LMFDB - Embedded newform 1368.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(1,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.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:	$10.9235349965$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{5} +2.56155 q^{7} +O(q^{10})$	$q-2.56155 q^{5} +2.56155 q^{7} -1.43845 q^{11} -5.12311 q^{13} +5.68466 q^{17} +1.00000 q^{19} -0.876894 q^{23} +1.56155 q^{25} -8.24621 q^{29} -2.00000 q^{31} -6.56155 q^{35} -8.00000 q^{37} -3.12311 q^{41} +2.56155 q^{43} -5.68466 q^{47} -0.438447 q^{49} +12.2462 q^{53} +3.68466 q^{55} -12.0000 q^{59} -5.68466 q^{61} +13.1231 q^{65} -10.2462 q^{67} +11.9309 q^{73} -3.68466 q^{77} -13.3693 q^{79} -4.00000 q^{83} -14.5616 q^{85} +6.00000 q^{89} -13.1231 q^{91} -2.56155 q^{95} -12.2462 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} + q^{7}+O(q^{10}) $ 2 * q - q^5 + q^7	$ 2 q - q^{5} + q^{7} - 7 q^{11} - 2 q^{13} - q^{17} + 2 q^{19} - 10 q^{23} - q^{25} - 4 q^{31} - 9 q^{35} - 16 q^{37} + 2 q^{41} + q^{43} + q^{47} - 5 q^{49} + 8 q^{53} - 5 q^{55} - 24 q^{59} + q^{61} + 18 q^{65} - 4 q^{67} - 5 q^{73} + 5 q^{77} - 2 q^{79} - 8 q^{83} - 25 q^{85} + 12 q^{89} - 18 q^{91} - q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q - q^5 + q^7 - 7 * q^11 - 2 * q^13 - q^17 + 2 * q^19 - 10 * q^23 - q^25 - 4 * q^31 - 9 * q^35 - 16 * q^37 + 2 * q^41 + q^43 + q^47 - 5 * q^49 + 8 * q^53 - 5 * q^55 - 24 * q^59 + q^61 + 18 * q^65 - 4 * q^67 - 5 * q^73 + 5 * q^77 - 2 * q^79 - 8 * q^83 - 25 * q^85 + 12 * q^89 - 18 * q^91 - q^95 - 8 * 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
$5$	−2.56155	−1.14556	−0.572781	−	0.819709i	$-0.694135\pi$
$5$	−2.56155	−1.14556	−0.572781	+	0.819709i	$0.694135\pi$
$6$	0	0
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.43845	−0.433708	−0.216854	−	0.976204i	$-0.569580\pi$
$11$	−1.43845	−0.433708	−0.216854	+	0.976204i	$0.569580\pi$
$12$	0	0
$13$	−5.12311	−1.42089	−0.710447	−	0.703751i	$-0.751507\pi$
$13$	−5.12311	−1.42089	−0.710447	+	0.703751i	$0.751507\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.68466	1.37873	0.689366	−	0.724413i	$-0.257889\pi$
$17$	5.68466	1.37873	0.689366	+	0.724413i	$0.257889\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.876894	−0.182845	−0.0914226	−	0.995812i	$-0.529141\pi$
$23$	−0.876894	−0.182845	−0.0914226	+	0.995812i	$0.529141\pi$
$24$	0	0
$25$	1.56155	0.312311
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.24621	−1.53128	−0.765641	−	0.643268i	$-0.777578\pi$
$29$	−8.24621	−1.53128	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.56155	−1.10910
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.12311	−0.487747	−0.243874	−	0.969807i	$-0.578418\pi$
$41$	−3.12311	−0.487747	−0.243874	+	0.969807i	$0.578418\pi$
$42$	0	0
$43$	2.56155	0.390633	0.195317	−	0.980740i	$-0.437427\pi$
$43$	2.56155	0.390633	0.195317	+	0.980740i	$0.437427\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.68466	−0.829193	−0.414596	−	0.910005i	$-0.636077\pi$
$47$	−5.68466	−0.829193	−0.414596	+	0.910005i	$0.636077\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.2462	1.68215	0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462	1.68215	0.841073	+	0.540921i	$0.181924\pi$
$54$	0	0
$55$	3.68466	0.496839
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−5.68466	−0.727846	−0.363923	−	0.931429i	$-0.618563\pi$
$61$	−5.68466	−0.727846	−0.363923	+	0.931429i	$0.618563\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.1231	1.62772
$66$	0	0
$67$	−10.2462	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$67$	−10.2462	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	11.9309	1.39640	0.698201	−	0.715901i	$-0.253984\pi$
$73$	11.9309	1.39640	0.698201	+	0.715901i	$0.253984\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.68466	−0.419906
$78$	0	0
$79$	−13.3693	−1.50417	−0.752083	−	0.659069i	$-0.770951\pi$
$79$	−13.3693	−1.50417	−0.752083	+	0.659069i	$0.770951\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−14.5616	−1.57942
$86$	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
$91$	−13.1231	−1.37568
