LMFDB - Embedded newform 8712.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8712,2,Mod(1,8712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8712, 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("8712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8712.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:	$69.5656702409$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2904)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	8712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37228 q^{5} -2.37228 q^{7} +O(q^{10})$	$q+3.37228 q^{5} -2.37228 q^{7} -1.37228 q^{13} -7.37228 q^{17} -2.37228 q^{19} +8.74456 q^{23} +6.37228 q^{25} +1.37228 q^{29} +7.11684 q^{31} -8.00000 q^{35} -9.00000 q^{37} -8.11684 q^{41} -4.00000 q^{43} -8.74456 q^{47} -1.37228 q^{49} +4.11684 q^{53} -0.744563 q^{59} +12.3723 q^{61} -4.62772 q^{65} +7.11684 q^{67} -4.00000 q^{71} +5.11684 q^{73} +14.3723 q^{79} -3.25544 q^{83} -24.8614 q^{85} -10.1168 q^{89} +3.25544 q^{91} -8.00000 q^{95} -9.74456 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + q^{7}+O(q^{10}) $ 2 * q + q^5 + q^7	$ 2 q + q^{5} + q^{7} + 3 q^{13} - 9 q^{17} + q^{19} + 6 q^{23} + 7 q^{25} - 3 q^{29} - 3 q^{31} - 16 q^{35} - 18 q^{37} + q^{41} - 8 q^{43} - 6 q^{47} + 3 q^{49} - 9 q^{53} + 10 q^{59} + 19 q^{61} - 15 q^{65} - 3 q^{67} - 8 q^{71} - 7 q^{73} + 23 q^{79} - 18 q^{83} - 21 q^{85} - 3 q^{89} + 18 q^{91} - 16 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q + q^5 + q^7 + 3 * q^13 - 9 * q^17 + q^19 + 6 * q^23 + 7 * q^25 - 3 * q^29 - 3 * q^31 - 16 * q^35 - 18 * q^37 + q^41 - 8 * q^43 - 6 * q^47 + 3 * q^49 - 9 * q^53 + 10 * q^59 + 19 * q^61 - 15 * q^65 - 3 * q^67 - 8 * q^71 - 7 * q^73 + 23 * q^79 - 18 * q^83 - 21 * q^85 - 3 * q^89 + 18 * q^91 - 16 * 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
$5$	3.37228	1.50813	0.754065	−	0.656800i	$-0.228090\pi$
$5$	3.37228	1.50813	0.754065	+	0.656800i	$0.228090\pi$
$6$	0	0
$7$	−2.37228	−0.896638	−0.448319	−	0.893874i	$-0.647977\pi$
$7$	−2.37228	−0.896638	−0.448319	+	0.893874i	$0.647977\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.37228	−0.380602	−0.190301	−	0.981726i	$-0.560946\pi$
$13$	−1.37228	−0.380602	−0.190301	+	0.981726i	$0.560946\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.37228	−1.78804	−0.894020	−	0.448026i	$-0.852127\pi$
$17$	−7.37228	−1.78804	−0.894020	+	0.448026i	$0.852127\pi$
$18$	0	0
$19$	−2.37228	−0.544239	−0.272119	−	0.962264i	$-0.587725\pi$
$19$	−2.37228	−0.544239	−0.272119	+	0.962264i	$0.587725\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.74456	1.82337	0.911684	−	0.410893i	$-0.134783\pi$
$23$	8.74456	1.82337	0.911684	+	0.410893i	$0.134783\pi$
$24$	0	0
$25$	6.37228	1.27446
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.37228	0.254826	0.127413	−	0.991850i	$-0.459333\pi$
$29$	1.37228	0.254826	0.127413	+	0.991850i	$0.459333\pi$
$30$	0	0
$31$	7.11684	1.27822	0.639111	−	0.769114i	$-0.279302\pi$
$31$	7.11684	1.27822	0.639111	+	0.769114i	$0.279302\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.00000	−1.35225
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.11684	−1.26764	−0.633819	−	0.773481i	$-0.718514\pi$
$41$	−8.11684	−1.26764	−0.633819	+	0.773481i	$0.718514\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.74456	−1.27553	−0.637763	−	0.770233i	$-0.720140\pi$
$47$	−8.74456	−1.27553	−0.637763	+	0.770233i	$0.720140\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.11684	0.565492	0.282746	−	0.959195i	$-0.408755\pi$
$53$	4.11684	0.565492	0.282746	+	0.959195i	$0.408755\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.744563	−0.0969338	−0.0484669	−	0.998825i	$-0.515434\pi$
$59$	−0.744563	−0.0969338	−0.0484669	+	0.998825i	$0.515434\pi$
$60$	0	0
$61$	12.3723	1.58411	0.792054	−	0.610451i	$-0.209012\pi$
$61$	12.3723	1.58411	0.792054	+	0.610451i	$0.209012\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.62772	−0.573998
$66$	0	0
$67$	7.11684	0.869461	0.434730	−	0.900561i	$-0.356844\pi$
$67$	7.11684	0.869461	0.434730	+	0.900561i	$0.356844\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	5.11684	0.598881	0.299441	−	0.954115i	$-0.403200\pi$
