LMFDB - Embedded newform 5445.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
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:	no (minimal twist has level 1815)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} +1.00000 q^{5} -1.73205 q^{7} +O(q^{10})$	$q-2.00000 q^{4} +1.00000 q^{5} -1.73205 q^{7} +4.00000 q^{16} -3.46410 q^{17} +5.19615 q^{19} -2.00000 q^{20} -6.00000 q^{23} +1.00000 q^{25} +3.46410 q^{28} +6.92820 q^{29} +1.00000 q^{31} -1.73205 q^{35} -5.00000 q^{37} +3.46410 q^{41} -10.3923 q^{43} +12.0000 q^{47} -4.00000 q^{49} -6.00000 q^{53} +12.1244 q^{61} -8.00000 q^{64} -5.00000 q^{67} +6.92820 q^{68} +6.00000 q^{71} +1.73205 q^{73} -10.3923 q^{76} +15.5885 q^{79} +4.00000 q^{80} -6.92820 q^{83} -3.46410 q^{85} +6.00000 q^{89} +12.0000 q^{92} +5.19615 q^{95} -13.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 2 q^{5}+O(q^{10}) $ 2 * q - 4 * q^4 + 2 * q^5	$ 2 q - 4 q^{4} + 2 q^{5} + 8 q^{16} - 4 q^{20} - 12 q^{23} + 2 q^{25} + 2 q^{31} - 10 q^{37} + 24 q^{47} - 8 q^{49} - 12 q^{53} - 16 q^{64} - 10 q^{67} + 12 q^{71} + 8 q^{80} + 12 q^{89} + 24 q^{92} - 26 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 2 * q^5 + 8 * q^16 - 4 * q^20 - 12 * q^23 + 2 * q^25 + 2 * q^31 - 10 * q^37 + 24 * q^47 - 8 * q^49 - 12 * q^53 - 16 * q^64 - 10 * q^67 + 12 * q^71 + 8 * q^80 + 12 * q^89 + 24 * q^92 - 26 * 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.73205	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−1.73205	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	5.19615	1.19208	0.596040	−	0.802955i	$-0.296740\pi$
$19$	5.19615	1.19208	0.596040	+	0.802955i	$0.296740\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	3.46410	0.654654
$29$	6.92820	1.28654	0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820	1.28654	0.643268	+	0.765641i	$0.277578\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.73205	−0.292770
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	−10.3923	−1.58481	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	−10.3923	−1.58481	−0.792406	+	0.609994i	$0.791172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	12.1244	1.55236	0.776182	−	0.630509i	$-0.217154\pi$
$61$	12.1244	1.55236	0.776182	+	0.630509i	$0.217154\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	6.92820	0.840168
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	1.73205	0.202721	0.101361	−	0.994850i	$-0.467680\pi$
$73$	1.73205	0.202721	0.101361	+	0.994850i	$0.467680\pi$
$74$	0	0
$75$	0	0
$76$	−10.3923	−1.19208
$77$	0	0
$78$	0	0
$79$	15.5885	1.75384	0.876919	−	0.480638i	$-0.159595\pi$
$79$	15.5885	1.75384	0.876919	+	0.480638i	$0.159595\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	0	0
$83$	−6.92820	−0.760469	−0.380235	−	0.924890i	$-0.624157\pi$
$83$	−6.92820	−0.760469	−0.380235	+	0.924890i	$0.624157\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	12.0000	1.25109
$93$	0	0
$94$	0	0
$95$	5.19615	0.533114
$96$	0	0
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	0	0
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.92820	−0.669775	−0.334887	−	0.942258i	$-0.608698\pi$
$107$	−6.92820	−0.669775	−0.334887	+	0.942258i	$0.608698\pi$
$108$	0	0
$109$	−12.1244	−1.16130	−0.580651	−	0.814152i	$-0.697202\pi$
$109$	−12.1244	−1.16130	−0.580651	+	0.814152i	$0.697202\pi$
$110$	0	0
$111$	0	0
$112$	−6.92820	−0.654654
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	−13.8564	−1.28654
$117$	0	0
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	−2.00000	−0.179605
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.73205	−0.153695	−0.0768473	−	0.997043i	$-0.524485\pi$