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.56155	−0.262810
$96$	0	0
$97$	−12.2462	−1.24341	−0.621707	−	0.783250i	$-0.713561\pi$
$97$	−12.2462	−1.24341	−0.621707	+	0.783250i	$0.713561\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.12311	0.111753	0.0558766	−	0.998438i	$-0.482205\pi$
$101$	1.12311	0.111753	0.0558766	+	0.998438i	$0.482205\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.3693	−1.09911	−0.549557	−	0.835456i	$-0.685203\pi$
$107$	−11.3693	−1.09911	−0.549557	+	0.835456i	$0.685203\pi$
$108$	0	0
$109$	14.2462	1.36454	0.682270	−	0.731101i	$-0.260993\pi$
$109$	14.2462	1.36454	0.682270	+	0.731101i	$0.260993\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	2.24621	0.209460
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.5616	1.33486
$120$	0	0
$121$	−8.93087	−0.811897
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.80776	0.787790
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.31534	−0.726515	−0.363257	−	0.931689i	$-0.618335\pi$
$131$	−8.31534	−0.726515	−0.363257	+	0.931689i	$0.618335\pi$
$132$	0	0
$133$	2.56155	0.222115
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.80776	−0.581627	−0.290813	−	0.956780i	$-0.593926\pi$
$137$	−6.80776	−0.581627	−0.290813	+	0.956780i	$0.593926\pi$
$138$	0	0
$139$	12.8078	1.08634	0.543170	−	0.839623i	$-0.317224\pi$
$139$	12.8078	1.08634	0.543170	+	0.839623i	$0.317224\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7.36932	0.616253
$144$	0	0
$145$	21.1231	1.75418
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.8078	1.37695	0.688473	−	0.725262i	$-0.258281\pi$
$149$	16.8078	1.37695	0.688473	+	0.725262i	$0.258281\pi$
$150$	0	0
$151$	−8.87689	−0.722391	−0.361196	−	0.932490i	$-0.617631\pi$
$151$	−8.87689	−0.722391	−0.361196	+	0.932490i	$0.617631\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.12311	0.411498
$156$	0	0
$157$	4.24621	0.338885	0.169442	−	0.985540i	$-0.445803\pi$
$157$	4.24621	0.338885	0.169442	+	0.985540i	$0.445803\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.24621	−0.177026
$162$	0	0
$163$	14.2462	1.11585	0.557925	−	0.829892i	$-0.311598\pi$
$163$	14.2462	1.11585	0.557925	+	0.829892i	$0.311598\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.1231	1.32503	0.662513	−	0.749051i	$-0.269490\pi$
$167$	17.1231	1.32503	0.662513	+	0.749051i	$0.269490\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.2462	−0.931062	−0.465531	−	0.885032i	$-0.654137\pi$
$173$	−12.2462	−0.931062	−0.465531	+	0.885032i	$0.654137\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.87689	0.514003	0.257002	−	0.966411i	$-0.417265\pi$
$179$	6.87689	0.514003	0.257002	+	0.966411i	$0.417265\pi$
$180$	0	0
$181$	6.87689	0.511156	0.255578	−	0.966789i	$-0.417734\pi$
$181$	6.87689	0.511156	0.255578	+	0.966789i	$0.417734\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	20.4924	1.50663
$186$	0	0
$187$	−8.17708	−0.597967
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.6847	0.990187	0.495094	−	0.868840i	$-0.335134\pi$
$191$	13.6847	0.990187	0.495094	+	0.868840i	$0.335134\pi$
$192$	0	0
$193$	27.1231	1.95236	0.976182	−	0.216954i	$-0.0696120\pi$
$193$	27.1231	1.95236	0.976182	+	0.216954i	$0.0696120\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−23.3693	−1.66499	−0.832497	−	0.554029i	$-0.813090\pi$
$197$	−23.3693	−1.66499	−0.832497	+	0.554029i	$0.813090\pi$
$198$	0	0
$199$	−25.9309	−1.83819	−0.919095	−	0.394035i	$-0.871079\pi$
$199$	−25.9309	−1.83819	−0.919095	+	0.394035i	$0.871079\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−21.1231	−1.48255
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.43845	−0.0994995
$210$	0	0
$211$	22.2462	1.53149	0.765746	−	0.643143i	$-0.222370\pi$
$211$	22.2462	1.53149	0.765746	+	0.643143i	$0.222370\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.56155	−0.447494
$216$	0	0
$217$	−5.12311	−0.347779
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−29.1231	−1.95903
$222$	0	0
$223$	13.3693	0.895276	0.447638	−	0.894215i	$-0.352265\pi$
$223$	13.3693	0.895276	0.447638	+	0.894215i	$0.352265\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.1231	1.13650	0.568250	−	0.822856i	$-0.307621\pi$