$73$	5.11684	0.598881	0.299441	+	0.954115i	$0.403200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.3723	1.61701	0.808504	−	0.588491i	$-0.200278\pi$
$79$	14.3723	1.61701	0.808504	+	0.588491i	$0.200278\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.25544	−0.357331	−0.178665	−	0.983910i	$-0.557178\pi$
$83$	−3.25544	−0.357331	−0.178665	+	0.983910i	$0.557178\pi$
$84$	0	0
$85$	−24.8614	−2.69660
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.1168	−1.07238	−0.536192	−	0.844096i	$-0.680138\pi$
$89$	−10.1168	−1.07238	−0.536192	+	0.844096i	$0.680138\pi$
$90$	0	0
$91$	3.25544	0.341263
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−9.74456	−0.989410	−0.494705	−	0.869061i	$-0.664724\pi$
$97$	−9.74456	−0.989410	−0.494705	+	0.869061i	$0.664724\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.48913	−0.347181	−0.173590	−	0.984818i	$-0.555537\pi$
$101$	−3.48913	−0.347181	−0.173590	+	0.984818i	$0.555537\pi$
$102$	0	0
$103$	−13.6277	−1.34278	−0.671390	−	0.741105i	$-0.734302\pi$
$103$	−13.6277	−1.34278	−0.671390	+	0.741105i	$0.734302\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.74456	−0.845369	−0.422684	−	0.906277i	$-0.638912\pi$
$107$	−8.74456	−0.845369	−0.422684	+	0.906277i	$0.638912\pi$
$108$	0	0
$109$	17.7446	1.69962	0.849810	−	0.527088i	$-0.176716\pi$
$109$	17.7446	1.69962	0.849810	+	0.527088i	$0.176716\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.3723	−1.63425	−0.817123	−	0.576463i	$-0.804433\pi$
$113$	−17.3723	−1.63425	−0.817123	+	0.576463i	$0.804433\pi$
$114$	0	0
$115$	29.4891	2.74988
$116$	0	0
$117$	0	0
$118$	0	0
$119$	17.4891	1.60323
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	4.62772	0.413916
$126$	0	0
$127$	−1.62772	−0.144437	−0.0722183	−	0.997389i	$-0.523008\pi$
$127$	−1.62772	−0.144437	−0.0722183	+	0.997389i	$0.523008\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.2337	−1.24360	−0.621802	−	0.783175i	$-0.713599\pi$
$131$	−14.2337	−1.24360	−0.621802	+	0.783175i	$0.713599\pi$
$132$	0	0
$133$	5.62772	0.487985
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.7446	−1.25971	−0.629857	−	0.776712i	$-0.716886\pi$
$137$	−14.7446	−1.25971	−0.629857	+	0.776712i	$0.716886\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	4.62772	0.384311
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.3723	−1.58704	−0.793520	−	0.608545i	$-0.791754\pi$
$149$	−19.3723	−1.58704	−0.793520	+	0.608545i	$0.791754\pi$
$150$	0	0
$151$	13.4891	1.09773	0.548865	−	0.835911i	$-0.315060\pi$
$151$	13.4891	1.09773	0.548865	+	0.835911i	$0.315060\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	24.0000	1.92773
$156$	0	0
$157$	−5.86141	−0.467791	−0.233896	−	0.972262i	$-0.575147\pi$
$157$	−5.86141	−0.467791	−0.233896	+	0.972262i	$0.575147\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−20.7446	−1.63490
$162$	0	0
$163$	−23.1168	−1.81065	−0.905325	−	0.424718i	$-0.860373\pi$
$163$	−23.1168	−1.81065	−0.905325	+	0.424718i	$0.860373\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.25544	−0.251913	−0.125957	−	0.992036i	$-0.540200\pi$
$167$	−3.25544	−0.251913	−0.125957	+	0.992036i	$0.540200\pi$
$168$	0	0
$169$	−11.1168	−0.855142
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.74456	−0.512780	−0.256390	−	0.966573i	$-0.582533\pi$
$173$	−6.74456	−0.512780	−0.256390	+	0.966573i	$0.582533\pi$
$174$	0	0
$175$	−15.1168	−1.14273
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.7446	1.25155	0.625774	−	0.780005i	$-0.284783\pi$
$179$	16.7446	1.25155	0.625774	+	0.780005i	$0.284783\pi$
$180$	0	0
$181$	−9.74456	−0.724308	−0.362154	−	0.932118i	$-0.617958\pi$
$181$	−9.74456	−0.724308	−0.362154	+	0.932118i	$0.617958\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−30.3505	−2.23142
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−21.9783	−1.58203	−0.791015	−	0.611797i	$-0.790447\pi$
$193$	−21.9783	−1.58203	−0.791015	+	0.611797i	$0.790447\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.3723	−1.38022	−0.690109	−	0.723705i	$-0.742437\pi$
$197$	−19.3723	−1.38022	−0.690109	+	0.723705i	$0.742437\pi$
