LMFDB - Embedded newform 8712.2.a.cf.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:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.31342$ 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)

$f(q)$	$=$	$q-1.31342 q^{5} +3.04547 q^{7} +O(q^{10})$	$q-1.31342 q^{5} +3.04547 q^{7} +1.31342 q^{13} -6.27492 q^{17} -3.04547 q^{19} -3.27492 q^{25} +6.27492 q^{29} +5.27492 q^{31} -4.00000 q^{35} +5.54983 q^{37} +2.27492 q^{41} -0.837253 q^{43} -12.1819 q^{47} +2.27492 q^{49} -5.61478 q^{53} -5.25370 q^{59} +3.04547 q^{61} -1.72508 q^{65} -7.82475 q^{67} +6.92820 q^{71} -3.88273 q^{73} +8.29917 q^{79} -12.0000 q^{83} +8.24163 q^{85} +1.31342 q^{89} +4.00000 q^{91} +4.00000 q^{95} -15.5498 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 10 q^{17} + 2 q^{25} + 10 q^{29} + 6 q^{31} - 16 q^{35} - 8 q^{37} - 6 q^{41} - 6 q^{49} - 22 q^{65} + 14 q^{67} - 48 q^{83} + 16 q^{91} + 16 q^{95} - 32 q^{97}+O(q^{100}) $ 4 * q - 10 * q^17 + 2 * q^25 + 10 * q^29 + 6 * q^31 - 16 * q^35 - 8 * q^37 - 6 * q^41 - 6 * q^49 - 22 * q^65 + 14 * q^67 - 48 * q^83 + 16 * q^91 + 16 * q^95 - 32 * 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
$5$	−1.31342	−0.587381	−0.293691	−	0.955901i	$-0.594884\pi$
$5$	−1.31342	−0.587381	−0.293691	+	0.955901i	$0.594884\pi$
$6$	0	0
$7$	3.04547	1.15108	0.575541	−	0.817773i	$-0.304792\pi$
$7$	3.04547	1.15108	0.575541	+	0.817773i	$0.304792\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.31342	0.364278	0.182139	−	0.983273i	$-0.441698\pi$
$13$	1.31342	0.364278	0.182139	+	0.983273i	$0.441698\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.27492	−1.52189	−0.760945	−	0.648816i	$-0.775265\pi$
$17$	−6.27492	−1.52189	−0.760945	+	0.648816i	$0.775265\pi$
$18$	0	0
$19$	−3.04547	−0.698680	−0.349340	−	0.936996i	$-0.613594\pi$
$19$	−3.04547	−0.698680	−0.349340	+	0.936996i	$0.613594\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−3.27492	−0.654983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.27492	1.16522	0.582611	−	0.812751i	$-0.302031\pi$
$29$	6.27492	1.16522	0.582611	+	0.812751i	$0.302031\pi$
$30$	0	0
$31$	5.27492	0.947403	0.473702	−	0.880685i	$-0.342918\pi$
$31$	5.27492	0.947403	0.473702	+	0.880685i	$0.342918\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	5.54983	0.912387	0.456194	−	0.889881i	$-0.349212\pi$
$37$	5.54983	0.912387	0.456194	+	0.889881i	$0.349212\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.27492	0.355282	0.177641	−	0.984095i	$-0.443153\pi$
$41$	2.27492	0.355282	0.177641	+	0.984095i	$0.443153\pi$
$42$	0	0
$43$	−0.837253	−0.127680	−0.0638400	−	0.997960i	$-0.520335\pi$
$43$	−0.837253	−0.127680	−0.0638400	+	0.997960i	$0.520335\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.1819	−1.77691	−0.888456	−	0.458961i	$-0.848222\pi$
$47$	−12.1819	−1.77691	−0.888456	+	0.458961i	$0.848222\pi$
$48$	0	0
$49$	2.27492	0.324988
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.61478	−0.771249	−0.385625	−	0.922656i	$-0.626014\pi$
$53$	−5.61478	−0.771249	−0.385625	+	0.922656i	$0.626014\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.25370	−0.683973	−0.341986	−	0.939705i	$-0.611100\pi$
$59$	−5.25370	−0.683973	−0.341986	+	0.939705i	$0.611100\pi$
$60$	0	0
$61$	3.04547	0.389933	0.194967	−	0.980810i	$-0.437540\pi$
$61$	3.04547	0.389933	0.194967	+	0.980810i	$0.437540\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.72508	−0.213970
$66$	0	0
$67$	−7.82475	−0.955946	−0.477973	−	0.878375i	$-0.658628\pi$
$67$	−7.82475	−0.955946	−0.477973	+	0.878375i	$0.658628\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	−3.88273	−0.454439	−0.227219	−	0.973844i	$-0.572963\pi$
$73$	−3.88273	−0.454439	−0.227219	+	0.973844i	$0.572963\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.29917	0.933730	0.466865	−	0.884329i	$-0.345383\pi$
$79$	8.29917	0.933730	0.466865	+	0.884329i	$0.345383\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	8.24163	0.893930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.31342	0.139223	0.0696113	−	0.997574i	$-0.477824\pi$