$227$	17.1231	1.13650	0.568250	+	0.822856i	$0.307621\pi$
$228$	0	0
$229$	−0.561553	−0.0371085	−0.0185542	−	0.999828i	$-0.505906\pi$
$229$	−0.561553	−0.0371085	−0.0185542	+	0.999828i	$0.505906\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.31534	0.151683	0.0758415	−	0.997120i	$-0.475836\pi$
$233$	2.31534	0.151683	0.0758415	+	0.997120i	$0.475836\pi$
$234$	0	0
$235$	14.5616	0.949891
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.8078	−1.73405	−0.867025	−	0.498265i	$-0.833971\pi$
$239$	−26.8078	−1.73405	−0.867025	+	0.498265i	$0.833971\pi$
$240$	0	0
$241$	13.3693	0.861193	0.430597	−	0.902544i	$-0.358303\pi$
$241$	13.3693	0.861193	0.430597	+	0.902544i	$0.358303\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.12311	0.0717526
$246$	0	0
$247$	−5.12311	−0.325975
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.8078	−0.808419	−0.404209	−	0.914666i	$-0.632453\pi$
$251$	−12.8078	−0.808419	−0.404209	+	0.914666i	$0.632453\pi$
$252$	0	0
$253$	1.26137	0.0793014
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.87689	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$257$	−4.87689	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$258$	0	0
$259$	−20.4924	−1.27334
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.31534	0.142770	0.0713850	−	0.997449i	$-0.477258\pi$
$263$	2.31534	0.142770	0.0713850	+	0.997449i	$0.477258\pi$
$264$	0	0
$265$	−31.3693	−1.92700
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.24621	−0.135452
$276$	0	0
$277$	−6.31534	−0.379452	−0.189726	−	0.981837i	$-0.560760\pi$
$277$	−6.31534	−0.379452	−0.189726	+	0.981837i	$0.560760\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.75379	0.462552	0.231276	−	0.972888i	$-0.425710\pi$
$281$	7.75379	0.462552	0.231276	+	0.972888i	$0.425710\pi$
$282$	0	0
$283$	16.3153	0.969846	0.484923	−	0.874557i	$-0.338848\pi$
$283$	16.3153	0.969846	0.484923	+	0.874557i	$0.338848\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	15.3153	0.900902
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.12311	−0.416136	−0.208068	−	0.978114i	$-0.566718\pi$
$293$	−7.12311	−0.416136	−0.208068	+	0.978114i	$0.566718\pi$
$294$	0	0
$295$	30.7386	1.78967
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.49242	0.259804
$300$	0	0
$301$	6.56155	0.378202
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.5616	0.833792
$306$	0	0
$307$	28.4924	1.62615	0.813074	−	0.582160i	$-0.197792\pi$
$307$	28.4924	1.62615	0.813074	+	0.582160i	$0.197792\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.93087	−0.449718	−0.224859	−	0.974391i	$-0.572192\pi$
$311$	−7.93087	−0.449718	−0.224859	+	0.974391i	$0.572192\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.1231	1.52339	0.761693	−	0.647938i	$-0.224369\pi$
$317$	27.1231	1.52339	0.761693	+	0.647938i	$0.224369\pi$
$318$	0	0
$319$	11.8617	0.664130
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.68466	0.316303
$324$	0	0
$325$	−8.00000	−0.443760
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−14.5616	−0.802804
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	26.2462	1.43398
$336$	0	0
$337$	−28.7386	−1.56549	−0.782747	−	0.622341i	$-0.786182\pi$
$337$	−28.7386	−1.56549	−0.782747	+	0.622341i	$0.786182\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.87689	0.155793
$342$	0	0
$343$	−19.0540	−1.02882
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.4233	1.84794	0.923970	−	0.382466i	$-0.124925\pi$
$347$	34.4233	1.84794	0.923970	+	0.382466i	$0.124925\pi$
$348$	0	0
$349$	29.6847	1.58898	0.794492	−	0.607275i	$-0.207737\pi$
$349$	29.6847	1.58898	0.794492	+	0.607275i	$0.207737\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.7538	0.625591	0.312796	−	0.949820i	$-0.398735\pi$
$353$	11.7538	0.625591	0.312796	+	0.949820i	$0.398735\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.56155	−0.240750	−0.120375	−	0.992729i	$-0.538410\pi$
$359$	−4.56155	−0.240750	−0.120375	+	0.992729i	$0.538410\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−30.5616	−1.59966
$366$	0	0
$367$	−14.2462	−0.743646	−0.371823	−	0.928304i	$-0.621267\pi$