$198$	0	0
$199$	−6.37228	−0.451719	−0.225860	−	0.974160i	$-0.572519\pi$
$199$	−6.37228	−0.451719	−0.225860	+	0.974160i	$0.572519\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.25544	−0.228487
$204$	0	0
$205$	−27.3723	−1.91176
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.1168	−0.765315	−0.382658	−	0.923890i	$-0.624991\pi$
$211$	−11.1168	−0.765315	−0.382658	+	0.923890i	$0.624991\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−13.4891	−0.919951
$216$	0	0
$217$	−16.8832	−1.14610
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.1168	0.680533
$222$	0	0
$223$	27.1168	1.81588	0.907939	−	0.419102i	$-0.137655\pi$
$223$	27.1168	1.81588	0.907939	+	0.419102i	$0.137655\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	−6.62772	−0.437972	−0.218986	−	0.975728i	$-0.570275\pi$
$229$	−6.62772	−0.437972	−0.218986	+	0.975728i	$0.570275\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.8614	1.49770	0.748850	−	0.662739i	$-0.230606\pi$
$233$	22.8614	1.49770	0.748850	+	0.662739i	$0.230606\pi$
$234$	0	0
$235$	−29.4891	−1.92366
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.2337	1.69692	0.848458	−	0.529263i	$-0.177531\pi$
$239$	26.2337	1.69692	0.848458	+	0.529263i	$0.177531\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.62772	−0.295654
$246$	0	0
$247$	3.25544	0.207139
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0.744563	0.0469964	0.0234982	−	0.999724i	$-0.492520\pi$
$251$	0.744563	0.0469964	0.0234982	+	0.999724i	$0.492520\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.88316	−0.491738	−0.245869	−	0.969303i	$-0.579073\pi$
$257$	−7.88316	−0.491738	−0.245869	+	0.969303i	$0.579073\pi$
$258$	0	0
$259$	21.3505	1.32666
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.2337	0.877687	0.438843	−	0.898564i	$-0.355388\pi$
$263$	14.2337	0.877687	0.438843	+	0.898564i	$0.355388\pi$
$264$	0	0
$265$	13.8832	0.852835
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.11684	0.251008	0.125504	−	0.992093i	$-0.459945\pi$
$269$	4.11684	0.251008	0.125504	+	0.992093i	$0.459945\pi$
$270$	0	0
$271$	−13.4891	−0.819406	−0.409703	−	0.912219i	$-0.634368\pi$
$271$	−13.4891	−0.819406	−0.409703	+	0.912219i	$0.634368\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.2337	0.674967	0.337483	−	0.941331i	$-0.390424\pi$
$277$	11.2337	0.674967	0.337483	+	0.941331i	$0.390424\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.25544	−0.0748931	−0.0374466	−	0.999299i	$-0.511922\pi$
$281$	−1.25544	−0.0748931	−0.0374466	+	0.999299i	$0.511922\pi$
$282$	0	0
$283$	24.6060	1.46267	0.731337	−	0.682017i	$-0.238897\pi$
$283$	24.6060	1.46267	0.731337	+	0.682017i	$0.238897\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	19.2554	1.13661
$288$	0	0
$289$	37.3505	2.19709
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.6060	0.911710	0.455855	−	0.890054i	$-0.349333\pi$
$293$	15.6060	0.911710	0.455855	+	0.890054i	$0.349333\pi$
$294$	0	0
$295$	−2.51087	−0.146189
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	9.48913	0.546944
$302$	0	0
$303$	0	0
$304$	0	0
$305$	41.7228	2.38904
$306$	0	0
$307$	−23.8614	−1.36184	−0.680921	−	0.732357i	$-0.738420\pi$
$307$	−23.8614	−1.36184	−0.680921	+	0.732357i	$0.738420\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	9.37228	0.529753	0.264876	−	0.964282i	$-0.414669\pi$
$313$	9.37228	0.529753	0.264876	+	0.964282i	$0.414669\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.7446	−1.05280	−0.526400	−	0.850237i	$-0.676458\pi$
$317$	−18.7446	−1.05280	−0.526400	+	0.850237i	$0.676458\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	17.4891	0.973121
$324$	0	0
$325$	−8.74456	−0.485061
$326$	0	0
$327$	0	0
$328$	0	0
$329$	20.7446	1.14368
$330$	0	0
$331$	−8.60597	−0.473027	−0.236513	−	0.971628i	$-0.576005\pi$
$331$	−8.60597	−0.473027	−0.236513	+	0.971628i	$0.576005\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	24.0000	1.31126
$336$	0	0
$337$	−11.7446	−0.639767	−0.319884	−	0.947457i	$-0.603644\pi$
$337$	−11.7446	−0.639767	−0.319884	+	0.947457i	$0.603644\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.8614	1.07242
