LMFDB - Embedded newform 8712.2.a.bj.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:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 968)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+2.23607 q^{5} +1.23607 q^{7} +O(q^{10})$	$q+2.23607 q^{5} +1.23607 q^{7} -6.23607 q^{13} +1.00000 q^{17} -7.70820 q^{19} -5.23607 q^{23} +6.23607 q^{29} +9.23607 q^{31} +2.76393 q^{35} +4.23607 q^{37} -1.47214 q^{41} -1.52786 q^{43} +7.70820 q^{47} -5.47214 q^{49} -9.18034 q^{53} +14.4721 q^{59} +2.94427 q^{61} -13.9443 q^{65} -5.23607 q^{67} +4.00000 q^{71} -12.4721 q^{73} +2.76393 q^{79} -10.1803 q^{83} +2.23607 q^{85} -14.4164 q^{89} -7.70820 q^{91} -17.2361 q^{95} -5.94427 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^7	$ 2 q - 2 q^{7} - 8 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{23} + 8 q^{29} + 14 q^{31} + 10 q^{35} + 4 q^{37} + 6 q^{41} - 12 q^{43} + 2 q^{47} - 2 q^{49} + 4 q^{53} + 20 q^{59} - 12 q^{61} - 10 q^{65} - 6 q^{67} + 8 q^{71} - 16 q^{73} + 10 q^{79} + 2 q^{83} - 2 q^{89} - 2 q^{91} - 30 q^{95} + 6 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 8 * q^13 + 2 * q^17 - 2 * q^19 - 6 * q^23 + 8 * q^29 + 14 * q^31 + 10 * q^35 + 4 * q^37 + 6 * q^41 - 12 * q^43 + 2 * q^47 - 2 * q^49 + 4 * q^53 + 20 * q^59 - 12 * q^61 - 10 * q^65 - 6 * q^67 + 8 * q^71 - 16 * q^73 + 10 * q^79 + 2 * q^83 - 2 * q^89 - 2 * q^91 - 30 * q^95 + 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0	0
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−7.70820	−1.76838	−0.884192	−	0.467124i	$-0.845290\pi$
$19$	−7.70820	−1.76838	−0.884192	+	0.467124i	$0.845290\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.23607	−1.09180	−0.545898	−	0.837852i	$-0.683811\pi$
$23$	−5.23607	−1.09180	−0.545898	+	0.837852i	$0.683811\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.23607	1.15801	0.579004	−	0.815324i	$-0.303441\pi$
$29$	6.23607	1.15801	0.579004	+	0.815324i	$0.303441\pi$
$30$	0	0
$31$	9.23607	1.65885	0.829423	−	0.558620i	$-0.188669\pi$
$31$	9.23607	1.65885	0.829423	+	0.558620i	$0.188669\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.76393	0.467190
$36$	0	0
$37$	4.23607	0.696405	0.348203	−	0.937419i	$-0.386792\pi$
$37$	4.23607	0.696405	0.348203	+	0.937419i	$0.386792\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.47214	−0.229909	−0.114955	−	0.993371i	$-0.536672\pi$
$41$	−1.47214	−0.229909	−0.114955	+	0.993371i	$0.536672\pi$
$42$	0	0
$43$	−1.52786	−0.232997	−0.116499	−	0.993191i	$-0.537167\pi$
$43$	−1.52786	−0.232997	−0.116499	+	0.993191i	$0.537167\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.70820	1.12436	0.562179	−	0.827016i	$-0.309963\pi$
$47$	7.70820	1.12436	0.562179	+	0.827016i	$0.309963\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.18034	−1.26102	−0.630508	−	0.776182i	$-0.717154\pi$
$53$	−9.18034	−1.26102	−0.630508	+	0.776182i	$0.717154\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.4721	1.88411	0.942056	−	0.335456i	$-0.108890\pi$
$59$	14.4721	1.88411	0.942056	+	0.335456i	$0.108890\pi$
$60$	0	0
$61$	2.94427	0.376975	0.188488	−	0.982076i	$-0.439641\pi$
$61$	2.94427	0.376975	0.188488	+	0.982076i	$0.439641\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13.9443	−1.72957
$66$	0	0
$67$	−5.23607	−0.639688	−0.319844	−	0.947470i	$-0.603630\pi$
$67$	−5.23607	−0.639688	−0.319844	+	0.947470i	$0.603630\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−12.4721	−1.45975	−0.729877	−	0.683579i	$-0.760422\pi$
$73$	−12.4721	−1.45975	−0.729877	+	0.683579i	$0.760422\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.76393	0.310967	0.155483	−	0.987839i	$-0.450307\pi$
$79$	2.76393	0.310967	0.155483	+	0.987839i	$0.450307\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.1803	−1.11744	−0.558719	−	0.829357i	$-0.688707\pi$
$83$	−10.1803	−1.11744	−0.558719	+	0.829357i	$0.688707\pi$
$84$	0	0
$85$	2.23607	0.242536
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.4164	−1.52814	−0.764068	−	0.645136i	$-0.776801\pi$
$89$	−14.4164	−1.52814	−0.764068	+	0.645136i	$0.776801\pi$
$90$	0	0
$91$	−7.70820	−0.808039
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−17.2361	−1.76838
$96$	0	0
$97$	−5.94427	−0.603549	−0.301775	−	0.953379i	$-0.597579\pi$