$127$	−1.73205	−0.153695	−0.0768473	+	0.997043i	$0.524485\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	−9.00000	−0.780399
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−3.46410	−0.293821	−0.146911	−	0.989150i	$-0.546933\pi$
$139$	−3.46410	−0.293821	−0.146911	+	0.989150i	$0.546933\pi$
$140$	3.46410	0.292770
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.92820	0.575356
$146$	0	0
$147$	0	0
$148$	10.0000	0.821995
$149$	24.2487	1.98653	0.993266	−	0.115857i	$-0.0369614\pi$
$149$	24.2487	1.98653	0.993266	+	0.115857i	$0.0369614\pi$
$150$	0	0
$151$	−17.3205	−1.40952	−0.704761	−	0.709444i	$-0.748946\pi$
$151$	−17.3205	−1.40952	−0.704761	+	0.709444i	$0.748946\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−5.00000	−0.399043	−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000	−0.399043	−0.199522	+	0.979893i	$0.563939\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	10.3923	0.819028
$162$	0	0
$163$	−13.0000	−1.01824	−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000	−1.01824	−0.509119	+	0.860696i	$0.670029\pi$
$164$	−6.92820	−0.541002
$165$	0	0
$166$	0	0
$167$	3.46410	0.268060	0.134030	−	0.990977i	$-0.457208\pi$
$167$	3.46410	0.268060	0.134030	+	0.990977i	$0.457208\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	20.7846	1.58481
$173$	−10.3923	−0.790112	−0.395056	−	0.918657i	$-0.629275\pi$
$173$	−10.3923	−0.790112	−0.395056	+	0.918657i	$0.629275\pi$
$174$	0	0
$175$	−1.73205	−0.130931
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.00000	−0.367607
$186$	0	0
$187$	0	0
$188$	−24.0000	−1.75038
$189$	0	0
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−1.73205	−0.124676	−0.0623379	−	0.998055i	$-0.519856\pi$
$193$	−1.73205	−0.124676	−0.0623379	+	0.998055i	$0.519856\pi$
$194$	0	0
$195$	0	0
$196$	8.00000	0.571429
$197$	−6.92820	−0.493614	−0.246807	−	0.969065i	$-0.579381\pi$
$197$	−6.92820	−0.493614	−0.246807	+	0.969065i	$0.579381\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−12.0000	−0.842235
$204$	0	0
$205$	3.46410	0.241943
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	19.0526	1.31163	0.655816	−	0.754921i	$-0.272325\pi$
$211$	19.0526	1.31163	0.655816	+	0.754921i	$0.272325\pi$
$212$	12.0000	0.824163
$213$	0	0
$214$	0	0
$215$	−10.3923	−0.708749
$216$	0	0
$217$	−1.73205	−0.117579
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	27.7128	1.83936	0.919682	−	0.392664i	$-0.128446\pi$
$227$	27.7128	1.83936	0.919682	+	0.392664i	$0.128446\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.8564	0.907763	0.453882	−	0.891062i	$-0.350039\pi$
$233$	13.8564	0.907763	0.453882	+	0.891062i	$0.350039\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	0	0
$241$	−6.92820	−0.446285	−0.223142	−	0.974786i	$-0.571631\pi$
$241$	−6.92820	−0.446285	−0.223142	+	0.974786i	$0.571631\pi$
$242$	0	0
$243$	0	0
$244$	−24.2487	−1.55236
$245$	−4.00000	−0.255551
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	8.66025	0.538122
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.3923	−0.640817	−0.320408	−	0.947279i	$-0.603820\pi$
$263$	−10.3923	−0.640817	−0.320408	+	0.947279i	$0.603820\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	10.0000	0.610847
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	10.3923	0.631288	0.315644	−	0.948878i	$-0.397780\pi$
$271$	10.3923	0.631288	0.315644	+	0.948878i	$0.397780\pi$
$272$	−13.8564	−0.840168
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−25.9808	−1.56103	−0.780516	−	0.625135i	$-0.785044\pi$