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.2337	1.19357	0.596783	−	0.802402i	$-0.296445\pi$
$347$	22.2337	1.19357	0.596783	+	0.802402i	$0.296445\pi$
$348$	0	0
$349$	−3.00000	−0.160586	−0.0802932	−	0.996771i	$-0.525586\pi$
$349$	−3.00000	−0.160586	−0.0802932	+	0.996771i	$0.525586\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.6277	−1.09790	−0.548951	−	0.835854i	$-0.684973\pi$
$353$	−20.6277	−1.09790	−0.548951	+	0.835854i	$0.684973\pi$
$354$	0	0
$355$	−13.4891	−0.715928
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.48913	−0.500817	−0.250408	−	0.968140i	$-0.580565\pi$
$359$	−9.48913	−0.500817	−0.250408	+	0.968140i	$0.580565\pi$
$360$	0	0
$361$	−13.3723	−0.703804
$362$	0	0
$363$	0	0
$364$	0	0
$365$	17.2554	0.903191
$366$	0	0
$367$	−10.5109	−0.548663	−0.274332	−	0.961635i	$-0.588457\pi$
$367$	−10.5109	−0.548663	−0.274332	+	0.961635i	$0.588457\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.76631	−0.507042
$372$	0	0
$373$	−5.11684	−0.264940	−0.132470	−	0.991187i	$-0.542291\pi$
$373$	−5.11684	−0.264940	−0.132470	+	0.991187i	$0.542291\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.88316	−0.0969875
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.2337	−0.727307	−0.363654	−	0.931534i	$-0.618471\pi$
$383$	−14.2337	−0.727307	−0.363654	+	0.931534i	$0.618471\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.8614	−0.753503	−0.376752	−	0.926314i	$-0.622959\pi$
$389$	−14.8614	−0.753503	−0.376752	+	0.926314i	$0.622959\pi$
$390$	0	0
$391$	−64.4674	−3.26026
$392$	0	0
$393$	0	0
$394$	0	0
$395$	48.4674	2.43866
$396$	0	0
$397$	5.51087	0.276583	0.138291	−	0.990392i	$-0.455839\pi$
$397$	5.51087	0.276583	0.138291	+	0.990392i	$0.455839\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.6060	−0.979075	−0.489538	−	0.871982i	$-0.662834\pi$
$401$	−19.6060	−0.979075	−0.489538	+	0.871982i	$0.662834\pi$
$402$	0	0
$403$	−9.76631	−0.486495
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	12.2554	0.605992	0.302996	−	0.952992i	$-0.402013\pi$
$409$	12.2554	0.605992	0.302996	+	0.952992i	$0.402013\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.76631	0.0869145
$414$	0	0
$415$	−10.9783	−0.538901
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.23369	−0.109123	−0.0545614	−	0.998510i	$-0.517376\pi$
$419$	−2.23369	−0.109123	−0.0545614	+	0.998510i	$0.517376\pi$
$420$	0	0
$421$	−1.13859	−0.0554916	−0.0277458	−	0.999615i	$-0.508833\pi$
$421$	−1.13859	−0.0554916	−0.0277458	+	0.999615i	$0.508833\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−46.9783	−2.27878
$426$	0	0
$427$	−29.3505	−1.42037
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.7446	0.613884	0.306942	−	0.951728i	$-0.400694\pi$
$431$	12.7446	0.613884	0.306942	+	0.951728i	$0.400694\pi$
$432$	0	0
$433$	28.4891	1.36910	0.684550	−	0.728966i	$-0.259999\pi$
$433$	28.4891	1.36910	0.684550	+	0.728966i	$0.259999\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20.7446	−0.992347
$438$	0	0
$439$	26.0951	1.24545	0.622725	−	0.782440i	$-0.286025\pi$
$439$	26.0951	1.24545	0.622725	+	0.782440i	$0.286025\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	−34.1168	−1.61729
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.13859	−0.336891	−0.168446	−	0.985711i	$-0.553875\pi$
$449$	−7.13859	−0.336891	−0.168446	+	0.985711i	$0.553875\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.9783	0.514668
$456$	0	0
$457$	35.0951	1.64168	0.820840	−	0.571159i	$-0.193506\pi$
$457$	35.0951	1.64168	0.820840	+	0.571159i	$0.193506\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−24.1168	−1.12323	−0.561617	−	0.827398i	$-0.689820\pi$
$461$	−24.1168	−1.12323	−0.561617	+	0.827398i	$0.689820\pi$
$462$	0	0
$463$	29.4891	1.37048	0.685238	−	0.728319i	$-0.259698\pi$
$463$	29.4891	1.37048	0.685238	+	0.728319i	$0.259698\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.4891	−0.809300	−0.404650	−	0.914472i	$-0.632607\pi$
$467$	−17.4891	−0.809300	−0.404650	+	0.914472i	$0.632607\pi$
$468$	0	0
$469$	−16.8832	−0.779592
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−15.1168	−0.693608