$97$	−5.94427	−0.603549	−0.301775	+	0.953379i	$0.597579\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.7082	−1.13187	−0.565937	−	0.824448i	$-0.691486\pi$
$107$	−11.7082	−1.13187	−0.565937	+	0.824448i	$0.691486\pi$
$108$	0	0
$109$	1.76393	0.168954	0.0844770	−	0.996425i	$-0.473078\pi$
$109$	1.76393	0.168954	0.0844770	+	0.996425i	$0.473078\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.05573	−0.193387	−0.0966933	−	0.995314i	$-0.530827\pi$
$113$	−2.05573	−0.193387	−0.0966933	+	0.995314i	$0.530827\pi$
$114$	0	0
$115$	−11.7082	−1.09180
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.23607	0.113310
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−9.52786	−0.845461	−0.422731	−	0.906255i	$-0.638928\pi$
$127$	−9.52786	−0.845461	−0.422731	+	0.906255i	$0.638928\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−9.52786	−0.826171
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.4721	−1.40731	−0.703655	−	0.710542i	$-0.748450\pi$
$137$	−16.4721	−1.40731	−0.703655	+	0.710542i	$0.748450\pi$
$138$	0	0
$139$	−10.7639	−0.912985	−0.456492	−	0.889727i	$-0.650894\pi$
$139$	−10.7639	−0.912985	−0.456492	+	0.889727i	$0.650894\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	13.9443	1.15801
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.76393	−0.472200	−0.236100	−	0.971729i	$-0.575869\pi$
$149$	−5.76393	−0.472200	−0.236100	+	0.971729i	$0.575869\pi$
$150$	0	0
$151$	−19.4164	−1.58008	−0.790042	−	0.613052i	$-0.789942\pi$
$151$	−19.4164	−1.58008	−0.790042	+	0.613052i	$0.789942\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	20.6525	1.65885
$156$	0	0
$157$	−3.52786	−0.281554	−0.140777	−	0.990041i	$-0.544960\pi$
$157$	−3.52786	−0.281554	−0.140777	+	0.990041i	$0.544960\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.47214	−0.510076
$162$	0	0
$163$	−7.70820	−0.603753	−0.301877	−	0.953347i	$-0.597613\pi$
$163$	−7.70820	−0.603753	−0.301877	+	0.953347i	$0.597613\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.4164	−1.19296	−0.596479	−	0.802629i	$-0.703434\pi$
$167$	−15.4164	−1.19296	−0.596479	+	0.802629i	$0.703434\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.05573	−0.0802655	−0.0401328	−	0.999194i	$-0.512778\pi$
$173$	−1.05573	−0.0802655	−0.0401328	+	0.999194i	$0.512778\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.47214	−0.483750	−0.241875	−	0.970307i	$-0.577762\pi$
$179$	−6.47214	−0.483750	−0.241875	+	0.970307i	$0.577762\pi$
$180$	0	0
$181$	−0.708204	−0.0526404	−0.0263202	−	0.999654i	$-0.508379\pi$
$181$	−0.708204	−0.0526404	−0.0263202	+	0.999654i	$0.508379\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.47214	0.696405
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.4721	−1.04717	−0.523584	−	0.851974i	$-0.675405\pi$
$191$	−14.4721	−1.04717	−0.523584	+	0.851974i	$0.675405\pi$
$192$	0	0
$193$	−13.9443	−1.00373	−0.501865	−	0.864946i	$-0.667353\pi$
$193$	−13.9443	−1.00373	−0.501865	+	0.864946i	$0.667353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.6525	0.972699	0.486349	−	0.873764i	$-0.338328\pi$
$197$	13.6525	0.972699	0.486349	+	0.873764i	$0.338328\pi$
$198$	0	0
$199$	−9.52786	−0.675412	−0.337706	−	0.941252i	$-0.609651\pi$
$199$	−9.52786	−0.675412	−0.337706	+	0.941252i	$0.609651\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.70820	0.541010
$204$	0	0
$205$	−3.29180	−0.229909
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	24.9443	1.71723	0.858617	−	0.512617i	$-0.171324\pi$
$211$	24.9443	1.71723	0.858617	+	0.512617i	$0.171324\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.41641	−0.232997
$216$	0	0
$217$	11.4164	0.774996
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.23607	−0.419483
$222$	0	0
$223$	−8.94427	−0.598953	−0.299476	−	0.954104i	$-0.596812\pi$
$223$	−8.94427	−0.598953	−0.299476	+	0.954104i	$0.596812\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.47214	0.429571	0.214785	−	0.976661i	$-0.431095\pi$
$227$	6.47214	0.429571	0.214785	+	0.976661i	$0.431095\pi$
$228$	0	0
$229$	10.7082	0.707618	0.353809	−	0.935318i	$-0.384886\pi$
$229$	10.7082	0.707618	0.353809	+	0.935318i	$0.384886\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.0000	−0.720634	−0.360317	−	0.932830i	$-0.617331\pi$
