LMFDB - Embedded newform 976.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [976,2,Mod(1,976)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(976, 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("976.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 976 = 2^{4} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	976.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:	$7.79339923728$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 488)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	976.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.41421 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{5} +2.41421 q^{7} -3.00000 q^{9} -3.24264 q^{11} -0.171573 q^{13} -7.65685 q^{17} +4.82843 q^{19} +1.24264 q^{23} -4.00000 q^{25} -4.00000 q^{29} -7.65685 q^{31} -2.41421 q^{35} +3.65685 q^{37} +2.65685 q^{41} +7.65685 q^{43} +3.00000 q^{45} -9.65685 q^{47} -1.17157 q^{49} -9.31371 q^{53} +3.24264 q^{55} +6.89949 q^{59} -1.00000 q^{61} -7.24264 q^{63} +0.171573 q^{65} +1.92893 q^{67} -14.0000 q^{71} -13.4853 q^{73} -7.82843 q^{77} -2.75736 q^{79} +9.00000 q^{81} -15.3137 q^{83} +7.65685 q^{85} +16.9706 q^{89} -0.414214 q^{91} -4.82843 q^{95} +10.0000 q^{97} +9.72792 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7 - 6 * q^9	$ 2 q - 2 q^{5} + 2 q^{7} - 6 q^{9} + 2 q^{11} - 6 q^{13} - 4 q^{17} + 4 q^{19} - 6 q^{23} - 8 q^{25} - 8 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{37} - 6 q^{41} + 4 q^{43} + 6 q^{45} - 8 q^{47} - 8 q^{49} + 4 q^{53} - 2 q^{55} - 6 q^{59} - 2 q^{61} - 6 q^{63} + 6 q^{65} + 18 q^{67} - 28 q^{71} - 10 q^{73} - 10 q^{77} - 14 q^{79} + 18 q^{81} - 8 q^{83} + 4 q^{85} + 2 q^{91} - 4 q^{95} + 20 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 - 6 * q^9 + 2 * q^11 - 6 * q^13 - 4 * q^17 + 4 * q^19 - 6 * q^23 - 8 * q^25 - 8 * q^29 - 4 * q^31 - 2 * q^35 - 4 * q^37 - 6 * q^41 + 4 * q^43 + 6 * q^45 - 8 * q^47 - 8 * q^49 + 4 * q^53 - 2 * q^55 - 6 * q^59 - 2 * q^61 - 6 * q^63 + 6 * q^65 + 18 * q^67 - 28 * q^71 - 10 * q^73 - 10 * q^77 - 14 * q^79 + 18 * q^81 - 8 * q^83 + 4 * q^85 + 2 * q^91 - 4 * q^95 + 20 * q^97 - 6 * q^99

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		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	2.41421	0.912487	0.456243	−	0.889855i	$-0.349195\pi$
$7$	2.41421	0.912487	0.456243	+	0.889855i	$0.349195\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−3.24264	−0.977693	−0.488846	−	0.872370i	$-0.662582\pi$
$11$	−3.24264	−0.977693	−0.488846	+	0.872370i	$0.662582\pi$
$12$	0	0
$13$	−0.171573	−0.0475858	−0.0237929	−	0.999717i	$-0.507574\pi$
$13$	−0.171573	−0.0475858	−0.0237929	+	0.999717i	$0.507574\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0	0
$19$	4.82843	1.10772	0.553859	−	0.832611i	$-0.313155\pi$
$19$	4.82843	1.10772	0.553859	+	0.832611i	$0.313155\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.24264	0.259108	0.129554	−	0.991572i	$-0.458645\pi$
$23$	1.24264	0.259108	0.129554	+	0.991572i	$0.458645\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−7.65685	−1.37521	−0.687606	−	0.726084i	$-0.741338\pi$
$31$	−7.65685	−1.37521	−0.687606	+	0.726084i	$0.741338\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.41421	−0.408077
$36$	0	0
$37$	3.65685	0.601183	0.300592	−	0.953753i	$-0.402816\pi$
$37$	3.65685	0.601183	0.300592	+	0.953753i	$0.402816\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.65685	0.414931	0.207465	−	0.978242i	$-0.433479\pi$
$41$	2.65685	0.414931	0.207465	+	0.978242i	$0.433479\pi$
$42$	0	0
$43$	7.65685	1.16766	0.583830	−	0.811876i	$-0.301554\pi$
$43$	7.65685	1.16766	0.583830	+	0.811876i	$0.301554\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	−9.65685	−1.40860	−0.704298	−	0.709904i	$-0.748738\pi$
$47$	−9.65685	−1.40860	−0.704298	+	0.709904i	$0.748738\pi$
$48$	0	0
$49$	−1.17157	−0.167368
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.31371	−1.27934	−0.639668	−	0.768651i	$-0.720928\pi$
$53$	−9.31371	−1.27934	−0.639668	+	0.768651i	$0.720928\pi$
$54$	0	0
$55$	3.24264	0.437238
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.89949	0.898238	0.449119	−	0.893472i	$-0.351738\pi$
$59$	6.89949	0.898238	0.449119	+	0.893472i	$0.351738\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	−7.24264	−0.912487
$64$	0	0
$65$	0.171573	0.0212810
$66$	0	0
$67$	1.92893	0.235657	0.117828	−	0.993034i	$-0.462407\pi$