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.7446	−0.582314	−0.291157	−	0.956675i	$-0.594040\pi$
$479$	−12.7446	−0.582314	−0.291157	+	0.956675i	$0.594040\pi$
$480$	0	0
$481$	12.3505	0.563136
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−32.8614	−1.49216
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.744563	0.0336016	0.0168008	−	0.999859i	$-0.494652\pi$
$491$	0.744563	0.0336016	0.0168008	+	0.999859i	$0.494652\pi$
$492$	0	0
$493$	−10.1168	−0.455640
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.48913	0.425645
$498$	0	0
$499$	28.6060	1.28058	0.640290	−	0.768134i	$-0.278814\pi$
$499$	28.6060	1.28058	0.640290	+	0.768134i	$0.278814\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.97825	−0.311145	−0.155572	−	0.987824i	$-0.549722\pi$
$503$	−6.97825	−0.311145	−0.155572	+	0.987824i	$0.549722\pi$
$504$	0	0
$505$	−11.7663	−0.523594
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−12.1386	−0.536980
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−45.9565	−2.02509
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.5109	−0.723355	−0.361677	−	0.932303i	$-0.617796\pi$
$521$	−16.5109	−0.723355	−0.361677	+	0.932303i	$0.617796\pi$
$522$	0	0
$523$	−26.3723	−1.15318	−0.576590	−	0.817034i	$-0.695617\pi$
$523$	−26.3723	−1.15318	−0.576590	+	0.817034i	$0.695617\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−52.4674	−2.28551
$528$	0	0
$529$	53.4674	2.32467
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.1386	0.482466
$534$	0	0
$535$	−29.4891	−1.27493
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	59.8397	2.56325
$546$	0	0
$547$	−14.9783	−0.640424	−0.320212	−	0.947346i	$-0.603754\pi$
$547$	−14.9783	−0.640424	−0.320212	+	0.947346i	$0.603754\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.25544	−0.138686
$552$	0	0
$553$	−34.0951	−1.44987
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.2337	−1.36579	−0.682893	−	0.730519i	$-0.739278\pi$
$557$	−32.2337	−1.36579	−0.682893	+	0.730519i	$0.739278\pi$
$558$	0	0
$559$	5.48913	0.232165
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−34.9783	−1.47416	−0.737079	−	0.675807i	$-0.763795\pi$
$563$	−34.9783	−1.47416	−0.737079	+	0.675807i	$0.763795\pi$
$564$	0	0
$565$	−58.5842	−2.46466
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−33.2554	−1.39414	−0.697070	−	0.717003i	$-0.745513\pi$
$569$	−33.2554	−1.39414	−0.697070	+	0.717003i	$0.745513\pi$
$570$	0	0
$571$	−9.62772	−0.402907	−0.201454	−	0.979498i	$-0.564567\pi$
$571$	−9.62772	−0.402907	−0.201454	+	0.979498i	$0.564567\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	55.7228	2.32380
$576$	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
$580$	0	0
$581$	7.72281	0.320396
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.4891	−1.54734	−0.773671	−	0.633588i	$-0.781582\pi$
$587$	−37.4891	−1.54734	−0.773671	+	0.633588i	$0.781582\pi$
$588$	0	0
$589$	−16.8832	−0.695658
$590$	0	0
$591$	0	0
$592$	0	0
$593$	33.3723	1.37044	0.685218	−	0.728338i	$-0.259707\pi$
$593$	33.3723	1.37044	0.685218	+	0.728338i	$0.259707\pi$
$594$	0	0
$595$	58.9783	2.41787
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−12.4891	−0.509442	−0.254721	−	0.967015i	$-0.581984\pi$
$601$	−12.4891	−0.509442	−0.254721	+	0.967015i	$0.581984\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	14.4891	0.585210	0.292605	−	0.956233i	$-0.405478\pi$
$613$	14.4891	0.585210	0.292605	+	0.956233i	$0.405478\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.3505	0.738765	0.369382	−	0.929278i	$-0.379569\pi$
$617$	18.3505	0.738765	0.369382	+	0.929278i	$0.379569\pi$
$618$	0	0
$619$	9.48913	0.381400	0.190700	−	0.981648i	$-0.438924\pi$
$619$	9.48913	0.381400	0.190700	+	0.981648i	$0.438924\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	−16.2554	−0.650217
$626$	0	0
$627$	0	0
$628$	0	0
$629$	66.3505	2.64557
$630$	0	0
$631$	2.51087	0.0999563	0.0499782	−	0.998750i	$-0.484085\pi$
$631$	2.51087	0.0999563	0.0499782	+	0.998750i	$0.484085\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.48913	−0.217829
$636$	0	0
$637$	1.88316	0.0746134