$233$	−11.0000	−0.720634	−0.360317	+	0.932830i	$0.617331\pi$
$234$	0	0
$235$	17.2361	1.12436
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.18034	0.141034	0.0705172	−	0.997511i	$-0.477535\pi$
$239$	2.18034	0.141034	0.0705172	+	0.997511i	$0.477535\pi$
$240$	0	0
$241$	−12.4721	−0.803401	−0.401700	−	0.915771i	$-0.631581\pi$
$241$	−12.4721	−0.803401	−0.401700	+	0.915771i	$0.631581\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−12.2361	−0.781734
$246$	0	0
$247$	48.0689	3.05855
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.2361	1.59289	0.796443	−	0.604713i	$-0.206712\pi$
$251$	25.2361	1.59289	0.796443	+	0.604713i	$0.206712\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.4721	1.46415	0.732076	−	0.681223i	$-0.238552\pi$
$257$	23.4721	1.46415	0.732076	+	0.681223i	$0.238552\pi$
$258$	0	0
$259$	5.23607	0.325353
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.6525	−1.52014	−0.760068	−	0.649843i	$-0.774835\pi$
$263$	−24.6525	−1.52014	−0.760068	+	0.649843i	$0.774835\pi$
$264$	0	0
$265$	−20.5279	−1.26102
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.29180	−0.200704	−0.100352	−	0.994952i	$-0.531997\pi$
$269$	−3.29180	−0.200704	−0.100352	+	0.994952i	$0.531997\pi$
$270$	0	0
$271$	5.88854	0.357704	0.178852	−	0.983876i	$-0.442762\pi$
$271$	5.88854	0.357704	0.178852	+	0.983876i	$0.442762\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.1246	0.848666	0.424333	−	0.905506i	$-0.360509\pi$
$277$	14.1246	0.848666	0.424333	+	0.905506i	$0.360509\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.4721	−1.45989	−0.729943	−	0.683508i	$-0.760453\pi$
$281$	−24.4721	−1.45989	−0.729943	+	0.683508i	$0.760453\pi$
$282$	0	0
$283$	15.4164	0.916410	0.458205	−	0.888846i	$-0.348492\pi$
$283$	15.4164	0.916410	0.458205	+	0.888846i	$0.348492\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.81966	−0.107411
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.7082	−1.09294	−0.546472	−	0.837477i	$-0.684030\pi$
$293$	−18.7082	−1.09294	−0.546472	+	0.837477i	$0.684030\pi$
$294$	0	0
$295$	32.3607	1.88411
$296$	0	0
$297$	0	0
$298$	0	0
$299$	32.6525	1.88834
$300$	0	0
$301$	−1.88854	−0.108854
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.58359	0.376975
$306$	0	0
$307$	−1.81966	−0.103853	−0.0519267	−	0.998651i	$-0.516536\pi$
$307$	−1.81966	−0.103853	−0.0519267	+	0.998651i	$0.516536\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.4721	1.27428	0.637139	−	0.770749i	$-0.280118\pi$
$311$	22.4721	1.27428	0.637139	+	0.770749i	$0.280118\pi$
$312$	0	0
$313$	−9.94427	−0.562083	−0.281042	−	0.959696i	$-0.590680\pi$
$313$	−9.94427	−0.562083	−0.281042	+	0.959696i	$0.590680\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.8885	1.11705	0.558526	−	0.829487i	$-0.311367\pi$
$317$	19.8885	1.11705	0.558526	+	0.829487i	$0.311367\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.70820	−0.428896
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.52786	0.525288
$330$	0	0
$331$	2.47214	0.135881	0.0679404	−	0.997689i	$-0.478357\pi$
$331$	2.47214	0.135881	0.0679404	+	0.997689i	$0.478357\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.7082	−0.639688
$336$	0	0
$337$	−20.4164	−1.11215	−0.556076	−	0.831131i	$-0.687694\pi$
$337$	−20.4164	−1.11215	−0.556076	+	0.831131i	$0.687694\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−15.4164	−0.832408
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.4164	0.827596	0.413798	−	0.910369i	$-0.364202\pi$
$347$	15.4164	0.827596	0.413798	+	0.910369i	$0.364202\pi$
$348$	0	0
$349$	3.65248	0.195513	0.0977563	−	0.995210i	$-0.468833\pi$
$349$	3.65248	0.195513	0.0977563	+	0.995210i	$0.468833\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.0000	−1.22417	−0.612083	−	0.790793i	$-0.709668\pi$
$353$	−23.0000	−1.22417	−0.612083	+	0.790793i	$0.709668\pi$
$354$	0	0
$355$	8.94427	0.474713
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.2361	−0.909685	−0.454842	−	0.890572i	$-0.650304\pi$
$359$	−17.2361	−0.909685	−0.454842	+	0.890572i	$0.650304\pi$
$360$	0	0
$361$	40.4164	2.12718
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−27.8885	−1.45975
$366$	0	0
$367$	18.1803	0.949006	0.474503	−	0.880254i	$-0.342628\pi$
