi$
$702$	0	0
$703$	−3.46410	−0.130651
$704$	0	0
$705$	0	0
$706$	0	0
$707$	41.5692	1.56337
$708$	0	0
$709$	−12.3205	−0.462706	−0.231353	−	0.972870i	$-0.574315\pi$
$709$	−12.3205	−0.462706	−0.231353	+	0.972870i	$0.574315\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	26.1436	0.979085
$714$	0	0
$715$	2.39230	0.0894671
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.92820	−0.183791	−0.0918955	−	0.995769i	$-0.529293\pi$
$719$	−4.92820	−0.183791	−0.0918955	+	0.995769i	$0.529293\pi$
$720$	0	0
$721$	10.1436	0.377767
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−38.1051	−1.41519
$726$	0	0
$727$	31.1769	1.15629	0.578144	−	0.815935i	$-0.303777\pi$
$727$	31.1769	1.15629	0.578144	+	0.815935i	$0.303777\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−65.7128	−2.43048
$732$	0	0
$733$	12.1436	0.448534	0.224267	−	0.974528i	$-0.428001\pi$
$733$	12.1436	0.448534	0.224267	+	0.974528i	$0.428001\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.9282	−0.549887
$738$	0	0
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.07180	−0.332812	−0.166406	−	0.986057i	$-0.553216\pi$
$743$	−9.07180	−0.332812	−0.166406	+	0.986057i	$0.553216\pi$
$744$	0	0
$745$	−3.28719	−0.120433
$746$	0	0
$747$	0	0
$748$	0	0
$749$	65.5692	2.39585
$750$	0	0
$751$	17.3205	0.632034	0.316017	−	0.948753i	$-0.397654\pi$
$751$	17.3205	0.632034	0.316017	+	0.948753i	$0.397654\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.00000	0.145575
$756$	0	0
$757$	−8.14359	−0.295984	−0.147992	−	0.988989i	$-0.547281\pi$
$757$	−8.14359	−0.295984	−0.147992	+	0.988989i	$0.547281\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−38.8038	−1.40664	−0.703319	−	0.710874i	$-0.748300\pi$
$761$	−38.8038	−1.40664	−0.703319	+	0.710874i	$0.748300\pi$
$762$	0	0
$763$	8.53590	0.309020
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35.7128	1.28951
$768$	0	0
$769$	31.7846	1.14618	0.573091	−	0.819492i	$-0.305744\pi$
$769$	31.7846	1.14618	0.573091	+	0.819492i	$0.305744\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.7321	1.14132	0.570661	−	0.821186i	$-0.306687\pi$
$773$	31.7321	1.14132	0.570661	+	0.821186i	$0.306687\pi$
$774$	0	0
$775$	−14.4308	−0.518369
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.71281	0.133025
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.12436	−0.147205
$786$	0	0
$787$	12.5359	0.446857	0.223428	−	0.974720i	$-0.428275\pi$
$787$	12.5359	0.446857	0.223428	+	0.974720i	$0.428275\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	15.7846	0.560528
$794$	0	0
$795$	0	0
$796$	0	0
$797$	48.5167	1.71855	0.859274	−	0.511515i	$-0.170916\pi$
$797$	48.5167	1.71855	0.859274	+	0.511515i	$0.170916\pi$
$798$	0	0
$799$	−39.7128	−1.40494
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.00000	0.0705785
$804$	0	0
$805$	−8.28719	−0.292085
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.2679	0.923532	0.461766	−	0.887002i	$-0.347216\pi$
$809$	26.2679	0.923532	0.461766	+	0.887002i	$0.347216\pi$
$810$	0	0
$811$	−49.5692	−1.74061	−0.870305	−	0.492513i	$-0.836079\pi$
$811$	−49.5692	−1.74061	−0.870305	+	0.492513i	$0.836079\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.14359	0.214937
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.6603	−0.930449	−0.465225	−	0.885193i	$-0.654026\pi$
$821$	−26.6603	−0.930449	−0.465225	+	0.885193i	$0.654026\pi$
$822$	0	0
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.8564	−0.551381	−0.275691	−	0.961246i	$-0.588907\pi$
$827$	−15.8564	−0.551381	−0.275691	+	0.961246i	$0.588907\pi$
$828$	0	0
$829$	−3.85641	−0.133939	−0.0669693	−	0.997755i	$-0.521333\pi$
$829$	−3.85641	−0.133939	−0.0669693	+	0.997755i	$0.521333\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	28.6603	0.993019