$89$	1.31342	0.139223	0.0696113	+	0.997574i	$0.477824\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−15.5498	−1.57885	−0.789423	−	0.613849i	$-0.789620\pi$
$97$	−15.5498	−1.57885	−0.789423	+	0.613849i	$0.789620\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.54983	−0.850740	−0.425370	−	0.905019i	$-0.639856\pi$
$101$	−8.54983	−0.850740	−0.425370	+	0.905019i	$0.639856\pi$
$102$	0	0
$103$	1.27492	0.125621	0.0628107	−	0.998025i	$-0.479994\pi$
$103$	1.27492	0.125621	0.0628107	+	0.998025i	$0.479994\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.0997	−1.26639	−0.633196	−	0.773991i	$-0.718257\pi$
$107$	−13.0997	−1.26639	−0.633196	+	0.773991i	$0.718257\pi$
$108$	0	0
$109$	20.0049	1.91612	0.958061	−	0.286564i	$-0.0925133\pi$
$109$	20.0049	1.91612	0.958061	+	0.286564i	$0.0925133\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.61478	−0.528194	−0.264097	−	0.964496i	$-0.585074\pi$
$113$	−5.61478	−0.528194	−0.264097	+	0.964496i	$0.585074\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−19.1101	−1.75182
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8685	0.972106
$126$	0	0
$127$	−9.13642	−0.810727	−0.405363	−	0.914156i	$-0.632855\pi$
$127$	−9.13642	−0.810727	−0.405363	+	0.914156i	$0.632855\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	−9.27492	−0.804237
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.8564	1.18383	0.591916	−	0.805999i	$-0.298372\pi$
$137$	13.8564	1.18383	0.591916	+	0.805999i	$0.298372\pi$
$138$	0	0
$139$	0.837253	0.0710149	0.0355075	−	0.999369i	$-0.488695\pi$
$139$	0.837253	0.0710149	0.0355075	+	0.999369i	$0.488695\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−8.24163	−0.684430
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.2749	−1.16945	−0.584723	−	0.811233i	$-0.698797\pi$
$149$	−14.2749	−1.16945	−0.584723	+	0.811233i	$0.698797\pi$
$150$	0	0
$151$	−13.0192	−1.05948	−0.529742	−	0.848159i	$-0.677711\pi$
$151$	−13.0192	−1.05948	−0.529742	+	0.848159i	$0.677711\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.92820	−0.556487
$156$	0	0
$157$	−9.82475	−0.784101	−0.392050	−	0.919944i	$-0.628234\pi$
$157$	−9.82475	−0.784101	−0.392050	+	0.919944i	$0.628234\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13.2749	1.03977	0.519886	−	0.854236i	$-0.325974\pi$
$163$	13.2749	1.03977	0.519886	+	0.854236i	$0.325974\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.09967	0.394624	0.197312	−	0.980341i	$-0.436779\pi$
$167$	5.09967	0.394624	0.197312	+	0.980341i	$0.436779\pi$
$168$	0	0
$169$	−11.2749	−0.867301
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.54983	0.345918	0.172959	−	0.984929i	$-0.444667\pi$
$173$	4.54983	0.345918	0.172959	+	0.984929i	$0.444667\pi$
$174$	0	0
$175$	−9.97368	−0.753939
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.4356	1.30320	0.651599	−	0.758564i	$-0.274099\pi$
$179$	17.4356	1.30320	0.651599	+	0.758564i	$0.274099\pi$
$180$	0	0
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.28929	−0.535919
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.7846	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$191$	20.7846	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$192$	0	0
$193$	0.894797	0.0644089	0.0322045	−	0.999481i	$-0.489747\pi$
$193$	0.894797	0.0644089	0.0322045	+	0.999481i	$0.489747\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.7251	−1.26286	−0.631430	−	0.775433i	$-0.717532\pi$
$197$	−17.7251	−1.26286	−0.631430	+	0.775433i	$0.717532\pi$
$198$	0	0
$199$	13.2749	0.941034	0.470517	−	0.882391i	$-0.344067\pi$
$199$	13.2749	0.941034	0.470517	+	0.882391i	$0.344067\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	19.1101	1.34127
$204$	0	0
$205$	−2.98793	−0.208686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.37097	−0.0943813	−0.0471907	−	0.998886i	$-0.515027\pi$
$211$	−1.37097	−0.0943813	−0.0471907	+	0.998886i	$0.515027\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.09967	0.0749968
$216$	0	0
$217$	16.0646	1.09054