$67$	1.92893	0.235657	0.117828	+	0.993034i	$0.462407\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	−13.4853	−1.57833	−0.789166	−	0.614179i	$-0.789487\pi$
$73$	−13.4853	−1.57833	−0.789166	+	0.614179i	$0.789487\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.82843	−0.892132
$78$	0	0
$79$	−2.75736	−0.310227	−0.155114	−	0.987897i	$-0.549574\pi$
$79$	−2.75736	−0.310227	−0.155114	+	0.987897i	$0.549574\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−15.3137	−1.68090	−0.840449	−	0.541891i	$-0.817709\pi$
$83$	−15.3137	−1.68090	−0.840449	+	0.541891i	$0.817709\pi$
$84$	0	0
$85$	7.65685	0.830502
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.9706	1.79888	0.899438	−	0.437048i	$-0.143976\pi$
$89$	16.9706	1.79888	0.899438	+	0.437048i	$0.143976\pi$
$90$	0	0
$91$	−0.414214	−0.0434214
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.82843	−0.495386
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	9.72792	0.977693
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	4.82843	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$103$	4.82843	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.17157	0.306608	0.153304	−	0.988179i	$-0.451009\pi$
$107$	3.17157	0.306608	0.153304	+	0.988179i	$0.451009\pi$
$108$	0	0
$109$	−17.4853	−1.67479	−0.837393	−	0.546601i	$-0.815921\pi$
$109$	−17.4853	−1.67479	−0.837393	+	0.546601i	$0.815921\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.65685	0.249936	0.124968	−	0.992161i	$-0.460117\pi$
$113$	2.65685	0.249936	0.124968	+	0.992161i	$0.460117\pi$
$114$	0	0
$115$	−1.24264	−0.115877
$116$	0	0
$117$	0.514719	0.0475858
$118$	0	0
$119$	−18.4853	−1.69454
$120$	0	0
$121$	−0.485281	−0.0441165
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.4853	1.26558	0.632792	−	0.774321i	$-0.281909\pi$
$131$	14.4853	1.26558	0.632792	+	0.774321i	$0.281909\pi$
$132$	0	0
$133$	11.6569	1.01078
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.82843	−0.327085	−0.163542	−	0.986536i	$-0.552292\pi$
$137$	−3.82843	−0.327085	−0.163542	+	0.986536i	$0.552292\pi$
$138$	0	0
$139$	−11.7279	−0.994749	−0.497375	−	0.867536i	$-0.665703\pi$
$139$	−11.7279	−0.994749	−0.497375	+	0.867536i	$0.665703\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.556349	0.0465243
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.1421	1.07665	0.538323	−	0.842739i	$-0.319058\pi$
$149$	13.1421	1.07665	0.538323	+	0.842739i	$0.319058\pi$
$150$	0	0
$151$	19.2426	1.56594	0.782972	−	0.622057i	$-0.213703\pi$
$151$	19.2426	1.56594	0.782972	+	0.622057i	$0.213703\pi$
$152$	0	0
$153$	22.9706	1.85706
$154$	0	0
$155$	7.65685	0.615013
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	6.34315	0.496834	0.248417	−	0.968653i	$-0.420090\pi$
$163$	6.34315	0.496834	0.248417	+	0.968653i	$0.420090\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.51472	−0.117212	−0.0586062	−	0.998281i	$-0.518666\pi$
$167$	−1.51472	−0.117212	−0.0586062	+	0.998281i	$0.518666\pi$
$168$	0	0
$169$	−12.9706	−0.997736
$170$	0	0
$171$	−14.4853	−1.10772
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−9.65685	−0.729990
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.8284	1.25782	0.628908	−	0.777480i	$-0.283502\pi$
$179$	16.8284	1.25782	0.628908	+	0.777480i	$0.283502\pi$
$180$	0	0
$181$	10.3431	0.768800	0.384400	−	0.923167i	$-0.374408\pi$
$181$	10.3431	0.768800	0.384400	+	0.923167i	$0.374408\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.65685	−0.268857
$186$	0	0
$187$	24.8284	1.81563
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.24264	0.234629	0.117315	−	0.993095i	$-0.462571\pi$
$191$	3.24264	0.234629	0.117315	+	0.993095i	$0.462571\pi$
$192$	0	0
$193$	−2.34315	−0.168663	−0.0843317	−	0.996438i	$-0.526876\pi$
$193$	−2.34315	−0.168663	−0.0843317	+	0.996438i	$0.526876\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.65685	0.616775	0.308388	−	0.951261i	$-0.400211\pi$
$197$	8.65685	0.616775	0.308388	+	0.951261i	$0.400211\pi$
$198$	0	0
$199$	−13.7990	−0.978184	−0.489092	−	0.872232i	$-0.662672\pi$
$199$	−13.7990	−0.978184	−0.489092	+	0.872232i	$0.662672\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.65685	−0.677778
$204$	0	0
$205$	−2.65685	−0.185563
$206$	0	0
$207$	−3.72792	−0.259108
$208$	0	0
$209$	−15.6569	−1.08301
$210$	0	0
$211$	9.31371	0.641182	0.320591	−	0.947218i	$-0.396118\pi$
$211$	9.31371	0.641182	0.320591	+	0.947218i	$0.396118\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.65685	−0.522193