$277$	−25.9808	−1.56103	−0.780516	+	0.625135i	$0.785044\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.1769	1.85986	0.929929	−	0.367738i	$-0.119868\pi$
$281$	31.1769	1.85986	0.929929	+	0.367738i	$0.119868\pi$
$282$	0	0
$283$	−15.5885	−0.926638	−0.463319	−	0.886192i	$-0.653342\pi$
$283$	−15.5885	−0.926638	−0.463319	+	0.886192i	$0.653342\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	−3.46410	−0.202721
$293$	−31.1769	−1.82137	−0.910687	−	0.413096i	$-0.864447\pi$
$293$	−31.1769	−1.82137	−0.910687	+	0.413096i	$0.864447\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	18.0000	1.03750
$302$	0	0
$303$	0	0
$304$	20.7846	1.19208
$305$	12.1244	0.694239
$306$	0	0
$307$	8.66025	0.494267	0.247133	−	0.968981i	$-0.420511\pi$
$307$	8.66025	0.494267	0.247133	+	0.968981i	$0.420511\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	0	0
$316$	−31.1769	−1.75384
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	0	0
$320$	−8.00000	−0.447214
$321$	0	0
$322$	0	0
$323$	−18.0000	−1.00155
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−20.7846	−1.14589
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	13.8564	0.760469
$333$	0	0
$334$	0	0
$335$	−5.00000	−0.273179
$336$	0	0
$337$	29.4449	1.60396	0.801982	−	0.597348i	$-0.203779\pi$
$337$	29.4449	1.60396	0.801982	+	0.597348i	$0.203779\pi$
$338$	0	0
$339$	0	0
$340$	6.92820	0.375735
$341$	0	0
$342$	0	0
$343$	19.0526	1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.3205	0.929814	0.464907	−	0.885360i	$-0.346088\pi$
$347$	17.3205	0.929814	0.464907	+	0.885360i	$0.346088\pi$
$348$	0	0
$349$	−5.19615	−0.278144	−0.139072	−	0.990282i	$-0.544412\pi$
$349$	−5.19615	−0.278144	−0.139072	+	0.990282i	$0.544412\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	−12.0000	−0.635999
$357$	0	0
$358$	0	0
$359$	−24.2487	−1.27980	−0.639899	−	0.768459i	$-0.721024\pi$
$359$	−24.2487	−1.27980	−0.639899	+	0.768459i	$0.721024\pi$
$360$	0	0
$361$	8.00000	0.421053
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.73205	0.0906597
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	−24.0000	−1.25109
$369$	0	0
$370$	0	0
$371$	10.3923	0.539542
$372$	0	0
$373$	19.0526	0.986504	0.493252	−	0.869886i	$-0.335808\pi$
$373$	19.0526	0.986504	0.493252	+	0.869886i	$0.335808\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	−10.3923	−0.533114
$381$	0	0
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	26.0000	1.31995
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	20.7846	1.05112
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.5885	0.784340
$396$	0	0
$397$	−19.0000	−0.953583	−0.476791	−	0.879017i	$-0.658200\pi$
$397$	−19.0000	−0.953583	−0.476791	+	0.879017i	$0.658200\pi$
$398$	0	0
$399$	0	0
$400$	4.00000	0.200000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	27.7128	1.37876
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−19.0526	−0.942088	−0.471044	−	0.882110i	$-0.656123\pi$
$409$	−19.0526	−0.942088	−0.471044	+	0.882110i	$0.656123\pi$
$410$	0	0
$411$	0	0
$412$	26.0000	1.28093
$413$	0	0
$414$	0	0
$415$	−6.92820	−0.340092
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.46410	−0.168034
$426$	0	0
$427$	−21.0000	−1.01626
$428$	13.8564	0.669775
$429$	0	0
$430$	0	0
$431$	−38.1051	−1.83546	−0.917729	−	0.397206i	$-0.869980\pi$
$431$	−38.1051	−1.83546	−0.917729	+	0.397206i	$0.869980\pi$
$432$	0	0
$433$	−7.00000	−0.336399	−0.168199	−	0.985753i	$-0.553795\pi$
$433$	−7.00000	−0.336399	−0.168199	+	0.985753i	$0.553795\pi$
$434$	0	0
$435$	0	0
$436$	24.2487	1.16130
$437$	−31.1769	−1.49139
$438$	0	0
$439$	−15.5885	−0.743996	−0.371998	−	0.928233i	$-0.621327\pi$