$367$	18.1803	0.949006	0.474503	+	0.880254i	$0.342628\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.3475	−0.589134
$372$	0	0
$373$	−35.8885	−1.85824	−0.929119	−	0.369780i	$-0.879433\pi$
$373$	−35.8885	−1.85824	−0.929119	+	0.369780i	$0.879433\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−38.8885	−2.00286
$378$	0	0
$379$	−20.9443	−1.07583	−0.537917	−	0.842997i	$-0.680789\pi$
$379$	−20.9443	−1.07583	−0.537917	+	0.842997i	$0.680789\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.41641	0.378961	0.189480	−	0.981885i	$-0.439320\pi$
$383$	7.41641	0.378961	0.189480	+	0.981885i	$0.439320\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.2361	1.33022	0.665111	−	0.746745i	$-0.268384\pi$
$389$	26.2361	1.33022	0.665111	+	0.746745i	$0.268384\pi$
$390$	0	0
$391$	−5.23607	−0.264799
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.18034	0.310967
$396$	0	0
$397$	−15.7639	−0.791169	−0.395585	−	0.918430i	$-0.629458\pi$
$397$	−15.7639	−0.791169	−0.395585	+	0.918430i	$0.629458\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.4721	1.17214	0.586071	−	0.810260i	$-0.300674\pi$
$401$	23.4721	1.17214	0.586071	+	0.810260i	$0.300674\pi$
$402$	0	0
$403$	−57.5967	−2.86910
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−34.3050	−1.69627	−0.848135	−	0.529780i	$-0.822275\pi$
$409$	−34.3050	−1.69627	−0.848135	+	0.529780i	$0.822275\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	17.8885	0.880238
$414$	0	0
$415$	−22.7639	−1.11744
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.81966	0.0888962	0.0444481	−	0.999012i	$-0.485847\pi$
$419$	1.81966	0.0888962	0.0444481	+	0.999012i	$0.485847\pi$
$420$	0	0
$421$	−31.1803	−1.51964	−0.759818	−	0.650135i	$-0.774712\pi$
$421$	−31.1803	−1.51964	−0.759818	+	0.650135i	$0.774712\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.63932	0.176119
$428$	0	0
$429$	0	0
$430$	0	0
$431$	41.3050	1.98959	0.994795	−	0.101899i	$-0.0324918\pi$
$431$	41.3050	1.98959	0.994795	+	0.101899i	$0.0324918\pi$
$432$	0	0
$433$	−9.00000	−0.432512	−0.216256	−	0.976337i	$-0.569385\pi$
$433$	−9.00000	−0.432512	−0.216256	+	0.976337i	$0.569385\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	40.3607	1.93071
$438$	0	0
$439$	17.2361	0.822633	0.411316	−	0.911493i	$-0.365069\pi$
$439$	17.2361	0.822633	0.411316	+	0.911493i	$0.365069\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.583592	0.0277273	0.0138636	−	0.999904i	$-0.495587\pi$
$443$	0.583592	0.0277273	0.0138636	+	0.999904i	$0.495587\pi$
$444$	0	0
$445$	−32.2361	−1.52814
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.88854	−0.419476	−0.209738	−	0.977758i	$-0.567261\pi$
$449$	−8.88854	−0.419476	−0.209738	+	0.977758i	$0.567261\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.2361	−0.808039
$456$	0	0
$457$	−8.05573	−0.376831	−0.188416	−	0.982089i	$-0.560335\pi$
$457$	−8.05573	−0.376831	−0.188416	+	0.982089i	$0.560335\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.8197	−0.597071	−0.298536	−	0.954399i	$-0.596498\pi$
$461$	−12.8197	−0.597071	−0.298536	+	0.954399i	$0.596498\pi$
$462$	0	0
$463$	−18.8328	−0.875235	−0.437618	−	0.899161i	$-0.644178\pi$
$463$	−18.8328	−0.875235	−0.437618	+	0.899161i	$0.644178\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.5279	0.625995	0.312997	−	0.949754i	$-0.398667\pi$
$467$	13.5279	0.625995	0.312997	+	0.949754i	$0.398667\pi$
$468$	0	0
$469$	−6.47214	−0.298855
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.8328	1.40879	0.704394	−	0.709810i	$-0.251219\pi$
$479$	30.8328	1.40879	0.704394	+	0.709810i	$0.251219\pi$
$480$	0	0
$481$	−26.4164	−1.20448
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.2918	−0.603549
$486$	0	0
$487$	29.5967	1.34116	0.670578	−	0.741839i	$-0.266046\pi$
$487$	29.5967	1.34116	0.670578	+	0.741839i	$0.266046\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.81966	−0.0821201	−0.0410601	−	0.999157i	$-0.513073\pi$
$491$	−1.81966	−0.0821201	−0.0410601	+	0.999157i	$0.513073\pi$
$492$	0	0
$493$	6.23607	0.280858
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.94427	0.221781
$498$	0	0
$499$	−21.3050	−0.953741	−0.476870	−	0.878974i	$-0.658229\pi$
$499$	−21.3050	−0.953741	−0.476870	+	0.878974i	$0.658229\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.5967	0.784600	0.392300	−	0.919837i	$-0.371679\pi$