$367$	−14.2462	−0.743646	−0.371823	+	0.928304i	$0.621267\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	31.3693	1.62861
$372$	0	0
$373$	−3.50758	−0.181615	−0.0908077	−	0.995868i	$-0.528945\pi$
$373$	−3.50758	−0.181615	−0.0908077	+	0.995868i	$0.528945\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	42.2462	2.17579
$378$	0	0
$379$	−29.1231	−1.49595	−0.747977	−	0.663725i	$-0.768975\pi$
$379$	−29.1231	−1.49595	−0.747977	+	0.663725i	$0.768975\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.1231	−1.07934	−0.539670	−	0.841877i	$-0.681451\pi$
$383$	−21.1231	−1.07934	−0.539670	+	0.841877i	$0.681451\pi$
$384$	0	0
$385$	9.43845	0.481028
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.93087	0.503515	0.251758	−	0.967790i	$-0.418991\pi$
$389$	9.93087	0.503515	0.251758	+	0.967790i	$0.418991\pi$
$390$	0	0
$391$	−4.98485	−0.252094
$392$	0	0
$393$	0	0
$394$	0	0
$395$	34.2462	1.72311
$396$	0	0
$397$	−10.3153	−0.517712	−0.258856	−	0.965916i	$-0.583346\pi$
$397$	−10.3153	−0.517712	−0.258856	+	0.965916i	$0.583346\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.3693	1.26688	0.633442	−	0.773790i	$-0.281642\pi$
$401$	25.3693	1.26688	0.633442	+	0.773790i	$0.281642\pi$
$402$	0	0
$403$	10.2462	0.510400
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.5076	0.570409
$408$	0	0
$409$	16.2462	0.803323	0.401662	−	0.915788i	$-0.368433\pi$
$409$	16.2462	0.803323	0.401662	+	0.915788i	$0.368433\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−30.7386	−1.51255
$414$	0	0
$415$	10.2462	0.502967
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.7386	−1.11085	−0.555427	−	0.831565i	$-0.687445\pi$
$419$	−22.7386	−1.11085	−0.555427	+	0.831565i	$0.687445\pi$
$420$	0	0
$421$	13.1231	0.639581	0.319791	−	0.947488i	$-0.396387\pi$
$421$	13.1231	0.639581	0.319791	+	0.947488i	$0.396387\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.87689	0.430593
$426$	0	0
$427$	−14.5616	−0.704683
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−29.6155	−1.42653	−0.713265	−	0.700895i	$-0.752784\pi$
$431$	−29.6155	−1.42653	−0.713265	+	0.700895i	$0.752784\pi$
$432$	0	0
$433$	8.24621	0.396288	0.198144	−	0.980173i	$-0.436509\pi$
$433$	8.24621	0.396288	0.198144	+	0.980173i	$0.436509\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.876894	−0.0419475
$438$	0	0
$439$	11.7538	0.560978	0.280489	−	0.959857i	$-0.409503\pi$
$439$	11.7538	0.560978	0.280489	+	0.959857i	$0.409503\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−32.8078	−1.55874	−0.779372	−	0.626562i	$-0.784462\pi$
$443$	−32.8078	−1.55874	−0.779372	+	0.626562i	$0.784462\pi$
$444$	0	0
$445$	−15.3693	−0.728575
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.6307	−0.501693	−0.250846	−	0.968027i	$-0.580709\pi$
$449$	−10.6307	−0.501693	−0.250846	+	0.968027i	$0.580709\pi$
$450$	0	0
$451$	4.49242	0.211540
$452$	0	0
$453$	0	0
$454$	0	0
$455$	33.6155	1.57592
$456$	0	0
$457$	−2.94602	−0.137809	−0.0689046	−	0.997623i	$-0.521950\pi$
$457$	−2.94602	−0.137809	−0.0689046	+	0.997623i	$0.521950\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.3002	0.992049	0.496024	−	0.868309i	$-0.334793\pi$
$461$	21.3002	0.992049	0.496024	+	0.868309i	$0.334793\pi$
$462$	0	0
$463$	11.1922	0.520147	0.260074	−	0.965589i	$-0.416253\pi$
$463$	11.1922	0.520147	0.260074	+	0.965589i	$0.416253\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.9309	0.829742	0.414871	−	0.909880i	$-0.363827\pi$
$467$	17.9309	0.829742	0.414871	+	0.909880i	$0.363827\pi$
$468$	0	0
$469$	−26.2462	−1.21194
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.68466	−0.169421
$474$	0	0
$475$	1.56155	0.0716490
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.36932	0.428095	0.214048	−	0.976823i	$-0.431335\pi$
$479$	9.36932	0.428095	0.214048	+	0.976823i	$0.431335\pi$
$480$	0	0
$481$	40.9848	1.86875
$482$	0	0
$483$	0	0
$484$	0	0
$485$	31.3693	1.42441
$486$	0	0
$487$	−24.8769	−1.12728	−0.563640	−	0.826021i	$-0.690599\pi$
$487$	−24.8769	−1.12728	−0.563640	+	0.826021i	$0.690599\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−34.7386	−1.56773	−0.783866	−	0.620930i	$-0.786755\pi$