$834$	0	0
$835$	4.82309	0.166910
$836$	0	0
$837$	0	0
$838$	0	0
$839$	36.9282	1.27490	0.637452	−	0.770490i	$-0.279989\pi$
$839$	36.9282	1.27490	0.637452	+	0.770490i	$0.279989\pi$
$840$	0	0
$841$	30.7846	1.06154
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.85641	0.0638623
$846$	0	0
$847$	24.2487	0.833196
$848$	0	0
$849$	0	0
$850$	0	0
$851$	57.7128	1.97837
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.58846	−0.122579	−0.0612897	−	0.998120i	$-0.519521\pi$
$857$	−3.58846	−0.122579	−0.0612897	+	0.998120i	$0.519521\pi$
$858$	0	0
$859$	1.85641	0.0633398	0.0316699	−	0.999498i	$-0.489917\pi$
$859$	1.85641	0.0633398	0.0316699	+	0.999498i	$0.489917\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.0718	0.649212	0.324606	−	0.945849i	$-0.394768\pi$
$863$	19.0718	0.649212	0.324606	+	0.945849i	$0.394768\pi$
$864$	0	0
$865$	−1.14359	−0.0388833
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.9282	0.506405
$870$	0	0
$871$	−33.3205	−1.12902
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.21539	0.311537
$876$	0	0
$877$	−14.3205	−0.483569	−0.241785	−	0.970330i	$-0.577733\pi$
$877$	−14.3205	−0.483569	−0.241785	+	0.970330i	$0.577733\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18.6410	−0.628032	−0.314016	−	0.949418i	$-0.601675\pi$
$881$	−18.6410	−0.628032	−0.314016	+	0.949418i	$0.601675\pi$
$882$	0	0
$883$	50.9282	1.71387	0.856935	−	0.515424i	$-0.172366\pi$
$883$	50.9282	1.71387	0.856935	+	0.515424i	$0.172366\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.6410	−0.827364	−0.413682	−	0.910421i	$-0.635758\pi$
$887$	−24.6410	−0.827364	−0.413682	+	0.910421i	$0.635758\pi$
$888$	0	0
$889$	−25.8564	−0.867196
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.71281	0.124245
$894$	0	0
$895$	1.85641	0.0620528
$896$	0	0
$897$	0	0
$898$	0	0
$899$	22.6410	0.755120
$900$	0	0
$901$	16.7846	0.559176
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.32051	−0.176860
$906$	0	0
$907$	−4.78461	−0.158870	−0.0794352	−	0.996840i	$-0.525312\pi$
$907$	−4.78461	−0.158870	−0.0794352	+	0.996840i	$0.525312\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46.7846	1.55004	0.775022	−	0.631935i	$-0.217739\pi$
$911$	46.7846	1.55004	0.775022	+	0.631935i	$0.217739\pi$
$912$	0	0
$913$	21.8564	0.723341
$914$	0	0
$915$	0	0
$916$	0	0
$917$	72.4974	2.39408
$918$	0	0
$919$	35.1769	1.16038	0.580190	−	0.814481i	$-0.302978\pi$
$919$	35.1769	1.16038	0.580190	+	0.814481i	$0.302978\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.92820	−0.293875
$924$	0	0
$925$	−31.8564	−1.04743
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.41154	−0.144738	−0.0723690	−	0.997378i	$-0.523056\pi$
$929$	−4.41154	−0.144738	−0.0723690	+	0.997378i	$0.523056\pi$
$930$	0	0
$931$	−2.67949	−0.0878168
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.07180	0.100458
$936$	0	0
$937$	−50.5692	−1.65202	−0.826012	−	0.563652i	$-0.809396\pi$
$937$	−50.5692	−1.65202	−0.826012	+	0.563652i	$0.809396\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.1244	−0.590837	−0.295419	−	0.955368i	$-0.595459\pi$
$941$	−18.1244	−0.590837	−0.295419	+	0.955368i	$0.595459\pi$
$942$	0	0
$943$	−61.8564	−2.01432
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.9282	−0.420110	−0.210055	−	0.977690i	$-0.567364\pi$
$947$	−12.9282	−0.420110	−0.210055	+	0.977690i	$0.567364\pi$
$948$	0	0
$949$	4.46410	0.144911
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13.9808	0.452881	0.226441	−	0.974025i	$-0.427291\pi$
$953$	13.9808	0.452881	0.226441	+	0.974025i	$0.427291\pi$
$954$	0	0
$955$	−2.10512	−0.0681200
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.5692	−0.373590