$503$	17.5967	0.784600	0.392300	+	0.919837i	$0.371679\pi$
$504$	0	0
$505$	4.47214	0.199007
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.9443	−0.839690	−0.419845	−	0.907596i	$-0.637916\pi$
$509$	−18.9443	−0.839690	−0.419845	+	0.907596i	$0.637916\pi$
$510$	0	0
$511$	−15.4164	−0.681982
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.4164	−0.587784	−0.293892	−	0.955839i	$-0.594951\pi$
$521$	−13.4164	−0.587784	−0.293892	+	0.955839i	$0.594951\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.23607	0.402329
$528$	0	0
$529$	4.41641	0.192018
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.18034	0.397645
$534$	0	0
$535$	−26.1803	−1.13187
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.4721	0.536219	0.268110	−	0.963388i	$-0.413601\pi$
$541$	12.4721	0.536219	0.268110	+	0.963388i	$0.413601\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.94427	0.168954
$546$	0	0
$547$	9.52786	0.407382	0.203691	−	0.979035i	$-0.434706\pi$
$547$	9.52786	0.407382	0.203691	+	0.979035i	$0.434706\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−48.0689	−2.04780
$552$	0	0
$553$	3.41641	0.145280
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.111456	0.00472255	0.00236127	−	0.999997i	$-0.499248\pi$
$557$	0.111456	0.00472255	0.00236127	+	0.999997i	$0.499248\pi$
$558$	0	0
$559$	9.52786	0.402986
$560$	0	0
$561$	0	0
$562$	0	0
$563$	42.1803	1.77769	0.888845	−	0.458209i	$-0.151509\pi$
$563$	42.1803	1.77769	0.888845	+	0.458209i	$0.151509\pi$
$564$	0	0
$565$	−4.59675	−0.193387
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.3607	−0.769720	−0.384860	−	0.922975i	$-0.625750\pi$
$569$	−18.3607	−0.769720	−0.384860	+	0.922975i	$0.625750\pi$
$570$	0	0
$571$	17.2361	0.721307	0.360653	−	0.932700i	$-0.382554\pi$
$571$	17.2361	0.721307	0.360653	+	0.932700i	$0.382554\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	28.3050	1.17835	0.589175	−	0.808005i	$-0.299453\pi$
$577$	28.3050	1.17835	0.589175	+	0.808005i	$0.299453\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.5836	−0.522055
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.6525	1.34771	0.673856	−	0.738863i	$-0.264637\pi$
$587$	32.6525	1.34771	0.673856	+	0.738863i	$0.264637\pi$
$588$	0	0
$589$	−71.1935	−2.93348
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.41641	0.181360	0.0906801	−	0.995880i	$-0.471096\pi$
$593$	4.41641	0.181360	0.0906801	+	0.995880i	$0.471096\pi$
$594$	0	0
$595$	2.76393	0.113310
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.70820	0.314949	0.157474	−	0.987523i	$-0.449665\pi$
$599$	7.70820	0.314949	0.157474	+	0.987523i	$0.449665\pi$
$600$	0	0
$601$	38.7771	1.58175	0.790875	−	0.611977i	$-0.209626\pi$
$601$	38.7771	1.58175	0.790875	+	0.611977i	$0.209626\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.65248	−0.351193	−0.175597	−	0.984462i	$-0.556185\pi$
$607$	−8.65248	−0.351193	−0.175597	+	0.984462i	$0.556185\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−48.0689	−1.94466
$612$	0	0
$613$	−2.59675	−0.104882	−0.0524408	−	0.998624i	$-0.516700\pi$
$613$	−2.59675	−0.104882	−0.0524408	+	0.998624i	$0.516700\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.3607	1.18202	0.591008	−	0.806666i	$-0.298730\pi$
$617$	29.3607	1.18202	0.591008	+	0.806666i	$0.298730\pi$
$618$	0	0
$619$	−20.0689	−0.806637	−0.403318	−	0.915060i	$-0.632143\pi$
$619$	−20.0689	−0.806637	−0.403318	+	0.915060i	$0.632143\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−17.8197	−0.713930
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.23607	0.168903
$630$	0	0
$631$	16.2918	0.648566	0.324283	−	0.945960i	$-0.394877\pi$
$631$	16.2918	0.648566	0.324283	+	0.945960i	$0.394877\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−21.3050	−0.845461
$636$	0	0
$637$	34.1246	1.35207
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.47214	0.295132	0.147566	−	0.989052i	$-0.452856\pi$
$641$	7.47214	0.295132	0.147566	+	0.989052i	$0.452856\pi$
$642$	0	0
$643$	2.18034	0.0859842	0.0429921	−	0.999075i	$-0.486311\pi$
$643$	2.18034	0.0859842	0.0429921	+	0.999075i	$0.486311\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.11146	0.0830099	0.0415050	−	0.999138i	$-0.486785\pi$