$439$	−15.5885	−0.743996	−0.371998	+	0.928233i	$0.621327\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	0	0
$448$	13.8564	0.654654
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.6410	−1.62044	−0.810219	−	0.586127i	$-0.800652\pi$
$457$	−34.6410	−1.62044	−0.810219	+	0.586127i	$0.800652\pi$
$458$	0	0
$459$	0	0
$460$	12.0000	0.559503
$461$	−17.3205	−0.806696	−0.403348	−	0.915047i	$-0.632154\pi$
$461$	−17.3205	−0.806696	−0.403348	+	0.915047i	$0.632154\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	27.7128	1.28654
$465$	0	0
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	8.66025	0.399893
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.19615	0.238416
$476$	−12.0000	−0.550019
$477$	0	0
$478$	0	0
$479$	27.7128	1.26623	0.633115	−	0.774057i	$-0.281776\pi$
$479$	27.7128	1.26623	0.633115	+	0.774057i	$0.281776\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.0000	−0.590300
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	24.2487	1.09433	0.547165	−	0.837025i	$-0.315707\pi$
$491$	24.2487	1.09433	0.547165	+	0.837025i	$0.315707\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	−10.3923	−0.466159
$498$	0	0
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	−2.00000	−0.0894427
$501$	0	0
$502$	0	0
$503$	34.6410	1.54457	0.772283	−	0.635278i	$-0.219115\pi$
$503$	34.6410	1.54457	0.772283	+	0.635278i	$0.219115\pi$
$504$	0	0
$505$	−13.8564	−0.616602
$506$	0	0
$507$	0	0
$508$	3.46410	0.153695
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.0000	−0.572848
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	−29.4449	−1.28753	−0.643767	−	0.765222i	$-0.722629\pi$
$523$	−29.4449	−1.28753	−0.643767	+	0.765222i	$0.722629\pi$
$524$	−6.92820	−0.302660
$525$	0	0
$526$	0	0
$527$	−3.46410	−0.150899
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	18.0000	0.780399
$533$	0	0
$534$	0	0
$535$	−6.92820	−0.299532
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.1244	−0.519350
$546$	0	0
$547$	31.1769	1.33303	0.666514	−	0.745492i	$-0.267786\pi$
$547$	31.1769	1.33303	0.666514	+	0.745492i	$0.267786\pi$
$548$	−24.0000	−1.02523
$549$	0	0
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	−27.0000	−1.14816
$554$	0	0
$555$	0	0
$556$	6.92820	0.293821
$557$	3.46410	0.146779	0.0733893	−	0.997303i	$-0.476618\pi$
$557$	3.46410	0.146779	0.0733893	+	0.997303i	$0.476618\pi$
$558$	0	0
$559$	0	0
$560$	−6.92820	−0.292770
$561$	0	0
$562$	0	0
$563$	34.6410	1.45994	0.729972	−	0.683477i	$-0.239533\pi$
$563$	34.6410	1.45994	0.729972	+	0.683477i	$0.239533\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.7846	−0.871336	−0.435668	−	0.900107i	$-0.643488\pi$
$569$	−20.7846	−0.871336	−0.435668	+	0.900107i	$0.643488\pi$
$570$	0	0
$571$	−19.0526	−0.797325	−0.398662	−	0.917098i	$-0.630525\pi$
$571$	−19.0526	−0.797325	−0.398662	+	0.917098i	$0.630525\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.00000	−0.250217
$576$	0	0
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	0	0
$579$	0	0
$580$	−13.8564	−0.575356
$581$	12.0000	0.497844
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	5.19615	0.214104
$590$	0	0
$591$	0	0
$592$	−20.0000	−0.821995
$593$	−10.3923	−0.426761	−0.213380	−	0.976969i	$-0.568447\pi$
$593$	−10.3923	−0.426761	−0.213380	+	0.976969i	$0.568447\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	−48.4974	−1.98653
$597$	0	0
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−22.5167	−0.918474	−0.459237	−	0.888314i	$-0.651877\pi$
$601$	−22.5167	−0.918474	−0.459237	+	0.888314i	$0.651877\pi$
$602$	0	0
$603$	0	0
$604$	34.6410	1.40952
$605$	0	0
$606$	0	0
$607$	3.46410	0.140604	0.0703018	−	0.997526i	$-0.477604\pi$