$216$	0	0
$217$	−18.4853	−1.25486
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.31371	0.0883696
$222$	0	0
$223$	25.2426	1.69037	0.845186	−	0.534472i	$-0.179490\pi$
$223$	25.2426	1.69037	0.845186	+	0.534472i	$0.179490\pi$
$224$	0	0
$225$	12.0000	0.800000
$226$	0	0
$227$	13.5858	0.901720	0.450860	−	0.892595i	$-0.351117\pi$
$227$	13.5858	0.901720	0.450860	+	0.892595i	$0.351117\pi$
$228$	0	0
$229$	−14.6569	−0.968552	−0.484276	−	0.874915i	$-0.660917\pi$
$229$	−14.6569	−0.968552	−0.484276	+	0.874915i	$0.660917\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.9706	−1.63588	−0.817938	−	0.575306i	$-0.804883\pi$
$233$	−24.9706	−1.63588	−0.817938	+	0.575306i	$0.804883\pi$
$234$	0	0
$235$	9.65685	0.629944
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.31371	−0.473084	−0.236542	−	0.971621i	$-0.576014\pi$
$239$	−7.31371	−0.473084	−0.236542	+	0.971621i	$0.576014\pi$
$240$	0	0
$241$	1.82843	0.117779	0.0588897	−	0.998264i	$-0.481244\pi$
$241$	1.82843	0.117779	0.0588897	+	0.998264i	$0.481244\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.17157	0.0748490
$246$	0	0
$247$	−0.828427	−0.0527116
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.68629	−0.169557	−0.0847786	−	0.996400i	$-0.527018\pi$
$251$	−2.68629	−0.169557	−0.0847786	+	0.996400i	$0.527018\pi$
$252$	0	0
$253$	−4.02944	−0.253329
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	8.82843	0.548572
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	8.82843	0.544384	0.272192	−	0.962243i	$-0.412251\pi$
$263$	8.82843	0.544384	0.272192	+	0.962243i	$0.412251\pi$
$264$	0	0
$265$	9.31371	0.572137
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	7.31371	0.444276	0.222138	−	0.975015i	$-0.428696\pi$
$271$	7.31371	0.444276	0.222138	+	0.975015i	$0.428696\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.9706	0.782154
$276$	0	0
$277$	19.3137	1.16045	0.580224	−	0.814457i	$-0.302965\pi$
$277$	19.3137	1.16045	0.580224	+	0.814457i	$0.302965\pi$
$278$	0	0
$279$	22.9706	1.37521
$280$	0	0
$281$	−5.65685	−0.337460	−0.168730	−	0.985662i	$-0.553967\pi$
$281$	−5.65685	−0.337460	−0.168730	+	0.985662i	$0.553967\pi$
$282$	0	0
$283$	−7.31371	−0.434755	−0.217377	−	0.976088i	$-0.569750\pi$
$283$	−7.31371	−0.434755	−0.217377	+	0.976088i	$0.569750\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.41421	0.378619
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−28.6274	−1.67243	−0.836216	−	0.548401i	$-0.815237\pi$
$293$	−28.6274	−1.67243	−0.836216	+	0.548401i	$0.815237\pi$
$294$	0	0
$295$	−6.89949	−0.401704
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.213203	−0.0123299
$300$	0	0
$301$	18.4853	1.06547
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.00000	0.0572598
$306$	0	0
$307$	16.8995	0.964505	0.482253	−	0.876032i	$-0.339819\pi$
$307$	16.8995	0.964505	0.482253	+	0.876032i	$0.339819\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.41421	0.477126	0.238563	−	0.971127i	$-0.423324\pi$
$311$	8.41421	0.477126	0.238563	+	0.971127i	$0.423324\pi$
$312$	0	0
$313$	1.31371	0.0742552	0.0371276	−	0.999311i	$-0.488179\pi$
$313$	1.31371	0.0742552	0.0371276	+	0.999311i	$0.488179\pi$
$314$	0	0
$315$	7.24264	0.408077
$316$	0	0
$317$	1.31371	0.0737852	0.0368926	−	0.999319i	$-0.488254\pi$
$317$	1.31371	0.0737852	0.0368926	+	0.999319i	$0.488254\pi$
$318$	0	0
$319$	12.9706	0.726212
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−36.9706	−2.05710
$324$	0	0
$325$	0.686292	0.0380686
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−23.3137	−1.28533
$330$	0	0
$331$	22.0711	1.21314	0.606568	−	0.795032i	$-0.292546\pi$
$331$	22.0711	1.21314	0.606568	+	0.795032i	$0.292546\pi$
$332$	0	0
$333$	−10.9706	−0.601183
$334$	0	0
$335$	−1.92893	−0.105389
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.8284	1.34453
$342$	0	0
$343$	−19.7279	−1.06521
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.1716	1.24392	0.621958	−	0.783051i	$-0.286338\pi$
$347$	23.1716	1.24392	0.621958	+	0.783051i	$0.286338\pi$
$348$	0	0
$349$	−34.2843	−1.83519	−0.917597	−	0.397511i	$-0.869874\pi$
$349$	−34.2843	−1.83519	−0.917597	+	0.397511i	$0.869874\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.82843	0.203767	0.101883	−	0.994796i	$-0.467513\pi$
$353$	3.82843	0.203767	0.101883	+	0.994796i	$0.467513\pi$
$354$	0	0
$355$	14.0000	0.743043
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−34.9706	−1.84568	−0.922838	−	0.385189i	$-0.874136\pi$