$647$	2.11146	0.0830099	0.0415050	+	0.999138i	$0.486785\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.3050	0.911993	0.455997	−	0.889982i	$-0.349283\pi$
$653$	23.3050	0.911993	0.455997	+	0.889982i	$0.349283\pi$
$654$	0	0
$655$	17.8885	0.698963
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.81966	−0.382520	−0.191260	−	0.981539i	$-0.561257\pi$
$659$	−9.81966	−0.382520	−0.191260	+	0.981539i	$0.561257\pi$
$660$	0	0
$661$	0.819660	0.0318811	0.0159405	−	0.999873i	$-0.494926\pi$
$661$	0.819660	0.0318811	0.0159405	+	0.999873i	$0.494926\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−21.3050	−0.826171
$666$	0	0
$667$	−32.6525	−1.26431
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	11.5279	0.444367	0.222183	−	0.975005i	$-0.428682\pi$
$673$	11.5279	0.444367	0.222183	+	0.975005i	$0.428682\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.2918	−0.587711	−0.293856	−	0.955850i	$-0.594939\pi$
$677$	−15.2918	−0.587711	−0.293856	+	0.955850i	$0.594939\pi$
$678$	0	0
$679$	−7.34752	−0.281972
$680$	0	0
$681$	0	0
$682$	0	0
$683$	47.1246	1.80317	0.901587	−	0.432599i	$-0.142403\pi$
$683$	47.1246	1.80317	0.901587	+	0.432599i	$0.142403\pi$
$684$	0	0
$685$	−36.8328	−1.40731
$686$	0	0
$687$	0	0
$688$	0	0
$689$	57.2492	2.18102
$690$	0	0
$691$	−7.41641	−0.282133	−0.141067	−	0.990000i	$-0.545053\pi$
$691$	−7.41641	−0.282133	−0.141067	+	0.990000i	$0.545053\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.0689	−0.912985
$696$	0	0
$697$	−1.47214	−0.0557611
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.34752	0.164204	0.0821019	−	0.996624i	$-0.473837\pi$
$701$	4.34752	0.164204	0.0821019	+	0.996624i	$0.473837\pi$
$702$	0	0
$703$	−32.6525	−1.23151
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.47214	0.0929742
$708$	0	0
$709$	31.8885	1.19760	0.598800	−	0.800899i	$-0.295645\pi$
$709$	31.8885	1.19760	0.598800	+	0.800899i	$0.295645\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−48.3607	−1.81112
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.5410	0.691463	0.345732	−	0.938333i	$-0.387631\pi$
$719$	18.5410	0.691463	0.345732	+	0.938333i	$0.387631\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−19.7082	−0.730937	−0.365468	−	0.930824i	$-0.619091\pi$
$727$	−19.7082	−0.730937	−0.365468	+	0.930824i	$0.619091\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.52786	−0.0565101
$732$	0	0
$733$	−15.7639	−0.582254	−0.291127	−	0.956684i	$-0.594030\pi$
$733$	−15.7639	−0.582254	−0.291127	+	0.956684i	$0.594030\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	27.7082	1.01926	0.509631	−	0.860393i	$-0.329782\pi$
$739$	27.7082	1.01926	0.509631	+	0.860393i	$0.329782\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38.5410	1.41393	0.706966	−	0.707247i	$-0.250063\pi$
$743$	38.5410	1.41393	0.706966	+	0.707247i	$0.250063\pi$
$744$	0	0
$745$	−12.8885	−0.472200
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−14.4721	−0.528800
$750$	0	0
$751$	−10.1115	−0.368972	−0.184486	−	0.982835i	$-0.559062\pi$
$751$	−10.1115	−0.368972	−0.184486	+	0.982835i	$0.559062\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−43.4164	−1.58008
$756$	0	0
$757$	15.6525	0.568899	0.284449	−	0.958691i	$-0.408189\pi$
$757$	15.6525	0.568899	0.284449	+	0.958691i	$0.408189\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.5279	−0.744134	−0.372067	−	0.928206i	$-0.621351\pi$
$761$	−20.5279	−0.744134	−0.372067	+	0.928206i	$0.621351\pi$
$762$	0	0
$763$	2.18034	0.0789336
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−90.2492	−3.25871
$768$	0	0
$769$	16.8885	0.609016	0.304508	−	0.952510i	$-0.401508\pi$
$769$	16.8885	0.609016	0.304508	+	0.952510i	$0.401508\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−22.0000	−0.791285	−0.395643	−	0.918405i	$-0.629478\pi$
$773$	−22.0000	−0.791285	−0.395643	+	0.918405i	$0.629478\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.3475	0.406567
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.88854	−0.281554
$786$	0	0
$787$	−55.7771	−1.98824	−0.994119	−	0.108291i	$-0.965462\pi$
$787$	−55.7771	−1.98824	−0.994119	+	0.108291i	$0.965462\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.54102	−0.0903483
$792$	0	0
$793$	−18.3607	−0.652007