$607$	3.46410	0.140604	0.0703018	+	0.997526i	$0.477604\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	29.4449	1.18927	0.594633	−	0.803997i	$-0.297297\pi$
$613$	29.4449	1.18927	0.594633	+	0.803997i	$0.297297\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	0	0
$623$	−10.3923	−0.416359
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	10.0000	0.399043
$629$	17.3205	0.690614
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.73205	−0.0687343
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	−20.7846	−0.819028
$645$	0	0
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	26.0000	1.01824
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	0	0
$655$	3.46410	0.135354
$656$	13.8564	0.541002
$657$	0	0
$658$	0	0
$659$	−10.3923	−0.404827	−0.202413	−	0.979300i	$-0.564878\pi$
$659$	−10.3923	−0.404827	−0.202413	+	0.979300i	$0.564878\pi$
$660$	0	0
$661$	49.0000	1.90588	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	49.0000	1.90588	0.952940	+	0.303160i	$0.0980418\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.00000	−0.349005
$666$	0	0
$667$	−41.5692	−1.60957
$668$	−6.92820	−0.268060
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	15.5885	0.600891	0.300445	−	0.953799i	$-0.402865\pi$
$673$	15.5885	0.600891	0.300445	+	0.953799i	$0.402865\pi$
$674$	0	0
$675$	0	0
$676$	26.0000	1.00000
$677$	20.7846	0.798817	0.399409	−	0.916773i	$-0.369215\pi$
$677$	20.7846	0.798817	0.399409	+	0.916773i	$0.369215\pi$
$678$	0	0
$679$	22.5167	0.864110
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−42.0000	−1.60709	−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000	−1.60709	−0.803543	+	0.595247i	$0.797054\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	0	0
$688$	−41.5692	−1.58481
$689$	0	0
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	20.7846	0.790112
$693$	0	0
$694$	0	0
$695$	−3.46410	−0.131401
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	0	0
$700$	3.46410	0.130931
$701$	24.2487	0.915861	0.457931	−	0.888988i	$-0.348591\pi$
$701$	24.2487	0.915861	0.457931	+	0.888988i	$0.348591\pi$
$702$	0	0
$703$	−25.9808	−0.979883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.0000	0.902613
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	0	0
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	22.5167	0.838564
$722$	0	0
$723$	0	0
$724$	14.0000	0.520306
$725$	6.92820	0.257307
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	36.0000	1.33151
$732$	0	0
$733$	−27.7128	−1.02360	−0.511798	−	0.859106i	$-0.671020\pi$
$733$	−27.7128	−1.02360	−0.511798	+	0.859106i	$0.671020\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	15.5885	0.573431	0.286715	−	0.958016i	$-0.407437\pi$
$739$	15.5885	0.573431	0.286715	+	0.958016i	$0.407437\pi$
$740$	10.0000	0.367607
$741$	0	0
$742$	0	0
$743$	−20.7846	−0.762513	−0.381257	−	0.924469i	$-0.624509\pi$
$743$	−20.7846	−0.762513	−0.381257	+	0.924469i	$0.624509\pi$
$744$	0	0
$745$	24.2487	0.888404
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−11.0000	−0.401396	−0.200698	−	0.979653i	$-0.564321\pi$
$751$	−11.0000	−0.401396	−0.200698	+	0.979653i	$0.564321\pi$
$752$	48.0000	1.75038
$753$	0	0
$754$	0	0
$755$	−17.3205	−0.630358
$756$	0	0
$757$	13.0000	0.472493	0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000	0.472493	0.236247	+	0.971693i	$0.424083\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.46410	−0.125574	−0.0627868	−	0.998027i	$-0.519999\pi$
$761$	−3.46410	−0.125574	−0.0627868	+	0.998027i	$0.519999\pi$
$762$	0	0
$763$	21.0000	0.760251
$764$	48.0000	1.73658