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.24163	−0.554392
$222$	0	0
$223$	−0.175248	−0.0117355	−0.00586775	−	0.999983i	$-0.501868\pi$
$223$	−0.175248	−0.0117355	−0.00586775	+	0.999983i	$0.501868\pi$
$224$	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
$229$	0.824752	0.0545011	0.0272506	−	0.999629i	$-0.491325\pi$
$229$	0.824752	0.0545011	0.0272506	+	0.999629i	$0.491325\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.8248	−1.49530	−0.747650	−	0.664093i	$-0.768818\pi$
$233$	−22.8248	−1.49530	−0.747650	+	0.664093i	$0.768818\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.0997	1.36482	0.682412	−	0.730968i	$-0.260931\pi$
$239$	21.0997	1.36482	0.682412	+	0.730968i	$0.260931\pi$
$240$	0	0
$241$	−10.5074	−0.676841	−0.338420	−	0.940995i	$-0.609893\pi$
$241$	−10.5074	−0.676841	−0.338420	+	0.940995i	$0.609893\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.98793	−0.190892
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−26.0383	−1.64352	−0.821762	−	0.569831i	$-0.807009\pi$
$251$	−26.0383	−1.64352	−0.821762	+	0.569831i	$0.807009\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	29.9786	1.87001	0.935006	−	0.354631i	$-0.115394\pi$
$257$	29.9786	1.87001	0.935006	+	0.354631i	$0.115394\pi$
$258$	0	0
$259$	16.9019	1.05023
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.0997	−1.54771	−0.773856	−	0.633362i	$-0.781675\pi$
$263$	−25.0997	−1.54771	−0.773856	+	0.633362i	$0.781675\pi$
$264$	0	0
$265$	7.37459	0.453017
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.6531	1.92992	0.964961	−	0.262392i	$-0.0845114\pi$
$269$	31.6531	1.92992	0.964961	+	0.262392i	$0.0845114\pi$
$270$	0	0
$271$	26.8756	1.63257	0.816287	−	0.577647i	$-0.196029\pi$
$271$	26.8756	1.63257	0.816287	+	0.577647i	$0.196029\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.9140	0.836008	0.418004	−	0.908445i	$-0.362730\pi$
$277$	13.9140	0.836008	0.418004	+	0.908445i	$0.362730\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.45017	−0.444440	−0.222220	−	0.974997i	$-0.571330\pi$
$281$	−7.45017	−0.444440	−0.222220	+	0.974997i	$0.571330\pi$
$282$	0	0
$283$	−16.0646	−0.954943	−0.477472	−	0.878647i	$-0.658447\pi$
$283$	−16.0646	−0.954943	−0.477472	+	0.878647i	$0.658447\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.92820	0.408959
$288$	0	0
$289$	22.3746	1.31615
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.3746	−1.13188	−0.565938	−	0.824448i	$-0.691486\pi$
$293$	−19.3746	−1.13188	−0.565938	+	0.824448i	$0.691486\pi$
$294$	0	0
$295$	6.90033	0.401753
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−2.54983	−0.146970
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	−34.3375	−1.95974	−0.979872	−	0.199628i	$-0.936026\pi$
$307$	−34.3375	−1.95974	−0.979872	+	0.199628i	$0.936026\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.4356	−0.988682	−0.494341	−	0.869268i	$-0.664591\pi$
$311$	−17.4356	−0.988682	−0.494341	+	0.869268i	$0.664591\pi$
$312$	0	0
$313$	−4.82475	−0.272711	−0.136356	−	0.990660i	$-0.543539\pi$
$313$	−4.82475	−0.272711	−0.136356	+	0.990660i	$0.543539\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.60271	−0.483176	−0.241588	−	0.970379i	$-0.577668\pi$
$317$	−8.60271	−0.483176	−0.241588	+	0.970379i	$0.577668\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	19.1101	1.06331
$324$	0	0
$325$	−4.30136	−0.238596
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−37.0997	−2.04537
$330$	0	0
$331$	30.3746	1.66954	0.834769	−	0.550600i	$-0.185601\pi$
$331$	30.3746	1.66954	0.834769	+	0.550600i	$0.185601\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.2772	0.561504
$336$	0	0
$337$	−32.1868	−1.75333	−0.876663	−	0.481104i	$-0.840236\pi$
$337$	−32.1868	−1.75333	−0.876663	+	0.481104i	$0.840236\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−14.3901	−0.776993
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	−4.35890	−0.233327	−0.116663	−	0.993172i	$-0.537220\pi$
$349$	−4.35890	−0.233327	−0.116663	+	0.993172i	$0.537220\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.4235	−1.08703	−0.543517	−	0.839398i	$-0.682908\pi$