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.9443	−1.52116	−0.760582	−	0.649242i	$-0.775086\pi$
$797$	−42.9443	−1.52116	−0.760582	+	0.649242i	$0.775086\pi$
$798$	0	0
$799$	7.70820	0.272697
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−14.4721	−0.510076
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.3607	−0.645527	−0.322764	−	0.946480i	$-0.604612\pi$
$809$	−18.3607	−0.645527	−0.322764	+	0.946480i	$0.604612\pi$
$810$	0	0
$811$	−13.5279	−0.475028	−0.237514	−	0.971384i	$-0.576332\pi$
$811$	−13.5279	−0.475028	−0.237514	+	0.971384i	$0.576332\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17.2361	−0.603753
$816$	0	0
$817$	11.7771	0.412028
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.0557	0.455648	0.227824	−	0.973702i	$-0.426839\pi$
$821$	13.0557	0.455648	0.227824	+	0.973702i	$0.426839\pi$
$822$	0	0
$823$	−1.52786	−0.0532580	−0.0266290	−	0.999645i	$-0.508477\pi$
$823$	−1.52786	−0.0532580	−0.0266290	+	0.999645i	$0.508477\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.70820	0.268041	0.134020	−	0.990979i	$-0.457211\pi$
$827$	7.70820	0.268041	0.134020	+	0.990979i	$0.457211\pi$
$828$	0	0
$829$	19.6525	0.682559	0.341279	−	0.939962i	$-0.389140\pi$
$829$	19.6525	0.682559	0.341279	+	0.939962i	$0.389140\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.47214	−0.189598
$834$	0	0
$835$	−34.4721	−1.19296
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.8197	0.615203	0.307601	−	0.951515i	$-0.400474\pi$
$839$	17.8197	0.615203	0.307601	+	0.951515i	$0.400474\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	0	0
$843$	0	0
$844$	0	0
$845$	57.8885	1.99143
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−22.1803	−0.760332
$852$	0	0
$853$	36.5967	1.25305	0.626524	−	0.779402i	$-0.284477\pi$
$853$	36.5967	1.25305	0.626524	+	0.779402i	$0.284477\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	−27.4164	−0.935436	−0.467718	−	0.883878i	$-0.654924\pi$
$859$	−27.4164	−0.935436	−0.467718	+	0.883878i	$0.654924\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13.5967	−0.462839	−0.231419	−	0.972854i	$-0.574337\pi$
$863$	−13.5967	−0.462839	−0.231419	+	0.972854i	$0.574337\pi$
$864$	0	0
$865$	−2.36068	−0.0802655
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	32.6525	1.10639
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−13.8197	−0.467190
$876$	0	0
$877$	3.29180	0.111156	0.0555780	−	0.998454i	$-0.482300\pi$
$877$	3.29180	0.111156	0.0555780	+	0.998454i	$0.482300\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13.4721	−0.453888	−0.226944	−	0.973908i	$-0.572873\pi$
$881$	−13.4721	−0.453888	−0.226944	+	0.973908i	$0.572873\pi$
$882$	0	0
$883$	24.9443	0.839442	0.419721	−	0.907653i	$-0.362128\pi$
$883$	24.9443	0.839442	0.419721	+	0.907653i	$0.362128\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.5967	−0.993762	−0.496881	−	0.867819i	$-0.665521\pi$
$887$	−29.5967	−0.993762	−0.496881	+	0.867819i	$0.665521\pi$
$888$	0	0
$889$	−11.7771	−0.394991
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−59.4164	−1.98829
$894$	0	0
$895$	−14.4721	−0.483750
$896$	0	0
$897$	0	0
$898$	0	0
$899$	57.5967	1.92096
$900$	0	0
$901$	−9.18034	−0.305841
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.58359	−0.0526404
$906$	0	0
$907$	−5.88854	−0.195526	−0.0977629	−	0.995210i	$-0.531169\pi$
$907$	−5.88854	−0.195526	−0.0977629	+	0.995210i	$0.531169\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.88854	0.326548
$918$	0	0
$919$	47.4164	1.56412	0.782061	−	0.623201i	$-0.214168\pi$
$919$	47.4164	1.56412	0.782061	+	0.623201i	$0.214168\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−24.9443	−0.821051
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35.9443	−1.17929	−0.589647	−	0.807661i	$-0.700733\pi$
$929$	−35.9443	−1.17929	−0.589647	+	0.807661i	$0.700733\pi$
$930$	0	0
$931$	42.1803	1.38240
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−48.4164	−1.58170	−0.790848	−	0.612013i	$-0.790360\pi$
$937$	−48.4164	−1.58170	−0.790848	+	0.612013i	$0.790360\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−47.0689	−1.53440	−0.767201	−	0.641407i	$-0.778351\pi$
$941$	−47.0689	−1.53440	−0.767201	+	0.641407i	$0.778351\pi$
$942$	0	0
$943$	7.70820	0.251014