$491$	−34.7386	−1.56773	−0.783866	+	0.620930i	$0.786755\pi$
$492$	0	0
$493$	−46.8769	−2.11123
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	36.1771	1.61951	0.809754	−	0.586769i	$-0.199600\pi$
$499$	36.1771	1.61951	0.809754	+	0.586769i	$0.199600\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.6155	0.696262	0.348131	−	0.937446i	$-0.386816\pi$
$503$	15.6155	0.696262	0.348131	+	0.937446i	$0.386816\pi$
$504$	0	0
$505$	−2.87689	−0.128020
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	0	0
$511$	30.5616	1.35196
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.3693	0.677253
$516$	0	0
$517$	8.17708	0.359628
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.6155	1.38510	0.692551	−	0.721369i	$-0.256487\pi$
$521$	31.6155	1.38510	0.692551	+	0.721369i	$0.256487\pi$
$522$	0	0
$523$	−7.36932	−0.322238	−0.161119	−	0.986935i	$-0.551510\pi$
$523$	−7.36932	−0.322238	−0.161119	+	0.986935i	$0.551510\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.3693	−0.495255
$528$	0	0
$529$	−22.2311	−0.966568
$530$	0	0
$531$	0	0
$532$	0	0
$533$	16.0000	0.693037
$534$	0	0
$535$	29.1231	1.25910
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.630683	0.0271654
$540$	0	0
$541$	2.80776	0.120715	0.0603576	−	0.998177i	$-0.480776\pi$
$541$	2.80776	0.120715	0.0603576	+	0.998177i	$0.480776\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−36.4924	−1.56316
$546$	0	0
$547$	−41.6155	−1.77935	−0.889676	−	0.456593i	$-0.849070\pi$
$547$	−41.6155	−1.77935	−0.889676	+	0.456593i	$0.849070\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.24621	−0.351300
$552$	0	0
$553$	−34.2462	−1.45630
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−41.4384	−1.75580	−0.877902	−	0.478841i	$-0.841057\pi$
$557$	−41.4384	−1.75580	−0.877902	+	0.478841i	$0.841057\pi$
$558$	0	0
$559$	−13.1231	−0.555048
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−5.12311	−0.215531
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.4924	−0.607554	−0.303777	−	0.952743i	$-0.598248\pi$
$569$	−14.4924	−0.607554	−0.303777	+	0.952743i	$0.598248\pi$
$570$	0	0
$571$	−42.7386	−1.78856	−0.894278	−	0.447512i	$-0.852310\pi$
$571$	−42.7386	−1.78856	−0.894278	+	0.447512i	$0.852310\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.36932	−0.0571045
$576$	0	0
$577$	−7.43845	−0.309667	−0.154833	−	0.987941i	$-0.549484\pi$
$577$	−7.43845	−0.309667	−0.154833	+	0.987941i	$0.549484\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10.2462	−0.425084
$582$	0	0
$583$	−17.6155	−0.729561
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.0540	−0.786442	−0.393221	−	0.919444i	$-0.628639\pi$
$587$	−19.0540	−0.786442	−0.393221	+	0.919444i	$0.628639\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5.50758	−0.226169	−0.113085	−	0.993585i	$-0.536073\pi$
$593$	−5.50758	−0.226169	−0.113085	+	0.993585i	$0.536073\pi$
$594$	0	0
$595$	−37.3002	−1.52916
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−48.4924	−1.98135	−0.990673	−	0.136258i	$-0.956492\pi$
$599$	−48.4924	−1.98135	−0.990673	+	0.136258i	$0.956492\pi$
$600$	0	0
$601$	−17.8617	−0.728596	−0.364298	−	0.931283i	$-0.618691\pi$
$601$	−17.8617	−0.728596	−0.364298	+	0.931283i	$0.618691\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22.8769	0.930078
$606$	0	0
$607$	42.9848	1.74470	0.872351	−	0.488881i	$-0.162595\pi$
$607$	42.9848	1.74470	0.872351	+	0.488881i	$0.162595\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.1231	1.17819
$612$	0	0
$613$	−47.7926	−1.93033	−0.965163	−	0.261651i	$-0.915733\pi$
$613$	−47.7926	−1.93033	−0.965163	+	0.261651i	$0.915733\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−39.3002	−1.58217	−0.791083	−	0.611709i	$-0.790482\pi$
$617$	−39.3002	−1.58217	−0.791083	+	0.611709i	$0.790482\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15.3693	0.615759
$624$	0	0
$625$	−30.3693	−1.21477
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−45.4773	−1.81330
$630$	0	0
$631$	−5.93087	−0.236104	−0.118052	−	0.993007i	$-0.537665\pi$
$631$	−5.93087	−0.236104	−0.118052	+	0.993007i	$0.537665\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	46.1080	1.82974