$359$	−34.9706	−1.84568	−0.922838	+	0.385189i	$0.874136\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.4853	0.705852
$366$	0	0
$367$	−12.1421	−0.633814	−0.316907	−	0.948457i	$-0.602644\pi$
$367$	−12.1421	−0.633814	−0.316907	+	0.948457i	$0.602644\pi$
$368$	0	0
$369$	−7.97056	−0.414931
$370$	0	0
$371$	−22.4853	−1.16738
$372$	0	0
$373$	−2.34315	−0.121323	−0.0606617	−	0.998158i	$-0.519321\pi$
$373$	−2.34315	−0.121323	−0.0606617	+	0.998158i	$0.519321\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.686292	0.0353458
$378$	0	0
$379$	−25.6569	−1.31790	−0.658952	−	0.752185i	$-0.729000\pi$
$379$	−25.6569	−1.31790	−0.658952	+	0.752185i	$0.729000\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.5858	−1.00079	−0.500393	−	0.865798i	$-0.666811\pi$
$383$	−19.5858	−1.00079	−0.500393	+	0.865798i	$0.666811\pi$
$384$	0	0
$385$	7.82843	0.398974
$386$	0	0
$387$	−22.9706	−1.16766
$388$	0	0
$389$	6.97056	0.353422	0.176711	−	0.984263i	$-0.443454\pi$
$389$	6.97056	0.353422	0.176711	+	0.984263i	$0.443454\pi$
$390$	0	0
$391$	−9.51472	−0.481180
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.75736	0.138738
$396$	0	0
$397$	−28.6274	−1.43677	−0.718384	−	0.695646i	$-0.755118\pi$
$397$	−28.6274	−1.43677	−0.718384	+	0.695646i	$0.755118\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.6863	−0.633523	−0.316762	−	0.948505i	$-0.602596\pi$
$401$	−12.6863	−0.633523	−0.316762	+	0.948505i	$0.602596\pi$
$402$	0	0
$403$	1.31371	0.0654405
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	−11.8579	−0.587773
$408$	0	0
$409$	14.9706	0.740247	0.370123	−	0.928983i	$-0.379315\pi$
$409$	14.9706	0.740247	0.370123	+	0.928983i	$0.379315\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.6569	0.819630
$414$	0	0
$415$	15.3137	0.751720
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	−23.6569	−1.15296	−0.576482	−	0.817110i	$-0.695575\pi$
$421$	−23.6569	−1.15296	−0.576482	+	0.817110i	$0.695575\pi$
$422$	0	0
$423$	28.9706	1.40860
$424$	0	0
$425$	30.6274	1.48565
$426$	0	0
$427$	−2.41421	−0.116832
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.4853	−1.66110	−0.830549	−	0.556946i	$-0.811973\pi$
$431$	−34.4853	−1.66110	−0.830549	+	0.556946i	$0.811973\pi$
$432$	0	0
$433$	0.686292	0.0329811	0.0164905	−	0.999864i	$-0.494751\pi$
$433$	0.686292	0.0329811	0.0164905	+	0.999864i	$0.494751\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000	0.287019
$438$	0	0
$439$	−18.4853	−0.882254	−0.441127	−	0.897445i	$-0.645421\pi$
$439$	−18.4853	−0.882254	−0.441127	+	0.897445i	$0.645421\pi$
$440$	0	0
$441$	3.51472	0.167368
$442$	0	0
$443$	28.2843	1.34383	0.671913	−	0.740630i	$-0.265473\pi$
$443$	28.2843	1.34383	0.671913	+	0.740630i	$0.265473\pi$
$444$	0	0
$445$	−16.9706	−0.804482
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.48528	−0.0700948	−0.0350474	−	0.999386i	$-0.511158\pi$
$449$	−1.48528	−0.0700948	−0.0350474	+	0.999386i	$0.511158\pi$
$450$	0	0
$451$	−8.61522	−0.405675
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.414214	0.0194186
$456$	0	0
$457$	−14.6863	−0.686996	−0.343498	−	0.939153i	$-0.611612\pi$
$457$	−14.6863	−0.686996	−0.343498	+	0.939153i	$0.611612\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.31371	0.433783	0.216891	−	0.976196i	$-0.430408\pi$
$461$	9.31371	0.433783	0.216891	+	0.976196i	$0.430408\pi$
$462$	0	0
$463$	−9.65685	−0.448792	−0.224396	−	0.974498i	$-0.572041\pi$
$463$	−9.65685	−0.448792	−0.224396	+	0.974498i	$0.572041\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.7574	1.05308	0.526542	−	0.850149i	$-0.323488\pi$
$467$	22.7574	1.05308	0.526542	+	0.850149i	$0.323488\pi$
$468$	0	0
$469$	4.65685	0.215034
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.8284	−1.14161
$474$	0	0
$475$	−19.3137	−0.886174
$476$	0	0
$477$	27.9411	1.27934
$478$	0	0
$479$	−7.85786	−0.359035	−0.179517	−	0.983755i	$-0.557454\pi$
$479$	−7.85786	−0.359035	−0.179517	+	0.983755i	$0.557454\pi$
$480$	0	0
$481$	−0.627417	−0.0286078
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	2.48528	0.112619	0.0563094	−	0.998413i	$-0.482067\pi$
$487$	2.48528	0.112619	0.0563094	+	0.998413i	$0.482067\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−42.4853	−1.91733	−0.958667	−	0.284531i	$-0.908162\pi$
$491$	−42.4853	−1.91733	−0.958667	+	0.284531i	$0.908162\pi$
$492$	0	0
$493$	30.6274	1.37939
$494$	0	0
$495$	−9.72792	−0.437238