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.87539	0.288411	0.144206	−	0.989548i	$-0.453937\pi$
$947$	8.87539	0.288411	0.144206	+	0.989548i	$0.453937\pi$
$948$	0	0
$949$	77.7771	2.52475
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.7771	0.543463	0.271732	−	0.962373i	$-0.412404\pi$
$953$	16.7771	0.543463	0.271732	+	0.962373i	$0.412404\pi$
$954$	0	0
$955$	−32.3607	−1.04717
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.3607	−0.657481
$960$	0	0
$961$	54.3050	1.75177
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−31.1803	−1.00373
$966$	0	0
$967$	−53.9574	−1.73515	−0.867577	−	0.497303i	$-0.834324\pi$
$967$	−53.9574	−1.73515	−0.867577	+	0.497303i	$0.834324\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	45.2361	1.45169	0.725847	−	0.687856i	$-0.241448\pi$
$971$	45.2361	1.45169	0.725847	+	0.687856i	$0.241448\pi$
$972$	0	0
$973$	−13.3050	−0.426537
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.3050	1.48143	0.740713	−	0.671821i	$-0.234488\pi$
$977$	46.3050	1.48143	0.740713	+	0.671821i	$0.234488\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18.1115	0.577666	0.288833	−	0.957380i	$-0.406733\pi$
$983$	18.1115	0.577666	0.288833	+	0.957380i	$0.406733\pi$
$984$	0	0
$985$	30.5279	0.972699
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	5.88854	0.187056	0.0935279	−	0.995617i	$-0.470186\pi$
$991$	5.88854	0.187056	0.0935279	+	0.995617i	$0.470186\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−21.3050	−0.675412
$996$	0	0
$997$	25.1803	0.797469	0.398735	−	0.917066i	$-0.369449\pi$
$997$	25.1803	0.797469	0.398735	+	0.917066i	$0.369449\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8712.2.a.bj.1.2		2
3.2	odd	2		968.2.a.f.1.2	✓	2
11.10	odd	2		8712.2.a.bk.1.2		2
12.11	even	2		1936.2.a.x.1.1		2
24.5	odd	2		7744.2.a.ct.1.1		2
24.11	even	2		7744.2.a.bs.1.2		2
33.2	even	10		968.2.i.l.81.1		4
33.5	odd	10		968.2.i.m.729.1		4
33.8	even	10		968.2.i.b.9.1		4
33.14	odd	10		968.2.i.a.9.1		4
33.17	even	10		968.2.i.l.729.1		4
33.20	odd	10		968.2.i.m.81.1		4
33.26	odd	10		968.2.i.a.753.1		4
33.29	even	10		968.2.i.b.753.1		4
33.32	even	2		968.2.a.g.1.2	yes	2
132.131	odd	2		1936.2.a.w.1.1		2
264.131	odd	2		7744.2.a.br.1.2		2
264.197	even	2		7744.2.a.cu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
968.2.a.f.1.2	✓	2	3.2	odd	2
968.2.a.g.1.2	yes	2	33.32	even	2
968.2.i.a.9.1		4	33.14	odd	10
968.2.i.a.753.1		4	33.26	odd	10
968.2.i.b.9.1		4	33.8	even	10
968.2.i.b.753.1		4	33.29	even	10
968.2.i.l.81.1		4	33.2	even	10
968.2.i.l.729.1		4	33.17	even	10
968.2.i.m.81.1		4	33.20	odd	10
968.2.i.m.729.1		4	33.5	odd	10
1936.2.a.w.1.1		2	132.131	odd	2
1936.2.a.x.1.1		2	12.11	even	2
7744.2.a.br.1.2		2	264.131	odd	2
7744.2.a.bs.1.2		2	24.11	even	2
7744.2.a.ct.1.1		2	24.5	odd	2
7744.2.a.cu.1.1		2	264.197	even	2
8712.2.a.bj.1.2		2	1.1	even	1	trivial
8712.2.a.bk.1.2		2	11.10	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)

\(f(q)\)	\(=\)	\(q+2.23607 q^{5} +1.23607 q^{7} +O(q^{10})\)	\(q+2.23607 q^{5} +1.23607 q^{7} -6.23607 q^{13} +1.00000 q^{17} -7.70820 q^{19} -5.23607 q^{23} +6.23607 q^{29} +9.23607 q^{31} +2.76393 q^{35} +4.23607 q^{37} -1.47214 q^{41} -1.52786 q^{43} +7.70820 q^{47} -5.47214 q^{49} -9.18034 q^{53} +14.4721 q^{59} +2.94427 q^{61} -13.9443 q^{65} -5.23607 q^{67} +4.00000 q^{71} -12.4721 q^{73} +2.76393 q^{79} -10.1803 q^{83} +2.23607 q^{85} -14.4164 q^{89} -7.70820 q^{91} -17.2361 q^{95} -5.94427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^7	\( 2 q - 2 q^{7} - 8 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{23} + 8 q^{29} + 14 q^{31} + 10 q^{35} + 4 q^{37} + 6 q^{41} - 12 q^{43} + 2 q^{47} - 2 q^{49} + 4 q^{53} + 20 q^{59} - 12 q^{61} - 10 q^{65} - 6 q^{67} + 8 q^{71} - 16 q^{73} + 10 q^{79} + 2 q^{83} - 2 q^{89} - 2 q^{91} - 30 q^{95} + 6 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 8 * q^13 + 2 * q^17 - 2 * q^19 - 6 * q^23 + 8 * q^29 + 14 * q^31 + 10 * q^35 + 4 * q^37 + 6 * q^41 - 12 * q^43 + 2 * q^47 - 2 * q^49 + 4 * q^53 + 20 * q^59 - 12 * q^61 - 10 * q^65 - 6 * q^67 + 8 * q^71 - 16 * q^73 + 10 * q^79 + 2 * q^83 - 2 * q^89 - 2 * q^91 - 30 * q^95 + 6 * q^97

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