$636$	0	0
$637$	2.24621	0.0889981
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.73863	−0.345155	−0.172578	−	0.984996i	$-0.555210\pi$
$641$	−8.73863	−0.345155	−0.172578	+	0.984996i	$0.555210\pi$
$642$	0	0
$643$	47.0540	1.85563	0.927814	−	0.373044i	$-0.121686\pi$
$643$	47.0540	1.85563	0.927814	+	0.373044i	$0.121686\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.19224	0.0468716	0.0234358	−	0.999725i	$-0.492539\pi$
$647$	1.19224	0.0468716	0.0234358	+	0.999725i	$0.492539\pi$
$648$	0	0
$649$	17.2614	0.677568
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.3002	0.520477	0.260238	−	0.965544i	$-0.416199\pi$
$653$	13.3002	0.520477	0.260238	+	0.965544i	$0.416199\pi$
$654$	0	0
$655$	21.3002	0.832267
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.36932	0.131250	0.0656250	−	0.997844i	$-0.479096\pi$
$659$	3.36932	0.131250	0.0656250	+	0.997844i	$0.479096\pi$
$660$	0	0
$661$	6.73863	0.262102	0.131051	−	0.991376i	$-0.458165\pi$
$661$	6.73863	0.262102	0.131051	+	0.991376i	$0.458165\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.56155	−0.254446
$666$	0	0
$667$	7.23106	0.279988
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.17708	0.315673
$672$	0	0
$673$	25.8617	0.996897	0.498448	−	0.866919i	$-0.333903\pi$
$673$	25.8617	0.996897	0.498448	+	0.866919i	$0.333903\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.0000	−1.46046	−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000	−1.46046	−0.730229	+	0.683202i	$0.760587\pi$
$678$	0	0
$679$	−31.3693	−1.20384
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−47.8617	−1.83138	−0.915689	−	0.401887i	$-0.868354\pi$
$683$	−47.8617	−1.83138	−0.915689	+	0.401887i	$0.868354\pi$
$684$	0	0
$685$	17.4384	0.666289
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−62.7386	−2.39015
$690$	0	0
$691$	49.9309	1.89946	0.949730	−	0.313070i	$-0.101358\pi$
$691$	49.9309	1.89946	0.949730	+	0.313070i	$0.101358\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−32.8078	−1.24447
$696$	0	0
$697$	−17.7538	−0.672473
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−14.8769	−0.561893	−0.280946	−	0.959723i	$-0.590648\pi$
$701$	−14.8769	−0.561893	−0.280946	+	0.959723i	$0.590648\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.87689	0.108197
$708$	0	0
$709$	42.4924	1.59584	0.797918	−	0.602766i	$-0.205935\pi$
$709$	42.4924	1.59584	0.797918	+	0.602766i	$0.205935\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.75379	0.0656799
$714$	0	0
$715$	−18.8769	−0.705956
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24.4233	0.910835	0.455418	−	0.890278i	$-0.349490\pi$
$719$	24.4233	0.910835	0.455418	+	0.890278i	$0.349490\pi$
$720$	0	0
$721$	−15.3693	−0.572383
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.8769	−0.478236
$726$	0	0
$727$	9.43845	0.350053	0.175026	−	0.984564i	$-0.443999\pi$
$727$	9.43845	0.350053	0.175026	+	0.984564i	$0.443999\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.5616	0.538578
$732$	0	0
$733$	14.4924	0.535290	0.267645	−	0.963518i	$-0.413755\pi$
$733$	14.4924	0.535290	0.267645	+	0.963518i	$0.413755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.7386	0.542905
$738$	0	0
$739$	−20.1771	−0.742226	−0.371113	−	0.928588i	$-0.621024\pi$
$739$	−20.1771	−0.742226	−0.371113	+	0.928588i	$0.621024\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.50758	−0.275426	−0.137713	−	0.990472i	$-0.543975\pi$
$743$	−7.50758	−0.275426	−0.137713	+	0.990472i	$0.543975\pi$
$744$	0	0
$745$	−43.0540	−1.57738
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−29.1231	−1.06414
$750$	0	0
$751$	−7.61553	−0.277895	−0.138947	−	0.990300i	$-0.544372\pi$
$751$	−7.61553	−0.277895	−0.138947	+	0.990300i	$0.544372\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.7386	0.827544
$756$	0	0
$757$	5.05398	0.183690	0.0918449	−	0.995773i	$-0.470724\pi$
$757$	5.05398	0.183690	0.0918449	+	0.995773i	$0.470724\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.6695	1.11177	0.555884	−	0.831260i	$-0.312380\pi$
$761$	30.6695	1.11177	0.555884	+	0.831260i	$0.312380\pi$
$762$	0	0
$763$	36.4924	1.32111
$764$	0	0
$765$	0	0