$496$	0	0
$497$	−33.7990	−1.51609
$498$	0	0
$499$	2.89949	0.129799	0.0648996	−	0.997892i	$-0.479327\pi$
$499$	2.89949	0.129799	0.0648996	+	0.997892i	$0.479327\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−32.2843	−1.43948	−0.719742	−	0.694241i	$-0.755740\pi$
$503$	−32.2843	−1.43948	−0.719742	+	0.694241i	$0.755740\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.6274	−1.62348	−0.811741	−	0.584018i	$-0.801480\pi$
$509$	−36.6274	−1.62348	−0.811741	+	0.584018i	$0.801480\pi$
$510$	0	0
$511$	−32.5563	−1.44021
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.82843	−0.212766
$516$	0	0
$517$	31.3137	1.37718
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.6569	1.56216	0.781078	−	0.624434i	$-0.214670\pi$
$521$	35.6569	1.56216	0.781078	+	0.624434i	$0.214670\pi$
$522$	0	0
$523$	0.0710678	0.00310758	0.00155379	−	0.999999i	$-0.499505\pi$
$523$	0.0710678	0.00310758	0.00155379	+	0.999999i	$0.499505\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	58.6274	2.55385
$528$	0	0
$529$	−21.4558	−0.932863
$530$	0	0
$531$	−20.6985	−0.898238
$532$	0	0
$533$	−0.455844	−0.0197448
$534$	0	0
$535$	−3.17157	−0.137119
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.79899	0.163634
$540$	0	0
$541$	40.2843	1.73196	0.865978	−	0.500082i	$-0.166697\pi$
$541$	40.2843	1.73196	0.865978	+	0.500082i	$0.166697\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	17.4853	0.748987
$546$	0	0
$547$	−20.2132	−0.864254	−0.432127	−	0.901813i	$-0.642237\pi$
$547$	−20.2132	−0.864254	−0.432127	+	0.901813i	$0.642237\pi$
$548$	0	0
$549$	3.00000	0.128037
$550$	0	0
$551$	−19.3137	−0.822792
$552$	0	0
$553$	−6.65685	−0.283078
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.6863	−0.707021	−0.353510	−	0.935431i	$-0.615012\pi$
$557$	−16.6863	−0.707021	−0.353510	+	0.935431i	$0.615012\pi$
$558$	0	0
$559$	−1.31371	−0.0555639
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−32.9706	−1.38954	−0.694772	−	0.719230i	$-0.744495\pi$
$563$	−32.9706	−1.38954	−0.694772	+	0.719230i	$0.744495\pi$
$564$	0	0
$565$	−2.65685	−0.111775
$566$	0	0
$567$	21.7279	0.912487
$568$	0	0
$569$	5.82843	0.244340	0.122170	−	0.992509i	$-0.461015\pi$
$569$	5.82843	0.244340	0.122170	+	0.992509i	$0.461015\pi$
$570$	0	0
$571$	25.7990	1.07965	0.539827	−	0.841776i	$-0.318490\pi$
$571$	25.7990	1.07965	0.539827	+	0.841776i	$0.318490\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.97056	−0.207287
$576$	0	0
$577$	29.6569	1.23463	0.617315	−	0.786716i	$-0.288220\pi$
$577$	29.6569	1.23463	0.617315	+	0.786716i	$0.288220\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−36.9706	−1.53380
$582$	0	0
$583$	30.2010	1.25080
$584$	0	0
$585$	−0.514719	−0.0212810
$586$	0	0
$587$	9.31371	0.384418	0.192209	−	0.981354i	$-0.438435\pi$
$587$	9.31371	0.384418	0.192209	+	0.981354i	$0.438435\pi$
$588$	0	0
$589$	−36.9706	−1.52335
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.2843	1.16150	0.580748	−	0.814083i	$-0.302760\pi$
$593$	28.2843	1.16150	0.580748	+	0.814083i	$0.302760\pi$
$594$	0	0
$595$	18.4853	0.757823
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.3848	0.955476	0.477738	−	0.878502i	$-0.341457\pi$
$599$	23.3848	0.955476	0.477738	+	0.878502i	$0.341457\pi$
$600$	0	0
$601$	−11.9706	−0.488289	−0.244145	−	0.969739i	$-0.578507\pi$
$601$	−11.9706	−0.488289	−0.244145	+	0.969739i	$0.578507\pi$
$602$	0	0
$603$	−5.78680	−0.235657
$604$	0	0
$605$	0.485281	0.0197295
$606$	0	0
$607$	36.4264	1.47850	0.739251	−	0.673430i	$-0.235180\pi$
$607$	36.4264	1.47850	0.739251	+	0.673430i	$0.235180\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.65685	0.0670291
$612$	0	0
$613$	13.3137	0.537736	0.268868	−	0.963177i	$-0.413351\pi$
$613$	13.3137	0.537736	0.268868	+	0.963177i	$0.413351\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.6569	−1.43549	−0.717745	−	0.696306i	$-0.754826\pi$
$617$	−35.6569	−1.43549	−0.717745	+	0.696306i	$0.754826\pi$
$618$	0	0
$619$	−9.65685	−0.388142	−0.194071	−	0.980988i	$-0.562169\pi$
$619$	−9.65685	−0.388142	−0.194071	+	0.980988i	$0.562169\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	40.9706	1.64145
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.0000	−1.11643
$630$	0	0
$631$	−33.5269	−1.33469	−0.667343	−	0.744751i	$-0.732568\pi$
$631$	−33.5269	−1.33469	−0.667343	+	0.744751i	$0.732568\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.65685	−0.224485
$636$	0	0
$637$	0.201010	0.00796431