$353$	−20.4235	−1.08703	−0.543517	+	0.839398i	$0.682908\pi$
$354$	0	0
$355$	−9.09967	−0.482960
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.09967	−0.480262	−0.240131	−	0.970740i	$-0.577190\pi$
$359$	−9.09967	−0.480262	−0.240131	+	0.970740i	$0.577190\pi$
$360$	0	0
$361$	−9.72508	−0.511846
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.09967	0.266929
$366$	0	0
$367$	−13.0997	−0.683797	−0.341899	−	0.939737i	$-0.611070\pi$
$367$	−13.0997	−0.683797	−0.341899	+	0.939737i	$0.611070\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17.0997	−0.887771
$372$	0	0
$373$	24.6673	1.27723	0.638613	−	0.769528i	$-0.279508\pi$
$373$	24.6673	1.27723	0.638613	+	0.769528i	$0.279508\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.24163	0.424465
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.5309	−0.793593	−0.396796	−	0.917907i	$-0.629878\pi$
$383$	−15.5309	−0.793593	−0.396796	+	0.917907i	$0.629878\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.91613	0.502768	0.251384	−	0.967887i	$-0.419114\pi$
$389$	9.91613	0.502768	0.251384	+	0.967887i	$0.419114\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.9003	−0.548455
$396$	0	0
$397$	−37.5498	−1.88457	−0.942286	−	0.334809i	$-0.891328\pi$
$397$	−37.5498	−1.88457	−0.942286	+	0.334809i	$0.891328\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.98793	0.149210	0.0746051	−	0.997213i	$-0.476230\pi$
$401$	2.98793	0.149210	0.0746051	+	0.997213i	$0.476230\pi$
$402$	0	0
$403$	6.92820	0.345118
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−36.6032	−1.80991	−0.904957	−	0.425503i	$-0.860097\pi$
$409$	−36.6032	−1.80991	−0.904957	+	0.425503i	$0.860097\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−16.0000	−0.787309
$414$	0	0
$415$	15.7611	0.773681
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13.8564	−0.676930	−0.338465	−	0.940979i	$-0.609908\pi$
$419$	−13.8564	−0.676930	−0.338465	+	0.940979i	$0.609908\pi$
$420$	0	0
$421$	13.3746	0.651837	0.325919	−	0.945398i	$-0.394326\pi$
$421$	13.3746	0.651837	0.325919	+	0.945398i	$0.394326\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	20.5498	0.996813
$426$	0	0
$427$	9.27492	0.448845
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−34.6495	−1.66515	−0.832574	−	0.553913i	$-0.813134\pi$
$433$	−34.6495	−1.66515	−0.832574	+	0.553913i	$0.813134\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	9.13642	0.436058	0.218029	−	0.975942i	$-0.430037\pi$
$439$	9.13642	0.436058	0.218029	+	0.975942i	$0.430037\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.5309	−0.737896	−0.368948	−	0.929450i	$-0.620282\pi$
$443$	−15.5309	−0.737896	−0.368948	+	0.929450i	$0.620282\pi$
$444$	0	0
$445$	−1.72508	−0.0817768
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−39.5336	−1.86571	−0.932854	−	0.360256i	$-0.882690\pi$
$449$	−39.5336	−1.86571	−0.932854	+	0.360256i	$0.882690\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.25370	−0.246297
$456$	0	0
$457$	−32.6054	−1.52522	−0.762609	−	0.646860i	$-0.776082\pi$
$457$	−32.6054	−1.52522	−0.762609	+	0.646860i	$0.776082\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.27492	−0.292252	−0.146126	−	0.989266i	$-0.546680\pi$
$461$	−6.27492	−0.292252	−0.146126	+	0.989266i	$0.546680\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.5457	1.69113	0.845567	−	0.533870i	$-0.179263\pi$
$467$	36.5457	1.69113	0.845567	+	0.533870i	$0.179263\pi$
$468$	0	0
$469$	−23.8301	−1.10037
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	9.97368	0.457624
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.0997	−1.32960	−0.664799	−	0.747022i	$-0.731483\pi$
$479$	−29.0997	−1.32960	−0.664799	+	0.747022i	$0.731483\pi$
$480$	0	0
$481$	7.28929	0.332363
$482$	0	0
$483$	0	0
$484$	0	0
$485$	20.4235	0.927385
$486$	0	0
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.0997	0.591180	0.295590	−	0.955315i	$-0.404484\pi$
$491$	13.0997	0.591180	0.295590	+	0.955315i	$0.404484\pi$
$492$	0	0
$493$	−39.3746	−1.77334