$766$	0	0
$767$	61.4773	2.21982
$768$	0	0
$769$	19.3002	0.695983	0.347991	−	0.937498i	$-0.386864\pi$
$769$	19.3002	0.695983	0.347991	+	0.937498i	$0.386864\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	23.6155	0.849392	0.424696	−	0.905336i	$-0.360381\pi$
$773$	23.6155	0.849392	0.424696	+	0.905336i	$0.360381\pi$
$774$	0	0
$775$	−3.12311	−0.112185
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.12311	−0.111897
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.8769	−0.388213
$786$	0	0
$787$	−6.24621	−0.222653	−0.111327	−	0.993784i	$-0.535510\pi$
$787$	−6.24621	−0.222653	−0.111327	+	0.993784i	$0.535510\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.12311	0.182157
$792$	0	0
$793$	29.1231	1.03419
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.384472	−0.0136187	−0.00680935	−	0.999977i	$-0.502167\pi$
$797$	−0.384472	−0.0136187	−0.00680935	+	0.999977i	$0.502167\pi$
$798$	0	0
$799$	−32.3153	−1.14323
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17.1619	−0.605631
$804$	0	0
$805$	5.75379	0.202794
$806$	0	0
$807$	0	0
$808$	0	0
$809$	45.0540	1.58401	0.792007	−	0.610512i	$-0.209036\pi$
$809$	45.0540	1.58401	0.792007	+	0.610512i	$0.209036\pi$
$810$	0	0
$811$	−26.8769	−0.943775	−0.471888	−	0.881659i	$-0.656427\pi$
$811$	−26.8769	−0.943775	−0.471888	+	0.881659i	$0.656427\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−36.4924	−1.27827
$816$	0	0
$817$	2.56155	0.0896174
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.0540	1.36299	0.681497	−	0.731821i	$-0.261329\pi$
$821$	39.0540	1.36299	0.681497	+	0.731821i	$0.261329\pi$
$822$	0	0
$823$	13.3002	0.463615	0.231808	−	0.972762i	$-0.425536\pi$
$823$	13.3002	0.463615	0.231808	+	0.972762i	$0.425536\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.7386	−1.20798	−0.603990	−	0.796992i	$-0.706423\pi$
$827$	−34.7386	−1.20798	−0.603990	+	0.796992i	$0.706423\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.49242	−0.0863573
$834$	0	0
$835$	−43.8617	−1.51790
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.86174	−0.271417	−0.135709	−	0.990749i	$-0.543331\pi$
$839$	−7.86174	−0.271417	−0.135709	+	0.990749i	$0.543331\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−33.9309	−1.16726
$846$	0	0
$847$	−22.8769	−0.786059
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.01515	0.240476
$852$	0	0
$853$	−56.7386	−1.94269	−0.971347	−	0.237666i	$-0.923618\pi$
$853$	−56.7386	−1.94269	−0.971347	+	0.237666i	$0.923618\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.8617	1.42997	0.714985	−	0.699140i	$-0.246434\pi$
$857$	41.8617	1.42997	0.714985	+	0.699140i	$0.246434\pi$
$858$	0	0
$859$	25.3002	0.863231	0.431616	−	0.902058i	$-0.357944\pi$
$859$	25.3002	0.863231	0.431616	+	0.902058i	$0.357944\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	31.3693	1.06659
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19.2311	0.652369
$870$	0	0
$871$	52.4924	1.77864
$872$	0	0
$873$	0	0
$874$	0	0
$875$	22.5616	0.762720
$876$	0	0
$877$	−47.3693	−1.59955	−0.799774	−	0.600301i	$-0.795047\pi$
$877$	−47.3693	−1.59955	−0.799774	+	0.600301i	$0.795047\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.3002	−0.785003	−0.392502	−	0.919751i	$-0.628390\pi$
$881$	−23.3002	−0.785003	−0.392502	+	0.919751i	$0.628390\pi$
$882$	0	0
$883$	−47.6847	−1.60472	−0.802358	−	0.596843i	$-0.796422\pi$
$883$	−47.6847	−1.60472	−0.802358	+	0.596843i	$0.796422\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.2311	1.04864	0.524318	−	0.851522i	$-0.324320\pi$
$887$	31.2311	1.04864	0.524318	+	0.851522i	$0.324320\pi$
$888$	0	0
$889$	−46.1080	−1.54641
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.68466	−0.190230
$894$	0	0
$895$	−17.6155	−0.588822
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16.4924	0.550053
$900$	0	0
$901$	69.6155	2.31923
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.6155	−0.585560
$906$	0	0
$907$	−34.8769	−1.15807	−0.579034	−	0.815303i	$-0.696570\pi$
$907$	−34.8769	−1.15807	−0.579034	+	0.815303i	$0.696570\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.12311	−0.0372101	−0.0186051	−	0.999827i	$-0.505923\pi$