$960$	0	0
$961$	−22.4256	−0.723407
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.09103	0.0351216
$966$	0	0
$967$	−9.32051	−0.299727	−0.149864	−	0.988707i	$-0.547883\pi$
$967$	−9.32051	−0.299727	−0.149864	+	0.988707i	$0.547883\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.928203	0.0297875	0.0148937	−	0.999889i	$-0.495259\pi$
$971$	0.928203	0.0297875	0.0148937	+	0.999889i	$0.495259\pi$
$972$	0	0
$973$	65.5692	2.10205
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.0718	0.546175	0.273088	−	0.961989i	$-0.411955\pi$
$977$	17.0718	0.546175	0.273088	+	0.961989i	$0.411955\pi$
$978$	0	0
$979$	10.3923	0.332140
$980$	0	0
$981$	0	0
$982$	0	0
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	3.92820	0.125163
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−102.354	−3.25466
$990$	0	0
$991$	−46.1051	−1.46458	−0.732289	−	0.680994i	$-0.761548\pi$
$991$	−46.1051	−1.46458	−0.732289	+	0.680994i	$0.761548\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.07180	0.0339782
$996$	0	0
$997$	−48.1769	−1.52578	−0.762889	−	0.646529i	$-0.776220\pi$
$997$	−48.1769	−1.52578	−0.762889	+	0.646529i	$0.776220\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.a.h.1.1	yes	2
3.2	odd	2		648.2.a.e.1.2	✓	2
4.3	odd	2		1296.2.a.q.1.1		2
8.3	odd	2		5184.2.a.bi.1.2		2
8.5	even	2		5184.2.a.bg.1.2		2
9.2	odd	6		648.2.i.j.433.1		4
9.4	even	3		648.2.i.i.217.2		4
9.5	odd	6		648.2.i.j.217.1		4
9.7	even	3		648.2.i.i.433.2		4
12.11	even	2		1296.2.a.m.1.2		2
24.5	odd	2		5184.2.a.cb.1.1		2
24.11	even	2		5184.2.a.bz.1.1		2
36.7	odd	6		1296.2.i.r.433.2		4
36.11	even	6		1296.2.i.t.433.1		4
36.23	even	6		1296.2.i.t.865.1		4
36.31	odd	6		1296.2.i.r.865.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.a.e.1.2	✓	2	3.2	odd	2
648.2.a.h.1.1	yes	2	1.1	even	1	trivial
648.2.i.i.217.2		4	9.4	even	3
648.2.i.i.433.2		4	9.7	even	3
648.2.i.j.217.1		4	9.5	odd	6
648.2.i.j.433.1		4	9.2	odd	6
1296.2.a.m.1.2		2	12.11	even	2
1296.2.a.q.1.1		2	4.3	odd	2
1296.2.i.r.433.2		4	36.7	odd	6
1296.2.i.r.865.2		4	36.31	odd	6
1296.2.i.t.433.1		4	36.11	even	6
1296.2.i.t.865.1		4	36.23	even	6
5184.2.a.bg.1.2		2	8.5	even	2
5184.2.a.bi.1.2		2	8.3	odd	2
5184.2.a.bz.1.1		2	24.11	even	2
5184.2.a.cb.1.1		2	24.5	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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q+0.267949 q^{5} -3.46410 q^{7} +O(q^{10})\)	\(q+0.267949 q^{5} -3.46410 q^{7} +2.00000 q^{11} +4.46410 q^{13} +5.73205 q^{17} -0.535898 q^{19} +8.92820 q^{23} -4.92820 q^{25} +7.73205 q^{29} +2.92820 q^{31} -0.928203 q^{35} +6.46410 q^{37} -6.92820 q^{41} -11.4641 q^{43} -6.92820 q^{47} +5.00000 q^{49} +2.92820 q^{53} +0.535898 q^{55} +8.00000 q^{59} +3.53590 q^{61} +1.19615 q^{65} -7.46410 q^{67} -2.00000 q^{71} +1.00000 q^{73} -6.92820 q^{77} +7.46410 q^{79} +10.9282 q^{83} +1.53590 q^{85} +5.19615 q^{89} -15.4641 q^{91} -0.143594 q^{95} +15.8564 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5}+O(q^{10}) \) 2 * q + 4 * q^5	\( 2 q + 4 q^{5} + 4 q^{11} + 2 q^{13} + 8 q^{17} - 8 q^{19} + 4 q^{23} + 4 q^{25} + 12 q^{29} - 8 q^{31} + 12 q^{35} + 6 q^{37} - 16 q^{43} + 10 q^{49} - 8 q^{53} + 8 q^{55} + 16 q^{59} + 14 q^{61} - 8 q^{65} - 8 q^{67} - 4 q^{71} + 2 q^{73} + 8 q^{79} + 8 q^{83} + 10 q^{85} - 24 q^{91} - 28 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 + 4 * q^11 + 2 * q^13 + 8 * q^17 - 8 * q^19 + 4 * q^23 + 4 * q^25 + 12 * q^29 - 8 * q^31 + 12 * q^35 + 6 * q^37 - 16 * q^43 + 10 * q^49 - 8 * q^53 + 8 * q^55 + 16 * q^59 + 14 * q^61 - 8 * q^65 - 8 * q^67 - 4 * q^71 + 2 * q^73 + 8 * q^79 + 8 * q^83 + 10 * q^85 - 24 * q^91 - 28 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more