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21.0997	0.946449
$498$	0	0
$499$	8.17525	0.365974	0.182987	−	0.983115i	$-0.441423\pi$
$499$	8.17525	0.365974	0.182987	+	0.983115i	$0.441423\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.90033	−0.129319	−0.0646597	−	0.997907i	$-0.520596\pi$
$503$	−2.90033	−0.129319	−0.0646597	+	0.997907i	$0.520596\pi$
$504$	0	0
$505$	11.2296	0.499709
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−19.1101	−0.847040	−0.423520	−	0.905887i	$-0.639206\pi$
$509$	−19.1101	−0.847040	−0.423520	+	0.905887i	$0.639206\pi$
$510$	0	0
$511$	−11.8248	−0.523096
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.67451	−0.0737876
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.6893	−0.994036	−0.497018	−	0.867740i	$-0.665572\pi$
$521$	−22.6893	−0.994036	−0.497018	+	0.867740i	$0.665572\pi$
$522$	0	0
$523$	23.8301	1.04202	0.521008	−	0.853552i	$-0.325556\pi$
$523$	23.8301	1.04202	0.521008	+	0.853552i	$0.325556\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.0997	−1.44184
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.98793	0.129422
$534$	0	0
$535$	17.2054	0.743855
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.4356	0.749615	0.374807	−	0.927103i	$-0.377709\pi$
$541$	17.4356	0.749615	0.374807	+	0.927103i	$0.377709\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−26.2749	−1.12549
$546$	0	0
$547$	23.5265	1.00592	0.502961	−	0.864309i	$-0.332244\pi$
$547$	23.5265	1.00592	0.502961	+	0.864309i	$0.332244\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19.1101	−0.814118
$552$	0	0
$553$	25.2749	1.07480
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.7492	1.81134	0.905670	−	0.423983i	$-0.139368\pi$
$557$	42.7492	1.81134	0.905670	+	0.423983i	$0.139368\pi$
$558$	0	0
$559$	−1.09967	−0.0465110
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.1993	−1.27275	−0.636375	−	0.771380i	$-0.719567\pi$
$563$	−30.1993	−1.27275	−0.636375	+	0.771380i	$0.719567\pi$
$564$	0	0
$565$	7.37459	0.310251
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.45017	0.312327	0.156164	−	0.987731i	$-0.450087\pi$
$569$	7.45017	0.312327	0.156164	+	0.987731i	$0.450087\pi$
$570$	0	0
$571$	−19.4136	−0.812436	−0.406218	−	0.913776i	$-0.633153\pi$
$571$	−19.4136	−0.812436	−0.406218	+	0.913776i	$0.633153\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.0997	−1.16980	−0.584902	−	0.811104i	$-0.698867\pi$
$577$	−28.0997	−1.16980	−0.584902	+	0.811104i	$0.698867\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−36.5457	−1.51617
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.0383	1.07472	0.537358	−	0.843354i	$-0.319422\pi$
$587$	26.0383	1.07472	0.537358	+	0.843354i	$0.319422\pi$
$588$	0	0
$589$	−16.0646	−0.661931
$590$	0	0
$591$	0	0
$592$	0	0
$593$	20.4743	0.840777	0.420388	−	0.907344i	$-0.361894\pi$
$593$	20.4743	0.840777	0.420388	+	0.907344i	$0.361894\pi$
$594$	0	0
$595$	25.0997	1.02899
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.83289	−0.360902	−0.180451	−	0.983584i	$-0.557756\pi$
$599$	−8.83289	−0.360902	−0.180451	+	0.983584i	$0.557756\pi$
$600$	0	0
$601$	−18.1002	−0.738323	−0.369162	−	0.929365i	$-0.620355\pi$
$601$	−18.1002	−0.738323	−0.369162	+	0.929365i	$0.620355\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.837253	0.0339831	0.0169915	−	0.999856i	$-0.494591\pi$
$607$	0.837253	0.0339831	0.0169915	+	0.999856i	$0.494591\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	−8.77534	−0.354433	−0.177216	−	0.984172i	$-0.556709\pi$
$613$	−8.77534	−0.354433	−0.177216	+	0.984172i	$0.556709\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.6772	1.03373	0.516863	−	0.856068i	$-0.327100\pi$
$617$	25.6772	1.03373	0.516863	+	0.856068i	$0.327100\pi$
$618$	0	0
$619$	25.0997	1.00884	0.504420	−	0.863458i	$-0.331706\pi$
$619$	25.0997	1.00884	0.504420	+	0.863458i	$0.331706\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	2.09967	0.0839868
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−34.8248	−1.38855
$630$	0	0
$631$	13.0997	0.521490	0.260745	−	0.965408i	$-0.416032\pi$