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.86141	0.350005	0.175002	−	0.984568i	$-0.444007\pi$
$641$	8.86141	0.350005	0.175002	+	0.984568i	$0.444007\pi$
$642$	0	0
$643$	−48.3288	−1.90590	−0.952951	−	0.303126i	$-0.901970\pi$
$643$	−48.3288	−1.90590	−0.952951	+	0.303126i	$0.901970\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.02175	−0.0401691	−0.0200846	−	0.999798i	$-0.506394\pi$
$647$	−1.02175	−0.0401691	−0.0200846	+	0.999798i	$0.506394\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	−48.0000	−1.87552
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−19.7228	−0.768292	−0.384146	−	0.923272i	$-0.625504\pi$
$659$	−19.7228	−0.768292	−0.384146	+	0.923272i	$0.625504\pi$
$660$	0	0
$661$	23.4674	0.912775	0.456388	−	0.889781i	$-0.349143\pi$
$661$	23.4674	0.912775	0.456388	+	0.889781i	$0.349143\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	18.9783	0.735945
$666$	0	0
$667$	12.0000	0.464642
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	24.3723	0.939482	0.469741	−	0.882804i	$-0.344347\pi$
$673$	24.3723	0.939482	0.469741	+	0.882804i	$0.344347\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.1168	0.542554	0.271277	−	0.962501i	$-0.412554\pi$
$677$	14.1168	0.542554	0.271277	+	0.962501i	$0.412554\pi$
$678$	0	0
$679$	23.1168	0.887143
$680$	0	0
$681$	0	0
$682$	0	0
$683$	49.2119	1.88304	0.941521	−	0.336954i	$-0.109397\pi$
$683$	49.2119	1.88304	0.941521	+	0.336954i	$0.109397\pi$
$684$	0	0
$685$	−49.7228	−1.89981
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.64947	−0.215228
$690$	0	0
$691$	13.6277	0.518423	0.259211	−	0.965821i	$-0.416537\pi$
$691$	13.6277	0.518423	0.259211	+	0.965821i	$0.416537\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−26.9783	−1.02334
$696$	0	0
$697$	59.8397	2.26659
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−16.1168	−0.608725	−0.304362	−	0.952556i	$-0.598443\pi$
$701$	−16.1168	−0.608725	−0.304362	+	0.952556i	$0.598443\pi$
$702$	0	0
$703$	21.3505	0.805251
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.27719	0.311296
$708$	0	0
$709$	34.4674	1.29445	0.647225	−	0.762299i	$-0.275930\pi$
$709$	34.4674	1.29445	0.647225	+	0.762299i	$0.275930\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	62.2337	2.33067
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.7446	−0.773642	−0.386821	−	0.922155i	$-0.626427\pi$
$719$	−20.7446	−0.773642	−0.386821	+	0.922155i	$0.626427\pi$
$720$	0	0
$721$	32.3288	1.20399
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.74456	0.324765
$726$	0	0
$727$	−44.4674	−1.64920	−0.824602	−	0.565714i	$-0.808601\pi$
$727$	−44.4674	−1.64920	−0.824602	+	0.565714i	$0.808601\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29.4891	1.09069
$732$	0	0
$733$	−45.3723	−1.67586	−0.837932	−	0.545775i	$-0.816235\pi$
$733$	−45.3723	−1.67586	−0.837932	+	0.545775i	$0.816235\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−41.3505	−1.52110	−0.760552	−	0.649277i	$-0.775072\pi$
$739$	−41.3505	−1.52110	−0.760552	+	0.649277i	$0.775072\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.4891	−1.08185	−0.540926	−	0.841070i	$-0.681926\pi$
$743$	−29.4891	−1.08185	−0.540926	+	0.841070i	$0.681926\pi$
$744$	0	0
$745$	−65.3288	−2.39346
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20.7446	0.757990
$750$	0	0
$751$	28.6060	1.04385	0.521923	−	0.852992i	$-0.325215\pi$
$751$	28.6060	1.04385	0.521923	+	0.852992i	$0.325215\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	45.4891	1.65552
$756$	0	0
$757$	34.7228	1.26202	0.631011	−	0.775774i	$-0.282640\pi$
$757$	34.7228	1.26202	0.631011	+	0.775774i	$0.282640\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−14.3505	−0.520206	−0.260103	−	0.965581i	$-0.583757\pi$
$761$	−14.3505	−0.520206	−0.260103	+	0.965581i	$0.583757\pi$
$762$	0	0
$763$	−42.0951	−1.52394
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.02175	0.0368932
$768$	0	0
$769$	−29.9783	−1.08104	−0.540522	−	0.841330i	$-0.681773\pi$
$769$	−29.9783	−1.08104	−0.540522	+	0.841330i	$0.681773\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13.7228	−0.493575	−0.246788	−	0.969070i	$-0.579375\pi$
$773$	−13.7228	−0.493575	−0.246788	+	0.969070i	$0.579375\pi$