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−1.73205	−0.0624593	−0.0312297	−	0.999512i	$-0.509942\pi$
$769$	−1.73205	−0.0624593	−0.0312297	+	0.999512i	$0.509942\pi$
$770$	0	0
$771$	0	0
$772$	3.46410	0.124676
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.0000	0.644917
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−16.0000	−0.571429
$785$	−5.00000	−0.178458
$786$	0	0
$787$	24.2487	0.864373	0.432187	−	0.901784i	$-0.357742\pi$
$787$	24.2487	0.864373	0.432187	+	0.901784i	$0.357742\pi$
$788$	13.8564	0.493614
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	22.0000	0.779769
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−41.5692	−1.47061
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	10.3923	0.366281
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.1051	−1.33970	−0.669852	−	0.742494i	$-0.733643\pi$
$809$	−38.1051	−1.33970	−0.669852	+	0.742494i	$0.733643\pi$
$810$	0	0
$811$	−29.4449	−1.03395	−0.516975	−	0.856001i	$-0.672942\pi$
$811$	−29.4449	−1.03395	−0.516975	+	0.856001i	$0.672942\pi$
$812$	24.0000	0.842235
$813$	0	0
$814$	0	0
$815$	−13.0000	−0.455370
$816$	0	0
$817$	−54.0000	−1.88922
$818$	0	0
$819$	0	0
$820$	−6.92820	−0.241943
$821$	10.3923	0.362694	0.181347	−	0.983419i	$-0.441954\pi$
$821$	10.3923	0.362694	0.181347	+	0.983419i	$0.441954\pi$
$822$	0	0
$823$	23.0000	0.801730	0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000	0.801730	0.400865	+	0.916137i	$0.368710\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.5692	−1.44550	−0.722752	−	0.691108i	$-0.757123\pi$
$827$	−41.5692	−1.44550	−0.722752	+	0.691108i	$0.757123\pi$
$828$	0	0
$829$	−55.0000	−1.91023	−0.955114	−	0.296237i	$-0.904268\pi$
$829$	−55.0000	−1.91023	−0.955114	+	0.296237i	$0.904268\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13.8564	0.480096
$834$	0	0
$835$	3.46410	0.119880
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	0	0
$843$	0	0
$844$	−38.1051	−1.31163
$845$	−13.0000	−0.447214
$846$	0	0
$847$	0	0
$848$	−24.0000	−0.824163
$849$	0	0
$850$	0	0
$851$	30.0000	1.02839
$852$	0	0
$853$	43.3013	1.48261	0.741304	−	0.671170i	$-0.234208\pi$
$853$	43.3013	1.48261	0.741304	+	0.671170i	$0.234208\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.3923	−0.354994	−0.177497	−	0.984121i	$-0.556800\pi$
$857$	−10.3923	−0.354994	−0.177497	+	0.984121i	$0.556800\pi$
$858$	0	0
$859$	23.0000	0.784750	0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000	0.784750	0.392375	+	0.919805i	$0.371654\pi$
$860$	20.7846	0.708749
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−10.3923	−0.353349
$866$	0	0
$867$	0	0
$868$	3.46410	0.117579
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.73205	−0.0585540
$876$	0	0
$877$	29.4449	0.994282	0.497141	−	0.867670i	$-0.334383\pi$
$877$	29.4449	0.994282	0.497141	+	0.867670i	$0.334383\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	−11.0000	−0.370179	−0.185090	−	0.982722i	$-0.559258\pi$
$883$	−11.0000	−0.370179	−0.185090	+	0.982722i	$0.559258\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.3205	−0.581566	−0.290783	−	0.956789i	$-0.593916\pi$
$887$	−17.3205	−0.581566	−0.290783	+	0.956789i	$0.593916\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	0	0
$891$	0	0
$892$	14.0000	0.468755
$893$	62.3538	2.08659
$894$	0	0
$895$	−12.0000	−0.401116
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.92820	0.231069
$900$	0	0
$901$	20.7846	0.692436
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.00000	−0.232688
$906$	0	0
$907$	−1.00000	−0.0332045	−0.0166022	−	0.999862i	$-0.505285\pi$
$907$	−1.00000	−0.0332045	−0.0166022	+	0.999862i	$0.505285\pi$