$631$	13.0997	0.521490	0.260745	+	0.965408i	$0.416032\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.0000	0.476205
$636$	0	0
$637$	2.98793	0.118386
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27.3517	−1.08033	−0.540164	−	0.841560i	$-0.681638\pi$
$641$	−27.3517	−1.08033	−0.540164	+	0.841560i	$0.681638\pi$
$642$	0	0
$643$	36.9244	1.45616	0.728078	−	0.685494i	$-0.240414\pi$
$643$	36.9244	1.45616	0.728078	+	0.685494i	$0.240414\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.7611	0.619632	0.309816	−	0.950796i	$-0.399732\pi$
$647$	15.7611	0.619632	0.309816	+	0.950796i	$0.399732\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.92820	0.271122	0.135561	−	0.990769i	$-0.456716\pi$
$653$	6.92820	0.271122	0.135561	+	0.990769i	$0.456716\pi$
$654$	0	0
$655$	21.0148	0.821116
$656$	0	0
$657$	0	0
$658$	0	0
$659$	47.2990	1.84251	0.921254	−	0.388962i	$-0.127166\pi$
$659$	47.2990	1.84251	0.921254	+	0.388962i	$0.127166\pi$
$660$	0	0
$661$	−41.1993	−1.60247	−0.801234	−	0.598351i	$-0.795823\pi$
$661$	−41.1993	−1.60247	−0.801234	+	0.598351i	$0.795823\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.1819	0.472394
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−5.55724	−0.214216	−0.107108	−	0.994247i	$-0.534159\pi$
$673$	−5.55724	−0.214216	−0.107108	+	0.994247i	$0.534159\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.72508	−0.373765	−0.186883	−	0.982382i	$-0.559838\pi$
$677$	−9.72508	−0.373765	−0.186883	+	0.982382i	$0.559838\pi$
$678$	0	0
$679$	−47.3566	−1.81738
$680$	0	0
$681$	0	0
$682$	0	0
$683$	31.2920	1.19736	0.598678	−	0.800990i	$-0.295693\pi$
$683$	31.2920	1.19736	0.598678	+	0.800990i	$0.295693\pi$
$684$	0	0
$685$	−18.1993	−0.695361
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7.37459	−0.280949
$690$	0	0
$691$	43.8248	1.66717	0.833586	−	0.552390i	$-0.186284\pi$
$691$	43.8248	1.66717	0.833586	+	0.552390i	$0.186284\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.09967	−0.0417128
$696$	0	0
$697$	−14.2749	−0.540701
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24.4743	−0.924380	−0.462190	−	0.886781i	$-0.652936\pi$
$701$	−24.4743	−0.924380	−0.462190	+	0.886781i	$0.652936\pi$
$702$	0	0
$703$	−16.9019	−0.637467
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.0383	−0.979271
$708$	0	0
$709$	31.0997	1.16797	0.583986	−	0.811764i	$-0.301492\pi$
$709$	31.0997	1.16797	0.583986	+	0.811764i	$0.301492\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22.4591	−0.837584	−0.418792	−	0.908082i	$-0.637546\pi$
$719$	−22.4591	−0.837584	−0.418792	+	0.908082i	$0.637546\pi$
$720$	0	0
$721$	3.88273	0.144600
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−20.5498	−0.763202
$726$	0	0
$727$	36.0000	1.33517	0.667583	−	0.744535i	$-0.267329\pi$
$727$	36.0000	1.33517	0.667583	+	0.744535i	$0.267329\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.25370	0.194315
$732$	0	0
$733$	3.94027	0.145537	0.0727686	−	0.997349i	$-0.476817\pi$
$733$	3.94027	0.145537	0.0727686	+	0.997349i	$0.476817\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.4811	0.753408	0.376704	−	0.926334i	$-0.377057\pi$
$739$	20.4811	0.753408	0.376704	+	0.926334i	$0.377057\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−47.2990	−1.73523	−0.867616	−	0.497235i	$-0.834349\pi$
$743$	−47.2990	−1.73523	−0.867616	+	0.497235i	$0.834349\pi$
$744$	0	0
$745$	18.7490	0.686911
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−39.8947	−1.45772
$750$	0	0
$751$	−44.9244	−1.63932	−0.819658	−	0.572854i	$-0.805836\pi$
$751$	−44.9244	−1.63932	−0.819658	+	0.572854i	$0.805836\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.0997	0.622321
$756$	0	0
$757$	4.09967	0.149005	0.0745025	−	0.997221i	$-0.476263\pi$
$757$	4.09967	0.149005	0.0745025	+	0.997221i	$0.476263\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.17525	−0.187603	−0.0938013	−	0.995591i	$-0.529902\pi$
$761$	−5.17525	−0.187603	−0.0938013	+	0.995591i	$0.529902\pi$
$762$	0	0
$763$	60.9244	2.20561