$638$	0	0
$639$	42.0000	1.66149
$640$	0	0
$641$	−25.3137	−0.999831	−0.499916	−	0.866074i	$-0.666636\pi$
$641$	−25.3137	−0.999831	−0.499916	+	0.866074i	$0.666636\pi$
$642$	0	0
$643$	−20.3431	−0.802255	−0.401128	−	0.916022i	$-0.631382\pi$
$643$	−20.3431	−0.802255	−0.401128	+	0.916022i	$0.631382\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.75736	−0.344287	−0.172144	−	0.985072i	$-0.555069\pi$
$647$	−8.75736	−0.344287	−0.172144	+	0.985072i	$0.555069\pi$
$648$	0	0
$649$	−22.3726	−0.878201
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−40.6274	−1.58987	−0.794937	−	0.606692i	$-0.792496\pi$
$653$	−40.6274	−1.58987	−0.794937	+	0.606692i	$0.792496\pi$
$654$	0	0
$655$	−14.4853	−0.565987
$656$	0	0
$657$	40.4558	1.57833
$658$	0	0
$659$	−37.6569	−1.46690	−0.733451	−	0.679742i	$-0.762092\pi$
$659$	−37.6569	−1.46690	−0.733451	+	0.679742i	$0.762092\pi$
$660$	0	0
$661$	−23.3137	−0.906798	−0.453399	−	0.891308i	$-0.649789\pi$
$661$	−23.3137	−0.906798	−0.453399	+	0.891308i	$0.649789\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11.6569	−0.452033
$666$	0	0
$667$	−4.97056	−0.192461
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.24264	0.125181
$672$	0	0
$673$	23.6569	0.911905	0.455952	−	0.890004i	$-0.349299\pi$
$673$	23.6569	0.911905	0.455952	+	0.890004i	$0.349299\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.686292	−0.0263763	−0.0131882	−	0.999913i	$-0.504198\pi$
$677$	−0.686292	−0.0263763	−0.0131882	+	0.999913i	$0.504198\pi$
$678$	0	0
$679$	24.1421	0.926490
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.34315	−0.242714	−0.121357	−	0.992609i	$-0.538725\pi$
$683$	−6.34315	−0.242714	−0.121357	+	0.992609i	$0.538725\pi$
$684$	0	0
$685$	3.82843	0.146277
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.59798	0.0608782
$690$	0	0
$691$	−10.3431	−0.393472	−0.196736	−	0.980457i	$-0.563034\pi$
$691$	−10.3431	−0.393472	−0.196736	+	0.980457i	$0.563034\pi$
$692$	0	0
$693$	23.4853	0.892132
$694$	0	0
$695$	11.7279	0.444865
$696$	0	0
$697$	−20.3431	−0.770552
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.9706	−1.16974	−0.584871	−	0.811126i	$-0.698855\pi$
$701$	−30.9706	−1.16974	−0.584871	+	0.811126i	$0.698855\pi$
$702$	0	0
$703$	17.6569	0.665941
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.4853	0.544775
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	8.27208	0.310227
$712$	0	0
$713$	−9.51472	−0.356329
$714$	0	0
$715$	−0.556349	−0.0208063
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.62742	−0.0979861	−0.0489931	−	0.998799i	$-0.515601\pi$
$719$	−2.62742	−0.0979861	−0.0489931	+	0.998799i	$0.515601\pi$
$720$	0	0
$721$	11.6569	0.434124
$722$	0	0
$723$	0	0
$724$	0	0
$725$	16.0000	0.594225
$726$	0	0
$727$	47.3137	1.75477	0.877384	−	0.479789i	$-0.159287\pi$
$727$	47.3137	1.75477	0.877384	+	0.479789i	$0.159287\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−58.6274	−2.16841
$732$	0	0
$733$	−38.1127	−1.40772	−0.703862	−	0.710336i	$-0.748543\pi$
$733$	−38.1127	−1.40772	−0.703862	+	0.710336i	$0.748543\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.25483	−0.230400
$738$	0	0
$739$	−48.3553	−1.77878	−0.889390	−	0.457149i	$-0.848870\pi$
$739$	−48.3553	−1.77878	−0.889390	+	0.457149i	$0.848870\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.58579	0.351668	0.175834	−	0.984420i	$-0.443738\pi$
$743$	9.58579	0.351668	0.175834	+	0.984420i	$0.443738\pi$
$744$	0	0
$745$	−13.1421	−0.481491
$746$	0	0
$747$	45.9411	1.68090
$748$	0	0
$749$	7.65685	0.279775
$750$	0	0
$751$	−41.1127	−1.50022	−0.750112	−	0.661311i	$-0.770000\pi$
$751$	−41.1127	−1.50022	−0.750112	+	0.661311i	$0.770000\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−19.2426	−0.700311
$756$	0	0
$757$	18.6569	0.678095	0.339047	−	0.940769i	$-0.389895\pi$
$757$	18.6569	0.678095	0.339047	+	0.940769i	$0.389895\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.2843	−0.735304	−0.367652	−	0.929963i	$-0.619838\pi$
$761$	−20.2843	−0.735304	−0.367652	+	0.929963i	$0.619838\pi$
$762$	0	0
$763$	−42.2132	−1.52822
$764$	0	0
$765$	−22.9706	−0.830502
$766$	0	0
$767$	−1.18377	−0.0427433
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40.6274	1.46127	0.730633	−	0.682770i	$-0.239225\pi$
$773$	40.6274	1.46127	0.730633	+	0.682770i	$0.239225\pi$
$774$	0	0
$775$	30.6274	1.10017
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.8284	0.459626
$780$	0	0
$781$	45.3970	1.62443