$911$	−1.12311	−0.0372101	−0.0186051	+	0.999827i	$0.505923\pi$
$912$	0	0
$913$	5.75379	0.190423
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.3002	−0.703394
$918$	0	0
$919$	44.4924	1.46767	0.733835	−	0.679328i	$-0.237729\pi$
$919$	44.4924	1.46767	0.733835	+	0.679328i	$0.237729\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−12.4924	−0.410748
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−0.438447	−0.0143695
$932$	0	0
$933$	0	0
$934$	0	0
$935$	20.9460	0.685008
$936$	0	0
$937$	−16.5616	−0.541042	−0.270521	−	0.962714i	$-0.587196\pi$
$937$	−16.5616	−0.541042	−0.270521	+	0.962714i	$0.587196\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	2.73863	0.0891822
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	−61.1231	−1.98414
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.6155	0.376264	0.188132	−	0.982144i	$-0.439757\pi$
$953$	11.6155	0.376264	0.188132	+	0.982144i	$0.439757\pi$
$954$	0	0
$955$	−35.0540	−1.13432
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17.4384	−0.563117
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−69.4773	−2.23655
$966$	0	0
$967$	−26.7386	−0.859856	−0.429928	−	0.902863i	$-0.641461\pi$
$967$	−26.7386	−0.859856	−0.429928	+	0.902863i	$0.641461\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	32.8078	1.05177
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.98485	−0.223465	−0.111732	−	0.993738i	$-0.535640\pi$
$977$	−6.98485	−0.223465	−0.111732	+	0.993738i	$0.535640\pi$
$978$	0	0
$979$	−8.63068	−0.275838
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−30.7386	−0.980410	−0.490205	−	0.871607i	$-0.663078\pi$
$983$	−30.7386	−0.980410	−0.490205	+	0.871607i	$0.663078\pi$
$984$	0	0
$985$	59.8617	1.90735
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.24621	−0.0714254
$990$	0	0
$991$	−21.8617	−0.694461	−0.347231	−	0.937780i	$-0.612878\pi$
$991$	−21.8617	−0.694461	−0.347231	+	0.937780i	$0.612878\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	66.4233	2.10576
$996$	0	0
$997$	−55.7926	−1.76697	−0.883485	−	0.468460i	$-0.844809\pi$
$997$	−55.7926	−1.76697	−0.883485	+	0.468460i	$0.844809\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.a.k.1.1		2
3.2	odd	2		456.2.a.f.1.2	✓	2
4.3	odd	2		2736.2.a.z.1.1		2
12.11	even	2		912.2.a.m.1.2		2
24.5	odd	2		3648.2.a.bl.1.1		2
24.11	even	2		3648.2.a.br.1.1		2
57.56	even	2		8664.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.a.f.1.2	✓	2	3.2	odd	2
912.2.a.m.1.2		2	12.11	even	2
1368.2.a.k.1.1		2	1.1	even	1	trivial
2736.2.a.z.1.1		2	4.3	odd	2
3648.2.a.bl.1.1		2	24.5	odd	2
3648.2.a.br.1.1		2	24.11	even	2
8664.2.a.r.1.2		2	57.56	even	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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{5} +2.56155 q^{7} +O(q^{10})\)	\(q-2.56155 q^{5} +2.56155 q^{7} -1.43845 q^{11} -5.12311 q^{13} +5.68466 q^{17} +1.00000 q^{19} -0.876894 q^{23} +1.56155 q^{25} -8.24621 q^{29} -2.00000 q^{31} -6.56155 q^{35} -8.00000 q^{37} -3.12311 q^{41} +2.56155 q^{43} -5.68466 q^{47} -0.438447 q^{49} +12.2462 q^{53} +3.68466 q^{55} -12.0000 q^{59} -5.68466 q^{61} +13.1231 q^{65} -10.2462 q^{67} +11.9309 q^{73} -3.68466 q^{77} -13.3693 q^{79} -4.00000 q^{83} -14.5616 q^{85} +6.00000 q^{89} -13.1231 q^{91} -2.56155 q^{95} -12.2462 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} + q^{7}+O(q^{10}) \) 2 * q - q^5 + q^7	\( 2 q - q^{5} + q^{7} - 7 q^{11} - 2 q^{13} - q^{17} + 2 q^{19} - 10 q^{23} - q^{25} - 4 q^{31} - 9 q^{35} - 16 q^{37} + 2 q^{41} + q^{43} + q^{47} - 5 q^{49} + 8 q^{53} - 5 q^{55} - 24 q^{59} + q^{61} + 18 q^{65} - 4 q^{67} - 5 q^{73} + 5 q^{77} - 2 q^{79} - 8 q^{83} - 25 q^{85} + 12 q^{89} - 18 q^{91} - q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q - q^5 + q^7 - 7 * q^11 - 2 * q^13 - q^17 + 2 * q^19 - 10 * q^23 - q^25 - 4 * q^31 - 9 * q^35 - 16 * q^37 + 2 * q^41 + q^43 + q^47 - 5 * q^49 + 8 * q^53 - 5 * q^55 - 24 * q^59 + q^61 + 18 * q^65 - 4 * q^67 - 5 * q^73 + 5 * q^77 - 2 * q^79 - 8 * q^83 - 25 * q^85 + 12 * q^89 - 18 * q^91 - q^95 - 8 * q^97

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