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.90033	−0.249157
$768$	0	0
$769$	12.9616	0.467408	0.233704	−	0.972308i	$-0.424915\pi$
$769$	12.9616	0.467408	0.233704	+	0.972308i	$0.424915\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.60271	0.309418	0.154709	−	0.987960i	$-0.450556\pi$
$773$	8.60271	0.309418	0.154709	+	0.987960i	$0.450556\pi$
$774$	0	0
$775$	−17.2749	−0.620533
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.92820	−0.248229
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.9041	0.460566
$786$	0	0
$787$	40.7320	1.45194	0.725969	−	0.687728i	$-0.241392\pi$
$787$	40.7320	1.45194	0.725969	+	0.687728i	$0.241392\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17.0997	−0.607994
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.92820	−0.245410	−0.122705	−	0.992443i	$-0.539157\pi$
$797$	−6.92820	−0.245410	−0.122705	+	0.992443i	$0.539157\pi$
$798$	0	0
$799$	76.4404	2.70427
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.54983	−0.300596	−0.150298	−	0.988641i	$-0.548023\pi$
$809$	−8.54983	−0.300596	−0.150298	+	0.988641i	$0.548023\pi$
$810$	0	0
$811$	−34.3375	−1.20575	−0.602876	−	0.797835i	$-0.705979\pi$
$811$	−34.3375	−1.20575	−0.602876	+	0.797835i	$0.705979\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17.4356	−0.610742
$816$	0	0
$817$	2.54983	0.0892074
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.6495	−0.476371	−0.238185	−	0.971220i	$-0.576553\pi$
$821$	−13.6495	−0.476371	−0.238185	+	0.971220i	$0.576553\pi$
$822$	0	0
$823$	31.4743	1.09712	0.548562	−	0.836110i	$-0.315176\pi$
$823$	31.4743	1.09712	0.548562	+	0.836110i	$0.315176\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.0997	−0.872801	−0.436401	−	0.899753i	$-0.643747\pi$
$827$	−25.0997	−0.872801	−0.436401	+	0.899753i	$0.643747\pi$
$828$	0	0
$829$	2.64950	0.0920211	0.0460105	−	0.998941i	$-0.485349\pi$
$829$	2.64950	0.0920211	0.0460105	+	0.998941i	$0.485349\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14.2749	−0.494597
$834$	0	0
$835$	−6.69803	−0.231795
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.92820	0.239188	0.119594	−	0.992823i	$-0.461841\pi$
$839$	6.92820	0.239188	0.119594	+	0.992823i	$0.461841\pi$
$840$	0	0
$841$	10.3746	0.357744
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.8087	0.509436
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−37.4405	−1.28194	−0.640969	−	0.767567i	$-0.721467\pi$
$853$	−37.4405	−1.28194	−0.640969	+	0.767567i	$0.721467\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.7492	−0.640460	−0.320230	−	0.947340i	$-0.603760\pi$
$857$	−18.7492	−0.640460	−0.320230	+	0.947340i	$0.603760\pi$
$858$	0	0
$859$	5.62541	0.191937	0.0959683	−	0.995384i	$-0.469405\pi$
$859$	5.62541	0.191937	0.0959683	+	0.995384i	$0.469405\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.1101	−0.650515	−0.325258	−	0.945625i	$-0.605451\pi$
$863$	−19.1101	−0.650515	−0.325258	+	0.945625i	$0.605451\pi$
$864$	0	0
$865$	−5.97586	−0.203185
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−10.2772	−0.348230
$872$	0	0
$873$	0	0
$874$	0	0
$875$	33.0997	1.11897
$876$	0	0
$877$	26.8180	0.905580	0.452790	−	0.891617i	$-0.350429\pi$
$877$	26.8180	0.905580	0.452790	+	0.891617i	$0.350429\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−4.89261	−0.164836	−0.0824182	−	0.996598i	$-0.526264\pi$
$881$	−4.89261	−0.164836	−0.0824182	+	0.996598i	$0.526264\pi$
$882$	0	0
$883$	10.7251	0.360928	0.180464	−	0.983582i	$-0.442240\pi$
$883$	10.7251	0.360928	0.180464	+	0.983582i	$0.442240\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.0997	0.574151	0.287075	−	0.957908i	$-0.407317\pi$
$887$	17.0997	0.574151	0.287075	+	0.957908i	$0.407317\pi$
$888$	0	0
$889$	−27.8248	−0.933212
$890$	0	0
$891$	0	0
$892$	0	0
$893$	37.0997	1.24149
$894$	0	0
$895$	−22.9003	−0.765474
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.0997	1.10394
$900$	0	0
$901$	35.2323	1.17376
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.31342	−0.0436597
$906$	0	0
$907$	−42.0241	−1.39539	−0.697693	−	0.716396i	$-0.745790\pi$