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	54.9706	1.95949	0.979744	−	0.200252i	$-0.0641760\pi$
$787$	54.9706	1.95949	0.979744	+	0.200252i	$0.0641760\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.41421	0.228063
$792$	0	0
$793$	0.171573	0.00609273
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.6274	1.47452	0.737259	−	0.675610i	$-0.236120\pi$
$797$	41.6274	1.47452	0.737259	+	0.675610i	$0.236120\pi$
$798$	0	0
$799$	73.9411	2.61585
$800$	0	0
$801$	−50.9117	−1.79888
$802$	0	0
$803$	43.7279	1.54312
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−43.8284	−1.54093	−0.770463	−	0.637485i	$-0.779975\pi$
$809$	−43.8284	−1.54093	−0.770463	+	0.637485i	$0.779975\pi$
$810$	0	0
$811$	−10.6152	−0.372751	−0.186375	−	0.982479i	$-0.559674\pi$
$811$	−10.6152	−0.372751	−0.186375	+	0.982479i	$0.559674\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.34315	−0.222191
$816$	0	0
$817$	36.9706	1.29344
$818$	0	0
$819$	1.24264	0.0434214
$820$	0	0
$821$	−35.5980	−1.24238	−0.621189	−	0.783661i	$-0.713350\pi$
$821$	−35.5980	−1.24238	−0.621189	+	0.783661i	$0.713350\pi$
$822$	0	0
$823$	41.3137	1.44011	0.720053	−	0.693919i	$-0.244118\pi$
$823$	41.3137	1.44011	0.720053	+	0.693919i	$0.244118\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.4558	1.37201	0.686007	−	0.727595i	$-0.259362\pi$
$827$	39.4558	1.37201	0.686007	+	0.727595i	$0.259362\pi$
$828$	0	0
$829$	6.51472	0.226266	0.113133	−	0.993580i	$-0.463911\pi$
$829$	6.51472	0.226266	0.113133	+	0.993580i	$0.463911\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.97056	0.310812
$834$	0	0
$835$	1.51472	0.0524190
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.5147	0.604675	0.302338	−	0.953201i	$-0.402233\pi$
$839$	17.5147	0.604675	0.302338	+	0.953201i	$0.402233\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.9706	0.446201
$846$	0	0
$847$	−1.17157	−0.0402557
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.54416	0.155772
$852$	0	0
$853$	45.6274	1.56225	0.781127	−	0.624373i	$-0.214645\pi$
$853$	45.6274	1.56225	0.781127	+	0.624373i	$0.214645\pi$
$854$	0	0
$855$	14.4853	0.495386
$856$	0	0
$857$	41.9706	1.43369	0.716844	−	0.697234i	$-0.245586\pi$
$857$	41.9706	1.43369	0.716844	+	0.697234i	$0.245586\pi$
$858$	0	0
$859$	−38.6274	−1.31795	−0.658975	−	0.752165i	$-0.729010\pi$
$859$	−38.6274	−1.31795	−0.658975	+	0.752165i	$0.729010\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.3137	−0.385123	−0.192562	−	0.981285i	$-0.561680\pi$
$863$	−11.3137	−0.385123	−0.192562	+	0.981285i	$0.561680\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.94113	0.303307
$870$	0	0
$871$	−0.330952	−0.0112139
$872$	0	0
$873$	−30.0000	−1.01535
$874$	0	0
$875$	21.7279	0.734538
$876$	0	0
$877$	−56.5685	−1.91018	−0.955092	−	0.296309i	$-0.904244\pi$
$877$	−56.5685	−1.91018	−0.955092	+	0.296309i	$0.904244\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.17157	0.0731621	0.0365811	−	0.999331i	$-0.488353\pi$
$881$	2.17157	0.0731621	0.0365811	+	0.999331i	$0.488353\pi$
$882$	0	0
$883$	6.89949	0.232186	0.116093	−	0.993238i	$-0.462963\pi$
$883$	6.89949	0.232186	0.116093	+	0.993238i	$0.462963\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.0294	0.571793	0.285896	−	0.958261i	$-0.407709\pi$
$887$	17.0294	0.571793	0.285896	+	0.958261i	$0.407709\pi$
$888$	0	0
$889$	13.6569	0.458036
$890$	0	0
$891$	−29.1838	−0.977693
$892$	0	0
$893$	−46.6274	−1.56033
$894$	0	0
$895$	−16.8284	−0.562512
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.6274	1.02148
$900$	0	0
$901$	71.3137	2.37580
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.3431	−0.343818
$906$	0	0
$907$	19.9411	0.662134	0.331067	−	0.943607i	$-0.392591\pi$
$907$	19.9411	0.662134	0.331067	+	0.943607i	$0.392591\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	1.65685	0.0548940	0.0274470	−	0.999623i	$-0.491262\pi$
$911$	1.65685	0.0548940	0.0274470	+	0.999623i	$0.491262\pi$
$912$	0	0
$913$	49.6569	1.64340
$914$	0	0
$915$	0	0
$916$	0	0
$917$	34.9706	1.15483
$918$	0	0
$919$	−21.6569	−0.714394	−0.357197	−	0.934029i	$-0.616267\pi$
$919$	−21.6569	−0.714394	−0.357197	+	0.934029i	$0.616267\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.40202	0.0790635
$924$	0	0
$925$	−14.6274	−0.480947
$926$	0	0
$927$	−14.4853	−0.475759
$928$	0	0
$929$	20.3137	0.666471	0.333236	−	0.942844i	$-0.391860\pi$
$929$	20.3137	0.666471	0.333236	+	0.942844i	$0.391860\pi$