$774$	0	0
$775$	45.3505	1.62904
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.2554	0.689898
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.7663	−0.705490
$786$	0	0
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	41.2119	1.46533
$792$	0	0
$793$	−16.9783	−0.602915
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.4891	0.406966	0.203483	−	0.979079i	$-0.434774\pi$
$797$	11.4891	0.406966	0.203483	+	0.979079i	$0.434774\pi$
$798$	0	0
$799$	64.4674	2.28069
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−69.9565	−2.46564
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−13.7228	−0.482468	−0.241234	−	0.970467i	$-0.577552\pi$
$809$	−13.7228	−0.482468	−0.241234	+	0.970467i	$0.577552\pi$
$810$	0	0
$811$	−15.1168	−0.530824	−0.265412	−	0.964135i	$-0.585508\pi$
$811$	−15.1168	−0.530824	−0.265412	+	0.964135i	$0.585508\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−77.9565	−2.73070
$816$	0	0
$817$	9.48913	0.331982
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.4891	1.09898	0.549489	−	0.835501i	$-0.314822\pi$
$821$	31.4891	1.09898	0.549489	+	0.835501i	$0.314822\pi$
$822$	0	0
$823$	6.37228	0.222124	0.111062	−	0.993813i	$-0.464575\pi$
$823$	6.37228	0.222124	0.111062	+	0.993813i	$0.464575\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−20.7446	−0.721359	−0.360680	−	0.932690i	$-0.617455\pi$
$827$	−20.7446	−0.721359	−0.360680	+	0.932690i	$0.617455\pi$
$828$	0	0
$829$	−10.7663	−0.373929	−0.186965	−	0.982367i	$-0.559865\pi$
$829$	−10.7663	−0.373929	−0.186965	+	0.982367i	$0.559865\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10.1168	0.350528
$834$	0	0
$835$	−10.9783	−0.379918
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.4891	−0.879982	−0.439991	−	0.898002i	$-0.645018\pi$
$839$	−25.4891	−0.879982	−0.439991	+	0.898002i	$0.645018\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−37.4891	−1.28967
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−78.7011	−2.69784
$852$	0	0
$853$	−25.9783	−0.889478	−0.444739	−	0.895660i	$-0.646704\pi$
$853$	−25.9783	−0.889478	−0.444739	+	0.895660i	$0.646704\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.25544	−0.0428849	−0.0214425	−	0.999770i	$-0.506826\pi$
$857$	−1.25544	−0.0428849	−0.0214425	+	0.999770i	$0.506826\pi$
$858$	0	0
$859$	35.1168	1.19817	0.599086	−	0.800685i	$-0.295531\pi$
$859$	35.1168	1.19817	0.599086	+	0.800685i	$0.295531\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.7446	0.706153	0.353077	−	0.935594i	$-0.385136\pi$
$863$	20.7446	0.706153	0.353077	+	0.935594i	$0.385136\pi$
$864$	0	0
$865$	−22.7446	−0.773338
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−9.76631	−0.330919
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−10.9783	−0.371133
$876$	0	0
$877$	33.0000	1.11433	0.557165	−	0.830402i	$-0.311889\pi$
$877$	33.0000	1.11433	0.557165	+	0.830402i	$0.311889\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.86141	−0.0964032	−0.0482016	−	0.998838i	$-0.515349\pi$
$881$	−2.86141	−0.0964032	−0.0482016	+	0.998838i	$0.515349\pi$
$882$	0	0
$883$	49.3505	1.66078	0.830389	−	0.557184i	$-0.188118\pi$
$883$	49.3505	1.66078	0.830389	+	0.557184i	$0.188118\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.7446	0.696534	0.348267	−	0.937395i	$-0.386770\pi$
$887$	20.7446	0.696534	0.348267	+	0.937395i	$0.386770\pi$
$888$	0	0
$889$	3.86141	0.129507
$890$	0	0
$891$	0	0
$892$	0	0
$893$	20.7446	0.694190
$894$	0	0
$895$	56.4674	1.88750
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.76631	0.325725
$900$	0	0
$901$	−30.3505	−1.01112
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−32.8614	−1.09235
$906$	0	0
$907$	−44.6060	−1.48112	−0.740558	−	0.671992i	$-0.765439\pi$
$907$	−44.6060	−1.48112	−0.740558	+	0.671992i	$0.765439\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	52.4674	1.73832	0.869161	−	0.494529i	$-0.164660\pi$
$911$	52.4674	1.73832	0.869161	+	0.494529i	$0.164660\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.7663	1.11506
$918$	0	0
$919$	−10.0951	−0.333006	−0.166503	−	0.986041i	$-0.553248\pi$