$907$	−42.0241	−1.39539	−0.697693	+	0.716396i	$0.745790\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−48.7276	−1.60913
$918$	0	0
$919$	−48.1939	−1.58977	−0.794885	−	0.606760i	$-0.792469\pi$
$919$	−48.1939	−1.58977	−0.794885	+	0.606760i	$0.792469\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.09967	0.299519
$924$	0	0
$925$	−18.1752	−0.597598
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.9068	1.21087	0.605436	−	0.795894i	$-0.292999\pi$
$929$	36.9068	1.21087	0.605436	+	0.795894i	$0.292999\pi$
$930$	0	0
$931$	−6.92820	−0.227063
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.4405	1.22313	0.611564	−	0.791195i	$-0.290541\pi$
$937$	37.4405	1.22313	0.611564	+	0.791195i	$0.290541\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.4743	1.05863	0.529315	−	0.848425i	$-0.322449\pi$
$941$	32.4743	1.05863	0.529315	+	0.848425i	$0.322449\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.9430	0.908025	0.454013	−	0.890995i	$-0.349992\pi$
$947$	27.9430	0.908025	0.454013	+	0.890995i	$0.349992\pi$
$948$	0	0
$949$	−5.09967	−0.165542
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29.1752	−0.945079	−0.472539	−	0.881309i	$-0.656663\pi$
$953$	−29.1752	−0.945079	−0.472539	+	0.881309i	$0.656663\pi$
$954$	0	0
$955$	−27.2990	−0.883375
$956$	0	0
$957$	0	0
$958$	0	0
$959$	42.1993	1.36269
$960$	0	0
$961$	−3.17525	−0.102427
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.17525	−0.0378326
$966$	0	0
$967$	−3.88273	−0.124860	−0.0624301	−	0.998049i	$-0.519885\pi$
$967$	−3.88273	−0.124860	−0.0624301	+	0.998049i	$0.519885\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.7994	1.34141	0.670703	−	0.741726i	$-0.265992\pi$
$971$	41.7994	1.34141	0.670703	+	0.741726i	$0.265992\pi$
$972$	0	0
$973$	2.54983	0.0817439
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.3994	0.844591	0.422296	−	0.906458i	$-0.361224\pi$
$977$	26.3994	0.844591	0.422296	+	0.906458i	$0.361224\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.1484	1.44001	0.720005	−	0.693969i	$-0.244139\pi$
$983$	45.1484	1.44001	0.720005	+	0.693969i	$0.244139\pi$
$984$	0	0
$985$	23.2806	0.741780
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−14.1993	−0.451057	−0.225528	−	0.974237i	$-0.572411\pi$
$991$	−14.1993	−0.451057	−0.225528	+	0.974237i	$0.572411\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17.4356	−0.552746
$996$	0	0
$997$	−14.6361	−0.463530	−0.231765	−	0.972772i	$-0.574450\pi$
$997$	−14.6361	−0.463530	−0.231765	+	0.972772i	$0.574450\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.cf.1.2	✓	4
3.2	odd	2		8712.2.a.cg.1.3	yes	4
11.10	odd	2		8712.2.a.cg.1.2	yes	4
33.32	even	2	inner	8712.2.a.cf.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8712.2.a.cf.1.2	✓	4	1.1	even	1	trivial
8712.2.a.cf.1.3	yes	4	33.32	even	2	inner
8712.2.a.cg.1.2	yes	4	11.10	odd	2
8712.2.a.cg.1.3	yes	4	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:	\( N \)	\(=\)	\( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8712.a (trivial)

\(f(q)\)	\(=\)	\(q-1.31342 q^{5} +3.04547 q^{7} +O(q^{10})\)	\(q-1.31342 q^{5} +3.04547 q^{7} +1.31342 q^{13} -6.27492 q^{17} -3.04547 q^{19} -3.27492 q^{25} +6.27492 q^{29} +5.27492 q^{31} -4.00000 q^{35} +5.54983 q^{37} +2.27492 q^{41} -0.837253 q^{43} -12.1819 q^{47} +2.27492 q^{49} -5.61478 q^{53} -5.25370 q^{59} +3.04547 q^{61} -1.72508 q^{65} -7.82475 q^{67} +6.92820 q^{71} -3.88273 q^{73} +8.29917 q^{79} -12.0000 q^{83} +8.24163 q^{85} +1.31342 q^{89} +4.00000 q^{91} +4.00000 q^{95} -15.5498 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 10 q^{17} + 2 q^{25} + 10 q^{29} + 6 q^{31} - 16 q^{35} - 8 q^{37} - 6 q^{41} - 6 q^{49} - 22 q^{65} + 14 q^{67} - 48 q^{83} + 16 q^{91} + 16 q^{95} - 32 q^{97}+O(q^{100}) \) 4 * q - 10 * q^17 + 2 * q^25 + 10 * q^29 + 6 * q^31 - 16 * q^35 - 8 * q^37 - 6 * q^41 - 6 * q^49 - 22 * q^65 + 14 * q^67 - 48 * q^83 + 16 * q^91 + 16 * q^95 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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