$930$	0	0
$931$	−5.65685	−0.185396
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.8284	−0.811976
$936$	0	0
$937$	27.9706	0.913758	0.456879	−	0.889529i	$-0.348967\pi$
$937$	27.9706	0.913758	0.456879	+	0.889529i	$0.348967\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−12.6863	−0.413561	−0.206781	−	0.978387i	$-0.566299\pi$
$941$	−12.6863	−0.413561	−0.206781	+	0.978387i	$0.566299\pi$
$942$	0	0
$943$	3.30152	0.107512
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.0711	0.327266	0.163633	−	0.986521i	$-0.447679\pi$
$947$	10.0711	0.327266	0.163633	+	0.986521i	$0.447679\pi$
$948$	0	0
$949$	2.31371	0.0751062
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.970563	−0.0314396	−0.0157198	−	0.999876i	$-0.505004\pi$
$953$	−0.970563	−0.0314396	−0.0157198	+	0.999876i	$0.505004\pi$
$954$	0	0
$955$	−3.24264	−0.104929
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.24264	−0.298460
$960$	0	0
$961$	27.6274	0.891207
$962$	0	0
$963$	−9.51472	−0.306608
$964$	0	0
$965$	2.34315	0.0754285
$966$	0	0
$967$	−51.5980	−1.65928	−0.829640	−	0.558299i	$-0.811454\pi$
$967$	−51.5980	−1.65928	−0.829640	+	0.558299i	$0.811454\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.4558	0.881100	0.440550	−	0.897728i	$-0.354783\pi$
$971$	27.4558	0.881100	0.440550	+	0.897728i	$0.354783\pi$
$972$	0	0
$973$	−28.3137	−0.907696
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.9411	1.53377	0.766886	−	0.641783i	$-0.221805\pi$
$977$	47.9411	1.53377	0.766886	+	0.641783i	$0.221805\pi$
$978$	0	0
$979$	−55.0294	−1.75875
$980$	0	0
$981$	52.4558	1.67479
$982$	0	0
$983$	−5.02944	−0.160414	−0.0802071	−	0.996778i	$-0.525558\pi$
$983$	−5.02944	−0.160414	−0.0802071	+	0.996778i	$0.525558\pi$
$984$	0	0
$985$	−8.65685	−0.275830
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.51472	0.302550
$990$	0	0
$991$	26.6274	0.845848	0.422924	−	0.906165i	$-0.361004\pi$
$991$	26.6274	0.845848	0.422924	+	0.906165i	$0.361004\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13.7990	0.437457
$996$	0	0
$997$	1.71573	0.0543377	0.0271688	−	0.999631i	$-0.491351\pi$
$997$	1.71573	0.0543377	0.0271688	+	0.999631i	$0.491351\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	976.2.a.e.1.2		2
3.2	odd	2		8784.2.a.bh.1.2		2
4.3	odd	2		488.2.a.a.1.1	✓	2
8.3	odd	2		3904.2.a.m.1.1		2
8.5	even	2		3904.2.a.n.1.2		2
12.11	even	2		4392.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
488.2.a.a.1.1	✓	2	4.3	odd	2
976.2.a.e.1.2		2	1.1	even	1	trivial
3904.2.a.m.1.1		2	8.3	odd	2
3904.2.a.n.1.2		2	8.5	even	2
4392.2.a.h.1.1		2	12.11	even	2
8784.2.a.bh.1.2		2	3.2	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:	\( N \)	\(=\)	\( 976 = 2^{4} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	976.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.41421 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{5} +2.41421 q^{7} -3.00000 q^{9} -3.24264 q^{11} -0.171573 q^{13} -7.65685 q^{17} +4.82843 q^{19} +1.24264 q^{23} -4.00000 q^{25} -4.00000 q^{29} -7.65685 q^{31} -2.41421 q^{35} +3.65685 q^{37} +2.65685 q^{41} +7.65685 q^{43} +3.00000 q^{45} -9.65685 q^{47} -1.17157 q^{49} -9.31371 q^{53} +3.24264 q^{55} +6.89949 q^{59} -1.00000 q^{61} -7.24264 q^{63} +0.171573 q^{65} +1.92893 q^{67} -14.0000 q^{71} -13.4853 q^{73} -7.82843 q^{77} -2.75736 q^{79} +9.00000 q^{81} -15.3137 q^{83} +7.65685 q^{85} +16.9706 q^{89} -0.414214 q^{91} -4.82843 q^{95} +10.0000 q^{97} +9.72792 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7 - 6 * q^9	\( 2 q - 2 q^{5} + 2 q^{7} - 6 q^{9} + 2 q^{11} - 6 q^{13} - 4 q^{17} + 4 q^{19} - 6 q^{23} - 8 q^{25} - 8 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{37} - 6 q^{41} + 4 q^{43} + 6 q^{45} - 8 q^{47} - 8 q^{49} + 4 q^{53} - 2 q^{55} - 6 q^{59} - 2 q^{61} - 6 q^{63} + 6 q^{65} + 18 q^{67} - 28 q^{71} - 10 q^{73} - 10 q^{77} - 14 q^{79} + 18 q^{81} - 8 q^{83} + 4 q^{85} + 2 q^{91} - 4 q^{95} + 20 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 - 6 * q^9 + 2 * q^11 - 6 * q^13 - 4 * q^17 + 4 * q^19 - 6 * q^23 - 8 * q^25 - 8 * q^29 - 4 * q^31 - 2 * q^35 - 4 * q^37 - 6 * q^41 + 4 * q^43 + 6 * q^45 - 8 * q^47 - 8 * q^49 + 4 * q^53 - 2 * q^55 - 6 * q^59 - 2 * q^61 - 6 * q^63 + 6 * q^65 + 18 * q^67 - 28 * q^71 - 10 * q^73 - 10 * q^77 - 14 * q^79 + 18 * q^81 - 8 * q^83 + 4 * q^85 + 2 * q^91 - 4 * q^95 + 20 * q^97 - 6 * q^99

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