$908$	−55.4256	−1.83936
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	28.0000	0.925146
$917$	−6.00000	−0.198137
$918$	0	0
$919$	−15.5885	−0.514216	−0.257108	−	0.966383i	$-0.582770\pi$
$919$	−15.5885	−0.514216	−0.257108	+	0.966383i	$0.582770\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−5.00000	−0.164399
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−20.7846	−0.681188
$932$	−27.7128	−0.907763
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−29.4449	−0.961922	−0.480961	−	0.876742i	$-0.659712\pi$
$937$	−29.4449	−0.961922	−0.480961	+	0.876742i	$0.659712\pi$
$938$	0	0
$939$	0	0
$940$	−24.0000	−0.782794
$941$	24.2487	0.790485	0.395243	−	0.918577i	$-0.370660\pi$
$941$	24.2487	0.790485	0.395243	+	0.918577i	$0.370660\pi$
$942$	0	0
$943$	−20.7846	−0.676840
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.3205	−0.561066	−0.280533	−	0.959844i	$-0.590511\pi$
$953$	−17.3205	−0.561066	−0.280533	+	0.959844i	$0.590511\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	41.5692	1.34444
$957$	0	0
$958$	0	0
$959$	−20.7846	−0.671170
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	13.8564	0.446285
$965$	−1.73205	−0.0557567
$966$	0	0
$967$	−43.3013	−1.39247	−0.696237	−	0.717812i	$-0.745144\pi$
$967$	−43.3013	−1.39247	−0.696237	+	0.717812i	$0.745144\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	6.00000	0.192351
$974$	0	0
$975$	0	0
$976$	48.4974	1.55236
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	0	0
$980$	8.00000	0.255551
$981$	0	0
$982$	0	0
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	0	0
$985$	−6.92820	−0.220751
$986$	0	0
$987$	0	0
$988$	0	0
$989$	62.3538	1.98274
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.0000	−0.348723
$996$	0	0
$997$	−8.66025	−0.274273	−0.137136	−	0.990552i	$-0.543790\pi$
$997$	−8.66025	−0.274273	−0.137136	+	0.990552i	$0.543790\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.q.1.1		2
3.2	odd	2		1815.2.a.g.1.1	✓	2
11.10	odd	2	inner	5445.2.a.q.1.2		2
15.14	odd	2		9075.2.a.bl.1.2		2
33.32	even	2		1815.2.a.g.1.2	yes	2
165.164	even	2		9075.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.2.a.g.1.1	✓	2	3.2	odd	2
1815.2.a.g.1.2	yes	2	33.32	even	2
5445.2.a.q.1.1		2	1.1	even	1	trivial
5445.2.a.q.1.2		2	11.10	odd	2	inner
9075.2.a.bl.1.1		2	165.164	even	2
9075.2.a.bl.1.2		2	15.14	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} +1.00000 q^{5} -1.73205 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} +1.00000 q^{5} -1.73205 q^{7} +4.00000 q^{16} -3.46410 q^{17} +5.19615 q^{19} -2.00000 q^{20} -6.00000 q^{23} +1.00000 q^{25} +3.46410 q^{28} +6.92820 q^{29} +1.00000 q^{31} -1.73205 q^{35} -5.00000 q^{37} +3.46410 q^{41} -10.3923 q^{43} +12.0000 q^{47} -4.00000 q^{49} -6.00000 q^{53} +12.1244 q^{61} -8.00000 q^{64} -5.00000 q^{67} +6.92820 q^{68} +6.00000 q^{71} +1.73205 q^{73} -10.3923 q^{76} +15.5885 q^{79} +4.00000 q^{80} -6.92820 q^{83} -3.46410 q^{85} +6.00000 q^{89} +12.0000 q^{92} +5.19615 q^{95} -13.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 2 q^{5}+O(q^{10}) \) 2 * q - 4 * q^4 + 2 * q^5	\( 2 q - 4 q^{4} + 2 q^{5} + 8 q^{16} - 4 q^{20} - 12 q^{23} + 2 q^{25} + 2 q^{31} - 10 q^{37} + 24 q^{47} - 8 q^{49} - 12 q^{53} - 16 q^{64} - 10 q^{67} + 12 q^{71} + 8 q^{80} + 12 q^{89} + 24 q^{92} - 26 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 2 * q^5 + 8 * q^16 - 4 * q^20 - 12 * q^23 + 2 * q^25 + 2 * q^31 - 10 * q^37 + 24 * q^47 - 8 * q^49 - 12 * q^53 - 16 * q^64 - 10 * q^67 + 12 * q^71 + 8 * q^80 + 12 * q^89 + 24 * q^92 - 26 * q^97

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