$919$	−10.0951	−0.333006	−0.166503	+	0.986041i	$0.553248\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.48913	0.180677
$924$	0	0
$925$	−57.3505	−1.88567
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.39403	−0.0785456	−0.0392728	−	0.999229i	$-0.512504\pi$
$929$	−2.39403	−0.0785456	−0.0392728	+	0.999229i	$0.512504\pi$
$930$	0	0
$931$	3.25544	0.106693
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33.4674	1.09333	0.546666	−	0.837351i	$-0.315897\pi$
$937$	33.4674	1.09333	0.546666	+	0.837351i	$0.315897\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42.8614	1.39724	0.698621	−	0.715492i	$-0.253797\pi$
$941$	42.8614	1.39724	0.698621	+	0.715492i	$0.253797\pi$
$942$	0	0
$943$	−70.9783	−2.31137
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.2554	0.885683	0.442841	−	0.896600i	$-0.353970\pi$
$947$	27.2554	0.885683	0.442841	+	0.896600i	$0.353970\pi$
$948$	0	0
$949$	−7.02175	−0.227936
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.8614	0.351835	0.175918	−	0.984405i	$-0.443711\pi$
$953$	10.8614	0.351835	0.175918	+	0.984405i	$0.443711\pi$
$954$	0	0
$955$	−13.4891	−0.436498
$956$	0	0
$957$	0	0
$958$	0	0
$959$	34.9783	1.12951
$960$	0	0
$961$	19.6495	0.633854
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−74.1168	−2.38591
$966$	0	0
$967$	−14.3723	−0.462181	−0.231091	−	0.972932i	$-0.574229\pi$
$967$	−14.3723	−0.462181	−0.231091	+	0.972932i	$0.574229\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.76631	0.313416	0.156708	−	0.987645i	$-0.449912\pi$
$971$	9.76631	0.313416	0.156708	+	0.987645i	$0.449912\pi$
$972$	0	0
$973$	18.9783	0.608415
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.3723	−0.427817	−0.213909	−	0.976854i	$-0.568619\pi$
$977$	−13.3723	−0.427817	−0.213909	+	0.976854i	$0.568619\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−50.2337	−1.60221	−0.801103	−	0.598527i	$-0.795753\pi$
$983$	−50.2337	−1.60221	−0.801103	+	0.598527i	$0.795753\pi$
$984$	0	0
$985$	−65.3288	−2.08155
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−34.9783	−1.11224
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−21.4891	−0.681251
$996$	0	0
$997$	−0.489125	−0.0154908	−0.00774538	−	0.999970i	$-0.502465\pi$
$997$	−0.489125	−0.0154908	−0.00774538	+	0.999970i	$0.502465\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.bq.1.2		2
3.2	odd	2		2904.2.a.u.1.1	yes	2
11.10	odd	2		8712.2.a.bl.1.2		2
12.11	even	2		5808.2.a.bo.1.1		2
33.32	even	2		2904.2.a.t.1.1	✓	2
132.131	odd	2		5808.2.a.bp.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2904.2.a.t.1.1	✓	2	33.32	even	2
2904.2.a.u.1.1	yes	2	3.2	odd	2
5808.2.a.bo.1.1		2	12.11	even	2
5808.2.a.bp.1.1		2	132.131	odd	2
8712.2.a.bl.1.2		2	11.10	odd	2
8712.2.a.bq.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q+3.37228 q^{5} -2.37228 q^{7} +O(q^{10})\)	\(q+3.37228 q^{5} -2.37228 q^{7} -1.37228 q^{13} -7.37228 q^{17} -2.37228 q^{19} +8.74456 q^{23} +6.37228 q^{25} +1.37228 q^{29} +7.11684 q^{31} -8.00000 q^{35} -9.00000 q^{37} -8.11684 q^{41} -4.00000 q^{43} -8.74456 q^{47} -1.37228 q^{49} +4.11684 q^{53} -0.744563 q^{59} +12.3723 q^{61} -4.62772 q^{65} +7.11684 q^{67} -4.00000 q^{71} +5.11684 q^{73} +14.3723 q^{79} -3.25544 q^{83} -24.8614 q^{85} -10.1168 q^{89} +3.25544 q^{91} -8.00000 q^{95} -9.74456 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + q^{7}+O(q^{10}) \) 2 * q + q^5 + q^7	\( 2 q + q^{5} + q^{7} + 3 q^{13} - 9 q^{17} + q^{19} + 6 q^{23} + 7 q^{25} - 3 q^{29} - 3 q^{31} - 16 q^{35} - 18 q^{37} + q^{41} - 8 q^{43} - 6 q^{47} + 3 q^{49} - 9 q^{53} + 10 q^{59} + 19 q^{61} - 15 q^{65} - 3 q^{67} - 8 q^{71} - 7 q^{73} + 23 q^{79} - 18 q^{83} - 21 q^{85} - 3 q^{89} + 18 q^{91} - 16 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q + q^5 + q^7 + 3 * q^13 - 9 * q^17 + q^19 + 6 * q^23 + 7 * q^25 - 3 * q^29 - 3 * q^31 - 16 * q^35 - 18 * q^37 + q^41 - 8 * q^43 - 6 * q^47 + 3 * q^49 - 9 * q^53 + 10 * q^59 + 19 * q^61 - 15 * q^65 - 3 * q^67 - 8 * q^71 - 7 * q^73 + 23 * q^79 - 18 * q^83 - 21 * q^85 - 3 * q^89 + 18 * q^91 - 16 * q^95 - 8 * q^97

Introduction

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